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+ +Russell Mendonca\* Carnegie Mellon University + +Oleh Rybkin\* University of Pennsylvania + +Kostas Daniilidis University of Pennsylvania + +Danijar Hafner University of Toronto + +Deepak Pathak Carnegie Mellon University + +# Abstract + +How can artificial agents learn to solve many diverse tasks in complex visual environments without any supervision? We decompose this question into two challenges: discovering new goals and learning to reliably achieve them. Our proposed agent, Latent Explorer Achiever (LEXA), addresses both challenges by learning a world model from image inputs and using it to train an explorer and an achiever policy via imagined rollouts. Unlike prior methods that explore by reaching previously visited states, the explorer plans to discover unseen surprising states through foresight, which are then used as diverse targets for the achiever to practice. After the unsupervised phase, LEXA solves tasks specified as goal images zero-shot without any additional learning. LEXA substantially outperforms previous approaches to unsupervised goal reaching, both on prior benchmarks and on a new challenging benchmark with 40 test tasks spanning across four robotic manipulation and locomotion domains. LEXA further achieves goals that require interacting with multiple objects in sequence. + +# 1 Introduction + +How can we build an agent that learns to solve hundreds of tasks in complex visual environments, such as rearranging objects with a robot arm or completing chores in a kitchen? While traditional reinforcement learning (RL) has been successful for individual tasks, it requires a substantial amount of human effort for every new task. Specifying task rewards requires domain knowledge, access to object positions, is timeconsuming, and prone to human errors. Moreover, traditional RL would require environment interaction to explore and practice in the environment for every new task. Instead, we approach learning hundreds of tasks through the paradigm of unsupervised goal-conditioned RL, where the agent learns many diverse skills in the environment in the complete absence of supervision, to later solve tasks via user-specified goal images immediately without further training [2, 26, 40]. + +Challenges Exploring the environment and learning to solve many different tasks is substantially more challenging than traditional RL with a dense reward function or learning from + +![](images/e9bbfaebb889bb3a54b1b8ad20dd535aa53cac17ad26150666466df10a82aed2.jpg) +Figure 1: LEXA learns a world model without any supervision, and leverages it to train two policies in imagination. The explorer finds new images and the achiever learns to reliably reach them. Once trained, the achiever reaches user-specified goals zero-shot without further training at test time. + +![](images/ab01d515c5043880effb1e5122d6556dd9c7a184f2bc430aef795af4ecd91478.jpg) +Figure 2: We benchmark LEXA across four visual control environments. A representative sample of the test-time goals is shown here. RoboYoga features complex locomotion and precise control of high-dimensional agents, RoboBins manipulation with multiple objects, and RoboKitchen a variety of diverse tasks that require complex control strategies such as opening a cabinet. + +expert demonstrations. Existing methods are limited to simple tasks, such as picking or pushing a puck [13, 32, 37] or controlling simple 2D robots [50]. The key challenge in improving the performance of unsupervised RL is exploration. In particular, previous approaches explore by either revisiting previously seen rare goals [14, 18, 55] or sampling goals from a generative model [32, 37]. However, in both these approaches, the policy as well as the generative model are trained on previously visited states from the replay buffer, and hence the sampled goals are either within or near the frontier of agent’s experience. Ideally, we would like the agent to discover goals much beyond its frontier for efficient exploration, but how does an agent generate goals that it is yet to encounter? This is an open question not just for AI but for cognitive science too [42]. + +Approach To rectify this issue, we leverage a learned world model to train a separate explorer and achiever policy in imagination. Instead of randomly sampling or generating goals, our explorer policy discovers distant goals by first planning a sequence of actions optimized in imagination of the world model to find novel states with high expected information gain [30, 43, 44]. It then executes those imagined actions in the environment to discover interesting states without the need to generate them. Note these actions are likely to lead the agent to states which are several steps outside the frontier because otherwise the model wouldn’t have had high uncertainty or information gain. Finally, these discovered states are used as diverse targets for the achiever to practice. We train the achiever from on-policy imagination rollouts within the world model and without relying on experience relabeling, therefore leveraging foresight over hindsight. After this unsupervised training phase, the achiever solves tasks specified as goal images zero-shot without any additional learning at deployment. Unlike in the conventional RL paradigm [31, 47], our method is trained once and then used to achieve several tasks at test time without any supervision during training or testing. + +Contributions We introduce Latent Explorer Achiever (LEXA), an unsupervised goal reaching agent that trains an explorer and an achiever within a shared world model. At training, LEXA unlocks diverse data for goal reaching in environments where exploration is nontrivial. At test time, the achiever solves challenging locomotion and manipulation tasks provided as user-specified goal images. Our contributions are summarized as follows: + +• We propose to learn separate explorer and achiever policies as an approach to overcome the exploration problem of unsupervised goal-conditioned RL. +• We show that forward-looking exploration by planning with a learned world model substantially outperforms previous strategies for goal exploration. +• To evaluate on challenging tasks, we introduce a new goal reaching benchmark with a total of 40 +diverse goal images across 4 different robot locomotion and manipulation environments. +• LEXA outperforms prior methods, being the first to show success in the Kitchen robotic manipulation environment, and achieves goal images where multiple objects need to be moved. + +![](images/261b4f4a6881305424cba99bcd987917803949b8856632917f699dbdedfd6295.jpg) +Figure 3: Latent Explorer Achiever (LEXA) learns a general world model that is used to train an explorer and a goal achiever policy. The explorer (left) is trained on imagined latent state rollouts of the world model $s _ { t : T }$ to maximize the disagreement objective $r _ { t } ^ { e } = \mathrm { V a r } ( \bar { s ^ { \prime } } )$ . The goal achiever (right) is conditioned on a goal $g$ and is also trained on imagined rollouts to minimize a distance function $d ( s _ { t } , e _ { g } )$ . Goals are sampled randomly from replay buffer images. For training a temporal distance, we use the imagined rollouts of the achiever and predict the number of time steps between each two states. By combining forward-looking exploration and data-efficient training of the achiever, LEXA provides a simple and powerful solution for unsupervised reinforcement learning. + +# 2 Latent Explorer Achiever (LEXA) + +Our aim is to build an agent that can achieve arbitrary user-specified goals after learning in the environment without any supervision. This presents two challenges - collecting trajectories that contain diverse goals and learning to achieve these goals when specified as a goal image. We introduce a simple solution based on a world model and imagination training that addresses both challenges. The world model represents the agent’s current knowledge about the environment and is used for training two policies, the explorer and the achiever. To explore novel situations, we construct an estimate of which states the world model is still uncertain about. To achieve goals, we train the goal-conditioned achiever in imagination, using the images found so far as unsupervised goals. At test time, the achiever is deployed to reach user-specified goals. The training procedure is in Algorithm 1. + +# 2.1 World Model + +To efficiently predict potential outcomes of future actions in environments with high-dimensional image inputs, we leverage a Recurrent State Space Model (RSSM) [23] that learns to predict forward using compact model states that facilitate planning [7, 51]. In contrast to predicting forward in image space, the model states enable efficient parallel planning with a large batch size and can reduce accumulating errors [39]. The world model consists of the following components: + +$$ +{ \begin{array} { l l l } { e _ { t } = \operatorname { e n c } _ { \phi } ( x _ { t } ) } & { { \mathrm { P o s t e r i o r : } } } & { \ q _ { \phi } ( s _ { t } \mid s _ { t - 1 } , a _ { t - 1 } , e _ { t } ) } \\ { p _ { \phi } ( s _ { t } \mid s _ { t - 1 } , a _ { t - 1 } ) } & { { \mathrm { I m a g e ~ d e c o d e r : } } } & { \ p _ { \phi } ( x _ { t } \mid s _ { t } ) } \end{array} } +$$ + +The model states $s _ { t }$ contain a deterministic component $h _ { t }$ and a stochastic component $z _ { t }$ with diagonalcovariance Gaussian distribution. $h _ { t }$ is the recurrent state of a Gated Recurrent Unit (GRU) [11]. The encoder and decoder are convolutional neural networks (CNNs) and the remaining components are multi-layer perceptrons (MLPs). The world model is trained end-to-end by optimizing the evidence lower bound (ELBO) via stochastic backpropagation [28, 38] with the Adam optimizer [27]. + +# 2.2 Explorer + +To efficiently explore, we seek out surprising states imagined by the world model [6, 41, 43, 44, 46], as opposed to retrospectively exploring by revisiting previously novel states [4, 5, 8, 34]. As the world model can predict model states that correspond to unseen situations in the environment, the imagined trajectories contain more novel goals, compared to model-free exploration that is limited to the replay buffer. To collect informative novel trajectories in the environment, we train an exploration + +1: initialize: World model $\mathcal { M }$ , Replay buffer $\mathcal { D }$ , Explorer $\pi ^ { \mathrm { e } } ( a _ { t } \mid z _ { t } )$ , Achiever $\pi ^ { \mathbf { g } } ( a _ { t } \mid z _ { t } , g )$ +2: while exploring do +3: Train $\mathcal { M }$ on $\mathcal { D }$ +4: Train $\pi ^ { \mathrm { e } }$ in imagination of $\mathcal { M }$ to maximize exploration rewards $\textstyle \sum _ { t } r _ { t } ^ { \mathrm { e } }$ . +5: Train $\pi ^ { \mathrm { g } }$ in imagination of $\mathcal { M }$ to maximize $\textstyle \sum _ { t } r _ { t } ^ { \mathrm { g } } ( z _ { t } , g )$ for images $g \sim \mathcal { D }$ . +6: (Optional) Train $d ( z _ { i } , z _ { j } )$ to predict distances $j - i$ on the imagination data from last step. +7: Deploy $\pi ^ { \mathrm { e } }$ in the environment to explore and grow $\mathcal { D }$ . +8: Deploy $\pi ^ { \mathrm { g } }$ in the environment to achieve a goal image $g \sim \mathcal { D }$ to grow $\mathcal { D }$ . +9: end while +10: while evaluating do +11: given: Evaluation goal $g$ +12: Deploy $\pi ^ { \mathrm { g } }$ in the world to reach $g$ . +13: end while + +policy $\pi ^ { e }$ from the model states $s _ { t }$ in imagination of the world model to maximize an exploration reward: + +$$ +{ \mathrm { E x p l o r e r : } } \qquad \pi ^ { e } ( a _ { t } \mid s _ { t } ) \qquad { \mathrm { E x p l o r e r ~ V a l u e : } } \qquad v ^ { e } ( s _ { t } ) +$$ + +To explore the most informative model states, we estimate the epistemic uncertainty as a disagreement of an ensemble of transition functions. We train an ensemble of 1-step models to predict the next model state from the current model state. The ensemble model is trained alongside the world model on model states produced by the encoder $q _ { \phi }$ . Because the ensemble models are initialized at random, they will differ, especially for inputs that they have not been trained on [29, 36]: + +$$ +\mathrm { E n s e m b l e : } \quad f ( s _ { t } , \theta ^ { k } ) = \hat { z } _ { t + 1 } ^ { k } \quad \mathrm { f o r } \quad k = 1 . . K +$$ + +Leveraging the ensemble, we estimate the epistemic uncertainty as the ensemble disagreement. The exploration reward is the variance of the ensemble predictions averaged across dimension of the model state, which approximates the expected information gain [3, 43]: + +$$ +r _ { t } ^ { \mathrm { e } } ( s _ { t } ) \doteq \frac { 1 } { N } \sum _ { n } \operatorname { V a r } _ { \{ \mathrm { k } \} } \left[ f ( s _ { t } , \theta _ { k } ) \right] _ { n } +$$ + +The explorer $\pi ^ { e }$ maximizes the sum of future exploration rewards $\boldsymbol { r } _ { t } ^ { e }$ using the Dreamer algorithm [24], which considers long-term rewards into the future by maximizing $\lambda$ -returns under a learned value function. As a result, the explorer is trained to seek out situations are as informative as possible from imagined latent trajectories of the world model, and is periodically deployed in the environment to add novel trajectories to the replay buffer, so the world model and goal achiever policy can improve. + +# 2.3 Achiever + +To leverage the knowledge obtained by exploration for learning to reach goals, we train a goal achiever policy $\pi ^ { g }$ that receives a model state and a goal as input. Our aim is to train a general policy that is capable of reaching many diverse goals. To achieve this in a data-efficient way, it is crucial that environment trajectories that were collected with one goal in mind are reused to also learn how to reach other goals. While prior work addressed this by goal relabeling which makes off-policy policy optimization a necessity [2], we instead leverage past trajectories via the world model trained on them that lets us generate an unlimited amount of new imagined trajectories for training the goal achiever on-policy in imagination. This simplifies policy optimization and can improve stability, while still sharing all collected experience across many goals. + +$$ +\pi ^ { g } ( a _ { t } \mid s _ { t } , e _ { g } ) \qquad \mathrm { A c h i e v e r ~ V a l u e } ; \qquad v ^ { g } ( s _ { t } , e _ { g } ) +$$ + +To train the goal achiever, we sample a goal image $x _ { g }$ from the replay buffer and compute its embedding $e _ { g } = \mathrm { e n c } _ { \phi } ( x _ { g } )$ . The achiever aims to maximize an unsupervised goal-reaching reward $r ^ { g } ( s _ { t } , e _ { g } )$ . We discuss different choices for this reward in Section 2.4. We again use the Dreamer algorithm [24] for training, where now the value function also receives the goal embedding as input. In addition to imagination training, it can also be important to perform practice trials with the goal achiever in the true environment, so that any model inaccuracies along the goal reaching trajectories may be corrected. To perform practice trials, we sample a goal from the replay buffer and execute the goal achiever policy for that goal in the environment. These trials are interleaved with exploration episodes collected by the exploration policy in equal proportion. We note that the goal achiever learning is entirely unsupervised because the practice goals are simply images the agent encountered through exploration or during previous practice trails. + +![](images/8833211f2fc46aac053a0c19271c2751e3ee461c189b8c3a201cf26bcb97a95e.jpg) +Figure 4: Successful LEXA trajectories. When given a goal image from the test set, LEXA’s achiever is used in the environment to reach that image. On RoboKitchen, LEXA manipulates up to three different objects together from a single goal image (kettle, light switch, and cabinet). On RoboBins, LEXA performs temporally extended tasks such as picking and placing two objects in a row. + +# 2.4 Latent Distances + +Training the achiever policy requires us to define a goal achievement reward $r ^ { g } ( s _ { t } , e _ { g } )$ that measures how close the latent state $s _ { t }$ should be considered to the goal $e _ { g }$ . One simple measure is the cosine distance in the latent space obtained by inputting image observations into the world-model. However, such a distance function brings visually similar states together even if they could be farther apart in temporal manner as measured by actions needed to reach from one to other. This bias makes this suitable only to scenarios where most of pixels in the observations are directly controllable, e.g., trying to arrange robot’s body in certain shape, such as RoboYoga poses in Figure 2. However, many environments contain agent as well as the world, such as manipulation involves interacting with objects that are not directly controllable. The cosine distance would try matching the entire goal image, and thus places a large weight on both matching the robot and object positions with the desired goal. Since the robot position is directly controllable it is much easier to match, but this metric overly focuses on it, yielding poor policies that ignore objects. We address this is by using the number of timesteps it takes to move from one image to another as a distance measure [25, 26]. This ignores large changes in robot position, since these can be completed in very few steps, and will instead focus more on the objects. This temporal cost function can be learned purely in imagination rollouts from our world model allowing as much data as needed without taking any steps in the real world. + +Cosine Distance To use cosine distance with LEXA, for a latent state $s _ { t }$ , and a goal embedding $e ^ { g }$ , we use the latent inference network $q$ to infer $s ^ { g }$ , and define the reward as the cosine similarity [54]: + +$$ +r _ { t } ^ { g } ( s _ { t } , e _ { g } ) \doteq \sum _ { i } \overline { { s } } _ { t i } \overline { { s } } _ { g i } , \quad \mathrm { w h e r e } \quad \overline { { s } } _ { t } = s _ { t } / \| s _ { t } \| _ { 2 } , \quad \overline { { s } } _ { g } = s _ { g } / \| s _ { g } \| _ { 2 } , +$$ + +i.e. the cosine of the angle between the two vectors $s _ { t } , s _ { g }$ in the $N -$ dimensional latent space. + +Temporal Distance To use temporal distances with LEXA, we train a neural network $d$ to predict the number of time steps between two embeddings. We train it by sampling pairs of states $s _ { t }$ , $s _ { t + k }$ from an imagined rollout of the achiever and predicting the distance $k$ . We implement the temporal distance in terms of predicted image embeddings $\boldsymbol { \hat { e } } _ { t + k }$ in order to remove extra recurrent information: + +Predicted embedding: $\mathrm { e m b } ( s _ { t } ) = \hat { e } _ { t } \approx e _ { t } { \quad } \mathrm { T e m p o r a l ~ d i s t a n c e } ; \quad d _ { \omega } ( \hat { e } _ { t } , \hat { e } _ { t + k } ) \approx k / H ,$ where $H$ is the maximum distance equal to the imagination horizon. Training distance function only on imagination data from the same trajectory would cause it to predict poor distance to far away states coming from other trajectories, such as images that are impossible to reach during one episode. In order to incorporate learning signal from such far-away goals, we include them by sampling images from a different trajectory. We annotate these negative samples with the maximum possible distance, so that the agent always prefers images that were seen in the same trajectory. + +![](images/b02095cdb02ada544f65d40a4cf2610dc2db333769a1afaf5a6454034a90d495.jpg) +Figure 5: Coincidental goal success achieved during the unsupervised exploration phase. The forwardlooking explorer policy of LEXA results in substantially better coverage compared to SkewFit, a popular method for goal based exploration. + +$$ +r _ { t } ^ { g } ( s _ { t } , e _ { g } ) = - d _ { \omega } ( \hat { e } _ { t } , e _ { g } ) , \quad \mathrm { w h e r e } \quad \hat { e } _ { t } = \mathrm { e m b } ( s _ { t } ) , \quad e _ { g } = \mathrm { e n c } _ { \phi } ( x _ { g } ) +$$ + +The learned distance function depends on the training data policy. However, as the policy becomes more competent, the distance estimates will be closer to the optimal number of time steps to reach a particular goal, and the policy converges to the optimal solution [25]. LEXA always uses the latest data to train the distance function using imagination, ensuring that the convergence is fast. + +# 3 Experiments + +Our evaluation focuses on the following scientific questions: + +1. Does LEXA outperform prior work on previous benchmarks and a new challenging benchmark? +2. How does forward-looking exploration of goals compare to previous goal exploration strategies? +3. How does the distance function affect the ability to reach goals in different types of environments? +4. Can we train one general LEXA to control different robots across visually distinct environments? +5. What components of LEXA are important for performance? + +We evaluate LEXA on prior benchmarks used by SkewFit [37], DISCERN [50], and Plan2Explore [43] in Section 3.3. Since these benchmarks are largely saturated, we also introduce a new challenging benchmark shown in Figure 2. We evaluate LEXA on this benchmark is Section 3.2. + +# 3.1 Experimental setup + +As not many prior methods have shown success on reaching diverse goals from image inputs, we perform an apples-to-apples comparison by implementing the baselines using the same world model and policy optimization as our method: + +• SkewFit SkewFit [37] uses model-free hindsight experience replay and explores by sampling goals from the latent space of a variational autoencoder [28, 38]. Being one of the state-of-the-art agents, we use the original implementation that does not use a world model or explorer policy. +• DDL Dynamic Distance Learning [25] trains a temporal distance function similar to our method. Following the original algorithm, DDL uses greedy exploration and trains the distance function on the replay buffer instead of in imagination. +• DIAYN Diversity is All You Need [15] learns a latent skill space and uses mutual information between skills and reached states as the objective. We augment DIAYN with our explorer policy and train a learned skill predictor to obtain a skill for a given test image [12]. +• GCSL Goal-Conditioned Supervised Learning [20] trains the goal policy on replay buffer goals and mimics the actions that previously led to the goal. We also augment GCSL with our explorer policy, as we found no learning success without it. + +Our new benchmark defines goal images for a diverse set of four existing environments as follows: + +![](images/1310ecf4c91afe8a97fdd0a2ff16844020c1a947729ce1b05b3d97a0c5844f2a.jpg) +Figure 6: Evaluation of goal reaching agents on our four benchmarks. A single agent is trained from images without rewards and then evaluated on reaching goal images from the test set (see Figure 1). Both LEXA agents solve many of the tasks and significantly outperform prior work. SkewFit and DLL struggle with exploration, while DIAYN and GCSL use our explorer but still are not able to learn a good downstream policy. Refer table 1 for final success percentage (averaged across tasks) for each method and benchmark domain. + +• RoboYoga We use the walker and quadruped domains of the DeepMind Control Suite [48] to define the RoboYoga benchmark, consisting of 12 goal images that correspond to different body poses for each of the two environments, such as lying down, standing up, and balancing. • RoboBins Based on MetaWorld [53], we create a scene with a Sawyer robotic arm, two bins, and two blocks of different colors. The goal images specify tasks that include reaching, manipulating only one block, and manipulating both blocks. • RoboKitchen The last benchmark involves the challenging kitchen environment from [22], where a franka robot can interact with various objects including a burner, light switch, sliding cabinet, hinge cabinet, microwave, or kettle. The goal images we include describe tasks that require interacting with only one object, as well as interacting with two objects. + +# 3.2 Performance on New Benchmark + +We show the results on our main benchmark in Figure 6 and include heatmaps that show per-task success on each of the evaluation tasks from the benchmarks in the Appendix. Further, we report success averaged across tasks for each domain at the end of training in Table 1. We visualize example successful trajectory executions for tasks that require manipulating multiple objects in Fig. 4. + +RoboYoga The environments in this benchmark are directly controllable since they contain no other objects except the robot. We recall that for such settings we expect the cosine distance to be effective, as perceptual distance is quite accurate. Training is thus faster compared to using learned temporal distances, where the metric is learned from scratch. From Table 1 and Figure 6 we see that this is indeed the case for these environments (Walker and Quadruped), as LEXA with the cosine metric outperforms all prior approaches. Furthermore with temporal distances LEXA makes better progress compared to prior work on a much larger number of goals as can be seen from the per-task performance (Figures ??, ??), even though average success over goals looks similar to that of DDL. + +RoboBins This environment involves interaction with block objects, and thus is not directly controllable, and so we expect LEXA to perform better with the temporal distance metric. From Table 1 and + +
MethodKitchenRoboBinsQuadrupedWalker
DDL0.0035.4222.5040.00
DIAYN0.0013.6913.810.28
GCSL0.007.9415.831.11
SkewFit0.2315.775.520.01
LEXA + Temporal (Ours)37.5069.4431.3936.72
LEXA + Cosine (Ours)6.0245.8356.1173.06
+ +Table 1: Performance on our new challenging benchmark, spanning across the four domains shown in Figure 2. The number are goal success rates, averaged over test goals within each environment. + +![](images/1ef7c10b47d9d98a8e1a64e538f880461c2ee0273a63c335f48dbd65f5cbaaed.jpg) +Figure 7: Success rates on RoboBin. In line with the prior literature, previous methods are successful at reaching and sometimes pushing. LEXA pushes the state-of-the-art by picking and placing multiple objects to reach challenging goal images. Analogous heat maps for the other domains are included in the appendix. + +Figure 6, we see that LEXA gets higher average success than all prior approaches. Further from the per-task performance in 7, LEXA with the temporal distance metric is the only approach that makes progress on all goals in the benchmark. The main difference in performance between using temporal and cosine distance can be seen in the tasks involving two blocks, which are the most complex tasks in this environment (the last 3 columns of the per-task plot). The best performing prior method is DDL which solves reaching, and can perform simple pushing tasks. This method performs poorly due to poor exploration, as shown in Figure 5. We see that while other prior methods make some progress on reaching, they fail on harder tasks. + +RoboKitchen This benchmark involves diverse objects that require different manipulation behavior. From Table 1 and Figure 6 and ?? we find that LEXA with temporal distance is able to learn multiple RoboKitchen tasks, some of which require sequentially completing 2 tasks in the environment. All prior methods barely make progress due to the challenging nature of this benchmark, and furthermore using the cosine distance function makes very limited progress. The gap in performance between using the two distance functions is much larger in this environment compared to RoboBins since there are many more objects and they are not as clearly visible as the blocks. + +# Single Agent Across All Environments + +In the previous sections we have shown that our approach can achieve diverse goals in different environments. However, we trained a new agent for every new environment, which doesn’t scale well to large numbers of environments. Thus we investigate if we can train a train a single agent across four environments in the benchmark. From Figure ?? we see that our approach with learned temporal distance is able to make progress on tasks from RoboKitchen, RoboBins Reaching, RoboBins Pick & Place and Walker, while the best prior method on the single-environment tasks (DDL) mainly solves walker tasks and reaching from RoboBin. + +# 3.3 Performance on Prior benchmarks + +To further verify the results obtained on our benchmark, we evaluate LEXA on previously used benchmarks. We observe that LEXA significantly outperforms prior work on these benchmarks, and is often close to the optimal policy. Additional details are provided in ??????. + +SkewFit Benchmark SkewFit [37] introduces a robotic manipulation benchmark for unsupervised methods with simple tasks like planar pushing or picking. We evaluate on this benchmark in Table 2. + +Table 2: Goal distance for SkewFit goals [37]. + +
MethodPusherPickup
RIG [32]7.7cm3.7cm
RIG + HER [2]7.5cm3.5cm
Skew-Fit [37]4.9cm1.8cm
LEXA + Temporal2.3cm1.4cm
+ +Baseline results are taken from [37]. LEXA significantly outperforms prior work on these tasks. Pushing and picking up blocks from images is largely solved and future work can focus on harder benchmarks such as those introduced in our paper. + +DISCERN Benchmark We attempted to replicate the tasks described in [50] that are based on simple two-dimensional robots [48]. While the original tasks are not released, we followed the procedure for generating the goals described in the paper. Despite following the exact procedure, we were not able to obtain similar goals to the ones used in the original paper. Nevertheless, we show the goal completion percentage results obtained with our reproduced evaluation compared to DISCERN results from the original paper. LEXA results were obtained with early stopping. In Table 3 we see that our agent solves many of the tasks in this benchmark. + +Table 3: Success for DISCERN goals [50]. + +
TaskLEXADISCERN
Cup84.0%76.5%
Cartpole35.9%21.3%
Finger40.9%21.8%
Pendulum79.1%75.7%
Pointmass83.2%49.6%
Reacher100.0%87.1%
+ +Plan2Explore Benchmark We provide a comparison on the standard reward-based DM control tasks [48] in Table 4. To compare on this benchmark, we create goal images that correspond to the reward functions. This setup is arguably harder for our agent, but is much more practical. Note our agent never observes the reward function and only observes the goal at test time. Plan2Explore adapts to new tasks but it needs the reward function to be known at test time, while DrQV2 is an oracle agent that observes the reward at training time. Baseline results are taken from [43, 52]. LEXA results were obtained with early stopping. LEXA outperforms Plan2Explore on most tasks and even performs comparably to state of the art oracle agents (DrQ, DrQv2, Dreamer) that use true task rewards during training. + +Table 4: Zero-shot return on P2E tasks [43]. + +
Task Zero-ShotLEXA [P2EDrQv2 X
Walker Stand957331968
Hopper Stand840841957
Cartpole Balance886950989
Cartpole Bal. Sparse996860983
Pendulum Swing Up788792837
Cup Catch969962909
Reacher Hard93766970
+ +# 3.4 Analysis + +Prior work Most work we compared against struggles with exploration, such as SkewFit and DLL methods. DIAYN is augmented with our explorer, but still fails to leverage the exploration data to learn a diverse set of skills. GCSL struggles to fit the exploration data and produces behavior that does not solve the task, perhaps because the exploration data is too diverse. We observed that all baselines make progress on the simple reaching, but struggle with other tasks. We have experimented with several versions and improvements to the baselines and report the best obtained performance. + +Ablation of different components We ablated components of LEXA on the RoboBins environment in Figure 8. Using a separate explorer policy crucial as without it the agent does not discover the more interesting tasks. Without negative sampling the agent learns slower, perhaps because the distance function doesn’t produce reasonable outputs when queried on images that are more than horizon length apart. Training the distance function with real data converges to slightly lower success than using imagination data, since real data is sampled in an off-policy manner due to its limited quantity. + +![](images/e74e622cfc1a92b2076d60433d82e8c9de54acaa1e72a8a8ad12dea1ef790bc6.jpg) +Figure 8: Ablations on RoboBins. A separate explorer is crucial for most tasks. Training temporal distance on negative samples speeds up learning, and both negative sampling and training in imagination as opposed to real data are important for the hardest tasks. + +Exploration performance Due to importance of exploration, we further examine the diversity of the data collected during training. We log the instances where the agent coincidentally solves an evaluation task during exploration, for the RoboKitchen and RoboBins environments. In Figure 5, we see that our method encounters harder tasks involving multiple objects much more often. + +# 4 Related Work + +Learning to Achieve Goals The problem of learning to reach many different goals has been commonly addressed with model-free methods that learn a single goal-conditioned policy [2, 26, 40]. Recent work has combined these approaches with various ways to generate training goals, such as asymmetric self-play [33, 45] or by sampling goals of intermediate difficulty [14, 18]. These approaches can achieve remarkable performance in simulated robotic domains, however, they focus on the settings where the agent can directly perceive the low-dimensional environment state. + +A few works have attempted to scale these model-free methods to visual goals by using contrastive [50] or reconstructive [32, 37] representation learning. However, these approaches struggle to perform meaningful exploration as no clear reward signal is available to guide the agent toward solving interesting tasks. Some works [10, 49] avoid this challenge by using a large dataset of interesting behaviors. Other works [37, 55] attempt to explore by generating goals similar to those that have already been seen, but do not try to explore truly novel states. + +A particularly relevant set of approaches used model-based methods to achieve goals via planning [13, 17] or learning model-regularized policies [35]. However, these approaches are limited by short planning horizons. In contrast, we learn long-horizon goal-conditioned value functions which allows us to solve more challenging tasks. More generally, most of the above approaches are limited by simplistic exploration, while our method leverages model imagination to search for novel states, which significantly improves exploration and in turn the downstream capabilities of the agent. + +Learning Distance Functions A crucial challenge for visual goal reaching is the choice of the reward or the cost function for the goal achieving policy. Several approaches use representation learning to create a distance in the feature space [9, 32, 50, 51]. However, this naive distance may not be most reflective of how hard a particular goal is to reach. One line of research has proposed using the mutual information between the current state and the goal as the distance metric [1, 12, 15, 21], however, it remains to be seen whether this approach can scale to more complex tasks. + +Other works proposed temporal distances that measure the amount of time it takes to reach the goal. One approach is to learn the distance with approximate dynamic programming using Q-learning methods [16, 19, 26]. Our distance function is most similar to Hartikainen et al. [25], who learn a temporal distance with supervised learning on recent policy experience. In contrast to [25], we always train the distance on-policy in imagination, and we further integrate this achiever policy into our latent explorer achiever framework to discover novel goals for the achiever to practice on. + +# 5 Conclusion + +We presented Latent Explorer Achiever (LEXA), an agent for unsupervised RL that explores its environment, learns to achieve the discovered goals, and solves image-based tasks in a zero-shot way. By planning for novelty in imagination, LEXA prospectively explores to discover meaningful behaviors in substantially more diverse environments than considered by prior work. Further, LEXA is able to solve challenging downstream tasks specified as images without any supervision such as rewards or demonstrations. By proposing a challenging benchmark and the first agent to achieve meaningful performance on these tasks, we hope to stimulate future research on unsupervised agents, which we believe are fundamentally more scalable than traditional agents that require a human to design the tasks and rewards for learning. + +Many challenges remain for building unsupervised agents. Many tasks in our benchmark are still unsolved and there remains room for progress on the algorithmic side both for the world model and policy optimization. Further, it is important to demonstrate the benefits of unsupervised agents on real-world systems to verify their scalability. Finally, for widespread adoption, it is crucial to consider the problem of goal specification and design methods that act on goals that are easy to specify, such as via natural language. We believe LEXA will enable future work to tackle these goals effectively. + +Acknowledgements We thank Ben Eysenbach, Stephen Tian, Sergey Levine, Dinesh Jayaraman, Karl Pertsch, Ed Hu and the members of GRASP lab and Pathak lab for insightful discussions. We also thank Murtaza Dalal and Chuning Zhu for help with MuJoCo environments. Finally, DP would like to thank Laura Schulz, Josh Tenenbaum and Alison Gopnik for seeding the idea of goal-setting in children, and Pulkit Agrawal for several crucial discussions (over gelato) since then. OR and KD were supported by ARL DCIST CRA W911NF-17-2-0181, ONR N00014-17-1-2093, and by Honda Research Institute. This work was partially supported by GoodAI Research Award and DARPA Machine Common Sense grant. + +References [1] J. Achiam, H. Edwards, D. Amodei, and P. Abbeel. Variational option discovery algorithms. arXiv preprint arXiv:1807.10299, 2018. 10 [2] M. Andrychowicz, F. Wolski, A. Ray, J. Schneider, R. Fong, P. Welinder, B. McGrew, J. 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Advances in Neural Information Processing Systems, 33, 2020. 2, 10 \ No newline at end of file diff --git a/parse/train/6vWuYzkp8d/6vWuYzkp8d_content_list.json b/parse/train/6vWuYzkp8d/6vWuYzkp8d_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..3023ae2f901ae8ea41f7913fe35f1e8fb9b9ffd8 --- /dev/null +++ b/parse/train/6vWuYzkp8d/6vWuYzkp8d_content_list.json @@ -0,0 +1,1184 @@ +[ + { + "type": "text", + "text": "Discovering and Achieving Goals via World Models ", + "text_level": 1, + "bbox": [ + 186, + 122, + 810, + 148 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Russell Mendonca\\* Carnegie Mellon University ", + "bbox": [ + 266, + 200, + 450, + 232 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Oleh Rybkin\\* University of Pennsylvania ", + "bbox": [ + 552, + 200, + 732, + 232 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Kostas Daniilidis University of Pennsylvania ", + "bbox": [ + 205, + 253, + 385, + 285 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Danijar Hafner University of Toronto ", + "bbox": [ + 424, + 253, + 568, + 285 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deepak Pathak Carnegie Mellon University ", + "bbox": [ + 606, + 253, + 792, + 285 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 308, + 535, + 324 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "How can artificial agents learn to solve many diverse tasks in complex visual environments without any supervision? We decompose this question into two challenges: discovering new goals and learning to reliably achieve them. Our proposed agent, Latent Explorer Achiever (LEXA), addresses both challenges by learning a world model from image inputs and using it to train an explorer and an achiever policy via imagined rollouts. Unlike prior methods that explore by reaching previously visited states, the explorer plans to discover unseen surprising states through foresight, which are then used as diverse targets for the achiever to practice. After the unsupervised phase, LEXA solves tasks specified as goal images zero-shot without any additional learning. LEXA substantially outperforms previous approaches to unsupervised goal reaching, both on prior benchmarks and on a new challenging benchmark with 40 test tasks spanning across four robotic manipulation and locomotion domains. LEXA further achieves goals that require interacting with multiple objects in sequence. ", + "bbox": [ + 232, + 335, + 766, + 529 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 549, + 310, + 566 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "How can we build an agent that learns to solve hundreds of tasks in complex visual environments, such as rearranging objects with a robot arm or completing chores in a kitchen? While traditional reinforcement learning (RL) has been successful for individual tasks, it requires a substantial amount of human effort for every new task. Specifying task rewards requires domain knowledge, access to object positions, is timeconsuming, and prone to human errors. Moreover, traditional RL would require environment interaction to explore and practice in the environment for every new task. Instead, we approach learning hundreds of tasks through the paradigm of unsupervised goal-conditioned RL, where the agent learns many diverse skills in the environment in the complete absence of supervision, to later solve tasks via user-specified goal images immediately without further training [2, 26, 40]. ", + "bbox": [ + 174, + 575, + 485, + 837 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Challenges Exploring the environment and learning to solve many different tasks is substantially more challenging than traditional RL with a dense reward function or learning from ", + "bbox": [ + 174, + 840, + 485, + 895 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/e9bbfaebb889bb3a54b1b8ad20dd535aa53cac17ad26150666466df10a82aed2.jpg", + "image_caption": [ + "Figure 1: LEXA learns a world model without any supervision, and leverages it to train two policies in imagination. The explorer finds new images and the achiever learns to reliably reach them. Once trained, the achiever reaches user-specified goals zero-shot without further training at test time. " + ], + "image_footnote": [], + "bbox": [ + 501, + 554, + 825, + 797 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/ab01d515c5043880effb1e5122d6556dd9c7a184f2bc430aef795af4ecd91478.jpg", + "image_caption": [ + "Figure 2: We benchmark LEXA across four visual control environments. A representative sample of the test-time goals is shown here. RoboYoga features complex locomotion and precise control of high-dimensional agents, RoboBins manipulation with multiple objects, and RoboKitchen a variety of diverse tasks that require complex control strategies such as opening a cabinet. " + ], + "image_footnote": [], + "bbox": [ + 173, + 49, + 823, + 343 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "expert demonstrations. Existing methods are limited to simple tasks, such as picking or pushing a puck [13, 32, 37] or controlling simple 2D robots [50]. The key challenge in improving the performance of unsupervised RL is exploration. In particular, previous approaches explore by either revisiting previously seen rare goals [14, 18, 55] or sampling goals from a generative model [32, 37]. However, in both these approaches, the policy as well as the generative model are trained on previously visited states from the replay buffer, and hence the sampled goals are either within or near the frontier of agent’s experience. Ideally, we would like the agent to discover goals much beyond its frontier for efficient exploration, but how does an agent generate goals that it is yet to encounter? This is an open question not just for AI but for cognitive science too [42]. ", + "bbox": [ + 173, + 415, + 825, + 540 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Approach To rectify this issue, we leverage a learned world model to train a separate explorer and achiever policy in imagination. Instead of randomly sampling or generating goals, our explorer policy discovers distant goals by first planning a sequence of actions optimized in imagination of the world model to find novel states with high expected information gain [30, 43, 44]. It then executes those imagined actions in the environment to discover interesting states without the need to generate them. Note these actions are likely to lead the agent to states which are several steps outside the frontier because otherwise the model wouldn’t have had high uncertainty or information gain. Finally, these discovered states are used as diverse targets for the achiever to practice. We train the achiever from on-policy imagination rollouts within the world model and without relying on experience relabeling, therefore leveraging foresight over hindsight. After this unsupervised training phase, the achiever solves tasks specified as goal images zero-shot without any additional learning at deployment. Unlike in the conventional RL paradigm [31, 47], our method is trained once and then used to achieve several tasks at test time without any supervision during training or testing. ", + "bbox": [ + 174, + 542, + 825, + 722 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions We introduce Latent Explorer Achiever (LEXA), an unsupervised goal reaching agent that trains an explorer and an achiever within a shared world model. At training, LEXA unlocks diverse data for goal reaching in environments where exploration is nontrivial. At test time, the achiever solves challenging locomotion and manipulation tasks provided as user-specified goal images. Our contributions are summarized as follows: ", + "bbox": [ + 174, + 726, + 825, + 795 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We propose to learn separate explorer and achiever policies as an approach to overcome the exploration problem of unsupervised goal-conditioned RL. \n• We show that forward-looking exploration by planning with a learned world model substantially outperforms previous strategies for goal exploration. \n• To evaluate on challenging tasks, we introduce a new goal reaching benchmark with a total of 40 \ndiverse goal images across 4 different robot locomotion and manipulation environments. \n• LEXA outperforms prior methods, being the first to show success in the Kitchen robotic manipulation environment, and achieves goal images where multiple objects need to be moved. ", + "bbox": [ + 179, + 797, + 826, + 909 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/261b4f4a6881305424cba99bcd987917803949b8856632917f699dbdedfd6295.jpg", + "image_caption": [ + "Figure 3: Latent Explorer Achiever (LEXA) learns a general world model that is used to train an explorer and a goal achiever policy. The explorer (left) is trained on imagined latent state rollouts of the world model $s _ { t : T }$ to maximize the disagreement objective $r _ { t } ^ { e } = \\mathrm { V a r } ( \\bar { s ^ { \\prime } } )$ . The goal achiever (right) is conditioned on a goal $g$ and is also trained on imagined rollouts to minimize a distance function $d ( s _ { t } , e _ { g } )$ . Goals are sampled randomly from replay buffer images. For training a temporal distance, we use the imagined rollouts of the achiever and predict the number of time steps between each two states. By combining forward-looking exploration and data-efficient training of the achiever, LEXA provides a simple and powerful solution for unsupervised reinforcement learning. " + ], + "image_footnote": [], + "bbox": [ + 178, + 88, + 820, + 299 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Latent Explorer Achiever (LEXA) ", + "text_level": 1, + "bbox": [ + 174, + 424, + 493, + 443 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our aim is to build an agent that can achieve arbitrary user-specified goals after learning in the environment without any supervision. This presents two challenges - collecting trajectories that contain diverse goals and learning to achieve these goals when specified as a goal image. We introduce a simple solution based on a world model and imagination training that addresses both challenges. The world model represents the agent’s current knowledge about the environment and is used for training two policies, the explorer and the achiever. To explore novel situations, we construct an estimate of which states the world model is still uncertain about. To achieve goals, we train the goal-conditioned achiever in imagination, using the images found so far as unsupervised goals. At test time, the achiever is deployed to reach user-specified goals. The training procedure is in Algorithm 1. ", + "bbox": [ + 173, + 457, + 826, + 583 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 World Model ", + "text_level": 1, + "bbox": [ + 174, + 599, + 305, + 614 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To efficiently predict potential outcomes of future actions in environments with high-dimensional image inputs, we leverage a Recurrent State Space Model (RSSM) [23] that learns to predict forward using compact model states that facilitate planning [7, 51]. In contrast to predicting forward in image space, the model states enable efficient parallel planning with a large batch size and can reduce accumulating errors [39]. The world model consists of the following components: ", + "bbox": [ + 174, + 625, + 825, + 694 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/58e48f6a2376d8a71828658c0a6a6f47419cb58a5a06de85dde3e53c525e8e41.jpg", + "text": "$$\n{ \\begin{array} { l l l } { e _ { t } = \\operatorname { e n c } _ { \\phi } ( x _ { t } ) } & { { \\mathrm { P o s t e r i o r : } } } & { \\ q _ { \\phi } ( s _ { t } \\mid s _ { t - 1 } , a _ { t - 1 } , e _ { t } ) } \\\\ { p _ { \\phi } ( s _ { t } \\mid s _ { t - 1 } , a _ { t - 1 } ) } & { { \\mathrm { I m a g e ~ d e c o d e r : } } } & { \\ p _ { \\phi } ( x _ { t } \\mid s _ { t } ) } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 320, + 700, + 763, + 738 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The model states $s _ { t }$ contain a deterministic component $h _ { t }$ and a stochastic component $z _ { t }$ with diagonalcovariance Gaussian distribution. $h _ { t }$ is the recurrent state of a Gated Recurrent Unit (GRU) [11]. The encoder and decoder are convolutional neural networks (CNNs) and the remaining components are multi-layer perceptrons (MLPs). The world model is trained end-to-end by optimizing the evidence lower bound (ELBO) via stochastic backpropagation [28, 38] with the Adam optimizer [27]. ", + "bbox": [ + 173, + 738, + 825, + 809 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Explorer ", + "text_level": 1, + "bbox": [ + 174, + 816, + 274, + 832 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To efficiently explore, we seek out surprising states imagined by the world model [6, 41, 43, 44, 46], as opposed to retrospectively exploring by revisiting previously novel states [4, 5, 8, 34]. As the world model can predict model states that correspond to unseen situations in the environment, the imagined trajectories contain more novel goals, compared to model-free exploration that is limited to the replay buffer. To collect informative novel trajectories in the environment, we train an exploration ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1: initialize: World model $\\mathcal { M }$ , Replay buffer $\\mathcal { D }$ , Explorer $\\pi ^ { \\mathrm { e } } ( a _ { t } \\mid z _ { t } )$ , Achiever $\\pi ^ { \\mathbf { g } } ( a _ { t } \\mid z _ { t } , g )$ \n2: while exploring do \n3: Train $\\mathcal { M }$ on $\\mathcal { D }$ \n4: Train $\\pi ^ { \\mathrm { e } }$ in imagination of $\\mathcal { M }$ to maximize exploration rewards $\\textstyle \\sum _ { t } r _ { t } ^ { \\mathrm { e } }$ . \n5: Train $\\pi ^ { \\mathrm { g } }$ in imagination of $\\mathcal { M }$ to maximize $\\textstyle \\sum _ { t } r _ { t } ^ { \\mathrm { g } } ( z _ { t } , g )$ for images $g \\sim \\mathcal { D }$ . \n6: (Optional) Train $d ( z _ { i } , z _ { j } )$ to predict distances $j - i$ on the imagination data from last step. \n7: Deploy $\\pi ^ { \\mathrm { e } }$ in the environment to explore and grow $\\mathcal { D }$ . \n8: Deploy $\\pi ^ { \\mathrm { g } }$ in the environment to achieve a goal image $g \\sim \\mathcal { D }$ to grow $\\mathcal { D }$ . \n9: end while \n10: while evaluating do \n11: given: Evaluation goal $g$ \n12: Deploy $\\pi ^ { \\mathrm { g } }$ in the world to reach $g$ . \n13: end while ", + "bbox": [ + 178, + 112, + 808, + 319 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "policy $\\pi ^ { e }$ from the model states $s _ { t }$ in imagination of the world model to maximize an exploration reward: ", + "bbox": [ + 173, + 328, + 823, + 358 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7ba64821b38f462d0dca465160657471479c64f31661612c406c54c58e2ee0f1.jpg", + "text": "$$\n{ \\mathrm { E x p l o r e r : } } \\qquad \\pi ^ { e } ( a _ { t } \\mid s _ { t } ) \\qquad { \\mathrm { E x p l o r e r ~ V a l u e : } } \\qquad v ^ { e } ( s _ { t } )\n$$", + "text_format": "latex", + "bbox": [ + 308, + 362, + 689, + 380 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To explore the most informative model states, we estimate the epistemic uncertainty as a disagreement of an ensemble of transition functions. We train an ensemble of 1-step models to predict the next model state from the current model state. The ensemble model is trained alongside the world model on model states produced by the encoder $q _ { \\phi }$ . Because the ensemble models are initialized at random, they will differ, especially for inputs that they have not been trained on [29, 36]: ", + "bbox": [ + 173, + 381, + 825, + 452 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9eaf94fbc2a8cb25b1753cea7aae4403ee0ce1fa7e767fda661a545d5f5b9b04.jpg", + "text": "$$\n\\mathrm { E n s e m b l e : } \\quad f ( s _ { t } , \\theta ^ { k } ) = \\hat { z } _ { t + 1 } ^ { k } \\quad \\mathrm { f o r } \\quad k = 1 . . K\n$$", + "text_format": "latex", + "bbox": [ + 341, + 455, + 655, + 474 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Leveraging the ensemble, we estimate the epistemic uncertainty as the ensemble disagreement. The exploration reward is the variance of the ensemble predictions averaged across dimension of the model state, which approximates the expected information gain [3, 43]: ", + "bbox": [ + 173, + 491, + 825, + 534 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e563d21593e05baec600751267cd9bf05f0136899c02a8bd8b7c13d11096854c.jpg", + "text": "$$\nr _ { t } ^ { \\mathrm { e } } ( s _ { t } ) \\doteq \\frac { 1 } { N } \\sum _ { n } \\operatorname { V a r } _ { \\{ \\mathrm { k } \\} } \\left[ f ( s _ { t } , \\theta _ { k } ) \\right] _ { n }\n$$", + "text_format": "latex", + "bbox": [ + 377, + 536, + 620, + 573 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The explorer $\\pi ^ { e }$ maximizes the sum of future exploration rewards $\\boldsymbol { r } _ { t } ^ { e }$ using the Dreamer algorithm [24], which considers long-term rewards into the future by maximizing $\\lambda$ -returns under a learned value function. As a result, the explorer is trained to seek out situations are as informative as possible from imagined latent trajectories of the world model, and is periodically deployed in the environment to add novel trajectories to the replay buffer, so the world model and goal achiever policy can improve. ", + "bbox": [ + 173, + 574, + 825, + 643 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 Achiever ", + "text_level": 1, + "bbox": [ + 173, + 655, + 276, + 669 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To leverage the knowledge obtained by exploration for learning to reach goals, we train a goal achiever policy $\\pi ^ { g }$ that receives a model state and a goal as input. Our aim is to train a general policy that is capable of reaching many diverse goals. To achieve this in a data-efficient way, it is crucial that environment trajectories that were collected with one goal in mind are reused to also learn how to reach other goals. While prior work addressed this by goal relabeling which makes off-policy policy optimization a necessity [2], we instead leverage past trajectories via the world model trained on them that lets us generate an unlimited amount of new imagined trajectories for training the goal achiever on-policy in imagination. This simplifies policy optimization and can improve stability, while still sharing all collected experience across many goals. ", + "bbox": [ + 173, + 675, + 825, + 801 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/09edeeeee38f4f3ef02c2b840b130771ce97529492066b7d71e0d48de6e85f00.jpg", + "text": "$$\n\\pi ^ { g } ( a _ { t } \\mid s _ { t } , e _ { g } ) \\qquad \\mathrm { A c h i e v e r ~ V a l u e } ; \\qquad v ^ { g } ( s _ { t } , e _ { g } )\n$$", + "text_format": "latex", + "bbox": [ + 369, + 806, + 704, + 824 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To train the goal achiever, we sample a goal image $x _ { g }$ from the replay buffer and compute its embedding $e _ { g } = \\mathrm { e n c } _ { \\phi } ( x _ { g } )$ . The achiever aims to maximize an unsupervised goal-reaching reward $r ^ { g } ( s _ { t } , e _ { g } )$ . We discuss different choices for this reward in Section 2.4. We again use the Dreamer algorithm [24] for training, where now the value function also receives the goal embedding as input. In addition to imagination training, it can also be important to perform practice trials with the goal achiever in the true environment, so that any model inaccuracies along the goal reaching trajectories may be corrected. To perform practice trials, we sample a goal from the replay buffer and execute the goal achiever policy for that goal in the environment. These trials are interleaved with exploration episodes collected by the exploration policy in equal proportion. We note that the goal achiever learning is entirely unsupervised because the practice goals are simply images the agent encountered through exploration or during previous practice trails. ", + "bbox": [ + 173, + 825, + 825, + 912 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/8833211f2fc46aac053a0c19271c2751e3ee461c189b8c3a201cf26bcb97a95e.jpg", + "image_caption": [ + "Figure 4: Successful LEXA trajectories. When given a goal image from the test set, LEXA’s achiever is used in the environment to reach that image. On RoboKitchen, LEXA manipulates up to three different objects together from a single goal image (kettle, light switch, and cabinet). On RoboBins, LEXA performs temporally extended tasks such as picking and placing two objects in a row. " + ], + "image_footnote": [], + "bbox": [ + 174, + 89, + 825, + 256 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 330, + 825, + 401 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.4 Latent Distances ", + "text_level": 1, + "bbox": [ + 174, + 420, + 330, + 435 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Training the achiever policy requires us to define a goal achievement reward $r ^ { g } ( s _ { t } , e _ { g } )$ that measures how close the latent state $s _ { t }$ should be considered to the goal $e _ { g }$ . One simple measure is the cosine distance in the latent space obtained by inputting image observations into the world-model. However, such a distance function brings visually similar states together even if they could be farther apart in temporal manner as measured by actions needed to reach from one to other. This bias makes this suitable only to scenarios where most of pixels in the observations are directly controllable, e.g., trying to arrange robot’s body in certain shape, such as RoboYoga poses in Figure 2. However, many environments contain agent as well as the world, such as manipulation involves interacting with objects that are not directly controllable. The cosine distance would try matching the entire goal image, and thus places a large weight on both matching the robot and object positions with the desired goal. Since the robot position is directly controllable it is much easier to match, but this metric overly focuses on it, yielding poor policies that ignore objects. We address this is by using the number of timesteps it takes to move from one image to another as a distance measure [25, 26]. This ignores large changes in robot position, since these can be completed in very few steps, and will instead focus more on the objects. This temporal cost function can be learned purely in imagination rollouts from our world model allowing as much data as needed without taking any steps in the real world. ", + "bbox": [ + 173, + 448, + 825, + 669 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Cosine Distance To use cosine distance with LEXA, for a latent state $s _ { t }$ , and a goal embedding $e ^ { g }$ , we use the latent inference network $q$ to infer $s ^ { g }$ , and define the reward as the cosine similarity [54]: ", + "bbox": [ + 174, + 675, + 825, + 704 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e88f6979b53255a72be4873ad55353e204741ed389c04e6460ef8ffeee08d936.jpg", + "text": "$$\nr _ { t } ^ { g } ( s _ { t } , e _ { g } ) \\doteq \\sum _ { i } \\overline { { s } } _ { t i } \\overline { { s } } _ { g i } , \\quad \\mathrm { w h e r e } \\quad \\overline { { s } } _ { t } = s _ { t } / \\| s _ { t } \\| _ { 2 } , \\quad \\overline { { s } } _ { g } = s _ { g } / \\| s _ { g } \\| _ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 271, + 712, + 725, + 744 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "i.e. the cosine of the angle between the two vectors $s _ { t } , s _ { g }$ in the $N -$ dimensional latent space. ", + "bbox": [ + 171, + 747, + 785, + 763 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Temporal Distance To use temporal distances with LEXA, we train a neural network $d$ to predict the number of time steps between two embeddings. We train it by sampling pairs of states $s _ { t }$ , $s _ { t + k }$ from an imagined rollout of the achiever and predicting the distance $k$ . We implement the temporal distance in terms of predicted image embeddings $\\boldsymbol { \\hat { e } } _ { t + k }$ in order to remove extra recurrent information: ", + "bbox": [ + 173, + 768, + 825, + 825 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Predicted embedding: $\\mathrm { e m b } ( s _ { t } ) = \\hat { e } _ { t } \\approx e _ { t } { \\quad } \\mathrm { T e m p o r a l ~ d i s t a n c e } ; \\quad d _ { \\omega } ( \\hat { e } _ { t } , \\hat { e } _ { t + k } ) \\approx k / H ,$ where $H$ is the maximum distance equal to the imagination horizon. Training distance function only on imagination data from the same trajectory would cause it to predict poor distance to far away states coming from other trajectories, such as images that are impossible to reach during one episode. In order to incorporate learning signal from such far-away goals, we include them by sampling images from a different trajectory. We annotate these negative samples with the maximum possible distance, so that the agent always prefers images that were seen in the same trajectory. ", + "bbox": [ + 176, + 835, + 774, + 852 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 856, + 823, + 911 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/b02095cdb02ada544f65d40a4cf2610dc2db333769a1afaf5a6454034a90d495.jpg", + "image_caption": [ + "Figure 5: Coincidental goal success achieved during the unsupervised exploration phase. The forwardlooking explorer policy of LEXA results in substantially better coverage compared to SkewFit, a popular method for goal based exploration. " + ], + "image_footnote": [], + "bbox": [ + 174, + 90, + 820, + 265 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 168, + 328, + 825, + 357 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/3852fdf70971e5fa1441eba39e645683b4ca9de4a9fbc1050f24b0f3ed4a5bdf.jpg", + "text": "$$\nr _ { t } ^ { g } ( s _ { t } , e _ { g } ) = - d _ { \\omega } ( \\hat { e } _ { t } , e _ { g } ) , \\quad \\mathrm { w h e r e } \\quad \\hat { e } _ { t } = \\mathrm { e m b } ( s _ { t } ) , \\quad e _ { g } = \\mathrm { e n c } _ { \\phi } ( x _ { g } )\n$$", + "text_format": "latex", + "bbox": [ + 267, + 363, + 730, + 382 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The learned distance function depends on the training data policy. However, as the policy becomes more competent, the distance estimates will be closer to the optimal number of time steps to reach a particular goal, and the policy converges to the optimal solution [25]. LEXA always uses the latest data to train the distance function using imagination, ensuring that the convergence is fast. ", + "bbox": [ + 174, + 385, + 825, + 440 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 457, + 313, + 474 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our evaluation focuses on the following scientific questions: ", + "bbox": [ + 173, + 486, + 570, + 502 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1. Does LEXA outperform prior work on previous benchmarks and a new challenging benchmark? \n2. How does forward-looking exploration of goals compare to previous goal exploration strategies? \n3. How does the distance function affect the ability to reach goals in different types of environments? \n4. Can we train one general LEXA to control different robots across visually distinct environments? \n5. What components of LEXA are important for performance? ", + "bbox": [ + 173, + 506, + 826, + 588 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate LEXA on prior benchmarks used by SkewFit [37], DISCERN [50], and Plan2Explore [43] in Section 3.3. Since these benchmarks are largely saturated, we also introduce a new challenging benchmark shown in Figure 2. We evaluate LEXA on this benchmark is Section 3.2. ", + "bbox": [ + 173, + 592, + 825, + 635 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 Experimental setup ", + "text_level": 1, + "bbox": [ + 174, + 647, + 349, + 662 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As not many prior methods have shown success on reaching diverse goals from image inputs, we perform an apples-to-apples comparison by implementing the baselines using the same world model and policy optimization as our method: ", + "bbox": [ + 174, + 671, + 825, + 713 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "• SkewFit SkewFit [37] uses model-free hindsight experience replay and explores by sampling goals from the latent space of a variational autoencoder [28, 38]. Being one of the state-of-the-art agents, we use the original implementation that does not use a world model or explorer policy. \n• DDL Dynamic Distance Learning [25] trains a temporal distance function similar to our method. Following the original algorithm, DDL uses greedy exploration and trains the distance function on the replay buffer instead of in imagination. \n• DIAYN Diversity is All You Need [15] learns a latent skill space and uses mutual information between skills and reached states as the objective. We augment DIAYN with our explorer policy and train a learned skill predictor to obtain a skill for a given test image [12]. \n• GCSL Goal-Conditioned Supervised Learning [20] trains the goal policy on replay buffer goals and mimics the actions that previously led to the goal. We also augment GCSL with our explorer policy, as we found no learning success without it. ", + "bbox": [ + 179, + 717, + 826, + 892 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our new benchmark defines goal images for a diverse set of four existing environments as follows: ", + "bbox": [ + 171, + 897, + 818, + 911 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/1310ecf4c91afe8a97fdd0a2ff16844020c1a947729ce1b05b3d97a0c5844f2a.jpg", + "image_caption": [ + "Figure 6: Evaluation of goal reaching agents on our four benchmarks. A single agent is trained from images without rewards and then evaluated on reaching goal images from the test set (see Figure 1). Both LEXA agents solve many of the tasks and significantly outperform prior work. SkewFit and DLL struggle with exploration, while DIAYN and GCSL use our explorer but still are not able to learn a good downstream policy. Refer table 1 for final success percentage (averaged across tasks) for each method and benchmark domain. " + ], + "image_footnote": [], + "bbox": [ + 176, + 89, + 820, + 242 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "• RoboYoga We use the walker and quadruped domains of the DeepMind Control Suite [48] to define the RoboYoga benchmark, consisting of 12 goal images that correspond to different body poses for each of the two environments, such as lying down, standing up, and balancing. • RoboBins Based on MetaWorld [53], we create a scene with a Sawyer robotic arm, two bins, and two blocks of different colors. The goal images specify tasks that include reaching, manipulating only one block, and manipulating both blocks. • RoboKitchen The last benchmark involves the challenging kitchen environment from [22], where a franka robot can interact with various objects including a burner, light switch, sliding cabinet, hinge cabinet, microwave, or kettle. The goal images we include describe tasks that require interacting with only one object, as well as interacting with two objects. ", + "bbox": [ + 179, + 340, + 826, + 487 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 Performance on New Benchmark ", + "text_level": 1, + "bbox": [ + 176, + 502, + 442, + 517 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We show the results on our main benchmark in Figure 6 and include heatmaps that show per-task success on each of the evaluation tasks from the benchmarks in the Appendix. Further, we report success averaged across tasks for each domain at the end of training in Table 1. We visualize example successful trajectory executions for tasks that require manipulating multiple objects in Fig. 4. ", + "bbox": [ + 174, + 525, + 825, + 582 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "RoboYoga The environments in this benchmark are directly controllable since they contain no other objects except the robot. We recall that for such settings we expect the cosine distance to be effective, as perceptual distance is quite accurate. Training is thus faster compared to using learned temporal distances, where the metric is learned from scratch. From Table 1 and Figure 6 we see that this is indeed the case for these environments (Walker and Quadruped), as LEXA with the cosine metric outperforms all prior approaches. Furthermore with temporal distances LEXA makes better progress compared to prior work on a much larger number of goals as can be seen from the per-task performance (Figures ??, ??), even though average success over goals looks similar to that of DDL. ", + "bbox": [ + 173, + 584, + 825, + 695 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "RoboBins This environment involves interaction with block objects, and thus is not directly controllable, and so we expect LEXA to perform better with the temporal distance metric. From Table 1 and ", + "bbox": [ + 174, + 699, + 825, + 728 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/a408384474e00ab844d1211531dfdf5c3c91a48468085be2d8527eb2438c6fb6.jpg", + "table_caption": [], + "table_footnote": [ + "Table 1: Performance on our new challenging benchmark, spanning across the four domains shown in Figure 2. The number are goal success rates, averaged over test goals within each environment. " + ], + "table_body": "
MethodKitchenRoboBinsQuadrupedWalker
DDL0.0035.4222.5040.00
DIAYN0.0013.6913.810.28
GCSL0.007.9415.831.11
SkewFit0.2315.775.520.01
LEXA + Temporal (Ours)37.5069.4431.3936.72
LEXA + Cosine (Ours)6.0245.8356.1173.06
", + "bbox": [ + 232, + 738, + 756, + 867 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/1ef7c10b47d9d98a8e1a64e538f880461c2ee0273a63c335f48dbd65f5cbaaed.jpg", + "image_caption": [ + "Figure 7: Success rates on RoboBin. In line with the prior literature, previous methods are successful at reaching and sometimes pushing. LEXA pushes the state-of-the-art by picking and placing multiple objects to reach challenging goal images. Analogous heat maps for the other domains are included in the appendix. " + ], + "image_footnote": [], + "bbox": [ + 272, + 88, + 725, + 272 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 6, we see that LEXA gets higher average success than all prior approaches. Further from the per-task performance in 7, LEXA with the temporal distance metric is the only approach that makes progress on all goals in the benchmark. The main difference in performance between using temporal and cosine distance can be seen in the tasks involving two blocks, which are the most complex tasks in this environment (the last 3 columns of the per-task plot). The best performing prior method is DDL which solves reaching, and can perform simple pushing tasks. This method performs poorly due to poor exploration, as shown in Figure 5. We see that while other prior methods make some progress on reaching, they fail on harder tasks. ", + "bbox": [ + 173, + 347, + 825, + 458 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "RoboKitchen This benchmark involves diverse objects that require different manipulation behavior. From Table 1 and Figure 6 and ?? we find that LEXA with temporal distance is able to learn multiple RoboKitchen tasks, some of which require sequentially completing 2 tasks in the environment. All prior methods barely make progress due to the challenging nature of this benchmark, and furthermore using the cosine distance function makes very limited progress. The gap in performance between using the two distance functions is much larger in this environment compared to RoboBins since there are many more objects and they are not as clearly visible as the blocks. ", + "bbox": [ + 174, + 463, + 825, + 560 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Single Agent Across All Environments ", + "text_level": 1, + "bbox": [ + 176, + 564, + 442, + 579 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In the previous sections we have shown that our approach can achieve diverse goals in different environments. However, we trained a new agent for every new environment, which doesn’t scale well to large numbers of environments. Thus we investigate if we can train a train a single agent across four environments in the benchmark. From Figure ?? we see that our approach with learned temporal distance is able to make progress on tasks from RoboKitchen, RoboBins Reaching, RoboBins Pick & Place and Walker, while the best prior method on the single-environment tasks (DDL) mainly solves walker tasks and reaching from RoboBin. ", + "bbox": [ + 173, + 583, + 825, + 681 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.3 Performance on Prior benchmarks ", + "text_level": 1, + "bbox": [ + 176, + 699, + 455, + 713 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To further verify the results obtained on our benchmark, we evaluate LEXA on previously used benchmarks. We observe that LEXA significantly outperforms prior work on these benchmarks, and is often close to the optimal policy. Additional details are provided in ??????. ", + "bbox": [ + 174, + 723, + 517, + 806 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "SkewFit Benchmark SkewFit [37] introduces a robotic manipulation benchmark for unsupervised methods with simple tasks like planar pushing or picking. We evaluate on this benchmark in Table 2. ", + "bbox": [ + 173, + 811, + 517, + 866 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/618a99a8eebbc909bcc8480480e11f5c2c2e3cdb46ac0a839edabb1385d30fc5.jpg", + "table_caption": [ + "Table 2: Goal distance for SkewFit goals [37]. " + ], + "table_footnote": [], + "table_body": "
MethodPusherPickup
RIG [32]7.7cm3.7cm
RIG + HER [2]7.5cm3.5cm
Skew-Fit [37]4.9cm1.8cm
LEXA + Temporal2.3cm1.4cm
", + "bbox": [ + 529, + 748, + 825, + 842 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Baseline results are taken from [37]. LEXA significantly outperforms prior work on these tasks. Pushing and picking up blocks from images is largely solved and future work can focus on harder benchmarks such as those introduced in our paper. ", + "bbox": [ + 173, + 867, + 826, + 909 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "DISCERN Benchmark We attempted to replicate the tasks described in [50] that are based on simple two-dimensional robots [48]. While the original tasks are not released, we followed the procedure for generating the goals described in the paper. Despite following the exact procedure, we were not able to obtain similar goals to the ones used in the original paper. Nevertheless, we show the goal completion percentage results obtained with our reproduced evaluation compared to DISCERN results from the original paper. LEXA results were obtained with early stopping. In Table 3 we see that our agent solves many of the tasks in this benchmark. ", + "bbox": [ + 174, + 92, + 517, + 270 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/9dbe6942689ae58e63d21aca585954939140886d90393a2f8da727367c842a83.jpg", + "table_caption": [ + "Table 3: Success for DISCERN goals [50]. " + ], + "table_footnote": [], + "table_body": "
TaskLEXADISCERN
Cup84.0%76.5%
Cartpole35.9%21.3%
Finger40.9%21.8%
Pendulum79.1%75.7%
Pointmass83.2%49.6%
Reacher100.0%87.1%
", + "bbox": [ + 529, + 114, + 823, + 238 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Plan2Explore Benchmark We provide a comparison on the standard reward-based DM control tasks [48] in Table 4. To compare on this benchmark, we create goal images that correspond to the reward functions. This setup is arguably harder for our agent, but is much more practical. Note our agent never observes the reward function and only observes the goal at test time. Plan2Explore adapts to new tasks but it needs the reward function to be known at test time, while DrQV2 is an oracle agent that observes the reward at training time. Baseline results are taken from [43, 52]. LEXA results were obtained with early stopping. LEXA outperforms Plan2Explore on most tasks and even performs comparably to state of the art oracle agents (DrQ, DrQv2, Dreamer) that use true task rewards during training. ", + "bbox": [ + 174, + 275, + 517, + 473 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/51d5fc875ba9f8aa14d97f8359720c5d4e5ca2112bacb09ed20a084ec9bc5fc4.jpg", + "table_caption": [ + "Table 4: Zero-shot return on P2E tasks [43]. " + ], + "table_footnote": [], + "table_body": "
Task Zero-ShotLEXA [P2EDrQv2 X
Walker Stand957331968
Hopper Stand840841957
Cartpole Balance886950989
Cartpole Bal. Sparse996860983
Pendulum Swing Up788792837
Cup Catch969962909
Reacher Hard93766970
", + "bbox": [ + 529, + 299, + 823, + 452 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 468, + 718, + 481 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3.4 Analysis ", + "text_level": 1, + "bbox": [ + 174, + 496, + 271, + 510 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Prior work Most work we compared against struggles with exploration, such as SkewFit and DLL methods. DIAYN is augmented with our explorer, but still fails to leverage the exploration data to learn a diverse set of skills. GCSL struggles to fit the exploration data and produces behavior that does not solve the task, perhaps because the exploration data is too diverse. We observed that all baselines make progress on the simple reaching, but struggle with other tasks. We have experimented with several versions and improvements to the baselines and report the best obtained performance. ", + "bbox": [ + 174, + 518, + 517, + 683 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Ablation of different components We ablated components of LEXA on the RoboBins environment in Figure 8. Using a separate explorer policy crucial as without it the agent does not discover the more interesting tasks. Without negative sampling the agent learns slower, perhaps because the distance function doesn’t produce reasonable outputs when queried on images that are more than horizon length apart. Training the distance function with real data converges to slightly lower success than using imagination data, since real data is sampled in an off-policy manner due to its limited quantity. ", + "bbox": [ + 174, + 686, + 517, + 852 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/e74e622cfc1a92b2076d60433d82e8c9de54acaa1e72a8a8ad12dea1ef790bc6.jpg", + "image_caption": [ + "Figure 8: Ablations on RoboBins. A separate explorer is crucial for most tasks. Training temporal distance on negative samples speeds up learning, and both negative sampling and training in imagination as opposed to real data are important for the hardest tasks. " + ], + "image_footnote": [], + "bbox": [ + 534, + 522, + 807, + 731 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Exploration performance Due to importance of exploration, we further examine the diversity of the data collected during training. We log the instances where the agent coincidentally solves an evaluation task during exploration, for the RoboKitchen and RoboBins environments. In Figure 5, we see that our method encounters harder tasks involving multiple objects much more often. ", + "bbox": [ + 174, + 856, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 89, + 321, + 106 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Learning to Achieve Goals The problem of learning to reach many different goals has been commonly addressed with model-free methods that learn a single goal-conditioned policy [2, 26, 40]. Recent work has combined these approaches with various ways to generate training goals, such as asymmetric self-play [33, 45] or by sampling goals of intermediate difficulty [14, 18]. These approaches can achieve remarkable performance in simulated robotic domains, however, they focus on the settings where the agent can directly perceive the low-dimensional environment state. ", + "bbox": [ + 174, + 117, + 825, + 199 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A few works have attempted to scale these model-free methods to visual goals by using contrastive [50] or reconstructive [32, 37] representation learning. However, these approaches struggle to perform meaningful exploration as no clear reward signal is available to guide the agent toward solving interesting tasks. Some works [10, 49] avoid this challenge by using a large dataset of interesting behaviors. Other works [37, 55] attempt to explore by generating goals similar to those that have already been seen, but do not try to explore truly novel states. ", + "bbox": [ + 174, + 202, + 825, + 285 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A particularly relevant set of approaches used model-based methods to achieve goals via planning [13, 17] or learning model-regularized policies [35]. However, these approaches are limited by short planning horizons. In contrast, we learn long-horizon goal-conditioned value functions which allows us to solve more challenging tasks. More generally, most of the above approaches are limited by simplistic exploration, while our method leverages model imagination to search for novel states, which significantly improves exploration and in turn the downstream capabilities of the agent. ", + "bbox": [ + 174, + 287, + 825, + 371 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Learning Distance Functions A crucial challenge for visual goal reaching is the choice of the reward or the cost function for the goal achieving policy. Several approaches use representation learning to create a distance in the feature space [9, 32, 50, 51]. However, this naive distance may not be most reflective of how hard a particular goal is to reach. One line of research has proposed using the mutual information between the current state and the goal as the distance metric [1, 12, 15, 21], however, it remains to be seen whether this approach can scale to more complex tasks. ", + "bbox": [ + 173, + 373, + 825, + 457 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Other works proposed temporal distances that measure the amount of time it takes to reach the goal. One approach is to learn the distance with approximate dynamic programming using Q-learning methods [16, 19, 26]. Our distance function is most similar to Hartikainen et al. [25], who learn a temporal distance with supervised learning on recent policy experience. In contrast to [25], we always train the distance on-policy in imagination, and we further integrate this achiever policy into our latent explorer achiever framework to discover novel goals for the achiever to practice on. ", + "bbox": [ + 174, + 458, + 825, + 541 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 556, + 299, + 573 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We presented Latent Explorer Achiever (LEXA), an agent for unsupervised RL that explores its environment, learns to achieve the discovered goals, and solves image-based tasks in a zero-shot way. By planning for novelty in imagination, LEXA prospectively explores to discover meaningful behaviors in substantially more diverse environments than considered by prior work. Further, LEXA is able to solve challenging downstream tasks specified as images without any supervision such as rewards or demonstrations. By proposing a challenging benchmark and the first agent to achieve meaningful performance on these tasks, we hope to stimulate future research on unsupervised agents, which we believe are fundamentally more scalable than traditional agents that require a human to design the tasks and rewards for learning. ", + "bbox": [ + 174, + 579, + 825, + 704 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Many challenges remain for building unsupervised agents. Many tasks in our benchmark are still unsolved and there remains room for progress on the algorithmic side both for the world model and policy optimization. Further, it is important to demonstrate the benefits of unsupervised agents on real-world systems to verify their scalability. Finally, for widespread adoption, it is crucial to consider the problem of goal specification and design methods that act on goals that are easy to specify, such as via natural language. We believe LEXA will enable future work to tackle these goals effectively. ", + "bbox": [ + 174, + 705, + 825, + 789 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgements We thank Ben Eysenbach, Stephen Tian, Sergey Levine, Dinesh Jayaraman, Karl Pertsch, Ed Hu and the members of GRASP lab and Pathak lab for insightful discussions. We also thank Murtaza Dalal and Chuning Zhu for help with MuJoCo environments. Finally, DP would like to thank Laura Schulz, Josh Tenenbaum and Alison Gopnik for seeding the idea of goal-setting in children, and Pulkit Agrawal for several crucial discussions (over gelato) since then. OR and KD were supported by ARL DCIST CRA W911NF-17-2-0181, ONR N00014-17-1-2093, and by Honda Research Institute. This work was partially supported by GoodAI Research Award and DARPA Machine Common Sense grant. ", + "bbox": [ + 174, + 800, + 825, + 911 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References [1] J. Achiam, H. Edwards, D. 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It then executes those", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "imagined actions in the environment to discover interesting states without the need to generate them.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "Note these actions are likely to lead the agent to states which are several steps outside the frontier", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "because otherwise the model wouldn’t have had high uncertainty or information gain. 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At training, LEXA", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 596, + 507, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 507, + 610 + ], + "score": 1.0, + "content": "unlocks diverse data for goal reaching in environments where exploration is nontrivial. At test time,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "the achiever solves challenging locomotion and manipulation tasks provided as user-specified goal", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 619, + 325, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 325, + 631 + ], + "score": 1.0, + "content": "images. Our contributions are summarized as follows:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 573, + 507, + 631 + ] + }, + { + "type": "list", + "bbox": [ + 110, + 632, + 506, + 720 + ], + "lines": [ + { + "bbox": [ + 109, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 109, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "• We propose to learn separate explorer and achiever policies as an approach to overcome the", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 117, + 643, + 356, + 655 + ], + "spans": [ + { + "bbox": [ + 117, + 643, + 356, + 655 + ], + "score": 1.0, + "content": "exploration problem of unsupervised goal-conditioned RL.", + "type": "text" + } + ], + "index": 35, + "is_list_end_line": true + }, + { + "bbox": [ + 111, + 653, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 111, + 653, + 504, + 667 + ], + "score": 1.0, + "content": "• We show that forward-looking exploration by planning with a learned world model substantially", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 666, + 330, + 677 + ], + "spans": [ + { + "bbox": [ + 118, + 666, + 330, + 677 + ], + "score": 1.0, + "content": "outperforms previous strategies for goal exploration.", + "type": "text" + } + ], + "index": 37, + "is_list_end_line": true + }, + { + "bbox": [ + 111, + 675, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 111, + 675, + 506, + 689 + ], + "score": 1.0, + "content": "• To evaluate on challenging tasks, we introduce a new goal reaching benchmark with a total of 40", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 686, + 475, + 700 + ], + "spans": [ + { + "bbox": [ + 115, + 686, + 475, + 700 + ], + "score": 1.0, + "content": "diverse goal images across 4 different robot locomotion and manipulation environments.", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 109, + 697, + 507, + 711 + ], + "spans": [ + { + "bbox": [ + 109, + 697, + 507, + 711 + ], + "score": 1.0, + "content": "• LEXA outperforms prior methods, being the first to show success in the Kitchen robotic manipu-", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 708, + 471, + 721 + ], + "spans": [ + { + "bbox": [ + 118, + 708, + 471, + 721 + ], + "score": 1.0, + "content": "lation environment, and achieves goal images where multiple objects need to be moved.", + "type": "text" + } + ], + "index": 41, + "is_list_end_line": true + } + ], + "index": 37.5, + "bbox_fs": [ + 109, + 631, + 507, + 721 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 70, + 502, + 237 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 70, + 502, + 237 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 70, + 502, + 237 + ], + "spans": [ + { + "bbox": [ + 109, + 70, + 502, + 237 + ], + "score": 0.972, + "type": "image", + "image_path": "261b4f4a6881305424cba99bcd987917803949b8856632917f699dbdedfd6295.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 70, + 502, + 125.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 125.66666666666666, + 502, + 181.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 181.33333333333331, + 502, + 236.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 242, + 506, + 331 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "score": 1.0, + "content": "Figure 3: Latent Explorer Achiever (LEXA) learns a general world model that is used to train an", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "explorer and a goal achiever policy. The explorer (left) is trained on imagined latent state rollouts of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 172, + 277 + ], + "score": 1.0, + "content": "the world model", + "type": "text" + }, + { + "bbox": [ + 173, + 266, + 190, + 276 + ], + "score": 0.88, + "content": "s _ { t : T }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 264, + 349, + 277 + ], + "score": 1.0, + "content": "to maximize the disagreement objective", + "type": "text" + }, + { + "bbox": [ + 349, + 264, + 402, + 276 + ], + "score": 0.93, + "content": "r _ { t } ^ { e } = \\mathrm { V a r } ( \\bar { s ^ { \\prime } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 264, + 506, + 277 + ], + "score": 1.0, + "content": ". The goal achiever (right)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 205, + 288 + ], + "score": 1.0, + "content": "is conditioned on a goal", + "type": "text" + }, + { + "bbox": [ + 205, + 277, + 212, + 287 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "and is also trained on imagined rollouts to minimize a distance function", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 286, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 107, + 286, + 142, + 299 + ], + "score": 0.92, + "content": "d ( s _ { t } , e _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 286, + 506, + 300 + ], + "score": 1.0, + "content": ". Goals are sampled randomly from replay buffer images. For training a temporal distance,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "we use the imagined rollouts of the achiever and predict the number of time steps between each two", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "states. By combining forward-looking exploration and data-efficient training of the achiever, LEXA", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 318, + 434, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 434, + 333 + ], + "score": 1.0, + "content": "provides a simple and powerful solution for unsupervised reinforcement learning.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "title", + "bbox": [ + 107, + 336, + 302, + 351 + ], + "lines": [ + { + "bbox": [ + 104, + 336, + 303, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 336, + 303, + 354 + ], + "score": 1.0, + "content": "2 Latent Explorer Achiever (LEXA)", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 506, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "Our aim is to build an agent that can achieve arbitrary user-specified goals after learning in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "environment without any supervision. This presents two challenges - collecting trajectories that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "contain diverse goals and learning to achieve these goals when specified as a goal image. We introduce", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 394, + 507, + 410 + ], + "spans": [ + { + "bbox": [ + 104, + 394, + 507, + 410 + ], + "score": 1.0, + "content": "a simple solution based on a world model and imagination training that addresses both challenges.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "The world model represents the agent’s current knowledge about the environment and is used for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "training two policies, the explorer and the achiever. To explore novel situations, we construct an", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "estimate of which states the world model is still uncertain about. To achieve goals, we train the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "goal-conditioned achiever in imagination, using the images found so far as unsupervised goals. At test", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 450, + 507, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 507, + 464 + ], + "score": 1.0, + "content": "time, the achiever is deployed to reach user-specified goals. The training procedure is in Algorithm 1.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 475, + 187, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 188, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 188, + 488 + ], + "score": 1.0, + "content": "2.1 World Model", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "To efficiently predict potential outcomes of future actions in environments with high-dimensional", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "image inputs, we leverage a Recurrent State Space Model (RSSM) [23] that learns to predict forward", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "using compact model states that facilitate planning [7, 51]. 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The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "score": 1.0, + "content": "encoder and decoder are convolutional neural networks (CNNs) and the remaining components are", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "multi-layer perceptrons (MLPs). The world model is trained end-to-end by optimizing the evidence", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 628, + 477, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 477, + 642 + ], + "score": 1.0, + "content": "lower bound (ELBO) via stochastic backpropagation [28, 38] with the Adam optimizer [27].", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 647, + 168, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 171, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 171, + 662 + ], + "score": 1.0, + "content": "2.2 Explorer", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "To efficiently explore, we seek out surprising states imagined by the world model [6, 41, 43, 44, 46],", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "as opposed to retrospectively exploring by revisiting previously novel states [4, 5, 8, 34]. As the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "world model can predict model states that correspond to unseen situations in the environment, the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "imagined trajectories contain more novel goals, compared to model-free exploration that is limited to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "the replay buffer. 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The explorer (left) is trained on imagined latent state rollouts of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 172, + 277 + ], + "score": 1.0, + "content": "the world model", + "type": "text" + }, + { + "bbox": [ + 173, + 266, + 190, + 276 + ], + "score": 0.88, + "content": "s _ { t : T }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 264, + 349, + 277 + ], + "score": 1.0, + "content": "to maximize the disagreement objective", + "type": "text" + }, + { + "bbox": [ + 349, + 264, + 402, + 276 + ], + "score": 0.93, + "content": "r _ { t } ^ { e } = \\mathrm { V a r } ( \\bar { s ^ { \\prime } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 264, + 506, + 277 + ], + "score": 1.0, + "content": ". 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By combining forward-looking exploration and data-efficient training of the achiever, LEXA", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 318, + 434, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 434, + 333 + ], + "score": 1.0, + "content": "provides a simple and powerful solution for unsupervised reinforcement learning.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "title", + "bbox": [ + 107, + 336, + 302, + 351 + ], + "lines": [ + { + "bbox": [ + 104, + 336, + 303, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 336, + 303, + 354 + ], + "score": 1.0, + "content": "2 Latent Explorer Achiever (LEXA)", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 506, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "Our aim is to build an agent that can achieve arbitrary user-specified goals after learning in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "environment without any supervision. This presents two challenges - collecting trajectories that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "contain diverse goals and learning to achieve these goals when specified as a goal image. 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This simplifies policy optimization and can improve stability,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 354, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 354, + 636 + ], + "score": 1.0, + "content": "while still sharing all collected experience across many goals.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 639, + 431, + 653 + ], + "lines": [ + { + "bbox": [ + 226, + 639, + 431, + 653 + ], + "spans": [ + { + "bbox": [ + 226, + 639, + 431, + 653 + ], + "score": 0.81, + "content": "\\pi ^ { g } ( a _ { t } \\mid s _ { t } , e _ { g } ) \\qquad \\mathrm { A c h i e v e r ~ V a l u e } ; \\qquad v ^ { g } ( s _ { t } , e _ { g } )", + "type": "interline_equation", + "image_path": "09edeeeee38f4f3ef02c2b840b130771ce97529492066b7d71e0d48de6e85f00.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 226, + 639, + 431, + 653 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 325, + 667 + ], + "score": 1.0, + "content": "To train the goal achiever, we sample a goal image", + "type": "text" + }, + { + "bbox": [ + 325, + 656, + 337, + 666 + ], + "score": 0.86, + "content": "x _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "from the replay buffer and compute its", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 154, + 678 + ], + "score": 1.0, + "content": "embedding", + "type": "text" + }, + { + "bbox": [ + 154, + 665, + 215, + 677 + ], + "score": 0.92, + "content": "e _ { g } = \\mathrm { e n c } _ { \\phi } ( x _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 665, + 505, + 678 + ], + "score": 1.0, + "content": ". 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The", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "exploration reward is the variance of the ensemble predictions averaged across dimension of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 412, + 394, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 394, + 424 + ], + "score": 1.0, + "content": "model state, which approximates the expected information gain [3, 43]:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 390, + 505, + 424 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 425, + 380, + 454 + ], + "lines": [ + { + "bbox": [ + 231, + 425, + 380, + 454 + ], + "spans": [ + { + "bbox": [ + 231, + 425, + 380, + 454 + ], + "score": 0.95, + "content": "r _ { t } ^ { \\mathrm { e } } ( s _ { t } ) \\doteq \\frac { 1 } { N } \\sum _ { n } \\operatorname { V a r } _ { \\{ \\mathrm { k } \\} } \\left[ f ( s _ { t } , \\theta _ { k } ) \\right] _ { n }", + "type": "interline_equation", + "image_path": "e563d21593e05baec600751267cd9bf05f0136899c02a8bd8b7c13d11096854c.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 231, + 425, + 380, + 439.5 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 231, + 439.5, + 380, + 454.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 160, + 468 + ], + "score": 1.0, + "content": "The explorer", + "type": "text" + }, + { + "bbox": [ + 161, + 456, + 172, + 465 + ], + "score": 0.86, + "content": "\\pi ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 454, + 374, + 468 + ], + "score": 1.0, + "content": "maximizes the sum of future exploration rewards", + "type": "text" + }, + { + "bbox": [ + 374, + 456, + 384, + 467 + ], + "score": 0.88, + "content": "\\boldsymbol { r } _ { t } ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "using the Dreamer algorithm", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 399, + 478 + ], + "score": 1.0, + "content": "[24], which considers long-term rewards into the future by maximizing", + "type": "text" + }, + { + "bbox": [ + 399, + 466, + 406, + 476 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "-returns under a learned", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "value function. As a result, the explorer is trained to seek out situations are as informative as possible", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "from imagined latent trajectories of the world model, and is periodically deployed in the environment", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "to add novel trajectories to the replay buffer, so the world model and goal achiever policy can improve.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 454, + 506, + 511 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 519, + 169, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 170, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 170, + 532 + ], + "score": 1.0, + "content": "2.3 Achiever", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 535, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "To leverage the knowledge obtained by exploration for learning to reach goals, we train a goal", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 168, + 560 + ], + "score": 1.0, + "content": "achiever policy", + "type": "text" + }, + { + "bbox": [ + 168, + 547, + 180, + 557 + ], + "score": 0.85, + "content": "\\pi ^ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "that receives a model state and a goal as input. Our aim is to train a general policy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "that is capable of reaching many diverse goals. To achieve this in a data-efficient way, it is crucial", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "that environment trajectories that were collected with one goal in mind are reused to also learn how", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "score": 1.0, + "content": "to reach other goals. While prior work addressed this by goal relabeling which makes off-policy", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 591, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 603 + ], + "score": 1.0, + "content": "policy optimization a necessity [2], we instead leverage past trajectories via the world model trained", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "on them that lets us generate an unlimited amount of new imagined trajectories for training the goal", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 611, + 507, + 627 + ], + "spans": [ + { + "bbox": [ + 104, + 611, + 507, + 627 + ], + "score": 1.0, + "content": "achiever on-policy in imagination. This simplifies policy optimization and can improve stability,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 354, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 354, + 636 + ], + "score": 1.0, + "content": "while still sharing all collected experience across many goals.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 535, + 507, + 636 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 639, + 431, + 653 + ], + "lines": [ + { + "bbox": [ + 226, + 639, + 431, + 653 + ], + "spans": [ + { + "bbox": [ + 226, + 639, + 431, + 653 + ], + "score": 0.81, + "content": "\\pi ^ { g } ( a _ { t } \\mid s _ { t } , e _ { g } ) \\qquad \\mathrm { A c h i e v e r ~ V a l u e } ; \\qquad v ^ { g } ( s _ { t } , e _ { g } )", + "type": "interline_equation", + "image_path": "09edeeeee38f4f3ef02c2b840b130771ce97529492066b7d71e0d48de6e85f00.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 226, + 639, + 431, + 653 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 325, + 667 + ], + "score": 1.0, + "content": "To train the goal achiever, we sample a goal image", + "type": "text" + }, + { + "bbox": [ + 325, + 656, + 337, + 666 + ], + "score": 0.86, + "content": "x _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "from the replay buffer and compute its", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 154, + 678 + ], + "score": 1.0, + "content": "embedding", + "type": "text" + }, + { + "bbox": [ + 154, + 665, + 215, + 677 + ], + "score": 0.92, + "content": "e _ { g } = \\mathrm { e n c } _ { \\phi } ( x _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 665, + 505, + 678 + ], + "score": 1.0, + "content": ". The achiever aims to maximize an unsupervised goal-reaching reward", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 676, + 146, + 688 + ], + "score": 0.91, + "content": "r ^ { g } ( s _ { t } , e _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 676, + 506, + 689 + ], + "score": 1.0, + "content": ". We discuss different choices for this reward in Section 2.4. We again use the Dreamer", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 686, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 506, + 699 + ], + "score": 1.0, + "content": "algorithm [24] for training, where now the value function also receives the goal embedding as input.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "In addition to imagination training, it can also be important to perform practice trials with the goal", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "achiever in the true environment, so that any model inaccuracies along the goal reaching trajectories", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "may be corrected. To perform practice trials, we sample a goal from the replay buffer and execute the", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "goal achiever policy for that goal in the environment. These trials are interleaved with exploration", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "score": 1.0, + "content": "episodes collected by the exploration policy in equal proportion. We note that the goal achiever", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "learning is entirely unsupervised because the practice goals are simply images the agent encountered", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 307, + 322, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 322, + 318 + ], + "score": 1.0, + "content": "through exploration or during previous practice trails.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 653, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 71, + 505, + 203 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 71, + 505, + 203 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 71, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 107, + 71, + 505, + 203 + ], + "score": 0.959, + "type": "image", + "image_path": "8833211f2fc46aac053a0c19271c2751e3ee461c189b8c3a201cf26bcb97a95e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 71, + 505, + 115.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 115.0, + 505, + 159.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 159.0, + 505, + 203.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 506, + 253 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "Figure 4: Successful LEXA trajectories. When given a goal image from the test set, LEXA’s achiever", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "is used in the environment to reach that image. On RoboKitchen, LEXA manipulates up to three", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 230, + 507, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 507, + 244 + ], + "score": 1.0, + "content": "different objects together from a single goal image (kettle, light switch, and cabinet). On RoboBins,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 241, + 478, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 478, + 255 + ], + "score": 1.0, + "content": "LEXA performs temporally extended tasks such as picking and placing two objects in a row.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "may be corrected. To perform practice trials, we sample a goal from the replay buffer and execute the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "goal achiever policy for that goal in the environment. These trials are interleaved with exploration", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "score": 1.0, + "content": "episodes collected by the exploration policy in equal proportion. We note that the goal achiever", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "learning is entirely unsupervised because the practice goals are simply images the agent encountered", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 307, + 322, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 322, + 318 + ], + "score": 1.0, + "content": "through exploration or during previous practice trails.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 107, + 333, + 202, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 203, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 203, + 346 + ], + "score": 1.0, + "content": "2.4 Latent Distances", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 355, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 407, + 370 + ], + "score": 1.0, + "content": "Training the achiever policy requires us to define a goal achievement reward", + "type": "text" + }, + { + "bbox": [ + 408, + 355, + 447, + 367 + ], + "score": 0.93, + "content": "r ^ { g } ( s _ { t } , e _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 353, + 506, + 370 + ], + "score": 1.0, + "content": "that measures", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 210, + 379 + ], + "score": 1.0, + "content": "how close the latent state", + "type": "text" + }, + { + "bbox": [ + 210, + 368, + 219, + 377 + ], + "score": 0.86, + "content": "s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 366, + 353, + 379 + ], + "score": 1.0, + "content": "should be considered to the goal", + "type": "text" + }, + { + "bbox": [ + 353, + 367, + 363, + 378 + ], + "score": 0.85, + "content": "e _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 366, + 506, + 379 + ], + "score": 1.0, + "content": ". One simple measure is the cosine", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "distance in the latent space obtained by inputting image observations into the world-model. However,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "such a distance function brings visually similar states together even if they could be farther apart in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "temporal manner as measured by actions needed to reach from one to other. This bias makes this", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 408, + 507, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 507, + 424 + ], + "score": 1.0, + "content": "suitable only to scenarios where most of pixels in the observations are directly controllable, e.g.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "score": 1.0, + "content": "trying to arrange robot’s body in certain shape, such as RoboYoga poses in Figure 2. However, many", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "environments contain agent as well as the world, such as manipulation involves interacting with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "objects that are not directly controllable. The cosine distance would try matching the entire goal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 453, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 465 + ], + "score": 1.0, + "content": "image, and thus places a large weight on both matching the robot and object positions with the desired", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "goal. Since the robot position is directly controllable it is much easier to match, but this metric overly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "focuses on it, yielding poor policies that ignore objects. We address this is by using the number of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 485, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 499 + ], + "score": 1.0, + "content": "timesteps it takes to move from one image to another as a distance measure [25, 26]. This ignores", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "large changes in robot position, since these can be completed in very few steps, and will instead focus", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "more on the objects. This temporal cost function can be learned purely in imagination rollouts from", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 519, + 478, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 478, + 532 + ], + "score": 1.0, + "content": "our world model allowing as much data as needed without taking any steps in the real world.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 535, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 390, + 547 + ], + "score": 1.0, + "content": "Cosine Distance To use cosine distance with LEXA, for a latent state", + "type": "text" + }, + { + "bbox": [ + 390, + 538, + 399, + 546 + ], + "score": 0.85, + "content": "s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 534, + 492, + 547 + ], + "score": 1.0, + "content": ", and a goal embedding", + "type": "text" + }, + { + "bbox": [ + 492, + 536, + 502, + 545 + ], + "score": 0.8, + "content": "e ^ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 534, + 505, + 547 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 249, + 559 + ], + "score": 1.0, + "content": "we use the latent inference network", + "type": "text" + }, + { + "bbox": [ + 250, + 548, + 256, + 558 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 546, + 288, + 559 + ], + "score": 1.0, + "content": "to infer", + "type": "text" + }, + { + "bbox": [ + 289, + 547, + 299, + 556 + ], + "score": 0.86, + "content": "s ^ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 546, + 506, + 559 + ], + "score": 1.0, + "content": ", and define the reward as the cosine similarity [54]:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 166, + 564, + 444, + 590 + ], + "lines": [ + { + "bbox": [ + 166, + 564, + 444, + 590 + ], + "spans": [ + { + "bbox": [ + 166, + 564, + 444, + 590 + ], + "score": 0.91, + "content": "r _ { t } ^ { g } ( s _ { t } , e _ { g } ) \\doteq \\sum _ { i } \\overline { { s } } _ { t i } \\overline { { s } } _ { g i } , \\quad \\mathrm { w h e r e } \\quad \\overline { { s } } _ { t } = s _ { t } / \\| s _ { t } \\| _ { 2 } , \\quad \\overline { { s } } _ { g } = s _ { g } / \\| s _ { g } \\| _ { 2 } ,", + "type": "interline_equation", + "image_path": "e88f6979b53255a72be4873ad55353e204741ed389c04e6460ef8ffeee08d936.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 166, + 564, + 444, + 590 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 592, + 481, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 482, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 312, + 606 + ], + "score": 1.0, + "content": "i.e. the cosine of the angle between the two vectors", + "type": "text" + }, + { + "bbox": [ + 313, + 595, + 336, + 605 + ], + "score": 0.9, + "content": "s _ { t } , s _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 591, + 362, + 606 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 362, + 593, + 380, + 603 + ], + "score": 0.61, + "content": "N -", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 591, + 482, + 606 + ], + "score": 1.0, + "content": "dimensional latent space.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 457, + 622 + ], + "score": 1.0, + "content": "Temporal Distance To use temporal distances with LEXA, we train a neural network", + "type": "text" + }, + { + "bbox": [ + 457, + 610, + 464, + 619 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "to predict", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 619, + 504, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 471, + 634 + ], + "score": 1.0, + "content": "the number of time steps between two embeddings. We train it by sampling pairs of states", + "type": "text" + }, + { + "bbox": [ + 471, + 622, + 480, + 631 + ], + "score": 0.45, + "content": "s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 619, + 484, + 634 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 484, + 622, + 504, + 632 + ], + "score": 0.56, + "content": "s _ { t + k }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 380, + 643 + ], + "score": 1.0, + "content": "from an imagined rollout of the achiever and predicting the distance", + "type": "text" + }, + { + "bbox": [ + 380, + 631, + 387, + 641 + ], + "score": 0.75, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 631, + 505, + 643 + ], + "score": 1.0, + "content": ". We implement the temporal", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 642, + 507, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 300, + 655 + ], + "score": 1.0, + "content": "distance in terms of predicted image embeddings", + "type": "text" + }, + { + "bbox": [ + 300, + 642, + 320, + 654 + ], + "score": 0.93, + "content": "\\boldsymbol { \\hat { e } } _ { t + k }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 642, + 507, + 655 + ], + "score": 1.0, + "content": "in order to remove extra recurrent information:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 108, + 662, + 474, + 675 + ], + "lines": [ + { + "bbox": [ + 114, + 661, + 474, + 676 + ], + "spans": [ + { + "bbox": [ + 114, + 661, + 204, + 676 + ], + "score": 1.0, + "content": "Predicted embedding:", + "type": "text" + }, + { + "bbox": [ + 205, + 662, + 474, + 675 + ], + "score": 0.63, + "content": "\\mathrm { e m b } ( s _ { t } ) = \\hat { e } _ { t } \\approx e _ { t } { \\quad } \\mathrm { T e m p o r a l ~ d i s t a n c e } ; \\quad d _ { \\omega } ( \\hat { e } _ { t } , \\hat { e } _ { t + k } ) \\approx k / H ,", + "type": "inline_equation", + "image_path": "2a9f790a999cc35bd238617fefa28df50bee1b9ef233fbda5964339aa2f9983c.jpg" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 132, + 690 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 678, + 143, + 688 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "is the maximum distance equal to the imagination horizon. Training distance function only", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 690, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 700 + ], + "score": 1.0, + "content": "on imagination data from the same trajectory would cause it to predict poor distance to far away states", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "coming from other trajectories, such as images that are impossible to reach during one episode. In", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "score": 1.0, + "content": "order to incorporate learning signal from such far-away goals, we include them by sampling images", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 740, + 309, + 753 + ], + "spans": [ + { + "bbox": [ + 302, + 740, + 309, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 480, + 74, + 500, + 82 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 71, + 505, + 203 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 71, + 505, + 203 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 71, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 107, + 71, + 505, + 203 + ], + "score": 0.959, + "type": "image", + "image_path": "8833211f2fc46aac053a0c19271c2751e3ee461c189b8c3a201cf26bcb97a95e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 71, + 505, + 115.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 115.0, + 505, + 159.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 159.0, + 505, + 203.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 506, + 253 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "Figure 4: Successful LEXA trajectories. When given a goal image from the test set, LEXA’s achiever", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "is used in the environment to reach that image. On RoboKitchen, LEXA manipulates up to three", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 230, + 507, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 507, + 244 + ], + "score": 1.0, + "content": "different objects together from a single goal image (kettle, light switch, and cabinet). On RoboBins,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 241, + 478, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 478, + 255 + ], + "score": 1.0, + "content": "LEXA performs temporally extended tasks such as picking and placing two objects in a row.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 505, + 318 + ], + "lines": [], + "index": 9, + "bbox_fs": [ + 105, + 262, + 506, + 318 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 333, + 202, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 203, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 203, + 346 + ], + "score": 1.0, + "content": "2.4 Latent Distances", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 355, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 407, + 370 + ], + "score": 1.0, + "content": "Training the achiever policy requires us to define a goal achievement reward", + "type": "text" + }, + { + "bbox": [ + 408, + 355, + 447, + 367 + ], + "score": 0.93, + "content": "r ^ { g } ( s _ { t } , e _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 353, + 506, + 370 + ], + "score": 1.0, + "content": "that measures", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 210, + 379 + ], + "score": 1.0, + "content": "how close the latent state", + "type": "text" + }, + { + "bbox": [ + 210, + 368, + 219, + 377 + ], + "score": 0.86, + "content": "s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 366, + 353, + 379 + ], + "score": 1.0, + "content": "should be considered to the goal", + "type": "text" + }, + { + "bbox": [ + 353, + 367, + 363, + 378 + ], + "score": 0.85, + "content": "e _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 366, + 506, + 379 + ], + "score": 1.0, + "content": ". One simple measure is the cosine", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "distance in the latent space obtained by inputting image observations into the world-model. However,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "such a distance function brings visually similar states together even if they could be farther apart in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "temporal manner as measured by actions needed to reach from one to other. This bias makes this", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 408, + 507, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 507, + 424 + ], + "score": 1.0, + "content": "suitable only to scenarios where most of pixels in the observations are directly controllable, e.g.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "score": 1.0, + "content": "trying to arrange robot’s body in certain shape, such as RoboYoga poses in Figure 2. However, many", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "environments contain agent as well as the world, such as manipulation involves interacting with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "objects that are not directly controllable. The cosine distance would try matching the entire goal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 453, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 465 + ], + "score": 1.0, + "content": "image, and thus places a large weight on both matching the robot and object positions with the desired", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "goal. Since the robot position is directly controllable it is much easier to match, but this metric overly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "focuses on it, yielding poor policies that ignore objects. We address this is by using the number of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 485, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 499 + ], + "score": 1.0, + "content": "timesteps it takes to move from one image to another as a distance measure [25, 26]. This ignores", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "large changes in robot position, since these can be completed in very few steps, and will instead focus", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "more on the objects. This temporal cost function can be learned purely in imagination rollouts from", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 519, + 478, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 478, + 532 + ], + "score": 1.0, + "content": "our world model allowing as much data as needed without taking any steps in the real world.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 353, + 507, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 535, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 390, + 547 + ], + "score": 1.0, + "content": "Cosine Distance To use cosine distance with LEXA, for a latent state", + "type": "text" + }, + { + "bbox": [ + 390, + 538, + 399, + 546 + ], + "score": 0.85, + "content": "s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 534, + 492, + 547 + ], + "score": 1.0, + "content": ", and a goal embedding", + "type": "text" + }, + { + "bbox": [ + 492, + 536, + 502, + 545 + ], + "score": 0.8, + "content": "e ^ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 534, + 505, + 547 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 249, + 559 + ], + "score": 1.0, + "content": "we use the latent inference network", + "type": "text" + }, + { + "bbox": [ + 250, + 548, + 256, + 558 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 546, + 288, + 559 + ], + "score": 1.0, + "content": "to infer", + "type": "text" + }, + { + "bbox": [ + 289, + 547, + 299, + 556 + ], + "score": 0.86, + "content": "s ^ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 546, + 506, + 559 + ], + "score": 1.0, + "content": ", and define the reward as the cosine similarity [54]:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 534, + 506, + 559 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 166, + 564, + 444, + 590 + ], + "lines": [ + { + "bbox": [ + 166, + 564, + 444, + 590 + ], + "spans": [ + { + "bbox": [ + 166, + 564, + 444, + 590 + ], + "score": 0.91, + "content": "r _ { t } ^ { g } ( s _ { t } , e _ { g } ) \\doteq \\sum _ { i } \\overline { { s } } _ { t i } \\overline { { s } } _ { g i } , \\quad \\mathrm { w h e r e } \\quad \\overline { { s } } _ { t } = s _ { t } / \\| s _ { t } \\| _ { 2 } , \\quad \\overline { { s } } _ { g } = s _ { g } / \\| s _ { g } \\| _ { 2 } ,", + "type": "interline_equation", + "image_path": "e88f6979b53255a72be4873ad55353e204741ed389c04e6460ef8ffeee08d936.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 166, + 564, + 444, + 590 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 592, + 481, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 482, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 312, + 606 + ], + "score": 1.0, + "content": "i.e. the cosine of the angle between the two vectors", + "type": "text" + }, + { + "bbox": [ + 313, + 595, + 336, + 605 + ], + "score": 0.9, + "content": "s _ { t } , s _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 591, + 362, + 606 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 362, + 593, + 380, + 603 + ], + "score": 0.61, + "content": "N -", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 591, + 482, + 606 + ], + "score": 1.0, + "content": "dimensional latent space.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 591, + 482, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 457, + 622 + ], + "score": 1.0, + "content": "Temporal Distance To use temporal distances with LEXA, we train a neural network", + "type": "text" + }, + { + "bbox": [ + 457, + 610, + 464, + 619 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "to predict", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 619, + 504, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 471, + 634 + ], + "score": 1.0, + "content": "the number of time steps between two embeddings. We train it by sampling pairs of states", + "type": "text" + }, + { + "bbox": [ + 471, + 622, + 480, + 631 + ], + "score": 0.45, + "content": "s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 619, + 484, + 634 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 484, + 622, + 504, + 632 + ], + "score": 0.56, + "content": "s _ { t + k }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 380, + 643 + ], + "score": 1.0, + "content": "from an imagined rollout of the achiever and predicting the distance", + "type": "text" + }, + { + "bbox": [ + 380, + 631, + 387, + 641 + ], + "score": 0.75, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 631, + 505, + 643 + ], + "score": 1.0, + "content": ". We implement the temporal", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 642, + 507, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 300, + 655 + ], + "score": 1.0, + "content": "distance in terms of predicted image embeddings", + "type": "text" + }, + { + "bbox": [ + 300, + 642, + 320, + 654 + ], + "score": 0.93, + "content": "\\boldsymbol { \\hat { e } } _ { t + k }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 642, + 507, + 655 + ], + "score": 1.0, + "content": "in order to remove extra recurrent information:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 608, + 507, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 662, + 474, + 675 + ], + "lines": [ + { + "bbox": [ + 114, + 661, + 474, + 676 + ], + "spans": [ + { + "bbox": [ + 114, + 661, + 204, + 676 + ], + "score": 1.0, + "content": "Predicted embedding:", + "type": "text" + }, + { + "bbox": [ + 205, + 662, + 474, + 675 + ], + "score": 0.63, + "content": "\\mathrm { e m b } ( s _ { t } ) = \\hat { e } _ { t } \\approx e _ { t } { \\quad } \\mathrm { T e m p o r a l ~ d i s t a n c e } ; \\quad d _ { \\omega } ( \\hat { e } _ { t } , \\hat { e } _ { t + k } ) \\approx k / H ,", + "type": "inline_equation", + "image_path": "2a9f790a999cc35bd238617fefa28df50bee1b9ef233fbda5964339aa2f9983c.jpg" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 132, + 690 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 678, + 143, + 688 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "is the maximum distance equal to the imagination horizon. Training distance function only", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 690, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 700 + ], + "score": 1.0, + "content": "on imagination data from the same trajectory would cause it to predict poor distance to far away states", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "coming from other trajectories, such as images that are impossible to reach during one episode. In", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "score": 1.0, + "content": "order to incorporate learning signal from such far-away goals, we include them by sampling images", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 258, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 507, + 274 + ], + "score": 1.0, + "content": "from a different trajectory. We annotate these negative samples with the maximum possible distance,", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 271, + 415, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 415, + 284 + ], + "score": 1.0, + "content": "so that the agent always prefers images that were seen in the same trajectory.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 37, + "bbox_fs": [ + 114, + 661, + 474, + 676 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 504, + 722 + ], + "lines": [], + "index": 39.5, + "bbox_fs": [ + 105, + 678, + 506, + 725 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 72, + 502, + 210 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 72, + 502, + 210 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 72, + 502, + 210 + ], + "spans": [ + { + "bbox": [ + 107, + 72, + 502, + 210 + ], + "score": 0.953, + "type": "image", + "image_path": "b02095cdb02ada544f65d40a4cf2610dc2db333769a1afaf5a6454034a90d495.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 72, + 502, + 118.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 118.0, + 502, + 164.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 164.0, + 502, + 210.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 219, + 504, + 254 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "Figure 5: Coincidental goal success achieved during the unsupervised exploration phase. The forward-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "looking explorer policy of LEXA results in substantially better coverage compared to SkewFit, a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 242, + 281, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 281, + 254 + ], + "score": 1.0, + "content": "popular method for goal based exploration.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 103, + 260, + 505, + 283 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 507, + 274 + ], + "score": 1.0, + "content": "from a different trajectory. We annotate these negative samples with the maximum possible distance,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 271, + 415, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 415, + 284 + ], + "score": 1.0, + "content": "so that the agent always prefers images that were seen in the same trajectory.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 288, + 447, + 303 + ], + "lines": [ + { + "bbox": [ + 164, + 288, + 447, + 303 + ], + "spans": [ + { + "bbox": [ + 164, + 288, + 447, + 303 + ], + "score": 0.89, + "content": "r _ { t } ^ { g } ( s _ { t } , e _ { g } ) = - d _ { \\omega } ( \\hat { e } _ { t } , e _ { g } ) , \\quad \\mathrm { w h e r e } \\quad \\hat { e } _ { t } = \\mathrm { e m b } ( s _ { t } ) , \\quad e _ { g } = \\mathrm { e n c } _ { \\phi } ( x _ { g } )", + "type": "interline_equation", + "image_path": "3852fdf70971e5fa1441eba39e645683b4ca9de4a9fbc1050f24b0f3ed4a5bdf.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 164, + 288, + 447, + 303 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "The learned distance function depends on the training data policy. However, as the policy becomes", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "more competent, the distance estimates will be closer to the optimal number of time steps to reach a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 326, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 338 + ], + "score": 1.0, + "content": "particular goal, and the policy converges to the optimal solution [25]. LEXA always uses the latest", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 337, + 468, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 468, + 349 + ], + "score": 1.0, + "content": "data to train the distance function using imagination, ensuring that the convergence is fast.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 107, + 362, + 192, + 376 + ], + "lines": [ + { + "bbox": [ + 103, + 359, + 193, + 379 + ], + "spans": [ + { + "bbox": [ + 103, + 359, + 193, + 379 + ], + "score": 1.0, + "content": "3 Experiments", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 349, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 350, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 350, + 399 + ], + "score": 1.0, + "content": "Our evaluation focuses on the following scientific questions:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 401, + 506, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "1. Does LEXA outperform prior work on previous benchmarks and a new challenging benchmark?", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "score": 1.0, + "content": "2. How does forward-looking exploration of goals compare to previous goal exploration strategies?", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "3. How does the distance function affect the ability to reach goals in different types of environments?", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "score": 1.0, + "content": "4. Can we train one general LEXA to control different robots across visually distinct environments?", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 455, + 361, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 361, + 466 + ], + "score": 1.0, + "content": "5. What components of LEXA are important for performance?", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 469, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "score": 1.0, + "content": "We evaluate LEXA on prior benchmarks used by SkewFit [37], DISCERN [50], and Plan2Explore", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 478, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 493 + ], + "score": 1.0, + "content": "[43] in Section 3.3. Since these benchmarks are largely saturated, we also introduce a new challenging", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 491, + 446, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 446, + 503 + ], + "score": 1.0, + "content": "benchmark shown in Figure 2. We evaluate LEXA on this benchmark is Section 3.2.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 107, + 513, + 214, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 215, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 215, + 528 + ], + "score": 1.0, + "content": "3.1 Experimental setup", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "As not many prior methods have shown success on reaching diverse goals from image inputs, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "perform an apples-to-apples comparison by implementing the baselines using the same world model", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 554, + 266, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 266, + 565 + ], + "score": 1.0, + "content": "and policy optimization as our method:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 110, + 568, + 506, + 707 + ], + "lines": [ + { + "bbox": [ + 110, + 567, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 110, + 567, + 505, + 582 + ], + "score": 1.0, + "content": "• SkewFit SkewFit [37] uses model-free hindsight experience replay and explores by sampling", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 118, + 580, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 118, + 580, + 506, + 592 + ], + "score": 1.0, + "content": "goals from the latent space of a variational autoencoder [28, 38]. 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The forward-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "looking explorer policy of LEXA results in substantially better coverage compared to SkewFit, a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 242, + 281, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 281, + 254 + ], + "score": 1.0, + "content": "popular method for goal based exploration.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 103, + 260, + 505, + 283 + ], + "lines": [], + "index": 6.5, + "bbox_fs": [ + 105, + 258, + 507, + 284 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 288, + 447, + 303 + ], + "lines": [ + { + "bbox": [ + 164, + 288, + 447, + 303 + ], + "spans": [ + { + "bbox": [ + 164, + 288, + 447, + 303 + ], + "score": 0.89, + "content": "r _ { t } ^ { g } ( s _ { t } , e _ { g } ) = - d _ { \\omega } ( \\hat { e } _ { t } , e _ { g } ) , \\quad \\mathrm { w h e r e } \\quad \\hat { e } _ { t } = \\mathrm { e m b } ( s _ { t } ) , \\quad e _ { g } = \\mathrm { e n c } _ { \\phi } ( x _ { g } )", + "type": "interline_equation", + "image_path": "3852fdf70971e5fa1441eba39e645683b4ca9de4a9fbc1050f24b0f3ed4a5bdf.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 164, + 288, + 447, + 303 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "The learned distance function depends on the training data policy. 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Does LEXA outperform prior work on previous benchmarks and a new challenging benchmark?", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "score": 1.0, + "content": "2. How does forward-looking exploration of goals compare to previous goal exploration strategies?", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "3. How does the distance function affect the ability to reach goals in different types of environments?", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "score": 1.0, + "content": "4. Can we train one general LEXA to control different robots across visually distinct environments?", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 455, + 361, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 361, + 466 + ], + "score": 1.0, + "content": "5. What components of LEXA are important for performance?", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + } + ], + "index": 17, + "bbox_fs": [ + 105, + 401, + 506, + 466 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 469, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "score": 1.0, + "content": "We evaluate LEXA on prior benchmarks used by SkewFit [37], DISCERN [50], and Plan2Explore", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 478, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 493 + ], + "score": 1.0, + "content": "[43] in Section 3.3. 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MethodKitchenRoboBinsQuadrupedWalker
DDL0.0035.4222.5040.00
DIAYN0.0013.6913.810.28
GCSL0.007.9415.831.11
SkewFit0.2315.775.520.01
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The number are goal success rates, averaged over test goals within each environment.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + } + ], + "index": 36.25 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 167, + 70, + 444, + 216 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 167, + 70, + 444, + 216 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 167, + 70, + 444, + 216 + ], + "spans": [ + { + "bbox": [ + 167, + 70, + 444, + 216 + ], + "score": 0.958, + "type": "image", + "image_path": "1ef7c10b47d9d98a8e1a64e538f880461c2ee0273a63c335f48dbd65f5cbaaed.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 167, + 70, + 444, + 118.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 167, + 118.66666666666666, + 444, + 167.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 167, + 167.33333333333331, + 444, + 215.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 222, + 505, + 266 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "Figure 7: Success rates on RoboBin. In line with the prior literature, previous methods are successful", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "at reaching and sometimes pushing. LEXA pushes the state-of-the-art by picking and placing multiple", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 243, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 258 + ], + "score": 1.0, + "content": "objects to reach challenging goal images. 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Further from the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "per-task performance in 7, LEXA with the temporal distance metric is the only approach that makes", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "progress on all goals in the benchmark. The main difference in performance between using temporal", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "and cosine distance can be seen in the tasks involving two blocks, which are the most complex tasks", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "in this environment (the last 3 columns of the per-task plot). The best performing prior method is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "score": 1.0, + "content": "DDL which solves reaching, and can perform simple pushing tasks. This method performs poorly", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "due to poor exploration, as shown in Figure 5. We see that while other prior methods make some", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 353, + 295, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 295, + 363 + ], + "score": 1.0, + "content": "progress on reaching, they fail on harder tasks.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "RoboKitchen This benchmark involves diverse objects that require different manipulation behavior.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "From Table 1 and Figure 6 and ?? we find that LEXA with temporal distance is able to learn multiple", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "RoboKitchen tasks, some of which require sequentially completing 2 tasks in the environment. All", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "score": 1.0, + "content": "prior methods barely make progress due to the challenging nature of this benchmark, and furthermore", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "using the cosine distance function makes very limited progress. The gap in performance between", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "using the two distance functions is much larger in this environment compared to RoboBins since", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 433, + 414, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 414, + 444 + ], + "score": 1.0, + "content": "there are many more objects and they are not as clearly visible as the blocks.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 447, + 271, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 272, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 272, + 460 + ], + "score": 1.0, + "content": "Single Agent Across All Environments", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "In the previous sections we have shown that our approach can achieve diverse goals in different", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "environments. However, we trained a new agent for every new environment, which doesn’t scale well", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "to large numbers of environments. Thus we investigate if we can train a train a single agent across", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "four environments in the benchmark. From Figure ?? we see that our approach with learned temporal", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "distance is able to make progress on tasks from RoboKitchen, RoboBins Reaching, RoboBins Pick &", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "Place and Walker, while the best prior method on the single-environment tasks (DDL) mainly solves", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 529, + 274, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 274, + 540 + ], + "score": 1.0, + "content": "walker tasks and reaching from RoboBin.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 108, + 554, + 279, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 279, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 279, + 566 + ], + "score": 1.0, + "content": "3.3 Performance on Prior benchmarks", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 317, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 573, + 318, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 318, + 585 + ], + "score": 1.0, + "content": "To further verify the results obtained on our bench-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 584, + 318, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 318, + 596 + ], + "score": 1.0, + "content": "mark, we evaluate LEXA on previously used bench-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 318, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 318, + 608 + ], + "score": 1.0, + "content": "marks. We observe that LEXA significantly outper-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 606, + 317, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 317, + 618 + ], + "score": 1.0, + "content": "forms prior work on these benchmarks, and is often", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 617, + 317, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 317, + 629 + ], + "score": 1.0, + "content": "close to the optimal policy. Additional details are", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 628, + 190, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 190, + 638 + ], + "score": 1.0, + "content": "provided in ??????.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 317, + 686 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 317, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 317, + 655 + ], + "score": 1.0, + "content": "SkewFit Benchmark SkewFit [37] introduces a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 654, + 317, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 317, + 666 + ], + "score": 1.0, + "content": "robotic manipulation benchmark for unsupervised", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 664, + 317, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 317, + 677 + ], + "score": 1.0, + "content": "methods with simple tasks like planar pushing or", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 676, + 318, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 318, + 687 + ], + "score": 1.0, + "content": "picking. We evaluate on this benchmark in Table 2.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "table", + "bbox": [ + 324, + 593, + 505, + 667 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 326, + 576, + 503, + 587 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 324, + 573, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 324, + 573, + 505, + 589 + ], + "score": 1.0, + "content": "Table 2: Goal distance for SkewFit goals [37].", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "table_body", + "bbox": [ + 324, + 593, + 505, + 667 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 324, + 593, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 324, + 593, + 505, + 667 + ], + "score": 0.977, + "html": "
MethodPusherPickup
RIG [32]7.7cm3.7cm
RIG + HER [2]7.5cm3.5cm
Skew-Fit [37]4.9cm1.8cm
LEXA + Temporal2.3cm1.4cm
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LEXA significantly outperforms prior work on these tasks.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 696, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 506, + 711 + ], + "score": 1.0, + "content": "Pushing and picking up blocks from images is largely solved and future work can focus on harder", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 708, + 309, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 309, + 721 + ], + "score": 1.0, + "content": "benchmarks such as those introduced in our paper.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 167, + 70, + 444, + 216 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 167, + 70, + 444, + 216 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 167, + 70, + 444, + 216 + ], + "spans": [ + { + "bbox": [ + 167, + 70, + 444, + 216 + ], + "score": 0.958, + "type": "image", + "image_path": "1ef7c10b47d9d98a8e1a64e538f880461c2ee0273a63c335f48dbd65f5cbaaed.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 167, + 70, + 444, + 118.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 167, + 118.66666666666666, + 444, + 167.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 167, + 167.33333333333331, + 444, + 215.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 222, + 505, + 266 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "Figure 7: Success rates on RoboBin. In line with the prior literature, previous methods are successful", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "at reaching and sometimes pushing. LEXA pushes the state-of-the-art by picking and placing multiple", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 243, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 258 + ], + "score": 1.0, + "content": "objects to reach challenging goal images. 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Further from the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "per-task performance in 7, LEXA with the temporal distance metric is the only approach that makes", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "progress on all goals in the benchmark. The main difference in performance between using temporal", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "and cosine distance can be seen in the tasks involving two blocks, which are the most complex tasks", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "in this environment (the last 3 columns of the per-task plot). The best performing prior method is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "score": 1.0, + "content": "DDL which solves reaching, and can perform simple pushing tasks. This method performs poorly", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "due to poor exploration, as shown in Figure 5. We see that while other prior methods make some", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 353, + 295, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 295, + 363 + ], + "score": 1.0, + "content": "progress on reaching, they fail on harder tasks.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 275, + 506, + 363 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "RoboKitchen This benchmark involves diverse objects that require different manipulation behavior.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "From Table 1 and Figure 6 and ?? we find that LEXA with temporal distance is able to learn multiple", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "RoboKitchen tasks, some of which require sequentially completing 2 tasks in the environment. All", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "score": 1.0, + "content": "prior methods barely make progress due to the challenging nature of this benchmark, and furthermore", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "using the cosine distance function makes very limited progress. The gap in performance between", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "using the two distance functions is much larger in this environment compared to RoboBins since", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 433, + 414, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 414, + 444 + ], + "score": 1.0, + "content": "there are many more objects and they are not as clearly visible as the blocks.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 366, + 506, + 444 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 447, + 271, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 272, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 272, + 460 + ], + "score": 1.0, + "content": "Single Agent Across All Environments", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "In the previous sections we have shown that our approach can achieve diverse goals in different", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "environments. However, we trained a new agent for every new environment, which doesn’t scale well", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "to large numbers of environments. Thus we investigate if we can train a train a single agent across", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "four environments in the benchmark. From Figure ?? we see that our approach with learned temporal", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "distance is able to make progress on tasks from RoboKitchen, RoboBins Reaching, RoboBins Pick &", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "Place and Walker, while the best prior method on the single-environment tasks (DDL) mainly solves", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 529, + 274, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 274, + 540 + ], + "score": 1.0, + "content": "walker tasks and reaching from RoboBin.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 462, + 506, + 540 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 554, + 279, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 279, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 279, + 566 + ], + "score": 1.0, + "content": "3.3 Performance on Prior benchmarks", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 317, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 573, + 318, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 318, + 585 + ], + "score": 1.0, + "content": "To further verify the results obtained on our bench-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 584, + 318, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 318, + 596 + ], + "score": 1.0, + "content": "mark, we evaluate LEXA on previously used bench-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 318, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 318, + 608 + ], + "score": 1.0, + "content": "marks. We observe that LEXA significantly outper-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 606, + 317, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 317, + 618 + ], + "score": 1.0, + "content": "forms prior work on these benchmarks, and is often", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 617, + 317, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 317, + 629 + ], + "score": 1.0, + "content": "close to the optimal policy. Additional details are", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 628, + 190, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 190, + 638 + ], + "score": 1.0, + "content": "provided in ??????.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 573, + 318, + 638 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 317, + 686 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 317, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 317, + 655 + ], + "score": 1.0, + "content": "SkewFit Benchmark SkewFit [37] introduces a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 654, + 317, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 317, + 666 + ], + "score": 1.0, + "content": "robotic manipulation benchmark for unsupervised", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 664, + 317, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 317, + 677 + ], + "score": 1.0, + "content": "methods with simple tasks like planar pushing or", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 676, + 318, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 318, + 687 + ], + "score": 1.0, + "content": "picking. We evaluate on this benchmark in Table 2.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 106, + 642, + 318, + 687 + ] + }, + { + "type": "table", + "bbox": [ + 324, + 593, + 505, + 667 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 326, + 576, + 503, + 587 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 324, + 573, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 324, + 573, + 505, + 589 + ], + "score": 1.0, + "content": "Table 2: Goal distance for SkewFit goals [37].", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "table_body", + "bbox": [ + 324, + 593, + 505, + 667 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 324, + 593, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 324, + 593, + 505, + 667 + ], + "score": 0.977, + "html": "
MethodPusherPickup
RIG [32]7.7cm3.7cm
RIG + HER [2]7.5cm3.5cm
Skew-Fit [37]4.9cm1.8cm
LEXA + Temporal2.3cm1.4cm
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LEXA significantly outperforms prior work on these tasks.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 696, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 506, + 711 + ], + "score": 1.0, + "content": "Pushing and picking up blocks from images is largely solved and future work can focus on harder", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 708, + 309, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 309, + 721 + ], + "score": 1.0, + "content": "benchmarks such as those introduced in our paper.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 686, + 506, + 721 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 317, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 317, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 317, + 84 + ], + "score": 1.0, + "content": "DISCERN Benchmark We attempted to replicate", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 317, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 317, + 96 + ], + "score": 1.0, + "content": "the tasks described in [50] that are based on simple", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 94, + 317, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 317, + 106 + ], + "score": 1.0, + "content": "two-dimensional robots [48]. While the original tasks", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 317, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 317, + 118 + ], + "score": 1.0, + "content": "are not released, we followed the procedure for gener-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 318, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 318, + 129 + ], + "score": 1.0, + "content": "ating the goals described in the paper. Despite follow-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 317, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 317, + 139 + ], + "score": 1.0, + "content": "ing the exact procedure, we were not able to obtain", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 318, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 318, + 150 + ], + "score": 1.0, + "content": "similar goals to the ones used in the original paper.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 318, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 318, + 162 + ], + "score": 1.0, + "content": "Nevertheless, we show the goal completion percent-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 161, + 317, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 317, + 172 + ], + "score": 1.0, + "content": "age results obtained with our reproduced evaluation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 170, + 318, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 318, + 183 + ], + "score": 1.0, + "content": "compared to DISCERN results from the original pa-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 181, + 318, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 318, + 195 + ], + "score": 1.0, + "content": "per. LEXA results were obtained with early stopping.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 192, + 317, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 317, + 205 + ], + "score": 1.0, + "content": "In Table 3 we see that our agent solves many of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 204, + 205, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 205, + 214 + ], + "score": 1.0, + "content": "tasks in this benchmark.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6 + }, + { + "type": "table", + "bbox": [ + 324, + 91, + 504, + 189 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 328, + 74, + 500, + 86 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 326, + 72, + 501, + 87 + ], + "spans": [ + { + "bbox": [ + 326, + 72, + 501, + 87 + ], + "score": 1.0, + "content": "Table 3: Success for DISCERN goals [50].", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "table_body", + "bbox": [ + 324, + 91, + 504, + 189 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 324, + 91, + 504, + 189 + ], + "spans": [ + { + "bbox": [ + 324, + 91, + 504, + 189 + ], + "score": 0.979, + "html": "
TaskLEXADISCERN
Cup84.0%76.5%
Cartpole35.9%21.3%
Finger40.9%21.8%
Pendulum79.1%75.7%
Pointmass83.2%49.6%
Reacher100.0%87.1%
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Task Zero-ShotLEXA [P2EDrQv2 X
Walker Stand957331968
Hopper Stand840841957
Cartpole Balance886950989
Cartpole Bal. Sparse996860983
Pendulum Swing Up788792837
Cup Catch969962909
Reacher Hard93766970
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Using a separate explorer policy crucial", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 577, + 318, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 318, + 589 + ], + "score": 1.0, + "content": "as without it the agent does not discover the more in-", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 588, + 317, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 317, + 600 + ], + "score": 1.0, + "content": "teresting tasks. 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A separate", + "type": "text" + } + ], + "index": 85 + }, + { + "bbox": [ + 323, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 323, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "explorer is crucial for most tasks. 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We log the instances where the agent coincidentally solves an", + "type": "text" + } + ], + "index": 93 + }, + { + "bbox": [ + 106, + 700, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 507, + 712 + ], + "score": 1.0, + "content": "evaluation task during exploration, for the RoboKitchen and RoboBins environments. In Figure 5,", + "type": "text" + } + ], + "index": 94 + }, + { + "bbox": [ + 105, + 711, + 476, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 476, + 723 + ], + "score": 1.0, + "content": "we see that our method encounters harder tasks involving multiple objects much more often.", + "type": "text" + } + ], + "index": 95 + } + ], + "index": 93.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 317, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 317, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 317, + 84 + ], + "score": 1.0, + "content": "DISCERN Benchmark We attempted to replicate", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 317, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 317, + 96 + ], + "score": 1.0, + "content": "the tasks described in [50] that are based on simple", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 94, + 317, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 317, + 106 + ], + "score": 1.0, + "content": "two-dimensional robots [48]. While the original tasks", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 317, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 317, + 118 + ], + "score": 1.0, + "content": "are not released, we followed the procedure for gener-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 318, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 318, + 129 + ], + "score": 1.0, + "content": "ating the goals described in the paper. Despite follow-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 317, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 317, + 139 + ], + "score": 1.0, + "content": "ing the exact procedure, we were not able to obtain", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 318, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 318, + 150 + ], + "score": 1.0, + "content": "similar goals to the ones used in the original paper.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 318, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 318, + 162 + ], + "score": 1.0, + "content": "Nevertheless, we show the goal completion percent-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 161, + 317, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 317, + 172 + ], + "score": 1.0, + "content": "age results obtained with our reproduced evaluation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 170, + 318, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 318, + 183 + ], + "score": 1.0, + "content": "compared to DISCERN results from the original pa-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 181, + 318, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 318, + 195 + ], + "score": 1.0, + "content": "per. LEXA results were obtained with early stopping.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 192, + 317, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 317, + 205 + ], + "score": 1.0, + "content": "In Table 3 we see that our agent solves many of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 204, + 205, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 205, + 214 + ], + "score": 1.0, + "content": "tasks in this benchmark.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 72, + 318, + 214 + ] + }, + { + "type": "table", + "bbox": [ + 324, + 91, + 504, + 189 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 328, + 74, + 500, + 86 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 326, + 72, + 501, + 87 + ], + "spans": [ + { + "bbox": [ + 326, + 72, + 501, + 87 + ], + "score": 1.0, + "content": "Table 3: Success for DISCERN goals [50].", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "table_body", + "bbox": [ + 324, + 91, + 504, + 189 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 324, + 91, + 504, + 189 + ], + "spans": [ + { + "bbox": [ + 324, + 91, + 504, + 189 + ], + "score": 0.979, + "html": "
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To compare on this benchmark, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 250, + 317, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 317, + 262 + ], + "score": 1.0, + "content": "create goal images that correspond to the reward func-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 261, + 317, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 317, + 273 + ], + "score": 1.0, + "content": "tions. This setup is arguably harder for our agent, but", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 272, + 318, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 318, + 284 + ], + "score": 1.0, + "content": "is much more practical. Note our agent never ob-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 283, + 317, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 317, + 294 + ], + "score": 1.0, + "content": "serves the reward function and only observes the goal", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 294, + 317, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 317, + 305 + ], + "score": 1.0, + "content": "at test time. Plan2Explore adapts to new tasks but it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 305, + 318, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 318, + 316 + ], + "score": 1.0, + "content": "needs the reward function to be known at test time,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 316, + 317, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 317, + 327 + ], + "score": 1.0, + "content": "while DrQV2 is an oracle agent that observes the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 327, + 317, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 317, + 338 + ], + "score": 1.0, + "content": "reward at training time. Baseline results are taken", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 337, + 317, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 317, + 349 + ], + "score": 1.0, + "content": "from [43, 52]. 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Task Zero-ShotLEXA [P2EDrQv2 X
Walker Stand957331968
Hopper Stand840841957
Cartpole Balance886950989
Cartpole Bal. Sparse996860983
Pendulum Swing Up788792837
Cup Catch969962909
Reacher Hard93766970
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Using a separate explorer policy crucial", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 577, + 318, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 318, + 589 + ], + "score": 1.0, + "content": "as without it the agent does not discover the more in-", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 588, + 317, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 317, + 600 + ], + "score": 1.0, + "content": "teresting tasks. 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We log the instances where the agent coincidentally solves an", + "type": "text" + } + ], + "index": 93 + }, + { + "bbox": [ + 106, + 700, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 507, + 712 + ], + "score": 1.0, + "content": "evaluation task during exploration, for the RoboKitchen and RoboBins environments. In Figure 5,", + "type": "text" + } + ], + "index": 94 + }, + { + "bbox": [ + 105, + 711, + 476, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 476, + 723 + ], + "score": 1.0, + "content": "we see that our method encounters harder tasks involving multiple objects much more often.", + "type": "text" + } + ], + "index": 95 + } + ], + "index": 93.5, + "bbox_fs": [ + 105, + 678, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 197, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 198, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 198, + 85 + ], + "score": 1.0, + "content": "4 Related Work", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 93, + 505, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 92, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 105 + ], + "score": 1.0, + "content": "Learning to Achieve Goals The problem of learning to reach many different goals has been", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "commonly addressed with model-free methods that learn a single goal-conditioned policy [2, 26, 40].", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 128 + ], + "score": 1.0, + "content": "Recent work has combined these approaches with various ways to generate training goals, such", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "as asymmetric self-play [33, 45] or by sampling goals of intermediate difficulty [14, 18]. These", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "approaches can achieve remarkable performance in simulated robotic domains, however, they focus", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 477, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 477, + 159 + ], + "score": 1.0, + "content": "on the settings where the agent can directly perceive the low-dimensional environment state.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "A few works have attempted to scale these model-free methods to visual goals by using contrastive", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "[50] or reconstructive [32, 37] representation learning. However, these approaches struggle to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "perform meaningful exploration as no clear reward signal is available to guide the agent toward", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 194, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 205 + ], + "score": 1.0, + "content": "solving interesting tasks. Some works [10, 49] avoid this challenge by using a large dataset of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "interesting behaviors. Other works [37, 55] attempt to explore by generating goals similar to those", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 392, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 392, + 227 + ], + "score": 1.0, + "content": "that have already been seen, but do not try to explore truly novel states.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 505, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 242 + ], + "score": 1.0, + "content": "A particularly relevant set of approaches used model-based methods to achieve goals via planning", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "[13, 17] or learning model-regularized policies [35]. However, these approaches are limited by short", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "planning horizons. In contrast, we learn long-horizon goal-conditioned value functions which allows", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 262, + 504, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 504, + 273 + ], + "score": 1.0, + "content": "us to solve more challenging tasks. More generally, most of the above approaches are limited by", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 271, + 507, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 507, + 285 + ], + "score": 1.0, + "content": "simplistic exploration, while our method leverages model imagination to search for novel states,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 282, + 483, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 483, + 295 + ], + "score": 1.0, + "content": "which significantly improves exploration and in turn the downstream capabilities of the agent.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 296, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "Learning Distance Functions A crucial challenge for visual goal reaching is the choice of the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "reward or the cost function for the goal achieving policy. Several approaches use representation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "learning to create a distance in the feature space [9, 32, 50, 51]. However, this naive distance may not", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "be most reflective of how hard a particular goal is to reach. One line of research has proposed using", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "the mutual information between the current state and the goal as the distance metric [1, 12, 15, 21],", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 350, + 453, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 453, + 362 + ], + "score": 1.0, + "content": "however, it remains to be seen whether this approach can scale to more complex tasks.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "Other works proposed temporal distances that measure the amount of time it takes to reach the goal.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "One approach is to learn the distance with approximate dynamic programming using Q-learning", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "methods [16, 19, 26]. Our distance function is most similar to Hartikainen et al. [25], who learn", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "a temporal distance with supervised learning on recent policy experience. In contrast to [25], we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "always train the distance on-policy in imagination, and we further integrate this achiever policy into", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 417, + 482, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 482, + 431 + ], + "score": 1.0, + "content": "our latent explorer achiever framework to discover novel goals for the achiever to practice on.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 107, + 441, + 183, + 454 + ], + "lines": [ + { + "bbox": [ + 104, + 438, + 185, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 185, + 457 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 459, + 504, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 504, + 471 + ], + "score": 1.0, + "content": "We presented Latent Explorer Achiever (LEXA), an agent for unsupervised RL that explores its", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "environment, learns to achieve the discovered goals, and solves image-based tasks in a zero-shot", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "way. By planning for novelty in imagination, LEXA prospectively explores to discover meaningful", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "behaviors in substantially more diverse environments than considered by prior work. Further, LEXA", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "is able to solve challenging downstream tasks specified as images without any supervision such as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "rewards or demonstrations. By proposing a challenging benchmark and the first agent to achieve", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "meaningful performance on these tasks, we hope to stimulate future research on unsupervised agents,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "score": 1.0, + "content": "which we believe are fundamentally more scalable than traditional agents that require a human to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 545, + 274, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 274, + 560 + ], + "score": 1.0, + "content": "design the tasks and rewards for learning.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "Many challenges remain for building unsupervised agents. Many tasks in our benchmark are still", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "unsolved and there remains room for progress on the algorithmic side both for the world model and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "policy optimization. Further, it is important to demonstrate the benefits of unsupervised agents on", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 593, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 604 + ], + "score": 1.0, + "content": "real-world systems to verify their scalability. Finally, for widespread adoption, it is crucial to consider", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "the problem of goal specification and design methods that act on goals that are easy to specify, such", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 613, + 504, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 504, + 627 + ], + "score": 1.0, + "content": "as via natural language. We believe LEXA will enable future work to tackle these goals effectively.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 507, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 507, + 648 + ], + "score": 1.0, + "content": "Acknowledgements We thank Ben Eysenbach, Stephen Tian, Sergey Levine, Dinesh Jayaraman,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "score": 1.0, + "content": "Karl Pertsch, Ed Hu and the members of GRASP lab and Pathak lab for insightful discussions. We", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "also thank Murtaza Dalal and Chuning Zhu for help with MuJoCo environments. Finally, DP would", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 665, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 681 + ], + "score": 1.0, + "content": "like to thank Laura Schulz, Josh Tenenbaum and Alison Gopnik for seeding the idea of goal-setting", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "in children, and Pulkit Agrawal for several crucial discussions (over gelato) since then. OR and KD", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "were supported by ARL DCIST CRA W911NF-17-2-0181, ONR N00014-17-1-2093, and by Honda", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "Research Institute. This work was partially supported by GoodAI Research Award and DARPA", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 710, + 235, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 235, + 723 + ], + "score": 1.0, + "content": "Machine Common Sense grant.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 50.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 742, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 197, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 198, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 198, + 85 + ], + "score": 1.0, + "content": "4 Related Work", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 93, + 505, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 92, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 105 + ], + "score": 1.0, + "content": "Learning to Achieve Goals The problem of learning to reach many different goals has been", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "commonly addressed with model-free methods that learn a single goal-conditioned policy [2, 26, 40].", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 128 + ], + "score": 1.0, + "content": "Recent work has combined these approaches with various ways to generate training goals, such", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "as asymmetric self-play [33, 45] or by sampling goals of intermediate difficulty [14, 18]. These", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "approaches can achieve remarkable performance in simulated robotic domains, however, they focus", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 477, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 477, + 159 + ], + "score": 1.0, + "content": "on the settings where the agent can directly perceive the low-dimensional environment state.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 92, + 506, + 159 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "A few works have attempted to scale these model-free methods to visual goals by using contrastive", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "[50] or reconstructive [32, 37] representation learning. However, these approaches struggle to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "perform meaningful exploration as no clear reward signal is available to guide the agent toward", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 194, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 205 + ], + "score": 1.0, + "content": "solving interesting tasks. Some works [10, 49] avoid this challenge by using a large dataset of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "interesting behaviors. Other works [37, 55] attempt to explore by generating goals similar to those", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 392, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 392, + 227 + ], + "score": 1.0, + "content": "that have already been seen, but do not try to explore truly novel states.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 160, + 505, + 227 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 505, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 242 + ], + "score": 1.0, + "content": "A particularly relevant set of approaches used model-based methods to achieve goals via planning", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "[13, 17] or learning model-regularized policies [35]. However, these approaches are limited by short", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "planning horizons. In contrast, we learn long-horizon goal-conditioned value functions which allows", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 262, + 504, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 504, + 273 + ], + "score": 1.0, + "content": "us to solve more challenging tasks. More generally, most of the above approaches are limited by", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 271, + 507, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 507, + 285 + ], + "score": 1.0, + "content": "simplistic exploration, while our method leverages model imagination to search for novel states,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 282, + 483, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 483, + 295 + ], + "score": 1.0, + "content": "which significantly improves exploration and in turn the downstream capabilities of the agent.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 226, + 507, + 295 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 296, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "Learning Distance Functions A crucial challenge for visual goal reaching is the choice of the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "reward or the cost function for the goal achieving policy. Several approaches use representation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "learning to create a distance in the feature space [9, 32, 50, 51]. However, this naive distance may not", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "be most reflective of how hard a particular goal is to reach. One line of research has proposed using", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "the mutual information between the current state and the goal as the distance metric [1, 12, 15, 21],", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 350, + 453, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 453, + 362 + ], + "score": 1.0, + "content": "however, it remains to be seen whether this approach can scale to more complex tasks.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 295, + 506, + 362 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "Other works proposed temporal distances that measure the amount of time it takes to reach the goal.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "One approach is to learn the distance with approximate dynamic programming using Q-learning", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "methods [16, 19, 26]. Our distance function is most similar to Hartikainen et al. [25], who learn", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "a temporal distance with supervised learning on recent policy experience. In contrast to [25], we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "always train the distance on-policy in imagination, and we further integrate this achiever policy into", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 417, + 482, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 482, + 431 + ], + "score": 1.0, + "content": "our latent explorer achiever framework to discover novel goals for the achiever to practice on.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 363, + 506, + 431 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 441, + 183, + 454 + ], + "lines": [ + { + "bbox": [ + 104, + 438, + 185, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 185, + 457 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 459, + 504, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 504, + 471 + ], + "score": 1.0, + "content": "We presented Latent Explorer Achiever (LEXA), an agent for unsupervised RL that explores its", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "environment, learns to achieve the discovered goals, and solves image-based tasks in a zero-shot", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "way. By planning for novelty in imagination, LEXA prospectively explores to discover meaningful", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "behaviors in substantially more diverse environments than considered by prior work. Further, LEXA", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "is able to solve challenging downstream tasks specified as images without any supervision such as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "rewards or demonstrations. By proposing a challenging benchmark and the first agent to achieve", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "meaningful performance on these tasks, we hope to stimulate future research on unsupervised agents,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "score": 1.0, + "content": "which we believe are fundamentally more scalable than traditional agents that require a human to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 545, + 274, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 274, + 560 + ], + "score": 1.0, + "content": "design the tasks and rewards for learning.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 459, + 506, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "Many challenges remain for building unsupervised agents. Many tasks in our benchmark are still", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "unsolved and there remains room for progress on the algorithmic side both for the world model and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "policy optimization. Further, it is important to demonstrate the benefits of unsupervised agents on", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 593, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 604 + ], + "score": 1.0, + "content": "real-world systems to verify their scalability. Finally, for widespread adoption, it is crucial to consider", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "the problem of goal specification and design methods that act on goals that are easy to specify, such", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 613, + 504, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 504, + 627 + ], + "score": 1.0, + "content": "as via natural language. We believe LEXA will enable future work to tackle these goals effectively.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 559, + 506, + 627 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 507, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 507, + 648 + ], + "score": 1.0, + "content": "Acknowledgements We thank Ben Eysenbach, Stephen Tian, Sergey Levine, Dinesh Jayaraman,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "score": 1.0, + "content": "Karl Pertsch, Ed Hu and the members of GRASP lab and Pathak lab for insightful discussions. We", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "also thank Murtaza Dalal and Chuning Zhu for help with MuJoCo environments. Finally, DP would", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 665, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 681 + ], + "score": 1.0, + "content": "like to thank Laura Schulz, Josh Tenenbaum and Alison Gopnik for seeding the idea of goal-setting", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "in children, and Pulkit Agrawal for several crucial discussions (over gelato) since then. OR and KD", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "were supported by ARL DCIST CRA W911NF-17-2-0181, ONR N00014-17-1-2093, and by Honda", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "Research Institute. 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initialize: World model M,Replay buffer D,Explorer πe(at | zt),Achiever π(at| Zt, g)
1: 2:while exploring do
3:Train Mon D
4:Train πe in imagination of M to maximize exploration rewards ∑t re.
5:Train πg in imagination of M to maximize ∑t rε(zt, g) for images g ~ D.
6:(Optional) Train d(zi, zj） to predict distances j -i on the imagination data from last step.
7:Deploy πe in the environment to explore and grow D.
8: 9:Deploy πg in the environment to achieve a goal image g ~ D to grow D.
end while 10: while evaluating do
given: Evaluation goal g
11: 12:Deploy πg in the world to reach g.
13: end while
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DDL0.0035.4222.5040.00
DIAYN0.0013.6913.810.28
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SkewFit0.2315.775.520.01
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MethodPusherPickup
RIG [32]7.7cm3.7cm
RIG + HER [2]7.5cm3.5cm
Skew-Fit [37]4.9cm1.8cm
LEXA + Temporal2.3cm1.4cm
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TaskLEXADISCERN
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Cartpole35.9%21.3%
Finger40.9%21.8%
Pendulum79.1%75.7%
Pointmass83.2%49.6%
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Task Zero-ShotLEXA [P2EDrQv2 X
Walker Stand957331968
Hopper Stand840841957
Cartpole Balance886950989
Cartpole Bal. Sparse996860983
Pendulum Swing Up788792837
Cup Catch969962909
Reacher Hard93766970
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0000000000000000000000000000000000000000..c73a3cb964102ce2e5434cae8e1bd2b1b0c9ba17 --- /dev/null +++ b/parse/train/B1nLkl-0Z/B1nLkl-0Z.md @@ -0,0 +1,368 @@ +# LEARNING GAUSSIAN POLICIES FROM SMOOTHED ACTION VALUE FUNCTIONS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +State-action value functions (i.e., Q-values) are ubiquitous in reinforcement learning (RL), giving rise to popular algorithms such as SARSA and Q-learning. We propose a new notion of action value defined by a Gaussian smoothed version of the expected Q-value. We show that such smoothed Q-values still satisfy a Bellman equation, making them learnable from experience sampled from an environment. Moreover, the gradients of expected reward with respect to the mean and covariance of a parameterized Gaussian policy can be recovered from the gradient and Hessian of the smoothed Q-value function. Based on these relationships we develop new algorithms for training a Gaussian policy directly from a learned smoothed Q-value approximator. Our approach is amenable to proximal optimization techniques by augmenting the objective with a penalty on KLdivergence from a previous policy. We find that the ability to learn both a mean and covariance during training allows this approach to achieve much better results on standard continuous control benchmarks. + +# 1 INTRODUCTION + +Model-free reinforcement learning algorithms often alternate between two concurrent but interacting processes: (1) policy evaluation, where an action value function (i.e., a Q-value) is updated to obtain a better estimate of the return associated with taking a specific action, and (2) policy improvement, where the policy is updated aiming to maximize the current value function. In the past, different notions of Q-value have led to distinct but important families of RL methods. For example, SARSA (Rummery & Niranjan, 1994; Sutton & Barto, 1998; Van Seijen et al., 2009) uses the expected Q-value, defined as the expected return of following the current policy. Q-learning (Watkins, 1989) exploits a hard-max notion of Q-value, defined as the expected return of following an optimal policy. Soft Q-learning (Haarnoja et al., 2017) and PCL (Nachum et al., 2017a) both use a soft-max form of Q-value, defined as the future return of following an optimal entropy regularized policy. Clearly, the choice of Q-value function has a considerable effect on the resulting algorithm; for example, restricting the types of policies that can be expressed, and determining the type of exploration that can be naturally applied. + +In this work we introduce a new notion of action value: the smoothed action value function ${ \tilde { Q } } ^ { \pi }$ . Unlike previous notions, which associate a value with a specific action at each state, the smoothed Qvalue associates a value with a specific distribution over actions. In particular, the smoothed Q-value of a state-action pair $( s , a )$ is defined as the expected return of first taking an action sampled from a normal distribution $N ( \dot { a } , \Sigma ( s ) )$ , centered at $a$ , then following actions sampled from the current policy thereafter. In this way, the smoothed Q-value can also be interpreted as a Gaussian-smoothed or noisy version of the expected Q-value. + +We show that smoothed Q-values possess a number of interesting properties that make them attractive for use in RL algorithms. For one, the smoothed Q-values satisfy a single-step Bellman consistency, which allows bootstrapping to be used to train a function approximator. Secondly, for Gaussian policies, the standard optimization objective (expected return) can be expressed in terms of smoothed Q-values. Moreover, the gradient of this objective with respect to the mean and covariance of the Gaussian policy is equivalent to the gradient and the Hessian of the smoothed Q-value function, which allows one to derive updates to the policy parameters by having access to the derivatives of a sufficiently accurate smoothed Q-value function. + +This observation leads us to propose an algorithm called Smoothie, which in the spirit of (Deep) Deterministic Policy Gradient (DDPG) (Silver et al., 2014; Lillicrap et al., 2016), trains a policy using the derivatives of a trained (smoothed) Q-value function, thus avoiding the high-variance of stochastic updates used in standard policy gradient algorithms (Williams & Peng, 1991; Konda & Tsitsiklis, 2000). Unlike DDPG, which is well-known to have poor exploratory behavior (Haarnoja et al., 2017), the approach we develop is able to utilize a non-deterministic Gaussian policy parameterized by both a mean and a covariance, thus allowing the policy to be exploratory by default and alleviating the need for excessive hyperparameter tuning. + +Furthermore, we show that Smoothie can be easily adapted to incorporate proximal policy optimization techniques by augmenting the objective with a penalty on KL-divergence from a previous version of the policy. The inclusion of a KL-penalty is not feasible in the standard DDPG algorithm, but we show that it is possible with our formulation, and it significantly improves stability and overall performance. On standard continuous control benchmarks, our results are competitive with or exceed state-of-the-art, especially for more difficult tasks in the low-data regime. + +# 2 NOTATION & BACKGROUND + +We consider the standard model-free RL framework, where an agent interacts with a stochastic black-box environment by sequentially observing the state of the environment, emitting an action, and receiving a reward feedback; the goal is to find an agent that achieves maximal cumulative discounted reward. This problem can be expressed in terms of a Markov decision process (MDP) that consists of a state space $s$ and an action space $\mathcal { A }$ , where at iteration $t$ the agent encounters a state $s _ { t } ~ \in ~ S$ and emits an action $a _ { t } \in \mathcal A$ , after which the environment returns a scalar reward $\boldsymbol { r } _ { t } \sim R ( s _ { t } , \boldsymbol { a } _ { t } )$ and places the agent in a new state $s _ { t + 1 } \sim P ( s _ { t } , a _ { t } )$ . + +We model the behavior of the agent using a stochastic policy $\pi$ that produces a distribution over feasible actions at each state $s$ as $\pi ( a \mid s )$ . The optimization objective (expected discounted return), as a function of the policy, can then be expressed in terms of the expected action value function $Q ^ { \pi } ( s , a )$ by, + +$$ +O _ { \tt E R } ( \pi ) = \int _ { \cal S } \int _ { \cal A } \pi ( a \mid s ) Q ^ { \pi } ( s , a ) \mathrm { d } a \mathrm { d } \rho ^ { \pi } ( s ) , +$$ + +where $\rho ^ { \pi } ( s )$ is the stationary distribution of the states under $\pi$ , and $Q ^ { \pi } ( s , a )$ is recursively defined using the Bellman equation, + +$$ +Q ^ { \pi } ( s , a ) = \mathbb { E } _ { r , s ^ { \prime } } \left[ r + \gamma \int _ { \mathcal { A } } Q ^ { \pi } ( s ^ { \prime } , a ^ { \prime } ) \pi ( a ^ { \prime } \mid s ^ { \prime } ) \mathrm { d } a \right] \mathrm { , } +$$ + +where $\gamma \in [ 0 , 1 ]$ is the discount factor. For brevity, we will often suppress explicit denotation of the sampling distribution $R$ over immediate rewards and the distribution $P$ over state transitions. + +The policy gradient theorem (Sutton et al., 2000) expresses the gradient of $O _ { \mathrm { E R } } ( \pi _ { \theta } )$ w.r.t. $\theta$ , the tunable parameters of a policy $\pi _ { \theta }$ , as, + +$$ +\begin{array} { r c l } { { \nabla _ { \theta } { \cal O } _ { \mathrm { E R } } ( \pi _ { \theta } ) } } & { { = } } & { { \displaystyle \int _ { S } \int _ { { \cal A } } \nabla _ { \theta } \pi _ { \theta } ( a \mid s ) Q ^ { \pi } ( s , a ) \mathrm { d } a \mathrm { d } \rho ^ { \pi } ( s ) } } \\ { { } } & { { = } } & { { \displaystyle \int _ { S } \mathbb { E } _ { a \sim \pi _ { \theta } ( a \mid s ) } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a \mid s ) Q ^ { \pi } ( s , a ) \right] \mathrm { d } \rho ^ { \pi } ( s ) . } } \end{array} +$$ + +Many reinforcement learning algorithms, including policy gradient and actor-critic variants, trade off variance and bias when estimating the random variable inside the expectation in (4); for example, by attempting to estimate $Q ^ { \pi } ( s , a )$ accurately using function approximation. In the simplest scenario, an unbiased estimate of $Q ^ { \pi } ( s , a )$ is formed by accumulating discounted rewards from each state forward using a single Monte Carlo sample. + +In this paper, we focus on multivariate Gaussian policies over continuous action spaces, $\mathcal { A } \equiv \mathbb { R } ^ { d _ { a } }$ . We represent the observed state of the MDP as a $d _ { s }$ -dimensional feature vector $\bar { \Phi } ( s ) \in \mathbb { R } ^ { d _ { s } }$ , and parametrize the Gaussian policy by a mean and covariance function, respectively $\mu ( \boldsymbol { s } ) : \mathbb { R } ^ { d _ { \boldsymbol { s } } } \mathbb { R } ^ { d _ { a } }$ and $\Sigma ( s ) : \mathbb { R } ^ { d _ { s } } \mathbb { R } ^ { d _ { a } } \times \mathbb { R } ^ { \tilde { d } _ { a } }$ . These map the observed state of the environment to a Gaussian distribution, + +$$ +\pi ( a | s ) = N ( a | \mu ( s ) , \Sigma ( s ) ) = | 2 \pi \Sigma ( s ) | ^ { - 1 / 2 } \exp \left\{ - \frac { 1 } { 2 } \| a - \mu ( s ) \| _ { \Sigma ( s ) ^ { - 1 } } ^ { 2 } \right\} , +$$ + +where $\| v \| _ { A } ^ { 2 } = v ^ { \mathsf { T } } A v$ . Below we develop new RL training methods for this family of parametric policies, but some of the ideas presented may generalize to other families of policies as well. We begin the formulation by reviewing some prior work on learning Gaussian policies. + +# 2.1 DETERMINISTIC POLICY GRADIENT + +Silver et al. (2014) present a new formulation of the policy gradient, called the deterministic policy gradient, for the family of Gaussian policies in the limit where the policy covariance approaches zero. In such a scenario, the policy becomes deterministic because sampling from the policy always returns the Gaussian mean. The key observation of (Silver et al., 2014) is that under a deterministic policy $\pi \equiv ( \mu , \Sigma \to 0 )$ , one can estimate the expected future return from a state $s$ as, + +$$ +\operatorname * { l i m } _ { \Sigma \to 0 } \int _ { A } \pi ( a \mid s ) Q ^ { \pi } ( s , a ) \mathrm { d } a = Q ^ { \pi } ( s , \mu ( s ) ) . +$$ + +Then, one can express the gradient of the optimization objective (expected discounted return) for a parameterized $\pi _ { \boldsymbol { \theta } } \equiv \mu _ { \boldsymbol { \theta } }$ as, + +$$ +\nabla _ { \theta } { \cal O } _ { \mathrm { E R } } ( \pi _ { \theta } ) = \int _ { S } \nabla _ { \theta } Q ^ { \pi } ( s , \mu _ { \theta } ( s ) ) \mathrm { d } \rho ^ { \pi } ( s ) = \int _ { S } \frac { \partial Q ^ { \pi } ( s , a ) } { \partial a } | _ { a = \mu _ { \theta } ( s ) } \nabla _ { \theta } \mu _ { \theta } ( s ) \mathrm { d } \rho ^ { \pi } ( s ) . +$$ + +This can be thought of as a characterization of the policy gradient theorem for deterministic policies. + +In the limit of $\Sigma 0$ , one can also re-express the Bellman equation (2) as, + +$$ +Q ^ { \pi } ( s , a ) = \mathbb { E } _ { r , s ^ { \prime } } \left[ r + Q ^ { \pi } ( s ^ { \prime } , \mu ( s ^ { \prime } ) ) \right] . +$$ + +Therefore, a value function approximator $Q _ { w } ^ { \pi }$ can be optimized by minimizing the Bellman error, + +$$ +E ( w ) = \sum _ { ( s , a , r , s ^ { \prime } ) \in \mathcal { D } } ( Q _ { w } ^ { \pi } ( s , a ) - r - \gamma Q _ { w } ^ { \pi } ( s ^ { \prime } , \mu ( s ^ { \prime } ) ) ^ { 2 } , +$$ + +for transitions $( s , a , r , s ^ { \prime } )$ sampled from a dataset $\mathcal { D }$ of interactions of the agent with the environment. Algorithms like DDPG (Lillicrap et al., 2016) alternate between improving the value function by gradient descent on (9), and improving the policy based on (7). + +In practice, to gain better sample efficiency, Degris et al. (2012) and Silver et al. (2014) replace the on-policy state distribution $\rho ^ { \pi } ( s )$ in (7) with an off-policy distribution $\rho ^ { \beta } ( s )$ based on a replay buffer. After this substitution, the policy gradient identity in (7) does not hold exactly, however, prior work finds that this works well in practice and improves sample efficiency. We also adopt a similar approximation in our method to make use of off-policy data. + +# 3 SMOOTHED ACTION VALUE FUNCTIONS + +In this paper, we introduce smoothed action value functions, the gradients of which provide an effective signal for optimizing the parameters of a Gaussian policy. Our notion of smoothed Qvalues, denoted $\tilde { Q } ^ { \pi } ( s , a )$ , differs from ordinary Q-values $Q ^ { \pi } ( s , a )$ in that smoothed Q-values do not assume the first action of the agent is fully specified, but rather they assume that only the mean of the distribution of the first action is known. Hence, to compute $\tilde { Q } ^ { \pi } ( s , a )$ , one has to perform an expectation of $Q ^ { \pi } ( s , { \tilde { a } } )$ for actions $\tilde { a }$ drawn in the vicinity of $a$ . More formally, smoothed action values are defined as, + +$$ +\tilde { Q } ^ { \pi } ( s , a ) = \int _ { A } N ( \tilde { a } | a , \Sigma ( s ) ) Q ^ { \pi } ( s , \tilde { a } ) \mathrm { d } \tilde { a } . +$$ + +With this definition of ${ \tilde { Q } } ^ { \pi }$ , one can re-express the expected reward objective for a Gaussian policy $\pi \equiv ( \mu , \Sigma )$ as, + +$$ +O _ { \mathrm { E R } } ( \pi ) = \int _ { S } \tilde { Q } ^ { \pi } ( s , \mu ( s ) ) \mathrm { d } \rho ^ { \pi } ( s ) . +$$ + +The insight that differentiates this approach from prior work including Heess et al. (2015); Ciosek & Whiteson (2017) is that instead of learning a function approximator for $Q ^ { \pi } ( s , a )$ and then drawing samples to approximate the expectation in (10) and its derivative, we directly learn a function approximator for $\bar { Q } ^ { \pi } ( s , a )$ . + +The key observation that enables direct bootstrapping of smoothed $\mathrm { Q }$ -values, $\tilde { Q } ^ { \pi } ( s , a )$ , is that their form allows a notion of Bellman consistency. First, note that for Gaussian policies $\pi \equiv ( \mu , \Sigma )$ we have + +$$ +Q ^ { \pi } ( s , a ) = \mathbb { E } _ { r , s ^ { \prime } } [ r + \gamma \tilde { Q } ^ { \pi } ( s ^ { \prime } , \mu ( s ^ { \prime } ) ) ] . +$$ + +Then, combining (10) and (12), one can derive the following one-step Bellman equation for smoothed Q-values, + +$$ +\tilde { Q } ^ { \pi } ( s , a ) = \int _ { \cal A } N ( \tilde { a } \mid a , \Sigma ( s ) ) \mathbb { E } _ { \tilde { r } , \tilde { s } ^ { \prime } } \left[ \tilde { r } + \gamma \tilde { Q } ^ { \pi } ( \tilde { s } ^ { \prime } , \mu ( \tilde { s } ^ { \prime } ) ) \right] \mathrm { d } \tilde { a } , +$$ + +where $\tilde { r }$ and ${ \tilde { s } } ^ { \prime }$ are sampled from $R ( s , { \tilde { a } } )$ and $P ( s , { \tilde { a } } )$ . Below, we elaborate on how one can make use of the derivatives of ${ \tilde { Q } } ^ { \pi }$ to learn $\mu$ and $\Sigma$ , and how the Bellman equation in (13) enables direct optimization of ${ \tilde { Q } } ^ { \pi }$ . + +# 3.1 POLICY IMPROVEMENT - OPTIMIZING $\left( \mu _ { \theta } , \Sigma _ { \phi } \right)$ GIVEN ${ \tilde { Q } } ^ { \pi }$ + +We parameterize a Gaussian policy $\pi _ { \boldsymbol { \theta } , \boldsymbol { \phi } } \equiv ( \mu _ { \boldsymbol { \theta } } , \Sigma _ { \boldsymbol { \phi } } )$ in terms of two sets of parameters $\theta$ and $\phi$ for the mean and the covariance. The gradient of the objective $w . r . t .$ . mean parameters follows from the policy gradient theorem and is almost identical to (7), + +$$ +\nabla _ { \theta } { \cal O } _ { \mathrm { E R } } ( \pi _ { \theta , \phi } ) = \int _ { S } \frac { \partial \tilde { Q } ^ { \pi } ( s , a ) } { \partial a } \big | _ { a = \mu _ { \theta } ( s ) } \nabla _ { \theta } \mu _ { \theta } ( s ) \mathrm { d } \rho ^ { \pi } ( s ) . +$$ + +Estimating the derivative of the objective w.r.t. covariance parameters is not as straightforward, since ${ \tilde { Q } } ^ { \pi }$ is not a direct function of $\Sigma$ . However, a key observation of this work is that the second derivative of ${ \tilde { Q } } ^ { \pi } w . r . t .$ . actions is sufficient to exactly compute the derivative of $\tilde { Q } ^ { \pi } w . r . t . \Sigma$ , + +$$ +\frac { \partial \tilde { Q } ^ { \pi } ( s , a ) } { \partial \Sigma ( s ) } = \frac { 1 } { 2 } \cdot \frac { \partial ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) } { \partial a ^ { 2 } } . +$$ + +A proof of this identity is provided in the Appendix. The proof may be easily derived by expressing both sides of the equation using standard matrix calculus like $\begin{array} { r } { \frac { \partial } { \partial A } | A | ^ { - 1 / 2 } = - \frac { 1 } { 2 } | A | ^ { - 1 / 2 } \bar { A } ^ { - 1 } } \end{array}$ and $\begin{array} { r } { \frac { \partial } { \partial A } | | v | | _ { A ^ { - 1 } } ^ { 2 } = - A ^ { - 1 } v v ^ { T } A ^ { - 1 } } \end{array}$ . + +Then, the full derivative w.r.t. $\phi$ takes the form, + +$$ +\nabla _ { \phi } { \cal O } _ { \mathrm { E R } } ( \pi _ { \theta , \phi } ) = \frac { 1 } { 2 } \int _ { S } \frac { \partial ^ { 2 } { \tilde { Q } } ^ { \pi } ( s , a ) } { \partial a ^ { 2 } } \big | _ { a = \mu _ { \theta } ( s ) } \nabla _ { \phi } \Sigma _ { \phi } ( s ) \mathrm { d } \rho ^ { \pi } ( s ) . +$$ + +# 3.2 POLICY EVALUATION - OPTIMIZING $\tilde { Q } _ { w } ^ { \pi }$ GIVEN $( \mu , \Sigma )$ + +We can think of two ways to optimize $\tilde { Q } _ { w } ^ { \pi }$ . The first approach leverages (10) to update ${ \tilde { Q } } ^ { \pi }$ based on expected Q-value function $Q ^ { \pi }$ . In such an approach, one trains a parameterized $Q _ { w } ^ { \pi }$ to approximate the standard expected Q-value function $Q ^ { \pi }$ using standard methods (see e.g., Rummery $\&$ Niranjan (1994); Sutton & Barto (1998); Van Seijen et al. (2009)). Then, one fits $\tilde { Q } _ { w } ^ { \pi }$ based on $Q _ { w } ^ { \pi }$ . In particular, given transitions $( s , a , r , s ^ { \prime } )$ sampled from interactions with the environment, one can train $Q _ { w } ^ { \pi }$ to minimize the Bellman error $( Q _ { w } ^ { \pi } ( s , a ) - r - \gamma Q _ { w } ^ { \pi } ( s ^ { \prime } , a ^ { \prime } ) ) ^ { 2 }$ where $a ^ { \prime } \sim N ( \mu ( s ^ { \prime } ) , \Sigma ( s ^ { \prime } ) )$ . Then, $\tilde { Q } _ { w } ^ { \pi }$ can be optimized to minimize the squared error $( \tilde { Q } _ { w } ^ { \pi } ( s , a ) - \mathbb { E } _ { \tilde { a } } Q _ { w } ^ { \pi } ( s , \tilde { a } ) ) ^ { 2 }$ where $\tilde { a } \sim$ $N ( a , \Sigma ( s ) )$ , using several samples. When the target values in these residuals are treated as fixed (i.e., using a target network), such a training procedure will achieve a fixed point when $\tilde { Q } _ { w } ^ { \pi } ( s , a )$ satisfies the recursion in the Bellman equation (10). + +The second approach requires a single function approximator for $\tilde { Q } _ { w } ^ { \pi } ( s , a )$ , resulting in a simpler implementation, and thus we use this approach in our experimental evaluation. Suppose one has access to a tuple $( s , \tilde { a } , \tilde { r } , \tilde { s } ^ { \prime } )$ sampled from a replay buffer with knowledge of the sampling probability $q ( \tilde { a } \mid s )$ (possibly unnormalized). Then assuming that this sampling distribution has a full support, we draw a phantom action $a \sim N ( \tilde { a } , \Sigma ( s ) )$ and optimize $\tilde { Q } _ { w } ^ { \pi } ( s , a )$ by minimizing a weighted Bellman error $\begin{array} { r } { \frac { 1 } { q ( \tilde { a } | s ) } ( \tilde { Q } _ { w } ^ { \pi } ( s , a ) - \tilde { r } - \gamma \tilde { Q } _ { w } ^ { \pi } ( \tilde { s } ^ { \prime } , \mu ( \tilde { s } ^ { \prime } ) ) ^ { 2 } } \end{array}$ . For a specific pair of state and action $( s , a )$ the + +expected value of the objective is, + +$$ +E ( w \mid ( s , a ) ) ~ = ~ \mathbb { E } _ { q ( \bar { a } \mid s ) , \bar { r } , \bar { s } ^ { \prime } } \left[ \frac { N ( a \mid \tilde { a } , \Sigma ( s ) ) } { q ( \tilde { a } \mid s ) } ( \tilde { Q } _ { w } ^ { \pi } ( s , a ) - \tilde { r } - \gamma \tilde { Q } _ { w } ^ { \pi } ( \tilde { s } ^ { \prime } , \mu ( \tilde { s } ^ { \prime } ) ) ) ^ { 2 } \right] ~ . +$$ + +Note that $N ( a | \tilde { a } , \Sigma ( s ) ) = N ( \tilde { a } | a , \Sigma ( s ) )$ . Therefore, when the target value $\tilde { r } + \gamma \tilde { Q } _ { w } ^ { \pi } ( \tilde { s } ^ { \prime } , \mu ( \tilde { s } ^ { \prime } ) )$ is treated as fixed (e.g., when using target networks) this training procedure reaches an optimum when $\tilde { Q } _ { w } ^ { \pi } ( s , a )$ satisfies the recursion in the Bellman equation (13). + +In practice, we find that it is unnecessary to keep track of the probabilities $q ( \tilde { a } \mid s )$ , and assume the replay buffer provides a near-uniform distribution of actions conditioned on states. Other recent work has also benefited from ignoring or heavily damping importance weights (Munos et al., 2016; Wang et al., 2017; Schulman et al., 2017). However, it is possible when interacting with the environment to save the probability of sampled actions along with their transitions, and thus have access to $q ( \tilde { \boldsymbol { a } } \mid \boldsymbol { s } ) \approx N ( \tilde { \boldsymbol { a } } \mid \bar { \mu } _ { \mathrm { o l d } } ( \boldsymbol { s } ) , \dot { \Sigma _ { \mathrm { o l d } } } ( \boldsymbol { s } ) )$ . + +# 3.3 PROXIMAL POLICY OPTIMIZATION + +Policy gradient algorithms are notoriously unstable, particularly in continuous control problems. Such instability has motivated the development of trust region methods that attempt to mitigate the issue by constraining each gradient step to lie within a trust region (Schulman et al., 2015), or augmenting the expected reward objective with a penalty on KL-divergence from a previous policy (Nachum et al., 2017b; Schulman et al., 2017; Azar et al., 2012). These stabilizing techniques have thus far not been applicable to algorithms like DDPG, since the policy is deterministic. The formulation we propose in this paper, however, is easily amenable to trust region optimization. Specifically, we may augment the objective (11) with a penalty + +$$ +{ \cal O } _ { \mathrm { T R } } ( \pi ) = { \cal O } _ { \mathrm { E R } } ( \pi ) - \lambda \int _ { \cal S } \mathrm { K L } ( \pi \parallel \pi _ { \mathrm { o l d } } ) \mathrm { d } \rho ^ { \pi } ( s ) , +$$ + +where $\pi _ { \mathrm { o l d } } \equiv \left( \mu _ { \mathrm { o l d } } , \Sigma _ { \mathrm { o l d } } \right)$ is a previous parameterization of the policy. The optimization is straightforward, since the KL-divergence of two Gaussians can be expressed analytically. + +# 4 RELATED WORK + +This paper follows a long line of work that uses Q-value functions to stably learn a policy, which in the past has been used to either approximate expected (Rummery & Niranjan, 1994; Van Seijen et al., 2009; Gu et al., 2017) or optimal (Watkins, 1989; Silver et al., 2014; Nachum et al., 2017a; Haarnoja et al., 2017; Metz et al., 2017) future value. + +Work that is most similar to what we present are methods that exploit gradient information from the Q-value function to train a policy. Deterministic policy gradient (Silver et al., 2014) is perhaps the best known of these. The method we propose can be interpreted as a generalization of the deterministic policy gradient. Indeed, if one takes the limit of the policy covariance $\Sigma ( s )$ as it goes to 0, the proposed Q-value function becomes the deterministic value function of DDPG, and the updates for training the Q-value approximator and the policy mean are identical. + +Stochastic Value Gradient (SVG) (Heess et al., 2015) also trains stochastic policies using an update that is similar to DDPG (i.e., SVG(0) with replay). The key differences with our approach are that SVG does not provide an update for the covariance, and the mean update in SVG estimates the gradient with a noisy Monte Carlo sample, which we avoid by estimating the smoothed $\mathrm { Q }$ -value function. Although a covariance update could be derived using the same reparameterization trick as in the mean update, that would also require a noisy Monte Carlo estimate. Methods for updating the covariance along the gradient of expected reward are essential for applying the subsequent trust region and proximal policy techniques. + +More recently, Ciosek & Whiteson (2017) introduced expected policy gradients (EPG), a generalization of DDPG that provides updates for the mean and covariance of a stochastic Gaussian policy using gradients of an estimated Q-value function. In that work, the expected Q-value used in standard policy gradient algorithms such as SARSA (Sutton & Barto, 1998; Rummery & Niranjan, 1994; Van Seijen et al., 2009) is estimated. The updates in EPG therefore require approximating an integral of the expected Q-value function. Our analogous process directly estimates an integral (via the smoothed Q-value function) and avoids approximate integrals, thereby making the updates simpler. Moreover, while Ciosek & Whiteson (2017) rely on a quadratic Taylor expansion of the estimated Q-value function, we instead rely on the strength of neural network function approximators to directly estimate the smoothed Q-value function. + +The novel training scheme we propose for learning the covariance of a Gaussian policy relies on properties of Gaussian integrals (Bonnet, 1964; Price, 1958). Similar identities have been used in the past to derive updates for variational auto-encoders (Kingma & Welling, 2014) and Gaussian back-propagation (Rezende et al., 2014). + +Finally, the perspective presented in this paper, where Q-values represent the averaged return of a distribution of actions rather than a single action, is distinct from recent advances in distributional RL (Bellemare et al., 2017). Those approaches focus on the distribution of returns of a single action, whereas we consider the single average return of a distribution of actions. Although we restrict our attention in this paper to Gaussian policies, an interesting topic for further investigation is to study the applicability of this new perspective to a wider class of policy distributions. + +# 5 EXPERIMENTS + +We utilize the insights from Section 3 to introduce a new RL algorithm, Smoothie. Smoothie maintains a parameterized $\tilde { Q } _ { w } ^ { \pi }$ trained via the procedure described in Section 3.2. It then uses the gradient and Hessian of this approximation to train a Gaussian policy $\mu _ { \theta } , \Sigma _ { \phi }$ using the updates stated in (14) and (16). See Algorithm 1 for a simplified pseudocode of our algorithm. + +# Algorithm 1 Smoothie + +
Algorithm1Smoothie Input: Environment ENV,learning rates 7π, nQ,discount factor y,KL-penalty 入,l
number of training steps N, target network lag T. Initialize0,,w,set0'=0,Φ'=,w'=w.
fori=OtoN-1do
/ Collect experience
Sample action a ~ N(μe(s),∑(s)) and apply to ENV to yield r and s'. Insert transition (s,a,r,s') to replay buffer.
// Train μ,∑
Sample batch{(k,@k,Tk,S)}1 fromreplay buffer aQ(ska) Compute gradients gk =
da la=μe(sk)*
²(s@） Compute Hessians Hk = a² la=μe(sk)*
Compute KL-penalties KLk =KL(μθ,Σ𝜙llμe,Σ).
Compute updates 1B
1B
△=B 1 ∑k=1 Updateθ←θ+ηπ△0,←+ηπ△.
// Train Qπ
Sample batch {(sk,k,Tk,S)}1 from replay bufer. Sample phantom actions ak ~ N(ák,Σ(sk)). Compute lossL(u)=∑1((s)-r-qQ(s,ue()2.
+ +We perform a number of evaluations of Smoothie compared to DDPG. We choose DDPG as a baseline because it (1) utilizes gradient information of a Q-value approximator, much like our algorithm; and (2) is a standard algorithm well-known to have achieve good, sample-efficient performance on continuous control benchmarks. + +# 5.1 A SYNTHETIC TASK + +To evaluate Smoothie we begin with a simple synthetic task which allows us to study its behavior in a restricted setting. We devised a simple single-action one-shot environment in which the reward function is a mixture of two Gaussians, one better than the other (see Figure 1 (Right)). We initialize the policy mean to be centered on the worse of the two Gaussians. We plot the learnable policy mean and standard deviation during training for Smoothie and DDPG in Figure 1 (Left). Smoothie learns both the mean and variance, while DDPG learns only the mean and the variance plotted is the exploratory noise, whose scale is kept fixed during training. + +As expected we observe that DDPG cannot escape the local optimum. At the beginning of training it exhibits some movement away from the local optimum (likely due to the initial noisy approximation given by $Q _ { w } ^ { \pi } \mathrm { . }$ ), it is unable to progress very far from the initial mean. Note that this is not an issue of exploration. The exploration scale is high enough that $Q _ { w } ^ { \pi }$ is aware of the better Gaussian. The issue is in the update for $\mu _ { \theta }$ , which is only with regard to the derivative of $Q _ { w } ^ { \pi }$ at the current mean. + +On the other hand, we find Smoothie is successfully able to solve the task. This is because the smoothed reward function approximated by $\tilde { Q } _ { w } ^ { \pi }$ has a derivative which clearly points $\mu _ { \theta }$ towards the better Gaussian. We also observe that Smoothie is able to suitably adjust the covariance $\Sigma _ { \phi }$ during training. Initially, $\Sigma _ { \phi }$ decreases due to the concavity of the smoothed reward function. As a region of convexity is entered, it begins to increase, before again decreasing to near-zero as $\mu _ { \theta }$ approaches the global optimum. + +![](images/3e62a205722715b4eae018cb617c9164cd435c130be75b6bf1c3addc835c777e.jpg) +Figure 1: Left: The learnable policy mean and standard deviation during training for Smoothie and DDPG on a simple one-shot synthetic task. The standard deviation for DDPG is the exploratory noise kept constant during training. Right: The reward function for the synthetic task along with its Gaussian-smoothed version. We find that Smoothie can successfully escape the lower-reward local optimum. We also notice Smoothie increases and decreases its policy variance as the convexity/concavity of the smoothed reward function changes. + +# 5.2 CONTINUOUS CONTROL + +We now turn our attention to standard continuous control benchmarks available on OpenAI Gym (Brockman et al., 2016) utilizing the MuJoCo environment (Todorov et al., 2012). + +Our implementations utilize feed forward neural networks for policy and Q-values. We parameterize the covariance $\Sigma _ { \phi }$ as a diagonal given by $e ^ { \phi }$ . The exploration for DDPG is determined by an Ornstein-Uhlenbeck process (Uhlenbeck & Ornstein, 1930; Lillicrap et al., 2016). Additional implementation details are provided in the Appendix. + +![](images/f21887a3716c42d41ffbbba35b3c52cf13877d7b56bb86c24b7339f4a95daf67.jpg) +Figure 2: Results of Smoothie, DDPG, and TRPO on continuous control benchmarks. The $\mathbf { X }$ -axis is in millions of environment steps. Each plot shows the average reward and standard deviation clipped at the min and max of six randomly seeded runs after choosing best hyperparameters. We see that Smoothie is competitive with DDPG even when DDPG uses a hyperparameter-tuned noise scale, and Smoothie learns the optimal noise scale (the covariance) during training. Moreoever, we observe significant advantages in terms of final reward performance, especially in the more difficult tasks like Hopper, Walker2d, and Humanoid. Across all tasks, TRPO is not sufficiently sampleefficient to provide a competitive baseline. + +We compare the results of Smoothie and DDPG in Figure 2. For each task we performed a hyperparameter search over actor learning rate, critic learning rate and reward scale, and plot the average of six runs for the best hyperparameters. For DDPG we extended the hyperparameter search to also consider the scale and damping of exploratory noise provided by the Ornstein-Uhlenbeck process. Smoothie, on the other hand, contains an additional hyperparameter to determine the weight on KL-penalty. + +Despite DDPG having the advantage of its exploration decided by a hyperparameter search while Smoothie must learn its exploration without supervision, we find that Smoothie performs competitively or better across all tasks, exhibiting a slight advantage in Swimmer and Ant, while showing more dramatic improvements in Hopper, Walker2d, and Humanoid. The improvement is especially dramatic for Hopper, where the average reward is doubled. We also highlight the results for Humanoid, which as far as we know, are the best published results for a method that only trains on the order of millions of environment steps. In contrast, TRPO, which to the best of our knowledge is the only other algorithm which can achieve better performance, requires on the order of tens of millions of environment steps to achieve comparable reward. This gives added evidence to the benefits of using a learnable covariance and not restricting a policy to be deterministic. + +Empirically, we found the introduction of a KL-penalty to improve performance of Smoothie, especially on harder tasks. We present a comparison of results of Smoothie with and without the KL-penalty on the four harder tasks in Figure 3. A KL-penalty to encourage stability is not possible in DDPG. Thus, our algorithm provides a much needed solution to the inherent instability in DDPG training. + +![](images/31e9c2cf700c5e3ed94cc8e3ab513890a7ad5439462138652b2f32dd4b21ec5f.jpg) +Figure 3: Results of Smoothie with and without a KL-penalty. The $\mathbf { X } ^ { } -$ -axis is in millions of environment steps. We observe benefits of using a proximal policy optimization method, especially in Hopper and Humanoid, where the performance improvement is significant without sacrificing sample efficiency. + +# 6 CONCLUSION + +We have presented a new Q-value function, ${ \tilde { Q } } ^ { \pi }$ , that is a Gaussian-smoothed version of the standard expected Q-value, $Q ^ { \pi }$ . The advantage of using ${ \tilde { Q } } ^ { \pi }$ over $Q ^ { \pi }$ is that its gradient and Hessian possess an intimate relationship with the gradient of expected reward with respect to mean and covariance of a Gaussian policy. The resulting algorithm, Smoothie, is able to successfully learn both mean and covariance during training, leading to performance that can match or surpass that of DDPG, especially when incorporating a penalty on divergence from a previous policy. + +The success of ${ \tilde { Q } } ^ { \pi }$ is encouraging. Intuitively it may be argued that learning ${ \tilde { Q } } ^ { \pi }$ is more sensible than learning $Q ^ { \pi }$ . The smoothed Q-values by definition make the true reward surface smoother, thus possibly easier to learn; moreover the smoothed Q-values have a more direct relationship with the expected discounted return objective. We encourage future work to further investigate these claims as well as techniques to apply the underlying motivations for ${ \tilde { Q } } ^ { \pi }$ to other types of policies. + +# REFERENCES + +Mohammad Gheshlaghi Azar, Vicenc¸ Gomez, and Hilbert J Kappen. Dynamic policy programming. ´ JMLR, 13, 2012. + +Marc G Bellemare, Will Dabney, and Remi Munos. A distributional perspective on reinforcement ´ learning. In ICML, pp. 449–458, 2017. + +Georges Bonnet. Transformations des signaux aleatoires a travers les systemes non lin ´ eaires sans ´ memoire. ´ Annals of Telecommunications, 19(9):203–220, 1964. + +Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. OpenAI Gym. arXiv:1606.01540, 2016. + +Kamil Ciosek and Shimon Whiteson. Expected policy gradients. arXiv preprint arXiv:1706.05374, 2017. + +Thomas Degris, Martha White, and Richard S Sutton. Off-policy actor-critic. ICML, 2012. + +Shixiang Gu, Timothy Lillicrap, Zoubin Ghahramani, Richard E Turner, Bernhard Scholkopf, and ¨ Sergey Levine. Interpolated policy gradient: Merging on-policy and off-policy gradient estimation for deep reinforcement learning. NIPS, 2017. + +Tuomas Haarnoja, Haoran Tang, Pieter Abbeel, and Sergey Levine. Reinforcement learning with deep energy-based policies. ICML, 2017. + +Nicolas Heess, Gregory Wayne, David Silver, Tim Lillicrap, Tom Erez, and Yuval Tassa. Learning continuous control policies by stochastic value gradients. In NIPS, 2015. + +Diederik P Kingma and Max Welling. Auto-encoding variational bayes. ICLR, 2014. + +Vijay R Konda and John N Tsitsiklis. Actor-critic algorithms, 2000. + +Timothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. ICLR, 2016. + +Luke Metz, Julian Ibarz, Navdeep Jaitly, and James Davidson. Discrete sequential prediction of continuous actions for deep RL. CoRR, abs/1705.05035, 2017. URL http://arxiv.org/ abs/1705.05035. + +Remi Munos, Tom Stepleton, Anna Harutyunyan, and Marc Bellemare. Safe and efficient off-policy ´ reinforcement learning. In NIPS, 2016. + +Ofir Nachum, Mohammad Norouzi, Kelvin Xu, and Dale Schuurmans. Bridging the gap between value and policy based reinforcement learning. NIPS, 2017a. + +Ofir Nachum, Mohammad Norouzi, Kelvin Xu, and Dale Schuurmans. Trust-pcl: An off-policy trust region method for continuous control. arXiv preprint arXiv:1707.01891, 2017b. + +Robert Price. A useful theorem for nonlinear devices having gaussian inputs. IRE Transactions on Information Theory, 4(2):69–72, 1958. + +Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In International Conference on Machine Learning, pp. 1278–1286, 2014. + +Gavin A Rummery and Mahesan Niranjan. On-line Q-learning using connectionist systems, volume 37. 1994. + +John Schulman, Sergey Levine, Pieter Abbeel, Michael Jordan, and Philipp Moritz. Trust region policy optimization. In ICML, 2015. + +John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017. + +David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin Riedmiller. Deterministic policy gradient algorithms. In ICML, 2014. + +Richard S. Sutton and Andrew G. Barto. Introduction to Reinforcement Learning. MIT Press, 1998. + +Richard S Sutton, David A McAllester, Satinder P Singh, and Yishay Mansour. Policy gradient methods for reinforcement learning with function approximation. NIPS, 2000. + +Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In Intelligent Robots and Systems (IROS), 2012 IEEE/RSJ International Conference on, pp. 5026– 5033. IEEE, 2012. + +George E Uhlenbeck and Leonard S Ornstein. On the theory of the brownian motion. Physical review, 36(5):823, 1930. + +Harm Van Seijen, Hado Van Hasselt, Shimon Whiteson, and Marco Wiering. A theoretical and empirical analysis of expected sarsa. In Adaptive Dynamic Programming and Reinforcement Learning, 2009. ADPRL’09. IEEE Symposium on, pp. 177–184. IEEE, 2009. + +Ziyu Wang, Victor Bapst, Nicolas Heess, Volodymyr Mnih, Remi Munos, Koray Kavukcuoglu, and Nando de Freitas. Sample efficient actor-critic with experience replay. ICLR, 2017. + +Christopher John Cornish Hellaby Watkins. Learning from delayed rewards. PhD thesis, University of Cambridge England, 1989. + +Ronald J Williams and Jing Peng. Function optimization using connectionist reinforcement learning algorithms. Connection Science, 1991. + +# A PROOF OF EQUATION (15) + +We note that similar identities for Gaussian integrals exist in the literature (Price, 1958; Rezende et al., 2014) and point the reader to these works for further information. + +The specific identity we state may be derived using standard matrix calculus. We make use of the fact that + +$$ +\frac { \partial } { \partial A } | A | ^ { - 1 / 2 } = - \frac { 1 } { 2 } | A | ^ { - 3 / 2 } \frac { \partial } { \partial A } | A | = - \frac { 1 } { 2 } | A | ^ { - 1 / 2 } A ^ { - 1 } , +$$ + +and for symmetric $A$ , + +$$ +\frac \partial { \partial A } | | \boldsymbol { v } | | _ { A ^ { - 1 } } ^ { 2 } = - A ^ { - 1 } { \boldsymbol { v } } { \boldsymbol { v } } ^ { T } A ^ { - 1 } . +$$ + +We omit $s$ from $\Sigma ( s )$ in the following equations for succinctness. The LHS of (15) is + +$$ +\begin{array} { l } { \displaystyle \int _ { A } Q ^ { \pi } ( s , \tilde { a } ) \frac { \partial } { \partial \Sigma } N ( \tilde { a } | a , \Sigma ) \mathrm { d } \tilde { a } } \\ { \displaystyle = \int _ { A } Q ^ { \pi } ( s , \tilde { a } ) \exp \left\{ - \frac { 1 } { 2 } | | \tilde { a } - a | | _ { \Sigma ^ { - 1 } } ^ { 2 } \right\} \left( \frac { \partial } { \partial \Sigma } | 2 \pi \Sigma | ^ { - 1 / 2 } - \frac { 1 } { 2 } | 2 \pi \Sigma | ^ { - 1 / 2 } \frac { \partial } { \partial \Sigma } | | \tilde { a } - a | | _ { \Sigma ^ { - 1 } } ^ { 2 } \right) \mathrm { d } \tilde { a } } \\ { \displaystyle \qquad = \frac { 1 } { 2 } \int _ { A } Q ^ { \pi } ( s , \tilde { a } ) N ( \tilde { a } | a , \Sigma ) \left( - \Sigma ^ { - 1 } + \Sigma ^ { - 1 } ( \tilde { a } - a ) ( \tilde { a } - a ) ^ { T } \Sigma ^ { - 1 } \right) \mathrm { d } \tilde { a } . \quad ( 2 \pi \Sigma ) \mathrm { d } \tilde { a } , } \end{array} +$$ + +Meanwhile, towards tackling the RHS of (15) we note that + +$$ +\frac { \partial \tilde { Q } ^ { \pi } ( s , a ) } { \partial a } = \int _ { \cal A } Q ^ { \pi } ( s , \tilde { a } ) N ( \tilde { a } | a , \Sigma ) \Sigma ^ { - 1 } ( \tilde { a } - a ) \mathrm { d } \tilde { a } . +$$ + +Thus we have + +$$ +\frac { \partial ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) } { \partial a ^ { 2 } } = \int _ { \mathcal { A } } Q ^ { \pi } ( s , \tilde { a } ) \left( \Sigma ^ { - 1 } ( \tilde { a } - a ) \frac { \partial } { \partial a } N ( \tilde { a } | a , \Sigma ) + N ( \tilde { a } | a , \Sigma ) \frac { \partial } { \partial a } \Sigma ^ { - 1 } ( \tilde { a } - a ) \right) \mathrm { d } \tilde { a } . +$$ + +$$ += \int _ { \mathcal { A } } Q ^ { \pi } ( s , \tilde { a } ) N ( \tilde { a } | a , \Sigma ) ( \Sigma ^ { - 1 } ( \tilde { a } - a ) ( \tilde { a } - a ) ^ { T } \Sigma ^ { - 1 } - \Sigma ^ { - 1 } ) \mathrm { d } \tilde { a } . +$$ + +# B COMPATIBLE FUNCTION APPROXIMATION + +A function approximator $\tilde { Q } _ { w } ^ { \pi }$ of ${ \tilde { Q } } ^ { \pi }$ should be sufficiently accurate so that updates for $\mu _ { \theta } , \Sigma _ { \phi }$ are not affected by substituting $\frac { \partial \tilde { Q } _ { w } ^ { \pi } ( s , a ) } { \partial a }$ and $\frac { \partial ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) } { \partial a ^ { 2 } }$ for $\frac { \partial \tilde { Q } ^ { \pi } ( s , a ) } { \partial a }$ and $\frac { \partial ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) } { \partial a ^ { 2 } }$ , respectively. + +We claim that a $\tilde { Q } _ { w } ^ { \pi }$ is compatible with respect to $\mu _ { \theta }$ if + +1 $\left. \nabla _ { a } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \right| _ { a = \mu _ { \theta } ( s ) } = \nabla _ { \theta } \mu _ { \theta } ( s ) ^ { T } w ,$ +2. $\begin{array} { r } { \nabla _ { w } \int _ { \mathcal { S } } \Big ( \nabla _ { a } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } - \nabla _ { a } \tilde { Q } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } \Big ) ^ { 2 } \mathrm { d } \rho ^ { \pi } ( s ) = 0 } \end{array}$ (i.e., $w$ minimizes the expected squared error of the gradients). + +Additionally, $\tilde { Q } _ { w } ^ { \pi }$ is compatible with respect to $\Sigma _ { \phi }$ if + +1. $\nabla _ { a } ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } = \nabla _ { \phi } \Sigma _ { \phi } ( s ) ^ { T } w ,$ +2. $\begin{array} { r } { \nabla _ { w } \int _ { \mathcal { S } } \left( \nabla _ { a } ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big \vert _ { a = \mu _ { \theta } ( s ) } - \nabla _ { a } ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) \big \vert _ { a = \mu _ { \theta } ( s ) } \right) ^ { 2 } \mathrm { d } \rho ^ { \pi } ( s ) = 0 } \end{array}$ (i.e., $w$ minimizes the expected squared error of the Hessians). + +One possible parameterization of $\tilde { Q } _ { w } ^ { \pi }$ may be achieved by taking $w = [ w _ { 0 } , w _ { 1 } , w _ { 2 } ]$ and parameterizing + +$$ +\begin{array} { r } { \tilde { Q } _ { w } ^ { \pi } ( s , a ) = V _ { w _ { 0 } } ( s ) + ( a - \mu _ { \theta } ( s ) ) ^ { T } \nabla _ { \theta } \mu _ { \theta } ( s ) ^ { T } w _ { 1 } + ( a - \mu _ { \theta } ( s ) ) ^ { T } \nabla _ { \phi } \Sigma _ { \phi } ( s ) ^ { T } w _ { 2 } ( a - \mu _ { \theta } ( s ) ) . } \end{array} +$$ + +
HyperparameterRangeSampling
actor learningrate critic learning rate[1e-6,1e-3] [1e-6,1e-3]log
reward scale[0.01,0.3]l0g
OU damping[1e-4,1e-3]l0g
OU stddev10g
入[1e-3,1.0]log
discount factor[1e-6, 4e-2] 0.995log
target network lag0.01fixed
batch size128fixed
clipping on gradients of Qfixed
4.0fixed
num gradient updates per observation1fixed
Huber loss clipping1.0fixed
+ +Proof. We shall show how the conditions stated for compatibility with respect to $\Sigma _ { \phi }$ are sufficient. The reasoning for $\mu _ { \theta }$ follows via a similar argument. We also refer the reader to Silver et al. (2014) which includes a similar procedure for showing compatibility. + +From the second condition for compatibility with respect to $\Sigma _ { \phi }$ we have + +$$ +\int _ { S } \left( \nabla _ { a } ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } - \nabla _ { a } ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } \right) \nabla _ { w } \left( \nabla _ { a } ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } \right) \mathrm { d } \rho ^ { \pi } ( s ) = 0 . +$$ + +We may combine this with the first condition to find + +$$ +\int _ { S } \nabla _ { a } ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } \nabla _ { \phi } \Sigma _ { \phi } ( s ) \mathrm { d } \rho ^ { \pi } ( s ) = \int _ { S } \nabla _ { a } ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } \nabla _ { \phi } \Sigma _ { \phi } ( s ) \mathrm { d } \rho ^ { \pi } ( s ) , +$$ + +which is the desired property for compatibility. + +While it is reassuring to know that there exists a class of function approximators which are compatible, this fact is largely ignored in practice. Not only is the class of compatible functions heavily restricted in terms of expressiveness, due to the first set of conditions for $\mu _ { \theta } , \Sigma _ { \phi }$ , it is also impossible to satisfy the second set of conditions without access to derivative and Hessian information of the true ${ \tilde { Q } } ^ { \pi }$ . This problem is also present in DDPG, and we feel this issue merits additional investigation in future work. + +# C IMPLEMENTATION DETAILS + +We utilize feed forward networks for both policy and Q-value approximator. For $\mu _ { \boldsymbol { \theta } } ( s )$ we use two hidden layers of dimensions (400, 300) and relu activation functions. For $\tilde { Q } _ { w } ^ { \pi } ( s , a )$ and $Q _ { w } ^ { \pi } ( s , a )$ we first embed the state into a 400 dimensional vector using a fully-connected layer and tanh nonlinearity. We then concatenate the embedded state with $a$ and pass the result through a 1-hidden layer neural network of dimension 300 with tanh activations. We use a diagonal $\Sigma _ { \phi } ^ { \bar { } } ( s ) = e ^ { \phi }$ for Smoothie, with $\phi$ initialized to $- 1$ . + +To find optimal hyperparameters we perform a 100-trial random search over the hyperparameters specified in Table 1. The OU exploration parameters only apply to DDPG. The $\lambda$ coefficient on KL-penalty only applies to Smoothie with a KL-penalty. \ No newline at end of file diff --git a/parse/train/B1nLkl-0Z/B1nLkl-0Z_content_list.json b/parse/train/B1nLkl-0Z/B1nLkl-0Z_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..3a0f50d889dded47d97b01e04c2fc6ab0adf2d51 --- /dev/null +++ b/parse/train/B1nLkl-0Z/B1nLkl-0Z_content_list.json @@ -0,0 +1,1818 @@ +[ + { + "type": "text", + "text": "LEARNING GAUSSIAN POLICIES FROM SMOOTHED ACTION VALUE FUNCTIONS ", + "text_level": 1, + "bbox": [ + 174, + 99, + 656, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "State-action value functions (i.e., Q-values) are ubiquitous in reinforcement learning (RL), giving rise to popular algorithms such as SARSA and Q-learning. We propose a new notion of action value defined by a Gaussian smoothed version of the expected Q-value. We show that such smoothed Q-values still satisfy a Bellman equation, making them learnable from experience sampled from an environment. Moreover, the gradients of expected reward with respect to the mean and covariance of a parameterized Gaussian policy can be recovered from the gradient and Hessian of the smoothed Q-value function. Based on these relationships we develop new algorithms for training a Gaussian policy directly from a learned smoothed Q-value approximator. Our approach is amenable to proximal optimization techniques by augmenting the objective with a penalty on KLdivergence from a previous policy. We find that the ability to learn both a mean and covariance during training allows this approach to achieve much better results on standard continuous control benchmarks. ", + "bbox": [ + 233, + 266, + 764, + 460 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 487, + 336, + 503 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Model-free reinforcement learning algorithms often alternate between two concurrent but interacting processes: (1) policy evaluation, where an action value function (i.e., a Q-value) is updated to obtain a better estimate of the return associated with taking a specific action, and (2) policy improvement, where the policy is updated aiming to maximize the current value function. In the past, different notions of Q-value have led to distinct but important families of RL methods. For example, SARSA (Rummery & Niranjan, 1994; Sutton & Barto, 1998; Van Seijen et al., 2009) uses the expected Q-value, defined as the expected return of following the current policy. Q-learning (Watkins, 1989) exploits a hard-max notion of Q-value, defined as the expected return of following an optimal policy. Soft Q-learning (Haarnoja et al., 2017) and PCL (Nachum et al., 2017a) both use a soft-max form of Q-value, defined as the future return of following an optimal entropy regularized policy. Clearly, the choice of Q-value function has a considerable effect on the resulting algorithm; for example, restricting the types of policies that can be expressed, and determining the type of exploration that can be naturally applied. ", + "bbox": [ + 174, + 518, + 825, + 699 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work we introduce a new notion of action value: the smoothed action value function ${ \\tilde { Q } } ^ { \\pi }$ . Unlike previous notions, which associate a value with a specific action at each state, the smoothed Qvalue associates a value with a specific distribution over actions. In particular, the smoothed Q-value of a state-action pair $( s , a )$ is defined as the expected return of first taking an action sampled from a normal distribution $N ( \\dot { a } , \\Sigma ( s ) )$ , centered at $a$ , then following actions sampled from the current policy thereafter. In this way, the smoothed Q-value can also be interpreted as a Gaussian-smoothed or noisy version of the expected Q-value. ", + "bbox": [ + 174, + 708, + 825, + 805 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We show that smoothed Q-values possess a number of interesting properties that make them attractive for use in RL algorithms. For one, the smoothed Q-values satisfy a single-step Bellman consistency, which allows bootstrapping to be used to train a function approximator. Secondly, for Gaussian policies, the standard optimization objective (expected return) can be expressed in terms of smoothed Q-values. Moreover, the gradient of this objective with respect to the mean and covariance of the Gaussian policy is equivalent to the gradient and the Hessian of the smoothed Q-value function, which allows one to derive updates to the policy parameters by having access to the derivatives of a sufficiently accurate smoothed Q-value function. ", + "bbox": [ + 174, + 813, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This observation leads us to propose an algorithm called Smoothie, which in the spirit of (Deep) Deterministic Policy Gradient (DDPG) (Silver et al., 2014; Lillicrap et al., 2016), trains a policy using the derivatives of a trained (smoothed) Q-value function, thus avoiding the high-variance of stochastic updates used in standard policy gradient algorithms (Williams & Peng, 1991; Konda & Tsitsiklis, 2000). Unlike DDPG, which is well-known to have poor exploratory behavior (Haarnoja et al., 2017), the approach we develop is able to utilize a non-deterministic Gaussian policy parameterized by both a mean and a covariance, thus allowing the policy to be exploratory by default and alleviating the need for excessive hyperparameter tuning. ", + "bbox": [ + 173, + 103, + 825, + 215 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Furthermore, we show that Smoothie can be easily adapted to incorporate proximal policy optimization techniques by augmenting the objective with a penalty on KL-divergence from a previous version of the policy. The inclusion of a KL-penalty is not feasible in the standard DDPG algorithm, but we show that it is possible with our formulation, and it significantly improves stability and overall performance. On standard continuous control benchmarks, our results are competitive with or exceed state-of-the-art, especially for more difficult tasks in the low-data regime. ", + "bbox": [ + 174, + 222, + 825, + 305 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 NOTATION & BACKGROUND", + "text_level": 1, + "bbox": [ + 176, + 324, + 439, + 342 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We consider the standard model-free RL framework, where an agent interacts with a stochastic black-box environment by sequentially observing the state of the environment, emitting an action, and receiving a reward feedback; the goal is to find an agent that achieves maximal cumulative discounted reward. This problem can be expressed in terms of a Markov decision process (MDP) that consists of a state space $s$ and an action space $\\mathcal { A }$ , where at iteration $t$ the agent encounters a state $s _ { t } ~ \\in ~ S$ and emits an action $a _ { t } \\in \\mathcal A$ , after which the environment returns a scalar reward $\\boldsymbol { r } _ { t } \\sim R ( s _ { t } , \\boldsymbol { a } _ { t } )$ and places the agent in a new state $s _ { t + 1 } \\sim P ( s _ { t } , a _ { t } )$ . ", + "bbox": [ + 173, + 354, + 825, + 454 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We model the behavior of the agent using a stochastic policy $\\pi$ that produces a distribution over feasible actions at each state $s$ as $\\pi ( a \\mid s )$ . The optimization objective (expected discounted return), as a function of the policy, can then be expressed in terms of the expected action value function $Q ^ { \\pi } ( s , a )$ by, ", + "bbox": [ + 174, + 459, + 825, + 515 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/c8cd7a89e53b1d3793cf1570c985f6d22970baa4281b62cb5b9aea8f11e54055.jpg", + "text": "$$\nO _ { \\tt E R } ( \\pi ) = \\int _ { \\cal S } \\int _ { \\cal A } \\pi ( a \\mid s ) Q ^ { \\pi } ( s , a ) \\mathrm { d } a \\mathrm { d } \\rho ^ { \\pi } ( s ) ,\n$$", + "text_format": "latex", + "bbox": [ + 339, + 513, + 656, + 547 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\rho ^ { \\pi } ( s )$ is the stationary distribution of the states under $\\pi$ , and $Q ^ { \\pi } ( s , a )$ is recursively defined using the Bellman equation, ", + "bbox": [ + 176, + 547, + 825, + 575 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/4935471ef55f310e3e24877e897a38f4f10940cefaaf19c3e367eff3cb69679d.jpg", + "text": "$$\nQ ^ { \\pi } ( s , a ) = \\mathbb { E } _ { r , s ^ { \\prime } } \\left[ r + \\gamma \\int _ { \\mathcal { A } } Q ^ { \\pi } ( s ^ { \\prime } , a ^ { \\prime } ) \\pi ( a ^ { \\prime } \\mid s ^ { \\prime } ) \\mathrm { d } a \\right] \\mathrm { , }\n$$", + "text_format": "latex", + "bbox": [ + 320, + 577, + 676, + 612 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\gamma \\in [ 0 , 1 ]$ is the discount factor. For brevity, we will often suppress explicit denotation of the sampling distribution $R$ over immediate rewards and the distribution $P$ over state transitions. ", + "bbox": [ + 176, + 614, + 821, + 642 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The policy gradient theorem (Sutton et al., 2000) expresses the gradient of $O _ { \\mathrm { E R } } ( \\pi _ { \\theta } )$ w.r.t. $\\theta$ , the tunable parameters of a policy $\\pi _ { \\theta }$ , as, ", + "bbox": [ + 174, + 648, + 821, + 678 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/0a901f690d4f358615e33840232298f5dac446623f98273090e286de6d11634e.jpg", + "text": "$$\n\\begin{array} { r c l } { { \\nabla _ { \\theta } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta } ) } } & { { = } } & { { \\displaystyle \\int _ { S } \\int _ { { \\cal A } } \\nabla _ { \\theta } \\pi _ { \\theta } ( a \\mid s ) Q ^ { \\pi } ( s , a ) \\mathrm { d } a \\mathrm { d } \\rho ^ { \\pi } ( s ) } } \\\\ { { } } & { { = } } & { { \\displaystyle \\int _ { S } \\mathbb { E } _ { a \\sim \\pi _ { \\theta } ( a \\mid s ) } \\left[ \\nabla _ { \\theta } \\log \\pi _ { \\theta } ( a \\mid s ) Q ^ { \\pi } ( s , a ) \\right] \\mathrm { d } \\rho ^ { \\pi } ( s ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 679, + 727, + 747 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Many reinforcement learning algorithms, including policy gradient and actor-critic variants, trade off variance and bias when estimating the random variable inside the expectation in (4); for example, by attempting to estimate $Q ^ { \\pi } ( s , a )$ accurately using function approximation. In the simplest scenario, an unbiased estimate of $Q ^ { \\pi } ( s , a )$ is formed by accumulating discounted rewards from each state forward using a single Monte Carlo sample. ", + "bbox": [ + 176, + 746, + 825, + 816 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we focus on multivariate Gaussian policies over continuous action spaces, $\\mathcal { A } \\equiv \\mathbb { R } ^ { d _ { a } }$ . We represent the observed state of the MDP as a $d _ { s }$ -dimensional feature vector $\\bar { \\Phi } ( s ) \\in \\mathbb { R } ^ { d _ { s } }$ , and parametrize the Gaussian policy by a mean and covariance function, respectively $\\mu ( \\boldsymbol { s } ) : \\mathbb { R } ^ { d _ { \\boldsymbol { s } } } \\mathbb { R } ^ { d _ { a } }$ and $\\Sigma ( s ) : \\mathbb { R } ^ { d _ { s } } \\mathbb { R } ^ { d _ { a } } \\times \\mathbb { R } ^ { \\tilde { d } _ { a } }$ . These map the observed state of the environment to a Gaussian distribution, ", + "bbox": [ + 173, + 821, + 825, + 893 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/30e174d88444c4e6a80ccbede2e10a2cc7673ac23a5d544af7e72083cd2e3091.jpg", + "text": "$$\n\\pi ( a | s ) = N ( a | \\mu ( s ) , \\Sigma ( s ) ) = | 2 \\pi \\Sigma ( s ) | ^ { - 1 / 2 } \\exp \\left\\{ - \\frac { 1 } { 2 } \\| a - \\mu ( s ) \\| _ { \\Sigma ( s ) ^ { - 1 } } ^ { 2 } \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 236, + 893, + 759, + 929 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\| v \\| _ { A } ^ { 2 } = v ^ { \\mathsf { T } } A v$ . Below we develop new RL training methods for this family of parametric policies, but some of the ideas presented may generalize to other families of policies as well. We begin the formulation by reviewing some prior work on learning Gaussian policies. ", + "bbox": [ + 174, + 102, + 825, + 146 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 DETERMINISTIC POLICY GRADIENT ", + "text_level": 1, + "bbox": [ + 176, + 162, + 464, + 176 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Silver et al. (2014) present a new formulation of the policy gradient, called the deterministic policy gradient, for the family of Gaussian policies in the limit where the policy covariance approaches zero. In such a scenario, the policy becomes deterministic because sampling from the policy always returns the Gaussian mean. The key observation of (Silver et al., 2014) is that under a deterministic policy $\\pi \\equiv ( \\mu , \\Sigma \\to 0 )$ , one can estimate the expected future return from a state $s$ as, ", + "bbox": [ + 173, + 188, + 825, + 258 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/5bbd75b4612abde13afa02d3c4fd27ad64c7f82ad3480efe30ec063b6a7aeef2.jpg", + "text": "$$\n\\operatorname * { l i m } _ { \\Sigma \\to 0 } \\int _ { A } \\pi ( a \\mid s ) Q ^ { \\pi } ( s , a ) \\mathrm { d } a = Q ^ { \\pi } ( s , \\mu ( s ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 263, + 651, + 297 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Then, one can express the gradient of the optimization objective (expected discounted return) for a parameterized $\\pi _ { \\boldsymbol { \\theta } } \\equiv \\mu _ { \\boldsymbol { \\theta } }$ as, ", + "bbox": [ + 174, + 301, + 823, + 330 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/edb5b4ed43a4f1a0e669d90a2efa86e348e4bfe334dbf8cd55ff8c9d7fefcff8.jpg", + "text": "$$\n\\nabla _ { \\theta } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta } ) = \\int _ { S } \\nabla _ { \\theta } Q ^ { \\pi } ( s , \\mu _ { \\theta } ( s ) ) \\mathrm { d } \\rho ^ { \\pi } ( s ) = \\int _ { S } \\frac { \\partial Q ^ { \\pi } ( s , a ) } { \\partial a } | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .\n$$", + "text_format": "latex", + "bbox": [ + 194, + 335, + 784, + 369 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This can be thought of as a characterization of the policy gradient theorem for deterministic policies. ", + "bbox": [ + 173, + 373, + 823, + 388 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the limit of $\\Sigma 0$ , one can also re-express the Bellman equation (2) as, ", + "bbox": [ + 176, + 395, + 666, + 410 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2f00868dda5fac9664e78a8235cb650a820dc280c48e8bb4598ed26563080e3e.jpg", + "text": "$$\nQ ^ { \\pi } ( s , a ) = \\mathbb { E } _ { r , s ^ { \\prime } } \\left[ r + Q ^ { \\pi } ( s ^ { \\prime } , \\mu ( s ^ { \\prime } ) ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 414, + 630, + 433 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Therefore, a value function approximator $Q _ { w } ^ { \\pi }$ can be optimized by minimizing the Bellman error, ", + "bbox": [ + 174, + 438, + 805, + 453 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/33f14e5df15606e8aea149929ac6bcadca43b1040e98716946a23e09f91a9fe2.jpg", + "text": "$$\nE ( w ) = \\sum _ { ( s , a , r , s ^ { \\prime } ) \\in \\mathcal { D } } ( Q _ { w } ^ { \\pi } ( s , a ) - r - \\gamma Q _ { w } ^ { \\pi } ( s ^ { \\prime } , \\mu ( s ^ { \\prime } ) ) ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 316, + 457, + 679, + 494 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "for transitions $( s , a , r , s ^ { \\prime } )$ sampled from a dataset $\\mathcal { D }$ of interactions of the agent with the environment. Algorithms like DDPG (Lillicrap et al., 2016) alternate between improving the value function by gradient descent on (9), and improving the policy based on (7). ", + "bbox": [ + 174, + 500, + 825, + 542 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In practice, to gain better sample efficiency, Degris et al. (2012) and Silver et al. (2014) replace the on-policy state distribution $\\rho ^ { \\pi } ( s )$ in (7) with an off-policy distribution $\\rho ^ { \\beta } ( s )$ based on a replay buffer. After this substitution, the policy gradient identity in (7) does not hold exactly, however, prior work finds that this works well in practice and improves sample efficiency. We also adopt a similar approximation in our method to make use of off-policy data. ", + "bbox": [ + 173, + 547, + 825, + 619 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 SMOOTHED ACTION VALUE FUNCTIONS ", + "text_level": 1, + "bbox": [ + 174, + 638, + 540, + 655 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this paper, we introduce smoothed action value functions, the gradients of which provide an effective signal for optimizing the parameters of a Gaussian policy. Our notion of smoothed Qvalues, denoted $\\tilde { Q } ^ { \\pi } ( s , a )$ , differs from ordinary Q-values $Q ^ { \\pi } ( s , a )$ in that smoothed Q-values do not assume the first action of the agent is fully specified, but rather they assume that only the mean of the distribution of the first action is known. Hence, to compute $\\tilde { Q } ^ { \\pi } ( s , a )$ , one has to perform an expectation of $Q ^ { \\pi } ( s , { \\tilde { a } } )$ for actions $\\tilde { a }$ drawn in the vicinity of $a$ . More formally, smoothed action values are defined as, ", + "bbox": [ + 173, + 669, + 825, + 770 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8713600fb539a4a0fd028c7442c7fd55bcda9b1e7c291fef1adceb76fdfeb0f3.jpg", + "text": "$$\n\\tilde { Q } ^ { \\pi } ( s , a ) = \\int _ { A } N ( \\tilde { a } | a , \\Sigma ( s ) ) Q ^ { \\pi } ( s , \\tilde { a } ) \\mathrm { d } \\tilde { a } .\n$$", + "text_format": "latex", + "bbox": [ + 348, + 767, + 648, + 801 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "With this definition of ${ \\tilde { Q } } ^ { \\pi }$ , one can re-express the expected reward objective for a Gaussian policy $\\pi \\equiv ( \\mu , \\Sigma )$ as, ", + "bbox": [ + 174, + 804, + 821, + 833 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/746ea57183920be259738989cf19f6bf70a85484858cb13172e34626faa37835.jpg", + "text": "$$\nO _ { \\mathrm { E R } } ( \\pi ) = \\int _ { S } \\tilde { Q } ^ { \\pi } ( s , \\mu ( s ) ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 832, + 612, + 864 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The insight that differentiates this approach from prior work including Heess et al. (2015); Ciosek & Whiteson (2017) is that instead of learning a function approximator for $Q ^ { \\pi } ( s , a )$ and then drawing samples to approximate the expectation in (10) and its derivative, we directly learn a function approximator for $\\bar { Q } ^ { \\pi } ( s , a )$ . ", + "bbox": [ + 174, + 866, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The key observation that enables direct bootstrapping of smoothed $\\mathrm { Q }$ -values, $\\tilde { Q } ^ { \\pi } ( s , a )$ , is that their form allows a notion of Bellman consistency. First, note that for Gaussian policies $\\pi \\equiv ( \\mu , \\Sigma )$ we have ", + "bbox": [ + 174, + 103, + 825, + 143 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/21a3aedaa734f3c25543a623fa9c74af9b5602d1607eca5b4b1dda92b52eb8f7.jpg", + "text": "$$\nQ ^ { \\pi } ( s , a ) = \\mathbb { E } _ { r , s ^ { \\prime } } [ r + \\gamma \\tilde { Q } ^ { \\pi } ( s ^ { \\prime } , \\mu ( s ^ { \\prime } ) ) ] .\n$$", + "text_format": "latex", + "bbox": [ + 369, + 142, + 629, + 161 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Then, combining (10) and (12), one can derive the following one-step Bellman equation for smoothed Q-values, ", + "bbox": [ + 171, + 165, + 825, + 195 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/fe18d88621a98303e4f2b541b01df7598dfd6b3292ac4b2c5c26bfe54af28ffb.jpg", + "text": "$$\n\\tilde { Q } ^ { \\pi } ( s , a ) = \\int _ { \\cal A } N ( \\tilde { a } \\mid a , \\Sigma ( s ) ) \\mathbb { E } _ { \\tilde { r } , \\tilde { s } ^ { \\prime } } \\left[ \\tilde { r } + \\gamma \\tilde { Q } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) \\right] \\mathrm { d } \\tilde { a } ,\n$$", + "text_format": "latex", + "bbox": [ + 292, + 199, + 702, + 233 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\tilde { r }$ and ${ \\tilde { s } } ^ { \\prime }$ are sampled from $R ( s , { \\tilde { a } } )$ and $P ( s , { \\tilde { a } } )$ . Below, we elaborate on how one can make use of the derivatives of ${ \\tilde { Q } } ^ { \\pi }$ to learn $\\mu$ and $\\Sigma$ , and how the Bellman equation in (13) enables direct optimization of ${ \\tilde { Q } } ^ { \\pi }$ . ", + "bbox": [ + 174, + 239, + 826, + 286 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 POLICY IMPROVEMENT - OPTIMIZING $\\left( \\mu _ { \\theta } , \\Sigma _ { \\phi } \\right)$ GIVEN ${ \\tilde { Q } } ^ { \\pi }$ ", + "text_level": 1, + "bbox": [ + 173, + 303, + 616, + 320 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We parameterize a Gaussian policy $\\pi _ { \\boldsymbol { \\theta } , \\boldsymbol { \\phi } } \\equiv ( \\mu _ { \\boldsymbol { \\theta } } , \\Sigma _ { \\boldsymbol { \\phi } } )$ in terms of two sets of parameters $\\theta$ and $\\phi$ for the mean and the covariance. The gradient of the objective $w . r . t .$ . mean parameters follows from the policy gradient theorem and is almost identical to (7), ", + "bbox": [ + 173, + 330, + 826, + 375 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6d9aef1b0a5509659a282090238279c9da22df639f2282e7c6aa94387ffb0566.jpg", + "text": "$$\n\\nabla _ { \\theta } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta , \\phi } ) = \\int _ { S } \\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a } \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .\n$$", + "text_format": "latex", + "bbox": [ + 307, + 380, + 691, + 417 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Estimating the derivative of the objective w.r.t. covariance parameters is not as straightforward, since ${ \\tilde { Q } } ^ { \\pi }$ is not a direct function of $\\Sigma$ . However, a key observation of this work is that the second derivative of ${ \\tilde { Q } } ^ { \\pi } w . r . t .$ . actions is sufficient to exactly compute the derivative of $\\tilde { Q } ^ { \\pi } w . r . t . \\Sigma$ , ", + "bbox": [ + 173, + 429, + 825, + 477 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5b5740281c05060410f4bd717038d516178564d8e3793b5af4abb32a9f57cbf7.jpg", + "text": "$$\n\\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial \\Sigma ( s ) } = \\frac { 1 } { 2 } \\cdot \\frac { \\partial ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 482, + 601, + 518 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A proof of this identity is provided in the Appendix. The proof may be easily derived by expressing both sides of the equation using standard matrix calculus like $\\begin{array} { r } { \\frac { \\partial } { \\partial A } | A | ^ { - 1 / 2 } = - \\frac { 1 } { 2 } | A | ^ { - 1 / 2 } \\bar { A } ^ { - 1 } } \\end{array}$ and $\\begin{array} { r } { \\frac { \\partial } { \\partial A } | | v | | _ { A ^ { - 1 } } ^ { 2 } = - A ^ { - 1 } v v ^ { T } A ^ { - 1 } } \\end{array}$ . ", + "bbox": [ + 173, + 525, + 825, + 571 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Then, the full derivative w.r.t. $\\phi$ takes the form, ", + "bbox": [ + 174, + 578, + 482, + 593 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f41c8adcf7c19ddab43a7d258f46d084293a60a0cb7aa0628a9bb7df0f7b6d52.jpg", + "text": "$$\n\\nabla _ { \\phi } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta , \\phi } ) = \\frac { 1 } { 2 } \\int _ { S } \\frac { \\partial ^ { 2 } { \\tilde { Q } } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } } \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .\n$$", + "text_format": "latex", + "bbox": [ + 294, + 599, + 702, + 636 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 POLICY EVALUATION - OPTIMIZING $\\tilde { Q } _ { w } ^ { \\pi }$ GIVEN $( \\mu , \\Sigma )$ ", + "text_level": 1, + "bbox": [ + 174, + 651, + 589, + 669 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We can think of two ways to optimize $\\tilde { Q } _ { w } ^ { \\pi }$ . The first approach leverages (10) to update ${ \\tilde { Q } } ^ { \\pi }$ based on expected Q-value function $Q ^ { \\pi }$ . In such an approach, one trains a parameterized $Q _ { w } ^ { \\pi }$ to approximate the standard expected Q-value function $Q ^ { \\pi }$ using standard methods (see e.g., Rummery $\\&$ Niranjan (1994); Sutton & Barto (1998); Van Seijen et al. (2009)). Then, one fits $\\tilde { Q } _ { w } ^ { \\pi }$ based on $Q _ { w } ^ { \\pi }$ . In particular, given transitions $( s , a , r , s ^ { \\prime } )$ sampled from interactions with the environment, one can train $Q _ { w } ^ { \\pi }$ to minimize the Bellman error $( Q _ { w } ^ { \\pi } ( s , a ) - r - \\gamma Q _ { w } ^ { \\pi } ( s ^ { \\prime } , a ^ { \\prime } ) ) ^ { 2 }$ where $a ^ { \\prime } \\sim N ( \\mu ( s ^ { \\prime } ) , \\Sigma ( s ^ { \\prime } ) )$ . Then, $\\tilde { Q } _ { w } ^ { \\pi }$ can be optimized to minimize the squared error $( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\mathbb { E } _ { \\tilde { a } } Q _ { w } ^ { \\pi } ( s , \\tilde { a } ) ) ^ { 2 }$ where $\\tilde { a } \\sim$ $N ( a , \\Sigma ( s ) )$ , using several samples. When the target values in these residuals are treated as fixed (i.e., using a target network), such a training procedure will achieve a fixed point when $\\tilde { Q } _ { w } ^ { \\pi } ( s , a )$ satisfies the recursion in the Bellman equation (10). ", + "bbox": [ + 173, + 679, + 825, + 828 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The second approach requires a single function approximator for $\\tilde { Q } _ { w } ^ { \\pi } ( s , a )$ , resulting in a simpler implementation, and thus we use this approach in our experimental evaluation. Suppose one has access to a tuple $( s , \\tilde { a } , \\tilde { r } , \\tilde { s } ^ { \\prime } )$ sampled from a replay buffer with knowledge of the sampling probability $q ( \\tilde { a } \\mid s )$ (possibly unnormalized). Then assuming that this sampling distribution has a full support, we draw a phantom action $a \\sim N ( \\tilde { a } , \\Sigma ( s ) )$ and optimize $\\tilde { Q } _ { w } ^ { \\pi } ( s , a )$ by minimizing a weighted Bellman error $\\begin{array} { r } { \\frac { 1 } { q ( \\tilde { a } | s ) } ( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\tilde { r } - \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) ^ { 2 } } \\end{array}$ . For a specific pair of state and action $( s , a )$ the ", + "bbox": [ + 173, + 834, + 825, + 926 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "expected value of the objective is, ", + "bbox": [ + 173, + 103, + 398, + 118 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c76fa0ee66173ea378e379d523736869626d6fb2a9f21c00ecef6ee7a64e1c84.jpg", + "text": "$$\nE ( w \\mid ( s , a ) ) ~ = ~ \\mathbb { E } _ { q ( \\bar { a } \\mid s ) , \\bar { r } , \\bar { s } ^ { \\prime } } \\left[ \\frac { N ( a \\mid \\tilde { a } , \\Sigma ( s ) ) } { q ( \\tilde { a } \\mid s ) } ( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\tilde { r } - \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) ) ^ { 2 } \\right] ~ .\n$$", + "text_format": "latex", + "bbox": [ + 207, + 125, + 763, + 160 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that $N ( a | \\tilde { a } , \\Sigma ( s ) ) = N ( \\tilde { a } | a , \\Sigma ( s ) )$ . Therefore, when the target value $\\tilde { r } + \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) )$ is treated as fixed (e.g., when using target networks) this training procedure reaches an optimum when $\\tilde { Q } _ { w } ^ { \\pi } ( s , a )$ satisfies the recursion in the Bellman equation (13). ", + "bbox": [ + 174, + 166, + 821, + 212 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In practice, we find that it is unnecessary to keep track of the probabilities $q ( \\tilde { a } \\mid s )$ , and assume the replay buffer provides a near-uniform distribution of actions conditioned on states. Other recent work has also benefited from ignoring or heavily damping importance weights (Munos et al., 2016; Wang et al., 2017; Schulman et al., 2017). However, it is possible when interacting with the environment to save the probability of sampled actions along with their transitions, and thus have access to $q ( \\tilde { \\boldsymbol { a } } \\mid \\boldsymbol { s } ) \\approx N ( \\tilde { \\boldsymbol { a } } \\mid \\bar { \\mu } _ { \\mathrm { o l d } } ( \\boldsymbol { s } ) , \\dot { \\Sigma _ { \\mathrm { o l d } } } ( \\boldsymbol { s } ) )$ . ", + "bbox": [ + 173, + 218, + 825, + 304 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 PROXIMAL POLICY OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 319, + 455, + 333 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Policy gradient algorithms are notoriously unstable, particularly in continuous control problems. Such instability has motivated the development of trust region methods that attempt to mitigate the issue by constraining each gradient step to lie within a trust region (Schulman et al., 2015), or augmenting the expected reward objective with a penalty on KL-divergence from a previous policy (Nachum et al., 2017b; Schulman et al., 2017; Azar et al., 2012). These stabilizing techniques have thus far not been applicable to algorithms like DDPG, since the policy is deterministic. The formulation we propose in this paper, however, is easily amenable to trust region optimization. Specifically, we may augment the objective (11) with a penalty ", + "bbox": [ + 173, + 344, + 825, + 457 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ce671f74d9ac9a165e85048a8cd34aa0136ad30376cb6d7e13762ea2132990ef.jpg", + "text": "$$\n{ \\cal O } _ { \\mathrm { T R } } ( \\pi ) = { \\cal O } _ { \\mathrm { E R } } ( \\pi ) - \\lambda \\int _ { \\cal S } \\mathrm { K L } ( \\pi \\parallel \\pi _ { \\mathrm { o l d } } ) \\mathrm { d } \\rho ^ { \\pi } ( s ) ,\n$$", + "text_format": "latex", + "bbox": [ + 336, + 462, + 660, + 496 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\pi _ { \\mathrm { o l d } } \\equiv \\left( \\mu _ { \\mathrm { o l d } } , \\Sigma _ { \\mathrm { o l d } } \\right)$ is a previous parameterization of the policy. The optimization is straightforward, since the KL-divergence of two Gaussians can be expressed analytically. ", + "bbox": [ + 171, + 502, + 821, + 531 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 550, + 344, + 566 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This paper follows a long line of work that uses Q-value functions to stably learn a policy, which in the past has been used to either approximate expected (Rummery & Niranjan, 1994; Van Seijen et al., 2009; Gu et al., 2017) or optimal (Watkins, 1989; Silver et al., 2014; Nachum et al., 2017a; Haarnoja et al., 2017; Metz et al., 2017) future value. ", + "bbox": [ + 174, + 582, + 825, + 638 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Work that is most similar to what we present are methods that exploit gradient information from the Q-value function to train a policy. Deterministic policy gradient (Silver et al., 2014) is perhaps the best known of these. The method we propose can be interpreted as a generalization of the deterministic policy gradient. Indeed, if one takes the limit of the policy covariance $\\Sigma ( s )$ as it goes to 0, the proposed Q-value function becomes the deterministic value function of DDPG, and the updates for training the Q-value approximator and the policy mean are identical. ", + "bbox": [ + 174, + 645, + 825, + 728 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Stochastic Value Gradient (SVG) (Heess et al., 2015) also trains stochastic policies using an update that is similar to DDPG (i.e., SVG(0) with replay). The key differences with our approach are that SVG does not provide an update for the covariance, and the mean update in SVG estimates the gradient with a noisy Monte Carlo sample, which we avoid by estimating the smoothed $\\mathrm { Q }$ -value function. Although a covariance update could be derived using the same reparameterization trick as in the mean update, that would also require a noisy Monte Carlo estimate. Methods for updating the covariance along the gradient of expected reward are essential for applying the subsequent trust region and proximal policy techniques. ", + "bbox": [ + 173, + 734, + 825, + 848 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "More recently, Ciosek & Whiteson (2017) introduced expected policy gradients (EPG), a generalization of DDPG that provides updates for the mean and covariance of a stochastic Gaussian policy using gradients of an estimated Q-value function. In that work, the expected Q-value used in standard policy gradient algorithms such as SARSA (Sutton & Barto, 1998; Rummery & Niranjan, 1994; Van Seijen et al., 2009) is estimated. The updates in EPG therefore require approximating an integral of the expected Q-value function. Our analogous process directly estimates an integral (via the smoothed Q-value function) and avoids approximate integrals, thereby making the updates simpler. Moreover, while Ciosek & Whiteson (2017) rely on a quadratic Taylor expansion of the estimated Q-value function, we instead rely on the strength of neural network function approximators to directly estimate the smoothed Q-value function. ", + "bbox": [ + 174, + 853, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 174 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The novel training scheme we propose for learning the covariance of a Gaussian policy relies on properties of Gaussian integrals (Bonnet, 1964; Price, 1958). Similar identities have been used in the past to derive updates for variational auto-encoders (Kingma & Welling, 2014) and Gaussian back-propagation (Rezende et al., 2014). ", + "bbox": [ + 174, + 180, + 823, + 236 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Finally, the perspective presented in this paper, where Q-values represent the averaged return of a distribution of actions rather than a single action, is distinct from recent advances in distributional RL (Bellemare et al., 2017). Those approaches focus on the distribution of returns of a single action, whereas we consider the single average return of a distribution of actions. Although we restrict our attention in this paper to Gaussian policies, an interesting topic for further investigation is to study the applicability of this new perspective to a wider class of policy distributions. ", + "bbox": [ + 174, + 243, + 825, + 327 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 359, + 326, + 376 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We utilize the insights from Section 3 to introduce a new RL algorithm, Smoothie. Smoothie maintains a parameterized $\\tilde { Q } _ { w } ^ { \\pi }$ trained via the procedure described in Section 3.2. It then uses the gradient and Hessian of this approximation to train a Gaussian policy $\\mu _ { \\theta } , \\Sigma _ { \\phi }$ using the updates stated in (14) and (16). See Algorithm 1 for a simplified pseudocode of our algorithm. ", + "bbox": [ + 174, + 398, + 825, + 458 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithm 1 Smoothie ", + "text_level": 1, + "bbox": [ + 174, + 484, + 328, + 498 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/eb3fa1f9a495240797e692645eace753d0aebfc6f323140585dd7b69aaea1d55.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Algorithm1Smoothie Input: Environment ENV,learning rates 7π, nQ,discount factor y,KL-penalty 入,l
number of training steps N, target network lag T. Initialize0,,w,set0'=0,Φ'=,w'=w.
fori=OtoN-1do
/ Collect experience
Sample action a ~ N(μe(s),∑(s)) and apply to ENV to yield r and s'. Insert transition (s,a,r,s') to replay buffer.
// Train μ,∑
Sample batch{(k,@k,Tk,S)}1 fromreplay buffer aQ(ska) Compute gradients gk =
da la=μe(sk)*
²(s@） Compute Hessians Hk = a² la=μe(sk)*
Compute KL-penalties KLk =KL(μθ,Σ𝜙llμe,Σ).
Compute updates 1B
1B
△=B 1 ∑k=1 Updateθ←θ+ηπ△0,←+ηπ△.
// Train Qπ
Sample batch {(sk,k,Tk,S)}1 from replay bufer. Sample phantom actions ak ~ N(ák,Σ(sk)). Compute lossL(u)=∑1((s)-r-qQ(s,ue()2.
", + "bbox": [ + 174, + 489, + 740, + 919 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We perform a number of evaluations of Smoothie compared to DDPG. We choose DDPG as a baseline because it (1) utilizes gradient information of a Q-value approximator, much like our algorithm; and (2) is a standard algorithm well-known to have achieve good, sample-efficient performance on continuous control benchmarks. ", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 A SYNTHETIC TASK ", + "text_level": 1, + "bbox": [ + 176, + 180, + 354, + 194 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To evaluate Smoothie we begin with a simple synthetic task which allows us to study its behavior in a restricted setting. We devised a simple single-action one-shot environment in which the reward function is a mixture of two Gaussians, one better than the other (see Figure 1 (Right)). We initialize the policy mean to be centered on the worse of the two Gaussians. We plot the learnable policy mean and standard deviation during training for Smoothie and DDPG in Figure 1 (Left). Smoothie learns both the mean and variance, while DDPG learns only the mean and the variance plotted is the exploratory noise, whose scale is kept fixed during training. ", + "bbox": [ + 174, + 208, + 825, + 306 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As expected we observe that DDPG cannot escape the local optimum. At the beginning of training it exhibits some movement away from the local optimum (likely due to the initial noisy approximation given by $Q _ { w } ^ { \\pi } \\mathrm { . }$ ), it is unable to progress very far from the initial mean. Note that this is not an issue of exploration. The exploration scale is high enough that $Q _ { w } ^ { \\pi }$ is aware of the better Gaussian. The issue is in the update for $\\mu _ { \\theta }$ , which is only with regard to the derivative of $Q _ { w } ^ { \\pi }$ at the current mean. ", + "bbox": [ + 174, + 313, + 825, + 382 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "On the other hand, we find Smoothie is successfully able to solve the task. This is because the smoothed reward function approximated by $\\tilde { Q } _ { w } ^ { \\pi }$ has a derivative which clearly points $\\mu _ { \\theta }$ towards the better Gaussian. We also observe that Smoothie is able to suitably adjust the covariance $\\Sigma _ { \\phi }$ during training. Initially, $\\Sigma _ { \\phi }$ decreases due to the concavity of the smoothed reward function. As a region of convexity is entered, it begins to increase, before again decreasing to near-zero as $\\mu _ { \\theta }$ approaches the global optimum. ", + "bbox": [ + 173, + 388, + 825, + 474 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/3e62a205722715b4eae018cb617c9164cd435c130be75b6bf1c3addc835c777e.jpg", + "image_caption": [ + "Figure 1: Left: The learnable policy mean and standard deviation during training for Smoothie and DDPG on a simple one-shot synthetic task. The standard deviation for DDPG is the exploratory noise kept constant during training. Right: The reward function for the synthetic task along with its Gaussian-smoothed version. We find that Smoothie can successfully escape the lower-reward local optimum. We also notice Smoothie increases and decreases its policy variance as the convexity/concavity of the smoothed reward function changes. " + ], + "image_footnote": [], + "bbox": [ + 258, + 496, + 735, + 666 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 CONTINUOUS CONTROL ", + "text_level": 1, + "bbox": [ + 176, + 804, + 382, + 819 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We now turn our attention to standard continuous control benchmarks available on OpenAI Gym (Brockman et al., 2016) utilizing the MuJoCo environment (Todorov et al., 2012). ", + "bbox": [ + 176, + 833, + 823, + 861 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our implementations utilize feed forward neural networks for policy and Q-values. We parameterize the covariance $\\Sigma _ { \\phi }$ as a diagonal given by $e ^ { \\phi }$ . The exploration for DDPG is determined by an Ornstein-Uhlenbeck process (Uhlenbeck & Ornstein, 1930; Lillicrap et al., 2016). Additional implementation details are provided in the Appendix. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/f21887a3716c42d41ffbbba35b3c52cf13877d7b56bb86c24b7339f4a95daf67.jpg", + "image_caption": [ + "Figure 2: Results of Smoothie, DDPG, and TRPO on continuous control benchmarks. The $\\mathbf { X }$ -axis is in millions of environment steps. Each plot shows the average reward and standard deviation clipped at the min and max of six randomly seeded runs after choosing best hyperparameters. We see that Smoothie is competitive with DDPG even when DDPG uses a hyperparameter-tuned noise scale, and Smoothie learns the optimal noise scale (the covariance) during training. Moreoever, we observe significant advantages in terms of final reward performance, especially in the more difficult tasks like Hopper, Walker2d, and Humanoid. Across all tasks, TRPO is not sufficiently sampleefficient to provide a competitive baseline. " + ], + "image_footnote": [], + "bbox": [ + 215, + 102, + 779, + 390 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compare the results of Smoothie and DDPG in Figure 2. For each task we performed a hyperparameter search over actor learning rate, critic learning rate and reward scale, and plot the average of six runs for the best hyperparameters. For DDPG we extended the hyperparameter search to also consider the scale and damping of exploratory noise provided by the Ornstein-Uhlenbeck process. Smoothie, on the other hand, contains an additional hyperparameter to determine the weight on KL-penalty. ", + "bbox": [ + 174, + 617, + 825, + 700 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Despite DDPG having the advantage of its exploration decided by a hyperparameter search while Smoothie must learn its exploration without supervision, we find that Smoothie performs competitively or better across all tasks, exhibiting a slight advantage in Swimmer and Ant, while showing more dramatic improvements in Hopper, Walker2d, and Humanoid. The improvement is especially dramatic for Hopper, where the average reward is doubled. We also highlight the results for Humanoid, which as far as we know, are the best published results for a method that only trains on the order of millions of environment steps. In contrast, TRPO, which to the best of our knowledge is the only other algorithm which can achieve better performance, requires on the order of tens of millions of environment steps to achieve comparable reward. This gives added evidence to the benefits of using a learnable covariance and not restricting a policy to be deterministic. ", + "bbox": [ + 174, + 708, + 825, + 847 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Empirically, we found the introduction of a KL-penalty to improve performance of Smoothie, especially on harder tasks. We present a comparison of results of Smoothie with and without the KL-penalty on the four harder tasks in Figure 3. A KL-penalty to encourage stability is not possible in DDPG. Thus, our algorithm provides a much needed solution to the inherent instability in DDPG training. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/31e9c2cf700c5e3ed94cc8e3ab513890a7ad5439462138652b2f32dd4b21ec5f.jpg", + "image_caption": [ + "Figure 3: Results of Smoothie with and without a KL-penalty. The $\\mathbf { X } ^ { } -$ -axis is in millions of environment steps. We observe benefits of using a proximal policy optimization method, especially in Hopper and Humanoid, where the performance improvement is significant without sacrificing sample efficiency. " + ], + "image_footnote": [], + "bbox": [ + 199, + 102, + 795, + 237 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 340, + 318, + 357 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have presented a new Q-value function, ${ \\tilde { Q } } ^ { \\pi }$ , that is a Gaussian-smoothed version of the standard expected Q-value, $Q ^ { \\pi }$ . The advantage of using ${ \\tilde { Q } } ^ { \\pi }$ over $Q ^ { \\pi }$ is that its gradient and Hessian possess an intimate relationship with the gradient of expected reward with respect to mean and covariance of a Gaussian policy. The resulting algorithm, Smoothie, is able to successfully learn both mean and covariance during training, leading to performance that can match or surpass that of DDPG, especially when incorporating a penalty on divergence from a previous policy. ", + "bbox": [ + 173, + 371, + 825, + 458 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The success of ${ \\tilde { Q } } ^ { \\pi }$ is encouraging. Intuitively it may be argued that learning ${ \\tilde { Q } } ^ { \\pi }$ is more sensible than learning $Q ^ { \\pi }$ . The smoothed Q-values by definition make the true reward surface smoother, thus possibly easier to learn; moreover the smoothed Q-values have a more direct relationship with the expected discounted return objective. We encourage future work to further investigate these claims as well as techniques to apply the underlying motivations for ${ \\tilde { Q } } ^ { \\pi }$ to other types of policies. 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Function optimization using connectionist reinforcement learning algorithms. Connection Science, 1991. ", + "bbox": [ + 173, + 886, + 823, + 915 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A PROOF OF EQUATION (15) ", + "text_level": 1, + "bbox": [ + 176, + 102, + 426, + 118 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We note that similar identities for Gaussian integrals exist in the literature (Price, 1958; Rezende et al., 2014) and point the reader to these works for further information. ", + "bbox": [ + 173, + 132, + 823, + 162 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The specific identity we state may be derived using standard matrix calculus. We make use of the fact that ", + "bbox": [ + 174, + 167, + 823, + 195 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/abe39c2a97779def484e4eba142c60209fd9e9f36be5544140f1ba14e1bd2e4a.jpg", + "text": "$$\n\\frac { \\partial } { \\partial A } | A | ^ { - 1 / 2 } = - \\frac { 1 } { 2 } | A | ^ { - 3 / 2 } \\frac { \\partial } { \\partial A } | A | = - \\frac { 1 } { 2 } | A | ^ { - 1 / 2 } A ^ { - 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 320, + 191, + 678, + 223 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "and for symmetric $A$ , ", + "bbox": [ + 173, + 223, + 316, + 238 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/18034b9d4af3f0775f1fdad3b9f4dd9d28f0abf8f059744b1cc914ba9d87feaf.jpg", + "text": "$$\n\\frac \\partial { \\partial A } | | \\boldsymbol { v } | | _ { A ^ { - 1 } } ^ { 2 } = - A ^ { - 1 } { \\boldsymbol { v } } { \\boldsymbol { v } } ^ { T } A ^ { - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 395, + 236, + 602, + 266 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We omit $s$ from $\\Sigma ( s )$ in the following equations for succinctness. The LHS of (15) is ", + "bbox": [ + 173, + 267, + 732, + 284 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/6b52af45a4ad5887c919cf42e53a40ce44cfb2c6eb63716e3df1ab2d181c4ad4.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\int _ { A } Q ^ { \\pi } ( s , \\tilde { a } ) \\frac { \\partial } { \\partial \\Sigma } N ( \\tilde { a } | a , \\Sigma ) \\mathrm { d } \\tilde { a } } \\\\ { \\displaystyle = \\int _ { A } Q ^ { \\pi } ( s , \\tilde { a } ) \\exp \\left\\{ - \\frac { 1 } { 2 } | | \\tilde { a } - a | | _ { \\Sigma ^ { - 1 } } ^ { 2 } \\right\\} \\left( \\frac { \\partial } { \\partial \\Sigma } | 2 \\pi \\Sigma | ^ { - 1 / 2 } - \\frac { 1 } { 2 } | 2 \\pi \\Sigma | ^ { - 1 / 2 } \\frac { \\partial } { \\partial \\Sigma } | | \\tilde { a } - a | | _ { \\Sigma ^ { - 1 } } ^ { 2 } \\right) \\mathrm { d } \\tilde { a } } \\\\ { \\displaystyle \\qquad = \\frac { 1 } { 2 } \\int _ { A } Q ^ { \\pi } ( s , \\tilde { a } ) N ( \\tilde { a } | a , \\Sigma ) \\left( - \\Sigma ^ { - 1 } + \\Sigma ^ { - 1 } ( \\tilde { a } - a ) ( \\tilde { a } - a ) ^ { T } \\Sigma ^ { - 1 } \\right) \\mathrm { d } \\tilde { a } . \\quad ( 2 \\pi \\Sigma ) \\mathrm { d } \\tilde { a } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 191, + 290, + 807, + 396 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Meanwhile, towards tackling the RHS of (15) we note that ", + "bbox": [ + 173, + 404, + 560, + 420 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/97ad7296a6d1cc5351d9efa7676e3b98d2abdfd6dc40f2d5825b5cc8fba010c2.jpg", + "text": "$$\n\\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a } = \\int _ { \\cal A } Q ^ { \\pi } ( s , \\tilde { a } ) N ( \\tilde { a } | a , \\Sigma ) \\Sigma ^ { - 1 } ( \\tilde { a } - a ) \\mathrm { d } \\tilde { a } .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 422, + 673, + 460 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Thus we have ", + "bbox": [ + 174, + 463, + 267, + 478 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/f2b6236837facdae824c64e99a49ae86f2e0eb661f82effd25f97f310e1950d1.jpg", + "text": "$$\n\\frac { \\partial ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } } = \\int _ { \\mathcal { A } } Q ^ { \\pi } ( s , \\tilde { a } ) \\left( \\Sigma ^ { - 1 } ( \\tilde { a } - a ) \\frac { \\partial } { \\partial a } N ( \\tilde { a } | a , \\Sigma ) + N ( \\tilde { a } | a , \\Sigma ) \\frac { \\partial } { \\partial a } \\Sigma ^ { - 1 } ( \\tilde { a } - a ) \\right) \\mathrm { d } \\tilde { a } .\n$$", + "text_format": "latex", + "bbox": [ + 191, + 482, + 779, + 517 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/930d5d06c21a4bd3186c1ee7d89ae0dba775aea5ad5fb4f3ec4efe3838e6785c.jpg", + "text": "$$\n= \\int _ { \\mathcal { A } } Q ^ { \\pi } ( s , \\tilde { a } ) N ( \\tilde { a } | a , \\Sigma ) ( \\Sigma ^ { - 1 } ( \\tilde { a } - a ) ( \\tilde { a } - a ) ^ { T } \\Sigma ^ { - 1 } - \\Sigma ^ { - 1 } ) \\mathrm { d } \\tilde { a } .\n$$", + "text_format": "latex", + "bbox": [ + 287, + 522, + 709, + 556 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "B COMPATIBLE FUNCTION APPROXIMATION ", + "text_level": 1, + "bbox": [ + 173, + 590, + 560, + 608 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A function approximator $\\tilde { Q } _ { w } ^ { \\pi }$ of ${ \\tilde { Q } } ^ { \\pi }$ should be sufficiently accurate so that updates for $\\mu _ { \\theta } , \\Sigma _ { \\phi }$ are not affected by substituting $\\frac { \\partial \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) } { \\partial a }$ and $\\frac { \\partial ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } }$ for $\\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a }$ and $\\frac { \\partial ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } }$ , respectively. ", + "bbox": [ + 174, + 619, + 825, + 659 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We claim that a $\\tilde { Q } _ { w } ^ { \\pi }$ is compatible with respect to $\\mu _ { \\theta }$ if ", + "bbox": [ + 174, + 664, + 534, + 681 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "1 $\\left. \\nabla _ { a } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\right| _ { a = \\mu _ { \\theta } ( s ) } = \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) ^ { T } w ,$ \n2. $\\begin{array} { r } { \\nabla _ { w } \\int _ { \\mathcal { S } } \\Big ( \\nabla _ { a } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } - \\nabla _ { a } \\tilde { Q } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\Big ) ^ { 2 } \\mathrm { d } \\rho ^ { \\pi } ( s ) = 0 } \\end{array}$ (i.e., $w$ minimizes the expected squared error of the gradients). ", + "bbox": [ + 210, + 690, + 826, + 758 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Additionally, $\\tilde { Q } _ { w } ^ { \\pi }$ is compatible with respect to $\\Sigma _ { \\phi }$ if ", + "bbox": [ + 174, + 767, + 522, + 785 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "1. $\\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } = \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) ^ { T } w ,$ \n2. $\\begin{array} { r } { \\nabla _ { w } \\int _ { \\mathcal { S } } \\left( \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big \\vert _ { a = \\mu _ { \\theta } ( s ) } - \\nabla _ { a } ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) \\big \\vert _ { a = \\mu _ { \\theta } ( s ) } \\right) ^ { 2 } \\mathrm { d } \\rho ^ { \\pi } ( s ) = 0 } \\end{array}$ (i.e., $w$ minimizes the expected squared error of the Hessians). ", + "bbox": [ + 210, + 795, + 825, + 861 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "One possible parameterization of $\\tilde { Q } _ { w } ^ { \\pi }$ may be achieved by taking $w = [ w _ { 0 } , w _ { 1 } , w _ { 2 } ]$ and parameterizing ", + "bbox": [ + 174, + 871, + 823, + 902 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/b8e0b8ed230238272a76a4d3fc82242545e3918890d0fbb8ad9f69a4b959e7a1.jpg", + "text": "$$\n\\begin{array} { r } { \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) = V _ { w _ { 0 } } ( s ) + ( a - \\mu _ { \\theta } ( s ) ) ^ { T } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) ^ { T } w _ { 1 } + ( a - \\mu _ { \\theta } ( s ) ) ^ { T } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) ^ { T } w _ { 2 } ( a - \\mu _ { \\theta } ( s ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 179, + 906, + 789, + 926 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/c90628bc28b19c55096f492408d94d5ecf169eef242d75e9f07e24db6cf27564.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
HyperparameterRangeSampling
actor learningrate critic learning rate[1e-6,1e-3] [1e-6,1e-3]log
reward scale[0.01,0.3]l0g
OU damping[1e-4,1e-3]l0g
OU stddev10g
入[1e-3,1.0]log
discount factor[1e-6, 4e-2] 0.995log
target network lag0.01fixed
batch size128fixed
clipping on gradients of Qfixed
4.0fixed
num gradient updates per observation1fixed
Huber loss clipping1.0fixed
", + "bbox": [ + 271, + 101, + 722, + 285 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Proof. We shall show how the conditions stated for compatibility with respect to $\\Sigma _ { \\phi }$ are sufficient. The reasoning for $\\mu _ { \\theta }$ follows via a similar argument. We also refer the reader to Silver et al. (2014) which includes a similar procedure for showing compatibility. ", + "bbox": [ + 173, + 348, + 825, + 391 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "From the second condition for compatibility with respect to $\\Sigma _ { \\phi }$ we have ", + "bbox": [ + 174, + 397, + 648, + 412 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/482a65447c1c827eb4abc41f71898558ca37abec243b140aaac7a7ec6b1504e3.jpg", + "text": "$$\n\\int _ { S } \\left( \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } - \\nabla _ { a } ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\right) \\nabla _ { w } \\left( \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\right) \\mathrm { d } \\rho ^ { \\pi } ( s ) = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 183, + 420, + 784, + 452 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We may combine this with the first condition to find ", + "bbox": [ + 174, + 458, + 517, + 473 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/fead4ffc0f1712d604674150fb03c9293a85fb979a783002ea72c59f0f461aa1.jpg", + "text": "$$\n\\int _ { S } \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) = \\int _ { S } \\nabla _ { a } ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) ,\n$$", + "text_format": "latex", + "bbox": [ + 202, + 479, + 764, + 513 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "which is the desired property for compatibility. ", + "bbox": [ + 176, + 518, + 482, + 534 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "While it is reassuring to know that there exists a class of function approximators which are compatible, this fact is largely ignored in practice. Not only is the class of compatible functions heavily restricted in terms of expressiveness, due to the first set of conditions for $\\mu _ { \\theta } , \\Sigma _ { \\phi }$ , it is also impossible to satisfy the second set of conditions without access to derivative and Hessian information of the true ${ \\tilde { Q } } ^ { \\pi }$ . This problem is also present in DDPG, and we feel this issue merits additional investigation in future work. ", + "bbox": [ + 173, + 540, + 825, + 626 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 646, + 439, + 662 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We utilize feed forward networks for both policy and Q-value approximator. For $\\mu _ { \\boldsymbol { \\theta } } ( s )$ we use two hidden layers of dimensions (400, 300) and relu activation functions. For $\\tilde { Q } _ { w } ^ { \\pi } ( s , a )$ and $Q _ { w } ^ { \\pi } ( s , a )$ we first embed the state into a 400 dimensional vector using a fully-connected layer and tanh nonlinearity. We then concatenate the embedded state with $a$ and pass the result through a 1-hidden layer neural network of dimension 300 with tanh activations. We use a diagonal $\\Sigma _ { \\phi } ^ { \\bar { } } ( s ) = e ^ { \\phi }$ for Smoothie, with $\\phi$ initialized to $- 1$ . ", + "bbox": [ + 174, + 676, + 825, + 763 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "To find optimal hyperparameters we perform a 100-trial random search over the hyperparameters specified in Table 1. The OU exploration parameters only apply to DDPG. The $\\lambda$ coefficient on KL-penalty only applies to Smoothie with a KL-penalty. 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We find that the ability to learn both a mean", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 469, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 469, + 356 + ], + "score": 1.0, + "content": "and covariance during training allows this approach to achieve much better results", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 321, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 321, + 366 + ], + "score": 1.0, + "content": "on standard continuous control benchmarks.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5, + "bbox_fs": [ + 140, + 212, + 470, + 366 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 386, + 206, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 208, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 208, + 402 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "Model-free reinforcement learning algorithms often alternate between two concurrent but interact-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "ing processes: (1) policy evaluation, where an action value function (i.e., a Q-value) is updated to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "obtain a better estimate of the return associated with taking a specific action, and (2) policy im-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "provement, where the policy is updated aiming to maximize the current value function. In the past,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "different notions of Q-value have led to distinct but important families of RL methods. For example,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "SARSA (Rummery & Niranjan, 1994; Sutton & Barto, 1998; Van Seijen et al., 2009) uses the ex-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 505, + 491 + ], + "score": 1.0, + "content": "pected Q-value, defined as the expected return of following the current policy. Q-learning (Watkins,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "1989) exploits a hard-max notion of Q-value, defined as the expected return of following an optimal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "score": 1.0, + "content": "policy. Soft Q-learning (Haarnoja et al., 2017) and PCL (Nachum et al., 2017a) both use a soft-max", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 523 + ], + "score": 1.0, + "content": "form of Q-value, defined as the future return of following an optimal entropy regularized policy.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "Clearly, the choice of Q-value function has a considerable effect on the resulting algorithm; for ex-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "ample, restricting the types of policies that can be expressed, and determining the type of exploration", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 542, + 224, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 224, + 556 + ], + "score": 1.0, + "content": "that can be naturally applied.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 410, + 506, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 486, + 573 + ], + "score": 1.0, + "content": "In this work we introduce a new notion of action value: the smoothed action value function", + "type": "text" + }, + { + "bbox": [ + 487, + 559, + 501, + 573 + ], + "score": 0.89, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 560, + 505, + 573 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "Unlike previous notions, which associate a value with a specific action at each state, the smoothed Q-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "value associates a value with a specific distribution over actions. In particular, the smoothed Q-value", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 192, + 606 + ], + "score": 1.0, + "content": "of a state-action pair", + "type": "text" + }, + { + "bbox": [ + 193, + 594, + 215, + 605 + ], + "score": 0.9, + "content": "( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "is defined as the expected return of first taking an action sampled from", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 195, + 618 + ], + "score": 1.0, + "content": "a normal distribution", + "type": "text" + }, + { + "bbox": [ + 195, + 605, + 242, + 617 + ], + "score": 0.92, + "content": "N ( \\dot { a } , \\Sigma ( s ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 604, + 294, + 618 + ], + "score": 1.0, + "content": ", centered at", + "type": "text" + }, + { + "bbox": [ + 294, + 607, + 301, + 615 + ], + "score": 0.71, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 604, + 505, + 618 + ], + "score": 1.0, + "content": ", then following actions sampled from the current", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "policy thereafter. In this way, the smoothed Q-value can also be interpreted as a Gaussian-smoothed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 273, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 273, + 640 + ], + "score": 1.0, + "content": "or noisy version of the expected Q-value.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 559, + 505, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "We show that smoothed Q-values possess a number of interesting properties that make them at-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "tractive for use in RL algorithms. For one, the smoothed Q-values satisfy a single-step Bellman", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "consistency, which allows bootstrapping to be used to train a function approximator. Secondly, for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "Gaussian policies, the standard optimization objective (expected return) can be expressed in terms of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "smoothed Q-values. Moreover, the gradient of this objective with respect to the mean and covariance", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "of the Gaussian policy is equivalent to the gradient and the Hessian of the smoothed Q-value func-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "tion, which allows one to derive updates to the policy parameters by having access to the derivatives", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 721, + 320, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 320, + 733 + ], + "score": 1.0, + "content": "of a sufficiently accurate smoothed Q-value function.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 643, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "This observation leads us to propose an algorithm called Smoothie, which in the spirit of (Deep)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "Deterministic Policy Gradient (DDPG) (Silver et al., 2014; Lillicrap et al., 2016), trains a policy", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "using the derivatives of a trained (smoothed) Q-value function, thus avoiding the high-variance of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "stochastic updates used in standard policy gradient algorithms (Williams & Peng, 1991; Konda &", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "Tsitsiklis, 2000). Unlike DDPG, which is well-known to have poor exploratory behavior (Haarnoja", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "et al., 2017), the approach we develop is able to utilize a non-deterministic Gaussian policy param-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 163 + ], + "score": 1.0, + "content": "eterized by both a mean and a covariance, thus allowing the policy to be exploratory by default and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 336, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 336, + 172 + ], + "score": 1.0, + "content": "alleviating the need for excessive hyperparameter tuning.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 504, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 504, + 189 + ], + "score": 1.0, + "content": "Furthermore, we show that Smoothie can be easily adapted to incorporate proximal policy opti-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 504, + 199 + ], + "score": 1.0, + "content": "mization techniques by augmenting the objective with a penalty on KL-divergence from a previous", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 504, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 504, + 211 + ], + "score": 1.0, + "content": "version of the policy. The inclusion of a KL-penalty is not feasible in the standard DDPG algorithm,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 504, + 221 + ], + "score": 1.0, + "content": "but we show that it is possible with our formulation, and it significantly improves stability and over-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "all performance. On standard continuous control benchmarks, our results are competitive with or", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 430, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 430, + 244 + ], + "score": 1.0, + "content": "exceed state-of-the-art, especially for more difficult tasks in the low-data regime.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 108, + 257, + 269, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 271, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 271, + 273 + ], + "score": 1.0, + "content": "2 NOTATION & BACKGROUND", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "We consider the standard model-free RL framework, where an agent interacts with a stochastic", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "score": 1.0, + "content": "black-box environment by sequentially observing the state of the environment, emitting an action,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "and receiving a reward feedback; the goal is to find an agent that achieves maximal cumulative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "discounted reward. This problem can be expressed in terms of a Markov decision process (MDP)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 227, + 338 + ], + "score": 1.0, + "content": "that consists of a state space", + "type": "text" + }, + { + "bbox": [ + 227, + 326, + 236, + 336 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 326, + 322, + 338 + ], + "score": 1.0, + "content": "and an action space", + "type": "text" + }, + { + "bbox": [ + 322, + 326, + 331, + 336 + ], + "score": 0.75, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 326, + 411, + 338 + ], + "score": 1.0, + "content": ", where at iteration", + "type": "text" + }, + { + "bbox": [ + 411, + 327, + 417, + 336 + ], + "score": 0.66, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "the agent encounters", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 337, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 104, + 337, + 136, + 349 + ], + "score": 1.0, + "content": "a state", + "type": "text" + }, + { + "bbox": [ + 136, + 337, + 168, + 348 + ], + "score": 0.9, + "content": "s _ { t } ~ \\in ~ S", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 337, + 253, + 349 + ], + "score": 1.0, + "content": "and emits an action", + "type": "text" + }, + { + "bbox": [ + 253, + 337, + 286, + 348 + ], + "score": 0.91, + "content": "a _ { t } \\in \\mathcal A", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 337, + 506, + 349 + ], + "score": 1.0, + "content": ", after which the environment returns a scalar reward", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 347, + 381, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 165, + 360 + ], + "score": 0.92, + "content": "\\boldsymbol { r } _ { t } \\sim R ( s _ { t } , \\boldsymbol { a } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 347, + 307, + 361 + ], + "score": 1.0, + "content": "and places the agent in a new state", + "type": "text" + }, + { + "bbox": [ + 307, + 348, + 376, + 360 + ], + "score": 0.93, + "content": "s _ { t + 1 } \\sim P ( s _ { t } , a _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 347, + 381, + 361 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 361, + 377 + ], + "score": 1.0, + "content": "We model the behavior of the agent using a stochastic policy", + "type": "text" + }, + { + "bbox": [ + 361, + 367, + 369, + 375 + ], + "score": 0.74, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "that produces a distribution over", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 222, + 388 + ], + "score": 1.0, + "content": "feasible actions at each state", + "type": "text" + }, + { + "bbox": [ + 222, + 378, + 228, + 386 + ], + "score": 0.72, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 375, + 240, + 388 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 240, + 376, + 272, + 388 + ], + "score": 0.93, + "content": "\\pi ( a \\mid s )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 375, + 505, + 388 + ], + "score": 1.0, + "content": ". The optimization objective (expected discounted return),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "as a function of the policy, can then be expressed in terms of the expected action value function", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 395, + 160, + 412 + ], + "spans": [ + { + "bbox": [ + 107, + 397, + 143, + 410 + ], + "score": 0.93, + "content": "Q ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 395, + 160, + 412 + ], + "score": 1.0, + "content": "by,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 407, + 402, + 434 + ], + "lines": [ + { + "bbox": [ + 208, + 407, + 402, + 434 + ], + "spans": [ + { + "bbox": [ + 208, + 407, + 402, + 434 + ], + "score": 0.94, + "content": "O _ { \\tt E R } ( \\pi ) = \\int _ { \\cal S } \\int _ { \\cal A } \\pi ( a \\mid s ) Q ^ { \\pi } ( s , a ) \\mathrm { d } a \\mathrm { d } \\rho ^ { \\pi } ( s ) ,", + "type": "interline_equation", + "image_path": "c8cd7a89e53b1d3793cf1570c985f6d22970baa4281b62cb5b9aea8f11e54055.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 208, + 407, + 402, + 420.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 208, + 420.5, + 402, + 434.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 434, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 133, + 447 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 434, + 158, + 446 + ], + "score": 0.92, + "content": "\\rho ^ { \\pi } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 433, + 350, + 447 + ], + "score": 1.0, + "content": "is the stationary distribution of the states under", + "type": "text" + }, + { + "bbox": [ + 351, + 436, + 358, + 444 + ], + "score": 0.78, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 433, + 379, + 447 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 379, + 434, + 415, + 446 + ], + "score": 0.93, + "content": "Q ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "is recursively defined", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 445, + 221, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 221, + 458 + ], + "score": 1.0, + "content": "using the Bellman equation,", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 457, + 414, + 485 + ], + "lines": [ + { + "bbox": [ + 196, + 457, + 414, + 485 + ], + "spans": [ + { + "bbox": [ + 196, + 457, + 414, + 485 + ], + "score": 0.92, + "content": "Q ^ { \\pi } ( s , a ) = \\mathbb { E } _ { r , s ^ { \\prime } } \\left[ r + \\gamma \\int _ { \\mathcal { A } } Q ^ { \\pi } ( s ^ { \\prime } , a ^ { \\prime } ) \\pi ( a ^ { \\prime } \\mid s ^ { \\prime } ) \\mathrm { d } a \\right] \\mathrm { , }", + "type": "interline_equation", + "image_path": "4935471ef55f310e3e24877e897a38f4f10940cefaaf19c3e367eff3cb69679d.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 457, + 414, + 471.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 196, + 471.0, + 414, + 485.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 487, + 503, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 133, + 499 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 486, + 172, + 498 + ], + "score": 0.92, + "content": "\\gamma \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "is the discount factor. 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On standard continuous control benchmarks, our results are competitive with or", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 430, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 430, + 244 + ], + "score": 1.0, + "content": "exceed state-of-the-art, especially for more difficult tasks in the low-data regime.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 176, + 505, + 244 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 257, + 269, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 271, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 271, + 273 + ], + "score": 1.0, + "content": "2 NOTATION & BACKGROUND", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "We consider the standard model-free RL framework, where an agent interacts with a stochastic", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "score": 1.0, + "content": "black-box environment by sequentially observing the state of the environment, emitting an action,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "and receiving a reward feedback; the goal is to find an agent that achieves maximal cumulative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "discounted reward. 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Below we develop new RL training methods for this family of parametric", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "policies, but some of the ideas presented may generalize to other families of policies as well. We", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 440, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 440, + 118 + ], + "score": 1.0, + "content": "begin the formulation by reviewing some prior work on learning Gaussian policies.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 108, + 129, + 284, + 140 + ], + "lines": [ + { + "bbox": [ + 106, + 129, + 286, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 286, + 142 + ], + "score": 1.0, + "content": "2.1 DETERMINISTIC POLICY GRADIENT", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 149, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 163 + ], + "score": 1.0, + "content": "Silver et al. (2014) present a new formulation of the policy gradient, called the deterministic policy", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "score": 1.0, + "content": "gradient, for the family of Gaussian policies in the limit where the policy covariance approaches", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "score": 1.0, + "content": "zero. In such a scenario, the policy becomes deterministic because sampling from the policy always", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "returns the Gaussian mean. The key observation of (Silver et al., 2014) is that under a deterministic", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 449, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 134, + 207 + ], + "score": 1.0, + "content": "policy", + "type": "text" + }, + { + "bbox": [ + 134, + 194, + 199, + 206 + ], + "score": 0.92, + "content": "\\pi \\equiv ( \\mu , \\Sigma \\to 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 194, + 427, + 207 + ], + "score": 1.0, + "content": ", one can estimate the expected future return from a state", + "type": "text" + }, + { + "bbox": [ + 428, + 196, + 433, + 204 + ], + "score": 0.63, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 194, + 449, + 207 + ], + "score": 1.0, + "content": "as,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 209, + 399, + 236 + ], + "lines": [ + { + "bbox": [ + 212, + 209, + 399, 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"score": 1.0, + "content": "parameterized", + "type": "text" + }, + { + "bbox": [ + 165, + 252, + 200, + 262 + ], + "score": 0.9, + "content": "\\pi _ { \\boldsymbol { \\theta } } \\equiv \\mu _ { \\boldsymbol { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 250, + 216, + 263 + ], + "score": 1.0, + "content": "as,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 266, + 480, + 293 + ], + "lines": [ + { + "bbox": [ + 119, + 266, + 480, + 293 + ], + "spans": [ + { + "bbox": [ + 119, + 266, + 480, + 293 + ], + "score": 0.93, + "content": "\\nabla _ { \\theta } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta } ) = \\int _ { S } \\nabla _ { \\theta } Q ^ { \\pi } ( s , \\mu _ { \\theta } ( s ) ) \\mathrm { d } \\rho ^ { \\pi } ( s ) = \\int _ { S } \\frac { \\partial Q ^ { \\pi } ( s , a ) } { \\partial a } | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .", + "type": "interline_equation", + "image_path": "edb5b4ed43a4f1a0e669d90a2efa86e348e4bfe334dbf8cd55ff8c9d7fefcff8.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 119, + 266, + 480, + 293 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 296, + 504, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "This can be thought of as a characterization of the policy gradient theorem for deterministic policies.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 108, + 313, + 408, + 325 + ], + "lines": [ + { + "bbox": [ + 104, + 310, + 408, + 328 + ], + "spans": [ + { + "bbox": [ + 104, + 310, + 164, + 328 + ], + "score": 1.0, + "content": "In the limit of", + "type": "text" + }, + { + "bbox": [ + 164, + 314, + 193, + 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(2012) and Silver et al. (2014) replace", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 444, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 232, + 460 + ], + "score": 1.0, + "content": "the on-policy state distribution", + "type": "text" + }, + { + "bbox": [ + 232, + 446, + 256, + 458 + ], + "score": 0.92, + "content": "\\rho ^ { \\pi } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 444, + 406, + 460 + ], + "score": 1.0, + "content": "in (7) with an off-policy distribution", + "type": "text" + }, + { + "bbox": [ + 407, + 445, + 430, + 458 + ], + "score": 0.92, + "content": "\\rho ^ { \\beta } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 444, + 505, + 460 + ], + "score": 1.0, + "content": "based on a replay", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 455, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 506, + 471 + ], + "score": 1.0, + "content": "buffer. After this substitution, the policy gradient identity in (7) does not hold exactly, however,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 467, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 507, + 482 + ], + "score": 1.0, + "content": "prior work finds that this works well in practice and improves sample efficiency. We also adopt a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 479, + 379, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 379, + 491 + ], + "score": 1.0, + "content": "similar approximation in our method to make use of off-policy data.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 506, + 331, + 519 + ], + "lines": [ + { + "bbox": [ + 104, + 504, + 332, + 521 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 332, + 521 + ], + "score": 1.0, + "content": "3 SMOOTHED ACTION VALUE FUNCTIONS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 530, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "In this paper, we introduce smoothed action value functions, the gradients of which provide an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "effective signal for optimizing the parameters of a Gaussian policy. Our notion of smoothed Q-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 173, + 567 + ], + "score": 1.0, + "content": "values, denoted", + "type": "text" + }, + { + "bbox": [ + 173, + 553, + 209, + 567 + ], + "score": 0.93, + "content": "\\tilde { Q } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 554, + 342, + 567 + ], + "score": 1.0, + "content": ", differs from ordinary Q-values", + "type": "text" + }, + { + "bbox": [ + 343, + 554, + 379, + 566 + ], + "score": 0.93, + "content": "Q ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "in that smoothed Q-values do", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "not assume the first action of the agent is fully specified, but rather they assume that only the mean", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 374, + 590 + ], + "score": 1.0, + "content": "of the distribution of the first action is known. Hence, to compute", + "type": "text" + }, + { + "bbox": [ + 374, + 576, + 410, + 590 + ], + "score": 0.94, + "content": "\\tilde { Q } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 576, + 505, + 590 + ], + "score": 1.0, + "content": ", one has to perform an", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 167, + 601 + ], + "score": 1.0, + "content": "expectation of", + "type": "text" + }, + { + "bbox": [ + 167, + 588, + 203, + 601 + ], + "score": 0.93, + "content": "Q ^ { \\pi } ( s , { \\tilde { a } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 589, + 252, + 601 + ], + "score": 1.0, + "content": "for actions", + "type": "text" + }, + { + "bbox": [ + 252, + 590, + 258, + 599 + ], + "score": 0.81, + "content": "\\tilde { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 589, + 359, + 601 + ], + "score": 1.0, + "content": "drawn in the vicinity of", + "type": "text" + }, + { + "bbox": [ + 359, + 591, + 366, + 599 + ], + "score": 0.77, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 589, + 505, + 601 + ], + "score": 1.0, + "content": ". More formally, smoothed action", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 599, + 195, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 195, + 612 + ], + "score": 1.0, + "content": "values are defined as,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 608, + 397, + 635 + ], + "lines": [ + { + "bbox": [ + 213, + 608, + 397, + 635 + ], + "spans": [ + { + "bbox": [ + 213, + 608, + 397, + 635 + ], + "score": 0.95, + "content": "\\tilde { Q } ^ { \\pi } ( s , a ) = \\int _ { A } N ( \\tilde { a } | a , \\Sigma ( s ) ) Q ^ { \\pi } ( s , \\tilde { a } ) \\mathrm { d } \\tilde { a } .", + "type": "interline_equation", + "image_path": "8713600fb539a4a0fd028c7442c7fd55bcda9b1e7c291fef1adceb76fdfeb0f3.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 213, + 608, + 397, + 635 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 503, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 198, + 651 + ], + "score": 1.0, + "content": "With this definition of", + "type": "text" + }, + { + "bbox": [ + 199, + 636, + 212, + 649 + ], + "score": 0.91, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 636, + 505, + 651 + ], + "score": 1.0, + "content": ", one can re-express the expected reward objective for a Gaussian policy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 648, + 168, + 662 + ], + "spans": [ + { + "bbox": [ + 107, + 649, + 152, + 661 + ], + "score": 0.93, + "content": "\\pi \\equiv ( \\mu , \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 648, + 168, + 662 + ], + "score": 1.0, + "content": "as,", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 659, + 375, + 685 + ], + "lines": [ + { + "bbox": [ + 236, + 659, + 375, + 685 + ], + "spans": [ + { + "bbox": [ + 236, + 659, + 375, + 685 + ], + "score": 0.95, + "content": "O _ { \\mathrm { E R } } ( \\pi ) = \\int _ { S } \\tilde { Q } ^ { \\pi } ( s , \\mu ( s ) ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .", + "type": "interline_equation", + "image_path": "746ea57183920be259738989cf19f6bf70a85484858cb13172e34626faa37835.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 236, + 659, + 375, + 685 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 686, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 699 + ], + "score": 1.0, + "content": "The insight that differentiates this approach from prior work including Heess et al. (2015); Ciosek", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 697, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 405, + 710 + ], + "score": 1.0, + "content": "& Whiteson (2017) is that instead of learning a function approximator for", + "type": "text" + }, + { + "bbox": [ + 405, + 697, + 441, + 709 + ], + "score": 0.93, + "content": "Q ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 697, + 505, + 710 + ], + "score": 1.0, + "content": "and then draw-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "ing samples to approximate the expectation in (10) and its derivative, we directly learn a function", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 719, + 217, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 177, + 733 + ], + "score": 1.0, + "content": "approximator for", + "type": "text" + }, + { + "bbox": [ + 177, + 719, + 213, + 732 + ], + "score": 0.93, + "content": "\\bar { Q } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 721, + 217, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 134, + 96 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 81, + 196, + 95 + ], + "score": 0.93, + "content": "\\| v \\| _ { A } ^ { 2 } = v ^ { \\mathsf { T } } A v", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 81, + 506, + 96 + ], + "score": 1.0, + "content": ". Below we develop new RL training methods for this family of parametric", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "policies, but some of the ideas presented may generalize to other families of policies as well. We", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 440, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 440, + 118 + ], + "score": 1.0, + "content": "begin the formulation by reviewing some prior work on learning Gaussian policies.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 506, + 118 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 129, + 284, + 140 + ], + "lines": [ + { + "bbox": [ + 106, + 129, + 286, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 286, + 142 + ], + "score": 1.0, + "content": "2.1 DETERMINISTIC POLICY GRADIENT", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 149, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 163 + ], + "score": 1.0, + "content": "Silver et al. (2014) present a new formulation of the policy gradient, called the deterministic policy", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "score": 1.0, + "content": "gradient, for the family of Gaussian policies in the limit where the policy covariance approaches", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "score": 1.0, + "content": "zero. In such a scenario, the policy becomes deterministic because sampling from the policy always", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "returns the Gaussian mean. 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{ \\partial a } | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .", + "type": "interline_equation", + "image_path": "edb5b4ed43a4f1a0e669d90a2efa86e348e4bfe334dbf8cd55ff8c9d7fefcff8.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 119, + 266, + 480, + 293 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 296, + 504, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "This can be thought of as a characterization of the policy gradient theorem for deterministic policies.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 295, + 505, + 309 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 313, + 408, + 325 + ], + "lines": [ + { + "bbox": [ + 104, + 310, + 408, + 328 + ], + "spans": [ + { + "bbox": [ + 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0.92, + "content": "E ( w ) = \\sum _ { ( s , a , r , s ^ { \\prime } ) \\in \\mathcal { D } } ( Q _ { w } ^ { \\pi } ( s , a ) - r - \\gamma Q _ { w } ^ { \\pi } ( s ^ { \\prime } , \\mu ( s ^ { \\prime } ) ) ^ { 2 } ,", + "type": "interline_equation", + "image_path": "33f14e5df15606e8aea149929ac6bcadca43b1040e98716946a23e09f91a9fe2.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 362, + 416, + 377.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 194, + 377.0, + 416, + 392.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 164, + 408 + ], + "score": 1.0, + "content": "for transitions", + "type": "text" + }, + { + "bbox": [ + 164, + 396, + 207, + 408 + ], + "score": 0.92, + "content": "( s , a , r , s ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, 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+ ] + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "In practice, to gain better sample efficiency, Degris et al. (2012) and Silver et al. (2014) replace", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 444, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 232, + 460 + ], + "score": 1.0, + "content": "the on-policy state distribution", + "type": "text" + }, + { + "bbox": [ + 232, + 446, + 256, + 458 + ], + "score": 0.92, + "content": "\\rho ^ { \\pi } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 444, + 406, + 460 + ], + "score": 1.0, + "content": "in (7) with an off-policy distribution", + "type": "text" + }, + { + "bbox": [ + 407, + 445, + 430, + 458 + ], + "score": 0.92, + "content": "\\rho ^ { \\beta } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 444, + 505, + 460 + ], + "score": 1.0, + "content": "based on a replay", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 455, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 506, + 471 + ], + "score": 1.0, + "content": "buffer. After this substitution, the policy gradient identity in (7) does not hold exactly, however,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 467, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 507, + 482 + ], + "score": 1.0, + "content": "prior work finds that this works well in practice and improves sample efficiency. We also adopt a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 479, + 379, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 379, + 491 + ], + "score": 1.0, + "content": "similar approximation in our method to make use of off-policy data.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 435, + 507, + 491 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 506, + 331, + 519 + ], + "lines": [ + { + "bbox": [ + 104, + 504, + 332, + 521 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 332, + 521 + ], + "score": 1.0, + "content": "3 SMOOTHED ACTION VALUE FUNCTIONS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 530, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "In this paper, we introduce smoothed action value functions, the gradients of which provide an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "effective signal for optimizing the parameters of a Gaussian policy. Our notion of smoothed Q-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 173, + 567 + ], + "score": 1.0, + "content": "values, denoted", + "type": "text" + }, + { + "bbox": [ + 173, + 553, + 209, + 567 + ], + "score": 0.93, + "content": "\\tilde { Q } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 554, + 342, + 567 + ], + "score": 1.0, + "content": ", differs from ordinary Q-values", + "type": "text" + }, + { + "bbox": [ + 343, + 554, + 379, + 566 + ], + "score": 0.93, + "content": "Q ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "in that smoothed Q-values do", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "not assume the first action of the agent is fully specified, but rather they assume that only the mean", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 374, + 590 + ], + "score": 1.0, + "content": "of the distribution of the first action is known. Hence, to compute", + "type": "text" + }, + { + "bbox": [ + 374, + 576, + 410, + 590 + ], + "score": 0.94, + "content": "\\tilde { Q } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 576, + 505, + 590 + ], + "score": 1.0, + "content": ", one has to perform an", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 167, + 601 + ], + "score": 1.0, + "content": "expectation of", + "type": "text" + }, + { + "bbox": [ + 167, + 588, + 203, + 601 + ], + "score": 0.93, + "content": "Q ^ { \\pi } ( s , { \\tilde { a } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 589, + 252, + 601 + ], + "score": 1.0, + "content": "for actions", + "type": "text" + }, + { + "bbox": [ + 252, + 590, + 258, + 599 + ], + "score": 0.81, + "content": "\\tilde { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 589, + 359, + 601 + ], + "score": 1.0, + "content": "drawn in the vicinity of", + "type": "text" + }, + { + "bbox": [ + 359, + 591, + 366, + 599 + ], + "score": 0.77, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 589, + 505, + 601 + ], + "score": 1.0, + "content": ". More formally, smoothed action", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 599, + 195, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 195, + 612 + ], + "score": 1.0, + "content": "values are defined as,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 531, + 505, + 612 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 608, + 397, + 635 + ], + "lines": [ + { + "bbox": [ + 213, + 608, + 397, + 635 + ], + "spans": [ + { + "bbox": [ + 213, + 608, + 397, + 635 + ], + "score": 0.95, + "content": "\\tilde { Q } ^ { \\pi } ( s , a ) = \\int _ { A } N ( \\tilde { a } | a , \\Sigma ( s ) ) Q ^ { \\pi } ( s , \\tilde { a } ) \\mathrm { d } \\tilde { a } .", + "type": "interline_equation", + "image_path": "8713600fb539a4a0fd028c7442c7fd55bcda9b1e7c291fef1adceb76fdfeb0f3.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 213, + 608, + 397, + 635 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 503, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 198, + 651 + ], + "score": 1.0, + "content": "With this definition of", + "type": "text" + }, + { + "bbox": [ + 199, + 636, + 212, + 649 + ], + "score": 0.91, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 636, + 505, + 651 + ], + "score": 1.0, + "content": ", one can re-express the expected reward objective for a Gaussian policy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 648, + 168, + 662 + ], + "spans": [ + { + "bbox": [ + 107, + 649, + 152, + 661 + ], + "score": 0.93, + "content": "\\pi \\equiv ( \\mu , \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 648, + 168, + 662 + ], + "score": 1.0, + "content": "as,", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 636, + 505, + 662 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 659, + 375, + 685 + ], + "lines": [ + { + "bbox": [ + 236, + 659, + 375, + 685 + ], + "spans": [ + { + "bbox": [ + 236, + 659, + 375, + 685 + ], + "score": 0.95, + "content": "O _ { \\mathrm { E R } } ( \\pi ) = \\int _ { S } \\tilde { Q } ^ { \\pi } ( s , \\mu ( s ) ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .", + "type": "interline_equation", + "image_path": "746ea57183920be259738989cf19f6bf70a85484858cb13172e34626faa37835.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 236, + 659, + 375, + 685 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 686, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 699 + ], + "score": 1.0, + "content": "The insight that differentiates this approach from prior work including Heess et al. (2015); Ciosek", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 697, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 405, + 710 + ], + "score": 1.0, + "content": "& Whiteson (2017) is that instead of learning a function approximator for", + "type": "text" + }, + { + "bbox": [ + 405, + 697, + 441, + 709 + ], + "score": 0.93, + "content": "Q ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 697, + 505, + 710 + ], + "score": 1.0, + "content": "and then draw-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "ing samples to approximate the expectation in (10) and its derivative, we directly learn a function", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 719, + 217, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 177, + 733 + ], + "score": 1.0, + "content": "approximator for", + "type": "text" + }, + { + "bbox": [ + 177, + 719, + 213, + 732 + ], + "score": 0.93, + "content": "\\bar { Q } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 721, + 217, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 686, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 114 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 375, + 95 + ], + "score": 1.0, + "content": "The key observation that enables direct bootstrapping of smoothed", + "type": "text" + }, + { + "bbox": [ + 376, + 83, + 385, + 94 + ], + "score": 0.28, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 82, + 416, + 95 + ], + "score": 1.0, + "content": "-values,", + "type": "text" + }, + { + "bbox": [ + 417, + 81, + 453, + 95 + ], + "score": 0.93, + "content": "\\tilde { Q } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ", is that their", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 442, + 106 + ], + "score": 1.0, + "content": "form allows a notion of Bellman consistency. First, note that for Gaussian policies", + "type": "text" + }, + { + "bbox": [ + 442, + 94, + 489, + 106 + ], + "score": 0.93, + "content": "\\pi \\equiv ( \\mu , \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 129, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 129, + 117 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 113, + 385, + 128 + ], + "lines": [ + { + "bbox": [ + 226, + 113, + 385, + 128 + ], + "spans": [ + { + "bbox": [ + 226, + 113, + 385, + 128 + ], + "score": 0.94, + "content": "Q ^ { \\pi } ( s , a ) = \\mathbb { E } _ { r , s ^ { \\prime } } [ r + \\gamma \\tilde { Q } ^ { \\pi } ( s ^ { \\prime } , \\mu ( s ^ { \\prime } ) ) ] .", + "type": "interline_equation", + "image_path": "21a3aedaa734f3c25543a623fa9c74af9b5602d1607eca5b4b1dda92b52eb8f7.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 226, + 113, + 385, + 128 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 131, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 505, + 144 + ], + "score": 1.0, + "content": "Then, combining (10) and (12), one can derive the following one-step Bellman equation for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 189, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 189, + 155 + ], + "score": 1.0, + "content": "smoothed Q-values,", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 158, + 430, + 185 + ], + "lines": [ + { + "bbox": [ + 179, + 158, + 430, + 185 + ], + "spans": [ + { + "bbox": [ + 179, + 158, + 430, + 185 + ], + "score": 0.92, + "content": "\\tilde { Q } ^ { \\pi } ( s , a ) = \\int _ { \\cal A } N ( \\tilde { a } \\mid a , \\Sigma ( s ) ) \\mathbb { E } _ { \\tilde { r } , \\tilde { s } ^ { \\prime } } \\left[ \\tilde { r } + \\gamma \\tilde { Q } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) \\right] \\mathrm { d } \\tilde { a } ,", + "type": "interline_equation", + "image_path": "fe18d88621a98303e4f2b541b01df7598dfd6b3292ac4b2c5c26bfe54af28ffb.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 179, + 158, + 430, + 185 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 190, + 506, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 190, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 133, + 203 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 191, + 140, + 200 + ], + "score": 0.81, + "content": "\\tilde { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 190, + 159, + 203 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 159, + 191, + 167, + 200 + ], + "score": 0.86, + "content": "{ \\tilde { s } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 190, + 243, + 203 + ], + "score": 1.0, + "content": "are sampled from", + "type": "text" + }, + { + "bbox": [ + 243, + 190, + 273, + 203 + ], + "score": 0.94, + "content": "R ( s , { \\tilde { a } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 190, + 293, + 203 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 190, + 324, + 203 + ], + "score": 0.94, + "content": "P ( s , { \\tilde { a } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 190, + 505, + 203 + ], + "score": 1.0, + "content": ". Below, we elaborate on how one can make", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 205, + 216 + ], + "score": 1.0, + "content": "use of the derivatives of", + "type": "text" + }, + { + "bbox": [ + 205, + 201, + 219, + 214 + ], + "score": 0.92, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 201, + 254, + 216 + ], + "score": 1.0, + "content": "to learn", + "type": "text" + }, + { + "bbox": [ + 254, + 205, + 261, + 215 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 201, + 280, + 216 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 280, + 204, + 288, + 213 + ], + "score": 0.82, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 201, + 506, + 216 + ], + "score": 1.0, + "content": ", and how the Bellman equation in (13) enables direct", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 214, + 188, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 170, + 227 + ], + "score": 1.0, + "content": "optimization of", + "type": "text" + }, + { + "bbox": [ + 171, + 214, + 184, + 227 + ], + "score": 0.91, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 214, + 188, + 227 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 106, + 240, + 377, + 254 + ], + "lines": [ + { + "bbox": [ + 104, + 239, + 377, + 257 + ], + "spans": [ + { + "bbox": [ + 104, + 239, + 294, + 257 + ], + "score": 1.0, + "content": "3.1 POLICY IMPROVEMENT - OPTIMIZING", + "type": "text" + }, + { + "bbox": [ + 294, + 241, + 330, + 254 + ], + "score": 0.85, + "content": "\\left( \\mu _ { \\theta } , \\Sigma _ { \\phi } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 239, + 362, + 257 + ], + "score": 1.0, + "content": "GIVEN", + "type": "text" + }, + { + "bbox": [ + 363, + 240, + 377, + 254 + ], + "score": 0.79, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 506, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 250, + 276 + ], + "score": 1.0, + "content": "We parameterize a Gaussian policy", + "type": "text" + }, + { + "bbox": [ + 250, + 263, + 317, + 275 + ], + "score": 0.92, + "content": "\\pi _ { \\boldsymbol { \\theta } , \\boldsymbol { \\phi } } \\equiv ( \\mu _ { \\boldsymbol { \\theta } } , \\Sigma _ { \\boldsymbol { \\phi } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 262, + 458, + 276 + ], + "score": 1.0, + "content": "in terms of two sets of parameters", + "type": "text" + }, + { + "bbox": [ + 458, + 264, + 464, + 273 + ], + "score": 0.83, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 262, + 482, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 483, + 264, + 489, + 274 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 262, + 506, + 276 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 344, + 286 + ], + "score": 1.0, + "content": "the mean and the covariance. The gradient of the objective", + "type": "text" + }, + { + "bbox": [ + 344, + 275, + 363, + 284 + ], + "score": 0.28, + "content": "w . r . t .", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 273, + 505, + 286 + ], + "score": 1.0, + "content": ". mean parameters follows from the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 285, + 323, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 323, + 297 + ], + "score": 1.0, + "content": "policy gradient theorem and is almost identical to (7),", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 301, + 423, + 331 + ], + "lines": [ + { + "bbox": [ + 188, + 301, + 423, + 331 + ], + "spans": [ + { + "bbox": [ + 188, + 301, + 423, + 331 + ], + "score": 0.95, + "content": "\\nabla _ { \\theta } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta , \\phi } ) = \\int _ { S } \\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a } \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .", + "type": "interline_equation", + "image_path": "6d9aef1b0a5509659a282090238279c9da22df639f2282e7c6aa94387ffb0566.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 188, + 301, + 423, + 316.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 188, + 316.0, + 423, + 331.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Estimating the derivative of the objective w.r.t. covariance parameters is not as straightforward, since", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 352, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 107, + 352, + 121, + 365 + ], + "score": 0.89, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 352, + 222, + 366 + ], + "score": 1.0, + "content": "is not a direct function of", + "type": "text" + }, + { + "bbox": [ + 222, + 354, + 230, + 363 + ], + "score": 0.79, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 352, + 506, + 366 + ], + "score": 1.0, + "content": ". However, a key observation of this work is that the second derivative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 364, + 429, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 117, + 379 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 364, + 152, + 378 + ], + "score": 0.89, + "content": "{ \\tilde { Q } } ^ { \\pi } w . r . t .", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 364, + 379, + 379 + ], + "score": 1.0, + "content": ". actions is sufficient to exactly compute the derivative of", + "type": "text" + }, + { + "bbox": [ + 379, + 364, + 424, + 378 + ], + "score": 0.89, + "content": "\\tilde { Q } ^ { \\pi } w . r . t . \\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 364, + 429, + 379 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 382, + 368, + 411 + ], + "lines": [ + { + "bbox": [ + 243, + 382, + 368, + 411 + ], + "spans": [ + { + "bbox": [ + 243, + 382, + 368, + 411 + ], + "score": 0.95, + "content": "\\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial \\Sigma ( s ) } = \\frac { 1 } { 2 } \\cdot \\frac { \\partial ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } } .", + "type": "interline_equation", + "image_path": "5b5740281c05060410f4bd717038d516178564d8e3793b5af4abb32a9f57cbf7.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 243, + 382, + 368, + 396.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 243, + 396.5, + 368, + 411.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 104, + 414, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 414, + 506, + 430 + ], + "score": 1.0, + "content": "A proof of this identity is provided in the Appendix. The proof may be easily derived by expressing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 426, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 426, + 360, + 442 + ], + "score": 1.0, + "content": "both sides of the equation using standard matrix calculus like", + "type": "text" + }, + { + "bbox": [ + 361, + 427, + 486, + 442 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { \\partial } { \\partial A } | A | ^ { - 1 / 2 } = - \\frac { 1 } { 2 } | A | ^ { - 1 / 2 } \\bar { A } ^ { - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 426, + 506, + 442 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 436, + 232, + 455 + ], + "spans": [ + { + "bbox": [ + 107, + 440, + 227, + 455 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { \\partial } { \\partial A } | | v | | _ { A ^ { - 1 } } ^ { 2 } = - A ^ { - 1 } v v ^ { T } A ^ { - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 436, + 232, + 455 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 295, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 296, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 226, + 471 + ], + "score": 1.0, + "content": "Then, the full derivative w.r.t.", + "type": "text" + }, + { + "bbox": [ + 226, + 459, + 233, + 470 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 457, + 296, + 471 + ], + "score": 1.0, + "content": "takes the form,", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 475, + 430, + 504 + ], + "lines": [ + { + "bbox": [ + 180, + 475, + 430, + 504 + ], + "spans": [ + { + "bbox": [ + 180, + 475, + 430, + 504 + ], + "score": 0.94, + "content": "\\nabla _ { \\phi } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta , \\phi } ) = \\frac { 1 } { 2 } \\int _ { S } \\frac { \\partial ^ { 2 } { \\tilde { Q } } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } } \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .", + "type": "interline_equation", + "image_path": "f41c8adcf7c19ddab43a7d258f46d084293a60a0cb7aa0628a9bb7df0f7b6d52.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 180, + 475, + 430, + 484.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 180, + 484.6666666666667, + 430, + 494.33333333333337 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 180, + 494.33333333333337, + 430, + 504.00000000000006 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 516, + 361, + 530 + ], + "lines": [ + { + "bbox": [ + 104, + 513, + 360, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 285, + 533 + ], + "score": 1.0, + "content": "3.2 POLICY EVALUATION - OPTIMIZING", + "type": "text" + }, + { + "bbox": [ + 286, + 516, + 301, + 530 + ], + "score": 0.88, + "content": "\\tilde { Q } _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 513, + 334, + 533 + ], + "score": 1.0, + "content": "GIVEN", + "type": "text" + }, + { + "bbox": [ + 334, + 518, + 360, + 530 + ], + "score": 0.9, + "content": "( \\mu , \\Sigma )", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 505, + 656 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 259, + 555 + ], + "score": 1.0, + "content": "We can think of two ways to optimize", + "type": "text" + }, + { + "bbox": [ + 259, + 539, + 275, + 552 + ], + "score": 0.91, + "content": "\\tilde { Q } _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 539, + 452, + 555 + ], + "score": 1.0, + "content": ". The first approach leverages (10) to update", + "type": "text" + }, + { + "bbox": [ + 452, + 539, + 466, + 552 + ], + "score": 0.91, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 539, + 506, + 555 + ], + "score": 1.0, + "content": "based on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 550, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 214, + 565 + ], + "score": 1.0, + "content": "expected Q-value function", + "type": "text" + }, + { + "bbox": [ + 214, + 552, + 228, + 563 + ], + "score": 0.89, + "content": "Q ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 550, + 425, + 565 + ], + "score": 1.0, + "content": ". In such an approach, one trains a parameterized", + "type": "text" + }, + { + "bbox": [ + 426, + 552, + 441, + 563 + ], + "score": 0.9, + "content": "Q _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 550, + 506, + 565 + ], + "score": 1.0, + "content": "to approximate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 265, + 576 + ], + "score": 1.0, + "content": "the standard expected Q-value function", + "type": "text" + }, + { + "bbox": [ + 265, + 563, + 279, + 574 + ], + "score": 0.89, + "content": "Q ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 562, + 457, + 576 + ], + "score": 1.0, + "content": "using standard methods (see e.g., Rummery", + "type": "text" + }, + { + "bbox": [ + 458, + 563, + 466, + 572 + ], + "score": 0.56, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "Niranjan", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 573, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 412, + 589 + ], + "score": 1.0, + "content": "(1994); Sutton & Barto (1998); Van Seijen et al. 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In", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 585, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 585, + 222, + 599 + ], + "score": 1.0, + "content": "particular, given transitions", + "type": "text" + }, + { + "bbox": [ + 222, + 586, + 265, + 597 + ], + "score": 0.93, + "content": "( s , a , r , s ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 585, + 506, + 599 + ], + "score": 1.0, + "content": "sampled from interactions with the environment, one can", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 595, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 127, + 611 + ], + "score": 1.0, + "content": "train", + "type": "text" + }, + { + "bbox": [ + 127, + 597, + 142, + 608 + ], + "score": 0.87, + "content": "Q _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 595, + 266, + 611 + ], + "score": 1.0, + "content": "to minimize the Bellman error", + "type": "text" + }, + { + "bbox": [ + 266, + 596, + 386, + 609 + ], + "score": 0.9, + "content": "( Q _ { w } ^ { \\pi } ( s , a ) - r - \\gamma Q _ { w } ^ { \\pi } ( s ^ { \\prime } , a ^ { \\prime } ) ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 595, + 414, + 611 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 414, + 596, + 501, + 609 + ], + "score": 0.92, + "content": "a ^ { \\prime } \\sim N ( \\mu ( s ^ { \\prime } ) , \\Sigma ( s ^ { \\prime } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 595, + 506, + 611 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 607, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 607, + 132, + 624 + ], + "score": 1.0, + "content": "Then,", + "type": "text" + }, + { + "bbox": [ + 133, + 608, + 148, + 621 + ], + "score": 0.87, + "content": "\\tilde { Q } _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 607, + 346, + 624 + ], + "score": 1.0, + "content": "can be optimized to minimize the squared error", + "type": "text" + }, + { + "bbox": [ + 347, + 609, + 456, + 621 + ], + "score": 0.91, + "content": "( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\mathbb { E } _ { \\tilde { a } } Q _ { w } ^ { \\pi } ( s , \\tilde { a } ) ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 607, + 486, + 624 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 486, + 610, + 505, + 620 + ], + "score": 0.85, + "content": "\\tilde { a } \\sim", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 153, + 632 + ], + "score": 0.92, + "content": "N ( a , \\Sigma ( s ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 620, + 506, + 633 + ], + "score": 1.0, + "content": ", using several samples. When the target values in these residuals are treated as fixed", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 631, + 504, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 466, + 647 + ], + "score": 1.0, + "content": "(i.e., using a target network), such a training procedure will achieve a fixed point when", + "type": "text" + }, + { + "bbox": [ + 466, + 631, + 504, + 645 + ], + "score": 0.93, + "content": "\\tilde { Q } _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 644, + 313, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 313, + 655 + ], + "score": 1.0, + "content": "satisfies the recursion in the Bellman equation (10).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 661, + 505, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 507, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 374, + 677 + ], + "score": 1.0, + "content": "The second approach requires a single function approximator for", + "type": "text" + }, + { + "bbox": [ + 374, + 660, + 411, + 674 + ], + "score": 0.93, + "content": "\\tilde { Q } _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 661, + 507, + 677 + ], + "score": 1.0, + "content": ", resulting in a simpler", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 674, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 505, + 686 + ], + "score": 1.0, + "content": "implementation, and thus we use this approach in our experimental evaluation. Suppose one has ac-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 165, + 697 + ], + "score": 1.0, + "content": "cess to a tuple", + "type": "text" + }, + { + "bbox": [ + 165, + 684, + 209, + 696 + ], + "score": 0.93, + "content": "( s , \\tilde { a } , \\tilde { r } , \\tilde { s } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 684, + 505, + 697 + ], + "score": 1.0, + "content": "sampled from a replay buffer with knowledge of the sampling probability", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 695, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 107, + 695, + 138, + 708 + ], + "score": 0.91, + "content": "q ( \\tilde { a } \\mid s )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 695, + 505, + 708 + ], + "score": 1.0, + "content": "(possibly unnormalized). Then assuming that this sampling distribution has a full support,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 707, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 214, + 721 + ], + "score": 1.0, + "content": "we draw a phantom action", + "type": "text" + }, + { + "bbox": [ + 215, + 707, + 280, + 720 + ], + "score": 0.91, + "content": "a \\sim N ( \\tilde { a } , \\Sigma ( s ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 707, + 336, + 721 + ], + "score": 1.0, + "content": "and optimize", + "type": "text" + }, + { + "bbox": [ + 337, + 707, + 374, + 720 + ], + "score": 0.95, + "content": "\\tilde { Q } _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 707, + 505, + 721 + ], + "score": 1.0, + "content": "by minimizing a weighted Bell-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 718, + 507, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 149, + 735 + ], + "score": 1.0, + "content": "man error", + "type": "text" + }, + { + "bbox": [ + 149, + 719, + 308, + 735 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { q ( \\tilde { a } | s ) } ( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\tilde { r } - \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 718, + 466, + 735 + ], + "score": 1.0, + "content": ". 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First, note that for Gaussian policies", + "type": "text" + }, + { + "bbox": [ + 442, + 94, + 489, + 106 + ], + "score": 0.93, + "content": "\\pi \\equiv ( \\mu , \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 129, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 129, + 117 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 505, + 117 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 113, + 385, + 128 + ], + "lines": [ + { + "bbox": [ + 226, + 113, + 385, + 128 + ], + "spans": [ + { + "bbox": [ + 226, + 113, + 385, + 128 + ], + "score": 0.94, + "content": "Q ^ { \\pi } ( s , a ) = \\mathbb { E } _ { r , s ^ { \\prime } } [ r + \\gamma \\tilde { Q } ^ { \\pi } ( s ^ { \\prime } , \\mu ( s ^ { \\prime } ) ) ] .", + "type": "interline_equation", + "image_path": "21a3aedaa734f3c25543a623fa9c74af9b5602d1607eca5b4b1dda92b52eb8f7.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 226, + 113, + 385, + 128 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 131, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 505, + 144 + ], + "score": 1.0, + "content": "Then, combining (10) and (12), one can derive the following one-step Bellman equation for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 189, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 189, + 155 + ], + "score": 1.0, + "content": "smoothed Q-values,", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 131, + 505, + 155 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 158, + 430, + 185 + ], + "lines": [ + { + "bbox": [ + 179, + 158, + 430, + 185 + ], + "spans": [ + { + "bbox": [ + 179, + 158, + 430, + 185 + ], + "score": 0.92, + "content": "\\tilde { Q } ^ { \\pi } ( s , a ) = \\int _ { \\cal A } N ( \\tilde { a } \\mid a , \\Sigma ( s ) ) \\mathbb { E } _ { \\tilde { r } , \\tilde { s } ^ { \\prime } } \\left[ \\tilde { r } + \\gamma \\tilde { Q } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) \\right] \\mathrm { d } \\tilde { a } ,", + "type": "interline_equation", + "image_path": "fe18d88621a98303e4f2b541b01df7598dfd6b3292ac4b2c5c26bfe54af28ffb.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 179, + 158, + 430, + 185 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 190, + 506, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 190, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 133, + 203 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 191, + 140, + 200 + ], + "score": 0.81, + "content": "\\tilde { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 190, + 159, + 203 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 159, + 191, + 167, + 200 + ], + "score": 0.86, + "content": "{ \\tilde { s } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 190, + 243, + 203 + ], + "score": 1.0, + "content": "are sampled from", + "type": "text" + }, + { + "bbox": [ + 243, + 190, + 273, + 203 + ], + "score": 0.94, + "content": "R ( s , { \\tilde { a } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 190, + 293, + 203 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 190, + 324, + 203 + ], + "score": 0.94, + "content": "P ( s , { \\tilde { a } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 190, + 505, + 203 + ], + "score": 1.0, + "content": ". Below, we elaborate on how one can make", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 205, + 216 + ], + "score": 1.0, + "content": "use of the derivatives of", + "type": "text" + }, + { + "bbox": [ + 205, + 201, + 219, + 214 + ], + "score": 0.92, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 201, + 254, + 216 + ], + "score": 1.0, + "content": "to learn", + "type": "text" + }, + { + "bbox": [ + 254, + 205, + 261, + 215 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 201, + 280, + 216 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 280, + 204, + 288, + 213 + ], + "score": 0.82, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 201, + 506, + 216 + ], + "score": 1.0, + "content": ", and how the Bellman equation in (13) enables direct", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 214, + 188, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 170, + 227 + ], + "score": 1.0, + "content": "optimization of", + "type": "text" + }, + { + "bbox": [ + 171, + 214, + 184, + 227 + ], + "score": 0.91, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 214, + 188, + 227 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 190, + 506, + 227 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 240, + 377, + 254 + ], + "lines": [ + { + "bbox": [ + 104, + 239, + 377, + 257 + ], + "spans": [ + { + "bbox": [ + 104, + 239, + 294, + 257 + ], + "score": 1.0, + "content": "3.1 POLICY IMPROVEMENT - OPTIMIZING", + "type": "text" + }, + { + "bbox": [ + 294, + 241, + 330, + 254 + ], + "score": 0.85, + "content": "\\left( \\mu _ { \\theta } , \\Sigma _ { \\phi } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 239, + 362, + 257 + ], + "score": 1.0, + "content": "GIVEN", + "type": "text" + }, + { + "bbox": [ + 363, + 240, + 377, + 254 + ], + "score": 0.79, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 506, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 250, + 276 + ], + "score": 1.0, + "content": "We parameterize a Gaussian policy", + "type": "text" + }, + { + "bbox": [ + 250, + 263, + 317, + 275 + ], + "score": 0.92, + "content": "\\pi _ { \\boldsymbol { \\theta } , \\boldsymbol { \\phi } } \\equiv ( \\mu _ { \\boldsymbol { \\theta } } , \\Sigma _ { \\boldsymbol { \\phi } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 262, + 458, + 276 + ], + "score": 1.0, + "content": "in terms of two sets of parameters", + "type": "text" + }, + { + "bbox": [ + 458, + 264, + 464, + 273 + ], + "score": 0.83, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 262, + 482, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 483, + 264, + 489, + 274 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 262, + 506, + 276 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 344, + 286 + ], + "score": 1.0, + "content": "the mean and the covariance. The gradient of the objective", + "type": "text" + }, + { + "bbox": [ + 344, + 275, + 363, + 284 + ], + "score": 0.28, + "content": "w . r . t .", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 273, + 505, + 286 + ], + "score": 1.0, + "content": ". mean parameters follows from the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 285, + 323, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 323, + 297 + ], + "score": 1.0, + "content": "policy gradient theorem and is almost identical to (7),", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 262, + 506, + 297 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 301, + 423, + 331 + ], + "lines": [ + { + "bbox": [ + 188, + 301, + 423, + 331 + ], + "spans": [ + { + "bbox": [ + 188, + 301, + 423, + 331 + ], + "score": 0.95, + "content": "\\nabla _ { \\theta } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta , \\phi } ) = \\int _ { S } \\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a } \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .", + "type": "interline_equation", + "image_path": "6d9aef1b0a5509659a282090238279c9da22df639f2282e7c6aa94387ffb0566.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 188, + 301, + 423, + 316.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 188, + 316.0, + 423, + 331.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Estimating the derivative of the objective w.r.t. covariance parameters is not as straightforward, since", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 352, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 107, + 352, + 121, + 365 + ], + "score": 0.89, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 352, + 222, + 366 + ], + "score": 1.0, + "content": "is not a direct function of", + "type": "text" + }, + { + "bbox": [ + 222, + 354, + 230, + 363 + ], + "score": 0.79, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 352, + 506, + 366 + ], + "score": 1.0, + "content": ". However, a key observation of this work is that the second derivative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 364, + 429, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 117, + 379 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 364, + 152, + 378 + ], + "score": 0.89, + "content": "{ \\tilde { Q } } ^ { \\pi } w . r . t .", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 364, + 379, + 379 + ], + "score": 1.0, + "content": ". actions is sufficient to exactly compute the derivative of", + "type": "text" + }, + { + "bbox": [ + 379, + 364, + 424, + 378 + ], + "score": 0.89, + "content": "\\tilde { Q } ^ { \\pi } w . r . t . \\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 364, + 429, + 379 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 340, + 506, + 379 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 382, + 368, + 411 + ], + "lines": [ + { + "bbox": [ + 243, + 382, + 368, + 411 + ], + "spans": [ + { + "bbox": [ + 243, + 382, + 368, + 411 + ], + "score": 0.95, + "content": "\\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial \\Sigma ( s ) } = \\frac { 1 } { 2 } \\cdot \\frac { \\partial ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } } .", + "type": "interline_equation", + "image_path": "5b5740281c05060410f4bd717038d516178564d8e3793b5af4abb32a9f57cbf7.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 243, + 382, + 368, + 396.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 243, + 396.5, + 368, + 411.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 104, + 414, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 414, + 506, + 430 + ], + "score": 1.0, + "content": "A proof of this identity is provided in the Appendix. The proof may be easily derived by expressing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 426, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 426, + 360, + 442 + ], + "score": 1.0, + "content": "both sides of the equation using standard matrix calculus like", + "type": "text" + }, + { + "bbox": [ + 361, + 427, + 486, + 442 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { \\partial } { \\partial A } | A | ^ { - 1 / 2 } = - \\frac { 1 } { 2 } | A | ^ { - 1 / 2 } \\bar { A } ^ { - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 426, + 506, + 442 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 436, + 232, + 455 + ], + "spans": [ + { + "bbox": [ + 107, + 440, + 227, + 455 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { \\partial } { \\partial A } | | v | | _ { A ^ { - 1 } } ^ { 2 } = - A ^ { - 1 } v v ^ { T } A ^ { - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 436, + 232, + 455 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 414, + 506, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 295, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 296, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 226, + 471 + ], + "score": 1.0, + "content": "Then, the full derivative w.r.t.", + "type": "text" + }, + { + "bbox": [ + 226, + 459, + 233, + 470 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 457, + 296, + 471 + ], + "score": 1.0, + "content": "takes the form,", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 457, + 296, + 471 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 475, + 430, + 504 + ], + "lines": [ + { + "bbox": [ + 180, + 475, + 430, + 504 + ], + "spans": [ + { + "bbox": [ + 180, + 475, + 430, + 504 + ], + "score": 0.94, + "content": "\\nabla _ { \\phi } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta , \\phi } ) = \\frac { 1 } { 2 } \\int _ { S } \\frac { \\partial ^ { 2 } { \\tilde { Q } } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } } \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .", + "type": "interline_equation", + "image_path": "f41c8adcf7c19ddab43a7d258f46d084293a60a0cb7aa0628a9bb7df0f7b6d52.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 180, + 475, + 430, + 484.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 180, + 484.6666666666667, + 430, + 494.33333333333337 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 180, + 494.33333333333337, + 430, + 504.00000000000006 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 516, + 361, + 530 + ], + "lines": [ + { + "bbox": [ + 104, + 513, + 360, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 285, + 533 + ], + "score": 1.0, + "content": "3.2 POLICY EVALUATION - OPTIMIZING", + "type": "text" + }, + { + "bbox": [ + 286, + 516, + 301, + 530 + ], + "score": 0.88, + "content": "\\tilde { Q } _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 513, + 334, + 533 + ], + "score": 1.0, + "content": "GIVEN", + "type": "text" + }, + { + "bbox": [ + 334, + 518, + 360, + 530 + ], + "score": 0.9, + "content": "( \\mu , \\Sigma )", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 505, + 656 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 259, + 555 + ], + "score": 1.0, + "content": "We can think of two ways to optimize", + "type": "text" + }, + { + "bbox": [ + 259, + 539, + 275, + 552 + ], + "score": 0.91, + "content": "\\tilde { Q } _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 539, + 452, + 555 + ], + "score": 1.0, + "content": ". The first approach leverages (10) to update", + "type": "text" + }, + { + "bbox": [ + 452, + 539, + 466, + 552 + ], + "score": 0.91, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 539, + 506, + 555 + ], + "score": 1.0, + "content": "based on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 550, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 214, + 565 + ], + "score": 1.0, + "content": "expected Q-value function", + "type": "text" + }, + { + "bbox": [ + 214, + 552, + 228, + 563 + ], + "score": 0.89, + "content": "Q ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 550, + 425, + 565 + ], + "score": 1.0, + "content": ". In such an approach, one trains a parameterized", + "type": "text" + }, + { + "bbox": [ + 426, + 552, + 441, + 563 + ], + "score": 0.9, + "content": "Q _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 550, + 506, + 565 + ], + "score": 1.0, + "content": "to approximate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 265, + 576 + ], + "score": 1.0, + "content": "the standard expected Q-value function", + "type": "text" + }, + { + "bbox": [ + 265, + 563, + 279, + 574 + ], + "score": 0.89, + "content": "Q ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 562, + 457, + 576 + ], + "score": 1.0, + "content": "using standard methods (see e.g., Rummery", + "type": "text" + }, + { + "bbox": [ + 458, + 563, + 466, + 572 + ], + "score": 0.56, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "Niranjan", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 573, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 412, + 589 + ], + "score": 1.0, + "content": "(1994); Sutton & Barto (1998); Van Seijen et al. (2009)). Then, one fits", + "type": "text" + }, + { + "bbox": [ + 412, + 573, + 428, + 587 + ], + "score": 0.91, + "content": "\\tilde { Q } _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 573, + 470, + 589 + ], + "score": 1.0, + "content": "based on", + "type": "text" + }, + { + "bbox": [ + 470, + 575, + 486, + 587 + ], + "score": 0.9, + "content": "Q _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 573, + 506, + 589 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 585, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 585, + 222, + 599 + ], + "score": 1.0, + "content": "particular, given transitions", + "type": "text" + }, + { + "bbox": [ + 222, + 586, + 265, + 597 + ], + "score": 0.93, + "content": "( s , a , r , s ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 585, + 506, + 599 + ], + "score": 1.0, + "content": "sampled from interactions with the environment, one can", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 595, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 127, + 611 + ], + "score": 1.0, + "content": "train", + "type": "text" + }, + { + "bbox": [ + 127, + 597, + 142, + 608 + ], + "score": 0.87, + "content": "Q _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 595, + 266, + 611 + ], + "score": 1.0, + "content": "to minimize the Bellman error", + "type": "text" + }, + { + "bbox": [ + 266, + 596, + 386, + 609 + ], + "score": 0.9, + "content": "( Q _ { w } ^ { \\pi } ( s , a ) - r - \\gamma Q _ { w } ^ { \\pi } ( s ^ { \\prime } , a ^ { \\prime } ) ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 595, + 414, + 611 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 414, + 596, + 501, + 609 + ], + "score": 0.92, + "content": "a ^ { \\prime } \\sim N ( \\mu ( s ^ { \\prime } ) , \\Sigma ( s ^ { \\prime } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 595, + 506, + 611 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 607, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 607, + 132, + 624 + ], + "score": 1.0, + "content": "Then,", + "type": "text" + }, + { + "bbox": [ + 133, + 608, + 148, + 621 + ], + "score": 0.87, + "content": "\\tilde { Q } _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 607, + 346, + 624 + ], + "score": 1.0, + "content": "can be optimized to minimize the squared error", + "type": "text" + }, + { + "bbox": [ + 347, + 609, + 456, + 621 + ], + "score": 0.91, + "content": "( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\mathbb { E } _ { \\tilde { a } } Q _ { w } ^ { \\pi } ( s , \\tilde { a } ) ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 607, + 486, + 624 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 486, + 610, + 505, + 620 + ], + "score": 0.85, + "content": "\\tilde { a } \\sim", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 153, + 632 + ], + "score": 0.92, + "content": "N ( a , \\Sigma ( s ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 620, + 506, + 633 + ], + "score": 1.0, + "content": ", using several samples. When the target values in these residuals are treated as fixed", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 631, + 504, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 466, + 647 + ], + "score": 1.0, + "content": "(i.e., using a target network), such a training procedure will achieve a fixed point when", + "type": "text" + }, + { + "bbox": [ + 466, + 631, + 504, + 645 + ], + "score": 0.93, + "content": "\\tilde { Q } _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 644, + 313, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 313, + 655 + ], + "score": 1.0, + "content": "satisfies the recursion in the Bellman equation (10).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 539, + 506, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 661, + 505, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 507, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 374, + 677 + ], + "score": 1.0, + "content": "The second approach requires a single function approximator for", + "type": "text" + }, + { + "bbox": [ + 374, + 660, + 411, + 674 + ], + "score": 0.93, + "content": "\\tilde { Q } _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 661, + 507, + 677 + ], + "score": 1.0, + "content": ", resulting in a simpler", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 674, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 505, + 686 + ], + "score": 1.0, + "content": "implementation, and thus we use this approach in our experimental evaluation. Suppose one has ac-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 165, + 697 + ], + "score": 1.0, + "content": "cess to a tuple", + "type": "text" + }, + { + "bbox": [ + 165, + 684, + 209, + 696 + ], + "score": 0.93, + "content": "( s , \\tilde { a } , \\tilde { r } , \\tilde { s } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 684, + 505, + 697 + ], + "score": 1.0, + "content": "sampled from a replay buffer with knowledge of the sampling probability", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 695, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 107, + 695, + 138, + 708 + ], + "score": 0.91, + "content": "q ( \\tilde { a } \\mid s )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 695, + 505, + 708 + ], + "score": 1.0, + "content": "(possibly unnormalized). Then assuming that this sampling distribution has a full support,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 707, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 214, + 721 + ], + "score": 1.0, + "content": "we draw a phantom action", + "type": "text" + }, + { + "bbox": [ + 215, + 707, + 280, + 720 + ], + "score": 0.91, + "content": "a \\sim N ( \\tilde { a } , \\Sigma ( s ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 707, + 336, + 721 + ], + "score": 1.0, + "content": "and optimize", + "type": "text" + }, + { + "bbox": [ + 337, + 707, + 374, + 720 + ], + "score": 0.95, + "content": "\\tilde { Q } _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 707, + 505, + 721 + ], + "score": 1.0, + "content": "by minimizing a weighted Bell-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 718, + 507, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 149, + 735 + ], + "score": 1.0, + "content": "man error", + "type": "text" + }, + { + "bbox": [ + 149, + 719, + 308, + 735 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { q ( \\tilde { a } | s ) } ( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\tilde { r } - \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 718, + 466, + 735 + ], + "score": 1.0, + "content": ". For a specific pair of state and action", + "type": "text" + }, + { + "bbox": [ + 466, + 720, + 488, + 732 + ], + "score": 0.91, + "content": "( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 718, + 507, + 735 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 660, + 507, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 244, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 244, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 244, + 96 + ], + "score": 1.0, + "content": "expected value of the objective is,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 99, + 467, + 127 + ], + "lines": [ + { + "bbox": [ + 127, + 99, + 467, + 127 + ], + "spans": [ + { + "bbox": [ + 127, + 99, + 467, + 127 + ], + "score": 0.93, + "content": "E ( w \\mid ( s , a ) ) ~ = ~ \\mathbb { E } _ { q ( \\bar { a } \\mid s ) , \\bar { r } , \\bar { s } ^ { \\prime } } \\left[ \\frac { N ( a \\mid \\tilde { a } , \\Sigma ( s ) ) } { q ( \\tilde { a } \\mid s ) } ( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\tilde { r } - \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) ) ^ { 2 } \\right] ~ .", + "type": "interline_equation", + "image_path": "c76fa0ee66173ea378e379d523736869626d6fb2a9f21c00ecef6ee7a64e1c84.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 127, + 99, + 467, + 108.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 127, + 108.33333333333333, + 467, + 117.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 127, + 117.66666666666666, + 467, + 126.99999999999999 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 503, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 147, + 147 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 147, + 133, + 272, + 145 + ], + "score": 0.91, + "content": "N ( a | \\tilde { a } , \\Sigma ( s ) ) = N ( \\tilde { a } | a , \\Sigma ( s ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 131, + 414, + 147 + ], + "score": 1.0, + "content": ". Therefore, when the target value", + "type": "text" + }, + { + "bbox": [ + 414, + 131, + 493, + 145 + ], + "score": 0.93, + "content": "\\tilde { r } + \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 131, + 506, + 147 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "score": 1.0, + "content": "treated as fixed (e.g., when using target networks) this training procedure reaches an optimum when", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 155, + 353, + 170 + ], + "spans": [ + { + "bbox": [ + 107, + 155, + 144, + 169 + ], + "score": 0.92, + "content": "\\tilde { Q } _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 155, + 353, + 170 + ], + "score": 1.0, + "content": "satisfies the recursion in the Bellman equation (13).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 173, + 505, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 417, + 186 + ], + "score": 1.0, + "content": "In practice, we find that it is unnecessary to keep track of the probabilities", + "type": "text" + }, + { + "bbox": [ + 418, + 173, + 450, + 186 + ], + "score": 0.93, + "content": "q ( \\tilde { a } \\mid s )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 173, + 505, + 186 + ], + "score": 1.0, + "content": ", and assume", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 185, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 196 + ], + "score": 1.0, + "content": "the replay buffer provides a near-uniform distribution of actions conditioned on states. Other recent", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "score": 1.0, + "content": "work has also benefited from ignoring or heavily damping importance weights (Munos et al., 2016;", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 206, + 504, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 504, + 218 + ], + "score": 1.0, + "content": "Wang et al., 2017; Schulman et al., 2017). However, it is possible when interacting with the envi-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 218, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 229 + ], + "score": 1.0, + "content": "ronment to save the probability of sampled actions along with their transitions, and thus have access", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 256, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 117, + 242 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 228, + 252, + 241 + ], + "score": 0.91, + "content": "q ( \\tilde { \\boldsymbol { a } } \\mid \\boldsymbol { s } ) \\approx N ( \\tilde { \\boldsymbol { a } } \\mid \\bar { \\mu } _ { \\mathrm { o l d } } ( \\boldsymbol { s } ) , \\dot { \\Sigma _ { \\mathrm { o l d } } } ( \\boldsymbol { s } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 228, + 256, + 242 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 253, + 279, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 251, + 281, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 281, + 266 + ], + "score": 1.0, + "content": "3.3 PROXIMAL POLICY OPTIMIZATION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 273, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "Policy gradient algorithms are notoriously unstable, particularly in continuous control problems.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "Such instability has motivated the development of trust region methods that attempt to mitigate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "the issue by constraining each gradient step to lie within a trust region (Schulman et al., 2015),", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "or augmenting the expected reward objective with a penalty on KL-divergence from a previous", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "policy (Nachum et al., 2017b; Schulman et al., 2017; Azar et al., 2012). These stabilizing techniques", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "have thus far not been applicable to algorithms like DDPG, since the policy is deterministic. The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "formulation we propose in this paper, however, is easily amenable to trust region optimization.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 350, + 360, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 360, + 364 + ], + "score": 1.0, + "content": "Specifically, we may augment the objective (11) with a penalty", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 366, + 404, + 393 + ], + "lines": [ + { + "bbox": [ + 206, + 366, + 404, + 393 + ], + "spans": [ + { + "bbox": [ + 206, + 366, + 404, + 393 + ], + "score": 0.93, + "content": "{ \\cal O } _ { \\mathrm { T R } } ( \\pi ) = { \\cal O } _ { \\mathrm { E R } } ( \\pi ) - \\lambda \\int _ { \\cal S } \\mathrm { K L } ( \\pi \\parallel \\pi _ { \\mathrm { o l d } } ) \\mathrm { d } \\rho ^ { \\pi } ( s ) ,", + "type": "interline_equation", + "image_path": "ce671f74d9ac9a165e85048a8cd34aa0136ad30376cb6d7e13762ea2132990ef.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 206, + 366, + 404, + 379.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 206, + 379.5, + 404, + 393.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 398, + 503, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 504, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 133, + 411 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 398, + 208, + 410 + ], + "score": 0.92, + "content": "\\pi _ { \\mathrm { o l d } } \\equiv \\left( \\mu _ { \\mathrm { o l d } } , \\Sigma _ { \\mathrm { o l d } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 398, + 504, + 411 + ], + "score": 1.0, + "content": "is a previous parameterization of the policy. The optimization is straight-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 408, + 434, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 434, + 422 + ], + "score": 1.0, + "content": "forward, since the KL-divergence of two Gaussians can be expressed analytically.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 108, + 436, + 211, + 449 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 213, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 213, + 451 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "This paper follows a long line of work that uses Q-value functions to stably learn a policy, which", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "in the past has been used to either approximate expected (Rummery & Niranjan, 1994; Van Seijen", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "et al., 2009; Gu et al., 2017) or optimal (Watkins, 1989; Silver et al., 2014; Nachum et al., 2017a;", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 494, + 321, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 321, + 506 + ], + "score": 1.0, + "content": "Haarnoja et al., 2017; Metz et al., 2017) future value.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "Work that is most similar to what we present are methods that exploit gradient information from", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "the Q-value function to train a policy. Deterministic policy gradient (Silver et al., 2014) is perhaps", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "the best known of these. The method we propose can be interpreted as a generalization of the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 443, + 557 + ], + "score": 1.0, + "content": "deterministic policy gradient. Indeed, if one takes the limit of the policy covariance", + "type": "text" + }, + { + "bbox": [ + 443, + 544, + 464, + 556 + ], + "score": 0.9, + "content": "\\Sigma ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 543, + 505, + 557 + ], + "score": 1.0, + "content": "as it goes", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "to 0, the proposed Q-value function becomes the deterministic value function of DDPG, and the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 566, + 428, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 428, + 579 + ], + "score": 1.0, + "content": "updates for training the Q-value approximator and the policy mean are identical.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "Stochastic Value Gradient (SVG) (Heess et al., 2015) also trains stochastic policies using an update", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "that is similar to DDPG (i.e., SVG(0) with replay). The key differences with our approach are that", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "SVG does not provide an update for the covariance, and the mean update in SVG estimates the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 470, + 628 + ], + "score": 1.0, + "content": "gradient with a noisy Monte Carlo sample, which we avoid by estimating the smoothed", + "type": "text" + }, + { + "bbox": [ + 470, + 617, + 479, + 627 + ], + "score": 0.34, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "-value", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "function. Although a covariance update could be derived using the same reparameterization trick as", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 636, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 104, + 636, + 505, + 652 + ], + "score": 1.0, + "content": "in the mean update, that would also require a noisy Monte Carlo estimate. Methods for updating", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "the covariance along the gradient of expected reward are essential for applying the subsequent trust", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 660, + 264, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 264, + 673 + ], + "score": 1.0, + "content": "region and proximal policy techniques.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "More recently, Ciosek & Whiteson (2017) introduced expected policy gradients (EPG), a general-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "score": 1.0, + "content": "ization of DDPG that provides updates for the mean and covariance of a stochastic Gaussian policy", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "using gradients of an estimated Q-value function. In that work, the expected Q-value used in stan-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "dard policy gradient algorithms such as SARSA (Sutton & Barto, 1998; Rummery & Niranjan,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "1994; Van Seijen et al., 2009) is estimated. The updates in EPG therefore require approximating", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 244, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 244, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 244, + 96 + ], + "score": 1.0, + "content": "expected value of the objective is,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 82, + 244, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 99, + 467, + 127 + ], + "lines": [ + { + "bbox": [ + 127, + 99, + 467, + 127 + ], + "spans": [ + { + "bbox": [ + 127, + 99, + 467, + 127 + ], + "score": 0.93, + "content": "E ( w \\mid ( s , a ) ) ~ = ~ \\mathbb { E } _ { q ( \\bar { a } \\mid s ) , \\bar { r } , \\bar { s } ^ { \\prime } } \\left[ \\frac { N ( a \\mid \\tilde { a } , \\Sigma ( s ) ) } { q ( \\tilde { a } \\mid s ) } ( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\tilde { r } - \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) ) ^ { 2 } \\right] ~ .", + "type": "interline_equation", + "image_path": "c76fa0ee66173ea378e379d523736869626d6fb2a9f21c00ecef6ee7a64e1c84.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 127, + 99, + 467, + 108.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 127, + 108.33333333333333, + 467, + 117.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 127, + 117.66666666666666, + 467, + 126.99999999999999 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 503, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 147, + 147 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 147, + 133, + 272, + 145 + ], + "score": 0.91, + "content": "N ( a | \\tilde { a } , \\Sigma ( s ) ) = N ( \\tilde { a } | a , \\Sigma ( s ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 131, + 414, + 147 + ], + "score": 1.0, + "content": ". Therefore, when the target value", + "type": "text" + }, + { + "bbox": [ + 414, + 131, + 493, + 145 + ], + "score": 0.93, + "content": "\\tilde { r } + \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 131, + 506, + 147 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "score": 1.0, + "content": "treated as fixed (e.g., when using target networks) this training procedure reaches an optimum when", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 155, + 353, + 170 + ], + "spans": [ + { + "bbox": [ + 107, + 155, + 144, + 169 + ], + "score": 0.92, + "content": "\\tilde { Q } _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 155, + 353, + 170 + ], + "score": 1.0, + "content": "satisfies the recursion in the Bellman equation (13).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 131, + 506, + 170 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 173, + 505, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 417, + 186 + ], + "score": 1.0, + "content": "In practice, we find that it is unnecessary to keep track of the probabilities", + "type": "text" + }, + { + "bbox": [ + 418, + 173, + 450, + 186 + ], + "score": 0.93, + "content": "q ( \\tilde { a } \\mid s )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 173, + 505, + 186 + ], + "score": 1.0, + "content": ", and assume", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 185, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 196 + ], + "score": 1.0, + "content": "the replay buffer provides a near-uniform distribution of actions conditioned on states. Other recent", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "score": 1.0, + "content": "work has also benefited from ignoring or heavily damping importance weights (Munos et al., 2016;", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 206, + 504, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 504, + 218 + ], + "score": 1.0, + "content": "Wang et al., 2017; Schulman et al., 2017). However, it is possible when interacting with the envi-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 218, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 229 + ], + "score": 1.0, + "content": "ronment to save the probability of sampled actions along with their transitions, and thus have access", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 256, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 117, + 242 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 228, + 252, + 241 + ], + "score": 0.91, + "content": "q ( \\tilde { \\boldsymbol { a } } \\mid \\boldsymbol { s } ) \\approx N ( \\tilde { \\boldsymbol { a } } \\mid \\bar { \\mu } _ { \\mathrm { o l d } } ( \\boldsymbol { s } ) , \\dot { \\Sigma _ { \\mathrm { o l d } } } ( \\boldsymbol { s } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 228, + 256, + 242 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 173, + 505, + 242 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 253, + 279, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 251, + 281, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 281, + 266 + ], + "score": 1.0, + "content": "3.3 PROXIMAL POLICY OPTIMIZATION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 273, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "Policy gradient algorithms are notoriously unstable, particularly in continuous control problems.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "Such instability has motivated the development of trust region methods that attempt to mitigate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "the issue by constraining each gradient step to lie within a trust region (Schulman et al., 2015),", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "or augmenting the expected reward objective with a penalty on KL-divergence from a previous", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "policy (Nachum et al., 2017b; Schulman et al., 2017; Azar et al., 2012). These stabilizing techniques", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "have thus far not been applicable to algorithms like DDPG, since the policy is deterministic. The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "formulation we propose in this paper, however, is easily amenable to trust region optimization.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 350, + 360, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 360, + 364 + ], + "score": 1.0, + "content": "Specifically, we may augment the objective (11) with a penalty", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 273, + 505, + 364 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 366, + 404, + 393 + ], + "lines": [ + { + "bbox": [ + 206, + 366, + 404, + 393 + ], + "spans": [ + { + "bbox": [ + 206, + 366, + 404, + 393 + ], + "score": 0.93, + "content": "{ \\cal O } _ { \\mathrm { T R } } ( \\pi ) = { \\cal O } _ { \\mathrm { E R } } ( \\pi ) - \\lambda \\int _ { \\cal S } \\mathrm { K L } ( \\pi \\parallel \\pi _ { \\mathrm { o l d } } ) \\mathrm { d } \\rho ^ { \\pi } ( s ) ,", + "type": "interline_equation", + "image_path": "ce671f74d9ac9a165e85048a8cd34aa0136ad30376cb6d7e13762ea2132990ef.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 206, + 366, + 404, + 379.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 206, + 379.5, + 404, + 393.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 398, + 503, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 504, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 133, + 411 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 398, + 208, + 410 + ], + "score": 0.92, + "content": "\\pi _ { \\mathrm { o l d } } \\equiv \\left( \\mu _ { \\mathrm { o l d } } , \\Sigma _ { \\mathrm { o l d } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 398, + 504, + 411 + ], + "score": 1.0, + "content": "is a previous parameterization of the policy. The optimization is straight-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 408, + 434, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 434, + 422 + ], + "score": 1.0, + "content": "forward, since the KL-divergence of two Gaussians can be expressed analytically.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 398, + 504, + 422 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 436, + 211, + 449 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 213, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 213, + 451 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "This paper follows a long line of work that uses Q-value functions to stably learn a policy, which", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "in the past has been used to either approximate expected (Rummery & Niranjan, 1994; Van Seijen", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "et al., 2009; Gu et al., 2017) or optimal (Watkins, 1989; Silver et al., 2014; Nachum et al., 2017a;", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 494, + 321, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 321, + 506 + ], + "score": 1.0, + "content": "Haarnoja et al., 2017; Metz et al., 2017) future value.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 461, + 506, + 506 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "Work that is most similar to what we present are methods that exploit gradient information from", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "the Q-value function to train a policy. Deterministic policy gradient (Silver et al., 2014) is perhaps", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "the best known of these. The method we propose can be interpreted as a generalization of the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 443, + 557 + ], + "score": 1.0, + "content": "deterministic policy gradient. Indeed, if one takes the limit of the policy covariance", + "type": "text" + }, + { + "bbox": [ + 443, + 544, + 464, + 556 + ], + "score": 0.9, + "content": "\\Sigma ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 543, + 505, + 557 + ], + "score": 1.0, + "content": "as it goes", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "to 0, the proposed Q-value function becomes the deterministic value function of DDPG, and the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 566, + 428, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 428, + 579 + ], + "score": 1.0, + "content": "updates for training the Q-value approximator and the policy mean are identical.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 511, + 506, + 579 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "Stochastic Value Gradient (SVG) (Heess et al., 2015) also trains stochastic policies using an update", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "that is similar to DDPG (i.e., SVG(0) with replay). The key differences with our approach are that", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "SVG does not provide an update for the covariance, and the mean update in SVG estimates the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 470, + 628 + ], + "score": 1.0, + "content": "gradient with a noisy Monte Carlo sample, which we avoid by estimating the smoothed", + "type": "text" + }, + { + "bbox": [ + 470, + 617, + 479, + 627 + ], + "score": 0.34, + "content": "\\mathrm { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "-value", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "function. Although a covariance update could be derived using the same reparameterization trick as", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 636, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 104, + 636, + 505, + 652 + ], + "score": 1.0, + "content": "in the mean update, that would also require a noisy Monte Carlo estimate. Methods for updating", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "the covariance along the gradient of expected reward are essential for applying the subsequent trust", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 660, + 264, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 264, + 673 + ], + "score": 1.0, + "content": "region and proximal policy techniques.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 583, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "More recently, Ciosek & Whiteson (2017) introduced expected policy gradients (EPG), a general-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "score": 1.0, + "content": "ization of DDPG that provides updates for the mean and covariance of a stochastic Gaussian policy", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "using gradients of an estimated Q-value function. In that work, the expected Q-value used in stan-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "dard policy gradient algorithms such as SARSA (Sutton & Barto, 1998; Rummery & Niranjan,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "1994; Van Seijen et al., 2009) is estimated. The updates in EPG therefore require approximating", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "an integral of the expected Q-value function. Our analogous process directly estimates an integral", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "(via the smoothed Q-value function) and avoids approximate integrals, thereby making the updates", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "simpler. 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Our analogous process directly estimates an integral", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "(via the smoothed Q-value function) and avoids approximate integrals, thereby making the updates", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "simpler. Moreover, while Ciosek & Whiteson (2017) rely on a quadratic Taylor expansion of the es-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "timated Q-value function, we instead rely on the strength of neural network function approximators", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 313, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 313, + 138 + ], + "score": 1.0, + "content": "to directly estimate the smoothed Q-value function.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 504, + 187 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "The novel training scheme we propose for learning the covariance of a Gaussian policy relies on", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "properties of Gaussian integrals (Bonnet, 1964; Price, 1958). Similar identities have been used in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "score": 1.0, + "content": "the past to derive updates for variational auto-encoders (Kingma & Welling, 2014) and Gaussian", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 271, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 271, + 188 + ], + "score": 1.0, + "content": "back-propagation (Rezende et al., 2014).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Finally, the perspective presented in this paper, where Q-values represent the averaged return of a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "distribution of actions rather than a single action, is distinct from recent advances in distributional", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "RL (Bellemare et al., 2017). Those approaches focus on the distribution of returns of a single action,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "whereas we consider the single average return of a distribution of actions. Although we restrict our", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "attention in this paper to Gaussian policies, an interesting topic for further investigation is to study", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 424, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 424, + 261 + ], + "score": 1.0, + "content": "the applicability of this new perspective to a wider class of policy distributions.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 285, + 200, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 202, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 202, + 300 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "We utilize the insights from Section 3 to introduce a new RL algorithm, Smoothie. Smoothie main-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 327, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 192, + 343 + ], + "score": 1.0, + "content": "tains a parameterized", + "type": "text" + }, + { + "bbox": [ + 192, + 327, + 208, + 341 + ], + "score": 0.91, + "content": "\\tilde { Q } _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 328, + 506, + 343 + ], + "score": 1.0, + "content": "trained via the procedure described in Section 3.2. It then uses the gradient", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 348, + 353 + ], + "score": 1.0, + "content": "and Hessian of this approximation to train a Gaussian policy", + "type": "text" + }, + { + "bbox": [ + 349, + 340, + 378, + 353 + ], + "score": 0.62, + "content": "\\mu _ { \\theta } , \\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "using the updates stated in (14)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 352, + 397, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 397, + 363 + ], + "score": 1.0, + "content": "and (16). See Algorithm 1 for a simplified pseudocode of our algorithm.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 384, + 201, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 383, + 201, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 201, + 397 + ], + "score": 1.0, + "content": "Algorithm 1 Smoothie", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "table", + "bbox": [ + 107, + 388, + 453, + 728 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 388, + 453, + 728 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 388, + 453, + 728 + ], + "spans": [ + { + "bbox": [ + 107, + 388, + 453, + 728 + ], + "score": 0.528, + "html": "
Algorithm1Smoothie Input: Environment ENV,learning rates 7π, nQ,discount factor y,KL-penalty 入,l
number of training steps N, target network lag T. Initialize0,,w,set0'=0,Φ'=,w'=w.
fori=OtoN-1do
/ Collect experience
Sample action a ~ N(μe(s),∑(s)) and apply to ENV to yield r and s'. Insert transition (s,a,r,s') to replay buffer.
// Train μ,∑
Sample batch{(k,@k,Tk,S)}1 fromreplay buffer aQ(ska) Compute gradients gk =
da la=μe(sk)*
²(s@） Compute Hessians Hk = a² la=μe(sk)*
Compute KL-penalties KLk =KL(μθ,Σ𝜙llμe,Σ).
Compute updates 1B
1B
△=B 1 ∑k=1 Updateθ←θ+ηπ△0,←+ηπ△.
// Train Qπ
Sample batch {(sk,k,Tk,S)}1 from replay bufer. Sample phantom actions ak ~ N(ák,Σ(sk)). Compute lossL(u)=∑1((s)-r-qQ(s,ue()2.
", + "type": "table", + "image_path": "eb3fa1f9a495240797e692645eace753d0aebfc6f323140585dd7b69aaea1d55.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 107, + 388, + 453, + 501.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 107, + 501.3333333333333, + 453, + 614.6666666666666 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 107, + 614.6666666666666, + 453, + 728.0 + ], + "spans": [], + "index": 23 + } + ] + } + ], + "index": 22 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 82, + 506, + 138 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 504, + 187 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "The novel training scheme we propose for learning the covariance of a Gaussian policy relies on", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "properties of Gaussian integrals (Bonnet, 1964; Price, 1958). Similar identities have been used in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "score": 1.0, + "content": "the past to derive updates for variational auto-encoders (Kingma & Welling, 2014) and Gaussian", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 271, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 271, + 188 + ], + "score": 1.0, + "content": "back-propagation (Rezende et al., 2014).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 144, + 506, + 188 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Finally, the perspective presented in this paper, where Q-values represent the averaged return of a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "distribution of actions rather than a single action, is distinct from recent advances in distributional", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "RL (Bellemare et al., 2017). Those approaches focus on the distribution of returns of a single action,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "whereas we consider the single average return of a distribution of actions. Although we restrict our", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "attention in this paper to Gaussian policies, an interesting topic for further investigation is to study", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 424, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 424, + 261 + ], + "score": 1.0, + "content": "the applicability of this new perspective to a wider class of policy distributions.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 192, + 506, + 261 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 285, + 200, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 202, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 202, + 300 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "We utilize the insights from Section 3 to introduce a new RL algorithm, Smoothie. Smoothie main-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 327, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 192, + 343 + ], + "score": 1.0, + "content": "tains a parameterized", + "type": "text" + }, + { + "bbox": [ + 192, + 327, + 208, + 341 + ], + "score": 0.91, + "content": "\\tilde { Q } _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 328, + 506, + 343 + ], + "score": 1.0, + "content": "trained via the procedure described in Section 3.2. It then uses the gradient", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 348, + 353 + ], + "score": 1.0, + "content": "and Hessian of this approximation to train a Gaussian policy", + "type": "text" + }, + { + "bbox": [ + 349, + 340, + 378, + 353 + ], + "score": 0.62, + "content": "\\mu _ { \\theta } , \\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "using the updates stated in (14)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 352, + 397, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 397, + 363 + ], + "score": 1.0, + "content": "and (16). See Algorithm 1 for a simplified pseudocode of our algorithm.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 317, + 506, + 363 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 384, + 201, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 383, + 201, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 201, + 397 + ], + "score": 1.0, + "content": "Algorithm 1 Smoothie", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "table", + "bbox": [ + 107, + 388, + 453, + 728 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 388, + 453, + 728 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 388, + 453, + 728 + ], + "spans": [ + { + "bbox": [ + 107, + 388, + 453, + 728 + ], + "score": 0.528, + "html": "
Algorithm1Smoothie Input: Environment ENV,learning rates 7π, nQ,discount factor y,KL-penalty 入,l
number of training steps N, target network lag T. Initialize0,,w,set0'=0,Φ'=,w'=w.
fori=OtoN-1do
/ Collect experience
Sample action a ~ N(μe(s),∑(s)) and apply to ENV to yield r and s'. Insert transition (s,a,r,s') to replay buffer.
// Train μ,∑
Sample batch{(k,@k,Tk,S)}1 fromreplay buffer aQ(ska) Compute gradients gk =
da la=μe(sk)*
²(s@） Compute Hessians Hk = a² la=μe(sk)*
Compute KL-penalties KLk =KL(μθ,Σ𝜙llμe,Σ).
Compute updates 1B
1B
△=B 1 ∑k=1 Updateθ←θ+ηπ△0,←+ηπ△.
// Train Qπ
Sample batch {(sk,k,Tk,S)}1 from replay bufer. Sample phantom actions ak ~ N(ák,Σ(sk)). Compute lossL(u)=∑1((s)-r-qQ(s,ue()2.
", + "type": "table", + "image_path": "eb3fa1f9a495240797e692645eace753d0aebfc6f323140585dd7b69aaea1d55.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 107, + 388, + 453, + 501.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 107, + 501.3333333333333, + 453, + 614.6666666666666 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 107, + 614.6666666666666, + 453, + 728.0 + ], + "spans": [], + "index": 23 + } + ] + } + ], + "index": 22 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "We perform a number of evaluations of Smoothie compared to DDPG. We choose DDPG as a base-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "line because it (1) utilizes gradient information of a Q-value approximator, much like our algorithm;", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "and (2) is a standard algorithm well-known to have achieve good, sample-efficient performance on", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 236, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 236, + 126 + ], + "score": 1.0, + "content": "continuous control benchmarks.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 143, + 217, + 154 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 218, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 218, + 156 + ], + "score": 1.0, + "content": "5.1 A SYNTHETIC TASK", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "To evaluate Smoothie we begin with a simple synthetic task which allows us to study its behavior", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "in a restricted setting. We devised a simple single-action one-shot environment in which the reward", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "function is a mixture of two Gaussians, one better than the other (see Figure 1 (Right)). We initialize", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 197, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 212 + ], + "score": 1.0, + "content": "the policy mean to be centered on the worse of the two Gaussians. We plot the learnable policy", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "mean and standard deviation during training for Smoothie and DDPG in Figure 1 (Left). Smoothie", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "learns both the mean and variance, while DDPG learns only the mean and the variance plotted is the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 231, + 346, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 346, + 245 + ], + "score": 1.0, + "content": "exploratory noise, whose scale is kept fixed during training.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "As expected we observe that DDPG cannot escape the local optimum. At the beginning of training it", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "exhibits some movement away from the local optimum (likely due to the initial noisy approximation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 269, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 144, + 283 + ], + "score": 1.0, + "content": "given by", + "type": "text" + }, + { + "bbox": [ + 144, + 270, + 160, + 282 + ], + "score": 0.87, + "content": "Q _ { w } ^ { \\pi } \\mathrm { . }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 269, + 506, + 283 + ], + "score": 1.0, + "content": "), it is unable to progress very far from the initial mean. Note that this is not an issue", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 339, + 294 + ], + "score": 1.0, + "content": "of exploration. The exploration scale is high enough that", + "type": "text" + }, + { + "bbox": [ + 339, + 281, + 354, + 293 + ], + "score": 0.9, + "content": "Q _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "is aware of the better Gaussian. The", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 502, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 206, + 306 + ], + "score": 1.0, + "content": "issue is in the update for", + "type": "text" + }, + { + "bbox": [ + 206, + 293, + 218, + 303 + ], + "score": 0.85, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 291, + 403, + 306 + ], + "score": 1.0, + "content": ", which is only with regard to the derivative of", + "type": "text" + }, + { + "bbox": [ + 403, + 292, + 418, + 304 + ], + "score": 0.9, + "content": "Q _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 291, + 502, + 306 + ], + "score": 1.0, + "content": "at the current mean.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "On the other hand, we find Smoothie is successfully able to solve the task. This is because the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 319, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 320, + 281, + 335 + ], + "score": 1.0, + "content": "smoothed reward function approximated by", + "type": "text" + }, + { + "bbox": [ + 282, + 319, + 297, + 333 + ], + "score": 0.91, + "content": "\\tilde { Q } _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 320, + 444, + 335 + ], + "score": 1.0, + "content": "has a derivative which clearly points", + "type": "text" + }, + { + "bbox": [ + 444, + 323, + 456, + 333 + ], + "score": 0.84, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 320, + 506, + 335 + ], + "score": 1.0, + "content": "towards the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 461, + 345 + ], + "score": 1.0, + "content": "better Gaussian. We also observe that Smoothie is able to suitably adjust the covariance", + "type": "text" + }, + { + "bbox": [ + 461, + 332, + 475, + 344 + ], + "score": 0.89, + "content": "\\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 330, + 505, + 345 + ], + "score": 1.0, + "content": "during", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 180, + 356 + ], + "score": 1.0, + "content": "training. Initially,", + "type": "text" + }, + { + "bbox": [ + 180, + 343, + 194, + 355 + ], + "score": 0.89, + "content": "\\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "decreases due to the concavity of the smoothed reward function. As a region", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 444, + 367 + ], + "score": 1.0, + "content": "of convexity is entered, it begins to increase, before again decreasing to near-zero as", + "type": "text" + }, + { + "bbox": [ + 444, + 356, + 456, + 366 + ], + "score": 0.82, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "approaches", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 364, + 190, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 190, + 378 + ], + "score": 1.0, + "content": "the global optimum.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "image", + "bbox": [ + 158, + 393, + 450, + 528 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 158, + 393, + 450, + 528 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 158, + 393, + 450, + 528 + ], + "spans": [ + { + "bbox": [ + 158, + 393, + 450, + 528 + ], + "score": 0.971, + "type": "image", + "image_path": "3e62a205722715b4eae018cb617c9164cd435c130be75b6bf1c3addc835c777e.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 158, + 393, + 450, + 438.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 158, + 438.0, + 450, + 483.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 158, + 483.0, + 450, + 528.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 546, + 505, + 613 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "Figure 1: Left: The learnable policy mean and standard deviation during training for Smoothie and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 555, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 571 + ], + "score": 1.0, + "content": "DDPG on a simple one-shot synthetic task. The standard deviation for DDPG is the exploratory", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "noise kept constant during training. Right: The reward function for the synthetic task along with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "its Gaussian-smoothed version. We find that Smoothie can successfully escape the lower-reward", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "score": 1.0, + "content": "local optimum. We also notice Smoothie increases and decreases its policy variance as the convex-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 601, + 329, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 329, + 614 + ], + "score": 1.0, + "content": "ity/concavity of the smoothed reward function changes.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + } + ], + "index": 26.25 + }, + { + "type": "title", + "bbox": [ + 108, + 637, + 234, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 236, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 236, + 651 + ], + "score": 1.0, + "content": "5.2 CONTINUOUS CONTROL", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 660, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "We now turn our attention to standard continuous control benchmarks available on OpenAI", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 671, + 458, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 458, + 683 + ], + "score": 1.0, + "content": "Gym (Brockman et al., 2016) utilizing the MuJoCo environment (Todorov et al., 2012).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Our implementations utilize feed forward neural networks for policy and Q-values. We parame-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 193, + 712 + ], + "score": 1.0, + "content": "terize the covariance", + "type": "text" + }, + { + "bbox": [ + 193, + 699, + 207, + 711 + ], + "score": 0.89, + "content": "\\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 698, + 304, + 712 + ], + "score": 1.0, + "content": "as a diagonal given by", + "type": "text" + }, + { + "bbox": [ + 304, + 698, + 315, + 708 + ], + "score": 0.87, + "content": "e ^ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ". The exploration for DDPG is determined by", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "an Ornstein-Uhlenbeck process (Uhlenbeck & Ornstein, 1930; Lillicrap et al., 2016). Additional", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 721, + 321, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 321, + 732 + ], + "score": 1.0, + "content": "implementation details are provided in the Appendix.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 7 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "We perform a number of evaluations of Smoothie compared to DDPG. We choose DDPG as a base-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "line because it (1) utilizes gradient information of a Q-value approximator, much like our algorithm;", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "and (2) is a standard algorithm well-known to have achieve good, sample-efficient performance on", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 236, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 236, + 126 + ], + "score": 1.0, + "content": "continuous control benchmarks.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 83, + 506, + 126 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 143, + 217, + 154 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 218, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 218, + 156 + ], + "score": 1.0, + "content": "5.1 A SYNTHETIC TASK", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "To evaluate Smoothie we begin with a simple synthetic task which allows us to study its behavior", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "in a restricted setting. We devised a simple single-action one-shot environment in which the reward", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "function is a mixture of two Gaussians, one better than the other (see Figure 1 (Right)). We initialize", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 197, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 212 + ], + "score": 1.0, + "content": "the policy mean to be centered on the worse of the two Gaussians. We plot the learnable policy", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "mean and standard deviation during training for Smoothie and DDPG in Figure 1 (Left). Smoothie", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "learns both the mean and variance, while DDPG learns only the mean and the variance plotted is the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 231, + 346, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 346, + 245 + ], + "score": 1.0, + "content": "exploratory noise, whose scale is kept fixed during training.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 165, + 505, + 245 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "As expected we observe that DDPG cannot escape the local optimum. At the beginning of training it", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "exhibits some movement away from the local optimum (likely due to the initial noisy approximation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 269, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 144, + 283 + ], + "score": 1.0, + "content": "given by", + "type": "text" + }, + { + "bbox": [ + 144, + 270, + 160, + 282 + ], + "score": 0.87, + "content": "Q _ { w } ^ { \\pi } \\mathrm { . }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 269, + 506, + 283 + ], + "score": 1.0, + "content": "), it is unable to progress very far from the initial mean. Note that this is not an issue", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 339, + 294 + ], + "score": 1.0, + "content": "of exploration. The exploration scale is high enough that", + "type": "text" + }, + { + "bbox": [ + 339, + 281, + 354, + 293 + ], + "score": 0.9, + "content": "Q _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "is aware of the better Gaussian. The", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 502, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 206, + 306 + ], + "score": 1.0, + "content": "issue is in the update for", + "type": "text" + }, + { + "bbox": [ + 206, + 293, + 218, + 303 + ], + "score": 0.85, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 291, + 403, + 306 + ], + "score": 1.0, + "content": ", which is only with regard to the derivative of", + "type": "text" + }, + { + "bbox": [ + 403, + 292, + 418, + 304 + ], + "score": 0.9, + "content": "Q _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 291, + 502, + 306 + ], + "score": 1.0, + "content": "at the current mean.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 248, + 506, + 306 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "On the other hand, we find Smoothie is successfully able to solve the task. This is because the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 319, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 320, + 281, + 335 + ], + "score": 1.0, + "content": "smoothed reward function approximated by", + "type": "text" + }, + { + "bbox": [ + 282, + 319, + 297, + 333 + ], + "score": 0.91, + "content": "\\tilde { Q } _ { w } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 320, + 444, + 335 + ], + "score": 1.0, + "content": "has a derivative which clearly points", + "type": "text" + }, + { + "bbox": [ + 444, + 323, + 456, + 333 + ], + "score": 0.84, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 320, + 506, + 335 + ], + "score": 1.0, + "content": "towards the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 461, + 345 + ], + "score": 1.0, + "content": "better Gaussian. We also observe that Smoothie is able to suitably adjust the covariance", + "type": "text" + }, + { + "bbox": [ + 461, + 332, + 475, + 344 + ], + "score": 0.89, + "content": "\\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 330, + 505, + 345 + ], + "score": 1.0, + "content": "during", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 180, + 356 + ], + "score": 1.0, + "content": "training. Initially,", + "type": "text" + }, + { + "bbox": [ + 180, + 343, + 194, + 355 + ], + "score": 0.89, + "content": "\\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "decreases due to the concavity of the smoothed reward function. As a region", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 444, + 367 + ], + "score": 1.0, + "content": "of convexity is entered, it begins to increase, before again decreasing to near-zero as", + "type": "text" + }, + { + "bbox": [ + 444, + 356, + 456, + 366 + ], + "score": 0.82, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "approaches", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 364, + 190, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 190, + 378 + ], + "score": 1.0, + "content": "the global optimum.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 309, + 506, + 378 + ] + }, + { + "type": "image", + "bbox": [ + 158, + 393, + 450, + 528 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 158, + 393, + 450, + 528 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 158, + 393, + 450, + 528 + ], + "spans": [ + { + "bbox": [ + 158, + 393, + 450, + 528 + ], + "score": 0.971, + "type": "image", + "image_path": "3e62a205722715b4eae018cb617c9164cd435c130be75b6bf1c3addc835c777e.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 158, + 393, + 450, + 438.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 158, + 438.0, + 450, + 483.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 158, + 483.0, + 450, + 528.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 546, + 505, + 613 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "Figure 1: Left: The learnable policy mean and standard deviation during training for Smoothie and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 555, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 571 + ], + "score": 1.0, + "content": "DDPG on a simple one-shot synthetic task. The standard deviation for DDPG is the exploratory", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "noise kept constant during training. Right: The reward function for the synthetic task along with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "its Gaussian-smoothed version. We find that Smoothie can successfully escape the lower-reward", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "score": 1.0, + "content": "local optimum. We also notice Smoothie increases and decreases its policy variance as the convex-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 601, + 329, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 329, + 614 + ], + "score": 1.0, + "content": "ity/concavity of the smoothed reward function changes.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + } + ], + "index": 26.25 + }, + { + "type": "title", + "bbox": [ + 108, + 637, + 234, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 236, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 236, + 651 + ], + "score": 1.0, + "content": "5.2 CONTINUOUS CONTROL", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 660, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "We now turn our attention to standard continuous control benchmarks available on OpenAI", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 671, + 458, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 458, + 683 + ], + "score": 1.0, + "content": "Gym (Brockman et al., 2016) utilizing the MuJoCo environment (Todorov et al., 2012).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 106, + 659, + 505, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Our implementations utilize feed forward neural networks for policy and Q-values. We parame-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 193, + 712 + ], + "score": 1.0, + "content": "terize the covariance", + "type": "text" + }, + { + "bbox": [ + 193, + 699, + 207, + 711 + ], + "score": 0.89, + "content": "\\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 698, + 304, + 712 + ], + "score": 1.0, + "content": "as a diagonal given by", + "type": "text" + }, + { + "bbox": [ + 304, + 698, + 315, + 708 + ], + "score": 0.87, + "content": "e ^ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ". The exploration for DDPG is determined by", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "an Ornstein-Uhlenbeck process (Uhlenbeck & Ornstein, 1930; Lillicrap et al., 2016). Additional", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 721, + 321, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 321, + 732 + ], + "score": 1.0, + "content": "implementation details are provided in the Appendix.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 132, + 81, + 477, + 309 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 132, + 81, + 477, + 309 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 81, + 477, + 309 + ], + "spans": [ + { + "bbox": [ + 132, + 81, + 477, + 309 + ], + "score": 0.977, + "type": "image", + "image_path": "f21887a3716c42d41ffbbba35b3c52cf13877d7b56bb86c24b7339f4a95daf67.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 132, + 81, + 477, + 157.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 132, + 157.0, + 477, + 233.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 132, + 233.0, + 477, + 309.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 326, + 505, + 415 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 328, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 478, + 339 + ], + "score": 1.0, + "content": "Figure 2: Results of Smoothie, DDPG, and TRPO on continuous control benchmarks. The", + "type": "text" + }, + { + "bbox": [ + 478, + 329, + 484, + 338 + ], + "score": 0.28, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 328, + 505, + 339 + ], + "score": 1.0, + "content": "-axis", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "is in millions of environment steps. Each plot shows the average reward and standard deviation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "clipped at the min and max of six randomly seeded runs after choosing best hyperparameters. We", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "see that Smoothie is competitive with DDPG even when DDPG uses a hyperparameter-tuned noise", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "scale, and Smoothie learns the optimal noise scale (the covariance) during training. Moreoever, we", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "observe significant advantages in terms of final reward performance, especially in the more difficult", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "tasks like Hopper, Walker2d, and Humanoid. Across all tasks, TRPO is not sufficiently sample-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 404, + 278, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 278, + 416 + ], + "score": 1.0, + "content": "efficient to provide a competitive baseline.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "We compare the results of Smoothie and DDPG in Figure 2. For each task we performed a hyper-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "parameter search over actor learning rate, critic learning rate and reward scale, and plot the average", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "of six runs for the best hyperparameters. For DDPG we extended the hyperparameter search to also", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "consider the scale and damping of exploratory noise provided by the Ornstein-Uhlenbeck process.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "Smoothie, on the other hand, contains an additional hyperparameter to determine the weight on", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 543, + 158, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 158, + 559 + ], + "score": 1.0, + "content": "KL-penalty.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "Despite DDPG having the advantage of its exploration decided by a hyperparameter search while", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "Smoothie must learn its exploration without supervision, we find that Smoothie performs competi-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 582, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 597 + ], + "score": 1.0, + "content": "tively or better across all tasks, exhibiting a slight advantage in Swimmer and Ant, while showing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "more dramatic improvements in Hopper, Walker2d, and Humanoid. The improvement is especially", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "dramatic for Hopper, where the average reward is doubled. We also highlight the results for Hu-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "manoid, which as far as we know, are the best published results for a method that only trains on the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "order of millions of environment steps. In contrast, TRPO, which to the best of our knowledge is the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "only other algorithm which can achieve better performance, requires on the order of tens of millions", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "of environment steps to achieve comparable reward. This gives added evidence to the benefits of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 660, + 410, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 410, + 672 + ], + "score": 1.0, + "content": "using a learnable covariance and not restricting a policy to be deterministic.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Empirically, we found the introduction of a KL-penalty to improve performance of Smoothie, es-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "pecially on harder tasks. We present a comparison of results of Smoothie with and without the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "KL-penalty on the four harder tasks in Figure 3. A KL-penalty to encourage stability is not possible", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "in DDPG. Thus, our algorithm provides a much needed solution to the inherent instability in DDPG", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 719, + 143, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 143, + 735 + ], + "score": 1.0, + "content": "training.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 132, + 81, + 477, + 309 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 132, + 81, + 477, + 309 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 81, + 477, + 309 + ], + "spans": [ + { + "bbox": [ + 132, + 81, + 477, + 309 + ], + "score": 0.977, + "type": "image", + "image_path": "f21887a3716c42d41ffbbba35b3c52cf13877d7b56bb86c24b7339f4a95daf67.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 132, + 81, + 477, + 157.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 132, + 157.0, + 477, + 233.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 132, + 233.0, + 477, + 309.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 326, + 505, + 415 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 328, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 478, + 339 + ], + "score": 1.0, + "content": "Figure 2: Results of Smoothie, DDPG, and TRPO on continuous control benchmarks. The", + "type": "text" + }, + { + "bbox": [ + 478, + 329, + 484, + 338 + ], + "score": 0.28, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 328, + 505, + 339 + ], + "score": 1.0, + "content": "-axis", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "is in millions of environment steps. Each plot shows the average reward and standard deviation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "clipped at the min and max of six randomly seeded runs after choosing best hyperparameters. We", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "see that Smoothie is competitive with DDPG even when DDPG uses a hyperparameter-tuned noise", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "scale, and Smoothie learns the optimal noise scale (the covariance) during training. Moreoever, we", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "observe significant advantages in terms of final reward performance, especially in the more difficult", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "tasks like Hopper, Walker2d, and Humanoid. Across all tasks, TRPO is not sufficiently sample-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 404, + 278, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 278, + 416 + ], + "score": 1.0, + "content": "efficient to provide a competitive baseline.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "We compare the results of Smoothie and DDPG in Figure 2. For each task we performed a hyper-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "parameter search over actor learning rate, critic learning rate and reward scale, and plot the average", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "of six runs for the best hyperparameters. For DDPG we extended the hyperparameter search to also", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "consider the scale and damping of exploratory noise provided by the Ornstein-Uhlenbeck process.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "Smoothie, on the other hand, contains an additional hyperparameter to determine the weight on", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 543, + 158, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 158, + 559 + ], + "score": 1.0, + "content": "KL-penalty.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 488, + 506, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "Despite DDPG having the advantage of its exploration decided by a hyperparameter search while", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "Smoothie must learn its exploration without supervision, we find that Smoothie performs competi-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 582, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 597 + ], + "score": 1.0, + "content": "tively or better across all tasks, exhibiting a slight advantage in Swimmer and Ant, while showing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "more dramatic improvements in Hopper, Walker2d, and Humanoid. The improvement is especially", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "dramatic for Hopper, where the average reward is doubled. We also highlight the results for Hu-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "manoid, which as far as we know, are the best published results for a method that only trains on the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "order of millions of environment steps. In contrast, TRPO, which to the best of our knowledge is the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "only other algorithm which can achieve better performance, requires on the order of tens of millions", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "of environment steps to achieve comparable reward. 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We observe benefits of using a proximal policy optimization method, especially", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "score": 1.0, + "content": "in Hopper and Humanoid, where the performance improvement is significant without sacrificing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 238, + 181, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 181, + 253 + ], + "score": 1.0, + "content": "sample efficiency.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 270, + 195, + 283 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 197, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 197, + 286 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 294, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 294, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 280, + 308 + ], + "score": 1.0, + "content": "We have presented a new Q-value function,", + "type": "text" + }, + { + "bbox": [ + 281, + 294, + 295, + 307 + ], + "score": 0.9, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 295, + 505, + 308 + ], + "score": 1.0, + "content": ", that is a Gaussian-smoothed version of the standard", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 306, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 181, + 321 + ], + "score": 1.0, + "content": "expected Q-value,", + "type": "text" + }, + { + "bbox": [ + 181, + 308, + 195, + 319 + ], + "score": 0.88, + "content": "Q ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 307, + 296, + 321 + ], + "score": 1.0, + "content": ". The advantage of using", + "type": "text" + }, + { + "bbox": [ + 296, + 306, + 311, + 320 + ], + "score": 0.91, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 307, + 331, + 321 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 332, + 308, + 346, + 319 + ], + "score": 0.89, + "content": "Q ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "is that its gradient and Hessian possess", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "an intimate relationship with the gradient of expected reward with respect to mean and covariance", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "of a Gaussian policy. The resulting algorithm, Smoothie, is able to successfully learn both mean", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "and covariance during training, leading to performance that can match or surpass that of DDPG,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 351, + 421, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 421, + 366 + ], + "score": 1.0, + "content": "especially when incorporating a penalty on divergence from a previous policy.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 505, + 427 + ], + "lines": [ + { + "bbox": [ + 105, + 368, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 169, + 384 + ], + "score": 1.0, + "content": "The success of", + "type": "text" + }, + { + "bbox": [ + 169, + 369, + 183, + 381 + ], + "score": 0.91, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 369, + 420, + 384 + ], + "score": 1.0, + "content": "is encouraging. Intuitively it may be argued that learning", + "type": "text" + }, + { + "bbox": [ + 421, + 368, + 435, + 382 + ], + "score": 0.9, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 369, + 505, + 384 + ], + "score": 1.0, + "content": "is more sensible", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 381, + 504, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 160, + 393 + ], + "score": 1.0, + "content": "than learning", + "type": "text" + }, + { + "bbox": [ + 161, + 381, + 175, + 393 + ], + "score": 0.88, + "content": "Q ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 381, + 504, + 393 + ], + "score": 1.0, + "content": ". The smoothed Q-values by definition make the true reward surface smoother, thus", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "score": 1.0, + "content": "possibly easier to learn; moreover the smoothed Q-values have a more direct relationship with the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "expected discounted return objective. 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We encourage future work to further investigate these claims", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 414, + 470, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 351, + 428 + ], + "score": 1.0, + "content": "as well as techniques to apply the underlying motivations for", + "type": "text" + }, + { + "bbox": [ + 351, + 414, + 366, + 426 + ], + "score": 0.91, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 414, + 470, + 428 + ], + "score": 1.0, + "content": "to other types of policies.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 368, + 506, + 428 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 444, + 175, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 176, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 176, + 457 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 504, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 477 + ], + "score": 1.0, + "content": "Mohammad Gheshlaghi Azar, Vicenc¸ Gomez, and Hilbert J Kappen. 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We make use of the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 144, + 142, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 142, + 156 + ], + "score": 1.0, + "content": "fact that", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 152, + 415, + 177 + ], + "lines": [ + { + "bbox": [ + 196, + 152, + 415, + 177 + ], + "spans": [ + { + "bbox": [ + 196, + 152, + 415, + 177 + ], + "score": 0.92, + "content": "\\frac { \\partial } { \\partial A } | A | ^ { - 1 / 2 } = - \\frac { 1 } { 2 } | A | ^ { - 3 / 2 } \\frac { \\partial } { \\partial A } | A | = - \\frac { 1 } { 2 } | A | ^ { - 1 / 2 } A ^ { - 1 } ,", + "type": "interline_equation", + "image_path": "abe39c2a97779def484e4eba142c60209fd9e9f36be5544140f1ba14e1bd2e4a.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 196, + 152, + 415, + 177 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 177, + 194, + 189 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 194, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 182, + 190 + ], + "score": 1.0, + "content": "and for symmetric", + "type": "text" + }, + { + "bbox": [ + 182, + 178, + 190, + 187 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 176, + 194, + 190 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 187, + 369, + 211 + ], + "lines": [ + { + "bbox": [ + 242, + 187, + 369, + 211 + ], + "spans": [ + { + "bbox": [ + 242, + 187, + 369, + 211 + ], + "score": 0.93, + "content": "\\frac \\partial { \\partial A } | | \\boldsymbol { v } | | _ { A ^ { - 1 } } ^ { 2 } = - A ^ { - 1 } { \\boldsymbol { v } } { \\boldsymbol { v } } ^ { T } A ^ { - 1 } .", + "type": "interline_equation", + "image_path": "18034b9d4af3f0775f1fdad3b9f4dd9d28f0abf8f059744b1cc914ba9d87feaf.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 242, + 187, + 369, + 211 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 212, + 448, + 225 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 449, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 143, + 226 + ], + "score": 1.0, + "content": "We omit", + "type": "text" + }, + { + "bbox": [ + 143, + 215, + 149, + 222 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 212, + 172, + 226 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 172, + 213, + 192, + 225 + ], + "score": 0.92, + "content": "\\Sigma ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 212, + 449, + 226 + ], + "score": 1.0, + "content": "in the following equations for succinctness. The LHS of (15) is", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 230, + 494, + 314 + ], + "lines": [ + { + "bbox": [ + 117, + 230, + 494, + 314 + ], + "spans": [ + { + "bbox": [ + 117, + 230, + 494, + 314 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle \\int _ { A } Q ^ { \\pi } ( s , \\tilde { a } ) \\frac { \\partial } { \\partial \\Sigma } N ( \\tilde { a } | a , \\Sigma ) \\mathrm { d } \\tilde { a } } \\\\ { \\displaystyle = \\int _ { A } Q ^ { \\pi } ( s , \\tilde { a } ) \\exp \\left\\{ - \\frac { 1 } { 2 } | | \\tilde { a } - a | | _ { \\Sigma ^ { - 1 } } ^ { 2 } \\right\\} \\left( \\frac { \\partial } { \\partial \\Sigma } | 2 \\pi \\Sigma | ^ { - 1 / 2 } - \\frac { 1 } { 2 } | 2 \\pi \\Sigma | ^ { - 1 / 2 } \\frac { \\partial } { \\partial \\Sigma } | | \\tilde { a } - a | | _ { \\Sigma ^ { - 1 } } ^ { 2 } \\right) \\mathrm { d } \\tilde { a } } \\\\ { \\displaystyle 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0 } } ( s ) + ( a - \\mu _ { \\theta } ( s ) ) ^ { T } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) ^ { T } w _ { 1 } + ( a - \\mu _ { \\theta } ( s ) ) ^ { T } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) ^ { T } w _ { 2 } ( a - \\mu _ { \\theta } ( s ) ) . } \\end{array}", + "type": "interline_equation", + "image_path": "b8e0b8ed230238272a76a4d3fc82242545e3918890d0fbb8ad9f69a4b959e7a1.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 110, + 718, + 483, + 734 + ], + "spans": [], + "index": 30 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 166, + 80, + 442, + 226 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 166, + 80, + 442, + 226 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 80, + 442, + 226 + ], + "spans": [ + { + "bbox": [ + 166, + 80, + 442, + 226 + ], + "score": 0.965, + "html": "
HyperparameterRangeSampling
actor learningrate critic learning rate[1e-6,1e-3] [1e-6,1e-3]log
reward scale[0.01,0.3]l0g
OU damping[1e-4,1e-3]l0g
OU stddev10g
入[1e-3,1.0]log
discount factor[1e-6, 4e-2] 0.995log
target network lag0.01fixed
batch size128fixed
clipping on gradients of Qfixed
4.0fixed
num gradient updates per observation1fixed
Huber loss clipping1.0fixed
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(2014)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 298, + 356, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 356, + 312 + ], + "score": 1.0, + "content": "which includes a similar procedure for showing compatibility.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 397, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 397, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 346, + 330 + ], + "score": 1.0, + "content": "From the second condition for compatibility with respect to", + "type": "text" + }, + { + "bbox": [ + 347, + 316, + 360, + 328 + ], + "score": 0.9, + "content": "\\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 314, + 397, + 330 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 333, + 480, + 358 + ], + "lines": [ + { + "bbox": [ + 112, + 333, + 480, + 358 + ], + "spans": [ + { + "bbox": [ + 112, + 333, + 480, + 358 + ], + "score": 0.9, + "content": "\\int _ { S } \\left( \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } - \\nabla _ { a } ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\right) \\nabla _ { w } \\left( \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\right) \\mathrm { d } \\rho ^ { \\pi } ( s ) = 0 .", + "type": "interline_equation", + "image_path": "482a65447c1c827eb4abc41f71898558ca37abec243b140aaac7a7ec6b1504e3.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 112, + 333, + 480, + 358 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 317, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 317, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 317, + 376 + ], + "score": 1.0, + "content": "We may combine this with the first condition to find", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 380, + 468, + 407 + ], + "lines": [ + { + "bbox": [ + 124, + 380, + 468, + 407 + ], + "spans": [ + { + "bbox": [ + 124, + 380, + 468, + 407 + ], + "score": 0.93, + "content": "\\int _ { S } \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) = \\int _ { S } \\nabla _ { a } ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) ,", + "type": "interline_equation", + "image_path": "fead4ffc0f1712d604674150fb03c9293a85fb979a783002ea72c59f0f461aa1.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 124, + 380, + 468, + 389.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 124, + 389.0, + 468, + 398.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 124, + 398.0, + 468, + 407.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 411, + 295, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 296, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 296, + 425 + ], + "score": 1.0, + "content": "which is the desired property for compatibility.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 504, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 504, + 440 + ], + "score": 1.0, + "content": "While it is reassuring to know that there exists a class of function approximators which are compat-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "ible, this fact is largely ignored in practice. 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This problem is also present in DDPG, and we feel this issue merits additional investigation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 484, + 169, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 169, + 496 + ], + "score": 1.0, + "content": "in future work.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 512, + 269, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 270, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 270, + 527 + ], + "score": 1.0, + "content": "C IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 432, + 550 + ], + "score": 1.0, + "content": "We utilize feed forward networks for both policy and Q-value approximator. For", + "type": "text" + }, + { + "bbox": [ + 432, + 537, + 456, + 549 + ], + "score": 0.91, + "content": "\\mu _ { \\boldsymbol { \\theta } } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "we use two", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 548, + 504, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 408, + 563 + ], + "score": 1.0, + "content": "hidden layers of dimensions (400, 300) and relu activation functions. For", + "type": "text" + }, + { + "bbox": [ + 409, + 548, + 447, + 562 + ], + "score": 0.93, + "content": "\\tilde { Q } _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 549, + 466, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 466, + 550, + 504, + 562 + ], + "score": 0.92, + "content": "Q _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "we first embed the state into a 400 dimensional vector using a fully-connected layer and tanh non-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 337, + 584 + ], + "score": 1.0, + "content": "linearity. We then concatenate the embedded state with", + "type": "text" + }, + { + "bbox": [ + 337, + 574, + 344, + 582 + ], + "score": 0.69, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 572, + 504, + 584 + ], + "score": 1.0, + "content": "and pass the result through a 1-hidden", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 438, + 595 + ], + "score": 1.0, + "content": "layer neural network of dimension 300 with tanh activations. We use a diagonal", + "type": "text" + }, + { + "bbox": [ + 438, + 582, + 489, + 595 + ], + "score": 0.93, + "content": "\\Sigma _ { \\phi } ^ { \\bar { } } ( s ) = e ^ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 594, + 248, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 169, + 605 + ], + "score": 1.0, + "content": "Smoothie, with", + "type": "text" + }, + { + "bbox": [ + 170, + 594, + 177, + 605 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 595, + 231, + 605 + ], + "score": 1.0, + "content": "initialized to", + "type": "text" + }, + { + "bbox": [ + 231, + 594, + 245, + 604 + ], + "score": 0.79, + "content": "- 1", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 595, + 248, + 605 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "To find optimal hyperparameters we perform a 100-trial random search over the hyperparameters", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 438, + 635 + ], + "score": 1.0, + "content": "specified in Table 1. The OU exploration parameters only apply to DDPG. The", + "type": "text" + }, + { + "bbox": [ + 438, + 622, + 445, + 632 + ], + "score": 0.77, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "coefficient on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 632, + 334, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 334, + 645 + ], + "score": 1.0, + "content": "KL-penalty only applies to Smoothie with a KL-penalty.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + } + ], + "page_idx": 11, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "12", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 412, + 504, + 421 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 166, + 80, + 442, + 226 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 166, + 80, + 442, + 226 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 80, + 442, + 226 + ], + "spans": [ + { + "bbox": [ + 166, + 80, + 442, + 226 + ], + "score": 0.965, + "html": "
HyperparameterRangeSampling
actor learningrate critic learning rate[1e-6,1e-3] [1e-6,1e-3]log
reward scale[0.01,0.3]l0g
OU damping[1e-4,1e-3]l0g
OU stddev10g
入[1e-3,1.0]log
discount factor[1e-6, 4e-2] 0.995log
target network lag0.01fixed
batch size128fixed
clipping on gradients of Qfixed
4.0fixed
num gradient updates per observation1fixed
Huber loss clipping1.0fixed
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We shall show how the conditions stated for compatibility with respect to", + "type": "text" + }, + { + "bbox": [ + 433, + 277, + 447, + 289 + ], + "score": 0.89, + "content": "\\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 277, + 504, + 289 + ], + "score": 1.0, + "content": "are sufficient.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 180, + 300 + ], + "score": 1.0, + "content": "The reasoning for", + "type": "text" + }, + { + "bbox": [ + 180, + 290, + 191, + 299 + ], + "score": 0.84, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "follows via a similar argument. We also refer the reader to Silver et al. (2014)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 298, + 356, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 356, + 312 + ], + "score": 1.0, + "content": "which includes a similar procedure for showing compatibility.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 277, + 505, + 312 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 397, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 397, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 346, + 330 + ], + "score": 1.0, + "content": "From the second condition for compatibility with respect to", + "type": "text" + }, + { + "bbox": [ + 347, + 316, + 360, + 328 + ], + "score": 0.9, + "content": "\\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 314, + 397, + 330 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 314, + 397, + 330 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 333, + 480, + 358 + ], + "lines": [ + { + "bbox": [ + 112, + 333, + 480, + 358 + ], + "spans": [ + { + "bbox": [ + 112, + 333, + 480, + 358 + ], + "score": 0.9, + "content": "\\int _ { S } \\left( \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } - \\nabla _ { a } ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\right) \\nabla _ { w } \\left( \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\right) \\mathrm { d } \\rho ^ { \\pi } ( s ) = 0 .", + "type": "interline_equation", + "image_path": "482a65447c1c827eb4abc41f71898558ca37abec243b140aaac7a7ec6b1504e3.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 112, + 333, + 480, + 358 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 317, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 317, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 317, + 376 + ], + "score": 1.0, + "content": "We may combine this with the first condition to find", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 363, + 317, + 376 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 380, + 468, + 407 + ], + "lines": [ + { + "bbox": [ + 124, + 380, + 468, + 407 + ], + "spans": [ + { + "bbox": [ + 124, + 380, + 468, + 407 + ], + "score": 0.93, + "content": "\\int _ { S } \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) = \\int _ { S } \\nabla _ { a } ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) ,", + "type": "interline_equation", + "image_path": "fead4ffc0f1712d604674150fb03c9293a85fb979a783002ea72c59f0f461aa1.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 124, + 380, + 468, + 389.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 124, + 389.0, + 468, + 398.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 124, + 398.0, + 468, + 407.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 411, + 295, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 296, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 296, + 425 + ], + "score": 1.0, + "content": "which is the desired property for compatibility.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 410, + 296, + 425 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 504, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 504, + 440 + ], + "score": 1.0, + "content": "While it is reassuring to know that there exists a class of function approximators which are compat-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "ible, this fact is largely ignored in practice. Not only is the class of compatible functions heavily", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 393, + 464 + ], + "score": 1.0, + "content": "restricted in terms of expressiveness, due to the first set of conditions for", + "type": "text" + }, + { + "bbox": [ + 393, + 451, + 421, + 462 + ], + "score": 0.86, + "content": "\\mu _ { \\theta } , \\Sigma _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 451, + 505, + 464 + ], + "score": 1.0, + "content": ", it is also impossible", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "to satisfy the second set of conditions without access to derivative and Hessian information of the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 472, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 124, + 487 + ], + "score": 1.0, + "content": "true", + "type": "text" + }, + { + "bbox": [ + 124, + 472, + 138, + 485 + ], + "score": 0.9, + "content": "{ \\tilde { Q } } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 472, + 505, + 487 + ], + "score": 1.0, + "content": ". This problem is also present in DDPG, and we feel this issue merits additional investigation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 484, + 169, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 169, + 496 + ], + "score": 1.0, + "content": "in future work.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 429, + 505, + 496 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 512, + 269, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 270, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 270, + 527 + ], + "score": 1.0, + "content": "C IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 432, + 550 + ], + "score": 1.0, + "content": "We utilize feed forward networks for both policy and Q-value approximator. For", + "type": "text" + }, + { + "bbox": [ + 432, + 537, + 456, + 549 + ], + "score": 0.91, + "content": "\\mu _ { \\boldsymbol { \\theta } } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "we use two", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 548, + 504, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 408, + 563 + ], + "score": 1.0, + "content": "hidden layers of dimensions (400, 300) and relu activation functions. For", + "type": "text" + }, + { + "bbox": [ + 409, + 548, + 447, + 562 + ], + "score": 0.93, + "content": "\\tilde { Q } _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 549, + 466, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 466, + 550, + 504, + 562 + ], + "score": 0.92, + "content": "Q _ { w } ^ { \\pi } ( s , a )", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "we first embed the state into a 400 dimensional vector using a fully-connected layer and tanh non-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 337, + 584 + ], + "score": 1.0, + "content": "linearity. We then concatenate the embedded state with", + "type": "text" + }, + { + "bbox": [ + 337, + 574, + 344, + 582 + ], + "score": 0.69, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 572, + 504, + 584 + ], + "score": 1.0, + "content": "and pass the result through a 1-hidden", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 438, + 595 + ], + "score": 1.0, + "content": "layer neural network of dimension 300 with tanh activations. We use a diagonal", + "type": "text" + }, + { + "bbox": [ + 438, + 582, + 489, + 595 + ], + "score": 0.93, + "content": "\\Sigma _ { \\phi } ^ { \\bar { } } ( s ) = e ^ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 594, + 248, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 169, + 605 + ], + "score": 1.0, + "content": "Smoothie, with", + "type": "text" + }, + { + "bbox": [ + 170, + 594, + 177, + 605 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 595, + 231, + 605 + ], + "score": 1.0, + "content": "initialized to", + "type": "text" + }, + { + "bbox": [ + 231, + 594, + 245, + 604 + ], + "score": 0.79, + "content": "- 1", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 595, + 248, + 605 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 537, + 505, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "To find optimal hyperparameters we perform a 100-trial random search over the hyperparameters", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 438, + 635 + ], + "score": 1.0, + "content": "specified in Table 1. The OU exploration parameters only apply to DDPG. The", + "type": "text" + }, + { + "bbox": [ + 438, + 622, + 445, + 632 + ], + "score": 0.77, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "coefficient on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 632, + 334, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 334, + 645 + ], + "score": 1.0, + "content": "KL-penalty only applies to Smoothie with a KL-penalty.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 610, + 506, + 645 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/B1nLkl-0Z/B1nLkl-0Z_model.json b/parse/train/B1nLkl-0Z/B1nLkl-0Z_model.json new file mode 100644 index 0000000000000000000000000000000000000000..e6601ea94db3ce0798800c473456cb2d8f6fea24 --- /dev/null +++ b/parse/train/B1nLkl-0Z/B1nLkl-0Z_model.json @@ -0,0 +1,18993 @@ +[ + { + 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Algorithm1Smoothie Input: Environment ENV,learning rates 7π, nQ,discount factor y,KL-penalty 入,l
number of training steps N, target network lag T. Initialize0,,w,set0'=0,Φ'=,w'=w.
fori=OtoN-1do
/ Collect experience
Sample action a ~ N(μe(s),∑(s)) and apply to ENV to yield r and s'. Insert transition (s,a,r,s') to replay buffer.
// Train μ,∑
Sample batch{(k,@k,Tk,S)}1 fromreplay buffer aQ(ska) Compute gradients gk =
da la=μe(sk)*
²(s@） Compute Hessians Hk = a² la=μe(sk)*
Compute KL-penalties KLk =KL(μθ,Σ𝜙llμe,Σ).
Compute updates 1B
1B
△=B 1 ∑k=1 Updateθ←θ+ηπ△0,←+ηπ△.
// Train Qπ
Sample batch {(sk,k,Tk,S)}1 from replay bufer. Sample phantom actions ak ~ N(ák,Σ(sk)). Compute lossL(u)=∑1((s)-r-qQ(s,ue()2.
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HyperparameterRangeSampling
actor learningrate critic learning rate[1e-6,1e-3] [1e-6,1e-3]log
reward scale[0.01,0.3]l0g
OU damping[1e-4,1e-3]l0g
OU stddev10g
入[1e-3,1.0]log
discount factor[1e-6, 4e-2] 0.995log
target network lag0.01fixed
batch size128fixed
clipping on gradients of Qfixed
4.0fixed
num gradient updates per observation1fixed
Huber loss clipping1.0fixed
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end of file diff --git a/parse/train/BJe-DsC5Fm/BJe-DsC5Fm.md b/parse/train/BJe-DsC5Fm/BJe-DsC5Fm.md new file mode 100644 index 0000000000000000000000000000000000000000..560f5883c785dfd1ced807042a415f8aa5458df1 --- /dev/null +++ b/parse/train/BJe-DsC5Fm/BJe-DsC5Fm.md @@ -0,0 +1,645 @@ +# SIGNSGD VIA ZEROTH-ORDER ORACLE + +Sijia Liu† Pin-Yu Chen† Xiangyi Chen‡ Mingyi Hong‡ †MIT-IBM Watson AI Lab, IBM Research ‡University of Minnesota, Twin Cities + +# ABSTRACT + +In this paper, we design and analyze a new zeroth-order (ZO) stochastic optimization algorithm, ZO-signSGD, which enjoys dual advantages of gradient-free operations and signSGD. The latter requires only the sign information of gradient estimates but is able to achieve a comparable or even better convergence speed√ than SGD-type algorithms. Our study shows that ZO-signSGD requires √ √ $\sqrt { d }$ times more iterations than signSGD, leading to a convergence rate of $O ( { \sqrt { d } } / { \sqrt { T } } )$ under some mild conditions, where $d$ is the number of optimization variables, and $T$ is the number of iterations. In addition, we analyze the effects of different types of gradient estimators on the convergence of ZO-signSGD, and propose several variants of ZO-signSGD with $O ( { \sqrt { d } } { \bar { / } } { \sqrt { T } } )$ convergence rate. On the application side we explore the connection between ZO-signSGD and black-box adversarial attacks in robust deep learning. Our empirical evaluations on image classification datasets MNIST and CIFAR-10 demonstrate the superior performance of ZO-signSGD on the generation of adversarial examples from black-box neural networks. + +# 1 INTRODUCTION + +Zeroth-order (gradient-free) optimization has attracted an increasing amount of attention for solving machine learning (ML) problems in scenarios where explicit expressions for the gradients are difficult or infeasible to obtain. One recent application of great interest is to generate prediction-evasive adversarial examples, e.g., crafted images with imperceptible perturbations to deceive a well-trained image classifier into misclassification. However, the black-box optimization nature limits the practical design of adversarial examples, where internal configurations and operating mechanism of public ML systems (e.g., Google Cloud Vision API) are not revealed to practitioners and the only mode of interaction with the system is via submitting inputs and receiving the corresponding predicted outputs (Papernot et al., 2017; Liu et al., 2017; Chen et al., 2017; Tu et al., 2018; Ilyas et al., 2018b; Cheng et al., 2018; Bhagoji et al., 2018). It was observed in both white-box and black-box settings1 that simply leveraging the sign information of gradient estimates of an attacking loss can achieve superior empirical performance in generating adversarial examples (Goodfellow et al., 2015; Madry et al., 2018; Ilyas et al., 2018a). Spurred by that, this paper proposes a zeroth-order (ZO) sign-based descent algorithm (we call it ‘ZO-signSGD’) for solving black-box optimization problems, e.g. design of black-box adversarial examples. The convergence behavior and algorithmic stability of the proposed ZO-signSGD algorithm are carefully studied in both theory and practice. + +In the first-order setting, a sign-based stochastic gradient descent method, known as signSGD, was analyzed by (Bernstein et al., 2018; Balles & Hennig, 2017). It was shown in (Bernstein et al., 2018) that signSGD not only reduces the per iteration cost of communicating gradients, but also could yield a faster empirical convergence speed than SGD (Kinga & Adam, 2015). That is because although the sign operation compresses the gradient using a single bit, it could mitigate the negative effect of√ extremely components of gradient noise. Theoretically, signSGD achieves $O ( 1 / \sqrt { T } )$ convergence rate under the condition of a sufficiently large mini-batch size, where $T$ denotes the total number of iterations. The work in (Balles & Hennig, 2017) established a connection between signSGD and Adam with restrictive convex analysis. Prior to (Bernstein et al., 2018; Balles & Hennig, 2017), although signSGD was not formally defined, the fast gradient sign method (Goodfellow et al., 2015) to generate white-box adversarial examples actually obeys the algorithmic protocol of signSGD. The effectiveness of signSGD has been witnessed by robust adversarial training of deep neural networks (DNNs) (Madry et al., 2018). Given the advantages of signSGD, one may wonder if it can be generalized for ZO optimization and what the corresponding convergence rate is. In this paper, we answer these questions affirmatively. + +Contributions We summarize our key contributions as follows. + +• We propose a new ZO algorithm, ‘ZO-signSGD’, and rigorously prove its convergence rate√ √ of $O ( { \sqrt { d } } / { \sqrt { T } } )$ under mild conditions. Our established convergence analysis applies to both mini-batch sampling schemes with and without replacement. In particular, the ZO sign-based gradient descent algorithm can be treated as a special case in our proposed ZO-signSGD algorithm. We carefully study the effects of different types of gradient estimators on the convergence of ZO-signSGD, and propose three variants of ZO-signSGD for both centralized and distributed ZO optimization. +• We conduct extensive synthetic experiments to thoroughly benchmark the performance of ZO-signSGD and to investigate its parameter sensitivity. We also demonstrate the superior performance of ZO-signSGD for generating adversarial examples from black-box DNNs. + +Related work Other types of ZO algorithms have been developed for convex and nonconvex optimization, where the full gradient is approximated via a random or deterministic gradient estimate (Jamieson et al., 2012; Nesterov & Spokoiny, 2015; Ghadimi & Lan, 2013; Duchi et al., 2015; Gao et al., 2014; Shamir, 2017; Hajinezhad et al., 2017; Ghadimi et al., 2016; Lian et al., 2016; Liu et al., 2018b;c; Wang et al., 2018). Examples include ZO-SGD (Ghadimi & Lan, 2013), ZO stochastic coordinate descent (ZO-SCD) (Lian et al., 2016), and ZO stochastic variance reduced gradient descent (ZO-SVRG) (Liu et al., 2018c;a; Gu et al., 2016). Both ZO-SGD and ZO-SCD can√ √ achieve $O ( { \sqrt { d } } / { \sqrt { T } } )$ convergence rate. And ZO-SVRG can further improve the iteration complexity to $O ( d / T )$ but suffers from an increase of function query complexity due to the additional variance reduced step, known as ‘gradient blending’ (Liu et al., 2018c), compared to ZO-SGD. The existing work showed that ZO algorithms align with the iteration complexity of their first-order counterparts up to a slowdown effect in terms of a small-degree polynomial of the problem size $d$ . + +# 2 SIGNSGD & ITS CONNECTION TO ADVERSARIAL MACHINE LEARNING + +In this section, we provide a background on signSGD, together with the problem setup of our interest. In particular, we show that the commonly-used methods for generating adversarial attacks fall into the framework of signSGD. + +Preliminaries on signSGD Consider a nonconvex finite-sum problem of the form + +$$ +\begin{array} { r l } { \underset { \mathbf { x } } { \mathrm { m i n i m i z e } } } & { { } f ( \mathbf { x } ) : = ( 1 / n ) \sum _ { i = 1 } ^ { n } f _ { i } ( \mathbf { x } ) , } \end{array} +$$ + +where $\mathbf { x } \in \mathbb { R } ^ { d }$ are optimization variables, and $\{ f _ { i } \}$ are $n$ individual nonconvex cost functions. The finite-sum form (1) encompasses many ML problems, ranging from generalized linear models to neural networks. If the gradients of $\{ f _ { i } \}$ are available, then problem (1) can be solved by many first-order methods such as SGD, SCD, and signSGD. The method of our interest is signSGD, which differs from SGD and SCD, takes the sign of gradient (or its estimate) as the descent direction. It was recently shown in (Bernstein et al., 2018) that signSGD is quite robust to gradient noise and yields fast empirical convergence. + +Algorithm 1 provides a generic sign-based gradient descent framework that encapsulates different variants of signSGD. In Algorithm 1, GradEstimate(·) signifies a general gradient estimation procedure, which adopts either a stochastic gradient estimate in the first-order setting (Bernstein et al., 2018) or a function difference based random gradient estimate in the ZO setting (Nesterov & Spokoiny, 2015; Duchi et al., 2015). We call the ZO variant of signSGD ‘ZO-signSGD’, which will be elaborated on in Sec. 3. + +Adversarial attacks meet signSGD It is now widely known that ML models (e.g., deep neural networks) are vulnerable to adversarial attacks, which craft inputs (e.g., images) with imperceptible perturbations to cause incorrect classification (Szegedy et al., 2013; Goodfellow et al., 2015; Kurakin et al., 2017; Lin et al., 2019). The resulting inputs crafted by adversaries are known as adversarial examples. Investigating adversarial examples not only helps to understand the limitation of learning models, but also provides opportunities to improve the models’ robustness (Papernot et al., 2016; + +# Algorithm 1 Generic sign-based gradient descent + +1: Input: learning rate $\{ \delta _ { k } \}$ , initial value $\mathbf { x } _ { \mathrm { 0 } }$ , and number of iterations $T$ +2: for $k = 0 , 1 , \ldots , T - 1$ do +3: $\hat { \mathbf { g } } _ { k } \longleftarrow \mathrm { G r a d E s t i m a t e } ( \mathbf { x } _ { k } )$ # applies to both first and zeroth order gradient estimates +4: sign-gradient update + +$$ +\begin{array} { r } { { \bf x } _ { k + 1 } = { \bf x } _ { k } - \delta _ { k } \mathrm { s i g n } ( \hat { \bf g } _ { k } ) , \quad \mathrm { w h e r e ~ s i g n ( { \bf x } ) ~ t a k e s ~ e l e m e n t - w i s e ~ s i g n s ~ o f ~ { \bf x } ~ } } \end{array} +$$ + +5: end for + +Athalye et al., 2018; Madry et al., 2018). In what follows, we show that the generation of adversarial examples in (Goodfellow et al., 2015; Kurakin et al., 2017) can be interpreted through signSGD. + +Let $\mathbf { x } _ { \mathrm { 0 } }$ denote the natural (legitimate) input of an ML model associated with the true label $t _ { 0 }$ , and $\mathbf { x } ^ { \prime } = \mathbf { x } _ { 0 } + \boldsymbol { \delta }$ be the adversarial example to be designed, where $\pmb { \delta }$ are adversarial perturbations. If $f ( \mathbf { x } , t _ { 0 } )$ is the training loss of a learning model, then the goal of (white-box) adversarial attack is to find minimal perturbation $\delta$ that is sufficient to mislead the learning model, namely, to maximize the loss $f ( \mathbf { x } _ { 0 } ^ { \mathbf { \hat { \alpha } } } + \mathbf { \delta } \mathbf { \delta } , t _ { 0 } )$ . Taking the first-order approximation of $\bar { f ( \mathbf { x } ^ { \prime } , t _ { 0 } ) }$ around $\mathbf { x } _ { \mathrm { 0 } }$ , we obtain $f ( \mathbf { x } ^ { \prime } , t _ { 0 } ) \approx f ( \mathbf { x } _ { 0 } , t _ { 0 } ) + \langle \nabla _ { x } f ( \mathbf { x } _ { 0 } , t _ { 0 } ) , \delta \rangle$ . By constraining the strength of perturbation in the $\ell _ { \infty }$ ball of small radius $\epsilon$ (i.e., $\| \pmb { \delta } \| _ { \infty } \leq \epsilon )$ , the linear approximation of $f ( \mathbf { x } ^ { \prime } , t _ { 0 } )$ is then maximized at $\pmb { \delta } = \epsilon \mathrm { s i g n } ( \nabla _ { \pmb { x } } f ( \mathbf { x } _ { 0 } , t _ { 0 } ) )$ (Shaham et al., 2018). Therefore, generation of adversarial examples proposed in (Goodfellow et al., 2015) obeys the sign-gradient update rule in (2), + +$$ +\mathbf { x } ^ { \prime } = \mathbf { x } _ { 0 } - \epsilon \mathrm { s i g n } ( - \nabla _ { x } f ( \mathbf { x } _ { 0 } , t _ { 0 } ) ) . +$$ + +Such a connection between adversarial example generation and signSGD also holds in other attacks, e.g., the iterative target attack method (Kurakin et al., 2017). Similarly, a so-called black-box attack (Ilyas et al., 2018a; Bhagoji et al., 2018) is associated with our proposed ZO-signSGD algorithm. + +# 3 ZO-SIGNSGD FOR BLACK-BOX OPTIMIZATION + +One limitation of signSGD (Bernstein et al., 2018) is the need of first-order information, i.e., stochastic gradients. However, there exists a large practical demand for solving ML problems where explicit expressions of the gradients are difficult or infeasible to obtain, e.g., the generation of adversarial examples from black-box neural networks as discussed in Sec. 1 and 2. + +Gradient estimation via ZO oracle In the ZO setting where the first-order information is unavailable, the gradient estimator at Step 3 of Algorithm 1 has only access to function values of $\{ f _ { i } ( \mathbf { x } ) \}$ given a query point x. Based on that, we construct a ZO gradient estimate through a forward difference of two function values (Nesterov & Spokoiny, 2015; Gao et al., 2014; Duchi et al., 2015). In Algorithm 1, GradEstimate $\left( \mathbf { x } \right)$ is then specified as + +$$ +\mathrm { G r a d E s t i m a t e } ( \mathbf { x } ) = \frac { 1 } { b q } \sum _ { i \in \mathcal { T } _ { k } } \sum _ { j = 1 } ^ { q } \hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } _ { i , j } ) , \hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } _ { i , j } ) : = \frac { d \big [ f _ { i } ( \mathbf { x } + \mu \mathbf { u } _ { i , j } ) - f _ { i } ( \mathbf { x } ) \big ] } { \mu } \mathbf { u } _ { i , j } , +$$ + +where $\mathbf { x } = \mathbf { x } _ { k }$ in Algorithm 1, $\mathcal { T } _ { k }$ is a mini-batch of size $\left| \mathcal { T } _ { k } \right| = b$ , $\{ \mathbf { u } _ { i , j } \} _ { j = 1 } ^ { q }$ are i.i.d. random directions drawn from a uniform distribution over a unit sphere, and $\hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } _ { i , j } )$ gives a two-point based random gradient estimate with direction $\mathbf { u } _ { i , j }$ and smoothing parameter $\mu > 0$ . We remark that the random direction vectors in (3) can also be drawn from the standard Gaussian distribution (Nesterov & Spokoiny, 2015). However, the uniform distribution could be more useful in practice since it is defined in a bounded space rather than the whole real space required for Gaussian. + +We highlight that unlike the first-order stochastic gradient estimate, the ZO gradient estimate (3) is a biased approximation to the true gradient of $f$ . Instead, it becomes unbiased to the gradient of the randomized smoothing function $\bar { f _ { \mu } }$ (Duchi et al., 2012; Gao et al., 2014), + +$$ +f _ { \mu } ( \mathbf { x } ) = \mathbb { E } _ { \mathbf { v } } [ f ( \mathbf { x } + \mu \mathbf { v } ) ] = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathbb { E } _ { \mathbf { v } } [ f _ { i } ( \mathbf { x } + \mu \mathbf { v } ) ] = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } f _ { i , \mu } ( \mathbf { x } ) , +$$ + +where $f _ { i , \mu }$ gives the randomized smoothing version of $f _ { i }$ , and the random variable $\mathbf { v }$ follows a uniform distribution over the unit Euclidean ball. Clearly, there exists a gap between a ZO gradient estimate and the true gradient of $f$ , but as will be evident later, such a gap can be measured through the smoothing function $f _ { \mu }$ . + +Motivations of ZO-signSGD. Compared to SGD-type methods, the fast empirical convergence of signSGD and ZO-signSGD has been shown in the application of generating white-box and black-box adversarial examples (Goodfellow et al., 2015; Madry et al., 2018; Ilyas et al., 2018a). As mentioned in (Bernstein et al., 2018), the sign operation could mitigate the negative effect of (coordinate-wise) gradient noise of large variance. Recall that the ZO gradient estimate is a biased approximation to the true gradient, and thus, could suffer from having larger noise variance than (first-order) stochastic gradients. In this context, one could benefit from ZO-signSGD due to its robustness to gradient noise. In Appendix 1, we provide two concrete examples (Fig. A1 and Fig. A2) to confirm the aforementioned analysis. In Fig. A1, we show the robustness of ZO-signSGD against sparse noise perturbation through a toy quadratic optimization problem, originally introduced in (Bernstein et al., 2018) to motivate the fast convergence of signSGD against SGD. In Fig. A2, we show that gradient estimation via ZO oracle indeed encounters gradient noise of large variance. Thus, taking the sign of a gradient estimate might scale down the extremely noisy components. + +ZO-signSGD & technical challenges beyond signSGD Algorithm 1 becomes ZO-signSGD as the ZO gradient estimate (3) is applied. We note that the extension from first order to ZO is nontrivial, as the proposed ZO-signSGD algorithm yields three key differences to signSGD. + +First, ZO-signSGD has milder assumption on the choice of mini-batch sampling. Recall that signSGD in (Bernstein et al., 2018) achieves $O ( 1 / \sqrt { T } )$ convergence rate given the condition that the mini-batch size is sufficiently large, $b = O ( T )$ . However, this condition only becomes true when the mini-batch sample is randomly selected from $[ n ]$ with replacement, which is unusual when $n \leq T$ . Here $[ n ]$ represents the integer set $\{ 1 , 2 , \ldots , n \}$ . And signSGD fails to cover signGD when $b = n$ , since sampling with replacement leads to $\mathcal { T } _ { k } \neq [ n ]$ even if $b = n$ . In the proposed ZO-signSGD algorithm, we will relax the assumption on mini-batch sampling. + +Second, in ZO-signSGD both the ZO gradient estimator and the sign operator give rise to approximation errors to the true gradient. Although the statistical properties of ZO gradient estimates can be acquired with the aid of the randomized smoothing function (4), the use of mini-batch sampling without replacement introduces extra difficulty to bound the variance of ZO gradient estimates since mini-batch samples are no longer independent. Moreover, the sign-based descent algorithm evaluates the convergence error in the $\ell _ { 1 }$ -norm geometry, leading to a mismatch with the $\ell _ { 2 }$ -norm based gradient variance. Besides translating the the gradient norm from $\ell _ { 1 }$ to $\ell _ { 2 }$ , the probabilistic convergence method (Ghadimi & Lan, 2013) is used to bound the eventual convergence error of ZO-signSGD. + +Finally, beyond the standard ZO gradient estimator (3), we will cover multiple variants of ZOsignSGD for centralized or distributed optimization. + +# 4 CONVERGENCE ANALYSIS OF ZO-SIGNSGD + +In this section, we begin by stating assumptions used in our analysis. We then derive the convergence rate of ZO-signSGD for nonconvex optimization. Assumptions of problem (1) are listed as follows. + +A1: Functions $\{ f _ { i } \}$ have $L$ -Lipschitz continuous gradients, where $L \in ( 0 , \infty )$ . + +A2: At time $k$ , the gradient of $f _ { i }$ is upper bounded by $\| \nabla f _ { i } ( \mathbf { x } _ { k } ) \| _ { 2 } \leq \sigma$ for $i \in [ n ]$ + +Both A1 and A2 are the standard assumptions used in nonconvex optimization literature (Bernstein et al., 2018; Reddi et al., 2018; Chen et al., 2018). A1 implies the $L$ -smoothness of $f _ { i }$ , namely, for any $\mathbf { x }$ and $\mathbf { y }$ we obtain $f _ { i } ( \mathbf { x } ) - f _ { i } ( \mathbf { y } ) \leq \langle \nabla f _ { i } ( \mathbf { y } ) , \mathbf { x } - \mathbf { y } \rangle + ( L / 2 ) \| \mathbf { x } - \mathbf { y } \| _ { i } ^ { 2 }$ . A2 implies the bounded variance of $\nabla f _ { i }$ in (Bernstein et al., 2018, Assumption 3), namely, $\begin{array} { r } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \| \nabla f _ { i } ( \mathbf { x } ) - \nabla f ( \mathbf { x } ) \| _ { 2 } ^ { 2 } \leq } \end{array}$ $4 \sigma ^ { 2 }$ , where we have used the fact that $\| \nabla f ( \mathbf { x } ) \| _ { 2 } \leq \sigma$ under A2. Throughout the paper, we assume that problem (1) is solvable, namely, $f ( \mathbf { x } ^ { * } ) > - \infty$ where $x ^ { * }$ is an optimal solution. + +We recall that Algorithm 1 becomes ZO-signSGD when the gradient estimation step (3) is applied. For nonconvex problems, the convergence of an algorithm is typically measured by stationarity, e.g., using $\| \nabla f ( \mathbf { x } ) \| _ { 2 } ^ { 2 }$ in SGD (Ghadimi $\&$ Lan, 2013) and $\| \nabla f ( \mathbf { x } ) \bar { \| _ { 1 } }$ in signSGD (Bernstein et al., 2018). For the latter, the $\ell _ { 1 }$ geometry is met when quantifying the stochasticity through the (non-linear) sign operation. Different from signSGD, ZO-signSGD only obtains a biased estimate to the true gradient. In Proposition 1, we bypass such a bias by leveraging the randomized smoothing technique used for ZO optimization (Gao et al., 2014; Nesterov & Spokoiny, 2015; Duchi et al., 2015). + +Proposition 1 Under A1, the outputs $\{ \mathbf { x } _ { k } \} _ { k = 0 } ^ { T - 1 }$ of ZO-signSGD, i.e., Algorithm 1 with (3), satisfies $\sum _ { k = 0 } ^ { T - 1 } ( \delta _ { k } \mathbb { E } [ \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } ] ) \leq \mathbb { E } [ f _ { \mu } ( \mathbf { x } _ { 0 } ) - f _ { \mu } ( \mathbf { x } _ { T } ) ] + \sum _ { k = 0 } ^ { T - 1 } \left[ 2 \delta _ { k } \sqrt { d } \sqrt { \mathbb { E } [ \| \hat { \mathbf { g } } _ { k } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } ^ { 2 } ] } \right] + \frac { d L } { 2 } \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } ^ { 2 } ,$ where the expectation is taken with respect to all the randomness of ZO-signSGD, $f _ { \mu }$ is the randomized smoothing function of $f$ in (4), and $\hat { \bf g } _ { k } =$ GradEstimate $\left( \mathbf { x } _ { k } \right)$ in (3). + +# Proof: See Appendix 2. + +In Proposition 1, the rationale behind introducing the smoothing function $f _ { \mu }$ is that $\nabla f _ { \mu } ( \mathbf { x } _ { k } )$ is the mean of ZO gradient estimate $\hat { \bf g } _ { k }$ . And thus, the convergence of ZO-signSGD is now linked with the variance of $\hat { \bf g } _ { k }$ , i.e., $\mathbb { E } [ \| \hat { \mathbf { g } } _ { k } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } ^ { 2 } ]$ . This crucial relationship presented in Proposition 1 holds for a general class of signSGD-type algorithms that use different ZO gradient estimators. Spurred by (5), we next investigate the second-order moment of $\hat { \bf g } _ { k }$ in Proposition 2. + +Proposition 2 Under A1 and $A 2$ , the variance of $Z O$ gradient estimate $\hat { \bf g } _ { k }$ is upper bounded by + +$$ +\mathbb { E } \left[ \Vert \hat { \mathbf { g } } _ { k } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \Vert _ { 2 } ^ { 2 } \right] \leq \frac { 4 \alpha _ { b } ( q + 1 ) } { b q } \sigma ^ { 2 } + \frac { ( 2 \alpha _ { b } + \beta _ { b } ) } { b q } C ( d , \mu ) , +$$ + +where $C ( d , \mu ) : = 2 d \sigma ^ { 2 } + \mu ^ { 2 } L ^ { 2 } d ^ { 2 } / 2 .$ . In (6), $\alpha _ { b }$ and $\beta _ { b }$ are Boolean variables depending on the choice of mini-batch sampling, + +where $I ( x > a )$ is the indicator function of $x$ with respect to the constraint $x > a$ , and $I ( x > a ) = 1$ if $x > a$ and 0 otherwise. + +# Proof: See Appendix 3. + +Compared to the variance bound $( \sigma ^ { 2 } / b )$ of the stochastic gradient estimate of $f$ in signSGD (Bernstein et al., 2018), Proposition 2 provides a general result for the ZO gradient estimate $\hat { \bf g } _ { k }$ . It is clear that the bound in (6) contains two parts: $\begin{array} { r } { h _ { 1 } : = \frac { 4 \alpha _ { b } ( q + 1 ) } { b q } \sigma ^ { 2 } } \end{array}$ 4αb(q+1) σ2 and h2 := (2αb+βb)bq C(d, µ), where the former $h _ { 1 } = O ( \sigma ^ { 2 } / b )$ characterizes the reduced variance (using $b$ mini-batch samples) for the stochastic gradient estimate of the smoothing function $f _ { \mu }$ , and the latter $h _ { 2 } = { O } \left( { C ( d , \mu ) } / { ( b q ) } \right)$ reveals the dimension-dependent variance induced by ZO gradient estimate using $b$ mini-batch samples and $q$ random directions. If a stochastic gradient estimate of $f$ is used in signSGD, then $h _ { 2 }$ is eliminated and the variance bound in (6) is reduced to $( \sigma ^ { 2 } / b )$ . + +Furthermore, Proposition 2 covers mini-batch sampling with and without replacement, while signSGD only considers the former case. For the latter case, Proposition 2 implies that if $b = n$ (i.e., $\mathcal { T } _ { k } = [ n ]$ for ZO-signGD), then the variance $\mathbb { E } \left[ \lVert \hat { \mathbf { g } } _ { k } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \rVert _ { 2 } ^ { 2 } \right]$ is reduced to $O ( C ( d , \mu ) / ( n q ) )$ , corresponding to $\alpha _ { b } = 0$ and $\beta _ { b } = 1$ in (7). In the other extreme case of $b = 1$ , both the studied mini-batch schemes become identical, corresponding to $\alpha _ { b } = 1$ and $\beta _ { b } = 0$ . Proposition 2 also implies that the use of large $b$ and $q$ reduces the variance of the gradient estimate, and will further improve the convergence rate. + +With the aid of Proposition 1 and 2, we can then show the convergence rate of ZO-signSGD in terms of stationarity of the original function $f$ . The remaining difficulty is how to bound the gap between $f$ and its smoothed version $f _ { \mu }$ . It has been shown in (Gao et al., 2014; Nesterov & Spokoiny, 2015) that there exists a tight relationship between $f _ { \mu }$ and $f$ given the fact that the former is a convolution of the latter and the density function of a random perturbation $\mathbf { v }$ in (4). We demonstrate the convergence rate of ZO-signSGD in Theorem 1. + +Theorem 1 Under A1 and $A 2$ , if we randomly pick $\mathbf { x } _ { R }$ from $\{ \mathbf { x } _ { k } \} _ { k = 0 } ^ { T - 1 }$ with probability $P ( R = k ) =$ $\frac { \delta _ { k } } { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } }$ , then the convergence rate of ZO-signSGD is given by + +$$ +\begin{array} { r } { \mathbb { E } \left[ \| \nabla f ( \mathbf { x } _ { R } ) \| _ { 2 } \right] \leq \frac { \sqrt { 2 } \left( f ( \mathbf { x } _ { 0 } ) - f ^ { * } + \mu ^ { 2 } L \right) } { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } } + \frac { d L } { \sqrt { 2 } } \frac { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } ^ { 2 } } { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } } + \frac { \mu L d } { \sqrt { 2 } } } \\ { + \frac { 2 \sqrt { 2 } \sqrt { d } \sqrt { 4 \alpha _ { b } ( q + 1 ) \sigma ^ { 2 } + C ( d , \mu ) ( 2 \alpha _ { b } + \beta _ { b } ) } } { \sqrt { b q } } , } \end{array} +$$ + +where $f ^ { * }$ denotes the minimum value. + +# Proof: See Appendix 4. + +In Theorem 1, we translate the gradient norm from $\ell _ { 1 }$ to $\ell _ { 2 }$ , and adopt a probabilistic output $\mathbf { x } _ { R }$ (Ghadimi & Lan, 2013; Lei et al., 2017) to avoid exhaustive search over $\left\{ { \bf x } _ { k } \right\}$ for $\operatorname* { m i n } _ { k }$ $\| \nabla f ( \mathbf { x } _ { k } ) \| _ { 2 }$ . Note that the convergence rate of ZO-signSGD relies on the learning rate $\delta _ { k }$ , the problem size $d$ , the smoothing parameter $\mu$ , the mini-batch size $b$ , and the number of random perturbations $q$ for ZO gradient estimation. We next obtain explicit dependence on these parameters by specifying Theorem 1. + +If $\begin{array} { r } { \delta _ { k } = \delta = O ( \frac { 1 } { \sqrt { d T } } ) } \end{array}$ and $\begin{array} { r } { \mu = O ( \frac { 1 } { \sqrt { d T } } ) } \end{array}$ , then the convergence in (8) simplifies to + +$$ +\mathbb { E } \left[ \Vert \nabla f ( \mathbf { x } _ { R } ) \Vert _ { 2 } \right] \leq O \left( \frac { \sqrt { d } } { \sqrt { T } } + \frac { \sqrt { d } \sqrt { \alpha _ { b } q + ( \alpha _ { b } + \beta _ { b } ) d } } { \sqrt { b q } } \right) , +$$ + +where $\alpha _ { b }$ and $\beta _ { b }$ were defined in (7), and $1 \leq ( \alpha _ { b } + \beta _ { b } ) \leq 2$ . We provide several key insights on the convergence rate of ZO-signSGD through (9). + +First, the convergence rate of ZO-signSGD is measured through $\| \nabla f ( \mathbf { x } _ { R } ) \| _ { 2 }$ rather than its squared counterpart $\| \nabla \bar { f } ( \mathbf { x } _ { R } ) \| _ { 2 } ^ { 2 }$ , where the latter was used in measuring the convergence of ZO-SGD. We recall from (Ghadimi & Lan, 2013, Theorem 3.2 & Corollary 3.3) that ZO-SGD yields the convergence error $\mathbb { E } \left[ \lVert \nabla f ( \mathbf { x } _ { R } ) \rVert _ { 2 } ^ { 2 } \right] \leq O ( \frac { \sqrt { d } } { \sqrt { T } } )$ . Since $\| \nabla f ( \mathbf { x } _ { R } ) \| _ { 2 } ^ { 2 } \leq \| \nabla f ( \mathbf { x } _ { R } ) \| _ { 2 }$ as $\| \nabla f ( \mathbf { x } _ { R } ) \| _ { 2 } ^ { 2 } \leq 1$ , the convergence of ZO-signSGD meets a stricter criterion than that of ZO-SGD. The possible downside of ZO-signSGD is that it suffers an additional error of order $\begin{array} { r } { O ( \frac { \sqrt { d } } { \sqrt { b } } + \frac { d } { \sqrt { b q } } ) } \end{array}$ in the worst case. The aforementioned results imply that ZO-signSGD could only converge to a neighborhood of a stationary point but with a fast convergence speed. Here the size of the neighborhood is controlled by the mini-batch size $b$ and the number of random direction vectors $q$ . + +Also, our convergence analysis applies to mini-batch sampling both with and without replacement. When b ∈ [1, n), ZO-signSGD achieves O( √dT $\begin{array} { r } { O ( \frac { \sqrt { d } } { \sqrt { T } } + \frac { \sqrt { d } } { \sqrt { b } } + \frac { d } { \sqrt { b q } } ) } \end{array}$ convergence rate regardless of the choice of mini-batch sampling. When $b = n$ , it is known from (9) that the use of mini-batch without replacement recovers ZO-signGD, yielding the convergence rate $\begin{array} { r } { O ( \frac { \sqrt { d } } { \sqrt { T } } + \frac { d } { \sqrt { n q } } ) } \end{array}$ . By contrast, the use of mini-batch with replacement leads to the worse convergence rate $\begin{array} { r } { O ( \frac { \sqrt { d } } { \sqrt { T } } + \frac { d } { \sqrt { n q } } + \frac { \sqrt { d } } { \sqrt { n } } ) } \end{array}$ . Clearly, as $b = n$ and $n < T$ , ZO-signSGD using mini-batch with replacement fails to achieve the rate $O ( \frac { \sqrt { d } } { \sqrt { T } } )$ regardless of the choice of $q$ . By contrast, ZO-signSGD using mini-batch without replacement recovers $O ( \frac { \sqrt { d } } { \sqrt { T } } )$ as $\begin{array} { r } { q \ = \ O ( \frac { d T } { n } ) } \end{array}$ . When $b > n$ , ZO-signSGD is restricted to using mini-batch sampling with replacement. Similar to signSGD (Bernstein et al., 2018), we can obtain $O ( \frac { \sqrt { d } } { \sqrt { T } } )$ convergence rate as $b = O ( T )$ and $\begin{array} { r } { q = O ( \frac { d T } { n } ) } \end{array}$ , where the dependence on $q$ is induced by the use of ZO gradient estimation. + +# 5 VARIANTS OF ZO-SIGNSGD + +Here we study three variants of ZO-signSGD, where the gradient will be estimated using a) the central difference of function values, b) the sign of ZO gradient estimates with majority vote, or c) the sign of ZO gradient estimates with majority vote for distributed optimization. That is, + +$$ +\begin{array} { l } { { \displaystyle a \mathrm { ) ~ G r a d E s t i m a t e ( \mathbf { x } ) } = \frac { 1 } { b q } \sum _ { i \in \mathbb { Z } _ { k } } \sum _ { j = 1 } ^ { q } \frac { d \left[ f _ { i } \left( \mathbf { x } + \mu \mathbf { u } _ { i , j } \right) - f _ { i } \left( \mathbf { x } - \mu \mathbf { u } _ { i , j } \right) \right] \mathbf { u } _ { i , j } } { 2 \mu } } \ ~ } \\ { { \displaystyle b \mathrm { ) ~ G r a d E s t i m a t e ( \mathbf { x } ) } = \frac { 1 } { b q } \sum _ { i \in \mathbb { Z } _ { k } } \sum _ { j = 1 } ^ { q } \mathrm { s i g n } \left( \hat { \nabla } f _ { i } \left( \mathbf { x } ; \mathbf { u } _ { i , j } \right) \right) , } \ ~ } \\ { { \displaystyle c \mathrm { ) ~ G r a d E s t i m a t e ( \mathbf { x } ) } = \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \mathrm { s i g n } \left( \frac { 1 } { b _ { m } q } \sum _ { i \in \mathbb { Z } _ { m , k } } \sum _ { j = 1 } ^ { q } \hat { \nabla } f _ { i } \left( \mathbf { x } ; \mathbf { u } _ { i , j } \right) \right) , } \ ~ } \end{array} +$$ + +where $\{ \mathbf { u } _ { i , j } \}$ and $\hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } _ { i , j } )$ have been defined in (3). The gradient estimator (12) is proposed for istrigent mization over a stahas only access to twork that consisdata (in terms of f i $M$ agents and vidual costs ral pr with $m \in [ M ]$ $n _ { m }$ $n _ { m }$ $\{ f _ { i } \} )$ $\textstyle \sum _ { m = 1 } ^ { M } n _ { m } =$ $n$ , and the mini-batch per agent satisfies $b _ { m } = | \mathcal { I } _ { m , k } | \in [ 1 , n _ { m } ]$ . According to (12), the central processor receives 1-bit compressed gradients $\begin{array} { r } { \mathrm { s i g n } \left( \frac { 1 } { b _ { m } q } \sum _ { i \in \mathcal { T } _ { m , k } } \sum _ { j = 1 } ^ { q } \hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } _ { i , j } ) \right) } \end{array}$ from $M$ agents and then performs the sign-based descent (2) and sends its 1-bit update back to every agent. + +The ZO gradient estimator (10) was used in (Shamir, 2017) for bandit convex optimization and in (Ilyas et al., 2018a) for designing black-box adversarial attacks. Compared to the form of forward difference (3), the central difference (10) requires $b ( q - 1 )$ times more function queries in gradient estimation. At the cost of more function queries, one may wonder if the convergence rate of ZO-signSGD can be further improved. + +Corollary 1 Suppose that the conditions in Theorem 1 hold, ZO-signSGD with gradient estimator (10) yields the same convergence rate of ZO-signSGD that uses the estimator (3). + +Proof: Recall that Proposition 1 is independent of specific forms of gradient estimators, and thus holds for (10). Although Proposition 2 relies on the second-order moments of each gradient estimator, we prove that under A1 and A2, both (3) and (10) maintain the same statistical properties. As a result, Proposition 2 and Theorem 1 also hold for (10); see more details in Appendix 5.  + +We next study the gradient estimator (11), whose sign is equivalent to the majority vote (i.e., the element-wise median) of signs of individual gradient estimates $\{ \hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } _ { i , j } ) \}$ . It was shown in (Bernstein et al., 2018) that signSGD with majority vote has a better convergence rate under additional assumptions of unimodal symmetric noise distribution of coordinate-wise gradient estimates. In Corollary 2, we show that such a speed-up in convergence can also be achieved by ZO-signSGD with majority vote, which we refer to as ‘ZO-M-signSGD’. + +Corollary 2 Suppose that the conditions in Theorem $^ { l }$ hold, and the distribution of gradient noise is unimodal and symmetric. Then, ZO-M-signSGD with $\begin{array} { r } { \delta _ { k } = { O } \left( \frac { 1 } { \sqrt { d T } } \right) } \end{array}$ and $\begin{array} { r } { \mu = O ( \frac { 1 } { \sqrt { d T } } ) } \end{array}$ yields + +$$ +\begin{array} { r } { \mathbb { E } \left[ \| \nabla f ( \mathbf { x } _ { R } ) \| _ { 2 } \right] = O \left( \sqrt { d } / \sqrt { T } + d / \sqrt { b q } \right) . } \end{array} +$$ + +Proof: See Appendix 6. + +We recall from Theorem 1 that under the same parameter setting of Corollary 2, ZO-signSGD yields $\begin{array} { r } { O ( \frac { \sqrt { d } } { \sqrt { T } } + \frac { \sqrt { d } } { \sqrt { b } } + \frac { d } { \sqrt { b q } } ) } \end{array}$ convergence rate in the worst case. It is clear from (13) that the error correction term of order √d is eliminated in ZO-M-signSGD. Such an improvement in convergence is achieved under the condition of unimodal symmetric gradient noise. We remark that different from the stochastic gradient noise studied in (Bernstein et al., 2018), the ZO gradient estimation noise could violate this assumption. For example, in a scalar case, if the gradient estimate $g$ follows the distribution where $g = 1$ with probability 0.9, $g = - 1 0$ with probability 0.1, then $\mathbb { E } [ g ] < 0$ and $\mathrm { s i g n } ( \mathbb { E } [ g ] ) < 0$ . However, $\mathbb { E } [ \mathrm { s i g n } ( g ) ] > 0$ . This implies that without the assumption of symmetry, the sign of gradient estimates with majority vote $( { \mathbb E } [ \mathrm { s i g n } ( g ) ] )$ can be in the opposite direction of the sign of averaged gradients $( \mathrm { s i g n } ( \mathbb { E } [ g ] ) )$ . Our results in the next section show that ZO-M-signSGD may not outperform ZO-signSGD. + +Lastly, we focus on the gradient estimator (12), whose sign can be interpreted as the major vote of $M$ distributed agents about the sign of the true gradient (Bernstein et al., 2018). The resulting variant of ZO-signSGD is called ‘ZO-D-signSGD’, and its convergence rate is illustrated in Corollary 3. Compared to ZO-M-signSGD for centralized optimization, ZO-D-signSGD suffers an extra error correction term $O ( \frac { \sqrt { d } } { \sqrt { n } } )$ in the distributed setting. It is also worth mentioning that if $M = n$ and $q = 1$ , then the gradient estimator (12) reduces to (11) with $\mathcal { T } _ { k } = [ n ]$ . In this case, Corollary 2 and 3 reach a consensus on $\begin{array} { r } { O ( \frac { \sqrt { d } } { \sqrt { T } } + \frac { d } { \sqrt { n } } ) } \end{array}$ convergence error. + +Corollary 3 Suppose that the conditions in Corollary 2 hold. ZO-M-signSGD with $\begin{array} { r } { \boldsymbol { b } _ { m } = \left\lfloor \frac { n } { M } \right\rfloor } \end{array}$ $\begin{array} { r } { \delta _ { k } = O ( \frac { 1 } { \sqrt { d T } } ) } \end{array}$ and $\begin{array} { r } { \mu = O ( \frac { 1 } { \sqrt { d T } } ) } \end{array}$ yields + +$$ +\mathbb { E } \left[ \| \nabla f ( \mathbf { x } _ { R } ) \| _ { 2 } \right] = O \left( \sqrt { d } / \sqrt { T } + \sqrt { d } / \sqrt { n } + d / \sqrt { n q } \right) . +$$ + +Proof: See Appendix 7. + +![](images/337bfc286984c7305cf44de134c83cbe2f9274c987c25327a863c940c6c1ba76.jpg) +Figure 1: Performance comparison of ZO-signSGD, ZO-M-signSGD, ZO-SGD, ZO-SCD, signSGD and SGD under a synthetic dataset. The solid line represents the loss/accuracy averaged over 10 independent trials with random initialization, and the shaded region indicates the standard deviation of results over random trials. (a)-(b): Training loss and test accuracy versus iterations. (c)-(d): Effects of mini-batch size $q$ and number of random direction vectors $q$ on the convergence of studied algorithms. Here (c) presents the training loss versus iterations, and (d) is the heat map of the final loss for different values of $^ { b }$ and $q$ . (e)-(f): Effects of problem size $d$ . Here (e) shows the final training loss versus $d$ , and (f) presents the convergence trajectory when $d \in \{ 2 0 0 , 4 0 0 \}$ . + +# 6 EXPERIMENTS + +In this section, we empirically show the effectiveness of ZO-signSGD, and validate its convergence behavior on both synthetic and real-world datasets such as MNIST and CIFAR-10. For the synthetic experiment, we study the problem of binary classification in the least squared formulation. For the real-world application, we design adversarial examples from black-box neural networks as mentioned in Sec. 2. Throughout this section, we compare ZO-signSGD and its variants with SGD, signSGD (Bernstein et al., 2018), ZO-SGD (Ghadimi & Lan, 2013), and ZO-SCD (Lian et al., 2016). + +Binary classification We consider the least squared problem with a nonconvex loss function (Xu et al., 2017; Liu et al., 2018b) $\begin{array} { r } { \operatorname* { m i n } _ { \mathbf { x } \in \mathbb { R } ^ { d } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( y _ { i } - 1 / ( 1 + e ^ { - \mathbf { a } _ { i } ^ { T } \mathbf { x } } ) ) ^ { 2 } } \end{array}$ , which satisfies Assumption A2 by letting $\sigma = \operatorname* { m a x } _ { i } \{ 2 \lVert \mathbf { a } _ { i } \rVert _ { 2 } \}$ . Here instead of using the conventional cost function of logistic regression (a convex function), the considered least squared formulation is introduced to align with our nonconvex theoretical analysis. For generating the synthetic dataset, we randomly draw samples $\left\{ \mathbf { a } _ { i } \right\}$ from $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ , and obtain the label $y _ { i } = 1$ if $1 / \dot { ( 1 + e ^ { - \mathbf { a } _ { i } ^ { T } \mathbf { x } } ) } > 0 . 5$ and 0 otherwise. The number of training samples $\left\{ \mathbf { a } _ { i } , y _ { i } \right\}$ is set by $n = 2 0 0 0$ against 200 testing samples. We find the best constant learning rate for algorithms via a greedy search over $\eta \in [ 0 . 0 0 1 , 0 . 1 ]$ (see Appendix 8.1 for more details), and we choose the smoothing parameter $\mu = 1 0 / \sqrt { T d }$ . Unless specified otherwise, let $b = q = 1 0$ , $T = 5 0 0 0$ and $d = 1 0 0$ . + +In Fig. 1, we report the training loss, the test accuracy, as well as the effects of algorithmic parameters on the convergence of the studied algorithms. We observe from Fig. 1-(a) and (b) that ZO-signSGD outperforms other ZO algorithms, and signSGD yields the best convergence performance once the first-order information is available. In Fig. 1-(c) and (d), we observe that the convergence performance of ZO algorithms is improved as $b$ and $q$ increase. In particular, ZO-signSGD and ZO-M-signSGD at $b = q = 3 0$ approach to the best result provided by signSGD. In Fig. 1-(e) and (f), the convergence of all algorithms degrades as the problem size $d$ increases. However, ZO-signSGD and ZO-M-signSGD converge faster than ZO-SGD and ZO-SCD. In Fig. 2, we demonstrate the convergence trajectory of different variants of ZO-signSGD for $b \in \{ 4 0 , 4 0 0 \}$ . To make a fair comparison between ZOsignSGD and ZO-D-signSGD, let each of $M = 4 0$ agents use a mini-batch of size $b / M$ . As we can see, ZO-signSGD outperforms ZO-M-signSGD and ZO-D-signSGD. And the convergence is improved as the mini-batch size increases. However, we observe that in all examples, ZO-signSGD and its variants converge to moderate accuracy much faster than ZO-SGD, only within a few tens of iterations. + +![](images/d06c7d33a9b1cb45fe96ad3bf4d31117fbae253be352e11dcb7b8e189b8be217.jpg) +Figure 2: Training loss (left) and testing accuracy (right) of ZO-signSGD, ZO-M-signSGD, ZO-D-signSGD and ZO-SGD versus iterations. + +![](images/0dcd6aa27fddfbaa2c914d807914b8186ef985663cc7baa8b27cd6648a025220.jpg) +Figure 3: Black-box attacking loss versus iterations. The solid marker indicates the iteration number that finds the first successful adversarial example, and its loss corresponds to the squared $\ell _ { 2 }$ distortion. + +Generating black-box adversarial examples Here we study adversarial robustness by generating adversarial examples from a black-box image classifier trained by a deep neural network (DNN) model; see details on problem formulation in Appendix 8.2. We recall from Sec. 2 that the task of black-box adversarial attack falls within the category of ZO optimization as one can only access to the input-output relation of the DNN while crafting adversarial examples. + +The DNN models trained on MNIST and CIFAR-10 (Carlini & Wagner, 2017) are performed as the zeroth-order oracle2. We select one image from each class of MNIST and CIFAR-10 and separately implement black-box attacks using the same attacking loss function (see Appendix 8.2) but with different ZO optimization algorithms (ZO-SGD, ZO-signSGD and ZO-M-signSGD). We also set the same parameters for each method, i.e., $\mu = 0 . 0 1$ , $q = 9$ , and $\delta = 0 . 0 5$ for MNIST and $\delta = 0 . 0 0 0 5$ for CIFAR-10, to accommodate to the dimension factor $d$ . Moreover, we benchmark their performance with the natural evolution strategy (NES) based two-point gradient estimator in (Ilyas et al., 2018a) for solving the same attacking loss function, where the sign of gradient estimate is also used in the descent step. We call the resulting black-box attack generation method ‘ZO-NES’. Similar to (10), NES computes the ZO gradient estimate using the central difference of two function values. Thus, one iteration of ZO-NES requires $2 q$ function queries and thus we set $q = 5$ to align with the number of function queries used in other ZO methods. All methods use the the same natural image as the initial point for finding adversarial examples. + +Table 1: Iteration comparison of attacking black-box DNN on MNIST (image ID 2). + +
Iteration04080 120160 200 240280 312 356
ZO-SGD7770777777
Classified as4 4
Iteration 0 40 80 120 145 202 240280 320 359
ZO-signSGD070007000
Classified as4044 4 279 320 3604 4 4
Iteration0 80 120 142 200 240
ZO-M-signSGD7777777777
Classified as44 4 4 4 4
Iteration0 40 80 120 160 200 258 287 321 359
ZO-NES7710777077
Classified as4 4 4
+ +Fig. 3 shows the plots of black-box attacking loss versus iterations (more results are shown in Appendix 8.3). We find that ZO-signSGD usually takes significantly less iterations than other methods to find the first successful adversarial example with a similar attacking loss. For MNIST, the average iteration over all attacked images in Table A1 to find the first successful adversarial example is 184 for ZO-SGD, 103 for ZO-signSGD, 151 for ZO-M-signSGD, and 227 for ZO-NES. Their corresponding average $\ell _ { 2 }$ distortion is 2.345 for ZO-SGD, 2.381 for ZO-signSGD, 2.418 for ZO-MsignSGD, and 2.488 for ZO-NES. For CIFAR-10, the average iteration over all attacked images in Table A2 to find the first successful adversarial example is 302 for ZO-SGD, 250 for ZO-signSGD, 389 for ZO-M-signSGD, and 363 for ZO-NES. Their corresponding average $\ell _ { 2 }$ distortion is 0.177 for ZO-SGD, 0.208 for ZO-signSGD, 0.219 for ZO-M-signSGD, and 0.235 for ZO-NES. As a visual illustration, we compare the adversarial examples of a hand-written digit “1” of each attacking method at different iterations in Table 1, corresponding to Fig. 3-(a). As we can see, ZO-signSGD and ZO-M-signSGD can reduce roughly $5 4 \%$ of iterations (around 600 less model queries) than ZO-SGD to find the first successful adversarial example. Given the first successful adversarial example, we observe that ZO-signSGD yields slightly higher $\ell _ { 2 }$ distortion than ZO-SGD. This is not surprising since Theorem 1 suggests that ZO-signSGD might not converge to a solution of very high accuracy but it can converge to moderate accuracy sufficient for black-box attacks at a very fast speed. Note that the first successful adversarial examples generated by different ZO methods are all visually similar to the original ones but lead to different top-1 predictions; see more results in Appendix 8.3. In addition, we observe that ZO-NES is not as effective as ZO-signSGD in either query efficiency (given by the number of iterations to achieve the first successful attack) or attack distortion. Thus, compared to ZO-NES, ZO-signSGD offers a provable and an efficient black-box adversarial attacking method. + +# 7 CONCLUSION + +Motivated by the impressive convergence behavior of (first-order) signSGD and the empirical success in crafting adversarial examples from black-box ML models, in this paper we rigorously prove the√ √ $O ( \sqrt { d } / \sqrt { T } )$ convergence rate of ZO-signSGD and its variants under mild conditions. Compared to signSGD, ZO-signSGD suffers a slowdown (proportional to the problem size $d$ ) in convergence rate, however, it enjoys the gradient-free advantages. Compared to other ZO algorithms, we corroborate the superior performance of ZO-signSGD on both synthetic and real-word datasets, particularly for its application to black-box adversarial attacks. In the future, we would like to generalize our analysis to nonsmooth and nonconvex constrained optimization problems. + +# ACKNOWLEDGMENTS + +This work was supported by the MIT-IBM Watson AI Lab. Mingyi Hong and Xiangyi Chen are supported partly by an NSF grant CMMI-1727757,and by an AFOSR grant 15RT0767. + +# REFERENCES + +A. Athalye, N. Carlini, and D. Wagner. Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. arXiv preprint arXiv:1802.00420, 2018. +L. Balles and P. Hennig. Dissecting adam: The sign, magnitude and variance of stochastic gradients. arXiv preprint arXiv:1705.07774, 2017. +J. Bernstein, Y.-X. Wang, K. Azizzadenesheli, and A. Anandkumar. signsgd: compressed optimisation for non-convex problems. arXiv preprint arXiv:1802.04434, 2018. +A. N. Bhagoji, W. He, B. Li, and D. Song. Practical black-box attacks on deep neural networks using efficient query mechanisms. 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Similar to (Bernstein et al., 2018, Figure A.1), we assume that the ZO gradient estimate of $f ( \mathbf { x } )$ and its first-order gradient $\nabla f ( \mathbf { x } ) = \mathbf { x }$ suffer from a sparse noise vector $\mathbf { v }$ , where $v _ { 1 } \in \overline { { \mathcal { N } ( 0 , 1 0 0 ^ { 2 } ) } }$ , and $v _ { i } = 0$ for $i \geq 2$ . As a result, the used descent direction at iteration $t$ is given by $\hat { \nabla } f ( \mathbf { x } _ { t } ) + \mathbf { v }$ or $\nabla f ( \mathbf { x } _ { t } ) + \mathbf { v }$ . Fig. A1 presents the convergence performance of 5 algorithms: SGD, signSGD, ZO-SGD, ZO-signSGD and its variant using the central difference based gradient estimator (10). Here we tune a constant learning rate finding 0.001 best for SGD and ZO-SGD and 0.01 best for signSGD and its ZO variants. As we can see, sign-based first-order and ZO algorithms converge much faster than the stochastic gradient-based descent algorithms. This is not surprising since the presence of extremely noisy component $v _ { 1 }$ leads to an inaccurate gradient value, and thus degrades the convergence of SGD and ZO-SGD. By contrast, the sign information is more robust to outliers and thus leads to better convergence performance of sign SGD and its variants. We also note that the convergence trajectory of ZO-signSGD using the gradient estimator (10) coincides with that using the gradient estimator (3) given by the forward difference of two function values. + +![](images/3a02060f85eca3a9b9d8126e85684d6a6fade1b1de8a7098c000772a2df7bcc7.jpg) +Figure A1: Comparison of different gradient-based and gradient sign-based first-order and ZO algorithms in the example of sparse noise perturbation. The solid line represents the loss averaged over 10 independent trials with random initialization, and the shaded region indicates the standard deviation of results over random trials. Left: Loss value against iterations for SGD, signSGD, ZO-SGD, ZO-signSGD and ZO-signSGD using the central difference based gradient estimator (10). Right: Local regions to highlight the effect of the gradient estimators (3) and (10) on the convergence of ZO-signSGD. + +# 1.2 STATISTICS OF GRADIENT ESTIMATES + +The intuition behind why ZO-signSGD could outperform ZO-SGD is that the sign operation can mitigate the negative effect of (coordinate-wise) gradient noise of large variance. To confirm this point, we examine the coordinate-wise variance of gradient noises during an entire training run of the binary classifier provided in the first experiment of Sec. 6. At each iteration, we perform an additional 100 random trials to obtain the statistics of gradient estimates. In Fig. A2-(a), we present the $\ell _ { 1 }$ norm of the mean of gradient estimates (over 100 trials) versus the number of iterations. As we can see, both signSGD and ZO-signSGD outperform SGD and ZO-SGD, evidenced by a fast decrease of the $\ell _ { 1 }$ norm of gradient estimate. In Fig. A2-(b), we present the coordinate-wise gradient noise variance (over 100 trails at each coordinate) against the number of iterations. It is not surprising that compared to first-order methods, ZO methods suffer gradient noise of larger variance. In this scenario, we could benefit from ZO-signSGD since taking the sign of gradient estimates might scale down extremely noisy components. Indeed, we observe a significant decrease of the noise variance while performing ZO-signSGD compared to ZO-SGD. + +![](images/a3d507387e8be5d4c8da524fed61e074d8dd332bfd27e24747d6b46669e9e7a1.jpg) +Figure A2: Statistics of gradient estimates during an entire training run of the binary classifier provided in the first experiment of Sec. 6. a) The $\ell _ { 1 }$ norm of the mean of gradient estimates versus iteration. b) Coordinate-wise gradient noise variance versus iteration. The solid line represents the variance averaged over all coordinates, and the shaded region indicates the corresponding standard deviation with respect to all coordinates at each iteration. + +# 2 PROOF OF PROPOSITION 1 + +Based on the definition of the smoothing function $f _ { \mu }$ , for any $\mathbf { x }$ and $\mathbf { y }$ we have + +$$ +\begin{array} { r l } & { \| \nabla f _ { \mu } ( x ) - \nabla f _ { \mu } ( y ) \| _ { 2 } = \| \mathbb { E } _ { \mathbf { v } } [ \nabla _ { \mathbf { x } } f ( \mathbf { x } + \mu \mathbf { v } ) - \nabla _ { \mathbf { y } } f ( \mathbf { y } + \mu \mathbf { v } ) ] \| _ { 2 } } \\ & { \qquad \le \mathbb { E } [ \| \nabla _ { \mathbf { x } } f ( \mathbf { x } + \mu \mathbf { v } ) - \nabla _ { \mathbf { y } } f ( \mathbf { y } + \mu \mathbf { v } ) \| _ { 2 } ] \le L \| \mathbf { x } - \mathbf { y } \| _ { 2 } , } \end{array} +$$ + +where the first inequality holds due to Jensen’s inequality, and the second inequality holds due to A1. It is known from (15) that $f _ { \mu }$ has $L$ -Lipschitz continuous gradient. + +By the $L$ -smoothness of $f _ { \mu }$ , we obtain that + +$$ +\begin{array} { l } { f _ { \mu } ( { \mathbf x } _ { k + 1 } ) \leq f _ { \mu } ( { \mathbf x } _ { k } ) + \langle \nabla f _ { \mu } ( { \mathbf x } _ { k } ) , { \mathbf x } _ { k + 1 } - { \mathbf x } _ { k } \rangle + \displaystyle \frac { L } { 2 } \| { \mathbf x } _ { k + 1 } - { \mathbf x } _ { k } \| _ { 2 } ^ { 2 } } \\ { \displaystyle \stackrel { ( 2 ) } { = } f _ { \mu } ( { \mathbf x } _ { k } ) - \delta _ { k } \langle \nabla f _ { \mu } ( { \mathbf x } _ { k } ) , \mathrm { s i g n } ( \hat { \mathbf g } _ { k } ) \rangle + \displaystyle \frac { L } { 2 } \delta _ { k } ^ { 2 } \| \mathrm { s i g n } ( \hat { \mathbf g } _ { k } ) \| _ { 2 } ^ { 2 } } \\ { \displaystyle = f _ { \mu } ( { \mathbf x } _ { k } ) - \delta _ { k } \| \nabla f _ { \mu } ( { \mathbf x } _ { k } ) \| _ { 1 } + \frac { \delta _ { k } ^ { 2 } d L } { 2 } } \\ { \displaystyle \quad \quad + 2 \delta _ { k } \displaystyle \sum _ { i = 1 } ^ { d } \vert ( \nabla f _ { \mu } ( { \mathbf x } _ { k } ) ) _ { i } \vert Z [ \mathrm { s i g n } ( \hat { \mathbf g } _ { k , i } ) \neq \mathrm { s i g n } ( ( \nabla f _ { \mu } ( { \mathbf x } _ { k } ) ) _ { i } ) ] , } \end{array} +$$ + +where $( \nabla f _ { \mu } ( \mathbf { x } ) ) _ { i }$ denotes the ith element of $\nabla f _ { \mu } ( \mathbf { x } )$ . + +Taking expectation for both sides of (16), we obtain that + +$$ +\begin{array} { l } { \displaystyle \mathbb { E } [ f _ { \mu } ( \mathbf { x } _ { k + 1 } ) - f _ { \mu } ( \mathbf { x } _ { k } ) ] \leq - \delta _ { k } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } + \frac { \delta _ { k } ^ { 2 } d L } { 2 } } \\ { \displaystyle + \ 2 \delta _ { k } \sum _ { i = 1 } ^ { d } \vert ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { i } \vert \mathrm { P r o b } \left[ \mathrm { s i g n } ( \hat { \mathbf { g } } _ { k , i } ) \neq \mathrm { s i g n } ( ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { i } ) \right] . } \end{array} +$$ + +Similar to (Bernstein et al., 2018, Theorem 1), we relax Prob $[ \mathrm { s i g n } ( \hat { \mathbf { g } } _ { k , i } ) \neq \mathrm { s i g n } ( ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { i } ) ]$ by Markov’s inequality, + +$$ +\begin{array} { r l } & { \mathrm { P r o b } \left[ \mathrm { s i g n } ( \hat { \bf g } _ { k , i } ) \neq \mathrm { s i g n } ( ( \nabla f _ { \mu } ( { \bf x } _ { k } ) ) _ { i } ) \right] \leq \mathrm { P r o b } [ \vert \hat { \bf g } _ { k , i } - ( \nabla f _ { \mu } ( { \bf x } _ { k } ) ) _ { i } \vert \ge \vert ( \nabla f _ { \mu } ( { \bf x } _ { k } ) ) _ { i } \vert ] } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \end{array} +$$ + +Substituting (18) into (17), we obtain + +$$ +\begin{array} { r l } { \displaystyle \mathbb { E } [ f _ { \mu } ( \mathbf { x } _ { k + 1 } ) - f _ { \mu } ( \mathbf { x } _ { k } ) ] \leq - \delta _ { k } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } + \frac { \delta _ { k } ^ { 2 } d L } { 2 } + 2 \delta _ { k } \displaystyle \sum _ { i = 1 } ^ { d } \mathbb { E } [ \| \hat { \mathbf { g } } _ { k , i } - ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { i } \| ] } & { } \\ { \displaystyle = - \delta _ { k } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } + \frac { \delta _ { k } ^ { 2 } d L } { 2 } + 2 \delta _ { k } \mathbb { E } [ \| \hat { \mathbf { g } } _ { k } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } ] } & { } \\ { \displaystyle \leq - \delta _ { k } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } + \frac { \delta _ { k } ^ { 2 } d L } { 2 } + 2 \delta _ { k } \sqrt { d } \mathbb { E } [ \| \hat { \mathbf { g } } _ { k } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } ] } & { } \\ { \displaystyle \leq - \delta _ { k } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } + \frac { \delta _ { k } ^ { 2 } d L } { 2 } + 2 \delta _ { k } \sqrt { d } \sqrt { \mathbb { E } [ \| \hat { \mathbf { g } } _ { k } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } ] } , } & { } \end{array} +$$ + +where the second inequality holds due to $\| \mathbf { x } \| _ { 1 } \leq { \sqrt { d } } \| \mathbf { x } \| _ { 2 }$ , and the last inequality holds by applying Jensen’s inequality to the concave function $\sqrt { \cdot }$ , i.e., $\begin{array} { r } { \mathbb { E } [ \| \hat { \mathbf { g } } _ { k } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } ] \le \sqrt { \mathbb { E } [ \| \hat { \mathbf { g } } _ { k } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } ^ { 2 } ] } } \end{array}$ . + +Taking sum of both sides of (19), we then obtain (5). + +# 3 PROOF OF PROPOSITION 2 + +We recall from (3) that + +$$ +\hat { \mathbf { g } } _ { k } = \frac { 1 } { b } \sum _ { i \in \mathcal { T } _ { k } } \hat { \nabla } f _ { i } ( \mathbf { x } _ { k } ) , \hat { \nabla } f _ { i } ( \mathbf { x } _ { k } ) = \frac { 1 } { q } \sum _ { j = 1 } ^ { q } \hat { \nabla } f _ { i } ( \mathbf { x } _ { k } ; \mathbf { u } _ { i , j } ) . +$$ + +Let $\mathbf { z } _ { i } : = \hat { \nabla } f _ { i } ( \mathbf { x } _ { k } ) - \nabla f _ { \mu } ( \mathbf { x } _ { k } )$ and $\mathbf { z } _ { i , j } = \hat { \nabla } f _ { i } ( \mathbf { x } _ { k } ; \mathbf { u } _ { i , j } ) - \nabla f _ { \mu } ( \mathbf { x } _ { k } )$ . Thus, + +$$ +{ \hat { \mathbf { g } } } _ { k } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) = { \frac { 1 } { b } } \sum _ { i \in { \mathcal { T } } _ { k } } \mathbf { z } _ { i } = { \frac { 1 } { b q } } \sum _ { i \in { \mathcal { T } } _ { k } } \sum _ { j = 1 } ^ { q } \mathbf { z } _ { i , j } +$$ + +where there are two sources of randomness: a) minibatch sampling $i \in \mathcal { Z } _ { k }$ , and b) the random direction sampling $\mathbf { u } = \mathbf { u } _ { i , j }$ . Note that these two sources of randomness are independent, and the random direction samples $\{ \bar { \mathbf { u } } _ { i , j } \}$ are i.i.d.. + +Next, we discuss two types of mini-batch sampling: a) mini-batch samples without replacement, and b) minibatch samples with replacement. + +Suppose that $\mathcal { T } _ { k }$ is a uniform random subset of $[ n ]$ (no replacement), motivated by (Lei et al., 2017, Lemma A.1) we introduce a new variable $W _ { i } = I ( i \in \mathcal { T } _ { k } )$ , where $I$ is an indicator function, and $I ( i \in \mathcal { T } _ { k } ) = 1$ if $i \in \mathcal { Z } _ { k }$ , and 0 otherwise. As a result, we have + +$$ +\mathbb { E } [ W _ { i } ^ { 2 } ] = \mathbb { E } [ W _ { i } ] = \frac { b } { n } , \quad \mathbb { E } [ W _ { i } W _ { j } ] = \frac { b ( b - 1 ) } { n ( n - 1 ) } , \ i \neq j . +$$ + +From (21), the variance of $\hat { \bf g } _ { k }$ is given by + +$$ +\begin{array} { r l } { { \mathbb { E } [ \| \displaystyle \frac { 1 } { b } \displaystyle \sum _ { i = 1 } ^ { n } \mathbb { Z } _ { a } \| { \mathbb { Z } } _ { 2 } ] ^ { 2 } ] } } & { = \frac { 1 } { b ^ { 2 } } ( \displaystyle \sum _ { i = 1 } ^ { n } \big ( { \mathbb { E } } _ { i } \langle z _ { k } | W _ { i } ^ { 2 } | { \mathbb { E } } _ { u } \| { \mathbb { Z } } _ { i } \| { \mathbb { Z } } _ { 2 } | { \mathbb { Z } } _ { 1 } \big ) + \displaystyle \sum _ { i \neq j } { \mathbb { E } } _ { i , j \in { \mathbb { Z } } _ { k } } \big \langle W _ { i } W _ { j } \big | | { \mathbb { Z } } _ { \mathbb { Z } _ { 1 } } \big | { \mathbb { Z } } _ { u } \big | { \mathbb { Z } } _ { 2 } \big | ) ) } \\ & { = \displaystyle \frac { 1 } { b ^ { 2 } } ( \displaystyle \frac { b } { n } \displaystyle \sum _ { i = 1 } ^ { n } \mathbb { E } _ { \mathbb { Z } } \| [ z _ { 1 } | { \mathbb { Z } } _ { 2 } ] \| + \displaystyle \frac { b ( b - 1 ) } { n ( n - 1 ) } \| \displaystyle \sum _ { i = 1 } ^ { n } \mathbb { E } _ { \mathbb { Z } } \| z _ { \mathbb { Z } ^ { i } } \| { \mathbb { Z } } _ { 1 } \| _ { 2 } ^ { 2 } - \displaystyle \frac { b ( b - 1 ) } { n ( n - 1 ) } \displaystyle \sum _ { i = 1 } ^ { n } \| \mathbb { E } _ { \mathbb { Z } } \| z _ { \mathbb { Z } ^ { i } } \| { \mathbb { Z } } _ { 2 } ^ { 2 } ) } \\ & { \overset { ( a ) } { = } \displaystyle \frac { 1 } { b n } \displaystyle \sum _ { i = 1 } ^ { n } \mathbb { E } _ { \mathbb { Z } } \| [ z _ { 1 } | { \mathbb { Z } } _ { i } ] \| ^ { 2 } - \displaystyle \frac { b } { b ( n - 1 ) n } \displaystyle \sum _ { i = 1 } ^ { n } \| \mathbb { E } _ { \mathbb { Z } } \| z _ { i } \| { \mathbb { Z } } _ { 2 } ^ { 2 } } \\ & = \displaystyle \frac { 1 } { b n } \displaystyle \sum _ { i = 1 } ^ { n } \mathbb { E } _ { \mathbb { Z } } \| [ z _ { 1 } | { \mathbb { Z } } _ { 2 } ] - \displaystyle \frac \end{array} +$$ + +In (23), the equality (a) holds since $\mathbb { E } _ { \mathbf { u } } [ \mathbf { z } _ { i } ] = \nabla f _ { i , \mu } ( \mathbf { x } _ { k } ) - \nabla f _ { \mu } ( \mathbf { x } _ { k } )$ and $\begin{array} { r } { f _ { \mu } ( \mathbf { x } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } f _ { i , \mu } ( \mathbf { x } ) } \end{array}$ from (4), where we have used the fact that $\mathbb { E } _ { \mathbf { u } } [ \hat { \nabla } f _ { i } ( \mathbf { x } _ { k } ) ] = \hat { \nabla } f _ { i , \mu } ( \mathbf { x } _ { k } )$ (Liu et al., 2018c, Lemma. 1), and recall that $f _ { i , \mu }$ denotes the smoothing function of $f _ { i }$ . The above implies that $\begin{array} { r } { \sum _ { i = 1 } ^ { n } { \mathbb { E } } _ { \mathbf { u } } [ { \mathbf { z } } _ { i } ] = \sum _ { i } \nabla f _ { i , \mu } ( \mathbf { x } _ { k } ) - n \nabla f _ { \mu } ( \mathbf { x } _ { k } ) = \dot { \mathbf { 0 } } } \end{array}$ And the equality (b) holds due to $\mathbb { E } _ { \mathbf { u } } [ \| \mathbf { z } _ { i } \| _ { 2 } ^ { 2 } ] - \| \mathbb { E } _ { \mathbf { u } } [ \mathbf { z } _ { i } ] \| _ { 2 } ^ { 2 } = \mathbb { E } _ { \mathbf { u } } \| \mathbf { z } _ { i } - \mathbb { E } _ { \mathbf { u } } [ \mathbf { z } _ { i } ] \| _ { 2 } ^ { 2 }$ . + +On the other hand, suppose that the mini-batch $\mathcal { T } _ { k }$ contains i.i.d. samples (namely, with replacement), the vectors $\left\{ \mathbf { z } _ { i } \right\}$ are then i.i.d. under both mini-batch sampling and random direction sampling. Therefore, we obtain that + +$$ +\mathbb { E } \left[ \left. \frac { 1 } { b } \sum _ { i \in \mathcal { I } _ { k } } \mathbf { z } _ { i } \right. _ { 2 } ^ { 2 } \right] = \frac { 1 } { b ^ { 2 } } \left[ \mathbb { E } \left[ \sum _ { i \in \mathcal { I } _ { k } } \| \mathbf { z } _ { i } \| _ { 2 } ^ { 2 } \right] + \mathbb { E } \left[ \sum _ { i \neq j , i , j \in \mathcal { I } _ { k } } \langle \mathbf { z } _ { i } , \mathbf { z } _ { j } \rangle \right] \right] +$$ + +where the second equality holds since $\begin{array} { r } { \mathbb { E } _ { i \in \mathcal { T } _ { k } , \mathbf { u } } [ \mathbf { z } _ { i } ] = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathbb { E } _ { \mathbf { u } } [ \mathbf { z } _ { i } ] = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \nabla f _ { i , \mu } ( \mathbf { x } _ { k } ) - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) = \mathbf { 0 } } \end{array}$ . Combining both (23) and (24), we obtain that + +$$ +\mathbb { E } \left[ \left. \frac { 1 } { b } \sum _ { i \in \mathcal { I } _ { k } } \mathbf { z } _ { i } \right. _ { 2 } ^ { 2 } \right] \leq \frac { \alpha _ { b } } { b n } \sum _ { i = 1 } ^ { n } \mathbb { E } _ { \mathbf { u } } [ \left. \mathbf { z } _ { i } \right. _ { 2 } ^ { 2 } ] + \frac { \beta _ { b } } { b n } \sum _ { i = 1 } ^ { n } \mathbb { E } _ { \mathbf { u } } [ \left. \hat { \nabla } f _ { i } ( \mathbf { x } _ { k } ) - \nabla f _ { i , \mu } ( \mathbf { x } _ { k } ) \right. _ { 2 } ^ { 2 } ] . +$$ + +In (25), $\alpha _ { b } \ = \ 1$ and $\beta _ { b } ~ = ~ 0$ if the mini-batch contains i.i.d. samples from $[ n ]$ with replacement, and $\alpha _ { b } = I ( b < n )$ and $\beta _ { b } = I ( b > 1 )$ if samples are randomly selected without replacement. + +In (25), we next bound $\begin{array} { r } { \frac { 1 } { n } \sum _ { i } \mathbb { E } _ { \mathbf { u } } [ \| \mathbf { z } _ { i } \| _ { 2 } ^ { 2 } ] } \end{array}$ , + +$$ +\begin{array} { r l r } { \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathbb { E } _ { \mathbf { u } } [ \| \mathbf { z } _ { i } \| _ { 2 } ^ { 2 } ] } & { \stackrel { ( \mathrm { 2 1 } ) } { = } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathbb { E } _ { \mathbf { u } } \left\| \frac { 1 } { q } \sum _ { j = 1 } ^ { q } \mathbf { z } _ { i , j } \right\| _ { 2 } ^ { 2 } = \frac { 1 } { n q ^ { 2 } } \sum _ { i } \left[ \sum _ { j } \mathbb { E } _ { \mathbf { u } } [ \| \mathbf { z } _ { i , j } \| _ { 2 } ^ { 2 } ] + \sum _ { j \neq k } \langle \mathbb { E } _ { \mathbf { u } } [ \mathbf { z } _ { i , j } ] , \mathbb { E } _ { \mathbf { u } } [ \mathbf { z } _ { i , k } ] \rangle \right] } & \\ & { } & { \displaystyle = \frac { 1 } { n q ^ { 2 } } \sum _ { i } \left[ q \mathbb { E } _ { \mathbf { u } } [ \| \mathbf { z } _ { i , 1 } \| _ { 2 } ^ { 2 } ] + ( q ^ { 2 } - q ) \| \mathbb { E } _ { \mathbf { u } } [ \mathbf { z } _ { i , 1 } ] \| _ { 2 } ^ { 2 } \right] } & \\ & { } & { \displaystyle = \frac { 1 } { q n } \sum _ { i } \mathbb { E } _ { \mathbf { u } } [ \| \mathbf { z } _ { i , 1 } \| _ { 2 } ^ { 2 } ] + \frac { q - 1 } { q n } \sum _ { i } \| \mathbb { E } _ { \mathbf { u } } [ \mathbf { z } _ { i , 1 } ] \| _ { 2 } ^ { 2 } , } & { ( 2 } \end{array} +$$ + +where we have used the facts that $\mathbb { E } _ { \mathbf { u } } [ \mathbf { z } _ { i , j } ] = \mathbb { E } _ { \mathbf { u } } [ \mathbf { z } _ { i , 1 } ]$ and $\mathbb { E } _ { \mathbf { u } } [ \| \mathbf { z } _ { i , j } \| _ { 2 } ^ { 2 } ] = \mathbb { E } _ { \mathbf { u } } [ \| \mathbf { z } _ { i , 1 } \| _ { 2 } ^ { 2 } ]$ for any $j$ since random direction vectors $\{ \mathbf { u } _ { i , j } \} _ { j = 1 } ^ { q }$ are i.i.d. samples. + +In (26), we further bound $\begin{array} { r } { \sum _ { i } \mathbb { E } _ { \mathbf { u } } [ \| \mathbf { z } _ { i , 1 } \| _ { 2 } ^ { 2 } ] } \end{array}$ , + +$$ +\begin{array} { r l r } { { \frac { 1 } { n } \sum _ { i } \mathbb { E } _ { \mathbf { u } } [ \| \mathbf { z } _ { i , 1 } \| _ { 2 } ^ { 2 } ] = \frac { 1 } { n } \sum _ { i } \mathbb { E } _ { \mathbf { u } } \| \hat { \nabla } f _ { i } ( \mathbf { x } _ { k } ; \mathbf { u } _ { i , 1 } ) - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } ^ { 2 } } } \\ & { } & { \quad = \frac { 1 } { n } \sum _ { i } \mathbb { E } _ { \mathbf { u } } [ \| \hat { \nabla } f _ { i } - \nabla f _ { i , \mu } + \nabla f _ { i , \mu } - \nabla f _ { \mu } \| _ { 2 } ^ { 2 } ] } \\ & { } & { \quad \leq \displaystyle \frac { 2 } { n } \sum _ { i } \mathbb { E } _ { \mathbf { u } } [ \| \hat { \nabla } f _ { i } - \nabla f _ { i , \mu } \| _ { 2 } ^ { 2 } ] + \frac { 2 } { n } \sum _ { i } \| \nabla f _ { i , \mu } - \nabla f _ { \mu } \| _ { 2 } ^ { 2 } , } \end{array} +$$ + +where for ease of notation, let $\hat { \nabla } f _ { i } : = \hat { \nabla } f _ { i } ( \mathbf x _ { k } ; \mathbf u _ { i , 1 } )$ , $\hat { \nabla } f _ { i , \mu } : = \hat { \nabla } f _ { i , \mu } ( \mathbf x _ { k } )$ and $\nabla f _ { \mu } : = \nabla f _ { \mu } ( \mathbf { x } _ { k } )$ . According to (Liu et al., 2018c, Lemma 1), the first term at RHS of (27) yields + +$$ +\mathbb { E } _ { \mathbf { u } } [ \| \hat { \nabla } f _ { i } - \nabla f _ { i , \mu } \| _ { 2 } ^ { 2 } ] \le 2 d \| \nabla f _ { i } \| _ { 2 } ^ { 2 } + \frac { \mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } \le 2 d \sigma ^ { 2 } + \frac { \mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } : = C ( d , \mu ) , +$$ + +where the last inequality holds due to A2. Based on the definition of $f _ { \mu }$ , the second term at RHS of (27) yields + +$$ +\begin{array} { r l r } { { \displaystyle \frac { 1 } { n } \sum _ { i } \| \nabla f _ { i , \mu } - \nabla f _ { \mu } \| _ { 2 } ^ { 2 } = \frac { 1 } { n } \sum _ { i } \| \mathbb { E } _ { \mathbf { v } } \big [ \nabla f _ { i } ( \mathbf { x } _ { k } + \mu \mathbf { v } ) - \nabla f ( \mathbf { x } _ { k } + \mu \mathbf { v } ) \big ] \| _ { 2 } ^ { 2 } } } \\ & { } & { \leq \frac { 1 } { n } \sum _ { i } \mathbb { E } _ { \mathbf { v } } \big [ \| \nabla f _ { i } ( \mathbf { x } _ { k } + \mu \mathbf { v } ) - \nabla f ( \mathbf { x } _ { k } + \mu \mathbf { v } ) \| _ { 2 } ^ { 2 } \big ] \leq 4 \sigma ^ { 2 } , } \end{array} +$$ + +where we have used the Jensen’s inequality and $\begin{array} { r } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \| \nabla f _ { i } ( \mathbf { x } ) - \nabla f ( \mathbf { x } ) \| _ { 2 } ^ { 2 } \leq 4 \sigma ^ { 2 } } \end{array}$ under A2. + +Substituting (28) and (29) into (27), we have + +$$ +\frac { 1 } { n } \sum _ { i } \mathbb { E } _ { \mathbf { u } } [ \| \mathbf { z } _ { i , 1 } \| _ { 2 } ^ { 2 } ] = \frac { 1 } { n } \sum _ { i } \mathbb { E } _ { \mathbf { u } } [ \| \hat { \nabla } f _ { i } - \nabla f _ { \mu } \| _ { 2 } ^ { 2 } ] \leq 4 d \sigma ^ { 2 } + \mu ^ { 2 } L ^ { 2 } d ^ { 2 } + 8 \sigma ^ { 2 } = 2 C ( d , \mu ) + 8 \sigma ^ { 2 } , +$$ + +where $C ( d , \mu )$ was defined in (28). + +We are now ready to bound (26). Based on $\begin{array} { r } { \frac { 1 } { n } \sum _ { i } \| \mathbb { E } _ { { \mathbf { u } } } [ { \mathbf { z } } _ { i , 1 } ] \| _ { 2 } ^ { 2 } = \frac { 1 } { n } \sum _ { i } \| \nabla f _ { i , \mu } - \nabla f _ { \mu } \| _ { 2 } ^ { 2 } \leq 4 \sigma ^ { 2 } } \end{array}$ from (29), and substituting (30) into (26), we obtain that + +$$ +\frac { 1 } { n } \sum _ { i } \mathbb { E } _ { \mathbf { u } } [ \left. \mathbf { z } _ { i } \right. _ { 2 } ^ { 2 } ] \leq \frac { 2 C ( d , \mu ) + 8 \sigma ^ { 2 } } { q } + \frac { 4 ( q - 1 ) \sigma ^ { 2 } } { q } . +$$ + +In (25), we also need to bound $\mathbb { E } _ { \mathbf { u } } [ \| \hat { \nabla } f _ { i } ( \mathbf { x } _ { k } ) - \nabla f _ { i , \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } ^ { 2 } ]$ + +$$ +\begin{array} { r l } & { \quad \mathbb { E } _ { \mathbf { u } } \left[ \left\| \hat { \nabla } f _ { i } ( \mathbf { x } _ { k } ) - \nabla f _ { i , \mu } ( \mathbf { x } _ { k } ) \right\| _ { 2 } ^ { 2 } \right] \stackrel { ( 2 0 ) } { = } \mathbb { E } _ { \mathbf { u } } \left[ \left\| \frac { 1 } { q } \sum _ { j = 1 } ^ { q } \left( \hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } _ { i , j } ) - \nabla f _ { i , \mu } ( \mathbf { x } ) \right) \right\| _ { 2 } ^ { 2 } \right] } \\ & { \stackrel { ( a ) } { = } \frac { 1 } { q } \mathbb { E } _ { \mathbf { u } } \left[ \left\| \hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } _ { i , 1 } ) - \nabla f _ { i , \mu } ( \mathbf { x } ) \right\| _ { 2 } ^ { 2 } \right] \stackrel { ( 2 8 ) } { \leq } \frac { 1 } { q } \left( 2 d \sigma ^ { 2 } + \frac { \mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } \right) = \frac { C ( d , \mu ) } { q } , } \end{array} +$$ + +where the equality (a) holds since $\mathbb { E } _ { \mathbf { u } } [ \hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } _ { i , j } ) ] \ = \ \nabla f _ { i , \mu } ( \mathbf { x } )$ for any $j$ , given by (Liu et al., 2018c, Lemma 1). + +Substituting (31) and (32) into (25) + +$$ +\begin{array} { r } { \mathbb { E } \left[ \left. \hat { \mathbf { g } } _ { k } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \right. _ { 2 } ^ { 2 } \right] = \mathbb { E } \left[ \left. \displaystyle \frac { 1 } { b } \sum _ { i \in \mathcal { I } _ { k } } \mathbf { z } _ { i } \right. _ { 2 } ^ { 2 } \right] \leq \displaystyle \frac { \alpha _ { b } } { b } \displaystyle \frac { 2 C ( d , \mu ) + 4 \sigma ^ { 2 } + 4 \sigma ^ { 2 } q } { q } + \displaystyle \frac { \beta _ { b } C ( d , \mu ) } { b q } } \\ { = \displaystyle \frac { 4 \alpha _ { b } ( q + 1 ) } { b q } \sigma ^ { 2 } + \displaystyle \frac { C ( d , \mu ) } { b q } ( 2 \alpha _ { b } + \beta _ { b } ) . } \end{array} +$$ + +# 4 PROOF OF THEOREM 1 + +Substituting (6) into (5), we obtain + +$$ +\begin{array} { r l r } { { \mathbb { E } [ \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } ] \le \mathbb { E } [ f _ { \mu } ( \mathbf { x } _ { 0 } ) - f _ { \mu } ( \mathbf { x } _ { T } ) ] + \sum _ { k = 0 } ^ { T - 1 } [ 2 \delta _ { k } \frac { \sqrt { d } } { \sqrt { b q } } \sqrt { 4 \alpha _ { b } ( q + 1 ) \sigma ^ { 2 } + C ( d , \mu ) ( 2 \alpha _ { b } + \beta _ { b } ) } ] } } \\ & { } & { \quad + \frac { d L } { 2 } \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } ^ { 2 } . \eqno ( 3 4 . } \end{array} +$$ + +It is known from (Liu et al., 2018c, Lemma 1) that + +$$ +\| \nabla f ( \mathbf { x } ) \| _ { 2 } ^ { 2 } \leq 2 \| \nabla f _ { \mu } ( \mathbf { x } ) \| _ { 2 } ^ { 2 } + { \frac { \mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } } , \quad | f _ { \mu } ( \mathbf { x } ) - f ( \mathbf { x } ) | \leq { \frac { \mu ^ { 2 } L } { 2 } } . +$$ + +From (35) we have $\begin{array} { r } { f _ { \mu } ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } _ { 0 } ) \leq \frac { \mu ^ { 2 } L } { 2 } } \end{array}$ and $\begin{array} { r } { f ^ { * } - f _ { \mu } ^ { * } \le \frac { \mu ^ { 2 } L } { 2 } } \end{array}$ , where $f _ { \mu } ^ { * } = \mathrm { m i n } _ { \mathbf { x } } f _ { \mu } ( \mathbf { x } )$ and $f ^ { * } = \operatorname* { m i n } _ { \mathbf { x } } { f ( \mathbf { x } ) }$ . This yields $f _ { \mu } ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } _ { 0 } ) + f ^ { * } - f _ { \mu } ^ { * } \leq \mu ^ { 2 } L$ , and thus + +$$ +f _ { \mu } ( \mathbf { x } _ { 0 } ) - f _ { \mu } ( \mathbf { x } _ { T } ) \leq f _ { \mu } ( \mathbf { x } _ { 0 } ) - f _ { \mu } ^ { * } \leq ( f ( \mathbf { x } _ { 0 } ) - f ^ { * } ) + \mu ^ { 2 } L . +$$ + +Substituting (36) into (34), we obtain + +$$ +\begin{array} { r l } & { \mathbb { E } \left[ \displaystyle \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } \right] \leq \mathbb { E } [ f ( \mathbf { x } _ { 0 } ) - f ^ { * } ) ] + \mu ^ { 2 } L } \\ & { \qquad + \displaystyle \sum _ { k = 0 } ^ { T - 1 } \left[ 2 \delta _ { k } \frac { \sqrt { d } } { \sqrt { b q } } \sqrt { 4 \alpha _ { b } ( q + 1 ) \sigma ^ { 2 } + C ( d , \mu ) ( 2 \alpha _ { b } + \beta _ { b } ) } \right] + \frac { d L } { 2 } \displaystyle \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } ^ { 2 } . } \end{array} +$$ + +Due to $\| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } \leq \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 }$ and dividing $\textstyle \sum _ { k = 0 } ^ { T - 1 } \delta _ { k }$ for both sides of (37), we obtain that + +$$ +\begin{array} { r } { \mathbb { E } \left[ \displaystyle \sum _ { k = 0 } ^ { T - 1 } \frac { \delta _ { k } } { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } \right] \leq \frac { f ( \mathbf { x } _ { 0 } ) - f ^ { * } + \mu ^ { 2 } L } { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } } + \frac { 2 \sqrt { d } } { \sqrt { b q } } \sqrt { 4 \alpha _ { b } ( q + 1 ) \sigma ^ { 2 } + C ( d , \mu ) ( 2 \alpha _ { b } + \beta _ { b } ) } } \\ { + \frac { d L } { 2 } \frac { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } ^ { 2 } } { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } } . \qquad ( 3 \leq \alpha \leq \alpha ) . } \end{array} +$$ + +By introducing the random variable $R$ with probability $\begin{array} { r } { P ( R = k ) = \frac { \delta _ { k } } { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } } } \end{array}$ , we then obtain that + +$$ +\mathbb { E } \left[ \Vert \nabla f _ { \mu } ( \mathbf { x } _ { R } ) \Vert _ { 2 } \right] = \mathbb { E } \left[ \mathbb { E } _ { R } [ \Vert \nabla f _ { \mu } ( \mathbf { x } _ { R } ) \Vert _ { 2 } ] \right] = \mathbb { E } \left[ \sum _ { k = 0 } ^ { T - 1 } P ( R = k ) \Vert \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \Vert _ { 2 } \right] . +$$ + +Based on (35), we further have + +$$ +\begin{array} { r l } & { \displaystyle \mathbb { E } \left[ \| \nabla f ( \mathbf { x } _ { R } ) \| _ { 2 } \right] \leq \mathbb { E } \left[ \sqrt { 2 \| \nabla f _ { \mu } ( \mathbf { x } _ { R } ) \| _ { 2 } ^ { 2 } + \frac { \mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } } \right] } \\ & { \displaystyle \leq \sqrt { 2 } \mathbb { E } [ \| \nabla f _ { \mu } ( \mathbf { x } _ { R } ) \| _ { 2 } ] + \frac { \mu L d } { \sqrt { 2 } } , } \end{array} +$$ + +where we have used the fact that ${ \sqrt { a ^ { 2 } + b ^ { 2 } } } \leq ( a + b )$ for $a , b \geq 0$ . + +Substituting (38) and (39) into (40), we finally obtain (8). + +# 5 PROOF OF COROLLARY 1 + +Upon defining $\begin{array} { r } { \tilde { \nabla } f _ { i } ( \mathbf x ; \mathbf u ) = \frac { d \left[ f _ { i } ( \mathbf x + \mu \mathbf u ) - f _ { i } ( \mathbf x - \mu \mathbf u ) \right] \mathbf u } { 2 \mu } } \end{array}$ (against $\hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } )$ in (3)), our major task is to derive the first- and second-order moments of $\tilde { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } )$ . + +Given $\mathbf { x }$ , we first study the mean of $\tilde { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } )$ + +$$ +\begin{array} { r l } & { \mathbb { E } _ { \mathbf { u } } \left[ \tilde { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } ) \right] = \mathbb { E } _ { \mathbf { u } } \left[ \frac { d } { 2 \mu } ( f _ { i } ( \mathbf { x } + \mu \mathbf { u } ) - f _ { i } ( \mathbf { x } - \mu \mathbf { u } ) ) \mathbf { u } \right] } \\ & { = \mathbb { E } _ { \mathbf { u } } \left[ \frac { d } { 2 \mu } f _ { i } ( \mathbf { x } + \mu \mathbf { u } ) \mathbf { u } \right] + \mathbb { E } _ { \mathbf { u } } \left[ \frac { d } { 2 \mu } f _ { i } ( \mathbf { x } - \mu \mathbf { u } ) ( - \mathbf { u } ) \right] \stackrel { ( a ) } = \mathbb { E } _ { \mathbf { u } } \left[ \frac { d } { \mu } f _ { i } ( \mathbf { x } + \mu \mathbf { u } ) \mathbf { u } \right] } \\ & { \stackrel { ( b ) } = \mathbb { E } _ { \mathbf { u } } \left[ \frac { d } { \mu } \left( f _ { i } ( \mathbf { x } + \mu \mathbf { u } ) - f _ { i } ( \mathbf { x } ) \right) \mathbf { u } \right] = \mathbb { E } _ { \mathbf { u } } \left[ \hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } ) \right] \stackrel { ( c ) } = \nabla f _ { i , \mu } ( \mathbf { x } ) , } \end{array} +$$ + +where (a) holds since the distribution of $\mathbf { u }$ is symmetric around the origin, (b) holds since $\mathbb { E } [ { \mathbf { u } } ] = \mathbf { 0 }$ , and (c) holds since $\begin{array} { r } { \nabla f _ { i , \mu } ( \mathbf { x } ) = \mathbb { E } _ { \mathbf { u } } \left[ \frac { d } { \mu } f _ { i } ( \mathbf { x } + \mu \mathbf { u } ) \mathbf { u } \right] } \end{array}$ obtained from (Gao et al., 2014, Lemma 4.1). It is clear from (41) that $\mathbb { E } _ { \mathbf { u } } \left[ \tilde { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } ) \right] = \mathbb { E } _ { \mathbf { u } } \left[ \hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } ) \right]$ . + +We next study the second-order moment of $\tilde { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } )$ , + +$$ +\begin{array} { l } { \displaystyle \mathbb { E } _ { \mathbf { u } } [ \| \tilde { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } ) \| ^ { 2 } ] = \frac { d ^ { 2 } } { 4 \mu ^ { 2 } } \mathbb { E } _ { \mathbf { u } } [ ( f _ { i } ( \mathbf { x } + \mu \mathbf { u } ) - f _ { i } ( \mathbf { x } - \mu \mathbf { u } ) ) ^ { 2 } \| \mathbf { u } \| ^ { 2 } ] } \\ { \displaystyle \leq \frac { d ^ { 2 } } { 2 \mu ^ { 2 } } \mathbb { E } _ { \mathbf { u } } [ ( f _ { i } ( \mathbf { x } + \mu \mathbf { u } ) - f _ { i } ( \mathbf { x } ) ) ^ { 2 } + ( f _ { i } ( \mathbf { x } ) - f _ { i } ( \mathbf { x } - \mu \mathbf { u } ) ) ^ { 2 } ] } \\ { \displaystyle = \mathbb { E } _ { \mathbf { u } } [ \| \frac { d } { \mu } ( f _ { i } ( \mathbf { x } + \mu \mathbf { u } ) - f _ { i } ( \mathbf { x } ) ) \mathbf { u } \| ^ { 2 } ] = \mathbb { E } _ { \mathbf { u } } [ \| \hat { \nabla } f _ { i } ( \mathbf { x } ; \mathbf { u } ) \| ^ { 2 } ] , } \end{array} +$$ + +where we have used the fact that $\| \mathbf { u } \| _ { 2 } = 1$ . + +Based on (41) and (42), we can conclude that both (3) and (10) maintain the same statistical properties. Following the proofs of Proposition 2 and Theorem 1, it is not difficult to reach the convergence rate in (8).  + +# 6 PROOF OF COROLLARY 2 + +Let $\hat { \mathbf { g } } _ { k } ^ { i , j } : = \hat { \nabla } f _ { i } ( \mathbf x _ { k } ; \mathbf u _ { i , j } )$ . If we replace $\hat { \bf g } _ { k }$ with $\hat { \mathbf { g } } _ { k } ^ { i , j }$ in (18), we then have + +$$ +| ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) \iota | \mathrm { P r o b } \left[ \mathrm { s i g n } ( \hat { g } _ { k , l } ^ { i , j } ) \neq \mathrm { s i g n } ( ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) \iota ) \right] \leq \mathbb { E } [ | \hat { g } _ { k , l } ^ { i , j } - ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) \iota | ] , +$$ + +where $\hat { g } _ { k , l } ^ { i , j }$ is the lth coordinate of $\hat { \mathbf { g } } _ { k } ^ { i , j }$ + +By letting $b = 1$ and $q = 1$ in Proposition 2, we know that E $\big [ \lVert \hat { \mathbf { g } } _ { k } ^ { i , j } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \rVert _ { 2 } ^ { 2 } \big ]$ is upper bounded. Moreover, by Jensen’s inequality we have + +$$ +\begin{array} { r } { \mathbb { E } [ | \hat { g } _ { k , l } ^ { i , j } - ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) \ b { \imath } | ] \leq \sqrt { \mathbb { E } [ ( \hat { g } _ { k , l } ^ { i , j } - ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) \ b { \imath } ) ^ { 2 } ] } : = \xi _ { l } , } \end{array} +$$ + +where $\xi _ { l }$ is finite since $\mathbb { E } \left[ \lVert \hat { \mathbf { g } } _ { k } ^ { i , j } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \rVert _ { 2 } ^ { 2 } \right]$ is upper bounded. + +Substituting (44) into (43), we have + +$$ +\begin{array} { r l } & { | ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { l } | \operatorname { P r o b } \left[ \operatorname { s i g n } ( \hat { g } _ { k , l } ^ { i , j } ) \neq \operatorname { s i g n } ( ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { l } ) \right] \leq \xi _ { l } . } \end{array} +$$ + +With the new gradient estimate $\begin{array} { r } { \bar { \bf g } _ { k } = \sum _ { i \in \mathcal { T } _ { k } } \sum _ { j = 1 } ^ { q } \mathrm { s i g n } \big ( \hat { \bf g } _ { k } ^ { i , j } \big ) } \end{array}$ in (11), we require to bound + +$$ +\operatorname { P r o b } \left[ \operatorname { s i g n } ( { \bar { g } } _ { k , l } ) \neq \operatorname { s i g n } ( ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { l } ) \right] , +$$ + +where ${ \bar { g } } _ { k , l }$ is the $l$ th coordinate of $\bar { \bf g } _ { k }$ . + +We recall that $\hat { g } _ { k , l } ^ { i , j }$ is an unbiased stochastic approximation to gradient component $( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { l }$ with variance $\xi _ { l } ^ { 2 }$ . Under the assumption that the noise distribution is unimodal and symmetric, we know from (Bernstein et al., 2018, Lemma D1) that + +$$ +\mathrm { P r o b } \left[ \mathrm { s i g n } ( \hat { g } _ { k , l } ^ { i , j } ) \neq \mathrm { s i g n } ( ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { l } \right] : = Q \leq \left\{ \begin{array} { l l } { \frac { 2 } { 9 } \frac { 1 } { S ^ { 2 } } } & { S > \frac { 2 } { \sqrt { 3 } } } \\ { \frac { 1 } { 2 } - \frac { S } { 2 \sqrt { 3 } } } & { \mathrm { o t h e r w i s e } } \end{array} \right. < \frac { 1 } { 2 } , +$$ + +where $S : = | ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) \iota | / \xi _ { l }$ + +Let $Z$ count the number of estimates $\{ \hat { g } _ { k , l } ^ { i , j } \}$ yielding the correct sign of $( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { l }$ . Thus, the probability of error in (46) is equal to + +$$ +\operatorname { P r o b } \left[ \operatorname { s i g n } ( { \bar { g } } _ { k , l } ) \neq \operatorname { s i g n } ( ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { l } ) \right] = \operatorname { P r o b } \left[ Z \leq { \frac { b q } { 2 } } \right] . +$$ + +Following the proof of (Bernstein et al., 2018, Theorem 2b) under (47), it is not difficult to obtain that + +$$ +\operatorname { P r o b } \left[ Z \leq { \frac { b q } { 2 } } \right] \leq { \frac { 1 } { \sqrt { b q } S } } . +$$ + +That is, + +$$ +| ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) \iota | \operatorname { P r o b } \left[ \mathrm { s i g n } ( { \bar { g } } _ { k , l } ) \neq \mathrm { s i g n } ( ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) \iota ) \right] \leq { \frac { \xi _ { l } } { \sqrt { b q } } } . +$$ + +Replace $\hat { \bf g } _ { k }$ with $\bar { \bf g } _ { k }$ in (17), we obtain that + +$$ +\begin{array} { r l } { \mathbb { E } \{ f _ { \mu } ( \mathbf { x } _ { k + 1 } ) - f _ { \mu } ( \mathbf { x } _ { k } ) \} \leq - \delta _ { k } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } + \frac { \delta _ { k } ^ { 2 } d L } { 2 } } \\ & { \qquad + 2 \delta _ { k } \displaystyle \sum _ { l = 1 } ^ { d } | ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { l } | \mathrm { P r o b } \big [ \mathrm { s i g n } ( \bar { g } _ { k , l } ) \neq \mathrm { s i g n } ( ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { l } ) \big ] } \\ { \overset { ( \mathrm { s p } ) } { \leq } - \delta _ { k } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } + \frac { \delta _ { k } ^ { 2 } d L } { 2 } + 2 \delta _ { k } \frac { 1 } { \sqrt { N q } } \| \xi \| _ { 1 } } \\ & { \qquad \leq - \delta _ { k } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } + \frac { \delta _ { k } ^ { 2 } d L } { 2 } + 2 \delta _ { k } \frac { \sqrt { d } } { \sqrt { N q } } \sqrt { | \xi \| _ { 2 } ^ { 2 } } } \\ & { \overset { ( \mathrm { a } 4 ) } { = } - \delta _ { k } \| \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 1 } + \frac { \delta _ { k } ^ { 2 } d L } { 2 } + 2 \delta _ { k } \frac { \sqrt { d } } { \sqrt { N q } } \sqrt { \mathbb { E } [ \| \hat { g } _ { k } ^ { i , j } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } ^ { 2 } ] } . } \end{array} +$$ + +Compared (51) with (19), the standard deviation $\sqrt { \mathbb { E } [ \| \hat { \mathbf { g } } _ { k } ^ { i , j } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \| _ { 2 } ^ { 2 } ] }$ is reduced by the factor $1 / { \sqrt { b q } }$ . According to Proposition 2, let $b = q = 1$ , we obtain + +$$ +\begin{array} { r } { \mathbb { E } \left[ \| \hat { \mathbf { g } } _ { k } ^ { i , j } - \nabla f _ { \mu } ( { \mathbf { x } } _ { k } ) \| _ { 2 } ^ { 2 } \right] \leq 8 \alpha _ { b } \sigma ^ { 2 } + ( 2 \alpha _ { b } + \beta _ { b } ) C ( d , \mu ) . } \end{array} +$$ + +Based on (51)-(52) and following the proof of Theorem 1, we have + +$$ +\begin{array} { r l } & { \displaystyle \mathbb { E } \left[ \| \nabla f ( \mathbf { x } _ { R } ) \| _ { 2 } \right] \leq \sqrt { 2 } \frac { f ( \mathbf { x } _ { 0 } ) - f ^ { * } + \mu ^ { 2 } L } { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } } } \\ & { \displaystyle + \frac { 2 \sqrt { 2 } \sqrt { d } } { \sqrt { b q } } \sqrt { 8 \alpha _ { b } \sigma ^ { 2 } + C ( d , \mu ) ( 2 \alpha _ { b } + \beta _ { b } ) } + \frac { d L } { \sqrt { 2 } } \frac { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } ^ { 2 } } { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } } + \frac { \mu L d } { \sqrt { 2 } } , } \end{array} +$$ + +where $C ( d , \mu ) : = 2 d \sigma ^ { 2 } + \mu ^ { 2 } L ^ { 2 } d ^ { 2 } / 2$ . + +If $\begin{array} { r } { \mu = O ( \frac { 1 } { \sqrt { d T } } ) } \end{array}$ and $\begin{array} { r } { \delta _ { k } = { O } \left( \frac { 1 } { \sqrt { d T } } \right) } \end{array}$ , then the convergence rate simplifies to $\begin{array} { r } { O ( \frac { \sqrt { d } } { \sqrt { T } } + \frac { d } { \sqrt { b q } } ) } \end{array}$ . + +# 7 PROOF OF COROLLARY 3 + +Let $\begin{array} { r } { \hat { \mathbf { g } } _ { k } ^ { m } : = \frac { 1 } { b _ { m } q } \sum _ { i \in \mathcal { T } _ { m , k } } \sum _ { j = 1 } ^ { q } \hat { \nabla } f _ { i } ( \mathbf x _ { k } ; \mathbf u _ { i , j } ) } \end{array}$ and $\begin{array} { r } { \bar { \bf g } _ { k } = \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \mathrm { s i g n } ( \hat { \bf g } _ { k } ^ { m } ) } \end{array}$ . Following (43)-(50), we can similarly obtain that + +$$ +| ( \nabla f _ { \mu } ( { \bf x } _ { k } ) ) _ { l } | \mathrm { P r o b } [ \mathrm { s i g n } ( \bar { g } _ { k , l } ) \neq \mathrm { s i g n } ( ( \nabla f _ { \mu } ( { \bf x } _ { k } ) ) _ { l } ) ] \leq \frac { \xi _ { l } } { \sqrt { M } } , +$$ + +where $\xi _ { l } : = \sqrt { \mathbb { E } [ ( \hat { g } _ { k , l } ^ { m } - ( \nabla f _ { \mu } ( \mathbf { x } _ { k } ) ) _ { l } ) ^ { 2 } ] }$ . By mimicking the derivation in (51), we have + +$$ +\mathbb { E } [ f _ { \mu } ( { \mathbf x } _ { k + 1 } ) - f _ { \mu } ( { \mathbf x } _ { k } ) ] \leq - \delta _ { k } \| \nabla f _ { \mu } ( { \mathbf x } _ { k } ) \| _ { 1 } + \frac { \delta _ { k } ^ { 2 } d L } { 2 } + 2 \delta _ { k } \frac { \sqrt { d } } { \sqrt { M } } \sqrt { \mathbb { E } [ \| \hat { \mathbf g } _ { k } ^ { m } - \nabla f _ { \mu } ( { \mathbf x } _ { k } ) \| _ { 2 } ^ { 2 } ] } . +$$ + +According to Proposition 2, let $b = \lfloor n / M \rfloor$ , we obtain + +$$ +\mathbb { E } \left[ \Vert \hat { \mathbf { g } } _ { k } ^ { m } - \nabla f _ { \mu } ( \mathbf { x } _ { k } ) \Vert _ { 2 } ^ { 2 } \right] \leq \frac { 4 ( q + 1 ) } { b q } \sigma ^ { 2 } + \frac { 3 } { b q } C ( d , \mu ) . +$$ + +Based on (55)-(56) and following the proof of Theorem 1, we have + +$$ +\begin{array} { r l } & { \mathbb { E } \left[ \| \nabla f ( \mathbf { x } _ { R } ) \| _ { 2 } \right] \leq \sqrt { 2 } \frac { f ( \mathbf { x } _ { 0 } ) - f ^ { * } + \mu ^ { 2 } L } { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } } } \\ & { + \frac { 2 \sqrt { 2 } \sqrt { d } } { \sqrt { M b q } } \sqrt { 4 ( q + 1 ) \sigma ^ { 2 } + 3 C ( d , \mu ) } + \frac { d L } { \sqrt { 2 } } \frac { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } ^ { 2 } } { \sum _ { k = 0 } ^ { T - 1 } \delta _ { k } } + \frac { \mu L d } { \sqrt { 2 } } , } \end{array} +$$ + +where $C ( d , \mu ) : = 2 d \sigma ^ { 2 } + \mu ^ { 2 } L ^ { 2 } d ^ { 2 } / 2$ . + +Since $\begin{array} { r } { \mu = O ( \frac { 1 } { \sqrt { d T } } ) } \end{array}$ , $\begin{array} { r } { \delta _ { k } = { O } \left( \frac { 1 } { \sqrt { d T } } \right) } \end{array}$ and $b M \approx n$ , then the convergence rate simplifies to $\begin{array} { r } { O ( \frac { \sqrt { d } } { \sqrt { T } } + \frac { \sqrt { d } } { \sqrt { n } } + \frac { d } { \sqrt { n q } } ) } \end{array}$ + +# 8 SUPPLEMENTAL EXPERIMENTS + +# 8.1 SYNTHETIC EXPERIMENTS + +In Fig. A3, we demonstrate the effect of the learning rate $\delta$ on the training loss of SGD, signSGD, ZO-SGD, ZO-SCD, ZO-signSGD and ZO-M-signSGD. We observe that compared to the gradient-based algorithms (SGD, ZO-SGD and ZO-SCD), the gradient sign-based algorithms support a more flexible choice of learning rates (corresponding to less variance), since the sign operation has an normalization effect to reduce oscillations in convergence. We find $\delta = 0 . 1$ best for SGD, $\delta = 0 . 0 0 9$ best for signSGD, $\delta = 0 . 1$ best for ZOSGD and ZOSCD, $\bar { \delta } = 0 . 0 1 7 8$ best for ZOsignSGD, and $\delta = 0 . 0 5 0 1$ best for ZO-M-signSGD. + +![](images/b814eebc6dcda2f68f8e69b9576e2d7eccddf9b5d27d6725c23c28669304782f.jpg) +Figure A3: The training loss at the last iteration versus the constant learning rate $\delta \in [ 1 0 ^ { - 3 } , 0 . 1 ]$ . Here the solid line represents the loss averaged over 10 independent trials with random initialization, and the shaded region indicates the standard deviation of results over random trials. + +# 8.2 BLACK-BOX ATTACK FORMULATION + +The target DNN classifier $F = [ F _ { 1 } , F _ { 2 } , \dots , F _ { K } ]$ takes an image as an input and produces the classification predictions (here the probability distribution) of $K$ image classes, where $F _ { k }$ denotes the prediction of class $k$ Given $F$ , an adversarial example $\mathbf { x }$ of a legitimate example $\mathbf { x } _ { 0 }$ means it is visually similar to $\mathbf { x } _ { 0 }$ but will yield a different top-1 prediction than $\mathbf { x } _ { 0 }$ . + +Let $\left( \mathbf { x } _ { 0 } , t _ { 0 } \right)$ denote a legitimate image $\mathbf { x } _ { 0 }$ and its groundtruth label $t _ { 0 } ~ \in ~ \{ 1 , 2 , \ldots , K \}$ . Without loss of generality, we assume the pixel value range of $\mathbf { x } _ { 0 }$ lies within $[ - 0 . 5 , 0 . 5 ] ^ { d }$ . By using the tanh transformation on the adversarial image $\mathbf { x }$ such that $\mathbf { x } = \operatorname { t a n h } ( \mathbf { w } ) / 2$ , where $\mathbf { w } \in \mathbb { R } ^ { d }$ , we adopt the untargeted black-box attacking loss designed in (Chen et al., 2017), which is defined as + +$$ +\underset { \mathbf { w } \in \mathbb { R } ^ { d } } { \mathrm { m i n i m i z e ~ } } c \cdot \mathbf { m a x } \{ \log F _ { t _ { 0 } } ( \operatorname { t a n h } ( \mathbf { w } ) / 2 ) - \underset { j \neq t _ { 0 } } { \mathrm { m a x } } \log F _ { j } ( \operatorname { t a n h } ( \mathbf { w } ) / 2 ) , 0 \} + \| \operatorname { t a n h } ( \mathbf { w } ) / 2 - \mathbf { x } _ { 0 } \| _ { 2 } ^ { 2 } , +$$ + +where $c$ is a regularization coefficient and $\mathbf { x } ~ = ~ \operatorname { t a n h } ( \mathbf { w } ) / 2$ ensures $\mathbf { x }$ lies within the valid image space $[ - 0 . 5 , 0 . 5 ] ^ { d }$ . The first term in the attacking objective is a hinge loss function that penalizes the top-1 prediction of $\mathbf { x }$ being $t _ { 0 }$ . The $\log F ( \cdot )$ operation preserves the class prediction ranking and better handles numerical stability. The second term encourages the visual similarity between $\mathbf { x }$ and $\mathbf { x } _ { 0 }$ through penalizing their squared $\ell _ { 2 }$ difference (i.e., the squared distortion). In our experiment, we set $c = 1$ for MNIST and $c = 0 . 1$ for CIFAR-10. We also note that different from the use of signed gradients in existing black-box attacks (e.g., Ilyas et al. (2018a); Bhagoji et al. (2018)) due to the explicit $\ell _ { \infty }$ perturbation constraint, the use and benefit of ZO-signSGD for solving (58) in our experiment are non-trivial since the attacking objective does not impose any $\ell _ { \infty }$ constraint. + +# 8.3 ADDITIONAL BLACK-BOX ATTACKING RESULTS ON MNIST AND CIFAR-10 + +Table A1: First successful adversarial examples attacking black-box DNN on MNIST. + +
Image ID3238444152134 1777
Original0123456780
Classified as02345689
ZO-SGD Classified as 640723956704
ZO-signSGD672395670A59354
Classified as6
ZO-M-signSGD672395670A
Classified as64593504
ZO-NES612395670A
Classified as6493504
+ +Table A2: First successful adversarial examples attacking black-box DNN on CIFAR-10. + +
Image ID1062568363317128
Original Classified asH airplaneL automobileN bird心 catYO deerdog# frog百 horse福 ship50 truck
ZO-SGD Classified as1 birdL truckDD truckdogMUOE horsecatcatdogtruck53 automobile
ZO-signSGD Classified as1 birdL truckN truckdogMV horsecatE catE dog福 truckautomobile
ZO-M-signSGD Classified asH bird园 truckNE truckdogYUO horse面一 automobile5
ZO-NESHRHL心YUCEcatcatdog心eautomobile
Classified asbirdtrucktruckdoghorsecatcatdogautomobileautomobile
+ +Table A3: Iteration comparison of attacking black-box DNN on MNIST (image ID 177). + +
Iteration03060 90 120162 181 210 241 271
ZO-SGD888888888
Classified as8 888 8 8 0 0 0 0
Iteration0 30 60 90
8 ZO-signSGD114 151 180 210 241 271
888888888
Classified as Iteration 0 30 60 90 120 1508 8 8 8 0 0 0 0 0 180 198 240 271
D ZO-M-signSGD888888888
Classified as 8 8 8 Iteration 0 408 8 8 8 0 0 0 160 200 240 280 320 349
80 120
ZO-NES
Classified as 8 8 88888888808 8 8 8 8 8 8 0
+ +![](images/381d47c27c2eda8a8de00f0e9d4b1bbc6823a776b98903402cc7eff9d6b53489.jpg) +Figure A4: Additional plots of black-box attacking loss versus iteration on MNIST. The solid marker indicates the iteration number that finds the first successful adversarial example, and its loss corresponds to the squared $\ell _ { 2 }$ distortion. + +![](images/6a0c2dfadd5b7081e9e79e0bf1d80841626873ae671e09a2abbd1fc48108a36d.jpg) +Figure A5: Additional plots of black-box attacking loss versus iteration on CIFAR-10. The solid marker indicates the iteration number that finds the first successful adversarial example, and its loss corresponds to the squared $\ell _ { 2 }$ distortion. \ No newline at end of file diff --git a/parse/train/BJe-DsC5Fm/BJe-DsC5Fm_content_list.json b/parse/train/BJe-DsC5Fm/BJe-DsC5Fm_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..0493102ad439152e420b234c425df246f844c4c0 --- /dev/null +++ b/parse/train/BJe-DsC5Fm/BJe-DsC5Fm_content_list.json @@ -0,0 +1,2888 @@ +[ + { + "type": "text", + "text": "SIGNSGD VIA ZEROTH-ORDER ORACLE ", + "text_level": 1, + "bbox": [ + 176, + 98, + 658, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sijia Liu† Pin-Yu Chen† Xiangyi Chen‡ Mingyi Hong‡ †MIT-IBM Watson AI Lab, IBM Research ‡University of Minnesota, Twin Cities ", + "bbox": [ + 183, + 143, + 609, + 189 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 198, + 544, + 213 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we design and analyze a new zeroth-order (ZO) stochastic optimization algorithm, ZO-signSGD, which enjoys dual advantages of gradient-free operations and signSGD. The latter requires only the sign information of gradient estimates but is able to achieve a comparable or even better convergence speed√ than SGD-type algorithms. Our study shows that ZO-signSGD requires √ √ $\\sqrt { d }$ times more iterations than signSGD, leading to a convergence rate of $O ( { \\sqrt { d } } / { \\sqrt { T } } )$ under some mild conditions, where $d$ is the number of optimization variables, and $T$ is the number of iterations. In addition, we analyze the effects of different types of gradient estimators on the convergence of ZO-signSGD, and propose several variants of ZO-signSGD with $O ( { \\sqrt { d } } { \\bar { / } } { \\sqrt { T } } )$ convergence rate. On the application side we explore the connection between ZO-signSGD and black-box adversarial attacks in robust deep learning. Our empirical evaluations on image classification datasets MNIST and CIFAR-10 demonstrate the superior performance of ZO-signSGD on the generation of adversarial examples from black-box neural networks. ", + "bbox": [ + 233, + 232, + 766, + 431 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 459, + 336, + 476 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Zeroth-order (gradient-free) optimization has attracted an increasing amount of attention for solving machine learning (ML) problems in scenarios where explicit expressions for the gradients are difficult or infeasible to obtain. One recent application of great interest is to generate prediction-evasive adversarial examples, e.g., crafted images with imperceptible perturbations to deceive a well-trained image classifier into misclassification. However, the black-box optimization nature limits the practical design of adversarial examples, where internal configurations and operating mechanism of public ML systems (e.g., Google Cloud Vision API) are not revealed to practitioners and the only mode of interaction with the system is via submitting inputs and receiving the corresponding predicted outputs (Papernot et al., 2017; Liu et al., 2017; Chen et al., 2017; Tu et al., 2018; Ilyas et al., 2018b; Cheng et al., 2018; Bhagoji et al., 2018). It was observed in both white-box and black-box settings1 that simply leveraging the sign information of gradient estimates of an attacking loss can achieve superior empirical performance in generating adversarial examples (Goodfellow et al., 2015; Madry et al., 2018; Ilyas et al., 2018a). Spurred by that, this paper proposes a zeroth-order (ZO) sign-based descent algorithm (we call it ‘ZO-signSGD’) for solving black-box optimization problems, e.g. design of black-box adversarial examples. The convergence behavior and algorithmic stability of the proposed ZO-signSGD algorithm are carefully studied in both theory and practice. ", + "bbox": [ + 174, + 484, + 825, + 705 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In the first-order setting, a sign-based stochastic gradient descent method, known as signSGD, was analyzed by (Bernstein et al., 2018; Balles & Hennig, 2017). It was shown in (Bernstein et al., 2018) that signSGD not only reduces the per iteration cost of communicating gradients, but also could yield a faster empirical convergence speed than SGD (Kinga & Adam, 2015). That is because although the sign operation compresses the gradient using a single bit, it could mitigate the negative effect of√ extremely components of gradient noise. Theoretically, signSGD achieves $O ( 1 / \\sqrt { T } )$ convergence rate under the condition of a sufficiently large mini-batch size, where $T$ denotes the total number of iterations. The work in (Balles & Hennig, 2017) established a connection between signSGD and Adam with restrictive convex analysis. Prior to (Bernstein et al., 2018; Balles & Hennig, 2017), although signSGD was not formally defined, the fast gradient sign method (Goodfellow et al., 2015) to generate white-box adversarial examples actually obeys the algorithmic protocol of signSGD. The effectiveness of signSGD has been witnessed by robust adversarial training of deep neural networks (DNNs) (Madry et al., 2018). Given the advantages of signSGD, one may wonder if it can be generalized for ZO optimization and what the corresponding convergence rate is. In this paper, we answer these questions affirmatively. ", + "bbox": [ + 173, + 713, + 825, + 893 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions We summarize our key contributions as follows. ", + "bbox": [ + 173, + 147, + 604, + 161 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We propose a new ZO algorithm, ‘ZO-signSGD’, and rigorously prove its convergence rate√ √ of $O ( { \\sqrt { d } } / { \\sqrt { T } } )$ under mild conditions. Our established convergence analysis applies to both mini-batch sampling schemes with and without replacement. In particular, the ZO sign-based gradient descent algorithm can be treated as a special case in our proposed ZO-signSGD algorithm. We carefully study the effects of different types of gradient estimators on the convergence of ZO-signSGD, and propose three variants of ZO-signSGD for both centralized and distributed ZO optimization. \n• We conduct extensive synthetic experiments to thoroughly benchmark the performance of ZO-signSGD and to investigate its parameter sensitivity. We also demonstrate the superior performance of ZO-signSGD for generating adversarial examples from black-box DNNs. ", + "bbox": [ + 215, + 164, + 825, + 334 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Related work Other types of ZO algorithms have been developed for convex and nonconvex optimization, where the full gradient is approximated via a random or deterministic gradient estimate (Jamieson et al., 2012; Nesterov & Spokoiny, 2015; Ghadimi & Lan, 2013; Duchi et al., 2015; Gao et al., 2014; Shamir, 2017; Hajinezhad et al., 2017; Ghadimi et al., 2016; Lian et al., 2016; Liu et al., 2018b;c; Wang et al., 2018). Examples include ZO-SGD (Ghadimi & Lan, 2013), ZO stochastic coordinate descent (ZO-SCD) (Lian et al., 2016), and ZO stochastic variance reduced gradient descent (ZO-SVRG) (Liu et al., 2018c;a; Gu et al., 2016). Both ZO-SGD and ZO-SCD can√ √ achieve $O ( { \\sqrt { d } } / { \\sqrt { T } } )$ convergence rate. And ZO-SVRG can further improve the iteration complexity to $O ( d / T )$ but suffers from an increase of function query complexity due to the additional variance reduced step, known as ‘gradient blending’ (Liu et al., 2018c), compared to ZO-SGD. The existing work showed that ZO algorithms align with the iteration complexity of their first-order counterparts up to a slowdown effect in terms of a small-degree polynomial of the problem size $d$ . ", + "bbox": [ + 173, + 345, + 826, + 515 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 SIGNSGD & ITS CONNECTION TO ADVERSARIAL MACHINE LEARNING ", + "text_level": 1, + "bbox": [ + 171, + 525, + 799, + 541 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this section, we provide a background on signSGD, together with the problem setup of our interest. In particular, we show that the commonly-used methods for generating adversarial attacks fall into the framework of signSGD. ", + "bbox": [ + 174, + 547, + 825, + 590 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Preliminaries on signSGD Consider a nonconvex finite-sum problem of the form ", + "bbox": [ + 173, + 594, + 723, + 609 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/f7772a7bebf0ff27113cbc3cf325c6031dfb67b835f0804c0f4e7c26390bd731.jpg", + "text": "$$\n\\begin{array} { r l } { \\underset { \\mathbf { x } } { \\mathrm { m i n i m i z e } } } & { { } f ( \\mathbf { x } ) : = ( 1 / n ) \\sum _ { i = 1 } ^ { n } f _ { i } ( \\mathbf { x } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 375, + 616, + 620, + 637 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\mathbf { x } \\in \\mathbb { R } ^ { d }$ are optimization variables, and $\\{ f _ { i } \\}$ are $n$ individual nonconvex cost functions. The finite-sum form (1) encompasses many ML problems, ranging from generalized linear models to neural networks. If the gradients of $\\{ f _ { i } \\}$ are available, then problem (1) can be solved by many first-order methods such as SGD, SCD, and signSGD. The method of our interest is signSGD, which differs from SGD and SCD, takes the sign of gradient (or its estimate) as the descent direction. It was recently shown in (Bernstein et al., 2018) that signSGD is quite robust to gradient noise and yields fast empirical convergence. ", + "bbox": [ + 174, + 645, + 825, + 743 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Algorithm 1 provides a generic sign-based gradient descent framework that encapsulates different variants of signSGD. In Algorithm 1, GradEstimate(·) signifies a general gradient estimation procedure, which adopts either a stochastic gradient estimate in the first-order setting (Bernstein et al., 2018) or a function difference based random gradient estimate in the ZO setting (Nesterov & Spokoiny, 2015; Duchi et al., 2015). We call the ZO variant of signSGD ‘ZO-signSGD’, which will be elaborated on in Sec. 3. ", + "bbox": [ + 173, + 750, + 825, + 833 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Adversarial attacks meet signSGD It is now widely known that ML models (e.g., deep neural networks) are vulnerable to adversarial attacks, which craft inputs (e.g., images) with imperceptible perturbations to cause incorrect classification (Szegedy et al., 2013; Goodfellow et al., 2015; Kurakin et al., 2017; Lin et al., 2019). The resulting inputs crafted by adversaries are known as adversarial examples. Investigating adversarial examples not only helps to understand the limitation of learning models, but also provides opportunities to improve the models’ robustness (Papernot et al., 2016; ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Algorithm 1 Generic sign-based gradient descent ", + "text_level": 1, + "bbox": [ + 176, + 103, + 500, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1: Input: learning rate $\\{ \\delta _ { k } \\}$ , initial value $\\mathbf { x } _ { \\mathrm { 0 } }$ , and number of iterations $T$ \n2: for $k = 0 , 1 , \\ldots , T - 1$ do \n3: $\\hat { \\mathbf { g } } _ { k } \\longleftarrow \\mathrm { G r a d E s t i m a t e } ( \\mathbf { x } _ { k } )$ # applies to both first and zeroth order gradient estimates \n4: sign-gradient update ", + "bbox": [ + 181, + 123, + 790, + 179 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/ab350cde9d4f0de74f2ca117be5408b1dcadec65d9bc4b78059882f2b882a15d.jpg", + "text": "$$\n\\begin{array} { r } { { \\bf x } _ { k + 1 } = { \\bf x } _ { k } - \\delta _ { k } \\mathrm { s i g n } ( \\hat { \\bf g } _ { k } ) , \\quad \\mathrm { w h e r e ~ s i g n ( { \\bf x } ) ~ t a k e s ~ e l e m e n t - w i s e ~ s i g n s ~ o f ~ { \\bf x } ~ } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 289, + 185, + 736, + 200 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "5: end for ", + "bbox": [ + 181, + 208, + 254, + 222 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Athalye et al., 2018; Madry et al., 2018). In what follows, we show that the generation of adversarial examples in (Goodfellow et al., 2015; Kurakin et al., 2017) can be interpreted through signSGD. ", + "bbox": [ + 171, + 252, + 823, + 281 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $\\mathbf { x } _ { \\mathrm { 0 } }$ denote the natural (legitimate) input of an ML model associated with the true label $t _ { 0 }$ , and $\\mathbf { x } ^ { \\prime } = \\mathbf { x } _ { 0 } + \\boldsymbol { \\delta }$ be the adversarial example to be designed, where $\\pmb { \\delta }$ are adversarial perturbations. If $f ( \\mathbf { x } , t _ { 0 } )$ is the training loss of a learning model, then the goal of (white-box) adversarial attack is to find minimal perturbation $\\delta$ that is sufficient to mislead the learning model, namely, to maximize the loss $f ( \\mathbf { x } _ { 0 } ^ { \\mathbf { \\hat { \\alpha } } } + \\mathbf { \\delta } \\mathbf { \\delta } , t _ { 0 } )$ . Taking the first-order approximation of $\\bar { f ( \\mathbf { x } ^ { \\prime } , t _ { 0 } ) }$ around $\\mathbf { x } _ { \\mathrm { 0 } }$ , we obtain $f ( \\mathbf { x } ^ { \\prime } , t _ { 0 } ) \\approx f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) + \\langle \\nabla _ { x } f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) , \\delta \\rangle$ . By constraining the strength of perturbation in the $\\ell _ { \\infty }$ ball of small radius $\\epsilon$ (i.e., $\\| \\pmb { \\delta } \\| _ { \\infty } \\leq \\epsilon )$ , the linear approximation of $f ( \\mathbf { x } ^ { \\prime } , t _ { 0 } )$ is then maximized at $\\pmb { \\delta } = \\epsilon \\mathrm { s i g n } ( \\nabla _ { \\pmb { x } } f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) )$ (Shaham et al., 2018). Therefore, generation of adversarial examples proposed in (Goodfellow et al., 2015) obeys the sign-gradient update rule in (2), ", + "bbox": [ + 173, + 285, + 825, + 405 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8c5e3d5b769882ff66b4b10e97e41df6c1ab4c2857b2414e472981dc9be25151.jpg", + "text": "$$\n\\mathbf { x } ^ { \\prime } = \\mathbf { x } _ { 0 } - \\epsilon \\mathrm { s i g n } ( - \\nabla _ { x } f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 406, + 602, + 422 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Such a connection between adversarial example generation and signSGD also holds in other attacks, e.g., the iterative target attack method (Kurakin et al., 2017). Similarly, a so-called black-box attack (Ilyas et al., 2018a; Bhagoji et al., 2018) is associated with our proposed ZO-signSGD algorithm. ", + "bbox": [ + 173, + 431, + 826, + 474 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 ZO-SIGNSGD FOR BLACK-BOX OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 174, + 484, + 602, + 502 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "One limitation of signSGD (Bernstein et al., 2018) is the need of first-order information, i.e., stochastic gradients. However, there exists a large practical demand for solving ML problems where explicit expressions of the gradients are difficult or infeasible to obtain, e.g., the generation of adversarial examples from black-box neural networks as discussed in Sec. 1 and 2. ", + "bbox": [ + 174, + 513, + 825, + 570 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Gradient estimation via ZO oracle In the ZO setting where the first-order information is unavailable, the gradient estimator at Step 3 of Algorithm 1 has only access to function values of $\\{ f _ { i } ( \\mathbf { x } ) \\}$ given a query point x. Based on that, we construct a ZO gradient estimate through a forward difference of two function values (Nesterov & Spokoiny, 2015; Gao et al., 2014; Duchi et al., 2015). In Algorithm 1, GradEstimate $\\left( \\mathbf { x } \\right)$ is then specified as ", + "bbox": [ + 173, + 575, + 826, + 640 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f917312d2f27baeae40650a8a5940d7151d7ff50f1a20268f41879b60fb5fc91.jpg", + "text": "$$\n\\mathrm { G r a d E s t i m a t e } ( \\mathbf { x } ) = \\frac { 1 } { b q } \\sum _ { i \\in \\mathcal { T } _ { k } } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) , \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) : = \\frac { d \\big [ f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } _ { i , j } ) - f _ { i } ( \\mathbf { x } ) \\big ] } { \\mu } \\mathbf { u } _ { i , j } ,\n$$", + "text_format": "latex", + "bbox": [ + 209, + 641, + 787, + 679 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbf { x } = \\mathbf { x } _ { k }$ in Algorithm 1, $\\mathcal { T } _ { k }$ is a mini-batch of size $\\left| \\mathcal { T } _ { k } \\right| = b$ , $\\{ \\mathbf { u } _ { i , j } \\} _ { j = 1 } ^ { q }$ are i.i.d. random directions drawn from a uniform distribution over a unit sphere, and $\\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } )$ gives a two-point based random gradient estimate with direction $\\mathbf { u } _ { i , j }$ and smoothing parameter $\\mu > 0$ . We remark that the random direction vectors in (3) can also be drawn from the standard Gaussian distribution (Nesterov & Spokoiny, 2015). However, the uniform distribution could be more useful in practice since it is defined in a bounded space rather than the whole real space required for Gaussian. ", + "bbox": [ + 173, + 686, + 825, + 776 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We highlight that unlike the first-order stochastic gradient estimate, the ZO gradient estimate (3) is a biased approximation to the true gradient of $f$ . Instead, it becomes unbiased to the gradient of the randomized smoothing function $\\bar { f _ { \\mu } }$ (Duchi et al., 2012; Gao et al., 2014), ", + "bbox": [ + 174, + 782, + 823, + 821 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f5642bc8d01d73ce03f6fb4a6a1f4b6e63e7ec1c1bdc75790c0cdf7b89f29e73.jpg", + "text": "$$\nf _ { \\mu } ( \\mathbf { x } ) = \\mathbb { E } _ { \\mathbf { v } } [ f ( \\mathbf { x } + \\mu \\mathbf { v } ) ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { v } } [ f _ { i } ( \\mathbf { x } + \\mu \\mathbf { v } ) ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } f _ { i , \\mu } ( \\mathbf { x } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 289, + 824, + 707, + 861 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $f _ { i , \\mu }$ gives the randomized smoothing version of $f _ { i }$ , and the random variable $\\mathbf { v }$ follows a uniform distribution over the unit Euclidean ball. Clearly, there exists a gap between a ZO gradient estimate and the true gradient of $f$ , but as will be evident later, such a gap can be measured through the smoothing function $f _ { \\mu }$ . ", + "bbox": [ + 174, + 867, + 825, + 925 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Motivations of ZO-signSGD. Compared to SGD-type methods, the fast empirical convergence of signSGD and ZO-signSGD has been shown in the application of generating white-box and black-box adversarial examples (Goodfellow et al., 2015; Madry et al., 2018; Ilyas et al., 2018a). As mentioned in (Bernstein et al., 2018), the sign operation could mitigate the negative effect of (coordinate-wise) gradient noise of large variance. Recall that the ZO gradient estimate is a biased approximation to the true gradient, and thus, could suffer from having larger noise variance than (first-order) stochastic gradients. In this context, one could benefit from ZO-signSGD due to its robustness to gradient noise. In Appendix 1, we provide two concrete examples (Fig. A1 and Fig. A2) to confirm the aforementioned analysis. In Fig. A1, we show the robustness of ZO-signSGD against sparse noise perturbation through a toy quadratic optimization problem, originally introduced in (Bernstein et al., 2018) to motivate the fast convergence of signSGD against SGD. In Fig. A2, we show that gradient estimation via ZO oracle indeed encounters gradient noise of large variance. Thus, taking the sign of a gradient estimate might scale down the extremely noisy components. ", + "bbox": [ + 174, + 103, + 825, + 284 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "ZO-signSGD & technical challenges beyond signSGD Algorithm 1 becomes ZO-signSGD as the ZO gradient estimate (3) is applied. We note that the extension from first order to ZO is nontrivial, as the proposed ZO-signSGD algorithm yields three key differences to signSGD. ", + "bbox": [ + 174, + 296, + 825, + 338 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "First, ZO-signSGD has milder assumption on the choice of mini-batch sampling. Recall that signSGD in (Bernstein et al., 2018) achieves $O ( 1 / \\sqrt { T } )$ convergence rate given the condition that the mini-batch size is sufficiently large, $b = O ( T )$ . However, this condition only becomes true when the mini-batch sample is randomly selected from $[ n ]$ with replacement, which is unusual when $n \\leq T$ . Here $[ n ]$ represents the integer set $\\{ 1 , 2 , \\ldots , n \\}$ . And signSGD fails to cover signGD when $b = n$ , since sampling with replacement leads to $\\mathcal { T } _ { k } \\neq [ n ]$ even if $b = n$ . In the proposed ZO-signSGD algorithm, we will relax the assumption on mini-batch sampling. ", + "bbox": [ + 174, + 344, + 825, + 444 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Second, in ZO-signSGD both the ZO gradient estimator and the sign operator give rise to approximation errors to the true gradient. Although the statistical properties of ZO gradient estimates can be acquired with the aid of the randomized smoothing function (4), the use of mini-batch sampling without replacement introduces extra difficulty to bound the variance of ZO gradient estimates since mini-batch samples are no longer independent. Moreover, the sign-based descent algorithm evaluates the convergence error in the $\\ell _ { 1 }$ -norm geometry, leading to a mismatch with the $\\ell _ { 2 }$ -norm based gradient variance. Besides translating the the gradient norm from $\\ell _ { 1 }$ to $\\ell _ { 2 }$ , the probabilistic convergence method (Ghadimi & Lan, 2013) is used to bound the eventual convergence error of ZO-signSGD. ", + "bbox": [ + 174, + 450, + 825, + 563 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Finally, beyond the standard ZO gradient estimator (3), we will cover multiple variants of ZOsignSGD for centralized or distributed optimization. ", + "bbox": [ + 176, + 569, + 821, + 598 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 CONVERGENCE ANALYSIS OF ZO-SIGNSGD ", + "text_level": 1, + "bbox": [ + 173, + 613, + 576, + 631 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we begin by stating assumptions used in our analysis. We then derive the convergence rate of ZO-signSGD for nonconvex optimization. Assumptions of problem (1) are listed as follows. ", + "bbox": [ + 171, + 643, + 823, + 672 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A1: Functions $\\{ f _ { i } \\}$ have $L$ -Lipschitz continuous gradients, where $L \\in ( 0 , \\infty )$ . ", + "bbox": [ + 171, + 679, + 691, + 694 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A2: At time $k$ , the gradient of $f _ { i }$ is upper bounded by $\\| \\nabla f _ { i } ( \\mathbf { x } _ { k } ) \\| _ { 2 } \\leq \\sigma$ for $i \\in [ n ]$ ", + "bbox": [ + 171, + 700, + 714, + 717 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Both A1 and A2 are the standard assumptions used in nonconvex optimization literature (Bernstein et al., 2018; Reddi et al., 2018; Chen et al., 2018). A1 implies the $L$ -smoothness of $f _ { i }$ , namely, for any $\\mathbf { x }$ and $\\mathbf { y }$ we obtain $f _ { i } ( \\mathbf { x } ) - f _ { i } ( \\mathbf { y } ) \\leq \\langle \\nabla f _ { i } ( \\mathbf { y } ) , \\mathbf { x } - \\mathbf { y } \\rangle + ( L / 2 ) \\| \\mathbf { x } - \\mathbf { y } \\| _ { i } ^ { 2 }$ . A2 implies the bounded variance of $\\nabla f _ { i }$ in (Bernstein et al., 2018, Assumption 3), namely, $\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\| \\nabla f _ { i } ( \\mathbf { x } ) - \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } \\leq } \\end{array}$ $4 \\sigma ^ { 2 }$ , where we have used the fact that $\\| \\nabla f ( \\mathbf { x } ) \\| _ { 2 } \\leq \\sigma$ under A2. Throughout the paper, we assume that problem (1) is solvable, namely, $f ( \\mathbf { x } ^ { * } ) > - \\infty$ where $x ^ { * }$ is an optimal solution. ", + "bbox": [ + 173, + 722, + 825, + 810 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We recall that Algorithm 1 becomes ZO-signSGD when the gradient estimation step (3) is applied. For nonconvex problems, the convergence of an algorithm is typically measured by stationarity, e.g., using $\\| \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 }$ in SGD (Ghadimi $\\&$ Lan, 2013) and $\\| \\nabla f ( \\mathbf { x } ) \\bar { \\| _ { 1 } }$ in signSGD (Bernstein et al., 2018). For the latter, the $\\ell _ { 1 }$ geometry is met when quantifying the stochasticity through the (non-linear) sign operation. Different from signSGD, ZO-signSGD only obtains a biased estimate to the true gradient. In Proposition 1, we bypass such a bias by leveraging the randomized smoothing technique used for ZO optimization (Gao et al., 2014; Nesterov & Spokoiny, 2015; Duchi et al., 2015). ", + "bbox": [ + 173, + 815, + 826, + 914 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proposition 1 Under A1, the outputs $\\{ \\mathbf { x } _ { k } \\} _ { k = 0 } ^ { T - 1 }$ of ZO-signSGD, i.e., Algorithm 1 with (3), satisfies $\\sum _ { k = 0 } ^ { T - 1 } ( \\delta _ { k } \\mathbb { E } [ \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } ] ) \\leq \\mathbb { E } [ f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f _ { \\mu } ( \\mathbf { x } _ { T } ) ] + \\sum _ { k = 0 } ^ { T - 1 } \\left[ 2 \\delta _ { k } \\sqrt { d } \\sqrt { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } ] } \\right] + \\frac { d L } { 2 } \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } ,$ where the expectation is taken with respect to all the randomness of ZO-signSGD, $f _ { \\mu }$ is the randomized smoothing function of $f$ in (4), and $\\hat { \\bf g } _ { k } =$ GradEstimate $\\left( \\mathbf { x } _ { k } \\right)$ in (3). ", + "bbox": [ + 174, + 101, + 820, + 159 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 174, + 826, + 203 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof: See Appendix 2. ", + "text_level": 1, + "bbox": [ + 174, + 212, + 331, + 227 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In Proposition 1, the rationale behind introducing the smoothing function $f _ { \\mu }$ is that $\\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )$ is the mean of ZO gradient estimate $\\hat { \\bf g } _ { k }$ . And thus, the convergence of ZO-signSGD is now linked with the variance of $\\hat { \\bf g } _ { k }$ , i.e., $\\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } ]$ . This crucial relationship presented in Proposition 1 holds for a general class of signSGD-type algorithms that use different ZO gradient estimators. Spurred by (5), we next investigate the second-order moment of $\\hat { \\bf g } _ { k }$ in Proposition 2. ", + "bbox": [ + 173, + 233, + 825, + 304 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 2 Under A1 and $A 2$ , the variance of $Z O$ gradient estimate $\\hat { \\bf g } _ { k }$ is upper bounded by ", + "bbox": [ + 166, + 311, + 797, + 327 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/5d3dc3e9cd898ce9ad0b6995d21887360a63337e2ddbf52f5eced7deba00b9f8.jpg", + "text": "$$\n\\mathbb { E } \\left[ \\Vert \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\Vert _ { 2 } ^ { 2 } \\right] \\leq \\frac { 4 \\alpha _ { b } ( q + 1 ) } { b q } \\sigma ^ { 2 } + \\frac { ( 2 \\alpha _ { b } + \\beta _ { b } ) } { b q } C ( d , \\mu ) ,\n$$", + "text_format": "latex", + "bbox": [ + 302, + 328, + 691, + 358 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $C ( d , \\mu ) : = 2 d \\sigma ^ { 2 } + \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } / 2 .$ . In (6), $\\alpha _ { b }$ and $\\beta _ { b }$ are Boolean variables depending on the choice of mini-batch sampling, ", + "bbox": [ + 174, + 359, + 826, + 388 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $I ( x > a )$ is the indicator function of $x$ with respect to the constraint $x > a$ , and $I ( x > a ) = 1$ if $x > a$ and 0 otherwise. ", + "bbox": [ + 174, + 422, + 828, + 450 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof: See Appendix 3. ", + "text_level": 1, + "bbox": [ + 174, + 462, + 331, + 477 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Compared to the variance bound $( \\sigma ^ { 2 } / b )$ of the stochastic gradient estimate of $f$ in signSGD (Bernstein et al., 2018), Proposition 2 provides a general result for the ZO gradient estimate $\\hat { \\bf g } _ { k }$ . It is clear that the bound in (6) contains two parts: $\\begin{array} { r } { h _ { 1 } : = \\frac { 4 \\alpha _ { b } ( q + 1 ) } { b q } \\sigma ^ { 2 } } \\end{array}$ 4αb(q+1) σ2 and h2 := (2αb+βb)bq C(d, µ), where the former $h _ { 1 } = O ( \\sigma ^ { 2 } / b )$ characterizes the reduced variance (using $b$ mini-batch samples) for the stochastic gradient estimate of the smoothing function $f _ { \\mu }$ , and the latter $h _ { 2 } = { O } \\left( { C ( d , \\mu ) } / { ( b q ) } \\right)$ reveals the dimension-dependent variance induced by ZO gradient estimate using $b$ mini-batch samples and $q$ random directions. If a stochastic gradient estimate of $f$ is used in signSGD, then $h _ { 2 }$ is eliminated and the variance bound in (6) is reduced to $( \\sigma ^ { 2 } / b )$ . ", + "bbox": [ + 173, + 482, + 825, + 602 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Furthermore, Proposition 2 covers mini-batch sampling with and without replacement, while signSGD only considers the former case. For the latter case, Proposition 2 implies that if $b = n$ (i.e., $\\mathcal { T } _ { k } = [ n ]$ for ZO-signGD), then the variance $\\mathbb { E } \\left[ \\lVert \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\rVert _ { 2 } ^ { 2 } \\right]$ is reduced to $O ( C ( d , \\mu ) / ( n q ) )$ , corresponding to $\\alpha _ { b } = 0$ and $\\beta _ { b } = 1$ in (7). In the other extreme case of $b = 1$ , both the studied mini-batch schemes become identical, corresponding to $\\alpha _ { b } = 1$ and $\\beta _ { b } = 0$ . Proposition 2 also implies that the use of large $b$ and $q$ reduces the variance of the gradient estimate, and will further improve the convergence rate. ", + "bbox": [ + 173, + 607, + 826, + 707 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "With the aid of Proposition 1 and 2, we can then show the convergence rate of ZO-signSGD in terms of stationarity of the original function $f$ . The remaining difficulty is how to bound the gap between $f$ and its smoothed version $f _ { \\mu }$ . It has been shown in (Gao et al., 2014; Nesterov & Spokoiny, 2015) that there exists a tight relationship between $f _ { \\mu }$ and $f$ given the fact that the former is a convolution of the latter and the density function of a random perturbation $\\mathbf { v }$ in (4). We demonstrate the convergence rate of ZO-signSGD in Theorem 1. ", + "bbox": [ + 173, + 713, + 825, + 797 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 1 Under A1 and $A 2$ , if we randomly pick $\\mathbf { x } _ { R }$ from $\\{ \\mathbf { x } _ { k } \\} _ { k = 0 } ^ { T - 1 }$ with probability $P ( R = k ) =$ $\\frac { \\delta _ { k } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } }$ , then the convergence rate of ZO-signSGD is given by ", + "bbox": [ + 173, + 806, + 823, + 840 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7f6e1d637d5354f82ff9708a5cdfce92f9760587ba34a808495b3619ca147ca9.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] \\leq \\frac { \\sqrt { 2 } \\left( f ( \\mathbf { x } _ { 0 } ) - f ^ { * } + \\mu ^ { 2 } L \\right) } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } + \\frac { d L } { \\sqrt { 2 } } \\frac { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } + \\frac { \\mu L d } { \\sqrt { 2 } } } \\\\ { + \\frac { 2 \\sqrt { 2 } \\sqrt { d } \\sqrt { 4 \\alpha _ { b } ( q + 1 ) \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } } { \\sqrt { b q } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 839, + 709, + 909 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $f ^ { * }$ denotes the minimum value. ", + "bbox": [ + 174, + 909, + 421, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof: See Appendix 4. ", + "text_level": 1, + "bbox": [ + 174, + 103, + 331, + 118 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Theorem 1, we translate the gradient norm from $\\ell _ { 1 }$ to $\\ell _ { 2 }$ , and adopt a probabilistic output $\\mathbf { x } _ { R }$ (Ghadimi & Lan, 2013; Lei et al., 2017) to avoid exhaustive search over $\\left\\{ { \\bf x } _ { k } \\right\\}$ for $\\operatorname* { m i n } _ { k }$ $\\| \\nabla f ( \\mathbf { x } _ { k } ) \\| _ { 2 }$ . Note that the convergence rate of ZO-signSGD relies on the learning rate $\\delta _ { k }$ , the problem size $d$ , the smoothing parameter $\\mu$ , the mini-batch size $b$ , and the number of random perturbations $q$ for ZO gradient estimation. We next obtain explicit dependence on these parameters by specifying Theorem 1. ", + "bbox": [ + 173, + 125, + 826, + 209 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "If $\\begin{array} { r } { \\delta _ { k } = \\delta = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}$ and $\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}$ , then the convergence in (8) simplifies to ", + "bbox": [ + 173, + 213, + 705, + 233 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/fe9bd1942e6e617fc5c3d518e014c47e389c97a383e2a966d618dbfbb36433dd.jpg", + "text": "$$\n\\mathbb { E } \\left[ \\Vert \\nabla f ( \\mathbf { x } _ { R } ) \\Vert _ { 2 } \\right] \\leq O \\left( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { \\sqrt { d } \\sqrt { \\alpha _ { b } q + ( \\alpha _ { b } + \\beta _ { b } ) d } } { \\sqrt { b q } } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 318, + 238, + 676, + 276 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\alpha _ { b }$ and $\\beta _ { b }$ were defined in (7), and $1 \\leq ( \\alpha _ { b } + \\beta _ { b } ) \\leq 2$ . We provide several key insights on the convergence rate of ZO-signSGD through (9). ", + "bbox": [ + 178, + 281, + 823, + 310 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "First, the convergence rate of ZO-signSGD is measured through $\\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 }$ rather than its squared counterpart $\\| \\nabla \\bar { f } ( \\mathbf { x } _ { R } ) \\| _ { 2 } ^ { 2 }$ , where the latter was used in measuring the convergence of ZO-SGD. We recall from (Ghadimi & Lan, 2013, Theorem 3.2 & Corollary 3.3) that ZO-SGD yields the convergence error $\\mathbb { E } \\left[ \\lVert \\nabla f ( \\mathbf { x } _ { R } ) \\rVert _ { 2 } ^ { 2 } \\right] \\leq O ( \\frac { \\sqrt { d } } { \\sqrt { T } } )$ . Since $\\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } ^ { 2 } \\leq \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 }$ as $\\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } ^ { 2 } \\leq 1$ , the convergence of ZO-signSGD meets a stricter criterion than that of ZO-SGD. The possible downside of ZO-signSGD is that it suffers an additional error of order $\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { b } } + \\frac { d } { \\sqrt { b q } } ) } \\end{array}$ in the worst case. The aforementioned results imply that ZO-signSGD could only converge to a neighborhood of a stationary point but with a fast convergence speed. Here the size of the neighborhood is controlled by the mini-batch size $b$ and the number of random direction vectors $q$ . ", + "bbox": [ + 173, + 315, + 826, + 454 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Also, our convergence analysis applies to mini-batch sampling both with and without replacement. When b ∈ [1, n), ZO-signSGD achieves O( √dT $\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { \\sqrt { d } } { \\sqrt { b } } + \\frac { d } { \\sqrt { b q } } ) } \\end{array}$ convergence rate regardless of the choice of mini-batch sampling. When $b = n$ , it is known from (9) that the use of mini-batch without replacement recovers ZO-signGD, yielding the convergence rate $\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { d } { \\sqrt { n q } } ) } \\end{array}$ . By contrast, the use of mini-batch with replacement leads to the worse convergence rate $\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { d } { \\sqrt { n q } } + \\frac { \\sqrt { d } } { \\sqrt { n } } ) } \\end{array}$ . Clearly, as $b = n$ and $n < T$ , ZO-signSGD using mini-batch with replacement fails to achieve the rate $O ( \\frac { \\sqrt { d } } { \\sqrt { T } } )$ regardless of the choice of $q$ . By contrast, ZO-signSGD using mini-batch without replacement recovers $O ( \\frac { \\sqrt { d } } { \\sqrt { T } } )$ as $\\begin{array} { r } { q \\ = \\ O ( \\frac { d T } { n } ) } \\end{array}$ . When $b > n$ , ZO-signSGD is restricted to using mini-batch sampling with replacement. Similar to signSGD (Bernstein et al., 2018), we can obtain $O ( \\frac { \\sqrt { d } } { \\sqrt { T } } )$ convergence rate as $b = O ( T )$ and $\\begin{array} { r } { q = O ( \\frac { d T } { n } ) } \\end{array}$ , where the dependence on $q$ is induced by the use of ZO gradient estimation. ", + "bbox": [ + 173, + 460, + 826, + 656 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 VARIANTS OF ZO-SIGNSGD ", + "text_level": 1, + "bbox": [ + 176, + 665, + 442, + 683 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here we study three variants of ZO-signSGD, where the gradient will be estimated using a) the central difference of function values, b) the sign of ZO gradient estimates with majority vote, or c) the sign of ZO gradient estimates with majority vote for distributed optimization. That is, ", + "bbox": [ + 174, + 690, + 825, + 731 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/dde518c17f209880984c162170618df28fa79730e04094ff4cee7b39d0553009.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle a \\mathrm { ) ~ G r a d E s t i m a t e ( \\mathbf { x } ) } = \\frac { 1 } { b q } \\sum _ { i \\in \\mathbb { Z } _ { k } } \\sum _ { j = 1 } ^ { q } \\frac { d \\left[ f _ { i } \\left( \\mathbf { x } + \\mu \\mathbf { u } _ { i , j } \\right) - f _ { i } \\left( \\mathbf { x } - \\mu \\mathbf { u } _ { i , j } \\right) \\right] \\mathbf { u } _ { i , j } } { 2 \\mu } } \\ ~ } \\\\ { { \\displaystyle b \\mathrm { ) ~ G r a d E s t i m a t e ( \\mathbf { x } ) } = \\frac { 1 } { b q } \\sum _ { i \\in \\mathbb { Z } _ { k } } \\sum _ { j = 1 } ^ { q } \\mathrm { s i g n } \\left( \\hat { \\nabla } f _ { i } \\left( \\mathbf { x } ; \\mathbf { u } _ { i , j } \\right) \\right) , } \\ ~ } \\\\ { { \\displaystyle c \\mathrm { ) ~ G r a d E s t i m a t e ( \\mathbf { x } ) } = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\mathrm { s i g n } \\left( \\frac { 1 } { b _ { m } q } \\sum _ { i \\in \\mathbb { Z } _ { m , k } } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } \\left( \\mathbf { x } ; \\mathbf { u } _ { i , j } \\right) \\right) , } \\ ~ } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 732, + 728, + 859 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\{ \\mathbf { u } _ { i , j } \\}$ and $\\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } )$ have been defined in (3). The gradient estimator (12) is proposed for istrigent mization over a stahas only access to twork that consisdata (in terms of f i $M$ agents and vidual costs ral pr with $m \\in [ M ]$ $n _ { m }$ $n _ { m }$ $\\{ f _ { i } \\} )$ $\\textstyle \\sum _ { m = 1 } ^ { M } n _ { m } =$ $n$ , and the mini-batch per agent satisfies $b _ { m } = | \\mathcal { I } _ { m , k } | \\in [ 1 , n _ { m } ]$ . According to (12), the central processor receives 1-bit compressed gradients $\\begin{array} { r } { \\mathrm { s i g n } \\left( \\frac { 1 } { b _ { m } q } \\sum _ { i \\in \\mathcal { T } _ { m , k } } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) \\right) } \\end{array}$ from $M$ agents and then performs the sign-based descent (2) and sends its 1-bit update back to every agent. ", + "bbox": [ + 174, + 864, + 825, + 925 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 101, + 825, + 138 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The ZO gradient estimator (10) was used in (Shamir, 2017) for bandit convex optimization and in (Ilyas et al., 2018a) for designing black-box adversarial attacks. Compared to the form of forward difference (3), the central difference (10) requires $b ( q - 1 )$ times more function queries in gradient estimation. At the cost of more function queries, one may wonder if the convergence rate of ZO-signSGD can be further improved. ", + "bbox": [ + 173, + 157, + 825, + 229 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Corollary 1 Suppose that the conditions in Theorem 1 hold, ZO-signSGD with gradient estimator (10) yields the same convergence rate of ZO-signSGD that uses the estimator (3). ", + "bbox": [ + 174, + 239, + 823, + 270 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proof: Recall that Proposition 1 is independent of specific forms of gradient estimators, and thus holds for (10). Although Proposition 2 relies on the second-order moments of each gradient estimator, we prove that under A1 and A2, both (3) and (10) maintain the same statistical properties. As a result, Proposition 2 and Theorem 1 also hold for (10); see more details in Appendix 5. \u0003 ", + "bbox": [ + 173, + 280, + 825, + 337 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We next study the gradient estimator (11), whose sign is equivalent to the majority vote (i.e., the element-wise median) of signs of individual gradient estimates $\\{ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) \\}$ . It was shown in (Bernstein et al., 2018) that signSGD with majority vote has a better convergence rate under additional assumptions of unimodal symmetric noise distribution of coordinate-wise gradient estimates. In Corollary 2, we show that such a speed-up in convergence can also be achieved by ZO-signSGD with majority vote, which we refer to as ‘ZO-M-signSGD’. ", + "bbox": [ + 173, + 343, + 825, + 430 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Corollary 2 Suppose that the conditions in Theorem $^ { l }$ hold, and the distribution of gradient noise is unimodal and symmetric. Then, ZO-M-signSGD with $\\begin{array} { r } { \\delta _ { k } = { O } \\left( \\frac { 1 } { \\sqrt { d T } } \\right) } \\end{array}$ and $\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}$ yields ", + "bbox": [ + 171, + 438, + 823, + 472 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/8575e4aabac7d2e123d0c00a53b2c9c3db6f8fadd43c6d73f0fcb25132162306.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] = O \\left( \\sqrt { d } / \\sqrt { T } + d / \\sqrt { b q } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 362, + 479, + 635, + 503 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proof: See Appendix 6. ", + "bbox": [ + 174, + 511, + 331, + 525 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We recall from Theorem 1 that under the same parameter setting of Corollary 2, ZO-signSGD yields $\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { \\sqrt { d } } { \\sqrt { b } } + \\frac { d } { \\sqrt { b q } } ) } \\end{array}$ convergence rate in the worst case. It is clear from (13) that the error correction term of order √d is eliminated in ZO-M-signSGD. Such an improvement in convergence is achieved under the condition of unimodal symmetric gradient noise. We remark that different from the stochastic gradient noise studied in (Bernstein et al., 2018), the ZO gradient estimation noise could violate this assumption. For example, in a scalar case, if the gradient estimate $g$ follows the distribution where $g = 1$ with probability 0.9, $g = - 1 0$ with probability 0.1, then $\\mathbb { E } [ g ] < 0$ and $\\mathrm { s i g n } ( \\mathbb { E } [ g ] ) < 0$ . However, $\\mathbb { E } [ \\mathrm { s i g n } ( g ) ] > 0$ . This implies that without the assumption of symmetry, the sign of gradient estimates with majority vote $( { \\mathbb E } [ \\mathrm { s i g n } ( g ) ] )$ can be in the opposite direction of the sign of averaged gradients $( \\mathrm { s i g n } ( \\mathbb { E } [ g ] ) )$ . Our results in the next section show that ZO-M-signSGD may not outperform ZO-signSGD. ", + "bbox": [ + 173, + 531, + 826, + 699 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Lastly, we focus on the gradient estimator (12), whose sign can be interpreted as the major vote of $M$ distributed agents about the sign of the true gradient (Bernstein et al., 2018). The resulting variant of ZO-signSGD is called ‘ZO-D-signSGD’, and its convergence rate is illustrated in Corollary 3. Compared to ZO-M-signSGD for centralized optimization, ZO-D-signSGD suffers an extra error correction term $O ( \\frac { \\sqrt { d } } { \\sqrt { n } } )$ in the distributed setting. It is also worth mentioning that if $M = n$ and $q = 1$ , then the gradient estimator (12) reduces to (11) with $\\mathcal { T } _ { k } = [ n ]$ . In this case, Corollary 2 and 3 reach a consensus on $\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { d } { \\sqrt { n } } ) } \\end{array}$ convergence error. ", + "bbox": [ + 173, + 705, + 826, + 819 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Corollary 3 Suppose that the conditions in Corollary 2 hold. ZO-M-signSGD with $\\begin{array} { r } { \\boldsymbol { b } _ { m } = \\left\\lfloor \\frac { n } { M } \\right\\rfloor } \\end{array}$ $\\begin{array} { r } { \\delta _ { k } = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}$ and $\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}$ yields ", + "bbox": [ + 171, + 832, + 823, + 868 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/83df8149263c7976b827f3ed2c450d7f8a03ee5433076a020efa044a4d4d2bd4.jpg", + "text": "$$\n\\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] = O \\left( \\sqrt { d } / \\sqrt { T } + \\sqrt { d } / \\sqrt { n } + d / \\sqrt { n q } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 875, + 669, + 900 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proof: See Appendix 7. ", + "bbox": [ + 174, + 915, + 333, + 930 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/337bfc286984c7305cf44de134c83cbe2f9274c987c25327a863c940c6c1ba76.jpg", + "image_caption": [ + "Figure 1: Performance comparison of ZO-signSGD, ZO-M-signSGD, ZO-SGD, ZO-SCD, signSGD and SGD under a synthetic dataset. The solid line represents the loss/accuracy averaged over 10 independent trials with random initialization, and the shaded region indicates the standard deviation of results over random trials. (a)-(b): Training loss and test accuracy versus iterations. (c)-(d): Effects of mini-batch size $q$ and number of random direction vectors $q$ on the convergence of studied algorithms. Here (c) presents the training loss versus iterations, and (d) is the heat map of the final loss for different values of $^ { b }$ and $q$ . (e)-(f): Effects of problem size $d$ . Here (e) shows the final training loss versus $d$ , and (f) presents the convergence trajectory when $d \\in \\{ 2 0 0 , 4 0 0 \\}$ . " + ], + "image_footnote": [], + "bbox": [ + 200, + 111, + 802, + 383 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 489, + 326, + 505 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we empirically show the effectiveness of ZO-signSGD, and validate its convergence behavior on both synthetic and real-world datasets such as MNIST and CIFAR-10. For the synthetic experiment, we study the problem of binary classification in the least squared formulation. For the real-world application, we design adversarial examples from black-box neural networks as mentioned in Sec. 2. Throughout this section, we compare ZO-signSGD and its variants with SGD, signSGD (Bernstein et al., 2018), ZO-SGD (Ghadimi & Lan, 2013), and ZO-SCD (Lian et al., 2016). ", + "bbox": [ + 173, + 512, + 825, + 595 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Binary classification We consider the least squared problem with a nonconvex loss function (Xu et al., 2017; Liu et al., 2018b) $\\begin{array} { r } { \\operatorname* { m i n } _ { \\mathbf { x } \\in \\mathbb { R } ^ { d } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } ( y _ { i } - 1 / ( 1 + e ^ { - \\mathbf { a } _ { i } ^ { T } \\mathbf { x } } ) ) ^ { 2 } } \\end{array}$ , which satisfies Assumption A2 by letting $\\sigma = \\operatorname* { m a x } _ { i } \\{ 2 \\lVert \\mathbf { a } _ { i } \\rVert _ { 2 } \\}$ . Here instead of using the conventional cost function of logistic regression (a convex function), the considered least squared formulation is introduced to align with our nonconvex theoretical analysis. For generating the synthetic dataset, we randomly draw samples $\\left\\{ \\mathbf { a } _ { i } \\right\\}$ from $\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )$ , and obtain the label $y _ { i } = 1$ if $1 / \\dot { ( 1 + e ^ { - \\mathbf { a } _ { i } ^ { T } \\mathbf { x } } ) } > 0 . 5$ and 0 otherwise. The number of training samples $\\left\\{ \\mathbf { a } _ { i } , y _ { i } \\right\\}$ is set by $n = 2 0 0 0$ against 200 testing samples. We find the best constant learning rate for algorithms via a greedy search over $\\eta \\in [ 0 . 0 0 1 , 0 . 1 ]$ (see Appendix 8.1 for more details), and we choose the smoothing parameter $\\mu = 1 0 / \\sqrt { T d }$ . Unless specified otherwise, let $b = q = 1 0$ , $T = 5 0 0 0$ and $d = 1 0 0$ . ", + "bbox": [ + 173, + 603, + 826, + 751 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Fig. 1, we report the training loss, the test accuracy, as well as the effects of algorithmic parameters on the convergence of the studied algorithms. We observe from Fig. 1-(a) and (b) that ZO-signSGD outperforms other ZO algorithms, and signSGD yields the best convergence performance once the first-order information is available. In Fig. 1-(c) and (d), we observe that the convergence performance of ZO algorithms is improved as $b$ and $q$ increase. In particular, ZO-signSGD and ZO-M-signSGD at $b = q = 3 0$ approach to the best result provided by signSGD. In Fig. 1-(e) and (f), the convergence of all algorithms degrades as the problem size $d$ increases. However, ZO-signSGD and ZO-M-signSGD converge faster than ZO-SGD and ZO-SCD. In Fig. 2, we demonstrate the convergence trajectory of different variants of ZO-signSGD for $b \\in \\{ 4 0 , 4 0 0 \\}$ . To make a fair comparison between ZOsignSGD and ZO-D-signSGD, let each of $M = 4 0$ agents use a mini-batch of size $b / M$ . As we can see, ZO-signSGD outperforms ZO-M-signSGD and ZO-D-signSGD. And the convergence is improved as the mini-batch size increases. However, we observe that in all examples, ZO-signSGD and its variants converge to moderate accuracy much faster than ZO-SGD, only within a few tens of iterations. ", + "bbox": [ + 173, + 757, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/d06c7d33a9b1cb45fe96ad3bf4d31117fbae253be352e11dcb7b8e189b8be217.jpg", + "image_caption": [ + "Figure 2: Training loss (left) and testing accuracy (right) of ZO-signSGD, ZO-M-signSGD, ZO-D-signSGD and ZO-SGD versus iterations. " + ], + "image_footnote": [], + "bbox": [ + 238, + 108, + 767, + 252 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/0dcd6aa27fddfbaa2c914d807914b8186ef985663cc7baa8b27cd6648a025220.jpg", + "image_caption": [ + "Figure 3: Black-box attacking loss versus iterations. The solid marker indicates the iteration number that finds the first successful adversarial example, and its loss corresponds to the squared $\\ell _ { 2 }$ distortion. " + ], + "image_footnote": [], + "bbox": [ + 236, + 323, + 736, + 617 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 671, + 821, + 699 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Generating black-box adversarial examples Here we study adversarial robustness by generating adversarial examples from a black-box image classifier trained by a deep neural network (DNN) model; see details on problem formulation in Appendix 8.2. We recall from Sec. 2 that the task of black-box adversarial attack falls within the category of ZO optimization as one can only access to the input-output relation of the DNN while crafting adversarial examples. ", + "bbox": [ + 174, + 708, + 825, + 779 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The DNN models trained on MNIST and CIFAR-10 (Carlini & Wagner, 2017) are performed as the zeroth-order oracle2. We select one image from each class of MNIST and CIFAR-10 and separately implement black-box attacks using the same attacking loss function (see Appendix 8.2) but with different ZO optimization algorithms (ZO-SGD, ZO-signSGD and ZO-M-signSGD). We also set the same parameters for each method, i.e., $\\mu = 0 . 0 1$ , $q = 9$ , and $\\delta = 0 . 0 5$ for MNIST and $\\delta = 0 . 0 0 0 5$ for CIFAR-10, to accommodate to the dimension factor $d$ . Moreover, we benchmark their performance with the natural evolution strategy (NES) based two-point gradient estimator in (Ilyas et al., 2018a) for solving the same attacking loss function, where the sign of gradient estimate is also used in the descent step. We call the resulting black-box attack generation method ‘ZO-NES’. Similar to (10), NES computes the ZO gradient estimate using the central difference of two function values. Thus, one iteration of ZO-NES requires $2 q$ function queries and thus we set $q = 5$ to align with the number of function queries used in other ZO methods. All methods use the the same natural image as the initial point for finding adversarial examples. ", + "bbox": [ + 174, + 785, + 825, + 897 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/4aebf266372cf9ce96379ec3df43fed31c03c889adb0fae526b3d4b953310157.jpg", + "table_caption": [ + "Table 1: Iteration comparison of attacking black-box DNN on MNIST (image ID 2). " + ], + "table_footnote": [], + "table_body": "
Iteration04080 120160 200 240280 312 356
ZO-SGD7770777777
Classified as4 4
Iteration 0 40 80 120 145 202 240280 320 359
ZO-signSGD070007000
Classified as4044 4 279 320 3604 4 4
Iteration0 80 120 142 200 240
ZO-M-signSGD7777777777
Classified as44 4 4 4 4
Iteration0 40 80 120 160 200 258 287 321 359
ZO-NES7710777077
Classified as4 4 4
", + "bbox": [ + 251, + 116, + 743, + 324 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 354, + 825, + 425 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Fig. 3 shows the plots of black-box attacking loss versus iterations (more results are shown in Appendix 8.3). We find that ZO-signSGD usually takes significantly less iterations than other methods to find the first successful adversarial example with a similar attacking loss. For MNIST, the average iteration over all attacked images in Table A1 to find the first successful adversarial example is 184 for ZO-SGD, 103 for ZO-signSGD, 151 for ZO-M-signSGD, and 227 for ZO-NES. Their corresponding average $\\ell _ { 2 }$ distortion is 2.345 for ZO-SGD, 2.381 for ZO-signSGD, 2.418 for ZO-MsignSGD, and 2.488 for ZO-NES. For CIFAR-10, the average iteration over all attacked images in Table A2 to find the first successful adversarial example is 302 for ZO-SGD, 250 for ZO-signSGD, 389 for ZO-M-signSGD, and 363 for ZO-NES. Their corresponding average $\\ell _ { 2 }$ distortion is 0.177 for ZO-SGD, 0.208 for ZO-signSGD, 0.219 for ZO-M-signSGD, and 0.235 for ZO-NES. As a visual illustration, we compare the adversarial examples of a hand-written digit “1” of each attacking method at different iterations in Table 1, corresponding to Fig. 3-(a). As we can see, ZO-signSGD and ZO-M-signSGD can reduce roughly $5 4 \\%$ of iterations (around 600 less model queries) than ZO-SGD to find the first successful adversarial example. Given the first successful adversarial example, we observe that ZO-signSGD yields slightly higher $\\ell _ { 2 }$ distortion than ZO-SGD. This is not surprising since Theorem 1 suggests that ZO-signSGD might not converge to a solution of very high accuracy but it can converge to moderate accuracy sufficient for black-box attacks at a very fast speed. Note that the first successful adversarial examples generated by different ZO methods are all visually similar to the original ones but lead to different top-1 predictions; see more results in Appendix 8.3. In addition, we observe that ZO-NES is not as effective as ZO-signSGD in either query efficiency (given by the number of iterations to achieve the first successful attack) or attack distortion. Thus, compared to ZO-NES, ZO-signSGD offers a provable and an efficient black-box adversarial attacking method. ", + "bbox": [ + 173, + 431, + 825, + 751 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 777, + 318, + 794 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Motivated by the impressive convergence behavior of (first-order) signSGD and the empirical success in crafting adversarial examples from black-box ML models, in this paper we rigorously prove the√ √ $O ( \\sqrt { d } / \\sqrt { T } )$ convergence rate of ZO-signSGD and its variants under mild conditions. Compared to signSGD, ZO-signSGD suffers a slowdown (proportional to the problem size $d$ ) in convergence rate, however, it enjoys the gradient-free advantages. Compared to other ZO algorithms, we corroborate the superior performance of ZO-signSGD on both synthetic and real-word datasets, particularly for its application to black-box adversarial attacks. In the future, we would like to generalize our analysis to nonsmooth and nonconvex constrained optimization problems. ", + "bbox": [ + 174, + 810, + 825, + 924 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 326, + 117 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "This work was supported by the MIT-IBM Watson AI Lab. Mingyi Hong and Xiangyi Chen are supported partly by an NSF grant CMMI-1727757,and by an AFOSR grant 15RT0767. ", + "bbox": [ + 171, + 127, + 823, + 157 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 176, + 287, + 191 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A. Athalye, N. Carlini, and D. Wagner. Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. arXiv preprint arXiv:1802.00420, 2018. \nL. Balles and P. Hennig. 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", + "bbox": [ + 171, + 200, + 828, + 926 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 59, + 828, + 848 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 103, + 263, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1 MOTIVATING EXAMPLES OF FAST CONVERGENCE OF ZO-SIGNSGD ", + "text_level": 1, + "bbox": [ + 173, + 143, + 769, + 161 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1.1 EXAMPLE OF SPARSE NOISE PERTURBATION ", + "text_level": 1, + "bbox": [ + 176, + 185, + 517, + 199 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We consider to minimize the function $\\begin{array} { r } { f ( \\mathbf { x } ) = \\frac { 1 } { 2 } \\| \\mathbf { x } \\| _ { 2 } ^ { 2 } } \\end{array}$ . Similar to (Bernstein et al., 2018, Figure A.1), we assume that the ZO gradient estimate of $f ( \\mathbf { x } )$ and its first-order gradient $\\nabla f ( \\mathbf { x } ) = \\mathbf { x }$ suffer from a sparse noise vector $\\mathbf { v }$ , where $v _ { 1 } \\in \\overline { { \\mathcal { N } ( 0 , 1 0 0 ^ { 2 } ) } }$ , and $v _ { i } = 0$ for $i \\geq 2$ . As a result, the used descent direction at iteration $t$ is given by $\\hat { \\nabla } f ( \\mathbf { x } _ { t } ) + \\mathbf { v }$ or $\\nabla f ( \\mathbf { x } _ { t } ) + \\mathbf { v }$ . Fig. A1 presents the convergence performance of 5 algorithms: SGD, signSGD, ZO-SGD, ZO-signSGD and its variant using the central difference based gradient estimator (10). Here we tune a constant learning rate finding 0.001 best for SGD and ZO-SGD and 0.01 best for signSGD and its ZO variants. As we can see, sign-based first-order and ZO algorithms converge much faster than the stochastic gradient-based descent algorithms. This is not surprising since the presence of extremely noisy component $v _ { 1 }$ leads to an inaccurate gradient value, and thus degrades the convergence of SGD and ZO-SGD. By contrast, the sign information is more robust to outliers and thus leads to better convergence performance of sign SGD and its variants. We also note that the convergence trajectory of ZO-signSGD using the gradient estimator (10) coincides with that using the gradient estimator (3) given by the forward difference of two function values. ", + "bbox": [ + 173, + 215, + 826, + 372 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/3a02060f85eca3a9b9d8126e85684d6a6fade1b1de8a7098c000772a2df7bcc7.jpg", + "image_caption": [ + "Figure A1: Comparison of different gradient-based and gradient sign-based first-order and ZO algorithms in the example of sparse noise perturbation. The solid line represents the loss averaged over 10 independent trials with random initialization, and the shaded region indicates the standard deviation of results over random trials. Left: Loss value against iterations for SGD, signSGD, ZO-SGD, ZO-signSGD and ZO-signSGD using the central difference based gradient estimator (10). Right: Local regions to highlight the effect of the gradient estimators (3) and (10) on the convergence of ZO-signSGD. " + ], + "image_footnote": [], + "bbox": [ + 173, + 421, + 805, + 590 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1.2 STATISTICS OF GRADIENT ESTIMATES ", + "text_level": 1, + "bbox": [ + 178, + 741, + 475, + 753 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The intuition behind why ZO-signSGD could outperform ZO-SGD is that the sign operation can mitigate the negative effect of (coordinate-wise) gradient noise of large variance. To confirm this point, we examine the coordinate-wise variance of gradient noises during an entire training run of the binary classifier provided in the first experiment of Sec. 6. At each iteration, we perform an additional 100 random trials to obtain the statistics of gradient estimates. In Fig. A2-(a), we present the $\\ell _ { 1 }$ norm of the mean of gradient estimates (over 100 trials) versus the number of iterations. As we can see, both signSGD and ZO-signSGD outperform SGD and ZO-SGD, evidenced by a fast decrease of the $\\ell _ { 1 }$ norm of gradient estimate. In Fig. A2-(b), we present the coordinate-wise gradient noise variance (over 100 trails at each coordinate) against the number of iterations. It is not surprising that compared to first-order methods, ZO methods suffer gradient noise of larger variance. In this scenario, we could benefit from ZO-signSGD since taking the sign of gradient estimates might scale down extremely noisy components. Indeed, we observe a significant decrease of the noise variance while performing ZO-signSGD compared to ZO-SGD. ", + "bbox": [ + 173, + 771, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/a3d507387e8be5d4c8da524fed61e074d8dd332bfd27e24747d6b46669e9e7a1.jpg", + "image_caption": [ + "Figure A2: Statistics of gradient estimates during an entire training run of the binary classifier provided in the first experiment of Sec. 6. a) The $\\ell _ { 1 }$ norm of the mean of gradient estimates versus iteration. b) Coordinate-wise gradient noise variance versus iteration. The solid line represents the variance averaged over all coordinates, and the shaded region indicates the corresponding standard deviation with respect to all coordinates at each iteration. " + ], + "image_footnote": [], + "bbox": [ + 173, + 109, + 826, + 306 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "2 PROOF OF PROPOSITION 1 ", + "text_level": 1, + "bbox": [ + 176, + 401, + 421, + 417 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Based on the definition of the smoothing function $f _ { \\mu }$ , for any $\\mathbf { x }$ and $\\mathbf { y }$ we have ", + "bbox": [ + 174, + 434, + 638, + 449 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/f8f0f55f634aa7894ddbe847b4abc256297a76b4836ff7f70592805d9a01a4eb.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| \\nabla f _ { \\mu } ( x ) - \\nabla f _ { \\mu } ( y ) \\| _ { 2 } = \\| \\mathbb { E } _ { \\mathbf { v } } [ \\nabla _ { \\mathbf { x } } f ( \\mathbf { x } + \\mu \\mathbf { v } ) - \\nabla _ { \\mathbf { y } } f ( \\mathbf { y } + \\mu \\mathbf { v } ) ] \\| _ { 2 } } \\\\ & { \\qquad \\le \\mathbb { E } [ \\| \\nabla _ { \\mathbf { x } } f ( \\mathbf { x } + \\mu \\mathbf { v } ) - \\nabla _ { \\mathbf { y } } f ( \\mathbf { y } + \\mu \\mathbf { v } ) \\| _ { 2 } ] \\le L \\| \\mathbf { x } - \\mathbf { y } \\| _ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 256, + 459, + 740, + 494 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where the first inequality holds due to Jensen’s inequality, and the second inequality holds due to A1. It is known from (15) that $f _ { \\mu }$ has $L$ -Lipschitz continuous gradient. ", + "bbox": [ + 171, + 505, + 823, + 531 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "By the $L$ -smoothness of $f _ { \\mu }$ , we obtain that ", + "bbox": [ + 174, + 537, + 428, + 551 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/f56eb13bde0271917a5118e0826e9d950a1310430c839225ea477dabcc21699b.jpg", + "text": "$$\n\\begin{array} { l } { f _ { \\mu } ( { \\mathbf x } _ { k + 1 } ) \\leq f _ { \\mu } ( { \\mathbf x } _ { k } ) + \\langle \\nabla f _ { \\mu } ( { \\mathbf x } _ { k } ) , { \\mathbf x } _ { k + 1 } - { \\mathbf x } _ { k } \\rangle + \\displaystyle \\frac { L } { 2 } \\| { \\mathbf x } _ { k + 1 } - { \\mathbf x } _ { k } \\| _ { 2 } ^ { 2 } } \\\\ { \\displaystyle \\stackrel { ( 2 ) } { = } f _ { \\mu } ( { \\mathbf x } _ { k } ) - \\delta _ { k } \\langle \\nabla f _ { \\mu } ( { \\mathbf x } _ { k } ) , \\mathrm { s i g n } ( \\hat { \\mathbf g } _ { k } ) \\rangle + \\displaystyle \\frac { L } { 2 } \\delta _ { k } ^ { 2 } \\| \\mathrm { s i g n } ( \\hat { \\mathbf g } _ { k } ) \\| _ { 2 } ^ { 2 } } \\\\ { \\displaystyle = f _ { \\mu } ( { \\mathbf x } _ { k } ) - \\delta _ { k } \\| \\nabla f _ { \\mu } ( { \\mathbf x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } } \\\\ { \\displaystyle \\quad \\quad + 2 \\delta _ { k } \\displaystyle \\sum _ { i = 1 } ^ { d } \\vert ( \\nabla f _ { \\mu } ( { \\mathbf x } _ { k } ) ) _ { i } \\vert Z [ \\mathrm { s i g n } ( \\hat { \\mathbf g } _ { k , i } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( { \\mathbf x } _ { k } ) ) _ { i } ) ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 274, + 564, + 725, + 689 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $( \\nabla f _ { \\mu } ( \\mathbf { x } ) ) _ { i }$ denotes the ith element of $\\nabla f _ { \\mu } ( \\mathbf { x } )$ . ", + "bbox": [ + 173, + 700, + 495, + 715 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Taking expectation for both sides of (16), we obtain that ", + "bbox": [ + 173, + 720, + 506, + 734 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e71c56baf4f50bd949662bae4f593c05e4824075db2d59c9b26162bfcc47c73a.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } [ f _ { \\mu } ( \\mathbf { x } _ { k + 1 } ) - f _ { \\mu } ( \\mathbf { x } _ { k } ) ] \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } } \\\\ { \\displaystyle + \\ 2 \\delta _ { k } \\sum _ { i = 1 } ^ { d } \\vert ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { i } \\vert \\mathrm { P r o b } \\left[ \\mathrm { s i g n } ( \\hat { \\mathbf { g } } _ { k , i } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { i } ) \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 222, + 746, + 776, + 816 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Similar to (Bernstein et al., 2018, Theorem 1), we relax Prob $[ \\mathrm { s i g n } ( \\hat { \\mathbf { g } } _ { k , i } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { i } ) ]$ by Markov’s inequality, ", + "bbox": [ + 173, + 834, + 828, + 863 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e2e99a6e48f2448296893dd04dc4e913c3199f48d9eb787a4103404ef0be2e6f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { P r o b } \\left[ \\mathrm { s i g n } ( \\hat { \\bf g } _ { k , i } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( { \\bf x } _ { k } ) ) _ { i } ) \\right] \\leq \\mathrm { P r o b } [ \\vert \\hat { \\bf g } _ { k , i } - ( \\nabla f _ { \\mu } ( { \\bf x } _ { k } ) ) _ { i } \\vert \\ge \\vert ( \\nabla f _ { \\mu } ( { \\bf x } _ { k } ) ) _ { i } \\vert ] } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 872, + 759, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Substituting (18) into (17), we obtain ", + "bbox": [ + 173, + 104, + 395, + 118 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/aa301521b8bfacb91453c40a7d5e3ed43b5110c07afafb37b2da1ec10909279a.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle \\mathbb { E } [ f _ { \\mu } ( \\mathbf { x } _ { k + 1 } ) - f _ { \\mu } ( \\mathbf { x } _ { k } ) ] \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\displaystyle \\sum _ { i = 1 } ^ { d } \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k , i } - ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { i } \\| ] } & { } \\\\ { \\displaystyle = - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } ] } & { } \\\\ { \\displaystyle \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\sqrt { d } \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ] } & { } \\\\ { \\displaystyle \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\sqrt { d } \\sqrt { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ] } , } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 123, + 767, + 253 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where the second inequality holds due to $\\| \\mathbf { x } \\| _ { 1 } \\leq { \\sqrt { d } } \\| \\mathbf { x } \\| _ { 2 }$ , and the last inequality holds by applying Jensen’s inequality to the concave function $\\sqrt { \\cdot }$ , i.e., $\\begin{array} { r } { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ] \\le \\sqrt { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } ] } } \\end{array}$ . ", + "bbox": [ + 173, + 257, + 823, + 289 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Taking sum of both sides of (19), we then obtain (5). ", + "bbox": [ + 174, + 292, + 485, + 308 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "3 PROOF OF PROPOSITION 2 ", + "text_level": 1, + "bbox": [ + 176, + 325, + 423, + 342 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We recall from (3) that ", + "bbox": [ + 173, + 356, + 310, + 369 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/a0de2e6a4a03357fc7b82b89d8e69737e665f1207f5ab1a4d303067515ff0fc2.jpg", + "text": "$$\n\\hat { \\mathbf { g } } _ { k } = \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { T } _ { k } } \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) , \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) = \\frac { 1 } { q } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ; \\mathbf { u } _ { i , j } ) .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 375, + 673, + 412 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Let $\\mathbf { z } _ { i } : = \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )$ and $\\mathbf { z } _ { i , j } = \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ; \\mathbf { u } _ { i , j } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )$ . Thus, ", + "bbox": [ + 173, + 419, + 637, + 435 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9dc517eed6652b72100d1d0e7c12acc932d5b253f96ebc5a9bc59e5acc808587.jpg", + "text": "$$\n{ \\hat { \\mathbf { g } } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) = { \\frac { 1 } { b } } \\sum _ { i \\in { \\mathcal { T } } _ { k } } \\mathbf { z } _ { i } = { \\frac { 1 } { b q } } \\sum _ { i \\in { \\mathcal { T } } _ { k } } \\sum _ { j = 1 } ^ { q } \\mathbf { z } _ { i , j }\n$$", + "text_format": "latex", + "bbox": [ + 354, + 440, + 642, + 479 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where there are two sources of randomness: a) minibatch sampling $i \\in \\mathcal { Z } _ { k }$ , and b) the random direction sampling $\\mathbf { u } = \\mathbf { u } _ { i , j }$ . Note that these two sources of randomness are independent, and the random direction samples $\\{ \\bar { \\mathbf { u } } _ { i , j } \\}$ are i.i.d.. ", + "bbox": [ + 173, + 483, + 826, + 522 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Next, we discuss two types of mini-batch sampling: a) mini-batch samples without replacement, and b) minibatch samples with replacement. ", + "bbox": [ + 173, + 529, + 825, + 555 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Suppose that $\\mathcal { T } _ { k }$ is a uniform random subset of $[ n ]$ (no replacement), motivated by (Lei et al., 2017, Lemma A.1) we introduce a new variable $W _ { i } = I ( i \\in \\mathcal { T } _ { k } )$ , where $I$ is an indicator function, and $I ( i \\in \\mathcal { T } _ { k } ) = 1$ if $i \\in \\mathcal { Z } _ { k }$ , and 0 otherwise. As a result, we have ", + "bbox": [ + 174, + 561, + 826, + 601 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/6bc998c9e7a0be95e00570ce0d0a8894c5e8614a8539b705ee0340c52e334cb1.jpg", + "text": "$$\n\\mathbb { E } [ W _ { i } ^ { 2 } ] = \\mathbb { E } [ W _ { i } ] = \\frac { b } { n } , \\quad \\mathbb { E } [ W _ { i } W _ { j } ] = \\frac { b ( b - 1 ) } { n ( n - 1 ) } , \\ i \\neq j .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 604, + 669, + 635 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "From (21), the variance of $\\hat { \\bf g } _ { k }$ is given by ", + "bbox": [ + 173, + 638, + 418, + 654 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9091acabac36783b1cd9ec55befab3b45e80bce0c89483dae6aefe1f849e1fed.jpg", + "text": "$$\n\\begin{array} { r l } { { \\mathbb { E } [ \\| \\displaystyle \\frac { 1 } { b } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { Z } _ { a } \\| { \\mathbb { Z } } _ { 2 } ] ^ { 2 } ] } } & { = \\frac { 1 } { b ^ { 2 } } ( \\displaystyle \\sum _ { i = 1 } ^ { n } \\big ( { \\mathbb { E } } _ { i } \\langle z _ { k } | W _ { i } ^ { 2 } | { \\mathbb { E } } _ { u } \\| { \\mathbb { Z } } _ { i } \\| { \\mathbb { Z } } _ { 2 } | { \\mathbb { Z } } _ { 1 } \\big ) + \\displaystyle \\sum _ { i \\neq j } { \\mathbb { E } } _ { i , j \\in { \\mathbb { Z } } _ { k } } \\big \\langle W _ { i } W _ { j } \\big | | { \\mathbb { Z } } _ { \\mathbb { Z } _ { 1 } } \\big | { \\mathbb { Z } } _ { u } \\big | { \\mathbb { Z } } _ { 2 } \\big | ) ) } \\\\ & { = \\displaystyle \\frac { 1 } { b ^ { 2 } } ( \\displaystyle \\frac { b } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| [ z _ { 1 } | { \\mathbb { Z } } _ { 2 } ] \\| + \\displaystyle \\frac { b ( b - 1 ) } { n ( n - 1 ) } \\| \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| z _ { \\mathbb { Z } ^ { i } } \\| { \\mathbb { Z } } _ { 1 } \\| _ { 2 } ^ { 2 } - \\displaystyle \\frac { b ( b - 1 ) } { n ( n - 1 ) } \\displaystyle \\sum _ { i = 1 } ^ { n } \\| \\mathbb { E } _ { \\mathbb { Z } } \\| z _ { \\mathbb { Z } ^ { i } } \\| { \\mathbb { Z } } _ { 2 } ^ { 2 } ) } \\\\ & { \\overset { ( a ) } { = } \\displaystyle \\frac { 1 } { b n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| [ z _ { 1 } | { \\mathbb { Z } } _ { i } ] \\| ^ { 2 } - \\displaystyle \\frac { b } { b ( n - 1 ) n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\| \\mathbb { E } _ { \\mathbb { Z } } \\| z _ { i } \\| { \\mathbb { Z } } _ { 2 } ^ { 2 } } \\\\ & = \\displaystyle \\frac { 1 } { b n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| [ z _ { 1 } | { \\mathbb { Z } } _ { 2 } ] - \\displaystyle \\frac \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 659, + 800, + 862 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In (23), the equality (a) holds since $\\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i } ] = \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )$ and $\\begin{array} { r } { f _ { \\mu } ( \\mathbf { x } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } f _ { i , \\mu } ( \\mathbf { x } ) } \\end{array}$ from (4), where we have used the fact that $\\mathbb { E } _ { \\mathbf { u } } [ \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) ] = \\hat { \\nabla } f _ { i , \\mu } ( \\mathbf { x } _ { k } )$ (Liu et al., 2018c, Lemma. 1), and recall that $f _ { i , \\mu }$ denotes the smoothing function of $f _ { i }$ . The above implies that $\\begin{array} { r } { \\sum _ { i = 1 } ^ { n } { \\mathbb { E } } _ { \\mathbf { u } } [ { \\mathbf { z } } _ { i } ] = \\sum _ { i } \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) - n \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) = \\dot { \\mathbf { 0 } } } \\end{array}$ And the equality (b) holds due to $\\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i } \\| _ { 2 } ^ { 2 } ] - \\| \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i } ] \\| _ { 2 } ^ { 2 } = \\mathbb { E } _ { \\mathbf { u } } \\| \\mathbf { z } _ { i } - \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i } ] \\| _ { 2 } ^ { 2 }$ . ", + "bbox": [ + 173, + 866, + 826, + 925 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "On the other hand, suppose that the mini-batch $\\mathcal { T } _ { k }$ contains i.i.d. samples (namely, with replacement), the vectors $\\left\\{ \\mathbf { z } _ { i } \\right\\}$ are then i.i.d. under both mini-batch sampling and random direction sampling. Therefore, we obtain that ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/f9c5e10f749d8e97abd7f71645e81e3df5882e45231522de3e6a2f7bf592ab17.jpg", + "text": "$$\n\\mathbb { E } \\left[ \\left. \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { I } _ { k } } \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } \\right] = \\frac { 1 } { b ^ { 2 } } \\left[ \\mathbb { E } \\left[ \\sum _ { i \\in \\mathcal { I } _ { k } } \\| \\mathbf { z } _ { i } \\| _ { 2 } ^ { 2 } \\right] + \\mathbb { E } \\left[ \\sum _ { i \\neq j , i , j \\in \\mathcal { I } _ { k } } \\langle \\mathbf { z } _ { i } , \\mathbf { z } _ { j } \\rangle \\right] \\right]\n$$", + "text_format": "latex", + "bbox": [ + 287, + 137, + 709, + 223 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where the second equality holds since $\\begin{array} { r } { \\mathbb { E } _ { i \\in \\mathcal { T } _ { k } , \\mathbf { u } } [ \\mathbf { z } _ { i } ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i } ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) = \\mathbf { 0 } } \\end{array}$ . Combining both (23) and (24), we obtain that ", + "bbox": [ + 173, + 228, + 826, + 265 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/c5ba89f03c612eab5032f479d5ddae6faa657cd98d15cd68af7a3eed7bb9452a.jpg", + "text": "$$\n\\mathbb { E } \\left[ \\left. \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { I } _ { k } } \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } \\right] \\leq \\frac { \\alpha _ { b } } { b n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\left. \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } ] + \\frac { \\beta _ { b } } { b n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\left. \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) \\right. _ { 2 } ^ { 2 } ] .\n$$", + "text_format": "latex", + "bbox": [ + 250, + 270, + 748, + 319 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In (25), $\\alpha _ { b } \\ = \\ 1$ and $\\beta _ { b } ~ = ~ 0$ if the mini-batch contains i.i.d. samples from $[ n ]$ with replacement, and $\\alpha _ { b } = I ( b < n )$ and $\\beta _ { b } = I ( b > 1 )$ if samples are randomly selected without replacement. ", + "bbox": [ + 173, + 324, + 826, + 352 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In (25), we next bound $\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i } \\| _ { 2 } ^ { 2 } ] } \\end{array}$ , ", + "bbox": [ + 173, + 356, + 416, + 372 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/429540654c6ed6179dac78719e1e53c5d31444856d8fdea88533a12a4be621f3.jpg", + "text": "$$\n\\begin{array} { r l r } { \\displaystyle \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i } \\| _ { 2 } ^ { 2 } ] } & { \\stackrel { ( \\mathrm { 2 1 } ) } { = } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } \\left\\| \\frac { 1 } { q } \\sum _ { j = 1 } ^ { q } \\mathbf { z } _ { i , j } \\right\\| _ { 2 } ^ { 2 } = \\frac { 1 } { n q ^ { 2 } } \\sum _ { i } \\left[ \\sum _ { j } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , j } \\| _ { 2 } ^ { 2 } ] + \\sum _ { j \\neq k } \\langle \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , j } ] , \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , k } ] \\rangle \\right] } & \\\\ & { } & { \\displaystyle = \\frac { 1 } { n q ^ { 2 } } \\sum _ { i } \\left[ q \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] + ( q ^ { 2 } - q ) \\| \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , 1 } ] \\| _ { 2 } ^ { 2 } \\right] } & \\\\ & { } & { \\displaystyle = \\frac { 1 } { q n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] + \\frac { q - 1 } { q n } \\sum _ { i } \\| \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , 1 } ] \\| _ { 2 } ^ { 2 } , } & { ( 2 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 378, + 810, + 492 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where we have used the facts that $\\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , j } ] = \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , 1 } ]$ and $\\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , j } \\| _ { 2 } ^ { 2 } ] = \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ]$ for any $j$ since random direction vectors $\\{ \\mathbf { u } _ { i , j } \\} _ { j = 1 } ^ { q }$ are i.i.d. samples. ", + "bbox": [ + 173, + 498, + 823, + 527 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In (26), we further bound $\\begin{array} { r } { \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] } \\end{array}$ , ", + "bbox": [ + 174, + 535, + 428, + 550 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/f4e59a000f669f461d13fb6d5d95bfa7d3cbde831e6d51c86043bfa63d28236c.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] = \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } \\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ; \\mathbf { u } _ { i , 1 } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { \\quad = \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { i , \\mu } + \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } ] } \\\\ & { } & { \\quad \\leq \\displaystyle \\frac { 2 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { i , \\mu } \\| _ { 2 } ^ { 2 } ] + \\frac { 2 } { n } \\sum _ { i } \\| \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 556, + 732, + 659 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where for ease of notation, let $\\hat { \\nabla } f _ { i } : = \\hat { \\nabla } f _ { i } ( \\mathbf x _ { k } ; \\mathbf u _ { i , 1 } )$ , $\\hat { \\nabla } f _ { i , \\mu } : = \\hat { \\nabla } f _ { i , \\mu } ( \\mathbf x _ { k } )$ and $\\nabla f _ { \\mu } : = \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )$ . According to (Liu et al., 2018c, Lemma 1), the first term at RHS of (27) yields ", + "bbox": [ + 171, + 665, + 825, + 694 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/c61b8704bee28cb58e0a56425b076129ae98c523a6039639c1cfc6803950d259.jpg", + "text": "$$\n\\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { i , \\mu } \\| _ { 2 } ^ { 2 } ] \\le 2 d \\| \\nabla f _ { i } \\| _ { 2 } ^ { 2 } + \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } \\le 2 d \\sigma ^ { 2 } + \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } : = C ( d , \\mu ) ,\n$$", + "text_format": "latex", + "bbox": [ + 258, + 699, + 738, + 729 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where the last inequality holds due to A2. Based on the definition of $f _ { \\mu }$ , the second term at RHS of (27) yields ", + "bbox": [ + 173, + 734, + 825, + 750 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/5ab99398c65b7b0f91278f84e8e10870ecbb4356c741b286cb0884855700c347.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\displaystyle \\frac { 1 } { n } \\sum _ { i } \\| \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } = \\frac { 1 } { n } \\sum _ { i } \\| \\mathbb { E } _ { \\mathbf { v } } \\big [ \\nabla f _ { i } ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) - \\nabla f ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) \\big ] \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { \\leq \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { v } } \\big [ \\| \\nabla f _ { i } ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) - \\nabla f ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) \\| _ { 2 } ^ { 2 } \\big ] \\leq 4 \\sigma ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 251, + 753, + 746, + 823 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where we have used the Jensen’s inequality and $\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\| \\nabla f _ { i } ( \\mathbf { x } ) - \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } \\leq 4 \\sigma ^ { 2 } } \\end{array}$ under A2. ", + "bbox": [ + 174, + 828, + 745, + 847 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Substituting (28) and (29) into (27), we have ", + "bbox": [ + 173, + 849, + 441, + 863 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/c0c76bbf2b2a8f13e7661b5e8ce2eb8b26e7be4f4698ec9489caca94e69db80b.jpg", + "text": "$$\n\\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] = \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } ] \\leq 4 d \\sigma ^ { 2 } + \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } + 8 \\sigma ^ { 2 } = 2 C ( d , \\mu ) + 8 \\sigma ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 199, + 869, + 772, + 904 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $C ( d , \\mu )$ was defined in (28). ", + "bbox": [ + 173, + 910, + 382, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We are now ready to bound (26). Based on $\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i } \\| \\mathbb { E } _ { { \\mathbf { u } } } [ { \\mathbf { z } } _ { i , 1 } ] \\| _ { 2 } ^ { 2 } = \\frac { 1 } { n } \\sum _ { i } \\| \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } \\leq 4 \\sigma ^ { 2 } } \\end{array}$ from (29), and substituting (30) into (26), we obtain that ", + "bbox": [ + 174, + 102, + 823, + 131 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/0ea78fcc459068eae77db4447740f4bd7db35ff004dcdb7228127fdfa1f5fe32.jpg", + "text": "$$\n\\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\left. \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } ] \\leq \\frac { 2 C ( d , \\mu ) + 8 \\sigma ^ { 2 } } { q } + \\frac { 4 ( q - 1 ) \\sigma ^ { 2 } } { q } .\n$$", + "text_format": "latex", + "bbox": [ + 338, + 136, + 658, + 171 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In (25), we also need to bound $\\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } ]$ ", + "bbox": [ + 171, + 179, + 545, + 196 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/179e0a4054d96abb65a10ae2188ed16786ccf17040b5a48d1224e146f6678b75.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathbb { E } _ { \\mathbf { u } } \\left[ \\left\\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) \\right\\| _ { 2 } ^ { 2 } \\right] \\stackrel { ( 2 0 ) } { = } \\mathbb { E } _ { \\mathbf { u } } \\left[ \\left\\| \\frac { 1 } { q } \\sum _ { j = 1 } ^ { q } \\left( \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } ) \\right) \\right\\| _ { 2 } ^ { 2 } \\right] } \\\\ & { \\stackrel { ( a ) } { = } \\frac { 1 } { q } \\mathbb { E } _ { \\mathbf { u } } \\left[ \\left\\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , 1 } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } ) \\right\\| _ { 2 } ^ { 2 } \\right] \\stackrel { ( 2 8 ) } { \\leq } \\frac { 1 } { q } \\left( 2 d \\sigma ^ { 2 } + \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } \\right) = \\frac { C ( d , \\mu ) } { q } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 200, + 756, + 281 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the equality (a) holds since $\\mathbb { E } _ { \\mathbf { u } } [ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) ] \\ = \\ \\nabla f _ { i , \\mu } ( \\mathbf { x } )$ for any $j$ , given by (Liu et al., 2018c, Lemma 1). ", + "bbox": [ + 173, + 286, + 828, + 315 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Substituting (31) and (32) into (25) ", + "bbox": [ + 173, + 320, + 383, + 335 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/a04b4d8a24a7fd507d407eecff4f8622c3f86e9709b383164671e8e3b53229d5.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\left[ \\left. \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\right. _ { 2 } ^ { 2 } \\right] = \\mathbb { E } \\left[ \\left. \\displaystyle \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { I } _ { k } } \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } \\right] \\leq \\displaystyle \\frac { \\alpha _ { b } } { b } \\displaystyle \\frac { 2 C ( d , \\mu ) + 4 \\sigma ^ { 2 } + 4 \\sigma ^ { 2 } q } { q } + \\displaystyle \\frac { \\beta _ { b } C ( d , \\mu ) } { b q } } \\\\ { = \\displaystyle \\frac { 4 \\alpha _ { b } ( q + 1 ) } { b q } \\sigma ^ { 2 } + \\displaystyle \\frac { C ( d , \\mu ) } { b q } ( 2 \\alpha _ { b } + \\beta _ { b } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 340, + 769, + 420 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "4 PROOF OF THEOREM 1 ", + "text_level": 1, + "bbox": [ + 174, + 457, + 392, + 473 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Substituting (6) into (5), we obtain ", + "bbox": [ + 174, + 487, + 382, + 501 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/31512d66f14787caef21934b4fa25ae5f63d3fc1af63adcb94df1e1c81e97b77.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\mathbb { E } [ \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } ] \\le \\mathbb { E } [ f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f _ { \\mu } ( \\mathbf { x } _ { T } ) ] + \\sum _ { k = 0 } ^ { T - 1 } [ 2 \\delta _ { k } \\frac { \\sqrt { d } } { \\sqrt { b q } } \\sqrt { 4 \\alpha _ { b } ( q + 1 ) \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } ] } } \\\\ & { } & { \\quad + \\frac { d L } { 2 } \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } . \\eqno ( 3 4 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 507, + 818, + 588 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "It is known from (Liu et al., 2018c, Lemma 1) that ", + "bbox": [ + 174, + 592, + 472, + 606 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/958ed7c8d1538a6d14eefff0ffbdebcbe62f85549793ce3bae714d90afa2b718.jpg", + "text": "$$\n\\| \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } \\leq 2 \\| \\nabla f _ { \\mu } ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } + { \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } } , \\quad | f _ { \\mu } ( \\mathbf { x } ) - f ( \\mathbf { x } ) | \\leq { \\frac { \\mu ^ { 2 } L } { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 295, + 611, + 700, + 641 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "From (35) we have $\\begin{array} { r } { f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f ( \\mathbf { x } _ { 0 } ) \\leq \\frac { \\mu ^ { 2 } L } { 2 } } \\end{array}$ and $\\begin{array} { r } { f ^ { * } - f _ { \\mu } ^ { * } \\le \\frac { \\mu ^ { 2 } L } { 2 } } \\end{array}$ , where $f _ { \\mu } ^ { * } = \\mathrm { m i n } _ { \\mathbf { x } } f _ { \\mu } ( \\mathbf { x } )$ and $f ^ { * } = \\operatorname* { m i n } _ { \\mathbf { x } } { f ( \\mathbf { x } ) }$ . This yields $f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f ( \\mathbf { x } _ { 0 } ) + f ^ { * } - f _ { \\mu } ^ { * } \\leq \\mu ^ { 2 } L$ , and thus ", + "bbox": [ + 171, + 647, + 826, + 680 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/cdc159917fff443ba2dbb8c96231210f166f8bad22cf3b2d04780308a5b953e8.jpg", + "text": "$$\nf _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f _ { \\mu } ( \\mathbf { x } _ { T } ) \\leq f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f _ { \\mu } ^ { * } \\leq ( f ( \\mathbf { x } _ { 0 } ) - f ^ { * } ) + \\mu ^ { 2 } L .\n$$", + "text_format": "latex", + "bbox": [ + 318, + 686, + 679, + 704 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Substituting (36) into (34), we obtain ", + "bbox": [ + 173, + 717, + 395, + 731 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/a0111f799b94ad45bfad46344737b8ad7056fe934f94cc6a9fd8c9862bcbeb17.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } \\left[ \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } \\right] \\leq \\mathbb { E } [ f ( \\mathbf { x } _ { 0 } ) - f ^ { * } ) ] + \\mu ^ { 2 } L } \\\\ & { \\qquad + \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\left[ 2 \\delta _ { k } \\frac { \\sqrt { d } } { \\sqrt { b q } } \\sqrt { 4 \\alpha _ { b } ( q + 1 ) \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } \\right] + \\frac { d L } { 2 } \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 737, + 789, + 819 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Due to $\\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } \\leq \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 }$ and dividing $\\textstyle \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k }$ for both sides of (37), we obtain that ", + "bbox": [ + 169, + 823, + 754, + 840 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/28aae7e2b5ac3dfc5a79ae8000028ac0baabb46709e4fea022ff74519fedbe96.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\left[ \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\frac { \\delta _ { k } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } \\right] \\leq \\frac { f ( \\mathbf { x } _ { 0 } ) - f ^ { * } + \\mu ^ { 2 } L } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } + \\frac { 2 \\sqrt { d } } { \\sqrt { b q } } \\sqrt { 4 \\alpha _ { b } ( q + 1 ) \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } } \\\\ { + \\frac { d L } { 2 } \\frac { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } . \\qquad ( 3 \\leq \\alpha \\leq \\alpha ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 847, + 808, + 925 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "By introducing the random variable $R$ with probability $\\begin{array} { r } { P ( R = k ) = \\frac { \\delta _ { k } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } } \\end{array}$ , we then obtain that ", + "bbox": [ + 173, + 102, + 759, + 125 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/78bcf7f600fecaf2996f75d98f9f54ac21a76360cca9b5ca21e45ea398ae0b63.jpg", + "text": "$$\n\\mathbb { E } \\left[ \\Vert \\nabla f _ { \\mu } ( \\mathbf { x } _ { R } ) \\Vert _ { 2 } \\right] = \\mathbb { E } \\left[ \\mathbb { E } _ { R } [ \\Vert \\nabla f _ { \\mu } ( \\mathbf { x } _ { R } ) \\Vert _ { 2 } ] \\right] = \\mathbb { E } \\left[ \\sum _ { k = 0 } ^ { T - 1 } P ( R = k ) \\Vert \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\Vert _ { 2 } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 258, + 130, + 738, + 169 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Based on (35), we further have ", + "bbox": [ + 173, + 172, + 359, + 186 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/9f8234e4878bdf820434807aa5e492348a6fc4e5d588c569284bfb089bdadcf9.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] \\leq \\mathbb { E } \\left[ \\sqrt { 2 \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { R } ) \\| _ { 2 } ^ { 2 } + \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } } \\right] } \\\\ & { \\displaystyle \\leq \\sqrt { 2 } \\mathbb { E } [ \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { R } ) \\| _ { 2 } ] + \\frac { \\mu L d } { \\sqrt { 2 } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 339, + 190, + 656, + 261 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where we have used the fact that ${ \\sqrt { a ^ { 2 } + b ^ { 2 } } } \\leq ( a + b )$ for $a , b \\geq 0$ . ", + "bbox": [ + 171, + 263, + 570, + 280 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Substituting (38) and (39) into (40), we finally obtain (8). ", + "bbox": [ + 176, + 285, + 511, + 300 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "5 PROOF OF COROLLARY 1 ", + "text_level": 1, + "bbox": [ + 176, + 318, + 413, + 334 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Upon defining $\\begin{array} { r } { \\tilde { \\nabla } f _ { i } ( \\mathbf x ; \\mathbf u ) = \\frac { d \\left[ f _ { i } ( \\mathbf x + \\mu \\mathbf u ) - f _ { i } ( \\mathbf x - \\mu \\mathbf u ) \\right] \\mathbf u } { 2 \\mu } } \\end{array}$ (against $\\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } )$ in (3)), our major task is to derive the first- and second-order moments of $\\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } )$ . ", + "bbox": [ + 176, + 345, + 823, + 380 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Given $\\mathbf { x }$ , we first study the mean of $\\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } )$ ", + "bbox": [ + 176, + 387, + 452, + 401 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/9e30ac888d77f90363953682eca8b9b621e50862fb8b6bec7ace9dc053c5a15a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { \\mathbf { u } } \\left[ \\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right] = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { 2 \\mu } ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ) \\mathbf { u } \\right] } \\\\ & { = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { 2 \\mu } f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) \\mathbf { u } \\right] + \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { 2 \\mu } f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ( - \\mathbf { u } ) \\right] \\stackrel { ( a ) } = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { \\mu } f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) \\mathbf { u } \\right] } \\\\ & { \\stackrel { ( b ) } = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { \\mu } \\left( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } ) \\right) \\mathbf { u } \\right] = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right] \\stackrel { ( c ) } = \\nabla f _ { i , \\mu } ( \\mathbf { x } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 406, + 743, + 503 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where (a) holds since the distribution of $\\mathbf { u }$ is symmetric around the origin, (b) holds since $\\mathbb { E } [ { \\mathbf { u } } ] = \\mathbf { 0 }$ , and (c) holds since $\\begin{array} { r } { \\nabla f _ { i , \\mu } ( \\mathbf { x } ) = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { \\mu } f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) \\mathbf { u } \\right] } \\end{array}$ obtained from (Gao et al., 2014, Lemma 4.1). It is clear from (41) that $\\mathbb { E } _ { \\mathbf { u } } \\left[ \\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right] = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right]$ . ", + "bbox": [ + 176, + 506, + 825, + 564 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We next study the second-order moment of $\\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } )$ , ", + "bbox": [ + 173, + 570, + 500, + 585 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/7dbbd99d84aa0c73702c9d8660c9265d9a019f4c9489e21e34bd17d87aeedf7b.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { \\mathbf { u } } [ \\| \\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\| ^ { 2 } ] = \\frac { d ^ { 2 } } { 4 \\mu ^ { 2 } } \\mathbb { E } _ { \\mathbf { u } } [ ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ) ^ { 2 } \\| \\mathbf { u } \\| ^ { 2 } ] } \\\\ { \\displaystyle \\leq \\frac { d ^ { 2 } } { 2 \\mu ^ { 2 } } \\mathbb { E } _ { \\mathbf { u } } [ ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } ) ) ^ { 2 } + ( f _ { i } ( \\mathbf { x } ) - f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ) ^ { 2 } ] } \\\\ { \\displaystyle = \\mathbb { E } _ { \\mathbf { u } } [ \\| \\frac { d } { \\mu } ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } ) ) \\mathbf { u } \\| ^ { 2 } ] = \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\| ^ { 2 } ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 590, + 712, + 693 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where we have used the fact that $\\| \\mathbf { u } \\| _ { 2 } = 1$ . ", + "bbox": [ + 174, + 696, + 433, + 712 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Based on (41) and (42), we can conclude that both (3) and (10) maintain the same statistical properties. Following the proofs of Proposition 2 and Theorem 1, it is not difficult to reach the convergence rate in (8). \u0003 ", + "bbox": [ + 173, + 717, + 823, + 743 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "6 PROOF OF COROLLARY 2 ", + "text_level": 1, + "bbox": [ + 176, + 762, + 413, + 779 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Let $\\hat { \\mathbf { g } } _ { k } ^ { i , j } : = \\hat { \\nabla } f _ { i } ( \\mathbf x _ { k } ; \\mathbf u _ { i , j } )$ . If we replace $\\hat { \\bf g } _ { k }$ with $\\hat { \\mathbf { g } } _ { k } ^ { i , j }$ in (18), we then have ", + "bbox": [ + 171, + 790, + 620, + 808 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/4154db6011aafdbb4b084150e14d8193c9cd0cbf79a37875b5d2e4338cc16275.jpg", + "text": "$$\n| ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) \\iota | \\mathrm { P r o b } \\left[ \\mathrm { s i g n } ( \\hat { g } _ { k , l } ^ { i , j } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) \\iota ) \\right] \\leq \\mathbb { E } [ | \\hat { g } _ { k , l } ^ { i , j } - ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) \\iota | ] ,\n$$", + "text_format": "latex", + "bbox": [ + 254, + 810, + 741, + 835 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $\\hat { g } _ { k , l } ^ { i , j }$ is the lth coordinate of $\\hat { \\mathbf { g } } _ { k } ^ { i , j }$ ", + "bbox": [ + 173, + 840, + 405, + 857 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "By letting $b = 1$ and $q = 1$ in Proposition 2, we know that E $\\big [ \\lVert \\hat { \\mathbf { g } } _ { k } ^ { i , j } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\rVert _ { 2 } ^ { 2 } \\big ]$ is upper bounded. Moreover, by Jensen’s inequality we have ", + "bbox": [ + 173, + 864, + 825, + 893 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/5702569bba6c65e5395be5a4db5228b858e6ce4d4bee43d00e1d317c8874a9f6.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } [ | \\hat { g } _ { k , l } ^ { i , j } - ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) \\ b { \\imath } | ] \\leq \\sqrt { \\mathbb { E } [ ( \\hat { g } _ { k , l } ^ { i , j } - ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) \\ b { \\imath } ) ^ { 2 } ] } : = \\xi _ { l } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 897, + 681, + 922 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $\\xi _ { l }$ is finite since $\\mathbb { E } \\left[ \\lVert \\hat { \\mathbf { g } } _ { k } ^ { i , j } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\rVert _ { 2 } ^ { 2 } \\right]$ is upper bounded. ", + "bbox": [ + 173, + 102, + 563, + 121 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Substituting (44) into (43), we have ", + "bbox": [ + 174, + 125, + 387, + 138 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/a3cd985f5e905c978b19ba8d0a73db325d57bc3d599ff996fdfd263731162b3e.jpg", + "text": "$$\n\\begin{array} { r l } & { | ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { l } | \\operatorname { P r o b } \\left[ \\operatorname { s i g n } ( \\hat { g } _ { k , l } ^ { i , j } ) \\neq \\operatorname { s i g n } ( ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { l } ) \\right] \\leq \\xi _ { l } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 142, + 678, + 166 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "With the new gradient estimate $\\begin{array} { r } { \\bar { \\bf g } _ { k } = \\sum _ { i \\in \\mathcal { T } _ { k } } \\sum _ { j = 1 } ^ { q } \\mathrm { s i g n } \\big ( \\hat { \\bf g } _ { k } ^ { i , j } \\big ) } \\end{array}$ in (11), we require to bound ", + "bbox": [ + 174, + 178, + 714, + 195 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/d997bede164817ee5e0339bfebd9d02dc647c5626fcf6ad2b9203399cea07176.jpg", + "text": "$$\n\\operatorname { P r o b } \\left[ \\operatorname { s i g n } ( { \\bar { g } } _ { k , l } ) \\neq \\operatorname { s i g n } ( ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { l } ) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 200, + 619, + 217 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where ${ \\bar { g } } _ { k , l }$ is the $l$ th coordinate of $\\bar { \\bf g } _ { k }$ . ", + "bbox": [ + 174, + 219, + 398, + 233 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We recall that $\\hat { g } _ { k , l } ^ { i , j }$ is an unbiased stochastic approximation to gradient component $( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { l }$ with variance $\\xi _ { l } ^ { 2 }$ . Under the assumption that the noise distribution is unimodal and symmetric, we know from (Bernstein et al., 2018, Lemma D1) that ", + "bbox": [ + 174, + 241, + 826, + 280 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/9003bc309c7733bf065643bd0bc307d3e301fdd347325c261c7737c6f47f9b2f.jpg", + "text": "$$\n\\mathrm { P r o b } \\left[ \\mathrm { s i g n } ( \\hat { g } _ { k , l } ^ { i , j } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { l } \\right] : = Q \\leq \\left\\{ \\begin{array} { l l } { \\frac { 2 } { 9 } \\frac { 1 } { S ^ { 2 } } } & { S > \\frac { 2 } { \\sqrt { 3 } } } \\\\ { \\frac { 1 } { 2 } - \\frac { S } { 2 \\sqrt { 3 } } } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. < \\frac { 1 } { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 253, + 284, + 741, + 321 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $S : = | ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) \\iota | / \\xi _ { l }$ ", + "bbox": [ + 173, + 325, + 352, + 340 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Let $Z$ count the number of estimates $\\{ \\hat { g } _ { k , l } ^ { i , j } \\}$ yielding the correct sign of $( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { l }$ . Thus, the probability of error in (46) is equal to ", + "bbox": [ + 173, + 347, + 826, + 375 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/41d26245669e9a9f155a92ce177531596f9002554ba590b158b8a06143f0b63d.jpg", + "text": "$$\n\\operatorname { P r o b } \\left[ \\operatorname { s i g n } ( { \\bar { g } } _ { k , l } ) \\neq \\operatorname { s i g n } ( ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { l } ) \\right] = \\operatorname { P r o b } \\left[ Z \\leq { \\frac { b q } { 2 } } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 316, + 377, + 681, + 410 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Following the proof of (Bernstein et al., 2018, Theorem 2b) under (47), it is not difficult to obtain that ", + "bbox": [ + 169, + 419, + 772, + 434 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/57bd981a1028f2d4b46047bff3be7f1935c87c23dac33ee3f00ad7489ff1aeb0.jpg", + "text": "$$\n\\operatorname { P r o b } \\left[ Z \\leq { \\frac { b q } { 2 } } \\right] \\leq { \\frac { 1 } { \\sqrt { b q } S } } .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 436, + 583, + 469 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "That is, ", + "bbox": [ + 173, + 472, + 220, + 484 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/0802f1a576fc660baba912f0b8519630e010e01449478042c093a3990f761ed7.jpg", + "text": "$$\n| ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) \\iota | \\operatorname { P r o b } \\left[ \\mathrm { s i g n } ( { \\bar { g } } _ { k , l } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) \\iota ) \\right] \\leq { \\frac { \\xi _ { l } } { \\sqrt { b q } } } .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 488, + 686, + 518 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Replace $\\hat { \\bf g } _ { k }$ with $\\bar { \\bf g } _ { k }$ in (17), we obtain that ", + "bbox": [ + 173, + 529, + 428, + 542 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/25f8d197455dab898dc5db34f6a85f4ad04fa984dac26f8a2cb14a675338d5a6.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathbb { E } \\{ f _ { \\mu } ( \\mathbf { x } _ { k + 1 } ) - f _ { \\mu } ( \\mathbf { x } _ { k } ) \\} \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } } \\\\ & { \\qquad + 2 \\delta _ { k } \\displaystyle \\sum _ { l = 1 } ^ { d } | ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { l } | \\mathrm { P r o b } \\big [ \\mathrm { s i g n } ( \\bar { g } _ { k , l } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { l } ) \\big ] } \\\\ { \\overset { ( \\mathrm { s p } ) } { \\leq } - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\frac { 1 } { \\sqrt { N q } } \\| \\xi \\| _ { 1 } } \\\\ & { \\qquad \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\frac { \\sqrt { d } } { \\sqrt { N q } } \\sqrt { | \\xi \\| _ { 2 } ^ { 2 } } } \\\\ & { \\overset { ( \\mathrm { a } 4 ) } { = } - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\frac { \\sqrt { d } } { \\sqrt { N q } } \\sqrt { \\mathbb { E } [ \\| \\hat { g } _ { k } ^ { i , j } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } ] } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 220, + 546, + 776, + 718 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Compared (51) with (19), the standard deviation $\\sqrt { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } ^ { i , j } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } ] }$ is reduced by the factor $1 / { \\sqrt { b q } }$ . According to Proposition 2, let $b = q = 1$ , we obtain ", + "bbox": [ + 173, + 728, + 826, + 762 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/779ecb3885339c53c9a70cb5329b2f2006656408de1a3b7b457dfbb2819ea033.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\left[ \\| \\hat { \\mathbf { g } } _ { k } ^ { i , j } - \\nabla f _ { \\mu } ( { \\mathbf { x } } _ { k } ) \\| _ { 2 } ^ { 2 } \\right] \\leq 8 \\alpha _ { b } \\sigma ^ { 2 } + ( 2 \\alpha _ { b } + \\beta _ { b } ) C ( d , \\mu ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 326, + 766, + 671, + 791 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Based on (51)-(52) and following the proof of Theorem 1, we have ", + "bbox": [ + 174, + 794, + 570, + 808 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/739ba13175e5d4c5afd75b75eddc12d3210c55296b5141b7e78e2acec038ff04.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] \\leq \\sqrt { 2 } \\frac { f ( \\mathbf { x } _ { 0 } ) - f ^ { * } + \\mu ^ { 2 } L } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } } \\\\ & { \\displaystyle + \\frac { 2 \\sqrt { 2 } \\sqrt { d } } { \\sqrt { b q } } \\sqrt { 8 \\alpha _ { b } \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } + \\frac { d L } { \\sqrt { 2 } } \\frac { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } + \\frac { \\mu L d } { \\sqrt { 2 } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 811, + 707, + 883 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $C ( d , \\mu ) : = 2 d \\sigma ^ { 2 } + \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } / 2$ . ", + "bbox": [ + 173, + 886, + 401, + 901 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "If $\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}$ and $\\begin{array} { r } { \\delta _ { k } = { O } \\left( \\frac { 1 } { \\sqrt { d T } } \\right) } \\end{array}$ , then the convergence rate simplifies to $\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { d } { \\sqrt { b q } } ) } \\end{array}$ . ", + "bbox": [ + 173, + 907, + 715, + 928 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "7 PROOF OF COROLLARY 3 ", + "text_level": 1, + "bbox": [ + 176, + 101, + 413, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Let $\\begin{array} { r } { \\hat { \\mathbf { g } } _ { k } ^ { m } : = \\frac { 1 } { b _ { m } q } \\sum _ { i \\in \\mathcal { T } _ { m , k } } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } ( \\mathbf x _ { k } ; \\mathbf u _ { i , j } ) } \\end{array}$ and $\\begin{array} { r } { \\bar { \\bf g } _ { k } = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\mathrm { s i g n } ( \\hat { \\bf g } _ { k } ^ { m } ) } \\end{array}$ . Following (43)-(50), we can similarly obtain that ", + "bbox": [ + 173, + 130, + 825, + 162 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/57de15381771dddf98a3ab37ecba2786aa40f157801fd8297536bd63d283ef7d.jpg", + "text": "$$\n| ( \\nabla f _ { \\mu } ( { \\bf x } _ { k } ) ) _ { l } | \\mathrm { P r o b } [ \\mathrm { s i g n } ( \\bar { g } _ { k , l } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( { \\bf x } _ { k } ) ) _ { l } ) ] \\leq \\frac { \\xi _ { l } } { \\sqrt { M } } ,\n$$", + "text_format": "latex", + "bbox": [ + 307, + 170, + 687, + 200 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\xi _ { l } : = \\sqrt { \\mathbb { E } [ ( \\hat { g } _ { k , l } ^ { m } - ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { l } ) ^ { 2 } ] }$ . By mimicking the derivation in (51), we have ", + "bbox": [ + 171, + 208, + 692, + 231 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/972b9307656309212abed7549440249226a2bbcc5a857078fd1185724a3f4555.jpg", + "text": "$$\n\\mathbb { E } [ f _ { \\mu } ( { \\mathbf x } _ { k + 1 } ) - f _ { \\mu } ( { \\mathbf x } _ { k } ) ] \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( { \\mathbf x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\frac { \\sqrt { d } } { \\sqrt { M } } \\sqrt { \\mathbb { E } [ \\| \\hat { \\mathbf g } _ { k } ^ { m } - \\nabla f _ { \\mu } ( { \\mathbf x } _ { k } ) \\| _ { 2 } ^ { 2 } ] } .\n$$", + "text_format": "latex", + "bbox": [ + 223, + 238, + 774, + 272 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "According to Proposition 2, let $b = \\lfloor n / M \\rfloor$ , we obtain ", + "bbox": [ + 173, + 286, + 498, + 301 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/27dd89d4d95d80730fe382303461d49a3ff5846ae5ceaf803ee4c405555c9b56.jpg", + "text": "$$\n\\mathbb { E } \\left[ \\Vert \\hat { \\mathbf { g } } _ { k } ^ { m } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\Vert _ { 2 } ^ { 2 } \\right] \\leq \\frac { 4 ( q + 1 ) } { b q } \\sigma ^ { 2 } + \\frac { 3 } { b q } C ( d , \\mu ) .\n$$", + "text_format": "latex", + "bbox": [ + 338, + 308, + 660, + 339 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Based on (55)-(56) and following the proof of Theorem 1, we have ", + "bbox": [ + 173, + 345, + 570, + 361 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/30ffb21d5ba7e5903d69c38a19c0454c61d0e7fbc97d4e1d0de9de2865bb25ef.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] \\leq \\sqrt { 2 } \\frac { f ( \\mathbf { x } _ { 0 } ) - f ^ { * } + \\mu ^ { 2 } L } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } } \\\\ & { + \\frac { 2 \\sqrt { 2 } \\sqrt { d } } { \\sqrt { M b q } } \\sqrt { 4 ( q + 1 ) \\sigma ^ { 2 } + 3 C ( d , \\mu ) } + \\frac { d L } { \\sqrt { 2 } } \\frac { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } + \\frac { \\mu L d } { \\sqrt { 2 } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 367, + 692, + 440 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $C ( d , \\mu ) : = 2 d \\sigma ^ { 2 } + \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } / 2$ . ", + "bbox": [ + 173, + 445, + 403, + 462 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Since $\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}$ , $\\begin{array} { r } { \\delta _ { k } = { O } \\left( \\frac { 1 } { \\sqrt { d T } } \\right) } \\end{array}$ and $b M \\approx n$ , then the convergence rate simplifies to $\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { \\sqrt { d } } { \\sqrt { n } } + \\frac { d } { \\sqrt { n q } } ) } \\end{array}$ ", + "bbox": [ + 174, + 467, + 825, + 494 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "8 SUPPLEMENTAL EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 518, + 465, + 536 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "8.1 SYNTHETIC EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 551, + 400, + 565 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In Fig. A3, we demonstrate the effect of the learning rate $\\delta$ on the training loss of SGD, signSGD, ZO-SGD, ZO-SCD, ZO-signSGD and ZO-M-signSGD. We observe that compared to the gradient-based algorithms (SGD, ZO-SGD and ZO-SCD), the gradient sign-based algorithms support a more flexible choice of learning rates (corresponding to less variance), since the sign operation has an normalization effect to reduce oscillations in convergence. We find $\\delta = 0 . 1$ best for SGD, $\\delta = 0 . 0 0 9$ best for signSGD, $\\delta = 0 . 1$ best for ZOSGD and ZOSCD, $\\bar { \\delta } = 0 . 0 1 7 8$ best for ZOsignSGD, and $\\delta = 0 . 0 5 0 1$ best for ZO-M-signSGD. ", + "bbox": [ + 173, + 575, + 826, + 654 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/b814eebc6dcda2f68f8e69b9576e2d7eccddf9b5d27d6725c23c28669304782f.jpg", + "image_caption": [ + "Figure A3: The training loss at the last iteration versus the constant learning rate $\\delta \\in [ 1 0 ^ { - 3 } , 0 . 1 ]$ . Here the solid line represents the loss averaged over 10 independent trials with random initialization, and the shaded region indicates the standard deviation of results over random trials. " + ], + "image_footnote": [], + "bbox": [ + 343, + 685, + 633, + 858 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "8.2 BLACK-BOX ATTACK FORMULATION ", + "text_level": 1, + "bbox": [ + 176, + 103, + 465, + 117 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "The target DNN classifier $F = [ F _ { 1 } , F _ { 2 } , \\dots , F _ { K } ]$ takes an image as an input and produces the classification predictions (here the probability distribution) of $K$ image classes, where $F _ { k }$ denotes the prediction of class $k$ Given $F$ , an adversarial example $\\mathbf { x }$ of a legitimate example $\\mathbf { x } _ { 0 }$ means it is visually similar to $\\mathbf { x } _ { 0 }$ but will yield a different top-1 prediction than $\\mathbf { x } _ { 0 }$ . ", + "bbox": [ + 174, + 127, + 825, + 180 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Let $\\left( \\mathbf { x } _ { 0 } , t _ { 0 } \\right)$ denote a legitimate image $\\mathbf { x } _ { 0 }$ and its groundtruth label $t _ { 0 } ~ \\in ~ \\{ 1 , 2 , \\ldots , K \\}$ . Without loss of generality, we assume the pixel value range of $\\mathbf { x } _ { 0 }$ lies within $[ - 0 . 5 , 0 . 5 ] ^ { d }$ . By using the tanh transformation on the adversarial image $\\mathbf { x }$ such that $\\mathbf { x } = \\operatorname { t a n h } ( \\mathbf { w } ) / 2$ , where $\\mathbf { w } \\in \\mathbb { R } ^ { d }$ , we adopt the untargeted black-box attacking loss designed in (Chen et al., 2017), which is defined as ", + "bbox": [ + 174, + 186, + 825, + 241 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/d70556820e201b3690cc0535eb214a8a153192ff06b7d2d5ef20639f0d085aed.jpg", + "text": "$$\n\\underset { \\mathbf { w } \\in \\mathbb { R } ^ { d } } { \\mathrm { m i n i m i z e ~ } } c \\cdot \\mathbf { m a x } \\{ \\log F _ { t _ { 0 } } ( \\operatorname { t a n h } ( \\mathbf { w } ) / 2 ) - \\underset { j \\neq t _ { 0 } } { \\mathrm { m a x } } \\log F _ { j } ( \\operatorname { t a n h } ( \\mathbf { w } ) / 2 ) , 0 \\} + \\| \\operatorname { t a n h } ( \\mathbf { w } ) / 2 - \\mathbf { x } _ { 0 } \\| _ { 2 } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 192, + 248, + 777, + 271 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $c$ is a regularization coefficient and $\\mathbf { x } ~ = ~ \\operatorname { t a n h } ( \\mathbf { w } ) / 2$ ensures $\\mathbf { x }$ lies within the valid image space $[ - 0 . 5 , 0 . 5 ] ^ { d }$ . The first term in the attacking objective is a hinge loss function that penalizes the top-1 prediction of $\\mathbf { x }$ being $t _ { 0 }$ . The $\\log F ( \\cdot )$ operation preserves the class prediction ranking and better handles numerical stability. The second term encourages the visual similarity between $\\mathbf { x }$ and $\\mathbf { x } _ { 0 }$ through penalizing their squared $\\ell _ { 2 }$ difference (i.e., the squared distortion). In our experiment, we set $c = 1$ for MNIST and $c = 0 . 1$ for CIFAR-10. We also note that different from the use of signed gradients in existing black-box attacks (e.g., Ilyas et al. (2018a); Bhagoji et al. (2018)) due to the explicit $\\ell _ { \\infty }$ perturbation constraint, the use and benefit of ZO-signSGD for solving (58) in our experiment are non-trivial since the attacking objective does not impose any $\\ell _ { \\infty }$ constraint. ", + "bbox": [ + 173, + 279, + 826, + 383 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "8.3 ADDITIONAL BLACK-BOX ATTACKING RESULTS ON MNIST AND CIFAR-10 ", + "text_level": 1, + "bbox": [ + 173, + 398, + 738, + 412 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/d723b5a11bfe4af37e4db7c1f31b08024cf154ce2f96bf12e87f78b3566c1330.jpg", + "table_caption": [ + "Table A1: First successful adversarial examples attacking black-box DNN on MNIST. " + ], + "table_footnote": [], + "table_body": "
Image ID3238444152134 1777
Original0123456780
Classified as02345689
ZO-SGD Classified as 640723956704
ZO-signSGD672395670A59354
Classified as6
ZO-M-signSGD672395670A
Classified as64593504
ZO-NES612395670A
Classified as6493504
", + "bbox": [ + 173, + 450, + 794, + 732 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/9d037effff3bcad01e9ee325e024e2855066ab2f70885499e03fa3ca0f2fe82b.jpg", + "table_caption": [ + "Table A2: First successful adversarial examples attacking black-box DNN on CIFAR-10. " + ], + "table_footnote": [], + "table_body": "
Image ID1062568363317128
Original Classified asH airplaneL automobileN bird心 catYO deerdog# frog百 horse福 ship50 truck
ZO-SGD Classified as1 birdL truckDD truckdogMUOE horsecatcatdogtruck53 automobile
ZO-signSGD Classified as1 birdL truckN truckdogMV horsecatE catE dog福 truckautomobile
ZO-M-signSGD Classified asH bird园 truckNE truckdogYUO horse面一 automobile5
ZO-NESHRHL心YUCEcatcatdog心eautomobile
Classified asbirdtrucktruckdoghorsecatcatdogautomobileautomobile
", + "bbox": [ + 173, + 186, + 825, + 436 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/107d5a4b671dd6a5e658960ae34083064fc8204b027fb3bdb593c9d844bbe3ce.jpg", + "table_caption": [ + "Table A3: Iteration comparison of attacking black-box DNN on MNIST (image ID 177). " + ], + "table_footnote": [], + "table_body": "
Iteration03060 90 120162 181 210 241 271
ZO-SGD888888888
Classified as8 888 8 8 0 0 0 0
Iteration0 30 60 90
8 ZO-signSGD114 151 180 210 241 271
888888888
Classified as Iteration 0 30 60 90 120 1508 8 8 8 0 0 0 0 0 180 198 240 271
D ZO-M-signSGD888888888
Classified as 8 8 8 Iteration 0 408 8 8 8 0 0 0 160 200 240 280 320 349
80 120
ZO-NES
Classified as 8 8 88888888808 8 8 8 8 8 8 0
", + "bbox": [ + 173, + 594, + 794, + 858 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/381d47c27c2eda8a8de00f0e9d4b1bbc6823a776b98903402cc7eff9d6b53489.jpg", + "image_caption": [ + "Figure A4: Additional plots of black-box attacking loss versus iteration on MNIST. The solid marker indicates the iteration number that finds the first successful adversarial example, and its loss corresponds to the squared $\\ell _ { 2 }$ distortion. " + ], + "image_footnote": [], + "bbox": [ + 202, + 155, + 767, + 830 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/6a0c2dfadd5b7081e9e79e0bf1d80841626873ae671e09a2abbd1fc48108a36d.jpg", + "image_caption": [ + "Figure A5: Additional plots of black-box attacking loss versus iteration on CIFAR-10. The solid marker indicates the iteration number that finds the first successful adversarial example, and its loss corresponds to the squared $\\ell _ { 2 }$ distortion. " + ], + "image_footnote": [], + "bbox": [ + 205, + 154, + 767, + 834 + ], + "page_idx": 23 + } +] \ No newline at end of file diff --git a/parse/train/BJe-DsC5Fm/BJe-DsC5Fm_middle.json b/parse/train/BJe-DsC5Fm/BJe-DsC5Fm_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..6d2708f938dc7b79392a0bdff112392b4fad9e70 --- /dev/null +++ b/parse/train/BJe-DsC5Fm/BJe-DsC5Fm_middle.json @@ -0,0 +1,64046 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 403, + 96 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 405, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 405, + 99 + ], + "score": 1.0, + "content": "SIGNSGD VIA ZEROTH-ORDER ORACLE", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 112, + 114, + 373, + 150 + ], + "lines": [ + { + "bbox": [ + 111, + 113, + 375, + 128 + ], + "spans": [ + { + "bbox": [ + 111, + 113, + 303, + 128 + ], + "score": 1.0, + "content": "Sijia Liu† Pin-Yu Chen† Xiangyi Chen‡", + "type": "text" + }, + { + "bbox": [ + 310, + 113, + 375, + 128 + ], + "score": 1.0, + "content": "Mingyi Hong‡", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 112, + 125, + 285, + 140 + ], + "spans": [ + { + "bbox": [ + 112, + 125, + 285, + 140 + ], + "score": 1.0, + "content": "†MIT-IBM Watson AI Lab, IBM Research", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 137, + 267, + 152 + ], + "spans": [ + { + "bbox": [ + 112, + 137, + 267, + 152 + ], + "score": 1.0, + "content": "‡University of Minnesota, Twin Cities", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 278, + 157, + 333, + 169 + ], + "lines": [ + { + "bbox": [ + 277, + 157, + 335, + 171 + ], + "spans": [ + { + "bbox": [ + 277, + 157, + 335, + 171 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 184, + 469, + 342 + ], + "lines": [ + { + "bbox": [ + 141, + 184, + 470, + 197 + ], + "spans": [ + { + "bbox": [ + 141, + 184, + 470, + 197 + ], + "score": 1.0, + "content": "In this paper, we design and analyze a new zeroth-order (ZO) stochastic opti-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 142, + 196, + 469, + 207 + ], + "spans": [ + { + "bbox": [ + 142, + 196, + 469, + 207 + ], + "score": 1.0, + "content": "mization algorithm, ZO-signSGD, which enjoys dual advantages of gradient-free", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 206, + 470, + 219 + ], + "spans": [ + { + "bbox": [ + 141, + 206, + 470, + 219 + ], + "score": 1.0, + "content": "operations and signSGD. 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In addition, we analyze the effects of different types of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 275, + 470, + 287 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 470, + 287 + ], + "score": 1.0, + "content": "gradient estimators on the convergence of ZO-signSGD, and propose several vari-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 140, + 285, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 140, + 285, + 248, + 300 + ], + "score": 1.0, + "content": "ants of ZO-signSGD with", + "type": "text" + }, + { + "bbox": [ + 249, + 285, + 299, + 299 + ], + "score": 0.93, + "content": "O ( { \\sqrt { d } } { \\bar { / } } { \\sqrt { T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 285, + 470, + 300 + ], + "score": 1.0, + "content": "convergence rate. 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Our empirical evaluations on image classification datasets", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 320, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 469, + 333 + ], + "score": 1.0, + "content": "MNIST and CIFAR-10 demonstrate the superior performance of ZO-signSGD on", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 331, + 431, + 343 + ], + "spans": [ + { + "bbox": [ + 142, + 331, + 431, + 343 + ], + "score": 1.0, + "content": "the generation of adversarial examples from black-box neural networks.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 364, + 206, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 208, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 208, + 380 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "score": 1.0, + "content": "Zeroth-order (gradient-free) optimization has attracted an increasing amount of attention for solving", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "machine learning (ML) problems in scenarios where explicit expressions for the gradients are difficult", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "or infeasible to obtain. One recent application of great interest is to generate prediction-evasive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "adversarial examples, e.g., crafted images with imperceptible perturbations to deceive a well-trained", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "image classifier into misclassification. However, the black-box optimization nature limits the practical", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "design of adversarial examples, where internal configurations and operating mechanism of public", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "score": 1.0, + "content": "ML systems (e.g., Google Cloud Vision API) are not revealed to practitioners and the only mode of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "interaction with the system is via submitting inputs and receiving the corresponding predicted outputs", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 470, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 506, + 486 + ], + "score": 1.0, + "content": "(Papernot et al., 2017; Liu et al., 2017; Chen et al., 2017; Tu et al., 2018; Ilyas et al., 2018b; Cheng", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "et al., 2018; Bhagoji et al., 2018). It was observed in both white-box and black-box settings1 that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "simply leveraging the sign information of gradient estimates of an attacking loss can achieve superior", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "empirical performance in generating adversarial examples (Goodfellow et al., 2015; Madry et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "2018; Ilyas et al., 2018a). Spurred by that, this paper proposes a zeroth-order (ZO) sign-based descent", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 525, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 539 + ], + "score": 1.0, + "content": "algorithm (we call it ‘ZO-signSGD’) for solving black-box optimization problems, e.g. design of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 536, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 550 + ], + "score": 1.0, + "content": "black-box adversarial examples. The convergence behavior and algorithmic stability of the proposed", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 548, + 399, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 399, + 560 + ], + "score": 1.0, + "content": "ZO-signSGD algorithm are carefully studied in both theory and practice.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "In the first-order setting, a sign-based stochastic gradient descent method, known as signSGD, was", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 576, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 506, + 588 + ], + "score": 1.0, + "content": "analyzed by (Bernstein et al., 2018; Balles & Hennig, 2017). It was shown in (Bernstein et al., 2018)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "that signSGD not only reduces the per iteration cost of communicating gradients, but also could yield", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 597, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 597, + 506, + 611 + ], + "score": 1.0, + "content": "a faster empirical convergence speed than SGD (Kinga & Adam, 2015). That is because although", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "score": 1.0, + "content": "the sign operation compresses the gradient using a single bit, it could mitigate the negative effect of√", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 619, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 619, + 408, + 635 + ], + "score": 1.0, + "content": "extremely components of gradient noise. Theoretically, signSGD achieves", + "type": "text" + }, + { + "bbox": [ + 409, + 619, + 451, + 633 + ], + "score": 0.93, + "content": "O ( 1 / \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 619, + 506, + 635 + ], + "score": 1.0, + "content": "convergence", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 383, + 644 + ], + "score": 1.0, + "content": "rate under the condition of a sufficiently large mini-batch size, where", + "type": "text" + }, + { + "bbox": [ + 384, + 632, + 393, + 642 + ], + "score": 0.72, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "denotes the total number of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 642, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 656 + ], + "score": 1.0, + "content": "iterations. The work in (Balles & Hennig, 2017) established a connection between signSGD and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 653, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 667 + ], + "score": 1.0, + "content": "Adam with restrictive convex analysis. Prior to (Bernstein et al., 2018; Balles & Hennig, 2017),", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "although signSGD was not formally defined, the fast gradient sign method (Goodfellow et al., 2015)", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "to generate white-box adversarial examples actually obeys the algorithmic protocol of signSGD.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "The effectiveness of signSGD has been witnessed by robust adversarial training of deep neural", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "networks (DNNs) (Madry et al., 2018). Given the advantages of signSGD, one may wonder if it can", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 42 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 115, + 721, + 479, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 719, + 480, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 719, + 480, + 733 + ], + "score": 1.0, + "content": "1‘white-box’ (vs ‘black-box’) implies whether the knowledge on the target model is known a priori.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 403, + 96 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 405, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 405, + 99 + ], + "score": 1.0, + "content": "SIGNSGD VIA ZEROTH-ORDER ORACLE", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 112, + 114, + 373, + 150 + ], + "lines": [ + { + "bbox": [ + 111, + 113, + 375, + 128 + ], + "spans": [ + { + "bbox": [ + 111, + 113, + 303, + 128 + ], + "score": 1.0, + "content": "Sijia Liu† Pin-Yu Chen† Xiangyi Chen‡", + "type": "text" + }, + { + "bbox": [ + 310, + 113, + 375, + 128 + ], + "score": 1.0, + "content": "Mingyi Hong‡", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 112, + 125, + 285, + 140 + ], + "spans": [ + { + "bbox": [ + 112, + 125, + 285, + 140 + ], + "score": 1.0, + "content": "†MIT-IBM Watson AI Lab, IBM Research", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 137, + 267, + 152 + ], + "spans": [ + { + "bbox": [ + 112, + 137, + 267, + 152 + ], + "score": 1.0, + "content": "‡University of Minnesota, Twin Cities", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 111, + 113, + 375, + 152 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 157, + 333, + 169 + ], + "lines": [ + { + "bbox": [ + 277, + 157, + 335, + 171 + ], + "spans": [ + { + "bbox": [ + 277, + 157, + 335, + 171 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 184, + 469, + 342 + ], + "lines": [ + { + "bbox": [ + 141, + 184, + 470, + 197 + ], + "spans": [ + { + "bbox": [ + 141, + 184, + 470, + 197 + ], + "score": 1.0, + "content": "In this paper, we design and analyze a new zeroth-order (ZO) stochastic opti-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 142, + 196, + 469, + 207 + ], + "spans": [ + { + "bbox": [ + 142, + 196, + 469, + 207 + ], + "score": 1.0, + "content": "mization algorithm, ZO-signSGD, which enjoys dual advantages of gradient-free", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 206, + 470, + 219 + ], + "spans": [ + { + "bbox": [ + 141, + 206, + 470, + 219 + ], + "score": 1.0, + "content": "operations and signSGD. The latter requires only the sign information of gradient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 217, + 469, + 230 + ], + "spans": [ + { + "bbox": [ + 141, + 217, + 469, + 230 + ], + "score": 1.0, + "content": "estimates but is able to achieve a comparable or even better convergence speed√", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 228, + 469, + 242 + ], + "spans": [ + { + "bbox": [ + 141, + 229, + 429, + 242 + ], + "score": 1.0, + "content": "than SGD-type algorithms. Our study shows that ZO-signSGD requires √ √", + "type": "text" + }, + { + "bbox": [ + 429, + 228, + 444, + 240 + ], + "score": 0.89, + "content": "\\sqrt { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 229, + 469, + 242 + ], + "score": 1.0, + "content": "times", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 140, + 240, + 470, + 254 + ], + "spans": [ + { + "bbox": [ + 140, + 242, + 392, + 254 + ], + "score": 1.0, + "content": "more iterations than signSGD, leading to a convergence rate of", + "type": "text" + }, + { + "bbox": [ + 393, + 240, + 443, + 254 + ], + "score": 0.93, + "content": "O ( { \\sqrt { d } } / { \\sqrt { T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 242, + 470, + 254 + ], + "score": 1.0, + "content": "under", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 252, + 470, + 265 + ], + "spans": [ + { + "bbox": [ + 141, + 252, + 262, + 265 + ], + "score": 1.0, + "content": "some mild conditions, where", + "type": "text" + }, + { + "bbox": [ + 262, + 253, + 268, + 263 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 252, + 450, + 265 + ], + "score": 1.0, + "content": "is the number of optimization variables, and", + "type": "text" + }, + { + "bbox": [ + 450, + 253, + 459, + 263 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 252, + 470, + 265 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 263, + 470, + 276 + ], + "spans": [ + { + "bbox": [ + 141, + 263, + 470, + 276 + ], + "score": 1.0, + "content": "the number of iterations. In addition, we analyze the effects of different types of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 275, + 470, + 287 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 470, + 287 + ], + "score": 1.0, + "content": "gradient estimators on the convergence of ZO-signSGD, and propose several vari-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 140, + 285, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 140, + 285, + 248, + 300 + ], + "score": 1.0, + "content": "ants of ZO-signSGD with", + "type": "text" + }, + { + "bbox": [ + 249, + 285, + 299, + 299 + ], + "score": 0.93, + "content": "O ( { \\sqrt { d } } { \\bar { / } } { \\sqrt { T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 285, + 470, + 300 + ], + "score": 1.0, + "content": "convergence rate. On the application side", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 298, + 469, + 310 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 469, + 310 + ], + "score": 1.0, + "content": "we explore the connection between ZO-signSGD and black-box adversarial attacks", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 309, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 309, + 469, + 321 + ], + "score": 1.0, + "content": "in robust deep learning. Our empirical evaluations on image classification datasets", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 320, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 469, + 333 + ], + "score": 1.0, + "content": "MNIST and CIFAR-10 demonstrate the superior performance of ZO-signSGD on", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 331, + 431, + 343 + ], + "spans": [ + { + "bbox": [ + 142, + 331, + 431, + 343 + ], + "score": 1.0, + "content": "the generation of adversarial examples from black-box neural networks.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5, + "bbox_fs": [ + 140, + 184, + 470, + 343 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 364, + 206, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 208, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 208, + 380 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "score": 1.0, + "content": "Zeroth-order (gradient-free) optimization has attracted an increasing amount of attention for solving", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "machine learning (ML) problems in scenarios where explicit expressions for the gradients are difficult", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "or infeasible to obtain. One recent application of great interest is to generate prediction-evasive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "adversarial examples, e.g., crafted images with imperceptible perturbations to deceive a well-trained", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "image classifier into misclassification. However, the black-box optimization nature limits the practical", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "design of adversarial examples, where internal configurations and operating mechanism of public", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "score": 1.0, + "content": "ML systems (e.g., Google Cloud Vision API) are not revealed to practitioners and the only mode of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "interaction with the system is via submitting inputs and receiving the corresponding predicted outputs", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 470, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 506, + 486 + ], + "score": 1.0, + "content": "(Papernot et al., 2017; Liu et al., 2017; Chen et al., 2017; Tu et al., 2018; Ilyas et al., 2018b; Cheng", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "et al., 2018; Bhagoji et al., 2018). It was observed in both white-box and black-box settings1 that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "simply leveraging the sign information of gradient estimates of an attacking loss can achieve superior", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "empirical performance in generating adversarial examples (Goodfellow et al., 2015; Madry et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "2018; Ilyas et al., 2018a). Spurred by that, this paper proposes a zeroth-order (ZO) sign-based descent", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 525, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 539 + ], + "score": 1.0, + "content": "algorithm (we call it ‘ZO-signSGD’) for solving black-box optimization problems, e.g. design of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 536, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 550 + ], + "score": 1.0, + "content": "black-box adversarial examples. The convergence behavior and algorithmic stability of the proposed", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 548, + 399, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 399, + 560 + ], + "score": 1.0, + "content": "ZO-signSGD algorithm are carefully studied in both theory and practice.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 382, + 506, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "In the first-order setting, a sign-based stochastic gradient descent method, known as signSGD, was", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 576, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 506, + 588 + ], + "score": 1.0, + "content": "analyzed by (Bernstein et al., 2018; Balles & Hennig, 2017). It was shown in (Bernstein et al., 2018)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "that signSGD not only reduces the per iteration cost of communicating gradients, but also could yield", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 597, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 597, + 506, + 611 + ], + "score": 1.0, + "content": "a faster empirical convergence speed than SGD (Kinga & Adam, 2015). That is because although", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "score": 1.0, + "content": "the sign operation compresses the gradient using a single bit, it could mitigate the negative effect of√", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 619, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 619, + 408, + 635 + ], + "score": 1.0, + "content": "extremely components of gradient noise. Theoretically, signSGD achieves", + "type": "text" + }, + { + "bbox": [ + 409, + 619, + 451, + 633 + ], + "score": 0.93, + "content": "O ( 1 / \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 619, + 506, + 635 + ], + "score": 1.0, + "content": "convergence", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 383, + 644 + ], + "score": 1.0, + "content": "rate under the condition of a sufficiently large mini-batch size, where", + "type": "text" + }, + { + "bbox": [ + 384, + 632, + 393, + 642 + ], + "score": 0.72, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "denotes the total number of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 642, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 656 + ], + "score": 1.0, + "content": "iterations. The work in (Balles & Hennig, 2017) established a connection between signSGD and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 653, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 667 + ], + "score": 1.0, + "content": "Adam with restrictive convex analysis. Prior to (Bernstein et al., 2018; Balles & Hennig, 2017),", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "although signSGD was not formally defined, the fast gradient sign method (Goodfellow et al., 2015)", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "to generate white-box adversarial examples actually obeys the algorithmic protocol of signSGD.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "The effectiveness of signSGD has been witnessed by robust adversarial training of deep neural", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "networks (DNNs) (Madry et al., 2018). Given the advantages of signSGD, one may wonder if it can", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "be generalized for ZO optimization and what the corresponding convergence rate is. In this paper, we", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 258, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 258, + 107 + ], + "score": 1.0, + "content": "answer these questions affirmatively.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 565, + 506, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "be generalized for ZO optimization and what the corresponding convergence rate is. In this paper, we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 258, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 258, + 107 + ], + "score": 1.0, + "content": "answer these questions affirmatively.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 117, + 370, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 115, + 371, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 371, + 130 + ], + "score": 1.0, + "content": "Contributions We summarize our key contributions as follows.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 132, + 130, + 505, + 265 + ], + "lines": [ + { + "bbox": [ + 131, + 129, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 131, + 129, + 505, + 145 + ], + "score": 1.0, + "content": "• We propose a new ZO algorithm, ‘ZO-signSGD’, and rigorously prove its convergence rate√ √", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 142, + 298, + 156 + ], + "spans": [ + { + "bbox": [ + 141, + 142, + 153, + 156 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 154, + 142, + 204, + 156 + ], + "score": 0.93, + "content": "O ( { \\sqrt { d } } / { \\sqrt { T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 142, + 298, + 156 + ], + "score": 1.0, + "content": "under mild conditions.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 139, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 139, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "Our established convergence analysis applies to both mini-batch sampling schemes with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 142, + 170, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 142, + 170, + 505, + 181 + ], + "score": 1.0, + "content": "and without replacement. In particular, the ZO sign-based gradient descent algorithm can be", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 180, + 403, + 193 + ], + "spans": [ + { + "bbox": [ + 141, + 180, + 403, + 193 + ], + "score": 1.0, + "content": "treated as a special case in our proposed ZO-signSGD algorithm.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 139, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 139, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "We carefully study the effects of different types of gradient estimators on the convergence of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 141, + 205, + 506, + 219 + ], + "score": 1.0, + "content": "ZO-signSGD, and propose three variants of ZO-signSGD for both centralized and distributed", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 218, + 213, + 229 + ], + "spans": [ + { + "bbox": [ + 142, + 218, + 213, + 229 + ], + "score": 1.0, + "content": "ZO optimization.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 136, + 231, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 136, + 231, + 506, + 245 + ], + "score": 1.0, + "content": "• We conduct extensive synthetic experiments to thoroughly benchmark the performance of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 243, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 506, + 257 + ], + "score": 1.0, + "content": "ZO-signSGD and to investigate its parameter sensitivity. We also demonstrate the superior", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 254, + 501, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 501, + 267 + ], + "score": 1.0, + "content": "performance of ZO-signSGD for generating adversarial examples from black-box DNNs.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 274, + 506, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "Related work Other types of ZO algorithms have been developed for convex and nonconvex", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 285, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 297 + ], + "score": 1.0, + "content": "optimization, where the full gradient is approximated via a random or deterministic gradient estimate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "(Jamieson et al., 2012; Nesterov & Spokoiny, 2015; Ghadimi & Lan, 2013; Duchi et al., 2015;", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "Gao et al., 2014; Shamir, 2017; Hajinezhad et al., 2017; Ghadimi et al., 2016; Lian et al., 2016;", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 330 + ], + "score": 1.0, + "content": "Liu et al., 2018b;c; Wang et al., 2018). Examples include ZO-SGD (Ghadimi & Lan, 2013), ZO", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "stochastic coordinate descent (ZO-SCD) (Lian et al., 2016), and ZO stochastic variance reduced", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "gradient descent (ZO-SVRG) (Liu et al., 2018c;a; Gu et al., 2016). Both ZO-SGD and ZO-SCD can√ √", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 138, + 366 + ], + "score": 1.0, + "content": "achieve", + "type": "text" + }, + { + "bbox": [ + 139, + 351, + 189, + 364 + ], + "score": 0.93, + "content": "O ( { \\sqrt { d } } / { \\sqrt { T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 351, + 506, + 366 + ], + "score": 1.0, + "content": "convergence rate. And ZO-SVRG can further improve the iteration complexity", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 117, + 376 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 363, + 151, + 375 + ], + "score": 0.93, + "content": "O ( d / T )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "but suffers from an increase of function query complexity due to the additional variance", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "reduced step, known as ‘gradient blending’ (Liu et al., 2018c), compared to ZO-SGD. The existing", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 384, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 505, + 399 + ], + "score": 1.0, + "content": "work showed that ZO algorithms align with the iteration complexity of their first-order counterparts", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 448, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 437, + 409 + ], + "score": 1.0, + "content": "up to a slowdown effect in terms of a small-degree polynomial of the problem size", + "type": "text" + }, + { + "bbox": [ + 438, + 397, + 444, + 406 + ], + "score": 0.63, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 396, + 448, + 409 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 105, + 416, + 489, + 429 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 489, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 489, + 431 + ], + "score": 1.0, + "content": "2 SIGNSGD & ITS CONNECTION TO ADVERSARIAL MACHINE LEARNING", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 507, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 507, + 447 + ], + "score": 1.0, + "content": "In this section, we provide a background on signSGD, together with the problem setup of our interest.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "In particular, we show that the commonly-used methods for generating adversarial attacks fall into", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 456, + 219, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 219, + 469 + ], + "score": 1.0, + "content": "the framework of signSGD.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 443, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 444, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 444, + 486 + ], + "score": 1.0, + "content": "Preliminaries on signSGD Consider a nonconvex finite-sum problem of the form", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 488, + 380, + 505 + ], + "lines": [ + { + "bbox": [ + 230, + 488, + 380, + 505 + ], + "spans": [ + { + "bbox": [ + 230, + 488, + 380, + 505 + ], + "score": 0.88, + "content": "\\begin{array} { r l } { \\underset { \\mathbf { x } } { \\mathrm { m i n i m i z e } } } & { { } f ( \\mathbf { x } ) : = ( 1 / n ) \\sum _ { i = 1 } ^ { n } f _ { i } ( \\mathbf { x } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "f7772a7bebf0ff27113cbc3cf325c6031dfb67b835f0804c0f4e7c26390bd731.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 230, + 488, + 380, + 505 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 133, + 525 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 511, + 164, + 523 + ], + "score": 0.92, + "content": "\\mathbf { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 510, + 292, + 525 + ], + "score": 1.0, + "content": "are optimization variables, and", + "type": "text" + }, + { + "bbox": [ + 293, + 512, + 311, + 524 + ], + "score": 0.92, + "content": "\\{ f _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 510, + 327, + 525 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 327, + 514, + 335, + 522 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "individual nonconvex cost functions. The", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "finite-sum form (1) encompasses many ML problems, ranging from generalized linear models to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 257, + 546 + ], + "score": 1.0, + "content": "neural networks. If the gradients of", + "type": "text" + }, + { + "bbox": [ + 257, + 533, + 276, + 546 + ], + "score": 0.92, + "content": "\\{ f _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "are available, then problem (1) can be solved by many", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "first-order methods such as SGD, SCD, and signSGD. The method of our interest is signSGD, which", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "differs from SGD and SCD, takes the sign of gradient (or its estimate) as the descent direction. It was", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "recently shown in (Bernstein et al., 2018) that signSGD is quite robust to gradient noise and yields", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 218, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 218, + 591 + ], + "score": 1.0, + "content": "fast empirical convergence.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 606 + ], + "score": 1.0, + "content": "Algorithm 1 provides a generic sign-based gradient descent framework that encapsulates different vari-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 605, + 507, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 507, + 619 + ], + "score": 1.0, + "content": "ants of signSGD. In Algorithm 1, GradEstimate(·) signifies a general gradient estimation procedure,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "which adopts either a stochastic gradient estimate in the first-order setting (Bernstein et al., 2018) or", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 628, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 640 + ], + "score": 1.0, + "content": "a function difference based random gradient estimate in the ZO setting (Nesterov & Spokoiny, 2015;", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "Duchi et al., 2015). We call the ZO variant of signSGD ‘ZO-signSGD’, which will be elaborated on", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 648, + 146, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 146, + 661 + ], + "score": 1.0, + "content": "in Sec. 3.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "Adversarial attacks meet signSGD It is now widely known that ML models (e.g., deep neural", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "networks) are vulnerable to adversarial attacks, which craft inputs (e.g., images) with imperceptible", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "perturbations to cause incorrect classification (Szegedy et al., 2013; Goodfellow et al., 2015; Kurakin", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "et al., 2017; Lin et al., 2019). The resulting inputs crafted by adversaries are known as adversarial", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "examples. Investigating adversarial examples not only helps to understand the limitation of learning", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "models, but also provides opportunities to improve the models’ robustness (Papernot et al., 2016;", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 107 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 117, + 370, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 115, + 371, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 371, + 130 + ], + "score": 1.0, + "content": "Contributions We summarize our key contributions as follows.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 106, + 115, + 371, + 130 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 130, + 505, + 265 + ], + "lines": [ + { + "bbox": [ + 131, + 129, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 131, + 129, + 505, + 145 + ], + "score": 1.0, + "content": "• We propose a new ZO algorithm, ‘ZO-signSGD’, and rigorously prove its convergence rate√ √", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 142, + 298, + 156 + ], + "spans": [ + { + "bbox": [ + 141, + 142, + 153, + 156 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 154, + 142, + 204, + 156 + ], + "score": 0.93, + "content": "O ( { \\sqrt { d } } / { \\sqrt { T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 142, + 298, + 156 + ], + "score": 1.0, + "content": "under mild conditions.", + "type": "text" + } + ], + "index": 4, + "is_list_end_line": true + }, + { + "bbox": [ + 139, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 139, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "Our established convergence analysis applies to both mini-batch sampling schemes with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 142, + 170, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 142, + 170, + 505, + 181 + ], + "score": 1.0, + "content": "and without replacement. In particular, the ZO sign-based gradient descent algorithm can be", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 180, + 403, + 193 + ], + "spans": [ + { + "bbox": [ + 141, + 180, + 403, + 193 + ], + "score": 1.0, + "content": "treated as a special case in our proposed ZO-signSGD algorithm.", + "type": "text" + } + ], + "index": 7, + "is_list_end_line": true + }, + { + "bbox": [ + 139, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 139, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "We carefully study the effects of different types of gradient estimators on the convergence of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 141, + 205, + 506, + 219 + ], + "score": 1.0, + "content": "ZO-signSGD, and propose three variants of ZO-signSGD for both centralized and distributed", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 218, + 213, + 229 + ], + "spans": [ + { + "bbox": [ + 142, + 218, + 213, + 229 + ], + "score": 1.0, + "content": "ZO optimization.", + "type": "text" + } + ], + "index": 10, + "is_list_end_line": true + }, + { + "bbox": [ + 136, + 231, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 136, + 231, + 506, + 245 + ], + "score": 1.0, + "content": "• We conduct extensive synthetic experiments to thoroughly benchmark the performance of", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 243, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 506, + 257 + ], + "score": 1.0, + "content": "ZO-signSGD and to investigate its parameter sensitivity. We also demonstrate the superior", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 254, + 501, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 501, + 267 + ], + "score": 1.0, + "content": "performance of ZO-signSGD for generating adversarial examples from black-box DNNs.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8, + "bbox_fs": [ + 131, + 129, + 506, + 267 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 274, + 506, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "Related work Other types of ZO algorithms have been developed for convex and nonconvex", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 285, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 297 + ], + "score": 1.0, + "content": "optimization, where the full gradient is approximated via a random or deterministic gradient estimate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "(Jamieson et al., 2012; Nesterov & Spokoiny, 2015; Ghadimi & Lan, 2013; Duchi et al., 2015;", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "Gao et al., 2014; Shamir, 2017; Hajinezhad et al., 2017; Ghadimi et al., 2016; Lian et al., 2016;", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 330 + ], + "score": 1.0, + "content": "Liu et al., 2018b;c; Wang et al., 2018). Examples include ZO-SGD (Ghadimi & Lan, 2013), ZO", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "stochastic coordinate descent (ZO-SCD) (Lian et al., 2016), and ZO stochastic variance reduced", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "gradient descent (ZO-SVRG) (Liu et al., 2018c;a; Gu et al., 2016). Both ZO-SGD and ZO-SCD can√ √", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 138, + 366 + ], + "score": 1.0, + "content": "achieve", + "type": "text" + }, + { + "bbox": [ + 139, + 351, + 189, + 364 + ], + "score": 0.93, + "content": "O ( { \\sqrt { d } } / { \\sqrt { T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 351, + 506, + 366 + ], + "score": 1.0, + "content": "convergence rate. And ZO-SVRG can further improve the iteration complexity", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 117, + 376 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 363, + 151, + 375 + ], + "score": 0.93, + "content": "O ( d / T )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "but suffers from an increase of function query complexity due to the additional variance", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "reduced step, known as ‘gradient blending’ (Liu et al., 2018c), compared to ZO-SGD. The existing", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 384, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 505, + 399 + ], + "score": 1.0, + "content": "work showed that ZO algorithms align with the iteration complexity of their first-order counterparts", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 448, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 437, + 409 + ], + "score": 1.0, + "content": "up to a slowdown effect in terms of a small-degree polynomial of the problem size", + "type": "text" + }, + { + "bbox": [ + 438, + 397, + 444, + 406 + ], + "score": 0.63, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 396, + 448, + 409 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 274, + 506, + 409 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 416, + 489, + 429 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 489, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 489, + 431 + ], + "score": 1.0, + "content": "2 SIGNSGD & ITS CONNECTION TO ADVERSARIAL MACHINE LEARNING", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 507, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 507, + 447 + ], + "score": 1.0, + "content": "In this section, we provide a background on signSGD, together with the problem setup of our interest.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "In particular, we show that the commonly-used methods for generating adversarial attacks fall into", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 456, + 219, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 219, + 469 + ], + "score": 1.0, + "content": "the framework of signSGD.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 433, + 507, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 443, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 444, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 444, + 486 + ], + "score": 1.0, + "content": "Preliminaries on signSGD Consider a nonconvex finite-sum problem of the form", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 469, + 444, + 486 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 488, + 380, + 505 + ], + "lines": [ + { + "bbox": [ + 230, + 488, + 380, + 505 + ], + "spans": [ + { + "bbox": [ + 230, + 488, + 380, + 505 + ], + "score": 0.88, + "content": "\\begin{array} { r l } { \\underset { \\mathbf { x } } { \\mathrm { m i n i m i z e } } } & { { } f ( \\mathbf { x } ) : = ( 1 / n ) \\sum _ { i = 1 } ^ { n } f _ { i } ( \\mathbf { x } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "f7772a7bebf0ff27113cbc3cf325c6031dfb67b835f0804c0f4e7c26390bd731.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 230, + 488, + 380, + 505 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 133, + 525 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 511, + 164, + 523 + ], + "score": 0.92, + "content": "\\mathbf { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 510, + 292, + 525 + ], + "score": 1.0, + "content": "are optimization variables, and", + "type": "text" + }, + { + "bbox": [ + 293, + 512, + 311, + 524 + ], + "score": 0.92, + "content": "\\{ f _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 510, + 327, + 525 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 327, + 514, + 335, + 522 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "individual nonconvex cost functions. The", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "finite-sum form (1) encompasses many ML problems, ranging from generalized linear models to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 257, + 546 + ], + "score": 1.0, + "content": "neural networks. If the gradients of", + "type": "text" + }, + { + "bbox": [ + 257, + 533, + 276, + 546 + ], + "score": 0.92, + "content": "\\{ f _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "are available, then problem (1) can be solved by many", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "first-order methods such as SGD, SCD, and signSGD. The method of our interest is signSGD, which", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "differs from SGD and SCD, takes the sign of gradient (or its estimate) as the descent direction. It was", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "recently shown in (Bernstein et al., 2018) that signSGD is quite robust to gradient noise and yields", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 218, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 218, + 591 + ], + "score": 1.0, + "content": "fast empirical convergence.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 510, + 506, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 606 + ], + "score": 1.0, + "content": "Algorithm 1 provides a generic sign-based gradient descent framework that encapsulates different vari-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 605, + 507, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 507, + 619 + ], + "score": 1.0, + "content": "ants of signSGD. In Algorithm 1, GradEstimate(·) signifies a general gradient estimation procedure,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "which adopts either a stochastic gradient estimate in the first-order setting (Bernstein et al., 2018) or", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 628, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 640 + ], + "score": 1.0, + "content": "a function difference based random gradient estimate in the ZO setting (Nesterov & Spokoiny, 2015;", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "Duchi et al., 2015). We call the ZO variant of signSGD ‘ZO-signSGD’, which will be elaborated on", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 648, + 146, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 146, + 661 + ], + "score": 1.0, + "content": "in Sec. 3.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 595, + 507, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "Adversarial attacks meet signSGD It is now widely known that ML models (e.g., deep neural", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "networks) are vulnerable to adversarial attacks, which craft inputs (e.g., images) with imperceptible", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "perturbations to cause incorrect classification (Szegedy et al., 2013; Goodfellow et al., 2015; Kurakin", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "et al., 2017; Lin et al., 2019). The resulting inputs crafted by adversaries are known as adversarial", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "examples. Investigating adversarial examples not only helps to understand the limitation of learning", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "models, but also provides opportunities to improve the models’ robustness (Papernot et al., 2016;", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 666, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 306, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 308, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 308, + 96 + ], + "score": 1.0, + "content": "Algorithm 1 Generic sign-based gradient descent", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 111, + 98, + 484, + 142 + ], + "lines": [ + { + "bbox": [ + 110, + 97, + 401, + 110 + ], + "spans": [ + { + "bbox": [ + 110, + 97, + 203, + 110 + ], + "score": 1.0, + "content": "1: Input: learning rate", + "type": "text" + }, + { + "bbox": [ + 203, + 98, + 223, + 110 + ], + "score": 0.91, + "content": "\\{ \\delta _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 97, + 276, + 110 + ], + "score": 1.0, + "content": ", initial value", + "type": "text" + }, + { + "bbox": [ + 276, + 100, + 288, + 110 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 97, + 392, + 110 + ], + "score": 1.0, + "content": ", and number of iterations", + "type": "text" + }, + { + "bbox": [ + 393, + 99, + 401, + 108 + ], + "score": 0.6, + "content": "T", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 110, + 108, + 235, + 122 + ], + "spans": [ + { + "bbox": [ + 110, + 108, + 138, + 122 + ], + "score": 1.0, + "content": "2: for", + "type": "text" + }, + { + "bbox": [ + 138, + 109, + 219, + 120 + ], + "score": 0.84, + "content": "k = 0 , 1 , \\ldots , T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 108, + 235, + 122 + ], + "score": 1.0, + "content": "do", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 110, + 119, + 485, + 133 + ], + "spans": [ + { + "bbox": [ + 110, + 120, + 122, + 132 + ], + "score": 1.0, + "content": "3:", + "type": "text" + }, + { + "bbox": [ + 138, + 121, + 252, + 132 + ], + "score": 0.43, + "content": "\\hat { \\mathbf { g } } _ { k } \\longleftarrow \\mathrm { G r a d E s t i m a t e } ( \\mathbf { x } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 119, + 485, + 133 + ], + "score": 1.0, + "content": "# applies to both first and zeroth order gradient estimates", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 110, + 130, + 223, + 142 + ], + "spans": [ + { + "bbox": [ + 110, + 130, + 122, + 142 + ], + "score": 1.0, + "content": "4:", + "type": "text" + }, + { + "bbox": [ + 137, + 130, + 223, + 142 + ], + "score": 1.0, + "content": "sign-gradient update", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 147, + 451, + 159 + ], + "lines": [ + { + "bbox": [ + 177, + 147, + 451, + 159 + ], + "spans": [ + { + "bbox": [ + 177, + 147, + 451, + 159 + ], + "score": 0.49, + "content": "\\begin{array} { r } { { \\bf x } _ { k + 1 } = { \\bf x } _ { k } - \\delta _ { k } \\mathrm { s i g n } ( \\hat { \\bf g } _ { k } ) , \\quad \\mathrm { w h e r e ~ s i g n ( { \\bf x } ) ~ t a k e s ~ e l e m e n t - w i s e ~ s i g n s ~ o f ~ { \\bf x } ~ } } \\end{array}", + "type": "interline_equation", + "image_path": "ab350cde9d4f0de74f2ca117be5408b1dcadec65d9bc4b78059882f2b882a15d.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 177, + 147, + 451, + 159 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 111, + 165, + 156, + 176 + ], + "lines": [ + { + "bbox": [ + 110, + 164, + 157, + 177 + ], + "spans": [ + { + "bbox": [ + 110, + 164, + 157, + 177 + ], + "score": 1.0, + "content": "5: end for", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 105, + 200, + 504, + 223 + ], + "lines": [ + { + "bbox": [ + 106, + 200, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 213 + ], + "score": 1.0, + "content": "Athalye et al., 2018; Madry et al., 2018). In what follows, we show that the generation of adversarial", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 210, + 493, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 493, + 225 + ], + "score": 1.0, + "content": "examples in (Goodfellow et al., 2015; Kurakin et al., 2017) can be interpreted through signSGD.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 122, + 240 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 229, + 134, + 238 + ], + "score": 0.82, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 227, + 474, + 240 + ], + "score": 1.0, + "content": "denote the natural (legitimate) input of an ML model associated with the true label", + "type": "text" + }, + { + "bbox": [ + 474, + 228, + 484, + 238 + ], + "score": 0.85, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 227, + 506, + 240 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 159, + 248 + ], + "score": 0.9, + "content": "\\mathbf { x } ^ { \\prime } = \\mathbf { x } _ { 0 } + \\boldsymbol { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 237, + 365, + 249 + ], + "score": 1.0, + "content": "be the adversarial example to be designed, where", + "type": "text" + }, + { + "bbox": [ + 366, + 238, + 372, + 247 + ], + "score": 0.77, + "content": "\\pmb { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "are adversarial perturbations. If", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 107, + 248, + 140, + 259 + ], + "score": 0.9, + "content": "f ( \\mathbf { x } , t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "is the training loss of a learning model, then the goal of (white-box) adversarial attack is to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 214, + 270 + ], + "score": 1.0, + "content": "find minimal perturbation", + "type": "text" + }, + { + "bbox": [ + 214, + 258, + 221, + 267 + ], + "score": 0.73, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 256, + 505, + 270 + ], + "score": 1.0, + "content": "that is sufficient to mislead the learning model, namely, to maximize", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 142, + 280 + ], + "score": 1.0, + "content": "the loss", + "type": "text" + }, + { + "bbox": [ + 142, + 267, + 199, + 279 + ], + "score": 0.91, + "content": "f ( \\mathbf { x } _ { 0 } ^ { \\mathbf { \\hat { \\alpha } } } + \\mathbf { \\delta } \\mathbf { \\delta } , t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 266, + 374, + 280 + ], + "score": 1.0, + "content": ". Taking the first-order approximation of", + "type": "text" + }, + { + "bbox": [ + 374, + 267, + 410, + 279 + ], + "score": 0.93, + "content": "\\bar { f ( \\mathbf { x } ^ { \\prime } , t _ { 0 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 266, + 444, + 280 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 444, + 269, + 456, + 279 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 266, + 505, + 280 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 277, + 504, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 275, + 290 + ], + "score": 0.89, + "content": "f ( \\mathbf { x } ^ { \\prime } , t _ { 0 } ) \\approx f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) + \\langle \\nabla _ { x } f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) , \\delta \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 277, + 489, + 291 + ], + "score": 1.0, + "content": ". By constraining the strength of perturbation in the", + "type": "text" + }, + { + "bbox": [ + 490, + 278, + 504, + 289 + ], + "score": 0.87, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 287, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 104, + 287, + 190, + 303 + ], + "score": 1.0, + "content": "ball of small radius", + "type": "text" + }, + { + "bbox": [ + 190, + 291, + 196, + 299 + ], + "score": 0.74, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 287, + 219, + 303 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 219, + 290, + 268, + 301 + ], + "score": 0.89, + "content": "\\| \\pmb { \\delta } \\| _ { \\infty } \\leq \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 287, + 388, + 303 + ], + "score": 1.0, + "content": ", the linear approximation of", + "type": "text" + }, + { + "bbox": [ + 389, + 289, + 424, + 301 + ], + "score": 0.93, + "content": "f ( \\mathbf { x } ^ { \\prime } , t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 287, + 506, + 303 + ], + "score": 1.0, + "content": "is then maximized", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 298, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 104, + 298, + 117, + 312 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 117, + 299, + 218, + 311 + ], + "score": 0.91, + "content": "\\pmb { \\delta } = \\epsilon \\mathrm { s i g n } ( \\nabla _ { \\pmb { x } } f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 298, + 506, + 312 + ], + "score": 1.0, + "content": "(Shaham et al., 2018). Therefore, generation of adversarial examples", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 309, + 429, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 429, + 322 + ], + "score": 1.0, + "content": "proposed in (Goodfellow et al., 2015) obeys the sign-gradient update rule in (2),", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 322, + 369, + 335 + ], + "lines": [ + { + "bbox": [ + 240, + 322, + 369, + 335 + ], + "spans": [ + { + "bbox": [ + 240, + 322, + 369, + 335 + ], + "score": 0.89, + "content": "\\mathbf { x } ^ { \\prime } = \\mathbf { x } _ { 0 } - \\epsilon \\mathrm { s i g n } ( - \\nabla _ { x } f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) ) .", + "type": "interline_equation", + "image_path": "8c5e3d5b769882ff66b4b10e97e41df6c1ab4c2857b2414e472981dc9be25151.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 240, + 322, + 369, + 335 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 506, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "Such a connection between adversarial example generation and signSGD also holds in other attacks,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 366 + ], + "score": 1.0, + "content": "e.g., the iterative target attack method (Kurakin et al., 2017). Similarly, a so-called black-box attack", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 364, + 495, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 495, + 378 + ], + "score": 1.0, + "content": "(Ilyas et al., 2018a; Bhagoji et al., 2018) is associated with our proposed ZO-signSGD algorithm.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 107, + 384, + 369, + 398 + ], + "lines": [ + { + "bbox": [ + 104, + 384, + 370, + 400 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 370, + 400 + ], + "score": 1.0, + "content": "3 ZO-SIGNSGD FOR BLACK-BOX OPTIMIZATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "One limitation of signSGD (Bernstein et al., 2018) is the need of first-order information, i.e., stochastic", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "score": 1.0, + "content": "gradients. However, there exists a large practical demand for solving ML problems where explicit", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "expressions of the gradients are difficult or infeasible to obtain, e.g., the generation of adversarial", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 391, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 391, + 452 + ], + "score": 1.0, + "content": "examples from black-box neural networks as discussed in Sec. 1 and 2.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 506, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 507, + 469 + ], + "score": 1.0, + "content": "Gradient estimation via ZO oracle In the ZO setting where the first-order information is unavail-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 471, + 480 + ], + "score": 1.0, + "content": "able, the gradient estimator at Step 3 of Algorithm 1 has only access to function values of", + "type": "text" + }, + { + "bbox": [ + 471, + 467, + 504, + 479 + ], + "score": 0.91, + "content": "\\{ f _ { i } ( \\mathbf { x } ) \\}", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 476, + 507, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 507, + 489 + ], + "score": 1.0, + "content": "given a query point x. Based on that, we construct a ZO gradient estimate through a forward differ-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "ence of two function values (Nesterov & Spokoiny, 2015; Gao et al., 2014; Duchi et al., 2015). In", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 496, + 315, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 221, + 509 + ], + "score": 1.0, + "content": "Algorithm 1, GradEstimate", + "type": "text" + }, + { + "bbox": [ + 222, + 497, + 235, + 508 + ], + "score": 0.64, + "content": "\\left( \\mathbf { x } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 496, + 315, + 509 + ], + "score": 1.0, + "content": "is then specified as", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 508, + 482, + 538 + ], + "lines": [ + { + "bbox": [ + 128, + 508, + 482, + 538 + ], + "spans": [ + { + "bbox": [ + 128, + 508, + 482, + 538 + ], + "score": 0.92, + "content": "\\mathrm { G r a d E s t i m a t e } ( \\mathbf { x } ) = \\frac { 1 } { b q } \\sum _ { i \\in \\mathcal { T } _ { k } } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) , \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) : = \\frac { d \\big [ f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } _ { i , j } ) - f _ { i } ( \\mathbf { x } ) \\big ] } { \\mu } \\mathbf { u } _ { i , j } ,", + "type": "interline_equation", + "image_path": "f917312d2f27baeae40650a8a5940d7151d7ff50f1a20268f41879b60fb5fc91.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 128, + 508, + 482, + 518.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 128, + 518.0, + 482, + 528.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 128, + 528.0, + 482, + 538.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 544, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 134, + 560 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 547, + 168, + 557 + ], + "score": 0.88, + "content": "\\mathbf { x } = \\mathbf { x } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 543, + 236, + 560 + ], + "score": 1.0, + "content": "in Algorithm 1,", + "type": "text" + }, + { + "bbox": [ + 236, + 546, + 248, + 557 + ], + "score": 0.8, + "content": "\\mathcal { T } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 543, + 346, + 560 + ], + "score": 1.0, + "content": "is a mini-batch of size", + "type": "text" + }, + { + "bbox": [ + 347, + 545, + 384, + 558 + ], + "score": 0.86, + "content": "\\left| \\mathcal { T } _ { k } \\right| = b", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 543, + 389, + 560 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 389, + 545, + 430, + 559 + ], + "score": 0.88, + "content": "\\{ \\mathbf { u } _ { i , j } \\} _ { j = 1 } ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 543, + 506, + 560 + ], + "score": 1.0, + "content": "are i.i.d. random", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 381, + 573 + ], + "score": 1.0, + "content": "directions drawn from a uniform distribution over a unit sphere, and", + "type": "text" + }, + { + "bbox": [ + 381, + 559, + 432, + 572 + ], + "score": 0.91, + "content": "\\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "gives a two-point", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 299, + 584 + ], + "score": 1.0, + "content": "based random gradient estimate with direction", + "type": "text" + }, + { + "bbox": [ + 299, + 573, + 316, + 583 + ], + "score": 0.89, + "content": "\\mathbf { u } _ { i , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 570, + 425, + 584 + ], + "score": 1.0, + "content": "and smoothing parameter", + "type": "text" + }, + { + "bbox": [ + 425, + 572, + 452, + 582 + ], + "score": 0.9, + "content": "\\mu > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 570, + 506, + 584 + ], + "score": 1.0, + "content": ". We remark", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "that the random direction vectors in (3) can also be drawn from the standard Gaussian distribution", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "(Nesterov & Spokoiny, 2015). However, the uniform distribution could be more useful in practice", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 604, + 477, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 477, + 616 + ], + "score": 1.0, + "content": "since it is defined in a bounded space rather than the whole real space required for Gaussian.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 504, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "score": 1.0, + "content": "We highlight that unlike the first-order stochastic gradient estimate, the ZO gradient estimate (3) is a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 287, + 642 + ], + "score": 1.0, + "content": "biased approximation to the true gradient of", + "type": "text" + }, + { + "bbox": [ + 288, + 630, + 294, + 641 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 629, + 505, + 642 + ], + "score": 1.0, + "content": ". Instead, it becomes unbiased to the gradient of the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 640, + 400, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 236, + 652 + ], + "score": 1.0, + "content": "randomized smoothing function", + "type": "text" + }, + { + "bbox": [ + 236, + 640, + 248, + 652 + ], + "score": 0.88, + "content": "\\bar { f _ { \\mu } }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 640, + 400, + 652 + ], + "score": 1.0, + "content": "(Duchi et al., 2012; Gao et al., 2014),", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 653, + 433, + 682 + ], + "lines": [ + { + "bbox": [ + 177, + 653, + 433, + 682 + ], + "spans": [ + { + "bbox": [ + 177, + 653, + 433, + 682 + ], + "score": 0.93, + "content": "f _ { \\mu } ( \\mathbf { x } ) = \\mathbb { E } _ { \\mathbf { v } } [ f ( \\mathbf { x } + \\mu \\mathbf { v } ) ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { v } } [ f _ { i } ( \\mathbf { x } + \\mu \\mathbf { v } ) ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } f _ { i , \\mu } ( \\mathbf { x } ) ,", + "type": "interline_equation", + "image_path": "f5642bc8d01d73ce03f6fb4a6a1f4b6e63e7ec1c1bdc75790c0cdf7b89f29e73.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 177, + 653, + 433, + 662.6666666666666 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 177, + 662.6666666666666, + 433, + 672.3333333333333 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 177, + 672.3333333333333, + 433, + 681.9999999999999 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 134, + 700 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 688, + 151, + 700 + ], + "score": 0.91, + "content": "f _ { i , \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 687, + 336, + 700 + ], + "score": 1.0, + "content": "gives the randomized smoothing version of", + "type": "text" + }, + { + "bbox": [ + 336, + 688, + 345, + 699 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 687, + 454, + 700 + ], + "score": 1.0, + "content": ", and the random variable", + "type": "text" + }, + { + "bbox": [ + 455, + 690, + 462, + 698 + ], + "score": 0.57, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "follows a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "uniform distribution over the unit Euclidean ball. 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\\delta _ { k } \\mathrm { s i g n } ( \\hat { \\bf g } _ { k } ) , \\quad \\mathrm { w h e r e ~ s i g n ( { \\bf x } ) ~ t a k e s ~ e l e m e n t - w i s e ~ s i g n s ~ o f ~ { \\bf x } ~ } } \\end{array}", + "type": "interline_equation", + "image_path": "ab350cde9d4f0de74f2ca117be5408b1dcadec65d9bc4b78059882f2b882a15d.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 177, + 147, + 451, + 159 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 111, + 165, + 156, + 176 + ], + "lines": [ + { + "bbox": [ + 110, + 164, + 157, + 177 + ], + "spans": [ + { + "bbox": [ + 110, + 164, + 157, + 177 + ], + "score": 1.0, + "content": "5: end for", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 110, + 164, + 157, + 177 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 200, + 504, + 223 + ], + "lines": [ + { + "bbox": [ + 106, + 200, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 213 + ], + "score": 1.0, + "content": "Athalye et al., 2018; Madry et al., 2018). In what follows, we show that the generation of adversarial", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 210, + 493, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 493, + 225 + ], + "score": 1.0, + "content": "examples in (Goodfellow et al., 2015; Kurakin et al., 2017) can be interpreted through signSGD.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 200, + 505, + 225 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 122, + 240 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 229, + 134, + 238 + ], + "score": 0.82, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 227, + 474, + 240 + ], + "score": 1.0, + "content": "denote the natural (legitimate) input of an ML model associated with the true label", + "type": "text" + }, + { + "bbox": [ + 474, + 228, + 484, + 238 + ], + "score": 0.85, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 227, + 506, + 240 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 159, + 248 + ], + "score": 0.9, + "content": "\\mathbf { x } ^ { \\prime } = \\mathbf { x } _ { 0 } + \\boldsymbol { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 237, + 365, + 249 + ], + "score": 1.0, + "content": "be the adversarial example to be designed, where", + "type": "text" + }, + { + "bbox": [ + 366, + 238, + 372, + 247 + ], + "score": 0.77, + "content": "\\pmb { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "are adversarial perturbations. If", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 107, + 248, + 140, + 259 + ], + "score": 0.9, + "content": "f ( \\mathbf { x } , t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "is the training loss of a learning model, then the goal of (white-box) adversarial attack is to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 214, + 270 + ], + "score": 1.0, + "content": "find minimal perturbation", + "type": "text" + }, + { + "bbox": [ + 214, + 258, + 221, + 267 + ], + "score": 0.73, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 256, + 505, + 270 + ], + "score": 1.0, + "content": "that is sufficient to mislead the learning model, namely, to maximize", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 142, + 280 + ], + "score": 1.0, + "content": "the loss", + "type": "text" + }, + { + "bbox": [ + 142, + 267, + 199, + 279 + ], + "score": 0.91, + "content": "f ( \\mathbf { x } _ { 0 } ^ { \\mathbf { \\hat { \\alpha } } } + \\mathbf { \\delta } \\mathbf { \\delta } , t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 266, + 374, + 280 + ], + "score": 1.0, + "content": ". Taking the first-order approximation of", + "type": "text" + }, + { + "bbox": [ + 374, + 267, + 410, + 279 + ], + "score": 0.93, + "content": "\\bar { f ( \\mathbf { x } ^ { \\prime } , t _ { 0 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 266, + 444, + 280 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 444, + 269, + 456, + 279 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 266, + 505, + 280 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 277, + 504, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 275, + 290 + ], + "score": 0.89, + "content": "f ( \\mathbf { x } ^ { \\prime } , t _ { 0 } ) \\approx f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) + \\langle \\nabla _ { x } f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) , \\delta \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 277, + 489, + 291 + ], + "score": 1.0, + "content": ". By constraining the strength of perturbation in the", + "type": "text" + }, + { + "bbox": [ + 490, + 278, + 504, + 289 + ], + "score": 0.87, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 287, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 104, + 287, + 190, + 303 + ], + "score": 1.0, + "content": "ball of small radius", + "type": "text" + }, + { + "bbox": [ + 190, + 291, + 196, + 299 + ], + "score": 0.74, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 287, + 219, + 303 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 219, + 290, + 268, + 301 + ], + "score": 0.89, + "content": "\\| \\pmb { \\delta } \\| _ { \\infty } \\leq \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 287, + 388, + 303 + ], + "score": 1.0, + "content": ", the linear approximation of", + "type": "text" + }, + { + "bbox": [ + 389, + 289, + 424, + 301 + ], + "score": 0.93, + "content": "f ( \\mathbf { x } ^ { \\prime } , t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 287, + 506, + 303 + ], + "score": 1.0, + "content": "is then maximized", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 298, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 104, + 298, + 117, + 312 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 117, + 299, + 218, + 311 + ], + "score": 0.91, + "content": "\\pmb { \\delta } = \\epsilon \\mathrm { s i g n } ( \\nabla _ { \\pmb { x } } f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 298, + 506, + 312 + ], + "score": 1.0, + "content": "(Shaham et al., 2018). Therefore, generation of adversarial examples", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 309, + 429, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 429, + 322 + ], + "score": 1.0, + "content": "proposed in (Goodfellow et al., 2015) obeys the sign-gradient update rule in (2),", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 227, + 506, + 322 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 322, + 369, + 335 + ], + "lines": [ + { + "bbox": [ + 240, + 322, + 369, + 335 + ], + "spans": [ + { + "bbox": [ + 240, + 322, + 369, + 335 + ], + "score": 0.89, + "content": "\\mathbf { x } ^ { \\prime } = \\mathbf { x } _ { 0 } - \\epsilon \\mathrm { s i g n } ( - \\nabla _ { x } f ( \\mathbf { x } _ { 0 } , t _ { 0 } ) ) .", + "type": "interline_equation", + "image_path": "8c5e3d5b769882ff66b4b10e97e41df6c1ab4c2857b2414e472981dc9be25151.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 240, + 322, + 369, + 335 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 506, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "Such a connection between adversarial example generation and signSGD also holds in other attacks,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 366 + ], + "score": 1.0, + "content": "e.g., the iterative target attack method (Kurakin et al., 2017). Similarly, a so-called black-box attack", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 364, + 495, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 495, + 378 + ], + "score": 1.0, + "content": "(Ilyas et al., 2018a; Bhagoji et al., 2018) is associated with our proposed ZO-signSGD algorithm.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 342, + 506, + 378 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 384, + 369, + 398 + ], + "lines": [ + { + "bbox": [ + 104, + 384, + 370, + 400 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 370, + 400 + ], + "score": 1.0, + "content": "3 ZO-SIGNSGD FOR BLACK-BOX OPTIMIZATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "One limitation of signSGD (Bernstein et al., 2018) is the need of first-order information, i.e., stochastic", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "score": 1.0, + "content": "gradients. However, there exists a large practical demand for solving ML problems where explicit", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "expressions of the gradients are difficult or infeasible to obtain, e.g., the generation of adversarial", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 391, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 391, + 452 + ], + "score": 1.0, + "content": "examples from black-box neural networks as discussed in Sec. 1 and 2.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 406, + 506, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 506, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 507, + 469 + ], + "score": 1.0, + "content": "Gradient estimation via ZO oracle In the ZO setting where the first-order information is unavail-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 471, + 480 + ], + "score": 1.0, + "content": "able, the gradient estimator at Step 3 of Algorithm 1 has only access to function values of", + "type": "text" + }, + { + "bbox": [ + 471, + 467, + 504, + 479 + ], + "score": 0.91, + "content": "\\{ f _ { i } ( \\mathbf { x } ) \\}", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 476, + 507, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 507, + 489 + ], + "score": 1.0, + "content": "given a query point x. Based on that, we construct a ZO gradient estimate through a forward differ-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "ence of two function values (Nesterov & Spokoiny, 2015; Gao et al., 2014; Duchi et al., 2015). In", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 496, + 315, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 221, + 509 + ], + "score": 1.0, + "content": "Algorithm 1, GradEstimate", + "type": "text" + }, + { + "bbox": [ + 222, + 497, + 235, + 508 + ], + "score": 0.64, + "content": "\\left( \\mathbf { x } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 496, + 315, + 509 + ], + "score": 1.0, + "content": "is then specified as", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 456, + 507, + 509 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 508, + 482, + 538 + ], + "lines": [ + { + "bbox": [ + 128, + 508, + 482, + 538 + ], + "spans": [ + { + "bbox": [ + 128, + 508, + 482, + 538 + ], + "score": 0.92, + "content": "\\mathrm { G r a d E s t i m a t e } ( \\mathbf { x } ) = \\frac { 1 } { b q } \\sum _ { i \\in \\mathcal { T } _ { k } } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) , \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) : = \\frac { d \\big [ f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } _ { i , j } ) - f _ { i } ( \\mathbf { x } ) \\big ] } { \\mu } \\mathbf { u } _ { i , j } ,", + "type": "interline_equation", + "image_path": "f917312d2f27baeae40650a8a5940d7151d7ff50f1a20268f41879b60fb5fc91.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 128, + 508, + 482, + 518.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 128, + 518.0, + 482, + 528.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 128, + 528.0, + 482, + 538.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 544, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 134, + 560 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 547, + 168, + 557 + ], + "score": 0.88, + "content": "\\mathbf { x } = \\mathbf { x } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 543, + 236, + 560 + ], + "score": 1.0, + "content": "in Algorithm 1,", + "type": "text" + }, + { + "bbox": [ + 236, + 546, + 248, + 557 + ], + "score": 0.8, + "content": "\\mathcal { T } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 543, + 346, + 560 + ], + "score": 1.0, + "content": "is a mini-batch of size", + "type": "text" + }, + { + "bbox": [ + 347, + 545, + 384, + 558 + ], + "score": 0.86, + "content": "\\left| \\mathcal { T } _ { k } \\right| = b", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 543, + 389, + 560 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 389, + 545, + 430, + 559 + ], + "score": 0.88, + "content": "\\{ \\mathbf { u } _ { i , j } \\} _ { j = 1 } ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 543, + 506, + 560 + ], + "score": 1.0, + "content": "are i.i.d. random", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 381, + 573 + ], + "score": 1.0, + "content": "directions drawn from a uniform distribution over a unit sphere, and", + "type": "text" + }, + { + "bbox": [ + 381, + 559, + 432, + 572 + ], + "score": 0.91, + "content": "\\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "gives a two-point", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 299, + 584 + ], + "score": 1.0, + "content": "based random gradient estimate with direction", + "type": "text" + }, + { + "bbox": [ + 299, + 573, + 316, + 583 + ], + "score": 0.89, + "content": "\\mathbf { u } _ { i , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 570, + 425, + 584 + ], + "score": 1.0, + "content": "and smoothing parameter", + "type": "text" + }, + { + "bbox": [ + 425, + 572, + 452, + 582 + ], + "score": 0.9, + "content": "\\mu > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 570, + 506, + 584 + ], + "score": 1.0, + "content": ". We remark", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "that the random direction vectors in (3) can also be drawn from the standard Gaussian distribution", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "(Nesterov & Spokoiny, 2015). However, the uniform distribution could be more useful in practice", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 604, + 477, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 477, + 616 + ], + "score": 1.0, + "content": "since it is defined in a bounded space rather than the whole real space required for Gaussian.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 543, + 506, + 616 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 504, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "score": 1.0, + "content": "We highlight that unlike the first-order stochastic gradient estimate, the ZO gradient estimate (3) is a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 287, + 642 + ], + "score": 1.0, + "content": "biased approximation to the true gradient of", + "type": "text" + }, + { + "bbox": [ + 288, + 630, + 294, + 641 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 629, + 505, + 642 + ], + "score": 1.0, + "content": ". Instead, it becomes unbiased to the gradient of the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 640, + 400, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 236, + 652 + ], + "score": 1.0, + "content": "randomized smoothing function", + "type": "text" + }, + { + "bbox": [ + 236, + 640, + 248, + 652 + ], + "score": 0.88, + "content": "\\bar { f _ { \\mu } }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 640, + 400, + 652 + ], + "score": 1.0, + "content": "(Duchi et al., 2012; Gao et al., 2014),", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 619, + 506, + 652 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 653, + 433, + 682 + ], + "lines": [ + { + "bbox": [ + 177, + 653, + 433, + 682 + ], + "spans": [ + { + "bbox": [ + 177, + 653, + 433, + 682 + ], + "score": 0.93, + "content": "f _ { \\mu } ( \\mathbf { x } ) = \\mathbb { E } _ { \\mathbf { v } } [ f ( \\mathbf { x } + \\mu \\mathbf { v } ) ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { v } } [ f _ { i } ( \\mathbf { x } + \\mu \\mathbf { v } ) ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } f _ { i , \\mu } ( \\mathbf { x } ) ,", + "type": "interline_equation", + "image_path": "f5642bc8d01d73ce03f6fb4a6a1f4b6e63e7ec1c1bdc75790c0cdf7b89f29e73.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 177, + 653, + 433, + 662.6666666666666 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 177, + 662.6666666666666, + 433, + 672.3333333333333 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 177, + 672.3333333333333, + 433, + 681.9999999999999 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 134, + 700 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 688, + 151, + 700 + ], + "score": 0.91, + "content": "f _ { i , \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 687, + 336, + 700 + ], + "score": 1.0, + "content": "gives the randomized smoothing version of", + "type": "text" + }, + { + "bbox": [ + 336, + 688, + 345, + 699 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 687, + 454, + 700 + ], + "score": 1.0, + "content": ", and the random variable", + "type": "text" + }, + { + "bbox": [ + 455, + 690, + 462, + 698 + ], + "score": 0.57, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "follows a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "uniform distribution over the unit Euclidean ball. Clearly, there exists a gap between a ZO gradient", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 238, + 722 + ], + "score": 1.0, + "content": "estimate and the true gradient of", + "type": "text" + }, + { + "bbox": [ + 238, + 711, + 244, + 721 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 710, + 505, + 722 + ], + "score": 1.0, + "content": ", but as will be evident later, such a gap can be measured through", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 718, + 217, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 201, + 735 + ], + "score": 1.0, + "content": "the smoothing function", + "type": "text" + }, + { + "bbox": [ + 202, + 721, + 213, + 733 + ], + "score": 0.89, + "content": "f _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 718, + 217, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 687, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Motivations of ZO-signSGD. Compared to SGD-type methods, the fast empirical convergence of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "signSGD and ZO-signSGD has been shown in the application of generating white-box and black-box", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "adversarial examples (Goodfellow et al., 2015; Madry et al., 2018; Ilyas et al., 2018a). As mentioned", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "in (Bernstein et al., 2018), the sign operation could mitigate the negative effect of (coordinate-wise)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "gradient noise of large variance. Recall that the ZO gradient estimate is a biased approximation to the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "score": 1.0, + "content": "true gradient, and thus, could suffer from having larger noise variance than (first-order) stochastic", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "gradients. In this context, one could benefit from ZO-signSGD due to its robustness to gradient", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "noise. In Appendix 1, we provide two concrete examples (Fig. A1 and Fig. A2) to confirm the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 185 + ], + "score": 1.0, + "content": "aforementioned analysis. In Fig. A1, we show the robustness of ZO-signSGD against sparse noise", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "perturbation through a toy quadratic optimization problem, originally introduced in (Bernstein et al.,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 190, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 206 + ], + "score": 1.0, + "content": "2018) to motivate the fast convergence of signSGD against SGD. In Fig. A2, we show that gradient", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "estimation via ZO oracle indeed encounters gradient noise of large variance. Thus, taking the sign of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 391, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 391, + 227 + ], + "score": 1.0, + "content": "a gradient estimate might scale down the extremely noisy components.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 505, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "score": 1.0, + "content": "ZO-signSGD & technical challenges beyond signSGD Algorithm 1 becomes ZO-signSGD as", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 507, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 507, + 259 + ], + "score": 1.0, + "content": "the ZO gradient estimate (3) is applied. We note that the extension from first order to ZO is nontrivial,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 257, + 432, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 432, + 270 + ], + "score": 1.0, + "content": "as the proposed ZO-signSGD algorithm yields three key differences to signSGD.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 273, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "First, ZO-signSGD has milder assumption on the choice of mini-batch sampling. Recall that signSGD", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 243, + 299 + ], + "score": 1.0, + "content": "in (Bernstein et al., 2018) achieves", + "type": "text" + }, + { + "bbox": [ + 243, + 285, + 284, + 298 + ], + "score": 0.93, + "content": "O ( 1 / \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "convergence rate given the condition that the mini-batch", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 204, + 309 + ], + "score": 1.0, + "content": "size is sufficiently large,", + "type": "text" + }, + { + "bbox": [ + 205, + 297, + 246, + 309 + ], + "score": 0.92, + "content": "b = O ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 297, + 505, + 309 + ], + "score": 1.0, + "content": ". However, this condition only becomes true when the mini-batch", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 504, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 248, + 321 + ], + "score": 1.0, + "content": "sample is randomly selected from", + "type": "text" + }, + { + "bbox": [ + 248, + 308, + 260, + 320 + ], + "score": 0.84, + "content": "[ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 307, + 434, + 321 + ], + "score": 1.0, + "content": "with replacement, which is unusual when", + "type": "text" + }, + { + "bbox": [ + 434, + 308, + 463, + 319 + ], + "score": 0.9, + "content": "n \\leq T", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 307, + 491, + 321 + ], + "score": 1.0, + "content": ". Here", + "type": "text" + }, + { + "bbox": [ + 491, + 308, + 504, + 320 + ], + "score": 0.78, + "content": "[ n ]", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 212, + 331 + ], + "score": 1.0, + "content": "represents the integer set", + "type": "text" + }, + { + "bbox": [ + 213, + 319, + 266, + 331 + ], + "score": 0.91, + "content": "\\{ 1 , 2 , \\ldots , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 318, + 450, + 331 + ], + "score": 1.0, + "content": ". And signSGD fails to cover signGD when", + "type": "text" + }, + { + "bbox": [ + 450, + 319, + 477, + 329 + ], + "score": 0.89, + "content": "b = n", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 318, + 506, + 331 + ], + "score": 1.0, + "content": ", since", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 329, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 248, + 343 + ], + "score": 1.0, + "content": "sampling with replacement leads to", + "type": "text" + }, + { + "bbox": [ + 249, + 330, + 285, + 342 + ], + "score": 0.92, + "content": "\\mathcal { T } _ { k } \\neq [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 329, + 315, + 343 + ], + "score": 1.0, + "content": "even if", + "type": "text" + }, + { + "bbox": [ + 316, + 330, + 340, + 340 + ], + "score": 0.89, + "content": "b = n", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 329, + 507, + 343 + ], + "score": 1.0, + "content": ". In the proposed ZO-signSGD algorithm,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 340, + 323, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 323, + 354 + ], + "score": 1.0, + "content": "we will relax the assumption on mini-batch sampling.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "Second, in ZO-signSGD both the ZO gradient estimator and the sign operator give rise to approxi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "mation errors to the true gradient. Although the statistical properties of ZO gradient estimates can", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "be acquired with the aid of the randomized smoothing function (4), the use of mini-batch sampling", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "without replacement introduces extra difficulty to bound the variance of ZO gradient estimates since", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "mini-batch samples are no longer independent. Moreover, the sign-based descent algorithm evaluates", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 216, + 425 + ], + "score": 1.0, + "content": "the convergence error in the", + "type": "text" + }, + { + "bbox": [ + 216, + 413, + 226, + 424 + ], + "score": 0.88, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 413, + 411, + 425 + ], + "score": 1.0, + "content": "-norm geometry, leading to a mismatch with the", + "type": "text" + }, + { + "bbox": [ + 412, + 412, + 421, + 424 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "-norm based gradient", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 422, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 344, + 437 + ], + "score": 1.0, + "content": "variance. Besides translating the the gradient norm from", + "type": "text" + }, + { + "bbox": [ + 344, + 424, + 354, + 434 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 422, + 367, + 437 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 367, + 424, + 377, + 434 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 422, + 506, + 437 + ], + "score": 1.0, + "content": ", the probabilistic convergence", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 434, + 497, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 497, + 447 + ], + "score": 1.0, + "content": "method (Ghadimi & Lan, 2013) is used to bound the eventual convergence error of ZO-signSGD.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 108, + 451, + 503, + 474 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "Finally, beyond the standard ZO gradient estimator (3), we will cover multiple variants of ZO-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 462, + 317, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 317, + 475 + ], + "score": 1.0, + "content": "signSGD for centralized or distributed optimization.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 106, + 486, + 353, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 353, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 353, + 501 + ], + "score": 1.0, + "content": "4 CONVERGENCE ANALYSIS OF ZO-SIGNSGD", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 105, + 510, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "In this section, we begin by stating assumptions used in our analysis. We then derive the convergence", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "rate of ZO-signSGD for nonconvex optimization. Assumptions of problem (1) are listed as follows.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 105, + 538, + 423, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 424, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 167, + 552 + ], + "score": 1.0, + "content": "A1: Functions", + "type": "text" + }, + { + "bbox": [ + 167, + 538, + 186, + 551 + ], + "score": 0.92, + "content": "\\{ f _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 537, + 208, + 552 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 208, + 539, + 216, + 549 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 537, + 372, + 552 + ], + "score": 1.0, + "content": "-Lipschitz continuous gradients, where", + "type": "text" + }, + { + "bbox": [ + 372, + 538, + 419, + 551 + ], + "score": 0.93, + "content": "L \\in ( 0 , \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 537, + 424, + 552 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 105, + 555, + 437, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 436, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 157, + 569 + ], + "score": 1.0, + "content": "A2: At time", + "type": "text" + }, + { + "bbox": [ + 157, + 556, + 164, + 566 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 554, + 228, + 569 + ], + "score": 1.0, + "content": ", the gradient of", + "type": "text" + }, + { + "bbox": [ + 228, + 556, + 238, + 567 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 554, + 323, + 569 + ], + "score": 1.0, + "content": "is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 323, + 555, + 393, + 568 + ], + "score": 0.91, + "content": "\\| \\nabla f _ { i } ( \\mathbf { x } _ { k } ) \\| _ { 2 } \\leq \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 554, + 408, + 569 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 408, + 555, + 436, + 568 + ], + "score": 0.86, + "content": "i \\in [ n ]", + "type": "inline_equation" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "Both A1 and A2 are the standard assumptions used in nonconvex optimization literature (Bernstein", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 373, + 596 + ], + "score": 1.0, + "content": "et al., 2018; Reddi et al., 2018; Chen et al., 2018). A1 implies the", + "type": "text" + }, + { + "bbox": [ + 373, + 584, + 381, + 593 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 583, + 443, + 596 + ], + "score": 1.0, + "content": "-smoothness of", + "type": "text" + }, + { + "bbox": [ + 444, + 583, + 453, + 595 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 583, + 506, + 596 + ], + "score": 1.0, + "content": ", namely, for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 122, + 608 + ], + "score": 1.0, + "content": "any", + "type": "text" + }, + { + "bbox": [ + 123, + 596, + 131, + 605 + ], + "score": 0.61, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 593, + 147, + 608 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 147, + 596, + 155, + 606 + ], + "score": 0.47, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 593, + 197, + 608 + ], + "score": 1.0, + "content": "we obtain", + "type": "text" + }, + { + "bbox": [ + 197, + 594, + 405, + 606 + ], + "score": 0.85, + "content": "f _ { i } ( \\mathbf { x } ) - f _ { i } ( \\mathbf { y } ) \\leq \\langle \\nabla f _ { i } ( \\mathbf { y } ) , \\mathbf { x } - \\mathbf { y } \\rangle + ( L / 2 ) \\| \\mathbf { x } - \\mathbf { y } \\| _ { i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 593, + 506, + 608 + ], + "score": 1.0, + "content": ". A2 implies the bounded", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 604, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 604, + 153, + 622 + ], + "score": 1.0, + "content": "variance of", + "type": "text" + }, + { + "bbox": [ + 153, + 606, + 171, + 618 + ], + "score": 0.91, + "content": "\\nabla f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 604, + 373, + 622 + ], + "score": 1.0, + "content": "in (Bernstein et al., 2018, Assumption 3), namely,", + "type": "text" + }, + { + "bbox": [ + 373, + 606, + 506, + 619 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\| \\nabla f _ { i } ( \\mathbf { x } ) - \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } \\leq } \\end{array}", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 617, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 123, + 629 + ], + "score": 0.87, + "content": "4 \\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 617, + 260, + 633 + ], + "score": 1.0, + "content": ", where we have used the fact that", + "type": "text" + }, + { + "bbox": [ + 261, + 618, + 323, + 631 + ], + "score": 0.9, + "content": "\\| \\nabla f ( \\mathbf { x } ) \\| _ { 2 } \\leq \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 617, + 506, + 633 + ], + "score": 1.0, + "content": "under A2. Throughout the paper, we assume", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 628, + 443, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 255, + 643 + ], + "score": 1.0, + "content": "that problem (1) is solvable, namely,", + "type": "text" + }, + { + "bbox": [ + 255, + 631, + 311, + 642 + ], + "score": 0.92, + "content": "f ( \\mathbf { x } ^ { * } ) > - \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 628, + 339, + 643 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 339, + 630, + 350, + 640 + ], + "score": 0.87, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 628, + 443, + 643 + ], + "score": 1.0, + "content": "is an optimal solution.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 106, + 646, + 506, + 724 + ], + "lines": [ + { + "bbox": [ + 105, + 646, + 507, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 507, + 660 + ], + "score": 1.0, + "content": "We recall that Algorithm 1 becomes ZO-signSGD when the gradient estimation step (3) is applied.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 657, + 507, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 507, + 672 + ], + "score": 1.0, + "content": "For nonconvex problems, the convergence of an algorithm is typically measured by stationarity, e.g.,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 668, + 507, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 130, + 682 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 131, + 668, + 174, + 681 + ], + "score": 0.93, + "content": "\\| \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 668, + 247, + 682 + ], + "score": 1.0, + "content": "in SGD (Ghadimi", + "type": "text" + }, + { + "bbox": [ + 247, + 669, + 256, + 679 + ], + "score": 0.3, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 668, + 319, + 682 + ], + "score": 1.0, + "content": "Lan, 2013) and", + "type": "text" + }, + { + "bbox": [ + 319, + 668, + 362, + 681 + ], + "score": 0.93, + "content": "\\| \\nabla f ( \\mathbf { x } ) \\bar { \\| _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 668, + 507, + 682 + ], + "score": 1.0, + "content": "in signSGD (Bernstein et al., 2018).", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 678, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 175, + 693 + ], + "score": 1.0, + "content": "For the latter, the", + "type": "text" + }, + { + "bbox": [ + 176, + 680, + 186, + 691 + ], + "score": 0.86, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 678, + 506, + 693 + ], + "score": 1.0, + "content": "geometry is met when quantifying the stochasticity through the (non-linear) sign", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 690, + 507, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 507, + 703 + ], + "score": 1.0, + "content": "operation. Different from signSGD, ZO-signSGD only obtains a biased estimate to the true gradient.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 701, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 506, + 714 + ], + "score": 1.0, + "content": "In Proposition 1, we bypass such a bias by leveraging the randomized smoothing technique used for", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 712, + 443, + 726 + ], + "spans": [ + { + "bbox": [ + 105, + 712, + 443, + 726 + ], + "score": 1.0, + "content": "ZO optimization (Gao et al., 2014; Nesterov & Spokoiny, 2015; Duchi et al., 2015).", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Motivations of ZO-signSGD. Compared to SGD-type methods, the fast empirical convergence of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "signSGD and ZO-signSGD has been shown in the application of generating white-box and black-box", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "adversarial examples (Goodfellow et al., 2015; Madry et al., 2018; Ilyas et al., 2018a). As mentioned", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "in (Bernstein et al., 2018), the sign operation could mitigate the negative effect of (coordinate-wise)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "gradient noise of large variance. Recall that the ZO gradient estimate is a biased approximation to the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "score": 1.0, + "content": "true gradient, and thus, could suffer from having larger noise variance than (first-order) stochastic", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "gradients. In this context, one could benefit from ZO-signSGD due to its robustness to gradient", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "noise. In Appendix 1, we provide two concrete examples (Fig. A1 and Fig. A2) to confirm the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 185 + ], + "score": 1.0, + "content": "aforementioned analysis. In Fig. A1, we show the robustness of ZO-signSGD against sparse noise", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "perturbation through a toy quadratic optimization problem, originally introduced in (Bernstein et al.,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 190, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 206 + ], + "score": 1.0, + "content": "2018) to motivate the fast convergence of signSGD against SGD. In Fig. A2, we show that gradient", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "estimation via ZO oracle indeed encounters gradient noise of large variance. Thus, taking the sign of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 391, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 391, + 227 + ], + "score": 1.0, + "content": "a gradient estimate might scale down the extremely noisy components.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 82, + 506, + 227 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 505, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "score": 1.0, + "content": "ZO-signSGD & technical challenges beyond signSGD Algorithm 1 becomes ZO-signSGD as", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 507, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 507, + 259 + ], + "score": 1.0, + "content": "the ZO gradient estimate (3) is applied. We note that the extension from first order to ZO is nontrivial,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 257, + 432, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 432, + 270 + ], + "score": 1.0, + "content": "as the proposed ZO-signSGD algorithm yields three key differences to signSGD.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 234, + 507, + 270 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 273, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "First, ZO-signSGD has milder assumption on the choice of mini-batch sampling. Recall that signSGD", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 243, + 299 + ], + "score": 1.0, + "content": "in (Bernstein et al., 2018) achieves", + "type": "text" + }, + { + "bbox": [ + 243, + 285, + 284, + 298 + ], + "score": 0.93, + "content": "O ( 1 / \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "convergence rate given the condition that the mini-batch", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 204, + 309 + ], + "score": 1.0, + "content": "size is sufficiently large,", + "type": "text" + }, + { + "bbox": [ + 205, + 297, + 246, + 309 + ], + "score": 0.92, + "content": "b = O ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 297, + 505, + 309 + ], + "score": 1.0, + "content": ". However, this condition only becomes true when the mini-batch", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 504, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 248, + 321 + ], + "score": 1.0, + "content": "sample is randomly selected from", + "type": "text" + }, + { + "bbox": [ + 248, + 308, + 260, + 320 + ], + "score": 0.84, + "content": "[ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 307, + 434, + 321 + ], + "score": 1.0, + "content": "with replacement, which is unusual when", + "type": "text" + }, + { + "bbox": [ + 434, + 308, + 463, + 319 + ], + "score": 0.9, + "content": "n \\leq T", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 307, + 491, + 321 + ], + "score": 1.0, + "content": ". Here", + "type": "text" + }, + { + "bbox": [ + 491, + 308, + 504, + 320 + ], + "score": 0.78, + "content": "[ n ]", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 212, + 331 + ], + "score": 1.0, + "content": "represents the integer set", + "type": "text" + }, + { + "bbox": [ + 213, + 319, + 266, + 331 + ], + "score": 0.91, + "content": "\\{ 1 , 2 , \\ldots , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 318, + 450, + 331 + ], + "score": 1.0, + "content": ". And signSGD fails to cover signGD when", + "type": "text" + }, + { + "bbox": [ + 450, + 319, + 477, + 329 + ], + "score": 0.89, + "content": "b = n", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 318, + 506, + 331 + ], + "score": 1.0, + "content": ", since", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 329, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 248, + 343 + ], + "score": 1.0, + "content": "sampling with replacement leads to", + "type": "text" + }, + { + "bbox": [ + 249, + 330, + 285, + 342 + ], + "score": 0.92, + "content": "\\mathcal { T } _ { k } \\neq [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 329, + 315, + 343 + ], + "score": 1.0, + "content": "even if", + "type": "text" + }, + { + "bbox": [ + 316, + 330, + 340, + 340 + ], + "score": 0.89, + "content": "b = n", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 329, + 507, + 343 + ], + "score": 1.0, + "content": ". In the proposed ZO-signSGD algorithm,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 340, + 323, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 323, + 354 + ], + "score": 1.0, + "content": "we will relax the assumption on mini-batch sampling.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 273, + 507, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "Second, in ZO-signSGD both the ZO gradient estimator and the sign operator give rise to approxi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "mation errors to the true gradient. Although the statistical properties of ZO gradient estimates can", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "be acquired with the aid of the randomized smoothing function (4), the use of mini-batch sampling", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "without replacement introduces extra difficulty to bound the variance of ZO gradient estimates since", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "mini-batch samples are no longer independent. Moreover, the sign-based descent algorithm evaluates", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 216, + 425 + ], + "score": 1.0, + "content": "the convergence error in the", + "type": "text" + }, + { + "bbox": [ + 216, + 413, + 226, + 424 + ], + "score": 0.88, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 413, + 411, + 425 + ], + "score": 1.0, + "content": "-norm geometry, leading to a mismatch with the", + "type": "text" + }, + { + "bbox": [ + 412, + 412, + 421, + 424 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "-norm based gradient", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 422, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 344, + 437 + ], + "score": 1.0, + "content": "variance. Besides translating the the gradient norm from", + "type": "text" + }, + { + "bbox": [ + 344, + 424, + 354, + 434 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 422, + 367, + 437 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 367, + 424, + 377, + 434 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 422, + 506, + 437 + ], + "score": 1.0, + "content": ", the probabilistic convergence", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 434, + 497, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 497, + 447 + ], + "score": 1.0, + "content": "method (Ghadimi & Lan, 2013) is used to bound the eventual convergence error of ZO-signSGD.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 358, + 506, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 451, + 503, + 474 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "Finally, beyond the standard ZO gradient estimator (3), we will cover multiple variants of ZO-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 462, + 317, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 317, + 475 + ], + "score": 1.0, + "content": "signSGD for centralized or distributed optimization.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 106, + 451, + 505, + 475 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 486, + 353, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 353, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 353, + 501 + ], + "score": 1.0, + "content": "4 CONVERGENCE ANALYSIS OF ZO-SIGNSGD", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 105, + 510, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "In this section, we begin by stating assumptions used in our analysis. We then derive the convergence", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "rate of ZO-signSGD for nonconvex optimization. Assumptions of problem (1) are listed as follows.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 510, + 505, + 534 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 538, + 423, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 424, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 167, + 552 + ], + "score": 1.0, + "content": "A1: Functions", + "type": "text" + }, + { + "bbox": [ + 167, + 538, + 186, + 551 + ], + "score": 0.92, + "content": "\\{ f _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 537, + 208, + 552 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 208, + 539, + 216, + 549 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 537, + 372, + 552 + ], + "score": 1.0, + "content": "-Lipschitz continuous gradients, where", + "type": "text" + }, + { + "bbox": [ + 372, + 538, + 419, + 551 + ], + "score": 0.93, + "content": "L \\in ( 0 , \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 537, + 424, + 552 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 537, + 424, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 555, + 437, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 436, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 157, + 569 + ], + "score": 1.0, + "content": "A2: At time", + "type": "text" + }, + { + "bbox": [ + 157, + 556, + 164, + 566 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 554, + 228, + 569 + ], + "score": 1.0, + "content": ", the gradient of", + "type": "text" + }, + { + "bbox": [ + 228, + 556, + 238, + 567 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 554, + 323, + 569 + ], + "score": 1.0, + "content": "is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 323, + 555, + 393, + 568 + ], + "score": 0.91, + "content": "\\| \\nabla f _ { i } ( \\mathbf { x } _ { k } ) \\| _ { 2 } \\leq \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 554, + 408, + 569 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 408, + 555, + 436, + 568 + ], + "score": 0.86, + "content": "i \\in [ n ]", + "type": "inline_equation" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 106, + 554, + 436, + 569 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "Both A1 and A2 are the standard assumptions used in nonconvex optimization literature (Bernstein", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 373, + 596 + ], + "score": 1.0, + "content": "et al., 2018; Reddi et al., 2018; Chen et al., 2018). A1 implies the", + "type": "text" + }, + { + "bbox": [ + 373, + 584, + 381, + 593 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 583, + 443, + 596 + ], + "score": 1.0, + "content": "-smoothness of", + "type": "text" + }, + { + "bbox": [ + 444, + 583, + 453, + 595 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 583, + 506, + 596 + ], + "score": 1.0, + "content": ", namely, for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 122, + 608 + ], + "score": 1.0, + "content": "any", + "type": "text" + }, + { + "bbox": [ + 123, + 596, + 131, + 605 + ], + "score": 0.61, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 593, + 147, + 608 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 147, + 596, + 155, + 606 + ], + "score": 0.47, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 593, + 197, + 608 + ], + "score": 1.0, + "content": "we obtain", + "type": "text" + }, + { + "bbox": [ + 197, + 594, + 405, + 606 + ], + "score": 0.85, + "content": "f _ { i } ( \\mathbf { x } ) - f _ { i } ( \\mathbf { y } ) \\leq \\langle \\nabla f _ { i } ( \\mathbf { y } ) , \\mathbf { x } - \\mathbf { y } \\rangle + ( L / 2 ) \\| \\mathbf { x } - \\mathbf { y } \\| _ { i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 593, + 506, + 608 + ], + "score": 1.0, + "content": ". A2 implies the bounded", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 604, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 604, + 153, + 622 + ], + "score": 1.0, + "content": "variance of", + "type": "text" + }, + { + "bbox": [ + 153, + 606, + 171, + 618 + ], + "score": 0.91, + "content": "\\nabla f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 604, + 373, + 622 + ], + "score": 1.0, + "content": "in (Bernstein et al., 2018, Assumption 3), namely,", + "type": "text" + }, + { + "bbox": [ + 373, + 606, + 506, + 619 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\| \\nabla f _ { i } ( \\mathbf { x } ) - \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } \\leq } \\end{array}", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 617, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 123, + 629 + ], + "score": 0.87, + "content": "4 \\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 617, + 260, + 633 + ], + "score": 1.0, + "content": ", where we have used the fact that", + "type": "text" + }, + { + "bbox": [ + 261, + 618, + 323, + 631 + ], + "score": 0.9, + "content": "\\| \\nabla f ( \\mathbf { x } ) \\| _ { 2 } \\leq \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 617, + 506, + 633 + ], + "score": 1.0, + "content": "under A2. Throughout the paper, we assume", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 628, + 443, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 255, + 643 + ], + "score": 1.0, + "content": "that problem (1) is solvable, namely,", + "type": "text" + }, + { + "bbox": [ + 255, + 631, + 311, + 642 + ], + "score": 0.92, + "content": "f ( \\mathbf { x } ^ { * } ) > - \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 628, + 339, + 643 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 339, + 630, + 350, + 640 + ], + "score": 0.87, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 628, + 443, + 643 + ], + "score": 1.0, + "content": "is an optimal solution.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 572, + 506, + 643 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 646, + 506, + 724 + ], + "lines": [ + { + "bbox": [ + 105, + 646, + 507, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 507, + 660 + ], + "score": 1.0, + "content": "We recall that Algorithm 1 becomes ZO-signSGD when the gradient estimation step (3) is applied.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 657, + 507, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 507, + 672 + ], + "score": 1.0, + "content": "For nonconvex problems, the convergence of an algorithm is typically measured by stationarity, e.g.,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 668, + 507, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 130, + 682 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 131, + 668, + 174, + 681 + ], + "score": 0.93, + "content": "\\| \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 668, + 247, + 682 + ], + "score": 1.0, + "content": "in SGD (Ghadimi", + "type": "text" + }, + { + "bbox": [ + 247, + 669, + 256, + 679 + ], + "score": 0.3, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 668, + 319, + 682 + ], + "score": 1.0, + "content": "Lan, 2013) and", + "type": "text" + }, + { + "bbox": [ + 319, + 668, + 362, + 681 + ], + "score": 0.93, + "content": "\\| \\nabla f ( \\mathbf { x } ) \\bar { \\| _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 668, + 507, + 682 + ], + "score": 1.0, + "content": "in signSGD (Bernstein et al., 2018).", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 678, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 175, + 693 + ], + "score": 1.0, + "content": "For the latter, the", + "type": "text" + }, + { + "bbox": [ + 176, + 680, + 186, + 691 + ], + "score": 0.86, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 678, + 506, + 693 + ], + "score": 1.0, + "content": "geometry is met when quantifying the stochasticity through the (non-linear) sign", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 690, + 507, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 507, + 703 + ], + "score": 1.0, + "content": "operation. Different from signSGD, ZO-signSGD only obtains a biased estimate to the true gradient.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 701, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 506, + 714 + ], + "score": 1.0, + "content": "In Proposition 1, we bypass such a bias by leveraging the randomized smoothing technique used for", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 712, + 443, + 726 + ], + "spans": [ + { + "bbox": [ + 105, + 712, + 443, + 726 + ], + "score": 1.0, + "content": "ZO optimization (Gao et al., 2014; Nesterov & Spokoiny, 2015; Duchi et al., 2015).", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 646, + 507, + 726 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 80, + 502, + 126 + ], + "lines": [ + { + "bbox": [ + 102, + 77, + 506, + 99 + ], + "spans": [ + { + "bbox": [ + 102, + 77, + 259, + 99 + ], + "score": 1.0, + "content": "Proposition 1 Under A1, the outputs", + "type": "text" + }, + { + "bbox": [ + 259, + 81, + 297, + 95 + ], + "score": 0.92, + "content": "\\{ \\mathbf { x } _ { k } \\} _ { k = 0 } ^ { T - 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For the latter case, Proposition 2 implies that if", + "type": "text" + }, + { + "bbox": [ + 421, + 493, + 446, + 503 + ], + "score": 0.89, + "content": "b = n", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 493, + 467, + 506 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 467, + 493, + 504, + 505 + ], + "score": 0.91, + "content": "\\mathcal { T } _ { k } = [ n ]", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 504, + 507, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 253, + 518 + ], + "score": 1.0, + "content": "for ZO-signGD), then the variance", + "type": "text" + }, + { + "bbox": [ + 253, + 504, + 345, + 517 + ], + "score": 0.91, + "content": "\\mathbb { E } \\left[ \\lVert \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\rVert _ { 2 } ^ { 2 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 504, + 403, + 518 + ], + "score": 1.0, + "content": "is reduced to", + "type": "text" + }, + { + "bbox": [ + 404, + 505, + 475, + 517 + ], + "score": 0.91, + "content": "O ( C ( d , \\mu ) / ( n q ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 504, + 507, + 518 + ], + "score": 1.0, + "content": ", corre-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 154, + 528 + ], + "score": 1.0, + "content": "sponding to", + "type": "text" + }, + { + "bbox": [ + 155, + 516, + 185, + 527 + ], + "score": 0.93, + "content": "\\alpha _ { b } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 515, + 201, + 528 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 202, + 516, + 231, + 527 + ], + "score": 0.92, + "content": "\\beta _ { b } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 515, + 369, + 528 + ], + "score": 1.0, + "content": "in (7). In the other extreme case of", + "type": "text" + }, + { + "bbox": [ + 369, + 516, + 392, + 526 + ], + "score": 0.9, + "content": "b = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 515, + 506, + 528 + ], + "score": 1.0, + "content": ", both the studied mini-batch", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 291, + 540 + ], + "score": 1.0, + "content": "schemes become identical, corresponding to", + "type": "text" + }, + { + "bbox": [ + 291, + 527, + 322, + 538 + ], + "score": 0.92, + "content": "\\alpha _ { b } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 526, + 342, + 540 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 342, + 527, + 372, + 538 + ], + "score": 0.93, + "content": "\\beta _ { b } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 526, + 506, + 540 + ], + "score": 1.0, + "content": ". Proposition 2 also implies that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 171, + 550 + ], + "score": 1.0, + "content": "the use of large", + "type": "text" + }, + { + "bbox": [ + 171, + 538, + 177, + 548 + ], + "score": 0.68, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 538, + 195, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 195, + 539, + 201, + 549 + ], + "score": 0.8, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "reduces the variance of the gradient estimate, and will further improve the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 550, + 178, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 178, + 561 + ], + "score": 1.0, + "content": "convergence rate.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "With the aid of Proposition 1 and 2, we can then show the convergence rate of ZO-signSGD in terms", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 577, + 504, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 256, + 589 + ], + "score": 1.0, + "content": "of stationarity of the original function", + "type": "text" + }, + { + "bbox": [ + 257, + 577, + 264, + 588 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 577, + 496, + 589 + ], + "score": 1.0, + "content": ". The remaining difficulty is how to bound the gap between", + "type": "text" + }, + { + "bbox": [ + 497, + 577, + 504, + 588 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 205, + 600 + ], + "score": 1.0, + "content": "and its smoothed version", + "type": "text" + }, + { + "bbox": [ + 206, + 588, + 217, + 600 + ], + "score": 0.89, + "content": "f _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 588, + 506, + 600 + ], + "score": 1.0, + "content": ". It has been shown in (Gao et al., 2014; Nesterov & Spokoiny, 2015) that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 263, + 611 + ], + "score": 1.0, + "content": "there exists a tight relationship between", + "type": "text" + }, + { + "bbox": [ + 263, + 599, + 275, + 610 + ], + "score": 0.89, + "content": "f _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 598, + 292, + 611 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 599, + 300, + 610 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "given the fact that the former is a convolution of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 333, + 622 + ], + "score": 1.0, + "content": "latter and the density function of a random perturbation", + "type": "text" + }, + { + "bbox": [ + 333, + 611, + 340, + 619 + ], + "score": 0.44, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "in (4). 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In the other extreme case of", + "type": "text" + }, + { + "bbox": [ + 369, + 516, + 392, + 526 + ], + "score": 0.9, + "content": "b = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 515, + 506, + 528 + ], + "score": 1.0, + "content": ", both the studied mini-batch", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 291, + 540 + ], + "score": 1.0, + "content": "schemes become identical, corresponding to", + "type": "text" + }, + { + "bbox": [ + 291, + 527, + 322, + 538 + ], + "score": 0.92, + "content": "\\alpha _ { b } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 526, + 342, + 540 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 342, + 527, + 372, + 538 + ], + "score": 0.93, + "content": "\\beta _ { b } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 526, + 506, + 540 + ], + "score": 1.0, + "content": ". Proposition 2 also implies that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 171, + 550 + ], + "score": 1.0, + "content": "the use of large", + "type": "text" + }, + { + "bbox": [ + 171, + 538, + 177, + 548 + ], + "score": 0.68, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 538, + 195, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 195, + 539, + 201, + 549 + ], + "score": 0.8, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "reduces the variance of the gradient estimate, and will further improve the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 550, + 178, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 178, + 561 + ], + "score": 1.0, + "content": "convergence rate.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 481, + 507, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "With the aid of Proposition 1 and 2, we can then show the convergence rate of ZO-signSGD in terms", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 577, + 504, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 256, + 589 + ], + "score": 1.0, + "content": "of stationarity of the original function", + "type": "text" + }, + { + "bbox": [ + 257, + 577, + 264, + 588 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 577, + 496, + 589 + ], + "score": 1.0, + "content": ". The remaining difficulty is how to bound the gap between", + "type": "text" + }, + { + "bbox": [ + 497, + 577, + 504, + 588 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 205, + 600 + ], + "score": 1.0, + "content": "and its smoothed version", + "type": "text" + }, + { + "bbox": [ + 206, + 588, + 217, + 600 + ], + "score": 0.89, + "content": "f _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 588, + 506, + 600 + ], + "score": 1.0, + "content": ". It has been shown in (Gao et al., 2014; Nesterov & Spokoiny, 2015) that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 263, + 611 + ], + "score": 1.0, + "content": "there exists a tight relationship between", + "type": "text" + }, + { + "bbox": [ + 263, + 599, + 275, + 610 + ], + "score": 0.89, + "content": "f _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 598, + 292, + 611 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 599, + 300, + 610 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "given the fact that the former is a convolution of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 333, + 622 + ], + "score": 1.0, + "content": "latter and the density function of a random perturbation", + "type": "text" + }, + { + "bbox": [ + 333, + 611, + 340, + 619 + ], + "score": 0.44, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "in (4). We demonstrate the convergence", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 620, + 249, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 249, + 632 + ], + "score": 1.0, + "content": "rate of ZO-signSGD in Theorem 1.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 565, + 506, + 632 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 639, + 504, + 666 + ], + "lines": [ + { + "bbox": [ + 102, + 635, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 102, + 635, + 214, + 659 + ], + "score": 1.0, + "content": "Theorem 1 Under A1 and", + "type": "text" + }, + { + "bbox": [ + 215, + 641, + 227, + 651 + ], + "score": 0.46, + "content": "A 2", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 635, + 309, + 659 + ], + "score": 1.0, + "content": ", if we randomly pick", + "type": "text" + }, + { + "bbox": [ + 310, + 642, + 324, + 652 + ], + "score": 0.77, + "content": "\\mathbf { x } _ { R }", + "type": 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We next obtain explicit dependence on these parameters by specifying", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 155, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 155, + 165 + ], + "score": 1.0, + "content": "Theorem 1.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 169, + 432, + 185 + ], + "lines": [ + { + "bbox": [ + 104, + 166, + 433, + 188 + ], + "spans": [ + { + "bbox": [ + 104, + 166, + 115, + 188 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 116, + 169, + 192, + 185 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\delta _ { k } = \\delta = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 166, + 210, + 188 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 210, + 169, + 265, + 185 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 166, + 433, + 188 + ], + "score": 1.0, + "content": ", then the convergence in (8) simplifies to", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 189, + 414, + 219 + ], + "lines": [ + { + "bbox": [ + 195, + 189, + 414, + 219 + ], + "spans": [ + { + "bbox": [ + 195, + 189, + 414, + 219 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left[ \\Vert \\nabla f ( \\mathbf { x } _ { R } ) \\Vert _ { 2 } \\right] \\leq O \\left( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { \\sqrt { d } \\sqrt { \\alpha _ { b } q + ( \\alpha _ { b } + \\beta _ { b } ) d } } { \\sqrt { b q } } \\right) ,", + "type": "interline_equation", + "image_path": "fe9bd1942e6e617fc5c3d518e014c47e389c97a383e2a966d618dbfbb36433dd.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 189, + 414, + 204.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 195, + 204.0, + 414, + 219.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 223, + 504, + 246 + ], + "lines": [ + { + "bbox": [ + 107, + 222, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 107, + 222, + 132, + 236 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 225, + 144, + 234 + ], + "score": 0.84, + "content": "\\alpha _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 222, + 162, + 236 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 224, + 173, + 235 + ], + "score": 0.89, + "content": "\\beta _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 222, + 270, + 236 + ], + "score": 1.0, + "content": "were defined in (7), and", + "type": "text" + }, + { + "bbox": [ + 270, + 223, + 348, + 235 + ], + "score": 0.92, + "content": "1 \\leq ( \\alpha _ { b } + \\beta _ { b } ) \\leq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 222, + 505, + 236 + ], + "score": 1.0, + "content": ". We provide several key insights on the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 234, + 293, + 247 + ], + "spans": [ + { + "bbox": [ + 107, + 234, + 293, + 247 + ], + "score": 1.0, + "content": "convergence rate of ZO-signSGD through (9).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 506, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 363, + 264 + ], + "score": 1.0, + "content": "First, the convergence rate of ZO-signSGD is measured through", + "type": "text" + }, + { + "bbox": [ + 363, + 251, + 412, + 263 + ], + "score": 0.93, + "content": "\\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "rather than its squared", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 155, + 275 + ], + "score": 1.0, + "content": "counterpart", + "type": "text" + }, + { + "bbox": [ + 156, + 262, + 205, + 274 + ], + "score": 0.93, + "content": "\\| \\nabla \\bar { f } ( \\mathbf { x } _ { R } ) \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 262, + 505, + 275 + ], + "score": 1.0, + "content": ", where the latter was used in measuring the convergence of ZO-SGD. We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 286 + ], + "score": 1.0, + "content": "recall from (Ghadimi & Lan, 2013, Theorem 3.2 & Corollary 3.3) that ZO-SGD yields the convergence", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 283, + 507, + 302 + ], + "spans": [ + { + "bbox": [ + 104, + 283, + 130, + 300 + ], + "score": 1.0, + "content": "error", + "type": "text" + }, + { + "bbox": [ + 131, + 284, + 245, + 302 + ], + "score": 0.92, + "content": "\\mathbb { E } \\left[ \\lVert \\nabla f ( \\mathbf { x } _ { R } ) \\rVert _ { 2 } ^ { 2 } \\right] \\leq O ( \\frac { \\sqrt { d } } { \\sqrt { T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 283, + 280, + 300 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 280, + 286, + 397, + 299 + ], + "score": 0.93, + "content": "\\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } ^ { 2 } \\leq \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 283, + 412, + 300 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 412, + 285, + 484, + 299 + ], + "score": 0.93, + "content": "\\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } ^ { 2 } \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 283, + 507, + 300 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "convergence of ZO-signSGD meets a stricter criterion than that of ZO-SGD. The possible downside", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 102, + 310, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 102, + 310, + 353, + 330 + ], + "score": 1.0, + "content": "of ZO-signSGD is that it suffers an additional error of order", + "type": "text" + }, + { + "bbox": [ + 353, + 311, + 411, + 329 + ], + "score": 0.94, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { b } } + \\frac { d } { \\sqrt { b q } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 310, + 505, + 330 + ], + "score": 1.0, + "content": "in the worst case. The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "aforementioned results imply that ZO-signSGD could only converge to a neighborhood of a stationary", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "point but with a fast convergence speed. 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We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 286 + ], + "score": 1.0, + "content": "recall from (Ghadimi & Lan, 2013, Theorem 3.2 & Corollary 3.3) that ZO-SGD yields the convergence", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 283, + 507, + 302 + ], + "spans": [ + { + "bbox": [ + 104, + 283, + 130, + 300 + ], + "score": 1.0, + "content": "error", + "type": "text" + }, + { + "bbox": [ + 131, + 284, + 245, + 302 + ], + "score": 0.92, + "content": "\\mathbb { E } \\left[ \\lVert \\nabla f ( \\mathbf { x } _ { R } ) \\rVert _ { 2 } ^ { 2 } \\right] \\leq O ( \\frac { \\sqrt { d } } { \\sqrt { T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 283, + 280, + 300 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 280, + 286, + 397, + 299 + ], + "score": 0.93, + "content": "\\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } ^ { 2 } \\leq \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 283, + 412, + 300 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 412, + 285, + 484, + 299 + ], + "score": 0.93, + "content": "\\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } ^ { 2 } \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 283, + 507, + 300 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "convergence of ZO-signSGD meets a stricter criterion than that of ZO-SGD. The possible downside", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 102, + 310, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 102, + 310, + 353, + 330 + ], + "score": 1.0, + "content": "of ZO-signSGD is that it suffers an additional error of order", + "type": "text" + }, + { + "bbox": [ + 353, + 311, + 411, + 329 + ], + "score": 0.94, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { b } } + \\frac { d } { \\sqrt { b q } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 310, + 505, + 330 + ], + "score": 1.0, + "content": "in the worst case. The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "aforementioned results imply that ZO-signSGD could only converge to a neighborhood of a stationary", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "point but with a fast convergence speed. Here the size of the neighborhood is controlled by the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 364, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 170, + 362 + ], + "score": 1.0, + "content": "mini-batch size", + "type": "text" + }, + { + "bbox": [ + 170, + 349, + 176, + 358 + ], + "score": 0.69, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 347, + 354, + 362 + ], + "score": 1.0, + "content": "and the number of random direction vectors", + "type": "text" + }, + { + "bbox": [ + 354, + 351, + 360, + 360 + ], + "score": 0.75, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 347, + 364, + 362 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 102, + 250, + 507, + 362 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 506, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "Also, our convergence analysis applies to mini-batch sampling both with and without replacement.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 102, + 374, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 102, + 374, + 307, + 396 + ], + "score": 1.0, + "content": "When b ∈ [1, n), ZO-signSGD achieves O( √dT", + "type": "text" + }, + { + "bbox": [ + 275, + 376, + 361, + 394 + ], + "score": 0.94, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { \\sqrt { d } } { \\sqrt { b } } + \\frac { d } { \\sqrt { b q } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "convergence rate regardless of the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 270, + 405 + ], + "score": 1.0, + "content": "choice of mini-batch sampling. When", + "type": "text" + }, + { + "bbox": [ + 271, + 393, + 299, + 403 + ], + "score": 0.87, + "content": "b = n", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 392, + 505, + 405 + ], + "score": 1.0, + "content": ", it is known from (9) that the use of mini-batch", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 102, + 403, + 508, + 423 + ], + "spans": [ + { + "bbox": [ + 102, + 404, + 392, + 423 + ], + "score": 1.0, + "content": "without replacement recovers ZO-signGD, yielding the convergence rate", + "type": "text" + }, + { + "bbox": [ + 392, + 403, + 452, + 421 + ], + "score": 0.93, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { d } { \\sqrt { n q } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 404, + 508, + 423 + ], + "score": 1.0, + "content": ". By contrast,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 102, + 420, + 504, + 440 + ], + "spans": [ + { + "bbox": [ + 102, + 420, + 415, + 440 + ], + "score": 1.0, + "content": "the use of mini-batch with replacement leads to the worse convergence rate", + "type": "text" + }, + { + "bbox": [ + 415, + 421, + 502, + 439 + ], + "score": 0.93, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { d } { \\sqrt { n q } } + \\frac { \\sqrt { d } } { \\sqrt { n } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 421, + 504, + 433 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 153, + 450 + ], + "score": 1.0, + "content": "Clearly, as", + "type": "text" + }, + { + "bbox": [ + 154, + 437, + 181, + 447 + ], + "score": 0.89, + "content": "b = n", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 436, + 200, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 437, + 231, + 448 + ], + "score": 0.89, + "content": "n < T", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 436, + 506, + 450 + ], + "score": 1.0, + "content": ", ZO-signSGD using mini-batch with replacement fails to achieve", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 448, + 507, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 139, + 466 + ], + "score": 1.0, + "content": "the rate", + "type": "text" + }, + { + "bbox": [ + 139, + 448, + 171, + 466 + ], + "score": 0.94, + "content": "O ( \\frac { \\sqrt { d } } { \\sqrt { T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 448, + 281, + 466 + ], + "score": 1.0, + "content": "regardless of the choice of", + "type": "text" + }, + { + "bbox": [ + 282, + 453, + 288, + 463 + ], + "score": 0.75, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 448, + 507, + 466 + ], + "score": 1.0, + "content": ". 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When", + "type": "text" + }, + { + "bbox": [ + 327, + 468, + 356, + 479 + ], + "score": 0.89, + "content": "b > n", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 463, + 508, + 484 + ], + "score": 1.0, + "content": ", ZO-signSGD is restricted to using", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 481, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 492 + ], + "score": 1.0, + "content": "mini-batch sampling with replacement. Similar to signSGD (Bernstein et al., 2018), we can obtain", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 492, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 107, + 492, + 138, + 510 + ], + "score": 0.93, + "content": "O ( \\frac { \\sqrt { d } } { \\sqrt { T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 492, + 222, + 509 + ], + "score": 1.0, + "content": "convergence rate as", + "type": "text" + }, + { + "bbox": [ + 222, + 494, + 264, + 507 + ], + "score": 0.93, + "content": "b = O ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 492, + 283, + 509 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 283, + 493, + 330, + 508 + ], + "score": 0.95, + "content": "\\begin{array} { r } { q = O ( \\frac { d T } { n } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 492, + 441, + 509 + ], + "score": 1.0, + "content": ", where the dependence on", + "type": "text" + }, + { + "bbox": [ + 441, + 497, + 447, + 506 + ], + "score": 0.78, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 492, + 506, + 509 + ], + "score": 1.0, + "content": "is induced by", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 245, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 245, + 521 + ], + "score": 1.0, + "content": "the use of ZO gradient estimation.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26, + "bbox_fs": [ + 102, + 364, + 508, + 521 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 527, + 271, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 272, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 272, + 543 + ], + "score": 1.0, + "content": "5 VARIANTS OF ZO-SIGNSGD", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "score": 1.0, + "content": "Here we study three variants of ZO-signSGD, where the gradient will be estimated using a) the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "central difference of function values, b) the sign of ZO gradient estimates with majority vote, or c)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 569, + 466, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 466, + 581 + ], + "score": 1.0, + "content": "the sign of ZO gradient estimates with majority vote for distributed optimization. That is,", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 548, + 506, + 581 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 580, + 446, + 681 + ], + "lines": [ + { + "bbox": [ + 163, + 580, + 446, + 681 + ], + "spans": [ + { + "bbox": [ + 163, + 580, + 446, + 681 + ], + "score": 0.95, + "content": "\\begin{array} { l } { { \\displaystyle a \\mathrm { ) ~ G r a d E s t i m a t e ( \\mathbf { x } ) } = \\frac { 1 } { b q } \\sum _ { i \\in \\mathbb { Z } _ { k } } \\sum _ { j = 1 } ^ { q } \\frac { d \\left[ f _ { i } \\left( \\mathbf { x } + \\mu \\mathbf { u } _ { i , j } \\right) - f _ { i } \\left( \\mathbf { x } - \\mu \\mathbf { u } _ { i , j } \\right) \\right] \\mathbf { u } _ { i , j } } { 2 \\mu } } \\ ~ } \\\\ { { \\displaystyle b \\mathrm { ) ~ G r a d E s t i m a t e ( \\mathbf { x } ) } = \\frac { 1 } { b q } \\sum _ { i \\in \\mathbb { Z } _ { k } } \\sum _ { j = 1 } ^ { q } \\mathrm { s i g n } \\left( \\hat { \\nabla } f _ { i } \\left( \\mathbf { x } ; \\mathbf { u } _ { i , j } \\right) \\right) , } \\ ~ } \\\\ { { \\displaystyle c \\mathrm { ) ~ G r a d E s t i m a t e ( \\mathbf { x } ) } = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\mathrm { s i g n } \\left( \\frac { 1 } { b _ { m } q } \\sum _ { i \\in \\mathbb { Z } _ { m , k } } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } \\left( \\mathbf { x } ; \\mathbf { u } _ { i , j } \\right) \\right) , } \\ ~ } \\end{array}", + "type": "interline_equation", + "image_path": "dde518c17f209880984c162170618df28fa79730e04094ff4cee7b39d0553009.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 163, + 580, + 446, + 613.6666666666666 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 163, + 613.6666666666666, + 446, + 647.3333333333333 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 163, + 647.3333333333333, + 446, + 680.9999999999999 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 685, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 684, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 133, + 698 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 685, + 160, + 698 + ], + "score": 0.93, + "content": "\\{ \\mathbf { u } _ { i , j } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 685, + 178, + 698 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 178, + 684, + 229, + 698 + ], + "score": 0.93, + "content": "\\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 685, + 506, + 698 + ], + "score": 1.0, + "content": "have been defined in (3). 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Compared to the form of forward", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 308, + 161 + ], + "score": 1.0, + "content": "difference (3), the central difference (10) requires", + "type": "text" + }, + { + "bbox": [ + 308, + 148, + 343, + 160 + ], + "score": 0.93, + "content": "b ( q - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 147, + 506, + 161 + ], + "score": 1.0, + "content": "times more function queries in gradient", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "score": 1.0, + "content": "estimation. At the cost of more function queries, one may wonder if the convergence rate of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 170, + 262, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 262, + 182 + ], + "score": 1.0, + "content": "ZO-signSGD can be further improved.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 190, + 504, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "score": 1.0, + "content": "Corollary 1 Suppose that the conditions in Theorem 1 hold, ZO-signSGD with gradient estimator", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 203, + 432, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 432, + 214 + ], + "score": 1.0, + "content": "(10) yields the same convergence rate of ZO-signSGD that uses the estimator (3).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "Proof: Recall that Proposition 1 is independent of specific forms of gradient estimators, and thus", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "score": 1.0, + "content": "holds for (10). Although Proposition 2 relies on the second-order moments of each gradient estimator,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 246, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 506, + 258 + ], + "score": 1.0, + "content": "we prove that under A1 and A2, both (3) and (10) maintain the same statistical properties. As a result,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 256, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 426, + 268 + ], + "score": 1.0, + "content": "Proposition 2 and Theorem 1 also hold for (10); see more details in Appendix 5.", + "type": "text" + }, + { + "bbox": [ + 493, + 256, + 506, + 267 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "score": 1.0, + "content": "We next study the gradient estimator (11), whose sign is equivalent to the majority vote (i.e., the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 284, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 369, + 298 + ], + "score": 1.0, + "content": "element-wise median) of signs of individual gradient estimates", + "type": "text" + }, + { + "bbox": [ + 370, + 284, + 430, + 298 + ], + "score": 0.94, + "content": "\\{ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 285, + 505, + 298 + ], + "score": 1.0, + "content": ". It was shown in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "(Bernstein et al., 2018) that signSGD with majority vote has a better convergence rate under additional", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "score": 1.0, + "content": "assumptions of unimodal symmetric noise distribution of coordinate-wise gradient estimates. In", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 318, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 333 + ], + "score": 1.0, + "content": "Corollary 2, we show that such a speed-up in convergence can also be achieved by ZO-signSGD with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 326, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 326, + 342 + ], + "score": 1.0, + "content": "majority vote, which we refer to as ‘ZO-M-signSGD’.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 105, + 347, + 504, + 374 + ], + "lines": [ + { + "bbox": [ + 107, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 107, + 348, + 317, + 360 + ], + "score": 1.0, + "content": "Corollary 2 Suppose that the conditions in Theorem", + "type": "text" + }, + { + "bbox": [ + 318, + 349, + 324, + 358 + ], + "score": 0.3, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "hold, and the distribution of gradient noise is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 358, + 482, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 321, + 376 + ], + "score": 1.0, + "content": "unimodal and symmetric. Then, ZO-M-signSGD with", + "type": "text" + }, + { + "bbox": [ + 322, + 359, + 379, + 375 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\delta _ { k } = { O } \\left( \\frac { 1 } { \\sqrt { d T } } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 358, + 398, + 376 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 399, + 359, + 453, + 375 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 358, + 482, + 376 + ], + "score": 1.0, + "content": "yields", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 380, + 389, + 399 + ], + "lines": [ + { + "bbox": [ + 222, + 380, + 389, + 399 + ], + "spans": [ + { + "bbox": [ + 222, + 380, + 389, + 399 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] = O \\left( \\sqrt { d } / \\sqrt { T } + d / \\sqrt { b q } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "8575e4aabac7d2e123d0c00a53b2c9c3db6f8fadd43c6d73f0fcb25132162306.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 222, + 380, + 389, + 399 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 203, + 416 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 204, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 204, + 418 + ], + "score": 1.0, + "content": "Proof: See Appendix 6.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 506, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "score": 1.0, + "content": "We recall from Theorem 1 that under the same parameter setting of Corollary 2, ZO-signSGD", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 432, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 133, + 450 + ], + "score": 1.0, + "content": "yields", + "type": "text" + }, + { + "bbox": [ + 133, + 433, + 219, + 451 + ], + "score": 0.94, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { \\sqrt { d } } { \\sqrt { b } } + \\frac { d } { \\sqrt { b q } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "convergence rate in the worst case. It is clear from (13) that the error", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 450, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 222, + 466 + ], + "score": 1.0, + "content": "correction term of order √d", + "type": "text" + }, + { + "bbox": [ + 219, + 452, + 505, + 467 + ], + "score": 1.0, + "content": "is eliminated in ZO-M-signSGD. Such an improvement in convergence", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "is achieved under the condition of unimodal symmetric gradient noise. We remark that different from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "score": 1.0, + "content": "the stochastic gradient noise studied in (Bernstein et al., 2018), the ZO gradient estimation noise", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 449, + 500 + ], + "score": 1.0, + "content": "could violate this assumption. For example, in a scalar case, if the gradient estimate", + "type": "text" + }, + { + "bbox": [ + 450, + 490, + 456, + 499 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "follows the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 183, + 511 + ], + "score": 1.0, + "content": "distribution where", + "type": "text" + }, + { + "bbox": [ + 184, + 499, + 210, + 510 + ], + "score": 0.91, + "content": "g = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 498, + 299, + 511 + ], + "score": 1.0, + "content": "with probability 0.9,", + "type": "text" + }, + { + "bbox": [ + 299, + 499, + 339, + 510 + ], + "score": 0.86, + "content": "g = - 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 498, + 447, + 511 + ], + "score": 1.0, + "content": "with probability 0.1, then", + "type": "text" + }, + { + "bbox": [ + 447, + 499, + 486, + 511 + ], + "score": 0.93, + "content": "\\mathbb { E } [ g ] < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 107, + 510, + 169, + 522 + ], + "score": 0.91, + "content": "\\mathrm { s i g n } ( \\mathbb { E } [ g ] ) < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 509, + 215, + 523 + ], + "score": 1.0, + "content": ". However,", + "type": "text" + }, + { + "bbox": [ + 215, + 510, + 277, + 522 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\mathrm { s i g n } ( g ) ] > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 509, + 506, + 523 + ], + "score": 1.0, + "content": ". This implies that without the assumption of symmetry,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 302, + 534 + ], + "score": 1.0, + "content": "the sign of gradient estimates with majority vote", + "type": "text" + }, + { + "bbox": [ + 302, + 520, + 349, + 532 + ], + "score": 0.9, + "content": "( { \\mathbb E } [ \\mathrm { s i g n } ( g ) ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "can be in the opposite direction of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 216, + 545 + ], + "score": 1.0, + "content": "sign of averaged gradients", + "type": "text" + }, + { + "bbox": [ + 217, + 531, + 265, + 543 + ], + "score": 0.89, + "content": "( \\mathrm { s i g n } ( \\mathbb { E } [ g ] ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 531, + 506, + 545 + ], + "score": 1.0, + "content": ". Our results in the next section show that ZO-M-signSGD", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 543, + 246, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 246, + 554 + ], + "score": 1.0, + "content": "may not outperform ZO-signSGD.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 559, + 506, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 504, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 492, + 572 + ], + "score": 1.0, + "content": "Lastly, we focus on the gradient estimator (12), whose sign can be interpreted as the major vote of", + "type": "text" + }, + { + "bbox": [ + 492, + 560, + 504, + 570 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "distributed agents about the sign of the true gradient (Bernstein et al., 2018). The resulting variant", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 580, + 507, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 507, + 594 + ], + "score": 1.0, + "content": "of ZO-signSGD is called ‘ZO-D-signSGD’, and its convergence rate is illustrated in Corollary 3.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "Compared to ZO-M-signSGD for centralized optimization, ZO-D-signSGD suffers an extra error", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 601, + 507, + 621 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 172, + 620 + ], + "score": 1.0, + "content": "correction term", + "type": "text" + }, + { + "bbox": [ + 172, + 603, + 203, + 621 + ], + "score": 0.94, + "content": "O ( \\frac { \\sqrt { d } } { \\sqrt { n } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 601, + 453, + 620 + ], + "score": 1.0, + "content": "in the distributed setting. It is also worth mentioning that if", + "type": "text" + }, + { + "bbox": [ + 453, + 606, + 486, + 616 + ], + "score": 0.9, + "content": "M = n", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 601, + 507, + 620 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 107, + 620, + 131, + 632 + ], + "score": 0.89, + "content": "q = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 619, + 343, + 633 + ], + "score": 1.0, + "content": ", then the gradient estimator (12) reduces to (11) with", + "type": "text" + }, + { + "bbox": [ + 343, + 619, + 380, + 632 + ], + "score": 0.92, + "content": "\\mathcal { T } _ { k } = [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 619, + 506, + 633 + ], + "score": 1.0, + "content": ". In this case, Corollary 2 and 3", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 102, + 629, + 331, + 653 + ], + "spans": [ + { + "bbox": [ + 102, + 629, + 193, + 653 + ], + "score": 1.0, + "content": "reach a consensus on", + "type": "text" + }, + { + "bbox": [ + 193, + 631, + 250, + 649 + ], + "score": 0.94, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { d } { \\sqrt { n } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 629, + 331, + 653 + ], + "score": 1.0, + "content": "convergence error.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 105, + 659, + 504, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 502, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 454, + 676 + ], + "score": 1.0, + "content": "Corollary 3 Suppose that the conditions in Corollary 2 hold. ZO-M-signSGD with", + "type": "text" + }, + { + "bbox": [ + 454, + 660, + 502, + 673 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\boldsymbol { b } _ { m } = \\left\\lfloor \\frac { n } { M } \\right\\rfloor } \\end{array}", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 671, + 267, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 672, + 165, + 688 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\delta _ { k } = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 671, + 183, + 689 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 184, + 672, + 238, + 688 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 671, + 267, + 689 + ], + "score": 1.0, + "content": "yields", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 693, + 410, + 713 + ], + "lines": [ + { + "bbox": [ + 200, + 693, + 410, + 713 + ], + "spans": [ + { + "bbox": [ + 200, + 693, + 410, + 713 + ], + "score": 0.92, + "content": "\\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] = O \\left( \\sqrt { d } / \\sqrt { T } + \\sqrt { d } / \\sqrt { n } + d / \\sqrt { n q } \\right) .", + "type": "interline_equation", + "image_path": "83df8149263c7976b827f3ed2c450d7f8a03ee5433076a020efa044a4d4d2bd4.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 200, + 693, + 410, + 713 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 725, + 204, + 737 + ], + "lines": [ + { + "bbox": [ + 105, + 723, + 204, + 739 + ], + "spans": [ + { + "bbox": [ + 105, + 723, + 204, + 739 + ], + "score": 1.0, + "content": "Proof: See Appendix 7.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 726, + 504, + 736 + ], + "lines": [ + { + "bbox": [ + 496, + 727, + 504, + 737 + ], + "spans": [ + { + "bbox": [ + 496, + 727, + 504, + 737 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 405, + 504, + 415 + ], + "lines": [ + { + "bbox": [ + 496, + 407, + 504, + 416 + ], + "spans": [ + { + "bbox": [ + 496, + 407, + 504, + 416 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 80, + 505, + 110 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 104, + 77, + 509, + 111 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 125, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "The ZO gradient estimator (10) was used in (Shamir, 2017) for bandit convex optimization and in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "(Ilyas et al., 2018a) for designing black-box adversarial attacks. Compared to the form of forward", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 308, + 161 + ], + "score": 1.0, + "content": "difference (3), the central difference (10) requires", + "type": "text" + }, + { + "bbox": [ + 308, + 148, + 343, + 160 + ], + "score": 0.93, + "content": "b ( q - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 147, + 506, + 161 + ], + "score": 1.0, + "content": "times more function queries in gradient", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "score": 1.0, + "content": "estimation. At the cost of more function queries, one may wonder if the convergence rate of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 170, + 262, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 262, + 182 + ], + "score": 1.0, + "content": "ZO-signSGD can be further improved.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 126, + 506, + 182 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 190, + 504, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "score": 1.0, + "content": "Corollary 1 Suppose that the conditions in Theorem 1 hold, ZO-signSGD with gradient estimator", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 203, + 432, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 432, + 214 + ], + "score": 1.0, + "content": "(10) yields the same convergence rate of ZO-signSGD that uses the estimator (3).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 190, + 505, + 214 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "Proof: Recall that Proposition 1 is independent of specific forms of gradient estimators, and thus", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "score": 1.0, + "content": "holds for (10). Although Proposition 2 relies on the second-order moments of each gradient estimator,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 246, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 506, + 258 + ], + "score": 1.0, + "content": "we prove that under A1 and A2, both (3) and (10) maintain the same statistical properties. As a result,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 256, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 426, + 268 + ], + "score": 1.0, + "content": "Proposition 2 and Theorem 1 also hold for (10); see more details in Appendix 5.", + "type": "text" + }, + { + "bbox": [ + 493, + 256, + 506, + 267 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 223, + 506, + 268 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "score": 1.0, + "content": "We next study the gradient estimator (11), whose sign is equivalent to the majority vote (i.e., the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 284, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 369, + 298 + ], + "score": 1.0, + "content": "element-wise median) of signs of individual gradient estimates", + "type": "text" + }, + { + "bbox": [ + 370, + 284, + 430, + 298 + ], + "score": 0.94, + "content": "\\{ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 285, + 505, + 298 + ], + "score": 1.0, + "content": ". It was shown in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "(Bernstein et al., 2018) that signSGD with majority vote has a better convergence rate under additional", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "score": 1.0, + "content": "assumptions of unimodal symmetric noise distribution of coordinate-wise gradient estimates. In", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 318, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 333 + ], + "score": 1.0, + "content": "Corollary 2, we show that such a speed-up in convergence can also be achieved by ZO-signSGD with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 326, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 326, + 342 + ], + "score": 1.0, + "content": "majority vote, which we refer to as ‘ZO-M-signSGD’.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 272, + 506, + 342 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 347, + 504, + 374 + ], + "lines": [ + { + "bbox": [ + 107, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 107, + 348, + 317, + 360 + ], + "score": 1.0, + "content": "Corollary 2 Suppose that the conditions in Theorem", + "type": "text" + }, + { + "bbox": [ + 318, + 349, + 324, + 358 + ], + "score": 0.3, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "hold, and the distribution of gradient noise is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 358, + 482, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 321, + 376 + ], + "score": 1.0, + "content": "unimodal and symmetric. Then, ZO-M-signSGD with", + "type": "text" + }, + { + "bbox": [ + 322, + 359, + 379, + 375 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\delta _ { k } = { O } \\left( \\frac { 1 } { \\sqrt { d T } } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 358, + 398, + 376 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 399, + 359, + 453, + 375 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 358, + 482, + 376 + ], + "score": 1.0, + "content": "yields", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 348, + 505, + 376 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 380, + 389, + 399 + ], + "lines": [ + { + "bbox": [ + 222, + 380, + 389, + 399 + ], + "spans": [ + { + "bbox": [ + 222, + 380, + 389, + 399 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] = O \\left( \\sqrt { d } / \\sqrt { T } + d / \\sqrt { b q } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "8575e4aabac7d2e123d0c00a53b2c9c3db6f8fadd43c6d73f0fcb25132162306.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 222, + 380, + 389, + 399 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 203, + 416 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 204, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 204, + 418 + ], + "score": 1.0, + "content": "Proof: See Appendix 6.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 404, + 204, + 418 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 506, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "score": 1.0, + "content": "We recall from Theorem 1 that under the same parameter setting of Corollary 2, ZO-signSGD", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 432, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 133, + 450 + ], + "score": 1.0, + "content": "yields", + "type": "text" + }, + { + "bbox": [ + 133, + 433, + 219, + 451 + ], + "score": 0.94, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { \\sqrt { d } } { \\sqrt { b } } + \\frac { d } { \\sqrt { b q } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "convergence rate in the worst case. It is clear from (13) that the error", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 450, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 222, + 466 + ], + "score": 1.0, + "content": "correction term of order √d", + "type": "text" + }, + { + "bbox": [ + 219, + 452, + 505, + 467 + ], + "score": 1.0, + "content": "is eliminated in ZO-M-signSGD. Such an improvement in convergence", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "is achieved under the condition of unimodal symmetric gradient noise. We remark that different from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "score": 1.0, + "content": "the stochastic gradient noise studied in (Bernstein et al., 2018), the ZO gradient estimation noise", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 449, + 500 + ], + "score": 1.0, + "content": "could violate this assumption. For example, in a scalar case, if the gradient estimate", + "type": "text" + }, + { + "bbox": [ + 450, + 490, + 456, + 499 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "follows the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 183, + 511 + ], + "score": 1.0, + "content": "distribution where", + "type": "text" + }, + { + "bbox": [ + 184, + 499, + 210, + 510 + ], + "score": 0.91, + "content": "g = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 498, + 299, + 511 + ], + "score": 1.0, + "content": "with probability 0.9,", + "type": "text" + }, + { + "bbox": [ + 299, + 499, + 339, + 510 + ], + "score": 0.86, + "content": "g = - 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 498, + 447, + 511 + ], + "score": 1.0, + "content": "with probability 0.1, then", + "type": "text" + }, + { + "bbox": [ + 447, + 499, + 486, + 511 + ], + "score": 0.93, + "content": "\\mathbb { E } [ g ] < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 107, + 510, + 169, + 522 + ], + "score": 0.91, + "content": "\\mathrm { s i g n } ( \\mathbb { E } [ g ] ) < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 509, + 215, + 523 + ], + "score": 1.0, + "content": ". However,", + "type": "text" + }, + { + "bbox": [ + 215, + 510, + 277, + 522 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\mathrm { s i g n } ( g ) ] > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 509, + 506, + 523 + ], + "score": 1.0, + "content": ". This implies that without the assumption of symmetry,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 302, + 534 + ], + "score": 1.0, + "content": "the sign of gradient estimates with majority vote", + "type": "text" + }, + { + "bbox": [ + 302, + 520, + 349, + 532 + ], + "score": 0.9, + "content": "( { \\mathbb E } [ \\mathrm { s i g n } ( g ) ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "can be in the opposite direction of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 216, + 545 + ], + "score": 1.0, + "content": "sign of averaged gradients", + "type": "text" + }, + { + "bbox": [ + 217, + 531, + 265, + 543 + ], + "score": 0.89, + "content": "( \\mathrm { s i g n } ( \\mathbb { E } [ g ] ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 531, + 506, + 545 + ], + "score": 1.0, + "content": ". Our results in the next section show that ZO-M-signSGD", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 543, + 246, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 246, + 554 + ], + "score": 1.0, + "content": "may not outperform ZO-signSGD.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 420, + 506, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 559, + 506, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 504, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 492, + 572 + ], + "score": 1.0, + "content": "Lastly, we focus on the gradient estimator (12), whose sign can be interpreted as the major vote of", + "type": "text" + }, + { + "bbox": [ + 492, + 560, + 504, + 570 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "distributed agents about the sign of the true gradient (Bernstein et al., 2018). The resulting variant", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 580, + 507, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 507, + 594 + ], + "score": 1.0, + "content": "of ZO-signSGD is called ‘ZO-D-signSGD’, and its convergence rate is illustrated in Corollary 3.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "Compared to ZO-M-signSGD for centralized optimization, ZO-D-signSGD suffers an extra error", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 601, + 507, + 621 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 172, + 620 + ], + "score": 1.0, + "content": "correction term", + "type": "text" + }, + { + "bbox": [ + 172, + 603, + 203, + 621 + ], + "score": 0.94, + "content": "O ( \\frac { \\sqrt { d } } { \\sqrt { n } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 601, + 453, + 620 + ], + "score": 1.0, + "content": "in the distributed setting. It is also worth mentioning that if", + "type": "text" + }, + { + "bbox": [ + 453, + 606, + 486, + 616 + ], + "score": 0.9, + "content": "M = n", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 601, + 507, + 620 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 107, + 620, + 131, + 632 + ], + "score": 0.89, + "content": "q = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 619, + 343, + 633 + ], + "score": 1.0, + "content": ", then the gradient estimator (12) reduces to (11) with", + "type": "text" + }, + { + "bbox": [ + 343, + 619, + 380, + 632 + ], + "score": 0.92, + "content": "\\mathcal { T } _ { k } = [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 619, + 506, + 633 + ], + "score": 1.0, + "content": ". In this case, Corollary 2 and 3", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 102, + 629, + 331, + 653 + ], + "spans": [ + { + "bbox": [ + 102, + 629, + 193, + 653 + ], + "score": 1.0, + "content": "reach a consensus on", + "type": "text" + }, + { + "bbox": [ + 193, + 631, + 250, + 649 + ], + "score": 0.94, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { d } { \\sqrt { n } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 629, + 331, + 653 + ], + "score": 1.0, + "content": "convergence error.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37, + "bbox_fs": [ + 102, + 559, + 507, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 659, + 504, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 502, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 454, + 676 + ], + "score": 1.0, + "content": "Corollary 3 Suppose that the conditions in Corollary 2 hold. ZO-M-signSGD with", + "type": "text" + }, + { + "bbox": [ + 454, + 660, + 502, + 673 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\boldsymbol { b } _ { m } = \\left\\lfloor \\frac { n } { M } \\right\\rfloor } \\end{array}", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 671, + 267, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 672, + 165, + 688 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\delta _ { k } = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 671, + 183, + 689 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 184, + 672, + 238, + 688 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 671, + 267, + 689 + ], + "score": 1.0, + "content": "yields", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 657, + 502, + 689 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 693, + 410, + 713 + ], + "lines": [ + { + "bbox": [ + 200, + 693, + 410, + 713 + ], + "spans": [ + { + "bbox": [ + 200, + 693, + 410, + 713 + ], + "score": 0.92, + "content": "\\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] = O \\left( \\sqrt { d } / \\sqrt { T } + \\sqrt { d } / \\sqrt { n } + d / \\sqrt { n q } \\right) .", + "type": "interline_equation", + "image_path": "83df8149263c7976b827f3ed2c450d7f8a03ee5433076a020efa044a4d4d2bd4.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 200, + 693, + 410, + 713 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 725, + 204, + 737 + ], + "lines": [ + { + "bbox": [ + 105, + 723, + 204, + 739 + ], + "spans": [ + { + "bbox": [ + 105, + 723, + 204, + 739 + ], + "score": 1.0, + "content": "Proof: See Appendix 7.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 723, + 204, + 739 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 123, + 88, + 491, + 304 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 123, + 88, + 491, + 304 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 123, + 88, + 491, + 304 + ], + "spans": [ + { + "bbox": [ + 123, + 88, + 491, + 304 + ], + "score": 0.978, + "type": "image", + "image_path": "337bfc286984c7305cf44de134c83cbe2f9274c987c25327a863c940c6c1ba76.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 123, + 88, + 491, + 160.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 123, + 160.0, + 491, + 232.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 123, + 232.0, + 491, + 304.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 307, + 506, + 378 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "Figure 1: Performance comparison of ZO-signSGD, ZO-M-signSGD, ZO-SGD, ZO-SCD, signSGD and SGD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "under a synthetic dataset. The solid line represents the loss/accuracy averaged over 10 independent trials with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "random initialization, and the shaded region indicates the standard deviation of results over random trials. (a)-(b):", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 336, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 412, + 350 + ], + "score": 1.0, + "content": "Training loss and test accuracy versus iterations. (c)-(d): Effects of mini-batch size", + "type": "text" + }, + { + "bbox": [ + 412, + 339, + 418, + 348 + ], + "score": 0.78, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 336, + 506, + 350 + ], + "score": 1.0, + "content": "and number of random", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 347, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 167, + 359 + ], + "score": 1.0, + "content": "direction vectors", + "type": "text" + }, + { + "bbox": [ + 168, + 349, + 173, + 358 + ], + "score": 0.77, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 347, + 506, + 359 + ], + "score": 1.0, + "content": "on the convergence of studied algorithms. Here (c) presents the training loss versus iterations,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 324, + 370 + ], + "score": 1.0, + "content": "and (d) is the heat map of the final loss for different values of", + "type": "text" + }, + { + "bbox": [ + 324, + 358, + 329, + 366 + ], + "score": 0.64, + "content": "^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 356, + 344, + 370 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 345, + 358, + 350, + 368 + ], + "score": 0.72, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 356, + 464, + 370 + ], + "score": 1.0, + "content": ". (e)-(f): Effects of problem size", + "type": "text" + }, + { + "bbox": [ + 465, + 358, + 470, + 366 + ], + "score": 0.6, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 356, + 506, + 370 + ], + "score": 1.0, + "content": ". Here (e)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 367, + 481, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 234, + 379 + ], + "score": 1.0, + "content": "shows the final training loss versus", + "type": "text" + }, + { + "bbox": [ + 234, + 368, + 240, + 376 + ], + "score": 0.73, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 367, + 419, + 379 + ], + "score": 1.0, + "content": ", and (f) presents the convergence trajectory when", + "type": "text" + }, + { + "bbox": [ + 420, + 367, + 477, + 378 + ], + "score": 0.94, + "content": "d \\in \\{ 2 0 0 , 4 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 367, + 481, + 379 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 388, + 200, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 201, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 201, + 402 + ], + "score": 1.0, + "content": "6 EXPERIMENTS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 505, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "In this section, we empirically show the effectiveness of ZO-signSGD, and validate its convergence", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 418, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 504, + 429 + ], + "score": 1.0, + "content": "behavior on both synthetic and real-world datasets such as MNIST and CIFAR-10. For the synthetic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "experiment, we study the problem of binary classification in the least squared formulation. For the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 440, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 451 + ], + "score": 1.0, + "content": "real-world application, we design adversarial examples from black-box neural networks as mentioned", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "in Sec. 2. Throughout this section, we compare ZO-signSGD and its variants with SGD, signSGD", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 461, + 473, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 473, + 473 + ], + "score": 1.0, + "content": "(Bernstein et al., 2018), ZO-SGD (Ghadimi & Lan, 2013), and ZO-SCD (Lian et al., 2016).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 506, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "Binary classification We consider the least squared problem with a nonconvex loss function (Xu", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 489, + 508, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 233, + 506 + ], + "score": 1.0, + "content": "et al., 2017; Liu et al., 2018b)", + "type": "text" + }, + { + "bbox": [ + 233, + 489, + 399, + 505 + ], + "score": 0.82, + "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { \\mathbf { x } \\in \\mathbb { R } ^ { d } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } ( y _ { i } - 1 / ( 1 + e ^ { - \\mathbf { a } _ { i } ^ { T } \\mathbf { x } } ) ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 489, + 508, + 506 + ], + "score": 1.0, + "content": ", which satisfies Assump-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 182, + 516 + ], + "score": 1.0, + "content": "tion A2 by letting", + "type": "text" + }, + { + "bbox": [ + 183, + 502, + 266, + 514 + ], + "score": 0.92, + "content": "\\sigma = \\operatorname* { m a x } _ { i } \\{ 2 \\lVert \\mathbf { a } _ { i } \\rVert _ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 502, + 506, + 516 + ], + "score": 1.0, + "content": ". Here instead of using the conventional cost function of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "logistic regression (a convex function), the considered least squared formulation is introduced to align", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "with our nonconvex theoretical analysis. For generating the synthetic dataset, we randomly draw", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 535, + 507, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 537, + 142, + 550 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 142, + 537, + 162, + 550 + ], + "score": 0.92, + "content": "\\left\\{ \\mathbf { a } _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 537, + 186, + 550 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 186, + 537, + 219, + 550 + ], + "score": 0.91, + "content": "\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 537, + 306, + 550 + ], + "score": 1.0, + "content": ", and obtain the label", + "type": "text" + }, + { + "bbox": [ + 307, + 537, + 336, + 549 + ], + "score": 0.9, + "content": "y _ { i } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 537, + 346, + 550 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 347, + 535, + 435, + 549 + ], + "score": 0.92, + "content": "1 / \\dot { ( 1 + e ^ { - \\mathbf { a } _ { i } ^ { T } \\mathbf { x } } ) } > 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 537, + 507, + 550 + ], + "score": 1.0, + "content": "and 0 otherwise.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 234, + 561 + ], + "score": 1.0, + "content": "The number of training samples", + "type": "text" + }, + { + "bbox": [ + 234, + 549, + 267, + 560 + ], + "score": 0.93, + "content": "\\left\\{ \\mathbf { a } _ { i } , y _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 547, + 302, + 561 + ], + "score": 1.0, + "content": "is set by", + "type": "text" + }, + { + "bbox": [ + 303, + 549, + 343, + 559 + ], + "score": 0.85, + "content": "n = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "against 200 testing samples. We find the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 367, + 573 + ], + "score": 1.0, + "content": "best constant learning rate for algorithms via a greedy search over", + "type": "text" + }, + { + "bbox": [ + 367, + 560, + 431, + 571 + ], + "score": 0.87, + "content": "\\eta \\in [ 0 . 0 0 1 , 0 . 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "(see Appendix 8.1", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 571, + 507, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 336, + 585 + ], + "score": 1.0, + "content": "for more details), and we choose the smoothing parameter", + "type": "text" + }, + { + "bbox": [ + 337, + 571, + 393, + 584 + ], + "score": 0.93, + "content": "\\mu = 1 0 / \\sqrt { T d }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 571, + 507, + 585 + ], + "score": 1.0, + "content": ". Unless specified otherwise,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 583, + 267, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 119, + 594 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 119, + 583, + 166, + 595 + ], + "score": 0.9, + "content": "b = q = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 583, + 169, + 594 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 169, + 583, + 212, + 594 + ], + "score": 0.88, + "content": "T = 5 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 583, + 229, + 594 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 230, + 583, + 264, + 593 + ], + "score": 0.91, + "content": "d = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 583, + 267, + 594 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "In Fig. 1, we report the training loss, the test accuracy, as well as the effects of algorithmic parameters", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "on the convergence of the studied algorithms. We observe from Fig. 1-(a) and (b) that ZO-signSGD", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "outperforms other ZO algorithms, and signSGD yields the best convergence performance once the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "first-order information is available. In Fig. 1-(c) and (d), we observe that the convergence performance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 236, + 657 + ], + "score": 1.0, + "content": "of ZO algorithms is improved as", + "type": "text" + }, + { + "bbox": [ + 236, + 644, + 242, + 654 + ], + "score": 0.7, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 643, + 259, + 657 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 260, + 645, + 266, + 655 + ], + "score": 0.82, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "increase. In particular, ZO-signSGD and ZO-M-signSGD at", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 107, + 655, + 154, + 666 + ], + "score": 0.92, + "content": "b = q = 3 0", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "approach to the best result provided by signSGD. In Fig. 1-(e) and (f), the convergence of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 278, + 678 + ], + "score": 1.0, + "content": "all algorithms degrades as the problem size", + "type": "text" + }, + { + "bbox": [ + 278, + 666, + 285, + 676 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "increases. However, ZO-signSGD and ZO-M-signSGD", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 675, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 506, + 691 + ], + "score": 1.0, + "content": "converge faster than ZO-SGD and ZO-SCD. In Fig. 2, we demonstrate the convergence trajectory", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 273, + 701 + ], + "score": 1.0, + "content": "of different variants of ZO-signSGD for", + "type": "text" + }, + { + "bbox": [ + 274, + 688, + 332, + 699 + ], + "score": 0.92, + "content": "b \\in \\{ 4 0 , 4 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 687, + 506, + 701 + ], + "score": 1.0, + "content": ". To make a fair comparison between ZO-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 281, + 712 + ], + "score": 1.0, + "content": "signSGD and ZO-D-signSGD, let each of", + "type": "text" + }, + { + "bbox": [ + 281, + 700, + 318, + 709 + ], + "score": 0.87, + "content": "M = 4 0", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 699, + 450, + 712 + ], + "score": 1.0, + "content": "agents use a mini-batch of size", + "type": "text" + }, + { + "bbox": [ + 450, + 699, + 470, + 711 + ], + "score": 0.82, + "content": "b / M", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 699, + 506, + 712 + ], + "score": 1.0, + "content": ". As we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "can see, ZO-signSGD outperforms ZO-M-signSGD and ZO-D-signSGD. And the convergence is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "improved as the mini-batch size increases. However, we observe that in all examples, ZO-signSGD", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 123, + 88, + 491, + 304 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 123, + 88, + 491, + 304 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 123, + 88, + 491, + 304 + ], + "spans": [ + { + "bbox": [ + 123, + 88, + 491, + 304 + ], + "score": 0.978, + "type": "image", + "image_path": "337bfc286984c7305cf44de134c83cbe2f9274c987c25327a863c940c6c1ba76.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 123, + 88, + 491, + 160.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 123, + 160.0, + 491, + 232.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 123, + 232.0, + 491, + 304.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 307, + 506, + 378 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "Figure 1: Performance comparison of ZO-signSGD, ZO-M-signSGD, ZO-SGD, ZO-SCD, signSGD and SGD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "under a synthetic dataset. The solid line represents the loss/accuracy averaged over 10 independent trials with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "random initialization, and the shaded region indicates the standard deviation of results over random trials. (a)-(b):", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 336, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 412, + 350 + ], + "score": 1.0, + "content": "Training loss and test accuracy versus iterations. (c)-(d): Effects of mini-batch size", + "type": "text" + }, + { + "bbox": [ + 412, + 339, + 418, + 348 + ], + "score": 0.78, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 336, + 506, + 350 + ], + "score": 1.0, + "content": "and number of random", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 347, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 167, + 359 + ], + "score": 1.0, + "content": "direction vectors", + "type": "text" + }, + { + "bbox": [ + 168, + 349, + 173, + 358 + ], + "score": 0.77, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 347, + 506, + 359 + ], + "score": 1.0, + "content": "on the convergence of studied algorithms. Here (c) presents the training loss versus iterations,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 324, + 370 + ], + "score": 1.0, + "content": "and (d) is the heat map of the final loss for different values of", + "type": "text" + }, + { + "bbox": [ + 324, + 358, + 329, + 366 + ], + "score": 0.64, + "content": "^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 356, + 344, + 370 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 345, + 358, + 350, + 368 + ], + "score": 0.72, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 356, + 464, + 370 + ], + "score": 1.0, + "content": ". (e)-(f): Effects of problem size", + "type": "text" + }, + { + "bbox": [ + 465, + 358, + 470, + 366 + ], + "score": 0.6, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 356, + 506, + 370 + ], + "score": 1.0, + "content": ". Here (e)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 367, + 481, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 234, + 379 + ], + "score": 1.0, + "content": "shows the final training loss versus", + "type": "text" + }, + { + "bbox": [ + 234, + 368, + 240, + 376 + ], + "score": 0.73, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 367, + 419, + 379 + ], + "score": 1.0, + "content": ", and (f) presents the convergence trajectory when", + "type": "text" + }, + { + "bbox": [ + 420, + 367, + 477, + 378 + ], + "score": 0.94, + "content": "d \\in \\{ 2 0 0 , 4 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 367, + 481, + 379 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 388, + 200, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 201, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 201, + 402 + ], + "score": 1.0, + "content": "6 EXPERIMENTS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 505, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "In this section, we empirically show the effectiveness of ZO-signSGD, and validate its convergence", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 418, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 504, + 429 + ], + "score": 1.0, + "content": "behavior on both synthetic and real-world datasets such as MNIST and CIFAR-10. For the synthetic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "experiment, we study the problem of binary classification in the least squared formulation. For the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 440, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 451 + ], + "score": 1.0, + "content": "real-world application, we design adversarial examples from black-box neural networks as mentioned", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "in Sec. 2. Throughout this section, we compare ZO-signSGD and its variants with SGD, signSGD", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 461, + 473, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 473, + 473 + ], + "score": 1.0, + "content": "(Bernstein et al., 2018), ZO-SGD (Ghadimi & Lan, 2013), and ZO-SCD (Lian et al., 2016).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 406, + 505, + 473 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 506, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "Binary classification We consider the least squared problem with a nonconvex loss function (Xu", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 489, + 508, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 233, + 506 + ], + "score": 1.0, + "content": "et al., 2017; Liu et al., 2018b)", + "type": "text" + }, + { + "bbox": [ + 233, + 489, + 399, + 505 + ], + "score": 0.82, + "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { \\mathbf { x } \\in \\mathbb { R } ^ { d } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } ( y _ { i } - 1 / ( 1 + e ^ { - \\mathbf { a } _ { i } ^ { T } \\mathbf { x } } ) ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 489, + 508, + 506 + ], + "score": 1.0, + "content": ", which satisfies Assump-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 182, + 516 + ], + "score": 1.0, + "content": "tion A2 by letting", + "type": "text" + }, + { + "bbox": [ + 183, + 502, + 266, + 514 + ], + "score": 0.92, + "content": "\\sigma = \\operatorname* { m a x } _ { i } \\{ 2 \\lVert \\mathbf { a } _ { i } \\rVert _ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 502, + 506, + 516 + ], + "score": 1.0, + "content": ". Here instead of using the conventional cost function of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "logistic regression (a convex function), the considered least squared formulation is introduced to align", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "with our nonconvex theoretical analysis. For generating the synthetic dataset, we randomly draw", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 535, + 507, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 537, + 142, + 550 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 142, + 537, + 162, + 550 + ], + "score": 0.92, + "content": "\\left\\{ \\mathbf { a } _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 537, + 186, + 550 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 186, + 537, + 219, + 550 + ], + "score": 0.91, + "content": "\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 537, + 306, + 550 + ], + "score": 1.0, + "content": ", and obtain the label", + "type": "text" + }, + { + "bbox": [ + 307, + 537, + 336, + 549 + ], + "score": 0.9, + "content": "y _ { i } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 537, + 346, + 550 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 347, + 535, + 435, + 549 + ], + "score": 0.92, + "content": "1 / \\dot { ( 1 + e ^ { - \\mathbf { a } _ { i } ^ { T } \\mathbf { x } } ) } > 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 537, + 507, + 550 + ], + "score": 1.0, + "content": "and 0 otherwise.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 234, + 561 + ], + "score": 1.0, + "content": "The number of training samples", + "type": "text" + }, + { + "bbox": [ + 234, + 549, + 267, + 560 + ], + "score": 0.93, + "content": "\\left\\{ \\mathbf { a } _ { i } , y _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 547, + 302, + 561 + ], + "score": 1.0, + "content": "is set by", + "type": "text" + }, + { + "bbox": [ + 303, + 549, + 343, + 559 + ], + "score": 0.85, + "content": "n = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "against 200 testing samples. We find the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 367, + 573 + ], + "score": 1.0, + "content": "best constant learning rate for algorithms via a greedy search over", + "type": "text" + }, + { + "bbox": [ + 367, + 560, + 431, + 571 + ], + "score": 0.87, + "content": "\\eta \\in [ 0 . 0 0 1 , 0 . 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "(see Appendix 8.1", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 571, + 507, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 336, + 585 + ], + "score": 1.0, + "content": "for more details), and we choose the smoothing parameter", + "type": "text" + }, + { + "bbox": [ + 337, + 571, + 393, + 584 + ], + "score": 0.93, + "content": "\\mu = 1 0 / \\sqrt { T d }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 571, + 507, + 585 + ], + "score": 1.0, + "content": ". Unless specified otherwise,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 583, + 267, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 119, + 594 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 119, + 583, + 166, + 595 + ], + "score": 0.9, + "content": "b = q = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 583, + 169, + 594 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 169, + 583, + 212, + 594 + ], + "score": 0.88, + "content": "T = 5 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 583, + 229, + 594 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 230, + 583, + 264, + 593 + ], + "score": 0.91, + "content": "d = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 583, + 267, + 594 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 477, + 508, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "In Fig. 1, we report the training loss, the test accuracy, as well as the effects of algorithmic parameters", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "on the convergence of the studied algorithms. We observe from Fig. 1-(a) and (b) that ZO-signSGD", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "outperforms other ZO algorithms, and signSGD yields the best convergence performance once the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "first-order information is available. In Fig. 1-(c) and (d), we observe that the convergence performance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 236, + 657 + ], + "score": 1.0, + "content": "of ZO algorithms is improved as", + "type": "text" + }, + { + "bbox": [ + 236, + 644, + 242, + 654 + ], + "score": 0.7, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 643, + 259, + 657 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 260, + 645, + 266, + 655 + ], + "score": 0.82, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "increase. In particular, ZO-signSGD and ZO-M-signSGD at", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 107, + 655, + 154, + 666 + ], + "score": 0.92, + "content": "b = q = 3 0", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "approach to the best result provided by signSGD. In Fig. 1-(e) and (f), the convergence of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 278, + 678 + ], + "score": 1.0, + "content": "all algorithms degrades as the problem size", + "type": "text" + }, + { + "bbox": [ + 278, + 666, + 285, + 676 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "increases. However, ZO-signSGD and ZO-M-signSGD", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 675, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 506, + 691 + ], + "score": 1.0, + "content": "converge faster than ZO-SGD and ZO-SCD. In Fig. 2, we demonstrate the convergence trajectory", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 273, + 701 + ], + "score": 1.0, + "content": "of different variants of ZO-signSGD for", + "type": "text" + }, + { + "bbox": [ + 274, + 688, + 332, + 699 + ], + "score": 0.92, + "content": "b \\in \\{ 4 0 , 4 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 687, + 506, + 701 + ], + "score": 1.0, + "content": ". To make a fair comparison between ZO-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 281, + 712 + ], + "score": 1.0, + "content": "signSGD and ZO-D-signSGD, let each of", + "type": "text" + }, + { + "bbox": [ + 281, + 700, + 318, + 709 + ], + "score": 0.87, + "content": "M = 4 0", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 699, + 450, + 712 + ], + "score": 1.0, + "content": "agents use a mini-batch of size", + "type": "text" + }, + { + "bbox": [ + 450, + 699, + 470, + 711 + ], + "score": 0.82, + "content": "b / M", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 699, + 506, + 712 + ], + "score": 1.0, + "content": ". As we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "can see, ZO-signSGD outperforms ZO-M-signSGD and ZO-D-signSGD. And the convergence is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "improved as the mini-batch size increases. However, we observe that in all examples, ZO-signSGD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "and its variants converge to moderate accuracy much faster than ZO-SGD, only within a few tens of", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 542, + 149, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 149, + 555 + ], + "score": 1.0, + "content": "iterations.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 600, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 146, + 86, + 470, + 200 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 146, + 86, + 470, + 200 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 146, + 86, + 470, + 200 + ], + "spans": [ + { + "bbox": [ + 146, + 86, + 470, + 200 + ], + "score": 0.969, + "type": "image", + "image_path": "d06c7d33a9b1cb45fe96ad3bf4d31117fbae253be352e11dcb7b8e189b8be217.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 146, + 86, + 470, + 124.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 146, + 124.0, + 470, + 162.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 146, + 162.0, + 470, + 200.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 212, + 505, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "Figure 2: Training loss (left) and testing accuracy (right) of ZO-signSGD, ZO-M-signSGD, ZO-D-signSGD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 222, + 220, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 220, + 234 + ], + "score": 1.0, + "content": "and ZO-SGD versus iterations.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "image", + "bbox": [ + 145, + 256, + 451, + 489 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 256, + 451, + 489 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 145, + 256, + 451, + 489 + ], + "spans": [ + { + "bbox": [ + 145, + 256, + 451, + 489 + ], + "score": 0.975, + "type": "image", + "image_path": "0dcd6aa27fddfbaa2c914d807914b8186ef985663cc7baa8b27cd6648a025220.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 145, + 256, + 451, + 333.6666666666667 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 145, + 333.6666666666667, + 451, + 411.33333333333337 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 145, + 411.33333333333337, + 451, + 489.00000000000006 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 496, + 504, + 516 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "Figure 3: Black-box attacking loss versus iterations. The solid marker indicates the iteration number that finds", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 505, + 441, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 391, + 517 + ], + "score": 1.0, + "content": "the first successful adversarial example, and its loss corresponds to the squared", + "type": "text" + }, + { + "bbox": [ + 391, + 506, + 401, + 516 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 505, + 441, + 517 + ], + "score": 1.0, + "content": "distortion.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 7.25 + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 503, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "and its variants converge to moderate accuracy much faster than ZO-SGD, only within a few tens of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 542, + 149, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 149, + 555 + ], + "score": 1.0, + "content": "iterations.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "Generating black-box adversarial examples Here we study adversarial robustness by generating", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "adversarial examples from a black-box image classifier trained by a deep neural network (DNN)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "model; see details on problem formulation in Appendix 8.2. 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Moreover, we benchmark their performance", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "with the natural evolution strategy (NES) based two-point gradient estimator in (Ilyas et al., 2018a)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "for solving the same attacking loss function, where the sign of gradient estimate is also used in the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 282, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 506, + 294 + ], + "score": 1.0, + "content": "descent step. We call the resulting black-box attack generation method ‘ZO-NES’. Similar to (10),", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 506, + 306 + ], + "score": 1.0, + "content": "NES computes the ZO gradient estimate using the central difference of two function values. Thus,", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 239, + 316 + ], + "score": 1.0, + "content": "one iteration of ZO-NES requires", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 240, + 304, + 251, + 315 + ], + "score": 0.73, + "content": "2 q", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 252, + 304, + 380, + 316 + ], + "score": 1.0, + "content": "function queries and thus we set", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 381, + 304, + 405, + 315 + ], + "score": 0.91, + "content": "q = 5", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 406, + 304, + 506, + 316 + ], + "score": 1.0, + "content": "to align with the number", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 314, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 327 + ], + "score": 1.0, + "content": "of function queries used in other ZO methods. All methods use the the same natural image as the", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 326, + 288, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 288, + 339 + ], + "score": 1.0, + "content": "initial point for finding adversarial examples.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 622, + 506, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 154, + 92, + 455, + 257 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 152, + 80, + 457, + 91 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 151, + 78, + 460, + 94 + ], + "spans": [ + { + "bbox": [ + 151, + 78, + 460, + 94 + ], + "score": 1.0, + "content": "Table 1: Iteration comparison of attacking black-box DNN on MNIST (image ID 2).", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 154, + 92, + 455, + 257 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 154, + 92, + 455, + 257 + ], + "spans": [ + { + "bbox": [ + 154, + 92, + 455, + 257 + ], + "score": 0.865, + "html": "
Iteration04080 120160 200 240280 312 356
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", + "type": "table", + "image_path": "4aebf266372cf9ce96379ec3df43fed31c03c889adb0fae526b3d4b953310157.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 154, + 92, + 455, + 147.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 154, + 147.0, + 455, + 202.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 154, + 202.0, + 455, + 257.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 106, + 282, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 506, + 294 + ], + "score": 1.0, + "content": "descent step. We call the resulting black-box attack generation method ‘ZO-NES’. Similar to (10),", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 506, + 306 + ], + "score": 1.0, + "content": "NES computes the ZO gradient estimate using the central difference of two function values. Thus,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 239, + 316 + ], + "score": 1.0, + "content": "one iteration of ZO-NES requires", + "type": "text" + }, + { + "bbox": [ + 240, + 304, + 251, + 315 + ], + "score": 0.73, + "content": "2 q", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 304, + 380, + 316 + ], + "score": 1.0, + "content": "function queries and thus we set", + "type": "text" + }, + { + "bbox": [ + 381, + 304, + 405, + 315 + ], + "score": 0.91, + "content": "q = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 304, + 506, + 316 + ], + "score": 1.0, + "content": "to align with the number", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 314, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 327 + ], + "score": 1.0, + "content": "of function queries used in other ZO methods. All methods use the the same natural image as the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 326, + 288, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 288, + 339 + ], + "score": 1.0, + "content": "initial point for finding adversarial examples.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "Fig. 3 shows the plots of black-box attacking loss versus iterations (more results are shown in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "Appendix 8.3). We find that ZO-signSGD usually takes significantly less iterations than other", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "methods to find the first successful adversarial example with a similar attacking loss. For MNIST, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "average iteration over all attacked images in Table A1 to find the first successful adversarial example", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "score": 1.0, + "content": "is 184 for ZO-SGD, 103 for ZO-signSGD, 151 for ZO-M-signSGD, and 227 for ZO-NES. Their", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 198, + 410 + ], + "score": 1.0, + "content": "corresponding average", + "type": "text" + }, + { + "bbox": [ + 198, + 398, + 208, + 408 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "distortion is 2.345 for ZO-SGD, 2.381 for ZO-signSGD, 2.418 for ZO-M-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "score": 1.0, + "content": "signSGD, and 2.488 for ZO-NES. For CIFAR-10, the average iteration over all attacked images in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "Table A2 to find the first successful adversarial example is 302 for ZO-SGD, 250 for ZO-signSGD,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 418, + 443 + ], + "score": 1.0, + "content": "389 for ZO-M-signSGD, and 363 for ZO-NES. Their corresponding average", + "type": "text" + }, + { + "bbox": [ + 419, + 431, + 429, + 441 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "distortion is 0.177", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "for ZO-SGD, 0.208 for ZO-signSGD, 0.219 for ZO-M-signSGD, and 0.235 for ZO-NES. As a visual", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "score": 1.0, + "content": "illustration, we compare the adversarial examples of a hand-written digit “1” of each attacking", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "method at different iterations in Table 1, corresponding to Fig. 3-(a). As we can see, ZO-signSGD and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 250, + 487 + ], + "score": 1.0, + "content": "ZO-M-signSGD can reduce roughly", + "type": "text" + }, + { + "bbox": [ + 250, + 474, + 270, + 485 + ], + "score": 0.89, + "content": "5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "of iterations (around 600 less model queries) than ZO-SGD", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "to find the first successful adversarial example. Given the first successful adversarial example, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 495, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 302, + 510 + ], + "score": 1.0, + "content": "observe that ZO-signSGD yields slightly higher", + "type": "text" + }, + { + "bbox": [ + 302, + 496, + 312, + 507 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 495, + 506, + 510 + ], + "score": 1.0, + "content": "distortion than ZO-SGD. This is not surprising", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 506, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 521 + ], + "score": 1.0, + "content": "since Theorem 1 suggests that ZO-signSGD might not converge to a solution of very high accuracy", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "but it can converge to moderate accuracy sufficient for black-box attacks at a very fast speed. Note", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "that the first successful adversarial examples generated by different ZO methods are all visually", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "similar to the original ones but lead to different top-1 predictions; see more results in Appendix 8.3.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 549, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 506, + 565 + ], + "score": 1.0, + "content": "In addition, we observe that ZO-NES is not as effective as ZO-signSGD in either query efficiency", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 561, + 507, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 561, + 507, + 574 + ], + "score": 1.0, + "content": "(given by the number of iterations to achieve the first successful attack) or attack distortion. Thus,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 572, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 506, + 587 + ], + "score": 1.0, + "content": "compared to ZO-NES, ZO-signSGD offers a provable and an efficient black-box adversarial attacking", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 583, + 142, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 142, + 595 + ], + "score": 1.0, + "content": "method.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 616, + 195, + 629 + ], + "lines": [ + { + "bbox": [ + 104, + 614, + 198, + 633 + ], + "spans": [ + { + "bbox": [ + 104, + 614, + 198, + 633 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 642, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "Motivated by the impressive convergence behavior of (first-order) signSGD and the empirical success", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "in crafting adversarial examples from black-box ML models, in this paper we rigorously prove the√ √", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 664, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 107, + 664, + 157, + 678 + ], + "score": 0.92, + "content": "O ( \\sqrt { d } / \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 664, + 506, + 679 + ], + "score": 1.0, + "content": "convergence rate of ZO-signSGD and its variants under mild conditions. Compared to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 414, + 690 + ], + "score": 1.0, + "content": "signSGD, ZO-signSGD suffers a slowdown (proportional to the problem size", + "type": "text" + }, + { + "bbox": [ + 414, + 677, + 421, + 687 + ], + "score": 0.53, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 676, + 507, + 690 + ], + "score": 1.0, + "content": ") in convergence rate,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "however, it enjoys the gradient-free advantages. Compared to other ZO algorithms, we corroborate", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "the superior performance of ZO-signSGD on both synthetic and real-word datasets, particularly for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "its application to black-box adversarial attacks. In the future, we would like to generalize our analysis", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 369, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 369, + 733 + ], + "score": 1.0, + "content": "to nonsmooth and nonconvex constrained optimization problems.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "10", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 154, + 92, + 455, + 257 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 152, + 80, + 457, + 91 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 151, + 78, + 460, + 94 + ], + "spans": [ + { + "bbox": [ + 151, + 78, + 460, + 94 + ], + "score": 1.0, + "content": "Table 1: Iteration comparison of attacking black-box DNN on MNIST (image ID 2).", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 154, + 92, + 455, + 257 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 154, + 92, + 455, + 257 + ], + "spans": [ + { + "bbox": [ + 154, + 92, + 455, + 257 + ], + "score": 0.865, + "html": "
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We find that ZO-signSGD usually takes significantly less iterations than other", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "methods to find the first successful adversarial example with a similar attacking loss. For MNIST, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "average iteration over all attacked images in Table A1 to find the first successful adversarial example", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "score": 1.0, + "content": "is 184 for ZO-SGD, 103 for ZO-signSGD, 151 for ZO-M-signSGD, and 227 for ZO-NES. Their", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 198, + 410 + ], + "score": 1.0, + "content": "corresponding average", + "type": "text" + }, + { + "bbox": [ + 198, + 398, + 208, + 408 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "distortion is 2.345 for ZO-SGD, 2.381 for ZO-signSGD, 2.418 for ZO-M-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "score": 1.0, + "content": "signSGD, and 2.488 for ZO-NES. For CIFAR-10, the average iteration over all attacked images in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "Table A2 to find the first successful adversarial example is 302 for ZO-SGD, 250 for ZO-signSGD,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 418, + 443 + ], + "score": 1.0, + "content": "389 for ZO-M-signSGD, and 363 for ZO-NES. Their corresponding average", + "type": "text" + }, + { + "bbox": [ + 419, + 431, + 429, + 441 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "distortion is 0.177", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "for ZO-SGD, 0.208 for ZO-signSGD, 0.219 for ZO-M-signSGD, and 0.235 for ZO-NES. As a visual", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "score": 1.0, + "content": "illustration, we compare the adversarial examples of a hand-written digit “1” of each attacking", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "method at different iterations in Table 1, corresponding to Fig. 3-(a). As we can see, ZO-signSGD and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 250, + 487 + ], + "score": 1.0, + "content": "ZO-M-signSGD can reduce roughly", + "type": "text" + }, + { + "bbox": [ + 250, + 474, + 270, + 485 + ], + "score": 0.89, + "content": "5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "of iterations (around 600 less model queries) than ZO-SGD", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "to find the first successful adversarial example. Given the first successful adversarial example, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 495, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 302, + 510 + ], + "score": 1.0, + "content": "observe that ZO-signSGD yields slightly higher", + "type": "text" + }, + { + "bbox": [ + 302, + 496, + 312, + 507 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 495, + 506, + 510 + ], + "score": 1.0, + "content": "distortion than ZO-SGD. This is not surprising", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 506, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 521 + ], + "score": 1.0, + "content": "since Theorem 1 suggests that ZO-signSGD might not converge to a solution of very high accuracy", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "but it can converge to moderate accuracy sufficient for black-box attacks at a very fast speed. Note", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "that the first successful adversarial examples generated by different ZO methods are all visually", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "similar to the original ones but lead to different top-1 predictions; see more results in Appendix 8.3.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 549, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 506, + 565 + ], + "score": 1.0, + "content": "In addition, we observe that ZO-NES is not as effective as ZO-signSGD in either query efficiency", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 561, + 507, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 561, + 507, + 574 + ], + "score": 1.0, + "content": "(given by the number of iterations to achieve the first successful attack) or attack distortion. Thus,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 572, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 506, + 587 + ], + "score": 1.0, + "content": "compared to ZO-NES, ZO-signSGD offers a provable and an efficient black-box adversarial attacking", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 583, + 142, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 142, + 595 + ], + "score": 1.0, + "content": "method.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 343, + 507, + 595 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 616, + 195, + 629 + ], + "lines": [ + { + "bbox": [ + 104, + 614, + 198, + 633 + ], + "spans": [ + { + "bbox": [ + 104, + 614, + 198, + 633 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 642, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "Motivated by the impressive convergence behavior of (first-order) signSGD and the empirical success", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "in crafting adversarial examples from black-box ML models, in this paper we rigorously prove the√ √", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 664, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 107, + 664, + 157, + 678 + ], + "score": 0.92, + "content": "O ( \\sqrt { d } / \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 664, + 506, + 679 + ], + "score": 1.0, + "content": "convergence rate of ZO-signSGD and its variants under mild conditions. Compared to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 414, + 690 + ], + "score": 1.0, + "content": "signSGD, ZO-signSGD suffers a slowdown (proportional to the problem size", + "type": "text" + }, + { + "bbox": [ + 414, + 677, + 421, + 687 + ], + "score": 0.53, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 676, + 507, + 690 + ], + "score": 1.0, + "content": ") in convergence rate,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "however, it enjoys the gradient-free advantages. Compared to other ZO algorithms, we corroborate", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "the superior performance of ZO-signSGD on both synthetic and real-word datasets, particularly for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "its application to black-box adversarial attacks. In the future, we would like to generalize our analysis", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 369, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 369, + 733 + ], + "score": 1.0, + "content": "to nonsmooth and nonconvex constrained optimization problems.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 642, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 83, + 200, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 83, + 200, + 94 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 200, + 94 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 101, + 504, + 125 + ], + "lines": [ + { + "bbox": [ + 105, + 101, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 505, + 114 + ], + "score": 1.0, + "content": "This work was supported by the MIT-IBM Watson AI Lab. 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Similar to (Bernstein et al., 2018, Figure A.1), we assume", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 182, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 220, + 195 + ], + "score": 1.0, + "content": "that the ZO gradient estimate of", + "type": "text" + }, + { + "bbox": [ + 221, + 183, + 239, + 193 + ], + "score": 0.9, + "content": "f ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 182, + 334, + 195 + ], + "score": 1.0, + "content": "and its first-order gradient", + "type": "text" + }, + { + "bbox": [ + 334, + 182, + 379, + 194 + ], + "score": 0.93, + "content": "\\nabla f ( \\mathbf { x } ) = \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 182, + 496, + 195 + ], + "score": 1.0, + "content": "suffer from a sparse noise vector", + "type": "text" + }, + { + "bbox": [ + 496, + 185, + 502, + 192 + ], + "score": 0.59, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 182, + 507, + 195 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 131, + 205 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 131, + 192, + 194, + 203 + ], + "score": 0.92, + "content": "v _ { 1 } \\in \\overline { { \\mathcal { N } ( 0 , 1 0 0 ^ { 2 } ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 191, + 213, + 205 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 213, + 194, + 239, + 203 + ], + "score": 0.89, + "content": "v _ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 191, + 252, + 205 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 253, + 193, + 274, + 203 + ], + "score": 0.9, + "content": "i \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 191, + 458, + 205 + ], + "score": 1.0, + "content": ". As a result, the used descent direction at iteration", + "type": "text" + }, + { + "bbox": [ + 458, + 194, + 462, + 202 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 202, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 154, + 215 + ], + "score": 0.91, + "content": "\\hat { \\nabla } f ( \\mathbf { x } _ { t } ) + \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 202, + 165, + 217 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 165, + 204, + 212, + 215 + ], + "score": 0.91, + "content": "\\nabla f ( \\mathbf { x } _ { t } ) + \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 202, + 506, + 217 + ], + "score": 1.0, + "content": ". Fig. A1 presents the convergence performance of 5 algorithms: SGD, signSGD,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "score": 1.0, + "content": "ZO-SGD, ZO-signSGD and its variant using the central difference based gradient estimator (10). Here we", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 225, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 235 + ], + "score": 1.0, + "content": "tune a constant learning rate finding 0.001 best for SGD and ZO-SGD and 0.01 best for signSGD and its ZO", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "variants. As we can see, sign-based first-order and ZO algorithms converge much faster than the stochastic", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 244, + 504, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 494, + 257 + ], + "score": 1.0, + "content": "gradient-based descent algorithms. This is not surprising since the presence of extremely noisy component", + "type": "text" + }, + { + "bbox": [ + 495, + 245, + 504, + 254 + ], + "score": 0.82, + "content": "v _ { 1 }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "leads to an inaccurate gradient value, and thus degrades the convergence of SGD and ZO-SGD. By contrast,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "the sign information is more robust to outliers and thus leads to better convergence performance of sign SGD", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "and its variants. We also note that the convergence trajectory of ZO-signSGD using the gradient estimator (10)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 284, + 492, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 492, + 296 + ], + "score": 1.0, + "content": "coincides with that using the gradient estimator (3) given by the forward difference of two function values.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8.5 + }, + { + "type": "image", + "bbox": [ + 106, + 334, + 493, + 468 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 334, + 493, + 468 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 334, + 493, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 493, + 468 + ], + "score": 0.967, + "type": "image", + "image_path": "3a02060f85eca3a9b9d8126e85684d6a6fade1b1de8a7098c000772a2df7bcc7.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 106, + 334, + 493, + 378.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 106, + 378.6666666666667, + 493, + 423.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 106, + 423.33333333333337, + 493, + 468.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 479, + 505, + 540 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "Figure A1: Comparison of different gradient-based and gradient sign-based first-order and ZO algorithms", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 490, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 501 + ], + "score": 1.0, + "content": "in the example of sparse noise perturbation. The solid line represents the loss averaged over 10 independent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "trials with random initialization, and the shaded region indicates the standard deviation of results over random", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "trials. Left: Loss value against iterations for SGD, signSGD, ZO-SGD, ZO-signSGD and ZO-signSGD using", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 520, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 531 + ], + "score": 1.0, + "content": "the central difference based gradient estimator (10). Right: Local regions to highlight the effect of the gradient", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 529, + 323, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 323, + 541 + ], + "score": 1.0, + "content": "estimators (3) and (10) on the convergence of ZO-signSGD.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + } + ], + "index": 18.25 + }, + { + "type": "title", + "bbox": [ + 109, + 587, + 291, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 292, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 292, + 599 + ], + "score": 1.0, + "content": "1.2 STATISTICS OF GRADIENT ESTIMATES", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "The intuition behind why ZO-signSGD could outperform ZO-SGD is that the sign operation can mitigate the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 622, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 506, + 633 + ], + "score": 1.0, + "content": "negative effect of (coordinate-wise) gradient noise of large variance. To confirm this point, we examine the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "score": 1.0, + "content": "coordinate-wise variance of gradient noises during an entire training run of the binary classifier provided in the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 642, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 653 + ], + "score": 1.0, + "content": "first experiment of Sec. 6. At each iteration, we perform an additional 100 random trials to obtain the statistics", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 293, + 664 + ], + "score": 1.0, + "content": "of gradient estimates. In Fig. A2-(a), we present the", + "type": "text" + }, + { + "bbox": [ + 293, + 652, + 302, + 661 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 651, + 506, + 664 + ], + "score": 1.0, + "content": "norm of the mean of gradient estimates (over 100 trials)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 662, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 506, + 673 + ], + "score": 1.0, + "content": "versus the number of iterations. As we can see, both signSGD and ZO-signSGD outperform SGD and ZO-SGD,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 672, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 232, + 683 + ], + "score": 1.0, + "content": "evidenced by a fast decrease of the", + "type": "text" + }, + { + "bbox": [ + 232, + 672, + 241, + 681 + ], + "score": 0.86, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 672, + 505, + 683 + ], + "score": 1.0, + "content": "norm of gradient estimate. In Fig. A2-(b), we present the coordinate-wise", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "gradient noise variance (over 100 trails at each coordinate) against the number of iterations. It is not surprising", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 505, + 703 + ], + "score": 1.0, + "content": "that compared to first-order methods, ZO methods suffer gradient noise of larger variance. In this scenario, we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 713 + ], + "score": 1.0, + "content": "could benefit from ZO-signSGD since taking the sign of gradient estimates might scale down extremely noisy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "components. Indeed, we observe a significant decrease of the noise variance while performing ZO-signSGD", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 721, + 191, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 191, + 732 + ], + "score": 1.0, + "content": "compared to ZO-SGD.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 161, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "score": 1.0, + "content": "APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 106, + 114, + 471, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 114, + 473, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 473, + 129 + ], + "score": 1.0, + "content": "1 MOTIVATING EXAMPLES OF FAST CONVERGENCE OF ZO-SIGNSGD", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 108, + 147, + 317, + 158 + ], + "lines": [ + { + "bbox": [ + 106, + 147, + 319, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 319, + 159 + ], + "score": 1.0, + "content": "1.1 EXAMPLE OF SPARSE NOISE PERTURBATION", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 506, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 240, + 185 + ], + "score": 1.0, + "content": "We consider to minimize the function", + "type": "text" + }, + { + "bbox": [ + 241, + 172, + 297, + 184 + ], + "score": 0.93, + "content": "\\begin{array} { r } { f ( \\mathbf { x } ) = \\frac { 1 } { 2 } \\| \\mathbf { x } \\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 172, + 506, + 185 + ], + "score": 1.0, + "content": ". Similar to (Bernstein et al., 2018, Figure A.1), we assume", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 182, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 220, + 195 + ], + "score": 1.0, + "content": "that the ZO gradient estimate of", + "type": "text" + }, + { + "bbox": [ + 221, + 183, + 239, + 193 + ], + "score": 0.9, + "content": "f ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 182, + 334, + 195 + ], + "score": 1.0, + "content": "and its first-order gradient", + "type": "text" + }, + { + "bbox": [ + 334, + 182, + 379, + 194 + ], + "score": 0.93, + "content": "\\nabla f ( \\mathbf { x } ) = \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 182, + 496, + 195 + ], + "score": 1.0, + "content": "suffer from a sparse noise vector", + "type": "text" + }, + { + "bbox": [ + 496, + 185, + 502, + 192 + ], + "score": 0.59, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 182, + 507, + 195 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 131, + 205 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 131, + 192, + 194, + 203 + ], + "score": 0.92, + "content": "v _ { 1 } \\in \\overline { { \\mathcal { N } ( 0 , 1 0 0 ^ { 2 } ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 191, + 213, + 205 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 213, + 194, + 239, + 203 + ], + "score": 0.89, + "content": "v _ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 191, + 252, + 205 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 253, + 193, + 274, + 203 + ], + "score": 0.9, + "content": "i \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 191, + 458, + 205 + ], + "score": 1.0, + "content": ". As a result, the used descent direction at iteration", + "type": "text" + }, + { + "bbox": [ + 458, + 194, + 462, + 202 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 202, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 154, + 215 + ], + "score": 0.91, + "content": "\\hat { \\nabla } f ( \\mathbf { x } _ { t } ) + \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 202, + 165, + 217 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 165, + 204, + 212, + 215 + ], + "score": 0.91, + "content": "\\nabla f ( \\mathbf { x } _ { t } ) + \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 202, + 506, + 217 + ], + "score": 1.0, + "content": ". Fig. A1 presents the convergence performance of 5 algorithms: SGD, signSGD,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "score": 1.0, + "content": "ZO-SGD, ZO-signSGD and its variant using the central difference based gradient estimator (10). Here we", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 225, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 235 + ], + "score": 1.0, + "content": "tune a constant learning rate finding 0.001 best for SGD and ZO-SGD and 0.01 best for signSGD and its ZO", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "variants. As we can see, sign-based first-order and ZO algorithms converge much faster than the stochastic", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 244, + 504, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 494, + 257 + ], + "score": 1.0, + "content": "gradient-based descent algorithms. This is not surprising since the presence of extremely noisy component", + "type": "text" + }, + { + "bbox": [ + 495, + 245, + 504, + 254 + ], + "score": 0.82, + "content": "v _ { 1 }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "leads to an inaccurate gradient value, and thus degrades the convergence of SGD and ZO-SGD. By contrast,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "the sign information is more robust to outliers and thus leads to better convergence performance of sign SGD", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "and its variants. We also note that the convergence trajectory of ZO-signSGD using the gradient estimator (10)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 284, + 492, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 492, + 296 + ], + "score": 1.0, + "content": "coincides with that using the gradient estimator (3) given by the forward difference of two function values.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 172, + 507, + 296 + ] + }, + { + "type": "image", + "bbox": [ + 106, + 334, + 493, + 468 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 334, + 493, + 468 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 334, + 493, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 493, + 468 + ], + "score": 0.967, + "type": "image", + "image_path": "3a02060f85eca3a9b9d8126e85684d6a6fade1b1de8a7098c000772a2df7bcc7.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 106, + 334, + 493, + 378.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 106, + 378.6666666666667, + 493, + 423.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 106, + 423.33333333333337, + 493, + 468.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 479, + 505, + 540 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "Figure A1: Comparison of different gradient-based and gradient sign-based first-order and ZO algorithms", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 490, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 501 + ], + "score": 1.0, + "content": "in the example of sparse noise perturbation. The solid line represents the loss averaged over 10 independent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "trials with random initialization, and the shaded region indicates the standard deviation of results over random", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "trials. Left: Loss value against iterations for SGD, signSGD, ZO-SGD, ZO-signSGD and ZO-signSGD using", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 520, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 531 + ], + "score": 1.0, + "content": "the central difference based gradient estimator (10). Right: Local regions to highlight the effect of the gradient", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 529, + 323, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 323, + 541 + ], + "score": 1.0, + "content": "estimators (3) and (10) on the convergence of ZO-signSGD.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + } + ], + "index": 18.25 + }, + { + "type": "title", + "bbox": [ + 109, + 587, + 291, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 292, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 292, + 599 + ], + "score": 1.0, + "content": "1.2 STATISTICS OF GRADIENT ESTIMATES", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "The intuition behind why ZO-signSGD could outperform ZO-SGD is that the sign operation can mitigate the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 622, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 506, + 633 + ], + "score": 1.0, + "content": "negative effect of (coordinate-wise) gradient noise of large variance. To confirm this point, we examine the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "score": 1.0, + "content": "coordinate-wise variance of gradient noises during an entire training run of the binary classifier provided in the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 642, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 653 + ], + "score": 1.0, + "content": "first experiment of Sec. 6. At each iteration, we perform an additional 100 random trials to obtain the statistics", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 293, + 664 + ], + "score": 1.0, + "content": "of gradient estimates. In Fig. A2-(a), we present the", + "type": "text" + }, + { + "bbox": [ + 293, + 652, + 302, + 661 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 651, + 506, + 664 + ], + "score": 1.0, + "content": "norm of the mean of gradient estimates (over 100 trials)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 662, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 506, + 673 + ], + "score": 1.0, + "content": "versus the number of iterations. As we can see, both signSGD and ZO-signSGD outperform SGD and ZO-SGD,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 672, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 232, + 683 + ], + "score": 1.0, + "content": "evidenced by a fast decrease of the", + "type": "text" + }, + { + "bbox": [ + 232, + 672, + 241, + 681 + ], + "score": 0.86, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 672, + 505, + 683 + ], + "score": 1.0, + "content": "norm of gradient estimate. In Fig. A2-(b), we present the coordinate-wise", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "gradient noise variance (over 100 trails at each coordinate) against the number of iterations. It is not surprising", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 505, + 703 + ], + "score": 1.0, + "content": "that compared to first-order methods, ZO methods suffer gradient noise of larger variance. In this scenario, we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 713 + ], + "score": 1.0, + "content": "could benefit from ZO-signSGD since taking the sign of gradient estimates might scale down extremely noisy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "components. Indeed, we observe a significant decrease of the noise variance while performing ZO-signSGD", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 721, + 191, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 191, + 732 + ], + "score": 1.0, + "content": "compared to ZO-SGD.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 612, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 87, + 506, + 243 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 87, + 506, + 243 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 87, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 87, + 506, + 243 + ], + "score": 0.97, + "type": "image", + "image_path": "a3d507387e8be5d4c8da524fed61e074d8dd332bfd27e24747d6b46669e9e7a1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 87, + 506, + 139.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 139.0, + 506, + 191.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 191.0, + 506, + 243.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 253, + 506, + 294 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "Figure A2: Statistics of gradient estimates during an entire training run of the binary classifier provided in the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 224, + 275 + ], + "score": 1.0, + "content": "first experiment of Sec. 6. a) The", + "type": "text" + }, + { + "bbox": [ + 225, + 263, + 234, + 273 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "norm of the mean of gradient estimates versus iteration. b) Coordinate-wise", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "gradient noise variance versus iteration. The solid line represents the variance averaged over all coordinates, and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 284, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 294 + ], + "score": 1.0, + "content": "the shaded region indicates the corresponding standard deviation with respect to all coordinates at each iteration.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 318, + 258, + 331 + ], + "lines": [ + { + "bbox": [ + 104, + 316, + 260, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 316, + 260, + 333 + ], + "score": 1.0, + "content": "2 PROOF OF PROPOSITION 1", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 391, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 343, + 391, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 287, + 357 + ], + "score": 1.0, + "content": "Based on the definition of the smoothing function", + "type": "text" + }, + { + "bbox": [ + 287, + 345, + 297, + 356 + ], + "score": 0.88, + "content": "f _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 343, + 328, + 357 + ], + "score": 1.0, + "content": ", for any", + "type": "text" + }, + { + "bbox": [ + 329, + 347, + 336, + 354 + ], + "score": 0.65, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 343, + 351, + 357 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 352, + 347, + 358, + 356 + ], + "score": 0.72, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 343, + 391, + 357 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 364, + 453, + 392 + ], + "lines": [ + { + "bbox": [ + 157, + 364, + 453, + 392 + ], + "spans": [ + { + "bbox": [ + 157, + 364, + 453, + 392 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\| \\nabla f _ { \\mu } ( x ) - \\nabla f _ { \\mu } ( y ) \\| _ { 2 } = \\| \\mathbb { E } _ { \\mathbf { v } } [ \\nabla _ { \\mathbf { x } } f ( \\mathbf { x } + \\mu \\mathbf { v } ) - \\nabla _ { \\mathbf { y } } f ( \\mathbf { y } + \\mu \\mathbf { v } ) ] \\| _ { 2 } } \\\\ & { \\qquad \\le \\mathbb { E } [ \\| \\nabla _ { \\mathbf { x } } f ( \\mathbf { x } + \\mu \\mathbf { v } ) - \\nabla _ { \\mathbf { y } } f ( \\mathbf { y } + \\mu \\mathbf { v } ) \\| _ { 2 } ] \\le L \\| \\mathbf { x } - \\mathbf { y } \\| _ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "f8f0f55f634aa7894ddbe847b4abc256297a76b4836ff7f70592805d9a01a4eb.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 157, + 364, + 453, + 373.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 157, + 373.3333333333333, + 453, + 382.66666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 157, + 382.66666666666663, + 453, + 391.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 400, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "where the first inequality holds due to Jensen’s inequality, and the second inequality holds due to A1. It is known", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 410, + 305, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 159, + 421 + ], + "score": 1.0, + "content": "from (15) that", + "type": "text" + }, + { + "bbox": [ + 159, + 411, + 169, + 421 + ], + "score": 0.89, + "content": "f _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 410, + 185, + 421 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 185, + 411, + 192, + 420 + ], + "score": 0.83, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 410, + 305, + 421 + ], + "score": 1.0, + "content": "-Lipschitz continuous gradient.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 262, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 263, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 132, + 438 + ], + "score": 1.0, + "content": "By the", + "type": "text" + }, + { + "bbox": [ + 133, + 427, + 140, + 435 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 426, + 195, + 438 + ], + "score": 1.0, + "content": "-smoothness of", + "type": "text" + }, + { + "bbox": [ + 196, + 426, + 206, + 437 + ], + "score": 0.89, + "content": "f _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 426, + 263, + 438 + ], + "score": 1.0, + "content": ", we obtain that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 447, + 444, + 546 + ], + "lines": [ + { + "bbox": [ + 168, + 447, + 444, + 546 + ], + "spans": [ + { + "bbox": [ + 168, + 447, + 444, + 546 + ], + "score": 0.95, + "content": "\\begin{array} { l } { f _ { \\mu } ( { \\mathbf x } _ { k + 1 } ) \\leq f _ { \\mu } ( { \\mathbf x } _ { k } ) + \\langle \\nabla f _ { \\mu } ( { \\mathbf x } _ { k } ) , { \\mathbf x } _ { k + 1 } - 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f _ { \\mu } ( \\mathbf { x } _ { k } ) ] \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } } \\\\ { \\displaystyle + \\ 2 \\delta _ { k } \\sum _ { i = 1 } ^ { d } \\vert ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { i } \\vert \\mathrm { P r o b } \\left[ \\mathrm { s i g n } ( \\hat { \\mathbf { g } } _ { k , i } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { i } ) \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "e71c56baf4f50bd949662bae4f593c05e4824075db2d59c9b26162bfcc47c73a.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 136, + 591, + 475, + 609.6666666666666 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 136, + 609.6666666666666, + 475, + 628.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 136, + 628.3333333333333, + 475, + 646.9999999999999 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 661, + 507, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 332, + 675 + ], + "score": 1.0, + "content": "Similar to (Bernstein et al., 2018, Theorem 1), we relax Prob", + "type": "text" + }, + { + "bbox": [ + 332, + 662, + 455, + 673 + ], + "score": 0.89, + "content": "[ \\mathrm { s i g n } ( \\hat { \\mathbf { g } } _ { k , i } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { i } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 659, + 506, + 675 + ], + "score": 1.0, + "content": "by Markov’s", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 670, + 148, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 670, + 148, + 685 + ], + "score": 1.0, + "content": "inequality,", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 659, + 506, + 685 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 691, + 465, + 732 + ], + "lines": [ + { + "bbox": [ + 145, + 691, + 465, + 732 + ], + "spans": [ + { + "bbox": [ + 145, + 691, + 465, + 732 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\mathrm { P r o b } \\left[ \\mathrm { s i g n } ( \\hat { \\bf g } _ { k , i } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( { \\bf x } _ { k } ) ) _ { i } ) \\right] \\leq \\mathrm { P r o b } [ \\vert \\hat { \\bf g } _ { k , i } - ( \\nabla f _ { \\mu } ( { \\bf x } _ { k } ) ) _ { i } \\vert \\ge \\vert ( \\nabla f _ { \\mu } ( { \\bf x } _ { k } ) ) _ { i } \\vert ] } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\end{array}", + "type": "interline_equation", + "image_path": "e2e99a6e48f2448296893dd04dc4e913c3199f48d9eb787a4103404ef0be2e6f.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 145, + 691, + 465, + 704.6666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 145, + 704.6666666666666, + 465, + 718.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 145, + 718.3333333333333, + 465, + 731.9999999999999 + ], + "spans": [], + "index": 27 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 83, + 242, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 242, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 242, + 95 + ], + "score": 1.0, + "content": "Substituting (18) into (17), we obtain", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 98, + 470, + 201 + ], + "lines": [ + { + "bbox": [ + 141, + 98, + 470, + 201 + ], + "spans": [ + { + "bbox": [ + 141, + 98, + 470, + 201 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { \\displaystyle \\mathbb { E } [ f _ { \\mu } ( \\mathbf { x } _ { k + 1 } ) - f _ { \\mu } ( \\mathbf { x } _ { k } ) ] \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\displaystyle \\sum _ { i = 1 } ^ { d } \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k , i } - ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { i } \\| ] } & { } \\\\ { \\displaystyle = - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } ] } & { } \\\\ { \\displaystyle \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\sqrt { d } \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ] } & { } \\\\ { \\displaystyle \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\sqrt { d } \\sqrt { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ] } , } & { } \\end{array}", + "type": "interline_equation", + "image_path": "aa301521b8bfacb91453c40a7d5e3ed43b5110c07afafb37b2da1ec10909279a.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 141, + 98, + 470, + 132.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 141, + 132.33333333333334, + 470, + 166.66666666666669 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 141, + 166.66666666666669, + 470, + 201.00000000000003 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 504, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 256, + 217 + ], + "score": 1.0, + "content": "where the second inequality holds due to", + "type": "text" + }, + { + "bbox": [ + 257, + 204, + 320, + 216 + ], + "score": 0.92, + "content": "\\| \\mathbf { x } \\| _ { 1 } \\leq { \\sqrt { d } } \\| \\mathbf { x } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 204, + 505, + 217 + ], + "score": 1.0, + "content": ", and the last inequality holds by applying Jensen’s", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 214, + 447, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 230, + 230 + ], + "score": 1.0, + "content": "inequality to the concave function", + "type": "text" + }, + { + "bbox": [ + 231, + 217, + 242, + 228 + ], + "score": 0.87, + "content": "\\sqrt { \\cdot }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 214, + 260, + 230 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 261, + 216, + 443, + 228 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ] \\le \\sqrt { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } ] } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 214, + 447, + 230 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 297, + 244 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 297, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 297, + 245 + ], + "score": 1.0, + "content": "Taking sum of both sides of (19), we then obtain (5).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 258, + 259, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 260, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 260, + 273 + ], + "score": 1.0, + "content": "3 PROOF OF PROPOSITION 2", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 190, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 191, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 191, + 294 + ], + "score": 1.0, + "content": "We recall from (3) that", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 297, + 412, + 327 + ], + "lines": [ + { + "bbox": [ + 199, + 297, + 412, + 327 + ], + "spans": [ + { + "bbox": [ + 199, + 297, + 412, + 327 + ], + "score": 0.92, + "content": "\\hat { \\mathbf { g } } _ { k } = \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { T } _ { k } } \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) , \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) = \\frac { 1 } { q } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ; \\mathbf { u } _ { i , j } ) .", + "type": "interline_equation", + "image_path": "a0de2e6a4a03357fc7b82b89d8e69737e665f1207f5ab1a4d303067515ff0fc2.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 297, + 412, + 312.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 199, + 312.0, + 412, + 327.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 332, + 390, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 391, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 120, + 348 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 120, + 332, + 223, + 345 + ], + "score": 0.93, + "content": "\\mathbf { z } _ { i } : = \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 330, + 239, + 348 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 332, + 363, + 345 + ], + "score": 0.94, + "content": "\\mathbf { z } _ { i , j } = \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ; \\mathbf { u } _ { i , j } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 330, + 391, + 348 + ], + "score": 1.0, + "content": ". 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Note that these two sources of randomness are independent, and the random direction samples", + "type": "text" + }, + { + "bbox": [ + 479, + 393, + 504, + 405 + ], + "score": 0.91, + "content": "\\{ \\bar { \\mathbf { u } } _ { i , j } \\}", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 403, + 141, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 141, + 414 + ], + "score": 1.0, + "content": "are i.i.d..", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "Next, we discuss two types of mini-batch sampling: a) mini-batch samples without replacement, and b) mini-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 429, + 226, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 226, + 441 + ], + "score": 1.0, + "content": "batch samples with replacement.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 506, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 154, + 457 + ], + "score": 1.0, + "content": "Suppose that", + "type": "text" + }, + { + "bbox": [ + 154, + 446, + 165, + 455 + ], + "score": 0.86, + "content": "\\mathcal { T } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 445, + 273, + 457 + ], + "score": 1.0, + "content": "is a uniform random subset of", + "type": "text" + }, + { + "bbox": [ + 274, + 446, + 285, + 456 + ], + "score": 0.86, + "content": "[ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "(no replacement), motivated by (Lei et al., 2017, Lemma A.1)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 455, + 507, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 212, + 468 + ], + "score": 1.0, + "content": "we introduce a new variable", + "type": "text" + }, + { + "bbox": [ + 212, + 456, + 273, + 466 + ], + "score": 0.91, + "content": "W _ { i } = I ( i \\in \\mathcal { T } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 455, + 301, + 468 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 302, + 456, + 307, + 464 + ], + "score": 0.82, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 455, + 414, + 468 + ], + "score": 1.0, + "content": "is an indicator function, and", + "type": "text" + }, + { + "bbox": [ + 414, + 456, + 468, + 466 + ], + "score": 0.92, + "content": "I ( i \\in \\mathcal { T } _ { k } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 455, + 477, + 468 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 477, + 456, + 502, + 465 + ], + "score": 0.91, + "content": "i \\in \\mathcal { Z } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 455, + 507, + 468 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 465, + 243, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 243, + 476 + ], + "score": 1.0, + "content": "and 0 otherwise. As a result, we have", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 479, + 410, + 503 + ], + "lines": [ + { + "bbox": [ + 200, + 479, + 410, + 503 + ], + "spans": [ + { + "bbox": [ + 200, + 479, + 410, + 503 + ], + "score": 0.93, + "content": "\\mathbb { E } [ W _ { i } ^ { 2 } ] = \\mathbb { E } [ W _ { i } ] = \\frac { b } { n } , \\quad \\mathbb { E } [ W _ { i } W _ { j } ] = \\frac { b ( b - 1 ) } { n ( n - 1 ) } , \\ i \\neq j .", + "type": "interline_equation", + "image_path": "6bc998c9e7a0be95e00570ce0d0a8894c5e8614a8539b705ee0340c52e334cb1.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 200, + 479, + 410, + 503 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 256, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 256, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 203, + 520 + ], + "score": 1.0, + "content": "From (21), the variance of", + "type": "text" + }, + { + "bbox": [ + 203, + 507, + 214, + 518 + ], + "score": 0.89, + "content": "\\hat { \\bf g } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 505, + 256, + 520 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 522, + 490, + 683 + ], + "lines": [ + { + "bbox": [ + 120, + 522, + 490, + 683 + ], + "spans": [ + { + "bbox": [ + 120, + 522, + 490, + 683 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { { \\mathbb { E } [ \\| \\displaystyle \\frac { 1 } { b } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { Z } _ { a } \\| { \\mathbb { Z } } _ { 2 } ] ^ { 2 } ] } } & { = \\frac { 1 } { b ^ { 2 } } ( \\displaystyle \\sum _ { i = 1 } ^ { n } \\big ( { \\mathbb { E } } _ { i } \\langle z _ { k } | W _ { i } ^ { 2 } | { \\mathbb { E } } _ { u } \\| { \\mathbb { Z } } _ { i } \\| { \\mathbb { Z } } _ { 2 } | { \\mathbb { Z } } _ { 1 } \\big ) + \\displaystyle \\sum _ { i \\neq j } { \\mathbb { E } } _ { i , j \\in { \\mathbb { Z } } _ { k } } \\big \\langle W _ { i } W _ { j } \\big | | { \\mathbb { Z } } _ { \\mathbb { Z } _ { 1 } } \\big | { \\mathbb { Z } } _ { u } \\big | { \\mathbb { Z } } _ { 2 } \\big | ) ) } \\\\ & { = \\displaystyle \\frac { 1 } { b ^ { 2 } } ( \\displaystyle \\frac { b } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| [ z _ { 1 } | { \\mathbb { Z } } _ { 2 } ] \\| + \\displaystyle \\frac { b ( b - 1 ) } { n ( n - 1 ) } \\| \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| z _ { \\mathbb { Z } ^ { i } } \\| { \\mathbb { Z } } _ { 1 } \\| _ { 2 } ^ { 2 } - \\displaystyle \\frac { b ( b - 1 ) } { n ( n - 1 ) } \\displaystyle \\sum _ { i = 1 } ^ { n } \\| \\mathbb { E } _ { \\mathbb { Z } } \\| z _ { \\mathbb { Z } ^ { i } } \\| { \\mathbb { Z } } _ { 2 } ^ { 2 } ) } \\\\ & { \\overset { ( a ) } { = } \\displaystyle \\frac { 1 } { b n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| [ z _ { 1 } | { \\mathbb { Z } } _ { i } ] \\| ^ { 2 } - \\displaystyle \\frac { b } { b ( n - 1 ) n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\| \\mathbb { E } _ { \\mathbb { Z } } \\| z _ { i } \\| { \\mathbb { Z } } _ { 2 } ^ { 2 } } \\\\ & = \\displaystyle \\frac { 1 } { b n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| [ z _ { 1 } | { \\mathbb { Z } } _ { 2 } ] - \\displaystyle \\frac \\end{array}", + "type": "interline_equation", + "image_path": "9091acabac36783b1cd9ec55befab3b45e80bce0c89483dae6aefe1f849e1fed.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 120, + 522, + 490, + 575.6666666666666 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 120, + 575.6666666666666, + 490, + 629.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 120, + 629.3333333333333, + 490, + 682.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 686, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 685, + 508, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 685, + 236, + 702 + ], + "score": 1.0, + "content": "In (23), the equality (a) holds since", + "type": "text" + }, + { + "bbox": [ + 236, + 687, + 358, + 699 + ], + "score": 0.9, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i } ] = \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) - 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f _ { \\mu } ( \\mathbf { x } _ { k } ) ] \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\displaystyle \\sum _ { i = 1 } ^ { d } \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k , i } - ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { i } \\| ] } & { } \\\\ { \\displaystyle = - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } ] } & { } \\\\ { \\displaystyle \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\sqrt { d } \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ] } & { } \\\\ { \\displaystyle \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\sqrt { d } \\sqrt { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ] } , } & { } \\end{array}", + "type": "interline_equation", + "image_path": "aa301521b8bfacb91453c40a7d5e3ed43b5110c07afafb37b2da1ec10909279a.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 141, + 98, + 470, + 132.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 141, + 132.33333333333334, + 470, + 166.66666666666669 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 141, + 166.66666666666669, + 470, + 201.00000000000003 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 504, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 256, + 217 + ], + "score": 1.0, + "content": "where the second inequality holds due to", + "type": "text" + }, + { + "bbox": [ + 257, + 204, + 320, + 216 + ], + "score": 0.92, + "content": "\\| \\mathbf { x } \\| _ { 1 } \\leq { \\sqrt { d } } \\| \\mathbf { x } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 204, + 505, + 217 + ], + "score": 1.0, + "content": ", and the last inequality holds by applying Jensen’s", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 214, + 447, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 230, + 230 + ], + "score": 1.0, + "content": "inequality to the concave function", + "type": "text" + }, + { + "bbox": [ + 231, + 217, + 242, + 228 + ], + "score": 0.87, + "content": "\\sqrt { \\cdot }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 214, + 260, + 230 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 261, + 216, + 443, + 228 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ] \\le \\sqrt { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } ] } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 214, + 447, + 230 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 204, + 505, + 230 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 297, + 244 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 297, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 297, + 245 + ], + "score": 1.0, + "content": "Taking sum of both sides of (19), we then obtain (5).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 232, + 297, + 245 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 258, + 259, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 260, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 260, + 273 + ], + "score": 1.0, + "content": "3 PROOF OF PROPOSITION 2", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 190, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 191, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 191, + 294 + ], + "score": 1.0, + "content": "We recall from (3) that", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 280, + 191, + 294 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 297, + 412, + 327 + ], + "lines": [ + { + "bbox": [ + 199, + 297, + 412, + 327 + ], + "spans": [ + { + "bbox": [ + 199, + 297, + 412, + 327 + ], + "score": 0.92, + "content": "\\hat { \\mathbf { g } } _ { k } = \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { T } _ { k } } \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) , \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) = \\frac { 1 } { q } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ; \\mathbf { u } _ { i , j } ) .", + "type": "interline_equation", + "image_path": "a0de2e6a4a03357fc7b82b89d8e69737e665f1207f5ab1a4d303067515ff0fc2.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 297, + 412, + 312.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 199, + 312.0, + 412, + 327.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 332, + 390, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 391, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 120, + 348 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 120, + 332, + 223, + 345 + ], + "score": 0.93, + "content": "\\mathbf { z } _ { i } : = \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 330, + 239, + 348 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 332, + 363, + 345 + ], + "score": 0.94, + "content": "\\mathbf { z } _ { i , j } = \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ; \\mathbf { u } _ { i , j } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 330, + 391, + 348 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 330, + 391, + 348 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 349, + 393, + 380 + ], + "lines": [ + { + "bbox": [ + 217, + 349, + 393, + 380 + ], + "spans": [ + { + "bbox": [ + 217, + 349, + 393, + 380 + ], + "score": 0.94, + "content": "{ \\hat { \\mathbf { g } } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) = { \\frac { 1 } { b } } \\sum _ { i \\in { \\mathcal { T } } _ { k } } \\mathbf { z } _ { i } = { \\frac { 1 } { b q } } \\sum _ { i \\in { \\mathcal { T } } _ { k } } \\sum _ { j = 1 } ^ { q } \\mathbf { z } _ { i , j }", + "type": "interline_equation", + "image_path": "9dc517eed6652b72100d1d0e7c12acc932d5b253f96ebc5a9bc59e5acc808587.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 217, + 349, + 393, + 364.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 217, + 364.5, + 393, + 380.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 506, + 414 + ], + "lines": [ + { + "bbox": [ + 104, + 381, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 104, + 381, + 343, + 396 + ], + "score": 1.0, + "content": "where there are two sources of randomness: a) minibatch sampling", + "type": "text" + }, + { + "bbox": [ + 343, + 384, + 367, + 393 + ], + "score": 0.91, + "content": "i \\in \\mathcal { Z } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 381, + 506, + 396 + ], + "score": 1.0, + "content": ", and b) the random direction sampling", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 393, + 504, + 406 + ], + "spans": [ + { + "bbox": [ + 107, + 394, + 140, + 405 + ], + "score": 0.89, + "content": "\\mathbf { u } = \\mathbf { u } _ { i , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 393, + 479, + 406 + ], + "score": 1.0, + "content": ". Note that these two sources of randomness are independent, and the random direction samples", + "type": "text" + }, + { + "bbox": [ + 479, + 393, + 504, + 405 + ], + "score": 0.91, + "content": "\\{ \\bar { \\mathbf { u } } _ { i , j } \\}", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 403, + 141, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 141, + 414 + ], + "score": 1.0, + "content": "are i.i.d..", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 381, + 506, + 414 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "Next, we discuss two types of mini-batch sampling: a) mini-batch samples without replacement, and b) mini-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 429, + 226, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 226, + 441 + ], + "score": 1.0, + "content": "batch samples with replacement.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 419, + 506, + 441 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 506, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 154, + 457 + ], + "score": 1.0, + "content": "Suppose that", + "type": "text" + }, + { + "bbox": [ + 154, + 446, + 165, + 455 + ], + "score": 0.86, + "content": "\\mathcal { T } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 445, + 273, + 457 + ], + "score": 1.0, + "content": "is a uniform random subset of", + "type": "text" + }, + { + "bbox": [ + 274, + 446, + 285, + 456 + ], + "score": 0.86, + "content": "[ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "(no replacement), motivated by (Lei et al., 2017, Lemma A.1)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 455, + 507, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 212, + 468 + ], + "score": 1.0, + "content": "we introduce a new variable", + "type": "text" + }, + { + "bbox": [ + 212, + 456, + 273, + 466 + ], + "score": 0.91, + "content": "W _ { i } = I ( i \\in \\mathcal { T } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 455, + 301, + 468 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 302, + 456, + 307, + 464 + ], + "score": 0.82, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 455, + 414, + 468 + ], + "score": 1.0, + "content": "is an indicator function, and", + "type": "text" + }, + { + "bbox": [ + 414, + 456, + 468, + 466 + ], + "score": 0.92, + "content": "I ( i \\in \\mathcal { T } _ { k } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 455, + 477, + 468 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 477, + 456, + 502, + 465 + ], + "score": 0.91, + "content": "i \\in \\mathcal { Z } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 455, + 507, + 468 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 465, + 243, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 243, + 476 + ], + "score": 1.0, + "content": "and 0 otherwise. As a result, we have", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 445, + 507, + 476 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 479, + 410, + 503 + ], + "lines": [ + { + "bbox": [ + 200, + 479, + 410, + 503 + ], + "spans": [ + { + "bbox": [ + 200, + 479, + 410, + 503 + ], + "score": 0.93, + "content": "\\mathbb { E } [ W _ { i } ^ { 2 } ] = \\mathbb { E } [ W _ { i } ] = \\frac { b } { n } , \\quad \\mathbb { E } [ W _ { i } W _ { j } ] = \\frac { b ( b - 1 ) } { n ( n - 1 ) } , \\ i \\neq j .", + "type": "interline_equation", + "image_path": "6bc998c9e7a0be95e00570ce0d0a8894c5e8614a8539b705ee0340c52e334cb1.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 200, + 479, + 410, + 503 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 256, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 256, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 203, + 520 + ], + "score": 1.0, + "content": "From (21), the variance of", + "type": "text" + }, + { + "bbox": [ + 203, + 507, + 214, + 518 + ], + "score": 0.89, + "content": "\\hat { \\bf g } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 505, + 256, + 520 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 505, + 256, + 520 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 522, + 490, + 683 + ], + "lines": [ + { + "bbox": [ + 120, + 522, + 490, + 683 + ], + "spans": [ + { + "bbox": [ + 120, + 522, + 490, + 683 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { { \\mathbb { E } [ \\| \\displaystyle \\frac { 1 } { b } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { Z } _ { a } \\| { \\mathbb { Z } } _ { 2 } ] ^ { 2 } ] } } & { = \\frac { 1 } { b ^ { 2 } } ( \\displaystyle \\sum _ { i = 1 } ^ { n } \\big ( { \\mathbb { E } } _ { i } \\langle z _ { k } | W _ { i } ^ { 2 } | { \\mathbb { E } } _ { u } \\| { \\mathbb { Z } } _ { i } \\| { \\mathbb { Z } } _ { 2 } | { \\mathbb { Z } } _ { 1 } \\big ) + \\displaystyle \\sum _ { i \\neq j } { \\mathbb { E } } _ { i , j \\in { \\mathbb { Z } } _ { k } } \\big \\langle W _ { i } W _ { j } \\big | | { \\mathbb { Z } } _ { \\mathbb { Z } _ { 1 } } \\big | { \\mathbb { Z } } _ { u } \\big | { \\mathbb { Z } } _ { 2 } \\big | ) ) } \\\\ & { = \\displaystyle \\frac { 1 } { b ^ { 2 } } ( \\displaystyle \\frac { b } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| [ z _ { 1 } | { \\mathbb { Z } } _ { 2 } ] \\| + \\displaystyle \\frac { b ( b - 1 ) } { n ( n - 1 ) } \\| \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| z _ { \\mathbb { Z } ^ { i } } \\| { \\mathbb { Z } } _ { 1 } \\| _ { 2 } ^ { 2 } - \\displaystyle \\frac { b ( b - 1 ) } { n ( n - 1 ) } \\displaystyle \\sum _ { i = 1 } ^ { n } \\| \\mathbb { E } _ { \\mathbb { Z } } \\| z _ { \\mathbb { Z } ^ { i } } \\| { \\mathbb { Z } } _ { 2 } ^ { 2 } ) } \\\\ & { \\overset { ( a ) } { = } \\displaystyle \\frac { 1 } { b n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| [ z _ { 1 } | { \\mathbb { Z } } _ { i } ] \\| ^ { 2 } - \\displaystyle \\frac { b } { b ( n - 1 ) n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\| \\mathbb { E } _ { \\mathbb { Z } } \\| z _ { i } \\| { \\mathbb { Z } } _ { 2 } ^ { 2 } } \\\\ & = \\displaystyle \\frac { 1 } { b n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbb { Z } } \\| [ z _ { 1 } | { \\mathbb { Z } } _ { 2 } ] - \\displaystyle \\frac \\end{array}", + "type": "interline_equation", + "image_path": "9091acabac36783b1cd9ec55befab3b45e80bce0c89483dae6aefe1f849e1fed.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 120, + 522, + 490, + 575.6666666666666 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 120, + 575.6666666666666, + 490, + 629.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 120, + 629.3333333333333, + 490, + 682.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 686, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 685, + 508, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 685, + 236, + 702 + ], + "score": 1.0, + "content": "In (23), the equality (a) holds since", + "type": "text" + }, + { + "bbox": [ + 236, + 687, + 358, + 699 + ], + "score": 0.9, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i } ] = \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 685, + 375, + 702 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 375, + 686, + 470, + 699 + ], + "score": 0.93, + "content": "\\begin{array} { r } { f _ { \\mu } ( \\mathbf { x } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } f _ { i , \\mu } ( \\mathbf { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 685, + 508, + 702 + ], + "score": 1.0, + "content": "from (4),", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 698, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 225, + 712 + ], + "score": 1.0, + "content": "where we have used the fact that", + "type": "text" + }, + { + "bbox": [ + 225, + 699, + 325, + 711 + ], + "score": 0.88, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) ] = \\hat { \\nabla } f _ { i , \\mu } ( \\mathbf { x } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 698, + 488, + 712 + ], + "score": 1.0, + "content": "(Liu et al., 2018c, Lemma. 1), and recall that", + "type": "text" + }, + { + "bbox": [ + 489, + 700, + 504, + 711 + ], + "score": 0.88, + "content": "f _ { i , \\mu }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 707, + 503, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 707, + 228, + 723 + ], + "score": 1.0, + "content": "denotes the smoothing function of", + "type": "text" + }, + { + "bbox": [ + 228, + 712, + 236, + 721 + ], + "score": 0.86, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 707, + 322, + 723 + ], + "score": 1.0, + "content": ". The above implies that", + "type": "text" + }, + { + "bbox": [ + 322, + 710, + 503, + 721 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\sum _ { i = 1 } ^ { n } { \\mathbb { E } } _ { \\mathbf { u } } [ { \\mathbf { z } } _ { i } ] = \\sum _ { i } \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) - n \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) = \\dot { \\mathbf { 0 } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 720, + 398, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 227, + 734 + ], + "score": 1.0, + "content": "And the equality (b) holds due to", + "type": "text" + }, + { + "bbox": [ + 227, + 721, + 394, + 732 + ], + "score": 0.9, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i } \\| _ { 2 } ^ { 2 } ] - \\| \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i } ] \\| _ { 2 } ^ { 2 } = \\mathbb { E } _ { \\mathbf { u } } \\| \\mathbf { z } _ { i } - \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i } ] \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 720, + 398, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 685, + 508, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 272, + 95 + ], + "score": 1.0, + "content": "On the other hand, suppose that the mini-batch", + "type": "text" + }, + { + "bbox": [ + 272, + 84, + 283, + 93 + ], + "score": 0.87, + "content": "\\mathcal { T } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "contains i.i.d. samples (namely, with replacement), the vectors", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 504, + 105 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 125, + 104 + ], + "score": 0.91, + "content": "\\left\\{ \\mathbf { z } _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 92, + 504, + 105 + ], + "score": 1.0, + "content": "are then i.i.d. under both mini-batch sampling and random direction sampling. Therefore, we obtain that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 109, + 434, + 177 + ], + "lines": [ + { + "bbox": [ + 176, + 109, + 434, + 177 + ], + "spans": [ + { + "bbox": [ + 176, + 109, + 434, + 177 + ], + "score": 0.94, + "content": "\\mathbb { E } \\left[ \\left. \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { I } _ { k } } \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } \\right] = \\frac { 1 } { b ^ { 2 } } \\left[ \\mathbb { E } \\left[ \\sum _ { i \\in \\mathcal { I } _ { k } } \\| \\mathbf { z } _ { i } \\| _ { 2 } ^ { 2 } \\right] + \\mathbb { E } \\left[ \\sum _ { i \\neq j , i , j \\in \\mathcal { I } _ { k } } \\langle \\mathbf { z } _ { i } , \\mathbf { z } _ { j } \\rangle \\right] \\right]", + "type": "interline_equation", + "image_path": "f9c5e10f749d8e97abd7f71645e81e3df5882e45231522de3e6a2f7bf592ab17.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 176, + 109, + 434, + 131.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 176, + 131.66666666666666, + 434, + 154.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 176, + 154.33333333333331, + 434, + 176.99999999999997 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 181, + 506, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 507, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 243, + 197 + ], + "score": 1.0, + "content": "where the second equality holds since", + "type": "text" + }, + { + "bbox": [ + 243, + 182, + 504, + 195 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbb { E } _ { i \\in \\mathcal { T } _ { k } , \\mathbf { u } } [ \\mathbf { z } _ { i } ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i } ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) = \\mathbf { 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 180, + 507, + 197 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 198, + 273, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 273, + 211 + ], + "score": 1.0, + "content": "Combining both (23) and (24), we obtain that", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 214, + 458, + 253 + ], + "lines": [ + { + "bbox": [ + 153, + 214, + 458, + 253 + ], + "spans": [ + { + "bbox": [ + 153, + 214, + 458, + 253 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left[ \\left. \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { I } _ { k } } \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } \\right] \\leq \\frac { \\alpha _ { b } } { b n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\left. \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } ] + \\frac { \\beta _ { b } } { b n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\left. \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) \\right. _ { 2 } ^ { 2 } ] .", + "type": "interline_equation", + "image_path": "c5ba89f03c612eab5032f479d5ddae6faa657cd98d15cd68af7a3eed7bb9452a.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 153, + 214, + 458, + 227.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 153, + 227.0, + 458, + 240.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 153, + 240.0, + 458, + 253.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 506, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 139, + 270 + ], + "score": 1.0, + "content": "In (25),", + "type": "text" + }, + { + "bbox": [ + 139, + 258, + 172, + 268 + ], + "score": 0.89, + "content": "\\alpha _ { b } \\ = \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 257, + 191, + 270 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 258, + 223, + 268 + ], + "score": 0.89, + "content": "\\beta _ { b } ~ = ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 257, + 404, + 270 + ], + "score": 1.0, + "content": "if the mini-batch contains i.i.d. samples from", + "type": "text" + }, + { + "bbox": [ + 405, + 258, + 416, + 269 + ], + "score": 0.86, + "content": "[ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "with replacement, and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 268, + 434, + 280 + ], + "spans": [ + { + "bbox": [ + 107, + 268, + 163, + 279 + ], + "score": 0.92, + "content": "\\alpha _ { b } = I ( b < n )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 268, + 180, + 280 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 180, + 268, + 235, + 279 + ], + "score": 0.9, + "content": "\\beta _ { b } = I ( b > 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 268, + 434, + 280 + ], + "score": 1.0, + "content": "if samples are randomly selected without replacement.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 255, + 295 + ], + "lines": [ + { + "bbox": [ + 104, + 281, + 256, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 281, + 191, + 297 + ], + "score": 1.0, + "content": "In (25), we next bound", + "type": "text" + }, + { + "bbox": [ + 191, + 282, + 252, + 296 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i } \\| _ { 2 } ^ { 2 } ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 281, + 256, + 297 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 300, + 496, + 390 + ], + "lines": [ + { + "bbox": [ + 115, + 300, + 496, + 390 + ], + "spans": [ + { + "bbox": [ + 115, + 300, + 496, + 390 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } { \\displaystyle \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i } \\| _ { 2 } ^ { 2 } ] } & { \\stackrel { ( \\mathrm { 2 1 } ) } { = } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } \\left\\| \\frac { 1 } { q } \\sum _ { j = 1 } ^ { q } \\mathbf { z } _ { i , j } \\right\\| _ { 2 } ^ { 2 } = \\frac { 1 } { n q ^ { 2 } } \\sum _ { i } \\left[ \\sum _ { j } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , j } \\| _ { 2 } ^ { 2 } ] + \\sum _ { j \\neq k } \\langle \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , j } ] , \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , k } ] \\rangle \\right] } & \\\\ & { } & { \\displaystyle = \\frac { 1 } { n q ^ { 2 } } \\sum _ { i } \\left[ q \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] + ( q ^ { 2 } - q ) \\| \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , 1 } ] \\| _ { 2 } ^ { 2 } \\right] } & \\\\ & { } & { \\displaystyle = \\frac { 1 } { q n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] + \\frac { q - 1 } { q n } \\sum _ { i } \\| \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , 1 } ] \\| _ { 2 } ^ { 2 } , } & { ( 2 } \\end{array}", + "type": "interline_equation", + "image_path": "429540654c6ed6179dac78719e1e53c5d31444856d8fdea88533a12a4be621f3.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 115, + 300, + 496, + 330.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 115, + 330.0, + 496, + 360.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 115, + 360.0, + 496, + 390.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 504, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 229, + 409 + ], + "score": 1.0, + "content": "where we have used the facts that", + "type": "text" + }, + { + "bbox": [ + 230, + 396, + 302, + 408 + ], + "score": 0.95, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , j } ] = \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 395, + 318, + 409 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 319, + 396, + 419, + 408 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , j } \\| _ { 2 } ^ { 2 } ] = \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 395, + 447, + 409 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 448, + 397, + 453, + 407 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 395, + 506, + 409 + ], + "score": 1.0, + "content": "since random", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 405, + 274, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 169, + 419 + ], + "score": 1.0, + "content": "direction vectors", + "type": "text" + }, + { + "bbox": [ + 169, + 406, + 207, + 419 + ], + "score": 0.94, + "content": "\\{ \\mathbf { u } _ { i , j } \\} _ { j = 1 } ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 405, + 274, + 419 + ], + "score": 1.0, + "content": "are i.i.d. samples.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 262, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 262, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 200, + 438 + ], + "score": 1.0, + "content": "In (26), we further bound", + "type": "text" + }, + { + "bbox": [ + 200, + 424, + 258, + 436 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 423, + 262, + 438 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 441, + 448, + 522 + ], + "lines": [ + { + "bbox": [ + 162, + 441, + 448, + 522 + ], + "spans": [ + { + "bbox": [ + 162, + 441, + 448, + 522 + ], + "score": 0.95, + "content": "\\begin{array} { r l r } { { \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] = \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } \\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ; \\mathbf { u } _ { i , 1 } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { \\quad = \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { i , \\mu } + \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } ] } \\\\ & { } & { \\quad \\leq \\displaystyle \\frac { 2 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { i , \\mu } \\| _ { 2 } ^ { 2 } ] + \\frac { 2 } { n } \\sum _ { i } \\| \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "f4e59a000f669f461d13fb6d5d95bfa7d3cbde831e6d51c86043bfa63d28236c.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 162, + 441, + 448, + 468.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 162, + 468.0, + 448, + 495.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 162, + 495.0, + 448, + 522.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 527, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 104, + 525, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 525, + 214, + 542 + ], + "score": 1.0, + "content": "where for ease of notation, let", + "type": "text" + }, + { + "bbox": [ + 214, + 527, + 297, + 540 + ], + "score": 0.71, + "content": "\\hat { \\nabla } f _ { i } : = \\hat { \\nabla } f _ { i } ( \\mathbf x _ { k } ; \\mathbf u _ { i , 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 525, + 301, + 542 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 302, + 527, + 378, + 540 + ], + "score": 0.75, + "content": "\\hat { \\nabla } f _ { i , \\mu } : = \\hat { \\nabla } f _ { i , \\mu } ( \\mathbf x _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 525, + 394, + 542 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 394, + 528, + 461, + 540 + ], + "score": 0.93, + "content": "\\nabla f _ { \\mu } : = \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 525, + 506, + 542 + ], + "score": 1.0, + "content": ". According", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 537, + 351, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 351, + 550 + ], + "score": 1.0, + "content": "to (Liu et al., 2018c, Lemma 1), the first term at RHS of (27) yields", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 554, + 452, + 578 + ], + "lines": [ + { + "bbox": [ + 158, + 554, + 452, + 578 + ], + "spans": [ + { + "bbox": [ + 158, + 554, + 452, + 578 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { i , \\mu } \\| _ { 2 } ^ { 2 } ] \\le 2 d \\| \\nabla f _ { i } \\| _ { 2 } ^ { 2 } + \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } \\le 2 d \\sigma ^ { 2 } + \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } : = C ( d , \\mu ) ,", + "type": "interline_equation", + "image_path": "c61b8704bee28cb58e0a56425b076129ae98c523a6039639c1cfc6803950d259.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 158, + 554, + 452, + 578 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 504, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 353, + 593 + ], + "score": 1.0, + "content": "where the last inequality holds due to A2. Based on the definition of", + "type": "text" + }, + { + "bbox": [ + 353, + 583, + 363, + 593 + ], + "score": 0.88, + "content": "f _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 581, + 504, + 593 + ], + "score": 1.0, + "content": ", the second term at RHS of (27) yields", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 597, + 457, + 652 + ], + "lines": [ + { + "bbox": [ + 154, + 597, + 457, + 652 + ], + "spans": [ + { + "bbox": [ + 154, + 597, + 457, + 652 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } { { \\displaystyle \\frac { 1 } { n } \\sum _ { i } \\| \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } = \\frac { 1 } { n } \\sum _ { i } \\| \\mathbb { E } _ { \\mathbf { v } } \\big [ \\nabla f _ { i } ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) - \\nabla f ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) \\big ] \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { \\leq \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { v } } \\big [ \\| \\nabla f _ { i } ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) - \\nabla f ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) \\| _ { 2 } ^ { 2 } \\big ] \\leq 4 \\sigma ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "5ab99398c65b7b0f91278f84e8e10870ecbb4356c741b286cb0884855700c347.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 154, + 597, + 457, + 615.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 154, + 615.3333333333334, + 457, + 633.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 154, + 633.6666666666667, + 457, + 652.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 456, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 457, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 280, + 671 + ], + "score": 1.0, + "content": "where we have used the Jensen’s inequality and", + "type": "text" + }, + { + "bbox": [ + 280, + 657, + 418, + 670 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\| \\nabla f _ { i } ( \\mathbf { x } ) - \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } \\leq 4 \\sigma ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 655, + 457, + 671 + ], + "score": 1.0, + "content": "under A2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 270, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 673, + 269, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 269, + 686 + ], + "score": 1.0, + "content": "Substituting (28) and (29) into (27), we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 689, + 473, + 716 + ], + "lines": [ + { + "bbox": [ + 122, + 689, + 473, + 716 + ], + "spans": [ + { + "bbox": [ + 122, + 689, + 473, + 716 + ], + "score": 0.92, + "content": "\\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] = \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } ] \\leq 4 d \\sigma ^ { 2 } + \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } + 8 \\sigma ^ { 2 } = 2 C ( d , \\mu ) + 8 \\sigma ^ { 2 } ,", + "type": "interline_equation", + "image_path": "c0c76bbf2b2a8f13e7661b5e8ce2eb8b26e7be4f4698ec9489caca94e69db80b.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 122, + 689, + 473, + 716 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 721, + 234, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 719, + 235, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 130, + 734 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 131, + 721, + 160, + 732 + ], + "score": 0.93, + "content": "C ( d , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 719, + 235, + 734 + ], + "score": 1.0, + "content": "was defined in (28).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "16", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 272, + 95 + ], + "score": 1.0, + "content": "On the other hand, suppose that the mini-batch", + "type": "text" + }, + { + "bbox": [ + 272, + 84, + 283, + 93 + ], + "score": 0.87, + "content": "\\mathcal { T } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "contains i.i.d. samples (namely, with replacement), the vectors", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 504, + 105 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 125, + 104 + ], + "score": 0.91, + "content": "\\left\\{ \\mathbf { z } _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 92, + 504, + 105 + ], + "score": 1.0, + "content": "are then i.i.d. under both mini-batch sampling and random direction sampling. Therefore, we obtain that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 83, + 505, + 105 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 109, + 434, + 177 + ], + "lines": [ + { + "bbox": [ + 176, + 109, + 434, + 177 + ], + "spans": [ + { + "bbox": [ + 176, + 109, + 434, + 177 + ], + "score": 0.94, + "content": "\\mathbb { E } \\left[ \\left. \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { I } _ { k } } \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } \\right] = \\frac { 1 } { b ^ { 2 } } \\left[ \\mathbb { E } \\left[ \\sum _ { i \\in \\mathcal { I } _ { k } } \\| \\mathbf { z } _ { i } \\| _ { 2 } ^ { 2 } \\right] + \\mathbb { E } \\left[ \\sum _ { i \\neq j , i , j \\in \\mathcal { I } _ { k } } \\langle \\mathbf { z } _ { i } , \\mathbf { z } _ { j } \\rangle \\right] \\right]", + "type": "interline_equation", + "image_path": "f9c5e10f749d8e97abd7f71645e81e3df5882e45231522de3e6a2f7bf592ab17.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 176, + 109, + 434, + 131.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 176, + 131.66666666666666, + 434, + 154.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 176, + 154.33333333333331, + 434, + 176.99999999999997 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 181, + 506, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 507, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 243, + 197 + ], + "score": 1.0, + "content": "where the second equality holds since", + "type": "text" + }, + { + "bbox": [ + 243, + 182, + 504, + 195 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbb { E } _ { i \\in \\mathcal { T } _ { k } , \\mathbf { u } } [ \\mathbf { z } _ { i } ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i } ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) = \\mathbf { 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 180, + 507, + 197 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 198, + 273, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 273, + 211 + ], + "score": 1.0, + "content": "Combining both (23) and (24), we obtain that", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 180, + 507, + 211 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 214, + 458, + 253 + ], + "lines": [ + { + "bbox": [ + 153, + 214, + 458, + 253 + ], + "spans": [ + { + "bbox": [ + 153, + 214, + 458, + 253 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left[ \\left. \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { I } _ { k } } \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } \\right] \\leq \\frac { \\alpha _ { b } } { b n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\left. \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } ] + \\frac { \\beta _ { b } } { b n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\left. \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) \\right. _ { 2 } ^ { 2 } ] .", + "type": "interline_equation", + "image_path": "c5ba89f03c612eab5032f479d5ddae6faa657cd98d15cd68af7a3eed7bb9452a.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 153, + 214, + 458, + 227.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 153, + 227.0, + 458, + 240.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 153, + 240.0, + 458, + 253.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 506, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 139, + 270 + ], + "score": 1.0, + "content": "In (25),", + "type": "text" + }, + { + "bbox": [ + 139, + 258, + 172, + 268 + ], + "score": 0.89, + "content": "\\alpha _ { b } \\ = \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 257, + 191, + 270 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 258, + 223, + 268 + ], + "score": 0.89, + "content": "\\beta _ { b } ~ = ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 257, + 404, + 270 + ], + "score": 1.0, + "content": "if the mini-batch contains i.i.d. samples from", + "type": "text" + }, + { + "bbox": [ + 405, + 258, + 416, + 269 + ], + "score": 0.86, + "content": "[ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "with replacement, and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 268, + 434, + 280 + ], + "spans": [ + { + "bbox": [ + 107, + 268, + 163, + 279 + ], + "score": 0.92, + "content": "\\alpha _ { b } = I ( b < n )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 268, + 180, + 280 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 180, + 268, + 235, + 279 + ], + "score": 0.9, + "content": "\\beta _ { b } = I ( b > 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 268, + 434, + 280 + ], + "score": 1.0, + "content": "if samples are randomly selected without replacement.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 257, + 505, + 280 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 255, + 295 + ], + "lines": [ + { + "bbox": [ + 104, + 281, + 256, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 281, + 191, + 297 + ], + "score": 1.0, + "content": "In (25), we next bound", + "type": "text" + }, + { + "bbox": [ + 191, + 282, + 252, + 296 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i } \\| _ { 2 } ^ { 2 } ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 281, + 256, + 297 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 281, + 256, + 297 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 300, + 496, + 390 + ], + "lines": [ + { + "bbox": [ + 115, + 300, + 496, + 390 + ], + "spans": [ + { + "bbox": [ + 115, + 300, + 496, + 390 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } { \\displaystyle \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i } \\| _ { 2 } ^ { 2 } ] } & { \\stackrel { ( \\mathrm { 2 1 } ) } { = } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { u } } \\left\\| \\frac { 1 } { q } \\sum _ { j = 1 } ^ { q } \\mathbf { z } _ { i , j } \\right\\| _ { 2 } ^ { 2 } = \\frac { 1 } { n q ^ { 2 } } \\sum _ { i } \\left[ \\sum _ { j } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , j } \\| _ { 2 } ^ { 2 } ] + \\sum _ { j \\neq k } \\langle \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , j } ] , \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , k } ] \\rangle \\right] } & \\\\ & { } & { \\displaystyle = \\frac { 1 } { n q ^ { 2 } } \\sum _ { i } \\left[ q \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] + ( q ^ { 2 } - q ) \\| \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , 1 } ] \\| _ { 2 } ^ { 2 } \\right] } & \\\\ & { } & { \\displaystyle = \\frac { 1 } { q n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] + \\frac { q - 1 } { q n } \\sum _ { i } \\| \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , 1 } ] \\| _ { 2 } ^ { 2 } , } & { ( 2 } \\end{array}", + "type": "interline_equation", + "image_path": "429540654c6ed6179dac78719e1e53c5d31444856d8fdea88533a12a4be621f3.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 115, + 300, + 496, + 330.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 115, + 330.0, + 496, + 360.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 115, + 360.0, + 496, + 390.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 504, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 229, + 409 + ], + "score": 1.0, + "content": "where we have used the facts that", + "type": "text" + }, + { + "bbox": [ + 230, + 396, + 302, + 408 + ], + "score": 0.95, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , j } ] = \\mathbb { E } _ { \\mathbf { u } } [ \\mathbf { z } _ { i , 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 395, + 318, + 409 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 319, + 396, + 419, + 408 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , j } \\| _ { 2 } ^ { 2 } ] = \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 395, + 447, + 409 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 448, + 397, + 453, + 407 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 395, + 506, + 409 + ], + "score": 1.0, + "content": "since random", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 405, + 274, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 169, + 419 + ], + "score": 1.0, + "content": "direction vectors", + "type": "text" + }, + { + "bbox": [ + 169, + 406, + 207, + 419 + ], + "score": 0.94, + "content": "\\{ \\mathbf { u } _ { i , j } \\} _ { j = 1 } ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 405, + 274, + 419 + ], + "score": 1.0, + "content": "are i.i.d. samples.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 395, + 506, + 419 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 262, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 262, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 200, + 438 + ], + "score": 1.0, + "content": "In (26), we further bound", + "type": "text" + }, + { + "bbox": [ + 200, + 424, + 258, + 436 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 423, + 262, + 438 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 423, + 262, + 438 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 441, + 448, + 522 + ], + "lines": [ + { + "bbox": [ + 162, + 441, + 448, + 522 + ], + "spans": [ + { + "bbox": [ + 162, + 441, + 448, + 522 + ], + "score": 0.95, + "content": "\\begin{array} { r l r } { { \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] = \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } \\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ; \\mathbf { u } _ { i , 1 } ) - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { \\quad = \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { i , \\mu } + \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } ] } \\\\ & { } & { \\quad \\leq \\displaystyle \\frac { 2 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { i , \\mu } \\| _ { 2 } ^ { 2 } ] + \\frac { 2 } { n } \\sum _ { i } \\| \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "f4e59a000f669f461d13fb6d5d95bfa7d3cbde831e6d51c86043bfa63d28236c.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 162, + 441, + 448, + 468.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 162, + 468.0, + 448, + 495.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 162, + 495.0, + 448, + 522.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 527, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 104, + 525, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 525, + 214, + 542 + ], + "score": 1.0, + "content": "where for ease of notation, let", + "type": "text" + }, + { + "bbox": [ + 214, + 527, + 297, + 540 + ], + "score": 0.71, + "content": "\\hat { \\nabla } f _ { i } : = \\hat { \\nabla } f _ { i } ( \\mathbf x _ { k } ; \\mathbf u _ { i , 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 525, + 301, + 542 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 302, + 527, + 378, + 540 + ], + "score": 0.75, + "content": "\\hat { \\nabla } f _ { i , \\mu } : = \\hat { \\nabla } f _ { i , \\mu } ( \\mathbf x _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 525, + 394, + 542 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 394, + 528, + 461, + 540 + ], + "score": 0.93, + "content": "\\nabla f _ { \\mu } : = \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 525, + 506, + 542 + ], + "score": 1.0, + "content": ". According", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 537, + 351, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 351, + 550 + ], + "score": 1.0, + "content": "to (Liu et al., 2018c, Lemma 1), the first term at RHS of (27) yields", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 525, + 506, + 550 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 554, + 452, + 578 + ], + "lines": [ + { + "bbox": [ + 158, + 554, + 452, + 578 + ], + "spans": [ + { + "bbox": [ + 158, + 554, + 452, + 578 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { i , \\mu } \\| _ { 2 } ^ { 2 } ] \\le 2 d \\| \\nabla f _ { i } \\| _ { 2 } ^ { 2 } + \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } \\le 2 d \\sigma ^ { 2 } + \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } : = C ( d , \\mu ) ,", + "type": "interline_equation", + "image_path": "c61b8704bee28cb58e0a56425b076129ae98c523a6039639c1cfc6803950d259.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 158, + 554, + 452, + 578 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 504, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 353, + 593 + ], + "score": 1.0, + "content": "where the last inequality holds due to A2. Based on the definition of", + "type": "text" + }, + { + "bbox": [ + 353, + 583, + 363, + 593 + ], + "score": 0.88, + "content": "f _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 581, + 504, + 593 + ], + "score": 1.0, + "content": ", the second term at RHS of (27) yields", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 581, + 504, + 593 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 597, + 457, + 652 + ], + "lines": [ + { + "bbox": [ + 154, + 597, + 457, + 652 + ], + "spans": [ + { + "bbox": [ + 154, + 597, + 457, + 652 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } { { \\displaystyle \\frac { 1 } { n } \\sum _ { i } \\| \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } = \\frac { 1 } { n } \\sum _ { i } \\| \\mathbb { E } _ { \\mathbf { v } } \\big [ \\nabla f _ { i } ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) - \\nabla f ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) \\big ] \\| _ { 2 } ^ { 2 } } } \\\\ & { } & { \\leq \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { v } } \\big [ \\| \\nabla f _ { i } ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) - \\nabla f ( \\mathbf { x } _ { k } + \\mu \\mathbf { v } ) \\| _ { 2 } ^ { 2 } \\big ] \\leq 4 \\sigma ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "5ab99398c65b7b0f91278f84e8e10870ecbb4356c741b286cb0884855700c347.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 154, + 597, + 457, + 615.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 154, + 615.3333333333334, + 457, + 633.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 154, + 633.6666666666667, + 457, + 652.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 456, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 457, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 280, + 671 + ], + "score": 1.0, + "content": "where we have used the Jensen’s inequality and", + "type": "text" + }, + { + "bbox": [ + 280, + 657, + 418, + 670 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\| \\nabla f _ { i } ( \\mathbf { x } ) - \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } \\leq 4 \\sigma ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 655, + 457, + 671 + ], + "score": 1.0, + "content": "under A2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 655, + 457, + 671 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 270, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 673, + 269, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 269, + 686 + ], + "score": 1.0, + "content": "Substituting (28) and (29) into (27), we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 673, + 269, + 686 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 689, + 473, + 716 + ], + "lines": [ + { + "bbox": [ + 122, + 689, + 473, + 716 + ], + "spans": [ + { + "bbox": [ + 122, + 689, + 473, + 716 + ], + "score": 0.92, + "content": "\\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\mathbf { z } _ { i , 1 } \\| _ { 2 } ^ { 2 } ] = \\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } ] \\leq 4 d \\sigma ^ { 2 } + \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } + 8 \\sigma ^ { 2 } = 2 C ( d , \\mu ) + 8 \\sigma ^ { 2 } ,", + "type": "interline_equation", + "image_path": "c0c76bbf2b2a8f13e7661b5e8ce2eb8b26e7be4f4698ec9489caca94e69db80b.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 122, + 689, + 473, + 716 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 721, + 234, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 719, + 235, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 130, + 734 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 131, + 721, + 160, + 732 + ], + "score": 0.93, + "content": "C ( d , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 719, + 235, + 734 + ], + "score": 1.0, + "content": "was defined in (28).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 719, + 235, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 504, + 104 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 258, + 96 + ], + "score": 1.0, + "content": "We are now ready to bound (26). Based on", + "type": "text" + }, + { + "bbox": [ + 258, + 82, + 450, + 95 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i } \\| \\mathbb { E } _ { { \\mathbf { u } } } [ { \\mathbf { z } } _ { i , 1 } ] \\| _ { 2 } ^ { 2 } = \\frac { 1 } { n } \\sum _ { i } \\| \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } \\leq 4 \\sigma ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "from (29), and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 257, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 257, + 105 + ], + "score": 1.0, + "content": "substituting (30) into (26), we obtain that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 108, + 403, + 136 + ], + "lines": [ + { + "bbox": [ + 207, + 108, + 403, + 136 + ], + "spans": [ + { + "bbox": [ + 207, + 108, + 403, + 136 + ], + "score": 0.91, + "content": "\\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\left. \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } ] \\leq \\frac { 2 C ( d , \\mu ) + 8 \\sigma ^ { 2 } } { q } + \\frac { 4 ( q - 1 ) \\sigma ^ { 2 } } { q } .", + "type": "interline_equation", + "image_path": "0ea78fcc459068eae77db4447740f4bd7db35ff004dcdb7228127fdfa1f5fe32.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 207, + 108, + 403, + 136 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 142, + 334, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 333, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 218, + 157 + ], + "score": 1.0, + "content": "In (25), we also need to bound", + "type": "text" + }, + { + "bbox": [ + 218, + 142, + 333, + 155 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } ]", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 159, + 463, + 223 + ], + "lines": [ + { + "bbox": [ + 146, + 159, + 463, + 223 + ], + "spans": [ + { + "bbox": [ + 146, + 159, + 463, + 223 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\quad \\mathbb { E } _ { \\mathbf { u } } \\left[ \\left\\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) \\right\\| _ { 2 } ^ { 2 } \\right] \\stackrel { ( 2 0 ) } { = } \\mathbb { E } _ { \\mathbf { u } } \\left[ \\left\\| \\frac { 1 } { q } \\sum _ { j = 1 } ^ { q } \\left( \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } ) \\right) \\right\\| _ { 2 } ^ { 2 } \\right] } \\\\ & { \\stackrel { ( a ) } { = } \\frac { 1 } { q } \\mathbb { E } _ { \\mathbf { u } } \\left[ \\left\\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , 1 } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } ) \\right\\| _ { 2 } ^ { 2 } \\right] \\stackrel { ( 2 8 ) } { \\leq } \\frac { 1 } { q } \\left( 2 d \\sigma ^ { 2 } + \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } \\right) = \\frac { C ( d , \\mu ) } { q } , } \\end{array}", + "type": "interline_equation", + "image_path": "179e0a4054d96abb65a10ae2188ed16786ccf17040b5a48d1224e146f6678b75.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 146, + 159, + 463, + 180.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 146, + 180.33333333333334, + 463, + 201.66666666666669 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 146, + 201.66666666666669, + 463, + 223.00000000000003 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 507, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 507, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 239, + 241 + ], + "score": 1.0, + "content": "where the equality (a) holds since", + "type": "text" + }, + { + "bbox": [ + 239, + 227, + 355, + 240 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) ] \\ = \\ \\nabla f _ { i , \\mu } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 227, + 388, + 241 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 388, + 230, + 394, + 239 + ], + "score": 0.77, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 227, + 507, + 241 + ], + "score": 1.0, + "content": ", given by (Liu et al., 2018c,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 237, + 148, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 148, + 251 + ], + "score": 1.0, + "content": "Lemma 1).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 235, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 235, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 235, + 267 + ], + "score": 1.0, + "content": "Substituting (31) and (32) into (25)", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 270, + 471, + 333 + ], + "lines": [ + { + "bbox": [ + 138, + 270, + 471, + 333 + ], + "spans": [ + { + "bbox": [ + 138, + 270, + 471, + 333 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbb { E } \\left[ \\left. \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\right. _ { 2 } ^ { 2 } \\right] = \\mathbb { E } \\left[ \\left. \\displaystyle \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { I } _ { k } } \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } \\right] \\leq \\displaystyle \\frac { \\alpha _ { b } } { b } \\displaystyle \\frac { 2 C ( d , \\mu ) + 4 \\sigma ^ { 2 } + 4 \\sigma ^ { 2 } q } { q } + \\displaystyle \\frac { \\beta _ { b } C ( d , \\mu ) } { b q } } \\\\ { = \\displaystyle \\frac { 4 \\alpha _ { b } ( q + 1 ) } { b q } \\sigma ^ { 2 } + \\displaystyle \\frac { C ( d , \\mu ) } { b q } ( 2 \\alpha _ { b } + \\beta _ { b } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "a04b4d8a24a7fd507d407eecff4f8622c3f86e9709b383164671e8e3b53229d5.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 138, + 270, + 471, + 291.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 138, + 291.0, + 471, + 312.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 138, + 312.0, + 471, + 333.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 362, + 240, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 241, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 241, + 377 + ], + "score": 1.0, + "content": "4 PROOF OF THEOREM 1", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 234, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 233, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 233, + 398 + ], + "score": 1.0, + "content": "Substituting (6) into (5), we obtain", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 402, + 501, + 466 + ], + "lines": [ + { + "bbox": [ + 111, + 402, + 501, + 466 + ], + "spans": [ + { + "bbox": [ + 111, + 402, + 501, + 466 + ], + "score": 0.92, + "content": "\\begin{array} { r l r } { { \\mathbb { E } [ \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } ] \\le \\mathbb { E } [ f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f _ { \\mu } ( \\mathbf { x } _ { T } ) ] + \\sum _ { k = 0 } ^ { T - 1 } [ 2 \\delta _ { k } \\frac { \\sqrt { d } } { \\sqrt { b q } } \\sqrt { 4 \\alpha _ { b } ( q + 1 ) \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } ] } } \\\\ & { } & { \\quad + \\frac { d L } { 2 } \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } . \\eqno ( 3 4 . } \\end{array}", + "type": "interline_equation", + "image_path": "31512d66f14787caef21934b4fa25ae5f63d3fc1af63adcb94df1e1c81e97b77.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 111, + 402, + 501, + 423.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 111, + 423.3333333333333, + 501, + 444.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 111, + 444.66666666666663, + 501, + 465.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 289, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 289, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 289, + 481 + ], + "score": 1.0, + "content": "It is known from (Liu et al., 2018c, Lemma 1) that", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 484, + 429, + 508 + ], + "lines": [ + { + "bbox": [ + 181, + 484, + 429, + 508 + ], + "spans": [ + { + "bbox": [ + 181, + 484, + 429, + 508 + ], + "score": 0.9, + "content": "\\| \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } \\leq 2 \\| \\nabla f _ { \\mu } ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } + { \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } } , \\quad | f _ { \\mu } ( \\mathbf { x } ) - f ( \\mathbf { x } ) | \\leq { \\frac { \\mu ^ { 2 } L } { 2 } } .", + "type": "interline_equation", + "image_path": "958ed7c8d1538a6d14eefff0ffbdebcbe62f85549793ce3bae714d90afa2b718.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 181, + 484, + 429, + 508 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 513, + 506, + 539 + ], + "lines": [ + { + "bbox": [ + 101, + 510, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 101, + 510, + 174, + 532 + ], + "score": 1.0, + "content": "From (35) we have", + "type": "text" + }, + { + "bbox": [ + 174, + 513, + 259, + 527 + ], + "score": 0.94, + "content": "\\begin{array} { r } { f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f ( \\mathbf { x } _ { 0 } ) \\leq \\frac { \\mu ^ { 2 } L } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 510, + 275, + 532 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 275, + 513, + 331, + 528 + ], + "score": 0.94, + "content": "\\begin{array} { r } { f ^ { * } - f _ { \\mu } ^ { * } \\le \\frac { \\mu ^ { 2 } L } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 510, + 358, + 532 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 358, + 515, + 424, + 527 + ], + "score": 0.92, + "content": "f _ { \\mu } ^ { * } = \\mathrm { m i n } _ { \\mathbf { x } } f _ { \\mu } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 510, + 440, + 532 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 440, + 515, + 503, + 527 + ], + "score": 0.92, + "content": "f ^ { * } = \\operatorname* { m i n } _ { \\mathbf { x } } { f ( \\mathbf { x } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 510, + 506, + 532 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 525, + 317, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 148, + 541 + ], + "score": 1.0, + "content": "This yields", + "type": "text" + }, + { + "bbox": [ + 149, + 527, + 280, + 539 + ], + "score": 0.88, + "content": "f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f ( \\mathbf { x } _ { 0 } ) + f ^ { * } - f _ { \\mu } ^ { * } \\leq \\mu ^ { 2 } L", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 525, + 317, + 541 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 544, + 416, + 558 + ], + "lines": [ + { + "bbox": [ + 195, + 544, + 416, + 558 + ], + "spans": [ + { + "bbox": [ + 195, + 544, + 416, + 558 + ], + "score": 0.88, + "content": "f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f _ { \\mu } ( \\mathbf { x } _ { T } ) \\leq f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f _ { \\mu } ^ { * } \\leq ( f ( \\mathbf { x } _ { 0 } ) - f ^ { * } ) + \\mu ^ { 2 } L .", + "type": "interline_equation", + "image_path": "cdc159917fff443ba2dbb8c96231210f166f8bad22cf3b2d04780308a5b953e8.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 195, + 544, + 416, + 558 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 242, + 579 + ], + "lines": [ + { + "bbox": [ + 107, + 568, + 242, + 580 + ], + "spans": [ + { + "bbox": [ + 107, + 568, + 242, + 580 + ], + "score": 1.0, + "content": "Substituting (36) into (34), we obtain", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 584, + 483, + 649 + ], + "lines": [ + { + "bbox": [ + 113, + 584, + 483, + 649 + ], + "spans": [ + { + "bbox": [ + 113, + 584, + 483, + 649 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } \\left[ \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } \\right] \\leq \\mathbb { E } [ f ( \\mathbf { x } _ { 0 } ) - f ^ { * } ) ] + \\mu ^ { 2 } L } \\\\ & { \\qquad + \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\left[ 2 \\delta _ { k } \\frac { \\sqrt { d } } { \\sqrt { b q } } \\sqrt { 4 \\alpha _ { b } ( q + 1 ) \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } \\right] + \\frac { d L } { 2 } \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "a0111f799b94ad45bfad46344737b8ad7056fe934f94cc6a9fd8c9862bcbeb17.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 113, + 584, + 483, + 605.6666666666666 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 113, + 605.6666666666666, + 483, + 627.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 113, + 627.3333333333333, + 483, + 648.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 652, + 462, + 666 + ], + "lines": [ + { + "bbox": [ + 103, + 649, + 464, + 670 + ], + "spans": [ + { + "bbox": [ + 103, + 649, + 133, + 670 + ], + "score": 1.0, + "content": "Due to", + "type": "text" + }, + { + "bbox": [ + 133, + 654, + 242, + 666 + ], + "score": 0.93, + "content": "\\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } \\leq \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 649, + 290, + 670 + ], + "score": 1.0, + "content": "and dividing", + "type": "text" + }, + { + "bbox": [ + 290, + 653, + 326, + 666 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 649, + 464, + 670 + ], + "score": 1.0, + "content": "for both sides of (37), we obtain that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 671, + 495, + 733 + ], + "lines": [ + { + "bbox": [ + 115, + 671, + 495, + 733 + ], + "spans": [ + { + "bbox": [ + 115, + 671, + 495, + 733 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbb { E } \\left[ \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\frac { \\delta _ { k } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } \\right] \\leq \\frac { f ( \\mathbf { x } _ { 0 } ) - f ^ { * } + \\mu ^ { 2 } L } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } + \\frac { 2 \\sqrt { d } } { \\sqrt { b q } } \\sqrt { 4 \\alpha _ { b } ( q + 1 ) \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } } \\\\ { + \\frac { d L } { 2 } \\frac { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } . \\qquad ( 3 \\leq \\alpha \\leq \\alpha ) . } \\end{array}", + "type": "interline_equation", + "image_path": "28aae7e2b5ac3dfc5a79ae8000028ac0baabb46709e4fea022ff74519fedbe96.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 115, + 671, + 495, + 691.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 115, + 691.6666666666666, + 495, + 712.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 115, + 712.3333333333333, + 495, + 732.9999999999999 + ], + "spans": [], + "index": 30 + } + ] + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 336, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 496, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 496, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 504, + 104 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 258, + 96 + ], + "score": 1.0, + "content": "We are now ready to bound (26). Based on", + "type": "text" + }, + { + "bbox": [ + 258, + 82, + 450, + 95 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i } \\| \\mathbb { E } _ { { \\mathbf { u } } } [ { \\mathbf { z } } _ { i , 1 } ] \\| _ { 2 } ^ { 2 } = \\frac { 1 } { n } \\sum _ { i } \\| \\nabla f _ { i , \\mu } - \\nabla f _ { \\mu } \\| _ { 2 } ^ { 2 } \\leq 4 \\sigma ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "from (29), and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 257, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 257, + 105 + ], + "score": 1.0, + "content": "substituting (30) into (26), we obtain that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 105 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 108, + 403, + 136 + ], + "lines": [ + { + "bbox": [ + 207, + 108, + 403, + 136 + ], + "spans": [ + { + "bbox": [ + 207, + 108, + 403, + 136 + ], + "score": 0.91, + "content": "\\frac { 1 } { n } \\sum _ { i } \\mathbb { E } _ { \\mathbf { u } } [ \\left. \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } ] \\leq \\frac { 2 C ( d , \\mu ) + 8 \\sigma ^ { 2 } } { q } + \\frac { 4 ( q - 1 ) \\sigma ^ { 2 } } { q } .", + "type": "interline_equation", + "image_path": "0ea78fcc459068eae77db4447740f4bd7db35ff004dcdb7228127fdfa1f5fe32.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 207, + 108, + 403, + 136 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 142, + 334, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 333, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 218, + 157 + ], + "score": 1.0, + "content": "In (25), we also need to bound", + "type": "text" + }, + { + "bbox": [ + 218, + 142, + 333, + 155 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } ]", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 142, + 333, + 157 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 159, + 463, + 223 + ], + "lines": [ + { + "bbox": [ + 146, + 159, + 463, + 223 + ], + "spans": [ + { + "bbox": [ + 146, + 159, + 463, + 223 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\quad \\mathbb { E } _ { \\mathbf { u } } \\left[ \\left\\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } _ { k } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } _ { k } ) \\right\\| _ { 2 } ^ { 2 } \\right] \\stackrel { ( 2 0 ) } { = } \\mathbb { E } _ { \\mathbf { u } } \\left[ \\left\\| \\frac { 1 } { q } \\sum _ { j = 1 } ^ { q } \\left( \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } ) \\right) \\right\\| _ { 2 } ^ { 2 } \\right] } \\\\ & { \\stackrel { ( a ) } { = } \\frac { 1 } { q } \\mathbb { E } _ { \\mathbf { u } } \\left[ \\left\\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , 1 } ) - \\nabla f _ { i , \\mu } ( \\mathbf { x } ) \\right\\| _ { 2 } ^ { 2 } \\right] \\stackrel { ( 2 8 ) } { \\leq } \\frac { 1 } { q } \\left( 2 d \\sigma ^ { 2 } + \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } \\right) = \\frac { C ( d , \\mu ) } { q } , } \\end{array}", + "type": "interline_equation", + "image_path": "179e0a4054d96abb65a10ae2188ed16786ccf17040b5a48d1224e146f6678b75.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 146, + 159, + 463, + 180.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 146, + 180.33333333333334, + 463, + 201.66666666666669 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 146, + 201.66666666666669, + 463, + 223.00000000000003 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 507, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 507, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 239, + 241 + ], + "score": 1.0, + "content": "where the equality (a) holds since", + "type": "text" + }, + { + "bbox": [ + 239, + 227, + 355, + 240 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mathbf { u } } [ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } _ { i , j } ) ] \\ = \\ \\nabla f _ { i , \\mu } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 227, + 388, + 241 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 388, + 230, + 394, + 239 + ], + "score": 0.77, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 227, + 507, + 241 + ], + "score": 1.0, + "content": ", given by (Liu et al., 2018c,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 237, + 148, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 148, + 251 + ], + "score": 1.0, + "content": "Lemma 1).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 227, + 507, + 251 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 235, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 235, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 235, + 267 + ], + "score": 1.0, + "content": "Substituting (31) and (32) into (25)", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 253, + 235, + 267 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 270, + 471, + 333 + ], + "lines": [ + { + "bbox": [ + 138, + 270, + 471, + 333 + ], + "spans": [ + { + "bbox": [ + 138, + 270, + 471, + 333 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbb { E } \\left[ \\left. \\hat { \\mathbf { g } } _ { k } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\right. _ { 2 } ^ { 2 } \\right] = \\mathbb { E } \\left[ \\left. \\displaystyle \\frac { 1 } { b } \\sum _ { i \\in \\mathcal { I } _ { k } } \\mathbf { z } _ { i } \\right. _ { 2 } ^ { 2 } \\right] \\leq \\displaystyle \\frac { \\alpha _ { b } } { b } \\displaystyle \\frac { 2 C ( d , \\mu ) + 4 \\sigma ^ { 2 } + 4 \\sigma ^ { 2 } q } { q } + \\displaystyle \\frac { \\beta _ { b } C ( d , \\mu ) } { b q } } \\\\ { = \\displaystyle \\frac { 4 \\alpha _ { b } ( q + 1 ) } { b q } \\sigma ^ { 2 } + \\displaystyle \\frac { C ( d , \\mu ) } { b q } ( 2 \\alpha _ { b } + \\beta _ { b } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "a04b4d8a24a7fd507d407eecff4f8622c3f86e9709b383164671e8e3b53229d5.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 138, + 270, + 471, + 291.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 138, + 291.0, + 471, + 312.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 138, + 312.0, + 471, + 333.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 362, + 240, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 241, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 241, + 377 + ], + "score": 1.0, + "content": "4 PROOF OF THEOREM 1", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 234, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 233, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 233, + 398 + ], + "score": 1.0, + "content": "Substituting (6) into (5), we obtain", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 386, + 233, + 398 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 402, + 501, + 466 + ], + "lines": [ + { + "bbox": [ + 111, + 402, + 501, + 466 + ], + "spans": [ + { + "bbox": [ + 111, + 402, + 501, + 466 + ], + "score": 0.92, + "content": "\\begin{array} { r l r } { { \\mathbb { E } [ \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } ] \\le \\mathbb { E } [ f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f _ { \\mu } ( \\mathbf { x } _ { T } ) ] + \\sum _ { k = 0 } ^ { T - 1 } [ 2 \\delta _ { k } \\frac { \\sqrt { d } } { \\sqrt { b q } } \\sqrt { 4 \\alpha _ { b } ( q + 1 ) \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } ] } } \\\\ & { } & { \\quad + \\frac { d L } { 2 } \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } . \\eqno ( 3 4 . } \\end{array}", + "type": "interline_equation", + "image_path": "31512d66f14787caef21934b4fa25ae5f63d3fc1af63adcb94df1e1c81e97b77.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 111, + 402, + 501, + 423.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 111, + 423.3333333333333, + 501, + 444.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 111, + 444.66666666666663, + 501, + 465.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 289, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 289, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 289, + 481 + ], + "score": 1.0, + "content": "It is known from (Liu et al., 2018c, Lemma 1) that", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 469, + 289, + 481 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 484, + 429, + 508 + ], + "lines": [ + { + "bbox": [ + 181, + 484, + 429, + 508 + ], + "spans": [ + { + "bbox": [ + 181, + 484, + 429, + 508 + ], + "score": 0.9, + "content": "\\| \\nabla f ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } \\leq 2 \\| \\nabla f _ { \\mu } ( \\mathbf { x } ) \\| _ { 2 } ^ { 2 } + { \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } } , \\quad | f _ { \\mu } ( \\mathbf { x } ) - f ( \\mathbf { x } ) | \\leq { \\frac { \\mu ^ { 2 } L } { 2 } } .", + "type": "interline_equation", + "image_path": "958ed7c8d1538a6d14eefff0ffbdebcbe62f85549793ce3bae714d90afa2b718.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 181, + 484, + 429, + 508 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 513, + 506, + 539 + ], + "lines": [ + { + "bbox": [ + 101, + 510, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 101, + 510, + 174, + 532 + ], + "score": 1.0, + "content": "From (35) we have", + "type": "text" + }, + { + "bbox": [ + 174, + 513, + 259, + 527 + ], + "score": 0.94, + "content": "\\begin{array} { r } { f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f ( \\mathbf { x } _ { 0 } ) \\leq \\frac { \\mu ^ { 2 } L } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 510, + 275, + 532 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 275, + 513, + 331, + 528 + ], + "score": 0.94, + "content": "\\begin{array} { r } { f ^ { * } - f _ { \\mu } ^ { * } \\le \\frac { \\mu ^ { 2 } L } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 510, + 358, + 532 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 358, + 515, + 424, + 527 + ], + "score": 0.92, + "content": "f _ { \\mu } ^ { * } = \\mathrm { m i n } _ { \\mathbf { x } } f _ { \\mu } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 510, + 440, + 532 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 440, + 515, + 503, + 527 + ], + "score": 0.92, + "content": "f ^ { * } = \\operatorname* { m i n } _ { \\mathbf { x } } { f ( \\mathbf { x } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 510, + 506, + 532 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 525, + 317, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 148, + 541 + ], + "score": 1.0, + "content": "This yields", + "type": "text" + }, + { + "bbox": [ + 149, + 527, + 280, + 539 + ], + "score": 0.88, + "content": "f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f ( \\mathbf { x } _ { 0 } ) + f ^ { * } - f _ { \\mu } ^ { * } \\leq \\mu ^ { 2 } L", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 525, + 317, + 541 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 101, + 510, + 506, + 541 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 544, + 416, + 558 + ], + "lines": [ + { + "bbox": [ + 195, + 544, + 416, + 558 + ], + "spans": [ + { + "bbox": [ + 195, + 544, + 416, + 558 + ], + "score": 0.88, + "content": "f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f _ { \\mu } ( \\mathbf { x } _ { T } ) \\leq f _ { \\mu } ( \\mathbf { x } _ { 0 } ) - f _ { \\mu } ^ { * } \\leq ( f ( \\mathbf { x } _ { 0 } ) - f ^ { * } ) + \\mu ^ { 2 } L .", + "type": "interline_equation", + "image_path": "cdc159917fff443ba2dbb8c96231210f166f8bad22cf3b2d04780308a5b953e8.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 195, + 544, + 416, + 558 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 242, + 579 + ], + "lines": [ + { + "bbox": [ + 107, + 568, + 242, + 580 + ], + "spans": [ + { + "bbox": [ + 107, + 568, + 242, + 580 + ], + "score": 1.0, + "content": "Substituting (36) into (34), we obtain", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 107, + 568, + 242, + 580 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 584, + 483, + 649 + ], + "lines": [ + { + "bbox": [ + 113, + 584, + 483, + 649 + ], + "spans": [ + { + "bbox": [ + 113, + 584, + 483, + 649 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } \\left[ \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } \\right] \\leq \\mathbb { E } [ f ( \\mathbf { x } _ { 0 } ) - f ^ { * } ) ] + \\mu ^ { 2 } L } \\\\ & { \\qquad + \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\left[ 2 \\delta _ { k } \\frac { \\sqrt { d } } { \\sqrt { b q } } \\sqrt { 4 \\alpha _ { b } ( q + 1 ) \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } \\right] + \\frac { d L } { 2 } \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "a0111f799b94ad45bfad46344737b8ad7056fe934f94cc6a9fd8c9862bcbeb17.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 113, + 584, + 483, + 605.6666666666666 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 113, + 605.6666666666666, + 483, + 627.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 113, + 627.3333333333333, + 483, + 648.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 652, + 462, + 666 + ], + "lines": [ + { + "bbox": [ + 103, + 649, + 464, + 670 + ], + "spans": [ + { + "bbox": [ + 103, + 649, + 133, + 670 + ], + "score": 1.0, + "content": "Due to", + "type": "text" + }, + { + "bbox": [ + 133, + 654, + 242, + 666 + ], + "score": 0.93, + "content": "\\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } \\leq \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 649, + 290, + 670 + ], + "score": 1.0, + "content": "and dividing", + "type": "text" + }, + { + "bbox": [ + 290, + 653, + 326, + 666 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 649, + 464, + 670 + ], + "score": 1.0, + "content": "for both sides of (37), we obtain that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 103, + 649, + 464, + 670 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 671, + 495, + 733 + ], + "lines": [ + { + "bbox": [ + 115, + 671, + 495, + 733 + ], + "spans": [ + { + "bbox": [ + 115, + 671, + 495, + 733 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbb { E } \\left[ \\displaystyle \\sum _ { k = 0 } ^ { T - 1 } \\frac { \\delta _ { k } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } \\right] \\leq \\frac { f ( \\mathbf { x } _ { 0 } ) - f ^ { * } + \\mu ^ { 2 } L } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } + \\frac { 2 \\sqrt { d } } { \\sqrt { b q } } \\sqrt { 4 \\alpha _ { b } ( q + 1 ) \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } } \\\\ { + \\frac { d L } { 2 } \\frac { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } . \\qquad ( 3 \\leq \\alpha \\leq \\alpha ) . } \\end{array}", + "type": "interline_equation", + "image_path": "28aae7e2b5ac3dfc5a79ae8000028ac0baabb46709e4fea022ff74519fedbe96.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 115, + 671, + 495, + 691.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 115, + 691.6666666666666, + 495, + 712.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 115, + 712.3333333333333, + 495, + 732.9999999999999 + ], + "spans": [], + "index": 30 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 465, + 99 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 465, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 236, + 104 + ], + "score": 1.0, + "content": "By introducing the random variable", + "type": "text" + }, + { + "bbox": [ + 237, + 84, + 245, + 92 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 82, + 306, + 104 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 306, + 81, + 390, + 100 + ], + "score": 0.92, + "content": "\\begin{array} { r } { P ( R = k ) = \\frac { \\delta _ { k } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 82, + 465, + 104 + ], + "score": 1.0, + "content": ", we then obtain that", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 103, + 452, + 134 + ], + "lines": [ + { + "bbox": [ + 158, + 103, + 452, + 134 + ], + "spans": [ + { + "bbox": [ + 158, + 103, + 452, + 134 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left[ \\Vert \\nabla f _ { \\mu } ( \\mathbf { x } _ { R } ) \\Vert _ { 2 } \\right] = \\mathbb { E } \\left[ \\mathbb { E } _ { R } [ \\Vert \\nabla f _ { \\mu } ( \\mathbf { x } _ { R } ) \\Vert _ { 2 } ] \\right] = \\mathbb { E } \\left[ \\sum _ { k = 0 } ^ { T - 1 } P ( R = k ) \\Vert \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\Vert _ { 2 } \\right] .", + "type": "interline_equation", + "image_path": "78bcf7f600fecaf2996f75d98f9f54ac21a76360cca9b5ca21e45ea398ae0b63.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 158, + 103, + 452, + 113.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 158, + 113.33333333333333, + 452, + 123.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 158, + 123.66666666666666, + 452, + 134.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 137, + 220, + 148 + ], + "lines": [ + { + "bbox": [ + 106, + 137, + 219, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 219, + 148 + ], + "score": 1.0, + "content": "Based on (35), we further have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 151, + 402, + 207 + ], + "lines": [ + { + "bbox": [ + 208, + 151, + 402, + 207 + ], + "spans": [ + { + "bbox": [ + 208, + 151, + 402, + 207 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\displaystyle \\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] \\leq \\mathbb { E } \\left[ \\sqrt { 2 \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { R } ) \\| _ { 2 } ^ { 2 } + \\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } } \\right] } \\\\ & { \\displaystyle \\leq \\sqrt { 2 } \\mathbb { E } [ \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { R } ) \\| _ { 2 } ] + \\frac { \\mu L d } { \\sqrt { 2 } } , } \\end{array}", + "type": "interline_equation", + "image_path": "9f8234e4878bdf820434807aa5e492348a6fc4e5d588c569284bfb089bdadcf9.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 208, + 151, + 402, + 169.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 208, + 169.66666666666666, + 402, + 188.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 208, + 188.33333333333331, + 402, + 206.99999999999997 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 209, + 349, + 222 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 350, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 225, + 223 + ], + "score": 1.0, + "content": "where we have used the fact that", + "type": "text" + }, + { + "bbox": [ + 226, + 210, + 302, + 222 + ], + "score": 0.92, + "content": "{ \\sqrt { a ^ { 2 } + b ^ { 2 } } } \\leq ( a + b )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 209, + 316, + 223 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 316, + 211, + 347, + 222 + ], + "score": 0.9, + "content": "a , b \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 209, + 350, + 223 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 108, + 226, + 313, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 314, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 314, + 239 + ], + "score": 1.0, + "content": "Substituting (38) and (39) into (40), we finally obtain (8).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 108, + 252, + 253, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 251, + 254, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 254, + 268 + ], + "score": 1.0, + "content": "5 PROOF OF COROLLARY 1", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 274, + 504, + 301 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 507, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 160, + 290 + ], + "score": 1.0, + "content": "Upon defining", + "type": "text" + }, + { + "bbox": [ + 161, + 275, + 295, + 289 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\tilde { \\nabla } f _ { i } ( \\mathbf x ; \\mathbf u ) = \\frac { d \\left[ f _ { i } ( \\mathbf x + \\mu \\mathbf u ) - f _ { i } ( \\mathbf x - \\mu \\mathbf u ) \\right] \\mathbf u } { 2 \\mu } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 270, + 328, + 290 + ], + "score": 1.0, + "content": "(against", + "type": "text" + }, + { + "bbox": [ + 328, + 275, + 367, + 288 + ], + "score": 0.91, + "content": "\\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 270, + 507, + 290 + ], + "score": 1.0, + "content": "in (3)), our major task is to derive the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 288, + 277, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 235, + 302 + ], + "score": 1.0, + "content": "first- and second-order moments of", + "type": "text" + }, + { + "bbox": [ + 235, + 289, + 273, + 301 + ], + "score": 0.91, + "content": "\\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 288, + 277, + 302 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 108, + 307, + 277, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 305, + 275, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 130, + 320 + ], + "score": 1.0, + "content": "Given", + "type": "text" + }, + { + "bbox": [ + 131, + 310, + 137, + 317 + ], + "score": 0.55, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 305, + 236, + 320 + ], + "score": 1.0, + "content": ", we first study the mean of", + "type": "text" + }, + { + "bbox": [ + 236, + 306, + 275, + 318 + ], + "score": 0.92, + "content": "\\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } )", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 322, + 455, + 399 + ], + "lines": [ + { + "bbox": [ + 156, + 322, + 455, + 399 + ], + "spans": [ + { + "bbox": [ + 156, + 322, + 455, + 399 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { \\mathbf { u } } \\left[ \\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right] = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { 2 \\mu } ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ) \\mathbf { u } \\right] } \\\\ & { = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { 2 \\mu } f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) \\mathbf { u } \\right] + \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { 2 \\mu } f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ( - \\mathbf { u } ) \\right] \\stackrel { ( a ) } = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { \\mu } f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) \\mathbf { u } \\right] } \\\\ & { \\stackrel { ( b ) } = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { \\mu } \\left( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } ) \\right) \\mathbf { u } \\right] = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right] \\stackrel { ( c ) } = \\nabla f _ { i , \\mu } ( \\mathbf { x } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "9e30ac888d77f90363953682eca8b9b621e50862fb8b6bec7ace9dc053c5a15a.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 156, + 322, + 455, + 347.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 156, + 347.6666666666667, + 455, + 373.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 156, + 373.33333333333337, + 455, + 399.00000000000006 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 401, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 255, + 414 + ], + "score": 1.0, + "content": "where (a) holds since the distribution of", + "type": "text" + }, + { + "bbox": [ + 255, + 403, + 262, + 411 + ], + "score": 0.62, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 399, + 437, + 414 + ], + "score": 1.0, + "content": "is symmetric around the origin, (b) holds since", + "type": "text" + }, + { + "bbox": [ + 437, + 401, + 474, + 413 + ], + "score": 0.94, + "content": "\\mathbb { E } [ { \\mathbf { u } } ] = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 399, + 506, + 414 + ], + "score": 1.0, + "content": ", and (c)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 411, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 150, + 429 + ], + "score": 1.0, + "content": "holds since", + "type": "text" + }, + { + "bbox": [ + 150, + 411, + 276, + 429 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nabla f _ { i , \\mu } ( \\mathbf { x } ) = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { \\mu } f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) \\mathbf { u } \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 412, + 506, + 429 + ], + "score": 1.0, + "content": "obtained from (Gao et al., 2014, Lemma 4.1). It is clear from", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 427, + 276, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 427, + 139, + 450 + ], + "score": 1.0, + "content": "(41) that", + "type": "text" + }, + { + "bbox": [ + 139, + 429, + 271, + 447 + ], + "score": 0.89, + "content": "\\mathbb { E } _ { \\mathbf { u } } \\left[ \\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right] = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 427, + 276, + 450 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 306, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 306, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 263, + 466 + ], + "score": 1.0, + "content": "We next study the second-order moment of", + "type": "text" + }, + { + "bbox": [ + 263, + 452, + 302, + 464 + ], + "score": 0.93, + "content": "\\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 452, + 306, + 466 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 468, + 436, + 549 + ], + "lines": [ + { + "bbox": [ + 173, + 468, + 436, + 549 + ], + "spans": [ + { + "bbox": [ + 173, + 468, + 436, + 549 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { \\mathbf { u } } [ \\| \\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\| ^ { 2 } ] = \\frac { d ^ { 2 } } { 4 \\mu ^ { 2 } } \\mathbb { E } _ { \\mathbf { u } } [ ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ) ^ { 2 } \\| \\mathbf { u } \\| ^ { 2 } ] } \\\\ { \\displaystyle \\leq \\frac { d ^ { 2 } } { 2 \\mu ^ { 2 } } \\mathbb { E } _ { \\mathbf { u } } [ ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } ) ) ^ { 2 } + ( f _ { i } ( \\mathbf { x } ) - f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ) ^ { 2 } ] } \\\\ { \\displaystyle = \\mathbb { E } _ { \\mathbf { u } } [ \\| \\frac { d } { \\mu } ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } ) ) \\mathbf { u } \\| ^ { 2 } ] = \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\| ^ { 2 } ] , } \\end{array}", + "type": "interline_equation", + "image_path": "7dbbd99d84aa0c73702c9d8660c9265d9a019f4c9489e21e34bd17d87aeedf7b.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 173, + 468, + 436, + 495.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 173, + 495.0, + 436, + 522.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 173, + 522.0, + 436, + 549.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 265, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 266, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 225, + 565 + ], + "score": 1.0, + "content": "where we have used the fact that", + "type": "text" + }, + { + "bbox": [ + 226, + 552, + 262, + 564 + ], + "score": 0.93, + "content": "\\| \\mathbf { u } \\| _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 551, + 266, + 565 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 504, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "score": 1.0, + "content": "Based on (41) and (42), we can conclude that both (3) and (10) maintain the same statistical properties. Following", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 453, + 590 + ], + "score": 1.0, + "content": "the proofs of Proposition 2 and Theorem 1, it is not difficult to reach the convergence rate in (8).", + "type": "text" + }, + { + "bbox": [ + 495, + 579, + 505, + 588 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 108, + 604, + 253, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 254, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 254, + 619 + ], + "score": 1.0, + "content": "6 PROOF OF COROLLARY 2", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 626, + 380, + 640 + ], + "lines": [ + { + "bbox": [ + 104, + 624, + 382, + 642 + ], + "spans": [ + { + "bbox": [ + 104, + 624, + 121, + 642 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 121, + 626, + 203, + 639 + ], + "score": 0.93, + "content": "\\hat { \\mathbf { g } } _ { k } ^ { i , j } : = \\hat { \\nabla } f _ { i } ( \\mathbf x _ { k } ; \\mathbf u _ { i , j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 624, + 255, + 642 + ], + "score": 1.0, + "content": ". 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\\frac { \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } } { 2 } } \\right] } \\\\ & { \\displaystyle \\leq \\sqrt { 2 } \\mathbb { E } [ \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { R } ) \\| _ { 2 } ] + \\frac { \\mu L d } { \\sqrt { 2 } } , } \\end{array}", + "type": "interline_equation", + "image_path": "9f8234e4878bdf820434807aa5e492348a6fc4e5d588c569284bfb089bdadcf9.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 208, + 151, + 402, + 169.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 208, + 169.66666666666666, + 402, + 188.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 208, + 188.33333333333331, + 402, + 206.99999999999997 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 209, + 349, + 222 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 350, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 225, + 223 + ], + "score": 1.0, + "content": "where we have used the fact that", + 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} & { \\mathbb { E } _ { \\mathbf { u } } \\left[ \\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right] = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { 2 \\mu } ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ) \\mathbf { u } \\right] } \\\\ & { = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { 2 \\mu } f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) \\mathbf { u } \\right] + \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { 2 \\mu } f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ( - \\mathbf { u } ) \\right] \\stackrel { ( a ) } = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { \\mu } f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) \\mathbf { u } \\right] } \\\\ & { \\stackrel { ( b ) } = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\frac { d } { \\mu } \\left( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } ) \\right) \\mathbf { u } \\right] = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right] \\stackrel { ( c ) } = \\nabla f _ { i , \\mu } ( \\mathbf { x } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "9e30ac888d77f90363953682eca8b9b621e50862fb8b6bec7ace9dc053c5a15a.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 156, + 322, + 455, + 347.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 156, + 347.6666666666667, + 455, + 373.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 156, + 373.33333333333337, + 455, + 399.00000000000006 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 401, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 255, + 414 + ], + "score": 1.0, + "content": "where (a) holds since the distribution of", + "type": 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\\mathbf { x } + \\mu \\mathbf { u } ) \\mathbf { u } \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 412, + 506, + 429 + ], + "score": 1.0, + "content": "obtained from (Gao et al., 2014, Lemma 4.1). It is clear from", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 427, + 276, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 427, + 139, + 450 + ], + "score": 1.0, + "content": "(41) that", + "type": "text" + }, + { + "bbox": [ + 139, + 429, + 271, + 447 + ], + "score": 0.89, + "content": "\\mathbb { E } _ { \\mathbf { u } } \\left[ \\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right] = \\mathbb { E } _ { \\mathbf { u } } \\left[ \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 427, + 276, + 450 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 399, + 506, + 450 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 306, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 306, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 263, + 466 + ], + "score": 1.0, + "content": "We next study the second-order moment of", + "type": "text" + }, + { + "bbox": [ + 263, + 452, + 302, + 464 + ], + "score": 0.93, + "content": "\\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 452, + 306, + 466 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 452, + 306, + 466 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 468, + 436, + 549 + ], + "lines": [ + { + "bbox": [ + 173, + 468, + 436, + 549 + ], + "spans": [ + { + "bbox": [ + 173, + 468, + 436, + 549 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { \\mathbf { u } } [ \\| \\tilde { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\| ^ { 2 } ] = \\frac { d ^ { 2 } } { 4 \\mu ^ { 2 } } \\mathbb { E } _ { \\mathbf { u } } [ ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ) ^ { 2 } \\| \\mathbf { u } \\| ^ { 2 } ] } \\\\ { \\displaystyle \\leq \\frac { d ^ { 2 } } { 2 \\mu ^ { 2 } } \\mathbb { E } _ { \\mathbf { u } } [ ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } ) ) ^ { 2 } + ( f _ { i } ( \\mathbf { x } ) - f _ { i } ( \\mathbf { x } - \\mu \\mathbf { u } ) ) ^ { 2 } ] } \\\\ { \\displaystyle = \\mathbb { E } _ { \\mathbf { u } } [ \\| \\frac { d } { \\mu } ( f _ { i } ( \\mathbf { x } + \\mu \\mathbf { u } ) - f _ { i } ( \\mathbf { x } ) ) \\mathbf { u } \\| ^ { 2 } ] = \\mathbb { E } _ { \\mathbf { u } } [ \\| \\hat { \\nabla } f _ { i } ( \\mathbf { x } ; \\mathbf { u } ) \\| ^ { 2 } ] , } \\end{array}", + "type": "interline_equation", + "image_path": "7dbbd99d84aa0c73702c9d8660c9265d9a019f4c9489e21e34bd17d87aeedf7b.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 173, + 468, + 436, + 495.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 173, + 495.0, + 436, + 522.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 173, + 522.0, + 436, + 549.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 265, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 266, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 225, + 565 + ], + "score": 1.0, + "content": "where we have used the fact that", + "type": "text" + }, + { + "bbox": [ + 226, + 552, + 262, + 564 + ], + "score": 0.93, + "content": "\\| \\mathbf { u } \\| _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 551, + 266, + 565 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 551, + 266, + 565 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 504, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "score": 1.0, + "content": "Based on (41) and (42), we can conclude that both (3) and (10) maintain the same statistical properties. Following", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 453, + 590 + ], + "score": 1.0, + "content": "the proofs of Proposition 2 and Theorem 1, it is not difficult to reach the convergence rate in (8).", + "type": "text" + }, + { + "bbox": [ + 495, + 579, + 505, + 588 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 567, + 506, + 590 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 604, + 253, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 254, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 254, + 619 + ], + "score": 1.0, + "content": "6 PROOF OF COROLLARY 2", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 626, + 380, + 640 + ], + "lines": [ + { + "bbox": [ + 104, + 624, + 382, + 642 + ], + "spans": [ + { + "bbox": [ + 104, + 624, + 121, + 642 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 121, + 626, + 203, + 639 + ], + "score": 0.93, + "content": "\\hat { \\mathbf { g } } _ { k } ^ { i , j } : = \\hat { \\nabla } f _ { i } ( \\mathbf x _ { k } ; \\mathbf u _ { i , j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 624, + 255, + 642 + ], + "score": 1.0, + "content": ". 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\\delta _ { k } \\| \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\frac { \\sqrt { d } } { \\sqrt { N q } } \\sqrt { \\mathbb { E } [ \\| \\hat { g } _ { k } ^ { i , j } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\| _ { 2 } ^ { 2 } ] } . } \\end{array}", + "type": "interline_equation", + "image_path": "25f8d197455dab898dc5db34f6a85f4ad04fa984dac26f8a2cb14a675338d5a6.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 135, + 433, + 475, + 478.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 135, + 478.3333333333333, + 475, + 523.6666666666666 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 135, + 523.6666666666666, + 475, + 569.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 506, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 507, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 290, + 596 + ], + "score": 1.0, + "content": "Compared (51) with (19), the standard deviation", + "type": "text" + }, + { + "bbox": [ + 290, + 577, + 384, + 595 + ], + "score": 0.91, + "content": "\\sqrt { \\mathbb { E } [ \\| \\hat { \\mathbf { g } } _ { k } ^ { i , j } - 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\\nabla f _ { \\mu } ( { \\mathbf { x } } _ { k } ) \\| _ { 2 } ^ { 2 } \\right] \\leq 8 \\alpha _ { b } \\sigma ^ { 2 } + ( 2 \\alpha _ { b } + \\beta _ { b } ) C ( d , \\mu ) . } \\end{array}", + "type": "interline_equation", + "image_path": "779ecb3885339c53c9a70cb5329b2f2006656408de1a3b7b457dfbb2819ea033.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 200, + 607, + 411, + 627 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 349, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 629, + 348, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 348, + 641 + ], + "score": 1.0, + "content": "Based on (51)-(52) and following the proof of Theorem 1, we have", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 629, + 348, + 641 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 643, + 433, + 700 + ], + "lines": [ + { + "bbox": [ + 175, + 643, + 433, + 700 + ], + "spans": [ + { + "bbox": [ + 175, + 643, + 433, + 700 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\displaystyle \\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] \\leq \\sqrt { 2 } \\frac { f ( \\mathbf { x } _ { 0 } ) - f ^ { * } + \\mu ^ { 2 } L } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } } \\\\ & { \\displaystyle + \\frac { 2 \\sqrt { 2 } \\sqrt { d } } { \\sqrt { b q } } \\sqrt { 8 \\alpha _ { b } \\sigma ^ { 2 } + C ( d , \\mu ) ( 2 \\alpha _ { b } + \\beta _ { b } ) } + \\frac { d L } { \\sqrt { 2 } } \\frac { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } + \\frac { \\mu L d } { \\sqrt { 2 } } , } \\end{array}", + "type": "interline_equation", + "image_path": "739ba13175e5d4c5afd75b75eddc12d3210c55296b5141b7e78e2acec038ff04.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 175, + 643, + 433, + 662.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 175, + 662.0, + 433, + 681.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 175, + 681.0, + 433, + 700.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 702, + 246, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 701, + 246, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 130, + 714 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 131, + 702, + 244, + 714 + ], + "score": 0.91, + "content": "C ( d , \\mu ) : = 2 d \\sigma ^ { 2 } + \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 701, + 246, + 714 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 701, + 246, + 714 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 719, + 438, + 735 + ], + "lines": [ + { + "bbox": [ + 104, + 717, + 439, + 738 + ], + "spans": [ + { + "bbox": [ + 104, + 717, + 114, + 738 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 115, + 720, + 165, + 735 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 717, + 182, + 738 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 182, + 720, + 236, + 735 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\delta _ { k } = { O } \\left( \\frac { 1 } { \\sqrt { d T } } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 717, + 379, + 738 + ], + "score": 1.0, + "content": ", then the convergence rate simplifies to", + "type": "text" + }, + { + "bbox": [ + 380, + 719, + 434, + 735 + ], + "score": 0.92, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { d } { \\sqrt { b q } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 717, + 439, + 738 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 717, + 439, + 738 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 80, + 253, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 255, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 255, + 96 + ], + "score": 1.0, + "content": "7 PROOF OF COROLLARY 3", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 129 + ], + "lines": [ + { + "bbox": [ + 102, + 100, + 509, + 122 + ], + "spans": [ + { + "bbox": [ + 102, + 100, + 121, + 122 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 121, + 104, + 279, + 120 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\hat { \\mathbf { g } } _ { k } ^ { m } : = \\frac { 1 } { b _ { m } q } \\sum _ { i \\in \\mathcal { T } _ { m , k } } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } ( \\mathbf x _ { k } ; \\mathbf u _ { i , j } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 100, + 297, + 122 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 297, + 104, + 395, + 118 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\bar { \\bf g } _ { k } = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\mathrm { s i g n } ( \\hat { \\bf g } _ { k } ^ { m } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 100, + 509, + 122 + ], + "score": 1.0, + "content": ". Following (43)-(50), we can", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 182, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 182, + 129 + ], + "score": 1.0, + "content": "similarly obtain that", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 135, + 421, + 159 + ], + "lines": [ + { + "bbox": [ + 188, + 135, + 421, + 159 + ], + "spans": [ + { + "bbox": [ + 188, + 135, + 421, + 159 + ], + "score": 0.89, + "content": "| ( \\nabla f _ { \\mu } ( { \\bf x } _ { k } ) ) _ { l } | \\mathrm { P r o b } [ \\mathrm { s i g n } ( \\bar { g } _ { k , l } ) \\neq \\mathrm { s i g n } ( ( \\nabla f _ { \\mu } ( { \\bf x } _ { k } ) ) _ { l } ) ] \\leq \\frac { \\xi _ { l } } { \\sqrt { M } } ,", + "type": "interline_equation", + "image_path": "57de15381771dddf98a3ab37ecba2786aa40f157801fd8297536bd63d283ef7d.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 188, + 135, + 421, + 159 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 165, + 424, + 183 + ], + "lines": [ + { + "bbox": [ + 103, + 164, + 427, + 184 + ], + "spans": [ + { + "bbox": [ + 103, + 164, + 131, + 184 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 131, + 165, + 254, + 184 + ], + "score": 0.94, + "content": "\\xi _ { l } : = \\sqrt { \\mathbb { E } [ ( \\hat { g } _ { k , l } ^ { m } - ( \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) ) _ { l } ) ^ { 2 } ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 164, + 427, + 184 + ], + "score": 1.0, + "content": ". 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\\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\Vert _ { 2 } ^ { 2 } \\right] \\leq \\frac { 4 ( q + 1 ) } { b q } \\sigma ^ { 2 } + \\frac { 3 } { b q } C ( d , \\mu ) .", + "type": "interline_equation", + "image_path": "27dd89d4d95d80730fe382303461d49a3ff5846ae5ceaf803ee4c405555c9b56.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 244, + 404, + 256.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 207, + 256.5, + 404, + 269.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 274, + 349, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 274, + 348, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 348, + 286 + ], + "score": 1.0, + "content": "Based on (55)-(56) and following the proof of Theorem 1, we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 291, + 424, + 349 + ], + "lines": [ + { + "bbox": [ + 185, + 291, + 424, + 349 + ], + "spans": [ + { + "bbox": [ + 185, + 291, + 424, + 349 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] \\leq \\sqrt { 2 } \\frac { f ( \\mathbf { x } _ { 0 } ) - f ^ { * } + \\mu ^ { 2 } L } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } } \\\\ & { + \\frac { 2 \\sqrt { 2 } \\sqrt { d } } { \\sqrt { M b q } } \\sqrt { 4 ( q + 1 ) \\sigma ^ { 2 } + 3 C ( d , \\mu ) } + \\frac { d L } { \\sqrt { 2 } } \\frac { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } + \\frac { \\mu L d } { \\sqrt { 2 } } , } \\end{array}", + "type": "interline_equation", + "image_path": "30ffb21d5ba7e5903d69c38a19c0454c61d0e7fbc97d4e1d0de9de2865bb25ef.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 291, + 424, + 305.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 185, + 305.5, + 424, + 320.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 185, + 320.0, + 424, + 334.5 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 185, + 334.5, + 424, + 349.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 247, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 247, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 130, + 367 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 131, + 353, + 244, + 366 + ], + "score": 0.9, + "content": "C ( d , \\mu ) : = 2 d \\sigma ^ { 2 } + \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 352, + 247, + 367 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 101, + 365, + 502, + 393 + ], + "spans": [ + { + "bbox": [ + 101, + 365, + 128, + 393 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 128, + 372, + 178, + 387 + ], + "score": 0.6, + "content": "\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 365, + 182, + 393 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 182, + 372, + 236, + 387 + ], + "score": 0.84, + "content": "\\begin{array} { r } { \\delta _ { k } = { O } \\left( \\frac { 1 } { \\sqrt { d T } } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 365, + 252, + 393 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 373, + 285, + 383 + ], + "score": 0.89, + "content": "b M \\approx n", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 365, + 424, + 393 + ], + "score": 1.0, + "content": ", then the convergence rate simplifies to", + "type": "text" + }, + { + "bbox": [ + 425, + 371, + 502, + 387 + ], + "score": 0.95, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { \\sqrt { d } } { \\sqrt { n } } + \\frac { d } { \\sqrt { n q } } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 411, + 285, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 286, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 286, + 426 + ], + "score": 1.0, + "content": "8 SUPPLEMENTAL EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 437, + 245, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 245, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 245, + 448 + ], + "score": 1.0, + "content": "8.1 SYNTHETIC EXPERIMENTS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 506, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 317, + 468 + ], + "score": 1.0, + "content": "In Fig. A3, we demonstrate the effect of the learning rate", + "type": "text" + }, + { + "bbox": [ + 317, + 458, + 323, + 466 + ], + "score": 0.75, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 457, + 506, + 468 + ], + "score": 1.0, + "content": "on the training loss of SGD, signSGD, ZO-SGD,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "ZO-SCD, ZO-signSGD and ZO-M-signSGD. We observe that compared to the gradient-based algorithms (SGD,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "ZO-SGD and ZO-SCD), the gradient sign-based algorithms support a more flexible choice of learning rates", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 487, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 499 + ], + "score": 1.0, + "content": "(corresponding to less variance), since the sign operation has an normalization effect to reduce oscillations", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 198, + 509 + ], + "score": 1.0, + "content": "in convergence. We find", + "type": "text" + }, + { + "bbox": [ + 199, + 497, + 230, + 507 + ], + "score": 0.9, + "content": "\\delta = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 497, + 284, + 509 + ], + "score": 1.0, + "content": "best for SGD,", + "type": "text" + }, + { + "bbox": [ + 284, + 497, + 324, + 507 + ], + "score": 0.88, + "content": "\\delta = 0 . 0 0 9", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 497, + 394, + 509 + ], + "score": 1.0, + "content": "best for signSGD,", + "type": "text" + }, + { + "bbox": [ + 394, + 497, + 425, + 507 + ], + "score": 0.89, + "content": "\\delta = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "best for ZOSGD and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 507, + 416, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 140, + 519 + ], + "score": 1.0, + "content": "ZOSCD,", + "type": "text" + }, + { + "bbox": [ + 141, + 507, + 184, + 517 + ], + "score": 0.88, + "content": "\\bar { \\delta } = 0 . 0 1 7 8", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 507, + 279, + 519 + ], + "score": 1.0, + "content": "best for ZOsignSGD, and", + "type": "text" + }, + { + "bbox": [ + 279, + 507, + 322, + 517 + ], + "score": 0.88, + "content": "\\delta = 0 . 0 5 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 507, + 416, + 519 + ], + "score": 1.0, + "content": "best for ZO-M-signSGD.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "image", + "bbox": [ + 210, + 543, + 388, + 680 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 210, + 543, + 388, + 680 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 210, + 543, + 388, + 680 + ], + "spans": [ + { + "bbox": [ + 210, + 543, + 388, + 680 + ], + "score": 0.97, + "type": "image", + "image_path": "b814eebc6dcda2f68f8e69b9576e2d7eccddf9b5d27d6725c23c28669304782f.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 210, + 543, + 388, + 555.4545454545455 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 210, + 555.4545454545455, + 388, + 567.909090909091 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 210, + 567.909090909091, + 388, + 580.3636363636365 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 210, + 580.3636363636365, + 388, + 592.818181818182 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 210, + 592.818181818182, + 388, + 605.2727272727275 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 210, + 605.2727272727275, + 388, + 617.727272727273 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 210, + 617.727272727273, + 388, + 630.1818181818185 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 210, + 630.1818181818185, + 388, + 642.636363636364 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 210, + 642.636363636364, + 388, + 655.0909090909095 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 210, + 655.0909090909095, + 388, + 667.545454545455 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 210, + 667.545454545455, + 388, + 680.0000000000005 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 691, + 506, + 722 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 409, + 702 + ], + "score": 1.0, + "content": "Figure A3: The training loss at the last iteration versus the constant learning rate", + "type": "text" + }, + { + "bbox": [ + 409, + 690, + 466, + 702 + ], + "score": 0.82, + "content": "\\delta \\in [ 1 0 ^ { - 3 } , 0 . 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 691, + 505, + 702 + ], + "score": 1.0, + "content": ". Here the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 702, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 505, + 712 + ], + "score": 1.0, + "content": "solid line represents the loss averaged over 10 independent trials with random initialization, and the shaded", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 712, + 353, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 712, + 353, + 722 + ], + "score": 1.0, + "content": "region indicates the standard deviation of results over random trials.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + } + ], + "index": 34.5 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 386, + 115, + 394 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 80, + 253, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 255, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 255, + 96 + ], + "score": 1.0, + "content": "7 PROOF OF COROLLARY 3", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 129 + ], + "lines": [ + { + "bbox": [ + 102, + 100, + 509, + 122 + ], + "spans": [ + { + "bbox": [ + 102, + 100, + 121, + 122 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 121, + 104, + 279, + 120 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\hat { \\mathbf { g } } _ { k } ^ { m } : = \\frac { 1 } { b _ { m } q } \\sum _ { i \\in \\mathcal { T } _ { m , k } } \\sum _ { j = 1 } ^ { q } \\hat { \\nabla } f _ { i } ( \\mathbf x _ { k } ; \\mathbf u _ { i , j } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 100, + 297, + 122 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 297, + 104, + 395, + 118 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\bar { \\bf g } _ { k } = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\mathrm { s i g n } ( \\hat { \\bf g } _ { k } ^ { m } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 100, + 509, + 122 + ], + "score": 1.0, + "content": ". 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By mimicking the derivation in (51), we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 103, + 164, + 427, + 184 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 189, + 474, + 216 + ], + "lines": [ + { + "bbox": [ + 137, + 189, + 474, + 216 + ], + "spans": [ + { + "bbox": [ + 137, + 189, + 474, + 216 + ], + "score": 0.91, + "content": "\\mathbb { E } [ f _ { \\mu } ( { \\mathbf x } _ { k + 1 } ) - f _ { \\mu } ( { \\mathbf x } _ { k } ) ] \\leq - \\delta _ { k } \\| \\nabla f _ { \\mu } ( { \\mathbf x } _ { k } ) \\| _ { 1 } + \\frac { \\delta _ { k } ^ { 2 } d L } { 2 } + 2 \\delta _ { k } \\frac { \\sqrt { d } } { \\sqrt { M } } \\sqrt { \\mathbb { E } [ \\| \\hat { \\mathbf g } _ { k } ^ { m } - \\nabla f _ { \\mu } ( { \\mathbf x } _ { k } ) \\| _ { 2 } ^ { 2 } ] } .", + "type": "interline_equation", + "image_path": "972b9307656309212abed7549440249226a2bbcc5a857078fd1185724a3f4555.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 137, + 189, + 474, + 198.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 137, + 198.0, + 474, + 207.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 137, + 207.0, + 474, + 216.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 305, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 305, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 218, + 240 + ], + "score": 1.0, + "content": "According to Proposition 2, let", + "type": "text" + }, + { + "bbox": [ + 219, + 228, + 263, + 239 + ], + "score": 0.93, + "content": "b = \\lfloor n / M \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 226, + 305, + 240 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 226, + 305, + 240 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 244, + 404, + 269 + ], + "lines": [ + { + "bbox": [ + 207, + 244, + 404, + 269 + ], + "spans": [ + { + "bbox": [ + 207, + 244, + 404, + 269 + ], + "score": 0.91, + "content": "\\mathbb { E } \\left[ \\Vert \\hat { \\mathbf { g } } _ { k } ^ { m } - \\nabla f _ { \\mu } ( \\mathbf { x } _ { k } ) \\Vert _ { 2 } ^ { 2 } \\right] \\leq \\frac { 4 ( q + 1 ) } { b q } \\sigma ^ { 2 } + \\frac { 3 } { b q } C ( d , \\mu ) .", + "type": "interline_equation", + "image_path": "27dd89d4d95d80730fe382303461d49a3ff5846ae5ceaf803ee4c405555c9b56.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 244, + 404, + 256.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 207, + 256.5, + 404, + 269.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 274, + 349, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 274, + 348, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 348, + 286 + ], + "score": 1.0, + "content": "Based on (55)-(56) and following the proof of Theorem 1, we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 274, + 348, + 286 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 291, + 424, + 349 + ], + "lines": [ + { + "bbox": [ + 185, + 291, + 424, + 349 + ], + "spans": [ + { + "bbox": [ + 185, + 291, + 424, + 349 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } \\left[ \\| \\nabla f ( \\mathbf { x } _ { R } ) \\| _ { 2 } \\right] \\leq \\sqrt { 2 } \\frac { f ( \\mathbf { x } _ { 0 } ) - f ^ { * } + \\mu ^ { 2 } L } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } } \\\\ & { + \\frac { 2 \\sqrt { 2 } \\sqrt { d } } { \\sqrt { M b q } } \\sqrt { 4 ( q + 1 ) \\sigma ^ { 2 } + 3 C ( d , \\mu ) } + \\frac { d L } { \\sqrt { 2 } } \\frac { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } ^ { 2 } } { \\sum _ { k = 0 } ^ { T - 1 } \\delta _ { k } } + \\frac { \\mu L d } { \\sqrt { 2 } } , } \\end{array}", + "type": "interline_equation", + "image_path": "30ffb21d5ba7e5903d69c38a19c0454c61d0e7fbc97d4e1d0de9de2865bb25ef.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 291, + 424, + 305.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 185, + 305.5, + 424, + 320.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 185, + 320.0, + 424, + 334.5 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 185, + 334.5, + 424, + 349.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 247, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 247, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 130, + 367 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 131, + 353, + 244, + 366 + ], + "score": 0.9, + "content": "C ( d , \\mu ) : = 2 d \\sigma ^ { 2 } + \\mu ^ { 2 } L ^ { 2 } d ^ { 2 } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 352, + 247, + 367 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 352, + 247, + 367 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 101, + 365, + 502, + 393 + ], + "spans": [ + { + "bbox": [ + 101, + 365, + 128, + 393 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 128, + 372, + 178, + 387 + ], + "score": 0.6, + "content": "\\begin{array} { r } { \\mu = O ( \\frac { 1 } { \\sqrt { d T } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 365, + 182, + 393 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 182, + 372, + 236, + 387 + ], + "score": 0.84, + "content": "\\begin{array} { r } { \\delta _ { k } = { O } \\left( \\frac { 1 } { \\sqrt { d T } } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 365, + 252, + 393 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 373, + 285, + 383 + ], + "score": 0.89, + "content": "b M \\approx n", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 365, + 424, + 393 + ], + "score": 1.0, + "content": ", then the convergence rate simplifies to", + "type": "text" + }, + { + "bbox": [ + 425, + 371, + 502, + 387 + ], + "score": 0.95, + "content": "\\begin{array} { r } { O ( \\frac { \\sqrt { d } } { \\sqrt { T } } + \\frac { \\sqrt { d } } { \\sqrt { n } } + \\frac { d } { \\sqrt { n q } } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 101, + 365, + 502, + 393 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 411, + 285, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 286, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 286, + 426 + ], + "score": 1.0, + "content": "8 SUPPLEMENTAL EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 437, + 245, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 245, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 245, + 448 + ], + "score": 1.0, + "content": "8.1 SYNTHETIC EXPERIMENTS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 506, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 317, + 468 + ], + "score": 1.0, + "content": "In Fig. A3, we demonstrate the effect of the learning rate", + "type": "text" + }, + { + "bbox": [ + 317, + 458, + 323, + 466 + ], + "score": 0.75, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 457, + 506, + 468 + ], + "score": 1.0, + "content": "on the training loss of SGD, signSGD, ZO-SGD,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "ZO-SCD, ZO-signSGD and ZO-M-signSGD. We observe that compared to the gradient-based algorithms (SGD,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "ZO-SGD and ZO-SCD), the gradient sign-based algorithms support a more flexible choice of learning rates", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 487, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 499 + ], + "score": 1.0, + "content": "(corresponding to less variance), since the sign operation has an normalization effect to reduce oscillations", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 198, + 509 + ], + "score": 1.0, + "content": "in convergence. 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b/parse/train/H1sUHgb0Z/images/fbcacc5d5c06972c935779e999a8bcd95976af593b9684ce4d51c3f22fae6cd2.jpg new file mode 100644 index 0000000000000000000000000000000000000000..b2588fad69d1535505db71f1a94d46a676262f98 --- /dev/null +++ b/parse/train/H1sUHgb0Z/images/fbcacc5d5c06972c935779e999a8bcd95976af593b9684ce4d51c3f22fae6cd2.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:274ad38438733ac3fc5eebc2b54f956e43f3419ea69892deadb86fecb3e5d626 +size 4912 diff --git a/parse/train/HJeOekHKwr/HJeOekHKwr.md b/parse/train/HJeOekHKwr/HJeOekHKwr.md new file mode 100644 index 0000000000000000000000000000000000000000..669b0c563b72ed9f60a6d04120f6f80227f5df1a --- /dev/null +++ b/parse/train/HJeOekHKwr/HJeOekHKwr.md @@ -0,0 +1,1235 @@ +# SMOOTHNESS AND STABILITY IN GANS + +Casey Chu +Stanford University +caseychu@stanford.edu + +Kentaro Minami Preferred Networks, Inc. minami@preferred.jp + +# Kenji Fukumizu + +The Institute of Statistical Mathematics / Preferred Networks, Inc. fukumizu@ism.ac.jp + +# ABSTRACT + +Generative adversarial networks, or GANs, commonly display unstable behavior during training. In this work, we develop a principled theoretical framework for understanding the stability of various types of GANs. In particular, we derive conditions that guarantee eventual stationarity of the generator when it is trained with gradient descent, conditions that must be satisfied by the divergence that is minimized by the GAN and the generator’s architecture. We find that existing GAN variants satisfy some, but not all, of these conditions. Using tools from convex analysis, optimal transport, and reproducing kernels, we construct a GAN that fulfills these conditions simultaneously. In the process, we explain and clarify the need for various existing GAN stabilization techniques, including Lipschitz constraints, gradient penalties, and smooth activation functions. + +# 1 INTRODUCTION: TAMING INSTABILITY WITH SMOOTHNESS + +Generative adversarial networks (Goodfellow et al., 2014), or GANs, are a powerful class of generative models defined through minimax game. GANs and their variants have shown impressive performance in synthesizing various types of datasets, especially natural images. Despite these successes, the training of GANs remains quite unstable in nature, and this instability remains difficult to understand theoretically. + +Since the introduction of GANs, there have been many techniques proposed to stabilize GANs training, including studies of new generator/discriminator architectures, loss functions, and regularization techniques. Notably, Arjovsky et al. (2017) proposed Wasserstein GAN (WGAN), which in principle avoids instability caused by mismatched generator and data distribution supports. In practice, this is enforced by Lipschitz constraints, which in turn motivated developments like gradient penalties (Gulrajani et al., 2017) and spectral normalization (Miyato et al., 2018). Indeed, these stabilization techniques have proven essential to achieving the latest state-of-the-art results (Karras et al., 2018; Brock et al., 2019). + +On the other hand, a solid theoretical understanding of training stability has not been established. Several empirical observations point to an incomplete understanding. For example, why does applying a gradient penalty together spectral norm seem to improve performance (Miyato et al., 2018), even though in principle they serve the same purpose? Why does applying only spectral normalization with the Wasserstein loss fail (Miyato, 2018), even though the analysis of Arjovsky et al. (2017) suggests it should be sufficient? Why is applying gradient penalties effective, even outside their original context of the Wasserstein GAN (Fedus et al., 2018)? + +In this work, we develop a framework to analyze the stability of GAN training that resolves these apparent contradictions and clarifies the roles of these regularization techniques. Our approach considers the smoothness of the loss function used. In optimization, smoothness is a well-known condition that ensures that gradient descent and its variants become stable (see e.g., Bertsekas (1999)). For example, the following well-known proposition is the starting point of our stability analysis: + +Proposition 1 (Bertsekas (1999), Proposition 1.2.3). Suppose $f : \mathbb { R } ^ { p } \mathbb { R }$ is $L$ -smooth and bounded below. Let $\begin{array} { r } { x _ { k + 1 } : = x _ { k } - \frac { 1 } { L } \dot { \nabla } \dot { f } ( x _ { k } ) } \end{array}$ . Then $| | \nabla f ( x _ { k } ) | | 0$ as $k \to \infty$ . + +This proposition says that under a smoothness condition on the function, gradient descent with a constant step size $\frac { \mathbf { i } } { L }$ approaches stationarity (i.e., the gradient norm approaches zero). This is a rather weak notion of convergence, as it does not guarantee that the iterates converge to a point, and even if the iterates do converge, the limit is a stationary point and not necessarily an minimizer. + +Nevertheless, empirically, not even this stationarity is satisfied by GANs, which are known to frequently destabilize and diverge during training. To diagnose this instability, we consider the smoothness of the GAN’s loss function. GANs are typically framed as minimax problems of the form + +$$ +\operatorname* { i n f } _ { \theta } \operatorname* { s u p } _ { \varphi } \mathcal { I } ( \mu _ { \theta } , \varphi ) , +$$ + +where $\mathcal { I }$ is a loss function that takes a generator distribution $\mu _ { \theta }$ and discriminator $\varphi$ , and $\theta \in \mathbb { R } ^ { p }$ denotes the parameters of the generator. Unfortunately, the minimax nature of this problem makes stability and convergence difficult to analyze. To make the analysis more tractable, we define $J ( \mu ) =$ $\operatorname* { s u p } _ { \varphi } \mathcal { I } ( \mu , \varphi )$ , so that (1) becomes simply + +$$ +\operatorname* { i n f } _ { \theta } J ( \mu _ { \theta } ) . +$$ + +This choice corresponds to the common assumption that the discriminator is allowed to reach optimality at every training step. Now, the GAN algorithm can be regarded as simply gradient descent on the $\mathbb { R } ^ { p } \to \mathbb { R }$ function $\theta \mapsto J ( \mu _ { \theta } )$ , which may be analyzed using Proposition 1. In particular, if this function $\theta \mapsto J ( \mu _ { \theta } )$ satisfies the smoothness assumption, then the GAN training should be stable in that it should approach stationarity under the assumption of an optimal discriminator. + +In the remainder of this paper, we investigate whether the smoothness assumption is satisfied for various GAN losses. Our analysis answers two questions: + +Q1. Which existing GAN losses, if any, satisfy the smoothness condition in Proposition 1? + +Q2. Are there choices of loss, regularization, or architecture that enforce smoothness in GANs? + +As results of our analysis, our contributions are as follows: + +1. We derive sufficient conditions for the GAN algorithm to be stationary under certain assumptions (Theorem 1). Our conditions relate to the smoothness of GAN loss used as well as the parameterization of the generator. +2. We show that most common GAN losses do not satisfy the all of the smoothness conditions, thereby corroborating their empirical instability. +3. We develop regularization techniques that enforce the smoothness conditions. These regularizers recover common GAN stabilization techniques such as gradient penalties and spectral normalization, thereby placing their use on a firmer theoretical foundation. +4. Our analysis provides several practical insights, suggesting for example the use of smooth activation functions, simultaneous spectral normalization and gradient penalties, and a particular learning rate for the generator. + +# 1.1 RELATED WORK + +Divergence minimization Our analysis regards the GAN algorithm as minimizing a divergence between the current generator distribution and the desired data distribution, under the assumption of an optimal discriminator at every training step. This perspective originates from the earliest GAN paper, in which Goodfellow et al. (2014) show that the original minimax GAN implicitly minimizes the Jensen–Shannon divergence. Since then, the community has introduced a large number of GAN or GAN-like variants that learn generative models by implicitly minimizing various divergences, including $f$ -divergences (Nowozin et al., 2016), Wasserstein distance (Arjovsky et al., 2017), and maximum-mean discrepancy (Li et al., 2015; Unterthiner et al., 2018). Meanwhile, the non-saturating GAN (Goodfellow et al., 2014) has been shown to minimize a certain Kullback– Leibler divergence (Arjovsky & Bottou, 2017). Several more theoretical works consider the topological, geometric, and convexity properties of divergence minimization (Arjovsky & Bottou, 2017; Liu et al., 2017; Bottou et al., 2018; Farnia & Tse, 2018; Chu et al., 2019), perspectives that we draw heavily upon. Sanjabi et al. (2018) also prove smoothness of GAN losses in the specific case of the regularized optimal transport loss. Their assumption for smoothness is entangled in that it involves a composite condition on generators and discriminators, while our analysis addresses them separately. + +Table 1: Common GANs, their corresponding loss functions, and their optimal discriminators. + +
Loss function J(μ)Optimal discriminator Φμ(x)
Minimax GANDjs(μ|l μo)μ(x）
Non-saturating GANDKL(¹μ+μo |μo)-110g（）+(@） μo(x）
Wasserstein GANW1(μ, μo)argmaXfeLip1 Ey~μ[f(y)] -Ey~μo[f(y)]
GMMN,Coulomb GAN1MMD²(μ, μo)Ey~μ[K(x,y)]-Ey~μo[K(x,y)]
IPM-GANIPMj(μ, μo)arg maxf∈F Ey~μ[f(y)] -Ey~μof(y)]
+ +Other approaches Even though many analyses, including ours, operate under the assumption of an optimal discriminator, this assumption is unrealistic in practice. Li et al. (2017b) contrast this optimal discriminator dynamics with first-order dynamics, which assumes that the generator and discriminator use alternating gradient updates and is what is used computationally. As this is a differing approach from ours, we only briefly mention some results in this area, which typically rely on game-theoretic notions (Kodali et al., 2017; Grnarova et al., 2018; Oliehoek et al., 2018) or local analysis (Nagarajan & Kolter, 2017; Mescheder et al., 2018). Some of these results rely on continuous dynamics approximations of gradient updates; in contrast, our work focuses on discrete dynamics. + +# 1.2 NOTATION + +Let $\bar { \mathbb { R } } : = \mathbb { R } \cup \{ \infty , - \infty \}$ . We let ${ \mathcal { P } } ( X )$ denote the set of all probability measures on a compact set $X \subseteq \mathbb { R } ^ { d }$ . We let ${ \mathcal { M } } ( X )$ and ${ \mathcal { C } } ( X )$ denote the dual pair consisting of the set of all finite signed measures on $X$ and the set of all continuous functions $X \to \mathbb { R }$ . For any statement $A$ , we let $\chi \{ A \}$ be 0 if $A$ is true and $\infty$ if $A$ is false. For a Euclidean vector $x$ , its Euclidean norm is denoted by $\| x \| _ { 2 }$ , and the operator norm of a matrix $A$ is denoted by $\| A \| _ { 2 }$ , i.e., $\begin{array} { r } { \| A \| _ { 2 } = \operatorname* { s u p } _ { \| x \| _ { 2 } \leq 1 } \| A x \| _ { 2 } / \| x \| _ { 2 } } \end{array}$ . A function $f : X \to Y$ between two metric spaces is $\alpha$ -Lipschitz if $d _ { Y } ( f ( x _ { 1 } ) , f ( x _ { 2 } ) ) \leq \alpha d _ { X } ( x _ { 1 } , x _ { 2 } ) ,$ . A function $f : \mathbb { R } ^ { d } \mathbb { R }$ is $\beta$ -smooth if its gradients are $\beta$ -Lipschitz, that is, for all $x , y \in \mathbb { R } ^ { d }$ , $\| \nabla f ( x ) - \nabla f ( y ) \| _ { 2 } \leq \beta \| x - y \| _ { 2 }$ . + +# 2 SMOOTHNESS OF GAN LOSSES + +This section presents Theorem 1, which provides concise criteria for the smoothness of GAN losses. + +In order to keep our analysis agnostic to the particular GAN used, let $J : { \mathcal { P } } ( X ) \to { \bar { \mathbb { R } } }$ be an arbitrary convex loss function, which takes a distribution over $X \subset \mathbb { R } ^ { d }$ and outputs a real number. Note that the typical minimax formulation of GANs can be recovered from just the loss function $J$ using convex duality. In particular, recall that the convex conjugate $J ^ { \star } : \bar { \mathcal { C } } ( X ) \bar { \mathbb { R } }$ of $J$ satisfies the following remarkable duality, known as the Fenchel–Moreau theorem: + +$$ +J ^ { \star } ( \varphi ) : = \operatorname* { s u p } _ { \mu \in { \mathcal M } ( X ) } \int \varphi ( x ) d \mu - J ( \mu ) , \qquad J ( \mu ) = \operatorname* { s u p } _ { \varphi \in { \mathcal C } ( X ) } \int \varphi ( x ) d \mu - J ^ { \star } ( \varphi ) . +$$ + +Based on this duality, minimizing $J$ can be framed as the minimax problem + +$$ +\operatorname* { i n f } _ { \mu \in { \mathcal P } ( X ) } J ( \mu ) = \operatorname* { i n f } _ { \mu \in { \mathcal P } ( X ) } \operatorname* { s u p } _ { \varphi \in { \mathcal C } ( X ) } \int \varphi ( x ) d \mu - J ^ { \star } ( \varphi ) : = \operatorname* { i n f } _ { \mu \in { \mathcal P } ( X ) } \operatorname* { s u p } _ { \varphi \in { \mathcal C } ( X ) } { \mathcal I } ( \mu , \varphi ) , +$$ + +recovering the well-known adversarial formulation of GANs. We now define the notion of an optimal discriminator for an arbitrary loss function $J$ , based on this convex duality: + +Definition 1 (Optimal discriminator). Let $J : \mathcal { M } ( X ) \bar { \mathbb { R } }$ be a convex, l.s.c., proper function. An optimal discriminator for a probability distribution ${ \boldsymbol { \mu } } \in { \mathcal { P } } ( { \boldsymbol { X } } )$ is a continuous function $\Phi _ { \mu } : X \to $ $\mathbb { R }$ that attains the maximum of the second equation in (3), i.e., $\begin{array} { r } { J ( \mu ) = \int \Phi _ { \mu } ( x ) d \mu - J ^ { \star } ( \dot { \Phi } _ { \mu } ) } \end{array}$ . + +This definition recovers the optimal discriminators of many existing GAN and GAN-like algorithms (Farnia & Tse, 2018; Chu et al., 2019), most notably those in Table 1. Our analysis will apply to any algorithm in this family of algorithms. See Appendix $\mathbf { B }$ for more details on this perspective. + +We also formalize the notion of a family of generators: + +Definition 2 (Family of generators). A family of generators is a set of pushforward probability measures $\{ \mu _ { \theta } = f _ { \theta \# } \omega : \theta \in \mathbb { R } ^ { p } \}$ , where $\omega$ is a fixed probability distribution on $Z$ (the latent variable) and $f _ { \theta } : Z \to X$ is a measurable function (the generator). + +Now, in light of Proposition 1, we are interested in the smoothness of the mapping $\theta \mapsto J ( \mu _ { \theta } )$ , which would guarantee the stationarity of gradient descent on this objective, which in turn implies stationarity of the GAN algorithm under the assumption of an optimal discriminator. The following theorem is our central result, which decomposes the smoothness of $\theta \mapsto J ( \mu _ { \theta } )$ into conditions on optimal discriminators and the family of generators. + +Theorem 1 (Smoothness decomposition for GANs). Let $J : \mathcal { M } ( X ) \bar { \mathbb { R } }$ be a convex function whose optimal discriminators $\Phi _ { \mu } : X \to \mathbb { R }$ satisfy the following regularity conditions: + +(D1) $x \mapsto \Phi _ { \mu } ( x )$ is $\alpha$ -Lipschitz, (D2) $x \mapsto \nabla _ { x } \Phi _ { \mu } ( x )$ is $\beta _ { 1 }$ -Lipschitz, (D3) $\mu \mapsto \nabla _ { x } \Phi _ { \mu } ( x )$ is $\beta _ { 2 }$ -Lipschitz w.r.t. the 1-Wasserstein distance. + +Also, let $\mu _ { \theta } = f _ { \theta \# } \omega$ be a family of generators that satisfies: + +(G1) $\theta \mapsto f _ { \theta } ( z )$ is $A$ -Lipschitz in expectation for $z \sim \omega$ , i.e., $\begin{array} { r } { \mathbb { E } _ { z \sim \omega } [ \| f _ { \theta _ { 1 } } ( z ) - f _ { \theta _ { 2 } } ( z ) \| _ { 2 } ] \ \le } \end{array}$ $A \| \theta _ { 1 } - \theta _ { 2 } \| _ { 2 }$ , and + +(G2) $\theta \ \mapsto \ D _ { \theta } f _ { \theta } ( z )$ is $B$ -Lipschitz in expectation for $z \sim \omega$ , i.e., $\mathbb { E } _ { z \sim \omega } [ \| D _ { \theta _ { 1 } } f _ { \theta _ { 1 } } ( z ) ~ -$ $D _ { \theta _ { 2 } } f _ { \theta _ { 2 } } ( z ) \lVert _ { 2 } \bigr \rVert \leq B \lVert \theta _ { 1 } - \theta _ { 2 } \rVert _ { 2 }$ . + +Then $ { \theta } \mapsto J ( { \mu } _ { { \theta } } )$ is $L$ -smooth, with $L = \alpha B + A ^ { 2 } ( \beta _ { 1 } + \beta _ { 2 } )$ . + +Theorem 1 connects the smoothness properties of the loss function $J$ with the smoothness properties of the optimal discriminator $\Phi _ { \mu }$ , and once paired with Proposition 1, it suggests a quantitative value $\textstyle { \frac { 1 } { L } }$ for a stable generator learning rate. In order to obtain claims of stability for practically sized learning rates, it is important to tightly bound the relevant constants. + +In Sections 4 to 6, we carefully analyze which GAN losses satisfy (D1), (D2), and (D3), and with what constants. We summarize our results in Table 2: it turns out that none of the listed losses, except for one, satisfy (D1), (D2), and (D3) simultaneously with a finite constant. The MMD-based loss satisfies the three conditions, but its constant for (D1) grows as $\alpha = O ( { \sqrt { d } } )$ , which is an unfavorable dependence on the data dimension $d$ that forces an unacceptably small learning rate. See for complete details of each condition. This failure of existing GANs to satisfy the stationarity conditions corroborates the observed instability of GANs. + +Table 2: Regularity of common GAN losses. + +
(D1)(D2)(D3)
Minimax GAN×X×
Non-saturating GAN Wasserstein GANXX？
√X
×？
IPMs MMD²*
+ +Theorem 1 decomposes smoothness into conditions on the generator and conditions on the discriminator, allowing a clean separation of concerns. In this paper, we focus on the discriminator conditions (D1), (D2), and (D3) and only provide an extremely simple example of a generator that satisfies (G1) and (G2), in Section 7. Because analysis of the generator conditions may become quite complicated and will vary with the choice of architecture considered (feedforward, convolutional, ResNet, etc.), we leave a detailed analysis of the generator conditions (G1) and (G2) as a promising avenue for future work. Indeed, such analyses may lead to new generator architectures or generator regularization techniques that stabilize GAN training. + +In this section, we present a generic regularization technique that imposes the three conditions sufficient for stable learning on an arbitrary loss function $J$ , thereby stabilizing training. In Section 2, we observe that the Wasserstein, IPM, and MMD losses respectively satisfy (D1), (D2), and (D3) individually, but not all of of them at the same time. Using techniques from convex analysis, we convert these three GAN losses into three regularizers that, when applied simultaneously, causes the resulting loss to satisfy all the three conditions. Here, we only outline the technique; the specifics of each case are deferred to Sections 4 to 6. + +We start with an arbitrary base loss function $J$ to be regularized. Next, we take an existing GAN loss that satisfies the desired regularity condition and convert it into a regularizer function $R$ : $\mathcal { M } ( X ) \bar { \mathbb { R } }$ . Then, we consider $J \oplus R$ , which denotes the inf-convolution defined as + +$$ +( J \oplus R ) ( \xi ) = \operatorname* { i n f } _ { \tilde { \xi } \in \mathcal { M } ( X ) } J ( \tilde { \xi } ) + R ( \xi - \tilde { \xi } ) . +$$ + +This new function $J \oplus R$ inherits the regularity of $R$ , making it a stable candidate as a GAN loss. Moreover, because the inf-convolution is a commutative operation, we can sequentially apply multiple regularizers $R _ { 1 } , R _ { 2 }$ , and $R _ { 3 }$ without destroying the added regularity. In particular, if we carefully choose functions $R _ { 1 }$ , $R _ { 2 }$ , and $R _ { 3 }$ , then $\tilde { J } = \bar { J } \oplus \bar { R _ { 1 } } \oplus R _ { 2 } \oplus \bar { R _ { 3 } }$ will satisfy (D1), (D2), and (D3) simultaneously. Moreover, under some technical assumptions, this composite function $\tilde { J }$ inherits the original minimizers of $J$ , making it a sensible GAN loss: + +Proposition 2 (Invariance of minimizers). Let $R _ { 1 } ( \xi ) : = \| \xi \| _ { \mathrm { K R } } , R _ { 2 } ( \xi ) : = \| \xi \| _ { S ^ { * } }$ , and $R _ { 3 } ( \xi ) : = $ $\frac { 1 } { 2 } \| \hat { \xi } \| _ { \mathcal { H } } ^ { 2 }$ be the three regularizers defined by (8), (12), and (19) respectively. Assume that $ { J } :$ $\bar { \mathcal { M } } ( X ) \bar { \mathbb { R } }$ has a unique minimizer at $\mu _ { 0 }$ with $J ( \mu _ { 0 } ) = 0 ;$ , and $J ( \mu ) \geq c \| \hat { \mu } - \hat { \mu } _ { 0 } \| _ { \mathcal { H } }$ for some $c > 0$ . Then the inf-convolution $\tilde { J } = J \oplus R _ { 1 } \oplus R _ { 2 } \oplus R _ { 3 }$ has a unique minimizer at $\mu _ { 0 }$ with $\tilde { J } ( \mu _ { 0 } ) = 0$ . + +The duality formulation (4) provides a practical method for minimizing this composite function. We leverage the duality relation $( J \oplus R _ { 1 } \oplus R _ { 2 } \oplus R _ { 3 } ) ^ { \star } = J ^ { \star } + R _ { 1 } ^ { \star } + R _ { 2 } ^ { \star } + R _ { 3 } ^ { \star }$ and apply (4): + +$$ +\begin{array} { l } { \underset { \mu } { \operatorname* { i n f } } \left( J \oplus R _ { 1 } \oplus R _ { 2 } \oplus R _ { 3 } \right) ( \mu ) = \underset { \mu \atop \varphi } { \operatorname* { i n f } } \underset { \varphi } { \operatorname* { s u p } } \int \varphi d \mu - J ^ { \star } ( \varphi ) - R _ { 1 } ^ { \star } ( \varphi ) - R _ { 2 } ^ { \star } ( \varphi ) - R _ { 3 } ^ { \star } ( \varphi ) } \\ { = \underset { \mu \atop \varphi } { \operatorname* { i n f } } \underset { \varphi } { \operatorname* { s u p } } \mathcal { I } ( \mu , \varphi ) - R _ { 1 } ^ { \star } ( \varphi ) - R _ { 2 } ^ { \star } ( \varphi ) - R _ { 3 } ^ { \star } ( \varphi ) . } \end{array} +$$ + +This minimax problem can be seen as a GAN whose discriminator objective has three added regularization terms. + +The concrete form of these regularizers are summarized in Table 3. Notably, we observe that we recover standard techniques for stabilizing GANs: + +• (D1) is enforced by Lipschitz constraints (i.e., spectral normalization) on the discriminator. +• (D2) is enforced by spectral normalization and a choice of Lipschitz, smooth activation functions for the discriminator. + +Table 3: Smoothness-inducing regularizers and their convex conjugates. To enforce a regularity condition on a loss function $J$ , we take a source loss that satisfies it and view it as a regularizer $R$ . We then consider the inf-convolution $J \oplus R$ , which corresponds to an added regularization term $R ^ { \star }$ on the discriminator $\varphi$ . These regularization terms correspond to existing GAN techniques. + +
PurposeSource lossR()R（）GAN techniques
(D1)W1|KRx{lφ|lLip ≤1}spectral norm
(D2)IPMs|/| s*x{∈s}smooth activations, spectral norm
(D3)MMD²gradient penalties
+ +• (D3) is enforced by gradient penalties on the discriminator. + +Our analysis therefore puts these regularization techniques on a firm theoretical foundation (Proposition 1 and Theorem 1) and provides insight into their function. + +# 4 ENFORCING (D1) WITH LIPSCHITZ CONSTRAINTS + +In this section, we show that enforcing (D1) leads to techniques and notions commonly used to stabilize GANs, including the Wasserstein distance, Lipschitz constraints and spectral normalization. Recall that (D1) demands that the optimal discriminator $\Phi _ { \mu }$ is Lipschitz: + +If $\Phi _ { \mu }$ is differentiable, this is equivalent to that the optimal discriminator has a gradient with bounded norm. This is a sensible criterion, since a discriminator whose gradient norm is too large may push the generator too hard and destabilize its training. + +To check (D1), the following proposition shows that it suffices to check whether $| J ( \mu ) - J ( \nu ) | \leq$ $\alpha W _ { 1 } ( \mu , \nu )$ for all distributions $\mu , \nu$ : + +Proposition 3. $( D l )$ holds if and only if $J$ is $\alpha$ -Lipschitz w.r.t. the Wasserstein-1 distance. + +Arjovsky et al. (2017) show that this property does not hold for common divergences based on the Kullback–Leibler or Jensen–Shannon divergence, while it does hold for the Wasserstein-1 distance. Indeed, it is this desirable property that motivates their introduction of the Wasserstein GAN. Framed in our context, their result is summarized as follows: + +Proposition 4. The minimax and non-saturating GAN losses do not satisfy $( D I )$ for some $\mu _ { 0 }$ + +Proposition 5. The Wasserstein GAN loss satisfies $( D l )$ with $\alpha = 1$ for any $\mu _ { 0 }$ . + +Our stability analysis therefore deepens the analysis of Arjovsky et al. (2017) and provides an alternative reason that the Wasserstein distance is desirable as a metric: it is part of a sufficient condition that ensures stationarity of gradient descent. + +# 4.1 FROM WASSERSTEIN DISTANCE TO LIPSCHITZ CONSTRAINTS + +Having identified the Wasserstein GAN loss as one that satisfies (D1), we next follow the strategy outlined in Section 3 to convert it into a regularizer for an arbitrary loss function. Towards this, we define the regularizer $R _ { 1 } : { \mathcal { M } } ( X ) \to { \bar { \mathbb { R } } }$ and compute its convex conjugate $R _ { 1 } ^ { \star } : { \mathcal { C } } ( X ) \to { \bar { \mathbb { R } } }$ : + +$$ +R _ { 1 } ( \xi ) : = \alpha \| \xi \| _ { \mathrm { K R } } = \alpha \operatorname* { s u p } _ { \stackrel { f \in \mathcal { C } ( X ) } { | | f | | _ { \mathrm { L i p } } \leq 1 } } \int f d \xi , \qquad R _ { 1 } ^ { \star } ( \varphi ) = \left\{ \begin{array} { l l } { 0 } & { \| \varphi \| _ { \mathrm { L i p } } \leq \alpha } \\ { \infty } & { \mathrm { o t h e r w i s e } . } \end{array} \right. +$$ + +This norm is the Kantorovich–Rubinstein norm (KR norm), which extends the Wasserstein-1 distance to $M ( X )$ ; it holds that $\| \mu - \nu \| _ { \mathrm { K R } } = W _ { 1 } ( \mu , \nu )$ for $\mu , \nu \in { \mathcal { P } } ( X )$ . Then, its inf-convolution with an arbitrary function inherits the Lipschitz property held by $R _ { 1 }$ : + +Proposition 6 (Pasch–Hausdorff). Let $J : \mathcal { M } ( X ) \bar { \mathbb { R } }$ be a function, and define ${ \tilde { J } } : = J \oplus R _ { 1 }$ . Then $\tilde { J }$ is $\alpha$ -Lipschitz w.r.t. the distance induced by the $K R$ norm, and hence the Wasserstein-1 distance when restricted to ${ \mathcal { P } } ( X )$ . + +Due to Proposition 3, we now obtain a transformed loss function $\tilde { J }$ that automatically satisfies (D1). This function is a generalization of the Pasch–Hausdorff envelope (see Chapter 9 in Rockafeller & Wets (1998)), also known as Lipschitz regularization or the McShane–Whitney extension (McShane, 1934; Whitney, 1934; Kirszbraun, 1934; Hiriart-Urruty, 1980). + +The convex conjugate computation in (8) shows that $\tilde { J }$ can be minimized in practice by imposing Lipschitz constraints on discriminators. Indeed, by (4), + +$$ +\operatorname* { i n f } _ { \mu } \big ( J \oplus \alpha \lVert \cdot \rVert _ { \mathrm { K R } } \big ) ( \mu ) = \operatorname* { i n f } _ { \mu } \operatorname* { s u p } _ { \varphi } \mathbb { E } _ { x \sim \mu } [ \varphi ( x ) ] - J ^ { \star } ( \varphi ) - \chi \{ \lVert \varphi \rVert _ { \mathrm { L i p } } \leq \alpha \} +$$ + +$$ += \operatorname* { i n f } _ { \mu } \operatorname* { s u p } _ { \varphi \colon \| \varphi \| _ { \mathrm { L i p } } \leq \alpha } \mathcal { I } ( \mu , \varphi ) . +$$ + +Farnia & Tse (2018) consider this loss $\tilde { J }$ in the special case of an $f$ -GAN with $J ( \mu ) = D _ { f } ( \mu , \mu _ { 0 } )$ ; they showed that minimizing $\tilde { J }$ corresponds to training a $f$ -GAN normally but constraining the discriminator to be $\alpha$ -Lipschitz. We show that this technique is in fact generic for any $J$ : minimizing the transformed loss can be achieved by training the GAN as normal, but imposing a Lipschitz constraint on the discriminator. + +Our analysis therefore justifies the use of Lipschitz constraints, such as spectral normalization (Miyato et al., 2018) and weight clipping (Arjovsky & Bottou, 2017), for general GAN losses. However, Theorem 1 also suggests that applying only Lipschitz constraints may not be enough to stabilize GANs, as (D1) alone does not ensure that the GAN objective is smooth. + +# 5 ENFORCING (D2) WITH DISCRIMINATOR SMOOTHNESS + +(D2) demands that the optimal discriminator $\Phi _ { \mu }$ is smooth: + +(D2) $x \mapsto \nabla \Phi _ { \mu } ( x )$ is $\beta _ { 1 }$ -Lipschitz for all $\mu \in { \mathcal { P } } ( X )$ , i.e., $\| \nabla \Phi _ { \mu } ( x ) - \nabla \Phi _ { \mu } ( y ) \| _ { 2 } \leq \beta _ { 1 } \| x - y \| _ { 2 }$ . + +Intuitively, this says that for a fixed generator $\mu$ , the optimal discriminator $\Phi _ { \mu }$ should not provide gradients that change too much spatially. + +Although the Wasserstein GAN loss (D1), we see that it, along with the minimax GAN and the non-saturating GAN, do not satisfy (D2): + +Proposition 7. The Wasserstein, minimax, and non-saturating GAN losses do not satisfy $( D 2 )$ for some $\mu _ { 0 }$ . + +We now construct a loss that by definition satisfies (D2). Let $s$ be the class of 1-smooth functions, that is, for which $\| \nabla f ( x ) - \nabla f ( y ) \| _ { 2 } \leq \| x - y \| _ { 2 }$ , and consider the integral probability metric (IPM) (Muller, 1997) w.r.t. ¨ $s$ , defined by + +$$ +\mathrm { I P M } _ { \cal S } ( \mu , \nu ) : = \operatorname* { s u p } _ { f \in { \cal S } } \int f d \mu - \int f d \nu . +$$ + +The optimal discriminator for the loss $\mathrm { I P M } _ { \cal S } ( \mu , \mu _ { 0 } )$ is the function that maximizes the supremum in the definition. This function by definition belongs to $s$ and therefore is 1-smooth. Hence, this IPM loss satisfies (D2) with $\beta _ { 1 } = 1$ by construction. + +# 5.1 FROM INTEGRAL PROBABILITY METRIC TO SMOOTH DISCRIMINATORS + +Having identified the IPM-based loss as one that satisfies (D2), we next follow the strategy outlined in Section 3 to convert it into a regularizer for an arbitrary loss function. To do so, we define a regularizer $R _ { 2 } : \mathcal { M } ( X ) \bar { \mathbb { R } }$ and compute its convex conjugate $R _ { 2 } ^ { \star } : { \mathcal { C } } ( X ) \to { \bar { \mathbb { R } } }$ : + +$$ +R _ { 2 } ( \xi ) : = \beta _ { 1 } \| \xi \| _ { \mathcal { S } ^ { * } } = \beta _ { 1 } \operatorname* { s u p } _ { f \in \mathcal { S } } \int f d \xi , \qquad R _ { 2 } ^ { \star } ( \varphi ) = \left\{ \begin{array} { l l } { 0 } & { \varphi \in \beta _ { 1 } S } \\ { \infty } & { \mathrm { o t h e r w i s e } . } \end{array} \right. +$$ + +The norm is the dual norm to $s$ , which extends the IPM to signed measures; it holds that $\mathrm { I P M } _ { \mathcal { S } } ( \mu , \nu ) = \| \mu - \nu \| _ { \mathcal { S } ^ { * } }$ for $\mu , \nu \in { \mathcal { P } } ( X )$ . Similar to the situation in the previous section, inf-convolution preserves the smoothness property of $R _ { 2 }$ : + +Proposition 8. Let $J : \mathcal { M } ( X ) \bar { \mathbb { R } }$ be a convex, proper, lower semicontinuous function, and define ${ \tilde { J } } : = J \oplus R _ { 2 }$ . Then the optimal discriminator for $\tilde { J }$ is $\beta _ { 1 }$ -smooth. + +Applying (4) and (12), we see that we can minimize this transformed loss function by restricting the family of discriminators to only $\beta _ { 1 }$ -smooth discriminators: + +$$ +\begin{array} { r l } & { \underset { \mu } { \operatorname* { i n f } } \left( J \oplus \beta _ { 1 } \lVert \cdot \rVert _ { \mathcal { S } ^ { * } } \right) ( \mu ) = \underset { \mu } { \operatorname* { i n f } } \ \underset { \varphi } { \operatorname* { s u p } } \mathbb { E } _ { x \sim \mu } [ \varphi ( x ) ] - J ^ { \star } ( \varphi ) - \chi \{ \varphi \in \beta _ { 1 } S \} } \\ & { \quad \quad = \underset { \mu } { \operatorname* { i n f } } \ \underset { \varphi \in \beta _ { 1 } S } { \operatorname* { s u p } } \mathcal { I } ( \mu , \varphi ) . } \end{array} +$$ + +In practice, we can enforce this by applying spectral normalization (Miyato et al., 2018) and using a Lipschitz, smooth activation function such as ELU (Clevert et al., 2016) or sigmoid. + +Proposition 9. Let $f : \mathbb { R } ^ { d } \mathbb { R }$ be a neural network consisting of k layers whose linear transformations have spectral norm 1 and whose activation functions are 1-Lipschitz and 1-smooth. Then $f$ is $k$ -smooth. + +# 6 ENFORCING (D3) WITH GRADIENT PENALTIES + +(D3) is the following smoothness condition: + +(D3) $\mu \mapsto \nabla \Phi _ { \mu } ( x )$ is $\beta _ { 2 }$ -Lipschitz for any $x \in X$ , i.e., $\| \nabla \Phi _ { \mu } ( x ) - \nabla \Phi _ { \nu } ( x ) \| _ { 2 } \leq \beta _ { 2 } W _ { 1 } ( \mu , \nu ) .$ + +(D3) requires that the gradients of the optimal discriminator do not change too rapidly in response to changes in $\mu$ . Indeed, if the discriminator’s gradients are too sensitive to changes in the generator, the generator may not be able to accurately follow those gradients as it updates itself using a finite step size. In finite-dimensional optimization of a function $f : { \mathbb { R } ^ { d } } \to { \mathbb { R } }$ , this condition is analogous to $f$ having a Lipschitz gradient. + +We now present an equivalent characterization of (D3) that is easier to check in practice. We define the Bregman divergence of a convex function $J : { \mathcal { M } } ( X ) \to { \bar { \mathbb { R } } }$ by + +$$ +\mathfrak { D } _ { J } ( \nu , \mu ) : = J ( \nu ) - J ( \mu ) - \int \Phi _ { \mu } ( x ) d ( \nu - \mu ) , +$$ + +where $\Phi _ { \mu }$ is the optimal discriminator for $J$ at $\mu$ . Then, (D3) is characterized in terms of the Bregman divergence and the KR norm as follows: + +Proposition 10. Let $J : \mathcal { M } ( X ) \mathbb { R }$ be a convex function. Then $J$ satisfies $( D 3 )$ if and only if $\begin{array} { r } { \mathfrak { D } _ { J } ( \nu , \mu ) \le \frac { \beta _ { 2 } } { 2 } \| \mu - \nu \| _ { \mathrm { K R } } ^ { 2 } } \end{array}$ for all $\mu , \nu \in \mathcal { M } ( X )$ . + +It is straightforward to compute the Bregman divergence corresponding to several popular GANs: + +$$ +\begin{array} { r l } & { \mathfrak { D } _ { D \mathrm { J S } } ( \cdot | | \mu _ { 0 } ) \big ( \nu , \mu \big ) = D _ { \mathrm { K L } } \big ( \frac { 1 } { 2 } \nu + \frac { 1 } { 2 } \mu _ { 0 } | | \frac { 1 } { 2 } \mu + \frac { 1 } { 2 } \mu _ { 0 } \big ) + \frac { 1 } { 2 } D _ { \mathrm { K L } } \big ( \nu | | \mu \big ) , } \\ & { \qquad \mathfrak { D } _ { D _ { \mathrm { K L } } \left( \frac { 1 } { 2 } \cdot + \frac { 1 } { 2 } \mu _ { 0 } \right. | | \mu _ { 0 } ) } \big ( \nu , \mu \big ) = D _ { \mathrm { K L } } \big ( \frac { 1 } { 2 } \nu + \frac { 1 } { 2 } \mu _ { 0 } | | \frac { 1 } { 2 } \mu + \frac { 1 } { 2 } \mu _ { 0 } \big ) , } \\ & { \qquad \mathfrak { D } _ { \frac { 1 } { 2 } \mathrm { M M D } ^ { 2 } ( \cdot , \mu _ { 0 } ) } \big ( \nu , \mu \big ) = \frac { 1 } { 2 } \mathrm { M M D } ^ { 2 } ( \nu , \mu ) . } \end{array} +$$ + +The first two Bregman divergences are not bounded above by $| | \mu - \nu | | _ { \mathrm { K R } } ^ { 2 }$ for reasons similar to those discussed in Section 4, and hence: + +Proposition 11. The minimax and non-saturating GAN losses do not satisfy $( D 3 )$ for some $\mu _ { 0 }$ . + +Even so, the Bregman divergence for the non-saturating loss is always less than that of the minimax GAN, suggesting that the non-saturating loss should be stable in more situations than the minimax GAN. On the other hand, the MMD-based loss (Li et al., 2015) does satisfy (D3) when its kernel is the Gaussian kernel $K ( x , y ) = e ^ { - \pi | | x - y | | ^ { 2 } }$ : + +Proposition 12. The MMD loss with Gaussian kernel satisfies $( D 3 )$ with $\beta _ { 2 } = 2 \pi$ for all $\mu _ { 0 }$ + +# 6.1 FROM MAXIMUM MEAN DISCREPANCY TO GRADIENT PENALTIES + +Having identified the MMD-based loss as one that satisfies (D3), we next follow the strategy outlined in Section 3 to convert it into a regularizer for an arbitrary loss function. To do so, we define the regularizer $R _ { 3 } : \mathcal { M } ( X ) \bar { \mathbb { R } }$ and compute its convex conjugate $R _ { 3 } ^ { \star } : { \mathcal { C } } ( X ) { \bar { \mathbb { R } } }$ : + +$$ +R _ { 3 } ( \xi ) : = \frac { \beta _ { 2 } } { 4 \pi } \| \hat { \xi } \| _ { \mathcal { H } } ^ { 2 } , \qquad R _ { 3 } ^ { \star } ( \varphi ) = \frac { \pi } { \beta _ { 2 } } \| \varphi \| _ { \mathcal { H } } ^ { 2 } . +$$ + +The norm is the norm of a reproducing kernel Hilbert space norm (RKHS) $\mathcal { H }$ with Gaussian kernel; this norm extends the MMD to signed measures, as it holds that $\mathrm { M M D } ( \mu , \nu ) = \| \hat { \mu } - \hat { \nu } \| _ { \mathcal { H } }$ for $\mu , \nu \in { \mathcal { P } } ( X )$ . Here, $\begin{array} { r } { \hat { \xi } = \int K ( x , \cdot ) \xi ( d x ) \in \mathcal { H } } \end{array}$ denotes the mean embedding of a signed measure $\xi \in \mathcal { M } ( X )$ ; we also adopt the convention that $| | \varphi | | _ { \mathcal { H } } = \infty$ if $\varphi \notin { \mathcal { H } }$ . Similar to the situation in the previous sections, inf-convolution preserves the smoothness property of $R _ { 3 }$ : + +Proposition 13 (Moreau–Yosida regularization). Suppose $J : \mathcal { M } ( X ) \bar { \mathbb { R } }$ is convex, and define ${ \tilde { J } } : = J \oplus R _ { 3 }$ . Then $\tilde { J }$ is convex, and $\begin{array} { r } { \mathfrak { D } _ { \widetilde { J } } ( \nu , \mu ) \le \frac { \beta _ { 2 } } { 2 } \vert \vert \mu - \nu \vert \vert _ { \mathrm { K R } } ^ { 2 } } \end{array}$ . + +By Proposition 10, this transformed loss function satisfies (D3), having inherited the regularity properties of the squared MMD. This transformed function is a generalization of Moreau–Yosida regularization or the Moreau envelope (see Chapter 1 in Rockafeller & Wets (1998)). It is well-known that in the case of a function $f : \mathbb { R } ^ { n } \mathbb { R }$ , this regularization results in a function with Lipschitz gradients, so it is unsurprising that this property carries over to the infinite-dimensional case. + +Applying (4) and (19), we see that the transformed loss function can be minimized as a GAN by implementing an RKHS squared norm penalty on the discriminator: + +$$ +\begin{array} { r l } & { \underset { \mu } { \operatorname* { i n f } } \big ( J \oplus \frac { \beta _ { 2 } } { 4 \pi } | | \cdot | | _ { \mathcal { H } } ^ { 2 } \big ) ( \mu ) = \underset { \mu } { \operatorname* { i n f } } \underset { \varphi } { \operatorname* { s u p } } \mathbb { E } _ { x \sim \mu } [ \varphi ( x ) ] - J ^ { \star } ( \varphi ) - \frac { \pi } { \beta _ { 2 } } | | \varphi | | _ { \mathcal { H } } ^ { 2 } . } \end{array} +$$ + +Computationally, the RKHS norm is difficult to evaluate. We propose taking advantage of the following infinite series representation of $| | f | | _ { \mathcal { H } } ^ { 2 }$ in terms of the derivatives of $f$ (Fasshauer & Ye, 2011; Novak et al., 2018): + +Proposition 14. Let $\mathcal { H }$ be an RKHS with the Gaussian kernel $K ( x , y ) = e ^ { - \pi | | x - y | | ^ { 2 } }$ . Then for $f \in \mathcal H$ , + +$$ +\begin{array} { r l } & { \| f \| _ { \mathcal H } ^ { 2 } = \displaystyle \sum _ { k = 0 } ^ { \infty } ( 4 \pi ) ^ { - k } \sum _ { \substack { k _ { 1 } + \cdots + k _ { d } = k } } \frac { 1 } { \prod _ { i = 1 } ^ { d } k _ { i } ! } \| \partial _ { x _ { 1 } } ^ { k _ { 1 } } \cdot \cdot \cdot \partial _ { x _ { d } } ^ { k _ { d } } f \| _ { L ^ { 2 } ( \mathbb R ^ { d } ) } ^ { 2 } } \\ & { \quad \quad = \| f \| _ { L ^ { 2 } ( \mathbb R ^ { d } ) } ^ { 2 } + \frac { 1 } { 4 \pi } \| \nabla f \| _ { L ^ { 2 } ( \mathbb R ^ { d } ) } ^ { 2 } + \frac { 1 } { 1 6 \pi ^ { 2 } } \| \nabla ^ { 2 } f \| _ { L ^ { 2 } ( \mathbb R ^ { d } ) } ^ { 2 } + o t h e r t e r m s . } \end{array} +$$ + +In an ideal world, we would use this expression as a penalty on the discriminator to enforce (D3). Of course, as an infinite series, this formulation is computationally impractical. However, the first two terms are very close to common GAN techniques like gradient penalties (Gulrajani et al., 2017) and penalizing the output of the discriminator (Karras et al., 2018). We therefore interpret these common practices as partially applying the penalty given by the RKHS norm squared, approximately enforcing (D3). We view the choice of only using the leading terms as a disadvantageous but practical necessity. + +Interestingly, according to our analysis, gradient penalties and spectral normalization are not interchangeable, even though both techniques were designed to constrain the Lipschitz constant of the discriminator. Instead, our analysis suggests that they serve different purposes: gradient penalties enforce the variational smoothness (D3), while spectral normalization enforces Lipschitz continuity (D1). This demystifies the puzzling observation of Miyato (2018) that GANs using only spectral normalization with a WGAN loss do not seem to train well; it also explains why using both spectral normalization and a gradient penalty is a reasonable strategy. It also motivates the use of gradient penalties applied to losses other than the Wasserstein loss (Fedus et al., 2018). + +# 7 VERIFYING THE THEORETICAL LEARNING RATE + +In this section, we empirically test the theoretical learning rate given by Theorem 1 and Proposition 1 as well as our regularization scheme (7) based on inf-convolutions. We approximately implement our composite regularization scheme (7) on a trivial base loss of $J ( \mu ) = \overset { \triangledown } { \chi } \bar { \{ \mu = \mu _ { 0 } \} }$ by alternating stochastic gradient steps on + +$$ +\operatorname* { i n f } _ { \mu } \operatorname* { s u p } _ { \varphi } \mathbb { E } _ { x \sim \mu } [ \varphi ( x ) ] - \mathbb { E } _ { x \sim \mu _ { 0 } } [ \varphi ( x ) ] - \frac { \pi } { \beta _ { 2 } } \mathbb { E } _ { x \sim \tilde { \mu } } \Big [ \varphi ( x ) ^ { 2 } + \frac { 1 } { 4 \pi } | | \nabla \varphi ( x ) | | ^ { 2 } \Big ] , +$$ + +where $\tilde { \mu }$ is a random interpolate between samples from $\mu$ and $\mu _ { 0 }$ , as used in Gulrajani et al. (2017). The regularization term is a truncation of the series for the squared RKHS norm (22) and approximately enforces (D3). The discriminator is a 7-layer convolutional neural network with spectral normalization1 and ELU activations, an architecture that enforces (D1) and (D2). We include a final scalar multiplication by $\alpha$ so that by Proposition 9, $\beta _ { 1 } = 7 \alpha$ . We take two discriminator steps for every generator step, to better approximate our assumption of an optimal discriminator. + +![](images/aa82e4dee4d350f0883d5b1945b97b4159438c424259c9b07cdff3a713b23542.jpg) +Figure 1: FID of simple particle generators with various learning rates. The horizontal axis shows the ratio $\gamma / \gamma _ { 0 }$ between the actual learning rate $\gamma$ and the theoretical learning rate $\gamma _ { 0 }$ suggested by Theorem 1 and Proposition 1. The vertical axis shows the FID after 100,000 SGD iterations. + +For the generator, we use an extremely simple particle-based generator which satisfies (G1) and (G2), in order to minimize the number of confounding factors in our experiment. Let $\omega$ be the discrete uniform distribution on $Z = \{ 1 , \ldots , N \}$ . For an $N \times d$ matrix $\theta$ and $z \in Z$ , define $f _ { \theta } : Z \to \mathbb { R } ^ { d }$ so that $f _ { \theta } ( z )$ is the $z$ th row of $\theta$ . The particle generator $\mu _ { \theta } = f _ { \theta \# } \omega$ satisfies (G1) with $\begin{array} { r } { A = \frac { 1 } { \sqrt { N } } } \end{array}$ , since + +$$ +\mathbb { E } _ { z } [ \| f _ { \theta } ( z ) - f _ { \theta ^ { \prime } } ( z ) \| _ { 2 } ] = \frac { 1 } { N } \sum _ { z = 1 } ^ { n } \| \theta _ { z } - \theta _ { z } ^ { \prime } \| _ { 2 } \leq \frac { 1 } { \sqrt { N } } \| \theta - \theta ^ { \prime } \| _ { \mathrm { F } } , +$$ + +and it satisfies (G2) with $B = 0$ , since $D _ { \theta } f _ { \theta } ( z )$ is constant w.r.t. $\theta$ . With this setup, Theorem 1 suggests a theoretical learning rate of + +$$ +\gamma _ { 0 } = \frac { 1 } { L } = \frac { 1 } { \alpha B + A ^ { 2 } ( \beta _ { 1 } + \beta _ { 2 } ) } = \frac { N } { 7 \alpha + \beta _ { 2 } } . +$$ + +We randomly generated hyperparameter settings for the Lipschitz constant $\alpha$ , the smoothness constant $\beta _ { 2 }$ , the number of particles $N$ , and the learning rate $\gamma$ . We trained each model for 100,000 steps on CIFAR-10 and evaluate each model using the Frechet Inception Distance (FID) of Heusel ´ et al. (2017). We hypothesize that stability is correlated with image quality; Figure 1 plots the FID for each hyperparameter setting in terms of the ratio of the true learning rate $\gamma$ and the theoretically motivated learning rate $\gamma _ { 0 }$ . We find that the best FID scores are obtained in the region where $\gamma / \gamma _ { 0 }$ is between 1 and 1000. For small learning rates $\gamma / \gamma _ { 0 } \ll 1$ , we observe that the convergence is too slow to make a reasonable progress on the objective, whereas as the learning rate gets larger $\gamma / \gamma _ { 0 } \gg 1$ , we observe a steady increase in FID, signalling unstable behavior. It also makes sense that learning rates slightly above the optimal rate produce good results, since our theoretical learning rate is a conservative lower bound. Note that our intention is to test our theory, not to generate good images, which is difficult due to our weak choice of generator. Overall, this experiment shows that our theory and regularization scheme are sensible. + +# 8 FUTURE WORK + +Inexact gradient descent In this paper, we employed several assumptions in order to regard the GAN algorithm as gradient descent. However, real-world GAN algorithms must be treated as “inexact” descent algorithms. As such, future work includes: (i) relaxing the optimal discriminator assumption (cf. Sanjabi et al. (2018)) or providing a stability result for discrete simultaneous gradient descent (cf. continuous time analysis in Nagarajan & Kolter (2017); Mescheder et al. (2018)), (ii) addressing stochastic approximations of gradients (i.e., SGD), and (iii) providing error bounds for the truncated gradient penalty used in (23). + +Generator architectures Another important direction of research is to seek more powerful generator architectures that satisfy our smoothness assumptions (G1) and (G2). In practice, generators are often implemented as deep neural networks, and involve some specific architectures such as deconvolution layers (Radford et al., 2015) and residual blocks (e.g., Gulrajani et al. (2017); Miyato et al. (2018)). In this paper, we did not provide results on the smoothness of general classes of generators, since our focus is to analyze stability properties influenced by the choice of loss function $J$ (and therefore optimal discriminators). However, our conditions (G1) and (G2) shed light on how to obtain smoothly parameterized neural networks, which is left for future work. + +# ACKNOWLEDGMENTS + +We would like to thank Kohei Hayashi, Katsuhiko Ishiguro, Masanori Koyama, Shin-ichi Maeda, Takeru Miyato, Masaki Watanabe, and Shoichiro Yamaguchi for helpful discussions. + +# REFERENCES + +Luigi Ambrosio, Nicola Gigli, and Giuseppe Savare.´ Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhauser, second edition, 2008.¨ + +Michael Arbel, Dougal Sutherland, Mikołaj Binkowski, and Arthur Gretton. On gradient regular- ´ izers for MMD GANs. In Advances in Neural Information Processing Systems, pp. 6700–6710, 2018. + +Martin Arjovsky and Leon Bottou. Towards principled methods for training generative adversarial networks. In International Conference on Learning Representations, 2017. + +Martin Arjovsky, Soumith Chintala, and Leon Bottou. 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In International Conference on Learning Representations, 2018. + +Cedrik Villani. ´ Optimal Transport: Old and New. Springer, 2009. + +Hassler Whitney. Analytic extensions of differentiable functions defined in closed sets. Transactions of the American Mathematical Society, 36(1):63–89, 1934. + +Xingyu Zhou. On the Fenchel duality between strong convexity and Lipschitz continuous gradient. arXiv preprint arXiv:1803.06573, 2018. + +# A INF-CONVOLUTION IN $\mathbb { R } ^ { d }$ + +To gain intuition on the inf-convolution, we present a finite-dimensional analogue of the techniques in Section 3. For simplicity of presentation, we will omit any regularity conditions (e.g., lower semicontinuity). We refer readers to Chapter 12 of Bauschke & Combettes (2011) for a detailed introduction. + +Let $J$ and $R$ be convex functions on $\mathbb { R } ^ { d }$ . The inf-convolution of $J$ and $R$ is a function $J \star R$ defined as + +$$ +( J \oplus R ) ( x ) : = \operatorname* { i n f } _ { z \in \mathbb { R } ^ { d } } J ( z ) + R ( x - z ) . +$$ + +The inf-convolution is often called the epigraphic sum since the epigraph of $J { \star R }$ coincides with the Minkowski sum of epigraphs of $J$ and $R$ , as Figure 2 illustrates. The inf-convolution is associative and commutative operation; that is, it is always true that $\left( J _ { 1 } \oplus J _ { 2 } \right) \oplus J _ { 3 } = J _ { 1 } \oplus ( J _ { 2 } \oplus J _ { 3 } ) = :$ $J _ { 1 } \oplus J _ { 2 } \oplus J _ { 3 }$ and $J _ { 1 } \oplus J _ { 2 } = J _ { 2 } \oplus J _ { 1 }$ . + +There are two important special cases of inf-convolutions: The first one is the Pasch–Hausdorff envelope $J _ { \alpha }$ , which is the inf-convolution between $J$ and $\alpha \| { \cdot } \| _ { 2 } ~ ( \alpha > 0 )$ . It is known that $J _ { \alpha }$ becomes $\alpha$ -Lipschitz. The second important example is the Moreau envelope $\begin{array} { r } { J ^ { \beta } = J \oplus \frac { 1 } { 2 \beta } \lVert \cdot \rVert _ { 2 } ^ { \overleftarrow { 2 } } } \end{array}$ i.e., the inf-convolution with the quadratic regularizer $\frac { 1 } { 2 \beta } \lVert \cdot \rVert _ { 2 } ^ { 2 }$ . The Moreau envelope $J ^ { \beta }$ is always differentiable, and the gradient of $J ^ { \beta }$ is $\beta$ -Lipschitz (thus $J ^ { \beta }$ is $\beta .$ -smooth). + +It is worth noting that the set of minimizers does not change after these two operations. More generally, we have the following result: + +Proposition 15. Let $J , R : \mathbb { R } ^ { d } \bar { \mathbb { R } }$ be proper and lower semicontinuous functions with $\operatorname* { m i n } J >$ $- \infty$ and $\operatorname* { m i n } R = 0$ . Suppose $R ( 0 ) = 0$ and $R ( x ) \geq \psi ( \| x \| _ { 2 } )$ for some increasing function $\psi : \mathbb { R } _ { \ge 0 } \mathbb { R }$ . Then, $\operatorname* { m i n } J \oplus R = \operatorname* { m i n } J$ and arg ${ \mathrm { m i n } } J \oplus { \cal R } = \arg \operatorname* { m i n } J$ . + +![](images/680218ae8fb4c9cdc0963df033e29a25d3a9ce21159c7615b779f72f7741abe0.jpg) +Figure 2: Illustration of inf-convolutions and their convex conjugates in $\mathbb { R }$ . Top row: Generally, inf-convolutions inherit the regularity of their parent functions. In this example, $J _ { 1 } ( x ) = \left| x \right|$ is 1-Lipschitz but not smooth, while $J _ { 2 } ( x ) = { x } ^ { 2 } / \dot { 2 }$ is 1-smooth but not Lipschitz. The inf-convolution $J _ { 1 } \oplus J _ { 2 }$ is the well-known Huber function, which is both 1-Lipschitz and 1-smooth. Bottom row: Convex conjugates of the functions in the top row. The conjugate of $J _ { 1 } \oplus J _ { 2 }$ is given as the sum of conjugates $J _ { 1 } ^ { \star } ( z ) = \chi \{ | z | \leq 1 \}$ and $J _ { 2 } ^ { \star } ( z ) \stackrel { = } { = } z ^ { 2 } / 2$ . + +To sum up, given a function $J$ , we can always construct a regularized alternative $J _ { \alpha } ^ { \beta }$ that is $\alpha$ - Lipschitz and $\beta$ -smooth and has the same minimizers as $J$ . + +The next question is how to implement the inf-convolution in GAN-like optimization problems. For this, it is convenient to consider the convex conjugate. Recall that the Fenchel–Moreau theorem says that there is a duality between a convex function $J$ and its convex conjugate $J ^ { \star }$ as $J ( x ) =$ $\begin{array} { r } { \operatorname* { s u p } _ { z \in \mathbb { R } ^ { d } } \langle x , z \rangle - J ^ { \star } ( z ) } \end{array}$ and $\begin{array} { r } { J ^ { \star } ( z ) = \operatorname* { s u p } _ { x \in \mathbb { R } ^ { d } } \langle x , z \rangle - J ( x ) } \end{array}$ . The important property is that the convex conjugate of the inf-convolution is the sum of convex conjugates, that is, we always have + +$$ +( J \oplus R ) ^ { \star } ( z ) = J ^ { \star } ( z ) + R ^ { \star } ( z ) . +$$ + +This property can be useful for implementing the regularized objective $J _ { \alpha } ^ { \beta }$ as follows. First, we can check that the convex conjugates of the norm and the squared norm are given as $( \| { \cdot } \| _ { 2 } ) ^ { \star } = \chi \{ \| { \cdot } \| \leq$ $1 \}$ and $\begin{array} { r } { ( \frac { 1 } { 2 } \| { \cdot } \| _ { 2 } ^ { 2 } ) ^ { \star } = \frac { 1 } { 2 } \| { \cdot } \| _ { 2 } ^ { 2 } } \end{array}$ . Hence, we have + +$$ +J _ { \alpha } ^ { \beta } ( \boldsymbol { x } ) : = \left( J \oplus \alpha \lVert \cdot \rVert _ { 2 } \oplus \frac { 1 } { 2 \beta } \lVert \cdot \rVert _ { 2 } ^ { 2 } \right) ( \boldsymbol { x } ) = \operatorname* { s u p } _ { \boldsymbol { z } : \ \lVert \boldsymbol { z } \rVert _ { 2 } \leq \alpha } \langle \boldsymbol { x } , \boldsymbol { z } \rangle - J ^ { \star } ( \boldsymbol { z } ) - \frac { \beta } { 2 } \lVert \boldsymbol { z } \rVert _ { 2 } ^ { 2 } , +$$ + +which means that minimizing $J _ { \alpha } ^ { \beta }$ can be recast in min-max problem with the norm clipping and $\ell _ { 2 }$ -regularization on the dual variable $z$ . + +# B COMMON GAN LOSSES + +For completeness and clarity, we explicitly write out the expressions for the losses listed in Table 1. For more detailed computations of optimal discriminators, see Chu et al. (2019); for more details on the convex duality interpretation, see Farnia & Tse (2018). + +Minimax GAN Goodfellow et al. (2014) originally proposed the minimax GAN and showed that the corresponding loss function for the minimax GAN is the Jensen–Shannon divergence, defined as + +$$ +J ( \mu ) : = D _ { \mathrm { J S } } ( \mu | | \mu _ { 0 } ) : = \frac 1 2 D _ { \mathrm { K L } } ( \mu | | \frac 1 2 \mu + \frac 1 2 \mu _ { 0 } ) + \frac 1 2 D _ { \mathrm { K L } } ( \mu _ { 0 } | | \frac 1 2 \mu + \frac 1 2 \mu _ { 0 } ) , +$$ + +where $\mu _ { 0 } \in { \mathcal { P } } ( X )$ is a fixed probability measure (usually the empirical measure of the data), and $D _ { \mathrm { K L } } ( \mu \lvert \lvert \nu \rangle )$ is the Kullback–Leibler divergence between $\mu$ and $\nu$ . The optimal discriminator in the sense of Definition 1 is given as + +$$ +\Phi _ { \mu } ( x ) = \frac { 1 } { 2 } \log \frac { d \mu } { d ( \mu + \mu _ { 0 } ) } ( x ) , +$$ + +where $\frac { d \mu } { d ( \mu + \mu _ { 0 } ) }$ is the Radon–Nikodym derivative. If $\mu$ and $\mu _ { 0 }$ have densities $\mu ( x )$ and $\mu _ { 0 } ( x )$ , then + +$$ +\frac { d \mu } { d ( \mu + \mu _ { 0 } ) } ( x ) = \frac { \mu ( x ) } { \mu ( x ) + \mu _ { 0 } ( x ) } , +$$ + +so our optimal discriminator matches that of Goodfellow et al. (2014) up to a constant factor and logarithm. To recover the minimax formulation, the convex duality (4) yields: + +$$ +\begin{array} { r l } & { \underset { \mu } { \operatorname* { i n f } } D _ { \mathrm { J S } } ( \mu , \mu _ { 0 } ) = \underset { \mu } { \operatorname* { i n f } } \operatorname* { s u p } _ { \varphi } \mathbb { E } _ { x \sim \mu } [ \varphi ( x ) ] - ( \underset { - 1 } { - } \frac { 1 } { 2 } \mathbb { E } _ { x \sim \mu _ { 0 } } [ \log ( 1 - e ^ { 2 \varphi ( x ) + \log 2 } ) ] - \frac { 1 } { 2 } \log 2 ) } \\ & { \qquad ( D _ { \mathrm { J S } } ( \cdot , \mu _ { 0 } ) ) ^ { \star } ( \varphi ) } \\ & { \qquad = \underset { \mu } { \operatorname* { i n f } } \underset { D } { \operatorname* { s u p } } \frac { 1 } { 2 } \mathbb { E } _ { x \sim \mu } [ \log ( 1 - D ( x ) ) ] + \frac { 1 } { 2 } \mathbb { E } _ { x \sim \mu _ { 0 } } [ \log D ( x ) ] , } \end{array} +$$ + +using the substitution $\begin{array} { r } { \varphi = \frac { 1 } { 2 } \log ( 1 - D ) - \frac { 1 } { 2 } \log 2 } \end{array}$ . + +Non-saturating GAN Goodfellow et al. (2014) also proposed the heuristic non-saturating GAN. Theorem 2.5 of Arjovsky & Bottou (2017) shows that the loss function minimized is + +$$ +J ( \mu ) : = D _ { \mathrm { K L } } ( \textstyle { \frac { 1 } { 2 } } \mu + \textstyle { \frac { 1 } { 2 } } \mu _ { 0 } \left| \right| \mu _ { 0 } ) = \frac { 1 } { 2 } D _ { \mathrm { K L } } ( \mu \left| \right| \mu _ { 0 } ) - D _ { \mathrm { J S } } ( \mu \left| \right| \mu _ { 0 } ) . +$$ + +The optimal discriminator is + +$$ +\Phi _ { \mu } ( x ) = - \frac { 1 } { 2 } \log \frac { d \mu _ { 0 } } { d ( \mu + \mu _ { 0 } ) } ( x ) . +$$ + +Wasserstein GAN Arjovsky et al. (2017) proposed the Wasserstein GAN, which minimizes the Wasserstein-1 distance between the input $\mu$ and a fixed measure $\mu _ { 0 }$ : + +$$ +J ( \mu ) : = W _ { 1 } ( \mu , \mu _ { 0 } ) : = \operatorname* { i n f } _ { \pi } \mathbb { E } _ { ( x , y ) \sim \pi } [ | | x - y | | ] , +$$ + +here the infimum is taken over all couplings $\pi$ , probability distributions over $X \times X$ who arginals are $\mu$ and $\mu _ { 0 }$ respectively. The optimal discriminator $\Phi _ { \mu }$ is called the Kantorovich pote al in the optimal transport literature (Villani, 2009). The convex duality (4) recover the Wasserste AN: + +$$ +\begin{array} { r l } & { \underset { \mu } { \operatorname* { i n f } } W _ { 1 } ( \mu , \mu _ { 0 } ) = \underset { \mu } { \operatorname* { i n f } } \operatorname* { s u p } _ { \varphi } \mathbb { E } _ { x \sim \mu } [ \varphi ( x ) ] - \underset { ( W _ { 1 } ( \cdot , \mu _ { 0 } ) ) ^ { \star } ( \varphi ) } { \underbrace { ( \mathbb { E } _ { x \sim \mu _ { 0 } } [ \varphi ( x ) ] + \chi \{ | | \varphi | | _ { \mathrm { L i p } } \leq 1 \} } } ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ( W _ { 1 } ( \cdot , \mu _ { 0 } ) ) ^ { \star } ( \varphi ) } \\ & { \underset { \quad \mu } { = } \underset { | | \varphi | | _ { \mathrm { L i p } } \leq 1 } { \operatorname* { i n f } } \mathbb { E } _ { x \sim \mu } [ \varphi ( x ) ] - \mathbb { E } _ { x \sim \mu _ { 0 } } [ \varphi ( x ) ] , } \end{array} +$$ + +an expression of Kantorovich–Rubinstein duality. The Lipschitz constraint on the discriminator is typically enforced by spectral normalization (Miyato et al., 2018), less frequently by weight clipping (Arjovsky et al., 2017), or heuristically by gradient penalties (Gulrajani et al., 2017) (although this work shows that gradient penalties may serve a different purpose altogether). + +Maximum mean discrepancy Given a positive definite kernel $K : X \times X \to \mathbb { R }$ , the maximum mean discrepancy (MMD, Gretton et al. (2012)) between $\mu$ and $\nu$ is defined by + +$$ +J ( \mu ) : = \frac { 1 } { 2 } \mathrm { M M D } _ { K } ^ { 2 } ( \mu , \nu ) : = \frac { 1 } { 2 } \int K ( x , y ) \left( \mu - \nu \right) ( d x ) \left( \mu - \nu \right) ( d y ) . +$$ + +where $( \mathcal { H } , \| \cdot \| _ { \mathcal { H } } )$ is the reproducing kernel Hilbert space (RKHS) for $K$ . The generative momentmatching network (GMMN, Li et al. (2015)) and the Coulomb GAN (Unterthiner et al., 2018) use the squared MMD as the loss function. The optimal discriminator in this case is + +$$ +\Phi _ { \mu } ( x ) = \mathbb { E } _ { y \sim \mu } [ K ( x , y ) ] - \mathbb { E } _ { y \sim \mu _ { 0 } } [ K ( x , y ) ] , +$$ + +which in constrast to other GANs, may be approximated by simple Monte Carlo, rather than an auxiliary optimization problem. + +Note that MMD-GANs (Li et al., 2017a; Arbel et al., 2018) minimize a modified version of the MMD, the Optimized MMD (Sriperumbudur et al., 2009; Arbel et al., 2018). These MMD-GANs are adversarial in a way that does not arise from convex duality, so our theory currently does not apply to these GANs. + +Integral probability metrics An integral probability metric (Muller, 1997) is defined by ¨ + +$$ +J ( \mu ) : = { \mathrm { I P M } } _ { \mathcal F } ( \mu , \mu _ { 0 } ) : = \operatorname* { s u p } _ { f \in \mathcal F } \int f d \mu - \int f d \mu _ { 0 } , +$$ + +where $\mathcal { F }$ is a class of functions. The optimal discriminator is the function that maximizes the supremum in the definition. The Wasserstein distance may be thought of as an IPM with $\mathcal { F }$ containing all 1-Lipschitz functions. The MMD may be thought of as an IPM with $\mathcal { F }$ all functions with RKHS norm at most 1, but no GANs based on MMD are actually trained this way, as it is difficult to constrain the discriminator to such functions. + +# C OPTIMAL DISCRIMINATORS ARE FUNCTIONAL DERIVATIVES + +Let $J : { \mathcal { P } } ( X ) \to { \bar { \mathbb { R } } }$ be a convex function. Recall the definition of the optimal discriminator (Definition 1): + +$$ +\Phi _ { \mu } \in \underset { \varphi \in \mathcal { C } ( X ) } { \arg \operatorname* { m a x } } \int \varphi d \mu - J ^ { \star } ( \varphi ) . +$$ + +This definition can be understood as an infinite dimensional analogue of subgradients. In fact, in finite-dimensional convex analysis, $z$ is a subgradient of $f : \mathbb { R } ^ { d } \bar { \bar { \mathbb { R } } }$ if and only if it can be written as $z \in \arg \operatorname* { m a x } _ { z ^ { \prime } } \langle z ^ { \prime } , x \rangle - f ^ { \star } ( z ^ { \prime } )$ . The calculus of subgradients shares many properties with the standard calculus of derivatives, such as chain rules (Rockafeller & Wets, 1998). This motivate us to investigate derivative-like features of optimal discriminators. + +We introduce the functional derivative, also known as the von Mises influence function: + +Definition 3 (Functional derivative). Let $J : { \mathcal { P } } ( X ) \to { \bar { \mathbb { R } } }$ be a function of probability measures. We say that a continuous function $\Phi _ { \mu } : X \to \mathbb { R }$ is a functional derivative of $J$ at $\mu$ if + +$$ +J ( \mu + \epsilon \xi ) = J ( \mu ) + \epsilon \int \Phi _ { \mu } d \xi + O ( \epsilon ^ { 2 } ) +$$ + +holds for any $\xi = \nu - \mu$ with $\nu \in { \mathcal { P } } ( X )$ . + +Under this definition, optimal discriminators are actually functional derivatives. + +Proposition 16 (Chu et al. (2019), Theorem 2). Let $J : \mathcal { M } ( X ) \bar { \mathbb { R } }$ be a proper, lower semicontinuous, and convex function. If there exists a maximizer $\Phi _ { \mu }$ of $\begin{array} { r } { \varphi \mapsto \int \varphi d \mu - J ^ { \star } ( \varphi ) } \end{array}$ , then $\Phi _ { \mu }$ is $a$ functional derivative of $J$ at $\mu$ in the sense of Definition 3. + +The following result relates the derivative of the loss function with the derivative of the optimal discriminator: + +Proposition 17 (Chu et al. (2019), Theorem 1). Let $J : { \mathcal { P } } ( X ) \to \mathbb { R }$ be continuously differentiable, in the sense that the functional derivative $\Phi _ { \mu }$ exists and $( \mu , \nu ) \mapsto \mathbb { E } _ { x \sim \nu } [ \Phi _ { \mu } ( x ) ]$ is continuous. Let $\theta \mapsto \mu _ { \theta }$ be continuous in the sense that ${ \frac { 1 } { \| h \| } } \big ( \mu _ { \theta + h } - \mu _ { \theta } \big )$ converges to a weak limit as $\| h \| 0$ . Then, we have + +$$ +\begin{array} { r } { \nabla _ { \theta } J ( \mu _ { \theta } ) = \nabla _ { \theta } \mathbb { E } _ { x \sim \mu _ { \theta } } [ \hat { \Phi } ( x ) ] , } \end{array} +$$ + +where $\hat { \Phi } = \Phi _ { \mu _ { \theta } }$ is treated as a function $X \to \mathbb { R }$ that is not dependent on $\theta$ . + +We use this important computational tool in many of our proofs. For the case of the generator model $\mu _ { \theta } = f _ { \theta \# } \omega$ , an important consequence of Proposition 17 is that + +$$ +\nabla _ { \theta } J ( \mu _ { \theta } ) = \nabla _ { \theta } \mathbb { E } _ { z \sim \omega } [ \hat { \Phi } ( f _ { \theta } ( z ) ) ] = \mathbb { E } _ { z \sim \omega } [ \nabla _ { x = f _ { \theta } ( z ) } \Phi _ { \mu } ( f _ { \theta } ( z ) ) \cdot D _ { \theta } f _ { \theta } ( z ) ] . +$$ + +We use this fact in the proof of Theorem 1. + +# D PROOFS FOR SECTIONS 1 AND 2 + +The following result is well known in the dynamical systems and the optimization literature. For the sake of completeness, we provide its proof. + +Proposition 1 (Bertsekas (1999), Proposition 1.2.3). Suppose $f : \mathbb { R } ^ { p } \mathbb { R }$ is $L$ -smooth and bounded below. Let $\begin{array} { r } { x _ { k + 1 } : = x _ { k } - \frac { 1 } { L } \dot { \nabla } \dot { f } ( x _ { k } ) } \end{array}$ . Then $| | \nabla f ( x _ { k } ) | | 0$ as $k \to \infty$ . + +Proof. It is known that if $f$ is $L$ -smooth, then + +$$ +f ( y ) \leq f ( x ) + \langle \nabla f ( x ) , y - x \rangle + { \frac { L } { 2 } } \vert \vert y - x \vert \vert ^ { 2 } +$$ + +holds for any $x , y \in \mathbb { R } ^ { d }$ (see e.g. Lemma 3.4 in Bubeck (2015)). Then, we have + +$$ +\begin{array} { l } { f ( x _ { k + 1 } ) \leq f ( x _ { k } ) + \langle \nabla f ( x _ { k } ) , x _ { k + 1 } - x _ { k } \rangle + \displaystyle \frac { L } { 2 } | | x _ { k + 1 } - x _ { k } | | ^ { 2 } } \\ { \leq f ( x _ { k } ) - \displaystyle \frac { 1 } { L } | | \nabla f ( x _ { k } ) | | ^ { 2 } + \displaystyle \frac { 1 } { 2 L } | | \nabla f ( x _ { k } ) | | ^ { 2 } } \\ { = f ( x _ { k } ) - \displaystyle \frac { 1 } { 2 L } | | \nabla f ( x _ { k } ) | | ^ { 2 } . } \end{array} +$$ + +Summing this inequality over $k$ , we have + +$$ +{ \frac { 1 } { 2 L } } \sum _ { k = 0 } ^ { n - 1 } | | \nabla f ( x _ { k } ) | | ^ { 2 } \leq f ( x _ { 0 } ) - f ( x _ { n } ) , +$$ + +from which we conclude that + +$$ +\operatorname* { m i n } _ { 0 \leq k \leq n - 1 } | | \nabla f ( x _ { k } ) | | ^ { 2 } \leq { \frac { 2 L } { n } } ( f ( x _ { 0 } ) - \operatorname* { i n f } _ { x } f ( x ) ) . +$$ + +Next, we move on to prove Theorem 1, which we restate here for readability. + +Theorem 1 (Smoothness decomposition for GANs). Let $J : \mathcal { M } ( X ) \bar { \mathbb { R } }$ be a convex function whose optimal discriminators $\Phi _ { \mu } : X \to \mathbb { R }$ satisfy the following regularity conditions: + +(D1) $x \mapsto \Phi _ { \mu } ( x )$ is $\alpha$ -Lipschitz, (D2) $x \mapsto \nabla _ { x } \Phi _ { \mu } ( x )$ is $\beta _ { 1 }$ -Lipschitz, (D3) $\mu \mapsto \nabla _ { x } \Phi _ { \mu } ( x )$ is $\beta _ { 2 }$ -Lipschitz w.r.t. the 1-Wasserstein distance. + +Also, let $\mu _ { \theta } = f _ { \theta \# } \omega$ be a family of generators that satisfies: + +(G1) $\theta \mapsto f _ { \theta } ( z )$ is $A$ -Lipschitz in expectation for $z \sim \omega$ , i.e., $\begin{array} { r } { \mathbb { E } _ { z \sim \omega } [ \| f _ { \theta _ { 1 } } ( z ) - f _ { \theta _ { 2 } } ( z ) \| _ { 2 } ] \ \le } \end{array}$ $A \| \theta _ { 1 } - \theta _ { 2 } \| _ { 2 }$ , and + +(G2) $\theta \ \mapsto \ D _ { \theta } f _ { \theta } ( z )$ is $B$ -Lipschitz in expectation for $z \sim \omega$ , i.e., $\mathbb { E } _ { z \sim \omega } [ \| D _ { \theta _ { 1 } } f _ { \theta _ { 1 } } ( z ) ~ -$ $D _ { \theta _ { 2 } } f _ { \theta _ { 2 } } ( z ) \lVert _ { 2 } \bigr \rVert \leq B \lVert \theta _ { 1 } - \theta _ { 2 } \rVert _ { 2 }$ . + +Then $ { \theta } \mapsto J ( { \mu } _ { { \theta } } )$ is $L$ -smooth, with $L = \alpha B + A ^ { 2 } ( \beta _ { 1 } + \beta _ { 2 } )$ . + +Intuitively, Theorem 1 can be understood as the chain rule. For simplicity, let us consider the smoothness of a composite function $D \circ G$ , where $D$ and $G$ are functions on $\mathbb { R }$ . A sufficient condition for $D \circ G$ to be smooth is that its second derivative is bounded. For this, suppose that (i) $D$ is $\alpha$ -Lipschitz and $\beta$ -smooth and (ii) $G$ is $A$ -Lipschitz and $B$ -smooth. By the chain rule, the second derivative is bounded as $( D \circ G ) ^ { \prime \prime } = ( D ^ { \prime } \circ G { \bar { G } } ) G ^ { \prime \prime } + ( D ^ { \prime \prime } \circ G ) ( G ^ { \prime } ) ^ { 2 } \leq \alpha B + A ^ { 2 } \beta$ , which has the same form as the consequence of Theorem 1. For GANs, we need somewhat more involved conditions; We need two types of smoothness (D2) and (D3) for the optimal discriminators. To this end, we utilize the functional gradient point of view that we explained in Appendix $\textrm { C }$ . + +Proof of Theorem 1. First, using the functional gradient interpretation of the optimal discriminator, we have + +$$ +\begin{array} { r l } & { \quad \| \nabla _ { \theta _ { 1 } } J ( \mu _ { \theta _ { 1 } } ) - \nabla _ { \theta _ { 2 } } J ( \mu _ { \theta _ { 2 } } ) \| } \\ & { = \big \| \mathbb { E } _ { z \sim \omega } \big [ \nabla \Phi _ { \mu _ { \theta _ { 1 } } } ( f _ { \theta _ { 1 } } ( z ) ) \cdot D _ { \theta _ { 1 } } f _ { \theta _ { 1 } } - \nabla \Phi _ { \mu _ { \theta _ { 2 } } } ( f _ { \theta _ { 2 } } ( z ) ) \cdot D _ { \theta _ { 2 } } f _ { \theta _ { 2 } } \big ] \big \| \qquad \mathrm { P r o p o s i t i o } } \\ & { = \big \| \mathbb { E } _ { z \sim \omega } \big [ \nabla \Phi _ { \mu _ { \theta _ { 1 } } } ( f _ { \theta _ { 1 } } ( z ) ) \cdot ( D _ { \theta _ { 1 } } f _ { \theta _ { 1 } } - D _ { \theta _ { 2 } } f _ { \theta _ { 2 } } ) } \\ & { \qquad + \big ( \nabla \Phi _ { \mu _ { \theta _ { 1 } } } ( f _ { \theta _ { 1 } } ( z ) ) - \nabla \Phi _ { \mu _ { \theta _ { 2 } } } ( f _ { \theta _ { 1 } } ( z ) ) \big ) \cdot D _ { \theta _ { 2 } } f _ { \theta _ { 2 } } } \\ & { \qquad + \big ( \nabla \Phi _ { \mu _ { \theta _ { 2 } } } ( f _ { \theta _ { 1 } } ( z ) ) - \nabla \Phi _ { \mu _ { \theta _ { 2 } } } ( f _ { \theta _ { 2 } } ( z ) ) \big ) \cdot D _ { \theta _ { 2 } } f _ { \theta _ { 2 } } \big ] \big \| } \\ & { \leq \big \| \mathbb { E } _ { z \sim \omega } \big [ \nabla \Phi _ { \mu _ { \theta _ { 1 } } } ( f _ { \theta _ { 1 } } ( z ) ) \cdot ( D _ { \theta _ { 1 } } f _ { \theta _ { 1 } } - D _ { \theta _ { 2 } } f _ { \theta _ { 2 } } ) \big ] \big \| } \\ & \quad + \big \| \mathbb { E } _ { z \sim \omega } \big [ \big ( \nabla \Phi _ { \mu _ { \theta _ { 1 } } } ( f _ { \theta _ { 1 } } ( z ) ) - \nabla \Phi _ { \mu _ { \theta _ { 2 } } } ( f _ { \theta _ { 1 } } ( z ) ) \big ) \cdot D _ { \theta _ { 2 } } f _ \theta _ \end{array} +$$ + +By assumption, there are bounded non-negative numbers $\alpha , \beta _ { 1 } , \beta _ { 2 } , A$ , and $B$ such that + +$$ +\begin{array} { r l } { ( \mathrm { D } 1 ^ { \prime } ) \quad } & { \alpha = \underset { \mu \in \mathcal { P } ( X ) } { \operatorname* { s u p } } \ \underset { x \sim \mu } { \operatorname* { s u p } } \left. \Phi _ { \mu } ( x ) \right. _ { 2 } , } \\ { ( \mathrm { D } 2 ^ { \prime } ) \quad } & { \underset { \mu \in \mathcal { P } ( X ) } { \operatorname* { s u p } } \ \Vert \nabla _ { x _ { 1 } } \Phi _ { \mu } ( x _ { 1 } ) - \nabla _ { x _ { 2 } } \Phi _ { \mu } ( x _ { 2 } ) \Vert _ { 2 } \leq \beta _ { 1 } \Vert x _ { 1 } - x _ { 2 } \Vert _ { 2 } } \\ { ( \mathrm { D } 3 ^ { \prime } ) \quad } & { \underset { x \in X } { \operatorname* { s u p } } \ \Vert \nabla _ { x } \Phi _ { \mu _ { 1 } } ( x ) - \nabla _ { x } \Phi _ { \mu _ { 2 } } ( x ) \Vert _ { 2 } \leq \beta _ { 2 } W _ { 1 } ( \mu _ { 1 } , \mu _ { 2 } ) , } \\ { ( \mathrm { G } 1 ^ { \prime } ) \quad } & { A = \mathbb { E } _ { z \sim \omega } \underset { \theta } { \operatorname* { s u p } } \left. D _ { \theta } f _ { \theta } ( z ) \right. _ { \mathrm { o p } } , \quad \mathrm { a n d } } \\ { ( \mathrm { G } 2 ^ { \prime } ) \quad } & { \mathbb { E } _ { z \sim \omega } \vert \vert D _ { \theta _ { 1 } } f _ { \theta _ { 1 } } ( z ) - D _ { \theta _ { 2 } } f _ { \theta _ { 2 } } ( z ) \vert \vert _ { \mathrm { o p } } \leq B \vert \vert \theta _ { 1 } - \theta _ { 2 } \vert \vert . } \end{array} +$$ + +Here, we wrote $\operatorname { s u p } _ { x \sim \mu } f ( x )$ for the essential supremum of $f$ w.r.t. $\mu$ . From $( \mathrm Ḋ 1 ^ Ḋ \prime Ḍ )$ and $( \mathbf { G } 2 ^ { \prime } )$ , the first term (a) is bounded as + +$$ +\begin{array} { r } { \left\| \mathbb { E } _ { z \sim \omega } [ \nabla \Phi _ { \mu _ { \theta _ { 1 } } } ( f _ { \theta _ { 1 } } ( z ) ) \cdot ( D _ { \theta _ { 1 } } f _ { \theta _ { 1 } } - D _ { \theta _ { 2 } } f _ { \theta _ { 2 } } ) ] \right\| \leq \alpha B \| \theta _ { 1 } - \theta _ { 2 } \| . } \end{array} +$$ + +From $( \mathrm Ḋ 3 Ḍ ^ { \prime } )$ ), $( \mathbf { G } 1 ^ { \prime } )$ , and the Cauchy-Schwarz inequality, the second term (b) is bounded as + +$$ +\begin{array} { r l } & { \quad \quad \Big | \Big | \mathbb { E } _ { z \sim \omega } [ \big ( \nabla \Phi _ { \mu _ { \theta _ { 1 } } } ( f _ { \theta _ { 1 } } ( z ) ) - \nabla \Phi _ { \mu _ { \theta _ { 2 } } } ( f _ { \theta _ { 1 } } ( z ) ) \big ) \cdot D _ { \theta _ { 2 } } f _ { \theta _ { 2 } } ] \Big | \Big | } \\ & { \le A \beta _ { 2 } W _ { 1 } ( \mu _ { \theta _ { 1 } } , \mu _ { \theta _ { 2 } } ) } \\ & { \le A \beta _ { 2 } \mathbb { E } _ { z \sim \omega } \| f _ { \theta _ { 1 } } ( z ) - f _ { \theta _ { 2 } } ( z ) \| \le A ^ { 2 } \beta _ { 2 } \| \theta _ { 1 } - \theta _ { 2 } \| , } \end{array} +$$ + +where the second inequality holds from the following optimal transport interpretation of $W _ { 1 }$ : + +$$ +W _ { 1 } ( \mu _ { \theta _ { 1 } } , \mu _ { \theta _ { 2 } } ) = \operatorname* { i n f } _ { \gamma \in { \mathcal { P } } ( X \times X ) ; \atop \operatorname { c o u p l i n g ~ o f } \mu _ { \theta _ { 1 } } \operatorname { a n d } \mu _ { \theta _ { 2 } } } \int \| x - y \| d \gamma \leq \mathbb { E } _ { z \sim \omega } \| f _ { \theta _ { 1 } } ( z ) - f _ { \theta _ { 2 } } ( z ) \| . +$$ + +Lastly, from $( \mathbf Ḋ 2 Ḍ ^ { \prime } )$ ), $( \mathbf { G } 1 ^ { \prime } )$ ) and the Cauchy-Schwarz inequality, the term (c) is bounded as + +$$ +\begin{array} { r } { \big | \mathbb { E } _ { z \sim \omega } \big [ \big ( \nabla \Phi _ { \mu _ { \theta _ { 2 } } } ( f _ { \theta _ { 1 } } ( z ) ) - \nabla \Phi _ { \mu _ { \theta _ { 2 } } } ( f _ { \theta _ { 2 } } ( z ) ) \big ) \cdot D _ { \theta _ { 2 } } f _ { \theta _ { 2 } } \big ] \big | \leq A \beta _ { 1 } \mathbb { E } _ { z \sim \omega } \big \| f _ { \theta _ { 1 } } ( z ) - f _ { \theta _ { 2 } } ( z ) \big \| \leq A ^ { 2 } \beta _ { 1 } \big \| \theta _ { 1 } - \theta _ { 2 } \big \| \theta _ { 2 } \big \| , } \end{array} +$$ + +Combining the above upper bounds for (a)–(c), we conclude that + +$$ +| | \nabla _ { \theta _ { 1 } } J ( \mu _ { \theta _ { 1 } } ) - \nabla _ { \theta _ { 2 } } J ( \mu _ { \theta _ { 2 } } ) | | \leq ( \alpha B + A ^ { 2 } ( \beta _ { 1 } + \beta _ { 2 } ) ) \| \theta _ { 1 } - \theta _ { 2 } \| . +$$ + +# E PROOFS FOR SECTION 3 + +In the finite-dimensional case, Proposition 15 says that taking inf-convolution with a “coercive” regularizer does not change the set of minimizers of the original objective. A similar invariance holds for GAN objectives, which are defined on infinite-dimensional space of signed measures, for the regularizers $R _ { 1 }$ , $R _ { 2 }$ and $R _ { 3 }$ introduced in Sections 4 to 6: + +Proposition 2 (Invariance of minimizers). Let $R _ { 1 } ( \xi ) : = \| \xi \| _ { \mathrm { K R } } , R _ { 2 } ( \xi ) : = \| \xi \| _ { S ^ { * } }$ , and $R _ { 3 } ( \xi ) : = $ $\frac { 1 } { 2 } \lVert \hat { \xi } \rVert _ { \mathcal { H } } ^ { 2 }$ be the three regularizers defined by (8), (12), and (19) respectively. Assume that $ { J } :$ $\bar { \mathcal { M } } ( \bar { X } ) \bar { \mathbb { R } }$ has a unique minimizer at $\mu _ { 0 }$ with $J ( \mu _ { 0 } ) = 0$ , and $J ( \mu ) \geq c \| \hat { \mu } - \hat { \mu } _ { 0 } \| _ { \mathcal { H } }$ for some $c > 0$ . Then the inf-convolution $\tilde { J } = J \oplus R _ { 1 } \oplus R _ { 2 } \oplus R _ { 3 }$ has a unique minimizer at $\mu _ { 0 }$ with $\tilde { J } ( \mu _ { 0 } ) = 0$ . + +Proof. In the following, we endow ${ \mathcal { M } } ( X )$ with the Gaussian RKHS norm $\| \widehat { \cdot } \| _ { \mathcal { H } }$ , and for convenience $\| \cdot \|$ will refer to this norm if not otherwise specified. To show the result, we apply Lemma 1 three times, first on $R _ { 3 }$ and $R _ { 1 }$ , and then again on $R _ { 3 } \oplus R _ { 1 }$ and $R _ { 2 }$ , and then finally on $R _ { 3 } \oplus R _ { 1 } \oplus R _ { 2 }$ and $J ( \cdot + \mu _ { 0 } )$ . The result follows from noting that $\tilde { J } ( \mu ) = [ R _ { 3 } \oplus R _ { 1 } \oplus R _ { 2 } \oplus J ( \cdot + \mu _ { 0 } ) ] ( \mu - \mu _ { 0 } )$ . + +In order to apply the lemma, it suffices to show that $R _ { 3 }$ is uniformly continuous on bounded sets and coercive, and that there exist constants $c _ { 1 }$ and $c _ { 2 }$ such that $R _ { 1 } ( \xi ) \overset { \cdot } { \geq } c _ { 1 } \| \xi \|$ and $R _ { 2 } ( \xi ) \geq c _ { 2 } \| \xi \| .$ R3 is uniformly continuous on bounded sets: suppose $\| \xi \| < C$ and $\| \xi ^ { \prime } \| < C$ ; then $| R _ { 3 } ( \xi ) - R _ { 3 } ( \xi ^ { \prime } ) | =$ $\begin{array} { r } { | \frac { 1 } { 2 } \| \xi \| ^ { 2 } - \frac { 1 } { 2 } \| \xi ^ { \prime } \| ^ { 2 } | = | \frac { 1 } { 2 } \langle \xi - \xi ^ { \prime } , \xi + \xi ^ { \prime } \rangle | \leq C \| \dot { \xi } - \xi ^ { \prime } \| } \end{array}$ by the Cauchy–Schwarz and triangle inequality. $R _ { 3 }$ is also coercive, as $R _ { 3 } ( \xi ) = \textstyle { \frac { 1 } { 2 } } \| \xi \| ^ { 2 } \to \infty$ when $\| \xi \| \infty$ . $R _ { 1 }$ satisfies $R _ { 1 } ( \xi ) \geq c _ { 1 } \| \xi \|$ by Lemma 7. $R _ { 2 }$ satisfies $R _ { 2 } ( \xi ) \geq c _ { 2 } ] | \xi | |$ by Lemma 5. □ + +Recall that a function $F$ is said to be coercive if $F ( \xi ) \to \infty$ when $\| \xi \| \infty$ + +Lemma 1. Suppose $F : \mathcal { M } ( X ) \mathbb { R }$ has a unique minimizer at 0 with $F ( 0 ) = 0$ , and is uniformly continuous on bounded sets and coercive. Suppose $G : \mathcal { M } ( X ) \bar { \mathbb { R } }$ has a unique minimizer at 0 with $G ( 0 ) = 0$ , and $G ( \cdot ) \geq c \| \cdot \|$ for some $c > 0$ . Then the inf-convolution $F \oplus G$ has a unique minimizer at 0 with $( F \oplus G ) ( 0 ) = 0$ , and is uniformly continuous on bounded sets, coercive, and real-valued. + +Proof. From Theorem 2.3 of Stromberg (1996), ¨ inf $F \oplus G = 0$ , and $0 \in \arg \operatorname* { m i n } F \oplus G$ . To show that 0 is the unique minimizer, let $\xi \in \arg \operatorname* { m i n } F \oplus G$ . From the definition of inf-convolution, for every $n$ there exists an $\xi _ { n } \in \mathcal { M } ( X )$ that satisfies + +$$ +\operatorname* { l i m } _ { n \to \infty } F ( \xi _ { n } ) + G ( \xi - \xi _ { n } ) = ( F \oplus G ) ( \xi ) = 0 . +$$ + +Since $F$ and $G$ are non-negative, $F ( \xi _ { n } ) \to 0$ and $G ( \xi - \xi _ { n } ) \to 0$ . By our assumption that $G ( \cdot ) \geq$ $c \| \cdot \|$ , the latter limit implies that $\| \xi - \xi _ { n } \| \to 0$ , which implies that $F ( \xi _ { n } ) \to F ( \xi )$ by continuity of $F$ . Comparing our two expressions for the limit of $F ( \xi _ { n } )$ , we find that $F ( \xi ) = 0$ , which implies that $\xi = 0$ , since $F$ has a unique minimizer at 0. Hence ar $\operatorname { m i n } F \oplus G = \{ 0 \}$ . + +From Theorem 2.10 of Stromberg (1996), ¨ $F \oplus G$ is uniformly continuous on bounded sets and real-valued. To show that $F \oplus G$ is coercive, we show the equivalent condition its sublevel sets are bounded (see Proposition 11.11 of Bauschke & Combettes (2011)). That is, we show that for all $a > 0$ , there exists a constant $b$ such that if $( F \oplus G ) ( \xi ) \leq a$ , then $\| \xi \| \le b$ . Let $a > 0$ , and let $\xi \in \mathcal { M } ( X )$ be such that $( F \oplus G ) ( \xi ) \leq a$ . Let $\epsilon > 0$ ; from the definition of inf-convolution, there exists a $\tilde { \xi } \in \mathcal { M } ( X )$ such that + +$$ +F ( { \tilde { \xi } } ) + G ( \xi - { \tilde { \xi } } ) \leq ( F \oplus G ) ( \xi ) + \epsilon \leq a + \epsilon . +$$ + +Because $F$ and $G$ are non-negative, we have that + +$$ +\begin{array} { r } { F ( \tilde { \xi } ) \le a + \epsilon , \qquad G ( \xi - \tilde { \xi } ) \le a + \epsilon . } \end{array} +$$ + +Using the fact that $F$ is coercive and hence its sublevel sets are bounded, let $b ^ { \prime }$ be such that if $F ( \nu ) \leq a + \epsilon$ , then $\| \nu \| \leq b ^ { \prime }$ . By the triangle inequality and our assumption that $G ( \cdot ) \geq c \| \cdot \|$ , + +$$ +\| \xi \| \leq \| \tilde { \xi } \| + \| \xi - \tilde { \xi } \| \leq b ^ { \prime } + \frac { G ( \xi - \tilde { \xi } ) } { c } \leq b ^ { \prime } + \frac { a + \epsilon } { c } . +$$ + +Because this constant is independent of $\xi$ , this shows that $F \oplus G$ is coercive. + +# F PROOFS FOR SECTION 4 + +Proposition 3. $( D l )$ holds if and only if $J$ is $\alpha$ -Lipschitz w.r.t. the Wasserstein-1 distance. + +Proof. First, we check the forward implication: + +$$ +| J ( \mu ) - J ( \nu ) | \leq \alpha W _ { 1 } ( \mu , \nu ) \quad \Rightarrow \quad \operatorname* { s u p } _ { x \sim \mu } \| \nabla _ { x } \Phi _ { \mu } ( x ) \| _ { 2 } \leq \alpha . +$$ + +From the duality between $L ^ { \infty }$ - and $L ^ { 1 }$ -norms, we have + +$$ +\operatorname* { s u p } _ { x \sim \mu } \| \nabla _ { x } \Phi _ { \mu } ( x ) \| _ { 2 } = \| \nabla _ { x } \Phi _ { \mu } ( \cdot ) \| _ { L ^ { \infty } ( \mu ; \mathbb { R } ^ { d } ) } = \operatorname* { s u p } _ { \| v \| _ { L ^ { 1 } ( \mu ; \mathbb { R } ^ { d } ) } = 1 } \int \nabla \Phi _ { \mu } ( x ) \cdot v ( x ) d \mu ( x ) . +$$ + +Below, we will abbreviate $\left\| \cdot \right\| _ { L ^ { 1 } \left( \mu ; \mathbb { R } ^ { d } \right) }$ as $\| \cdot \| _ { L ^ { 1 } }$ . We utilize the fact that the optimal discriminator $\Phi _ { \mu }$ is the functional gradient of $J$ in the sense of Definition 3. For any $t \geq 0$ , let $\mu _ { t } = ( \operatorname { i d } + t v ) _ { \# } \mu$ be the law of $x + t v ( x )$ with $x \sim \mu$ . Then, $\mu _ { t }$ converges weakly to $\mu ~ = ~ \mu _ { 0 }$ as $t 0$ since $W _ { 1 } ( \mu _ { t } , \mu ) \leq t \| v \| _ { L ^ { 1 } ( \mu ) } \to 0$ . Applying Proposition 17 to $J$ and $\mu _ { t }$ , we have + +$$ +\left. \frac { d } { d t } J ( \mu _ { t } ) \right| _ { t = 0 } = \left. \frac { d } { d t } \int \Phi _ { \mu } ( x + t v ( x ) ) d \mu ( x ) \right| _ { t = 0 } = \int \nabla \Phi _ { \mu } ( x ) \cdot v ( x ) d \mu . +$$ + +In particular, the right-hand side of (26) is bounded from above as + +$$ +\begin{array} { r l } & { \underset { | | \tau | _ { \xi } | , \xi _ { 1 } , \omega _ { \xi } = 1 } { \operatorname* { s u p } } \int \nabla \Phi _ { p } ( x ) \cdot v ( x ) d \mu ( x ) } \\ & { \quad = \underset { | | \tau | _ { \xi } | , \xi _ { 1 } | \neq 0 } { \operatorname* { s u p } } \frac { d } { d t } J ( \mu _ { t } ) \biggr | _ { t = 0 } } \\ & { \quad = \underset { | | \tau | _ { \xi } | , \xi _ { 1 } | , \xi _ { 0 } = 1 } { \operatorname* { s u p } } \frac { 1 } { \mathrm { i n s p } } \frac { J ( \mu _ { t } ) - J ( \mu ) } { t } } \\ & { \quad \le \underset { | | \tau | _ { \xi } | , \xi _ { 1 } | , \xi _ { 0 } = 1 } { \operatorname* { s u p } } \frac { \mathrm { i n s K } ( \mu _ { t } , \mu ) } { t } } \\ & { \quad \le \underset { | | \tau | _ { \xi } | , \xi _ { 1 } | , \xi _ { 0 } = 1 } { \operatorname* { s u p } } \frac { \mathrm { i n s K } ( \mu _ { t } , \mu ) } { t } } \\ & { \quad \le \underset { | | \tau | _ { \xi } | , \xi _ { 1 } | \neq 0 } { \operatorname* { s u p } } \frac { \mathrm { i n s K } ( \mu _ { t } , \mu _ { t } ) } { t } } \\ & { \quad = \alpha , } \end{array} +$$ + +which proves the first half of the statement. + +For the converse, we borrow some techniques from the optimal transport theory. Let $p > 1$ , and let $W _ { p } ( \mu , \nu )$ denote the Wasserstein- $p$ distance between $\mu$ and $\nu$ . Suppose that $\mu _ { t } : [ 0 , 1 ] \to { \mathcal { P } } ( X )$ is an ${ \mathcal { P } } _ { p } ( X )$ -absolutely continuous curve, that is, every $\mu _ { t }$ has a finite $p$ -moment and there exists $v \in L ^ { 1 } ( [ 0 , 1 ] )$ such that $\begin{array} { r } { W _ { p } ( \mu _ { s } , \mu _ { t } ) \le \int _ { s } ^ { t } v ( r ) d r } \end{array}$ holds for any $0 \leq s \leq t \leq 1$ . Then, the limit $\begin{array} { r } { | \mu ^ { \prime } | _ { W _ { p } } ( t ) : = \operatorname* { l i m } _ { h \to 0 } | h | ^ { - 1 } W _ { p } ( \mu _ { t + h } , \mu _ { t } ) } \end{array}$ exists for almost all $t \in [ 0 , 1 ]$ (see Ambrosio et al. (2008), Theorem 1.1.2). Such $| \mu ^ { \prime } | _ { W _ { p } }$ is called the metric derivative of $\mu _ { t }$ . + +Lemma 2 (Ambrosio et al. (2008), Theorem 8.3.1). Let $\mu _ { t } : [ 0 , 1 ] \to { \mathcal { P } } ( X )$ be an ${ \mathcal { P } } _ { p } ( X )$ -absolutely continuous curve, and let $| \mu ^ { \prime } | _ { W _ { p } } \in L ^ { 1 } ( [ 0 , 1 ] )$ be its metric derivative. Then, there exists $a$ vector field $v : ( x , t ) \mapsto v _ { t } ( x ) \in \mathbb { R } ^ { d }$ such that + +$$ +v _ { t } \in L ^ { p } ( \mu _ { t } ; X ) , \quad \| v _ { t } \| \leq | \mu ^ { \prime } | _ { W _ { p } } ( t ) +$$ + +for almost all $t \in [ 0 , 1 ]$ , and + +$$ +\int _ { 0 } ^ { 1 } \frac { d } { d t } \varphi ( x , t ) \mu _ { t } ( d x ) d t = - \int _ { 0 } ^ { 1 } \langle v _ { t } ( x ) , \nabla _ { x } \varphi ( x , t ) \rangle \mu _ { t } ( d x ) d t +$$ + +holds for any cylindrical function $\varphi ( x , t )$ . + +Given $p > 1$ , let $\mu _ { t }$ be a ${ \mathcal { P } } _ { p } ( X )$ -absolutely continuous curve such that $\mu _ { 0 } = \mu$ and $\mu _ { 1 } = \nu$ . Then + +$$ +\begin{array} { r l } { | J ( \mu ) - J ( \nu ) | = \displaystyle \left| \int _ { 0 } ^ { 1 } \frac { d } { d t } J ( \mu _ { t } ) d t \right| } & { } \\ & { = \displaystyle \left| \int _ { 0 } ^ { 1 } \int v _ { t } \cdot \nabla \Phi _ { \mu _ { t } } d \mu _ { t } d t \right| } \\ & { \leq \displaystyle \int _ { 0 } ^ { 1 } \| v _ { t } \| _ { L ^ { p } ( \mu _ { t } ) } \| \nabla \Phi _ { \mu _ { t } } \| _ { L ^ { q } ( \mu _ { t } ) } d t } \\ & { \leq \displaystyle \int _ { 0 } ^ { 1 } \| v _ { t } \| _ { L ^ { p } ( \mu _ { t } ) } d t \cdot \operatorname* { s u p } _ { \tilde { \mu } } \| \nabla \Phi _ { \tilde { \mu } } \| _ { L ^ { q } ( \mu ) } } \\ & { \leq \displaystyle \int _ { 0 } ^ { 1 } \| \mu _ { t } ^ { \prime } \| _ { W _ { p } } d t \cdot \operatorname* { s u p } \| \nabla \Phi _ { \tilde { \mu } } \| _ { L ^ { q } ( \mu ) } , } \end{array} +$$ + +where $| \mu _ { t } ^ { \prime } |$ is the metric derivative. The existence of $v _ { t }$ and the bound $\| v _ { t } \| _ { L ^ { p } ( \mu _ { t } ) } \leq | \mu _ { t } ^ { \prime } | _ { W _ { p } }$ is due to Lemma 2. Taking the infimum over all possible such curves, we find that + +$$ +\left| J ( \mu ) - J ( \nu ) \right| \leq W _ { p } ( \mu , \nu ) \cdot \operatorname* { s u p } _ { \tilde { \mu } } \left\| \nabla \Phi _ { \tilde { \mu } } \right\| _ { L ^ { q } ( \mu ) } . +$$ + +To conclude, we consider the limit $p 1$ . Using the fact that $q \mapsto | | f | | _ { L ^ { q } ( \mu ) }$ is increasing in $q$ , we have + +$$ +| J ( \mu ) - J ( \nu ) | \leq W _ { p } ( \mu , \nu ) \cdot \operatorname* { s u p } _ { \tilde { \mu } } \| \nabla \Phi _ { \tilde { \mu } } \| _ { L ^ { \infty } ( \mu ) } \leq \alpha W _ { p } ( \mu , \nu ) . +$$ + +Then + +$$ +\begin{array} { r l } & { | J ( \mu ) - J ( \nu ) | \leq \underset { p > 1 } { \operatorname* { i n f } } \alpha W _ { p } ( \mu , \nu ) } \\ & { \qquad = \alpha \underset { \pi \in \Pi ( \mu , \nu ) } { \operatorname* { i n f } } \underset { p > 1 } { \operatorname* { i n f } } | | x - y | | _ { L ^ { p } ( \pi ) } } \\ & { \qquad = \alpha \underset { \pi \in \Pi ( \mu , \nu ) } { \operatorname* { i n f } } | | x - y | | _ { L ^ { 1 } ( \pi ) } } \\ & { \qquad = \alpha W _ { 1 } ( \mu , \nu ) . } \end{array} +$$ + +Proposition 4. The minimax and non-saturating GAN losses do not satisfy $( D l )$ for some $\mu _ { 0 }$ + +Proposition 5. The Wasserstein GAN loss satisfies $( D l )$ with $\alpha = 1$ for any $\mu _ { 0 }$ + +Proof for Proposition 4 and Proposition 5. The counterexample for Proposition 4 is a slight simplification of Example 1 of Arjovsky et al. (2017). Let $\delta _ { x }$ denote the distribution that outputs $x \in \mathbb { R }$ with probability 1. Then, we have + +$$ +\begin{array} { r } { \mathrm { { \mathcal { I } } _ { J S } } ( \delta _ { x } \mid | \delta _ { 0 } ) = \left\{ \begin{array} { l l } { \log 2 } & { x \neq 0 } \\ { 0 } & { x = 0 , } \end{array} \right. \quad D _ { \mathrm { K L } } ( \frac 1 2 \delta _ { x } + \frac 1 2 \delta _ { 0 } \| \delta _ { 0 } ) = \left\{ \begin{array} { l l } { \infty } & { x \neq 0 } \\ { 0 } & { x = 0 , } \end{array} \right. \quad \mathrm { a n d } \quad W _ { 1 } ( \delta _ { x } , \delta _ { 0 } ) = | x | . } \end{array} +$$ + +Therefore, for $J ( \mu ) = D _ { \mathrm { J S } } ( \mu \vert \vert \delta _ { 0 } )$ , the inequality $| J ( \delta _ { x } ) - J ( \delta _ { 0 } ) | \leq \alpha W _ { 1 } ( \delta _ { x } , \delta _ { 0 } )$ cannot hold for sufficiently small $x \neq 0$ , because the left-hand side equals $\log 2$ while the right-hand side equals $\alpha | x |$ . For $\begin{array} { r } { \dot { \boldsymbol J } ( \mu ) = \dot { D _ { \mathrm { K L } } } ( \frac { 1 } { 2 } \mu + \frac { 1 } { 2 } \delta _ { 0 } | | \delta _ { 0 } ) } \end{array}$ , the inequality will not hold for any $x \neq 0$ . + +Next, we verify Proposition 5. For $J ( \mu ) = W _ { 1 } ( \mu , \mu _ { 0 } )$ , we have $J ( \mu ) - J ( \nu ) = W _ { 1 } ( \mu , \mu _ { 0 } ) -$ $W _ { 1 } ( \nu , \mu _ { 0 } ) \leq \dot { W } _ { 1 } ( \bar { \mu , \nu } )$ by the triangle inequality (see e.g., Section 6 in Villani (2009)). Reversing the roles of $\mu$ and $\nu$ , we can conclude that $| J ( \mu ) - J ( \nu ) | \leq W _ { 1 } ( \mu , \nu )$ . The result follows from Proposition 3. □ + +Proposition 6 (Pasch–Hausdorff). Let $J : \mathcal { M } ( X ) \bar { \mathbb { R } }$ be a function, and define ${ \tilde { J } } : = J \oplus R _ { 1 }$ . Then $\tilde { J }$ is $\alpha$ -Lipschitz w.r.t. the distance induced by the KR norm, and hence the Wasserstein- $I$ distance when restricted to ${ \mathcal { P } } ( X )$ . + +Proof. Observe that by the triangle inequality, + +$$ +\begin{array} { r l } & { \tilde { J } ( \mu ) - \tilde { J } ( \nu ) = \underset { \tilde { \nu } } { \operatorname* { s u p } } \operatorname* { i n f } J ( \tilde { \mu } ) + \alpha W _ { 1 } ( \mu , \tilde { \mu } ) - J ( \tilde { \nu } ) - \alpha W _ { 1 } ( \nu , \tilde { \nu } ) } \\ & { \qquad \leq \underset { \tilde { \nu } } { \operatorname* { s u p } } \alpha W _ { 1 } ( \mu , \tilde { \nu } ) - \alpha W _ { 1 } ( \nu , \tilde { \nu } ) } \\ & { \qquad \leq \underset { \tilde { \nu } } { \operatorname* { s u p } } \alpha W _ { 1 } ( \mu , \nu ) } \\ & { \qquad = \alpha W _ { 1 } ( \mu , \nu ) . } \end{array} +$$ + +Interchanging the roles of $\mu$ and $\nu$ completes the proof. + +Lemma 3. The convex conjugate of $\mu \mapsto \alpha \vert \vert \mu \vert \vert _ { \mathrm { K R } }$ is given as $\varphi \mapsto \chi \{ \| \varphi \| _ { \mathrm { L i p } } \leq \alpha \}$ . + +Proof. The convex conjugate is + +$$ +\begin{array} { r l } { \underset { \mu } { \operatorname* { s u p } } \left[ \int f d \mu - \alpha \lVert \mu \rVert _ { \mathbf { k R } } \right] = \underset { \mu } { \operatorname* { s u p } } \left[ \int f d \mu - \alpha \underset { \lVert \mathbf { k } \rVert _ { \mathbf { k } \oplus \mathbf { k } } } { \operatorname* { s u p } } \int g d \mu \right] } & { } \\ & { = \underset { \mu } { \operatorname* { s u p } } \left[ \int f d \mu - \underset { \lVert \mathbf { k } \rVert _ { \mathbf { k } \oplus \mathbf { k } } } { \operatorname* { s u p } } \int g d \mu \right] } \\ & { = \underset { \mu } { \operatorname* { s u p } } \left[ \underset { \lVert \mathbf { k } \rVert _ { \mathbf { k } \oplus \mathbf { k } } } { \operatorname* { i n f } } \leq \int f d \mu - \int g d \mu \right] } \\ & { = \underset { \lVert \mathbf { k } \oplus \mathbf { k } } { \operatorname* { i n f } } \frac { \operatorname* { s u p } } { \operatorname* { s u p } } \left[ \int f d \mu - \int g d \mu \right] } \\ & { = \underset { \lVert \mathbf { k } \oplus \mathbf { k } } { \operatorname* { i n f } } \frac { \operatorname* { s u p } } { \operatorname* { s u p } } } \\ & { = \underset { \lVert \mathbf { k } \oplus \mathbf { k } \oplus \mathbf { k } } { \operatorname* { i n f } } \times \mathcal { X } \{ f - g \} } \\ & { = \underset { \lVert \mathbf { k } \oplus \mathbf { k } \oplus \mathbf { x } } { \operatorname* { s u p } } \mathcal { X } \{ f - g \} } \\ & { = \mathcal { X } \{ \lVert \mathbf { k } \rVert _ { \mathbf { k } \oplus \mathbf { k } } \leq \alpha \} , } \end{array} +$$ + +where Sion’s minimax theorem ensures that we can swap the inf and the sup. + +# G PROOFS FOR SECTION 5 + +Proposition 7. The Wasserstein, minimax, and non-saturating GAN losses do not satisfy $( D 2 )$ for some $\mu _ { 0 }$ . + +Proof. First, we prove the statement for the Wasserstein GAN. Consider $J ( \mu ) = W _ { 1 } ( \mu , \delta _ { 0 } )$ evaluated at $\mu = \textstyle { \frac { 1 } { 2 } } \delta _ { - 1 } + \frac { 1 } { 2 } \delta _ { 1 }$ , a mixture of spikes at $\pm 1$ . The optimal discriminator of $J$ at this mixture is the Kantorovich potential that transfers $\mu$ to $\delta _ { 0 }$ , which is $\Phi _ { \mu } ( x ) = \vert x \vert$ . The gradient of this Kantorovich potential is discontinuous at 0, and hence not Lipschitz. + +For the minimax GAN $J ( \mu ) ~ = ~ { \cal D } _ { \mathrm { J S } } ( \mu | | \mu _ { 0 } )$ , let $\mu$ and $\mu _ { 0 }$ be probability distributions on $\mathbb { R }$ whose densities are $p ( x ) \propto \exp ( - | x | / 2 )$ and $p _ { 0 } ( x ) \ \propto \ \exp ( - | x | )$ , respectively (i.e., Laplace distributions). Then, the optimal discriminator at µ is given as Φµ(x) = 12 log p(x)p(x)+p0(x) $- \textstyle { \frac { 1 } { 2 } } \log ( 1 + 2 \exp ( - | x | / 2 ) )$ . By elementary calculations, we can see that this function is not differentiable at $x = 0$ , which implies that the minimax GAN does not satisfy (D2). For the non-saturating GAN, we obtain a similar result by swapping the role of $\mu$ and $\mu _ { 0 }$ . □ + +Proposition 8. Let $J : \mathcal { M } ( X ) \bar { \mathbb { R } }$ be a convex, proper, lower semicontinuous function, and define ${ \tilde { J } } : = J \oplus R _ { 2 }$ . Then the optimal discriminator for $\tilde { J }$ is $\beta _ { 1 }$ -smooth. + +Proof. $\tilde { J }$ is convex, proper, lower semicontinuous. Hence + +$$ +\tilde { J } ( \mu ) = ( J \oplus \lambda | | \cdot | | _ { \mathcal { F } ^ { * } } ) ( \mu ) = \operatorname* { s u p } _ { \varphi } \int \varphi d \mu - J ^ { \star } ( \varphi ) - \chi \{ \frac { 1 } { \lambda } \varphi \in \mathcal { F } \} . +$$ + +Then by the envelope theorem, $\frac { \delta J } { \delta \mu }$ is the $\varphi$ that maximizes the right-hand side, and hence $\frac { 1 } { \lambda } \frac { \delta J } { \delta \mu } \in$ $\mathcal { F }$ . □ + +Lemma 4. The convex conjugate of $\mu \mapsto \beta _ { 1 } \| \mu \| _ { S ^ { * } }$ is $\varphi \mapsto \chi \{ \varphi \in \beta _ { 1 } S \}$ . + +Proof. The proof of Lemma 4 is nearly identical to the proof of Lemma 3. + +Proposition 9. Let $f : \mathbb { R } ^ { d } \mathbb { R }$ be a neural network consisting of $k$ layers whose linear transformations have spectral norm 1 and whose activation functions are 1-Lipschitz and 1-smooth. Then $f$ is $k$ -smooth. + +Proof. In this section, let $\ell ( f )$ and $s ( f )$ denote the Lipschitz constant of $f$ and $\nabla f$ respectively. This inequality holds + +$$ +s ( f \circ g ) \leq s ( f ) \ell ( g ) ^ { 2 } + \ell ( f ) s ( g ) , +$$ + +since + +$$ +\begin{array} { r l } & { | d ( f \circ g ) ( x ) - d ( f \circ g ) ( y ) | | = | | d f ( g ( x ) ) d g ( x ) - d f ( g ( y ) ) d g ( y ) | | } \\ & { \qquad = | | d f ( g ( x ) ) d g ( x ) - d f ( g ( y ) ) d g ( x ) + d f ( g ( y ) ) d g ( x ) - d f ( g ( y ) ) d g ( y ) } \\ & { \qquad \le | | d f ( g ( x ) ) - d f ( g ( y ) ) | | | d g ( x ) | | + | | d f ( g ( y ) ) | | | d g ( x ) - d g ( y ) | | } \\ & { \qquad \le s ( f ) | | g ( x ) - g ( y ) | | \ell ( g ) + \ell ( f ) s ( g ) | | x - y | | } \\ & { \qquad \le \big ( s ( f ) \ell ( g ) ^ { 2 } + \ell ( f ) s ( g ) \big ) | | x - y | | . } \end{array} +$$ + +Let $\sigma$ be an elementwise activation function with $\ell ( \sigma ) = s ( \sigma ) = 1$ , and let $A$ be a linear layer with spectral norm 1. Then $\ell ( A ) = 1$ and $s ( A ) = 0$ , so + +$$ +\begin{array} { l } { { \ell ( \sigma \circ A ) \leq \ell ( \sigma ) \ell ( A ) = 1 } } \\ { { s ( \sigma \circ A ) \leq s ( \sigma ) \ell ( A ) ^ { 2 } + \ell ( \sigma ) s ( A ) = 1 . } } \end{array} +$$ + +We note that we can use the same value of $s ( \sigma )$ whether we consider $\sigma : \mathbb { R } \mathbb { R }$ or elementwise as $\sigma : \mathbb { R } ^ { d } \mathbb { R } ^ { d }$ , since for an elementwise $\sigma$ , we have + +$$ +\begin{array} { r l } & { | | d \sigma ( x ) - d \sigma ( y ) | | _ { 2 } = | | \mathrm { d i a g } \sigma ^ { \prime } ( x ) - \mathrm { d i a g } \sigma ^ { \prime } ( y ) | | _ { 2 } } \\ & { \qquad = | | \mathrm { d i a g } ( \sigma ^ { \prime } ( x ) - \sigma ^ { \prime } ( y ) ) | | _ { 2 } } \\ & { \qquad = \underset { i } { \operatorname* { m a x } } | \sigma ^ { \prime } ( x _ { i } ) - \sigma ^ { \prime } ( y _ { i } ) | } \\ & { \qquad \le \underset { i } { \operatorname* { m a x } } s ( \sigma ) | x _ { i } - y _ { i } | } \\ & { \qquad = s ( \sigma ) | | x - y | | _ { \infty } } \\ & { \qquad \le s ( \sigma ) | | x - y | | _ { 2 } . } \end{array} +$$ + +We apply the inequality (27) recursively for all $k$ layers to obtain that the entire network is $k$ - smooth. □ + +Lemma 5. Let $\mathcal { H }$ be an RKHS with the Gaussian kernel. Then $\begin{array} { r } { | | \hat { \mu } | | _ { \mathcal { H } } \leq \frac { \sqrt { 2 } } { \sigma ^ { 2 } } | | \mu | | _ { S ^ { * } } } \end{array}$ . + +Proof. It follows from $\begin{array} { r } { \frac { \partial f ( { \boldsymbol x } ) } { \partial { \boldsymbol x } _ { i } } = \langle f , \frac { \partial K ( { \boldsymbol x } , \cdot ) } { \partial x _ { i } } \rangle } \end{array}$ , ∂K(x,·) i that, for f ∈ H, + +$$ +\begin{array} { l } { \displaystyle \| \nabla f ( x ) - \nabla f ( y ) \| ^ { 2 } = \sum _ { i = 1 } ^ { d } \left| \frac { \partial f ( x ) } { \partial x _ { i } } - \frac { \partial f ( y ) } { \partial y _ { i } } \right| ^ { 2 } } \\ { \displaystyle \leq \sum _ { i = 1 } ^ { d } \| f \| _ { \mathcal H } ^ { 2 } \left\| \frac { \partial K ( x , \cdot ) } { \partial x _ { i } } - \frac { \partial K ( y , \cdot ) } { \partial y _ { i } } \right\| _ { \mathcal H } ^ { 2 } } \\ { \displaystyle = \| f \| _ { \mathcal H } ^ { 2 } \sum _ { i = 1 } ^ { d } \left\{ \frac { \partial ^ { 2 } K ( x , \tilde { x } ) } { \partial x _ { i } \partial \tilde { x } _ { i } } - 2 \frac { \partial ^ { 2 } K ( x , y ) } { \partial x _ { i } \partial y _ { i } } + \frac { \partial ^ { 2 } K ( y , \tilde { y } ) } { \partial y _ { i } \partial \tilde { y } _ { i } } \right\} } \\ { \displaystyle \leq \| f \| _ { \mathcal H } ^ { 2 } \cdot ( \Gamma ( x , y ) + \Gamma ( y , x ) ) , } \end{array} +$$ + +where $\tilde { x }$ and $\tilde { y }$ denote copies of $x$ and $y$ , and + +$$ +\Gamma ( x , y ) = \sum _ { i = 1 } ^ { d } \left| \frac { \partial ^ { 2 } K ( x , \tilde { x } ) } { \partial x _ { i } \partial \tilde { x } _ { i } } - \frac { \partial ^ { 2 } K ( x , y ) } { \partial x _ { i } \partial \tilde { x } _ { i } } \right| . +$$ + +From (30), we find that for a Gaussian kernel, + +$$ +\Gamma ( x , y ) = \frac { 1 } { \sigma ^ { 4 } } \exp \Bigl ( - \frac { \| x - y \| ^ { 2 } } { 2 \sigma ^ { 2 } } \Bigr ) \| x - y \| ^ { 2 } , +$$ + +and thus, + +$$ +\| \nabla f ( x ) - \nabla f ( y ) \| \leq { \frac { \sqrt { 2 } } { \sigma ^ { 2 } } } \| f \| _ { \mathcal H } \| x - y \| . +$$ + +This implies that + +$$ +\| \hat { \mu } \| _ { \mathcal H } = \operatorname* { s u p } _ { f \in \mathcal H , \| f \| _ { \mathcal H } \leq 1 } \int f d \mu \leq \operatorname* { s u p } _ { \| f \| _ { \mathcal S } \leq \frac { \sqrt { 2 } } { \sigma ^ { 2 } } } \int f d \mu = \frac { \sqrt { 2 } } { \sigma ^ { 2 } } \| \mu \| _ { \mathcal S ^ { * } } . +$$ + +# H PROOFS FOR SECTION 6 + +Lemma 6. The convex conjugate of $\mu \mapsto \frac { \lambda } { 2 } | | \mu | | _ { \mathrm { K R } } ^ { 2 }$ is $\begin{array} { r } { f \mapsto \frac { 1 } { 2 \lambda } | | f | | _ { \mathrm { L i p } } ^ { 2 } . } \end{array}$ + +Proof. This proof is generalized from the finite-dimensional case (Boyd & Vandenberghe, 2004). We can bound the convex conjugate from above by + +$$ +\begin{array} { r l } { { \operatorname* { s u p } _ { \mu } [ \int f d \mu - \frac { \lambda } { 2 } | | \mu | | _ { \mathrm { K R } } ^ { 2 } ] \leq \operatorname* { s u p } _ { \mu } [ | | f | | _ { \mathrm { L i p } } | | \mu | | _ { \mathrm { K R } } - \frac { \lambda } { 2 } | | \mu | | _ { \mathrm { K R } } ^ { 2 } ] } } \\ & { \leq \operatorname* { s u p } _ { z \in \mathbb { R } } [ | | f | | _ { \mathrm { L i p } } z - \frac { \lambda } { 2 } z ^ { 2 } ] } \\ & { = | | f | | _ { \mathrm { L i p } } \cdot \frac { | | f | | _ { \mathrm { L i p } } } { \lambda } - \frac { \lambda } { 2 } \cdot \frac { | | f | | _ { \mathrm { L i p } } ^ { 2 } } { \lambda ^ { 2 } } } \\ & { = \frac { 1 } { 2 \lambda } | | f | | _ { \mathrm { L i p } } ^ { 2 } . } \end{array} +$$ + +If $f$ is constant, then we may choose $\mu = 0$ to see that + +$$ +\operatorname* { s u p } _ { \mu } \left[ \int f d \mu - \frac { \lambda } { 2 } | | \mu | | _ { \mathrm { K R } } ^ { 2 } \right] \geq 0 = \frac { 1 } { 2 \lambda } | | f | | _ { \mathrm { L i p } } ^ { 2 } , +$$ + +and we are done. + +Otherwise, for $f ( x ) \neq f ( y )$ , let $\mu _ { x , y }$ be the signed measure given by + +$$ +\mu _ { x , y } = { \frac { | | f | | _ { \mathrm { L i p } } ^ { 2 } } { \lambda ( f ( x ) - f ( y ) ) } } ( \delta _ { x } - \delta _ { y } ) . +$$ + +Note that + +$$ +\begin{array} { l } { \displaystyle | | \mu _ { x , y } | | _ { \mathrm { K R } } = \operatorname* { s u p } _ { | | g | | _ { \mathrm { L i p } } \leq 1 } \int g d \mu _ { x , y } } \\ { = \frac { | | f | | _ { \mathrm { L i p } } ^ { 2 } } { \lambda ( f ( x ) - f ( y ) ) } \operatorname* { s u p } _ { | | g | | _ { \mathrm { L i p } } \leq 1 } \Big ( g ( x ) - g ( y ) \Big ) } \\ { = \frac { | | f | | _ { \mathrm { L i p } } ^ { 2 } } { \lambda ( f ( x ) - f ( y ) ) } \cdot d ( x , y ) } \end{array} +$$ + +and + +$$ +\int f d \mu _ { x , y } = { \frac { | | f | | _ { \mathrm { L i p } } ^ { 2 } } { \lambda ( f ( x ) - f ( y ) ) } } { \Big ( } f ( x ) - f ( y ) { \Big ) } = { \frac { 1 } { \lambda } } | | f | | _ { \mathrm { L i p } } ^ { 2 } . +$$ + +Then + +$$ +\begin{array} { r l } { \underset { \mu } { \operatorname* { s u p } } \left[ \int f d \mu - \frac { \lambda } { 2 } | | \mu | | _ { \mathrm { K R } } ^ { 2 } \right] \geq \underset { f ( x ) \neq f ( y ) } { \operatorname* { s u p } } \left[ \int f d \mu _ { x , y } - \frac { \lambda } { 2 } | | \mu _ { x , y } | | _ { \mathrm { K R } } ^ { 2 } \right] } & { } \\ { = \underset { f ( x ) \neq f ( y ) } { \operatorname* { s u p } } \left[ \frac { 1 } { \lambda } | | f | | _ { \mathrm { L i p } } ^ { 2 } - \frac { \lambda } { 2 } \big ( \frac { 1 } { \lambda } | | f | | _ { \mathrm { L i p } } ^ { 2 } \frac { d ( x , y ) } { f ( x ) - f ( y ) } \big ) ^ { 2 } \right] } & { } \\ { = \frac { 1 } { \lambda } | | f | | _ { \mathrm { L i p } } ^ { 2 } - \frac { \lambda } { 2 } \big ( \frac { 1 } { \lambda } | | f | | _ { \mathrm { L i p } } ^ { 2 } \frac { 1 } { | | f | | _ { \mathrm { L i p } } } \big ) ^ { 2 } } & { } \\ { = \frac { 1 } { 2 \lambda } | | f | | _ { \mathrm { L i p } } ^ { 2 } . } \end{array} +$$ + +Proposition 10. Let $J : \mathcal { M } ( X ) \mathbb { R }$ be a convex function. Then $J$ satisfies $( D 3 )$ if and only if $\begin{array} { r } { \mathfrak { D } _ { J } ( \nu , \mu ) \le \frac { \beta _ { 2 } } { 2 } \| \mu - \nu \| _ { \mathrm { K R } } ^ { 2 } } \end{array}$ for all $\mu , \nu \in \mathcal { M } ( X )$ . + +Proof. Recall the definition of the Bregman divergence: + +$$ +\mathfrak { D } _ { J } ( \nu , \mu ) : = J ( \nu ) - J ( \mu ) - \int \Phi _ { \mu } ( x ) d ( \nu - \mu ) . +$$ + +Proposition 10 claims that the optimal discriminator $\Phi _ { \mu }$ is $\beta _ { 2 }$ -smooth as a function of $\mu$ if and only if + +$$ +\forall \mu , \nu \in \mathcal { M } ( { \cal X } ) , \quad { \cal J } ( \nu ) - { \cal J } ( \mu ) - \int \Phi _ { \mu } ( x ) d ( \nu - \mu ) \leq \frac { \beta _ { 2 } } { 2 } \| \mu - \nu \| _ { \mathrm { K R } } ^ { 2 } . +$$ + +This proof is generalized from the finite-dimensional case (Zhou, 2018; Sidford, 2017). Suppose that $\| \nabla \Phi _ { \mu } ( x ) - \nabla \Phi _ { \nu } ( x ) \| _ { 2 } \leq \beta _ { 2 } \| \mu - \nu \| _ { \mathrm { K R } }$ holds for all $x$ . Let $\mu _ { t }$ $0 \leq t \leq 1$ ) be defined as a mixture between $\mu _ { t } = ( 1 - t ) \mu + t \nu$ so that + +$$ +\mu _ { 0 } = \mu , \quad \mu _ { 1 } = \nu , \quad \mathrm { a n d } \quad \| \mu _ { t } - \mu _ { s } \| _ { \mathrm { K R } } = | t - s | \cdot \| \mu - \nu \| _ { \mathrm { K R } } . +$$ + +Then, we have + +$$ +\begin{array} { r l } { \| \mathcal { D } _ { J } ( v , p ) \| = \bigg | \int _ { 0 } ^ { 1 } \frac { d } { d t } \hat { { \boldsymbol { H } } } ( \mu _ { t } ) \hat { { \boldsymbol { d } } } \boldsymbol { L } \bigg | \hat { \psi } - \int \hat { \mathbf { q } } _ { \mu } \hat { { \boldsymbol { d } } } ( v - \mu ) \bigg | } \\ & { = \bigg | \int _ { 0 } ^ { 1 } \int \Phi _ { \mu } , d ( v - p ) d \hbar - \int \Phi _ { \mu } d ( v - \mu ) \bigg | } \\ & { = \bigg | \int _ { 0 } ^ { 1 } \int ( \Phi _ { \mu \nu } - \Phi _ { \mu } ) d v - \mu ) d v - \mu \big | } \\ & { \leq \int _ { 0 } ^ { 1 } \Big | \| \Phi _ { \mu \nu } - \Phi _ { \mu } \| _ { \mathrm { L E } ^ { \infty } } \| \| \mu _ { t } \| = \nu | \mathrm { k i n } d t } \\ & { \leq \int _ { 0 } ^ { 1 } \hat { \mathcal { S } } _ { 2 } \| u _ { t } - \mu | \mathrm { s g n } \| u \| \mu - \nu \| \mathrm { k i n } d t } \\ & { \leq \int _ { 0 } ^ { 1 } \hat { \mathcal { S } } _ { 2 } \| u _ { t } - \nu \| _ { \mathrm { L E } ^ { \infty } } d u } \\ & { \leq \int _ { 0 } ^ { 1 } \hat { \mathcal { S } } _ { 2 } \| u _ { t } - \nu \| _ { \mathrm { L E } ^ { \infty } } ^ { 2 } d u } \\ & { = \frac { \mathcal { S } _ { 2 } } { \mathcal { S } } \| u _ { t } - \nu \| _ { \mathrm { L E } ^ { \infty } } ^ { 2 } . } \end{array} +$$ + +Since the choice of $\mu$ and $\nu$ is arbitrary, we have proved (28) by assuming (D2). + +Next, we move on to prove the converse. Before proceeding, note that the Bregman divergence ${ \mathfrak { D } } _ { J } ( \nu , \mu )$ is always non-negative. In fact, for any $\mu , \nu \in \mathcal { M } ( X )$ , we have + +$$ +\int \Phi _ { \mu } d ( \nu - \mu ) = \underbrace { \int \Phi _ { \mu } d \nu - J ^ { \star } ( \Phi _ { \mu } ) } _ { \leq J ( \nu ) } - \underbrace { \left[ \int \Phi _ { \mu } d \mu - J ^ { \star } ( \Phi _ { \mu } ) \right] } _ { = J ( \mu ) } +$$ + +$$ +\leq J ( \nu ) - J ( \mu ) , +$$ + +which implies $\mathfrak { D } _ { J } ( \nu , \mu ) \geq 0$ . + +Choose $\xi \in \mathcal { M } ( X )$ arbitrarily. If $\xi = \mu$ , the inequality $\| \nabla \Phi _ { \mu } - \nabla \Phi _ { \xi } \| _ { 2 } \leq \beta _ { 2 } \| \xi - \mu \| _ { \mathrm { K R } }$ is trivial, so we assume $\xi \neq \mu$ below. By the non-negativity of the Bregman divergence, we have + +$$ +\begin{array} { r l } { { 0 \leq \mathfrak { D } _ { J } ( \nu , \mu ) } } \\ & { = J ( \nu ) - [ J ( \mu ) + \int \Phi _ { \mu } d ( \nu - \mu ) ] } \\ & { \leq [ J ( \xi ) + \int \Phi _ { \xi } d ( \nu - \xi ) + \frac { \beta _ { 2 } } { 2 } | | \xi - \nu | | _ { \mathrm { K R } } ^ { 2 } ] - [ J ( \mu ) + \int \Phi _ { \mu } d ( \nu - \mu ) ] } \\ & { = J ( \xi ) - J ( \mu ) - \int \Phi _ { \mu } d ( \xi - \mu ) + \int [ \Phi _ { \xi } - \Phi _ { \mu } ] d ( \nu - \xi ) + \frac { \beta _ { 2 } } { 2 } | | \xi - \nu | | _ { \mathrm { K R } } ^ { 2 } . } \end{array} +$$ + +Since the above inequality holds for all $\nu \in \mathcal { M } ( X )$ , the last expression is still non-negative if we take the infimum over all $\nu$ . In particular, + +$$ +\begin{array} { l } { { \displaystyle \operatorname* { i n f } _ { \nu \in \mathcal { M } ( X ) } \int [ \Phi _ { \xi } - \Phi _ { \mu } ] d ( \nu - \xi ) + \frac { \beta _ { 2 } } { 2 } \| \xi - \nu \| _ { \mathrm { K R } } ^ { 2 } } } \\ { ~ = - \displaystyle \operatorname* { s u p } _ { \nu \in \mathcal { M } ( X ) } \left[ \int [ \Phi _ { \mu } - \Phi _ { \xi } ] d ( \nu - \xi ) - \frac { \beta _ { 2 } } { 2 } \| \xi - \nu \| _ { \mathrm { K R } } ^ { 2 } \right] } \\ { ~ = - \displaystyle \operatorname* { s u p } _ { \zeta \in \mathcal { M } ( X ) } \left[ \int [ \Phi _ { \mu } - \Phi _ { \xi } ] d \zeta - \frac { \beta _ { 2 } } { 2 } \| \zeta \| _ { \mathrm { K R } } ^ { 2 } \right] } \\ { ~ = - \displaystyle \frac { 1 } { 2 \beta _ { 2 } } \| \Phi _ { \mu } - \Phi _ { \xi } \| _ { \mathrm { L i p } } ^ { 2 } , } \end{array} +$$ + +using Lemma 6 for the last equality. + +Continuing from (29), we have + +$$ +0 \leq J ( \xi ) - J ( \mu ) - \int \Phi _ { \mu } d ( \xi - \mu ) - \frac { 1 } { 2 \beta _ { 2 } } \left. \Phi _ { \mu } - \Phi _ { \xi } \right. _ { \mathrm { L i p } } ^ { 2 } . +$$ + +Swapping the roles of $\mu$ and $\xi$ , we obtain a similar inequality. Adding both sides of thus obtained two inequalities, we obtain + +$$ +0 \leq \int [ \Phi _ { \xi } - \Phi _ { \mu } ] d ( \xi - \mu ) - \frac { 1 } { \beta _ { 2 } } \left\| \Phi _ { \mu } - \Phi _ { \xi } \right\| _ { \mathrm { L i p } } ^ { 2 } . +$$ + +Finally, we have + +$$ +\begin{array} { r } { \left\| \Phi _ { \mu } - \Phi _ { \xi } \right\| _ { \mathrm { L i p } } ^ { 2 } \leq \beta _ { 2 } \displaystyle \int [ \Phi _ { \xi } - \Phi _ { \mu } ] d ( \xi - \mu ) \leq \beta _ { 2 } \left\| \Phi _ { \mu } - \Phi _ { \xi } \right\| _ { \mathrm { L i p } } \left\| \xi - \mu \right\| _ { \mathrm { K R } } , } \end{array} +$$ + +and hence we have the desired result: + +$$ +\begin{array} { r } { \| \Phi _ { \mu } - \Phi _ { \xi } \| _ { \mathrm { L i p } } \leq \beta _ { 2 } \| \xi - \mu \| _ { \mathrm { K R } } . } \end{array} +$$ + +Proof. First, we note that + +$$ +\begin{array} { r l } & { \mathrm { M M D } _ { K } ^ { 2 } ( \mu , \nu ) = \displaystyle \int K ( x , y ) \left( \mu - \nu \right) ( d x ) \left( \mu - \nu \right) ( d y ) } \\ & { \qquad \le \left[ \displaystyle \operatorname* { s u p } _ { x \neq x ^ { \prime } } \frac { \int \left( K ( x , y ) - K ( x ^ { \prime } , y ) \right) \left( \mu - \nu \right) \left( d y \right) } { d ( x , x ^ { \prime } ) } \right] \| \mu - \nu \| _ { \mathrm { K R } } } \\ & { \qquad \le \left[ \displaystyle \operatorname* { s u p } _ { x \neq x ^ { \prime } , y \neq y ^ { \prime } } \frac { K ( x , y ) - K ( x , y ^ { \prime } ) - K ( x ^ { \prime } , y ) + K ( x ^ { \prime } , y ^ { \prime } ) } { d ( x , x ^ { \prime } ) d ( y , y ^ { \prime } ) } \right] \| \mu - \nu \| _ { \mathrm { K R } } ^ { 2 } . } \end{array} +$$ + +Next, we show that + +$$ +c = \operatorname* { s u p } _ { x \neq x ^ { \prime } , y \neq y ^ { \prime } } \frac { K ( x , y ) - K ( x , y ^ { \prime } ) - K ( x ^ { \prime } , y ) + K ( x ^ { \prime } , y ^ { \prime } ) } { d ( x , x ^ { \prime } ) d ( y , y ^ { \prime } ) } = \frac { 1 } { \sigma ^ { 2 } } . +$$ + +Because $K$ is differentiable, it suffices to check that the operator norm of $\partial _ { x _ { i } } \partial _ { y _ { j } } K ( x , y )$ is bounded. To see this, note that + +$$ +\begin{array} { l } { c = \displaystyle \operatorname* { s u p } _ { x \neq x ^ { \prime } } \frac { \mathrm { L i p } \left( K ( x , \cdot ) - K ( x ^ { \prime } , \cdot ) \right) } { | | x - x ^ { \prime } | | } } \\ { = \displaystyle \operatorname* { s u p } _ { x \neq x ^ { \prime } , y } \frac { | | \nabla _ { y } K ( x , y ) - \nabla _ { y } K ( x ^ { \prime } , y ) | | _ { 2 } } { | | x - x ^ { \prime } | | } } \\ { \leq \displaystyle \operatorname* { s u p } _ { x , y } | | \nabla _ { x } \nabla _ { y } K ( x , y ) | | _ { \mathrm { o p } } , } \end{array} +$$ + +where we obtained by the last equality by applying the vector-valued mean value theorem (Rudin, 1964) to $t \mapsto \nabla _ { y } K ( ( 1 - t ) x + t x ^ { \prime } , y )$ , thereby obtaining + +$$ +\begin{array} { r l } & { | | \nabla _ { y } K ( x , y ) - \nabla _ { y } K ( x ^ { \prime } , y ) | | _ { 2 } \leq | | \partial _ { t } \nabla _ { y } K ( ( 1 - t ) x + t x ^ { \prime } , y ) | | _ { 2 } } \\ & { \qquad = | | ( x ^ { \prime } - x ) \cdot \nabla _ { x } \nabla _ { y } K ( ( 1 - t ) x + t x ^ { \prime } , y ) | | _ { 2 } } \\ & { \qquad \leq | | x ^ { \prime } - x | | _ { 2 } | | \nabla _ { x } \nabla _ { y } K ( ( 1 - t ) x + t x ^ { \prime } , y ) | | _ { \mathrm { { o p } } } . } \end{array} +$$ + +Now + +$$ +\begin{array} { l } { \partial _ { x _ { i } } \partial _ { y _ { j } } K ( x , y ) = \partial _ { x _ { i } } \partial _ { y _ { j } } \exp \left( - \displaystyle \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { k } ( x _ { k } - y _ { k } ) ^ { 2 } \right) } \\ { = \partial _ { x _ { i } } \exp \left( - \displaystyle \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { k } ( x _ { k } - y _ { k } ) ^ { 2 } \right) \cdot - \displaystyle \frac { 1 } { \sigma ^ { 2 } } ( x _ { j } - y _ { j } ) } \\ { = \exp \left( - \displaystyle \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { k } ( x _ { k } - y _ { k } ) ^ { 2 } \right) \left[ \displaystyle \frac { 1 } { \sigma ^ { 4 } } ( x _ { i } - y _ { i } ) ( x _ { j } - y _ { j } ) - \displaystyle \frac { 1 } { \sigma ^ { 2 } } \delta _ { i j } \right] . } \end{array} +$$ + +This is the symmetric matrix + +$$ +\frac { 1 } { \sigma ^ { 2 } } \exp \Big ( { - \frac { | | x - y | | ^ { 2 } } { 2 \sigma ^ { 2 } } } \Big ) \Big [ \frac { 1 } { \sigma ^ { 2 } } ( x - y ) ( x - y ) ^ { T } - I \Big ] . +$$ + +By inspection, we see that $x - y$ is an eigenvector with eigenvalue $\begin{array} { r } { \frac { 1 } { \sigma ^ { 2 } } \exp ( { - \frac { | | x - y | | ^ { 2 } } { 2 \sigma ^ { 2 } } } ) ( \frac { 1 } { \sigma ^ { 2 } } | | x - y | | ^ { 2 } - } \end{array}$ 1), and the other eigenvectors are orthogonal to x − y with eigenvalues − 1σ2 exp(− ||x−y||22σ2 ). + +Setting $\begin{array} { r } { z = \frac { | | x - y | | ^ { 2 } } { 2 \sigma ^ { 2 } } } \end{array}$ and taking the absolute value of the eigenvalues, we see the maximum operator norm over all $x , y$ is equal to + +$$ +\operatorname* { m a x } \bigg \{ \operatorname* { s u p } _ { z \ge 0 } \frac { 1 } { \sigma ^ { 2 } } e ^ { - z } | 2 z - 1 | , \operatorname* { s u p } _ { z \ge 0 } \frac { 1 } { \sigma ^ { 2 } } e ^ { - z } \bigg \} = \frac { 1 } { \sigma ^ { 2 } } . +$$ + +Therefore + +$$ +c \leq { \frac { 1 } { \sigma ^ { 2 } } } . +$$ + +It is not strictly necessary for the proof to show equality. However, to show equality, let $u$ be a unit vector, and set $x ^ { \prime } = y ^ { \prime } = 0$ , and $x = y = t u$ . Then + +$$ +\begin{array} { r l } & { c \geq \underset { t \neq 0 } { \operatorname* { s u p } } \frac { K ( t u , t u ) - K ( t u , 0 ) - K ( 0 , t u ) + K ( 0 , 0 ) } { d ( t u , 0 ) d ( t u , 0 ) } } \\ & { = \underset { t \neq 0 } { \operatorname* { s u p } } \frac { 2 - 2 \exp ( - \frac { t ^ { 2 } } { 2 \sigma ^ { 2 } } ) } { { t ^ { 2 } } } } \\ & { \geq \underset { t \neq 0 } { \operatorname* { l i m } } \frac { 2 - 2 \exp ( - \frac { t ^ { 2 } } { 2 \sigma ^ { 2 } } ) } { { t ^ { 2 } } } } \\ & { = \underset { t \neq 0 } { \operatorname* { l i m } } \frac { - 2 \exp ( - \frac { t ^ { 2 } } { 2 \sigma ^ { 2 } } ) \cdot - \frac { 2 t } { 2 \sigma ^ { 2 } } } { 2 t } } \\ & { = \frac { 1 } { t \neq 0 } , } \end{array} +$$ + +where we used l’Hopital’s rule to evaluate the limit. ˆ + +Proposition 11. The minimax and non-saturating GAN losses do not satisfy $( D 3 )$ for some $\mu _ { 0 }$ + +Proof. For the minimax loss, the Bregman divergence is: + +$$ +\begin{array} { r l } { \mathfrak { D } _ { B \ni \{ \cdot \} \parallel \mu _ { 0 } } ( v , \mu ) = } & { \displaystyle \int \left[ \frac { 1 } { 2 } \log \frac { \mu _ { 0 } } { \frac { 1 } { 2 } \mu _ { 0 } + \frac { 1 } { 2 } } \ d \mu _ { 0 } + \frac { 1 } { 2 } \log \frac { v } { \frac { 1 } { 2 } \mu _ { 0 } + \frac { 1 } { 2 } \nu } d \nu \right] } \\ & { \quad - \displaystyle \int \left[ \frac { 1 } { 2 } \log \frac { \mu _ { 0 } } { \frac { 1 } { 2 } \mu _ { 0 } + \frac { 1 } { 2 } \mu } d \mu _ { 0 } + \frac { 1 } { 2 } \log \frac { \mu } { \frac { 1 } { 2 } \mu _ { 0 } + \frac { 1 } { 2 } \mu } d \mu \right] } \\ & { \quad - \displaystyle \int \frac { 1 } { 2 } \log \frac { \mu } { \mu _ { 0 } + \mu } d ( \nu - \mu ) } \\ & { = \frac { 1 } { 2 } \int \log \frac { \mu _ { 0 } + \mu } { \mu _ { 0 } + \nu } d \mu _ { 0 } + \frac { 1 } { 2 } \int \left[ \log \frac { \nu } { \frac { 1 } { 2 } \mu _ { 0 } + \frac { 1 } { 2 } \nu } - \log \frac { \mu } { \mu _ { 0 } + \mu } \right] d \nu + \frac { 1 } { 2 } \log \frac { 1 } { 2 } } \\ & { = \frac { 1 } { 2 } \int \log \frac { \mu _ { 0 } + \mu } { \mu _ { 0 } + \nu } d ( \mu _ { 0 } + \nu ) + \frac { 1 } { 2 } \int \log \frac { \nu } { \mu } d \nu } \\ & { = D _ { \mathrm { K L } } ( \frac { 1 } { 2 } \nu + \frac { 1 } { 2 } \mu _ { 0 } ) \parallel \frac { 1 } { 2 } \mu + \frac { 1 } { 2 } \mu _ { 0 } ) + \frac { 1 } { 2 } D _ { \mathrm { K L } } ( \nu | \mu ) , } \end{array} +$$ + +and for the non-saturating loss, the Bregman divergence is + +$$ +\begin{array} { l } { { \displaystyle \mathfrak { D } _ { N \mathrm { k } , ( \frac { 1 } { 2 } + \frac { 1 } { 2 } \mu _ { 0 } ) \| \mu _ { 0 } ) } ( \nu , \mu ) = \int \log \frac { \frac { 1 } { 2 } \mu _ { 0 } + \frac { 1 } { 2 } \nu } { \mu _ { 0 } } d ( \frac { 1 } { 2 } \nu + \frac { 1 } { 2 } \mu _ { 0 } ) } } \\ { { \displaystyle \phantom { \frac { 1 } { 2 } } - \int \log \frac { \frac { 1 } { 2 } \mu _ { 0 } + \frac { 1 } { 2 } \mu } { \mu _ { 0 } } d ( \frac { 1 } { 2 } \mu + \frac { 1 } { 2 } \mu _ { 0 } ) } } \\ { { \displaystyle \phantom { \frac { 1 } { 2 } } + \int \frac { 1 } { 2 } \log \frac { \mu _ { 0 } } { \mu _ { 0 } + \mu } d ( \nu - \mu ) } } \\ { { \displaystyle \phantom { \frac { 1 } { 2 } } = \frac { 1 } { 2 } \int \log \frac { \mu _ { 0 } + \nu } { \mu _ { 0 } + \mu } d \mu _ { 0 } + \frac { 1 } { 2 } \int \log \frac { \frac { 1 } { 2 } \mu _ { 0 } + \frac { 1 } { 2 } \nu } { \mu _ { 0 } + \mu } d \nu - \frac { 1 } { 2 } \log \frac { 1 } { 2 } } } \\ { { \displaystyle \phantom { \frac { 1 } { 2 } } = D _ { \mathrm { K L } } ( \frac { 1 } { 2 } \nu + \frac { 1 } { 2 } \mu _ { 0 } | | \frac { 1 } { 2 } \mu + \frac { 1 } { 2 } \mu _ { 0 } ) . } } \end{array} +$$ + +Choosing $\nu$ to be not absolutely continuous w.r.t. ${ \scriptstyle { \frac { 1 } { 2 } } } \mu + { \scriptstyle { \frac { 1 } { 2 } } } \mu _ { 0 }$ makes the Bregman divergence $\infty$ , which is sufficient to show that the Bregman divergence is not bounded by $| | \mu - \nu | | _ { \mathrm { K R } } ^ { 2 }$ . + +Proposition 12. The MMD loss with Gaussian kernel satisfies $( D 3 )$ with $\beta _ { 2 } = 2 \pi$ for all $\mu _ { 0 }$ . + +Proof. We work with the Gaussian kernel $K ( x , y ) = e ^ { - { \frac { | | x - y | | ^ { 2 } } { 2 \sigma ^ { 2 } } } }$ and use Proposition 10: + +$$ +\begin{array} { r l } { \mathfrak { D } _ { \frac { 1 } { 2 } \mathrm { M M D } ^ { 2 } ( \cdot , \cdot , \mu _ { 0 } ) } ( \nu , \mu ) = \displaystyle \frac { 1 } { 2 } \int K ( x , y ) \big ( \nu - \mu _ { 0 } ) ( d x ) \big ( \nu - \mu _ { 0 } ) ( d y ) } & { } \\ & { \qquad - \frac { 1 } { 2 } \int K ( x , y ) ( \mu - \mu _ { 0 } ) ( d x ) \big ( \mu - \mu _ { 0 } ) ( d y ) } \\ & { \qquad - \displaystyle \int ( K ( x , y ) \mu ( d y ) - K ( x , y ) \mu _ { 0 } ( d y ) \big ) \big ( \nu - \mu ) ( d x ) } \\ & { = \displaystyle \frac { 1 } { 2 } \int K ( x , y ) \big ( \nu - \mu ) ( d x ) \big ( \nu - \mu ) ( d y ) } \\ & { = \frac { 1 } { 2 } \mathrm { M M D } ^ { 2 } ( \mu , \nu ) } \\ & { \leq \frac { 1 } { 2 \pi ^ { 2 } } \| \mu - \nu \| _ { \mathrm { K R } } ^ { 2 } , } \end{array} +$$ + +where the last line is from Lemma 7. The result follows with $\sigma = ( 2 \pi ) ^ { - 1 / 2 }$ . + +The result actually applies more generally for the MMD loss with a differentiable kernel $K$ that satisfies $\begin{array} { r } { \beta _ { 2 } : = \operatorname* { s u p } _ { x , y } \bar { \| \nabla _ { x } \nabla _ { y } K ( \bar { x } , y ) \| _ { 2 } } < \infty } \end{array}$ . □ + +Proposition 13 (Moreau–Yosida regularization). Suppose $J : \mathcal { M } ( X ) \bar { \mathbb { R } }$ is convex, and define ${ \tilde { J } } : = J \oplus R _ { 3 }$ . Then $\tilde { J }$ is convex, and $\begin{array} { r } { \mathfrak { D } _ { \widetilde { J } } ( \nu , \mu ) \le \frac { \beta _ { 2 } } { 2 } \vert \vert \mu - \nu \vert \vert _ { \mathrm { K R } } ^ { 2 } } \end{array}$ . + +Proof. We work on the more general case of K(x, y) = (2πσ2)−d/2 exp(− kx−yk22σ2 ), and define + +$$ +\tilde { J } ( \mu ) = \operatorname * { i n f } _ { \tilde { \mu } } J ( \tilde { \mu } ) + \frac { c L } { 2 } \mathrm { M M D } _ { K } ^ { 2 } ( \mu , \tilde { \mu } ) +$$ + +for $c = \sigma ^ { 2 } ( 2 \pi \sigma ^ { 2 } ) ^ { d / 2 }$ . + +Let $\mu ^ { * }$ be the unique minimizer of the infimum, which exists because the function is a strongly convex. By the envelope theorem, we compute that + +$$ +\begin{array} { r l } & { \frac { d } { d \epsilon } \tilde { J } ( \mu + \epsilon \chi ) \Big | _ { \epsilon = 0 } = \frac { d } { d \epsilon } \frac { c L } { 2 } | | \mu + \epsilon \chi - \mu ^ { * } | | _ { \mathcal { H } } ^ { 2 } \Big | _ { \epsilon = 0 } } \\ & { \qquad = \frac { d } { d \epsilon } \frac { c L } { 2 } \big ( | | \mu - \mu ^ { * } | | _ { \mathcal { H } } ^ { 2 } + 2 \epsilon \langle \mu - \mu ^ { * } , \chi \rangle _ { \mathcal { H } } + \epsilon ^ { 2 } | | \chi | | _ { \mathcal { H } } ^ { 2 } \big ) \Big | _ { \epsilon = 0 } } \\ & { \qquad = c L \langle \mu - \mu ^ { * } , \chi \rangle _ { \mathcal { H } } } \\ & { \qquad = c L \int ( \mu - \mu ^ { * } ) d \chi , } \end{array} +$$ + +so $\begin{array} { r } { \frac { \delta \tilde { J } } { \delta \mu } = c L ( \mu - \mu ^ { * } ) } \end{array}$ + +Then + +$$ +\begin{array} { r l } & { \mathfrak { D } _ { \mathcal { J } } ( \nu , \mu ) = \mathcal { \tilde { J } } ( \nu ) - \mathcal { \tilde { J } } ( \mu ) - \displaystyle \int \frac { \delta \mathcal { \tilde { J } } } { \delta \mu } d ( \nu - \mu ) } \\ & { \qquad = \operatorname* { i n f } J ( \tilde { \nu } ) + \frac { c L } { 2 } \| \nu - \tilde { \nu } \| _ { \mathcal H } ^ { 2 } - \left[ J ( \mu ^ { * } ) + \frac { c L } { 2 } \| \mu - \mu ^ { * } \| _ { \mathcal H } ^ { 2 } \right] - c L \langle \mu - \mu ^ { * } , \nu - \mu \rangle _ { \mathcal H } } \\ & { \qquad \le J ( \mu ^ { * } ) + \frac { c L } { 2 } \| \nu - \mu ^ { * } \| _ { \mathcal H } ^ { 2 } - \left[ J ( \mu ^ { * } ) + \frac { c L } { 2 } \| \mu - \mu ^ { * } \| _ { \mathcal H } ^ { 2 } \right] - c L \langle \mu - \mu ^ { * } , \nu - \mu \rangle _ { \mathcal H } } \\ & { \qquad = \frac { c L } { 2 } \| \mu - \nu \| _ { \mathcal H } ^ { 2 } } \\ & { \qquad \le \frac { c L } { 2 } \cdot ( 2 \pi \sigma ^ { 2 } ) ^ { - d / 2 } \cdot \frac { 1 } { \sigma ^ { 2 } } \| \mu - \nu \| _ { \mathcal H } ^ { 2 } } \\ & { \qquad = \frac { L } { 2 } \| \mu - \nu \| _ { \mathbb K ^ { 0 } } ^ { 2 } , } \end{array} +$$ + +where we used Lemma 7 for the second-to-last line. The proposition follows for $\sigma = ( 2 \pi ) ^ { - 1 / 2 }$ + +Regarding this choice of $\sigma$ , it will turn out that in the general case, the dual penalty includes a numerically unfavorable factor of $( 2 \pi \sigma ^ { 2 } ) ^ { - d / 2 }$ , dependent on the dimension of the problem. In practical applications, such as image generation, $d$ can be quite large, making the accurate computation of $( \bar { 2 } \pi \sigma ^ { 2 } ) ^ { - d / 2 }$ completely infeasible. For numerical stability, we propose choosing the critical parameter $\sigma = ( 2 \pi ) ^ { - 1 / 2 }$ , which corresponds to the dimension-free kernel $K ( x , y ) = e ^ { - \pi | | x - y | | ^ { 2 } }$ . □ + +Lemma 8. Let $\mathcal { H }$ be an RKHS with a continuous kernel $K$ on a compact domain $X$ . The convex conjugate of $\mu \mapsto \frac { \lambda } { 2 } | | \mu | | _ { \mathcal { H } } ^ { 2 }$ is $\begin{array} { r } { f \mapsto \frac { 1 } { 2 \lambda } | | f | | _ { \mathcal { H } } ^ { 2 } + \chi \{ f \in \mathcal { H } \} } \end{array}$ . + +Proof. First we show an upper bound for $f \in \mathcal H$ : + +$$ +\begin{array} { l } { \displaystyle \operatorname* { s u p } _ { \mu } [ \int f d \mu - \frac \lambda 2 | | \mu | | _ { \mathcal H } ^ { 2 } ] = \operatorname* { s u p } _ { \mu } [ \langle f , \mu \rangle _ { \mathcal H } - \frac \lambda 2 | | \mu | | _ { \mathcal H } ^ { 2 } ] } \\ { \displaystyle \quad \leq \operatorname* { s u p } _ { \mu } [ | | f | | \mu | | \mu | ] _ { \mathcal H } - \frac \lambda 2 | | \mu | | _ { \mathcal H } ^ { 2 } ] } \\ { \displaystyle \quad \leq \operatorname* { s u p } _ { z \in \mathbb R } [ | | f | | \mu z - \frac \lambda 2 z ^ { 2 } ] } \\ { \displaystyle \quad = | | f | | _ { \mathcal H } \cdot \frac { | | f | | _ { \mathcal H } } { \lambda } - \frac { \lambda } { 2 } \cdot \frac { | | f | | _ { \mathcal H } ^ { 2 } } { \lambda ^ { 2 } } } \\ { \displaystyle \quad = \frac { 1 } { 2 \lambda } | | f | | _ { \mathcal H } ^ { 2 } . } \end{array} +$$ + +To derive a lower bound for general $f \in { \mathcal { C } } ( X )$ , we use the Mercer decomposition of the positive definite kernel $K$ , + +$$ +K ( x , y ) = \sum _ { i = 1 } ^ { \infty } \gamma _ { i } \phi _ { i } ( x ) \phi _ { i } ( y ) +$$ + +where $\gamma _ { i } \geq 0$ are eigenvalues and $\{ \phi _ { i } \} _ { i = 1 } ^ { \infty }$ is a complete orthonormal sequence of $L ^ { 2 } ( X ; d x )$ . It is well-known that the corresponding RKHS is given by + +$$ +\mathcal { H } = \left\{ g \in L ^ { 2 } ( X , d x ) \mid \sum _ { i = 1 } ^ { \infty } \frac { ( g , \phi _ { i } ) _ { L ^ { 2 } ( X ) } ^ { 2 } } { \gamma _ { i } } < \infty \right\} , +$$ + +and the norm of $g \in \mathcal { H }$ by + +$$ +\| g \| _ { \mathcal { H } } ^ { 2 } = \sum _ { j = 1 } ^ { \infty } \frac { ( g , \phi _ { i } ) _ { L ^ { 2 } ( X ) } ^ { 2 } } { \gamma _ { i } } . +$$ + +Let $f \in \mathcal { C } ( X ) \subset L ^ { 2 } ( X , d x )$ be an arbitrary function. We replace the supremum of the conjugate function + +$$ +\operatorname* { s u p } _ { \mu } \left[ \int f d \mu - \frac { \lambda } { 2 } | | \hat { \mu } | | _ { \mathcal { H } } ^ { 2 } \right] +$$ + +by $\mu \in { \mathcal { M } } ( X )$ that has a square-integrable Radon–Nykodim derivative with respect to the Lebesgue measure $d x$ , and then we have a lower bound. We use $\mu ( x )$ for the Radon–Nykodim derivative with slight abuse of notation. + +Suppose $\textstyle f ( x ) = \sum _ { j = 1 } ^ { \infty } a _ { j } \phi _ { j } ( x )$ and $\textstyle \mu ( x ) = \sum _ { j = 1 } ^ { \infty } b _ { j } \phi _ { j } ( x )$ are the expansion. Note that, since the kernel embedding is given by + +$$ +\hat { \mu } ( x ) = \int K ( x , y ) \mu ( y ) d y = \sum _ { j = 1 } ^ { \infty } \gamma _ { j } b _ { j } \phi _ { j } ( x ) , +$$ + +the above maximization is reduced to + +$$ +\operatorname * { s u p } _ { b _ { j } } \sum _ { j = 1 } ^ { \infty } a _ { j } b _ { j } - \frac { \lambda } { 2 } \gamma _ { j } b _ { j } ^ { 2 } . +$$ + +This is maximized when $b _ { j } = a _ { j } / ( \lambda \gamma _ { j } )$ , and the maximum value is + +$$ +\frac { 1 } { 2 \lambda } \sum _ { j = 1 } ^ { \infty } \frac { a _ { j } ^ { 2 } } { \gamma _ { j } } . +$$ + +If $f \in \mathcal H$ , this value is finite, and the lower bound of the conjugate given by is $\frac { 1 } { 2 \lambda } \| f \| _ { \mathcal { H } } ^ { 2 }$ , which is the same as the upper bound. If $f \notin { \mathcal { H } }$ , the value (32) is infinite, and the conjugate function takes $+ \infty$ . □ + +Proposition 14. Let $\mathcal { H }$ be an RKHS with the Gaussian kernel $K ( x , y ) = e ^ { - \pi | | x - y | | ^ { 2 } }$ . Then for $f \in \mathcal H$ , + +$$ +\begin{array} { r l } & { \displaystyle | | f | | _ { \mathcal { H } } ^ { 2 } = \sum _ { k = 0 } ^ { \infty } ( 4 \pi ) ^ { - k } \sum _ { \stackrel { k _ { 1 } + \cdots + k _ { d } = k } { k _ { 1 } \prod _ { i = 1 } ^ { d } k _ { i } ! } } | | \partial _ { x _ { 1 } } ^ { k _ { 1 } } \cdot \cdot \cdot \partial _ { x _ { d } } ^ { k _ { d } } f | | _ { L ^ { 2 } ( \mathbb { R } ^ { d } ) } ^ { 2 } } \\ & { \quad \quad = | | f | | _ { L ^ { 2 } ( \mathbb { R } ^ { d } ) } ^ { 2 } + \frac { 1 } { 4 \pi } | | \nabla f | | _ { L ^ { 2 } ( \mathbb { R } ^ { d } ) } ^ { 2 } + \frac { 1 } { 1 6 \pi ^ { 2 } } | | \nabla ^ { 2 } f | | _ { L ^ { 2 } ( \mathbb { R } ^ { d } ) } ^ { 2 } + o t h e r t e r m s . } \end{array} +$$ + +Proof. We consider the more general case of $K ( x , y ) = ( 2 \pi \sigma ^ { 2 } ) ^ { - d / 2 } e ^ { - | | x - y | | ^ { 2 } / 2 \sigma ^ { 2 } }$ , where we have + +$$ +\begin{array} { r l } & { \displaystyle | | f | | _ { \mathcal { H } } ^ { 2 } = \sum _ { k = 0 } ^ { \infty } ( \frac { 1 } { 2 } \sigma ^ { 2 } ) ^ { k } \sum _ { { k _ { 1 } + \cdots + k _ { d } = k } \atop = | | f | | _ { L ^ { 2 } ( \mathbb { R } ^ { d } ) } ^ { 2 } } \frac { 1 } { | | \nabla f | | _ { L ^ { 2 } ( \mathbb { R } ^ { d } ) } ^ { 2 } } | | \partial _ { x _ { 1 } } ^ { k _ { 1 } } \cdot \cdot \cdot \partial _ { x _ { d } } ^ { k _ { d } } f | | _ { L ^ { 2 } ( \mathbb { R } ^ { d } ) } ^ { 2 } } \\ & { \quad \quad \quad = | | f | | _ { L ^ { 2 } ( \mathbb { R } ^ { d } ) } ^ { 2 } + \frac { 1 } { 2 } \sigma ^ { 2 } | | \nabla f | | _ { L ^ { 2 } ( \mathbb { R } ^ { d } ) } ^ { 2 } + \frac { 1 } { 4 } \sigma ^ { 4 } | | \nabla ^ { 2 } f | | _ { L ^ { 2 } ( \mathbb { R } ^ { d } ) } ^ { 2 } + \mathrm { o t h e r ~ t e r m s } . } \end{array} +$$ + +Novak et al. (2018) prove this result for $\sigma = 1$ . We sketch the proof for the general case here. We use the Fourier transform convention that + +$$ +\hat { f } ( \boldsymbol { k } ) = ( 2 \pi ) ^ { - d / 2 } \int _ { \mathbb { R } ^ { d } } f ( \boldsymbol { x } ) e ^ { - i \boldsymbol { k } \cdot \boldsymbol { x } } d \boldsymbol { x } , \quad f ( \boldsymbol { x } ) = ( 2 \pi ) ^ { - d / 2 } \int _ { \mathbb { R } ^ { d } } \hat { f } ( \boldsymbol { k } ) e ^ { i \boldsymbol { k } \cdot \boldsymbol { x } } d \boldsymbol { k } . +$$ + +Consider the inner product + +$$ +\langle f , g \rangle = \int _ { \mathbb { R } ^ { d } } e ^ { \sigma ^ { 2 } | | k | | ^ { 2 } / 2 } \hat { f } ( k ) \overline { { \hat { g } ( k ) } } d k , +$$ + +defined for functions $| | f | | < \infty$ . Expanding the exponential in Taylor series, this inner product gives the equation for the norm in the proposition. By use of the Fourier inversion formula, it can be shown that $\langle f , K _ { x } \rangle = f ( x )$ , where + +$$ +K _ { x } ( y ) = ( 2 \pi \sigma ^ { 2 } ) ^ { - d / 2 } e ^ { - | | x - y | | ^ { 2 } / 2 \sigma ^ { 2 } } , +$$ + +so this is an RKHS with the Gaussian kernel. \ No newline at end of file diff --git a/parse/train/HJeOekHKwr/HJeOekHKwr_content_list.json b/parse/train/HJeOekHKwr/HJeOekHKwr_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..7d287db691c988d3f8815cd990e7f7a45bc1b34d --- /dev/null +++ b/parse/train/HJeOekHKwr/HJeOekHKwr_content_list.json @@ -0,0 +1,5717 @@ +[ + { + "type": "text", + "text": "SMOOTHNESS AND STABILITY IN GANS ", + "text_level": 1, + "bbox": [ + 176, + 99, + 658, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Casey Chu \nStanford University \ncaseychu@stanford.edu ", + "bbox": [ + 183, + 145, + 392, + 186 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Kentaro Minami Preferred Networks, Inc. minami@preferred.jp ", + "bbox": [ + 625, + 145, + 813, + 188 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Kenji Fukumizu ", + "text_level": 1, + "bbox": [ + 184, + 208, + 299, + 222 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The Institute of Statistical Mathematics / Preferred Networks, Inc. fukumizu@ism.ac.jp ", + "bbox": [ + 184, + 223, + 614, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 286, + 544, + 303 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Generative adversarial networks, or GANs, commonly display unstable behavior during training. In this work, we develop a principled theoretical framework for understanding the stability of various types of GANs. In particular, we derive conditions that guarantee eventual stationarity of the generator when it is trained with gradient descent, conditions that must be satisfied by the divergence that is minimized by the GAN and the generator’s architecture. We find that existing GAN variants satisfy some, but not all, of these conditions. Using tools from convex analysis, optimal transport, and reproducing kernels, we construct a GAN that fulfills these conditions simultaneously. In the process, we explain and clarify the need for various existing GAN stabilization techniques, including Lipschitz constraints, gradient penalties, and smooth activation functions. ", + "bbox": [ + 233, + 315, + 764, + 469 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION: TAMING INSTABILITY WITH SMOOTHNESS ", + "text_level": 1, + "bbox": [ + 178, + 493, + 692, + 508 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Generative adversarial networks (Goodfellow et al., 2014), or GANs, are a powerful class of generative models defined through minimax game. GANs and their variants have shown impressive performance in synthesizing various types of datasets, especially natural images. Despite these successes, the training of GANs remains quite unstable in nature, and this instability remains difficult to understand theoretically. ", + "bbox": [ + 174, + 523, + 825, + 593 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Since the introduction of GANs, there have been many techniques proposed to stabilize GANs training, including studies of new generator/discriminator architectures, loss functions, and regularization techniques. Notably, Arjovsky et al. (2017) proposed Wasserstein GAN (WGAN), which in principle avoids instability caused by mismatched generator and data distribution supports. In practice, this is enforced by Lipschitz constraints, which in turn motivated developments like gradient penalties (Gulrajani et al., 2017) and spectral normalization (Miyato et al., 2018). Indeed, these stabilization techniques have proven essential to achieving the latest state-of-the-art results (Karras et al., 2018; Brock et al., 2019). ", + "bbox": [ + 174, + 599, + 825, + 712 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "On the other hand, a solid theoretical understanding of training stability has not been established. Several empirical observations point to an incomplete understanding. For example, why does applying a gradient penalty together spectral norm seem to improve performance (Miyato et al., 2018), even though in principle they serve the same purpose? Why does applying only spectral normalization with the Wasserstein loss fail (Miyato, 2018), even though the analysis of Arjovsky et al. (2017) suggests it should be sufficient? Why is applying gradient penalties effective, even outside their original context of the Wasserstein GAN (Fedus et al., 2018)? ", + "bbox": [ + 174, + 718, + 825, + 815 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we develop a framework to analyze the stability of GAN training that resolves these apparent contradictions and clarifies the roles of these regularization techniques. Our approach considers the smoothness of the loss function used. In optimization, smoothness is a well-known condition that ensures that gradient descent and its variants become stable (see e.g., Bertsekas (1999)). For example, the following well-known proposition is the starting point of our stability analysis: ", + "bbox": [ + 174, + 823, + 825, + 892 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Proposition 1 (Bertsekas (1999), Proposition 1.2.3). Suppose $f : \\mathbb { R } ^ { p } \\mathbb { R }$ is $L$ -smooth and bounded below. Let $\\begin{array} { r } { x _ { k + 1 } : = x _ { k } - \\frac { 1 } { L } \\dot { \\nabla } \\dot { f } ( x _ { k } ) } \\end{array}$ . Then $| | \\nabla f ( x _ { k } ) | | 0$ as $k \\to \\infty$ . ", + "bbox": [ + 173, + 895, + 823, + 925 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This proposition says that under a smoothness condition on the function, gradient descent with a constant step size $\\frac { \\mathbf { i } } { L }$ approaches stationarity (i.e., the gradient norm approaches zero). This is a rather weak notion of convergence, as it does not guarantee that the iterates converge to a point, and even if the iterates do converge, the limit is a stationary point and not necessarily an minimizer. ", + "bbox": [ + 174, + 103, + 823, + 160 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Nevertheless, empirically, not even this stationarity is satisfied by GANs, which are known to frequently destabilize and diverge during training. To diagnose this instability, we consider the smoothness of the GAN’s loss function. GANs are typically framed as minimax problems of the form ", + "bbox": [ + 174, + 165, + 825, + 208 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/b263ffe4f4bc0564c57ae1535c9cc3193397bdbc281607ac8a876560293ef9f5.jpg", + "text": "$$\n\\operatorname* { i n f } _ { \\theta } \\operatorname* { s u p } _ { \\varphi } \\mathcal { I } ( \\mu _ { \\theta } , \\varphi ) ,\n$$", + "text_format": "latex", + "bbox": [ + 437, + 213, + 557, + 238 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\mathcal { I }$ is a loss function that takes a generator distribution $\\mu _ { \\theta }$ and discriminator $\\varphi$ , and $\\theta \\in \\mathbb { R } ^ { p }$ denotes the parameters of the generator. Unfortunately, the minimax nature of this problem makes stability and convergence difficult to analyze. To make the analysis more tractable, we define $J ( \\mu ) =$ $\\operatorname* { s u p } _ { \\varphi } \\mathcal { I } ( \\mu , \\varphi )$ , so that (1) becomes simply ", + "bbox": [ + 174, + 236, + 823, + 291 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/ef23c33984aaf9362a1047a22102199852bccfff2c1f0009208416645b663419.jpg", + "text": "$$\n\\operatorname* { i n f } _ { \\theta } J ( \\mu _ { \\theta } ) .\n$$", + "text_format": "latex", + "bbox": [ + 460, + 296, + 534, + 319 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This choice corresponds to the common assumption that the discriminator is allowed to reach optimality at every training step. Now, the GAN algorithm can be regarded as simply gradient descent on the $\\mathbb { R } ^ { p } \\to \\mathbb { R }$ function $\\theta \\mapsto J ( \\mu _ { \\theta } )$ , which may be analyzed using Proposition 1. In particular, if this function $\\theta \\mapsto J ( \\mu _ { \\theta } )$ satisfies the smoothness assumption, then the GAN training should be stable in that it should approach stationarity under the assumption of an optimal discriminator. ", + "bbox": [ + 174, + 321, + 825, + 393 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the remainder of this paper, we investigate whether the smoothness assumption is satisfied for various GAN losses. Our analysis answers two questions: ", + "bbox": [ + 176, + 400, + 820, + 428 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Q1. Which existing GAN losses, if any, satisfy the smoothness condition in Proposition 1? ", + "bbox": [ + 194, + 438, + 799, + 453 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Q2. Are there choices of loss, regularization, or architecture that enforce smoothness in GANs? ", + "bbox": [ + 196, + 457, + 823, + 472 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "As results of our analysis, our contributions are as follows: ", + "bbox": [ + 174, + 482, + 558, + 496 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We derive sufficient conditions for the GAN algorithm to be stationary under certain assumptions (Theorem 1). Our conditions relate to the smoothness of GAN loss used as well as the parameterization of the generator. \n2. We show that most common GAN losses do not satisfy the all of the smoothness conditions, thereby corroborating their empirical instability. \n3. We develop regularization techniques that enforce the smoothness conditions. These regularizers recover common GAN stabilization techniques such as gradient penalties and spectral normalization, thereby placing their use on a firmer theoretical foundation. \n4. Our analysis provides several practical insights, suggesting for example the use of smooth activation functions, simultaneous spectral normalization and gradient penalties, and a particular learning rate for the generator. ", + "bbox": [ + 210, + 507, + 825, + 674 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 689, + 326, + 704 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Divergence minimization Our analysis regards the GAN algorithm as minimizing a divergence between the current generator distribution and the desired data distribution, under the assumption of an optimal discriminator at every training step. This perspective originates from the earliest GAN paper, in which Goodfellow et al. (2014) show that the original minimax GAN implicitly minimizes the Jensen–Shannon divergence. Since then, the community has introduced a large number of GAN or GAN-like variants that learn generative models by implicitly minimizing various divergences, including $f$ -divergences (Nowozin et al., 2016), Wasserstein distance (Arjovsky et al., 2017), and maximum-mean discrepancy (Li et al., 2015; Unterthiner et al., 2018). Meanwhile, the non-saturating GAN (Goodfellow et al., 2014) has been shown to minimize a certain Kullback– Leibler divergence (Arjovsky & Bottou, 2017). Several more theoretical works consider the topological, geometric, and convexity properties of divergence minimization (Arjovsky & Bottou, 2017; Liu et al., 2017; Bottou et al., 2018; Farnia & Tse, 2018; Chu et al., 2019), perspectives that we draw heavily upon. Sanjabi et al. (2018) also prove smoothness of GAN losses in the specific case of the regularized optimal transport loss. Their assumption for smoothness is entangled in that it involves a composite condition on generators and discriminators, while our analysis addresses them separately. ", + "bbox": [ + 173, + 715, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "table", + "img_path": "images/b208f36dc85e726acc9bd2b11736e05618cc0c24ef75af5b086928b9f2e8d9ab.jpg", + "table_caption": [ + "Table 1: Common GANs, their corresponding loss functions, and their optimal discriminators. " + ], + "table_footnote": [], + "table_body": "
Loss function J(μ)Optimal discriminator Φμ(x)
Minimax GANDjs(μ|l μo)μ(x）
Non-saturating GANDKL(¹μ+μo |μo)-110g（）+(@） μo(x）
Wasserstein GANW1(μ, μo)argmaXfeLip1 Ey~μ[f(y)] -Ey~μo[f(y)]
GMMN,Coulomb GAN1MMD²(μ, μo)Ey~μ[K(x,y)]-Ey~μo[K(x,y)]
IPM-GANIPMj(μ, μo)arg maxf∈F Ey~μ[f(y)] -Ey~μof(y)]
", + "bbox": [ + 174, + 119, + 823, + 263 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Other approaches Even though many analyses, including ours, operate under the assumption of an optimal discriminator, this assumption is unrealistic in practice. Li et al. (2017b) contrast this optimal discriminator dynamics with first-order dynamics, which assumes that the generator and discriminator use alternating gradient updates and is what is used computationally. As this is a differing approach from ours, we only briefly mention some results in this area, which typically rely on game-theoretic notions (Kodali et al., 2017; Grnarova et al., 2018; Oliehoek et al., 2018) or local analysis (Nagarajan & Kolter, 2017; Mescheder et al., 2018). Some of these results rely on continuous dynamics approximations of gradient updates; in contrast, our work focuses on discrete dynamics. ", + "bbox": [ + 173, + 285, + 826, + 411 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1.2 NOTATION ", + "text_level": 1, + "bbox": [ + 174, + 426, + 289, + 440 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $\\bar { \\mathbb { R } } : = \\mathbb { R } \\cup \\{ \\infty , - \\infty \\}$ . We let ${ \\mathcal { P } } ( X )$ denote the set of all probability measures on a compact set $X \\subseteq \\mathbb { R } ^ { d }$ . We let ${ \\mathcal { M } } ( X )$ and ${ \\mathcal { C } } ( X )$ denote the dual pair consisting of the set of all finite signed measures on $X$ and the set of all continuous functions $X \\to \\mathbb { R }$ . For any statement $A$ , we let $\\chi \\{ A \\}$ be 0 if $A$ is true and $\\infty$ if $A$ is false. For a Euclidean vector $x$ , its Euclidean norm is denoted by $\\| x \\| _ { 2 }$ , and the operator norm of a matrix $A$ is denoted by $\\| A \\| _ { 2 }$ , i.e., $\\begin{array} { r } { \\| A \\| _ { 2 } = \\operatorname* { s u p } _ { \\| x \\| _ { 2 } \\leq 1 } \\| A x \\| _ { 2 } / \\| x \\| _ { 2 } } \\end{array}$ . A function $f : X \\to Y$ between two metric spaces is $\\alpha$ -Lipschitz if $d _ { Y } ( f ( x _ { 1 } ) , f ( x _ { 2 } ) ) \\leq \\alpha d _ { X } ( x _ { 1 } , x _ { 2 } ) ,$ . A function $f : \\mathbb { R } ^ { d } \\mathbb { R }$ is $\\beta$ -smooth if its gradients are $\\beta$ -Lipschitz, that is, for all $x , y \\in \\mathbb { R } ^ { d }$ , $\\| \\nabla f ( x ) - \\nabla f ( y ) \\| _ { 2 } \\leq \\beta \\| x - y \\| _ { 2 }$ . ", + "bbox": [ + 173, + 452, + 826, + 568 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 SMOOTHNESS OF GAN LOSSES ", + "text_level": 1, + "bbox": [ + 176, + 585, + 465, + 602 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This section presents Theorem 1, which provides concise criteria for the smoothness of GAN losses. ", + "bbox": [ + 169, + 614, + 823, + 631 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In order to keep our analysis agnostic to the particular GAN used, let $J : { \\mathcal { P } } ( X ) \\to { \\bar { \\mathbb { R } } }$ be an arbitrary convex loss function, which takes a distribution over $X \\subset \\mathbb { R } ^ { d }$ and outputs a real number. Note that the typical minimax formulation of GANs can be recovered from just the loss function $J$ using convex duality. In particular, recall that the convex conjugate $J ^ { \\star } : \\bar { \\mathcal { C } } ( X ) \\bar { \\mathbb { R } }$ of $J$ satisfies the following remarkable duality, known as the Fenchel–Moreau theorem: ", + "bbox": [ + 173, + 636, + 823, + 707 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d419ff9c13d7375a6986c9ea0ec904943dd4bf0878e440c354002daec5042605.jpg", + "text": "$$\nJ ^ { \\star } ( \\varphi ) : = \\operatorname* { s u p } _ { \\mu \\in { \\mathcal M } ( X ) } \\int \\varphi ( x ) d \\mu - J ( \\mu ) , \\qquad J ( \\mu ) = \\operatorname* { s u p } _ { \\varphi \\in { \\mathcal C } ( X ) } \\int \\varphi ( x ) d \\mu - J ^ { \\star } ( \\varphi ) .\n$$", + "text_format": "latex", + "bbox": [ + 230, + 708, + 766, + 744 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Based on this duality, minimizing $J$ can be framed as the minimax problem ", + "bbox": [ + 173, + 746, + 669, + 761 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/af2c8ee3c32c845370873ed3a058ec7d56de1bf07f8c2dd3e4c0722ff2d15f49.jpg", + "text": "$$\n\\operatorname* { i n f } _ { \\mu \\in { \\mathcal P } ( X ) } J ( \\mu ) = \\operatorname* { i n f } _ { \\mu \\in { \\mathcal P } ( X ) } \\operatorname* { s u p } _ { \\varphi \\in { \\mathcal C } ( X ) } \\int \\varphi ( x ) d \\mu - J ^ { \\star } ( \\varphi ) : = \\operatorname* { i n f } _ { \\mu \\in { \\mathcal P } ( X ) } \\operatorname* { s u p } _ { \\varphi \\in { \\mathcal C } ( X ) } { \\mathcal I } ( \\mu , \\varphi ) ,\n$$", + "text_format": "latex", + "bbox": [ + 233, + 762, + 763, + 797 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "recovering the well-known adversarial formulation of GANs. We now define the notion of an optimal discriminator for an arbitrary loss function $J$ , based on this convex duality: ", + "bbox": [ + 173, + 799, + 825, + 827 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 1 (Optimal discriminator). Let $J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ be a convex, l.s.c., proper function. An optimal discriminator for a probability distribution ${ \\boldsymbol { \\mu } } \\in { \\mathcal { P } } ( { \\boldsymbol { X } } )$ is a continuous function $\\Phi _ { \\mu } : X \\to $ $\\mathbb { R }$ that attains the maximum of the second equation in (3), i.e., $\\begin{array} { r } { J ( \\mu ) = \\int \\Phi _ { \\mu } ( x ) d \\mu - J ^ { \\star } ( \\dot { \\Phi } _ { \\mu } ) } \\end{array}$ . ", + "bbox": [ + 174, + 829, + 825, + 873 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This definition recovers the optimal discriminators of many existing GAN and GAN-like algorithms (Farnia & Tse, 2018; Chu et al., 2019), most notably those in Table 1. Our analysis will apply to any algorithm in this family of algorithms. See Appendix $\\mathbf { B }$ for more details on this perspective. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We also formalize the notion of a family of generators: ", + "bbox": [ + 174, + 103, + 534, + 118 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 2 (Family of generators). A family of generators is a set of pushforward probability measures $\\{ \\mu _ { \\theta } = f _ { \\theta \\# } \\omega : \\theta \\in \\mathbb { R } ^ { p } \\}$ , where $\\omega$ is a fixed probability distribution on $Z$ (the latent variable) and $f _ { \\theta } : Z \\to X$ is a measurable function (the generator). ", + "bbox": [ + 174, + 122, + 825, + 165 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Now, in light of Proposition 1, we are interested in the smoothness of the mapping $\\theta \\mapsto J ( \\mu _ { \\theta } )$ , which would guarantee the stationarity of gradient descent on this objective, which in turn implies stationarity of the GAN algorithm under the assumption of an optimal discriminator. The following theorem is our central result, which decomposes the smoothness of $\\theta \\mapsto J ( \\mu _ { \\theta } )$ into conditions on optimal discriminators and the family of generators. ", + "bbox": [ + 174, + 176, + 826, + 247 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 1 (Smoothness decomposition for GANs). Let $J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ be a convex function whose optimal discriminators $\\Phi _ { \\mu } : X \\to \\mathbb { R }$ satisfy the following regularity conditions: ", + "bbox": [ + 173, + 251, + 823, + 281 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(D1) $x \\mapsto \\Phi _ { \\mu } ( x )$ is $\\alpha$ -Lipschitz, (D2) $x \\mapsto \\nabla _ { x } \\Phi _ { \\mu } ( x )$ is $\\beta _ { 1 }$ -Lipschitz, (D3) $\\mu \\mapsto \\nabla _ { x } \\Phi _ { \\mu } ( x )$ is $\\beta _ { 2 }$ -Lipschitz w.r.t. the 1-Wasserstein distance. ", + "bbox": [ + 194, + 291, + 408, + 308 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 192, + 315, + 437, + 332 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 191, + 339, + 648, + 356 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Also, let $\\mu _ { \\theta } = f _ { \\theta \\# } \\omega$ be a family of generators that satisfies: ", + "bbox": [ + 171, + 367, + 566, + 383 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(G1) $\\theta \\mapsto f _ { \\theta } ( z )$ is $A$ -Lipschitz in expectation for $z \\sim \\omega$ , i.e., $\\begin{array} { r } { \\mathbb { E } _ { z \\sim \\omega } [ \\| f _ { \\theta _ { 1 } } ( z ) - f _ { \\theta _ { 2 } } ( z ) \\| _ { 2 } ] \\ \\le } \\end{array}$ $A \\| \\theta _ { 1 } - \\theta _ { 2 } \\| _ { 2 }$ , and ", + "bbox": [ + 186, + 393, + 825, + 424 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(G2) $\\theta \\ \\mapsto \\ D _ { \\theta } f _ { \\theta } ( z )$ is $B$ -Lipschitz in expectation for $z \\sim \\omega$ , i.e., $\\mathbb { E } _ { z \\sim \\omega } [ \\| D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } ( z ) ~ -$ $D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ( z ) \\lVert _ { 2 } \\bigr \\rVert \\leq B \\lVert \\theta _ { 1 } - \\theta _ { 2 } \\rVert _ { 2 }$ . ", + "bbox": [ + 191, + 430, + 823, + 462 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Then $ { \\theta } \\mapsto J ( { \\mu } _ { { \\theta } } )$ is $L$ -smooth, with $L = \\alpha B + A ^ { 2 } ( \\beta _ { 1 } + \\beta _ { 2 } )$ . ", + "bbox": [ + 173, + 472, + 575, + 488 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 1 connects the smoothness properties of the loss function $J$ with the smoothness properties of the optimal discriminator $\\Phi _ { \\mu }$ , and once paired with Proposition 1, it suggests a quantitative value $\\textstyle { \\frac { 1 } { L } }$ for a stable generator learning rate. In order to obtain claims of stability for practically sized learning rates, it is important to tightly bound the relevant constants. ", + "bbox": [ + 173, + 498, + 825, + 558 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In Sections 4 to 6, we carefully analyze which GAN losses satisfy (D1), (D2), and (D3), and with what constants. We summarize our results in Table 2: it turns out that none of the listed losses, except for one, satisfy (D1), (D2), and (D3) simultaneously with a finite constant. The MMD-based loss satisfies the three conditions, but its constant for (D1) grows as $\\alpha = O ( { \\sqrt { d } } )$ , which is an unfavorable dependence on the data dimension $d$ that forces an unacceptably small learning rate. See for complete details of each condition. This failure of existing GANs to satisfy the stationarity conditions corroborates the observed instability of GANs. ", + "bbox": [ + 173, + 563, + 825, + 662 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/b9ceade9e50a525c645cc621f8f6147e563dfcb5e0881bce3a2ff301dcdfa076.jpg", + "table_caption": [ + "Table 2: Regularity of common GAN losses. " + ], + "table_footnote": [], + "table_body": "
(D1)(D2)(D3)
Minimax GAN×X×
Non-saturating GAN Wasserstein GANXX？
√X
×？
IPMs MMD²*
", + "bbox": [ + 343, + 696, + 655, + 797 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 1 decomposes smoothness into conditions on the generator and conditions on the discriminator, allowing a clean separation of concerns. In this paper, we focus on the discriminator conditions (D1), (D2), and (D3) and only provide an extremely simple example of a generator that satisfies (G1) and (G2), in Section 7. Because analysis of the generator conditions may become quite complicated and will vary with the choice of architecture considered (feedforward, convolutional, ResNet, etc.), we leave a detailed analysis of the generator conditions (G1) and (G2) as a promising avenue for future work. Indeed, such analyses may lead to new generator architectures or generator regularization techniques that stabilize GAN training. ", + "bbox": [ + 173, + 811, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we present a generic regularization technique that imposes the three conditions sufficient for stable learning on an arbitrary loss function $J$ , thereby stabilizing training. In Section 2, we observe that the Wasserstein, IPM, and MMD losses respectively satisfy (D1), (D2), and (D3) individually, but not all of of them at the same time. Using techniques from convex analysis, we convert these three GAN losses into three regularizers that, when applied simultaneously, causes the resulting loss to satisfy all the three conditions. Here, we only outline the technique; the specifics of each case are deferred to Sections 4 to 6. ", + "bbox": [ + 173, + 132, + 825, + 231 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We start with an arbitrary base loss function $J$ to be regularized. Next, we take an existing GAN loss that satisfies the desired regularity condition and convert it into a regularizer function $R$ : $\\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ . Then, we consider $J \\oplus R$ , which denotes the inf-convolution defined as ", + "bbox": [ + 174, + 237, + 825, + 280 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f0186e0736f9b4233f82f5be43e4dbaba7c7fd4af6dc65957492b3dd1009ec20.jpg", + "text": "$$\n( J \\oplus R ) ( \\xi ) = \\operatorname* { i n f } _ { \\tilde { \\xi } \\in \\mathcal { M } ( X ) } J ( \\tilde { \\xi } ) + R ( \\xi - \\tilde { \\xi } ) .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 286, + 635, + 315 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This new function $J \\oplus R$ inherits the regularity of $R$ , making it a stable candidate as a GAN loss. Moreover, because the inf-convolution is a commutative operation, we can sequentially apply multiple regularizers $R _ { 1 } , R _ { 2 }$ , and $R _ { 3 }$ without destroying the added regularity. In particular, if we carefully choose functions $R _ { 1 }$ , $R _ { 2 }$ , and $R _ { 3 }$ , then $\\tilde { J } = \\bar { J } \\oplus \\bar { R _ { 1 } } \\oplus R _ { 2 } \\oplus \\bar { R _ { 3 } }$ will satisfy (D1), (D2), and (D3) simultaneously. Moreover, under some technical assumptions, this composite function $\\tilde { J }$ inherits the original minimizers of $J$ , making it a sensible GAN loss: ", + "bbox": [ + 173, + 320, + 825, + 409 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 2 (Invariance of minimizers). Let $R _ { 1 } ( \\xi ) : = \\| \\xi \\| _ { \\mathrm { K R } } , R _ { 2 } ( \\xi ) : = \\| \\xi \\| _ { S ^ { * } }$ , and $R _ { 3 } ( \\xi ) : = $ $\\frac { 1 } { 2 } \\| \\hat { \\xi } \\| _ { \\mathcal { H } } ^ { 2 }$ be the three regularizers defined by (8), (12), and (19) respectively. Assume that $ { J } :$ $\\bar { \\mathcal { M } } ( X ) \\bar { \\mathbb { R } }$ has a unique minimizer at $\\mu _ { 0 }$ with $J ( \\mu _ { 0 } ) = 0 ;$ , and $J ( \\mu ) \\geq c \\| \\hat { \\mu } - \\hat { \\mu } _ { 0 } \\| _ { \\mathcal { H } }$ for some $c > 0$ . Then the inf-convolution $\\tilde { J } = J \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus R _ { 3 }$ has a unique minimizer at $\\mu _ { 0 }$ with $\\tilde { J } ( \\mu _ { 0 } ) = 0$ . ", + "bbox": [ + 173, + 411, + 825, + 492 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The duality formulation (4) provides a practical method for minimizing this composite function. We leverage the duality relation $( J \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus R _ { 3 } ) ^ { \\star } = J ^ { \\star } + R _ { 1 } ^ { \\star } + R _ { 2 } ^ { \\star } + R _ { 3 } ^ { \\star }$ and apply (4): ", + "bbox": [ + 171, + 502, + 823, + 532 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8e90d785a28f709fc118a9b6728e042f010918444b52bd7f606aad06d672a944.jpg", + "text": "$$\n\\begin{array} { l } { \\underset { \\mu } { \\operatorname* { i n f } } \\left( J \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus R _ { 3 } \\right) ( \\mu ) = \\underset { \\mu \\atop \\varphi } { \\operatorname* { i n f } } \\underset { \\varphi } { \\operatorname* { s u p } } \\int \\varphi d \\mu - J ^ { \\star } ( \\varphi ) - R _ { 1 } ^ { \\star } ( \\varphi ) - R _ { 2 } ^ { \\star } ( \\varphi ) - R _ { 3 } ^ { \\star } ( \\varphi ) } \\\\ { = \\underset { \\mu \\atop \\varphi } { \\operatorname* { i n f } } \\underset { \\varphi } { \\operatorname* { s u p } } \\mathcal { I } ( \\mu , \\varphi ) - R _ { 1 } ^ { \\star } ( \\varphi ) - R _ { 2 } ^ { \\star } ( \\varphi ) - R _ { 3 } ^ { \\star } ( \\varphi ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 537, + 782, + 599 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This minimax problem can be seen as a GAN whose discriminator objective has three added regularization terms. ", + "bbox": [ + 171, + 604, + 823, + 633 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The concrete form of these regularizers are summarized in Table 3. Notably, we observe that we recover standard techniques for stabilizing GANs: ", + "bbox": [ + 171, + 638, + 823, + 669 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• (D1) is enforced by Lipschitz constraints (i.e., spectral normalization) on the discriminator. \n• (D2) is enforced by spectral normalization and a choice of Lipschitz, smooth activation functions for the discriminator. ", + "bbox": [ + 210, + 679, + 825, + 741 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/2ae6f8a1890e001271e3b3147014adf28e59e3f7a7900929ab67a3355927f338.jpg", + "table_caption": [ + "Table 3: Smoothness-inducing regularizers and their convex conjugates. To enforce a regularity condition on a loss function $J$ , we take a source loss that satisfies it and view it as a regularizer $R$ . We then consider the inf-convolution $J \\oplus R$ , which corresponds to an added regularization term $R ^ { \\star }$ on the discriminator $\\varphi$ . These regularization terms correspond to existing GAN techniques. " + ], + "table_footnote": [], + "table_body": "
PurposeSource lossR()R（）GAN techniques
(D1)W1|KRx{lφ|lLip ≤1}spectral norm
(D2)IPMs|/| s*x{∈s}smooth activations, spectral norm
(D3)MMD²gradient penalties
", + "bbox": [ + 199, + 821, + 797, + 921 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• (D3) is enforced by gradient penalties on the discriminator. ", + "bbox": [ + 215, + 103, + 629, + 119 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our analysis therefore puts these regularization techniques on a firm theoretical foundation (Proposition 1 and Theorem 1) and provides insight into their function. ", + "bbox": [ + 171, + 130, + 823, + 159 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 ENFORCING (D1) WITH LIPSCHITZ CONSTRAINTS ", + "text_level": 1, + "bbox": [ + 174, + 179, + 620, + 196 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we show that enforcing (D1) leads to techniques and notions commonly used to stabilize GANs, including the Wasserstein distance, Lipschitz constraints and spectral normalization. Recall that (D1) demands that the optimal discriminator $\\Phi _ { \\mu }$ is Lipschitz: ", + "bbox": [ + 173, + 210, + 825, + 253 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "If $\\Phi _ { \\mu }$ is differentiable, this is equivalent to that the optimal discriminator has a gradient with bounded norm. This is a sensible criterion, since a discriminator whose gradient norm is too large may push the generator too hard and destabilize its training. ", + "bbox": [ + 174, + 281, + 825, + 324 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To check (D1), the following proposition shows that it suffices to check whether $| J ( \\mu ) - J ( \\nu ) | \\leq$ $\\alpha W _ { 1 } ( \\mu , \\nu )$ for all distributions $\\mu , \\nu$ : ", + "bbox": [ + 173, + 330, + 823, + 359 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 3. $( D l )$ holds if and only if $J$ is $\\alpha$ -Lipschitz w.r.t. the Wasserstein-1 distance. ", + "bbox": [ + 174, + 363, + 764, + 380 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Arjovsky et al. (2017) show that this property does not hold for common divergences based on the Kullback–Leibler or Jensen–Shannon divergence, while it does hold for the Wasserstein-1 distance. Indeed, it is this desirable property that motivates their introduction of the Wasserstein GAN. Framed in our context, their result is summarized as follows: ", + "bbox": [ + 174, + 390, + 826, + 446 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 4. The minimax and non-saturating GAN losses do not satisfy $( D I )$ for some $\\mu _ { 0 }$ ", + "bbox": [ + 173, + 450, + 784, + 467 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 5. The Wasserstein GAN loss satisfies $( D l )$ with $\\alpha = 1$ for any $\\mu _ { 0 }$ . ", + "bbox": [ + 174, + 474, + 696, + 491 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our stability analysis therefore deepens the analysis of Arjovsky et al. (2017) and provides an alternative reason that the Wasserstein distance is desirable as a metric: it is part of a sufficient condition that ensures stationarity of gradient descent. ", + "bbox": [ + 176, + 501, + 821, + 545 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 FROM WASSERSTEIN DISTANCE TO LIPSCHITZ CONSTRAINTS ", + "text_level": 1, + "bbox": [ + 174, + 561, + 638, + 577 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Having identified the Wasserstein GAN loss as one that satisfies (D1), we next follow the strategy outlined in Section 3 to convert it into a regularizer for an arbitrary loss function. Towards this, we define the regularizer $R _ { 1 } : { \\mathcal { M } } ( X ) \\to { \\bar { \\mathbb { R } } }$ and compute its convex conjugate $R _ { 1 } ^ { \\star } : { \\mathcal { C } } ( X ) \\to { \\bar { \\mathbb { R } } }$ : ", + "bbox": [ + 174, + 587, + 825, + 631 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/a9edbd666998be9b0fbb67aaf0ea1ee87b8930cfbb6e02426f7888e012e513bb.jpg", + "text": "$$\nR _ { 1 } ( \\xi ) : = \\alpha \\| \\xi \\| _ { \\mathrm { K R } } = \\alpha \\operatorname* { s u p } _ { \\stackrel { f \\in \\mathcal { C } ( X ) } { | | f | | _ { \\mathrm { L i p } } \\leq 1 } } \\int f d \\xi , \\qquad R _ { 1 } ^ { \\star } ( \\varphi ) = \\left\\{ \\begin{array} { l l } { 0 } & { \\| \\varphi \\| _ { \\mathrm { L i p } } \\leq \\alpha } \\\\ { \\infty } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 250, + 638, + 746, + 688 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This norm is the Kantorovich–Rubinstein norm (KR norm), which extends the Wasserstein-1 distance to $M ( X )$ ; it holds that $\\| \\mu - \\nu \\| _ { \\mathrm { K R } } = W _ { 1 } ( \\mu , \\nu )$ for $\\mu , \\nu \\in { \\mathcal { P } } ( X )$ . Then, its inf-convolution with an arbitrary function inherits the Lipschitz property held by $R _ { 1 }$ : ", + "bbox": [ + 173, + 693, + 823, + 736 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 6 (Pasch–Hausdorff). Let $J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ be a function, and define ${ \\tilde { J } } : = J \\oplus R _ { 1 }$ . Then $\\tilde { J }$ is $\\alpha$ -Lipschitz w.r.t. the distance induced by the $K R$ norm, and hence the Wasserstein-1 distance when restricted to ${ \\mathcal { P } } ( X )$ . ", + "bbox": [ + 173, + 741, + 825, + 787 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Due to Proposition 3, we now obtain a transformed loss function $\\tilde { J }$ that automatically satisfies (D1). This function is a generalization of the Pasch–Hausdorff envelope (see Chapter 9 in Rockafeller & Wets (1998)), also known as Lipschitz regularization or the McShane–Whitney extension (McShane, 1934; Whitney, 1934; Kirszbraun, 1934; Hiriart-Urruty, 1980). ", + "bbox": [ + 173, + 799, + 825, + 858 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The convex conjugate computation in (8) shows that $\\tilde { J }$ can be minimized in practice by imposing Lipschitz constraints on discriminators. Indeed, by (4), ", + "bbox": [ + 171, + 864, + 825, + 895 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/1f786b38890752fa2cec078d5bf8a8700d87db41c4dd2d7880e1be7f1ec3c9f0.jpg", + "text": "$$\n\\operatorname* { i n f } _ { \\mu } \\big ( J \\oplus \\alpha \\lVert \\cdot \\rVert _ { \\mathrm { K R } } \\big ) ( \\mu ) = \\operatorname* { i n f } _ { \\mu } \\operatorname* { s u p } _ { \\varphi } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - J ^ { \\star } ( \\varphi ) - \\chi \\{ \\lVert \\varphi \\rVert _ { \\mathrm { L i p } } \\leq \\alpha \\}\n$$", + "text_format": "latex", + "bbox": [ + 258, + 900, + 738, + 928 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/48c83cfa1096e1351e435aa120e0b1819d6322798ed04d84c0ea8973d809ffff.jpg", + "text": "$$\n= \\operatorname* { i n f } _ { \\mu } \\operatorname* { s u p } _ { \\varphi \\colon \\| \\varphi \\| _ { \\mathrm { L i p } } \\leq \\alpha } \\mathcal { I } ( \\mu , \\varphi ) .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 102, + 583, + 131 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Farnia & Tse (2018) consider this loss $\\tilde { J }$ in the special case of an $f$ -GAN with $J ( \\mu ) = D _ { f } ( \\mu , \\mu _ { 0 } )$ ; they showed that minimizing $\\tilde { J }$ corresponds to training a $f$ -GAN normally but constraining the discriminator to be $\\alpha$ -Lipschitz. We show that this technique is in fact generic for any $J$ : minimizing the transformed loss can be achieved by training the GAN as normal, but imposing a Lipschitz constraint on the discriminator. ", + "bbox": [ + 173, + 133, + 825, + 208 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our analysis therefore justifies the use of Lipschitz constraints, such as spectral normalization (Miyato et al., 2018) and weight clipping (Arjovsky & Bottou, 2017), for general GAN losses. However, Theorem 1 also suggests that applying only Lipschitz constraints may not be enough to stabilize GANs, as (D1) alone does not ensure that the GAN objective is smooth. ", + "bbox": [ + 173, + 214, + 825, + 271 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 ENFORCING (D2) WITH DISCRIMINATOR SMOOTHNESS ", + "text_level": 1, + "bbox": [ + 174, + 290, + 660, + 306 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "(D2) demands that the optimal discriminator $\\Phi _ { \\mu }$ is smooth: ", + "bbox": [ + 173, + 320, + 563, + 335 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "(D2) $x \\mapsto \\nabla \\Phi _ { \\mu } ( x )$ is $\\beta _ { 1 }$ -Lipschitz for all $\\mu \\in { \\mathcal { P } } ( X )$ , i.e., $\\| \\nabla \\Phi _ { \\mu } ( x ) - \\nabla \\Phi _ { \\mu } ( y ) \\| _ { 2 } \\leq \\beta _ { 1 } \\| x - y \\| _ { 2 }$ . ", + "bbox": [ + 174, + 340, + 823, + 358 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Intuitively, this says that for a fixed generator $\\mu$ , the optimal discriminator $\\Phi _ { \\mu }$ should not provide gradients that change too much spatially. ", + "bbox": [ + 174, + 363, + 823, + 392 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Although the Wasserstein GAN loss (D1), we see that it, along with the minimax GAN and the non-saturating GAN, do not satisfy (D2): ", + "bbox": [ + 176, + 397, + 823, + 428 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proposition 7. The Wasserstein, minimax, and non-saturating GAN losses do not satisfy $( D 2 )$ for some $\\mu _ { 0 }$ . ", + "bbox": [ + 173, + 429, + 823, + 458 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We now construct a loss that by definition satisfies (D2). Let $s$ be the class of 1-smooth functions, that is, for which $\\| \\nabla f ( x ) - \\nabla f ( y ) \\| _ { 2 } \\leq \\| x - y \\| _ { 2 }$ , and consider the integral probability metric (IPM) (Muller, 1997) w.r.t. ¨ $s$ , defined by ", + "bbox": [ + 173, + 467, + 825, + 510 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/769d760952c9c8006b1a05ff9674099bf0e3352e9810fad4e6d6bdfe3031cf26.jpg", + "text": "$$\n\\mathrm { I P M } _ { \\cal S } ( \\mu , \\nu ) : = \\operatorname* { s u p } _ { f \\in { \\cal S } } \\int f d \\mu - \\int f d \\nu .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 511, + 630, + 546 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The optimal discriminator for the loss $\\mathrm { I P M } _ { \\cal S } ( \\mu , \\mu _ { 0 } )$ is the function that maximizes the supremum in the definition. This function by definition belongs to $s$ and therefore is 1-smooth. Hence, this IPM loss satisfies (D2) with $\\beta _ { 1 } = 1$ by construction. ", + "bbox": [ + 174, + 547, + 825, + 592 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 FROM INTEGRAL PROBABILITY METRIC TO SMOOTH DISCRIMINATORS ", + "text_level": 1, + "bbox": [ + 176, + 607, + 699, + 622 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Having identified the IPM-based loss as one that satisfies (D2), we next follow the strategy outlined in Section 3 to convert it into a regularizer for an arbitrary loss function. To do so, we define a regularizer $R _ { 2 } : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ and compute its convex conjugate $R _ { 2 } ^ { \\star } : { \\mathcal { C } } ( X ) \\to { \\bar { \\mathbb { R } } }$ : ", + "bbox": [ + 173, + 632, + 826, + 676 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/c847ebe43fd8513d821db7ca31e5d0425beaeca50d670cb05847801221f50035.jpg", + "text": "$$\nR _ { 2 } ( \\xi ) : = \\beta _ { 1 } \\| \\xi \\| _ { \\mathcal { S } ^ { * } } = \\beta _ { 1 } \\operatorname* { s u p } _ { f \\in \\mathcal { S } } \\int f d \\xi , \\qquad R _ { 2 } ^ { \\star } ( \\varphi ) = \\left\\{ \\begin{array} { l l } { 0 } & { \\varphi \\in \\beta _ { 1 } S } \\\\ { \\infty } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 266, + 679, + 727, + 714 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The norm is the dual norm to $s$ , which extends the IPM to signed measures; it holds that $\\mathrm { I P M } _ { \\mathcal { S } } ( \\mu , \\nu ) = \\| \\mu - \\nu \\| _ { \\mathcal { S } ^ { * } }$ for $\\mu , \\nu \\in { \\mathcal { P } } ( X )$ . Similar to the situation in the previous section, inf-convolution preserves the smoothness property of $R _ { 2 }$ : ", + "bbox": [ + 174, + 715, + 823, + 758 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proposition 8. Let $J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ be a convex, proper, lower semicontinuous function, and define ${ \\tilde { J } } : = J \\oplus R _ { 2 }$ . Then the optimal discriminator for $\\tilde { J }$ is $\\beta _ { 1 }$ -smooth. ", + "bbox": [ + 173, + 761, + 821, + 791 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Applying (4) and (12), we see that we can minimize this transformed loss function by restricting the family of discriminators to only $\\beta _ { 1 }$ -smooth discriminators: ", + "bbox": [ + 169, + 801, + 823, + 829 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/3dbec029e1c8e6d10328329528a5466f3fba9da73405f4ce630573ac39761102.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { \\mu } { \\operatorname* { i n f } } \\left( J \\oplus \\beta _ { 1 } \\lVert \\cdot \\rVert _ { \\mathcal { S } ^ { * } } \\right) ( \\mu ) = \\underset { \\mu } { \\operatorname* { i n f } } \\ \\underset { \\varphi } { \\operatorname* { s u p } } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - J ^ { \\star } ( \\varphi ) - \\chi \\{ \\varphi \\in \\beta _ { 1 } S \\} } \\\\ & { \\quad \\quad = \\underset { \\mu } { \\operatorname* { i n f } } \\ \\underset { \\varphi \\in \\beta _ { 1 } S } { \\operatorname* { s u p } } \\mathcal { I } ( \\mu , \\varphi ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 830, + 730, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In practice, we can enforce this by applying spectral normalization (Miyato et al., 2018) and using a Lipschitz, smooth activation function such as ELU (Clevert et al., 2016) or sigmoid. ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proposition 9. Let $f : \\mathbb { R } ^ { d } \\mathbb { R }$ be a neural network consisting of k layers whose linear transformations have spectral norm 1 and whose activation functions are 1-Lipschitz and 1-smooth. Then $f$ is $k$ -smooth. ", + "bbox": [ + 173, + 102, + 825, + 146 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 ENFORCING (D3) WITH GRADIENT PENALTIES ", + "text_level": 1, + "bbox": [ + 174, + 166, + 589, + 183 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(D3) is the following smoothness condition: ", + "bbox": [ + 176, + 196, + 464, + 212 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(D3) $\\mu \\mapsto \\nabla \\Phi _ { \\mu } ( x )$ is $\\beta _ { 2 }$ -Lipschitz for any $x \\in X$ , i.e., $\\| \\nabla \\Phi _ { \\mu } ( x ) - \\nabla \\Phi _ { \\nu } ( x ) \\| _ { 2 } \\leq \\beta _ { 2 } W _ { 1 } ( \\mu , \\nu ) .$ ", + "bbox": [ + 173, + 217, + 802, + 234 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(D3) requires that the gradients of the optimal discriminator do not change too rapidly in response to changes in $\\mu$ . Indeed, if the discriminator’s gradients are too sensitive to changes in the generator, the generator may not be able to accurately follow those gradients as it updates itself using a finite step size. In finite-dimensional optimization of a function $f : { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }$ , this condition is analogous to $f$ having a Lipschitz gradient. ", + "bbox": [ + 173, + 239, + 825, + 310 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We now present an equivalent characterization of (D3) that is easier to check in practice. We define the Bregman divergence of a convex function $J : { \\mathcal { M } } ( X ) \\to { \\bar { \\mathbb { R } } }$ by ", + "bbox": [ + 173, + 316, + 823, + 345 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/598edc74ea3d2ea830ac891bc02a235feaf388df814ff9d8710862c540c54926.jpg", + "text": "$$\n\\mathfrak { D } _ { J } ( \\nu , \\mu ) : = J ( \\nu ) - J ( \\mu ) - \\int \\Phi _ { \\mu } ( x ) d ( \\nu - \\mu ) ,\n$$", + "text_format": "latex", + "bbox": [ + 334, + 351, + 660, + 383 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $\\Phi _ { \\mu }$ is the optimal discriminator for $J$ at $\\mu$ . Then, (D3) is characterized in terms of the Bregman divergence and the KR norm as follows: ", + "bbox": [ + 171, + 388, + 821, + 416 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Proposition 10. Let $J : \\mathcal { M } ( X ) \\mathbb { R }$ be a convex function. Then $J$ satisfies $( D 3 )$ if and only if $\\begin{array} { r } { \\mathfrak { D } _ { J } ( \\nu , \\mu ) \\le \\frac { \\beta _ { 2 } } { 2 } \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } } \\end{array}$ for all $\\mu , \\nu \\in \\mathcal { M } ( X )$ . ", + "bbox": [ + 173, + 420, + 820, + 454 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "It is straightforward to compute the Bregman divergence corresponding to several popular GANs: ", + "bbox": [ + 171, + 463, + 810, + 478 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/e09bf18a7654237ad88536bfa1229c64ec24edf86328bcbfc3ebf62a42abde98.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathfrak { D } _ { D \\mathrm { J S } } ( \\cdot | | \\mu _ { 0 } ) \\big ( \\nu , \\mu \\big ) = D _ { \\mathrm { K L } } \\big ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } | | \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } \\big ) + \\frac { 1 } { 2 } D _ { \\mathrm { K L } } \\big ( \\nu | | \\mu \\big ) , } \\\\ & { \\qquad \\mathfrak { D } _ { D _ { \\mathrm { K L } } \\left( \\frac { 1 } { 2 } \\cdot + \\frac { 1 } { 2 } \\mu _ { 0 } \\right. | | \\mu _ { 0 } ) } \\big ( \\nu , \\mu \\big ) = D _ { \\mathrm { K L } } \\big ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } | | \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } \\big ) , } \\\\ & { \\qquad \\mathfrak { D } _ { \\frac { 1 } { 2 } \\mathrm { M M D } ^ { 2 } ( \\cdot , \\mu _ { 0 } ) } \\big ( \\nu , \\mu \\big ) = \\frac { 1 } { 2 } \\mathrm { M M D } ^ { 2 } ( \\nu , \\mu ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 272, + 481, + 722, + 547 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The first two Bregman divergences are not bounded above by $| | \\mu - \\nu | | _ { \\mathrm { K R } } ^ { 2 }$ for reasons similar to those discussed in Section 4, and hence: ", + "bbox": [ + 169, + 551, + 825, + 580 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Proposition 11. The minimax and non-saturating GAN losses do not satisfy $( D 3 )$ for some $\\mu _ { 0 }$ . ", + "bbox": [ + 173, + 584, + 794, + 599 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Even so, the Bregman divergence for the non-saturating loss is always less than that of the minimax GAN, suggesting that the non-saturating loss should be stable in more situations than the minimax GAN. On the other hand, the MMD-based loss (Li et al., 2015) does satisfy (D3) when its kernel is the Gaussian kernel $K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }$ : ", + "bbox": [ + 174, + 609, + 825, + 669 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Proposition 12. The MMD loss with Gaussian kernel satisfies $( D 3 )$ with $\\beta _ { 2 } = 2 \\pi$ for all $\\mu _ { 0 }$ ", + "bbox": [ + 169, + 672, + 779, + 689 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.1 FROM MAXIMUM MEAN DISCREPANCY TO GRADIENT PENALTIES ", + "text_level": 1, + "bbox": [ + 176, + 705, + 661, + 719 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Having identified the MMD-based loss as one that satisfies (D3), we next follow the strategy outlined in Section 3 to convert it into a regularizer for an arbitrary loss function. To do so, we define the regularizer $R _ { 3 } : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ and compute its convex conjugate $R _ { 3 } ^ { \\star } : { \\mathcal { C } } ( X ) { \\bar { \\mathbb { R } } }$ : ", + "bbox": [ + 173, + 729, + 825, + 773 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/0897a6672da3d502f8cb8b541af4a7e1d40a0ffd4529e66a7b6b57d5c9cd9812.jpg", + "text": "$$\nR _ { 3 } ( \\xi ) : = \\frac { \\beta _ { 2 } } { 4 \\pi } \\| \\hat { \\xi } \\| _ { \\mathcal { H } } ^ { 2 } , \\qquad R _ { 3 } ^ { \\star } ( \\varphi ) = \\frac { \\pi } { \\beta _ { 2 } } \\| \\varphi \\| _ { \\mathcal { H } } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 779, + 648, + 811 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The norm is the norm of a reproducing kernel Hilbert space norm (RKHS) $\\mathcal { H }$ with Gaussian kernel; this norm extends the MMD to signed measures, as it holds that $\\mathrm { M M D } ( \\mu , \\nu ) = \\| \\hat { \\mu } - \\hat { \\nu } \\| _ { \\mathcal { H } }$ for $\\mu , \\nu \\in { \\mathcal { P } } ( X )$ . Here, $\\begin{array} { r } { \\hat { \\xi } = \\int K ( x , \\cdot ) \\xi ( d x ) \\in \\mathcal { H } } \\end{array}$ denotes the mean embedding of a signed measure $\\xi \\in \\mathcal { M } ( X )$ ; we also adopt the convention that $| | \\varphi | | _ { \\mathcal { H } } = \\infty$ if $\\varphi \\notin { \\mathcal { H } }$ . Similar to the situation in the previous sections, inf-convolution preserves the smoothness property of $R _ { 3 }$ : ", + "bbox": [ + 173, + 816, + 825, + 890 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Proposition 13 (Moreau–Yosida regularization). Suppose $J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ is convex, and define ${ \\tilde { J } } : = J \\oplus R _ { 3 }$ . Then $\\tilde { J }$ is convex, and $\\begin{array} { r } { \\mathfrak { D } _ { \\widetilde { J } } ( \\nu , \\mu ) \\le \\frac { \\beta _ { 2 } } { 2 } \\vert \\vert \\mu - \\nu \\vert \\vert _ { \\mathrm { K R } } ^ { 2 } } \\end{array}$ . ", + "bbox": [ + 171, + 892, + 825, + 926 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "By Proposition 10, this transformed loss function satisfies (D3), having inherited the regularity properties of the squared MMD. This transformed function is a generalization of Moreau–Yosida regularization or the Moreau envelope (see Chapter 1 in Rockafeller & Wets (1998)). It is well-known that in the case of a function $f : \\mathbb { R } ^ { n } \\mathbb { R }$ , this regularization results in a function with Lipschitz gradients, so it is unsurprising that this property carries over to the infinite-dimensional case. ", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Applying (4) and (19), we see that the transformed loss function can be minimized as a GAN by implementing an RKHS squared norm penalty on the discriminator: ", + "bbox": [ + 171, + 180, + 821, + 208 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/8f42bad177c78a07078738f1e506df16de63825ff670a323ab5710499f8f01db.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { \\mu } { \\operatorname* { i n f } } \\big ( J \\oplus \\frac { \\beta _ { 2 } } { 4 \\pi } | | \\cdot | | _ { \\mathcal { H } } ^ { 2 } \\big ) ( \\mu ) = \\underset { \\mu } { \\operatorname* { i n f } } \\underset { \\varphi } { \\operatorname* { s u p } } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - J ^ { \\star } ( \\varphi ) - \\frac { \\pi } { \\beta _ { 2 } } | | \\varphi | | _ { \\mathcal { H } } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 274, + 215, + 723, + 244 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Computationally, the RKHS norm is difficult to evaluate. We propose taking advantage of the following infinite series representation of $| | f | | _ { \\mathcal { H } } ^ { 2 }$ in terms of the derivatives of $f$ (Fasshauer & Ye, 2011; Novak et al., 2018): ", + "bbox": [ + 174, + 257, + 825, + 300 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Proposition 14. Let $\\mathcal { H }$ be an RKHS with the Gaussian kernel $K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }$ . Then for $f \\in \\mathcal H$ , ", + "bbox": [ + 173, + 305, + 820, + 335 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/59037ab0f57840e6530aa46f9808ef8ff7fd922e2a85c37f1daf31a9c7319f40.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| f \\| _ { \\mathcal H } ^ { 2 } = \\displaystyle \\sum _ { k = 0 } ^ { \\infty } ( 4 \\pi ) ^ { - k } \\sum _ { \\substack { k _ { 1 } + \\cdots + k _ { d } = k } } \\frac { 1 } { \\prod _ { i = 1 } ^ { d } k _ { i } ! } \\| \\partial _ { x _ { 1 } } ^ { k _ { 1 } } \\cdot \\cdot \\cdot \\partial _ { x _ { d } } ^ { k _ { d } } f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } } \\\\ & { \\quad \\quad = \\| f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } + \\frac { 1 } { 4 \\pi } \\| \\nabla f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } + \\frac { 1 } { 1 6 \\pi ^ { 2 } } \\| \\nabla ^ { 2 } f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } + o t h e r t e r m s . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 342, + 746, + 409 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In an ideal world, we would use this expression as a penalty on the discriminator to enforce (D3). Of course, as an infinite series, this formulation is computationally impractical. However, the first two terms are very close to common GAN techniques like gradient penalties (Gulrajani et al., 2017) and penalizing the output of the discriminator (Karras et al., 2018). We therefore interpret these common practices as partially applying the penalty given by the RKHS norm squared, approximately enforcing (D3). We view the choice of only using the leading terms as a disadvantageous but practical necessity. ", + "bbox": [ + 173, + 420, + 825, + 518 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Interestingly, according to our analysis, gradient penalties and spectral normalization are not interchangeable, even though both techniques were designed to constrain the Lipschitz constant of the discriminator. Instead, our analysis suggests that they serve different purposes: gradient penalties enforce the variational smoothness (D3), while spectral normalization enforces Lipschitz continuity (D1). This demystifies the puzzling observation of Miyato (2018) that GANs using only spectral normalization with a WGAN loss do not seem to train well; it also explains why using both spectral normalization and a gradient penalty is a reasonable strategy. It also motivates the use of gradient penalties applied to losses other than the Wasserstein loss (Fedus et al., 2018). ", + "bbox": [ + 173, + 523, + 825, + 636 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 VERIFYING THE THEORETICAL LEARNING RATE ", + "text_level": 1, + "bbox": [ + 174, + 656, + 604, + 672 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this section, we empirically test the theoretical learning rate given by Theorem 1 and Proposition 1 as well as our regularization scheme (7) based on inf-convolutions. We approximately implement our composite regularization scheme (7) on a trivial base loss of $J ( \\mu ) = \\overset { \\triangledown } { \\chi } \\bar { \\{ \\mu = \\mu _ { 0 } \\} }$ by alternating stochastic gradient steps on ", + "bbox": [ + 174, + 688, + 825, + 744 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/6608ac516318ceaaef9bc1dfd61a9bb98a6b7ce6b9004b7250e3c937891c75d4.jpg", + "text": "$$\n\\operatorname* { i n f } _ { \\mu } \\operatorname* { s u p } _ { \\varphi } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - \\mathbb { E } _ { x \\sim \\mu _ { 0 } } [ \\varphi ( x ) ] - \\frac { \\pi } { \\beta _ { 2 } } \\mathbb { E } _ { x \\sim \\tilde { \\mu } } \\Big [ \\varphi ( x ) ^ { 2 } + \\frac { 1 } { 4 \\pi } | | \\nabla \\varphi ( x ) | | ^ { 2 } \\Big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 253, + 751, + 741, + 784 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "where $\\tilde { \\mu }$ is a random interpolate between samples from $\\mu$ and $\\mu _ { 0 }$ , as used in Gulrajani et al. (2017). The regularization term is a truncation of the series for the squared RKHS norm (22) and approximately enforces (D3). The discriminator is a 7-layer convolutional neural network with spectral normalization1 and ELU activations, an architecture that enforces (D1) and (D2). We include a final scalar multiplication by $\\alpha$ so that by Proposition 9, $\\beta _ { 1 } = 7 \\alpha$ . We take two discriminator steps for every generator step, to better approximate our assumption of an optimal discriminator. ", + "bbox": [ + 173, + 790, + 825, + 875 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/aa82e4dee4d350f0883d5b1945b97b4159438c424259c9b07cdff3a713b23542.jpg", + "image_caption": [ + "Figure 1: FID of simple particle generators with various learning rates. The horizontal axis shows the ratio $\\gamma / \\gamma _ { 0 }$ between the actual learning rate $\\gamma$ and the theoretical learning rate $\\gamma _ { 0 }$ suggested by Theorem 1 and Proposition 1. The vertical axis shows the FID after 100,000 SGD iterations. " + ], + "image_footnote": [], + "bbox": [ + 359, + 73, + 632, + 207 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "For the generator, we use an extremely simple particle-based generator which satisfies (G1) and (G2), in order to minimize the number of confounding factors in our experiment. Let $\\omega$ be the discrete uniform distribution on $Z = \\{ 1 , \\ldots , N \\}$ . For an $N \\times d$ matrix $\\theta$ and $z \\in Z$ , define $f _ { \\theta } : Z \\to \\mathbb { R } ^ { d }$ so that $f _ { \\theta } ( z )$ is the $z$ th row of $\\theta$ . The particle generator $\\mu _ { \\theta } = f _ { \\theta \\# } \\omega$ satisfies (G1) with $\\begin{array} { r } { A = \\frac { 1 } { \\sqrt { N } } } \\end{array}$ , since ", + "bbox": [ + 173, + 270, + 825, + 344 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/52f13f61aa10829968c9a63ae1239644677f04b95a7687f1bded26379b0e4dc1.jpg", + "text": "$$\n\\mathbb { E } _ { z } [ \\| f _ { \\theta } ( z ) - f _ { \\theta ^ { \\prime } } ( z ) \\| _ { 2 } ] = \\frac { 1 } { N } \\sum _ { z = 1 } ^ { n } \\| \\theta _ { z } - \\theta _ { z } ^ { \\prime } \\| _ { 2 } \\leq \\frac { 1 } { \\sqrt { N } } \\| \\theta - \\theta ^ { \\prime } \\| _ { \\mathrm { F } } ,\n$$", + "text_format": "latex", + "bbox": [ + 287, + 349, + 709, + 391 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "and it satisfies (G2) with $B = 0$ , since $D _ { \\theta } f _ { \\theta } ( z )$ is constant w.r.t. $\\theta$ . With this setup, Theorem 1 suggests a theoretical learning rate of ", + "bbox": [ + 176, + 393, + 821, + 422 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/fb75a3f293bf9879968664692240ce7767e6351916583ecca43b7d48297fbde0.jpg", + "text": "$$\n\\gamma _ { 0 } = \\frac { 1 } { L } = \\frac { 1 } { \\alpha B + A ^ { 2 } ( \\beta _ { 1 } + \\beta _ { 2 } ) } = \\frac { N } { 7 \\alpha + \\beta _ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 425, + 648, + 459 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We randomly generated hyperparameter settings for the Lipschitz constant $\\alpha$ , the smoothness constant $\\beta _ { 2 }$ , the number of particles $N$ , and the learning rate $\\gamma$ . We trained each model for 100,000 steps on CIFAR-10 and evaluate each model using the Frechet Inception Distance (FID) of Heusel ´ et al. (2017). We hypothesize that stability is correlated with image quality; Figure 1 plots the FID for each hyperparameter setting in terms of the ratio of the true learning rate $\\gamma$ and the theoretically motivated learning rate $\\gamma _ { 0 }$ . We find that the best FID scores are obtained in the region where $\\gamma / \\gamma _ { 0 }$ is between 1 and 1000. For small learning rates $\\gamma / \\gamma _ { 0 } \\ll 1$ , we observe that the convergence is too slow to make a reasonable progress on the objective, whereas as the learning rate gets larger $\\gamma / \\gamma _ { 0 } \\gg 1$ , we observe a steady increase in FID, signalling unstable behavior. It also makes sense that learning rates slightly above the optimal rate produce good results, since our theoretical learning rate is a conservative lower bound. Note that our intention is to test our theory, not to generate good images, which is difficult due to our weak choice of generator. Overall, this experiment shows that our theory and regularization scheme are sensible. ", + "bbox": [ + 173, + 468, + 825, + 648 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "8 FUTURE WORK ", + "text_level": 1, + "bbox": [ + 176, + 669, + 331, + 685 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Inexact gradient descent In this paper, we employed several assumptions in order to regard the GAN algorithm as gradient descent. However, real-world GAN algorithms must be treated as “inexact” descent algorithms. As such, future work includes: (i) relaxing the optimal discriminator assumption (cf. Sanjabi et al. (2018)) or providing a stability result for discrete simultaneous gradient descent (cf. continuous time analysis in Nagarajan & Kolter (2017); Mescheder et al. (2018)), (ii) addressing stochastic approximations of gradients (i.e., SGD), and (iii) providing error bounds for the truncated gradient penalty used in (23). ", + "bbox": [ + 173, + 699, + 825, + 797 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Generator architectures Another important direction of research is to seek more powerful generator architectures that satisfy our smoothness assumptions (G1) and (G2). In practice, generators are often implemented as deep neural networks, and involve some specific architectures such as deconvolution layers (Radford et al., 2015) and residual blocks (e.g., Gulrajani et al. (2017); Miyato et al. (2018)). In this paper, we did not provide results on the smoothness of general classes of generators, since our focus is to analyze stability properties influenced by the choice of loss function $J$ (and therefore optimal discriminators). However, our conditions (G1) and (G2) shed light on how to obtain smoothly parameterized neural networks, which is left for future work. 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Transactions of the American Mathematical Society, 36(1):63–89, 1934. ", + "bbox": [ + 174, + 459, + 823, + 488 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Xingyu Zhou. On the Fenchel duality between strong convexity and Lipschitz continuous gradient. arXiv preprint arXiv:1803.06573, 2018. ", + "bbox": [ + 174, + 497, + 823, + 526 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A INF-CONVOLUTION IN $\\mathbb { R } ^ { d }$ ", + "text_level": 1, + "bbox": [ + 176, + 551, + 421, + 569 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "To gain intuition on the inf-convolution, we present a finite-dimensional analogue of the techniques in Section 3. For simplicity of presentation, we will omit any regularity conditions (e.g., lower semicontinuity). We refer readers to Chapter 12 of Bauschke & Combettes (2011) for a detailed introduction. ", + "bbox": [ + 174, + 583, + 825, + 640 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Let $J$ and $R$ be convex functions on $\\mathbb { R } ^ { d }$ . The inf-convolution of $J$ and $R$ is a function $J \\star R$ defined as ", + "bbox": [ + 171, + 645, + 823, + 674 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/3d3ce40559c91980378de44bde7e9b81eed2676e55659d86974ae5137b4a21ea.jpg", + "text": "$$\n( J \\oplus R ) ( x ) : = \\operatorname* { i n f } _ { z \\in \\mathbb { R } ^ { d } } J ( z ) + R ( x - z ) .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 672, + 629, + 698 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The inf-convolution is often called the epigraphic sum since the epigraph of $J { \\star R }$ coincides with the Minkowski sum of epigraphs of $J$ and $R$ , as Figure 2 illustrates. The inf-convolution is associative and commutative operation; that is, it is always true that $\\left( J _ { 1 } \\oplus J _ { 2 } \\right) \\oplus J _ { 3 } = J _ { 1 } \\oplus ( J _ { 2 } \\oplus J _ { 3 } ) = :$ $J _ { 1 } \\oplus J _ { 2 } \\oplus J _ { 3 }$ and $J _ { 1 } \\oplus J _ { 2 } = J _ { 2 } \\oplus J _ { 1 }$ . ", + "bbox": [ + 173, + 700, + 825, + 757 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "There are two important special cases of inf-convolutions: The first one is the Pasch–Hausdorff envelope $J _ { \\alpha }$ , which is the inf-convolution between $J$ and $\\alpha \\| { \\cdot } \\| _ { 2 } ~ ( \\alpha > 0 )$ . It is known that $J _ { \\alpha }$ becomes $\\alpha$ -Lipschitz. The second important example is the Moreau envelope $\\begin{array} { r } { J ^ { \\beta } = J \\oplus \\frac { 1 } { 2 \\beta } \\lVert \\cdot \\rVert _ { 2 } ^ { \\overleftarrow { 2 } } } \\end{array}$ i.e., the inf-convolution with the quadratic regularizer $\\frac { 1 } { 2 \\beta } \\lVert \\cdot \\rVert _ { 2 } ^ { 2 }$ . The Moreau envelope $J ^ { \\beta }$ is always differentiable, and the gradient of $J ^ { \\beta }$ is $\\beta$ -Lipschitz (thus $J ^ { \\beta }$ is $\\beta .$ -smooth). ", + "bbox": [ + 173, + 762, + 825, + 843 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "It is worth noting that the set of minimizers does not change after these two operations. More generally, we have the following result: ", + "bbox": [ + 173, + 848, + 823, + 877 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proposition 15. Let $J , R : \\mathbb { R } ^ { d } \\bar { \\mathbb { R } }$ be proper and lower semicontinuous functions with $\\operatorname* { m i n } J >$ $- \\infty$ and $\\operatorname* { m i n } R = 0$ . Suppose $R ( 0 ) = 0$ and $R ( x ) \\geq \\psi ( \\| x \\| _ { 2 } )$ for some increasing function $\\psi : \\mathbb { R } _ { \\ge 0 } \\mathbb { R }$ . Then, $\\operatorname* { m i n } J \\oplus R = \\operatorname* { m i n } J$ and arg ${ \\mathrm { m i n } } J \\oplus { \\cal R } = \\arg \\operatorname* { m i n } J$ . ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/680218ae8fb4c9cdc0963df033e29a25d3a9ce21159c7615b779f72f7741abe0.jpg", + "image_caption": [ + "Figure 2: Illustration of inf-convolutions and their convex conjugates in $\\mathbb { R }$ . Top row: Generally, inf-convolutions inherit the regularity of their parent functions. In this example, $J _ { 1 } ( x ) = \\left| x \\right|$ is 1-Lipschitz but not smooth, while $J _ { 2 } ( x ) = { x } ^ { 2 } / \\dot { 2 }$ is 1-smooth but not Lipschitz. The inf-convolution $J _ { 1 } \\oplus J _ { 2 }$ is the well-known Huber function, which is both 1-Lipschitz and 1-smooth. Bottom row: Convex conjugates of the functions in the top row. The conjugate of $J _ { 1 } \\oplus J _ { 2 }$ is given as the sum of conjugates $J _ { 1 } ^ { \\star } ( z ) = \\chi \\{ | z | \\leq 1 \\}$ and $J _ { 2 } ^ { \\star } ( z ) \\stackrel { = } { = } z ^ { 2 } / 2$ . " + ], + "image_footnote": [], + "bbox": [ + 171, + 102, + 820, + 367 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "To sum up, given a function $J$ , we can always construct a regularized alternative $J _ { \\alpha } ^ { \\beta }$ that is $\\alpha$ - Lipschitz and $\\beta$ -smooth and has the same minimizers as $J$ . ", + "bbox": [ + 173, + 491, + 823, + 520 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The next question is how to implement the inf-convolution in GAN-like optimization problems. For this, it is convenient to consider the convex conjugate. Recall that the Fenchel–Moreau theorem says that there is a duality between a convex function $J$ and its convex conjugate $J ^ { \\star }$ as $J ( x ) =$ $\\begin{array} { r } { \\operatorname* { s u p } _ { z \\in \\mathbb { R } ^ { d } } \\langle x , z \\rangle - J ^ { \\star } ( z ) } \\end{array}$ and $\\begin{array} { r } { J ^ { \\star } ( z ) = \\operatorname* { s u p } _ { x \\in \\mathbb { R } ^ { d } } \\langle x , z \\rangle - J ( x ) } \\end{array}$ . The important property is that the convex conjugate of the inf-convolution is the sum of convex conjugates, that is, we always have ", + "bbox": [ + 173, + 526, + 825, + 597 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/3af6ad391e4ba7529eab54752c63d542b919b4d3e5e209f5c196bf3bb03f112a.jpg", + "text": "$$\n( J \\oplus R ) ^ { \\star } ( z ) = J ^ { \\star } ( z ) + R ^ { \\star } ( z ) .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 603, + 606, + 621 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This property can be useful for implementing the regularized objective $J _ { \\alpha } ^ { \\beta }$ as follows. First, we can check that the convex conjugates of the norm and the squared norm are given as $( \\| { \\cdot } \\| _ { 2 } ) ^ { \\star } = \\chi \\{ \\| { \\cdot } \\| \\leq$ $1 \\}$ and $\\begin{array} { r } { ( \\frac { 1 } { 2 } \\| { \\cdot } \\| _ { 2 } ^ { 2 } ) ^ { \\star } = \\frac { 1 } { 2 } \\| { \\cdot } \\| _ { 2 } ^ { 2 } } \\end{array}$ . Hence, we have ", + "bbox": [ + 174, + 627, + 825, + 671 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/3203b02b8dd5cb4b565b4b7db431b8f58c0ccd29b7f0654acee980ee0ba5d7f0.jpg", + "text": "$$\nJ _ { \\alpha } ^ { \\beta } ( \\boldsymbol { x } ) : = \\left( J \\oplus \\alpha \\lVert \\cdot \\rVert _ { 2 } \\oplus \\frac { 1 } { 2 \\beta } \\lVert \\cdot \\rVert _ { 2 } ^ { 2 } \\right) ( \\boldsymbol { x } ) = \\operatorname* { s u p } _ { \\boldsymbol { z } : \\ \\lVert \\boldsymbol { z } \\rVert _ { 2 } \\leq \\alpha } \\langle \\boldsymbol { x } , \\boldsymbol { z } \\rangle - J ^ { \\star } ( \\boldsymbol { z } ) - \\frac { \\beta } { 2 } \\lVert \\boldsymbol { z } \\rVert _ { 2 } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 245, + 678, + 751, + 714 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "which means that minimizing $J _ { \\alpha } ^ { \\beta }$ can be recast in min-max problem with the norm clipping and $\\ell _ { 2 }$ -regularization on the dual variable $z$ . ", + "bbox": [ + 173, + 723, + 825, + 752 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B COMMON GAN LOSSES ", + "text_level": 1, + "bbox": [ + 176, + 771, + 408, + 787 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For completeness and clarity, we explicitly write out the expressions for the losses listed in Table 1. For more detailed computations of optimal discriminators, see Chu et al. (2019); for more details on the convex duality interpretation, see Farnia & Tse (2018). ", + "bbox": [ + 174, + 803, + 823, + 847 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Minimax GAN Goodfellow et al. (2014) originally proposed the minimax GAN and showed that the corresponding loss function for the minimax GAN is the Jensen–Shannon divergence, defined as ", + "bbox": [ + 173, + 861, + 825, + 901 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/a0c34c7887bb0636bf4af4d19ba810794a8a6417233595f623ca6d0d37784883.jpg", + "text": "$$\nJ ( \\mu ) : = D _ { \\mathrm { J S } } ( \\mu | | \\mu _ { 0 } ) : = \\frac 1 2 D _ { \\mathrm { K L } } ( \\mu | | \\frac 1 2 \\mu + \\frac 1 2 \\mu _ { 0 } ) + \\frac 1 2 D _ { \\mathrm { K L } } ( \\mu _ { 0 } | | \\frac 1 2 \\mu + \\frac 1 2 \\mu _ { 0 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 248, + 898, + 748, + 929 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\mu _ { 0 } \\in { \\mathcal { P } } ( X )$ is a fixed probability measure (usually the empirical measure of the data), and $D _ { \\mathrm { K L } } ( \\mu \\lvert \\lvert \\nu \\rangle )$ is the Kullback–Leibler divergence between $\\mu$ and $\\nu$ . The optimal discriminator in the sense of Definition 1 is given as ", + "bbox": [ + 173, + 103, + 826, + 146 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/391c00d0c82cd0524efb7eaefe32c9c6ce29f52d1239a4ee623e9178b7db611e.jpg", + "text": "$$\n\\Phi _ { \\mu } ( x ) = \\frac { 1 } { 2 } \\log \\frac { d \\mu } { d ( \\mu + \\mu _ { 0 } ) } ( x ) ,\n$$", + "text_format": "latex", + "bbox": [ + 397, + 151, + 599, + 185 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $\\frac { d \\mu } { d ( \\mu + \\mu _ { 0 } ) }$ is the Radon–Nikodym derivative. If $\\mu$ and $\\mu _ { 0 }$ have densities $\\mu ( x )$ and $\\mu _ { 0 } ( x )$ , then ", + "bbox": [ + 174, + 193, + 818, + 215 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/2a5e955c4a26be2313305be76e5bc62c6756c4dfb140e2e0cd561f9f7aa34c60.jpg", + "text": "$$\n\\frac { d \\mu } { d ( \\mu + \\mu _ { 0 } ) } ( x ) = \\frac { \\mu ( x ) } { \\mu ( x ) + \\mu _ { 0 } ( x ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 390, + 217, + 607, + 252 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "so our optimal discriminator matches that of Goodfellow et al. (2014) up to a constant factor and logarithm. To recover the minimax formulation, the convex duality (4) yields: ", + "bbox": [ + 169, + 256, + 826, + 285 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/82088f721f2db240cc30973b14d6ee9f9358579e498c06d1ebbe4ef3db550b7a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { \\mu } { \\operatorname* { i n f } } D _ { \\mathrm { J S } } ( \\mu , \\mu _ { 0 } ) = \\underset { \\mu } { \\operatorname* { i n f } } \\operatorname* { s u p } _ { \\varphi } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - ( \\underset { - 1 } { - } \\frac { 1 } { 2 } \\mathbb { E } _ { x \\sim \\mu _ { 0 } } [ \\log ( 1 - e ^ { 2 \\varphi ( x ) + \\log 2 } ) ] - \\frac { 1 } { 2 } \\log 2 ) } \\\\ & { \\qquad ( D _ { \\mathrm { J S } } ( \\cdot , \\mu _ { 0 } ) ) ^ { \\star } ( \\varphi ) } \\\\ & { \\qquad = \\underset { \\mu } { \\operatorname* { i n f } } \\underset { D } { \\operatorname* { s u p } } \\frac { 1 } { 2 } \\mathbb { E } _ { x \\sim \\mu } [ \\log ( 1 - D ( x ) ) ] + \\frac { 1 } { 2 } \\mathbb { E } _ { x \\sim \\mu _ { 0 } } [ \\log D ( x ) ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 290, + 781, + 359 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "using the substitution $\\begin{array} { r } { \\varphi = \\frac { 1 } { 2 } \\log ( 1 - D ) - \\frac { 1 } { 2 } \\log 2 } \\end{array}$ . ", + "bbox": [ + 173, + 366, + 509, + 383 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Non-saturating GAN Goodfellow et al. (2014) also proposed the heuristic non-saturating GAN. Theorem 2.5 of Arjovsky & Bottou (2017) shows that the loss function minimized is ", + "bbox": [ + 173, + 396, + 823, + 426 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/8c1ccdabbdbf61614e6988f99b86baf8167999bb1e1782c100a1c43be5de023e.jpg", + "text": "$$\nJ ( \\mu ) : = D _ { \\mathrm { K L } } ( \\textstyle { \\frac { 1 } { 2 } } \\mu + \\textstyle { \\frac { 1 } { 2 } } \\mu _ { 0 } \\left| \\right| \\mu _ { 0 } ) = \\frac { 1 } { 2 } D _ { \\mathrm { K L } } ( \\mu \\left| \\right| \\mu _ { 0 } ) - D _ { \\mathrm { J S } } ( \\mu \\left| \\right| \\mu _ { 0 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 281, + 430, + 715, + 462 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The optimal discriminator is ", + "bbox": [ + 174, + 465, + 362, + 481 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/52a467e94009f670fa80a3749c276875b626ea4e9fdaee1d06309c3b2e1cf9de.jpg", + "text": "$$\n\\Phi _ { \\mu } ( x ) = - \\frac { 1 } { 2 } \\log \\frac { d \\mu _ { 0 } } { d ( \\mu + \\mu _ { 0 } ) } ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 486, + 607, + 520 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Wasserstein GAN Arjovsky et al. (2017) proposed the Wasserstein GAN, which minimizes the Wasserstein-1 distance between the input $\\mu$ and a fixed measure $\\mu _ { 0 }$ : ", + "bbox": [ + 173, + 531, + 821, + 560 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/970426816371fd5ae0d6d938b66ebfcc06531fc5376e45bbff92ef2bca6c72e0.jpg", + "text": "$$\nJ ( \\mu ) : = W _ { 1 } ( \\mu , \\mu _ { 0 } ) : = \\operatorname* { i n f } _ { \\pi } \\mathbb { E } _ { ( x , y ) \\sim \\pi } [ | | x - y | | ] ,\n$$", + "text_format": "latex", + "bbox": [ + 343, + 566, + 651, + 588 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "here the infimum is taken over all couplings $\\pi$ , probability distributions over $X \\times X$ who arginals are $\\mu$ and $\\mu _ { 0 }$ respectively. The optimal discriminator $\\Phi _ { \\mu }$ is called the Kantorovich pote al in the optimal transport literature (Villani, 2009). The convex duality (4) recover the Wasserste AN: ", + "bbox": [ + 187, + 593, + 803, + 647 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/5516f4401f74959561db17309ba4e68e84c2130ffb20c80f169608f8cb6ac1f5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { \\mu } { \\operatorname* { i n f } } W _ { 1 } ( \\mu , \\mu _ { 0 } ) = \\underset { \\mu } { \\operatorname* { i n f } } \\operatorname* { s u p } _ { \\varphi } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - \\underset { ( W _ { 1 } ( \\cdot , \\mu _ { 0 } ) ) ^ { \\star } ( \\varphi ) } { \\underbrace { ( \\mathbb { E } _ { x \\sim \\mu _ { 0 } } [ \\varphi ( x ) ] + \\chi \\{ | | \\varphi | | _ { \\mathrm { L i p } } \\leq 1 \\} } } ) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad ( W _ { 1 } ( \\cdot , \\mu _ { 0 } ) ) ^ { \\star } ( \\varphi ) } \\\\ & { \\underset { \\quad \\mu } { = } \\underset { | | \\varphi | | _ { \\mathrm { L i p } } \\leq 1 } { \\operatorname* { i n f } } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - \\mathbb { E } _ { x \\sim \\mu _ { 0 } } [ \\varphi ( x ) ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 645, + 743, + 714 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "an expression of Kantorovich–Rubinstein duality. The Lipschitz constraint on the discriminator is typically enforced by spectral normalization (Miyato et al., 2018), less frequently by weight clipping (Arjovsky et al., 2017), or heuristically by gradient penalties (Gulrajani et al., 2017) (although this work shows that gradient penalties may serve a different purpose altogether). ", + "bbox": [ + 173, + 715, + 826, + 773 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Maximum mean discrepancy Given a positive definite kernel $K : X \\times X \\to \\mathbb { R }$ , the maximum mean discrepancy (MMD, Gretton et al. (2012)) between $\\mu$ and $\\nu$ is defined by ", + "bbox": [ + 173, + 786, + 823, + 816 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/420be4b30b50817ff05d8c88d133ec16dae787bf4d0b77746380a86077e74d84.jpg", + "text": "$$\nJ ( \\mu ) : = \\frac { 1 } { 2 } \\mathrm { M M D } _ { K } ^ { 2 } ( \\mu , \\nu ) : = \\frac { 1 } { 2 } \\int K ( x , y ) \\left( \\mu - \\nu \\right) ( d x ) \\left( \\mu - \\nu \\right) ( d y ) .\n$$", + "text_format": "latex", + "bbox": [ + 271, + 820, + 727, + 854 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $( \\mathcal { H } , \\| \\cdot \\| _ { \\mathcal { H } } )$ is the reproducing kernel Hilbert space (RKHS) for $K$ . The generative momentmatching network (GMMN, Li et al. (2015)) and the Coulomb GAN (Unterthiner et al., 2018) use the squared MMD as the loss function. The optimal discriminator in this case is ", + "bbox": [ + 174, + 859, + 825, + 902 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/4ae0db597fb85278e39cc76bcbf033e8d165ac726f4c40579ac2cf2bdb579aaf.jpg", + "text": "$$\n\\Phi _ { \\mu } ( x ) = \\mathbb { E } _ { y \\sim \\mu } [ K ( x , y ) ] - \\mathbb { E } _ { y \\sim \\mu _ { 0 } } [ K ( x , y ) ] ,\n$$", + "text_format": "latex", + "bbox": [ + 349, + 907, + 645, + 926 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "which in constrast to other GANs, may be approximated by simple Monte Carlo, rather than an auxiliary optimization problem. ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Note that MMD-GANs (Li et al., 2017a; Arbel et al., 2018) minimize a modified version of the MMD, the Optimized MMD (Sriperumbudur et al., 2009; Arbel et al., 2018). These MMD-GANs are adversarial in a way that does not arise from convex duality, so our theory currently does not apply to these GANs. ", + "bbox": [ + 173, + 138, + 825, + 195 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Integral probability metrics An integral probability metric (Muller, 1997) is defined by ¨ ", + "bbox": [ + 168, + 209, + 769, + 224 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d9c66aa02e9ad93ee256bc6cf2a749fef1c18bec6b22b3f298e4c47fca0a62ba.jpg", + "text": "$$\nJ ( \\mu ) : = { \\mathrm { I P M } } _ { \\mathcal F } ( \\mu , \\mu _ { 0 } ) : = \\operatorname* { s u p } _ { f \\in \\mathcal F } \\int f d \\mu - \\int f d \\mu _ { 0 } ,\n$$", + "text_format": "latex", + "bbox": [ + 328, + 228, + 668, + 263 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\mathcal { F }$ is a class of functions. The optimal discriminator is the function that maximizes the supremum in the definition. The Wasserstein distance may be thought of as an IPM with $\\mathcal { F }$ containing all 1-Lipschitz functions. The MMD may be thought of as an IPM with $\\mathcal { F }$ all functions with RKHS norm at most 1, but no GANs based on MMD are actually trained this way, as it is difficult to constrain the discriminator to such functions. ", + "bbox": [ + 173, + 267, + 825, + 338 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C OPTIMAL DISCRIMINATORS ARE FUNCTIONAL DERIVATIVES ", + "text_level": 1, + "bbox": [ + 173, + 358, + 705, + 375 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Let $J : { \\mathcal { P } } ( X ) \\to { \\bar { \\mathbb { R } } }$ be a convex function. Recall the definition of the optimal discriminator (Definition 1): ", + "bbox": [ + 174, + 387, + 823, + 416 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/0425a38699615f046ccc2711c233b632644fa5cc145a06081b19306289009a72.jpg", + "text": "$$\n\\Phi _ { \\mu } \\in \\underset { \\varphi \\in \\mathcal { C } ( X ) } { \\arg \\operatorname* { m a x } } \\int \\varphi d \\mu - J ^ { \\star } ( \\varphi ) .\n$$", + "text_format": "latex", + "bbox": [ + 388, + 414, + 607, + 449 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "This definition can be understood as an infinite dimensional analogue of subgradients. In fact, in finite-dimensional convex analysis, $z$ is a subgradient of $f : \\mathbb { R } ^ { d } \\bar { \\bar { \\mathbb { R } } }$ if and only if it can be written as $z \\in \\arg \\operatorname* { m a x } _ { z ^ { \\prime } } \\langle z ^ { \\prime } , x \\rangle - f ^ { \\star } ( z ^ { \\prime } )$ . The calculus of subgradients shares many properties with the standard calculus of derivatives, such as chain rules (Rockafeller & Wets, 1998). This motivate us to investigate derivative-like features of optimal discriminators. ", + "bbox": [ + 173, + 450, + 825, + 522 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We introduce the functional derivative, also known as the von Mises influence function: ", + "bbox": [ + 174, + 527, + 746, + 542 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Definition 3 (Functional derivative). Let $J : { \\mathcal { P } } ( X ) \\to { \\bar { \\mathbb { R } } }$ be a function of probability measures. We say that a continuous function $\\Phi _ { \\mu } : X \\to \\mathbb { R }$ is a functional derivative of $J$ at $\\mu$ if ", + "bbox": [ + 174, + 545, + 828, + 575 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/cccb311ff9a9ed3028bbeda0d2ef2c291df3acbe789ec3f08ba949df41b1b02c.jpg", + "text": "$$\nJ ( \\mu + \\epsilon \\xi ) = J ( \\mu ) + \\epsilon \\int \\Phi _ { \\mu } d \\xi + O ( \\epsilon ^ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 362, + 580, + 635, + 613 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "holds for any $\\xi = \\nu - \\mu$ with $\\nu \\in { \\mathcal { P } } ( X )$ . ", + "bbox": [ + 176, + 616, + 444, + 632 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Under this definition, optimal discriminators are actually functional derivatives. ", + "bbox": [ + 176, + 642, + 694, + 657 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proposition 16 (Chu et al. (2019), Theorem 2). Let $J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ be a proper, lower semicontinuous, and convex function. If there exists a maximizer $\\Phi _ { \\mu }$ of $\\begin{array} { r } { \\varphi \\mapsto \\int \\varphi d \\mu - J ^ { \\star } ( \\varphi ) } \\end{array}$ , then $\\Phi _ { \\mu }$ is $a$ functional derivative of $J$ at $\\mu$ in the sense of Definition 3. ", + "bbox": [ + 173, + 660, + 825, + 703 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The following result relates the derivative of the loss function with the derivative of the optimal discriminator: ", + "bbox": [ + 173, + 713, + 823, + 742 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proposition 17 (Chu et al. (2019), Theorem 1). Let $J : { \\mathcal { P } } ( X ) \\to \\mathbb { R }$ be continuously differentiable, in the sense that the functional derivative $\\Phi _ { \\mu }$ exists and $( \\mu , \\nu ) \\mapsto \\mathbb { E } _ { x \\sim \\nu } [ \\Phi _ { \\mu } ( x ) ]$ is continuous. Let $\\theta \\mapsto \\mu _ { \\theta }$ be continuous in the sense that ${ \\frac { 1 } { \\| h \\| } } \\big ( \\mu _ { \\theta + h } - \\mu _ { \\theta } \\big )$ converges to a weak limit as $\\| h \\| 0$ . Then, we have ", + "bbox": [ + 173, + 744, + 825, + 805 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/6d29bb9bd38a668d1ef3f0182021573af1a64777fdf39d58b343cff3b30c4dce.jpg", + "text": "$$\n\\begin{array} { r } { \\nabla _ { \\theta } J ( \\mu _ { \\theta } ) = \\nabla _ { \\theta } \\mathbb { E } _ { x \\sim \\mu _ { \\theta } } [ \\hat { \\Phi } ( x ) ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 398, + 803, + 596, + 821 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\hat { \\Phi } = \\Phi _ { \\mu _ { \\theta } }$ is treated as a function $X \\to \\mathbb { R }$ that is not dependent on $\\theta$ . ", + "bbox": [ + 174, + 825, + 665, + 842 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We use this important computational tool in many of our proofs. For the case of the generator model $\\mu _ { \\theta } = f _ { \\theta \\# } \\omega$ , an important consequence of Proposition 17 is that ", + "bbox": [ + 173, + 851, + 823, + 881 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/b01369971264d010d0f1ca0c8a802a3880abf12852f36b9caebfa870780eb5cf.jpg", + "text": "$$\n\\nabla _ { \\theta } J ( \\mu _ { \\theta } ) = \\nabla _ { \\theta } \\mathbb { E } _ { z \\sim \\omega } [ \\hat { \\Phi } ( f _ { \\theta } ( z ) ) ] = \\mathbb { E } _ { z \\sim \\omega } [ \\nabla _ { x = f _ { \\theta } ( z ) } \\Phi _ { \\mu } ( f _ { \\theta } ( z ) ) \\cdot D _ { \\theta } f _ { \\theta } ( z ) ] .\n$$", + "text_format": "latex", + "bbox": [ + 254, + 885, + 743, + 906 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We use this fact in the proof of Theorem 1. ", + "bbox": [ + 176, + 909, + 455, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "D PROOFS FOR SECTIONS 1 AND 2 ", + "text_level": 1, + "bbox": [ + 176, + 102, + 477, + 118 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The following result is well known in the dynamical systems and the optimization literature. For the sake of completeness, we provide its proof. ", + "bbox": [ + 174, + 133, + 825, + 162 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proposition 1 (Bertsekas (1999), Proposition 1.2.3). Suppose $f : \\mathbb { R } ^ { p } \\mathbb { R }$ is $L$ -smooth and bounded below. Let $\\begin{array} { r } { x _ { k + 1 } : = x _ { k } - \\frac { 1 } { L } \\dot { \\nabla } \\dot { f } ( x _ { k } ) } \\end{array}$ . Then $| | \\nabla f ( x _ { k } ) | | 0$ as $k \\to \\infty$ . ", + "bbox": [ + 171, + 166, + 823, + 196 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. It is known that if $f$ is $L$ -smooth, then ", + "bbox": [ + 174, + 210, + 475, + 226 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/04b598abbb2b2e843ca8a02236e0fc063066bbf3bd08f3c1782bb06727cf55f6.jpg", + "text": "$$\nf ( y ) \\leq f ( x ) + \\langle \\nabla f ( x ) , y - x \\rangle + { \\frac { L } { 2 } } \\vert \\vert y - x \\vert \\vert ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 343, + 231, + 653, + 262 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "holds for any $x , y \\in \\mathbb { R } ^ { d }$ (see e.g. Lemma 3.4 in Bubeck (2015)). Then, we have ", + "bbox": [ + 173, + 270, + 696, + 286 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/3c1ab09ca79623922374e6055f12fdb81ba18cc42eef2d135115ffc660cf3e79.jpg", + "text": "$$\n\\begin{array} { l } { f ( x _ { k + 1 } ) \\leq f ( x _ { k } ) + \\langle \\nabla f ( x _ { k } ) , x _ { k + 1 } - x _ { k } \\rangle + \\displaystyle \\frac { L } { 2 } | | x _ { k + 1 } - x _ { k } | | ^ { 2 } } \\\\ { \\leq f ( x _ { k } ) - \\displaystyle \\frac { 1 } { L } | | \\nabla f ( x _ { k } ) | | ^ { 2 } + \\displaystyle \\frac { 1 } { 2 L } | | \\nabla f ( x _ { k } ) | | ^ { 2 } } \\\\ { = f ( x _ { k } ) - \\displaystyle \\frac { 1 } { 2 L } | | \\nabla f ( x _ { k } ) | | ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 289, + 291, + 707, + 385 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Summing this inequality over $k$ , we have ", + "bbox": [ + 174, + 388, + 444, + 404 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/9b7140047760529c336f3926badcbf8d81918dafbcc83b161ab0d6496e0b32b0.jpg", + "text": "$$\n{ \\frac { 1 } { 2 L } } \\sum _ { k = 0 } ^ { n - 1 } | | \\nabla f ( x _ { k } ) | | ^ { 2 } \\leq f ( x _ { 0 } ) - f ( x _ { n } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 367, + 410, + 629, + 454 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "from which we conclude that ", + "bbox": [ + 174, + 459, + 367, + 473 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/e741100bbf9ac9829320088ed9d021165bdd7d36bd8ade1ded38d179691e2a6c.jpg", + "text": "$$\n\\operatorname* { m i n } _ { 0 \\leq k \\leq n - 1 } | | \\nabla f ( x _ { k } ) | | ^ { 2 } \\leq { \\frac { 2 L } { n } } ( f ( x _ { 0 } ) - \\operatorname* { i n f } _ { x } f ( x ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 338, + 478, + 658, + 510 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Next, we move on to prove Theorem 1, which we restate here for readability. ", + "bbox": [ + 173, + 547, + 676, + 563 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Theorem 1 (Smoothness decomposition for GANs). Let $J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ be a convex function whose optimal discriminators $\\Phi _ { \\mu } : X \\to \\mathbb { R }$ satisfy the following regularity conditions: ", + "bbox": [ + 173, + 565, + 823, + 595 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "(D1) $x \\mapsto \\Phi _ { \\mu } ( x )$ is $\\alpha$ -Lipschitz, (D2) $x \\mapsto \\nabla _ { x } \\Phi _ { \\mu } ( x )$ is $\\beta _ { 1 }$ -Lipschitz, (D3) $\\mu \\mapsto \\nabla _ { x } \\Phi _ { \\mu } ( x )$ is $\\beta _ { 2 }$ -Lipschitz w.r.t. the 1-Wasserstein distance. ", + "bbox": [ + 192, + 606, + 410, + 622 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "", + "bbox": [ + 192, + 630, + 437, + 647 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "", + "bbox": [ + 191, + 654, + 648, + 670 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Also, let $\\mu _ { \\theta } = f _ { \\theta \\# } \\omega$ be a family of generators that satisfies: ", + "bbox": [ + 171, + 681, + 568, + 696 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "(G1) $\\theta \\mapsto f _ { \\theta } ( z )$ is $A$ -Lipschitz in expectation for $z \\sim \\omega$ , i.e., $\\begin{array} { r } { \\mathbb { E } _ { z \\sim \\omega } [ \\| f _ { \\theta _ { 1 } } ( z ) - f _ { \\theta _ { 2 } } ( z ) \\| _ { 2 } ] \\ \\le } \\end{array}$ $A \\| \\theta _ { 1 } - \\theta _ { 2 } \\| _ { 2 }$ , and ", + "bbox": [ + 186, + 707, + 825, + 738 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "(G2) $\\theta \\ \\mapsto \\ D _ { \\theta } f _ { \\theta } ( z )$ is $B$ -Lipschitz in expectation for $z \\sim \\omega$ , i.e., $\\mathbb { E } _ { z \\sim \\omega } [ \\| D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } ( z ) ~ -$ $D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ( z ) \\lVert _ { 2 } \\bigr \\rVert \\leq B \\lVert \\theta _ { 1 } - \\theta _ { 2 } \\rVert _ { 2 }$ . ", + "bbox": [ + 189, + 744, + 823, + 776 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Then $ { \\theta } \\mapsto J ( { \\mu } _ { { \\theta } } )$ is $L$ -smooth, with $L = \\alpha B + A ^ { 2 } ( \\beta _ { 1 } + \\beta _ { 2 } )$ . ", + "bbox": [ + 173, + 785, + 575, + 801 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Intuitively, Theorem 1 can be understood as the chain rule. For simplicity, let us consider the smoothness of a composite function $D \\circ G$ , where $D$ and $G$ are functions on $\\mathbb { R }$ . A sufficient condition for $D \\circ G$ to be smooth is that its second derivative is bounded. For this, suppose that (i) $D$ is $\\alpha$ -Lipschitz and $\\beta$ -smooth and (ii) $G$ is $A$ -Lipschitz and $B$ -smooth. By the chain rule, the second derivative is bounded as $( D \\circ G ) ^ { \\prime \\prime } = ( D ^ { \\prime } \\circ G { \\bar { G } } ) G ^ { \\prime \\prime } + ( D ^ { \\prime \\prime } \\circ G ) ( G ^ { \\prime } ) ^ { 2 } \\leq \\alpha B + A ^ { 2 } \\beta$ , which has the same form as the consequence of Theorem 1. For GANs, we need somewhat more involved conditions; We need two types of smoothness (D2) and (D3) for the optimal discriminators. To this end, we utilize the functional gradient point of view that we explained in Appendix $\\textrm { C }$ . ", + "bbox": [ + 173, + 810, + 825, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof of Theorem 1. First, using the functional gradient interpretation of the optimal discriminator, we have ", + "bbox": [ + 169, + 103, + 825, + 133 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/5d6846f5afd0a889d10a8f63b8ad24f5864426c0578a372be1e1158bf5a12cac.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\| \\nabla _ { \\theta _ { 1 } } J ( \\mu _ { \\theta _ { 1 } } ) - \\nabla _ { \\theta _ { 2 } } J ( \\mu _ { \\theta _ { 2 } } ) \\| } \\\\ & { = \\big \\| \\mathbb { E } _ { z \\sim \\omega } \\big [ \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\cdot D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 2 } } ( z ) ) \\cdot D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } \\big ] \\big \\| \\qquad \\mathrm { P r o p o s i t i o } } \\\\ & { = \\big \\| \\mathbb { E } _ { z \\sim \\omega } \\big [ \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\cdot ( D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } - D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ) } \\\\ & { \\qquad + \\big ( \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\big ) \\cdot D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } } \\\\ & { \\qquad + \\big ( \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 1 } } ( z ) ) - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 2 } } ( z ) ) \\big ) \\cdot D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } \\big ] \\big \\| } \\\\ & { \\leq \\big \\| \\mathbb { E } _ { z \\sim \\omega } \\big [ \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\cdot ( D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } - D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ) \\big ] \\big \\| } \\\\ & \\quad + \\big \\| \\mathbb { E } _ { z \\sim \\omega } \\big [ \\big ( \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\big ) \\cdot D _ { \\theta _ { 2 } } f _ \\theta _ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 210, + 138, + 753, + 306 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "By assumption, there are bounded non-negative numbers $\\alpha , \\beta _ { 1 } , \\beta _ { 2 } , A$ , and $B$ such that ", + "bbox": [ + 173, + 318, + 743, + 334 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/66130a1f4d58a3cf7848ba6b5cf00a669dc6ec7d81152c5144ec05b1abaff6f9.jpg", + "text": "$$\n\\begin{array} { r l } { ( \\mathrm { D } 1 ^ { \\prime } ) \\quad } & { \\alpha = \\underset { \\mu \\in \\mathcal { P } ( X ) } { \\operatorname* { s u p } } \\ \\underset { x \\sim \\mu } { \\operatorname* { s u p } } \\left. \\Phi _ { \\mu } ( x ) \\right. _ { 2 } , } \\\\ { ( \\mathrm { D } 2 ^ { \\prime } ) \\quad } & { \\underset { \\mu \\in \\mathcal { P } ( X ) } { \\operatorname* { s u p } } \\ \\Vert \\nabla _ { x _ { 1 } } \\Phi _ { \\mu } ( x _ { 1 } ) - \\nabla _ { x _ { 2 } } \\Phi _ { \\mu } ( x _ { 2 } ) \\Vert _ { 2 } \\leq \\beta _ { 1 } \\Vert x _ { 1 } - x _ { 2 } \\Vert _ { 2 } } \\\\ { ( \\mathrm { D } 3 ^ { \\prime } ) \\quad } & { \\underset { x \\in X } { \\operatorname* { s u p } } \\ \\Vert \\nabla _ { x } \\Phi _ { \\mu _ { 1 } } ( x ) - \\nabla _ { x } \\Phi _ { \\mu _ { 2 } } ( x ) \\Vert _ { 2 } \\leq \\beta _ { 2 } W _ { 1 } ( \\mu _ { 1 } , \\mu _ { 2 } ) , } \\\\ { ( \\mathrm { G } 1 ^ { \\prime } ) \\quad } & { A = \\mathbb { E } _ { z \\sim \\omega } \\underset { \\theta } { \\operatorname* { s u p } } \\left. D _ { \\theta } f _ { \\theta } ( z ) \\right. _ { \\mathrm { o p } } , \\quad \\mathrm { a n d } } \\\\ { ( \\mathrm { G } 2 ^ { \\prime } ) \\quad } & { \\mathbb { E } _ { z \\sim \\omega } \\vert \\vert D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } ( z ) - D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ( z ) \\vert \\vert _ { \\mathrm { o p } } \\leq B \\vert \\vert \\theta _ { 1 } - \\theta _ { 2 } \\vert \\vert . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 276, + 340, + 720, + 474 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Here, we wrote $\\operatorname { s u p } _ { x \\sim \\mu } f ( x )$ for the essential supremum of $f$ w.r.t. $\\mu$ . From $( \\mathrm Ḋ 1 ^ Ḋ \\prime Ḍ )$ and $( \\mathbf { G } 2 ^ { \\prime } )$ , the first term (a) is bounded as ", + "bbox": [ + 173, + 479, + 823, + 508 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/bbe47833a8db57f817ed0b6b7d66e979b6f1a95c39d2f673185c2ece87413d7c.jpg", + "text": "$$\n\\begin{array} { r } { \\left\\| \\mathbb { E } _ { z \\sim \\omega } [ \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\cdot ( D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } - D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ) ] \\right\\| \\leq \\alpha B \\| \\theta _ { 1 } - \\theta _ { 2 } \\| . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 515, + 712, + 535 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "From $( \\mathrm Ḋ 3 Ḍ ^ { \\prime } )$ ), $( \\mathbf { G } 1 ^ { \\prime } )$ , and the Cauchy-Schwarz inequality, the second term (b) is bounded as ", + "bbox": [ + 169, + 542, + 761, + 558 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/2a3a7df7b79a5b6df7ffe7a94bfb56bcbcadf0aea819e2af4b78baa7f69649d3.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\quad \\Big | \\Big | \\mathbb { E } _ { z \\sim \\omega } [ \\big ( \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\big ) \\cdot D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ] \\Big | \\Big | } \\\\ & { \\le A \\beta _ { 2 } W _ { 1 } ( \\mu _ { \\theta _ { 1 } } , \\mu _ { \\theta _ { 2 } } ) } \\\\ & { \\le A \\beta _ { 2 } \\mathbb { E } _ { z \\sim \\omega } \\| f _ { \\theta _ { 1 } } ( z ) - f _ { \\theta _ { 2 } } ( z ) \\| \\le A ^ { 2 } \\beta _ { 2 } \\| \\theta _ { 1 } - \\theta _ { 2 } \\| , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 563, + 691, + 626 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where the second inequality holds from the following optimal transport interpretation of $W _ { 1 }$ : ", + "bbox": [ + 171, + 630, + 781, + 646 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/fbd5fb9b74ee6e9b228026f10ab48ddac4080617e48275723aecd6ca8b724e39.jpg", + "text": "$$\nW _ { 1 } ( \\mu _ { \\theta _ { 1 } } , \\mu _ { \\theta _ { 2 } } ) = \\operatorname* { i n f } _ { \\gamma \\in { \\mathcal { P } } ( X \\times X ) ; \\atop \\operatorname { c o u p l i n g ~ o f } \\mu _ { \\theta _ { 1 } } \\operatorname { a n d } \\mu _ { \\theta _ { 2 } } } \\int \\| x - y \\| d \\gamma \\leq \\mathbb { E } _ { z \\sim \\omega } \\| f _ { \\theta _ { 1 } } ( z ) - f _ { \\theta _ { 2 } } ( z ) \\| .\n$$", + "text_format": "latex", + "bbox": [ + 243, + 652, + 754, + 695 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lastly, from $( \\mathbf Ḋ 2 Ḍ ^ { \\prime } )$ ), $( \\mathbf { G } 1 ^ { \\prime } )$ ) and the Cauchy-Schwarz inequality, the term (c) is bounded as ", + "bbox": [ + 168, + 704, + 758, + 719 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/33cbe646b307100018670861f91ce74903734573e13882d8f4b8c22ed388028f.jpg", + "text": "$$\n\\begin{array} { r } { \\big | \\mathbb { E } _ { z \\sim \\omega } \\big [ \\big ( \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 1 } } ( z ) ) - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 2 } } ( z ) ) \\big ) \\cdot D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } \\big ] \\big | \\leq A \\beta _ { 1 } \\mathbb { E } _ { z \\sim \\omega } \\big \\| f _ { \\theta _ { 1 } } ( z ) - f _ { \\theta _ { 2 } } ( z ) \\big \\| \\leq A ^ { 2 } \\beta _ { 1 } \\big \\| \\theta _ { 1 } - \\theta _ { 2 } \\big \\| \\theta _ { 2 } \\big \\| , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 726, + 839, + 746 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Combining the above upper bounds for (a)–(c), we conclude that ", + "bbox": [ + 173, + 752, + 606, + 767 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/e88746fac3fb4d23d6aa5a3ac86e0739bf5ca5149c4d9c6aace25be7e4d92e41.jpg", + "text": "$$\n| | \\nabla _ { \\theta _ { 1 } } J ( \\mu _ { \\theta _ { 1 } } ) - \\nabla _ { \\theta _ { 2 } } J ( \\mu _ { \\theta _ { 2 } } ) | | \\leq ( \\alpha B + A ^ { 2 } ( \\beta _ { 1 } + \\beta _ { 2 } ) ) \\| \\theta _ { 1 } - \\theta _ { 2 } \\| .\n$$", + "text_format": "latex", + "bbox": [ + 287, + 772, + 710, + 792 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "E PROOFS FOR SECTION 3 ", + "text_level": 1, + "bbox": [ + 174, + 835, + 408, + 852 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In the finite-dimensional case, Proposition 15 says that taking inf-convolution with a “coercive” regularizer does not change the set of minimizers of the original objective. A similar invariance holds for GAN objectives, which are defined on infinite-dimensional space of signed measures, for the regularizers $R _ { 1 }$ , $R _ { 2 }$ and $R _ { 3 }$ introduced in Sections 4 to 6: ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Proposition 2 (Invariance of minimizers). Let $R _ { 1 } ( \\xi ) : = \\| \\xi \\| _ { \\mathrm { K R } } , R _ { 2 } ( \\xi ) : = \\| \\xi \\| _ { S ^ { * } }$ , and $R _ { 3 } ( \\xi ) : = $ $\\frac { 1 } { 2 } \\lVert \\hat { \\xi } \\rVert _ { \\mathcal { H } } ^ { 2 }$ be the three regularizers defined by (8), (12), and (19) respectively. Assume that $ { J } :$ $\\bar { \\mathcal { M } } ( \\bar { X } ) \\bar { \\mathbb { R } }$ has a unique minimizer at $\\mu _ { 0 }$ with $J ( \\mu _ { 0 } ) = 0$ , and $J ( \\mu ) \\geq c \\| \\hat { \\mu } - \\hat { \\mu } _ { 0 } \\| _ { \\mathcal { H } }$ for some $c > 0$ . Then the inf-convolution $\\tilde { J } = J \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus R _ { 3 }$ has a unique minimizer at $\\mu _ { 0 }$ with $\\tilde { J } ( \\mu _ { 0 } ) = 0$ . ", + "bbox": [ + 173, + 102, + 825, + 183 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof. In the following, we endow ${ \\mathcal { M } } ( X )$ with the Gaussian RKHS norm $\\| \\widehat { \\cdot } \\| _ { \\mathcal { H } }$ , and for convenience $\\| \\cdot \\|$ will refer to this norm if not otherwise specified. To show the result, we apply Lemma 1 three times, first on $R _ { 3 }$ and $R _ { 1 }$ , and then again on $R _ { 3 } \\oplus R _ { 1 }$ and $R _ { 2 }$ , and then finally on $R _ { 3 } \\oplus R _ { 1 } \\oplus R _ { 2 }$ and $J ( \\cdot + \\mu _ { 0 } )$ . The result follows from noting that $\\tilde { J } ( \\mu ) = [ R _ { 3 } \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus J ( \\cdot + \\mu _ { 0 } ) ] ( \\mu - \\mu _ { 0 } )$ . ", + "bbox": [ + 173, + 203, + 825, + 263 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In order to apply the lemma, it suffices to show that $R _ { 3 }$ is uniformly continuous on bounded sets and coercive, and that there exist constants $c _ { 1 }$ and $c _ { 2 }$ such that $R _ { 1 } ( \\xi ) \\overset { \\cdot } { \\geq } c _ { 1 } \\| \\xi \\|$ and $R _ { 2 } ( \\xi ) \\geq c _ { 2 } \\| \\xi \\| .$ R3 is uniformly continuous on bounded sets: suppose $\\| \\xi \\| < C$ and $\\| \\xi ^ { \\prime } \\| < C$ ; then $| R _ { 3 } ( \\xi ) - R _ { 3 } ( \\xi ^ { \\prime } ) | =$ $\\begin{array} { r } { | \\frac { 1 } { 2 } \\| \\xi \\| ^ { 2 } - \\frac { 1 } { 2 } \\| \\xi ^ { \\prime } \\| ^ { 2 } | = | \\frac { 1 } { 2 } \\langle \\xi - \\xi ^ { \\prime } , \\xi + \\xi ^ { \\prime } \\rangle | \\leq C \\| \\dot { \\xi } - \\xi ^ { \\prime } \\| } \\end{array}$ by the Cauchy–Schwarz and triangle inequality. $R _ { 3 }$ is also coercive, as $R _ { 3 } ( \\xi ) = \\textstyle { \\frac { 1 } { 2 } } \\| \\xi \\| ^ { 2 } \\to \\infty$ when $\\| \\xi \\| \\infty$ . $R _ { 1 }$ satisfies $R _ { 1 } ( \\xi ) \\geq c _ { 1 } \\| \\xi \\|$ by Lemma 7. $R _ { 2 }$ satisfies $R _ { 2 } ( \\xi ) \\geq c _ { 2 } ] | \\xi | |$ by Lemma 5. □ ", + "bbox": [ + 173, + 268, + 825, + 357 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Recall that a function $F$ is said to be coercive if $F ( \\xi ) \\to \\infty$ when $\\| \\xi \\| \\infty$ ", + "bbox": [ + 173, + 376, + 678, + 392 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Lemma 1. Suppose $F : \\mathcal { M } ( X ) \\mathbb { R }$ has a unique minimizer at 0 with $F ( 0 ) = 0$ , and is uniformly continuous on bounded sets and coercive. Suppose $G : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ has a unique minimizer at 0 with $G ( 0 ) = 0$ , and $G ( \\cdot ) \\geq c \\| \\cdot \\|$ for some $c > 0$ . Then the inf-convolution $F \\oplus G$ has a unique minimizer at 0 with $( F \\oplus G ) ( 0 ) = 0$ , and is uniformly continuous on bounded sets, coercive, and real-valued. ", + "bbox": [ + 173, + 396, + 826, + 467 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof. From Theorem 2.3 of Stromberg (1996), ¨ inf $F \\oplus G = 0$ , and $0 \\in \\arg \\operatorname* { m i n } F \\oplus G$ . To show that 0 is the unique minimizer, let $\\xi \\in \\arg \\operatorname* { m i n } F \\oplus G$ . From the definition of inf-convolution, for every $n$ there exists an $\\xi _ { n } \\in \\mathcal { M } ( X )$ that satisfies ", + "bbox": [ + 173, + 488, + 825, + 531 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/d08101671fd19c6515046e9f0c6b929b8cd77fe866b9578aac73b12c77b9c6ba.jpg", + "text": "$$\n\\operatorname* { l i m } _ { n \\to \\infty } F ( \\xi _ { n } ) + G ( \\xi - \\xi _ { n } ) = ( F \\oplus G ) ( \\xi ) = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 540, + 651, + 564 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Since $F$ and $G$ are non-negative, $F ( \\xi _ { n } ) \\to 0$ and $G ( \\xi - \\xi _ { n } ) \\to 0$ . By our assumption that $G ( \\cdot ) \\geq$ $c \\| \\cdot \\|$ , the latter limit implies that $\\| \\xi - \\xi _ { n } \\| \\to 0$ , which implies that $F ( \\xi _ { n } ) \\to F ( \\xi )$ by continuity of $F$ . Comparing our two expressions for the limit of $F ( \\xi _ { n } )$ , we find that $F ( \\xi ) = 0$ , which implies that $\\xi = 0$ , since $F$ has a unique minimizer at 0. Hence ar $\\operatorname { m i n } F \\oplus G = \\{ 0 \\}$ . ", + "bbox": [ + 173, + 570, + 825, + 628 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "From Theorem 2.10 of Stromberg (1996), ¨ $F \\oplus G$ is uniformly continuous on bounded sets and real-valued. To show that $F \\oplus G$ is coercive, we show the equivalent condition its sublevel sets are bounded (see Proposition 11.11 of Bauschke & Combettes (2011)). That is, we show that for all $a > 0$ , there exists a constant $b$ such that if $( F \\oplus G ) ( \\xi ) \\leq a$ , then $\\| \\xi \\| \\le b$ . Let $a > 0$ , and let $\\xi \\in \\mathcal { M } ( X )$ be such that $( F \\oplus G ) ( \\xi ) \\leq a$ . Let $\\epsilon > 0$ ; from the definition of inf-convolution, there exists a $\\tilde { \\xi } \\in \\mathcal { M } ( X )$ such that ", + "bbox": [ + 173, + 633, + 825, + 722 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/424df517264799532d9f23baf1066f8faee11da85ead27b0973e1291f67898c3.jpg", + "text": "$$\nF ( { \\tilde { \\xi } } ) + G ( \\xi - { \\tilde { \\xi } } ) \\leq ( F \\oplus G ) ( \\xi ) + \\epsilon \\leq a + \\epsilon .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 729, + 651, + 750 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Because $F$ and $G$ are non-negative, we have that ", + "bbox": [ + 173, + 757, + 496, + 772 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/61a09dc600cf8db839d84f21488fc6883c7d3d7017f118bac61014000c1aba92.jpg", + "text": "$$\n\\begin{array} { r } { F ( \\tilde { \\xi } ) \\le a + \\epsilon , \\qquad G ( \\xi - \\tilde { \\xi } ) \\le a + \\epsilon . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 370, + 781, + 627, + 801 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Using the fact that $F$ is coercive and hence its sublevel sets are bounded, let $b ^ { \\prime }$ be such that if $F ( \\nu ) \\leq a + \\epsilon$ , then $\\| \\nu \\| \\leq b ^ { \\prime }$ . By the triangle inequality and our assumption that $G ( \\cdot ) \\geq c \\| \\cdot \\|$ , ", + "bbox": [ + 173, + 809, + 826, + 838 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/3352b7cd950b0647c77933f123e0effa951bd2e89f2990f8aaee690faa509c2e.jpg", + "text": "$$\n\\| \\xi \\| \\leq \\| \\tilde { \\xi } \\| + \\| \\xi - \\tilde { \\xi } \\| \\leq b ^ { \\prime } + \\frac { G ( \\xi - \\tilde { \\xi } ) } { c } \\leq b ^ { \\prime } + \\frac { a + \\epsilon } { c } .\n$$", + "text_format": "latex", + "bbox": [ + 318, + 847, + 679, + 881 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Because this constant is independent of $\\xi$ , this shows that $F \\oplus G$ is coercive. ", + "bbox": [ + 173, + 887, + 676, + 904 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "F PROOFS FOR SECTION 4 ", + "text_level": 1, + "bbox": [ + 176, + 102, + 408, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proposition 3. $( D l )$ holds if and only if $J$ is $\\alpha$ -Lipschitz w.r.t. the Wasserstein-1 distance. ", + "bbox": [ + 166, + 133, + 759, + 150 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. First, we check the forward implication: ", + "bbox": [ + 173, + 165, + 486, + 181 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/4ba48a0a9fa79d380ee7d7e5ef932b293a8c4d957627cf058da93ced8d0db7a9.jpg", + "text": "$$\n| J ( \\mu ) - J ( \\nu ) | \\leq \\alpha W _ { 1 } ( \\mu , \\nu ) \\quad \\Rightarrow \\quad \\operatorname* { s u p } _ { x \\sim \\mu } \\| \\nabla _ { x } \\Phi _ { \\mu } ( x ) \\| _ { 2 } \\leq \\alpha .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 188, + 699, + 215 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "From the duality between $L ^ { \\infty }$ - and $L ^ { 1 }$ -norms, we have ", + "bbox": [ + 173, + 222, + 537, + 238 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/bce10698f56c6d7b8d518b1aff51ab742c20e8422e5d0db052c0ca6b1ed60a78.jpg", + "text": "$$\n\\operatorname* { s u p } _ { x \\sim \\mu } \\| \\nabla _ { x } \\Phi _ { \\mu } ( x ) \\| _ { 2 } = \\| \\nabla _ { x } \\Phi _ { \\mu } ( \\cdot ) \\| _ { L ^ { \\infty } ( \\mu ; \\mathbb { R } ^ { d } ) } = \\operatorname* { s u p } _ { \\| v \\| _ { L ^ { 1 } ( \\mu ; \\mathbb { R } ^ { d } ) } = 1 } \\int \\nabla \\Phi _ { \\mu } ( x ) \\cdot v ( x ) d \\mu ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 205, + 244, + 763, + 284 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Below, we will abbreviate $\\left\\| \\cdot \\right\\| _ { L ^ { 1 } \\left( \\mu ; \\mathbb { R } ^ { d } \\right) }$ as $\\| \\cdot \\| _ { L ^ { 1 } }$ . We utilize the fact that the optimal discriminator $\\Phi _ { \\mu }$ is the functional gradient of $J$ in the sense of Definition 3. For any $t \\geq 0$ , let $\\mu _ { t } = ( \\operatorname { i d } + t v ) _ { \\# } \\mu$ be the law of $x + t v ( x )$ with $x \\sim \\mu$ . Then, $\\mu _ { t }$ converges weakly to $\\mu ~ = ~ \\mu _ { 0 }$ as $t 0$ since $W _ { 1 } ( \\mu _ { t } , \\mu ) \\leq t \\| v \\| _ { L ^ { 1 } ( \\mu ) } \\to 0$ . Applying Proposition 17 to $J$ and $\\mu _ { t }$ , we have ", + "bbox": [ + 173, + 291, + 825, + 352 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/df31be8534dbf0242987d9195988d6899635d56b6fd2b5cf52cb3ffa52af3d23.jpg", + "text": "$$\n\\left. \\frac { d } { d t } J ( \\mu _ { t } ) \\right| _ { t = 0 } = \\left. \\frac { d } { d t } \\int \\Phi _ { \\mu } ( x + t v ( x ) ) d \\mu ( x ) \\right| _ { t = 0 } = \\int \\nabla \\Phi _ { \\mu } ( x ) \\cdot v ( x ) d \\mu .\n$$", + "text_format": "latex", + "bbox": [ + 256, + 358, + 743, + 395 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In particular, the right-hand side of (26) is bounded from above as ", + "bbox": [ + 173, + 400, + 606, + 415 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/33dc821b69964865de637171b247649c9be73578bebce2c52983f90c6b5f3fef.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } , \\omega _ { \\xi } = 1 } { \\operatorname* { s u p } } \\int \\nabla \\Phi _ { p } ( x ) \\cdot v ( x ) d \\mu ( x ) } \\\\ & { \\quad = \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | \\neq 0 } { \\operatorname* { s u p } } \\frac { d } { d t } J ( \\mu _ { t } ) \\biggr | _ { t = 0 } } \\\\ & { \\quad = \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | , \\xi _ { 0 } = 1 } { \\operatorname* { s u p } } \\frac { 1 } { \\mathrm { i n s p } } \\frac { J ( \\mu _ { t } ) - J ( \\mu ) } { t } } \\\\ & { \\quad \\le \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | , \\xi _ { 0 } = 1 } { \\operatorname* { s u p } } \\frac { \\mathrm { i n s K } ( \\mu _ { t } , \\mu ) } { t } } \\\\ & { \\quad \\le \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | , \\xi _ { 0 } = 1 } { \\operatorname* { s u p } } \\frac { \\mathrm { i n s K } ( \\mu _ { t } , \\mu ) } { t } } \\\\ & { \\quad \\le \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | \\neq 0 } { \\operatorname* { s u p } } \\frac { \\mathrm { i n s K } ( \\mu _ { t } , \\mu _ { t } ) } { t } } \\\\ & { \\quad = \\alpha , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 374, + 422, + 624, + 640 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "which proves the first half of the statement. ", + "bbox": [ + 176, + 646, + 457, + 661 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For the converse, we borrow some techniques from the optimal transport theory. Let $p > 1$ , and let $W _ { p } ( \\mu , \\nu )$ denote the Wasserstein- $p$ distance between $\\mu$ and $\\nu$ . Suppose that $\\mu _ { t } : [ 0 , 1 ] \\to { \\mathcal { P } } ( X )$ is an ${ \\mathcal { P } } _ { p } ( X )$ -absolutely continuous curve, that is, every $\\mu _ { t }$ has a finite $p$ -moment and there exists $v \\in L ^ { 1 } ( [ 0 , 1 ] )$ such that $\\begin{array} { r } { W _ { p } ( \\mu _ { s } , \\mu _ { t } ) \\le \\int _ { s } ^ { t } v ( r ) d r } \\end{array}$ holds for any $0 \\leq s \\leq t \\leq 1$ . Then, the limit $\\begin{array} { r } { | \\mu ^ { \\prime } | _ { W _ { p } } ( t ) : = \\operatorname* { l i m } _ { h \\to 0 } | h | ^ { - 1 } W _ { p } ( \\mu _ { t + h } , \\mu _ { t } ) } \\end{array}$ exists for almost all $t \\in [ 0 , 1 ]$ (see Ambrosio et al. (2008), Theorem 1.1.2). Such $| \\mu ^ { \\prime } | _ { W _ { p } }$ is called the metric derivative of $\\mu _ { t }$ . ", + "bbox": [ + 173, + 666, + 826, + 758 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Lemma 2 (Ambrosio et al. (2008), Theorem 8.3.1). Let $\\mu _ { t } : [ 0 , 1 ] \\to { \\mathcal { P } } ( X )$ be an ${ \\mathcal { P } } _ { p } ( X )$ -absolutely continuous curve, and let $| \\mu ^ { \\prime } | _ { W _ { p } } \\in L ^ { 1 } ( [ 0 , 1 ] )$ be its metric derivative. Then, there exists $a$ vector field $v : ( x , t ) \\mapsto v _ { t } ( x ) \\in \\mathbb { R } ^ { d }$ such that ", + "bbox": [ + 173, + 765, + 825, + 813 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/e2be5c1f70caa38cb7c0e778a15d95d4973b7a6999d02d08cf815d469bd29735.jpg", + "text": "$$\nv _ { t } \\in L ^ { p } ( \\mu _ { t } ; X ) , \\quad \\| v _ { t } \\| \\leq | \\mu ^ { \\prime } | _ { W _ { p } } ( t )\n$$", + "text_format": "latex", + "bbox": [ + 375, + 819, + 622, + 838 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "for almost all $t \\in [ 0 , 1 ]$ , and ", + "bbox": [ + 171, + 843, + 362, + 859 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/4aaafaf271d067b4620942a845728017be0a0a7635312b68d15ba357edf0dece.jpg", + "text": "$$\n\\int _ { 0 } ^ { 1 } \\frac { d } { d t } \\varphi ( x , t ) \\mu _ { t } ( d x ) d t = - \\int _ { 0 } ^ { 1 } \\langle v _ { t } ( x ) , \\nabla _ { x } \\varphi ( x , t ) \\rangle \\mu _ { t } ( d x ) d t\n$$", + "text_format": "latex", + "bbox": [ + 299, + 867, + 697, + 901 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "holds for any cylindrical function $\\varphi ( x , t )$ . ", + "bbox": [ + 174, + 909, + 447, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Given $p > 1$ , let $\\mu _ { t }$ be a ${ \\mathcal { P } } _ { p } ( X )$ -absolutely continuous curve such that $\\mu _ { 0 } = \\mu$ and $\\mu _ { 1 } = \\nu$ . Then ", + "bbox": [ + 169, + 102, + 808, + 119 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/6618aeddfb7f6480ba989740001448a8536631caeddff4d3457316b3efd854be.jpg", + "text": "$$\n\\begin{array} { r l } { | J ( \\mu ) - J ( \\nu ) | = \\displaystyle \\left| \\int _ { 0 } ^ { 1 } \\frac { d } { d t } J ( \\mu _ { t } ) d t \\right| } & { } \\\\ & { = \\displaystyle \\left| \\int _ { 0 } ^ { 1 } \\int v _ { t } \\cdot \\nabla \\Phi _ { \\mu _ { t } } d \\mu _ { t } d t \\right| } \\\\ & { \\leq \\displaystyle \\int _ { 0 } ^ { 1 } \\| v _ { t } \\| _ { L ^ { p } ( \\mu _ { t } ) } \\| \\nabla \\Phi _ { \\mu _ { t } } \\| _ { L ^ { q } ( \\mu _ { t } ) } d t } \\\\ & { \\leq \\displaystyle \\int _ { 0 } ^ { 1 } \\| v _ { t } \\| _ { L ^ { p } ( \\mu _ { t } ) } d t \\cdot \\operatorname* { s u p } _ { \\tilde { \\mu } } \\| \\nabla \\Phi _ { \\tilde { \\mu } } \\| _ { L ^ { q } ( \\mu ) } } \\\\ & { \\leq \\displaystyle \\int _ { 0 } ^ { 1 } \\| \\mu _ { t } ^ { \\prime } \\| _ { W _ { p } } d t \\cdot \\operatorname* { s u p } \\| \\nabla \\Phi _ { \\tilde { \\mu } } \\| _ { L ^ { q } ( \\mu ) } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 127, + 679, + 314 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $| \\mu _ { t } ^ { \\prime } |$ is the metric derivative. The existence of $v _ { t }$ and the bound $\\| v _ { t } \\| _ { L ^ { p } ( \\mu _ { t } ) } \\leq | \\mu _ { t } ^ { \\prime } | _ { W _ { p } }$ is due to Lemma 2. Taking the infimum over all possible such curves, we find that ", + "bbox": [ + 173, + 320, + 823, + 349 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/6de71584022a380762c2f906989a49506d478a6b040ed96d41ab246dc7df10cc.jpg", + "text": "$$\n\\left| J ( \\mu ) - J ( \\nu ) \\right| \\leq W _ { p } ( \\mu , \\nu ) \\cdot \\operatorname* { s u p } _ { \\tilde { \\mu } } \\left\\| \\nabla \\Phi _ { \\tilde { \\mu } } \\right\\| _ { L ^ { q } ( \\mu ) } .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 357, + 655, + 385 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "To conclude, we consider the limit $p 1$ . Using the fact that $q \\mapsto | | f | | _ { L ^ { q } ( \\mu ) }$ is increasing in $q$ , we have ", + "bbox": [ + 173, + 401, + 825, + 429 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/ad0e60985024e2c0bcd7975a9fcee22cbe19f76256bc7f0b0f92a02a96c47c0a.jpg", + "text": "$$\n| J ( \\mu ) - J ( \\nu ) | \\leq W _ { p } ( \\mu , \\nu ) \\cdot \\operatorname* { s u p } _ { \\tilde { \\mu } } \\| \\nabla \\Phi _ { \\tilde { \\mu } } \\| _ { L ^ { \\infty } ( \\mu ) } \\leq \\alpha W _ { p } ( \\mu , \\nu ) .\n$$", + "text_format": "latex", + "bbox": [ + 292, + 430, + 702, + 458 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Then ", + "bbox": [ + 173, + 463, + 210, + 478 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/e3d32a4d3a86f82525e879b996d34e8793fcb1c27810d81fded46f11347d3369.jpg", + "text": "$$\n\\begin{array} { r l } & { | J ( \\mu ) - J ( \\nu ) | \\leq \\underset { p > 1 } { \\operatorname* { i n f } } \\alpha W _ { p } ( \\mu , \\nu ) } \\\\ & { \\qquad = \\alpha \\underset { \\pi \\in \\Pi ( \\mu , \\nu ) } { \\operatorname* { i n f } } \\underset { p > 1 } { \\operatorname* { i n f } } | | x - y | | _ { L ^ { p } ( \\pi ) } } \\\\ & { \\qquad = \\alpha \\underset { \\pi \\in \\Pi ( \\mu , \\nu ) } { \\operatorname* { i n f } } | | x - y | | _ { L ^ { 1 } ( \\pi ) } } \\\\ & { \\qquad = \\alpha W _ { 1 } ( \\mu , \\nu ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 476, + 653, + 574 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proposition 4. The minimax and non-saturating GAN losses do not satisfy $( D l )$ for some $\\mu _ { 0 }$ ", + "bbox": [ + 169, + 606, + 784, + 621 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proposition 5. The Wasserstein GAN loss satisfies $( D l )$ with $\\alpha = 1$ for any $\\mu _ { 0 }$ ", + "bbox": [ + 174, + 626, + 691, + 641 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof for Proposition 4 and Proposition 5. The counterexample for Proposition 4 is a slight simplification of Example 1 of Arjovsky et al. (2017). Let $\\delta _ { x }$ denote the distribution that outputs $x \\in \\mathbb { R }$ with probability 1. Then, we have ", + "bbox": [ + 173, + 661, + 825, + 704 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/c59a4163190a10d54bd5193b71621db4e4e12225fba1f5871f6077c4de423a44.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { { \\mathcal { I } } _ { J S } } ( \\delta _ { x } \\mid | \\delta _ { 0 } ) = \\left\\{ \\begin{array} { l l } { \\log 2 } & { x \\neq 0 } \\\\ { 0 } & { x = 0 , } \\end{array} \\right. \\quad D _ { \\mathrm { K L } } ( \\frac 1 2 \\delta _ { x } + \\frac 1 2 \\delta _ { 0 } \\| \\delta _ { 0 } ) = \\left\\{ \\begin{array} { l l } { \\infty } & { x \\neq 0 } \\\\ { 0 } & { x = 0 , } \\end{array} \\right. \\quad \\mathrm { a n d } \\quad W _ { 1 } ( \\delta _ { x } , \\delta _ { 0 } ) = | x | . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 714, + 833, + 750 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Therefore, for $J ( \\mu ) = D _ { \\mathrm { J S } } ( \\mu \\vert \\vert \\delta _ { 0 } )$ , the inequality $| J ( \\delta _ { x } ) - J ( \\delta _ { 0 } ) | \\leq \\alpha W _ { 1 } ( \\delta _ { x } , \\delta _ { 0 } )$ cannot hold for sufficiently small $x \\neq 0$ , because the left-hand side equals $\\log 2$ while the right-hand side equals $\\alpha | x |$ . For $\\begin{array} { r } { \\dot { \\boldsymbol J } ( \\mu ) = \\dot { D _ { \\mathrm { K L } } } ( \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\delta _ { 0 } | | \\delta _ { 0 } ) } \\end{array}$ , the inequality will not hold for any $x \\neq 0$ . ", + "bbox": [ + 173, + 757, + 826, + 803 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Next, we verify Proposition 5. For $J ( \\mu ) = W _ { 1 } ( \\mu , \\mu _ { 0 } )$ , we have $J ( \\mu ) - J ( \\nu ) = W _ { 1 } ( \\mu , \\mu _ { 0 } ) -$ $W _ { 1 } ( \\nu , \\mu _ { 0 } ) \\leq \\dot { W } _ { 1 } ( \\bar { \\mu , \\nu } )$ by the triangle inequality (see e.g., Section 6 in Villani (2009)). Reversing the roles of $\\mu$ and $\\nu$ , we can conclude that $| J ( \\mu ) - J ( \\nu ) | \\leq W _ { 1 } ( \\mu , \\nu )$ . The result follows from Proposition 3. □ ", + "bbox": [ + 173, + 806, + 825, + 864 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proposition 6 (Pasch–Hausdorff). Let $J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ be a function, and define ${ \\tilde { J } } : = J \\oplus R _ { 1 }$ . Then $\\tilde { J }$ is $\\alpha$ -Lipschitz w.r.t. the distance induced by the KR norm, and hence the Wasserstein- $I$ distance when restricted to ${ \\mathcal { P } } ( X )$ . ", + "bbox": [ + 174, + 877, + 825, + 925 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof. Observe that by the triangle inequality, ", + "bbox": [ + 173, + 103, + 478, + 118 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/ee6b6ea03e49120d023b0f7b980c52269b49b3033f95e25d2a92e4506b74909a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\tilde { J } ( \\mu ) - \\tilde { J } ( \\nu ) = \\underset { \\tilde { \\nu } } { \\operatorname* { s u p } } \\operatorname* { i n f } J ( \\tilde { \\mu } ) + \\alpha W _ { 1 } ( \\mu , \\tilde { \\mu } ) - J ( \\tilde { \\nu } ) - \\alpha W _ { 1 } ( \\nu , \\tilde { \\nu } ) } \\\\ & { \\qquad \\leq \\underset { \\tilde { \\nu } } { \\operatorname* { s u p } } \\alpha W _ { 1 } ( \\mu , \\tilde { \\nu } ) - \\alpha W _ { 1 } ( \\nu , \\tilde { \\nu } ) } \\\\ & { \\qquad \\leq \\underset { \\tilde { \\nu } } { \\operatorname* { s u p } } \\alpha W _ { 1 } ( \\mu , \\nu ) } \\\\ & { \\qquad = \\alpha W _ { 1 } ( \\mu , \\nu ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 125, + 714, + 226 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Interchanging the roles of $\\mu$ and $\\nu$ completes the proof. ", + "bbox": [ + 173, + 231, + 537, + 246 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Lemma 3. The convex conjugate of $\\mu \\mapsto \\alpha \\vert \\vert \\mu \\vert \\vert _ { \\mathrm { K R } }$ is given as $\\varphi \\mapsto \\chi \\{ \\| \\varphi \\| _ { \\mathrm { L i p } } \\leq \\alpha \\}$ . ", + "bbox": [ + 173, + 253, + 732, + 271 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof. The convex conjugate is ", + "bbox": [ + 173, + 285, + 383, + 300 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/adc43f1388ddac1e33920a12bacd8733153b91f1a2e43c62674348d6db14d4a8.jpg", + "text": "$$\n\\begin{array} { r l } { \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\alpha \\lVert \\mu \\rVert _ { \\mathbf { k R } } \\right] = \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\alpha \\underset { \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } } { \\operatorname* { s u p } } \\int g d \\mu \\right] } & { } \\\\ & { = \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\underset { \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } } { \\operatorname* { s u p } } \\int g d \\mu \\right] } \\\\ & { = \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\underset { \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } } { \\operatorname* { i n f } } \\leq \\int f d \\mu - \\int g d \\mu \\right] } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } } { \\operatorname* { i n f } } \\frac { \\operatorname* { s u p } } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\int g d \\mu \\right] } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } } { \\operatorname* { i n f } } \\frac { \\operatorname* { s u p } } { \\operatorname* { s u p } } } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } \\oplus \\mathbf { k } } { \\operatorname* { i n f } } \\times \\mathcal { X } \\{ f - g \\} } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } \\oplus \\mathbf { x } } { \\operatorname* { s u p } } \\mathcal { X } \\{ f - g \\} } \\\\ & { = \\mathcal { X } \\{ \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } \\leq \\alpha \\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 308, + 710, + 500 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where Sion’s minimax theorem ensures that we can swap the inf and the sup. ", + "bbox": [ + 173, + 503, + 678, + 518 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "G PROOFS FOR SECTION 5 ", + "text_level": 1, + "bbox": [ + 176, + 537, + 410, + 555 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proposition 7. The Wasserstein, minimax, and non-saturating GAN losses do not satisfy $( D 2 )$ for some $\\mu _ { 0 }$ . ", + "bbox": [ + 173, + 570, + 825, + 599 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof. First, we prove the statement for the Wasserstein GAN. Consider $J ( \\mu ) = W _ { 1 } ( \\mu , \\delta _ { 0 } )$ evaluated at $\\mu = \\textstyle { \\frac { 1 } { 2 } } \\delta _ { - 1 } + \\frac { 1 } { 2 } \\delta _ { 1 }$ , a mixture of spikes at $\\pm 1$ . The optimal discriminator of $J$ at this mixture is the Kantorovich potential that transfers $\\mu$ to $\\delta _ { 0 }$ , which is $\\Phi _ { \\mu } ( x ) = \\vert x \\vert$ . The gradient of this Kantorovich potential is discontinuous at 0, and hence not Lipschitz. ", + "bbox": [ + 173, + 614, + 825, + 672 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For the minimax GAN $J ( \\mu ) ~ = ~ { \\cal D } _ { \\mathrm { J S } } ( \\mu | | \\mu _ { 0 } )$ , let $\\mu$ and $\\mu _ { 0 }$ be probability distributions on $\\mathbb { R }$ whose densities are $p ( x ) \\propto \\exp ( - | x | / 2 )$ and $p _ { 0 } ( x ) \\ \\propto \\ \\exp ( - | x | )$ , respectively (i.e., Laplace distributions). Then, the optimal discriminator at µ is given as Φµ(x) = 12 log p(x)p(x)+p0(x) $- \\textstyle { \\frac { 1 } { 2 } } \\log ( 1 + 2 \\exp ( - | x | / 2 ) )$ . By elementary calculations, we can see that this function is not differentiable at $x = 0$ , which implies that the minimax GAN does not satisfy (D2). For the non-saturating GAN, we obtain a similar result by swapping the role of $\\mu$ and $\\mu _ { 0 }$ . □ ", + "bbox": [ + 173, + 678, + 825, + 772 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proposition 8. Let $J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ be a convex, proper, lower semicontinuous function, and define ${ \\tilde { J } } : = J \\oplus R _ { 2 }$ . Then the optimal discriminator for $\\tilde { J }$ is $\\beta _ { 1 }$ -smooth. ", + "bbox": [ + 173, + 779, + 821, + 810 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof. $\\tilde { J }$ is convex, proper, lower semicontinuous. Hence ", + "bbox": [ + 173, + 827, + 553, + 843 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/ddba0c050726f6bccaf74d400f7ff13bfc9a356ba8086af5187e4c36338269b1.jpg", + "text": "$$\n\\tilde { J } ( \\mu ) = ( J \\oplus \\lambda | | \\cdot | | _ { \\mathcal { F } ^ { * } } ) ( \\mu ) = \\operatorname* { s u p } _ { \\varphi } \\int \\varphi d \\mu - J ^ { \\star } ( \\varphi ) - \\chi \\{ \\frac { 1 } { \\lambda } \\varphi \\in \\mathcal { F } \\} .\n$$", + "text_format": "latex", + "bbox": [ + 274, + 849, + 723, + 883 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Then by the envelope theorem, $\\frac { \\delta J } { \\delta \\mu }$ is the $\\varphi$ that maximizes the right-hand side, and hence $\\frac { 1 } { \\lambda } \\frac { \\delta J } { \\delta \\mu } \\in$ $\\mathcal { F }$ . □ ", + "bbox": [ + 174, + 891, + 825, + 922 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Lemma 4. The convex conjugate of $\\mu \\mapsto \\beta _ { 1 } \\| \\mu \\| _ { S ^ { * } }$ is $\\varphi \\mapsto \\chi \\{ \\varphi \\in \\beta _ { 1 } S \\}$ . ", + "bbox": [ + 173, + 102, + 653, + 121 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. The proof of Lemma 4 is nearly identical to the proof of Lemma 3. ", + "bbox": [ + 174, + 132, + 660, + 148 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proposition 9. Let $f : \\mathbb { R } ^ { d } \\mathbb { R }$ be a neural network consisting of $k$ layers whose linear transformations have spectral norm 1 and whose activation functions are 1-Lipschitz and 1-smooth. Then $f$ is $k$ -smooth. ", + "bbox": [ + 173, + 155, + 825, + 198 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. In this section, let $\\ell ( f )$ and $s ( f )$ denote the Lipschitz constant of $f$ and $\\nabla f$ respectively. This inequality holds ", + "bbox": [ + 173, + 213, + 821, + 241 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/6ed7daf55421b16512cb0b5761cacea9fc9d60c3bf0357e9a3f8722800ea4a65.jpg", + "text": "$$\ns ( f \\circ g ) \\leq s ( f ) \\ell ( g ) ^ { 2 } + \\ell ( f ) s ( g ) ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 238, + 611, + 257 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "since ", + "bbox": [ + 173, + 258, + 210, + 272 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/c3837cabf66c3b66bf72825cdec9ae6dda833aca8ffeb10c68ec9c8962d161d0.jpg", + "text": "$$\n\\begin{array} { r l } & { | d ( f \\circ g ) ( x ) - d ( f \\circ g ) ( y ) | | = | | d f ( g ( x ) ) d g ( x ) - d f ( g ( y ) ) d g ( y ) | | } \\\\ & { \\qquad = | | d f ( g ( x ) ) d g ( x ) - d f ( g ( y ) ) d g ( x ) + d f ( g ( y ) ) d g ( x ) - d f ( g ( y ) ) d g ( y ) } \\\\ & { \\qquad \\le | | d f ( g ( x ) ) - d f ( g ( y ) ) | | | d g ( x ) | | + | | d f ( g ( y ) ) | | | d g ( x ) - d g ( y ) | | } \\\\ & { \\qquad \\le s ( f ) | | g ( x ) - g ( y ) | | \\ell ( g ) + \\ell ( f ) s ( g ) | | x - y | | } \\\\ & { \\qquad \\le \\big ( s ( f ) \\ell ( g ) ^ { 2 } + \\ell ( f ) s ( g ) \\big ) | | x - y | | . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 272, + 839, + 368 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Let $\\sigma$ be an elementwise activation function with $\\ell ( \\sigma ) = s ( \\sigma ) = 1$ , and let $A$ be a linear layer with spectral norm 1. Then $\\ell ( A ) = 1$ and $s ( A ) = 0$ , so ", + "bbox": [ + 173, + 377, + 823, + 406 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/5d73fa6e99ea3bb19d9b563de57bc7654f349873c7f32d5ac5785d08be633db4.jpg", + "text": "$$\n\\begin{array} { l } { { \\ell ( \\sigma \\circ A ) \\leq \\ell ( \\sigma ) \\ell ( A ) = 1 } } \\\\ { { s ( \\sigma \\circ A ) \\leq s ( \\sigma ) \\ell ( A ) ^ { 2 } + \\ell ( \\sigma ) s ( A ) = 1 . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 364, + 410, + 632, + 449 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We note that we can use the same value of $s ( \\sigma )$ whether we consider $\\sigma : \\mathbb { R } \\mathbb { R }$ or elementwise as $\\sigma : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }$ , since for an elementwise $\\sigma$ , we have ", + "bbox": [ + 173, + 453, + 823, + 482 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/c7eccebc73791bf2828f2a43d66c89987b4124486976dde8e2e52c4376b0538d.jpg", + "text": "$$\n\\begin{array} { r l } & { | | d \\sigma ( x ) - d \\sigma ( y ) | | _ { 2 } = | | \\mathrm { d i a g } \\sigma ^ { \\prime } ( x ) - \\mathrm { d i a g } \\sigma ^ { \\prime } ( y ) | | _ { 2 } } \\\\ & { \\qquad = | | \\mathrm { d i a g } ( \\sigma ^ { \\prime } ( x ) - \\sigma ^ { \\prime } ( y ) ) | | _ { 2 } } \\\\ & { \\qquad = \\underset { i } { \\operatorname* { m a x } } | \\sigma ^ { \\prime } ( x _ { i } ) - \\sigma ^ { \\prime } ( y _ { i } ) | } \\\\ & { \\qquad \\le \\underset { i } { \\operatorname* { m a x } } s ( \\sigma ) | x _ { i } - y _ { i } | } \\\\ & { \\qquad = s ( \\sigma ) | | x - y | | _ { \\infty } } \\\\ & { \\qquad \\le s ( \\sigma ) | | x - y | | _ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 328, + 484, + 666, + 608 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We apply the inequality (27) recursively for all $k$ layers to obtain that the entire network is $k$ - smooth. □ ", + "bbox": [ + 169, + 617, + 825, + 647 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Lemma 5. Let $\\mathcal { H }$ be an RKHS with the Gaussian kernel. Then $\\begin{array} { r } { | | \\hat { \\mu } | | _ { \\mathcal { H } } \\leq \\frac { \\sqrt { 2 } } { \\sigma ^ { 2 } } | | \\mu | | _ { S ^ { * } } } \\end{array}$ . ", + "bbox": [ + 171, + 652, + 720, + 671 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. It follows from $\\begin{array} { r } { \\frac { \\partial f ( { \\boldsymbol x } ) } { \\partial { \\boldsymbol x } _ { i } } = \\langle f , \\frac { \\partial K ( { \\boldsymbol x } , \\cdot ) } { \\partial x _ { i } } \\rangle } \\end{array}$ , ∂K(x,·) i that, for f ∈ H, ", + "bbox": [ + 173, + 684, + 571, + 707 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/f924ff17dd940a5e643bb380e6127e72d5fd536b4c31c94d92bef5d0cee23f29.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\| \\nabla f ( x ) - \\nabla f ( y ) \\| ^ { 2 } = \\sum _ { i = 1 } ^ { d } \\left| \\frac { \\partial f ( x ) } { \\partial x _ { i } } - \\frac { \\partial f ( y ) } { \\partial y _ { i } } \\right| ^ { 2 } } \\\\ { \\displaystyle \\leq \\sum _ { i = 1 } ^ { d } \\| f \\| _ { \\mathcal H } ^ { 2 } \\left\\| \\frac { \\partial K ( x , \\cdot ) } { \\partial x _ { i } } - \\frac { \\partial K ( y , \\cdot ) } { \\partial y _ { i } } \\right\\| _ { \\mathcal H } ^ { 2 } } \\\\ { \\displaystyle = \\| f \\| _ { \\mathcal H } ^ { 2 } \\sum _ { i = 1 } ^ { d } \\left\\{ \\frac { \\partial ^ { 2 } K ( x , \\tilde { x } ) } { \\partial x _ { i } \\partial \\tilde { x } _ { i } } - 2 \\frac { \\partial ^ { 2 } K ( x , y ) } { \\partial x _ { i } \\partial y _ { i } } + \\frac { \\partial ^ { 2 } K ( y , \\tilde { y } ) } { \\partial y _ { i } \\partial \\tilde { y } _ { i } } \\right\\} } \\\\ { \\displaystyle \\leq \\| f \\| _ { \\mathcal H } ^ { 2 } \\cdot ( \\Gamma ( x , y ) + \\Gamma ( y , x ) ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 712, + 759, + 863 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where $\\tilde { x }$ and $\\tilde { y }$ denote copies of $x$ and $y$ , and ", + "bbox": [ + 174, + 866, + 465, + 881 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/7085940f73378390bbf831149966e8072c3436e32ef90f1d1beb8fc5119f5b91.jpg", + "text": "$$\n\\Gamma ( x , y ) = \\sum _ { i = 1 } ^ { d } \\left| \\frac { \\partial ^ { 2 } K ( x , \\tilde { x } ) } { \\partial x _ { i } \\partial \\tilde { x } _ { i } } - \\frac { \\partial ^ { 2 } K ( x , y ) } { \\partial x _ { i } \\partial \\tilde { x } _ { i } } \\right| .\n$$", + "text_format": "latex", + "bbox": [ + 356, + 885, + 642, + 928 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "From (30), we find that for a Gaussian kernel, ", + "bbox": [ + 174, + 103, + 475, + 118 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/5b6febddd8961d29a19bc49927bd3b7a177d43c22e40ab0fe9f8cfec8d9edcaf.jpg", + "text": "$$\n\\Gamma ( x , y ) = \\frac { 1 } { \\sigma ^ { 4 } } \\exp \\Bigl ( - \\frac { \\| x - y \\| ^ { 2 } } { 2 \\sigma ^ { 2 } } \\Bigr ) \\| x - y \\| ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 354, + 126, + 642, + 159 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "and thus, ", + "bbox": [ + 173, + 166, + 235, + 180 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/4a6043292436b32c9ee054c39b5cfe7a5d9b11eeb61e9b78ca58045962738a6e.jpg", + "text": "$$\n\\| \\nabla f ( x ) - \\nabla f ( y ) \\| \\leq { \\frac { \\sqrt { 2 } } { \\sigma ^ { 2 } } } \\| f \\| _ { \\mathcal H } \\| x - y \\| .\n$$", + "text_format": "latex", + "bbox": [ + 361, + 180, + 635, + 213 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "This implies that ", + "bbox": [ + 173, + 224, + 287, + 241 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/079b85eee7b6bdf455f45d7ec4064145674604d2acc607913d3637370d218e0b.jpg", + "text": "$$\n\\| \\hat { \\mu } \\| _ { \\mathcal H } = \\operatorname* { s u p } _ { f \\in \\mathcal H , \\| f \\| _ { \\mathcal H } \\leq 1 } \\int f d \\mu \\leq \\operatorname* { s u p } _ { \\| f \\| _ { \\mathcal S } \\leq \\frac { \\sqrt { 2 } } { \\sigma ^ { 2 } } } \\int f d \\mu = \\frac { \\sqrt { 2 } } { \\sigma ^ { 2 } } \\| \\mu \\| _ { \\mathcal S ^ { * } } .\n$$", + "text_format": "latex", + "bbox": [ + 285, + 250, + 710, + 291 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "H PROOFS FOR SECTION 6 ", + "text_level": 1, + "bbox": [ + 174, + 337, + 411, + 353 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Lemma 6. The convex conjugate of $\\mu \\mapsto \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathrm { K R } } ^ { 2 }$ is $\\begin{array} { r } { f \\mapsto \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } . } \\end{array}$ ", + "bbox": [ + 171, + 367, + 633, + 388 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proof. This proof is generalized from the finite-dimensional case (Boyd & Vandenberghe, 2004). We can bound the convex conjugate from above by ", + "bbox": [ + 168, + 405, + 823, + 435 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/8823eec8192bdc74a2279c05288666ac79d22f70be16dcb0e96b5d0c2ca5221a.jpg", + "text": "$$\n\\begin{array} { r l } { { \\operatorname* { s u p } _ { \\mu } [ \\int f d \\mu - \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathrm { K R } } ^ { 2 } ] \\leq \\operatorname* { s u p } _ { \\mu } [ | | f | | _ { \\mathrm { L i p } } | | \\mu | | _ { \\mathrm { K R } } - \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathrm { K R } } ^ { 2 } ] } } \\\\ & { \\leq \\operatorname* { s u p } _ { z \\in \\mathbb { R } } [ | | f | | _ { \\mathrm { L i p } } z - \\frac { \\lambda } { 2 } z ^ { 2 } ] } \\\\ & { = | | f | | _ { \\mathrm { L i p } } \\cdot \\frac { | | f | | _ { \\mathrm { L i p } } } { \\lambda } - \\frac { \\lambda } { 2 } \\cdot \\frac { | | f | | _ { \\mathrm { L i p } } ^ { 2 } } { \\lambda ^ { 2 } } } \\\\ & { = \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 443, + 710, + 580 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "If $f$ is constant, then we may choose $\\mu = 0$ to see that ", + "bbox": [ + 173, + 593, + 531, + 608 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/7fcb34641616422f080f2cda7eb71a2225d9e82f8d531f530d40da0aead1286d.jpg", + "text": "$$\n\\operatorname* { s u p } _ { \\mu } \\left[ \\int f d \\mu - \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathrm { K R } } ^ { 2 } \\right] \\geq 0 = \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 344, + 614, + 651, + 650 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "and we are done. ", + "bbox": [ + 173, + 659, + 285, + 672 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Otherwise, for $f ( x ) \\neq f ( y )$ , let $\\mu _ { x , y }$ be the signed measure given by ", + "bbox": [ + 173, + 679, + 624, + 695 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/822b5e9b03fed8f8cdda16a30cdbd841423589e4d3a98f026662fa38fc7c24fd.jpg", + "text": "$$\n\\mu _ { x , y } = { \\frac { | | f | | _ { \\mathrm { L i p } } ^ { 2 } } { \\lambda ( f ( x ) - f ( y ) ) } } ( \\delta _ { x } - \\delta _ { y } ) .\n$$", + "text_format": "latex", + "bbox": [ + 382, + 703, + 614, + 741 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Note that ", + "bbox": [ + 173, + 747, + 238, + 762 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/ece08f2c70eb1396e03e768d775281e505f3daaa74282a4c29b9f86172f232c5.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle | | \\mu _ { x , y } | | _ { \\mathrm { K R } } = \\operatorname* { s u p } _ { | | g | | _ { \\mathrm { L i p } } \\leq 1 } \\int g d \\mu _ { x , y } } \\\\ { = \\frac { | | f | | _ { \\mathrm { L i p } } ^ { 2 } } { \\lambda ( f ( x ) - f ( y ) ) } \\operatorname* { s u p } _ { | | g | | _ { \\mathrm { L i p } } \\leq 1 } \\Big ( g ( x ) - g ( y ) \\Big ) } \\\\ { = \\frac { | | f | | _ { \\mathrm { L i p } } ^ { 2 } } { \\lambda ( f ( x ) - f ( y ) ) } \\cdot d ( x , y ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 315, + 760, + 679, + 877 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 880, + 200, + 893 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/80254740d87d38e8bd2806afe8f63643d678f40adf431d46040ad5a47854aed6.jpg", + "text": "$$\n\\int f d \\mu _ { x , y } = { \\frac { | | f | | _ { \\mathrm { L i p } } ^ { 2 } } { \\lambda ( f ( x ) - f ( y ) ) } } { \\Big ( } f ( x ) - f ( y ) { \\Big ) } = { \\frac { 1 } { \\lambda } } | | f | | _ { \\mathrm { L i p } } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 302, + 892, + 694, + 928 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Then ", + "bbox": [ + 173, + 103, + 210, + 117 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/447bf7e9c569a13dad90756baa47e5ef2d5a5a72497fbac85f0ba282309ee55c.jpg", + "text": "$$\n\\begin{array} { r l } { \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathrm { K R } } ^ { 2 } \\right] \\geq \\underset { f ( x ) \\neq f ( y ) } { \\operatorname* { s u p } } \\left[ \\int f d \\mu _ { x , y } - \\frac { \\lambda } { 2 } | | \\mu _ { x , y } | | _ { \\mathrm { K R } } ^ { 2 } \\right] } & { } \\\\ { = \\underset { f ( x ) \\neq f ( y ) } { \\operatorname* { s u p } } \\left[ \\frac { 1 } { \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } - \\frac { \\lambda } { 2 } \\big ( \\frac { 1 } { \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } \\frac { d ( x , y ) } { f ( x ) - f ( y ) } \\big ) ^ { 2 } \\right] } & { } \\\\ { = \\frac { 1 } { \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } - \\frac { \\lambda } { 2 } \\big ( \\frac { 1 } { \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } \\frac { 1 } { | | f | | _ { \\mathrm { L i p } } } \\big ) ^ { 2 } } & { } \\\\ { = \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 122, + 772, + 265 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proposition 10. Let $J : \\mathcal { M } ( X ) \\mathbb { R }$ be a convex function. Then $J$ satisfies $( D 3 )$ if and only if $\\begin{array} { r } { \\mathfrak { D } _ { J } ( \\nu , \\mu ) \\le \\frac { \\beta _ { 2 } } { 2 } \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } } \\end{array}$ for all $\\mu , \\nu \\in \\mathcal { M } ( X )$ . ", + "bbox": [ + 173, + 290, + 825, + 324 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proof. Recall the definition of the Bregman divergence: ", + "bbox": [ + 173, + 338, + 542, + 354 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/3437d8382bc25881b8b5386bb47cecdf6f5ad580966d471c5a4afcdd9ce36285.jpg", + "text": "$$\n\\mathfrak { D } _ { J } ( \\nu , \\mu ) : = J ( \\nu ) - J ( \\mu ) - \\int \\Phi _ { \\mu } ( x ) d ( \\nu - \\mu ) .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 359, + 660, + 393 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proposition 10 claims that the optimal discriminator $\\Phi _ { \\mu }$ is $\\beta _ { 2 }$ -smooth as a function of $\\mu$ if and only if ", + "bbox": [ + 169, + 398, + 828, + 426 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/7d81949a1b5f909314393f881304918939c59c0bd730d5c46791abcef596825f.jpg", + "text": "$$\n\\forall \\mu , \\nu \\in \\mathcal { M } ( { \\cal X } ) , \\quad { \\cal J } ( \\nu ) - { \\cal J } ( \\mu ) - \\int \\Phi _ { \\mu } ( x ) d ( \\nu - \\mu ) \\leq \\frac { \\beta _ { 2 } } { 2 } \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 256, + 422, + 740, + 455 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "This proof is generalized from the finite-dimensional case (Zhou, 2018; Sidford, 2017). Suppose that $\\| \\nabla \\Phi _ { \\mu } ( x ) - \\nabla \\Phi _ { \\nu } ( x ) \\| _ { 2 } \\leq \\beta _ { 2 } \\| \\mu - \\nu \\| _ { \\mathrm { K R } }$ holds for all $x$ . Let $\\mu _ { t }$ $0 \\leq t \\leq 1$ ) be defined as a mixture between $\\mu _ { t } = ( 1 - t ) \\mu + t \\nu$ so that ", + "bbox": [ + 174, + 465, + 826, + 508 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/073b099b59e22d148db45c2fe2542da63ffe209cdd6f9e5025157035bec0e46b.jpg", + "text": "$$\n\\mu _ { 0 } = \\mu , \\quad \\mu _ { 1 } = \\nu , \\quad \\mathrm { a n d } \\quad \\| \\mu _ { t } - \\mu _ { s } \\| _ { \\mathrm { K R } } = | t - s | \\cdot \\| \\mu - \\nu \\| _ { \\mathrm { K R } } .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 515, + 715, + 534 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Then, we have ", + "bbox": [ + 174, + 540, + 272, + 554 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/6ab74676b3e85cc18d0b0b0eccafa6af6324314e2fb84c81c4c3be1276cb94cb.jpg", + "text": "$$\n\\begin{array} { r l } { \\| \\mathcal { D } _ { J } ( v , p ) \\| = \\bigg | \\int _ { 0 } ^ { 1 } \\frac { d } { d t } \\hat { { \\boldsymbol { H } } } ( \\mu _ { t } ) \\hat { { \\boldsymbol { d } } } \\boldsymbol { L } \\bigg | \\hat { \\psi } - \\int \\hat { \\mathbf { q } } _ { \\mu } \\hat { { \\boldsymbol { d } } } ( v - \\mu ) \\bigg | } \\\\ & { = \\bigg | \\int _ { 0 } ^ { 1 } \\int \\Phi _ { \\mu } , d ( v - p ) d \\hbar - \\int \\Phi _ { \\mu } d ( v - \\mu ) \\bigg | } \\\\ & { = \\bigg | \\int _ { 0 } ^ { 1 } \\int ( \\Phi _ { \\mu \\nu } - \\Phi _ { \\mu } ) d v - \\mu ) d v - \\mu \\big | } \\\\ & { \\leq \\int _ { 0 } ^ { 1 } \\Big | \\| \\Phi _ { \\mu \\nu } - \\Phi _ { \\mu } \\| _ { \\mathrm { L E } ^ { \\infty } } \\| \\| \\mu _ { t } \\| = \\nu | \\mathrm { k i n } d t } \\\\ & { \\leq \\int _ { 0 } ^ { 1 } \\hat { \\mathcal { S } } _ { 2 } \\| u _ { t } - \\mu | \\mathrm { s g n } \\| u \\| \\mu - \\nu \\| \\mathrm { k i n } d t } \\\\ & { \\leq \\int _ { 0 } ^ { 1 } \\hat { \\mathcal { S } } _ { 2 } \\| u _ { t } - \\nu \\| _ { \\mathrm { L E } ^ { \\infty } } d u } \\\\ & { \\leq \\int _ { 0 } ^ { 1 } \\hat { \\mathcal { S } } _ { 2 } \\| u _ { t } - \\nu \\| _ { \\mathrm { L E } ^ { \\infty } } ^ { 2 } d u } \\\\ & { = \\frac { \\mathcal { S } _ { 2 } } { \\mathcal { S } } \\| u _ { t } - \\nu \\| _ { \\mathrm { L E } ^ { \\infty } } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 310, + 561, + 687, + 813 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Since the choice of $\\mu$ and $\\nu$ is arbitrary, we have proved (28) by assuming (D2). ", + "bbox": [ + 173, + 815, + 696, + 830 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Next, we move on to prove the converse. Before proceeding, note that the Bregman divergence ${ \\mathfrak { D } } _ { J } ( \\nu , \\mu )$ is always non-negative. In fact, for any $\\mu , \\nu \\in \\mathcal { M } ( X )$ , we have ", + "bbox": [ + 173, + 837, + 826, + 866 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/72d6c37f06f87e65748b814e032cfafeb225f3bff76a41d04a27ef753c964b46.jpg", + "text": "$$\n\\int \\Phi _ { \\mu } d ( \\nu - \\mu ) = \\underbrace { \\int \\Phi _ { \\mu } d \\nu - J ^ { \\star } ( \\Phi _ { \\mu } ) } _ { \\leq J ( \\nu ) } - \\underbrace { \\left[ \\int \\Phi _ { \\mu } d \\mu - J ^ { \\star } ( \\Phi _ { \\mu } ) \\right] } _ { = J ( \\mu ) }\n$$", + "text_format": "latex", + "bbox": [ + 284, + 872, + 712, + 928 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/abccbe16d4fe3ec153db55a1c980fdeab0488934f53e3fea6f5b9642be29dcf3.jpg", + "text": "$$\n\\leq J ( \\nu ) - J ( \\mu ) ,\n$$", + "text_format": "latex", + "bbox": [ + 390, + 102, + 500, + 119 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "which implies $\\mathfrak { D } _ { J } ( \\nu , \\mu ) \\geq 0$ . ", + "bbox": [ + 174, + 125, + 367, + 141 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Choose $\\xi \\in \\mathcal { M } ( X )$ arbitrarily. If $\\xi = \\mu$ , the inequality $\\| \\nabla \\Phi _ { \\mu } - \\nabla \\Phi _ { \\xi } \\| _ { 2 } \\leq \\beta _ { 2 } \\| \\xi - \\mu \\| _ { \\mathrm { K R } }$ is trivial, so we assume $\\xi \\neq \\mu$ below. By the non-negativity of the Bregman divergence, we have ", + "bbox": [ + 173, + 146, + 825, + 176 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/181dc6ada7b4ee912d5479f53cc558c5860d0e25fc92e3a09bbf11f96af7bb8e.jpg", + "text": "$$\n\\begin{array} { r l } { { 0 \\leq \\mathfrak { D } _ { J } ( \\nu , \\mu ) } } \\\\ & { = J ( \\nu ) - [ J ( \\mu ) + \\int \\Phi _ { \\mu } d ( \\nu - \\mu ) ] } \\\\ & { \\leq [ J ( \\xi ) + \\int \\Phi _ { \\xi } d ( \\nu - \\xi ) + \\frac { \\beta _ { 2 } } { 2 } | | \\xi - \\nu | | _ { \\mathrm { K R } } ^ { 2 } ] - [ J ( \\mu ) + \\int \\Phi _ { \\mu } d ( \\nu - \\mu ) ] } \\\\ & { = J ( \\xi ) - J ( \\mu ) - \\int \\Phi _ { \\mu } d ( \\xi - \\mu ) + \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\nu - \\xi ) + \\frac { \\beta _ { 2 } } { 2 } | | \\xi - \\nu | | _ { \\mathrm { K R } } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 180, + 761, + 304 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Since the above inequality holds for all $\\nu \\in \\mathcal { M } ( X )$ , the last expression is still non-negative if we take the infimum over all $\\nu$ . In particular, ", + "bbox": [ + 173, + 309, + 823, + 338 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/df828e39fb614b94ffcca73c3d69067f65c43486ec0876458e0ee22851b6e449.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\operatorname* { i n f } _ { \\nu \\in \\mathcal { M } ( X ) } \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\nu - \\xi ) + \\frac { \\beta _ { 2 } } { 2 } \\| \\xi - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } } } \\\\ { ~ = - \\displaystyle \\operatorname* { s u p } _ { \\nu \\in \\mathcal { M } ( X ) } \\left[ \\int [ \\Phi _ { \\mu } - \\Phi _ { \\xi } ] d ( \\nu - \\xi ) - \\frac { \\beta _ { 2 } } { 2 } \\| \\xi - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } \\right] } \\\\ { ~ = - \\displaystyle \\operatorname* { s u p } _ { \\zeta \\in \\mathcal { M } ( X ) } \\left[ \\int [ \\Phi _ { \\mu } - \\Phi _ { \\xi } ] d \\zeta - \\frac { \\beta _ { 2 } } { 2 } \\| \\zeta \\| _ { \\mathrm { K R } } ^ { 2 } \\right] } \\\\ { ~ = - \\displaystyle \\frac { 1 } { 2 \\beta _ { 2 } } \\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\| _ { \\mathrm { L i p } } ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 343, + 702, + 488 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "using Lemma 6 for the last equality. ", + "bbox": [ + 173, + 491, + 411, + 506 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Continuing from (29), we have ", + "bbox": [ + 173, + 512, + 379, + 527 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/61ee9e5826d63e35d7fb5078a547d921cb0ba068c7fc24c859a3d14240480b5e.jpg", + "text": "$$\n0 \\leq J ( \\xi ) - J ( \\mu ) - \\int \\Phi _ { \\mu } d ( \\xi - \\mu ) - \\frac { 1 } { 2 \\beta _ { 2 } } \\left. \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right. _ { \\mathrm { L i p } } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 302, + 532, + 696, + 566 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Swapping the roles of $\\mu$ and $\\xi$ , we obtain a similar inequality. Adding both sides of thus obtained two inequalities, we obtain ", + "bbox": [ + 173, + 570, + 821, + 599 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/7eca492bb5087690b5a524d9b13cb915b6dd4c8eff8bea400a7eaba1ec24ba64.jpg", + "text": "$$\n0 \\leq \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\xi - \\mu ) - \\frac { 1 } { \\beta _ { 2 } } \\left\\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right\\| _ { \\mathrm { L i p } } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 606, + 661, + 638 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Finally, we have ", + "bbox": [ + 173, + 643, + 282, + 657 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/cce09b6860735101984ff334234c8433e78c6c97dab782ec7cd89a1b4c5df44d.jpg", + "text": "$$\n\\begin{array} { r } { \\left\\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right\\| _ { \\mathrm { L i p } } ^ { 2 } \\leq \\beta _ { 2 } \\displaystyle \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\xi - \\mu ) \\leq \\beta _ { 2 } \\left\\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right\\| _ { \\mathrm { L i p } } \\left\\| \\xi - \\mu \\right\\| _ { \\mathrm { K R } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 664, + 750, + 696 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "and hence we have the desired result: ", + "bbox": [ + 174, + 702, + 419, + 715 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/8141b90d5613fb2a4449966f2b194c53b90495f0bf422a025dfa1e46e446d6b3.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\| _ { \\mathrm { L i p } } \\leq \\beta _ { 2 } \\| \\xi - \\mu \\| _ { \\mathrm { K R } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 390, + 722, + 606, + 742 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof. First, we note that ", + "bbox": [ + 174, + 801, + 344, + 815 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/0a09a5f92d9a9ff556ff1e4799710a4fcc5fb544a84cf2d414ac05beb4e7f6d3.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { M M D } _ { K } ^ { 2 } ( \\mu , \\nu ) = \\displaystyle \\int K ( x , y ) \\left( \\mu - \\nu \\right) ( d x ) \\left( \\mu - \\nu \\right) ( d y ) } \\\\ & { \\qquad \\le \\left[ \\displaystyle \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } } \\frac { \\int \\left( K ( x , y ) - K ( x ^ { \\prime } , y ) \\right) \\left( \\mu - \\nu \\right) \\left( d y \\right) } { d ( x , x ^ { \\prime } ) } \\right] \\| \\mu - \\nu \\| _ { \\mathrm { K R } } } \\\\ & { \\qquad \\le \\left[ \\displaystyle \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } , y \\neq y ^ { \\prime } } \\frac { K ( x , y ) - K ( x , y ^ { \\prime } ) - K ( x ^ { \\prime } , y ) + K ( x ^ { \\prime } , y ^ { \\prime } ) } { d ( x , x ^ { \\prime } ) d ( y , y ^ { \\prime } ) } \\right] \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 207, + 821, + 789, + 930 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Next, we show that ", + "bbox": [ + 174, + 103, + 302, + 117 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/716797c1ae632c17565dff4e7aeba7bb8a2079444acd04b9e422905103b06767.jpg", + "text": "$$\nc = \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } , y \\neq y ^ { \\prime } } \\frac { K ( x , y ) - K ( x , y ^ { \\prime } ) - K ( x ^ { \\prime } , y ) + K ( x ^ { \\prime } , y ^ { \\prime } ) } { d ( x , x ^ { \\prime } ) d ( y , y ^ { \\prime } ) } = \\frac { 1 } { \\sigma ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 279, + 117, + 717, + 154 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Because $K$ is differentiable, it suffices to check that the operator norm of $\\partial _ { x _ { i } } \\partial _ { y _ { j } } K ( x , y )$ is bounded. To see this, note that ", + "bbox": [ + 171, + 162, + 821, + 190 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/41e12e95c1cabe9cd78e4b9f7cae0757931efe39a87393c3fe63693f46721d20.jpg", + "text": "$$\n\\begin{array} { l } { c = \\displaystyle \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } } \\frac { \\mathrm { L i p } \\left( K ( x , \\cdot ) - K ( x ^ { \\prime } , \\cdot ) \\right) } { | | x - x ^ { \\prime } | | } } \\\\ { = \\displaystyle \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } , y } \\frac { | | \\nabla _ { y } K ( x , y ) - \\nabla _ { y } K ( x ^ { \\prime } , y ) | | _ { 2 } } { | | x - x ^ { \\prime } | | } } \\\\ { \\leq \\displaystyle \\operatorname* { s u p } _ { x , y } | | \\nabla _ { x } \\nabla _ { y } K ( x , y ) | | _ { \\mathrm { o p } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 357, + 186, + 637, + 290 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "where we obtained by the last equality by applying the vector-valued mean value theorem (Rudin, 1964) to $t \\mapsto \\nabla _ { y } K ( ( 1 - t ) x + t x ^ { \\prime } , y )$ , thereby obtaining ", + "bbox": [ + 171, + 289, + 823, + 318 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/ee58ffae8098cb0a6b2f8b875cb9bf9b9054c84eb7988728fde285c4eaaaab11.jpg", + "text": "$$\n\\begin{array} { r l } & { | | \\nabla _ { y } K ( x , y ) - \\nabla _ { y } K ( x ^ { \\prime } , y ) | | _ { 2 } \\leq | | \\partial _ { t } \\nabla _ { y } K ( ( 1 - t ) x + t x ^ { \\prime } , y ) | | _ { 2 } } \\\\ & { \\qquad = | | ( x ^ { \\prime } - x ) \\cdot \\nabla _ { x } \\nabla _ { y } K ( ( 1 - t ) x + t x ^ { \\prime } , y ) | | _ { 2 } } \\\\ & { \\qquad \\leq | | x ^ { \\prime } - x | | _ { 2 } | | \\nabla _ { x } \\nabla _ { y } K ( ( 1 - t ) x + t x ^ { \\prime } , y ) | | _ { \\mathrm { { o p } } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 319, + 753, + 378 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Now ", + "bbox": [ + 173, + 383, + 209, + 397 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/7e1393c628a4f1d2c0aa36280407613ac4d9c40355ec2cc72e29e955066f1df4.jpg", + "text": "$$\n\\begin{array} { l } { \\partial _ { x _ { i } } \\partial _ { y _ { j } } K ( x , y ) = \\partial _ { x _ { i } } \\partial _ { y _ { j } } \\exp \\left( - \\displaystyle \\frac { 1 } { 2 \\sigma ^ { 2 } } \\sum _ { k } ( x _ { k } - y _ { k } ) ^ { 2 } \\right) } \\\\ { = \\partial _ { x _ { i } } \\exp \\left( - \\displaystyle \\frac { 1 } { 2 \\sigma ^ { 2 } } \\sum _ { k } ( x _ { k } - y _ { k } ) ^ { 2 } \\right) \\cdot - \\displaystyle \\frac { 1 } { \\sigma ^ { 2 } } ( x _ { j } - y _ { j } ) } \\\\ { = \\exp \\left( - \\displaystyle \\frac { 1 } { 2 \\sigma ^ { 2 } } \\sum _ { k } ( x _ { k } - y _ { k } ) ^ { 2 } \\right) \\left[ \\displaystyle \\frac { 1 } { \\sigma ^ { 4 } } ( x _ { i } - y _ { i } ) ( x _ { j } - y _ { j } ) - \\displaystyle \\frac { 1 } { \\sigma ^ { 2 } } \\delta _ { i j } \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 228, + 396, + 767, + 511 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "This is the symmetric matrix ", + "bbox": [ + 174, + 511, + 364, + 525 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/e6d13c84d565034af6fb08a48cad498b26df72450f503e27875b43c0b4bb45c2.jpg", + "text": "$$\n\\frac { 1 } { \\sigma ^ { 2 } } \\exp \\Big ( { - \\frac { | | x - y | | ^ { 2 } } { 2 \\sigma ^ { 2 } } } \\Big ) \\Big [ \\frac { 1 } { \\sigma ^ { 2 } } ( x - y ) ( x - y ) ^ { T } - I \\Big ] .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 526, + 665, + 559 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "By inspection, we see that $x - y$ is an eigenvector with eigenvalue $\\begin{array} { r } { \\frac { 1 } { \\sigma ^ { 2 } } \\exp ( { - \\frac { | | x - y | | ^ { 2 } } { 2 \\sigma ^ { 2 } } } ) ( \\frac { 1 } { \\sigma ^ { 2 } } | | x - y | | ^ { 2 } - } \\end{array}$ 1), and the other eigenvectors are orthogonal to x − y with eigenvalues − 1σ2 exp(− ||x−y||22σ2 ). ", + "bbox": [ + 173, + 563, + 825, + 599 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Setting $\\begin{array} { r } { z = \\frac { | | x - y | | ^ { 2 } } { 2 \\sigma ^ { 2 } } } \\end{array}$ and taking the absolute value of the eigenvalues, we see the maximum operator norm over all $x , y$ is equal to ", + "bbox": [ + 178, + 604, + 821, + 638 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/b0964649def180ef259ab1b3372b6c936b942bb04fd129d0e2e27af19125700d.jpg", + "text": "$$\n\\operatorname* { m a x } \\bigg \\{ \\operatorname* { s u p } _ { z \\ge 0 } \\frac { 1 } { \\sigma ^ { 2 } } e ^ { - z } | 2 z - 1 | , \\operatorname* { s u p } _ { z \\ge 0 } \\frac { 1 } { \\sigma ^ { 2 } } e ^ { - z } \\bigg \\} = \\frac { 1 } { \\sigma ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 336, + 640, + 658, + 672 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Therefore ", + "bbox": [ + 173, + 674, + 241, + 689 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/71a174ec59ca6c4d5c117b7d40f9d71badc788cf2dd794d7ee75a4faf6e3fe44.jpg", + "text": "$$\nc \\leq { \\frac { 1 } { \\sigma ^ { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 470, + 684, + 527, + 714 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "It is not strictly necessary for the proof to show equality. However, to show equality, let $u$ be a unit vector, and set $x ^ { \\prime } = y ^ { \\prime } = 0$ , and $x = y = t u$ . Then ", + "bbox": [ + 173, + 719, + 823, + 748 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/a4134feb8370abe258458b2371f6fb5e993f6fbfac5fca607870342569d68ad3.jpg", + "text": "$$\n\\begin{array} { r l } & { c \\geq \\underset { t \\neq 0 } { \\operatorname* { s u p } } \\frac { K ( t u , t u ) - K ( t u , 0 ) - K ( 0 , t u ) + K ( 0 , 0 ) } { d ( t u , 0 ) d ( t u , 0 ) } } \\\\ & { = \\underset { t \\neq 0 } { \\operatorname* { s u p } } \\frac { 2 - 2 \\exp ( - \\frac { t ^ { 2 } } { 2 \\sigma ^ { 2 } } ) } { { t ^ { 2 } } } } \\\\ & { \\geq \\underset { t \\neq 0 } { \\operatorname* { l i m } } \\frac { 2 - 2 \\exp ( - \\frac { t ^ { 2 } } { 2 \\sigma ^ { 2 } } ) } { { t ^ { 2 } } } } \\\\ & { = \\underset { t \\neq 0 } { \\operatorname* { l i m } } \\frac { - 2 \\exp ( - \\frac { t ^ { 2 } } { 2 \\sigma ^ { 2 } } ) \\cdot - \\frac { 2 t } { 2 \\sigma ^ { 2 } } } { 2 t } } \\\\ & { = \\frac { 1 } { t \\neq 0 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 750, + 674, + 930 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "where we used l’Hopital’s rule to evaluate the limit. ˆ ", + "bbox": [ + 173, + 102, + 514, + 118 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proposition 11. The minimax and non-saturating GAN losses do not satisfy $( D 3 )$ for some $\\mu _ { 0 }$ ", + "bbox": [ + 173, + 126, + 794, + 142 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proof. For the minimax loss, the Bregman divergence is: ", + "bbox": [ + 173, + 156, + 549, + 171 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/ab011b3d7abc2c586431d4b326d7c8c5d5dba719d53a70a1f96be2e66fc544ea.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathfrak { D } _ { B \\ni \\{ \\cdot \\} \\parallel \\mu _ { 0 } } ( v , \\mu ) = } & { \\displaystyle \\int \\left[ \\frac { 1 } { 2 } \\log \\frac { \\mu _ { 0 } } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } } \\ d \\mu _ { 0 } + \\frac { 1 } { 2 } \\log \\frac { v } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } d \\nu \\right] } \\\\ & { \\quad - \\displaystyle \\int \\left[ \\frac { 1 } { 2 } \\log \\frac { \\mu _ { 0 } } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\mu } d \\mu _ { 0 } + \\frac { 1 } { 2 } \\log \\frac { \\mu } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\mu } d \\mu \\right] } \\\\ & { \\quad - \\displaystyle \\int \\frac { 1 } { 2 } \\log \\frac { \\mu } { \\mu _ { 0 } + \\mu } d ( \\nu - \\mu ) } \\\\ & { = \\frac { 1 } { 2 } \\int \\log \\frac { \\mu _ { 0 } + \\mu } { \\mu _ { 0 } + \\nu } d \\mu _ { 0 } + \\frac { 1 } { 2 } \\int \\left[ \\log \\frac { \\nu } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } - \\log \\frac { \\mu } { \\mu _ { 0 } + \\mu } \\right] d \\nu + \\frac { 1 } { 2 } \\log \\frac { 1 } { 2 } } \\\\ & { = \\frac { 1 } { 2 } \\int \\log \\frac { \\mu _ { 0 } + \\mu } { \\mu _ { 0 } + \\nu } d ( \\mu _ { 0 } + \\nu ) + \\frac { 1 } { 2 } \\int \\log \\frac { \\nu } { \\mu } d \\nu } \\\\ & { = D _ { \\mathrm { K L } } ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } ) \\parallel \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } ) + \\frac { 1 } { 2 } D _ { \\mathrm { K L } } ( \\nu | \\mu ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 176, + 818, + 383 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "and for the non-saturating loss, the Bregman divergence is ", + "bbox": [ + 171, + 386, + 558, + 401 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/624234490f122d9c8c8a64f264c0cffd9317857f4240f8be3fbea7675223677c.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\mathfrak { D } _ { N \\mathrm { k } , ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\mu _ { 0 } ) \\| \\mu _ { 0 } ) } ( \\nu , \\mu ) = \\int \\log \\frac { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } { \\mu _ { 0 } } d ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } ) } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } - \\int \\log \\frac { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\mu } { \\mu _ { 0 } } d ( \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } ) } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } + \\int \\frac { 1 } { 2 } \\log \\frac { \\mu _ { 0 } } { \\mu _ { 0 } + \\mu } d ( \\nu - \\mu ) } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } = \\frac { 1 } { 2 } \\int \\log \\frac { \\mu _ { 0 } + \\nu } { \\mu _ { 0 } + \\mu } d \\mu _ { 0 } + \\frac { 1 } { 2 } \\int \\log \\frac { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } { \\mu _ { 0 } + \\mu } d \\nu - \\frac { 1 } { 2 } \\log \\frac { 1 } { 2 } } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } = D _ { \\mathrm { K L } } ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } | | \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 407, + 782, + 570 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Choosing $\\nu$ to be not absolutely continuous w.r.t. ${ \\scriptstyle { \\frac { 1 } { 2 } } } \\mu + { \\scriptstyle { \\frac { 1 } { 2 } } } \\mu _ { 0 }$ makes the Bregman divergence $\\infty$ , which is sufficient to show that the Bregman divergence is not bounded by $| | \\mu - \\nu | | _ { \\mathrm { K R } } ^ { 2 }$ . ", + "bbox": [ + 173, + 575, + 825, + 608 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proposition 12. The MMD loss with Gaussian kernel satisfies $( D 3 )$ with $\\beta _ { 2 } = 2 \\pi$ for all $\\mu _ { 0 }$ . ", + "bbox": [ + 173, + 614, + 781, + 631 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proof. We work with the Gaussian kernel $K ( x , y ) = e ^ { - { \\frac { | | x - y | | ^ { 2 } } { 2 \\sigma ^ { 2 } } } }$ and use Proposition 10: ", + "bbox": [ + 173, + 645, + 751, + 667 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/8c6d7841abea1bee82f67bd624e7ca88be97bd4bfa59a83774faaae4385ae7e5.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathfrak { D } _ { \\frac { 1 } { 2 } \\mathrm { M M D } ^ { 2 } ( \\cdot , \\cdot , \\mu _ { 0 } ) } ( \\nu , \\mu ) = \\displaystyle \\frac { 1 } { 2 } \\int K ( x , y ) \\big ( \\nu - \\mu _ { 0 } ) ( d x ) \\big ( \\nu - \\mu _ { 0 } ) ( d y ) } & { } \\\\ & { \\qquad - \\frac { 1 } { 2 } \\int K ( x , y ) ( \\mu - \\mu _ { 0 } ) ( d x ) \\big ( \\mu - \\mu _ { 0 } ) ( d y ) } \\\\ & { \\qquad - \\displaystyle \\int ( K ( x , y ) \\mu ( d y ) - K ( x , y ) \\mu _ { 0 } ( d y ) \\big ) \\big ( \\nu - \\mu ) ( d x ) } \\\\ & { = \\displaystyle \\frac { 1 } { 2 } \\int K ( x , y ) \\big ( \\nu - \\mu ) ( d x ) \\big ( \\nu - \\mu ) ( d y ) } \\\\ & { = \\frac { 1 } { 2 } \\mathrm { M M D } ^ { 2 } ( \\mu , \\nu ) } \\\\ & { \\leq \\frac { 1 } { 2 \\pi ^ { 2 } } \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 674, + 761, + 867 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "where the last line is from Lemma 7. The result follows with $\\sigma = ( 2 \\pi ) ^ { - 1 / 2 }$ . ", + "bbox": [ + 173, + 872, + 674, + 890 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "The result actually applies more generally for the MMD loss with a differentiable kernel $K$ that satisfies $\\begin{array} { r } { \\beta _ { 2 } : = \\operatorname* { s u p } _ { x , y } \\bar { \\| \\nabla _ { x } \\nabla _ { y } K ( \\bar { x } , y ) \\| _ { 2 } } < \\infty } \\end{array}$ . □ ", + "bbox": [ + 171, + 895, + 828, + 926 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proposition 13 (Moreau–Yosida regularization). Suppose $J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }$ is convex, and define ${ \\tilde { J } } : = J \\oplus R _ { 3 }$ . Then $\\tilde { J }$ is convex, and $\\begin{array} { r } { \\mathfrak { D } _ { \\widetilde { J } } ( \\nu , \\mu ) \\le \\frac { \\beta _ { 2 } } { 2 } \\vert \\vert \\mu - \\nu \\vert \\vert _ { \\mathrm { K R } } ^ { 2 } } \\end{array}$ . ", + "bbox": [ + 171, + 102, + 825, + 137 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Proof. We work on the more general case of K(x, y) = (2πσ2)−d/2 exp(− kx−yk22σ2 ), and define ", + "bbox": [ + 171, + 190, + 802, + 212 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/fa50d72d8c9073b097d83aa6240cf0912e7cb04b02653c86e316076706e38cac.jpg", + "text": "$$\n\\tilde { J } ( \\mu ) = \\operatorname * { i n f } _ { \\tilde { \\mu } } J ( \\tilde { \\mu } ) + \\frac { c L } { 2 } \\mathrm { M M D } _ { K } ^ { 2 } ( \\mu , \\tilde { \\mu } )\n$$", + "text_format": "latex", + "bbox": [ + 372, + 229, + 624, + 261 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "for $c = \\sigma ^ { 2 } ( 2 \\pi \\sigma ^ { 2 } ) ^ { d / 2 }$ . ", + "bbox": [ + 173, + 281, + 318, + 297 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Let $\\mu ^ { * }$ be the unique minimizer of the infimum, which exists because the function is a strongly convex. By the envelope theorem, we compute that ", + "bbox": [ + 171, + 304, + 823, + 333 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/7ffd00c12dd3d738b41e3f3e443cb91be716fec8f159ea9e266b8d26f611c8df.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { d } { d \\epsilon } \\tilde { J } ( \\mu + \\epsilon \\chi ) \\Big | _ { \\epsilon = 0 } = \\frac { d } { d \\epsilon } \\frac { c L } { 2 } | | \\mu + \\epsilon \\chi - \\mu ^ { * } | | _ { \\mathcal { H } } ^ { 2 } \\Big | _ { \\epsilon = 0 } } \\\\ & { \\qquad = \\frac { d } { d \\epsilon } \\frac { c L } { 2 } \\big ( | | \\mu - \\mu ^ { * } | | _ { \\mathcal { H } } ^ { 2 } + 2 \\epsilon \\langle \\mu - \\mu ^ { * } , \\chi \\rangle _ { \\mathcal { H } } + \\epsilon ^ { 2 } | | \\chi | | _ { \\mathcal { H } } ^ { 2 } \\big ) \\Big | _ { \\epsilon = 0 } } \\\\ & { \\qquad = c L \\langle \\mu - \\mu ^ { * } , \\chi \\rangle _ { \\mathcal { H } } } \\\\ & { \\qquad = c L \\int ( \\mu - \\mu ^ { * } ) d \\chi , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 349, + 750, + 467 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "so $\\begin{array} { r } { \\frac { \\delta \\tilde { J } } { \\delta \\mu } = c L ( \\mu - \\mu ^ { * } ) } \\end{array}$ ", + "bbox": [ + 173, + 483, + 316, + 505 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Then ", + "bbox": [ + 173, + 510, + 210, + 523 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/6c3a00581fb70cabe3bb42b8516aef3b8aa0a2270f5120d88531855fe188a19a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathfrak { D } _ { \\mathcal { J } } ( \\nu , \\mu ) = \\mathcal { \\tilde { J } } ( \\nu ) - \\mathcal { \\tilde { J } } ( \\mu ) - \\displaystyle \\int \\frac { \\delta \\mathcal { \\tilde { J } } } { \\delta \\mu } d ( \\nu - \\mu ) } \\\\ & { \\qquad = \\operatorname* { i n f } J ( \\tilde { \\nu } ) + \\frac { c L } { 2 } \\| \\nu - \\tilde { \\nu } \\| _ { \\mathcal H } ^ { 2 } - \\left[ J ( \\mu ^ { * } ) + \\frac { c L } { 2 } \\| \\mu - \\mu ^ { * } \\| _ { \\mathcal H } ^ { 2 } \\right] - c L \\langle \\mu - \\mu ^ { * } , \\nu - \\mu \\rangle _ { \\mathcal H } } \\\\ & { \\qquad \\le J ( \\mu ^ { * } ) + \\frac { c L } { 2 } \\| \\nu - \\mu ^ { * } \\| _ { \\mathcal H } ^ { 2 } - \\left[ J ( \\mu ^ { * } ) + \\frac { c L } { 2 } \\| \\mu - \\mu ^ { * } \\| _ { \\mathcal H } ^ { 2 } \\right] - c L \\langle \\mu - \\mu ^ { * } , \\nu - \\mu \\rangle _ { \\mathcal H } } \\\\ & { \\qquad = \\frac { c L } { 2 } \\| \\mu - \\nu \\| _ { \\mathcal H } ^ { 2 } } \\\\ & { \\qquad \\le \\frac { c L } { 2 } \\cdot ( 2 \\pi \\sigma ^ { 2 } ) ^ { - d / 2 } \\cdot \\frac { 1 } { \\sigma ^ { 2 } } \\| \\mu - \\nu \\| _ { \\mathcal H } ^ { 2 } } \\\\ & { \\qquad = \\frac { L } { 2 } \\| \\mu - \\nu \\| _ { \\mathbb K ^ { 0 } } ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 194, + 541, + 802, + 733 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "where we used Lemma 7 for the second-to-last line. The proposition follows for $\\sigma = ( 2 \\pi ) ^ { - 1 / 2 }$ ", + "bbox": [ + 171, + 751, + 799, + 766 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Regarding this choice of $\\sigma$ , it will turn out that in the general case, the dual penalty includes a numerically unfavorable factor of $( 2 \\pi \\sigma ^ { 2 } ) ^ { - d / 2 }$ , dependent on the dimension of the problem. In practical applications, such as image generation, $d$ can be quite large, making the accurate computation of $( \\bar { 2 } \\pi \\sigma ^ { 2 } ) ^ { - d / 2 }$ completely infeasible. For numerical stability, we propose choosing the critical parameter $\\sigma = ( 2 \\pi ) ^ { - 1 / 2 }$ , which corresponds to the dimension-free kernel $K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }$ . □ ", + "bbox": [ + 173, + 772, + 825, + 849 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Lemma 8. Let $\\mathcal { H }$ be an RKHS with a continuous kernel $K$ on a compact domain $X$ . The convex conjugate of $\\mu \\mapsto \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathcal { H } } ^ { 2 }$ is $\\begin{array} { r } { f \\mapsto \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathcal { H } } ^ { 2 } + \\chi \\{ f \\in \\mathcal { H } \\} } \\end{array}$ . ", + "bbox": [ + 173, + 895, + 823, + 926 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Proof. First we show an upper bound for $f \\in \\mathcal H$ : ", + "bbox": [ + 173, + 103, + 496, + 119 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/cfeb19d9fe1087df3a7d5a9f63b774727f1173339edc9e4dd644357d389a913d.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { s u p } _ { \\mu } [ \\int f d \\mu - \\frac \\lambda 2 | | \\mu | | _ { \\mathcal H } ^ { 2 } ] = \\operatorname* { s u p } _ { \\mu } [ \\langle f , \\mu \\rangle _ { \\mathcal H } - \\frac \\lambda 2 | | \\mu | | _ { \\mathcal H } ^ { 2 } ] } \\\\ { \\displaystyle \\quad \\leq \\operatorname* { s u p } _ { \\mu } [ | | f | | \\mu | | \\mu | ] _ { \\mathcal H } - \\frac \\lambda 2 | | \\mu | | _ { \\mathcal H } ^ { 2 } ] } \\\\ { \\displaystyle \\quad \\leq \\operatorname* { s u p } _ { z \\in \\mathbb R } [ | | f | | \\mu z - \\frac \\lambda 2 z ^ { 2 } ] } \\\\ { \\displaystyle \\quad = | | f | | _ { \\mathcal H } \\cdot \\frac { | | f | | _ { \\mathcal H } } { \\lambda } - \\frac { \\lambda } { 2 } \\cdot \\frac { | | f | | _ { \\mathcal H } ^ { 2 } } { \\lambda ^ { 2 } } } \\\\ { \\displaystyle \\quad = \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathcal H } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 303, + 125, + 692, + 296 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "To derive a lower bound for general $f \\in { \\mathcal { C } } ( X )$ , we use the Mercer decomposition of the positive definite kernel $K$ , ", + "bbox": [ + 173, + 308, + 821, + 335 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/f73c3dff23878cabfa9851d9dc9c885ccc20f9c15583059c39242e1bd2092147.jpg", + "text": "$$\nK ( x , y ) = \\sum _ { i = 1 } ^ { \\infty } \\gamma _ { i } \\phi _ { i } ( x ) \\phi _ { i } ( y )\n$$", + "text_format": "latex", + "bbox": [ + 401, + 334, + 596, + 376 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "where $\\gamma _ { i } \\geq 0$ are eigenvalues and $\\{ \\phi _ { i } \\} _ { i = 1 } ^ { \\infty }$ is a complete orthonormal sequence of $L ^ { 2 } ( X ; d x )$ . It is well-known that the corresponding RKHS is given by ", + "bbox": [ + 173, + 381, + 825, + 410 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/a0b43008f029132cf6e1f4818b4b8095c38d1f36ace7f206aada2c77666db033.jpg", + "text": "$$\n\\mathcal { H } = \\left\\{ g \\in L ^ { 2 } ( X , d x ) \\mid \\sum _ { i = 1 } ^ { \\infty } \\frac { ( g , \\phi _ { i } ) _ { L ^ { 2 } ( X ) } ^ { 2 } } { \\gamma _ { i } } < \\infty \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 331, + 416, + 665, + 460 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "and the norm of $g \\in \\mathcal { H }$ by ", + "bbox": [ + 174, + 467, + 348, + 481 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/b400f2fc35ebd1d326e20b21252bf1468bdfbe1c7236951f737f3e3ce0b0afb3.jpg", + "text": "$$\n\\| g \\| _ { \\mathcal { H } } ^ { 2 } = \\sum _ { j = 1 } ^ { \\infty } \\frac { ( g , \\phi _ { i } ) _ { L ^ { 2 } ( X ) } ^ { 2 } } { \\gamma _ { i } } .\n$$", + "text_format": "latex", + "bbox": [ + 408, + 481, + 588, + 523 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Let $f \\in \\mathcal { C } ( X ) \\subset L ^ { 2 } ( X , d x )$ be an arbitrary function. We replace the supremum of the conjugate function ", + "bbox": [ + 173, + 535, + 825, + 563 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/ed785088a186fd54963882156734c2ebea8fdf870e931f808dd05275006ede84.jpg", + "text": "$$\n\\operatorname* { s u p } _ { \\mu } \\left[ \\int f d \\mu - \\frac { \\lambda } { 2 } | | \\hat { \\mu } | | _ { \\mathcal { H } } ^ { 2 } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 411, + 560, + 584, + 595 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "by $\\mu \\in { \\mathcal { M } } ( X )$ that has a square-integrable Radon–Nykodim derivative with respect to the Lebesgue measure $d x$ , and then we have a lower bound. We use $\\mu ( x )$ for the Radon–Nykodim derivative with slight abuse of notation. ", + "bbox": [ + 173, + 599, + 826, + 642 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Suppose $\\textstyle f ( x ) = \\sum _ { j = 1 } ^ { \\infty } a _ { j } \\phi _ { j } ( x )$ and $\\textstyle \\mu ( x ) = \\sum _ { j = 1 } ^ { \\infty } b _ { j } \\phi _ { j } ( x )$ are the expansion. Note that, since the kernel embedding is given by ", + "bbox": [ + 174, + 647, + 823, + 681 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/31e843bf819073f414ff1abba1634a0b9e791df02c541fb16eb868d729fb2510.jpg", + "text": "$$\n\\hat { \\mu } ( x ) = \\int K ( x , y ) \\mu ( y ) d y = \\sum _ { j = 1 } ^ { \\infty } \\gamma _ { j } b _ { j } \\phi _ { j } ( x ) ,\n$$", + "text_format": "latex", + "bbox": [ + 348, + 686, + 648, + 729 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "the above maximization is reduced to ", + "bbox": [ + 173, + 737, + 421, + 751 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/b461b3ba6bab32a9466d5688c6286da9addd520a2a168f184f158ce214cddf51.jpg", + "text": "$$\n\\operatorname * { s u p } _ { b _ { j } } \\sum _ { j = 1 } ^ { \\infty } a _ { j } b _ { j } - \\frac { \\lambda } { 2 } \\gamma _ { j } b _ { j } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 421, + 756, + 575, + 799 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "This is maximized when $b _ { j } = a _ { j } / ( \\lambda \\gamma _ { j } )$ , and the maximum value is ", + "bbox": [ + 173, + 806, + 617, + 823 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/f3a804b2de09c52c516df77c82bf614097d30c364b58a38786ad63679ce6fb94.jpg", + "text": "$$\n\\frac { 1 } { 2 \\lambda } \\sum _ { j = 1 } ^ { \\infty } \\frac { a _ { j } ^ { 2 } } { \\gamma _ { j } } .\n$$", + "text_format": "latex", + "bbox": [ + 459, + 829, + 539, + 872 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "If $f \\in \\mathcal H$ , this value is finite, and the lower bound of the conjugate given by is $\\frac { 1 } { 2 \\lambda } \\| f \\| _ { \\mathcal { H } } ^ { 2 }$ , which is the same as the upper bound. If $f \\notin { \\mathcal { H } }$ , the value (32) is infinite, and the conjugate function takes $+ \\infty$ . □ ", + "bbox": [ + 173, + 881, + 825, + 924 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Proposition 14. Let $\\mathcal { H }$ be an RKHS with the Gaussian kernel $K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }$ . Then for $f \\in \\mathcal H$ , ", + "bbox": [ + 168, + 102, + 825, + 133 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/4e5335bb7f6a79dc83592e1cc93b80593c6b89b9c6e21ce206fd1b2902fc1f58.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle | | f | | _ { \\mathcal { H } } ^ { 2 } = \\sum _ { k = 0 } ^ { \\infty } ( 4 \\pi ) ^ { - k } \\sum _ { \\stackrel { k _ { 1 } + \\cdots + k _ { d } = k } { k _ { 1 } \\prod _ { i = 1 } ^ { d } k _ { i } ! } } | | \\partial _ { x _ { 1 } } ^ { k _ { 1 } } \\cdot \\cdot \\cdot \\partial _ { x _ { d } } ^ { k _ { d } } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } \\\\ & { \\quad \\quad = | | f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 4 \\pi } | | \\nabla f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 1 6 \\pi ^ { 2 } } | | \\nabla ^ { 2 } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + o t h e r t e r m s . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 137, + 746, + 205 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Proof. We consider the more general case of $K ( x , y ) = ( 2 \\pi \\sigma ^ { 2 } ) ^ { - d / 2 } e ^ { - | | x - y | | ^ { 2 } / 2 \\sigma ^ { 2 } }$ , where we have ", + "bbox": [ + 171, + 217, + 823, + 236 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/e5b45c1eb9abda125f0ea4c739835f2877d44333f4eeae9e297115c0d5ae9312.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle | | f | | _ { \\mathcal { H } } ^ { 2 } = \\sum _ { k = 0 } ^ { \\infty } ( \\frac { 1 } { 2 } \\sigma ^ { 2 } ) ^ { k } \\sum _ { { k _ { 1 } + \\cdots + k _ { d } = k } \\atop = | | f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } \\frac { 1 } { | | \\nabla f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } | | \\partial _ { x _ { 1 } } ^ { k _ { 1 } } \\cdot \\cdot \\cdot \\partial _ { x _ { d } } ^ { k _ { d } } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } \\\\ & { \\quad \\quad \\quad = | | f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 2 } \\sigma ^ { 2 } | | \\nabla f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 4 } \\sigma ^ { 4 } | | \\nabla ^ { 2 } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\mathrm { o t h e r ~ t e r m s } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 242, + 750, + 309 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Novak et al. (2018) prove this result for $\\sigma = 1$ . We sketch the proof for the general case here. We use the Fourier transform convention that ", + "bbox": [ + 173, + 319, + 826, + 348 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/bdd526ae349d382f5be4edd161494406194d851fa0ee68dc437ee8d61cae6034.jpg", + "text": "$$\n\\hat { f } ( \\boldsymbol { k } ) = ( 2 \\pi ) ^ { - d / 2 } \\int _ { \\mathbb { R } ^ { d } } f ( \\boldsymbol { x } ) e ^ { - i \\boldsymbol { k } \\cdot \\boldsymbol { x } } d \\boldsymbol { x } , \\quad f ( \\boldsymbol { x } ) = ( 2 \\pi ) ^ { - d / 2 } \\int _ { \\mathbb { R } ^ { d } } \\hat { f } ( \\boldsymbol { k } ) e ^ { i \\boldsymbol { k } \\cdot \\boldsymbol { x } } d \\boldsymbol { k } .\n$$", + "text_format": "latex", + "bbox": [ + 245, + 352, + 750, + 386 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Consider the inner product ", + "bbox": [ + 173, + 398, + 352, + 414 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/e120d4868f2caf4db8d26813f760524a9c33c46f04f4c1d1f320c3a067b7dff4.jpg", + "text": "$$\n\\langle f , g \\rangle = \\int _ { \\mathbb { R } ^ { d } } e ^ { \\sigma ^ { 2 } | | k | | ^ { 2 } / 2 } \\hat { f } ( k ) \\overline { { \\hat { g } ( k ) } } d k ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 420, + 620, + 453 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "defined for functions $| | f | | < \\infty$ . Expanding the exponential in Taylor series, this inner product gives the equation for the norm in the proposition. By use of the Fourier inversion formula, it can be shown that $\\langle f , K _ { x } \\rangle = f ( x )$ , where ", + "bbox": [ + 173, + 459, + 825, + 502 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/c838bf10a41a4d619b0750db846400961ae55377bec74063bd8fc927d9798371.jpg", + "text": "$$\nK _ { x } ( y ) = ( 2 \\pi \\sigma ^ { 2 } ) ^ { - d / 2 } e ^ { - | | x - y | | ^ { 2 } / 2 \\sigma ^ { 2 } } ,\n$$", + "text_format": "latex", + "bbox": [ + 377, + 508, + 620, + 530 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "so this is an RKHS with the Gaussian kernel. 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GANs and their variants have shown impressive", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "performance in synthesizing various types of datasets, especially natural images. Despite these suc-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "cesses, the training of GANs remains quite unstable in nature, and this instability remains difficult", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 459, + 217, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 217, + 472 + ], + "score": 1.0, + "content": "to understand theoretically.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "Since the introduction of GANs, there have been many techniques proposed to stabilize GANs", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "training, including studies of new generator/discriminator architectures, loss functions, and regular-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "ization techniques. 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Our approach con-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 674, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 505, + 686 + ], + "score": 1.0, + "content": "siders the smoothness of the loss function used. In optimization, smoothness is a well-known con-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 684, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 698 + ], + "score": 1.0, + "content": "dition that ensures that gradient descent and its variants become stable (see e.g., Bertsekas (1999)).", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 696, + 492, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 492, + 708 + ], + "score": 1.0, + "content": "For example, the following well-known proposition is the starting point of our stability analysis:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 368, + 722 + ], + "score": 1.0, + "content": "Proposition 1 (Bertsekas (1999), Proposition 1.2.3). 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In this work, we develop a principled theoretical framework for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 273, + 470, + 286 + ], + "spans": [ + { + "bbox": [ + 141, + 273, + 470, + 286 + ], + "score": 1.0, + "content": "understanding the stability of various types of GANs. In particular, we derive", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 284, + 470, + 295 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 470, + 295 + ], + "score": 1.0, + "content": "conditions that guarantee eventual stationarity of the generator when it is trained", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 295, + 470, + 307 + ], + "spans": [ + { + "bbox": [ + 142, + 295, + 470, + 307 + ], + "score": 1.0, + "content": "with gradient descent, conditions that must be satisfied by the divergence that is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 305, + 470, + 319 + ], + "spans": [ + { + "bbox": [ + 141, + 305, + 470, + 319 + ], + "score": 1.0, + "content": "minimized by the GAN and the generator’s architecture. We find that existing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 317, + 470, + 329 + ], + "spans": [ + { + "bbox": [ + 142, + 317, + 470, + 329 + ], + "score": 1.0, + "content": "GAN variants satisfy some, but not all, of these conditions. Using tools from", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 328, + 470, + 339 + ], + "spans": [ + { + "bbox": [ + 141, + 328, + 470, + 339 + ], + "score": 1.0, + "content": "convex analysis, optimal transport, and reproducing kernels, we construct a GAN", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 339, + 469, + 351 + ], + "spans": [ + { + "bbox": [ + 141, + 339, + 469, + 351 + ], + "score": 1.0, + "content": "that fulfills these conditions simultaneously. In the process, we explain and clarify", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 350, + 469, + 362 + ], + "spans": [ + { + "bbox": [ + 142, + 350, + 469, + 362 + ], + "score": 1.0, + "content": "the need for various existing GAN stabilization techniques, including Lipschitz", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 361, + 398, + 373 + ], + "spans": [ + { + "bbox": [ + 141, + 361, + 398, + 373 + ], + "score": 1.0, + "content": "constraints, gradient penalties, and smooth activation functions.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16, + "bbox_fs": [ + 141, + 250, + 470, + 373 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 391, + 424, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 426, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 426, + 406 + ], + "score": 1.0, + "content": "1 INTRODUCTION: TAMING INSTABILITY WITH SMOOTHNESS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "Generative adversarial networks (Goodfellow et al., 2014), or GANs, are a powerful class of gen-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "erative models defined through minimax game. GANs and their variants have shown impressive", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "performance in synthesizing various types of datasets, especially natural images. Despite these suc-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "cesses, the training of GANs remains quite unstable in nature, and this instability remains difficult", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 459, + 217, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 217, + 472 + ], + "score": 1.0, + "content": "to understand theoretically.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 415, + 506, + 472 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "Since the introduction of GANs, there have been many techniques proposed to stabilize GANs", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "training, including studies of new generator/discriminator architectures, loss functions, and regular-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "ization techniques. Notably, Arjovsky et al. (2017) proposed Wasserstein GAN (WGAN), which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "in principle avoids instability caused by mismatched generator and data distribution supports. In", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "practice, this is enforced by Lipschitz constraints, which in turn motivated developments like gradi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "ent penalties (Gulrajani et al., 2017) and spectral normalization (Miyato et al., 2018). Indeed, these", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "score": 1.0, + "content": "stabilization techniques have proven essential to achieving the latest state-of-the-art results (Karras", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 553, + 235, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 235, + 565 + ], + "score": 1.0, + "content": "et al., 2018; Brock et al., 2019).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 476, + 506, + 565 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "On the other hand, a solid theoretical understanding of training stability has not been established.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "score": 1.0, + "content": "Several empirical observations point to an incomplete understanding. For example, why does apply-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "ing a gradient penalty together spectral norm seem to improve performance (Miyato et al., 2018),", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 603, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 506, + 615 + ], + "score": 1.0, + "content": "even though in principle they serve the same purpose? Why does applying only spectral normal-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "score": 1.0, + "content": "ization with the Wasserstein loss fail (Miyato, 2018), even though the analysis of Arjovsky et al.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "score": 1.0, + "content": "(2017) suggests it should be sufficient? Why is applying gradient penalties effective, even outside", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 635, + 376, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 376, + 648 + ], + "score": 1.0, + "content": "their original context of the Wasserstein GAN (Fedus et al., 2018)?", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 569, + 506, + 648 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 652, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 105, + 652, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 665 + ], + "score": 1.0, + "content": "In this work, we develop a framework to analyze the stability of GAN training that resolves these", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 663, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 676 + ], + "score": 1.0, + "content": "apparent contradictions and clarifies the roles of these regularization techniques. Our approach con-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 674, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 505, + 686 + ], + "score": 1.0, + "content": "siders the smoothness of the loss function used. In optimization, smoothness is a well-known con-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 684, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 698 + ], + "score": 1.0, + "content": "dition that ensures that gradient descent and its variants become stable (see e.g., Bertsekas (1999)).", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 696, + 492, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 492, + 708 + ], + "score": 1.0, + "content": "For example, the following well-known proposition is the starting point of our stability analysis:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 652, + 505, + 708 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 368, + 722 + ], + "score": 1.0, + "content": "Proposition 1 (Bertsekas (1999), Proposition 1.2.3). Suppose", + "type": "text" + }, + { + "bbox": [ + 369, + 709, + 431, + 721 + ], + "score": 0.89, + "content": "f : \\mathbb { R } ^ { p } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 709, + 444, + 722 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 444, + 710, + 452, + 720 + ], + "score": 0.68, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "-smooth and", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 719, + 429, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 187, + 733 + ], + "score": 1.0, + "content": "bounded below. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 720, + 288, + 734 + ], + "score": 0.93, + "content": "\\begin{array} { r } { x _ { k + 1 } : = x _ { k } - \\frac { 1 } { L } \\dot { \\nabla } \\dot { f } ( x _ { k } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 719, + 316, + 733 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 316, + 721, + 381, + 732 + ], + "score": 0.88, + "content": "| | \\nabla f ( x _ { k } ) | | 0", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 719, + 393, + 733 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 393, + 721, + 425, + 731 + ], + "score": 0.73, + "content": "k \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 719, + 429, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 709, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 95 + ], + "score": 1.0, + "content": "This proposition says that under a smoothness condition on the function, gradient descent with a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 182, + 106 + ], + "score": 1.0, + "content": "constant step size", + "type": "text" + }, + { + "bbox": [ + 182, + 93, + 191, + 106 + ], + "score": 0.87, + "content": "\\frac { \\mathbf { i } } { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "approaches stationarity (i.e., the gradient norm approaches zero). This is a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "rather weak notion of convergence, as it does not guarantee that the iterates converge to a point, and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 488, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 488, + 128 + ], + "score": 1.0, + "content": "even if the iterates do converge, the limit is a stationary point and not necessarily an minimizer.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 131, + 505, + 165 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 504, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 504, + 145 + ], + "score": 1.0, + "content": "Nevertheless, empirically, not even this stationarity is satisfied by GANs, which are known to fre-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "quently destabilize and diverge during training. To diagnose this instability, we consider the smooth-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 486, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 486, + 167 + ], + "score": 1.0, + "content": "ness of the GAN’s loss function. GANs are typically framed as minimax problems of the form", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 268, + 169, + 341, + 189 + ], + "lines": [ + { + "bbox": [ + 268, + 169, + 341, + 189 + ], + "spans": [ + { + "bbox": [ + 268, + 169, + 341, + 189 + ], + "score": 0.92, + "content": "\\operatorname* { i n f } _ { \\theta } \\operatorname* { s u p } _ { \\varphi } \\mathcal { I } ( \\mu _ { \\theta } , \\varphi ) ,", + "type": "interline_equation", + "image_path": "b263ffe4f4bc0564c57ae1535c9cc3193397bdbc281607ac8a876560293ef9f5.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 268, + 169, + 341, + 189 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 504, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 133, + 199 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 187, + 144, + 198 + ], + "score": 0.84, + "content": "\\mathcal { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 186, + 355, + 199 + ], + "score": 1.0, + "content": "is a loss function that takes a generator distribution", + "type": "text" + }, + { + "bbox": [ + 355, + 189, + 367, + 198 + ], + "score": 0.84, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 186, + 442, + 199 + ], + "score": 1.0, + "content": "and discriminator", + "type": "text" + }, + { + "bbox": [ + 442, + 189, + 450, + 198 + ], + "score": 0.81, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 186, + 471, + 199 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 472, + 187, + 504, + 197 + ], + "score": 0.9, + "content": "\\theta \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 198, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 209 + ], + "score": 1.0, + "content": "denotes the parameters of the generator. Unfortunately, the minimax nature of this problem makes", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 472, + 221 + ], + "score": 1.0, + "content": "stability and convergence difficult to analyze. To make the analysis more tractable, we define", + "type": "text" + }, + { + "bbox": [ + 472, + 208, + 505, + 221 + ], + "score": 0.92, + "content": "J ( \\mu ) =", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 219, + 277, + 233 + ], + "spans": [ + { + "bbox": [ + 107, + 219, + 163, + 233 + ], + "score": 0.93, + "content": "\\operatorname* { s u p } _ { \\varphi } \\mathcal { I } ( \\mu , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 220, + 277, + 233 + ], + "score": 1.0, + "content": ", so that (1) becomes simply", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 282, + 235, + 327, + 253 + ], + "lines": [ + { + "bbox": [ + 282, + 235, + 327, + 253 + ], + "spans": [ + { + "bbox": [ + 282, + 235, + 327, + 253 + ], + "score": 0.91, + "content": "\\operatorname* { i n f } _ { \\theta } J ( \\mu _ { \\theta } ) .", + "type": "interline_equation", + "image_path": "ef23c33984aaf9362a1047a22102199852bccfff2c1f0009208416645b663419.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 282, + 235, + 327, + 253 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 270 + ], + "score": 1.0, + "content": "This choice corresponds to the common assumption that the discriminator is allowed to reach opti-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "mality at every training step. Now, the GAN algorithm can be regarded as simply gradient descent", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 135, + 291 + ], + "score": 1.0, + "content": "on the", + "type": "text" + }, + { + "bbox": [ + 135, + 278, + 173, + 288 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { p } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 277, + 212, + 291 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 212, + 278, + 260, + 290 + ], + "score": 0.93, + "content": "\\theta \\mapsto J ( \\mu _ { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 277, + 505, + 291 + ], + "score": 1.0, + "content": ", which may be analyzed using Proposition 1. In particular,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 169, + 302 + ], + "score": 1.0, + "content": "if this function", + "type": "text" + }, + { + "bbox": [ + 169, + 289, + 217, + 301 + ], + "score": 0.93, + "content": "\\theta \\mapsto J ( \\mu _ { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 288, + 505, + 302 + ], + "score": 1.0, + "content": "satisfies the smoothness assumption, then the GAN training should be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 484, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 484, + 312 + ], + "score": 1.0, + "content": "stable in that it should approach stationarity under the assumption of an optimal discriminator.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 108, + 317, + 502, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 504, + 329 + ], + "score": 1.0, + "content": "In the remainder of this paper, we investigate whether the smoothness assumption is satisfied for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 328, + 338, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 338, + 339 + ], + "score": 1.0, + "content": "various GAN losses. Our analysis answers two questions:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 119, + 347, + 489, + 359 + ], + "lines": [ + { + "bbox": [ + 122, + 347, + 489, + 361 + ], + "spans": [ + { + "bbox": [ + 122, + 347, + 489, + 361 + ], + "score": 1.0, + "content": "Q1. Which existing GAN losses, if any, satisfy the smoothness condition in Proposition 1?", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 120, + 362, + 504, + 374 + ], + "lines": [ + { + "bbox": [ + 122, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 122, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "Q2. Are there choices of loss, regularization, or architecture that enforce smoothness in GANs?", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 342, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 343, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 343, + 394 + ], + "score": 1.0, + "content": "As results of our analysis, our contributions are as follows:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 129, + 402, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 129, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 129, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "1. We derive sufficient conditions for the GAN algorithm to be stationary under certain as-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 141, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "sumptions (Theorem 1). Our conditions relate to the smoothness of GAN loss used as well", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 424, + 304, + 437 + ], + "spans": [ + { + "bbox": [ + 141, + 424, + 304, + 437 + ], + "score": 1.0, + "content": "as the parameterization of the generator.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 128, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 128, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "2. We show that most common GAN losses do not satisfy the all of the smoothness conditions,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 449, + 336, + 462 + ], + "spans": [ + { + "bbox": [ + 141, + 449, + 336, + 462 + ], + "score": 1.0, + "content": "thereby corroborating their empirical instability.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 127, + 461, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 127, + 461, + 505, + 478 + ], + "score": 1.0, + "content": "3. We develop regularization techniques that enforce the smoothness conditions. These reg-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 142, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "ularizers recover common GAN stabilization techniques such as gradient penalties and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 486, + 475, + 498 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 475, + 498 + ], + "score": 1.0, + "content": "spectral normalization, thereby placing their use on a firmer theoretical foundation.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 128, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 128, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "4. Our analysis provides several practical insights, suggesting for example the use of smooth", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 513, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 142, + 513, + 505, + 523 + ], + "score": 1.0, + "content": "activation functions, simultaneous spectral normalization and gradient penalties, and a par-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 522, + 293, + 534 + ], + "spans": [ + { + "bbox": [ + 142, + 522, + 293, + 534 + ], + "score": 1.0, + "content": "ticular learning rate for the generator.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 546, + 200, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 202, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 202, + 559 + ], + "score": 1.0, + "content": "1.1 RELATED WORK", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 581 + ], + "score": 1.0, + "content": "Divergence minimization Our analysis regards the GAN algorithm as minimizing a divergence", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "between the current generator distribution and the desired data distribution, under the assumption of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "an optimal discriminator at every training step. This perspective originates from the earliest GAN", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "paper, in which Goodfellow et al. (2014) show that the original minimax GAN implicitly mini-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "mizes the Jensen–Shannon divergence. Since then, the community has introduced a large number", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "of GAN or GAN-like variants that learn generative models by implicitly minimizing various di-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 192, + 646 + ], + "score": 1.0, + "content": "vergences, including", + "type": "text" + }, + { + "bbox": [ + 193, + 633, + 200, + 645 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "-divergences (Nowozin et al., 2016), Wasserstein distance (Arjovsky et al.,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "2017), and maximum-mean discrepancy (Li et al., 2015; Unterthiner et al., 2018). Meanwhile, the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "non-saturating GAN (Goodfellow et al., 2014) has been shown to minimize a certain Kullback–", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "Leibler divergence (Arjovsky & Bottou, 2017). Several more theoretical works consider the topo-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "logical, geometric, and convexity properties of divergence minimization (Arjovsky & Bottou, 2017;", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Liu et al., 2017; Bottou et al., 2018; Farnia & Tse, 2018; Chu et al., 2019), perspectives that we draw", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "heavily upon. Sanjabi et al. (2018) also prove smoothness of GAN losses in the specific case of the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "regularized optimal transport loss. Their assumption for smoothness is entangled in that it involves a", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 721, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 733 + ], + "score": 1.0, + "content": "composite condition on generators and discriminators, while our analysis addresses them separately.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 42 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 95 + ], + "score": 1.0, + "content": "This proposition says that under a smoothness condition on the function, gradient descent with a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 182, + 106 + ], + "score": 1.0, + "content": "constant step size", + "type": "text" + }, + { + "bbox": [ + 182, + 93, + 191, + 106 + ], + "score": 0.87, + "content": "\\frac { \\mathbf { i } } { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "approaches stationarity (i.e., the gradient norm approaches zero). This is a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "rather weak notion of convergence, as it does not guarantee that the iterates converge to a point, and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 488, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 488, + 128 + ], + "score": 1.0, + "content": "even if the iterates do converge, the limit is a stationary point and not necessarily an minimizer.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 83, + 506, + 128 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 131, + 505, + 165 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 504, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 504, + 145 + ], + "score": 1.0, + "content": "Nevertheless, empirically, not even this stationarity is satisfied by GANs, which are known to fre-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "quently destabilize and diverge during training. To diagnose this instability, we consider the smooth-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 486, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 486, + 167 + ], + "score": 1.0, + "content": "ness of the GAN’s loss function. GANs are typically framed as minimax problems of the form", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 132, + 505, + 167 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 268, + 169, + 341, + 189 + ], + "lines": [ + { + "bbox": [ + 268, + 169, + 341, + 189 + ], + "spans": [ + { + "bbox": [ + 268, + 169, + 341, + 189 + ], + "score": 0.92, + "content": "\\operatorname* { i n f } _ { \\theta } \\operatorname* { s u p } _ { \\varphi } \\mathcal { I } ( \\mu _ { \\theta } , \\varphi ) ,", + "type": "interline_equation", + "image_path": "b263ffe4f4bc0564c57ae1535c9cc3193397bdbc281607ac8a876560293ef9f5.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 268, + 169, + 341, + 189 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 504, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 133, + 199 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 187, + 144, + 198 + ], + "score": 0.84, + "content": "\\mathcal { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 186, + 355, + 199 + ], + "score": 1.0, + "content": "is a loss function that takes a generator distribution", + "type": "text" + }, + { + "bbox": [ + 355, + 189, + 367, + 198 + ], + "score": 0.84, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 186, + 442, + 199 + ], + "score": 1.0, + "content": "and discriminator", + "type": "text" + }, + { + "bbox": [ + 442, + 189, + 450, + 198 + ], + "score": 0.81, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 186, + 471, + 199 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 472, + 187, + 504, + 197 + ], + "score": 0.9, + "content": "\\theta \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 198, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 209 + ], + "score": 1.0, + "content": "denotes the parameters of the generator. Unfortunately, the minimax nature of this problem makes", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 472, + 221 + ], + "score": 1.0, + "content": "stability and convergence difficult to analyze. To make the analysis more tractable, we define", + "type": "text" + }, + { + "bbox": [ + 472, + 208, + 505, + 221 + ], + "score": 0.92, + "content": "J ( \\mu ) =", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 219, + 277, + 233 + ], + "spans": [ + { + "bbox": [ + 107, + 219, + 163, + 233 + ], + "score": 0.93, + "content": "\\operatorname* { s u p } _ { \\varphi } \\mathcal { I } ( \\mu , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 220, + 277, + 233 + ], + "score": 1.0, + "content": ", so that (1) becomes simply", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 186, + 505, + 233 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 282, + 235, + 327, + 253 + ], + "lines": [ + { + "bbox": [ + 282, + 235, + 327, + 253 + ], + "spans": [ + { + "bbox": [ + 282, + 235, + 327, + 253 + ], + "score": 0.91, + "content": "\\operatorname* { i n f } _ { \\theta } J ( \\mu _ { \\theta } ) .", + "type": "interline_equation", + "image_path": "ef23c33984aaf9362a1047a22102199852bccfff2c1f0009208416645b663419.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 282, + 235, + 327, + 253 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 270 + ], + "score": 1.0, + "content": "This choice corresponds to the common assumption that the discriminator is allowed to reach opti-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "mality at every training step. Now, the GAN algorithm can be regarded as simply gradient descent", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 135, + 291 + ], + "score": 1.0, + "content": "on the", + "type": "text" + }, + { + "bbox": [ + 135, + 278, + 173, + 288 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { p } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 277, + 212, + 291 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 212, + 278, + 260, + 290 + ], + "score": 0.93, + "content": "\\theta \\mapsto J ( \\mu _ { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 277, + 505, + 291 + ], + "score": 1.0, + "content": ", which may be analyzed using Proposition 1. In particular,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 169, + 302 + ], + "score": 1.0, + "content": "if this function", + "type": "text" + }, + { + "bbox": [ + 169, + 289, + 217, + 301 + ], + "score": 0.93, + "content": "\\theta \\mapsto J ( \\mu _ { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 288, + 505, + 302 + ], + "score": 1.0, + "content": "satisfies the smoothness assumption, then the GAN training should be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 484, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 484, + 312 + ], + "score": 1.0, + "content": "stable in that it should approach stationarity under the assumption of an optimal discriminator.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 254, + 505, + 312 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 317, + 502, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 504, + 329 + ], + "score": 1.0, + "content": "In the remainder of this paper, we investigate whether the smoothness assumption is satisfied for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 328, + 338, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 338, + 339 + ], + "score": 1.0, + "content": "various GAN losses. Our analysis answers two questions:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 106, + 316, + 504, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 119, + 347, + 489, + 359 + ], + "lines": [ + { + "bbox": [ + 122, + 347, + 489, + 361 + ], + "spans": [ + { + "bbox": [ + 122, + 347, + 489, + 361 + ], + "score": 1.0, + "content": "Q1. Which existing GAN losses, if any, satisfy the smoothness condition in Proposition 1?", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 122, + 347, + 489, + 361 + ] + }, + { + "type": "text", + "bbox": [ + 120, + 362, + 504, + 374 + ], + "lines": [ + { + "bbox": [ + 122, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 122, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "Q2. Are there choices of loss, regularization, or architecture that enforce smoothness in GANs?", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 122, + 361, + 506, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 342, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 343, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 343, + 394 + ], + "score": 1.0, + "content": "As results of our analysis, our contributions are as follows:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 381, + 343, + 394 + ] + }, + { + "type": "list", + "bbox": [ + 129, + 402, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 129, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 129, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "1. We derive sufficient conditions for the GAN algorithm to be stationary under certain as-", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 141, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "sumptions (Theorem 1). Our conditions relate to the smoothness of GAN loss used as well", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 424, + 304, + 437 + ], + "spans": [ + { + "bbox": [ + 141, + 424, + 304, + 437 + ], + "score": 1.0, + "content": "as the parameterization of the generator.", + "type": "text" + } + ], + "index": 25, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 128, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "2. We show that most common GAN losses do not satisfy the all of the smoothness conditions,", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 449, + 336, + 462 + ], + "spans": [ + { + "bbox": [ + 141, + 449, + 336, + 462 + ], + "score": 1.0, + "content": "thereby corroborating their empirical instability.", + "type": "text" + } + ], + "index": 27, + "is_list_end_line": true + }, + { + "bbox": [ + 127, + 461, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 127, + 461, + 505, + 478 + ], + "score": 1.0, + "content": "3. We develop regularization techniques that enforce the smoothness conditions. These reg-", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 142, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "ularizers recover common GAN stabilization techniques such as gradient penalties and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 486, + 475, + 498 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 475, + 498 + ], + "score": 1.0, + "content": "spectral normalization, thereby placing their use on a firmer theoretical foundation.", + "type": "text" + } + ], + "index": 30, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 128, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "4. Our analysis provides several practical insights, suggesting for example the use of smooth", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 513, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 142, + 513, + 505, + 523 + ], + "score": 1.0, + "content": "activation functions, simultaneous spectral normalization and gradient penalties, and a par-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 522, + 293, + 534 + ], + "spans": [ + { + "bbox": [ + 142, + 522, + 293, + 534 + ], + "score": 1.0, + "content": "ticular learning rate for the generator.", + "type": "text" + } + ], + "index": 33, + "is_list_end_line": true + } + ], + "index": 28, + "bbox_fs": [ + 127, + 402, + 506, + 534 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 546, + 200, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 202, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 202, + 559 + ], + "score": 1.0, + "content": "1.1 RELATED WORK", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 581 + ], + "score": 1.0, + "content": "Divergence minimization Our analysis regards the GAN algorithm as minimizing a divergence", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "between the current generator distribution and the desired data distribution, under the assumption of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "an optimal discriminator at every training step. This perspective originates from the earliest GAN", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "paper, in which Goodfellow et al. (2014) show that the original minimax GAN implicitly mini-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "mizes the Jensen–Shannon divergence. Since then, the community has introduced a large number", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "of GAN or GAN-like variants that learn generative models by implicitly minimizing various di-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 192, + 646 + ], + "score": 1.0, + "content": "vergences, including", + "type": "text" + }, + { + "bbox": [ + 193, + 633, + 200, + 645 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "-divergences (Nowozin et al., 2016), Wasserstein distance (Arjovsky et al.,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "2017), and maximum-mean discrepancy (Li et al., 2015; Unterthiner et al., 2018). Meanwhile, the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "non-saturating GAN (Goodfellow et al., 2014) has been shown to minimize a certain Kullback–", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "Leibler divergence (Arjovsky & Bottou, 2017). Several more theoretical works consider the topo-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "logical, geometric, and convexity properties of divergence minimization (Arjovsky & Bottou, 2017;", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Liu et al., 2017; Bottou et al., 2018; Farnia & Tse, 2018; Chu et al., 2019), perspectives that we draw", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "heavily upon. Sanjabi et al. (2018) also prove smoothness of GAN losses in the specific case of the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "regularized optimal transport loss. Their assumption for smoothness is entangled in that it involves a", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 721, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 733 + ], + "score": 1.0, + "content": "composite condition on generators and discriminators, while our analysis addresses them separately.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 565, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 95, + 504, + 209 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 113, + 80, + 493, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 78, + 495, + 94 + ], + "spans": [ + { + "bbox": [ + 115, + 78, + 495, + 94 + ], + "score": 1.0, + "content": "Table 1: Common GANs, their corresponding loss functions, and their optimal discriminators.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 107, + 95, + 504, + 209 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 95, + 504, + 209 + ], + "spans": [ + { + "bbox": [ + 107, + 95, + 504, + 209 + ], + "score": 0.983, + "html": "
Loss function J(μ)Optimal discriminator Φμ(x)
Minimax GANDjs(μ|l μo)μ(x）
Non-saturating GANDKL(¹μ+μo |μo)-110g（）+(@） μo(x）
Wasserstein GANW1(μ, μo)argmaXfeLip1 Ey~μ[f(y)] -Ey~μo[f(y)]
GMMN,Coulomb GAN1MMD²(μ, μo)Ey~μ[K(x,y)]-Ey~μo[K(x,y)]
IPM-GANIPMj(μ, μo)arg maxf∈F Ey~μ[f(y)] -Ey~μof(y)]
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Note", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 525, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 472, + 541 + ], + "score": 1.0, + "content": "that the typical minimax formulation of GANs can be recovered from just the loss function", + "type": "text" + }, + { + "bbox": [ + 472, + 528, + 480, + 537 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 525, + 506, + 541 + ], + "score": 1.0, + "content": "using", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 362, + 551 + ], + "score": 1.0, + "content": "convex duality. 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Let", + "type": "text" + }, + { + "bbox": [ + 278, + 656, + 346, + 669 + ], + "score": 0.94, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 656, + 505, + 670 + ], + "score": 1.0, + "content": "be a convex, l.s.c., proper function. An", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 315, + 681 + ], + "score": 1.0, + "content": "optimal discriminator for a probability distribution", + "type": "text" + }, + { + "bbox": [ + 316, + 668, + 359, + 680 + ], + "score": 0.91, + "content": "{ \\boldsymbol { \\mu } } \\in { \\mathcal { P } } ( { \\boldsymbol { X } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 668, + 460, + 681 + ], + "score": 1.0, + "content": "is a continuous function", + "type": "text" + }, + { + "bbox": [ + 460, + 669, + 505, + 680 + ], + "score": 0.88, + "content": "\\Phi _ { \\mu } : X \\to ", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 679, + 485, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 115, + 689 + ], + "score": 0.7, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 679, + 356, + 693 + ], + "score": 1.0, + "content": "that attains the maximum of the second equation in (3), i.e.,", + "type": "text" + }, + { + "bbox": [ + 356, + 679, + 481, + 692 + ], + "score": 0.91, + "content": "\\begin{array} { r } { J ( \\mu ) = \\int \\Phi _ { \\mu } ( x ) d \\mu - J ^ { \\star } ( \\dot { \\Phi } _ { \\mu } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 679, + 485, + 693 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "This definition recovers the optimal discriminators of many existing GAN and GAN-like algorithms", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "score": 1.0, + "content": "(Farnia & Tse, 2018; Chu et al., 2019), most notably those in Table 1. Our analysis will apply to any", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 475, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 321, + 733 + ], + "score": 1.0, + "content": "algorithm in this family of algorithms. See Appendix", + "type": "text" + }, + { + "bbox": [ + 321, + 721, + 329, + 730 + ], + "score": 0.32, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 720, + 475, + 733 + ], + "score": 1.0, + "content": "for more details on this perspective.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 95, + 504, + 209 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 113, + 80, + 493, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 78, + 495, + 94 + ], + "spans": [ + { + "bbox": [ + 115, + 78, + 495, + 94 + ], + "score": 1.0, + "content": "Table 1: Common GANs, their corresponding loss functions, and their optimal discriminators.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 107, + 95, + 504, + 209 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 95, + 504, + 209 + ], + "spans": [ + { + "bbox": [ + 107, + 95, + 504, + 209 + ], + "score": 0.983, + "html": "
Loss function J(μ)Optimal discriminator Φμ(x)
Minimax GANDjs(μ|l μo)μ(x）
Non-saturating GANDKL(¹μ+μo |μo)-110g（）+(@） μo(x）
Wasserstein GANW1(μ, μo)argmaXfeLip1 Ey~μ[f(y)] -Ey~μo[f(y)]
GMMN,Coulomb GAN1MMD²(μ, μo)Ey~μ[K(x,y)]-Ey~μo[K(x,y)]
IPM-GANIPMj(μ, μo)arg maxf∈F Ey~μ[f(y)] -Ey~μof(y)]
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Li et al. (2017b) contrast this", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 249, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 261 + ], + "score": 1.0, + "content": "optimal discriminator dynamics with first-order dynamics, which assumes that the generator and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "score": 1.0, + "content": "discriminator use alternating gradient updates and is what is used computationally. 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In particular, recall that the convex conjugate", + "type": "text" + }, + { + "bbox": [ + 362, + 538, + 433, + 550 + ], + "score": 0.93, + "content": "J ^ { \\star } : \\bar { \\mathcal { C } } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 537, + 446, + 551 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 446, + 539, + 454, + 548 + ], + "score": 0.84, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "satisfies the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 548, + 390, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 390, + 562 + ], + "score": 1.0, + "content": "following remarkable duality, known as the Fenchel–Moreau theorem:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 503, + 506, + 562 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 561, + 469, + 590 + ], + "lines": [ + { + "bbox": [ + 141, + 561, + 469, + 590 + ], + "spans": [ + { + "bbox": [ + 141, + 561, + 469, + 590 + ], + "score": 0.9, + "content": "J ^ { \\star } ( \\varphi ) : = \\operatorname* { s u p } _ { \\mu \\in { \\mathcal M } ( X ) } \\int \\varphi ( x ) d \\mu - J ( \\mu ) , \\qquad J ( \\mu ) = \\operatorname* { s u p } _ { \\varphi \\in { \\mathcal C } ( X ) } \\int \\varphi ( x ) d \\mu - J ^ { \\star } ( \\varphi ) .", + "type": "interline_equation", + "image_path": "d419ff9c13d7375a6986c9ea0ec904943dd4bf0878e440c354002daec5042605.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 141, + 561, + 469, + 570.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 141, + 570.6666666666666, + 469, + 580.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 141, + 580.3333333333333, + 469, + 589.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 410, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 411, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 243, + 605 + ], + "score": 1.0, + "content": "Based on this duality, minimizing", + "type": "text" + }, + { + "bbox": [ + 243, + 592, + 251, + 601 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 589, + 411, + 605 + ], + "score": 1.0, + "content": "can be framed as the minimax problem", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 589, + 411, + 605 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 604, + 467, + 632 + ], + "lines": [ + { + "bbox": [ + 143, + 604, + 467, + 632 + ], + "spans": [ + { + "bbox": [ + 143, + 604, + 467, + 632 + ], + "score": 0.89, + "content": "\\operatorname* { i n f } _ { \\mu \\in { \\mathcal P } ( X ) } J ( \\mu ) = \\operatorname* { i n f } _ { \\mu \\in { \\mathcal P } ( X ) } \\operatorname* { s u p } _ { \\varphi \\in { \\mathcal C } ( X ) } \\int \\varphi ( x ) d \\mu - J ^ { \\star } ( \\varphi ) : = \\operatorname* { i n f } _ { \\mu \\in { \\mathcal P } ( X ) } \\operatorname* { s u p } _ { \\varphi \\in { \\mathcal C } ( X ) } { \\mathcal I } ( \\mu , \\varphi ) ,", + "type": "interline_equation", + "image_path": "af2c8ee3c32c845370873ed3a058ec7d56de1bf07f8c2dd3e4c0722ff2d15f49.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 143, + 604, + 467, + 632 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 504, + 645 + ], + "score": 1.0, + "content": "recovering the well-known adversarial formulation of GANs. 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Let", + "type": "text" + }, + { + "bbox": [ + 278, + 656, + 346, + 669 + ], + "score": 0.94, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 656, + 505, + 670 + ], + "score": 1.0, + "content": "be a convex, l.s.c., proper function. 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Our analysis will apply to any", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 475, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 321, + 733 + ], + "score": 1.0, + "content": "algorithm in this family of algorithms. See Appendix", + "type": "text" + }, + { + "bbox": [ + 321, + 721, + 329, + 730 + ], + "score": 0.32, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 720, + 475, + 733 + ], + "score": 1.0, + "content": "for more details on this perspective.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 698, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 327, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 328, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 328, + 96 + ], + "score": 1.0, + "content": "We also formalize the notion of a family of generators:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 97, + 505, + 131 + ], + "lines": [ + { + "bbox": [ + 105, + 96, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 505, + 111 + ], + "score": 1.0, + "content": "Definition 2 (Family of generators). A family of generators is a set of pushforward probability", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 109, + 506, + 121 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 147, + 121 + ], + "score": 1.0, + "content": "measures", + "type": "text" + }, + { + "bbox": [ + 148, + 109, + 254, + 121 + ], + "score": 0.9, + "content": "\\{ \\mu _ { \\theta } = f _ { \\theta \\# } \\omega : \\theta \\in \\mathbb { R } ^ { p } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 109, + 286, + 121 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 286, + 111, + 295, + 119 + ], + "score": 0.37, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 109, + 449, + 121 + ], + "score": 1.0, + "content": "is a fixed probability distribution on", + "type": "text" + }, + { + "bbox": [ + 449, + 109, + 458, + 119 + ], + "score": 0.76, + "content": "Z", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 109, + 506, + 121 + ], + "score": 1.0, + "content": "(the latent", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 119, + 381, + 132 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 163, + 132 + ], + "score": 1.0, + "content": "variable) and", + "type": "text" + }, + { + "bbox": [ + 163, + 120, + 214, + 131 + ], + "score": 0.9, + "content": "f _ { \\theta } : Z \\to X", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 119, + 381, + 132 + ], + "score": 1.0, + "content": "is a measurable function (the generator).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 140, + 506, + 196 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 451, + 154 + ], + "score": 1.0, + "content": "Now, in light of Proposition 1, we are interested in the smoothness of the mapping", + "type": "text" + }, + { + "bbox": [ + 452, + 141, + 501, + 153 + ], + "score": 0.92, + "content": "\\theta \\mapsto J ( \\mu _ { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 141, + 505, + 154 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 152, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 505, + 164 + ], + "score": 1.0, + "content": "which would guarantee the stationarity of gradient descent on this objective, which in turn implies", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 161, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 104, + 161, + 506, + 177 + ], + "score": 1.0, + "content": "stationarity of the GAN algorithm under the assumption of an optimal discriminator. The following", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 173, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 380, + 187 + ], + "score": 1.0, + "content": "theorem is our central result, which decomposes the smoothness of", + "type": "text" + }, + { + "bbox": [ + 380, + 173, + 428, + 186 + ], + "score": 0.93, + "content": "\\theta \\mapsto J ( \\mu _ { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 173, + 506, + 187 + ], + "score": 1.0, + "content": "into conditions on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 185, + 316, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 316, + 197 + ], + "score": 1.0, + "content": "optimal discriminators and the family of generators.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 504, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 342, + 213 + ], + "score": 1.0, + "content": "Theorem 1 (Smoothness decomposition for GANs). Let", + "type": "text" + }, + { + "bbox": [ + 342, + 199, + 416, + 212 + ], + "score": 0.92, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 198, + 505, + 213 + ], + "score": 1.0, + "content": "be a convex function", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 210, + 454, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 227, + 223 + ], + "score": 1.0, + "content": "whose optimal discriminators", + "type": "text" + }, + { + "bbox": [ + 228, + 211, + 281, + 223 + ], + "score": 0.92, + "content": "\\Phi _ { \\mu } : X \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 210, + 454, + 223 + ], + "score": 1.0, + "content": "satisfy the following regularity conditions:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 119, + 231, + 250, + 244 + ], + "lines": [ + { + "bbox": [ + 118, + 230, + 252, + 247 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This failure of existing GANs to satisfy the stationarity", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 515, + 339, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 339, + 526 + ], + "score": 1.0, + "content": "conditions corroborates the observed instability of GANs.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "table", + "bbox": [ + 210, + 552, + 401, + 632 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 215, + 536, + 395, + 549 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 215, + 536, + 396, + 550 + ], + "spans": [ + { + "bbox": [ + 215, + 536, + 396, + 550 + ], + "score": 1.0, + "content": "Table 2: Regularity of common GAN losses.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "table_body", + "bbox": [ + 210, + 552, + 401, + 632 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 210, + 552, + 401, + 632 + ], + "spans": [ + { + "bbox": [ + 210, + 552, + 401, + 632 + ], + "score": 0.975, + "html": "
(D1)(D2)(D3)
Minimax GAN×X×
Non-saturating GAN Wasserstein GANXX？
√X
×？
IPMs MMD²*
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In this paper, we focus on the discriminator", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "conditions (D1), (D2), and (D3) and only provide an extremely simple example of a generator that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "satisfies (G1) and (G2), in Section 7. 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The following", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 173, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 380, + 187 + ], + "score": 1.0, + "content": "theorem is our central result, which decomposes the smoothness of", + "type": "text" + }, + { + "bbox": [ + 380, + 173, + 428, + 186 + ], + "score": 0.93, + "content": "\\theta \\mapsto J ( \\mu _ { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 173, + 506, + 187 + ], + "score": 1.0, + "content": "into conditions on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 185, + 316, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 316, + 197 + ], + "score": 1.0, + "content": "optimal discriminators and the family of generators.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 141, + 506, + 197 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 504, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 342, + 213 + ], + "score": 1.0, + "content": "Theorem 1 (Smoothness decomposition for GANs). 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In order to obtain claims of stability for practically sized", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 430, + 381, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 381, + 442 + ], + "score": 1.0, + "content": "learning rates, it is important to tightly bound the relevant constants.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 395, + 505, + 442 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 460 + ], + "score": 1.0, + "content": "In Sections 4 to 6, we carefully analyze which GAN losses satisfy (D1), (D2), and (D3), and with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "what constants. We summarize our results in Table 2: it turns out that none of the listed losses,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "except for one, satisfy (D1), (D2), and (D3) simultaneously with a finite constant. The MMD-based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 394, + 494 + ], + "score": 1.0, + "content": "loss satisfies the three conditions, but its constant for (D1) grows as", + "type": "text" + }, + { + "bbox": [ + 394, + 480, + 449, + 493 + ], + "score": 0.94, + "content": "\\alpha = O ( { \\sqrt { d } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 480, + 506, + 494 + ], + "score": 1.0, + "content": ", which is an", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 493, + 504, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 300, + 504 + ], + "score": 1.0, + "content": "unfavorable dependence on the data dimension", + "type": "text" + }, + { + "bbox": [ + 301, + 493, + 308, + 502 + ], + "score": 0.74, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 493, + 504, + 504 + ], + "score": 1.0, + "content": "that forces an unacceptably small learning rate.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 502, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 517 + ], + "score": 1.0, + "content": "See for complete details of each condition. This failure of existing GANs to satisfy the stationarity", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 515, + 339, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 339, + 526 + ], + "score": 1.0, + "content": "conditions corroborates the observed instability of GANs.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 446, + 506, + 526 + ] + }, + { + "type": "table", + "bbox": [ + 210, + 552, + 401, + 632 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 215, + 536, + 395, + 549 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 215, + 536, + 396, + 550 + ], + "spans": [ + { + "bbox": [ + 215, + 536, + 396, + 550 + ], + "score": 1.0, + "content": "Table 2: Regularity of common GAN losses.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "table_body", + "bbox": [ + 210, + 552, + 401, + 632 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 210, + 552, + 401, + 632 + ], + "spans": [ + { + "bbox": [ + 210, + 552, + 401, + 632 + ], + "score": 0.975, + "html": "
(D1)(D2)(D3)
Minimax GAN×X×
Non-saturating GAN Wasserstein GANXX？
√X
×？
IPMs MMD²*
", + "type": "table", + "image_path": "b9ceade9e50a525c645cc621f8f6147e563dfcb5e0881bce3a2ff301dcdfa076.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 210, + 552, + 401, + 568.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 210, + 568.0, + 401, + 584.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 210, + 584.0, + 401, + 600.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 210, + 600.0, + 401, + 616.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 210, + 616.0, + 401, + 632.0 + ], + "spans": [], + "index": 36 + } + ] + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 504, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 504, + 656 + ], + "score": 1.0, + "content": "Theorem 1 decomposes smoothness into conditions on the generator and conditions on the dis-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "criminator, allowing a clean separation of concerns. In this paper, we focus on the discriminator", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "conditions (D1), (D2), and (D3) and only provide an extremely simple example of a generator that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "satisfies (G1) and (G2), in Section 7. Because analysis of the generator conditions may become quite", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "complicated and will vary with the choice of architecture considered (feedforward, convolutional,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 698, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 505, + 713 + ], + "score": 1.0, + "content": "ResNet, etc.), we leave a detailed analysis of the generator conditions (G1) and (G2) as a promising", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "avenue for future work. Indeed, such analyses may lead to new generator architectures or generator", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 721, + 322, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 322, + 734 + ], + "score": 1.0, + "content": "regularization techniques that stabilize GAN training.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 644, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "In this section, we present a generic regularization technique that imposes the three conditions suf-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 326, + 129 + ], + "score": 1.0, + "content": "ficient for stable learning on an arbitrary loss function", + "type": "text" + }, + { + "bbox": [ + 326, + 118, + 333, + 127 + ], + "score": 0.71, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 117, + 505, + 129 + ], + "score": 1.0, + "content": ", thereby stabilizing training. In Section 2,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "score": 1.0, + "content": "we observe that the Wasserstein, IPM, and MMD losses respectively satisfy (D1), (D2), and (D3)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "score": 1.0, + "content": "individually, but not all of of them at the same time. Using techniques from convex analysis, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "convert these three GAN losses into three regularizers that, when applied simultaneously, causes the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "score": 1.0, + "content": "resulting loss to satisfy all the three conditions. Here, we only outline the technique; the specifics of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 172, + 271, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 271, + 183 + ], + "score": 1.0, + "content": "each case are deferred to Sections 4 to 6.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 188, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 289, + 201 + ], + "score": 1.0, + "content": "We start with an arbitrary base loss function", + "type": "text" + }, + { + "bbox": [ + 290, + 189, + 297, + 199 + ], + "score": 0.84, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "to be regularized. Next, we take an existing GAN", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 200, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 487, + 211 + ], + "score": 1.0, + "content": "loss that satisfies the desired regularity condition and convert it into a regularizer function", + "type": "text" + }, + { + "bbox": [ + 487, + 200, + 497, + 210 + ], + "score": 0.51, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 200, + 505, + 211 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 210, + 455, + 223 + ], + "spans": [ + { + "bbox": [ + 107, + 210, + 159, + 223 + ], + "score": 0.93, + "content": "\\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 210, + 239, + 223 + ], + "score": 1.0, + "content": ". Then, we consider", + "type": "text" + }, + { + "bbox": [ + 240, + 211, + 267, + 221 + ], + "score": 0.9, + "content": "J \\oplus R", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 210, + 455, + 223 + ], + "score": 1.0, + "content": ", which denotes the inf-convolution defined as", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 227, + 389, + 250 + ], + "lines": [ + { + "bbox": [ + 222, + 227, + 389, + 250 + ], + "spans": [ + { + "bbox": [ + 222, + 227, + 389, + 250 + ], + "score": 0.94, + "content": "( J \\oplus R ) ( \\xi ) = \\operatorname* { i n f } _ { \\tilde { \\xi } \\in \\mathcal { M } ( X ) } J ( \\tilde { \\xi } ) + R ( \\xi - \\tilde { \\xi } ) .", + "type": "interline_equation", + "image_path": "f0186e0736f9b4233f82f5be43e4dbaba7c7fd4af6dc65957492b3dd1009ec20.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 222, + 227, + 389, + 250 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 182, + 267 + ], + "score": 1.0, + "content": "This new function", + "type": "text" + }, + { + "bbox": [ + 183, + 255, + 211, + 266 + ], + "score": 0.91, + "content": "J \\oplus R", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 255, + 313, + 267 + ], + "score": 1.0, + "content": "inherits the regularity of", + "type": "text" + }, + { + "bbox": [ + 313, + 255, + 322, + 265 + ], + "score": 0.82, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 255, + 505, + 267 + ], + "score": 1.0, + "content": ", making it a stable candidate as a GAN loss.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "Moreover, because the inf-convolution is a commutative operation, we can sequentially apply multi-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 169, + 290 + ], + "score": 1.0, + "content": "ple regularizers", + "type": "text" + }, + { + "bbox": [ + 169, + 277, + 199, + 288 + ], + "score": 0.34, + "content": "R _ { 1 } , R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 277, + 219, + 290 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 219, + 277, + 232, + 288 + ], + "score": 0.89, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "without destroying the added regularity. In particular, if we carefully", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 177, + 302 + ], + "score": 1.0, + "content": "choose functions", + "type": "text" + }, + { + "bbox": [ + 177, + 290, + 190, + 300 + ], + "score": 0.8, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 289, + 194, + 302 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 195, + 289, + 208, + 300 + ], + "score": 0.75, + "content": "R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 289, + 228, + 302 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 229, + 289, + 242, + 300 + ], + "score": 0.89, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 289, + 266, + 302 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 267, + 288, + 369, + 300 + ], + "score": 0.92, + "content": "\\tilde { J } = \\bar { J } \\oplus \\bar { R _ { 1 } } \\oplus R _ { 2 } \\oplus \\bar { R _ { 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "will satisfy (D1), (D2), and (D3)", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 462, + 314 + ], + "score": 1.0, + "content": "simultaneously. Moreover, under some technical assumptions, this composite function", + "type": "text" + }, + { + "bbox": [ + 463, + 300, + 471, + 311 + ], + "score": 0.86, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "inherits", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 312, + 351, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 213, + 326 + ], + "score": 1.0, + "content": "the original minimizers of", + "type": "text" + }, + { + "bbox": [ + 213, + 313, + 221, + 322 + ], + "score": 0.81, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 312, + 351, + 326 + ], + "score": 1.0, + "content": ", making it a sensible GAN loss:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 297, + 340 + ], + "score": 1.0, + "content": "Proposition 2 (Invariance of minimizers). Let", + "type": "text" + }, + { + "bbox": [ + 298, + 327, + 441, + 340 + ], + "score": 0.85, + "content": "R _ { 1 } ( \\xi ) : = \\| \\xi \\| _ { \\mathrm { K R } } , R _ { 2 } ( \\xi ) : = \\| \\xi \\| _ { S ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 325, + 464, + 340 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 464, + 327, + 505, + 339 + ], + "score": 0.9, + "content": "R _ { 3 } ( \\xi ) : = ", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 338, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 107, + 338, + 136, + 353 + ], + "score": 0.92, + "content": "\\frac { 1 } { 2 } \\| \\hat { \\xi } \\| _ { \\mathcal { H } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 339, + 487, + 354 + ], + "score": 1.0, + "content": "be the three regularizers defined by (8), (12), and (19) respectively. Assume that", + "type": "text" + }, + { + "bbox": [ + 487, + 340, + 505, + 351 + ], + "score": 0.67, + "content": " { J } :", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 351, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 107, + 352, + 162, + 365 + ], + "score": 0.9, + "content": "\\bar { \\mathcal { M } } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 351, + 273, + 366 + ], + "score": 1.0, + "content": "has a unique minimizer at", + "type": "text" + }, + { + "bbox": [ + 273, + 354, + 285, + 364 + ], + "score": 0.81, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 351, + 307, + 366 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 308, + 352, + 354, + 365 + ], + "score": 0.92, + "content": "J ( \\mu _ { 0 } ) = 0 ;", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 351, + 377, + 366 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 377, + 352, + 465, + 365 + ], + "score": 0.92, + "content": "J ( \\mu ) \\geq c \\| \\hat { \\mu } - \\hat { \\mu } _ { 0 } \\| _ { \\mathcal { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 351, + 505, + 366 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 135, + 376 + ], + "score": 0.87, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 365, + 246, + 378 + ], + "score": 1.0, + "content": ". Then the inf-convolution", + "type": "text" + }, + { + "bbox": [ + 247, + 365, + 356, + 377 + ], + "score": 0.9, + "content": "\\tilde { J } = J \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 365, + 471, + 378 + ], + "score": 1.0, + "content": "has a unique minimizer at", + "type": "text" + }, + { + "bbox": [ + 471, + 367, + 483, + 377 + ], + "score": 0.83, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 375, + 155, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 151, + 390 + ], + "score": 0.9, + "content": "\\tilde { J } ( \\mu _ { 0 } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 375, + 155, + 390 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 105, + 398, + 504, + 422 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "The duality formulation (4) provides a practical method for minimizing this composite function. We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 408, + 470, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 221, + 423 + ], + "score": 1.0, + "content": "leverage the duality relation", + "type": "text" + }, + { + "bbox": [ + 221, + 409, + 410, + 421 + ], + "score": 0.91, + "content": "( J \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus R _ { 3 } ) ^ { \\star } = J ^ { \\star } + R _ { 1 } ^ { \\star } + R _ { 2 } ^ { \\star } + R _ { 3 } ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 408, + 470, + 423 + ], + "score": 1.0, + "content": "and apply (4):", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 426, + 479, + 475 + ], + "lines": [ + { + "bbox": [ + 132, + 426, + 479, + 475 + ], + "spans": [ + { + "bbox": [ + 132, + 426, + 479, + 475 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\underset { \\mu } { \\operatorname* { i n f } } \\left( J \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus R _ { 3 } \\right) ( \\mu ) = \\underset { \\mu \\atop \\varphi } { \\operatorname* { i n f } } \\underset { \\varphi } { \\operatorname* { s u p } } \\int \\varphi d \\mu - J ^ { \\star } ( \\varphi ) - R _ { 1 } ^ { \\star } ( \\varphi ) - R _ { 2 } ^ { \\star } ( \\varphi ) - R _ { 3 } ^ { \\star } ( \\varphi ) } \\\\ { = \\underset { \\mu \\atop \\varphi } { \\operatorname* { i n f } } \\underset { \\varphi } { \\operatorname* { s u p } } \\mathcal { I } ( \\mu , \\varphi ) - R _ { 1 } ^ { \\star } ( \\varphi ) - R _ { 2 } ^ { \\star } ( \\varphi ) - R _ { 3 } ^ { \\star } ( \\varphi ) . } \\end{array}", + "type": "interline_equation", + "image_path": "8e90d785a28f709fc118a9b6728e042f010918444b52bd7f606aad06d672a944.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 132, + 426, + 479, + 442.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 132, + 442.3333333333333, + 479, + 458.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 132, + 458.66666666666663, + 479, + 474.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 479, + 504, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 492 + ], + "score": 1.0, + "content": "This minimax problem can be seen as a GAN whose discriminator objective has three added regu-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 489, + 174, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 174, + 502 + ], + "score": 1.0, + "content": "larization terms.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 105, + 506, + 504, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "The concrete form of these regularizers are summarized in Table 3. Notably, we observe that we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 518, + 309, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 309, + 531 + ], + "score": 1.0, + "content": "recover standard techniques for stabilizing GANs:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 129, + 538, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 131, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 131, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "• (D1) is enforced by Lipschitz constraints (i.e., spectral normalization) on the discrimi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 550, + 168, + 561 + ], + "spans": [ + { + "bbox": [ + 141, + 550, + 168, + 561 + ], + "score": 1.0, + "content": "nator.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 133, + 564, + 504, + 577 + ], + "spans": [ + { + "bbox": [ + 133, + 564, + 504, + 577 + ], + "score": 1.0, + "content": "• (D2) is enforced by spectral normalization and a choice of Lipschitz, smooth activation", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 575, + 268, + 587 + ], + "spans": [ + { + "bbox": [ + 141, + 575, + 268, + 587 + ], + "score": 1.0, + "content": "functions for the discriminator.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "table", + "bbox": [ + 122, + 651, + 488, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 602, + 504, + 648 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 601, + 504, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 504, + 615 + ], + "score": 1.0, + "content": "Table 3: Smoothness-inducing regularizers and their convex conjugates. To enforce a regularity", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 614, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 221, + 626 + ], + "score": 1.0, + "content": "condition on a loss function", + "type": "text" + }, + { + "bbox": [ + 221, + 614, + 229, + 624 + ], + "score": 0.78, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 614, + 492, + 626 + ], + "score": 1.0, + "content": ", we take a source loss that satisfies it and view it as a regularizer", + "type": "text" + }, + { + "bbox": [ + 492, + 614, + 501, + 623 + ], + "score": 0.73, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 614, + 504, + 626 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 624, + 504, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 256, + 637 + ], + "score": 1.0, + "content": "We then consider the inf-convolution", + "type": "text" + }, + { + "bbox": [ + 256, + 625, + 283, + 635 + ], + "score": 0.9, + "content": "J \\oplus R", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 624, + 490, + 637 + ], + "score": 1.0, + "content": ", which corresponds to an added regularization term", + "type": "text" + }, + { + "bbox": [ + 491, + 625, + 504, + 635 + ], + "score": 0.86, + "content": "R ^ { \\star }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 635, + 473, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 189, + 649 + ], + "score": 1.0, + "content": "on the discriminator", + "type": "text" + }, + { + "bbox": [ + 189, + 637, + 197, + 647 + ], + "score": 0.79, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 635, + 473, + 649 + ], + "score": 1.0, + "content": ". These regularization terms correspond to existing GAN techniques.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "table_body", + "bbox": [ + 122, + 651, + 488, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 651, + 488, + 730 + ], + "spans": [ + { + "bbox": [ + 122, + 651, + 488, + 730 + ], + "score": 0.981, + "html": "
PurposeSource lossR()R（）GAN techniques
(D1)W1|KRx{lφ|lLip ≤1}spectral norm
(D2)IPMs|/| s*x{∈s}smooth activations, spectral norm
(D3)MMD²gradient penalties
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In Section 2,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "score": 1.0, + "content": "we observe that the Wasserstein, IPM, and MMD losses respectively satisfy (D1), (D2), and (D3)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "score": 1.0, + "content": "individually, but not all of of them at the same time. Using techniques from convex analysis, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "convert these three GAN losses into three regularizers that, when applied simultaneously, causes the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "score": 1.0, + "content": "resulting loss to satisfy all the three conditions. Here, we only outline the technique; the specifics of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 172, + 271, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 271, + 183 + ], + "score": 1.0, + "content": "each case are deferred to Sections 4 to 6.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 105, + 506, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 188, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 289, + 201 + ], + "score": 1.0, + "content": "We start with an arbitrary base loss function", + "type": "text" + }, + { + "bbox": [ + 290, + 189, + 297, + 199 + ], + "score": 0.84, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "to be regularized. Next, we take an existing GAN", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 200, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 487, + 211 + ], + "score": 1.0, + "content": "loss that satisfies the desired regularity condition and convert it into a regularizer function", + "type": "text" + }, + { + "bbox": [ + 487, + 200, + 497, + 210 + ], + "score": 0.51, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 200, + 505, + 211 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 210, + 455, + 223 + ], + "spans": [ + { + "bbox": [ + 107, + 210, + 159, + 223 + ], + "score": 0.93, + "content": "\\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 210, + 239, + 223 + ], + "score": 1.0, + "content": ". Then, we consider", + "type": "text" + }, + { + "bbox": [ + 240, + 211, + 267, + 221 + ], + "score": 0.9, + "content": "J \\oplus R", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 210, + 455, + 223 + ], + "score": 1.0, + "content": ", which denotes the inf-convolution defined as", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 189, + 505, + 223 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 227, + 389, + 250 + ], + "lines": [ + { + "bbox": [ + 222, + 227, + 389, + 250 + ], + "spans": [ + { + "bbox": [ + 222, + 227, + 389, + 250 + ], + "score": 0.94, + "content": "( J \\oplus R ) ( \\xi ) = \\operatorname* { i n f } _ { \\tilde { \\xi } \\in \\mathcal { M } ( X ) } J ( \\tilde { \\xi } ) + R ( \\xi - \\tilde { \\xi } ) .", + "type": "interline_equation", + "image_path": "f0186e0736f9b4233f82f5be43e4dbaba7c7fd4af6dc65957492b3dd1009ec20.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 222, + 227, + 389, + 250 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 182, + 267 + ], + "score": 1.0, + "content": "This new function", + "type": "text" + }, + { + "bbox": [ + 183, + 255, + 211, + 266 + ], + "score": 0.91, + "content": "J \\oplus R", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 255, + 313, + 267 + ], + "score": 1.0, + "content": "inherits the regularity of", + "type": "text" + }, + { + "bbox": [ + 313, + 255, + 322, + 265 + ], + "score": 0.82, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 255, + 505, + 267 + ], + "score": 1.0, + "content": ", making it a stable candidate as a GAN loss.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "Moreover, because the inf-convolution is a commutative operation, we can sequentially apply multi-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 169, + 290 + ], + "score": 1.0, + "content": "ple regularizers", + "type": "text" + }, + { + "bbox": [ + 169, + 277, + 199, + 288 + ], + "score": 0.34, + "content": "R _ { 1 } , R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 277, + 219, + 290 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 219, + 277, + 232, + 288 + ], + "score": 0.89, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "without destroying the added regularity. In particular, if we carefully", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 177, + 302 + ], + "score": 1.0, + "content": "choose functions", + "type": "text" + }, + { + "bbox": [ + 177, + 290, + 190, + 300 + ], + "score": 0.8, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 289, + 194, + 302 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 195, + 289, + 208, + 300 + ], + "score": 0.75, + "content": "R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 289, + 228, + 302 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 229, + 289, + 242, + 300 + ], + "score": 0.89, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 289, + 266, + 302 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 267, + 288, + 369, + 300 + ], + "score": 0.92, + "content": "\\tilde { J } = \\bar { J } \\oplus \\bar { R _ { 1 } } \\oplus R _ { 2 } \\oplus \\bar { R _ { 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "will satisfy (D1), (D2), and (D3)", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 462, + 314 + ], + "score": 1.0, + "content": "simultaneously. Moreover, under some technical assumptions, this composite function", + "type": "text" + }, + { + "bbox": [ + 463, + 300, + 471, + 311 + ], + "score": 0.86, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "inherits", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 312, + 351, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 213, + 326 + ], + "score": 1.0, + "content": "the original minimizers of", + "type": "text" + }, + { + "bbox": [ + 213, + 313, + 221, + 322 + ], + "score": 0.81, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 312, + 351, + 326 + ], + "score": 1.0, + "content": ", making it a sensible GAN loss:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 255, + 505, + 326 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 297, + 340 + ], + "score": 1.0, + "content": "Proposition 2 (Invariance of minimizers). Let", + "type": "text" + }, + { + "bbox": [ + 298, + 327, + 441, + 340 + ], + "score": 0.85, + "content": "R _ { 1 } ( \\xi ) : = \\| \\xi \\| _ { \\mathrm { K R } } , R _ { 2 } ( \\xi ) : = \\| \\xi \\| _ { S ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 325, + 464, + 340 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 464, + 327, + 505, + 339 + ], + "score": 0.9, + "content": "R _ { 3 } ( \\xi ) : = ", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 338, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 107, + 338, + 136, + 353 + ], + "score": 0.92, + "content": "\\frac { 1 } { 2 } \\| \\hat { \\xi } \\| _ { \\mathcal { H } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 339, + 487, + 354 + ], + "score": 1.0, + "content": "be the three regularizers defined by (8), (12), and (19) respectively. Assume that", + "type": "text" + }, + { + "bbox": [ + 487, + 340, + 505, + 351 + ], + "score": 0.67, + "content": " { J } :", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 351, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 107, + 352, + 162, + 365 + ], + "score": 0.9, + "content": "\\bar { \\mathcal { M } } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 351, + 273, + 366 + ], + "score": 1.0, + "content": "has a unique minimizer at", + "type": "text" + }, + { + "bbox": [ + 273, + 354, + 285, + 364 + ], + "score": 0.81, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 351, + 307, + 366 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 308, + 352, + 354, + 365 + ], + "score": 0.92, + "content": "J ( \\mu _ { 0 } ) = 0 ;", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 351, + 377, + 366 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 377, + 352, + 465, + 365 + ], + "score": 0.92, + "content": "J ( \\mu ) \\geq c \\| \\hat { \\mu } - \\hat { \\mu } _ { 0 } \\| _ { \\mathcal { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 351, + 505, + 366 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 135, + 376 + ], + "score": 0.87, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 365, + 246, + 378 + ], + "score": 1.0, + "content": ". Then the inf-convolution", + "type": "text" + }, + { + "bbox": [ + 247, + 365, + 356, + 377 + ], + "score": 0.9, + "content": "\\tilde { J } = J \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 365, + 471, + 378 + ], + "score": 1.0, + "content": "has a unique minimizer at", + "type": "text" + }, + { + "bbox": [ + 471, + 367, + 483, + 377 + ], + "score": 0.83, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 375, + 155, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 151, + 390 + ], + "score": 0.9, + "content": "\\tilde { J } ( \\mu _ { 0 } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 375, + 155, + 390 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 325, + 505, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 398, + 504, + 422 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "The duality formulation (4) provides a practical method for minimizing this composite function. 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These regularization terms correspond to existing GAN techniques.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "table_body", + "bbox": [ + 122, + 651, + 488, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 651, + 488, + 730 + ], + "spans": [ + { + "bbox": [ + 122, + 651, + 488, + 730 + ], + "score": 0.981, + "html": "
PurposeSource lossR()R（）GAN techniques
(D1)W1|KRx{lφ|lLip ≤1}spectral norm
(D2)IPMs|/| s*x{∈s}smooth activations, spectral norm
(D3)MMD²gradient penalties
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This is a sensible criterion, since a discriminator whose gradient norm is too large may push", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 245, + 307, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 307, + 259 + ], + "score": 1.0, + "content": "the generator too hard and destabilize its training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 504, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 260, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 433, + 276 + ], + "score": 1.0, + "content": "To check (D1), the following proposition shows that it suffices to check whether", + "type": "text" + }, + { + "bbox": [ + 433, + 262, + 505, + 275 + ], + "score": 0.9, + "content": "| J ( \\mu ) - J ( \\nu ) | \\leq", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 273, + 253, + 287 + ], + "spans": [ + { + "bbox": [ + 107, + 273, + 151, + 286 + ], + "score": 0.93, + "content": "\\alpha W _ { 1 } ( \\mu , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 273, + 232, + 287 + ], + "score": 1.0, + "content": "for all distributions", + "type": "text" + }, + { + "bbox": [ + 232, + 276, + 249, + 285 + ], + "score": 0.83, + "content": "\\mu , \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 273, + 253, + 287 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 468, + 301 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 468, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 171, + 302 + ], + "score": 1.0, + "content": "Proposition 3.", + "type": "text" + }, + { + "bbox": [ + 171, + 289, + 190, + 300 + ], + "score": 0.43, + "content": "( D l )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 288, + 269, + 302 + ], + "score": 1.0, + "content": "holds if and only if", + "type": "text" + }, + { + "bbox": [ + 269, + 289, + 277, + 299 + ], + "score": 0.66, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 288, + 287, + 302 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 287, + 290, + 295, + 299 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 288, + 468, + 302 + ], + "score": 1.0, + "content": "-Lipschitz w.r.t. the Wasserstein-1 distance.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 506, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "Arjovsky et al. (2017) show that this property does not hold for common divergences based on the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "Kullback–Leibler or Jensen–Shannon divergence, while it does hold for the Wasserstein-1 distance.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 332, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 506, + 343 + ], + "score": 1.0, + "content": "Indeed, it is this desirable property that motivates their introduction of the Wasserstein GAN. Framed", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 343, + 318, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 318, + 355 + ], + "score": 1.0, + "content": "in our context, their result is summarized as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 480, + 370 + ], + "lines": [ + { + "bbox": [ + 104, + 355, + 478, + 373 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 408, + 373 + ], + "score": 1.0, + "content": "Proposition 4. The minimax and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 408, + 358, + 428, + 369 + ], + "score": 0.63, + "content": "( D I )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 355, + 466, + 373 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 467, + 360, + 478, + 370 + ], + "score": 0.76, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 426, + 389 + ], + "lines": [ + { + "bbox": [ + 104, + 374, + 427, + 393 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 312, + 393 + ], + "score": 1.0, + "content": "Proposition 5. The Wasserstein GAN loss satisfies", + "type": "text" + }, + { + "bbox": [ + 312, + 378, + 332, + 389 + ], + "score": 0.28, + "content": "( D l )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 374, + 353, + 393 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 353, + 378, + 379, + 388 + ], + "score": 0.82, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 374, + 411, + 393 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 411, + 379, + 423, + 389 + ], + "score": 0.63, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 374, + 427, + 393 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 108, + 397, + 503, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "Our stability analysis therefore deepens the analysis of Arjovsky et al. (2017) and provides an alter-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "score": 1.0, + "content": "native reason that the Wasserstein distance is desirable as a metric: it is part of a sufficient condition", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 421, + 284, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 284, + 432 + ], + "score": 1.0, + "content": "that ensures stationarity of gradient descent.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 107, + 445, + 391, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 392, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 392, + 458 + ], + "score": 1.0, + "content": "4.1 FROM WASSERSTEIN DISTANCE TO LIPSCHITZ CONSTRAINTS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 480 + ], + "score": 1.0, + "content": "Having identified the Wasserstein GAN loss as one that satisfies (D1), we next follow the strategy", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 477, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 489 + ], + "score": 1.0, + "content": "outlined in Section 3 to convert it into a regularizer for an arbitrary loss function. Towards this, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 487, + 476, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 194, + 501 + ], + "score": 1.0, + "content": "define the regularizer", + "type": "text" + }, + { + "bbox": [ + 194, + 488, + 267, + 500 + ], + "score": 0.93, + "content": "R _ { 1 } : { \\mathcal { M } } ( X ) \\to { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 487, + 405, + 501 + ], + "score": 1.0, + "content": "and compute its convex conjugate", + "type": "text" + }, + { + "bbox": [ + 405, + 488, + 472, + 500 + ], + "score": 0.93, + "content": "R _ { 1 } ^ { \\star } : { \\mathcal { C } } ( X ) \\to { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 487, + 476, + 501 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 506, + 457, + 545 + ], + "lines": [ + { + "bbox": [ + 153, + 506, + 457, + 545 + ], + "spans": [ + { + "bbox": [ + 153, + 506, + 457, + 545 + ], + "score": 0.93, + "content": "R _ { 1 } ( \\xi ) : = \\alpha \\| \\xi \\| _ { \\mathrm { K R } } = \\alpha \\operatorname* { s u p } _ { \\stackrel { f \\in \\mathcal { C } ( X ) } { | | f | | _ { \\mathrm { L i p } } \\leq 1 } } \\int f d \\xi , \\qquad R _ { 1 } ^ { \\star } ( \\varphi ) = \\left\\{ \\begin{array} { l l } { 0 } & { \\| \\varphi \\| _ { \\mathrm { L i p } } \\leq \\alpha } \\\\ { \\infty } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "a9edbd666998be9b0fbb67aaf0ea1ee87b8930cfbb6e02426f7888e012e513bb.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 153, + 506, + 457, + 519.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 153, + 519.0, + 457, + 532.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 153, + 532.0, + 457, + 545.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 504, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "This norm is the Kantorovich–Rubinstein norm (KR norm), which extends the Wasserstein-1", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 151, + 573 + ], + "score": 1.0, + "content": "distance to", + "type": "text" + }, + { + "bbox": [ + 152, + 561, + 181, + 573 + ], + "score": 0.92, + "content": "M ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 561, + 235, + 573 + ], + "score": 1.0, + "content": "; it holds that", + "type": "text" + }, + { + "bbox": [ + 235, + 561, + 331, + 573 + ], + "score": 0.92, + "content": "\\| \\mu - \\nu \\| _ { \\mathrm { K R } } = W _ { 1 } ( \\mu , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 561, + 346, + 573 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 347, + 561, + 399, + 573 + ], + "score": 0.92, + "content": "\\mu , \\nu \\in { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 561, + 505, + 573 + ], + "score": 1.0, + "content": ". Then, its inf-convolution", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 571, + 384, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 366, + 584 + ], + "score": 1.0, + "content": "with an arbitrary function inherits the Lipschitz property held by", + "type": "text" + }, + { + "bbox": [ + 366, + 573, + 379, + 583 + ], + "score": 0.87, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 571, + 384, + 584 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 587, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 261, + 601 + ], + "score": 1.0, + "content": "Proposition 6 (Pasch–Hausdorff). Let", + "type": "text" + }, + { + "bbox": [ + 262, + 587, + 329, + 600 + ], + "score": 0.89, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 587, + 427, + 601 + ], + "score": 1.0, + "content": "be a function, and define", + "type": "text" + }, + { + "bbox": [ + 428, + 587, + 478, + 599 + ], + "score": 0.92, + "content": "{ \\tilde { J } } : = J \\oplus R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 587, + 505, + 601 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 599, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 107, + 600, + 114, + 611 + ], + "score": 0.83, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 599, + 125, + 614 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 126, + 603, + 133, + 611 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 599, + 309, + 614 + ], + "score": 1.0, + "content": "-Lipschitz w.r.t. the distance induced by the", + "type": "text" + }, + { + "bbox": [ + 309, + 601, + 324, + 611 + ], + "score": 0.29, + "content": "K R", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 599, + 506, + 614 + ], + "score": 1.0, + "content": "norm, and hence the Wasserstein-1 distance", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 612, + 209, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 181, + 624 + ], + "score": 1.0, + "content": "when restricted to", + "type": "text" + }, + { + "bbox": [ + 181, + 613, + 206, + 624 + ], + "score": 0.92, + "content": "{ \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 612, + 209, + 624 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 365, + 648 + ], + "score": 1.0, + "content": "Due to Proposition 3, we now obtain a transformed loss function", + "type": "text" + }, + { + "bbox": [ + 365, + 633, + 373, + 645 + ], + "score": 0.85, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "that automatically satisfies (D1).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 645, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 659 + ], + "score": 1.0, + "content": "This function is a generalization of the Pasch–Hausdorff envelope (see Chapter 9 in Rockafeller &", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 656, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 670 + ], + "score": 1.0, + "content": "Wets (1998)), also known as Lipschitz regularization or the McShane–Whitney extension (McShane,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 667, + 360, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 360, + 681 + ], + "score": 1.0, + "content": "1934; Whitney, 1934; Kirszbraun, 1934; Hiriart-Urruty, 1980).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 105, + 685, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 104, + 682, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 682, + 322, + 701 + ], + "score": 1.0, + "content": "The convex conjugate computation in (8) shows that", + "type": "text" + }, + { + "bbox": [ + 322, + 684, + 330, + 696 + ], + "score": 0.86, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 682, + 506, + 701 + ], + "score": 1.0, + "content": "can be minimized in practice by imposing", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 696, + 328, + 709 + ], + "spans": [ + { + "bbox": [ + 106, + 696, + 328, + 709 + ], + "score": 1.0, + "content": "Lipschitz constraints on discriminators. 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This is a sensible criterion, since a discriminator whose gradient norm is too large may push", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 245, + 307, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 307, + 259 + ], + "score": 1.0, + "content": "the generator too hard and destabilize its training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 223, + 505, + 259 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 504, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 260, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 433, + 276 + ], + "score": 1.0, + "content": "To check (D1), the following proposition shows that it suffices to check whether", + "type": "text" + }, + { + "bbox": [ + 433, + 262, + 505, + 275 + ], + "score": 0.9, + "content": "| J ( \\mu ) - J ( \\nu ) | \\leq", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 273, + 253, + 287 + ], + "spans": [ + { + "bbox": [ + 107, + 273, + 151, + 286 + ], + "score": 0.93, + "content": "\\alpha W _ { 1 } ( \\mu , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 273, + 232, + 287 + ], + "score": 1.0, + "content": "for all distributions", + "type": "text" + }, + { + "bbox": [ + 232, + 276, + 249, + 285 + ], + "score": 0.83, + "content": "\\mu , \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 273, + 253, + 287 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 260, + 505, + 287 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 468, + 301 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 468, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 171, + 302 + ], + "score": 1.0, + "content": "Proposition 3.", + "type": "text" + }, + { + "bbox": [ + 171, + 289, + 190, + 300 + ], + "score": 0.43, + "content": "( D l )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 288, + 269, + 302 + ], + "score": 1.0, + "content": "holds if and only if", + "type": "text" + }, + { + "bbox": [ + 269, + 289, + 277, + 299 + ], + "score": 0.66, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 288, + 287, + 302 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 287, + 290, + 295, + 299 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 288, + 468, + 302 + ], + "score": 1.0, + "content": "-Lipschitz w.r.t. the Wasserstein-1 distance.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 288, + 468, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 506, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "Arjovsky et al. (2017) show that this property does not hold for common divergences based on the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "Kullback–Leibler or Jensen–Shannon divergence, while it does hold for the Wasserstein-1 distance.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 332, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 506, + 343 + ], + "score": 1.0, + "content": "Indeed, it is this desirable property that motivates their introduction of the Wasserstein GAN. Framed", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 343, + 318, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 318, + 355 + ], + "score": 1.0, + "content": "in our context, their result is summarized as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 310, + 506, + 355 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 480, + 370 + ], + "lines": [ + { + "bbox": [ + 104, + 355, + 478, + 373 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 408, + 373 + ], + "score": 1.0, + "content": "Proposition 4. The minimax and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 408, + 358, + 428, + 369 + ], + "score": 0.63, + "content": "( D I )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 355, + 466, + 373 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 467, + 360, + 478, + 370 + ], + "score": 0.76, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 355, + 478, + 373 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 426, + 389 + ], + "lines": [ + { + "bbox": [ + 104, + 374, + 427, + 393 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 312, + 393 + ], + "score": 1.0, + "content": "Proposition 5. The Wasserstein GAN loss satisfies", + "type": "text" + }, + { + "bbox": [ + 312, + 378, + 332, + 389 + ], + "score": 0.28, + "content": "( D l )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 374, + 353, + 393 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 353, + 378, + 379, + 388 + ], + "score": 0.82, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 374, + 411, + 393 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 411, + 379, + 423, + 389 + ], + "score": 0.63, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 374, + 427, + 393 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 374, + 427, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 397, + 503, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "Our stability analysis therefore deepens the analysis of Arjovsky et al. (2017) and provides an alter-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "score": 1.0, + "content": "native reason that the Wasserstein distance is desirable as a metric: it is part of a sufficient condition", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 421, + 284, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 284, + 432 + ], + "score": 1.0, + "content": "that ensures stationarity of gradient descent.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 398, + 505, + 432 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 445, + 391, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 392, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 392, + 458 + ], + "score": 1.0, + "content": "4.1 FROM WASSERSTEIN DISTANCE TO LIPSCHITZ CONSTRAINTS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 480 + ], + "score": 1.0, + "content": "Having identified the Wasserstein GAN loss as one that satisfies (D1), we next follow the strategy", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 477, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 489 + ], + "score": 1.0, + "content": "outlined in Section 3 to convert it into a regularizer for an arbitrary loss function. Towards this, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 487, + 476, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 194, + 501 + ], + "score": 1.0, + "content": "define the regularizer", + "type": "text" + }, + { + "bbox": [ + 194, + 488, + 267, + 500 + ], + "score": 0.93, + "content": "R _ { 1 } : { \\mathcal { M } } ( X ) \\to { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 487, + 405, + 501 + ], + "score": 1.0, + "content": "and compute its convex conjugate", + "type": "text" + }, + { + "bbox": [ + 405, + 488, + 472, + 500 + ], + "score": 0.93, + "content": "R _ { 1 } ^ { \\star } : { \\mathcal { C } } ( X ) \\to { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 487, + 476, + 501 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 464, + 506, + 501 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 506, + 457, + 545 + ], + "lines": [ + { + "bbox": [ + 153, + 506, + 457, + 545 + ], + "spans": [ + { + "bbox": [ + 153, + 506, + 457, + 545 + ], + "score": 0.93, + "content": "R _ { 1 } ( \\xi ) : = \\alpha \\| \\xi \\| _ { \\mathrm { K R } } = \\alpha \\operatorname* { s u p } _ { \\stackrel { f \\in \\mathcal { C } ( X ) } { | | f | | _ { \\mathrm { L i p } } \\leq 1 } } \\int f d \\xi , \\qquad R _ { 1 } ^ { \\star } ( \\varphi ) = \\left\\{ \\begin{array} { l l } { 0 } & { \\| \\varphi \\| _ { \\mathrm { L i p } } \\leq \\alpha } \\\\ { \\infty } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "a9edbd666998be9b0fbb67aaf0ea1ee87b8930cfbb6e02426f7888e012e513bb.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 153, + 506, + 457, + 519.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 153, + 519.0, + 457, + 532.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 153, + 532.0, + 457, + 545.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 504, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "This norm is the Kantorovich–Rubinstein norm (KR norm), which extends the Wasserstein-1", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 151, + 573 + ], + "score": 1.0, + "content": "distance to", + "type": "text" + }, + { + "bbox": [ + 152, + 561, + 181, + 573 + ], + "score": 0.92, + "content": "M ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 561, + 235, + 573 + ], + "score": 1.0, + "content": "; it holds that", + "type": "text" + }, + { + "bbox": [ + 235, + 561, + 331, + 573 + ], + "score": 0.92, + "content": "\\| \\mu - \\nu \\| _ { \\mathrm { K R } } = W _ { 1 } ( \\mu , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 561, + 346, + 573 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 347, + 561, + 399, + 573 + ], + "score": 0.92, + "content": "\\mu , \\nu \\in { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 561, + 505, + 573 + ], + "score": 1.0, + "content": ". Then, its inf-convolution", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 571, + 384, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 366, + 584 + ], + "score": 1.0, + "content": "with an arbitrary function inherits the Lipschitz property held by", + "type": "text" + }, + { + "bbox": [ + 366, + 573, + 379, + 583 + ], + "score": 0.87, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 571, + 384, + 584 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 549, + 505, + 584 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 587, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 261, + 601 + ], + "score": 1.0, + "content": "Proposition 6 (Pasch–Hausdorff). Let", + "type": "text" + }, + { + "bbox": [ + 262, + 587, + 329, + 600 + ], + "score": 0.89, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 587, + 427, + 601 + ], + "score": 1.0, + "content": "be a function, and define", + "type": "text" + }, + { + "bbox": [ + 428, + 587, + 478, + 599 + ], + "score": 0.92, + "content": "{ \\tilde { J } } : = J \\oplus R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 587, + 505, + 601 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 599, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 107, + 600, + 114, + 611 + ], + "score": 0.83, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 599, + 125, + 614 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 126, + 603, + 133, + 611 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 599, + 309, + 614 + ], + "score": 1.0, + "content": "-Lipschitz w.r.t. the distance induced by the", + "type": "text" + }, + { + "bbox": [ + 309, + 601, + 324, + 611 + ], + "score": 0.29, + "content": "K R", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 599, + 506, + 614 + ], + "score": 1.0, + "content": "norm, and hence the Wasserstein-1 distance", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 612, + 209, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 181, + 624 + ], + "score": 1.0, + "content": "when restricted to", + "type": "text" + }, + { + "bbox": [ + 181, + 613, + 206, + 624 + ], + "score": 0.92, + "content": "{ \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 612, + 209, + 624 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 587, + 506, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 365, + 648 + ], + "score": 1.0, + "content": "Due to Proposition 3, we now obtain a transformed loss function", + "type": "text" + }, + { + "bbox": [ + 365, + 633, + 373, + 645 + ], + "score": 0.85, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "that automatically satisfies (D1).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 645, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 659 + ], + "score": 1.0, + "content": "This function is a generalization of the Pasch–Hausdorff envelope (see Chapter 9 in Rockafeller &", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 656, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 670 + ], + "score": 1.0, + "content": "Wets (1998)), also known as Lipschitz regularization or the McShane–Whitney extension (McShane,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 667, + 360, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 360, + 681 + ], + "score": 1.0, + "content": "1934; Whitney, 1934; Kirszbraun, 1934; Hiriart-Urruty, 1980).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 633, + 506, + 681 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 685, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 104, + 682, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 682, + 322, + 701 + ], + "score": 1.0, + "content": "The convex conjugate computation in (8) shows that", + "type": "text" + }, + { + "bbox": [ + 322, + 684, + 330, + 696 + ], + "score": 0.86, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 682, + 506, + 701 + ], + "score": 1.0, + "content": "can be minimized in practice by imposing", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 696, + 328, + 709 + ], + "spans": [ + { + "bbox": [ + 106, + 696, + 328, + 709 + ], + "score": 1.0, + "content": "Lipschitz constraints on discriminators. Indeed, by (4),", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 682, + 506, + 709 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 713, + 452, + 735 + ], + "lines": [ + { + "bbox": [ + 158, + 713, + 452, + 735 + ], + "spans": [ + { + "bbox": [ + 158, + 713, + 452, + 735 + ], + "score": 0.91, + "content": "\\operatorname* { i n f } _ { \\mu } \\big ( J \\oplus \\alpha \\lVert \\cdot \\rVert _ { \\mathrm { K R } } \\big ) ( \\mu ) = \\operatorname* { i n f } _ { \\mu } \\operatorname* { s u p } _ { \\varphi } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - J ^ { \\star } ( \\varphi ) - \\chi \\{ \\lVert \\varphi \\rVert _ { \\mathrm { L i p } } \\leq \\alpha \\}", + "type": "interline_equation", + "image_path": "1f786b38890752fa2cec078d5bf8a8700d87db41c4dd2d7880e1be7f1ec3c9f0.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 158, + 713, + 452, + 735 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 245, + 81, + 357, + 104 + ], + "lines": [ + { + "bbox": [ + 245, + 81, + 357, + 104 + ], + "spans": [ + { + "bbox": [ + 245, + 81, + 357, + 104 + ], + "score": 0.92, + "content": "= \\operatorname* { i n f } _ { \\mu } \\operatorname* { s u p } _ { \\varphi \\colon \\| \\varphi \\| _ { \\mathrm { L i p } } \\leq \\alpha } \\mathcal { I } ( \\mu , \\varphi ) .", + "type": "interline_equation", + "image_path": "48c83cfa1096e1351e435aa120e0b1819d6322798ed04d84c0ea8973d809ffff.jpg" + } + ] + } + ], + "index": 0, + "virtual_lines": [ + { + "bbox": [ + 245, + 81, + 357, + 104 + ], + "spans": [], + "index": 0 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 505, + 165 + ], + "lines": [ + { + "bbox": [ + 104, + 105, + 506, + 122 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 262, + 122 + ], + "score": 1.0, + "content": "Farnia & Tse (2018) consider this loss", + "type": "text" + }, + { + "bbox": [ + 263, + 106, + 271, + 118 + ], + "score": 0.86, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 105, + 370, + 122 + ], + "score": 1.0, + "content": "in the special case of an", + "type": "text" + }, + { + "bbox": [ + 370, + 108, + 377, + 119 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 105, + 424, + 122 + ], + "score": 1.0, + "content": "-GAN with", + "type": "text" + }, + { + "bbox": [ + 424, + 107, + 501, + 120 + ], + "score": 0.93, + "content": "J ( \\mu ) = D _ { f } ( \\mu , \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 105, + 506, + 122 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 119, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 230, + 133 + ], + "score": 1.0, + "content": "they showed that minimizing", + "type": "text" + }, + { + "bbox": [ + 230, + 119, + 238, + 131 + ], + "score": 0.84, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 119, + 346, + 133 + ], + "score": 1.0, + "content": "corresponds to training a", + "type": "text" + }, + { + "bbox": [ + 347, + 121, + 353, + 132 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 119, + 505, + 133 + ], + "score": 1.0, + "content": "-GAN normally but constraining the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 130, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 183, + 145 + ], + "score": 1.0, + "content": "discriminator to be", + "type": "text" + }, + { + "bbox": [ + 183, + 133, + 191, + 142 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 130, + 444, + 145 + ], + "score": 1.0, + "content": "-Lipschitz. We show that this technique is in fact generic for any", + "type": "text" + }, + { + "bbox": [ + 444, + 132, + 452, + 142 + ], + "score": 0.79, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 130, + 506, + 145 + ], + "score": 1.0, + "content": ": minimizing", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 155 + ], + "score": 1.0, + "content": "the transformed loss can be achieved by training the GAN as normal, but imposing a Lipschitz", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 154, + 232, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 232, + 165 + ], + "score": 1.0, + "content": "constraint on the discriminator.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 170, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "Our analysis therefore justifies the use of Lipschitz constraints, such as spectral normalization (Miy-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "ato et al., 2018) and weight clipping (Arjovsky & Bottou, 2017), for general GAN losses. However,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Theorem 1 also suggests that applying only Lipschitz constraints may not be enough to stabilize", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 395, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 395, + 216 + ], + "score": 1.0, + "content": "GANs, as (D1) alone does not ensure that the GAN objective is smooth.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 107, + 230, + 404, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 405, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 405, + 245 + ], + "score": 1.0, + "content": "5 ENFORCING (D2) WITH DISCRIMINATOR SMOOTHNESS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 345, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 345, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 286, + 269 + ], + "score": 1.0, + "content": "(D2) demands that the optimal discriminator", + "type": "text" + }, + { + "bbox": [ + 287, + 255, + 300, + 267 + ], + "score": 0.9, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 253, + 345, + 269 + ], + "score": 1.0, + "content": "is smooth:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 504, + 284 + ], + "lines": [ + { + "bbox": [ + 108, + 269, + 504, + 285 + ], + "spans": [ + { + "bbox": [ + 108, + 269, + 133, + 285 + ], + "score": 1.0, + "content": "(D2)", + "type": "text" + }, + { + "bbox": [ + 133, + 271, + 189, + 284 + ], + "score": 0.93, + "content": "x \\mapsto \\nabla \\Phi _ { \\mu } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 269, + 200, + 285 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 200, + 272, + 211, + 283 + ], + "score": 0.88, + "content": "\\beta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 269, + 280, + 285 + ], + "score": 1.0, + "content": "-Lipschitz for all", + "type": "text" + }, + { + "bbox": [ + 280, + 271, + 323, + 284 + ], + "score": 0.92, + "content": "\\mu \\in { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 269, + 344, + 285 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 345, + 271, + 501, + 284 + ], + "score": 0.93, + "content": "\\| \\nabla \\Phi _ { \\mu } ( x ) - \\nabla \\Phi _ { \\mu } ( y ) \\| _ { 2 } \\leq \\beta _ { 1 } \\| x - y \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 269, + 504, + 285 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 504, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 294, + 302 + ], + "score": 1.0, + "content": "Intuitively, this says that for a fixed generator", + "type": "text" + }, + { + "bbox": [ + 294, + 291, + 302, + 300 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 288, + 411, + 302 + ], + "score": 1.0, + "content": ", the optimal discriminator", + "type": "text" + }, + { + "bbox": [ + 411, + 289, + 425, + 301 + ], + "score": 0.91, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 288, + 505, + 302 + ], + "score": 1.0, + "content": "should not provide", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 299, + 271, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 271, + 313 + ], + "score": 1.0, + "content": "gradients that change too much spatially.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 108, + 315, + 504, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 328 + ], + "score": 1.0, + "content": "Although the Wasserstein GAN loss (D1), we see that it, along with the minimax GAN and the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 327, + 273, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 273, + 340 + ], + "score": 1.0, + "content": "non-saturating GAN, do not satisfy (D2):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 504, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 339, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 469, + 354 + ], + "score": 1.0, + "content": "Proposition 7. The Wasserstein, minimax, and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 469, + 341, + 489, + 352 + ], + "score": 0.69, + "content": "( D 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 339, + 505, + 354 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 351, + 145, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 129, + 366 + ], + "score": 1.0, + "content": "some", + "type": "text" + }, + { + "bbox": [ + 130, + 354, + 141, + 363 + ], + "score": 0.79, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 351, + 145, + 366 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 353, + 384 + ], + "score": 1.0, + "content": "We now construct a loss that by definition satisfies (D2). Let", + "type": "text" + }, + { + "bbox": [ + 353, + 371, + 361, + 381 + ], + "score": 0.8, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "be the class of 1-smooth functions,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 381, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 179, + 395 + ], + "score": 1.0, + "content": "that is, for which", + "type": "text" + }, + { + "bbox": [ + 179, + 382, + 315, + 394 + ], + "score": 0.91, + "content": "\\| \\nabla f ( x ) - \\nabla f ( y ) \\| _ { 2 } \\leq \\| x - y \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 381, + 505, + 395 + ], + "score": 1.0, + "content": ", and consider the integral probability metric", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 393, + 273, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 216, + 405 + ], + "score": 1.0, + "content": "(IPM) (Muller, 1997) w.r.t. ¨", + "type": "text" + }, + { + "bbox": [ + 217, + 394, + 224, + 403 + ], + "score": 0.81, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 393, + 273, + 405 + ], + "score": 1.0, + "content": ", defined by", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 405, + 386, + 433 + ], + "lines": [ + { + "bbox": [ + 225, + 405, + 386, + 433 + ], + "spans": [ + { + "bbox": [ + 225, + 405, + 386, + 433 + ], + "score": 0.95, + "content": "\\mathrm { I P M } _ { \\cal S } ( \\mu , \\nu ) : = \\operatorname* { s u p } _ { f \\in { \\cal S } } \\int f d \\mu - \\int f d \\nu .", + "type": "interline_equation", + "image_path": "769d760952c9c8006b1a05ff9674099bf0e3352e9810fad4e6d6bdfe3031cf26.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 225, + 405, + 386, + 433 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 258, + 448 + ], + "score": 1.0, + "content": "The optimal discriminator for the loss", + "type": "text" + }, + { + "bbox": [ + 259, + 435, + 313, + 447 + ], + "score": 0.92, + "content": "\\mathrm { I P M } _ { \\cal S } ( \\mu , \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "is the function that maximizes the supremum in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 319, + 458 + ], + "score": 1.0, + "content": "the definition. This function by definition belongs to", + "type": "text" + }, + { + "bbox": [ + 319, + 447, + 327, + 456 + ], + "score": 0.84, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "and therefore is 1-smooth. Hence, this IPM", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 456, + 297, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 200, + 470 + ], + "score": 1.0, + "content": "loss satisfies (D2) with", + "type": "text" + }, + { + "bbox": [ + 200, + 457, + 230, + 468 + ], + "score": 0.91, + "content": "\\beta _ { 1 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 456, + 297, + 470 + ], + "score": 1.0, + "content": "by construction.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 481, + 428, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 429, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 429, + 494 + ], + "score": 1.0, + "content": "5.1 FROM INTEGRAL PROBABILITY METRIC TO SMOOTH DISCRIMINATORS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 501, + 506, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "Having identified the IPM-based loss as one that satisfies (D2), we next follow the strategy outlined", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "in Section 3 to convert it into a regularizer for an arbitrary loss function. To do so, we define a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 523, + 435, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 152, + 536 + ], + "score": 1.0, + "content": "regularizer", + "type": "text" + }, + { + "bbox": [ + 152, + 523, + 225, + 536 + ], + "score": 0.93, + "content": "R _ { 2 } : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 524, + 363, + 536 + ], + "score": 1.0, + "content": "and compute its convex conjugate", + "type": "text" + }, + { + "bbox": [ + 364, + 523, + 430, + 536 + ], + "score": 0.95, + "content": "R _ { 2 } ^ { \\star } : { \\mathcal { C } } ( X ) \\to { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 524, + 435, + 536 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 538, + 445, + 566 + ], + "lines": [ + { + "bbox": [ + 163, + 538, + 445, + 566 + ], + "spans": [ + { + "bbox": [ + 163, + 538, + 445, + 566 + ], + "score": 0.92, + "content": "R _ { 2 } ( \\xi ) : = \\beta _ { 1 } \\| \\xi \\| _ { \\mathcal { S } ^ { * } } = \\beta _ { 1 } \\operatorname* { s u p } _ { f \\in \\mathcal { S } } \\int f d \\xi , \\qquad R _ { 2 } ^ { \\star } ( \\varphi ) = \\left\\{ \\begin{array} { l l } { 0 } & { \\varphi \\in \\beta _ { 1 } S } \\\\ { \\infty } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "c847ebe43fd8513d821db7ca31e5d0425beaeca50d670cb05847801221f50035.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 163, + 538, + 445, + 566 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 504, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 244, + 580 + ], + "score": 1.0, + "content": "The norm is the dual norm to", + "type": "text" + }, + { + "bbox": [ + 244, + 568, + 252, + 578 + ], + "score": 0.8, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 567, + 506, + 580 + ], + "score": 1.0, + "content": ", which extends the IPM to signed measures; it holds that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 577, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 221, + 591 + ], + "score": 0.9, + "content": "\\mathrm { I P M } _ { \\mathcal { S } } ( \\mu , \\nu ) = \\| \\mu - \\nu \\| _ { \\mathcal { S } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 577, + 239, + 592 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 239, + 579, + 298, + 591 + ], + "score": 0.93, + "content": "\\mu , \\nu \\in { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 577, + 506, + 592 + ], + "score": 1.0, + "content": ". Similar to the situation in the previous section,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 589, + 339, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 321, + 602 + ], + "score": 1.0, + "content": "inf-convolution preserves the smoothness property of", + "type": "text" + }, + { + "bbox": [ + 321, + 590, + 334, + 601 + ], + "score": 0.89, + "content": "R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 589, + 339, + 602 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 503, + 627 + ], + "lines": [ + { + "bbox": [ + 104, + 601, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 185, + 616 + ], + "score": 1.0, + "content": "Proposition 8. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 602, + 253, + 615 + ], + "score": 0.92, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 601, + 505, + 616 + ], + "score": 1.0, + "content": "be a convex, proper, lower semicontinuous function, and define", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 613, + 371, + 629 + ], + "spans": [ + { + "bbox": [ + 107, + 614, + 160, + 627 + ], + "score": 0.91, + "content": "{ \\tilde { J } } : = J \\oplus R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 613, + 306, + 629 + ], + "score": 1.0, + "content": ". Then the optimal discriminator for", + "type": "text" + }, + { + "bbox": [ + 306, + 614, + 314, + 626 + ], + "score": 0.76, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 613, + 324, + 629 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 324, + 615, + 335, + 627 + ], + "score": 0.86, + "content": "\\beta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 613, + 371, + 629 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 104, + 635, + 504, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 504, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 504, + 648 + ], + "score": 1.0, + "content": "Applying (4) and (12), we see that we can minimize this transformed loss function by restricting the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 646, + 343, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 235, + 658 + ], + "score": 1.0, + "content": "family of discriminators to only", + "type": "text" + }, + { + "bbox": [ + 236, + 646, + 246, + 657 + ], + "score": 0.88, + "content": "\\beta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 646, + 343, + 658 + ], + "score": 1.0, + "content": "-smooth discriminators:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 658, + 447, + 704 + ], + "lines": [ + { + "bbox": [ + 163, + 658, + 447, + 704 + ], + "spans": [ + { + "bbox": [ + 163, + 658, + 447, + 704 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\underset { \\mu } { \\operatorname* { i n f } } \\left( J \\oplus \\beta _ { 1 } \\lVert \\cdot \\rVert _ { \\mathcal { S } ^ { * } } \\right) ( \\mu ) = \\underset { \\mu } { \\operatorname* { i n f } } \\ \\underset { \\varphi } { \\operatorname* { s u p } } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - J ^ { \\star } ( \\varphi ) - \\chi \\{ \\varphi \\in \\beta _ { 1 } S \\} } \\\\ & { \\quad \\quad = \\underset { \\mu } { \\operatorname* { i n f } } \\ \\underset { \\varphi \\in \\beta _ { 1 } S } { \\operatorname* { s u p } } \\mathcal { I } ( \\mu , \\varphi ) . } \\end{array}", + "type": "interline_equation", + "image_path": "3dbec029e1c8e6d10328329528a5466f3fba9da73405f4ce630573ac39761102.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 163, + 658, + 447, + 673.3333333333334 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 163, + 673.3333333333334, + 447, + 688.6666666666667 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 163, + 688.6666666666667, + 447, + 704.0000000000001 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "In practice, we can enforce this by applying spectral normalization (Miyato et al., 2018) and using a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 444, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 444, + 732 + ], + "score": 1.0, + "content": "Lipschitz, smooth activation function such as ELU (Clevert et al., 2016) or sigmoid.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 245, + 81, + 357, + 104 + ], + "lines": [ + { + "bbox": [ + 245, + 81, + 357, + 104 + ], + "spans": [ + { + "bbox": [ + 245, + 81, + 357, + 104 + ], + "score": 0.92, + "content": "= \\operatorname* { i n f } _ { \\mu } \\operatorname* { s u p } _ { \\varphi \\colon \\| \\varphi \\| _ { \\mathrm { L i p } } \\leq \\alpha } \\mathcal { I } ( \\mu , \\varphi ) .", + "type": "interline_equation", + "image_path": "48c83cfa1096e1351e435aa120e0b1819d6322798ed04d84c0ea8973d809ffff.jpg" + } + ] + } + ], + "index": 0, + "virtual_lines": [ + { + "bbox": [ + 245, + 81, + 357, + 104 + ], + "spans": [], + "index": 0 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 505, + 165 + ], + "lines": [ + { + "bbox": [ + 104, + 105, + 506, + 122 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 262, + 122 + ], + "score": 1.0, + "content": "Farnia & Tse (2018) consider this loss", + "type": "text" + }, + { + "bbox": [ + 263, + 106, + 271, + 118 + ], + "score": 0.86, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 105, + 370, + 122 + ], + "score": 1.0, + "content": "in the special case of an", + "type": "text" + }, + { + "bbox": [ + 370, + 108, + 377, + 119 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 105, + 424, + 122 + ], + "score": 1.0, + "content": "-GAN with", + "type": "text" + }, + { + "bbox": [ + 424, + 107, + 501, + 120 + ], + "score": 0.93, + "content": "J ( \\mu ) = D _ { f } ( \\mu , \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 105, + 506, + 122 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 119, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 230, + 133 + ], + "score": 1.0, + "content": "they showed that minimizing", + "type": "text" + }, + { + "bbox": [ + 230, + 119, + 238, + 131 + ], + "score": 0.84, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 119, + 346, + 133 + ], + "score": 1.0, + "content": "corresponds to training a", + "type": "text" + }, + { + "bbox": [ + 347, + 121, + 353, + 132 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 119, + 505, + 133 + ], + "score": 1.0, + "content": "-GAN normally but constraining the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 130, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 183, + 145 + ], + "score": 1.0, + "content": "discriminator to be", + "type": "text" + }, + { + "bbox": [ + 183, + 133, + 191, + 142 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 130, + 444, + 145 + ], + "score": 1.0, + "content": "-Lipschitz. We show that this technique is in fact generic for any", + "type": "text" + }, + { + "bbox": [ + 444, + 132, + 452, + 142 + ], + "score": 0.79, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 130, + 506, + 145 + ], + "score": 1.0, + "content": ": minimizing", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 155 + ], + "score": 1.0, + "content": "the transformed loss can be achieved by training the GAN as normal, but imposing a Lipschitz", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 154, + 232, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 232, + 165 + ], + "score": 1.0, + "content": "constraint on the discriminator.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 105, + 506, + 165 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 170, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "Our analysis therefore justifies the use of Lipschitz constraints, such as spectral normalization (Miy-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "ato et al., 2018) and weight clipping (Arjovsky & Bottou, 2017), for general GAN losses. However,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Theorem 1 also suggests that applying only Lipschitz constraints may not be enough to stabilize", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 395, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 395, + 216 + ], + "score": 1.0, + "content": "GANs, as (D1) alone does not ensure that the GAN objective is smooth.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 169, + 505, + 216 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 230, + 404, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 405, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 405, + 245 + ], + "score": 1.0, + "content": "5 ENFORCING (D2) WITH DISCRIMINATOR SMOOTHNESS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 345, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 345, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 286, + 269 + ], + "score": 1.0, + "content": "(D2) demands that the optimal discriminator", + "type": "text" + }, + { + "bbox": [ + 287, + 255, + 300, + 267 + ], + "score": 0.9, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 253, + 345, + 269 + ], + "score": 1.0, + "content": "is smooth:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 253, + 345, + 269 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 504, + 284 + ], + "lines": [ + { + "bbox": [ + 108, + 269, + 504, + 285 + ], + "spans": [ + { + "bbox": [ + 108, + 269, + 133, + 285 + ], + "score": 1.0, + "content": "(D2)", + "type": "text" + }, + { + "bbox": [ + 133, + 271, + 189, + 284 + ], + "score": 0.93, + "content": "x \\mapsto \\nabla \\Phi _ { \\mu } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 269, + 200, + 285 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 200, + 272, + 211, + 283 + ], + "score": 0.88, + "content": "\\beta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 269, + 280, + 285 + ], + "score": 1.0, + "content": "-Lipschitz for all", + "type": "text" + }, + { + "bbox": [ + 280, + 271, + 323, + 284 + ], + "score": 0.92, + "content": "\\mu \\in { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 269, + 344, + 285 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 345, + 271, + 501, + 284 + ], + "score": 0.93, + "content": "\\| \\nabla \\Phi _ { \\mu } ( x ) - \\nabla \\Phi _ { \\mu } ( y ) \\| _ { 2 } \\leq \\beta _ { 1 } \\| x - y \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 269, + 504, + 285 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 108, + 269, + 504, + 285 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 504, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 294, + 302 + ], + "score": 1.0, + "content": "Intuitively, this says that for a fixed generator", + "type": "text" + }, + { + "bbox": [ + 294, + 291, + 302, + 300 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 288, + 411, + 302 + ], + "score": 1.0, + "content": ", the optimal discriminator", + "type": "text" + }, + { + "bbox": [ + 411, + 289, + 425, + 301 + ], + "score": 0.91, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 288, + 505, + 302 + ], + "score": 1.0, + "content": "should not provide", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 299, + 271, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 271, + 313 + ], + "score": 1.0, + "content": "gradients that change too much spatially.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 288, + 505, + 313 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 315, + 504, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 328 + ], + "score": 1.0, + "content": "Although the Wasserstein GAN loss (D1), we see that it, along with the minimax GAN and the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 327, + 273, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 273, + 340 + ], + "score": 1.0, + "content": "non-saturating GAN, do not satisfy (D2):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 106, + 315, + 505, + 340 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 504, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 339, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 469, + 354 + ], + "score": 1.0, + "content": "Proposition 7. The Wasserstein, minimax, and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 469, + 341, + 489, + 352 + ], + "score": 0.69, + "content": "( D 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 339, + 505, + 354 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 351, + 145, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 129, + 366 + ], + "score": 1.0, + "content": "some", + "type": "text" + }, + { + "bbox": [ + 130, + 354, + 141, + 363 + ], + "score": 0.79, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 351, + 145, + 366 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 339, + 505, + 366 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 353, + 384 + ], + "score": 1.0, + "content": "We now construct a loss that by definition satisfies (D2). Let", + "type": "text" + }, + { + "bbox": [ + 353, + 371, + 361, + 381 + ], + "score": 0.8, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "be the class of 1-smooth functions,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 381, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 179, + 395 + ], + "score": 1.0, + "content": "that is, for which", + "type": "text" + }, + { + "bbox": [ + 179, + 382, + 315, + 394 + ], + "score": 0.91, + "content": "\\| \\nabla f ( x ) - \\nabla f ( y ) \\| _ { 2 } \\leq \\| x - y \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 381, + 505, + 395 + ], + "score": 1.0, + "content": ", and consider the integral probability metric", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 393, + 273, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 216, + 405 + ], + "score": 1.0, + "content": "(IPM) (Muller, 1997) w.r.t. ¨", + "type": "text" + }, + { + "bbox": [ + 217, + 394, + 224, + 403 + ], + "score": 0.81, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 393, + 273, + 405 + ], + "score": 1.0, + "content": ", defined by", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 370, + 505, + 405 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 405, + 386, + 433 + ], + "lines": [ + { + "bbox": [ + 225, + 405, + 386, + 433 + ], + "spans": [ + { + "bbox": [ + 225, + 405, + 386, + 433 + ], + "score": 0.95, + "content": "\\mathrm { I P M } _ { \\cal S } ( \\mu , \\nu ) : = \\operatorname* { s u p } _ { f \\in { \\cal S } } \\int f d \\mu - \\int f d \\nu .", + "type": "interline_equation", + "image_path": "769d760952c9c8006b1a05ff9674099bf0e3352e9810fad4e6d6bdfe3031cf26.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 225, + 405, + 386, + 433 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 258, + 448 + ], + "score": 1.0, + "content": "The optimal discriminator for the loss", + "type": "text" + }, + { + "bbox": [ + 259, + 435, + 313, + 447 + ], + "score": 0.92, + "content": "\\mathrm { I P M } _ { \\cal S } ( \\mu , \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "is the function that maximizes the supremum in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 319, + 458 + ], + "score": 1.0, + "content": "the definition. This function by definition belongs to", + "type": "text" + }, + { + "bbox": [ + 319, + 447, + 327, + 456 + ], + "score": 0.84, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "and therefore is 1-smooth. Hence, this IPM", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 456, + 297, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 200, + 470 + ], + "score": 1.0, + "content": "loss satisfies (D2) with", + "type": "text" + }, + { + "bbox": [ + 200, + 457, + 230, + 468 + ], + "score": 0.91, + "content": "\\beta _ { 1 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 456, + 297, + 470 + ], + "score": 1.0, + "content": "by construction.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 434, + 505, + 470 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 481, + 428, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 429, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 429, + 494 + ], + "score": 1.0, + "content": "5.1 FROM INTEGRAL PROBABILITY METRIC TO SMOOTH DISCRIMINATORS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 501, + 506, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "Having identified the IPM-based loss as one that satisfies (D2), we next follow the strategy outlined", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "in Section 3 to convert it into a regularizer for an arbitrary loss function. To do so, we define a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 523, + 435, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 152, + 536 + ], + "score": 1.0, + "content": "regularizer", + "type": "text" + }, + { + "bbox": [ + 152, + 523, + 225, + 536 + ], + "score": 0.93, + "content": "R _ { 2 } : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 524, + 363, + 536 + ], + "score": 1.0, + "content": "and compute its convex conjugate", + "type": "text" + }, + { + "bbox": [ + 364, + 523, + 430, + 536 + ], + "score": 0.95, + "content": "R _ { 2 } ^ { \\star } : { \\mathcal { C } } ( X ) \\to { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 524, + 435, + 536 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 501, + 506, + 536 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 538, + 445, + 566 + ], + "lines": [ + { + "bbox": [ + 163, + 538, + 445, + 566 + ], + "spans": [ + { + "bbox": [ + 163, + 538, + 445, + 566 + ], + "score": 0.92, + "content": "R _ { 2 } ( \\xi ) : = \\beta _ { 1 } \\| \\xi \\| _ { \\mathcal { S } ^ { * } } = \\beta _ { 1 } \\operatorname* { s u p } _ { f \\in \\mathcal { S } } \\int f d \\xi , \\qquad R _ { 2 } ^ { \\star } ( \\varphi ) = \\left\\{ \\begin{array} { l l } { 0 } & { \\varphi \\in \\beta _ { 1 } S } \\\\ { \\infty } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "c847ebe43fd8513d821db7ca31e5d0425beaeca50d670cb05847801221f50035.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 163, + 538, + 445, + 566 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 504, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 244, + 580 + ], + "score": 1.0, + "content": "The norm is the dual norm to", + "type": "text" + }, + { + "bbox": [ + 244, + 568, + 252, + 578 + ], + "score": 0.8, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 567, + 506, + 580 + ], + "score": 1.0, + "content": ", which extends the IPM to signed measures; it holds that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 577, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 221, + 591 + ], + "score": 0.9, + "content": "\\mathrm { I P M } _ { \\mathcal { S } } ( \\mu , \\nu ) = \\| \\mu - \\nu \\| _ { \\mathcal { S } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 577, + 239, + 592 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 239, + 579, + 298, + 591 + ], + "score": 0.93, + "content": "\\mu , \\nu \\in { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 577, + 506, + 592 + ], + "score": 1.0, + "content": ". Similar to the situation in the previous section,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 589, + 339, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 321, + 602 + ], + "score": 1.0, + "content": "inf-convolution preserves the smoothness property of", + "type": "text" + }, + { + "bbox": [ + 321, + 590, + 334, + 601 + ], + "score": 0.89, + "content": "R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 589, + 339, + 602 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 567, + 506, + 602 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 503, + 627 + ], + "lines": [ + { + "bbox": [ + 104, + 601, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 185, + 616 + ], + "score": 1.0, + "content": "Proposition 8. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 602, + 253, + 615 + ], + "score": 0.92, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 601, + 505, + 616 + ], + "score": 1.0, + "content": "be a convex, proper, lower semicontinuous function, and define", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 613, + 371, + 629 + ], + "spans": [ + { + "bbox": [ + 107, + 614, + 160, + 627 + ], + "score": 0.91, + "content": "{ \\tilde { J } } : = J \\oplus R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 613, + 306, + 629 + ], + "score": 1.0, + "content": ". Then the optimal discriminator for", + "type": "text" + }, + { + "bbox": [ + 306, + 614, + 314, + 626 + ], + "score": 0.76, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 613, + 324, + 629 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 324, + 615, + 335, + 627 + ], + "score": 0.86, + "content": "\\beta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 613, + 371, + 629 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 601, + 505, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 635, + 504, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 504, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 504, + 648 + ], + "score": 1.0, + "content": "Applying (4) and (12), we see that we can minimize this transformed loss function by restricting the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 646, + 343, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 235, + 658 + ], + "score": 1.0, + "content": "family of discriminators to only", + "type": "text" + }, + { + "bbox": [ + 236, + 646, + 246, + 657 + ], + "score": 0.88, + "content": "\\beta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 646, + 343, + 658 + ], + "score": 1.0, + "content": "-smooth discriminators:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 106, + 635, + 504, + 658 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 658, + 447, + 704 + ], + "lines": [ + { + "bbox": [ + 163, + 658, + 447, + 704 + ], + "spans": [ + { + "bbox": [ + 163, + 658, + 447, + 704 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\underset { \\mu } { \\operatorname* { i n f } } \\left( J \\oplus \\beta _ { 1 } \\lVert \\cdot \\rVert _ { \\mathcal { S } ^ { * } } \\right) ( \\mu ) = \\underset { \\mu } { \\operatorname* { i n f } } \\ \\underset { \\varphi } { \\operatorname* { s u p } } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - J ^ { \\star } ( \\varphi ) - \\chi \\{ \\varphi \\in \\beta _ { 1 } S \\} } \\\\ & { \\quad \\quad = \\underset { \\mu } { \\operatorname* { i n f } } \\ \\underset { \\varphi \\in \\beta _ { 1 } S } { \\operatorname* { s u p } } \\mathcal { I } ( \\mu , \\varphi ) . } \\end{array}", + "type": "interline_equation", + "image_path": "3dbec029e1c8e6d10328329528a5466f3fba9da73405f4ce630573ac39761102.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 163, + 658, + 447, + 673.3333333333334 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 163, + 673.3333333333334, + 447, + 688.6666666666667 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 163, + 688.6666666666667, + 447, + 704.0000000000001 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "In practice, we can enforce this by applying spectral normalization (Miyato et al., 2018) and using a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 444, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 444, + 732 + ], + "score": 1.0, + "content": "Lipschitz, smooth activation function such as ELU (Clevert et al., 2016) or sigmoid.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 709, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 187, + 95 + ], + "score": 1.0, + "content": "Proposition 9. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 82, + 239, + 94 + ], + "score": 0.91, + "content": "f : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 81, + 504, + 95 + ], + "score": 1.0, + "content": "be a neural network consisting of k layers whose linear transfor-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 496, + 106 + ], + "score": 1.0, + "content": "mations have spectral norm 1 and whose activation functions are 1-Lipschitz and 1-smooth. Then", + "type": "text" + }, + { + "bbox": [ + 497, + 94, + 504, + 105 + ], + "score": 0.72, + "content": "f", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 158, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 115, + 116 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 116, + 105, + 122, + 114 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 104, + 158, + 116 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 107, + 132, + 361, + 145 + ], + "lines": [ + { + "bbox": [ + 104, + 131, + 363, + 147 + ], + "spans": [ + { + "bbox": [ + 104, + 131, + 363, + 147 + ], + "score": 1.0, + "content": "6 ENFORCING (D3) WITH GRADIENT PENALTIES", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 108, + 156, + 284, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 285, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 285, + 170 + ], + "score": 1.0, + "content": "(D3) is the following smoothness condition:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 491, + 186 + ], + "lines": [ + { + "bbox": [ + 108, + 172, + 489, + 188 + ], + "spans": [ + { + "bbox": [ + 108, + 172, + 133, + 188 + ], + "score": 1.0, + "content": "(D3)", + "type": "text" + }, + { + "bbox": [ + 133, + 173, + 190, + 186 + ], + "score": 0.92, + "content": "\\mu \\mapsto \\nabla \\Phi _ { \\mu } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 172, + 200, + 188 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 200, + 174, + 211, + 185 + ], + "score": 0.85, + "content": "\\beta _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 172, + 283, + 188 + ], + "score": 1.0, + "content": "-Lipschitz for any", + "type": "text" + }, + { + "bbox": [ + 283, + 174, + 311, + 184 + ], + "score": 0.88, + "content": "x \\in X", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 172, + 332, + 188 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 332, + 173, + 489, + 186 + ], + "score": 0.91, + "content": "\\| \\nabla \\Phi _ { \\mu } ( x ) - \\nabla \\Phi _ { \\nu } ( x ) \\| _ { 2 } \\leq \\beta _ { 2 } W _ { 1 } ( \\mu , \\nu ) .", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 190, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 191, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 203 + ], + "score": 1.0, + "content": "(D3) requires that the gradients of the optimal discriminator do not change too rapidly in response", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 201, + 504, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 160, + 215 + ], + "score": 1.0, + "content": "to changes in", + "type": "text" + }, + { + "bbox": [ + 161, + 203, + 168, + 213 + ], + "score": 0.79, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 201, + 504, + 215 + ], + "score": 1.0, + "content": ". Indeed, if the discriminator’s gradients are too sensitive to changes in the generator,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 212, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 226 + ], + "score": 1.0, + "content": "the generator may not be able to accurately follow those gradients as it updates itself using a finite", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 221, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 104, + 221, + 342, + 238 + ], + "score": 1.0, + "content": "step size. In finite-dimensional optimization of a function", + "type": "text" + }, + { + "bbox": [ + 342, + 223, + 392, + 235 + ], + "score": 0.93, + "content": "f : { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 221, + 505, + 238 + ], + "score": 1.0, + "content": ", this condition is analogous", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 235, + 237, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 117, + 247 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 235, + 124, + 246 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 235, + 237, + 247 + ], + "score": 1.0, + "content": "having a Lipschitz gradient.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 251, + 504, + 274 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "We now present an equivalent characterization of (D3) that is easier to check in practice. We define", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 261, + 376, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 295, + 276 + ], + "score": 1.0, + "content": "the Bregman divergence of a convex function", + "type": "text" + }, + { + "bbox": [ + 295, + 262, + 362, + 274 + ], + "score": 0.92, + "content": "J : { \\mathcal { M } } ( X ) \\to { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 261, + 376, + 276 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 278, + 404, + 304 + ], + "lines": [ + { + "bbox": [ + 205, + 278, + 404, + 304 + ], + "spans": [ + { + "bbox": [ + 205, + 278, + 404, + 304 + ], + "score": 0.91, + "content": "\\mathfrak { D } _ { J } ( \\nu , \\mu ) : = J ( \\nu ) - J ( \\mu ) - \\int \\Phi _ { \\mu } ( x ) d ( \\nu - \\mu ) ,", + "type": "interline_equation", + "image_path": "598edc74ea3d2ea830ac891bc02a235feaf388df814ff9d8710862c540c54926.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 205, + 278, + 404, + 304 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 308, + 503, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 134, + 320 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 309, + 149, + 320 + ], + "score": 0.89, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 308, + 285, + 320 + ], + "score": 1.0, + "content": "is the optimal discriminator for", + "type": "text" + }, + { + "bbox": [ + 285, + 309, + 293, + 318 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 308, + 306, + 320 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 307, + 310, + 314, + 320 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 308, + 505, + 320 + ], + "score": 1.0, + "content": ". Then, (D3) is characterized in terms of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 318, + 309, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 309, + 331 + ], + "score": 1.0, + "content": "Bregman divergence and the KR norm as follows:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 502, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 193, + 348 + ], + "score": 1.0, + "content": "Proposition 10. Let", + "type": "text" + }, + { + "bbox": [ + 193, + 333, + 266, + 346 + ], + "score": 0.91, + "content": "J : \\mathcal { M } ( X ) \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 331, + 383, + 348 + ], + "score": 1.0, + "content": "be a convex function. 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The minimax and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 413, + 464, + 433, + 475 + ], + "score": 0.6, + "content": "( D 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 461, + 471, + 478 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 471, + 466, + 483, + 475 + ], + "score": 0.77, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 461, + 487, + 478 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "score": 1.0, + "content": "Even so, the Bregman divergence for the non-saturating loss is always less than that of the minimax", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "GAN, suggesting that the non-saturating loss should be stable in more situations than the minimax", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "GAN. On the other hand, the MMD-based loss (Li et al., 2015) does satisfy (D3) when its kernel is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 517, + 284, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 187, + 530 + ], + "score": 1.0, + "content": "the Gaussian kernel", + "type": "text" + }, + { + "bbox": [ + 188, + 517, + 279, + 532 + ], + "score": 0.93, + "content": "K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 518, + 284, + 530 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 104, + 533, + 477, + 546 + ], + "lines": [ + { + "bbox": [ + 104, + 532, + 474, + 549 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 358, + 549 + ], + "score": 1.0, + "content": "Proposition 12. The MMD loss with Gaussian kernel satisfies", + "type": "text" + }, + { + "bbox": [ + 359, + 534, + 378, + 545 + ], + "score": 0.66, + "content": "( D 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 532, + 399, + 549 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 399, + 534, + 435, + 545 + ], + "score": 0.89, + "content": "\\beta _ { 2 } = 2 \\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 532, + 463, + 549 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 464, + 535, + 474, + 546 + ], + "score": 0.83, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 559, + 405, + 570 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 405, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 405, + 571 + ], + "score": 1.0, + "content": "6.1 FROM MAXIMUM MEAN DISCREPANCY TO GRADIENT PENALTIES", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "Having identified the MMD-based loss as one that satisfies (D3), we next follow the strategy outlined", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "in Section 3 to convert it into a regularizer for an arbitrary loss function. To do so, we define the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 600, + 435, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 152, + 614 + ], + "score": 1.0, + "content": "regularizer", + "type": "text" + }, + { + "bbox": [ + 152, + 601, + 225, + 613 + ], + "score": 0.94, + "content": "R _ { 3 } : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 600, + 363, + 614 + ], + "score": 1.0, + "content": "and compute its convex conjugate", + "type": "text" + }, + { + "bbox": [ + 364, + 601, + 430, + 613 + ], + "score": 0.93, + "content": "R _ { 3 } ^ { \\star } : { \\mathcal { C } } ( X ) { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 600, + 435, + 614 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 617, + 397, + 643 + ], + "lines": [ + { + "bbox": [ + 214, + 617, + 397, + 643 + ], + "spans": [ + { + "bbox": [ + 214, + 617, + 397, + 643 + ], + "score": 0.93, + "content": "R _ { 3 } ( \\xi ) : = \\frac { \\beta _ { 2 } } { 4 \\pi } \\| \\hat { \\xi } \\| _ { \\mathcal { H } } ^ { 2 } , \\qquad R _ { 3 } ^ { \\star } ( \\varphi ) = \\frac { \\pi } { \\beta _ { 2 } } \\| \\varphi \\| _ { \\mathcal { H } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "0897a6672da3d502f8cb8b541af4a7e1d40a0ffd4529e66a7b6b57d5c9cd9812.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 214, + 617, + 397, + 643 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 647, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 432, + 660 + ], + "score": 1.0, + "content": "The norm is the norm of a reproducing kernel Hilbert space norm (RKHS)", + "type": "text" + }, + { + "bbox": [ + 433, + 648, + 442, + 657 + ], + "score": 0.79, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 646, + 505, + 660 + ], + "score": 1.0, + "content": "with Gaussian", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 656, + 504, + 672 + ], + "spans": [ + { + "bbox": [ + 104, + 656, + 397, + 672 + ], + "score": 1.0, + "content": "kernel; this norm extends the MMD to signed measures, as it holds that", + "type": "text" + }, + { + "bbox": [ + 397, + 658, + 504, + 671 + ], + "score": 0.9, + "content": "\\mathrm { M M D } ( \\mu , \\nu ) = \\| \\hat { \\mu } - \\hat { \\nu } \\| _ { \\mathcal { H } }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 669, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 120, + 684 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 120, + 670, + 174, + 683 + ], + "score": 0.9, + "content": "\\mu , \\nu \\in { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 669, + 203, + 684 + ], + "score": 1.0, + "content": ". Here,", + "type": "text" + }, + { + "bbox": [ + 203, + 669, + 306, + 684 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\hat { \\xi } = \\int K ( x , \\cdot ) \\xi ( d x ) \\in \\mathcal { H } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 669, + 506, + 684 + ], + "score": 1.0, + "content": "denotes the mean embedding of a signed measure", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 682, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 107, + 682, + 154, + 694 + ], + "score": 0.91, + "content": "\\xi \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 682, + 294, + 696 + ], + "score": 1.0, + "content": "; we also adopt the convention that", + "type": "text" + }, + { + "bbox": [ + 294, + 682, + 343, + 694 + ], + "score": 0.91, + "content": "| | \\varphi | | _ { \\mathcal { H } } = \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 682, + 353, + 696 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 353, + 682, + 381, + 694 + ], + "score": 0.91, + "content": "\\varphi \\notin { \\mathcal { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 682, + 505, + 696 + ], + "score": 1.0, + "content": ". Similar to the situation in the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 693, + 412, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 395, + 707 + ], + "score": 1.0, + "content": "previous sections, inf-convolution preserves the smoothness property of", + "type": "text" + }, + { + "bbox": [ + 395, + 694, + 408, + 704 + ], + "score": 0.88, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 693, + 412, + 707 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 105, + 707, + 505, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 706, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 344, + 721 + ], + "score": 1.0, + "content": "Proposition 13 (Moreau–Yosida regularization). Suppose", + "type": "text" + }, + { + "bbox": [ + 344, + 707, + 416, + 720 + ], + "score": 0.93, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 706, + 505, + 721 + ], + "score": 1.0, + "content": "is convex, and define", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 714, + 369, + 737 + ], + "spans": [ + { + "bbox": [ + 107, + 719, + 160, + 732 + ], + "score": 0.91, + "content": "{ \\tilde { J } } : = J \\oplus R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 714, + 187, + 737 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 187, + 82, + 239, + 94 + ], + "score": 0.91, + "content": "f : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 81, + 504, + 95 + ], + "score": 1.0, + "content": "be a neural network consisting of k layers whose linear transfor-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 496, + 106 + ], + "score": 1.0, + "content": "mations have spectral norm 1 and whose activation functions are 1-Lipschitz and 1-smooth. Then", + "type": "text" + }, + { + "bbox": [ + 497, + 94, + 504, + 105 + ], + "score": 0.72, + "content": "f", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 158, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 115, + 116 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 116, + 105, + 122, + 114 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 104, + 158, + 116 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 504, + 116 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 132, + 361, + 145 + ], + "lines": [ + { + "bbox": [ + 104, + 131, + 363, + 147 + ], + "spans": [ + { + "bbox": [ + 104, + 131, + 363, + 147 + ], + "score": 1.0, + "content": "6 ENFORCING (D3) WITH GRADIENT PENALTIES", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 108, + 156, + 284, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 285, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 285, + 170 + ], + "score": 1.0, + "content": "(D3) is the following smoothness condition:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 155, + 285, + 170 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 491, + 186 + ], + "lines": [ + { + "bbox": [ + 108, + 172, + 489, + 188 + ], + "spans": [ + { + "bbox": [ + 108, + 172, + 133, + 188 + ], + "score": 1.0, + "content": "(D3)", + "type": "text" + }, + { + "bbox": [ + 133, + 173, + 190, + 186 + ], + "score": 0.92, + "content": "\\mu \\mapsto \\nabla \\Phi _ { \\mu } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 172, + 200, + 188 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 200, + 174, + 211, + 185 + ], + "score": 0.85, + "content": "\\beta _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 172, + 283, + 188 + ], + "score": 1.0, + "content": "-Lipschitz for any", + "type": "text" + }, + { + "bbox": [ + 283, + 174, + 311, + 184 + ], + "score": 0.88, + "content": "x \\in X", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 172, + 332, + 188 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 332, + 173, + 489, + 186 + ], + "score": 0.91, + "content": "\\| \\nabla \\Phi _ { \\mu } ( x ) - \\nabla \\Phi _ { \\nu } ( x ) \\| _ { 2 } \\leq \\beta _ { 2 } W _ { 1 } ( \\mu , \\nu ) .", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 108, + 172, + 489, + 188 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 190, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 191, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 203 + ], + "score": 1.0, + "content": "(D3) requires that the gradients of the optimal discriminator do not change too rapidly in response", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 201, + 504, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 160, + 215 + ], + "score": 1.0, + "content": "to changes in", + "type": "text" + }, + { + "bbox": [ + 161, + 203, + 168, + 213 + ], + "score": 0.79, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 201, + 504, + 215 + ], + "score": 1.0, + "content": ". Indeed, if the discriminator’s gradients are too sensitive to changes in the generator,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 212, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 226 + ], + "score": 1.0, + "content": "the generator may not be able to accurately follow those gradients as it updates itself using a finite", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 221, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 104, + 221, + 342, + 238 + ], + "score": 1.0, + "content": "step size. In finite-dimensional optimization of a function", + "type": "text" + }, + { + "bbox": [ + 342, + 223, + 392, + 235 + ], + "score": 0.93, + "content": "f : { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 221, + 505, + 238 + ], + "score": 1.0, + "content": ", this condition is analogous", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 235, + 237, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 117, + 247 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 235, + 124, + 246 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 235, + 237, + 247 + ], + "score": 1.0, + "content": "having a Lipschitz gradient.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 104, + 191, + 505, + 247 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 251, + 504, + 274 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "We now present an equivalent characterization of (D3) that is easier to check in practice. We define", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 261, + 376, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 295, + 276 + ], + "score": 1.0, + "content": "the Bregman divergence of a convex function", + "type": "text" + }, + { + "bbox": [ + 295, + 262, + 362, + 274 + ], + "score": 0.92, + "content": "J : { \\mathcal { M } } ( X ) \\to { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 261, + 376, + 276 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 250, + 505, + 276 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 278, + 404, + 304 + ], + "lines": [ + { + "bbox": [ + 205, + 278, + 404, + 304 + ], + "spans": [ + { + "bbox": [ + 205, + 278, + 404, + 304 + ], + "score": 0.91, + "content": "\\mathfrak { D } _ { J } ( \\nu , \\mu ) : = J ( \\nu ) - J ( \\mu ) - \\int \\Phi _ { \\mu } ( x ) d ( \\nu - \\mu ) ,", + "type": "interline_equation", + "image_path": "598edc74ea3d2ea830ac891bc02a235feaf388df814ff9d8710862c540c54926.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 205, + 278, + 404, + 304 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 308, + 503, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 134, + 320 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 309, + 149, + 320 + ], + "score": 0.89, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 308, + 285, + 320 + ], + "score": 1.0, + "content": "is the optimal discriminator for", + "type": "text" + }, + { + "bbox": [ + 285, + 309, + 293, + 318 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 308, + 306, + 320 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 307, + 310, + 314, + 320 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 308, + 505, + 320 + ], + "score": 1.0, + "content": ". Then, (D3) is characterized in terms of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 318, + 309, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 309, + 331 + ], + "score": 1.0, + "content": "Bregman divergence and the KR norm as follows:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 106, + 308, + 505, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 502, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 193, + 348 + ], + "score": 1.0, + "content": "Proposition 10. Let", + "type": "text" + }, + { + "bbox": [ + 193, + 333, + 266, + 346 + ], + "score": 0.91, + "content": "J : \\mathcal { M } ( X ) \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 331, + 383, + 348 + ], + "score": 1.0, + "content": "be a convex function. Then", + "type": "text" + }, + { + "bbox": [ + 383, + 334, + 391, + 344 + ], + "score": 0.78, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 331, + 428, + 348 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 428, + 334, + 447, + 345 + ], + "score": 0.62, + "content": "( D 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 331, + 505, + 348 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 344, + 306, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 215, + 360 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathfrak { D } _ { J } ( \\nu , \\mu ) \\le \\frac { \\beta _ { 2 } } { 2 } \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 344, + 243, + 360 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 244, + 346, + 301, + 358 + ], + "score": 0.9, + "content": "\\mu , \\nu \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 344, + 306, + 360 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 331, + 505, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 367, + 496, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 498, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 498, + 381 + ], + "score": 1.0, + "content": "It is straightforward to compute the Bregman divergence corresponding to several popular GANs:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 366, + 498, + 381 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 381, + 442, + 434 + ], + "lines": [ + { + "bbox": [ + 167, + 381, + 442, + 434 + ], + "spans": [ + { + "bbox": [ + 167, + 381, + 442, + 434 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathfrak { D } _ { D \\mathrm { J S } } ( \\cdot | | \\mu _ { 0 } ) \\big ( \\nu , \\mu \\big ) = D _ { \\mathrm { K L } } \\big ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } | | \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } \\big ) + \\frac { 1 } { 2 } D _ { \\mathrm { K L } } \\big ( \\nu | | \\mu \\big ) , } \\\\ & { \\qquad \\mathfrak { D } _ { D _ { \\mathrm { K L } } \\left( \\frac { 1 } { 2 } \\cdot + \\frac { 1 } { 2 } \\mu _ { 0 } \\right. | | \\mu _ { 0 } ) } \\big ( \\nu , \\mu \\big ) = D _ { \\mathrm { K L } } \\big ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } | | \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } \\big ) , } \\\\ & { \\qquad \\mathfrak { D } _ { \\frac { 1 } { 2 } \\mathrm { M M D } ^ { 2 } ( \\cdot , \\mu _ { 0 } ) } \\big ( \\nu , \\mu \\big ) = \\frac { 1 } { 2 } \\mathrm { M M D } ^ { 2 } ( \\nu , \\mu ) . } \\end{array}", + "type": "interline_equation", + "image_path": "e09bf18a7654237ad88536bfa1229c64ec24edf86328bcbfc3ebf62a42abde98.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 167, + 381, + 442, + 398.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 167, + 398.6666666666667, + 442, + 416.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 167, + 416.33333333333337, + 442, + 434.00000000000006 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 437, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 349, + 452 + ], + "score": 1.0, + "content": "The first two Bregman divergences are not bounded above by", + "type": "text" + }, + { + "bbox": [ + 350, + 437, + 395, + 451 + ], + "score": 0.93, + "content": "| | \\mu - \\nu | | _ { \\mathrm { K R } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 436, + 506, + 452 + ], + "score": 1.0, + "content": "for reasons similar to those", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 448, + 245, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 245, + 461 + ], + "score": 1.0, + "content": "discussed in Section 4, and hence:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 436, + 506, + 461 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 463, + 486, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 487, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 413, + 478 + ], + "score": 1.0, + "content": "Proposition 11. The minimax and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 413, + 464, + 433, + 475 + ], + "score": 0.6, + "content": "( D 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 461, + 471, + 478 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 471, + 466, + 483, + 475 + ], + "score": 0.77, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 461, + 487, + 478 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 461, + 487, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "score": 1.0, + "content": "Even so, the Bregman divergence for the non-saturating loss is always less than that of the minimax", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "GAN, suggesting that the non-saturating loss should be stable in more situations than the minimax", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "GAN. On the other hand, the MMD-based loss (Li et al., 2015) does satisfy (D3) when its kernel is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 517, + 284, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 187, + 530 + ], + "score": 1.0, + "content": "the Gaussian kernel", + "type": "text" + }, + { + "bbox": [ + 188, + 517, + 279, + 532 + ], + "score": 0.93, + "content": "K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 518, + 284, + 530 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 485, + 505, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 533, + 477, + 546 + ], + "lines": [ + { + "bbox": [ + 104, + 532, + 474, + 549 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 358, + 549 + ], + "score": 1.0, + "content": "Proposition 12. The MMD loss with Gaussian kernel satisfies", + "type": "text" + }, + { + "bbox": [ + 359, + 534, + 378, + 545 + ], + "score": 0.66, + "content": "( D 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 532, + 399, + 549 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 399, + 534, + 435, + 545 + ], + "score": 0.89, + "content": "\\beta _ { 2 } = 2 \\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 532, + 463, + 549 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 464, + 535, + 474, + 546 + ], + "score": 0.83, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 532, + 474, + 549 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 559, + 405, + 570 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 405, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 405, + 571 + ], + "score": 1.0, + "content": "6.1 FROM MAXIMUM MEAN DISCREPANCY TO GRADIENT PENALTIES", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "Having identified the MMD-based loss as one that satisfies (D3), we next follow the strategy outlined", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "in Section 3 to convert it into a regularizer for an arbitrary loss function. To do so, we define the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 600, + 435, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 152, + 614 + ], + "score": 1.0, + "content": "regularizer", + "type": "text" + }, + { + "bbox": [ + 152, + 601, + 225, + 613 + ], + "score": 0.94, + "content": "R _ { 3 } : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 600, + 363, + 614 + ], + "score": 1.0, + "content": "and compute its convex conjugate", + "type": "text" + }, + { + "bbox": [ + 364, + 601, + 430, + 613 + ], + "score": 0.93, + "content": "R _ { 3 } ^ { \\star } : { \\mathcal { C } } ( X ) { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 600, + 435, + 614 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 579, + 505, + 614 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 617, + 397, + 643 + ], + "lines": [ + { + "bbox": [ + 214, + 617, + 397, + 643 + ], + "spans": [ + { + "bbox": [ + 214, + 617, + 397, + 643 + ], + "score": 0.93, + "content": "R _ { 3 } ( \\xi ) : = \\frac { \\beta _ { 2 } } { 4 \\pi } \\| \\hat { \\xi } \\| _ { \\mathcal { H } } ^ { 2 } , \\qquad R _ { 3 } ^ { \\star } ( \\varphi ) = \\frac { \\pi } { \\beta _ { 2 } } \\| \\varphi \\| _ { \\mathcal { H } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "0897a6672da3d502f8cb8b541af4a7e1d40a0ffd4529e66a7b6b57d5c9cd9812.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 214, + 617, + 397, + 643 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 647, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 432, + 660 + ], + "score": 1.0, + "content": "The norm is the norm of a reproducing kernel Hilbert space norm (RKHS)", + "type": "text" + }, + { + "bbox": [ + 433, + 648, + 442, + 657 + ], + "score": 0.79, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 646, + 505, + 660 + ], + "score": 1.0, + "content": "with Gaussian", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 656, + 504, + 672 + ], + "spans": [ + { + "bbox": [ + 104, + 656, + 397, + 672 + ], + "score": 1.0, + "content": "kernel; this norm extends the MMD to signed measures, as it holds that", + "type": "text" + }, + { + "bbox": [ + 397, + 658, + 504, + 671 + ], + "score": 0.9, + "content": "\\mathrm { M M D } ( \\mu , \\nu ) = \\| \\hat { \\mu } - \\hat { \\nu } \\| _ { \\mathcal { H } }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 669, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 120, + 684 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 120, + 670, + 174, + 683 + ], + "score": 0.9, + "content": "\\mu , \\nu \\in { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 669, + 203, + 684 + ], + "score": 1.0, + "content": ". Here,", + "type": "text" + }, + { + "bbox": [ + 203, + 669, + 306, + 684 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\hat { \\xi } = \\int K ( x , \\cdot ) \\xi ( d x ) \\in \\mathcal { H } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 669, + 506, + 684 + ], + "score": 1.0, + "content": "denotes the mean embedding of a signed measure", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 682, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 107, + 682, + 154, + 694 + ], + "score": 0.91, + "content": "\\xi \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 682, + 294, + 696 + ], + "score": 1.0, + "content": "; we also adopt the convention that", + "type": "text" + }, + { + "bbox": [ + 294, + 682, + 343, + 694 + ], + "score": 0.91, + "content": "| | \\varphi | | _ { \\mathcal { H } } = \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 682, + 353, + 696 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 353, + 682, + 381, + 694 + ], + "score": 0.91, + "content": "\\varphi \\notin { \\mathcal { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 682, + 505, + 696 + ], + "score": 1.0, + "content": ". Similar to the situation in the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 693, + 412, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 395, + 707 + ], + "score": 1.0, + "content": "previous sections, inf-convolution preserves the smoothness property of", + "type": "text" + }, + { + "bbox": [ + 395, + 694, + 408, + 704 + ], + "score": 0.88, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 693, + 412, + 707 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 646, + 506, + 707 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 707, + 505, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 706, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 344, + 721 + ], + "score": 1.0, + "content": "Proposition 13 (Moreau–Yosida regularization). Suppose", + "type": "text" + }, + { + "bbox": [ + 344, + 707, + 416, + 720 + ], + "score": 0.93, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 706, + 505, + 721 + ], + "score": 1.0, + "content": "is convex, and define", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 714, + 369, + 737 + ], + "spans": [ + { + "bbox": [ + 107, + 719, + 160, + 732 + ], + "score": 0.91, + "content": "{ \\tilde { J } } : = J \\oplus R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 714, + 187, + 737 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 188, + 719, + 196, + 731 + ], + "score": 0.84, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 714, + 255, + 737 + ], + "score": 1.0, + "content": "is convex, and", + "type": "text" + }, + { + "bbox": [ + 256, + 719, + 365, + 734 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathfrak { D } _ { \\widetilde { J } } ( \\nu , \\mu ) \\le \\frac { \\beta _ { 2 } } { 2 } \\vert \\vert \\mu - \\nu \\vert \\vert _ { \\mathrm { K R } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 714, + 369, + 737 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 706, + 505, + 737 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "By Proposition 10, this transformed loss function satisfies (D3), having inherited the regularity prop-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "erties of the squared MMD. This transformed function is a generalization of Moreau–Yosida regu-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "larization or the Moreau envelope (see Chapter 1 in Rockafeller & Wets (1998)). It is well-known", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 226, + 128 + ], + "score": 1.0, + "content": "that in the case of a function", + "type": "text" + }, + { + "bbox": [ + 226, + 115, + 281, + 127 + ], + "score": 0.92, + "content": "f : \\mathbb { R } ^ { n } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 114, + 506, + 128 + ], + "score": 1.0, + "content": ", this regularization results in a function with Lipschitz", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 478, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 478, + 139 + ], + "score": 1.0, + "content": "gradients, so it is unsurprising that this property carries over to the infinite-dimensional case.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 105, + 143, + 503, + 165 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "Applying (4) and (19), we see that the transformed loss function can be minimized as a GAN by", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 380, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 380, + 167 + ], + "score": 1.0, + "content": "implementing an RKHS squared norm penalty on the discriminator:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 171, + 443, + 194 + ], + "lines": [ + { + "bbox": [ + 168, + 171, + 443, + 194 + ], + "spans": [ + { + "bbox": [ + 168, + 171, + 443, + 194 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\underset { \\mu } { \\operatorname* { i n f } } \\big ( J \\oplus \\frac { \\beta _ { 2 } } { 4 \\pi } | | \\cdot | | _ { \\mathcal { H } } ^ { 2 } \\big ) ( \\mu ) = \\underset { \\mu } { \\operatorname* { i n f } } \\underset { \\varphi } { \\operatorname* { s u p } } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - J ^ { \\star } ( \\varphi ) - \\frac { \\pi } { \\beta _ { 2 } } | | \\varphi | | _ { \\mathcal { H } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "8f42bad177c78a07078738f1e506df16de63825ff670a323ab5710499f8f01db.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 168, + 171, + 443, + 194 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 217 + ], + "score": 1.0, + "content": "Computationally, the RKHS norm is difficult to evaluate. We propose taking advantage of the fol-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 215, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 261, + 230 + ], + "score": 1.0, + "content": "lowing infinite series representation of", + "type": "text" + }, + { + "bbox": [ + 261, + 215, + 286, + 228 + ], + "score": 0.92, + "content": "| | f | | _ { \\mathcal { H } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 215, + 401, + 230 + ], + "score": 1.0, + "content": "in terms of the derivatives of", + "type": "text" + }, + { + "bbox": [ + 402, + 216, + 409, + 227 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 215, + 506, + 230 + ], + "score": 1.0, + "content": "(Fasshauer & Ye, 2011;", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 226, + 188, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 188, + 239 + ], + "score": 1.0, + "content": "Novak et al., 2018):", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 242, + 502, + 266 + ], + "lines": [ + { + "bbox": [ + 104, + 240, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 193, + 257 + ], + "score": 1.0, + "content": "Proposition 14. Let", + "type": "text" + }, + { + "bbox": [ + 193, + 244, + 203, + 254 + ], + "score": 0.64, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 240, + 365, + 257 + ], + "score": 1.0, + "content": "be an RKHS with the Gaussian kernel", + "type": "text" + }, + { + "bbox": [ + 366, + 241, + 459, + 255 + ], + "score": 0.92, + "content": "K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 240, + 505, + 257 + ], + "score": 1.0, + "content": ". Then for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 252, + 139, + 268 + ], + "spans": [ + { + "bbox": [ + 107, + 255, + 134, + 266 + ], + "score": 0.9, + "content": "f \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 252, + 139, + 268 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 271, + 457, + 324 + ], + "lines": [ + { + "bbox": [ + 153, + 271, + 457, + 324 + ], + "spans": [ + { + "bbox": [ + 153, + 271, + 457, + 324 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\| f \\| _ { \\mathcal H } ^ { 2 } = \\displaystyle \\sum _ { k = 0 } ^ { \\infty } ( 4 \\pi ) ^ { - k } \\sum _ { \\substack { k _ { 1 } + \\cdots + k _ { d } = k } } \\frac { 1 } { \\prod _ { i = 1 } ^ { d } k _ { i } ! } \\| \\partial _ { x _ { 1 } } ^ { k _ { 1 } } \\cdot \\cdot \\cdot \\partial _ { x _ { d } } ^ { k _ { d } } f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } } \\\\ & { \\quad \\quad = \\| f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } + \\frac { 1 } { 4 \\pi } \\| \\nabla f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } + \\frac { 1 } { 1 6 \\pi ^ { 2 } } \\| \\nabla ^ { 2 } f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } + o t h e r t e r m s . } \\end{array}", + "type": "interline_equation", + "image_path": "59037ab0f57840e6530aa46f9808ef8ff7fd922e2a85c37f1daf31a9c7319f40.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 153, + 271, + 457, + 288.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 153, + 288.6666666666667, + 457, + 306.33333333333337 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 153, + 306.33333333333337, + 457, + 324.00000000000006 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "In an ideal world, we would use this expression as a penalty on the discriminator to enforce (D3). Of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 345, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 356 + ], + "score": 1.0, + "content": "course, as an infinite series, this formulation is computationally impractical. However, the first two", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "terms are very close to common GAN techniques like gradient penalties (Gulrajani et al., 2017) and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "penalizing the output of the discriminator (Karras et al., 2018). We therefore interpret these common", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "practices as partially applying the penalty given by the RKHS norm squared, approximately enforc-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "ing (D3). We view the choice of only using the leading terms as a disadvantageous but practical", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 399, + 147, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 147, + 412 + ], + "score": 1.0, + "content": "necessity.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 415, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "Interestingly, according to our analysis, gradient penalties and spectral normalization are not inter-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "changeable, even though both techniques were designed to constrain the Lipschitz constant of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "discriminator. Instead, our analysis suggests that they serve different purposes: gradient penalties", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 449, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 504, + 461 + ], + "score": 1.0, + "content": "enforce the variational smoothness (D3), while spectral normalization enforces Lipschitz continuity", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "(D1). This demystifies the puzzling observation of Miyato (2018) that GANs using only spectral", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "normalization with a WGAN loss do not seem to train well; it also explains why using both spectral", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 482, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 493 + ], + "score": 1.0, + "content": "normalization and a gradient penalty is a reasonable strategy. It also motivates the use of gradient", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 493, + 420, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 420, + 505 + ], + "score": 1.0, + "content": "penalties applied to losses other than the Wasserstein loss (Fedus et al., 2018).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 520, + 370, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 371, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 371, + 535 + ], + "score": 1.0, + "content": "7 VERIFYING THE THEORETICAL LEARNING RATE", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "In this section, we empirically test the theoretical learning rate given by Theorem 1 and Proposition 1", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 557, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 569 + ], + "score": 1.0, + "content": "as well as our regularization scheme (7) based on inf-convolutions. We approximately implement", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 365, + 581 + ], + "score": 1.0, + "content": "our composite regularization scheme (7) on a trivial base loss of", + "type": "text" + }, + { + "bbox": [ + 365, + 567, + 446, + 579 + ], + "score": 0.93, + "content": "J ( \\mu ) = \\overset { \\triangledown } { \\chi } \\bar { \\{ \\mu = \\mu _ { 0 } \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "by alternating", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 578, + 219, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 219, + 591 + ], + "score": 1.0, + "content": "stochastic gradient steps on", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 595, + 454, + 621 + ], + "lines": [ + { + "bbox": [ + 155, + 595, + 454, + 621 + ], + "spans": [ + { + "bbox": [ + 155, + 595, + 454, + 621 + ], + "score": 0.92, + "content": "\\operatorname* { i n f } _ { \\mu } \\operatorname* { s u p } _ { \\varphi } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - \\mathbb { E } _ { x \\sim \\mu _ { 0 } } [ \\varphi ( x ) ] - \\frac { \\pi } { \\beta _ { 2 } } \\mathbb { E } _ { x \\sim \\tilde { \\mu } } \\Big [ \\varphi ( x ) ^ { 2 } + \\frac { 1 } { 4 \\pi } | | \\nabla \\varphi ( x ) | | ^ { 2 } \\Big ] ,", + "type": "interline_equation", + "image_path": "6608ac516318ceaaef9bc1dfd61a9bb98a6b7ce6b9004b7250e3c937891c75d4.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 155, + 595, + 454, + 621 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 504, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 133, + 639 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 627, + 141, + 638 + ], + "score": 0.84, + "content": "\\tilde { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 627, + 328, + 639 + ], + "score": 1.0, + "content": "is a random interpolate between samples from", + "type": "text" + }, + { + "bbox": [ + 329, + 628, + 336, + 638 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 627, + 353, + 639 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 354, + 628, + 365, + 638 + ], + "score": 0.86, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 627, + 504, + 639 + ], + "score": 1.0, + "content": ", as used in Gulrajani et al. (2017).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "The regularization term is a truncation of the series for the squared RKHS norm (22) and approx-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 660 + ], + "score": 1.0, + "content": "imately enforces (D3). The discriminator is a 7-layer convolutional neural network with spectral", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "normalization1 and ELU activations, an architecture that enforces (D1) and (D2). We include a final", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 204, + 683 + ], + "score": 1.0, + "content": "scalar multiplication by", + "type": "text" + }, + { + "bbox": [ + 204, + 672, + 212, + 681 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 670, + 315, + 683 + ], + "score": 1.0, + "content": "so that by Proposition 9,", + "type": "text" + }, + { + "bbox": [ + 316, + 671, + 353, + 682 + ], + "score": 0.9, + "content": "\\beta _ { 1 } = 7 \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 670, + 506, + 683 + ], + "score": 1.0, + "content": ". We take two discriminator steps for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 682, + 457, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 457, + 695 + ], + "score": 1.0, + "content": "every generator step, to better approximate our assumption of an optimal discriminator.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 701, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 698, + 506, + 715 + ], + "spans": [ + { + "bbox": [ + 118, + 698, + 506, + 715 + ], + "score": 1.0, + "content": "1Whereas Miyato et al. (2018) divide each layer by the spectral norm of the convolutional kernel, we divide", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "by the spectral norm of the convolutional operator itself, computed using the same power iteration algorithm", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 473, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 473, + 734 + ], + "score": 1.0, + "content": "applied to the operator. This is so that the layers are truly 1-Lipschitz, which is critical for our theory.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "By Proposition 10, this transformed loss function satisfies (D3), having inherited the regularity prop-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "erties of the squared MMD. This transformed function is a generalization of Moreau–Yosida regu-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "larization or the Moreau envelope (see Chapter 1 in Rockafeller & Wets (1998)). It is well-known", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 226, + 128 + ], + "score": 1.0, + "content": "that in the case of a function", + "type": "text" + }, + { + "bbox": [ + 226, + 115, + 281, + 127 + ], + "score": 0.92, + "content": "f : \\mathbb { R } ^ { n } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 114, + 506, + 128 + ], + "score": 1.0, + "content": ", this regularization results in a function with Lipschitz", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 478, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 478, + 139 + ], + "score": 1.0, + "content": "gradients, so it is unsurprising that this property carries over to the infinite-dimensional case.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 81, + 506, + 139 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 143, + 503, + 165 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "Applying (4) and (19), we see that the transformed loss function can be minimized as a GAN by", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 380, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 380, + 167 + ], + "score": 1.0, + "content": "implementing an RKHS squared norm penalty on the discriminator:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 142, + 505, + 167 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 171, + 443, + 194 + ], + "lines": [ + { + "bbox": [ + 168, + 171, + 443, + 194 + ], + "spans": [ + { + "bbox": [ + 168, + 171, + 443, + 194 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\underset { \\mu } { \\operatorname* { i n f } } \\big ( J \\oplus \\frac { \\beta _ { 2 } } { 4 \\pi } | | \\cdot | | _ { \\mathcal { H } } ^ { 2 } \\big ) ( \\mu ) = \\underset { \\mu } { \\operatorname* { i n f } } \\underset { \\varphi } { \\operatorname* { s u p } } \\mathbb { E } _ { x \\sim \\mu } [ \\varphi ( x ) ] - J ^ { \\star } ( \\varphi ) - \\frac { \\pi } { \\beta _ { 2 } } | | \\varphi | | _ { \\mathcal { H } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "8f42bad177c78a07078738f1e506df16de63825ff670a323ab5710499f8f01db.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 168, + 171, + 443, + 194 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 217 + ], + "score": 1.0, + "content": "Computationally, the RKHS norm is difficult to evaluate. We propose taking advantage of the fol-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 215, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 261, + 230 + ], + "score": 1.0, + "content": "lowing infinite series representation of", + "type": "text" + }, + { + "bbox": [ + 261, + 215, + 286, + 228 + ], + "score": 0.92, + "content": "| | f | | _ { \\mathcal { H } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 215, + 401, + 230 + ], + "score": 1.0, + "content": "in terms of the derivatives of", + "type": "text" + }, + { + "bbox": [ + 402, + 216, + 409, + 227 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 215, + 506, + 230 + ], + "score": 1.0, + "content": "(Fasshauer & Ye, 2011;", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 226, + 188, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 188, + 239 + ], + "score": 1.0, + "content": "Novak et al., 2018):", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 203, + 506, + 239 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 242, + 502, + 266 + ], + "lines": [ + { + "bbox": [ + 104, + 240, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 193, + 257 + ], + "score": 1.0, + "content": "Proposition 14. Let", + "type": "text" + }, + { + "bbox": [ + 193, + 244, + 203, + 254 + ], + "score": 0.64, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 240, + 365, + 257 + ], + "score": 1.0, + "content": "be an RKHS with the Gaussian kernel", + "type": "text" + }, + { + "bbox": [ + 366, + 241, + 459, + 255 + ], + "score": 0.92, + "content": "K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 240, + 505, + 257 + ], + "score": 1.0, + "content": ". Then for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 252, + 139, + 268 + ], + "spans": [ + { + "bbox": [ + 107, + 255, + 134, + 266 + ], + "score": 0.9, + "content": "f \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 252, + 139, + 268 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 240, + 505, + 268 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 271, + 457, + 324 + ], + "lines": [ + { + "bbox": [ + 153, + 271, + 457, + 324 + ], + "spans": [ + { + "bbox": [ + 153, + 271, + 457, + 324 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\| f \\| _ { \\mathcal H } ^ { 2 } = \\displaystyle \\sum _ { k = 0 } ^ { \\infty } ( 4 \\pi ) ^ { - k } \\sum _ { \\substack { k _ { 1 } + \\cdots + k _ { d } = k } } \\frac { 1 } { \\prod _ { i = 1 } ^ { d } k _ { i } ! } \\| \\partial _ { x _ { 1 } } ^ { k _ { 1 } } \\cdot \\cdot \\cdot \\partial _ { x _ { d } } ^ { k _ { d } } f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } } \\\\ & { \\quad \\quad = \\| f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } + \\frac { 1 } { 4 \\pi } \\| \\nabla f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } + \\frac { 1 } { 1 6 \\pi ^ { 2 } } \\| \\nabla ^ { 2 } f \\| _ { L ^ { 2 } ( \\mathbb R ^ { d } ) } ^ { 2 } + o t h e r t e r m s . } \\end{array}", + "type": "interline_equation", + "image_path": "59037ab0f57840e6530aa46f9808ef8ff7fd922e2a85c37f1daf31a9c7319f40.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 153, + 271, + 457, + 288.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 153, + 288.6666666666667, + 457, + 306.33333333333337 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 153, + 306.33333333333337, + 457, + 324.00000000000006 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "In an ideal world, we would use this expression as a penalty on the discriminator to enforce (D3). Of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 345, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 356 + ], + "score": 1.0, + "content": "course, as an infinite series, this formulation is computationally impractical. However, the first two", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "terms are very close to common GAN techniques like gradient penalties (Gulrajani et al., 2017) and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "penalizing the output of the discriminator (Karras et al., 2018). We therefore interpret these common", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "practices as partially applying the penalty given by the RKHS norm squared, approximately enforc-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "ing (D3). We view the choice of only using the leading terms as a disadvantageous but practical", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 399, + 147, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 147, + 412 + ], + "score": 1.0, + "content": "necessity.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 333, + 506, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 415, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "Interestingly, according to our analysis, gradient penalties and spectral normalization are not inter-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "changeable, even though both techniques were designed to constrain the Lipschitz constant of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "discriminator. Instead, our analysis suggests that they serve different purposes: gradient penalties", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 449, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 504, + 461 + ], + "score": 1.0, + "content": "enforce the variational smoothness (D3), while spectral normalization enforces Lipschitz continuity", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "(D1). This demystifies the puzzling observation of Miyato (2018) that GANs using only spectral", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "normalization with a WGAN loss do not seem to train well; it also explains why using both spectral", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 482, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 493 + ], + "score": 1.0, + "content": "normalization and a gradient penalty is a reasonable strategy. It also motivates the use of gradient", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 493, + 420, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 420, + 505 + ], + "score": 1.0, + "content": "penalties applied to losses other than the Wasserstein loss (Fedus et al., 2018).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 416, + 506, + 505 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 520, + 370, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 371, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 371, + 535 + ], + "score": 1.0, + "content": "7 VERIFYING THE THEORETICAL LEARNING RATE", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "In this section, we empirically test the theoretical learning rate given by Theorem 1 and Proposition 1", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 557, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 569 + ], + "score": 1.0, + "content": "as well as our regularization scheme (7) based on inf-convolutions. 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We trained each model for 100,000", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "score": 1.0, + "content": "steps on CIFAR-10 and evaluate each model using the Frechet Inception Distance (FID) of Heusel ´", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "et al. (2017). We hypothesize that stability is correlated with image quality; Figure 1 plots the FID", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 412, + 428 + ], + "score": 1.0, + "content": "for each hyperparameter setting in terms of the ratio of the true learning rate", + "type": "text" + }, + { + "bbox": [ + 413, + 417, + 420, + 427 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "and the theoretically", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 425, + 504, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 201, + 441 + ], + "score": 1.0, + "content": "motivated learning rate", + "type": "text" + }, + { + "bbox": [ + 201, + 428, + 212, + 438 + ], + "score": 0.85, + "content": "\\gamma _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 425, + 482, + 441 + ], + "score": 1.0, + "content": ". We find that the best FID scores are obtained in the region where", + "type": "text" + }, + { + "bbox": [ + 482, + 426, + 504, + 439 + ], + "score": 0.9, + "content": "\\gamma / \\gamma _ { 0 }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 308, + 451 + ], + "score": 1.0, + "content": "is between 1 and 1000. For small learning rates", + "type": "text" + }, + { + "bbox": [ + 308, + 438, + 354, + 450 + ], + "score": 0.91, + "content": "\\gamma / \\gamma _ { 0 } \\ll 1", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 437, + 506, + 451 + ], + "score": 1.0, + "content": ", we observe that the convergence is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "too slow to make a reasonable progress on the objective, whereas as the learning rate gets larger", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 150, + 471 + ], + "score": 0.91, + "content": "\\gamma / \\gamma _ { 0 } \\gg 1", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 460, + 506, + 472 + ], + "score": 1.0, + "content": ", we observe a steady increase in FID, signalling unstable behavior. It also makes sense", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "score": 1.0, + "content": "that learning rates slightly above the optimal rate produce good results, since our theoretical learning", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 481, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 104, + 481, + 505, + 495 + ], + "score": 1.0, + "content": "rate is a conservative lower bound. Note that our intention is to test our theory, not to generate good", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "images, which is difficult due to our weak choice of generator. Overall, this experiment shows that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 504, + 308, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 308, + 515 + ], + "score": 1.0, + "content": "our theory and regularization scheme are sensible.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 530, + 203, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 205, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 205, + 545 + ], + "score": 1.0, + "content": "8 FUTURE WORK", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "Inexact gradient descent In this paper, we employed several assumptions in order to regard the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "GAN algorithm as gradient descent. However, real-world GAN algorithms must be treated as “in-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "exact” descent algorithms. As such, future work includes: (i) relaxing the optimal discriminator", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "assumption (cf. Sanjabi et al. (2018)) or providing a stability result for discrete simultaneous gradi-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "ent descent (cf. continuous time analysis in Nagarajan & Kolter (2017); Mescheder et al. 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In practice, generators", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "are often implemented as deep neural networks, and involve some specific architectures such as de-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "convolution layers (Radford et al., 2015) and residual blocks (e.g., Gulrajani et al. (2017); Miyato", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "et al. (2018)). 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With this setup, Theorem 1", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 324, + 258, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 258, + 335 + ], + "score": 1.0, + "content": "suggests a theoretical learning rate of", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 106, + 312, + 505, + 335 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 337, + 397, + 364 + ], + "lines": [ + { + "bbox": [ + 214, + 337, + 397, + 364 + ], + "spans": [ + { + "bbox": [ + 214, + 337, + 397, + 364 + ], + "score": 0.93, + "content": "\\gamma _ { 0 } = \\frac { 1 } { L } = \\frac { 1 } { \\alpha B + A ^ { 2 } ( \\beta _ { 1 } + \\beta _ { 2 } ) } = \\frac { N } { 7 \\alpha + \\beta _ { 2 } } .", + "type": "interline_equation", + "image_path": "fb75a3f293bf9879968664692240ce7767e6351916583ecca43b7d48297fbde0.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 214, + 337, + 397, + 350.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 214, + 350.5, + 397, + 364.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 408, + 385 + ], + "score": 1.0, + "content": "We randomly generated hyperparameter settings for the Lipschitz constant", + "type": "text" + }, + { + "bbox": [ + 409, + 374, + 416, + 382 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 371, + 505, + 385 + ], + "score": 1.0, + "content": ", the smoothness con-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 128, + 395 + ], + "score": 1.0, + "content": "stant", + "type": "text" + }, + { + "bbox": [ + 129, + 383, + 140, + 394 + ], + "score": 0.88, + "content": "\\beta _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 382, + 242, + 395 + ], + "score": 1.0, + "content": ", the number of particles", + "type": "text" + }, + { + "bbox": [ + 243, + 383, + 253, + 393 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 382, + 344, + 395 + ], + "score": 1.0, + "content": ", and the learning rate", + "type": "text" + }, + { + "bbox": [ + 344, + 385, + 352, + 394 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 382, + 505, + 395 + ], + "score": 1.0, + "content": ". We trained each model for 100,000", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "score": 1.0, + "content": "steps on CIFAR-10 and evaluate each model using the Frechet Inception Distance (FID) of Heusel ´", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "et al. (2017). We hypothesize that stability is correlated with image quality; Figure 1 plots the FID", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 412, + 428 + ], + "score": 1.0, + "content": "for each hyperparameter setting in terms of the ratio of the true learning rate", + "type": "text" + }, + { + "bbox": [ + 413, + 417, + 420, + 427 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "and the theoretically", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 425, + 504, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 201, + 441 + ], + "score": 1.0, + "content": "motivated learning rate", + "type": "text" + }, + { + "bbox": [ + 201, + 428, + 212, + 438 + ], + "score": 0.85, + "content": "\\gamma _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 425, + 482, + 441 + ], + "score": 1.0, + "content": ". We find that the best FID scores are obtained in the region where", + "type": "text" + }, + { + "bbox": [ + 482, + 426, + 504, + 439 + ], + "score": 0.9, + "content": "\\gamma / \\gamma _ { 0 }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 308, + 451 + ], + "score": 1.0, + "content": "is between 1 and 1000. For small learning rates", + "type": "text" + }, + { + "bbox": [ + 308, + 438, + 354, + 450 + ], + "score": 0.91, + "content": "\\gamma / \\gamma _ { 0 } \\ll 1", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 437, + 506, + 451 + ], + "score": 1.0, + "content": ", we observe that the convergence is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "too slow to make a reasonable progress on the objective, whereas as the learning rate gets larger", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 150, + 471 + ], + "score": 0.91, + "content": "\\gamma / \\gamma _ { 0 } \\gg 1", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 460, + 506, + 472 + ], + "score": 1.0, + "content": ", we observe a steady increase in FID, signalling unstable behavior. It also makes sense", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "score": 1.0, + "content": "that learning rates slightly above the optimal rate produce good results, since our theoretical learning", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 481, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 104, + 481, + 505, + 495 + ], + "score": 1.0, + "content": "rate is a conservative lower bound. Note that our intention is to test our theory, not to generate good", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "images, which is difficult due to our weak choice of generator. Overall, this experiment shows that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 504, + 308, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 308, + 515 + ], + "score": 1.0, + "content": "our theory and regularization scheme are sensible.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 371, + 506, + 515 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 530, + 203, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 205, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 205, + 545 + ], + "score": 1.0, + "content": "8 FUTURE WORK", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "Inexact gradient descent In this paper, we employed several assumptions in order to regard the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "GAN algorithm as gradient descent. However, real-world GAN algorithms must be treated as “in-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "exact” descent algorithms. As such, future work includes: (i) relaxing the optimal discriminator", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "assumption (cf. Sanjabi et al. (2018)) or providing a stability result for discrete simultaneous gradi-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "ent descent (cf. continuous time analysis in Nagarajan & Kolter (2017); Mescheder et al. (2018)),", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "(ii) addressing stochastic approximations of gradients (i.e., SGD), and (iii) providing error bounds", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 621, + 294, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 294, + 633 + ], + "score": 1.0, + "content": "for the truncated gradient penalty used in (23).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 554, + 505, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "Generator architectures Another important direction of research is to seek more powerful gen-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "erator architectures that satisfy our smoothness assumptions (G1) and (G2). In practice, generators", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "are often implemented as deep neural networks, and involve some specific architectures such as de-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "convolution layers (Radford et al., 2015) and residual blocks (e.g., Gulrajani et al. (2017); Miyato", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "et al. (2018)). In this paper, we did not provide results on the smoothness of general classes of gen-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 496, + 711 + ], + "score": 1.0, + "content": "erators, since our focus is to analyze stability properties influenced by the choice of loss function", + "type": "text" + }, + { + "bbox": [ + 496, + 699, + 504, + 709 + ], + "score": 0.76, + "content": "J", + "type": "inline_equation" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "(and therefore optimal discriminators). However, our conditions (G1) and (G2) shed light on how to", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 419, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 419, + 732 + ], + "score": 1.0, + "content": "obtain smoothly parameterized neural networks, which is left for future work.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 642, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 83, + 201, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 83, + 201, + 94 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 201, + 94 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 101, + 504, + 124 + ], + "lines": [ + { + "bbox": [ + 106, + 100, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 505, + 114 + ], + "score": 1.0, + "content": "We would like to thank Kohei Hayashi, Katsuhiko Ishiguro, Masanori Koyama, Shin-ichi Maeda,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 112, + 447, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 112, + 447, + 126 + ], + "score": 1.0, + "content": "Takeru Miyato, Masaki Watanabe, and Shoichiro Yamaguchi for helpful discussions.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 141, + 175, + 153 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 176, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 176, + 154 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 159, + 504, + 181 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "Luigi Ambrosio, Nicola Gigli, and Giuseppe Savare.´ Gradient Flows in Metric Spaces and in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 115, + 170, + 381, + 181 + ], + "spans": [ + { + "bbox": [ + 115, + 170, + 381, + 181 + ], + "score": 1.0, + "content": "Space of Probability Measures. 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On the Fenchel duality between strong convexity and Lipschitz continuous gradient.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 405, + 279, + 418 + ], + "spans": [ + { + "bbox": [ + 115, + 405, + 279, + 418 + ], + "score": 1.0, + "content": "arXiv preprint arXiv:1803.06573, 2018.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 108, + 437, + 258, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 258, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 243, + 453 + ], + "score": 1.0, + "content": "A INF-CONVOLUTION IN", + "type": "text" + }, + { + "bbox": [ + 243, + 437, + 258, + 449 + ], + "score": 0.82, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "To gain intuition on the inf-convolution, we present a finite-dimensional analogue of the techniques", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "in Section 3. For simplicity of presentation, we will omit any regularity conditions (e.g., lower", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 486, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 496 + ], + "score": 1.0, + "content": "semicontinuity). We refer readers to Chapter 12 of Bauschke & Combettes (2011) for a detailed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 496, + 160, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 160, + 507 + ], + "score": 1.0, + "content": "introduction.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 105, + 511, + 504, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 122, + 525 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 514, + 130, + 523 + ], + "score": 0.82, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 511, + 147, + 525 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 148, + 514, + 157, + 523 + ], + "score": 0.83, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 511, + 252, + 525 + ], + "score": 1.0, + "content": "be convex functions on", + "type": "text" + }, + { + "bbox": [ + 252, + 512, + 264, + 523 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 511, + 361, + 525 + ], + "score": 1.0, + "content": ". The inf-convolution of", + "type": "text" + }, + { + "bbox": [ + 362, + 514, + 369, + 523 + ], + "score": 0.84, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 511, + 387, + 525 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 387, + 514, + 396, + 523 + ], + "score": 0.85, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 511, + 448, + 525 + ], + "score": 1.0, + "content": "is a function", + "type": "text" + }, + { + "bbox": [ + 449, + 514, + 473, + 523 + ], + "score": 0.89, + "content": "J \\star R", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "defined", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 524, + 119, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 119, + 536 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 533, + 385, + 553 + ], + "lines": [ + { + "bbox": [ + 225, + 533, + 385, + 553 + ], + "spans": [ + { + "bbox": [ + 225, + 533, + 385, + 553 + ], + "score": 0.93, + "content": "( J \\oplus R ) ( x ) : = \\operatorname* { i n f } _ { z \\in \\mathbb { R } ^ { d } } J ( z ) + R ( x - z ) .", + "type": "interline_equation", + "image_path": "3d3ce40559c91980378de44bde7e9b81eed2676e55659d86974ae5137b4a21ea.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 225, + 533, + 385, + 553 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 407, + 568 + ], + "score": 1.0, + "content": "The inf-convolution is often called the epigraphic sum since the epigraph of", + "type": "text" + }, + { + "bbox": [ + 407, + 556, + 430, + 566 + ], + "score": 0.9, + "content": "J { \\star R }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "coincides with the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 237, + 578 + ], + "score": 1.0, + "content": "Minkowski sum of epigraphs of", + "type": "text" + }, + { + "bbox": [ + 237, + 567, + 245, + 576 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 566, + 263, + 578 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 263, + 567, + 272, + 577 + ], + "score": 0.82, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 566, + 505, + 578 + ], + "score": 1.0, + "content": ", as Figure 2 illustrates. The inf-convolution is associative", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 577, + 504, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 342, + 590 + ], + "score": 1.0, + "content": "and commutative operation; that is, it is always true that", + "type": "text" + }, + { + "bbox": [ + 342, + 577, + 504, + 590 + ], + "score": 0.89, + "content": "\\left( J _ { 1 } \\oplus J _ { 2 } \\right) \\oplus J _ { 3 } = J _ { 1 } \\oplus ( J _ { 2 } \\oplus J _ { 3 } ) = :", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 588, + 263, + 601 + ], + "spans": [ + { + "bbox": [ + 107, + 588, + 162, + 600 + ], + "score": 0.93, + "content": "J _ { 1 } \\oplus J _ { 2 } \\oplus J _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 589, + 180, + 601 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 181, + 588, + 259, + 600 + ], + "score": 0.91, + "content": "J _ { 1 } \\oplus J _ { 2 } = J _ { 2 } \\oplus J _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 589, + 263, + 601 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "There are two important special cases of inf-convolutions: The first one is the Pasch–Hausdorff", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 615, + 504, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 145, + 630 + ], + "score": 1.0, + "content": "envelope", + "type": "text" + }, + { + "bbox": [ + 146, + 617, + 158, + 627 + ], + "score": 0.87, + "content": "J _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 615, + 320, + 630 + ], + "score": 1.0, + "content": ", which is the inf-convolution between", + "type": "text" + }, + { + "bbox": [ + 320, + 617, + 328, + 627 + ], + "score": 0.81, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 615, + 348, + 630 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 349, + 616, + 413, + 628 + ], + "score": 0.5, + "content": "\\alpha \\| { \\cdot } \\| _ { 2 } ~ ( \\alpha > 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 615, + 491, + 630 + ], + "score": 1.0, + "content": ". It is known that", + "type": "text" + }, + { + "bbox": [ + 491, + 617, + 504, + 627 + ], + "score": 0.86, + "content": "J _ { \\alpha }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 623, + 502, + 644 + ], + "spans": [ + { + "bbox": [ + 104, + 623, + 144, + 644 + ], + "score": 1.0, + "content": "becomes", + "type": "text" + }, + { + "bbox": [ + 145, + 629, + 152, + 637 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 623, + 424, + 644 + ], + "score": 1.0, + "content": "-Lipschitz. The second important example is the Moreau envelope", + "type": "text" + }, + { + "bbox": [ + 425, + 626, + 502, + 641 + ], + "score": 0.92, + "content": "\\begin{array} { r } { J ^ { \\beta } = J \\oplus \\frac { 1 } { 2 \\beta } \\lVert \\cdot \\rVert _ { 2 } ^ { \\overleftarrow { 2 } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 640, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 104, + 640, + 325, + 657 + ], + "score": 1.0, + "content": "i.e., the inf-convolution with the quadratic regularizer", + "type": "text" + }, + { + "bbox": [ + 326, + 640, + 354, + 655 + ], + "score": 0.92, + "content": "\\frac { 1 } { 2 \\beta } \\lVert \\cdot \\rVert _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 640, + 451, + 657 + ], + "score": 1.0, + "content": ". The Moreau envelope", + "type": "text" + }, + { + "bbox": [ + 451, + 641, + 464, + 652 + ], + "score": 0.89, + "content": "J ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 640, + 506, + 657 + ], + "score": 1.0, + "content": "is always", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 408, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 242, + 668 + ], + "score": 1.0, + "content": "differentiable, and the gradient of", + "type": "text" + }, + { + "bbox": [ + 243, + 654, + 255, + 666 + ], + "score": 0.89, + "content": "J ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 654, + 266, + 668 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 266, + 656, + 273, + 667 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 654, + 337, + 668 + ], + "score": 1.0, + "content": "-Lipschitz (thus", + "type": "text" + }, + { + "bbox": [ + 338, + 655, + 350, + 666 + ], + "score": 0.88, + "content": "J ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 654, + 361, + 668 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 361, + 656, + 369, + 667 + ], + "score": 0.84, + "content": "\\beta .", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 654, + 408, + 668 + ], + "score": 1.0, + "content": "-smooth).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 672, + 504, + 695 + ], + "lines": [ + { + "bbox": [ + 104, + 671, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 505, + 686 + ], + "score": 1.0, + "content": "It is worth noting that the set of minimizers does not change after these two operations. More", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 684, + 266, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 266, + 695 + ], + "score": 1.0, + "content": "generally, we have the following result:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 192, + 712 + ], + "score": 1.0, + "content": "Proposition 15. Let", + "type": "text" + }, + { + "bbox": [ + 192, + 697, + 255, + 710 + ], + "score": 0.9, + "content": "J , R : \\mathbb { R } ^ { d } \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 697, + 466, + 712 + ], + "score": 1.0, + "content": "be proper and lower semicontinuous functions with", + "type": "text" + }, + { + "bbox": [ + 466, + 699, + 505, + 709 + ], + "score": 0.71, + "content": "\\operatorname* { m i n } J >", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 711, + 126, + 720 + ], + "score": 0.86, + "content": "- \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 709, + 145, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 146, + 710, + 196, + 720 + ], + "score": 0.72, + "content": "\\operatorname* { m i n } R = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 709, + 241, + 722 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 242, + 710, + 286, + 721 + ], + "score": 0.91, + "content": "R ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 709, + 308, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 308, + 710, + 382, + 721 + ], + "score": 0.9, + "content": "R ( x ) \\geq \\psi ( \\| x \\| _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "for some increasing function", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 720, + 413, + 733 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 163, + 733 + ], + "score": 0.92, + "content": "\\psi : \\mathbb { R } _ { \\ge 0 } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 720, + 192, + 733 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 193, + 721, + 275, + 731 + ], + "score": 0.89, + "content": "\\operatorname* { m i n } J \\oplus R = \\operatorname* { m i n } J", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 720, + 308, + 733 + ], + "score": 1.0, + "content": "and arg", + "type": "text" + }, + { + "bbox": [ + 308, + 721, + 409, + 732 + ], + "score": 0.8, + "content": "{ \\mathrm { m i n } } J \\oplus { \\cal R } = \\arg \\operatorname* { m i n } J", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 720, + 413, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Alec Radford, Luke Metz, and Soumith Chintala. 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On the Fenchel duality between strong convexity and Lipschitz continuous gradient.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 405, + 279, + 418 + ], + "spans": [ + { + "bbox": [ + 115, + 405, + 279, + 418 + ], + "score": 1.0, + "content": "arXiv preprint arXiv:1803.06573, 2018.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 393, + 504, + 418 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 437, + 258, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 258, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 243, + 453 + ], + "score": 1.0, + "content": "A INF-CONVOLUTION IN", + "type": "text" + }, + { + "bbox": [ + 243, + 437, + 258, + 449 + ], + "score": 0.82, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "To gain intuition on the inf-convolution, we present a finite-dimensional analogue of the techniques", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "in Section 3. For simplicity of presentation, we will omit any regularity conditions (e.g., lower", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 486, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 496 + ], + "score": 1.0, + "content": "semicontinuity). We refer readers to Chapter 12 of Bauschke & Combettes (2011) for a detailed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 496, + 160, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 160, + 507 + ], + "score": 1.0, + "content": "introduction.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 462, + 506, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 511, + 504, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 122, + 525 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 514, + 130, + 523 + ], + "score": 0.82, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 511, + 147, + 525 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 148, + 514, + 157, + 523 + ], + "score": 0.83, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 511, + 252, + 525 + ], + "score": 1.0, + "content": "be convex functions on", + "type": "text" + }, + { + "bbox": [ + 252, + 512, + 264, + 523 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 511, + 361, + 525 + ], + "score": 1.0, + "content": ". The inf-convolution of", + "type": "text" + }, + { + "bbox": [ + 362, + 514, + 369, + 523 + ], + "score": 0.84, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 511, + 387, + 525 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 387, + 514, + 396, + 523 + ], + "score": 0.85, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 511, + 448, + 525 + ], + "score": 1.0, + "content": "is a function", + "type": "text" + }, + { + "bbox": [ + 449, + 514, + 473, + 523 + ], + "score": 0.89, + "content": "J \\star R", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "defined", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 524, + 119, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 119, + 536 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 511, + 506, + 536 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 533, + 385, + 553 + ], + "lines": [ + { + "bbox": [ + 225, + 533, + 385, + 553 + ], + "spans": [ + { + "bbox": [ + 225, + 533, + 385, + 553 + ], + "score": 0.93, + "content": "( J \\oplus R ) ( x ) : = \\operatorname* { i n f } _ { z \\in \\mathbb { R } ^ { d } } J ( z ) + R ( x - z ) .", + "type": "interline_equation", + "image_path": "3d3ce40559c91980378de44bde7e9b81eed2676e55659d86974ae5137b4a21ea.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 225, + 533, + 385, + 553 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 407, + 568 + ], + "score": 1.0, + "content": "The inf-convolution is often called the epigraphic sum since the epigraph of", + "type": "text" + }, + { + "bbox": [ + 407, + 556, + 430, + 566 + ], + "score": 0.9, + "content": "J { \\star R }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "coincides with the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 237, + 578 + ], + "score": 1.0, + "content": "Minkowski sum of epigraphs of", + "type": "text" + }, + { + "bbox": [ + 237, + 567, + 245, + 576 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 566, + 263, + 578 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 263, + 567, + 272, + 577 + ], + "score": 0.82, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 566, + 505, + 578 + ], + "score": 1.0, + "content": ", as Figure 2 illustrates. The inf-convolution is associative", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 577, + 504, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 342, + 590 + ], + "score": 1.0, + "content": "and commutative operation; that is, it is always true that", + "type": "text" + }, + { + "bbox": [ + 342, + 577, + 504, + 590 + ], + "score": 0.89, + "content": "\\left( J _ { 1 } \\oplus J _ { 2 } \\right) \\oplus J _ { 3 } = J _ { 1 } \\oplus ( J _ { 2 } \\oplus J _ { 3 } ) = :", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 588, + 263, + 601 + ], + "spans": [ + { + "bbox": [ + 107, + 588, + 162, + 600 + ], + "score": 0.93, + "content": "J _ { 1 } \\oplus J _ { 2 } \\oplus J _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 589, + 180, + 601 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 181, + 588, + 259, + 600 + ], + "score": 0.91, + "content": "J _ { 1 } \\oplus J _ { 2 } = J _ { 2 } \\oplus J _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 589, + 263, + 601 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 555, + 505, + 601 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "There are two important special cases of inf-convolutions: The first one is the Pasch–Hausdorff", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 615, + 504, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 145, + 630 + ], + "score": 1.0, + "content": "envelope", + "type": "text" + }, + { + "bbox": [ + 146, + 617, + 158, + 627 + ], + "score": 0.87, + "content": "J _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 615, + 320, + 630 + ], + "score": 1.0, + "content": ", which is the inf-convolution between", + "type": "text" + }, + { + "bbox": [ + 320, + 617, + 328, + 627 + ], + "score": 0.81, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 615, + 348, + 630 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 349, + 616, + 413, + 628 + ], + "score": 0.5, + "content": "\\alpha \\| { \\cdot } \\| _ { 2 } ~ ( \\alpha > 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 615, + 491, + 630 + ], + "score": 1.0, + "content": ". It is known that", + "type": "text" + }, + { + "bbox": [ + 491, + 617, + 504, + 627 + ], + "score": 0.86, + "content": "J _ { \\alpha }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 623, + 502, + 644 + ], + "spans": [ + { + "bbox": [ + 104, + 623, + 144, + 644 + ], + "score": 1.0, + "content": "becomes", + "type": "text" + }, + { + "bbox": [ + 145, + 629, + 152, + 637 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 623, + 424, + 644 + ], + "score": 1.0, + "content": "-Lipschitz. The second important example is the Moreau envelope", + "type": "text" + }, + { + "bbox": [ + 425, + 626, + 502, + 641 + ], + "score": 0.92, + "content": "\\begin{array} { r } { J ^ { \\beta } = J \\oplus \\frac { 1 } { 2 \\beta } \\lVert \\cdot \\rVert _ { 2 } ^ { \\overleftarrow { 2 } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 640, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 104, + 640, + 325, + 657 + ], + "score": 1.0, + "content": "i.e., the inf-convolution with the quadratic regularizer", + "type": "text" + }, + { + "bbox": [ + 326, + 640, + 354, + 655 + ], + "score": 0.92, + "content": "\\frac { 1 } { 2 \\beta } \\lVert \\cdot \\rVert _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 640, + 451, + 657 + ], + "score": 1.0, + "content": ". The Moreau envelope", + "type": "text" + }, + { + "bbox": [ + 451, + 641, + 464, + 652 + ], + "score": 0.89, + "content": "J ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 640, + 506, + 657 + ], + "score": 1.0, + "content": "is always", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 408, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 242, + 668 + ], + "score": 1.0, + "content": "differentiable, and the gradient of", + "type": "text" + }, + { + "bbox": [ + 243, + 654, + 255, + 666 + ], + "score": 0.89, + "content": "J ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 654, + 266, + 668 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 266, + 656, + 273, + 667 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 654, + 337, + 668 + ], + "score": 1.0, + "content": "-Lipschitz (thus", + "type": "text" + }, + { + "bbox": [ + 338, + 655, + 350, + 666 + ], + "score": 0.88, + "content": "J ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 654, + 361, + 668 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 361, + 656, + 369, + 667 + ], + "score": 0.84, + "content": "\\beta .", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 654, + 408, + 668 + ], + "score": 1.0, + "content": "-smooth).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 605, + 506, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 672, + 504, + 695 + ], + "lines": [ + { + "bbox": [ + 104, + 671, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 505, + 686 + ], + "score": 1.0, + "content": "It is worth noting that the set of minimizers does not change after these two operations. More", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 684, + 266, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 266, + 695 + ], + "score": 1.0, + "content": "generally, we have the following result:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 671, + 505, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 192, + 712 + ], + "score": 1.0, + "content": "Proposition 15. Let", + "type": "text" + }, + { + "bbox": [ + 192, + 697, + 255, + 710 + ], + "score": 0.9, + "content": "J , R : \\mathbb { R } ^ { d } \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 697, + 466, + 712 + ], + "score": 1.0, + "content": "be proper and lower semicontinuous functions with", + "type": "text" + }, + { + "bbox": [ + 466, + 699, + 505, + 709 + ], + "score": 0.71, + "content": "\\operatorname* { m i n } J >", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 711, + 126, + 720 + ], + "score": 0.86, + "content": "- \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 709, + 145, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 146, + 710, + 196, + 720 + ], + "score": 0.72, + "content": "\\operatorname* { m i n } R = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 709, + 241, + 722 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 242, + 710, + 286, + 721 + ], + "score": 0.91, + "content": "R ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 709, + 308, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 308, + 710, + 382, + 721 + ], + "score": 0.9, + "content": "R ( x ) \\geq \\psi ( \\| x \\| _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "for some increasing function", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 720, + 413, + 733 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 163, + 733 + ], + "score": 0.92, + "content": "\\psi : \\mathbb { R } _ { \\ge 0 } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 720, + 192, + 733 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 193, + 721, + 275, + 731 + ], + "score": 0.89, + "content": "\\operatorname* { m i n } J \\oplus R = \\operatorname* { m i n } J", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 720, + 308, + 733 + ], + "score": 1.0, + "content": "and arg", + "type": "text" + }, + { + "bbox": [ + 308, + 721, + 409, + 732 + ], + "score": 0.8, + "content": "{ \\mathrm { m i n } } J \\oplus { \\cal R } = \\arg \\operatorname* { m i n } J", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 720, + 413, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 697, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 81, + 502, + 291 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 81, + 502, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 81, + 502, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 502, + 291 + ], + "score": 0.972, + "type": "image", + "image_path": "680218ae8fb4c9cdc0963df033e29a25d3a9ce21159c7615b779f72f7741abe0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 81, + 502, + 151.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 151.0, + 502, + 221.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 221.0, + 502, + 291.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 301, + 506, + 370 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 402, + 315 + ], + "score": 1.0, + "content": "Figure 2: Illustration of inf-convolutions and their convex conjugates in", + "type": "text" + }, + { + "bbox": [ + 403, + 302, + 411, + 312 + ], + "score": 0.78, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 301, + 505, + 315 + ], + "score": 1.0, + "content": ". Top row: Generally,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 441, + 326 + ], + "score": 1.0, + "content": "inf-convolutions inherit the regularity of their parent functions. In this example,", + "type": "text" + }, + { + "bbox": [ + 441, + 313, + 493, + 325 + ], + "score": 0.93, + "content": "J _ { 1 } ( x ) = \\left| x \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 243, + 336 + ], + "score": 1.0, + "content": "1-Lipschitz but not smooth, while", + "type": "text" + }, + { + "bbox": [ + 243, + 323, + 301, + 336 + ], + "score": 0.94, + "content": "J _ { 2 } ( x ) = { x } ^ { 2 } / \\dot { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "is 1-smooth but not Lipschitz. The inf-convolution", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 140, + 346 + ], + "score": 0.91, + "content": "J _ { 1 } \\oplus J _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "is the well-known Huber function, which is both 1-Lipschitz and 1-smooth. Bottom row:", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 381, + 358 + ], + "score": 1.0, + "content": "Convex conjugates of the functions in the top row. The conjugate of", + "type": "text" + }, + { + "bbox": [ + 381, + 346, + 414, + 357 + ], + "score": 0.92, + "content": "J _ { 1 } \\oplus J _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "is given as the sum of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 356, + 315, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 151, + 370 + ], + "score": 1.0, + "content": "conjugates", + "type": "text" + }, + { + "bbox": [ + 152, + 357, + 235, + 369 + ], + "score": 0.94, + "content": "J _ { 1 } ^ { \\star } ( z ) = \\chi \\{ | z | \\leq 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 356, + 253, + 370 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 356, + 311, + 369 + ], + "score": 0.94, + "content": "J _ { 2 } ^ { \\star } ( z ) \\stackrel { = } { = } z ^ { 2 } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 356, + 315, + 370 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 504, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 228, + 403 + ], + "score": 1.0, + "content": "To sum up, given a function", + "type": "text" + }, + { + "bbox": [ + 228, + 391, + 236, + 400 + ], + "score": 0.77, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 389, + 448, + 403 + ], + "score": 1.0, + "content": ", we can always construct a regularized alternative", + "type": "text" + }, + { + "bbox": [ + 448, + 389, + 461, + 402 + ], + "score": 0.9, + "content": "J _ { \\alpha } ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 389, + 492, + 403 + ], + "score": 1.0, + "content": "that is", + "type": "text" + }, + { + "bbox": [ + 493, + 392, + 500, + 400 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 400, + 345, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 163, + 413 + ], + "score": 1.0, + "content": "Lipschitz and", + "type": "text" + }, + { + "bbox": [ + 163, + 401, + 171, + 412 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 400, + 333, + 413 + ], + "score": 1.0, + "content": "-smooth and has the same minimizers as", + "type": "text" + }, + { + "bbox": [ + 333, + 401, + 340, + 411 + ], + "score": 0.8, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 400, + 345, + 413 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "The next question is how to implement the inf-convolution in GAN-like optimization problems. For", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "this, it is convenient to consider the convex conjugate. Recall that the Fenchel–Moreau theorem", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 330, + 452 + ], + "score": 1.0, + "content": "says that there is a duality between a convex function", + "type": "text" + }, + { + "bbox": [ + 330, + 441, + 338, + 450 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 439, + 444, + 452 + ], + "score": 1.0, + "content": "and its convex conjugate", + "type": "text" + }, + { + "bbox": [ + 444, + 440, + 457, + 450 + ], + "score": 0.86, + "content": "J ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 439, + 470, + 452 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 471, + 439, + 505, + 451 + ], + "score": 0.91, + "content": "J ( x ) =", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 202, + 463 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\operatorname* { s u p } _ { z \\in \\mathbb { R } ^ { d } } \\langle x , z \\rangle - J ^ { \\star } ( z ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 450, + 222, + 464 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 223, + 451, + 356, + 463 + ], + "score": 0.92, + "content": "\\begin{array} { r } { J ^ { \\star } ( z ) = \\operatorname* { s u p } _ { x \\in \\mathbb { R } ^ { d } } \\langle x , z \\rangle - J ( x ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 450, + 506, + 464 + ], + "score": 1.0, + "content": ". The important property is that the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 462, + 493, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 493, + 474 + ], + "score": 1.0, + "content": "convex conjugate of the inf-convolution is the sum of convex conjugates, that is, we always have", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 478, + 371, + 492 + ], + "lines": [ + { + "bbox": [ + 240, + 478, + 371, + 492 + ], + "spans": [ + { + "bbox": [ + 240, + 478, + 371, + 492 + ], + "score": 0.92, + "content": "( J \\oplus R ) ^ { \\star } ( z ) = J ^ { \\star } ( z ) + R ^ { \\star } ( z ) .", + "type": "interline_equation", + "image_path": "3af6ad391e4ba7529eab54752c63d542b919b4d3e5e209f5c196bf3bb03f112a.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 240, + 478, + 371, + 492 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 497, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 390, + 510 + ], + "score": 1.0, + "content": "This property can be useful for implementing the regularized objective", + "type": "text" + }, + { + "bbox": [ + 391, + 497, + 404, + 510 + ], + "score": 0.9, + "content": "J _ { \\alpha } ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 496, + 505, + 510 + ], + "score": 1.0, + "content": "as follows. First, we can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 506, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 506, + 426, + 523 + ], + "score": 1.0, + "content": "check that the convex conjugates of the norm and the squared norm are given as", + "type": "text" + }, + { + "bbox": [ + 426, + 508, + 505, + 521 + ], + "score": 0.91, + "content": "( \\| { \\cdot } \\| _ { 2 } ) ^ { \\star } = \\chi \\{ \\| { \\cdot } \\| \\leq", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 519, + 281, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 118, + 532 + ], + "score": 0.83, + "content": "1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 519, + 136, + 533 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 137, + 519, + 210, + 533 + ], + "score": 0.94, + "content": "\\begin{array} { r } { ( \\frac { 1 } { 2 } \\| { \\cdot } \\| _ { 2 } ^ { 2 } ) ^ { \\star } = \\frac { 1 } { 2 } \\| { \\cdot } \\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 519, + 281, + 533 + ], + "score": 1.0, + "content": ". Hence, we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 537, + 460, + 566 + ], + "lines": [ + { + "bbox": [ + 150, + 537, + 460, + 566 + ], + "spans": [ + { + "bbox": [ + 150, + 537, + 460, + 566 + ], + "score": 0.92, + "content": "J _ { \\alpha } ^ { \\beta } ( \\boldsymbol { x } ) : = \\left( J \\oplus \\alpha \\lVert \\cdot \\rVert _ { 2 } \\oplus \\frac { 1 } { 2 \\beta } \\lVert \\cdot \\rVert _ { 2 } ^ { 2 } \\right) ( \\boldsymbol { x } ) = \\operatorname* { s u p } _ { \\boldsymbol { z } : \\ \\lVert \\boldsymbol { z } \\rVert _ { 2 } \\leq \\alpha } \\langle \\boldsymbol { x } , \\boldsymbol { z } \\rangle - J ^ { \\star } ( \\boldsymbol { z } ) - \\frac { \\beta } { 2 } \\lVert \\boldsymbol { z } \\rVert _ { 2 } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "3203b02b8dd5cb4b565b4b7db431b8f58c0ccd29b7f0654acee980ee0ba5d7f0.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 150, + 537, + 460, + 546.6666666666666 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 150, + 546.6666666666666, + 460, + 556.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 150, + 556.3333333333333, + 460, + 565.9999999999999 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 231, + 587 + ], + "score": 1.0, + "content": "which means that minimizing", + "type": "text" + }, + { + "bbox": [ + 231, + 572, + 244, + 585 + ], + "score": 0.91, + "content": "J _ { \\alpha } ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 571, + 506, + 587 + ], + "score": 1.0, + "content": "can be recast in min-max problem with the norm clipping and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 584, + 267, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 116, + 595 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 584, + 257, + 596 + ], + "score": 1.0, + "content": "-regularization on the dual variable", + "type": "text" + }, + { + "bbox": [ + 257, + 587, + 263, + 594 + ], + "score": 0.79, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 584, + 267, + 596 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 611, + 250, + 624 + ], + "lines": [ + { + "bbox": [ + 104, + 610, + 251, + 627 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 251, + 627 + ], + "score": 1.0, + "content": "B COMMON GAN LOSSES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 636, + 504, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "For completeness and clarity, we explicitly write out the expressions for the losses listed in Table 1.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "For more detailed computations of optimal discriminators, see Chu et al. (2019); for more details on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 659, + 342, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 342, + 671 + ], + "score": 1.0, + "content": "the convex duality interpretation, see Farnia & Tse (2018).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 682, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 506, + 695 + ], + "score": 1.0, + "content": "Minimax GAN Goodfellow et al. (2014) originally proposed the minimax GAN and showed that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 693, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 506, + 706 + ], + "score": 1.0, + "content": "the corresponding loss function for the minimax GAN is the Jensen–Shannon divergence, defined", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 706, + 118, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 118, + 716 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 712, + 458, + 736 + ], + "lines": [ + { + "bbox": [ + 152, + 712, + 458, + 736 + ], + "spans": [ + { + "bbox": [ + 152, + 712, + 458, + 736 + ], + "score": 0.91, + "content": "J ( \\mu ) : = D _ { \\mathrm { J S } } ( \\mu | | \\mu _ { 0 } ) : = \\frac 1 2 D _ { \\mathrm { K L } } ( \\mu | | \\frac 1 2 \\mu + \\frac 1 2 \\mu _ { 0 } ) + \\frac 1 2 D _ { \\mathrm { K L } } ( \\mu _ { 0 } | | \\frac 1 2 \\mu + \\frac 1 2 \\mu _ { 0 } ) ,", + "type": "interline_equation", + "image_path": "a0c34c7887bb0636bf4af4d19ba810794a8a6417233595f623ca6d0d37784883.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 152, + 712, + 458, + 736 + ], + "spans": [], + "index": 32 + } + ] + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 81, + 502, + 291 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 81, + 502, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 81, + 502, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 502, + 291 + ], + "score": 0.972, + "type": "image", + "image_path": "680218ae8fb4c9cdc0963df033e29a25d3a9ce21159c7615b779f72f7741abe0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 81, + 502, + 151.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 151.0, + 502, + 221.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 221.0, + 502, + 291.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 301, + 506, + 370 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 402, + 315 + ], + "score": 1.0, + "content": "Figure 2: Illustration of inf-convolutions and their convex conjugates in", + "type": "text" + }, + { + "bbox": [ + 403, + 302, + 411, + 312 + ], + "score": 0.78, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 301, + 505, + 315 + ], + "score": 1.0, + "content": ". Top row: Generally,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 441, + 326 + ], + "score": 1.0, + "content": "inf-convolutions inherit the regularity of their parent functions. In this example,", + "type": "text" + }, + { + "bbox": [ + 441, + 313, + 493, + 325 + ], + "score": 0.93, + "content": "J _ { 1 } ( x ) = \\left| x \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 243, + 336 + ], + "score": 1.0, + "content": "1-Lipschitz but not smooth, while", + "type": "text" + }, + { + "bbox": [ + 243, + 323, + 301, + 336 + ], + "score": 0.94, + "content": "J _ { 2 } ( x ) = { x } ^ { 2 } / \\dot { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "is 1-smooth but not Lipschitz. The inf-convolution", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 140, + 346 + ], + "score": 0.91, + "content": "J _ { 1 } \\oplus J _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "is the well-known Huber function, which is both 1-Lipschitz and 1-smooth. Bottom row:", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 381, + 358 + ], + "score": 1.0, + "content": "Convex conjugates of the functions in the top row. The conjugate of", + "type": "text" + }, + { + "bbox": [ + 381, + 346, + 414, + 357 + ], + "score": 0.92, + "content": "J _ { 1 } \\oplus J _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "is given as the sum of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 356, + 315, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 151, + 370 + ], + "score": 1.0, + "content": "conjugates", + "type": "text" + }, + { + "bbox": [ + 152, + 357, + 235, + 369 + ], + "score": 0.94, + "content": "J _ { 1 } ^ { \\star } ( z ) = \\chi \\{ | z | \\leq 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 356, + 253, + 370 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 356, + 311, + 369 + ], + "score": 0.94, + "content": "J _ { 2 } ^ { \\star } ( z ) \\stackrel { = } { = } z ^ { 2 } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 356, + 315, + 370 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 504, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 228, + 403 + ], + "score": 1.0, + "content": "To sum up, given a function", + "type": "text" + }, + { + "bbox": [ + 228, + 391, + 236, + 400 + ], + "score": 0.77, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 389, + 448, + 403 + ], + "score": 1.0, + "content": ", we can always construct a regularized alternative", + "type": "text" + }, + { + "bbox": [ + 448, + 389, + 461, + 402 + ], + "score": 0.9, + "content": "J _ { \\alpha } ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 389, + 492, + 403 + ], + "score": 1.0, + "content": "that is", + "type": "text" + }, + { + "bbox": [ + 493, + 392, + 500, + 400 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 400, + 345, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 163, + 413 + ], + "score": 1.0, + "content": "Lipschitz and", + "type": "text" + }, + { + "bbox": [ + 163, + 401, + 171, + 412 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 400, + 333, + 413 + ], + "score": 1.0, + "content": "-smooth and has the same minimizers as", + "type": "text" + }, + { + "bbox": [ + 333, + 401, + 340, + 411 + ], + "score": 0.8, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 400, + 345, + 413 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 389, + 505, + 413 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "The next question is how to implement the inf-convolution in GAN-like optimization problems. For", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "this, it is convenient to consider the convex conjugate. Recall that the Fenchel–Moreau theorem", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 330, + 452 + ], + "score": 1.0, + "content": "says that there is a duality between a convex function", + "type": "text" + }, + { + "bbox": [ + 330, + 441, + 338, + 450 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 439, + 444, + 452 + ], + "score": 1.0, + "content": "and its convex conjugate", + "type": "text" + }, + { + "bbox": [ + 444, + 440, + 457, + 450 + ], + "score": 0.86, + "content": "J ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 439, + 470, + 452 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 471, + 439, + 505, + 451 + ], + "score": 0.91, + "content": "J ( x ) =", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 202, + 463 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\operatorname* { s u p } _ { z \\in \\mathbb { R } ^ { d } } \\langle x , z \\rangle - J ^ { \\star } ( z ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 450, + 222, + 464 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 223, + 451, + 356, + 463 + ], + "score": 0.92, + "content": "\\begin{array} { r } { J ^ { \\star } ( z ) = \\operatorname* { s u p } _ { x \\in \\mathbb { R } ^ { d } } \\langle x , z \\rangle - J ( x ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 450, + 506, + 464 + ], + "score": 1.0, + "content": ". The important property is that the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 462, + 493, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 493, + 474 + ], + "score": 1.0, + "content": "convex conjugate of the inf-convolution is the sum of convex conjugates, that is, we always have", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 418, + 506, + 474 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 478, + 371, + 492 + ], + "lines": [ + { + "bbox": [ + 240, + 478, + 371, + 492 + ], + "spans": [ + { + "bbox": [ + 240, + 478, + 371, + 492 + ], + "score": 0.92, + "content": "( J \\oplus R ) ^ { \\star } ( z ) = J ^ { \\star } ( z ) + R ^ { \\star } ( z ) .", + "type": "interline_equation", + "image_path": "3af6ad391e4ba7529eab54752c63d542b919b4d3e5e209f5c196bf3bb03f112a.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 240, + 478, + 371, + 492 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 497, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 390, + 510 + ], + "score": 1.0, + "content": "This property can be useful for implementing the regularized objective", + "type": "text" + }, + { + "bbox": [ + 391, + 497, + 404, + 510 + ], + "score": 0.9, + "content": "J _ { \\alpha } ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 496, + 505, + 510 + ], + "score": 1.0, + "content": "as follows. First, we can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 506, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 506, + 426, + 523 + ], + "score": 1.0, + "content": "check that the convex conjugates of the norm and the squared norm are given as", + "type": "text" + }, + { + "bbox": [ + 426, + 508, + 505, + 521 + ], + "score": 0.91, + "content": "( \\| { \\cdot } \\| _ { 2 } ) ^ { \\star } = \\chi \\{ \\| { \\cdot } \\| \\leq", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 519, + 281, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 118, + 532 + ], + "score": 0.83, + "content": "1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 519, + 136, + 533 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 137, + 519, + 210, + 533 + ], + "score": 0.94, + "content": "\\begin{array} { r } { ( \\frac { 1 } { 2 } \\| { \\cdot } \\| _ { 2 } ^ { 2 } ) ^ { \\star } = \\frac { 1 } { 2 } \\| { \\cdot } \\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 519, + 281, + 533 + ], + "score": 1.0, + "content": ". Hence, we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 496, + 505, + 533 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 537, + 460, + 566 + ], + "lines": [ + { + "bbox": [ + 150, + 537, + 460, + 566 + ], + "spans": [ + { + "bbox": [ + 150, + 537, + 460, + 566 + ], + "score": 0.92, + "content": "J _ { \\alpha } ^ { \\beta } ( \\boldsymbol { x } ) : = \\left( J \\oplus \\alpha \\lVert \\cdot \\rVert _ { 2 } \\oplus \\frac { 1 } { 2 \\beta } \\lVert \\cdot \\rVert _ { 2 } ^ { 2 } \\right) ( \\boldsymbol { x } ) = \\operatorname* { s u p } _ { \\boldsymbol { z } : \\ \\lVert \\boldsymbol { z } \\rVert _ { 2 } \\leq \\alpha } \\langle \\boldsymbol { x } , \\boldsymbol { z } \\rangle - J ^ { \\star } ( \\boldsymbol { z } ) - \\frac { \\beta } { 2 } \\lVert \\boldsymbol { z } \\rVert _ { 2 } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "3203b02b8dd5cb4b565b4b7db431b8f58c0ccd29b7f0654acee980ee0ba5d7f0.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 150, + 537, + 460, + 546.6666666666666 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 150, + 546.6666666666666, + 460, + 556.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 150, + 556.3333333333333, + 460, + 565.9999999999999 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 231, + 587 + ], + "score": 1.0, + "content": "which means that minimizing", + "type": "text" + }, + { + "bbox": [ + 231, + 572, + 244, + 585 + ], + "score": 0.91, + "content": "J _ { \\alpha } ^ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 571, + 506, + 587 + ], + "score": 1.0, + "content": "can be recast in min-max problem with the norm clipping and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 584, + 267, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 116, + 595 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 584, + 257, + 596 + ], + "score": 1.0, + "content": "-regularization on the dual variable", + "type": "text" + }, + { + "bbox": [ + 257, + 587, + 263, + 594 + ], + "score": 0.79, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 584, + 267, + 596 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 571, + 506, + 596 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 611, + 250, + 624 + ], + "lines": [ + { + "bbox": [ + 104, + 610, + 251, + 627 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 251, + 627 + ], + "score": 1.0, + "content": "B COMMON GAN LOSSES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 636, + 504, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "For completeness and clarity, we explicitly write out the expressions for the losses listed in Table 1.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "For more detailed computations of optimal discriminators, see Chu et al. (2019); for more details on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 659, + 342, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 342, + 671 + ], + "score": 1.0, + "content": "the convex duality interpretation, see Farnia & Tse (2018).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 637, + 505, + 671 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 682, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 506, + 695 + ], + "score": 1.0, + "content": "Minimax GAN Goodfellow et al. (2014) originally proposed the minimax GAN and showed that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 693, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 506, + 706 + ], + "score": 1.0, + "content": "the corresponding loss function for the minimax GAN is the Jensen–Shannon divergence, defined", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 706, + 118, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 118, + 716 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 682, + 506, + 716 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 712, + 458, + 736 + ], + "lines": [ + { + "bbox": [ + 152, + 712, + 458, + 736 + ], + "spans": [ + { + "bbox": [ + 152, + 712, + 458, + 736 + ], + "score": 0.91, + "content": "J ( \\mu ) : = D _ { \\mathrm { J S } } ( \\mu | | \\mu _ { 0 } ) : = \\frac 1 2 D _ { \\mathrm { K L } } ( \\mu | | \\frac 1 2 \\mu + \\frac 1 2 \\mu _ { 0 } ) + \\frac 1 2 D _ { \\mathrm { K L } } ( \\mu _ { 0 } | | \\frac 1 2 \\mu + \\frac 1 2 \\mu _ { 0 } ) ,", + "type": "interline_equation", + "image_path": "a0c34c7887bb0636bf4af4d19ba810794a8a6417233595f623ca6d0d37784883.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 152, + 712, + 458, + 736 + ], + "spans": [], + "index": 32 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 133, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 82, + 184, + 95 + ], + "score": 0.93, + "content": "\\mu _ { 0 } \\in { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "is a fixed probability measure (usually the empirical measure of the data), and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 155, + 106 + ], + "score": 0.92, + "content": "D _ { \\mathrm { K L } } ( \\mu \\lvert \\lvert \\nu \\rangle )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 93, + 335, + 106 + ], + "score": 1.0, + "content": "is the Kullback–Leibler divergence between", + "type": "text" + }, + { + "bbox": [ + 336, + 96, + 343, + 105 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 93, + 361, + 106 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 361, + 96, + 368, + 104 + ], + "score": 0.76, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 93, + 505, + 106 + ], + "score": 1.0, + "content": ". The optimal discriminator in the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 237, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 237, + 117 + ], + "score": 1.0, + "content": "sense of Definition 1 is given as", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 120, + 367, + 147 + ], + "lines": [ + { + "bbox": [ + 243, + 120, + 367, + 147 + ], + "spans": [ + { + "bbox": [ + 243, + 120, + 367, + 147 + ], + "score": 0.94, + "content": "\\Phi _ { \\mu } ( x ) = \\frac { 1 } { 2 } \\log \\frac { d \\mu } { d ( \\mu + \\mu _ { 0 } ) } ( x ) ,", + "type": "interline_equation", + "image_path": "391c00d0c82cd0524efb7eaefe32c9c6ce29f52d1239a4ee623e9178b7db611e.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 243, + 120, + 367, + 133.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 243, + 133.5, + 367, + 147.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 153, + 501, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 150, + 501, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 133, + 172 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 152, + 167, + 168 + ], + "score": 0.94, + "content": "\\frac { d \\mu } { d ( \\mu + \\mu _ { 0 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 150, + 317, + 172 + ], + "score": 1.0, + "content": "is the Radon–Nikodym derivative. 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(2014) up to a constant factor and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 214, + 419, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 419, + 227 + ], + "score": 1.0, + "content": "logarithm. 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Recall the definition of the optimal discriminator (Defini-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 317, + 140, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 140, + 333 + ], + "score": 1.0, + "content": "tion 1):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 328, + 372, + 356 + ], + "lines": [ + { + "bbox": [ + 238, + 328, + 372, + 356 + ], + "spans": [ + { + "bbox": [ + 238, + 328, + 372, + 356 + ], + "score": 0.94, + "content": "\\Phi _ { \\mu } \\in \\underset { \\varphi \\in \\mathcal { C } ( X ) } { \\arg \\operatorname* { m a x } } \\int \\varphi d \\mu - J ^ { \\star } ( \\varphi ) .", + "type": "interline_equation", + "image_path": "0425a38699615f046ccc2711c233b632644fa5cc145a06081b19306289009a72.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 238, + 328, + 372, + 342.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 238, + 342.0, + 372, + 356.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "This definition can be understood as an infinite dimensional analogue of subgradients. In fact, in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 249, + 381 + ], + "score": 1.0, + "content": "finite-dimensional convex analysis,", + "type": "text" + }, + { + "bbox": [ + 249, + 371, + 255, + 379 + ], + "score": 0.76, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 368, + 333, + 381 + ], + "score": 1.0, + "content": "is a subgradient of", + "type": "text" + }, + { + "bbox": [ + 333, + 369, + 383, + 380 + ], + "score": 0.92, + "content": "f : \\mathbb { R } ^ { d } \\bar { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "if and only if it can be written", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 380, + 504, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 118, + 392 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 119, + 380, + 246, + 392 + ], + "score": 0.89, + "content": "z \\in \\arg \\operatorname* { m a x } _ { z ^ { \\prime } } \\langle z ^ { \\prime } , x \\rangle - f ^ { \\star } ( z ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 380, + 504, + 392 + ], + "score": 1.0, + "content": ". The calculus of subgradients shares many properties with the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 390, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 404 + ], + "score": 1.0, + "content": "standard calculus of derivatives, such as chain rules (Rockafeller & Wets, 1998). This motivate us", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 361, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 361, + 415 + ], + "score": 1.0, + "content": "to investigate derivative-like features of optimal discriminators.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 457, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 458, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 458, + 431 + ], + "score": 1.0, + "content": "We introduce the functional derivative, also known as the von Mises influence function:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 507, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 504, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 272, + 445 + ], + "score": 1.0, + "content": "Definition 3 (Functional derivative). Let", + "type": "text" + }, + { + "bbox": [ + 272, + 432, + 335, + 445 + ], + "score": 0.93, + "content": "J : { \\mathcal { P } } ( X ) \\to { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 432, + 504, + 445 + ], + "score": 1.0, + "content": "be a function of probability measures. We", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 443, + 436, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 229, + 458 + ], + "score": 1.0, + "content": "say that a continuous function", + "type": "text" + }, + { + "bbox": [ + 230, + 444, + 282, + 456 + ], + "score": 0.92, + "content": "\\Phi _ { \\mu } : X \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 443, + 398, + 458 + ], + "score": 1.0, + "content": "is a functional derivative of", + "type": "text" + }, + { + "bbox": [ + 399, + 445, + 406, + 454 + ], + "score": 0.78, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 443, + 417, + 458 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 418, + 446, + 425, + 456 + ], + "score": 0.7, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 443, + 436, + 458 + ], + "score": 1.0, + "content": "if", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 460, + 389, + 486 + ], + "lines": [ + { + "bbox": [ + 222, + 460, + 389, + 486 + ], + "spans": [ + { + "bbox": [ + 222, + 460, + 389, + 486 + ], + "score": 0.94, + "content": "J ( \\mu + \\epsilon \\xi ) = J ( \\mu ) + \\epsilon \\int \\Phi _ { \\mu } d \\xi + O ( \\epsilon ^ { 2 } )", + "type": "interline_equation", + "image_path": "cccb311ff9a9ed3028bbeda0d2ef2c291df3acbe789ec3f08ba949df41b1b02c.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 222, + 460, + 389, + 486 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 488, + 272, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 272, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 161, + 502 + ], + "score": 1.0, + "content": "holds for any", + "type": "text" + }, + { + "bbox": [ + 162, + 489, + 205, + 501 + ], + "score": 0.91, + "content": "\\xi = \\nu - \\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 488, + 225, + 502 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 226, + 489, + 268, + 501 + ], + "score": 0.92, + "content": "\\nu \\in { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 488, + 272, + 502 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 108, + 509, + 425, + 521 + ], + "lines": [ + { + "bbox": [ + 106, + 508, + 426, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 426, + 523 + ], + "score": 1.0, + "content": "Under this definition, optimal discriminators are actually functional derivatives.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 523, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 319, + 536 + ], + "score": 1.0, + "content": "Proposition 16 (Chu et al. (2019), Theorem 2). Let", + "type": "text" + }, + { + "bbox": [ + 319, + 522, + 389, + 535 + ], + "score": 0.92, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "be a proper, lower semicon-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 504, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 335, + 548 + ], + "score": 1.0, + "content": "tinuous, and convex function. If there exists a maximizer", + "type": "text" + }, + { + "bbox": [ + 335, + 536, + 348, + 547 + ], + "score": 0.88, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 534, + 360, + 548 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 361, + 534, + 449, + 547 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\varphi \\mapsto \\int \\varphi d \\mu - J ^ { \\star } ( \\varphi ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 534, + 473, + 548 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 473, + 535, + 487, + 547 + ], + "score": 0.9, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 534, + 497, + 548 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 498, + 537, + 504, + 545 + ], + "score": 0.57, + "content": "a", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 546, + 342, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 202, + 558 + ], + "score": 1.0, + "content": "functional derivative of", + "type": "text" + }, + { + "bbox": [ + 202, + 546, + 210, + 556 + ], + "score": 0.64, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 546, + 221, + 558 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 221, + 547, + 228, + 557 + ], + "score": 0.69, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 546, + 342, + 558 + ], + "score": 1.0, + "content": "in the sense of Definition 3.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "The following result relates the derivative of the loss function with the derivative of the optimal", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 576, + 165, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 165, + 589 + ], + "score": 1.0, + "content": "discriminator:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 317, + 604 + ], + "score": 1.0, + "content": "Proposition 17 (Chu et al. (2019), Theorem 1). Let", + "type": "text" + }, + { + "bbox": [ + 317, + 591, + 380, + 603 + ], + "score": 0.89, + "content": "J : { \\mathcal { P } } ( X ) \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "be continuously differentiable,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 275, + 615 + ], + "score": 1.0, + "content": "in the sense that the functional derivative", + "type": "text" + }, + { + "bbox": [ + 276, + 603, + 289, + 614 + ], + "score": 0.89, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 601, + 334, + 615 + ], + "score": 1.0, + "content": "exists and", + "type": "text" + }, + { + "bbox": [ + 334, + 602, + 429, + 614 + ], + "score": 0.92, + "content": "( \\mu , \\nu ) \\mapsto \\mathbb { E } _ { x \\sim \\nu } [ \\Phi _ { \\mu } ( x ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "is continuous. Let", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 611, + 507, + 631 + ], + "spans": [ + { + "bbox": [ + 107, + 614, + 141, + 626 + ], + "score": 0.9, + "content": "\\theta \\mapsto \\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 611, + 272, + 631 + ], + "score": 1.0, + "content": "be continuous in the sense that", + "type": "text" + }, + { + "bbox": [ + 273, + 614, + 341, + 629 + ], + "score": 0.91, + "content": "{ \\frac { 1 } { \\| h \\| } } \\big ( \\mu _ { \\theta + h } - \\mu _ { \\theta } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 611, + 462, + 631 + ], + "score": 1.0, + "content": "converges to a weak limit as", + "type": "text" + }, + { + "bbox": [ + 462, + 614, + 501, + 626 + ], + "score": 0.92, + "content": "\\| h \\| 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 611, + 507, + 631 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 167, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 167, + 639 + ], + "score": 1.0, + "content": "Then, we have", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 636, + 365, + 651 + ], + "lines": [ + { + "bbox": [ + 244, + 636, + 365, + 651 + ], + "spans": [ + { + "bbox": [ + 244, + 636, + 365, + 651 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\nabla _ { \\theta } J ( \\mu _ { \\theta } ) = \\nabla _ { \\theta } \\mathbb { E } _ { x \\sim \\mu _ { \\theta } } [ \\hat { \\Phi } ( x ) ] , } \\end{array}", + "type": "interline_equation", + "image_path": "6d29bb9bd38a668d1ef3f0182021573af1a64777fdf39d58b343cff3b30c4dce.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 244, + 636, + 365, + 651 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 407, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 652, + 408, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 133, + 668 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 653, + 171, + 667 + ], + "score": 0.93, + "content": "\\hat { \\Phi } = \\Phi _ { \\mu _ { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 652, + 266, + 668 + ], + "score": 1.0, + "content": "is treated as a function", + "type": "text" + }, + { + "bbox": [ + 266, + 655, + 299, + 664 + ], + "score": 0.91, + "content": "X \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 652, + 398, + 668 + ], + "score": 1.0, + "content": "that is not dependent on", + "type": "text" + }, + { + "bbox": [ + 399, + 655, + 404, + 664 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 652, + 408, + 668 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 674, + 504, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 674, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 505, + 687 + ], + "score": 1.0, + "content": "We use this important computational tool in many of our proofs. For the case of the generator model", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 686, + 363, + 699 + ], + "spans": [ + { + "bbox": [ + 107, + 686, + 154, + 698 + ], + "score": 0.92, + "content": "\\mu _ { \\theta } = f _ { \\theta \\# } \\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 686, + 363, + 699 + ], + "score": 1.0, + "content": ", an important consequence of Proposition 17 is that", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 701, + 455, + 718 + ], + "lines": [ + { + "bbox": [ + 156, + 701, + 455, + 718 + ], + "spans": [ + { + "bbox": [ + 156, + 701, + 455, + 718 + ], + "score": 0.9, + "content": "\\nabla _ { \\theta } J ( \\mu _ { \\theta } ) = \\nabla _ { \\theta } \\mathbb { E } _ { z \\sim \\omega } [ \\hat { \\Phi } ( f _ { \\theta } ( z ) ) ] = \\mathbb { E } _ { z \\sim \\omega } [ \\nabla _ { x = f _ { \\theta } ( z ) } \\Phi _ { \\mu } ( f _ { \\theta } ( z ) ) \\cdot D _ { \\theta } f _ { \\theta } ( z ) ] .", + "type": "interline_equation", + "image_path": "b01369971264d010d0f1ca0c8a802a3880abf12852f36b9caebfa870780eb5cf.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 156, + 701, + 455, + 718 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 720, + 279, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 720, + 280, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 280, + 733 + ], + "score": 1.0, + "content": "We use this fact in the proof of Theorem 1.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "16", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "which in constrast to other GANs, may be approximated by simple Monte Carlo, rather than an", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 235, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 235, + 106 + ], + "score": 1.0, + "content": "auxiliary optimization problem.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 82, + 505, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 122 + ], + "score": 1.0, + "content": "Note that MMD-GANs (Li et al., 2017a; Arbel et al., 2018) minimize a modified version of the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 121, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 505, + 133 + ], + "score": 1.0, + "content": "MMD, the Optimized MMD (Sriperumbudur et al., 2009; Arbel et al., 2018). These MMD-GANs", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "are adversarial in a way that does not arise from convex duality, so our theory currently does not", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 196, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 196, + 155 + ], + "score": 1.0, + "content": "apply to these GANs.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 110, + 506, + 155 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 166, + 471, + 178 + ], + "lines": [ + { + "bbox": [ + 104, + 164, + 471, + 182 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 471, + 182 + ], + "score": 1.0, + "content": "Integral probability metrics An integral probability metric (Muller, 1997) is defined by ¨", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 164, + 471, + 182 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 181, + 409, + 209 + ], + "lines": [ + { + "bbox": [ + 201, + 181, + 409, + 209 + ], + "spans": [ + { + "bbox": [ + 201, + 181, + 409, + 209 + ], + "score": 0.93, + "content": "J ( \\mu ) : = { \\mathrm { I P M } } _ { \\mathcal F } ( \\mu , \\mu _ { 0 } ) : = \\operatorname* { s u p } _ { f \\in \\mathcal F } \\int f d \\mu - \\int f d \\mu _ { 0 } ,", + "type": "interline_equation", + "image_path": "d9c66aa02e9ad93ee256bc6cf2a749fef1c18bec6b22b3f298e4c47fca0a62ba.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 201, + 181, + 409, + 195.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 201, + 195.0, + 409, + 209.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 212, + 505, + 268 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 133, + 225 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 213, + 143, + 222 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "is a class of functions. The optimal discriminator is the function that maximizes the supre-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 449, + 236 + ], + "score": 1.0, + "content": "mum in the definition. The Wasserstein distance may be thought of as an IPM with", + "type": "text" + }, + { + "bbox": [ + 450, + 224, + 459, + 234 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "containing", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 393, + 246 + ], + "score": 1.0, + "content": "all 1-Lipschitz functions. The MMD may be thought of as an IPM with", + "type": "text" + }, + { + "bbox": [ + 393, + 235, + 403, + 244 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 235, + 505, + 246 + ], + "score": 1.0, + "content": "all functions with RKHS", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "norm at most 1, but no GANs based on MMD are actually trained this way, as it is difficult to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 256, + 289, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 289, + 269 + ], + "score": 1.0, + "content": "constrain the discriminator to such functions.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 212, + 505, + 269 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 284, + 432, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 433, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 433, + 299 + ], + "score": 1.0, + "content": "C OPTIMAL DISCRIMINATORS ARE FUNCTIONAL DERIVATIVES", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 504, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 122, + 321 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 307, + 185, + 320 + ], + "score": 0.92, + "content": "J : { \\mathcal { P } } ( X ) \\to { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 307, + 505, + 321 + ], + "score": 1.0, + "content": "be a convex function. Recall the definition of the optimal discriminator (Defini-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 317, + 140, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 140, + 333 + ], + "score": 1.0, + "content": "tion 1):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 307, + 505, + 333 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 328, + 372, + 356 + ], + "lines": [ + { + "bbox": [ + 238, + 328, + 372, + 356 + ], + "spans": [ + { + "bbox": [ + 238, + 328, + 372, + 356 + ], + "score": 0.94, + "content": "\\Phi _ { \\mu } \\in \\underset { \\varphi \\in \\mathcal { C } ( X ) } { \\arg \\operatorname* { m a x } } \\int \\varphi d \\mu - J ^ { \\star } ( \\varphi ) .", + "type": "interline_equation", + "image_path": "0425a38699615f046ccc2711c233b632644fa5cc145a06081b19306289009a72.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 238, + 328, + 372, + 342.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 238, + 342.0, + 372, + 356.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "This definition can be understood as an infinite dimensional analogue of subgradients. In fact, in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 249, + 381 + ], + "score": 1.0, + "content": "finite-dimensional convex analysis,", + "type": "text" + }, + { + "bbox": [ + 249, + 371, + 255, + 379 + ], + "score": 0.76, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 368, + 333, + 381 + ], + "score": 1.0, + "content": "is a subgradient of", + "type": "text" + }, + { + "bbox": [ + 333, + 369, + 383, + 380 + ], + "score": 0.92, + "content": "f : \\mathbb { R } ^ { d } \\bar { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "if and only if it can be written", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 380, + 504, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 118, + 392 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 119, + 380, + 246, + 392 + ], + "score": 0.89, + "content": "z \\in \\arg \\operatorname* { m a x } _ { z ^ { \\prime } } \\langle z ^ { \\prime } , x \\rangle - f ^ { \\star } ( z ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 380, + 504, + 392 + ], + "score": 1.0, + "content": ". The calculus of subgradients shares many properties with the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 390, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 404 + ], + "score": 1.0, + "content": "standard calculus of derivatives, such as chain rules (Rockafeller & Wets, 1998). This motivate us", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 361, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 361, + 415 + ], + "score": 1.0, + "content": "to investigate derivative-like features of optimal discriminators.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 357, + 506, + 415 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 457, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 458, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 458, + 431 + ], + "score": 1.0, + "content": "We introduce the functional derivative, also known as the von Mises influence function:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 417, + 458, + 431 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 507, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 504, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 272, + 445 + ], + "score": 1.0, + "content": "Definition 3 (Functional derivative). Let", + "type": "text" + }, + { + "bbox": [ + 272, + 432, + 335, + 445 + ], + "score": 0.93, + "content": "J : { \\mathcal { P } } ( X ) \\to { \\bar { \\mathbb { R } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 432, + 504, + 445 + ], + "score": 1.0, + "content": "be a function of probability measures. We", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 443, + 436, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 229, + 458 + ], + "score": 1.0, + "content": "say that a continuous function", + "type": "text" + }, + { + "bbox": [ + 230, + 444, + 282, + 456 + ], + "score": 0.92, + "content": "\\Phi _ { \\mu } : X \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 443, + 398, + 458 + ], + "score": 1.0, + "content": "is a functional derivative of", + "type": "text" + }, + { + "bbox": [ + 399, + 445, + 406, + 454 + ], + "score": 0.78, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 443, + 417, + 458 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 418, + 446, + 425, + 456 + ], + "score": 0.7, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 443, + 436, + 458 + ], + "score": 1.0, + "content": "if", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 432, + 504, + 458 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 460, + 389, + 486 + ], + "lines": [ + { + "bbox": [ + 222, + 460, + 389, + 486 + ], + "spans": [ + { + "bbox": [ + 222, + 460, + 389, + 486 + ], + "score": 0.94, + "content": "J ( \\mu + \\epsilon \\xi ) = J ( \\mu ) + \\epsilon \\int \\Phi _ { \\mu } d \\xi + O ( \\epsilon ^ { 2 } )", + "type": "interline_equation", + "image_path": "cccb311ff9a9ed3028bbeda0d2ef2c291df3acbe789ec3f08ba949df41b1b02c.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 222, + 460, + 389, + 486 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 488, + 272, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 272, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 161, + 502 + ], + "score": 1.0, + "content": "holds for any", + "type": "text" + }, + { + "bbox": [ + 162, + 489, + 205, + 501 + ], + "score": 0.91, + "content": "\\xi = \\nu - \\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 488, + 225, + 502 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 226, + 489, + 268, + 501 + ], + "score": 0.92, + "content": "\\nu \\in { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 488, + 272, + 502 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 488, + 272, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 509, + 425, + 521 + ], + "lines": [ + { + "bbox": [ + 106, + 508, + 426, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 426, + 523 + ], + "score": 1.0, + "content": "Under this definition, optimal discriminators are actually functional derivatives.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 508, + 426, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 523, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 319, + 536 + ], + "score": 1.0, + "content": "Proposition 16 (Chu et al. (2019), Theorem 2). Let", + "type": "text" + }, + { + "bbox": [ + 319, + 522, + 389, + 535 + ], + "score": 0.92, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "be a proper, lower semicon-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 504, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 335, + 548 + ], + "score": 1.0, + "content": "tinuous, and convex function. If there exists a maximizer", + "type": "text" + }, + { + "bbox": [ + 335, + 536, + 348, + 547 + ], + "score": 0.88, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 534, + 360, + 548 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 361, + 534, + 449, + 547 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\varphi \\mapsto \\int \\varphi d \\mu - J ^ { \\star } ( \\varphi ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 534, + 473, + 548 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 473, + 535, + 487, + 547 + ], + "score": 0.9, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 534, + 497, + 548 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 498, + 537, + 504, + 545 + ], + "score": 0.57, + "content": "a", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 546, + 342, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 202, + 558 + ], + "score": 1.0, + "content": "functional derivative of", + "type": "text" + }, + { + "bbox": [ + 202, + 546, + 210, + 556 + ], + "score": 0.64, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 546, + 221, + 558 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 221, + 547, + 228, + 557 + ], + "score": 0.69, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 546, + 342, + 558 + ], + "score": 1.0, + "content": "in the sense of Definition 3.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 522, + 505, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "The following result relates the derivative of the loss function with the derivative of the optimal", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 576, + 165, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 165, + 589 + ], + "score": 1.0, + "content": "discriminator:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 106, + 565, + 505, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 317, + 604 + ], + "score": 1.0, + "content": "Proposition 17 (Chu et al. (2019), Theorem 1). Let", + "type": "text" + }, + { + "bbox": [ + 317, + 591, + 380, + 603 + ], + "score": 0.89, + "content": "J : { \\mathcal { P } } ( X ) \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "be continuously differentiable,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 275, + 615 + ], + "score": 1.0, + "content": "in the sense that the functional derivative", + "type": "text" + }, + { + "bbox": [ + 276, + 603, + 289, + 614 + ], + "score": 0.89, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 601, + 334, + 615 + ], + "score": 1.0, + "content": "exists and", + "type": "text" + }, + { + "bbox": [ + 334, + 602, + 429, + 614 + ], + "score": 0.92, + "content": "( \\mu , \\nu ) \\mapsto \\mathbb { E } _ { x \\sim \\nu } [ \\Phi _ { \\mu } ( x ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "is continuous. Let", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 611, + 507, + 631 + ], + "spans": [ + { + "bbox": [ + 107, + 614, + 141, + 626 + ], + "score": 0.9, + "content": "\\theta \\mapsto \\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 611, + 272, + 631 + ], + "score": 1.0, + "content": "be continuous in the sense that", + "type": "text" + }, + { + "bbox": [ + 273, + 614, + 341, + 629 + ], + "score": 0.91, + "content": "{ \\frac { 1 } { \\| h \\| } } \\big ( \\mu _ { \\theta + h } - \\mu _ { \\theta } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 611, + 462, + 631 + ], + "score": 1.0, + "content": "converges to a weak limit as", + "type": "text" + }, + { + "bbox": [ + 462, + 614, + 501, + 626 + ], + "score": 0.92, + "content": "\\| h \\| 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 611, + 507, + 631 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 167, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 167, + 639 + ], + "score": 1.0, + "content": "Then, we have", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 590, + 507, + 639 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 636, + 365, + 651 + ], + "lines": [ + { + "bbox": [ + 244, + 636, + 365, + 651 + ], + "spans": [ + { + "bbox": [ + 244, + 636, + 365, + 651 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\nabla _ { \\theta } J ( \\mu _ { \\theta } ) = \\nabla _ { \\theta } \\mathbb { E } _ { x \\sim \\mu _ { \\theta } } [ \\hat { \\Phi } ( x ) ] , } \\end{array}", + "type": "interline_equation", + "image_path": "6d29bb9bd38a668d1ef3f0182021573af1a64777fdf39d58b343cff3b30c4dce.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 244, + 636, + 365, + 651 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 407, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 652, + 408, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 133, + 668 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 653, + 171, + 667 + ], + "score": 0.93, + "content": "\\hat { \\Phi } = \\Phi _ { \\mu _ { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 652, + 266, + 668 + ], + "score": 1.0, + "content": "is treated as a function", + "type": "text" + }, + { + "bbox": [ + 266, + 655, + 299, + 664 + ], + "score": 0.91, + "content": "X \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 652, + 398, + 668 + ], + "score": 1.0, + "content": "that is not dependent on", + "type": "text" + }, + { + "bbox": [ + 399, + 655, + 404, + 664 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 652, + 408, + 668 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 652, + 408, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 674, + 504, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 674, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 505, + 687 + ], + "score": 1.0, + "content": "We use this important computational tool in many of our proofs. For the case of the generator model", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 686, + 363, + 699 + ], + "spans": [ + { + "bbox": [ + 107, + 686, + 154, + 698 + ], + "score": 0.92, + "content": "\\mu _ { \\theta } = f _ { \\theta \\# } \\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 686, + 363, + 699 + ], + "score": 1.0, + "content": ", an important consequence of Proposition 17 is that", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 106, + 674, + 505, + 699 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 701, + 455, + 718 + ], + "lines": [ + { + "bbox": [ + 156, + 701, + 455, + 718 + ], + "spans": [ + { + "bbox": [ + 156, + 701, + 455, + 718 + ], + "score": 0.9, + "content": "\\nabla _ { \\theta } J ( \\mu _ { \\theta } ) = \\nabla _ { \\theta } \\mathbb { E } _ { z \\sim \\omega } [ \\hat { \\Phi } ( f _ { \\theta } ( z ) ) ] = \\mathbb { E } _ { z \\sim \\omega } [ \\nabla _ { x = f _ { \\theta } ( z ) } \\Phi _ { \\mu } ( f _ { \\theta } ( z ) ) \\cdot D _ { \\theta } f _ { \\theta } ( z ) ] .", + "type": "interline_equation", + "image_path": "b01369971264d010d0f1ca0c8a802a3880abf12852f36b9caebfa870780eb5cf.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 156, + 701, + 455, + 718 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 720, + 279, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 720, + 280, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 280, + 733 + ], + "score": 1.0, + "content": "We use this fact in the proof of Theorem 1.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44, + "bbox_fs": [ + 106, + 720, + 280, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 292, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 293, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 293, + 96 + ], + "score": 1.0, + "content": "D PROOFS FOR SECTIONS 1 AND 2", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 129 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "The following result is well known in the dynamical systems and the optimization literature. For the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 282, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 282, + 129 + ], + "score": 1.0, + "content": "sake of completeness, we provide its proof.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 105, + 132, + 504, + 156 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 368, + 144 + ], + "score": 1.0, + "content": "Proposition 1 (Bertsekas (1999), Proposition 1.2.3). Suppose", + "type": "text" + }, + { + "bbox": [ + 369, + 132, + 431, + 143 + ], + "score": 0.9, + "content": "f : \\mathbb { R } ^ { p } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 131, + 444, + 144 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 444, + 133, + 452, + 142 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 131, + 506, + 144 + ], + "score": 1.0, + "content": "-smooth and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 430, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 187, + 156 + ], + "score": 1.0, + "content": "bounded below. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 142, + 288, + 156 + ], + "score": 0.92, + "content": "\\begin{array} { r } { x _ { k + 1 } : = x _ { k } - \\frac { 1 } { L } \\dot { \\nabla } \\dot { f } ( x _ { k } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 142, + 315, + 156 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 316, + 144, + 381, + 155 + ], + "score": 0.91, + "content": "| | \\nabla f ( x _ { k } ) | | 0", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 142, + 393, + 156 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 393, + 144, + 425, + 154 + ], + "score": 0.84, + "content": "k \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 142, + 430, + 156 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 291, + 179 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 290, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 209, + 180 + ], + "score": 1.0, + "content": "Proof. It is known that if", + "type": "text" + }, + { + "bbox": [ + 209, + 168, + 217, + 179 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 167, + 227, + 180 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 227, + 168, + 235, + 177 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 167, + 290, + 180 + ], + "score": 1.0, + "content": "-smooth, then", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 183, + 400, + 208 + ], + "lines": [ + { + "bbox": [ + 210, + 183, + 400, + 208 + ], + "spans": [ + { + "bbox": [ + 210, + 183, + 400, + 208 + ], + "score": 0.93, + "content": "f ( y ) \\leq f ( x ) + \\langle \\nabla f ( x ) , y - x \\rangle + { \\frac { L } { 2 } } \\vert \\vert y - x \\vert \\vert ^ { 2 }", + "type": "interline_equation", + "image_path": "04b598abbb2b2e843ca8a02236e0fc063066bbf3bd08f3c1782bb06727cf55f6.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 210, + 183, + 400, + 208 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 214, + 426, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 427, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 161, + 228 + ], + "score": 1.0, + "content": "holds for any", + "type": "text" + }, + { + "bbox": [ + 162, + 214, + 202, + 226 + ], + "score": 0.93, + "content": "x , y \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 213, + 427, + 228 + ], + "score": 1.0, + "content": "(see e.g. Lemma 3.4 in Bubeck (2015)). Then, we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 231, + 433, + 305 + ], + "lines": [ + { + "bbox": [ + 177, + 231, + 433, + 305 + ], + "spans": [ + { + "bbox": [ + 177, + 231, + 433, + 305 + ], + "score": 0.94, + "content": "\\begin{array} { l } { f ( x _ { k + 1 } ) \\leq f ( x _ { k } ) + \\langle \\nabla f ( x _ { k } ) , x _ { k + 1 } - x _ { k } \\rangle + \\displaystyle \\frac { L } { 2 } | | x _ { k + 1 } - x _ { k } | | ^ { 2 } } \\\\ { \\leq f ( x _ { k } ) - \\displaystyle \\frac { 1 } { L } | | \\nabla f ( x _ { k } ) | | ^ { 2 } + \\displaystyle \\frac { 1 } { 2 L } | | \\nabla f ( x _ { k } ) | | ^ { 2 } } \\\\ { = f ( x _ { k } ) - \\displaystyle \\frac { 1 } { 2 L } | | \\nabla f ( x _ { k } ) | | ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "3c1ab09ca79623922374e6055f12fdb81ba18cc42eef2d135115ffc660cf3e79.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 177, + 231, + 433, + 255.66666666666666 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 177, + 255.66666666666666, + 433, + 280.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 177, + 280.3333333333333, + 433, + 305.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 272, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 273, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 227, + 322 + ], + "score": 1.0, + "content": "Summing this inequality over", + "type": "text" + }, + { + "bbox": [ + 227, + 309, + 233, + 318 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 307, + 273, + 322 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 325, + 385, + 360 + ], + "lines": [ + { + "bbox": [ + 225, + 325, + 385, + 360 + ], + "spans": [ + { + "bbox": [ + 225, + 325, + 385, + 360 + ], + "score": 0.94, + "content": "{ \\frac { 1 } { 2 L } } \\sum _ { k = 0 } ^ { n - 1 } | | \\nabla f ( x _ { k } ) | | ^ { 2 } \\leq f ( x _ { 0 } ) - f ( x _ { n } ) ,", + "type": "interline_equation", + "image_path": "9b7140047760529c336f3926badcbf8d81918dafbcc83b161ab0d6496e0b32b0.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 225, + 325, + 385, + 342.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 225, + 342.5, + 385, + 360.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 225, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 226, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 226, + 376 + ], + "score": 1.0, + "content": "from which we conclude that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 379, + 403, + 404 + ], + "lines": [ + { + "bbox": [ + 207, + 379, + 403, + 404 + ], + "spans": [ + { + "bbox": [ + 207, + 379, + 403, + 404 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { 0 \\leq k \\leq n - 1 } | | \\nabla f ( x _ { k } ) | | ^ { 2 } \\leq { \\frac { 2 L } { n } } ( f ( x _ { 0 } ) - \\operatorname* { i n f } _ { x } f ( x ) ) .", + "type": "interline_equation", + "image_path": "e741100bbf9ac9829320088ed9d021165bdd7d36bd8ade1ded38d179691e2a6c.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 207, + 379, + 403, + 404 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 414, + 446 + ], + "lines": [ + { + "bbox": [ + 104, + 432, + 415, + 449 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 415, + 449 + ], + "score": 1.0, + "content": "Next, we move on to prove Theorem 1, which we restate here for readability.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 504, + 472 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 342, + 462 + ], + "score": 1.0, + "content": "Theorem 1 (Smoothness decomposition for GANs). Let", + "type": "text" + }, + { + "bbox": [ + 342, + 448, + 416, + 461 + ], + "score": 0.92, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 447, + 506, + 462 + ], + "score": 1.0, + "content": "be a convex function", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 460, + 454, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 227, + 473 + ], + "score": 1.0, + "content": "whose optimal discriminators", + "type": "text" + }, + { + "bbox": [ + 228, + 461, + 281, + 473 + ], + "score": 0.91, + "content": "\\Phi _ { \\mu } : X \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 460, + 454, + 473 + ], + "score": 1.0, + "content": "satisfy the following regularity conditions:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 118, + 480, + 251, + 493 + ], + "lines": [ + { + "bbox": [ + 118, + 479, + 252, + 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"D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ( z ) \\lVert _ { 2 } \\bigr \\rVert \\leq B \\lVert \\theta _ { 1 } - \\theta _ { 2 } \\rVert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 601, + 269, + 615 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 352, + 635 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 352, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 129, + 636 + ], + "score": 1.0, + "content": "Then", + "type": "text" + }, + { + "bbox": [ + 129, + 623, + 175, + 635 + ], + "score": 0.93, + "content": " { \\theta } \\mapsto J ( { \\mu } _ { { \\theta } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 622, + 186, + 636 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 186, + 623, + 194, + 633 + ], + "score": 0.69, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 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For simplicity, let us consider the smooth-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 224, + 667 + ], + "score": 1.0, + "content": "ness of a composite function", + "type": "text" + }, + { + "bbox": [ + 224, + 655, + 251, + 665 + ], + "score": 0.89, + "content": "D \\circ G", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 655, + 282, + 667 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 282, + 655, + 292, + 665 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 655, + 310, + 667 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 655, + 320, + 665 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 655, + 388, + 667 + ], + "score": 1.0, + "content": "are functions on", + "type": "text" + }, + { + "bbox": [ + 388, + 655, + 396, + 665 + ], + "score": 0.81, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 655, + 506, + 667 + ], + "score": 1.0, + "content": ". A sufficient condition for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 107, + 666, + 130, + 676 + ], + "score": 0.85, + "content": "D \\circ G", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 665, + 436, + 678 + ], + "score": 1.0, + "content": "to be smooth is that its second derivative is bounded. 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For GANs, we need somewhat more involved conditions; We need", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "two types of smoothness (D2) and (D3) for the optimal discriminators. To this end, we utilize the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 721, + 376, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 364, + 733 + ], + "score": 1.0, + "content": "functional gradient point of view that we explained in Appendix", + "type": "text" + }, + { + "bbox": [ + 365, + 721, + 372, + 730 + ], + "score": 0.25, + "content": "\\textrm { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 721, + 376, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 410, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 496, + 411, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 496, + 411, + 505, + 421 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 292, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 293, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 293, + 96 + ], + "score": 1.0, + "content": "D PROOFS FOR SECTIONS 1 AND 2", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 129 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "The following result is well known in the dynamical systems and the optimization literature. For the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 282, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 282, + 129 + ], + "score": 1.0, + "content": "sake of completeness, we provide its proof.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 106, + 505, + 129 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 132, + 504, + 156 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 368, + 144 + ], + "score": 1.0, + "content": "Proposition 1 (Bertsekas (1999), Proposition 1.2.3). Suppose", + "type": "text" + }, + { + "bbox": [ + 369, + 132, + 431, + 143 + ], + "score": 0.9, + "content": "f : \\mathbb { R } ^ { p } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 131, + 444, + 144 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 444, + 133, + 452, + 142 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 131, + 506, + 144 + ], + "score": 1.0, + "content": "-smooth and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 430, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 187, + 156 + ], + "score": 1.0, + "content": "bounded below. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 142, + 288, + 156 + ], + "score": 0.92, + "content": "\\begin{array} { r } { x _ { k + 1 } : = x _ { k } - \\frac { 1 } { L } \\dot { \\nabla } \\dot { f } ( x _ { k } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 142, + 315, + 156 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 316, + 144, + 381, + 155 + ], + "score": 0.91, + "content": "| | \\nabla f ( x _ { k } ) | | 0", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 142, + 393, + 156 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 393, + 144, + 425, + 154 + ], + "score": 0.84, + "content": "k \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 142, + 430, + 156 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 131, + 506, + 156 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 291, + 179 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 290, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 209, + 180 + ], + "score": 1.0, + "content": "Proof. It is known that if", + "type": "text" + }, + { + "bbox": [ + 209, + 168, + 217, + 179 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 167, + 227, + 180 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 227, + 168, + 235, + 177 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 167, + 290, + 180 + ], + "score": 1.0, + "content": "-smooth, then", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 167, + 290, + 180 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 183, + 400, + 208 + ], + "lines": [ + { + "bbox": [ + 210, + 183, + 400, + 208 + ], + "spans": [ + { + "bbox": [ + 210, + 183, + 400, + 208 + ], + "score": 0.93, + "content": "f ( y ) \\leq f ( x ) + \\langle \\nabla f ( x ) , y - x \\rangle + { \\frac { L } { 2 } } \\vert \\vert y - x \\vert \\vert ^ { 2 }", + "type": "interline_equation", + "image_path": "04b598abbb2b2e843ca8a02236e0fc063066bbf3bd08f3c1782bb06727cf55f6.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 210, + 183, + 400, + 208 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 214, + 426, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 427, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 161, + 228 + ], + "score": 1.0, + "content": "holds for any", + "type": "text" + }, + { + "bbox": [ + 162, + 214, + 202, + 226 + ], + "score": 0.93, + "content": "x , y \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 213, + 427, + 228 + ], + "score": 1.0, + "content": "(see e.g. Lemma 3.4 in Bubeck (2015)). Then, we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 213, + 427, + 228 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 231, + 433, + 305 + ], + "lines": [ + { + "bbox": [ + 177, + 231, + 433, + 305 + ], + "spans": [ + { + "bbox": [ + 177, + 231, + 433, + 305 + ], + "score": 0.94, + "content": "\\begin{array} { l } { f ( x _ { k + 1 } ) \\leq f ( x _ { k } ) + \\langle \\nabla f ( x _ { k } ) , x _ { k + 1 } - x _ { k } \\rangle + \\displaystyle \\frac { L } { 2 } | | x _ { k + 1 } - x _ { k } | | ^ { 2 } } \\\\ { \\leq f ( x _ { k } ) - \\displaystyle \\frac { 1 } { L } | | \\nabla f ( x _ { k } ) | | ^ { 2 } + \\displaystyle \\frac { 1 } { 2 L } | | \\nabla f ( x _ { k } ) | | ^ { 2 } } \\\\ { = f ( x _ { k } ) - \\displaystyle \\frac { 1 } { 2 L } | | \\nabla f ( x _ { k } ) | | ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "3c1ab09ca79623922374e6055f12fdb81ba18cc42eef2d135115ffc660cf3e79.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 177, + 231, + 433, + 255.66666666666666 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 177, + 255.66666666666666, + 433, + 280.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 177, + 280.3333333333333, + 433, + 305.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 272, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 273, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 227, + 322 + ], + "score": 1.0, + "content": "Summing this inequality over", + "type": "text" + }, + { + "bbox": [ + 227, + 309, + 233, + 318 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 307, + 273, + 322 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 307, + 273, + 322 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 325, + 385, + 360 + ], + "lines": [ + { + "bbox": [ + 225, + 325, + 385, + 360 + ], + "spans": [ + { + "bbox": [ + 225, + 325, + 385, + 360 + ], + "score": 0.94, + "content": "{ \\frac { 1 } { 2 L } } \\sum _ { k = 0 } ^ { n - 1 } | | \\nabla f ( x _ { k } ) | | ^ { 2 } \\leq f ( x _ { 0 } ) - f ( x _ { n } ) ,", + "type": "interline_equation", + "image_path": "9b7140047760529c336f3926badcbf8d81918dafbcc83b161ab0d6496e0b32b0.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 225, + 325, + 385, + 342.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 225, + 342.5, + 385, + 360.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 225, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 226, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 226, + 376 + ], + "score": 1.0, + "content": "from which we conclude that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 363, + 226, + 376 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 379, + 403, + 404 + ], + "lines": [ + { + "bbox": [ + 207, + 379, + 403, + 404 + ], + "spans": [ + { + "bbox": [ + 207, + 379, + 403, + 404 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { 0 \\leq k \\leq n - 1 } | | \\nabla f ( x _ { k } ) | | ^ { 2 } \\leq { \\frac { 2 L } { n } } ( f ( x _ { 0 } ) - \\operatorname* { i n f } _ { x } f ( x ) ) .", + "type": "interline_equation", + "image_path": "e741100bbf9ac9829320088ed9d021165bdd7d36bd8ade1ded38d179691e2a6c.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 207, + 379, + 403, + 404 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 414, + 446 + ], + "lines": [ + { + "bbox": [ + 104, + 432, + 415, + 449 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 415, + 449 + ], + "score": 1.0, + "content": "Next, we move on to prove Theorem 1, which we restate here for readability.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 432, + 415, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 504, + 472 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 342, + 462 + ], + "score": 1.0, + "content": "Theorem 1 (Smoothness decomposition for GANs). Let", + "type": "text" + }, + { + "bbox": [ + 342, + 448, + 416, + 461 + ], + "score": 0.92, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 447, + 506, + 462 + ], + "score": 1.0, + "content": "be a convex function", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 460, + 454, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 227, + 473 + ], + "score": 1.0, + "content": "whose optimal discriminators", + "type": "text" + }, + { + "bbox": [ + 228, + 461, + 281, + 473 + ], + "score": 0.91, + "content": "\\Phi _ { \\mu } : X \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 460, + 454, + 473 + ], + "score": 1.0, + "content": "satisfy the following regularity conditions:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 106, + 447, + 506, + 473 + ] + }, + { + "type": "text", + "bbox": [ + 118, + 480, + 251, + 493 + ], + 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"score": 0.93, + "content": " { \\theta } \\mapsto J ( { \\mu } _ { { \\theta } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 622, + 186, + 636 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 186, + 623, + 194, + 633 + ], + "score": 0.69, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 622, + 249, + 636 + ], + "score": 1.0, + "content": "-smooth, with", + "type": "text" + }, + { + "bbox": [ + 249, + 622, + 349, + 635 + ], + "score": 0.91, + "content": "L = \\alpha B + A ^ { 2 } ( \\beta _ { 1 } + \\beta _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 622, + 352, + 636 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 622, + 352, + 636 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 642, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "Intuitively, Theorem 1 can be understood as the chain rule. For simplicity, let us consider the smooth-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 224, + 667 + ], + "score": 1.0, + "content": "ness of a composite function", + "type": "text" + }, + { + "bbox": [ + 224, + 655, + 251, + 665 + ], + "score": 0.89, + "content": "D \\circ G", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 655, + 282, + 667 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 282, + 655, + 292, + 665 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 655, + 310, + 667 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 655, + 320, + 665 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 655, + 388, + 667 + ], + "score": 1.0, + "content": "are functions on", + "type": "text" + }, + { + "bbox": [ + 388, + 655, + 396, + 665 + ], + "score": 0.81, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 655, + 506, + 667 + ], + "score": 1.0, + "content": ". A sufficient condition for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 107, + 666, + 130, + 676 + ], + "score": 0.85, + "content": "D \\circ G", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 665, + 436, + 678 + ], + "score": 1.0, + "content": "to be smooth is that its second derivative is bounded. 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By the chain rule, the second derivative is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 154, + 700 + ], + "score": 1.0, + "content": "bounded as", + "type": "text" + }, + { + "bbox": [ + 154, + 687, + 388, + 700 + ], + "score": 0.86, + "content": "( D \\circ G ) ^ { \\prime \\prime } = ( D ^ { \\prime } \\circ G { \\bar { G } } ) G ^ { \\prime \\prime } + ( D ^ { \\prime \\prime } \\circ G ) ( G ^ { \\prime } ) ^ { 2 } \\leq \\alpha B + A ^ { 2 } \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 687, + 506, + 700 + ], + "score": 1.0, + "content": ", which has the same form as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "the consequence of Theorem 1. For GANs, we need somewhat more involved conditions; We need", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "two types of smoothness (D2) and (D3) for the optimal discriminators. To this end, we utilize the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 721, + 376, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 364, + 733 + ], + "score": 1.0, + "content": "functional gradient point of view that we explained in Appendix", + "type": "text" + }, + { + "bbox": [ + 365, + 721, + 372, + 730 + ], + "score": 0.25, + "content": "\\textrm { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 721, + 376, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 643, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Proof of Theorem 1. First, using the functional gradient interpretation of the optimal discriminator,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 143, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 143, + 107 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 110, + 461, + 243 + ], + "lines": [ + { + "bbox": [ + 129, + 110, + 461, + 243 + ], + "spans": [ + { + "bbox": [ + 129, + 110, + 461, + 243 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad \\| \\nabla _ { \\theta _ { 1 } } J ( \\mu _ { \\theta _ { 1 } } ) - \\nabla _ { \\theta _ { 2 } } J ( \\mu _ { \\theta _ { 2 } } ) \\| } \\\\ & { = \\big \\| \\mathbb { E } _ { z \\sim \\omega } \\big [ \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\cdot D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 2 } } ( z ) ) \\cdot D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } \\big ] \\big \\| \\qquad \\mathrm { P r o p o s i t i o } } \\\\ & { = \\big \\| \\mathbb { E } _ { z \\sim \\omega } \\big [ \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\cdot ( D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } - D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ) } \\\\ & { \\qquad + \\big ( \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\big ) \\cdot D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } } \\\\ & { \\qquad + \\big ( \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 1 } } ( z ) ) - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 2 } } ( z ) ) \\big ) \\cdot D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } \\big ] \\big \\| } \\\\ & { \\leq \\big \\| \\mathbb { E } _ { z \\sim \\omega } \\big [ \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\cdot ( D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } - D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ) \\big ] \\big \\| } \\\\ & \\quad + \\big \\| \\mathbb { E } _ { z \\sim \\omega } \\big [ \\big ( \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\big ) \\cdot D _ { \\theta _ { 2 } } f _ \\theta _ \\end{array}", + "type": "interline_equation", + "image_path": "5d6846f5afd0a889d10a8f63b8ad24f5864426c0578a372be1e1158bf5a12cac.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 129, + 110, + 461, + 154.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 129, + 154.33333333333334, + 461, + 198.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 129, + 198.66666666666669, + 461, + 243.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 455, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 251, + 456, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 335, + 267 + ], + "score": 1.0, + "content": "By assumption, there are bounded non-negative numbers", + "type": "text" + }, + { + "bbox": [ + 335, + 253, + 386, + 264 + ], + "score": 0.86, + "content": "\\alpha , \\beta _ { 1 } , \\beta _ { 2 } , A", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 251, + 406, + 267 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 407, + 253, + 416, + 263 + ], + "score": 0.86, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 251, + 456, + 267 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 270, + 441, + 376 + ], + "lines": [ + { + "bbox": [ + 169, + 270, + 441, + 376 + ], + "spans": [ + { + "bbox": [ + 169, + 270, + 441, + 376 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { ( \\mathrm { D } 1 ^ { \\prime } ) \\quad } & { \\alpha = \\underset { \\mu \\in \\mathcal { P } ( X ) } { \\operatorname* { s u p } } \\ \\underset { x \\sim \\mu } { \\operatorname* { s u p } } \\left. \\Phi _ { \\mu } ( x ) \\right. _ { 2 } , } \\\\ { ( \\mathrm { D } 2 ^ { \\prime } ) \\quad } & { \\underset { \\mu \\in \\mathcal { P } ( X ) } { \\operatorname* { s u p } } \\ \\Vert \\nabla _ { x _ { 1 } } \\Phi _ { \\mu } ( x _ { 1 } ) - 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First, using the functional gradient interpretation of the optimal discriminator,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 143, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 143, + 107 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 107 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 110, + 461, + 243 + ], + "lines": [ + { + "bbox": [ + 129, + 110, + 461, + 243 + ], + "spans": [ + { + "bbox": [ + 129, + 110, + 461, + 243 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad \\| \\nabla _ { \\theta _ { 1 } } J ( \\mu _ { \\theta _ { 1 } } ) - \\nabla _ { \\theta _ { 2 } } J ( \\mu _ { \\theta _ { 2 } } ) \\| } \\\\ & { = \\big \\| \\mathbb { E } _ { z \\sim \\omega } \\big [ \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\cdot D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f _ { \\theta _ { 2 } } ( z ) ) \\cdot D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } \\big ] \\big \\| \\qquad \\mathrm { P r o p o s i t i o } } \\\\ & { = \\big \\| \\mathbb { E } _ { z \\sim \\omega } \\big [ \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\cdot ( D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } - 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D _ { \\theta } f _ { \\theta } ( z ) \\right. _ { \\mathrm { o p } } , \\quad \\mathrm { a n d } } \\\\ { ( \\mathrm { G } 2 ^ { \\prime } ) \\quad } & { \\mathbb { E } _ { z \\sim \\omega } \\vert \\vert D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } ( z ) - D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ( z ) \\vert \\vert _ { \\mathrm { o p } } \\leq B \\vert \\vert \\theta _ { 1 } - \\theta _ { 2 } \\vert \\vert . } \\end{array}", + "type": "interline_equation", + "image_path": "66130a1f4d58a3cf7848ba6b5cf00a669dc6ec7d81152c5144ec05b1abaff6f9.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 169, + 270, + 441, + 305.3333333333333 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 169, + 305.3333333333333, + 441, + 340.66666666666663 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 169, + 340.66666666666663, + 441, + 375.99999999999994 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 504, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 172, + 393 + ], + "score": 1.0, + "content": "Here, we wrote", + "type": "text" + }, + { + "bbox": [ + 172, + 380, + 225, + 394 + ], + "score": 0.93, + "content": "\\operatorname { s u p } _ { x \\sim \\mu } f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 379, + 351, + 393 + ], + "score": 1.0, + "content": "for the essential supremum of", + "type": "text" + }, + { + "bbox": [ + 351, + 381, + 358, + 392 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 379, + 383, + 393 + ], + "score": 1.0, + "content": "w.r.t.", + "type": "text" + }, + { + "bbox": [ + 383, + 383, + 390, + 392 + ], + "score": 0.79, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 379, + 421, + 393 + ], + "score": 1.0, + "content": ". 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D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ) ] \\right\\| \\leq \\alpha B \\| \\theta _ { 1 } - \\theta _ { 2 } \\| . } \\end{array}", + "type": "interline_equation", + "image_path": "bbe47833a8db57f817ed0b6b7d66e979b6f1a95c39d2f673185c2ece87413d7c.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 174, + 408, + 436, + 424 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 430, + 466, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 467, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 131, + 444 + ], + "score": 1.0, + "content": "From", + "type": "text" + }, + { + "bbox": [ + 131, + 431, + 151, + 442 + ], + "score": 0.62, + "content": "( \\mathrm Ḋ 3 Ḍ ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 429, + 157, + 444 + ], + "score": 1.0, + "content": "),", + "type": "text" + }, + { + "bbox": [ + 158, + 430, + 179, + 442 + ], + "score": 0.45, + "content": "( \\mathbf { G } 1 ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 429, + 467, + 444 + ], + "score": 1.0, + "content": ", and the Cauchy-Schwarz inequality, the second term (b) is bounded as", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 429, + 467, + 444 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 446, + 423, + 496 + ], + "lines": [ + { + "bbox": [ + 187, + 446, + 423, + 496 + ], + "spans": [ + { + "bbox": [ + 187, + 446, + 423, + 496 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\quad \\quad \\Big | \\Big | \\mathbb { E } _ { z \\sim \\omega } [ \\big ( \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) - 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\\nabla _ { \\theta _ { 2 } } J ( \\mu _ { \\theta _ { 2 } } ) | | \\leq ( \\alpha B + A ^ { 2 } ( \\beta _ { 1 } + \\beta _ { 2 } ) ) \\| \\theta _ { 1 } - \\theta _ { 2 } \\| .", + "type": "interline_equation", + "image_path": "e88746fac3fb4d23d6aa5a3ac86e0739bf5ca5149c4d9c6aace25be7e4d92e41.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 176, + 612, + 435, + 628 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 662, + 250, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 661, + 251, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 251, + 677 + ], + "score": 1.0, + "content": "E PROOFS FOR SECTION 3", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "In the finite-dimensional case, Proposition 15 says that taking inf-convolution with a “coercive”", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "regularizer does not change the set of minimizers of the original objective. 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Let", + "type": "text" + }, + { + "bbox": [ + 298, + 82, + 441, + 95 + ], + "score": 0.71, + "content": "R _ { 1 } ( \\xi ) : = \\| \\xi \\| _ { \\mathrm { K R } } , R _ { 2 } ( \\xi ) : = \\| \\xi \\| _ { S ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 81, + 464, + 95 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 464, + 82, + 505, + 95 + ], + "score": 0.9, + "content": "R _ { 3 } ( \\xi ) : = ", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 94, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 136, + 108 + ], + "score": 0.91, + "content": "\\frac { 1 } { 2 } \\lVert \\hat { \\xi } \\rVert _ { \\mathcal { H } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 94, + 487, + 109 + ], + "score": 1.0, + "content": "be the three regularizers defined by (8), (12), and (19) respectively. Assume that", + "type": "text" + }, + { + "bbox": [ + 487, + 96, + 505, + 106 + ], + "score": 0.6, + "content": " { J } :", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 106, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 107, + 107, + 162, + 120 + ], + "score": 0.92, + "content": "\\bar { \\mathcal { M } } ( \\bar { X } ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 106, + 273, + 122 + ], + "score": 1.0, + "content": "has a unique minimizer at", + "type": "text" + }, + { + "bbox": [ + 273, + 109, + 285, + 120 + ], + "score": 0.81, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 106, + 307, + 122 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 307, + 108, + 354, + 120 + ], + "score": 0.92, + "content": "J ( \\mu _ { 0 } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 106, + 377, + 122 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 377, + 108, + 465, + 120 + ], + "score": 0.92, + "content": "J ( \\mu ) \\geq c \\| \\hat { \\mu } - \\hat { \\mu } _ { 0 } \\| _ { \\mathcal { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 106, + 505, + 122 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 119, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 107, + 122, + 135, + 131 + ], + "score": 0.88, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 119, + 247, + 134 + ], + "score": 1.0, + "content": ". Then the inf-convolution", + "type": "text" + }, + { + "bbox": [ + 247, + 120, + 356, + 132 + ], + "score": 0.9, + "content": "\\tilde { J } = J \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 119, + 471, + 134 + ], + "score": 1.0, + "content": "has a unique minimizer at", + "type": "text" + }, + { + "bbox": [ + 471, + 123, + 483, + 133 + ], + "score": 0.82, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 119, + 506, + 134 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 131, + 155, + 146 + ], + "spans": [ + { + "bbox": [ + 107, + 132, + 150, + 145 + ], + "score": 0.92, + "content": "\\tilde { J } ( \\mu _ { 0 } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 131, + 155, + 146 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 161, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 252, + 175 + ], + "score": 1.0, + "content": "Proof. In the following, we endow", + "type": "text" + }, + { + "bbox": [ + 252, + 162, + 282, + 174 + ], + "score": 0.94, + "content": "{ \\mathcal { M } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 160, + 414, + 175 + ], + "score": 1.0, + "content": "with the Gaussian RKHS norm", + "type": "text" + }, + { + "bbox": [ + 415, + 162, + 438, + 174 + ], + "score": 0.91, + "content": "\\| \\widehat { \\cdot } \\| _ { \\mathcal { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 160, + 505, + 175 + ], + "score": 1.0, + "content": ", and for conve-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 135, + 185 + ], + "score": 1.0, + "content": "nience", + "type": "text" + }, + { + "bbox": [ + 135, + 173, + 152, + 185 + ], + "score": 0.91, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "will refer to this norm if not otherwise specified. To show the result, we apply Lemma 1", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 504, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 184, + 196 + ], + "score": 1.0, + "content": "three times, first on", + "type": "text" + }, + { + "bbox": [ + 185, + 185, + 198, + 195 + ], + "score": 0.89, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 183, + 215, + 196 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 215, + 184, + 228, + 195 + ], + "score": 0.89, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 183, + 303, + 196 + ], + "score": 1.0, + "content": ", and then again on", + "type": "text" + }, + { + "bbox": [ + 304, + 184, + 338, + 195 + ], + "score": 0.91, + "content": "R _ { 3 } \\oplus R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 183, + 355, + 196 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 356, + 185, + 369, + 195 + ], + "score": 0.87, + "content": "R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 183, + 447, + 196 + ], + "score": 1.0, + "content": ", and then finally on", + "type": "text" + }, + { + "bbox": [ + 448, + 184, + 504, + 195 + ], + "score": 0.91, + "content": "R _ { 3 } \\oplus R _ { 1 } \\oplus R _ { 2 }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 123, + 210 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 196, + 164, + 209 + ], + "score": 0.92, + "content": "J ( \\cdot + \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 195, + 311, + 210 + ], + "score": 1.0, + "content": ". The result follows from noting that", + "type": "text" + }, + { + "bbox": [ + 311, + 196, + 500, + 209 + ], + "score": 0.91, + "content": "\\tilde { J } ( \\mu ) = [ R _ { 3 } \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus J ( \\cdot + \\mu _ { 0 } ) ] ( \\mu - \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 195, + 504, + 210 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 213, + 505, + 283 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 311, + 226 + ], + "score": 1.0, + "content": "In order to apply the lemma, it suffices to show that", + "type": "text" + }, + { + "bbox": [ + 311, + 214, + 324, + 225 + ], + "score": 0.89, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "is uniformly continuous on bounded sets and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 263, + 238 + ], + "score": 1.0, + "content": "coercive, and that there exist constants", + "type": "text" + }, + { + "bbox": [ + 263, + 226, + 273, + 235 + ], + "score": 0.86, + "content": "c _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 223, + 291, + 238 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 291, + 226, + 301, + 235 + ], + "score": 0.84, + "content": "c _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 223, + 341, + 238 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 342, + 224, + 404, + 236 + ], + "score": 0.91, + "content": "R _ { 1 } ( \\xi ) \\overset { \\cdot } { \\geq } c _ { 1 } \\| \\xi \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 223, + 423, + 238 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 224, + 491, + 236 + ], + "score": 0.89, + "content": "R _ { 2 } ( \\xi ) \\geq c _ { 2 } \\| \\xi \\| .", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 223, + 506, + 238 + ], + "score": 1.0, + "content": "R3", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 307, + 248 + ], + "score": 1.0, + "content": "is uniformly continuous on bounded sets: suppose", + "type": "text" + }, + { + "bbox": [ + 307, + 235, + 344, + 247 + ], + "score": 0.91, + "content": "\\| \\xi \\| < C", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 235, + 362, + 248 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 362, + 236, + 401, + 248 + ], + "score": 0.9, + "content": "\\| \\xi ^ { \\prime } \\| < C", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 235, + 425, + 248 + ], + "score": 1.0, + "content": "; then", + "type": "text" + }, + { + "bbox": [ + 425, + 236, + 505, + 248 + ], + "score": 0.89, + "content": "| R _ { 3 } ( \\xi ) - R _ { 3 } ( \\xi ^ { \\prime } ) | =", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 244, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 312, + 259 + ], + "score": 0.89, + "content": "\\begin{array} { r } { | \\frac { 1 } { 2 } \\| \\xi \\| ^ { 2 } - \\frac { 1 } { 2 } \\| \\xi ^ { \\prime } \\| ^ { 2 } | = | \\frac { 1 } { 2 } \\langle \\xi - \\xi ^ { \\prime } , \\xi + \\xi ^ { \\prime } \\rangle | \\leq C \\| \\dot { \\xi } - \\xi ^ { \\prime } \\| } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 244, + 506, + 261 + ], + "score": 1.0, + "content": "by the Cauchy–Schwarz and triangle inequality.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 257, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 107, + 260, + 120, + 270 + ], + "score": 0.88, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 257, + 201, + 273 + ], + "score": 1.0, + "content": "is also coercive, as", + "type": "text" + }, + { + "bbox": [ + 202, + 259, + 298, + 271 + ], + "score": 0.88, + "content": "R _ { 3 } ( \\xi ) = \\textstyle { \\frac { 1 } { 2 } } \\| \\xi \\| ^ { 2 } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 257, + 325, + 273 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 325, + 259, + 369, + 271 + ], + "score": 0.9, + "content": "\\| \\xi \\| \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 257, + 375, + 273 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 375, + 259, + 388, + 270 + ], + "score": 0.87, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 257, + 425, + 273 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 426, + 259, + 490, + 271 + ], + "score": 0.92, + "content": "R _ { 1 } ( \\xi ) \\geq c _ { 1 } \\| \\xi \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 257, + 506, + 273 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 269, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 150, + 282 + ], + "score": 1.0, + "content": "Lemma 7.", + "type": "text" + }, + { + "bbox": [ + 150, + 271, + 163, + 281 + ], + "score": 0.83, + "content": "R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 269, + 199, + 282 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 199, + 271, + 261, + 282 + ], + "score": 0.93, + "content": "R _ { 2 } ( \\xi ) \\geq c _ { 2 } ] | \\xi | |", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 269, + 318, + 282 + ], + "score": 1.0, + "content": "by Lemma 5.", + "type": "text" + }, + { + "bbox": [ + 495, + 271, + 504, + 280 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 415, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 413, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 194, + 313 + ], + "score": 1.0, + "content": "Recall that a function", + "type": "text" + }, + { + "bbox": [ + 195, + 300, + 204, + 309 + ], + "score": 0.86, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 297, + 299, + 313 + ], + "score": 1.0, + "content": "is said to be coercive if", + "type": "text" + }, + { + "bbox": [ + 299, + 299, + 346, + 311 + ], + "score": 0.92, + "content": "F ( \\xi ) \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 297, + 372, + 313 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 372, + 299, + 413, + 311 + ], + "score": 0.9, + "content": "\\| \\xi \\| \\infty", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 506, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 190, + 328 + ], + "score": 1.0, + "content": "Lemma 1. Suppose", + "type": "text" + }, + { + "bbox": [ + 190, + 315, + 259, + 327 + ], + "score": 0.92, + "content": "F : \\mathcal { M } ( X ) \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 314, + 393, + 328 + ], + "score": 1.0, + "content": "has a unique minimizer at 0 with", + "type": "text" + }, + { + "bbox": [ + 393, + 315, + 433, + 327 + ], + "score": 0.91, + "content": "F ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 314, + 505, + 328 + ], + "score": 1.0, + "content": ", and is uniformly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 326, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 315, + 338 + ], + "score": 1.0, + "content": "continuous on bounded sets and coercive. Suppose", + "type": "text" + }, + { + "bbox": [ + 316, + 326, + 388, + 338 + ], + "score": 0.92, + "content": "G : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 326, + 506, + 338 + ], + "score": 1.0, + "content": "has a unique minimizer at 0", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 127, + 349 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 337, + 168, + 349 + ], + "score": 0.92, + "content": "G ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 336, + 191, + 349 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 191, + 337, + 246, + 348 + ], + "score": 0.92, + "content": "G ( \\cdot ) \\geq c \\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 336, + 285, + 349 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 286, + 338, + 311, + 348 + ], + "score": 0.88, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 336, + 419, + 349 + ], + "score": 1.0, + "content": ". Then the inf-convolution", + "type": "text" + }, + { + "bbox": [ + 419, + 337, + 449, + 348 + ], + "score": 0.9, + "content": "F \\oplus G", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "has a unique", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 189, + 361 + ], + "score": 1.0, + "content": "minimizer at 0 with", + "type": "text" + }, + { + "bbox": [ + 189, + 348, + 258, + 360 + ], + "score": 0.9, + "content": "( F \\oplus G ) ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 347, + 506, + 361 + ], + "score": 1.0, + "content": ", and is uniformly continuous on bounded sets, coercive, and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 359, + 156, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 156, + 369 + ], + "score": 1.0, + "content": "real-valued.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 387, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 315, + 401 + ], + "score": 1.0, + "content": "Proof. From Theorem 2.3 of Stromberg (1996), ¨ inf", + "type": "text" + }, + { + "bbox": [ + 315, + 388, + 363, + 398 + ], + "score": 0.91, + "content": "F \\oplus G = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 387, + 383, + 401 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 384, + 388, + 464, + 399 + ], + "score": 0.9, + "content": "0 \\in \\arg \\operatorname* { m i n } F \\oplus G", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 387, + 505, + 401 + ], + "score": 1.0, + "content": ". To show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 245, + 412 + ], + "score": 1.0, + "content": "that 0 is the unique minimizer, let", + "type": "text" + }, + { + "bbox": [ + 245, + 399, + 327, + 410 + ], + "score": 0.84, + "content": "\\xi \\in \\arg \\operatorname* { m i n } F \\oplus G", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 398, + 505, + 412 + ], + "score": 1.0, + "content": ". From the definition of inf-convolution, for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 304, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 131, + 421 + ], + "score": 1.0, + "content": "every", + "type": "text" + }, + { + "bbox": [ + 131, + 412, + 138, + 420 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 410, + 198, + 421 + ], + "score": 1.0, + "content": "there exists an", + "type": "text" + }, + { + "bbox": [ + 199, + 410, + 250, + 422 + ], + "score": 0.93, + "content": "\\xi _ { n } \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 410, + 304, + 421 + ], + "score": 1.0, + "content": "that satisfies", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 428, + 399, + 447 + ], + "lines": [ + { + "bbox": [ + 211, + 428, + 399, + 447 + ], + "spans": [ + { + "bbox": [ + 211, + 428, + 399, + 447 + ], + "score": 0.89, + "content": "\\operatorname* { l i m } _ { n \\to \\infty } F ( \\xi _ { n } ) + G ( \\xi - \\xi _ { n } ) = ( F \\oplus G ) ( \\xi ) = 0 .", + "type": "interline_equation", + "image_path": "d08101671fd19c6515046e9f0c6b929b8cd77fe866b9578aac73b12c77b9c6ba.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 211, + 428, + 399, + 447 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 131, + 466 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 454, + 140, + 463 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 453, + 159, + 466 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 159, + 454, + 168, + 464 + ], + "score": 0.84, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 453, + 241, + 466 + ], + "score": 1.0, + "content": "are non-negative,", + "type": "text" + }, + { + "bbox": [ + 241, + 453, + 289, + 465 + ], + "score": 0.9, + "content": "F ( \\xi _ { n } ) \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 453, + 307, + 466 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 308, + 453, + 372, + 465 + ], + "score": 0.92, + "content": "G ( \\xi - \\xi _ { n } ) \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 453, + 473, + 466 + ], + "score": 1.0, + "content": ". By our assumption that", + "type": "text" + }, + { + "bbox": [ + 474, + 453, + 505, + 465 + ], + "score": 0.89, + "content": "G ( \\cdot ) \\geq", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 107, + 464, + 128, + 477 + ], + "score": 0.89, + "content": "c \\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 464, + 242, + 477 + ], + "score": 1.0, + "content": ", the latter limit implies that", + "type": "text" + }, + { + "bbox": [ + 243, + 465, + 302, + 477 + ], + "score": 0.91, + "content": "\\| \\xi - \\xi _ { n } \\| \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 464, + 384, + 477 + ], + "score": 1.0, + "content": ", which implies that", + "type": "text" + }, + { + "bbox": [ + 384, + 464, + 447, + 476 + ], + "score": 0.91, + "content": "F ( \\xi _ { n } ) \\to F ( \\xi )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "by continuity", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 117, + 488 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 477, + 126, + 485 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 474, + 322, + 488 + ], + "score": 1.0, + "content": ". Comparing our two expressions for the limit of", + "type": "text" + }, + { + "bbox": [ + 322, + 475, + 348, + 487 + ], + "score": 0.92, + "content": "F ( \\xi _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 474, + 402, + 488 + ], + "score": 1.0, + "content": ", we find that", + "type": "text" + }, + { + "bbox": [ + 402, + 476, + 442, + 487 + ], + "score": 0.89, + "content": "F ( \\xi ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 474, + 505, + 488 + ], + "score": 1.0, + "content": ", which implies", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 485, + 424, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 124, + 500 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 487, + 148, + 498 + ], + "score": 0.9, + "content": "\\xi = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 485, + 175, + 500 + ], + "score": 1.0, + "content": ", since", + "type": "text" + }, + { + "bbox": [ + 175, + 487, + 184, + 496 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 485, + 341, + 500 + ], + "score": 1.0, + "content": "has a unique minimizer at 0. Hence ar", + "type": "text" + }, + { + "bbox": [ + 342, + 487, + 420, + 498 + ], + "score": 0.85, + "content": "\\operatorname { m i n } F \\oplus G = \\{ 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 485, + 424, + 500 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 283, + 516 + ], + "score": 1.0, + "content": "From Theorem 2.10 of Stromberg (1996), ¨", + "type": "text" + }, + { + "bbox": [ + 283, + 504, + 314, + 514 + ], + "score": 0.89, + "content": "F \\oplus G", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "is uniformly continuous on bounded sets and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 210, + 527 + ], + "score": 1.0, + "content": "real-valued. To show that", + "type": "text" + }, + { + "bbox": [ + 211, + 515, + 240, + 525 + ], + "score": 0.9, + "content": "F \\oplus G", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 513, + 506, + 527 + ], + "score": 1.0, + "content": "is coercive, we show the equivalent condition its sublevel sets are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "bounded (see Proposition 11.11 of Bauschke & Combettes (2011)). That is, we show that for all", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 107, + 537, + 134, + 546 + ], + "score": 0.88, + "content": "a > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 536, + 231, + 549 + ], + "score": 1.0, + "content": ", there exists a constant", + "type": "text" + }, + { + "bbox": [ + 231, + 537, + 237, + 546 + ], + "score": 0.77, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 536, + 288, + 549 + ], + "score": 1.0, + "content": "such that if", + "type": "text" + }, + { + "bbox": [ + 288, + 536, + 359, + 548 + ], + "score": 0.92, + "content": "( F \\oplus G ) ( \\xi ) \\leq a", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 536, + 384, + 549 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 384, + 536, + 420, + 548 + ], + "score": 0.91, + "content": "\\| \\xi \\| \\le b", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 536, + 443, + 549 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 443, + 537, + 470, + 547 + ], + "score": 0.89, + "content": "a > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 536, + 506, + 549 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 154, + 559 + ], + "score": 0.9, + "content": "\\xi \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 546, + 207, + 560 + ], + "score": 1.0, + "content": "be such that", + "type": "text" + }, + { + "bbox": [ + 208, + 547, + 276, + 559 + ], + "score": 0.92, + "content": "( F \\oplus G ) ( \\xi ) \\leq a", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 546, + 297, + 560 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 298, + 548, + 322, + 558 + ], + "score": 0.86, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "; from the definition of inf-convolution, there", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 559, + 225, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 138, + 571 + ], + "score": 1.0, + "content": "exists a", + "type": "text" + }, + { + "bbox": [ + 139, + 559, + 185, + 572 + ], + "score": 0.94, + "content": "\\tilde { \\xi } \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 560, + 225, + 571 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 578, + 399, + 594 + ], + "lines": [ + { + "bbox": [ + 211, + 578, + 399, + 594 + ], + "spans": [ + { + "bbox": [ + 211, + 578, + 399, + 594 + ], + "score": 0.9, + "content": "F ( { \\tilde { \\xi } } ) + G ( \\xi - { \\tilde { \\xi } } ) \\leq ( F \\oplus G ) ( \\xi ) + \\epsilon \\leq a + \\epsilon .", + "type": "interline_equation", + "image_path": "424df517264799532d9f23baf1066f8faee11da85ead27b0973e1291f67898c3.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 211, + 578, + 399, + 594 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 304, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 303, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 142, + 613 + ], + "score": 1.0, + "content": "Because", + "type": "text" + }, + { + "bbox": [ + 142, + 601, + 151, + 610 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 600, + 169, + 613 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 601, + 178, + 610 + ], + "score": 0.84, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 600, + 303, + 613 + ], + "score": 1.0, + "content": "are non-negative, we have that", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 619, + 384, + 635 + ], + "lines": [ + { + "bbox": [ + 227, + 619, + 384, + 635 + ], + "spans": [ + { + "bbox": [ + 227, + 619, + 384, + 635 + ], + "score": 0.9, + "content": "\\begin{array} { r } { F ( \\tilde { \\xi } ) \\le a + \\epsilon , \\qquad G ( \\xi - \\tilde { \\xi } ) \\le a + \\epsilon . } \\end{array}", + "type": "interline_equation", + "image_path": "61a09dc600cf8db839d84f21488fc6883c7d3d7017f118bac61014000c1aba92.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 227, + 619, + 384, + 635 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 641, + 506, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 640, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 187, + 654 + ], + "score": 1.0, + "content": "Using the fact that", + "type": "text" + }, + { + "bbox": [ + 187, + 642, + 197, + 651 + ], + "score": 0.86, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 640, + 431, + 654 + ], + "score": 1.0, + "content": "is coercive and hence its sublevel sets are bounded, let", + "type": "text" + }, + { + "bbox": [ + 432, + 641, + 440, + 651 + ], + "score": 0.85, + "content": "b ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 640, + 506, + 654 + ], + "score": 1.0, + "content": "be such that if", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 651, + 488, + 666 + ], + "spans": [ + { + "bbox": [ + 107, + 652, + 164, + 664 + ], + "score": 0.91, + "content": "F ( \\nu ) \\leq a + \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 651, + 187, + 666 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 187, + 652, + 224, + 664 + ], + "score": 0.93, + "content": "\\| \\nu \\| \\leq b ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 651, + 430, + 666 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 298, + 82, + 441, + 95 + ], + "score": 0.71, + "content": "R _ { 1 } ( \\xi ) : = \\| \\xi \\| _ { \\mathrm { K R } } , R _ { 2 } ( \\xi ) : = \\| \\xi \\| _ { S ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 81, + 464, + 95 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 464, + 82, + 505, + 95 + ], + "score": 0.9, + "content": "R _ { 3 } ( \\xi ) : = ", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 94, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 136, + 108 + ], + "score": 0.91, + "content": "\\frac { 1 } { 2 } \\lVert \\hat { \\xi } \\rVert _ { \\mathcal { H } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 94, + 487, + 109 + ], + "score": 1.0, + "content": "be the three regularizers defined by (8), (12), and (19) respectively. Assume that", + "type": "text" + }, + { + "bbox": [ + 487, + 96, + 505, + 106 + ], + "score": 0.6, + "content": " { J } :", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 106, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 107, + 107, + 162, + 120 + ], + "score": 0.92, + "content": "\\bar { \\mathcal { M } } ( \\bar { X } ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 106, + 273, + 122 + ], + "score": 1.0, + "content": "has a unique minimizer at", + "type": "text" + }, + { + "bbox": [ + 273, + 109, + 285, + 120 + ], + "score": 0.81, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 106, + 307, + 122 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 307, + 108, + 354, + 120 + ], + "score": 0.92, + "content": "J ( \\mu _ { 0 } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 106, + 377, + 122 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 377, + 108, + 465, + 120 + ], + "score": 0.92, + "content": "J ( \\mu ) \\geq c \\| \\hat { \\mu } - \\hat { \\mu } _ { 0 } \\| _ { \\mathcal { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 106, + 505, + 122 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 119, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 107, + 122, + 135, + 131 + ], + "score": 0.88, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 119, + 247, + 134 + ], + "score": 1.0, + "content": ". Then the inf-convolution", + "type": "text" + }, + { + "bbox": [ + 247, + 120, + 356, + 132 + ], + "score": 0.9, + "content": "\\tilde { J } = J \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 119, + 471, + 134 + ], + "score": 1.0, + "content": "has a unique minimizer at", + "type": "text" + }, + { + "bbox": [ + 471, + 123, + 483, + 133 + ], + "score": 0.82, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 119, + 506, + 134 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 131, + 155, + 146 + ], + "spans": [ + { + "bbox": [ + 107, + 132, + 150, + 145 + ], + "score": 0.92, + "content": "\\tilde { J } ( \\mu _ { 0 } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 131, + 155, + 146 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 81, + 506, + 146 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 161, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 252, + 175 + ], + "score": 1.0, + "content": "Proof. In the following, we endow", + "type": "text" + }, + { + "bbox": [ + 252, + 162, + 282, + 174 + ], + "score": 0.94, + "content": "{ \\mathcal { M } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 160, + 414, + 175 + ], + "score": 1.0, + "content": "with the Gaussian RKHS norm", + "type": "text" + }, + { + "bbox": [ + 415, + 162, + 438, + 174 + ], + "score": 0.91, + "content": "\\| \\widehat { \\cdot } \\| _ { \\mathcal { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 160, + 505, + 175 + ], + "score": 1.0, + "content": ", and for conve-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 135, + 185 + ], + "score": 1.0, + "content": "nience", + "type": "text" + }, + { + "bbox": [ + 135, + 173, + 152, + 185 + ], + "score": 0.91, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "will refer to this norm if not otherwise specified. To show the result, we apply Lemma 1", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 504, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 184, + 196 + ], + "score": 1.0, + "content": "three times, first on", + "type": "text" + }, + { + "bbox": [ + 185, + 185, + 198, + 195 + ], + "score": 0.89, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 183, + 215, + 196 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 215, + 184, + 228, + 195 + ], + "score": 0.89, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 183, + 303, + 196 + ], + "score": 1.0, + "content": ", and then again on", + "type": "text" + }, + { + "bbox": [ + 304, + 184, + 338, + 195 + ], + "score": 0.91, + "content": "R _ { 3 } \\oplus R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 183, + 355, + 196 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 356, + 185, + 369, + 195 + ], + "score": 0.87, + "content": "R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 183, + 447, + 196 + ], + "score": 1.0, + "content": ", and then finally on", + "type": "text" + }, + { + "bbox": [ + 448, + 184, + 504, + 195 + ], + "score": 0.91, + "content": "R _ { 3 } \\oplus R _ { 1 } \\oplus R _ { 2 }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 123, + 210 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 196, + 164, + 209 + ], + "score": 0.92, + "content": "J ( \\cdot + \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 195, + 311, + 210 + ], + "score": 1.0, + "content": ". The result follows from noting that", + "type": "text" + }, + { + "bbox": [ + 311, + 196, + 500, + 209 + ], + "score": 0.91, + "content": "\\tilde { J } ( \\mu ) = [ R _ { 3 } \\oplus R _ { 1 } \\oplus R _ { 2 } \\oplus J ( \\cdot + \\mu _ { 0 } ) ] ( \\mu - \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 195, + 504, + 210 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 160, + 505, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 213, + 505, + 283 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 311, + 226 + ], + "score": 1.0, + "content": "In order to apply the lemma, it suffices to show that", + "type": "text" + }, + { + "bbox": [ + 311, + 214, + 324, + 225 + ], + "score": 0.89, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "is uniformly continuous on bounded sets and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 263, + 238 + ], + "score": 1.0, + "content": "coercive, and that there exist constants", + "type": "text" + }, + { + "bbox": [ + 263, + 226, + 273, + 235 + ], + "score": 0.86, + "content": "c _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 223, + 291, + 238 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 291, + 226, + 301, + 235 + ], + "score": 0.84, + "content": "c _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 223, + 341, + 238 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 342, + 224, + 404, + 236 + ], + "score": 0.91, + "content": "R _ { 1 } ( \\xi ) \\overset { \\cdot } { \\geq } c _ { 1 } \\| \\xi \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 223, + 423, + 238 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 224, + 491, + 236 + ], + "score": 0.89, + "content": "R _ { 2 } ( \\xi ) \\geq c _ { 2 } \\| \\xi \\| .", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 223, + 506, + 238 + ], + "score": 1.0, + "content": "R3", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 307, + 248 + ], + "score": 1.0, + "content": "is uniformly continuous on bounded sets: suppose", + "type": "text" + }, + { + "bbox": [ + 307, + 235, + 344, + 247 + ], + "score": 0.91, + "content": "\\| \\xi \\| < C", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 235, + 362, + 248 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 362, + 236, + 401, + 248 + ], + "score": 0.9, + "content": "\\| \\xi ^ { \\prime } \\| < C", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 235, + 425, + 248 + ], + "score": 1.0, + "content": "; then", + "type": "text" + }, + { + "bbox": [ + 425, + 236, + 505, + 248 + ], + "score": 0.89, + "content": "| R _ { 3 } ( \\xi ) - R _ { 3 } ( \\xi ^ { \\prime } ) | =", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 244, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 312, + 259 + ], + "score": 0.89, + "content": "\\begin{array} { r } { | \\frac { 1 } { 2 } \\| \\xi \\| ^ { 2 } - \\frac { 1 } { 2 } \\| \\xi ^ { \\prime } \\| ^ { 2 } | = | \\frac { 1 } { 2 } \\langle \\xi - \\xi ^ { \\prime } , \\xi + \\xi ^ { \\prime } \\rangle | \\leq C \\| \\dot { \\xi } - \\xi ^ { \\prime } \\| } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 244, + 506, + 261 + ], + "score": 1.0, + "content": "by the Cauchy–Schwarz and triangle inequality.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 257, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 107, + 260, + 120, + 270 + ], + "score": 0.88, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 257, + 201, + 273 + ], + "score": 1.0, + "content": "is also coercive, as", + "type": "text" + }, + { + "bbox": [ + 202, + 259, + 298, + 271 + ], + "score": 0.88, + "content": "R _ { 3 } ( \\xi ) = \\textstyle { \\frac { 1 } { 2 } } \\| \\xi \\| ^ { 2 } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 257, + 325, + 273 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 325, + 259, + 369, + 271 + ], + "score": 0.9, + "content": "\\| \\xi \\| \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 257, + 375, + 273 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 375, + 259, + 388, + 270 + ], + "score": 0.87, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 257, + 425, + 273 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 426, + 259, + 490, + 271 + ], + "score": 0.92, + "content": "R _ { 1 } ( \\xi ) \\geq c _ { 1 } \\| \\xi \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 257, + 506, + 273 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 269, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 150, + 282 + ], + "score": 1.0, + "content": "Lemma 7.", + "type": "text" + }, + { + "bbox": [ + 150, + 271, + 163, + 281 + ], + "score": 0.83, + "content": "R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 269, + 199, + 282 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 199, + 271, + 261, + 282 + ], + "score": 0.93, + "content": "R _ { 2 } ( \\xi ) \\geq c _ { 2 } ] | \\xi | |", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 269, + 318, + 282 + ], + "score": 1.0, + "content": "by Lemma 5.", + "type": "text" + }, + { + "bbox": [ + 495, + 271, + 504, + 280 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 213, + 506, + 282 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 415, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 413, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 194, + 313 + ], + "score": 1.0, + "content": "Recall that a function", + "type": "text" + }, + { + "bbox": [ + 195, + 300, + 204, + 309 + ], + "score": 0.86, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 297, + 299, + 313 + ], + "score": 1.0, + "content": "is said to be coercive if", + "type": "text" + }, + { + "bbox": [ + 299, + 299, + 346, + 311 + ], + "score": 0.92, + "content": "F ( \\xi ) \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 297, + 372, + 313 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 372, + 299, + 413, + 311 + ], + "score": 0.9, + "content": "\\| \\xi \\| \\infty", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 297, + 413, + 313 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 506, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 190, + 328 + ], + "score": 1.0, + "content": "Lemma 1. Suppose", + "type": "text" + }, + { + "bbox": [ + 190, + 315, + 259, + 327 + ], + "score": 0.92, + "content": "F : \\mathcal { M } ( X ) \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 314, + 393, + 328 + ], + "score": 1.0, + "content": "has a unique minimizer at 0 with", + "type": "text" + }, + { + "bbox": [ + 393, + 315, + 433, + 327 + ], + "score": 0.91, + "content": "F ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 314, + 505, + 328 + ], + "score": 1.0, + "content": ", and is uniformly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 326, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 315, + 338 + ], + "score": 1.0, + "content": "continuous on bounded sets and coercive. Suppose", + "type": "text" + }, + { + "bbox": [ + 316, + 326, + 388, + 338 + ], + "score": 0.92, + "content": "G : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 326, + 506, + 338 + ], + "score": 1.0, + "content": "has a unique minimizer at 0", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 127, + 349 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 337, + 168, + 349 + ], + "score": 0.92, + "content": "G ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 336, + 191, + 349 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 191, + 337, + 246, + 348 + ], + "score": 0.92, + "content": "G ( \\cdot ) \\geq c \\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 336, + 285, + 349 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 286, + 338, + 311, + 348 + ], + "score": 0.88, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 336, + 419, + 349 + ], + "score": 1.0, + "content": ". Then the inf-convolution", + "type": "text" + }, + { + "bbox": [ + 419, + 337, + 449, + 348 + ], + "score": 0.9, + "content": "F \\oplus G", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "has a unique", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 189, + 361 + ], + "score": 1.0, + "content": "minimizer at 0 with", + "type": "text" + }, + { + "bbox": [ + 189, + 348, + 258, + 360 + ], + "score": 0.9, + "content": "( F \\oplus G ) ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 347, + 506, + 361 + ], + "score": 1.0, + "content": ", and is uniformly continuous on bounded sets, coercive, and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 359, + 156, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 156, + 369 + ], + "score": 1.0, + "content": "real-valued.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 314, + 506, + 369 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 387, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 315, + 401 + ], + "score": 1.0, + "content": "Proof. From Theorem 2.3 of Stromberg (1996), ¨ inf", + "type": "text" + }, + { + "bbox": [ + 315, + 388, + 363, + 398 + ], + "score": 0.91, + "content": "F \\oplus G = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 387, + 383, + 401 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 384, + 388, + 464, + 399 + ], + "score": 0.9, + "content": "0 \\in \\arg \\operatorname* { m i n } F \\oplus G", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 387, + 505, + 401 + ], + "score": 1.0, + "content": ". To show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 245, + 412 + ], + "score": 1.0, + "content": "that 0 is the unique minimizer, let", + "type": "text" + }, + { + "bbox": [ + 245, + 399, + 327, + 410 + ], + "score": 0.84, + "content": "\\xi \\in \\arg \\operatorname* { m i n } F \\oplus G", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 398, + 505, + 412 + ], + "score": 1.0, + "content": ". From the definition of inf-convolution, for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 304, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 131, + 421 + ], + "score": 1.0, + "content": "every", + "type": "text" + }, + { + "bbox": [ + 131, + 412, + 138, + 420 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 410, + 198, + 421 + ], + "score": 1.0, + "content": "there exists an", + "type": "text" + }, + { + "bbox": [ + 199, + 410, + 250, + 422 + ], + "score": 0.93, + "content": "\\xi _ { n } \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 410, + 304, + 421 + ], + "score": 1.0, + "content": "that satisfies", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 387, + 505, + 422 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 428, + 399, + 447 + ], + "lines": [ + { + "bbox": [ + 211, + 428, + 399, + 447 + ], + "spans": [ + { + "bbox": [ + 211, + 428, + 399, + 447 + ], + "score": 0.89, + "content": "\\operatorname* { l i m } _ { n \\to \\infty } F ( \\xi _ { n } ) + G ( \\xi - \\xi _ { n } ) = ( F \\oplus G ) ( \\xi ) = 0 .", + "type": "interline_equation", + "image_path": "d08101671fd19c6515046e9f0c6b929b8cd77fe866b9578aac73b12c77b9c6ba.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 211, + 428, + 399, + 447 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 131, + 466 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 454, + 140, + 463 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 453, + 159, + 466 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 159, + 454, + 168, + 464 + ], + "score": 0.84, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 453, + 241, + 466 + ], + "score": 1.0, + "content": "are non-negative,", + "type": "text" + }, + { + "bbox": [ + 241, + 453, + 289, + 465 + ], + "score": 0.9, + "content": "F ( \\xi _ { n } ) \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 453, + 307, + 466 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 308, + 453, + 372, + 465 + ], + "score": 0.92, + "content": "G ( \\xi - \\xi _ { n } ) \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 453, + 473, + 466 + ], + "score": 1.0, + "content": ". By our assumption that", + "type": "text" + }, + { + "bbox": [ + 474, + 453, + 505, + 465 + ], + "score": 0.89, + "content": "G ( \\cdot ) \\geq", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 107, + 464, + 128, + 477 + ], + "score": 0.89, + "content": "c \\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 464, + 242, + 477 + ], + "score": 1.0, + "content": ", the latter limit implies that", + "type": "text" + }, + { + "bbox": [ + 243, + 465, + 302, + 477 + ], + "score": 0.91, + "content": "\\| \\xi - \\xi _ { n } \\| \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 464, + 384, + 477 + ], + "score": 1.0, + "content": ", which implies that", + "type": "text" + }, + { + "bbox": [ + 384, + 464, + 447, + 476 + ], + "score": 0.91, + "content": "F ( \\xi _ { n } ) \\to F ( \\xi )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "by continuity", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 117, + 488 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 477, + 126, + 485 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 474, + 322, + 488 + ], + "score": 1.0, + "content": ". Comparing our two expressions for the limit of", + "type": "text" + }, + { + "bbox": [ + 322, + 475, + 348, + 487 + ], + "score": 0.92, + "content": "F ( \\xi _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 474, + 402, + 488 + ], + "score": 1.0, + "content": ", we find that", + "type": "text" + }, + { + "bbox": [ + 402, + 476, + 442, + 487 + ], + "score": 0.89, + "content": "F ( \\xi ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 474, + 505, + 488 + ], + "score": 1.0, + "content": ", which implies", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 485, + 424, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 124, + 500 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 487, + 148, + 498 + ], + "score": 0.9, + "content": "\\xi = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 485, + 175, + 500 + ], + "score": 1.0, + "content": ", since", + "type": "text" + }, + { + "bbox": [ + 175, + 487, + 184, + 496 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 485, + 341, + 500 + ], + "score": 1.0, + "content": "has a unique minimizer at 0. Hence ar", + "type": "text" + }, + { + "bbox": [ + 342, + 487, + 420, + 498 + ], + "score": 0.85, + "content": "\\operatorname { m i n } F \\oplus G = \\{ 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 485, + 424, + 500 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 453, + 505, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 283, + 516 + ], + "score": 1.0, + "content": "From Theorem 2.10 of Stromberg (1996), ¨", + "type": "text" + }, + { + "bbox": [ + 283, + 504, + 314, + 514 + ], + "score": 0.89, + "content": "F \\oplus G", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "is uniformly continuous on bounded sets and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 210, + 527 + ], + "score": 1.0, + "content": "real-valued. To show that", + "type": "text" + }, + { + "bbox": [ + 211, + 515, + 240, + 525 + ], + "score": 0.9, + "content": "F \\oplus G", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 513, + 506, + 527 + ], + "score": 1.0, + "content": "is coercive, we show the equivalent condition its sublevel sets are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "bounded (see Proposition 11.11 of Bauschke & Combettes (2011)). That is, we show that for all", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 107, + 537, + 134, + 546 + ], + "score": 0.88, + "content": "a > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 536, + 231, + 549 + ], + "score": 1.0, + "content": ", there exists a constant", + "type": "text" + }, + { + "bbox": [ + 231, + 537, + 237, + 546 + ], + "score": 0.77, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 536, + 288, + 549 + ], + "score": 1.0, + "content": "such that if", + "type": "text" + }, + { + "bbox": [ + 288, + 536, + 359, + 548 + ], + "score": 0.92, + "content": "( F \\oplus G ) ( \\xi ) \\leq a", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 536, + 384, + 549 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 384, + 536, + 420, + 548 + ], + "score": 0.91, + "content": "\\| \\xi \\| \\le b", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 536, + 443, + 549 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 443, + 537, + 470, + 547 + ], + "score": 0.89, + "content": "a > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 536, + 506, + 549 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 154, + 559 + ], + "score": 0.9, + "content": "\\xi \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 546, + 207, + 560 + ], + "score": 1.0, + "content": "be such that", + "type": "text" + }, + { + "bbox": [ + 208, + 547, + 276, + 559 + ], + "score": 0.92, + "content": "( F \\oplus G ) ( \\xi ) \\leq a", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 546, + 297, + 560 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 298, + 548, + 322, + 558 + ], + "score": 0.86, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "; from the definition of inf-convolution, there", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 559, + 225, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 138, + 571 + ], + "score": 1.0, + "content": "exists a", + "type": "text" + }, + { + "bbox": [ + 139, + 559, + 185, + 572 + ], + "score": 0.94, + "content": "\\tilde { \\xi } \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 560, + 225, + 571 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 503, + 506, + 572 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 578, + 399, + 594 + ], + "lines": [ + { + "bbox": [ + 211, + 578, + 399, + 594 + ], + "spans": [ + { + "bbox": [ + 211, + 578, + 399, + 594 + ], + "score": 0.9, + "content": "F ( { \\tilde { \\xi } } ) + G ( \\xi - { \\tilde { \\xi } } ) \\leq ( F \\oplus G ) ( \\xi ) + \\epsilon \\leq a + \\epsilon .", + "type": "interline_equation", + "image_path": "424df517264799532d9f23baf1066f8faee11da85ead27b0973e1291f67898c3.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 211, + 578, + 399, + 594 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 304, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 303, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 142, + 613 + ], + "score": 1.0, + "content": "Because", + "type": "text" + }, + { + "bbox": [ + 142, + 601, + 151, + 610 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 600, + 169, + 613 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 601, + 178, + 610 + ], + "score": 0.84, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 600, + 303, + 613 + ], + "score": 1.0, + "content": "are non-negative, we have that", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 600, + 303, + 613 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 619, + 384, + 635 + ], + "lines": [ + { + "bbox": [ + 227, + 619, + 384, + 635 + ], + "spans": [ + { + "bbox": [ + 227, + 619, + 384, + 635 + ], + "score": 0.9, + "content": "\\begin{array} { r } { F ( \\tilde { \\xi } ) \\le a + \\epsilon , \\qquad G ( \\xi - \\tilde { \\xi } ) \\le a + \\epsilon . } \\end{array}", + "type": "interline_equation", + "image_path": "61a09dc600cf8db839d84f21488fc6883c7d3d7017f118bac61014000c1aba92.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 227, + 619, + 384, + 635 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 641, + 506, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 640, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 187, + 654 + ], + "score": 1.0, + "content": "Using the fact that", + "type": "text" + }, + { + "bbox": [ + 187, + 642, + 197, + 651 + ], + "score": 0.86, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 640, + 431, + 654 + ], + "score": 1.0, + "content": "is coercive and hence its sublevel sets are bounded, let", + "type": "text" + }, + { + "bbox": [ + 432, + 641, + 440, + 651 + ], + "score": 0.85, + "content": "b ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 640, + 506, + 654 + ], + "score": 1.0, + "content": "be such that if", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 651, + 488, + 666 + ], + "spans": [ + { + "bbox": [ + 107, + 652, + 164, + 664 + ], + "score": 0.91, + "content": "F ( \\nu ) \\leq a + \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 651, + 187, + 666 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 187, + 652, + 224, + 664 + ], + "score": 0.93, + "content": "\\| \\nu \\| \\leq b ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 651, + 430, + 666 + ], + "score": 1.0, + "content": ". By the triangle inequality and our assumption that", + "type": "text" + }, + { + "bbox": [ + 431, + 652, + 483, + 664 + ], + "score": 0.93, + "content": "G ( \\cdot ) \\geq c \\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 651, + 488, + 666 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 106, + 640, + 506, + 666 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 671, + 416, + 698 + ], + "lines": [ + { + "bbox": [ + 195, + 671, + 416, + 698 + ], + "spans": [ + { + "bbox": [ + 195, + 671, + 416, + 698 + ], + "score": 0.91, + "content": "\\| \\xi \\| \\leq \\| \\tilde { \\xi } \\| + \\| \\xi - \\tilde { \\xi } \\| \\leq b ^ { \\prime } + \\frac { G ( \\xi - \\tilde { \\xi } ) } { c } \\leq b ^ { \\prime } + \\frac { a + \\epsilon } { c } .", + "type": "interline_equation", + "image_path": "3352b7cd950b0647c77933f123e0effa951bd2e89f2990f8aaee690faa509c2e.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 195, + 671, + 416, + 698 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 703, + 414, + 716 + ], + "lines": [ + { + "bbox": [ + 106, + 703, + 415, + 717 + ], + "spans": [ + { + "bbox": [ + 106, + 703, + 266, + 717 + ], + "score": 1.0, + "content": "Because this constant is independent of", + "type": "text" + }, + { + "bbox": [ + 266, + 704, + 272, + 715 + ], + "score": 0.83, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 703, + 337, + 717 + ], + "score": 1.0, + "content": ", this shows that", + "type": "text" + }, + { + "bbox": [ + 338, + 704, + 366, + 714 + ], + "score": 0.87, + "content": "F \\oplus G", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 703, + 415, + 717 + ], + "score": 1.0, + "content": "is coercive.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 106, + 703, + 415, + 717 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 250, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 251, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 251, + 96 + ], + "score": 1.0, + "content": "F PROOFS FOR SECTION 4", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 102, + 106, + 465, + 119 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 467, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 170, + 120 + ], + "score": 1.0, + "content": "Proposition 3.", + "type": "text" + }, + { + "bbox": [ + 171, + 107, + 191, + 118 + ], + "score": 0.62, + "content": "( D l )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 105, + 269, + 120 + ], + "score": 1.0, + "content": "holds if and only if", + "type": "text" + }, + { + "bbox": [ + 270, + 107, + 277, + 117 + ], + "score": 0.66, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 105, + 287, + 120 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 287, + 108, + 295, + 117 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 105, + 467, + 120 + ], + "score": 1.0, + "content": "-Lipschitz w.r.t. the Wasserstein-1 distance.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 298, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 299, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 299, + 146 + ], + "score": 1.0, + "content": "Proof. First, we check the forward implication:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 149, + 428, + 171 + ], + "lines": [ + { + "bbox": [ + 182, + 149, + 428, + 171 + ], + "spans": [ + { + "bbox": [ + 182, + 149, + 428, + 171 + ], + "score": 0.9, + "content": "| J ( \\mu ) - J ( \\nu ) | \\leq \\alpha W _ { 1 } ( \\mu , \\nu ) \\quad \\Rightarrow \\quad \\operatorname* { s u p } _ { x \\sim \\mu } \\| \\nabla _ { x } \\Phi _ { \\mu } ( x ) \\| _ { 2 } \\leq \\alpha .", + "type": "interline_equation", + "image_path": "4ba48a0a9fa79d380ee7d7e5ef932b293a8c4d957627cf058da93ced8d0db7a9.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 182, + 149, + 428, + 171 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 329, + 189 + ], + "lines": [ + { + "bbox": [ + 105, + 175, + 330, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 211, + 191 + ], + "score": 1.0, + "content": "From the duality between", + "type": "text" + }, + { + "bbox": [ + 211, + 178, + 227, + 187 + ], + "score": 0.86, + "content": "L ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 175, + 249, + 191 + ], + "score": 1.0, + "content": "- and", + "type": "text" + }, + { + "bbox": [ + 249, + 177, + 261, + 187 + ], + "score": 0.89, + "content": "L ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 175, + 330, + 191 + ], + "score": 1.0, + "content": "-norms, we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 194, + 467, + 225 + ], + "lines": [ + { + "bbox": [ + 126, + 194, + 467, + 225 + ], + "spans": [ + { + "bbox": [ + 126, + 194, + 467, + 225 + ], + "score": 0.93, + "content": "\\operatorname* { s u p } _ { x \\sim \\mu } \\| \\nabla _ { x } \\Phi _ { \\mu } ( x ) \\| _ { 2 } = \\| \\nabla _ { x } \\Phi _ { \\mu } ( \\cdot ) \\| _ { L ^ { \\infty } ( \\mu ; \\mathbb { R } ^ { d } ) } = \\operatorname* { s u p } _ { \\| v \\| _ { L ^ { 1 } ( \\mu ; \\mathbb { R } ^ { d } ) } = 1 } \\int \\nabla \\Phi _ { \\mu } ( x ) \\cdot v ( x ) d \\mu ( x ) .", + "type": "interline_equation", + "image_path": "bce10698f56c6d7b8d518b1aff51ab742c20e8422e5d0db052c0ca6b1ed60a78.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 126, + 194, + 467, + 204.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 126, + 204.33333333333334, + 467, + 214.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 126, + 214.66666666666669, + 467, + 225.00000000000003 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 104, + 230, + 507, + 246 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 215, + 246 + ], + "score": 1.0, + "content": "Below, we will abbreviate", + "type": "text" + }, + { + "bbox": [ + 215, + 231, + 262, + 245 + ], + "score": 0.92, + "content": "\\left\\| \\cdot \\right\\| _ { L ^ { 1 } \\left( \\mu ; \\mathbb { R } ^ { d } \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 230, + 276, + 246 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 276, + 231, + 299, + 244 + ], + "score": 0.9, + "content": "\\| \\cdot \\| _ { L ^ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 230, + 507, + 246 + ], + "score": 1.0, + "content": ". We utilize the fact that the optimal discriminator", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 242, + 504, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 120, + 256 + ], + "score": 0.9, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 242, + 234, + 258 + ], + "score": 1.0, + "content": "is the functional gradient of", + "type": "text" + }, + { + "bbox": [ + 235, + 245, + 242, + 254 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 242, + 389, + 258 + ], + "score": 1.0, + "content": "in the sense of Definition 3. For any", + "type": "text" + }, + { + "bbox": [ + 390, + 244, + 413, + 255 + ], + "score": 0.89, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 242, + 429, + 258 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 430, + 243, + 504, + 256 + ], + "score": 0.91, + "content": "\\mu _ { t } = ( \\operatorname { i d } + t v ) _ { \\# } \\mu", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 165, + 268 + ], + "score": 1.0, + "content": "be the law of", + "type": "text" + }, + { + "bbox": [ + 165, + 255, + 208, + 266 + ], + "score": 0.91, + "content": "x + t v ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 254, + 232, + 268 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 233, + 256, + 263, + 266 + ], + "score": 0.88, + "content": "x \\sim \\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 254, + 298, + 268 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 299, + 256, + 309, + 266 + ], + "score": 0.83, + "content": "\\mu _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 254, + 399, + 268 + ], + "score": 1.0, + "content": "converges weakly to", + "type": "text" + }, + { + "bbox": [ + 399, + 257, + 435, + 266 + ], + "score": 0.9, + "content": "\\mu ~ = ~ \\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 254, + 449, + 268 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 450, + 256, + 480, + 265 + ], + "score": 0.87, + "content": "t 0", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "since", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 264, + 415, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 223, + 279 + ], + "score": 0.9, + "content": "W _ { 1 } ( \\mu _ { t } , \\mu ) \\leq t \\| v \\| _ { L ^ { 1 } ( \\mu ) } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 264, + 338, + 280 + ], + "score": 1.0, + "content": ". Applying Proposition 17 to", + "type": "text" + }, + { + "bbox": [ + 339, + 266, + 346, + 276 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 264, + 364, + 280 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 364, + 267, + 375, + 277 + ], + "score": 0.86, + "content": "\\mu _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 264, + 415, + 280 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 284, + 455, + 313 + ], + "lines": [ + { + "bbox": [ + 157, + 284, + 455, + 313 + ], + "spans": [ + { + "bbox": [ + 157, + 284, + 455, + 313 + ], + "score": 0.92, + "content": "\\left. \\frac { d } { d t } J ( \\mu _ { t } ) \\right| _ { t = 0 } = \\left. \\frac { d } { d t } \\int \\Phi _ { \\mu } ( x + t v ( x ) ) d \\mu ( x ) \\right| _ { t = 0 } = \\int \\nabla \\Phi _ { \\mu } ( x ) \\cdot v ( x ) d \\mu .", + "type": "interline_equation", + "image_path": "df31be8534dbf0242987d9195988d6899635d56b6fd2b5cf52cb3ffa52af3d23.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 157, + 284, + 455, + 313 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 371, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 372, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 372, + 331 + ], + "score": 1.0, + "content": "In particular, the right-hand side of (26) is bounded from above as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 335, + 382, + 507 + ], + "lines": [ + { + "bbox": [ + 229, + 335, + 382, + 507 + ], + "spans": [ + { + "bbox": [ + 229, + 335, + 382, + 507 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } , \\omega _ { \\xi } = 1 } { \\operatorname* { s u p } } \\int \\nabla \\Phi _ { p } ( x ) \\cdot v ( x ) d \\mu ( x ) } \\\\ & { \\quad = \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | \\neq 0 } { \\operatorname* { s u p } } \\frac { d } { d t } J ( \\mu _ { t } ) \\biggr | _ { t = 0 } } \\\\ & { \\quad = \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | , \\xi _ { 0 } = 1 } { \\operatorname* { s u p } } \\frac { 1 } { \\mathrm { i n s p } } \\frac { J ( \\mu _ { t } ) - J ( \\mu ) } { t } } \\\\ & { \\quad \\le \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | , \\xi _ { 0 } = 1 } { \\operatorname* { s u p } } \\frac { \\mathrm { i n s K } ( \\mu _ { t } , \\mu ) } { t } } \\\\ & { \\quad \\le \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | , \\xi _ { 0 } = 1 } { \\operatorname* { s u p } } \\frac { \\mathrm { i n s K } ( \\mu _ { t } , \\mu ) } { t } } \\\\ & { \\quad \\le \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | \\neq 0 } { \\operatorname* { s u p } } \\frac { \\mathrm { i n s K } ( \\mu _ { t } , \\mu _ { t } ) } { t } } \\\\ & { \\quad = \\alpha , } \\end{array}", + "type": "interline_equation", + "image_path": "33dc821b69964865de637171b247649c9be73578bebce2c52983f90c6b5f3fef.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 229, + 335, + 382, + 421.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 229, + 421.0, + 382, + 507.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 512, + 280, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 512, + 281, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 281, + 524 + ], + "score": 1.0, + "content": "which proves the first half of the statement.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 506, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 446, + 542 + ], + "score": 1.0, + "content": "For the converse, we borrow some techniques from the optimal transport theory. Let", + "type": "text" + }, + { + "bbox": [ + 446, + 530, + 471, + 540 + ], + "score": 0.89, + "content": "p > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 528, + 506, + 542 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 540, + 504, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 145, + 552 + ], + "score": 0.92, + "content": "W _ { p } ( \\mu , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 540, + 243, + 553 + ], + "score": 1.0, + "content": "denote the Wasserstein-", + "type": "text" + }, + { + "bbox": [ + 244, + 542, + 250, + 552 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 540, + 324, + 553 + ], + "score": 1.0, + "content": "distance between", + "type": "text" + }, + { + "bbox": [ + 324, + 542, + 331, + 552 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 540, + 351, + 553 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 351, + 542, + 357, + 550 + ], + "score": 0.74, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 540, + 419, + 553 + ], + "score": 1.0, + "content": ". Suppose that", + "type": "text" + }, + { + "bbox": [ + 420, + 540, + 504, + 552 + ], + "score": 0.91, + "content": "\\mu _ { t } : [ 0 , 1 ] \\to { \\mathcal { P } } ( X )", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 128, + 564 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 129, + 551, + 158, + 564 + ], + "score": 0.92, + "content": "{ \\mathcal { P } } _ { p } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 551, + 334, + 564 + ], + "score": 1.0, + "content": "-absolutely continuous curve, that is, every", + "type": "text" + }, + { + "bbox": [ + 335, + 553, + 345, + 563 + ], + "score": 0.85, + "content": "\\mu _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 551, + 395, + 564 + ], + "score": 1.0, + "content": "has a finite", + "type": "text" + }, + { + "bbox": [ + 395, + 553, + 401, + 563 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "-moment and there exists", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 563, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 107, + 564, + 166, + 577 + ], + "score": 0.91, + "content": "v \\in L ^ { 1 } ( [ 0 , 1 ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 563, + 207, + 578 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 208, + 563, + 311, + 578 + ], + "score": 0.93, + "content": "\\begin{array} { r } { W _ { p } ( \\mu _ { s } , \\mu _ { t } ) \\le \\int _ { s } ^ { t } v ( r ) d r } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 563, + 370, + 578 + ], + "score": 1.0, + "content": "holds for any", + "type": "text" + }, + { + "bbox": [ + 370, + 565, + 436, + 576 + ], + "score": 0.91, + "content": "0 \\leq s \\leq t \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 563, + 506, + 578 + ], + "score": 1.0, + "content": ". Then, the limit", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 576, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 272, + 590 + ], + "score": 0.89, + "content": "\\begin{array} { r } { | \\mu ^ { \\prime } | _ { W _ { p } } ( t ) : = \\operatorname* { l i m } _ { h \\to 0 } | h | ^ { - 1 } W _ { p } ( \\mu _ { t + h } , \\mu _ { t } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 576, + 353, + 591 + ], + "score": 1.0, + "content": "exists for almost all", + "type": "text" + }, + { + "bbox": [ + 353, + 578, + 390, + 589 + ], + "score": 0.93, + "content": "t \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 576, + 506, + 591 + ], + "score": 1.0, + "content": "(see Ambrosio et al. (2008),", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 586, + 371, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 196, + 603 + ], + "score": 1.0, + "content": "Theorem 1.1.2). Such", + "type": "text" + }, + { + "bbox": [ + 197, + 589, + 223, + 601 + ], + "score": 0.9, + "content": "| \\mu ^ { \\prime } | _ { W _ { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 586, + 355, + 603 + ], + "score": 1.0, + "content": "is called the metric derivative of", + "type": "text" + }, + { + "bbox": [ + 356, + 591, + 366, + 600 + ], + "score": 0.85, + "content": "\\mu _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 586, + 371, + 603 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 606, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 327, + 620 + ], + "score": 1.0, + "content": "Lemma 2 (Ambrosio et al. (2008), Theorem 8.3.1). Let", + "type": "text" + }, + { + "bbox": [ + 327, + 607, + 406, + 619 + ], + "score": 0.88, + "content": "\\mu _ { t } : [ 0 , 1 ] \\to { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 606, + 430, + 620 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 430, + 607, + 459, + 619 + ], + "score": 0.93, + "content": "{ \\mathcal { P } } _ { p } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "-absolutely", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 212, + 632 + ], + "score": 1.0, + "content": "continuous curve, and let", + "type": "text" + }, + { + "bbox": [ + 212, + 618, + 293, + 632 + ], + "score": 0.93, + "content": "| \\mu ^ { \\prime } | _ { W _ { p } } \\in L ^ { 1 } ( [ 0 , 1 ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 618, + 469, + 632 + ], + "score": 1.0, + "content": "be its metric derivative. 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First, we check the forward implication:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 131, + 299, + 146 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 149, + 428, + 171 + ], + "lines": [ + { + "bbox": [ + 182, + 149, + 428, + 171 + ], + "spans": [ + { + "bbox": [ + 182, + 149, + 428, + 171 + ], + "score": 0.9, + "content": "| J ( \\mu ) - J ( \\nu ) | \\leq \\alpha W _ { 1 } ( \\mu , \\nu ) \\quad \\Rightarrow \\quad \\operatorname* { s u p } _ { x \\sim \\mu } \\| \\nabla _ { x } \\Phi _ { \\mu } ( x ) \\| _ { 2 } \\leq \\alpha .", + "type": "interline_equation", + "image_path": "4ba48a0a9fa79d380ee7d7e5ef932b293a8c4d957627cf058da93ced8d0db7a9.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 182, + 149, + 428, + 171 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 329, + 189 + ], + "lines": [ + { + "bbox": [ + 105, + 175, + 330, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 211, + 191 + ], + "score": 1.0, + "content": "From the duality between", + "type": "text" + }, + { + "bbox": [ + 211, + 178, + 227, + 187 + ], + "score": 0.86, + "content": "L ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 175, + 249, + 191 + ], + "score": 1.0, + "content": "- and", + "type": "text" + }, + { + "bbox": [ + 249, + 177, + 261, + 187 + ], + "score": 0.89, + "content": "L ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 175, + 330, + 191 + ], + "score": 1.0, + "content": "-norms, we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 175, + 330, + 191 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 194, + 467, + 225 + ], + "lines": [ + { + "bbox": [ + 126, + 194, + 467, + 225 + ], + "spans": [ + { + "bbox": [ + 126, + 194, + 467, + 225 + ], + "score": 0.93, + "content": "\\operatorname* { s u p } _ { x \\sim \\mu } \\| \\nabla _ { x } \\Phi _ { \\mu } ( x ) \\| _ { 2 } = \\| \\nabla _ { x } \\Phi _ { \\mu } ( \\cdot ) \\| _ { L ^ { \\infty } ( \\mu ; \\mathbb { R } ^ { d } ) } = \\operatorname* { s u p } _ { \\| v \\| _ { L ^ { 1 } ( \\mu ; \\mathbb { R } ^ { d } ) } = 1 } \\int \\nabla \\Phi _ { \\mu } ( x ) \\cdot v ( x ) d \\mu ( x ) .", + "type": "interline_equation", + "image_path": "bce10698f56c6d7b8d518b1aff51ab742c20e8422e5d0db052c0ca6b1ed60a78.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 126, + 194, + 467, + 204.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 126, + 204.33333333333334, + 467, + 214.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 126, + 214.66666666666669, + 467, + 225.00000000000003 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 104, + 230, + 507, + 246 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 215, + 246 + ], + "score": 1.0, + "content": "Below, we will abbreviate", + "type": "text" + }, + { + "bbox": [ + 215, + 231, + 262, + 245 + ], + "score": 0.92, + "content": "\\left\\| \\cdot \\right\\| _ { L ^ { 1 } \\left( \\mu ; \\mathbb { R } ^ { d } \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 230, + 276, + 246 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 276, + 231, + 299, + 244 + ], + "score": 0.9, + "content": "\\| \\cdot \\| _ { L ^ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 230, + 507, + 246 + ], + "score": 1.0, + "content": ". We utilize the fact that the optimal discriminator", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 242, + 504, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 120, + 256 + ], + "score": 0.9, + "content": "\\Phi _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 242, + 234, + 258 + ], + "score": 1.0, + "content": "is the functional gradient of", + "type": "text" + }, + { + "bbox": [ + 235, + 245, + 242, + 254 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 242, + 389, + 258 + ], + "score": 1.0, + "content": "in the sense of Definition 3. For any", + "type": "text" + }, + { + "bbox": [ + 390, + 244, + 413, + 255 + ], + "score": 0.89, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 242, + 429, + 258 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 430, + 243, + 504, + 256 + ], + "score": 0.91, + "content": "\\mu _ { t } = ( \\operatorname { i d } + t v ) _ { \\# } \\mu", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 165, + 268 + ], + "score": 1.0, + "content": "be the law of", + "type": "text" + }, + { + "bbox": [ + 165, + 255, + 208, + 266 + ], + "score": 0.91, + "content": "x + t v ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 254, + 232, + 268 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 233, + 256, + 263, + 266 + ], + "score": 0.88, + "content": "x \\sim \\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 254, + 298, + 268 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 299, + 256, + 309, + 266 + ], + "score": 0.83, + "content": "\\mu _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 254, + 399, + 268 + ], + "score": 1.0, + "content": "converges weakly to", + "type": "text" + }, + { + "bbox": [ + 399, + 257, + 435, + 266 + ], + "score": 0.9, + "content": "\\mu ~ = ~ \\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 254, + 449, + 268 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 450, + 256, + 480, + 265 + ], + "score": 0.87, + "content": "t 0", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "since", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 264, + 415, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 223, + 279 + ], + "score": 0.9, + "content": "W _ { 1 } ( \\mu _ { t } , \\mu ) \\leq t \\| v \\| _ { L ^ { 1 } ( \\mu ) } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 264, + 338, + 280 + ], + "score": 1.0, + "content": ". Applying Proposition 17 to", + "type": "text" + }, + { + "bbox": [ + 339, + 266, + 346, + 276 + ], + "score": 0.83, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 264, + 364, + 280 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 364, + 267, + 375, + 277 + ], + "score": 0.86, + "content": "\\mu _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 264, + 415, + 280 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 230, + 507, + 280 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 284, + 455, + 313 + ], + "lines": [ + { + "bbox": [ + 157, + 284, + 455, + 313 + ], + "spans": [ + { + "bbox": [ + 157, + 284, + 455, + 313 + ], + "score": 0.92, + "content": "\\left. \\frac { d } { d t } J ( \\mu _ { t } ) \\right| _ { t = 0 } = \\left. \\frac { d } { d t } \\int \\Phi _ { \\mu } ( x + t v ( x ) ) d \\mu ( x ) \\right| _ { t = 0 } = \\int \\nabla \\Phi _ { \\mu } ( x ) \\cdot v ( x ) d \\mu .", + "type": "interline_equation", + "image_path": "df31be8534dbf0242987d9195988d6899635d56b6fd2b5cf52cb3ffa52af3d23.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 157, + 284, + 455, + 313 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 371, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 372, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 372, + 331 + ], + "score": 1.0, + "content": "In particular, the right-hand side of (26) is bounded from above as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 316, + 372, + 331 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 335, + 382, + 507 + ], + "lines": [ + { + "bbox": [ + 229, + 335, + 382, + 507 + ], + "spans": [ + { + "bbox": [ + 229, + 335, + 382, + 507 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } , \\omega _ { \\xi } = 1 } { \\operatorname* { s u p } } \\int \\nabla \\Phi _ { p } ( x ) \\cdot v ( x ) d \\mu ( x ) } \\\\ & { \\quad = \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | \\neq 0 } { \\operatorname* { s u p } } \\frac { d } { d t } J ( \\mu _ { t } ) \\biggr | _ { t = 0 } } \\\\ & { \\quad = \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | , \\xi _ { 0 } = 1 } { \\operatorname* { s u p } } \\frac { 1 } { \\mathrm { i n s p } } \\frac { J ( \\mu _ { t } ) - J ( \\mu ) } { t } } \\\\ & { \\quad \\le \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | , \\xi _ { 0 } = 1 } { \\operatorname* { s u p } } \\frac { \\mathrm { i n s K } ( \\mu _ { t } , \\mu ) } { t } } \\\\ & { \\quad \\le \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | , \\xi _ { 0 } = 1 } { \\operatorname* { s u p } } \\frac { \\mathrm { i n s K } ( \\mu _ { t } , \\mu ) } { t } } \\\\ & { \\quad \\le \\underset { | | \\tau | _ { \\xi } | , \\xi _ { 1 } | \\neq 0 } { \\operatorname* { s u p } } \\frac { \\mathrm { i n s K } ( \\mu _ { t } , \\mu _ { t } ) } { t } } \\\\ & { \\quad = \\alpha , } \\end{array}", + "type": "interline_equation", + "image_path": "33dc821b69964865de637171b247649c9be73578bebce2c52983f90c6b5f3fef.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 229, + 335, + 382, + 421.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 229, + 421.0, + 382, + 507.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 512, + 280, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 512, + 281, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 281, + 524 + ], + "score": 1.0, + "content": "which proves the first half of the statement.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 512, + 281, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 506, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 446, + 542 + ], + "score": 1.0, + "content": "For the converse, we borrow some techniques from the optimal transport theory. Let", + "type": "text" + }, + { + "bbox": [ + 446, + 530, + 471, + 540 + ], + "score": 0.89, + "content": "p > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 528, + 506, + 542 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 540, + 504, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 145, + 552 + ], + "score": 0.92, + "content": "W _ { p } ( \\mu , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 540, + 243, + 553 + ], + "score": 1.0, + "content": "denote the Wasserstein-", + "type": "text" + }, + { + "bbox": [ + 244, + 542, + 250, + 552 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 540, + 324, + 553 + ], + "score": 1.0, + "content": "distance between", + "type": "text" + }, + { + "bbox": [ + 324, + 542, + 331, + 552 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 540, + 351, + 553 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 351, + 542, + 357, + 550 + ], + "score": 0.74, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 540, + 419, + 553 + ], + "score": 1.0, + "content": ". Suppose that", + "type": "text" + }, + { + "bbox": [ + 420, + 540, + 504, + 552 + ], + "score": 0.91, + "content": "\\mu _ { t } : [ 0 , 1 ] \\to { \\mathcal { P } } ( X )", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 128, + 564 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 129, + 551, + 158, + 564 + ], + "score": 0.92, + "content": "{ \\mathcal { P } } _ { p } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 551, + 334, + 564 + ], + "score": 1.0, + "content": "-absolutely continuous curve, that is, every", + "type": "text" + }, + { + "bbox": [ + 335, + 553, + 345, + 563 + ], + "score": 0.85, + "content": "\\mu _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 551, + 395, + 564 + ], + "score": 1.0, + "content": "has a finite", + "type": "text" + }, + { + "bbox": [ + 395, + 553, + 401, + 563 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "-moment and there exists", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 563, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 107, + 564, + 166, + 577 + ], + "score": 0.91, + "content": "v \\in L ^ { 1 } ( [ 0 , 1 ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 563, + 207, + 578 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 208, + 563, + 311, + 578 + ], + "score": 0.93, + "content": "\\begin{array} { r } { W _ { p } ( \\mu _ { s } , \\mu _ { t } ) \\le \\int _ { s } ^ { t } v ( r ) d r } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 563, + 370, + 578 + ], + "score": 1.0, + "content": "holds for any", + "type": "text" + }, + { + "bbox": [ + 370, + 565, + 436, + 576 + ], + "score": 0.91, + "content": "0 \\leq s \\leq t \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 563, + 506, + 578 + ], + "score": 1.0, + "content": ". Then, the limit", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 576, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 272, + 590 + ], + "score": 0.89, + "content": "\\begin{array} { r } { | \\mu ^ { \\prime } | _ { W _ { p } } ( t ) : = \\operatorname* { l i m } _ { h \\to 0 } | h | ^ { - 1 } W _ { p } ( \\mu _ { t + h } , \\mu _ { t } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 576, + 353, + 591 + ], + "score": 1.0, + "content": "exists for almost all", + "type": "text" + }, + { + "bbox": [ + 353, + 578, + 390, + 589 + ], + "score": 0.93, + "content": "t \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 576, + 506, + 591 + ], + "score": 1.0, + "content": "(see Ambrosio et al. (2008),", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 586, + 371, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 196, + 603 + ], + "score": 1.0, + "content": "Theorem 1.1.2). Such", + "type": "text" + }, + { + "bbox": [ + 197, + 589, + 223, + 601 + ], + "score": 0.9, + "content": "| \\mu ^ { \\prime } | _ { W _ { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 586, + 355, + 603 + ], + "score": 1.0, + "content": "is called the metric derivative of", + "type": "text" + }, + { + "bbox": [ + 356, + 591, + 366, + 600 + ], + "score": 0.85, + "content": "\\mu _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 586, + 371, + 603 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 528, + 506, + 603 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 606, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 327, + 620 + ], + "score": 1.0, + "content": "Lemma 2 (Ambrosio et al. (2008), Theorem 8.3.1). Let", + "type": "text" + }, + { + "bbox": [ + 327, + 607, + 406, + 619 + ], + "score": 0.88, + "content": "\\mu _ { t } : [ 0 , 1 ] \\to { \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 606, + 430, + 620 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 430, + 607, + 459, + 619 + ], + "score": 0.93, + "content": "{ \\mathcal { P } } _ { p } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "-absolutely", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 212, + 632 + ], + "score": 1.0, + "content": "continuous curve, and let", + "type": "text" + }, + { + "bbox": [ + 212, + 618, + 293, + 632 + ], + "score": 0.93, + "content": "| \\mu ^ { \\prime } | _ { W _ { p } } \\in L ^ { 1 } ( [ 0 , 1 ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 618, + 469, + 632 + ], + "score": 1.0, + "content": "be its metric derivative. 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Then", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 101, + 416, + 249 + ], + "lines": [ + { + "bbox": [ + 195, + 101, + 416, + 249 + ], + "spans": [ + { + "bbox": [ + 195, + 101, + 416, + 249 + ], + "score": 0.96, + "content": "\\begin{array} { r l } { | J ( \\mu ) - J ( \\nu ) | = \\displaystyle \\left| \\int _ { 0 } ^ { 1 } \\frac { d } { d t } J ( \\mu _ { t } ) d t \\right| } & { } \\\\ & { = \\displaystyle \\left| \\int _ { 0 } ^ { 1 } \\int v _ { t } \\cdot \\nabla \\Phi _ { \\mu _ { t } } d \\mu _ { t } d t \\right| } \\\\ & { \\leq \\displaystyle \\int _ { 0 } ^ { 1 } \\| v _ { t } \\| _ { L ^ { p } ( \\mu _ { t } ) } \\| \\nabla \\Phi _ { \\mu _ { t } } \\| _ { L ^ { q } ( \\mu _ { t } ) } d t } \\\\ & { \\leq \\displaystyle \\int _ { 0 } ^ { 1 } \\| v _ { t } \\| _ { L ^ { p } ( \\mu _ { t } ) } d t \\cdot \\operatorname* { s u p } _ { \\tilde { \\mu } } \\| \\nabla \\Phi _ { \\tilde { \\mu } } \\| _ { L ^ { q } ( \\mu ) } } \\\\ & { \\leq \\displaystyle \\int _ { 0 } ^ { 1 } \\| \\mu _ { t } ^ { \\prime } \\| _ { W _ { p } } d t \\cdot \\operatorname* { s u p } \\| \\nabla \\Phi _ { \\tilde { \\mu } } \\| _ { L ^ { q } ( \\mu ) } , } \\end{array}", + "type": "interline_equation", + "image_path": "6618aeddfb7f6480ba989740001448a8536631caeddff4d3457316b3efd854be.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 101, + 416, + 115.8 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 195, + 115.8, + 416, + 130.6 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 195, + 130.6, + 416, + 145.4 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 195, + 145.4, + 416, + 160.20000000000002 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 195, + 160.20000000000002, + 416, + 175.00000000000003 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 195, + 175.00000000000003, + 416, + 189.80000000000004 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 195, + 189.80000000000004, + 416, + 204.60000000000005 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 195, + 204.60000000000005, + 416, + 219.40000000000006 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 195, + 219.40000000000006, + 416, + 234.20000000000007 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 195, + 234.20000000000007, + 416, + 249.00000000000009 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 504, + 277 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 133, + 269 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 255, + 150, + 267 + ], + "score": 0.9, + "content": "| \\mu _ { t } ^ { \\prime } |", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 253, + 319, + 269 + ], + "score": 1.0, + "content": "is the metric derivative. The existence of", + "type": "text" + }, + { + "bbox": [ + 319, + 257, + 329, + 266 + ], + "score": 0.84, + "content": "v _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 253, + 390, + 269 + ], + "score": 1.0, + "content": "and the bound", + "type": "text" + }, + { + "bbox": [ + 390, + 255, + 476, + 268 + ], + "score": 0.94, + "content": "\\| v _ { t } \\| _ { L ^ { p } ( \\mu _ { t } ) } \\leq | \\mu _ { t } ^ { \\prime } | _ { W _ { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 253, + 506, + 269 + ], + "score": 1.0, + "content": "is due", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 265, + 410, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 410, + 278 + ], + "score": 1.0, + "content": "to Lemma 2. Taking the infimum over all possible such curves, we find that", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 283, + 401, + 305 + ], + "lines": [ + { + "bbox": [ + 209, + 283, + 401, + 305 + ], + "spans": [ + { + "bbox": [ + 209, + 283, + 401, + 305 + ], + "score": 0.92, + "content": "\\left| J ( \\mu ) - J ( \\nu ) \\right| \\leq W _ { p } ( \\mu , \\nu ) \\cdot \\operatorname* { s u p } _ { \\tilde { \\mu } } \\left\\| \\nabla \\Phi _ { \\tilde { \\mu } } \\right\\| _ { L ^ { q } ( \\mu ) } .", + "type": "interline_equation", + "image_path": "6de71584022a380762c2f906989a49506d478a6b040ed96d41ab246dc7df10cc.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 209, + 283, + 401, + 305 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 104, + 316, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 316, + 247, + 333 + ], + "score": 1.0, + "content": "To conclude, we consider the limit", + "type": "text" + }, + { + "bbox": [ + 248, + 320, + 275, + 330 + ], + "score": 0.91, + "content": "p 1", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 316, + 356, + 333 + ], + "score": 1.0, + "content": ". Using the fact that", + "type": "text" + }, + { + "bbox": [ + 356, + 318, + 416, + 332 + ], + "score": 0.93, + "content": "q \\mapsto | | f | | _ { L ^ { q } ( \\mu ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 316, + 481, + 333 + ], + "score": 1.0, + "content": "is increasing in", + "type": "text" + }, + { + "bbox": [ + 481, + 321, + 487, + 330 + ], + "score": 0.78, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 316, + 505, + 333 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 329, + 128, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 128, + 342 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 341, + 430, + 363 + ], + "lines": [ + { + "bbox": [ + 179, + 341, + 430, + 363 + ], + "spans": [ + { + "bbox": [ + 179, + 341, + 430, + 363 + ], + "score": 0.89, + "content": "| J ( \\mu ) - J ( \\nu ) | \\leq W _ { p } ( \\mu , \\nu ) \\cdot \\operatorname* { s u p } _ { \\tilde { \\mu } } \\| \\nabla \\Phi _ { \\tilde { \\mu } } \\| _ { L ^ { \\infty } ( \\mu ) } \\leq \\alpha W _ { p } ( \\mu , \\nu ) .", + "type": "interline_equation", + "image_path": "ad0e60985024e2c0bcd7975a9fcee22cbe19f76256bc7f0b0f92a02a96c47c0a.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 179, + 341, + 430, + 363 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 367, + 129, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 131, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 131, + 380 + ], + "score": 1.0, + "content": "Then", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 377, + 400, + 455 + ], + "lines": [ + { + "bbox": [ + 210, + 377, + 400, + 455 + ], + "spans": [ + { + "bbox": [ + 210, + 377, + 400, + 455 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { | J ( \\mu ) - J ( \\nu ) | \\leq \\underset { p > 1 } { \\operatorname* { i n f } } \\alpha W _ { p } ( \\mu , \\nu ) } \\\\ & { \\qquad = \\alpha \\underset { \\pi \\in \\Pi ( \\mu , \\nu ) } { \\operatorname* { i n f } } \\underset { p > 1 } { \\operatorname* { i n f } } | | x - y | | _ { L ^ { p } ( \\pi ) } } \\\\ & { \\qquad = \\alpha \\underset { \\pi \\in \\Pi ( \\mu , \\nu ) } { \\operatorname* { i n f } } | | x - y | | _ { L ^ { 1 } ( \\pi ) } } \\\\ & { \\qquad = \\alpha W _ { 1 } ( \\mu , \\nu ) . } \\end{array}", + "type": "interline_equation", + "image_path": "e3d32a4d3a86f82525e879b996d34e8793fcb1c27810d81fded46f11347d3369.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 210, + 377, + 400, + 392.6 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 210, + 392.6, + 400, + 408.20000000000005 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 210, + 408.20000000000005, + 400, + 423.80000000000007 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 210, + 423.80000000000007, + 400, + 439.4000000000001 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 210, + 439.4000000000001, + 400, + 455.0000000000001 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 480, + 480, + 492 + ], + "lines": [ + { + "bbox": [ + 104, + 478, + 478, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 478, + 408, + 496 + ], + "score": 1.0, + "content": "Proposition 4. The minimax and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 408, + 481, + 427, + 492 + ], + "score": 0.56, + "content": "( D l )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 478, + 466, + 496 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 467, + 483, + 478, + 492 + ], + "score": 0.75, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 423, + 508 + ], + "lines": [ + { + "bbox": [ + 104, + 493, + 423, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 312, + 511 + ], + "score": 1.0, + "content": "Proposition 5. The Wasserstein GAN loss satisfies", + "type": "text" + }, + { + "bbox": [ + 312, + 497, + 332, + 507 + ], + "score": 0.7, + "content": "( D l )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 493, + 353, + 511 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 353, + 497, + 379, + 507 + ], + "score": 0.85, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 493, + 411, + 511 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 411, + 498, + 423, + 508 + ], + "score": 0.73, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 504, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 504, + 537 + ], + "score": 1.0, + "content": "Proof for Proposition 4 and Proposition 5. The counterexample for Proposition 4 is a slight simpli-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 536, + 504, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 320, + 548 + ], + "score": 1.0, + "content": "fication of Example 1 of Arjovsky et al. (2017). Let", + "type": "text" + }, + { + "bbox": [ + 320, + 536, + 330, + 547 + ], + "score": 0.88, + "content": "\\delta _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 536, + 475, + 548 + ], + "score": 1.0, + "content": "denote the distribution that outputs", + "type": "text" + }, + { + "bbox": [ + 476, + 536, + 504, + 546 + ], + "score": 0.9, + "content": "x \\in \\mathbb { R }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 547, + 244, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 244, + 559 + ], + "score": 1.0, + "content": "with probability 1. Then, we have", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 566, + 510, + 594 + ], + "lines": [ + { + "bbox": [ + 111, + 566, + 510, + 594 + ], + "spans": [ + { + "bbox": [ + 111, + 566, + 510, + 594 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathrm { { \\mathcal { I } } _ { J S } } ( \\delta _ { x } \\mid | \\delta _ { 0 } ) = \\left\\{ \\begin{array} { l l } { \\log 2 } & { x \\neq 0 } \\\\ { 0 } & { x = 0 , } \\end{array} \\right. \\quad D _ { \\mathrm { K L } } ( \\frac 1 2 \\delta _ { x } + \\frac 1 2 \\delta _ { 0 } \\| \\delta _ { 0 } ) = \\left\\{ \\begin{array} { l l } { \\infty } & { x \\neq 0 } \\\\ { 0 } & { x = 0 , } \\end{array} \\right. \\quad \\mathrm { a n d } \\quad W _ { 1 } ( \\delta _ { x } , \\delta _ { 0 } ) = | x | . } \\end{array}", + "type": "interline_equation", + "image_path": "c59a4163190a10d54bd5193b71621db4e4e12225fba1f5871f6077c4de423a44.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 111, + 566, + 510, + 594 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 506, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 165, + 615 + ], + "score": 1.0, + "content": "Therefore, for", + "type": "text" + }, + { + "bbox": [ + 165, + 601, + 249, + 613 + ], + "score": 0.92, + "content": "J ( \\mu ) = D _ { \\mathrm { J S } } ( \\mu \\vert \\vert \\delta _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 600, + 310, + 615 + ], + "score": 1.0, + "content": ", the inequality", + "type": "text" + }, + { + "bbox": [ + 310, + 601, + 440, + 613 + ], + "score": 0.93, + "content": "| J ( \\delta _ { x } ) - J ( \\delta _ { 0 } ) | \\leq \\alpha W _ { 1 } ( \\delta _ { x } , \\delta _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "cannot hold for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 179, + 625 + ], + "score": 1.0, + "content": "sufficiently small", + "type": "text" + }, + { + "bbox": [ + 180, + 613, + 207, + 623 + ], + "score": 0.86, + "content": "x \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 612, + 349, + 625 + ], + "score": 1.0, + "content": ", because the left-hand side equals", + "type": "text" + }, + { + "bbox": [ + 349, + 614, + 370, + 624 + ], + "score": 0.53, + "content": "\\log 2", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "while the right-hand side equals", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 623, + 441, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 125, + 635 + ], + "score": 0.9, + "content": "\\alpha | x |", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 623, + 146, + 636 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 146, + 623, + 266, + 636 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\dot { \\boldsymbol J } ( \\mu ) = \\dot { D _ { \\mathrm { K L } } } ( \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\delta _ { 0 } | | \\delta _ { 0 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 623, + 411, + 636 + ], + "score": 1.0, + "content": ", the inequality will not hold for any", + "type": "text" + }, + { + "bbox": [ + 411, + 623, + 436, + 635 + ], + "score": 0.91, + "content": "x \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 623, + 441, + 636 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 639, + 505, + 685 + ], + "lines": [ + { + "bbox": [ + 104, + 638, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 104, + 638, + 255, + 654 + ], + "score": 1.0, + "content": "Next, we verify Proposition 5. For", + "type": "text" + }, + { + "bbox": [ + 255, + 640, + 336, + 652 + ], + "score": 0.92, + "content": "J ( \\mu ) = W _ { 1 } ( \\mu , \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 638, + 378, + 654 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 379, + 640, + 505, + 652 + ], + "score": 0.91, + "content": "J ( \\mu ) - J ( \\nu ) = W _ { 1 } ( \\mu , \\mu _ { 0 } ) -", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 649, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 107, + 651, + 201, + 663 + ], + "score": 0.92, + "content": "W _ { 1 } ( \\nu , \\mu _ { 0 } ) \\leq \\dot { W } _ { 1 } ( \\bar { \\mu , \\nu } )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 649, + 506, + 665 + ], + "score": 1.0, + "content": "by the triangle inequality (see e.g., Section 6 in Villani (2009)). 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Taking the infimum over all possible such curves, we find that", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 253, + 506, + 278 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 283, + 401, + 305 + ], + "lines": [ + { + "bbox": [ + 209, + 283, + 401, + 305 + ], + "spans": [ + { + "bbox": [ + 209, + 283, + 401, + 305 + ], + "score": 0.92, + "content": "\\left| J ( \\mu ) - J ( \\nu ) \\right| \\leq W _ { p } ( \\mu , \\nu ) \\cdot \\operatorname* { s u p } _ { \\tilde { \\mu } } \\left\\| \\nabla \\Phi _ { \\tilde { \\mu } } \\right\\| _ { L ^ { q } ( \\mu ) } .", + "type": "interline_equation", + "image_path": "6de71584022a380762c2f906989a49506d478a6b040ed96d41ab246dc7df10cc.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 209, + 283, + 401, + 305 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 104, + 316, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 316, + 247, + 333 + ], + "score": 1.0, + "content": "To conclude, we consider the limit", + "type": "text" + }, + { + "bbox": [ + 248, + 320, + 275, + 330 + ], + "score": 0.91, + "content": "p 1", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 316, + 356, + 333 + ], + "score": 1.0, + "content": ". 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The minimax and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 408, + 481, + 427, + 492 + ], + "score": 0.56, + "content": "( D l )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 478, + 466, + 496 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 467, + 483, + 478, + 492 + ], + "score": 0.75, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 478, + 478, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 423, + 508 + ], + "lines": [ + { + "bbox": [ + 104, + 493, + 423, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 312, + 511 + ], + "score": 1.0, + "content": "Proposition 5. The Wasserstein GAN loss satisfies", + "type": "text" + }, + { + "bbox": [ + 312, + 497, + 332, + 507 + ], + "score": 0.7, + "content": "( D l )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 493, + 353, + 511 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 353, + 497, + 379, + 507 + ], + "score": 0.85, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 493, + 411, + 511 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 411, + 498, + 423, + 508 + ], + "score": 0.73, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 493, + 423, + 511 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 504, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 504, + 537 + ], + "score": 1.0, + "content": "Proof for Proposition 4 and Proposition 5. The counterexample for Proposition 4 is a slight simpli-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 536, + 504, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 320, + 548 + ], + "score": 1.0, + "content": "fication of Example 1 of Arjovsky et al. (2017). Let", + "type": "text" + }, + { + "bbox": [ + 320, + 536, + 330, + 547 + ], + "score": 0.88, + "content": "\\delta _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 536, + 475, + 548 + ], + "score": 1.0, + "content": "denote the distribution that outputs", + "type": "text" + }, + { + "bbox": [ + 476, + 536, + 504, + 546 + ], + "score": 0.9, + "content": "x \\in \\mathbb { R }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 547, + 244, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 244, + 559 + ], + "score": 1.0, + "content": "with probability 1. Then, we have", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 524, + 504, + 559 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 566, + 510, + 594 + ], + "lines": [ + { + "bbox": [ + 111, + 566, + 510, + 594 + ], + "spans": [ + { + "bbox": [ + 111, + 566, + 510, + 594 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathrm { { \\mathcal { I } } _ { J S } } ( \\delta _ { x } \\mid | \\delta _ { 0 } ) = \\left\\{ \\begin{array} { l l } { \\log 2 } & { x \\neq 0 } \\\\ { 0 } & { x = 0 , } \\end{array} \\right. \\quad D _ { \\mathrm { K L } } ( \\frac 1 2 \\delta _ { x } + \\frac 1 2 \\delta _ { 0 } \\| \\delta _ { 0 } ) = \\left\\{ \\begin{array} { l l } { \\infty } & { x \\neq 0 } \\\\ { 0 } & { x = 0 , } \\end{array} \\right. \\quad \\mathrm { a n d } \\quad W _ { 1 } ( \\delta _ { x } , \\delta _ { 0 } ) = | x | . } \\end{array}", + "type": "interline_equation", + "image_path": "c59a4163190a10d54bd5193b71621db4e4e12225fba1f5871f6077c4de423a44.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 111, + 566, + 510, + 594 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 506, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 165, + 615 + ], + "score": 1.0, + "content": "Therefore, for", + "type": "text" + }, + { + "bbox": [ + 165, + 601, + 249, + 613 + ], + "score": 0.92, + "content": "J ( \\mu ) = D _ { \\mathrm { J S } } ( \\mu \\vert \\vert \\delta _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 600, + 310, + 615 + ], + "score": 1.0, + "content": ", the inequality", + "type": "text" + }, + { + "bbox": [ + 310, + 601, + 440, + 613 + ], + "score": 0.93, + "content": "| J ( \\delta _ { x } ) - J ( \\delta _ { 0 } ) | \\leq \\alpha W _ { 1 } ( \\delta _ { x } , \\delta _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "cannot hold for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 179, + 625 + ], + "score": 1.0, + "content": "sufficiently small", + "type": "text" + }, + { + "bbox": [ + 180, + 613, + 207, + 623 + ], + "score": 0.86, + "content": "x \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 612, + 349, + 625 + ], + "score": 1.0, + "content": ", because the left-hand side equals", + "type": "text" + }, + { + "bbox": [ + 349, + 614, + 370, + 624 + ], + "score": 0.53, + "content": "\\log 2", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "while the right-hand side equals", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 623, + 441, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 125, + 635 + ], + "score": 0.9, + "content": "\\alpha | x |", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 623, + 146, + 636 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 146, + 623, + 266, + 636 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\dot { \\boldsymbol J } ( \\mu ) = \\dot { D _ { \\mathrm { K L } } } ( \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\delta _ { 0 } | | \\delta _ { 0 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 623, + 411, + 636 + ], + "score": 1.0, + "content": ", the inequality will not hold for any", + "type": "text" + }, + { + "bbox": [ + 411, + 623, + 436, + 635 + ], + "score": 0.91, + "content": "x \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 623, + 441, + 636 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 600, + 506, + 636 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 639, + 505, + 685 + ], + "lines": [ + { + "bbox": [ + 104, + 638, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 104, + 638, + 255, + 654 + ], + "score": 1.0, + "content": "Next, we verify Proposition 5. For", + "type": "text" + }, + { + "bbox": [ + 255, + 640, + 336, + 652 + ], + "score": 0.92, + "content": "J ( \\mu ) = W _ { 1 } ( \\mu , \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 638, + 378, + 654 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 379, + 640, + 505, + 652 + ], + "score": 0.91, + "content": "J ( \\mu ) - J ( \\nu ) = W _ { 1 } ( \\mu , \\mu _ { 0 } ) -", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 649, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 107, + 651, + 201, + 663 + ], + "score": 0.92, + "content": "W _ { 1 } ( \\nu , \\mu _ { 0 } ) \\leq \\dot { W } _ { 1 } ( \\bar { \\mu , \\nu } )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 649, + 506, + 665 + ], + "score": 1.0, + "content": "by the triangle inequality (see e.g., Section 6 in Villani (2009)). Reversing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 156, + 675 + ], + "score": 1.0, + "content": "the roles of", + "type": "text" + }, + { + "bbox": [ + 156, + 664, + 164, + 673 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 661, + 183, + 675 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 183, + 664, + 190, + 672 + ], + "score": 0.77, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 661, + 285, + 675 + ], + "score": 1.0, + "content": ", we can conclude that", + "type": "text" + }, + { + "bbox": [ + 285, + 662, + 399, + 674 + ], + "score": 0.93, + "content": "| J ( \\mu ) - J ( \\nu ) | \\leq W _ { 1 } ( \\mu , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 661, + 506, + 675 + ], + "score": 1.0, + "content": ". The result follows from", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 672, + 504, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 165, + 686 + ], + "score": 1.0, + "content": "Proposition 3.", + "type": "text" + }, + { + "bbox": [ + 495, + 674, + 504, + 683 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 638, + 506, + 686 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 695, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 695, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 106, + 696, + 261, + 709 + ], + "score": 1.0, + "content": "Proposition 6 (Pasch–Hausdorff). Let", + "type": "text" + }, + { + "bbox": [ + 261, + 696, + 329, + 709 + ], + "score": 0.91, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 696, + 427, + 709 + ], + "score": 1.0, + "content": "be a function, and define", + "type": "text" + }, + { + "bbox": [ + 428, + 695, + 478, + 708 + ], + "score": 0.91, + "content": "{ \\tilde { J } } : = J \\oplus R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 696, + 505, + 709 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 707, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 114, + 720 + ], + "score": 0.82, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 707, + 125, + 723 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 126, + 711, + 133, + 720 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 707, + 462, + 723 + ], + "score": 1.0, + "content": "-Lipschitz w.r.t. the distance induced by the KR norm, and hence the Wasserstein-", + "type": "text" + }, + { + "bbox": [ + 462, + 710, + 468, + 720 + ], + "score": 0.34, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 707, + 506, + 723 + ], + "score": 1.0, + "content": "distance", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 720, + 209, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 181, + 733 + ], + "score": 1.0, + "content": "when restricted to", + "type": "text" + }, + { + "bbox": [ + 181, + 721, + 206, + 732 + ], + "score": 0.92, + "content": "{ \\mathcal { P } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 720, + 209, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 695, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 293, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 293, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 293, + 97 + ], + "score": 1.0, + "content": "Proof. Observe that by the triangle inequality,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 99, + 437, + 179 + ], + "lines": [ + { + "bbox": [ + 173, + 99, + 437, + 179 + ], + "spans": [ + { + "bbox": [ + 173, + 99, + 437, + 179 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\tilde { J } ( \\mu ) - \\tilde { J } ( \\nu ) = \\underset { \\tilde { \\nu } } { \\operatorname* { s u p } } \\operatorname* { i n f } J ( \\tilde { \\mu } ) + \\alpha W _ { 1 } ( \\mu , \\tilde { \\mu } ) - J ( \\tilde { \\nu } ) - \\alpha W _ { 1 } ( \\nu , \\tilde { \\nu } ) } \\\\ & { \\qquad \\leq \\underset { \\tilde { \\nu } } { \\operatorname* { s u p } } \\alpha W _ { 1 } ( \\mu , \\tilde { \\nu } ) - \\alpha W _ { 1 } ( \\nu , \\tilde { \\nu } ) } \\\\ & { \\qquad \\leq \\underset { \\tilde { \\nu } } { \\operatorname* { s u p } } \\alpha W _ { 1 } ( \\mu , \\nu ) } \\\\ & { \\qquad = \\alpha W _ { 1 } ( \\mu , \\nu ) . } \\end{array}", + "type": "interline_equation", + "image_path": "ee6b6ea03e49120d023b0f7b980c52269b49b3033f95e25d2a92e4506b74909a.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 173, + 99, + 437, + 125.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 173, + 125.66666666666667, + 437, + 152.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 173, + 152.33333333333334, + 437, + 179.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 183, + 329, + 195 + ], + "lines": [ + { + "bbox": [ + 106, + 182, + 329, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 212, + 197 + ], + "score": 1.0, + "content": "Interchanging the roles of", + "type": "text" + }, + { + "bbox": [ + 212, + 185, + 219, + 195 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 182, + 237, + 197 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 237, + 186, + 244, + 193 + ], + "score": 0.77, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 182, + 329, + 197 + ], + "score": 1.0, + "content": "completes the proof.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 201, + 448, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 200, + 449, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 253, + 216 + ], + "score": 1.0, + "content": "Lemma 3. The convex conjugate of", + "type": "text" + }, + { + "bbox": [ + 254, + 202, + 312, + 214 + ], + "score": 0.92, + "content": "\\mu \\mapsto \\alpha \\vert \\vert \\mu \\vert \\vert _ { \\mathrm { K R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 200, + 357, + 216 + ], + "score": 1.0, + "content": "is given as", + "type": "text" + }, + { + "bbox": [ + 358, + 201, + 445, + 215 + ], + "score": 0.87, + "content": "\\varphi \\mapsto \\chi \\{ \\| \\varphi \\| _ { \\mathrm { L i p } } \\leq \\alpha \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 200, + 449, + 216 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 235, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 236, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 236, + 240 + ], + "score": 1.0, + "content": "Proof. The convex conjugate is", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 244, + 435, + 396 + ], + "lines": [ + { + "bbox": [ + 173, + 244, + 435, + 396 + ], + "spans": [ + { + "bbox": [ + 173, + 244, + 435, + 396 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\alpha \\lVert \\mu \\rVert _ { \\mathbf { k R } } \\right] = \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\alpha \\underset { \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } } { \\operatorname* { s u p } } \\int g d \\mu \\right] } & { } \\\\ & { = \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\underset { \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } } { \\operatorname* { s u p } } \\int g d \\mu \\right] } \\\\ & { = \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\underset { \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } } { \\operatorname* { i n f } } \\leq \\int f d \\mu - \\int g d \\mu \\right] } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } } { \\operatorname* { i n f } } \\frac { \\operatorname* { s u p } } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\int g d \\mu \\right] } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } } { \\operatorname* { i n f } } \\frac { \\operatorname* { s u p } } { \\operatorname* { s u p } } } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } \\oplus \\mathbf { k } } { \\operatorname* { i n f } } \\times \\mathcal { X } \\{ f - g \\} } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } \\oplus \\mathbf { x } } { \\operatorname* { s u p } } \\mathcal { X } \\{ f - g \\} } \\\\ & { = \\mathcal { X } \\{ \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } \\leq \\alpha \\} , } \\end{array}", + "type": "interline_equation", + "image_path": "adc43f1388ddac1e33920a12bacd8733153b91f1a2e43c62674348d6db14d4a8.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 173, + 244, + 435, + 294.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 173, + 294.6666666666667, + 435, + 345.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 173, + 345.33333333333337, + 435, + 396.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 415, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 415, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 415, + 414 + ], + "score": 1.0, + "content": "where Sion’s minimax theorem ensures that we can swap the inf and the sup.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 426, + 251, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 253, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 253, + 442 + ], + "score": 1.0, + "content": "G PROOFS FOR SECTION 5", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 469, + 465 + ], + "score": 1.0, + "content": "Proposition 7. The Wasserstein, minimax, and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 469, + 453, + 489, + 464 + ], + "score": 0.67, + "content": "( D 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 462, + 145, + 478 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 129, + 478 + ], + "score": 1.0, + "content": "some", + "type": "text" + }, + { + "bbox": [ + 129, + 466, + 141, + 475 + ], + "score": 0.79, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 462, + 145, + 478 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 400, + 501 + ], + "score": 1.0, + "content": "Proof. First, we prove the statement for the Wasserstein GAN. Consider", + "type": "text" + }, + { + "bbox": [ + 400, + 488, + 477, + 500 + ], + "score": 0.92, + "content": "J ( \\mu ) = W _ { 1 } ( \\mu , \\delta _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 487, + 505, + 501 + ], + "score": 1.0, + "content": "evalu-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 136, + 511 + ], + "score": 1.0, + "content": "ated at", + "type": "text" + }, + { + "bbox": [ + 136, + 498, + 205, + 512 + ], + "score": 0.93, + "content": "\\mu = \\textstyle { \\frac { 1 } { 2 } } \\delta _ { - 1 } + \\frac { 1 } { 2 } \\delta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 498, + 298, + 511 + ], + "score": 1.0, + "content": ", a mixture of spikes at", + "type": "text" + }, + { + "bbox": [ + 299, + 500, + 312, + 510 + ], + "score": 0.85, + "content": "\\pm 1", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 498, + 435, + 511 + ], + "score": 1.0, + "content": ". The optimal discriminator of", + "type": "text" + }, + { + "bbox": [ + 436, + 500, + 443, + 509 + ], + "score": 0.81, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "at this mixture", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 281, + 523 + ], + "score": 1.0, + "content": "is the Kantorovich potential that transfers", + "type": "text" + }, + { + "bbox": [ + 281, + 512, + 289, + 522 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 510, + 302, + 523 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 302, + 511, + 312, + 522 + ], + "score": 0.87, + "content": "\\delta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 510, + 356, + 523 + ], + "score": 1.0, + "content": ", which is", + "type": "text" + }, + { + "bbox": [ + 357, + 510, + 412, + 522 + ], + "score": 0.93, + "content": "\\Phi _ { \\mu } ( x ) = \\vert x \\vert", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 510, + 505, + 523 + ], + "score": 1.0, + "content": ". The gradient of this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 521, + 382, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 382, + 533 + ], + "score": 1.0, + "content": "Kantorovich potential is discontinuous at 0, and hence not Lipschitz.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 207, + 551 + ], + "score": 1.0, + "content": "For the minimax GAN", + "type": "text" + }, + { + "bbox": [ + 208, + 537, + 299, + 549 + ], + "score": 0.9, + "content": "J ( \\mu ) ~ = ~ { \\cal D } _ { \\mathrm { J S } } ( \\mu | | \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 536, + 320, + 551 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 320, + 540, + 328, + 549 + ], + "score": 0.79, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 536, + 349, + 551 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 349, + 540, + 361, + 549 + ], + "score": 0.83, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 536, + 495, + 551 + ], + "score": 1.0, + "content": "be probability distributions on", + "type": "text" + }, + { + "bbox": [ + 495, + 538, + 504, + 548 + ], + "score": 0.77, + "content": "\\mathbb { R }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 191, + 562 + ], + "score": 1.0, + "content": "whose densities are", + "type": "text" + }, + { + "bbox": [ + 192, + 550, + 283, + 561 + ], + "score": 0.9, + "content": "p ( x ) \\propto \\exp ( - | x | / 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 549, + 304, + 562 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 305, + 550, + 390, + 561 + ], + "score": 0.91, + "content": "p _ { 0 } ( x ) \\ \\propto \\ \\exp ( - | x | )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 549, + 506, + 562 + ], + "score": 1.0, + "content": ", respectively (i.e., Laplace", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 103, + 558, + 493, + 579 + ], + "spans": [ + { + "bbox": [ + 103, + 558, + 493, + 579 + ], + "score": 1.0, + "content": "distributions). Then, the optimal discriminator at µ is given as Φµ(x) = 12 log p(x)p(x)+p0(x)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 218, + 590 + ], + "score": 0.9, + "content": "- \\textstyle { \\frac { 1 } { 2 } } \\log ( 1 + 2 \\exp ( - | x | / 2 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 576, + 506, + 590 + ], + "score": 1.0, + "content": ". By elementary calculations, we can see that this function is not differ-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 149, + 600 + ], + "score": 1.0, + "content": "entiable at", + "type": "text" + }, + { + "bbox": [ + 149, + 589, + 174, + 598 + ], + "score": 0.86, + "content": "x = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 587, + 505, + 600 + ], + "score": 1.0, + "content": ", which implies that the minimax GAN does not satisfy (D2). For the non-saturating", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 597, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 333, + 613 + ], + "score": 1.0, + "content": "GAN, we obtain a similar result by swapping the role of", + "type": "text" + }, + { + "bbox": [ + 333, + 601, + 340, + 611 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 597, + 358, + 613 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 358, + 601, + 370, + 611 + ], + "score": 0.85, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 597, + 373, + 613 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 493, + 601, + 505, + 607 + ], + "score": 0.965, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 503, + 642 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 185, + 631 + ], + "score": 1.0, + "content": "Proposition 8. Let", + "type": "text" + }, + { + "bbox": [ + 186, + 617, + 253, + 630 + ], + "score": 0.91, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 615, + 505, + 631 + ], + "score": 1.0, + "content": "be a convex, proper, lower semicontinuous function, and define", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 628, + 371, + 643 + ], + "spans": [ + { + "bbox": [ + 107, + 629, + 160, + 642 + ], + "score": 0.92, + "content": "{ \\tilde { J } } : = J \\oplus R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 628, + 306, + 643 + ], + "score": 1.0, + "content": ". Then the optimal discriminator for", + "type": "text" + }, + { + "bbox": [ + 306, + 629, + 314, + 641 + ], + "score": 0.82, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 628, + 324, + 643 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 324, + 630, + 335, + 642 + ], + "score": 0.87, + "content": "\\beta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 628, + 371, + 643 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 339, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 340, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 136, + 671 + ], + "score": 1.0, + "content": "Proof.", + "type": "text" + }, + { + "bbox": [ + 136, + 655, + 144, + 667 + ], + "score": 0.83, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 654, + 340, + 671 + ], + "score": 1.0, + "content": "is convex, proper, lower semicontinuous. Hence", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 673, + 443, + 700 + ], + "lines": [ + { + "bbox": [ + 168, + 673, + 443, + 700 + ], + "spans": [ + { + "bbox": [ + 168, + 673, + 443, + 700 + ], + "score": 0.94, + "content": "\\tilde { J } ( \\mu ) = ( J \\oplus \\lambda | | \\cdot | | _ { \\mathcal { F } ^ { * } } ) ( \\mu ) = \\operatorname* { s u p } _ { \\varphi } \\int \\varphi d \\mu - J ^ { \\star } ( \\varphi ) - \\chi \\{ \\frac { 1 } { \\lambda } \\varphi \\in \\mathcal { F } \\} .", + "type": "interline_equation", + "image_path": "ddba0c050726f6bccaf74d400f7ff13bfc9a356ba8086af5187e4c36338269b1.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 168, + 673, + 443, + 700 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 706, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 705, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 705, + 235, + 723 + ], + "score": 1.0, + "content": "Then by the envelope theorem,", + "type": "text" + }, + { + "bbox": [ + 236, + 707, + 248, + 722 + ], + "score": 0.92, + "content": "\\frac { \\delta J } { \\delta \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 705, + 274, + 723 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 275, + 709, + 282, + 720 + ], + "score": 0.82, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 705, + 474, + 723 + ], + "score": 1.0, + "content": "that maximizes the right-hand side, and hence", + "type": "text" + }, + { + "bbox": [ + 474, + 707, + 505, + 722 + ], + "score": 0.89, + "content": "\\frac { 1 } { \\lambda } \\frac { \\delta J } { \\delta \\mu } \\in", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 719, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 116, + 730 + ], + "score": 0.79, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 719, + 120, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 720, + 505, + 731 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + } + ], + "page_idx": 21, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 183, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 495, + 185, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 495, + 185, + 505, + 195 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 399, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 496, + 401, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 496, + 401, + 505, + 411 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 293, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 293, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 293, + 97 + ], + "score": 1.0, + "content": "Proof. Observe that by the triangle inequality,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 80, + 293, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 99, + 437, + 179 + ], + "lines": [ + { + "bbox": [ + 173, + 99, + 437, + 179 + ], + "spans": [ + { + "bbox": [ + 173, + 99, + 437, + 179 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\tilde { J } ( \\mu ) - \\tilde { J } ( \\nu ) = \\underset { \\tilde { \\nu } } { \\operatorname* { s u p } } \\operatorname* { i n f } J ( \\tilde { \\mu } ) + \\alpha W _ { 1 } ( \\mu , \\tilde { \\mu } ) - J ( \\tilde { \\nu } ) - \\alpha W _ { 1 } ( \\nu , \\tilde { \\nu } ) } \\\\ & { \\qquad \\leq \\underset { \\tilde { \\nu } } { \\operatorname* { s u p } } \\alpha W _ { 1 } ( \\mu , \\tilde { \\nu } ) - \\alpha W _ { 1 } ( \\nu , \\tilde { \\nu } ) } \\\\ & { \\qquad \\leq \\underset { \\tilde { \\nu } } { \\operatorname* { s u p } } \\alpha W _ { 1 } ( \\mu , \\nu ) } \\\\ & { \\qquad = \\alpha W _ { 1 } ( \\mu , \\nu ) . } \\end{array}", + "type": "interline_equation", + "image_path": "ee6b6ea03e49120d023b0f7b980c52269b49b3033f95e25d2a92e4506b74909a.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 173, + 99, + 437, + 125.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 173, + 125.66666666666667, + 437, + 152.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 173, + 152.33333333333334, + 437, + 179.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 183, + 329, + 195 + ], + "lines": [ + { + "bbox": [ + 106, + 182, + 329, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 212, + 197 + ], + "score": 1.0, + "content": "Interchanging the roles of", + "type": "text" + }, + { + "bbox": [ + 212, + 185, + 219, + 195 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 182, + 237, + 197 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 237, + 186, + 244, + 193 + ], + "score": 0.77, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 182, + 329, + 197 + ], + "score": 1.0, + "content": "completes the proof.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 106, + 182, + 329, + 197 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 201, + 448, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 200, + 449, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 253, + 216 + ], + "score": 1.0, + "content": "Lemma 3. The convex conjugate of", + "type": "text" + }, + { + "bbox": [ + 254, + 202, + 312, + 214 + ], + "score": 0.92, + "content": "\\mu \\mapsto \\alpha \\vert \\vert \\mu \\vert \\vert _ { \\mathrm { K R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 200, + 357, + 216 + ], + "score": 1.0, + "content": "is given as", + "type": "text" + }, + { + "bbox": [ + 358, + 201, + 445, + 215 + ], + "score": 0.87, + "content": "\\varphi \\mapsto \\chi \\{ \\| \\varphi \\| _ { \\mathrm { L i p } } \\leq \\alpha \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 200, + 449, + 216 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 200, + 449, + 216 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 235, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 236, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 236, + 240 + ], + "score": 1.0, + "content": "Proof. The convex conjugate is", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 226, + 236, + 240 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 244, + 435, + 396 + ], + "lines": [ + { + "bbox": [ + 173, + 244, + 435, + 396 + ], + "spans": [ + { + "bbox": [ + 173, + 244, + 435, + 396 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\alpha \\lVert \\mu \\rVert _ { \\mathbf { k R } } \\right] = \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\alpha \\underset { \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } } { \\operatorname* { s u p } } \\int g d \\mu \\right] } & { } \\\\ & { = \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\underset { \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } } { \\operatorname* { s u p } } \\int g d \\mu \\right] } \\\\ & { = \\underset { \\mu } { \\operatorname* { s u p } } \\left[ \\underset { \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } } { \\operatorname* { i n f } } \\leq \\int f d \\mu - \\int g d \\mu \\right] } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } } { \\operatorname* { i n f } } \\frac { \\operatorname* { s u p } } { \\operatorname* { s u p } } \\left[ \\int f d \\mu - \\int g d \\mu \\right] } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } } { \\operatorname* { i n f } } \\frac { \\operatorname* { s u p } } { \\operatorname* { s u p } } } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } \\oplus \\mathbf { k } } { \\operatorname* { i n f } } \\times \\mathcal { X } \\{ f - g \\} } \\\\ & { = \\underset { \\lVert \\mathbf { k } \\oplus \\mathbf { k } \\oplus \\mathbf { x } } { \\operatorname* { s u p } } \\mathcal { X } \\{ f - g \\} } \\\\ & { = \\mathcal { X } \\{ \\lVert \\mathbf { k } \\rVert _ { \\mathbf { k } \\oplus \\mathbf { k } } \\leq \\alpha \\} , } \\end{array}", + "type": "interline_equation", + "image_path": "adc43f1388ddac1e33920a12bacd8733153b91f1a2e43c62674348d6db14d4a8.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 173, + 244, + 435, + 294.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 173, + 294.6666666666667, + 435, + 345.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 173, + 345.33333333333337, + 435, + 396.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 415, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 415, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 415, + 414 + ], + "score": 1.0, + "content": "where Sion’s minimax theorem ensures that we can swap the inf and the sup.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 397, + 415, + 414 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 426, + 251, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 253, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 253, + 442 + ], + "score": 1.0, + "content": "G PROOFS FOR SECTION 5", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 469, + 465 + ], + "score": 1.0, + "content": "Proposition 7. The Wasserstein, minimax, and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 469, + 453, + 489, + 464 + ], + "score": 0.67, + "content": "( D 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 462, + 145, + 478 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 129, + 478 + ], + "score": 1.0, + "content": "some", + "type": "text" + }, + { + "bbox": [ + 129, + 466, + 141, + 475 + ], + "score": 0.79, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 462, + 145, + 478 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 451, + 505, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 400, + 501 + ], + "score": 1.0, + "content": "Proof. First, we prove the statement for the Wasserstein GAN. Consider", + "type": "text" + }, + { + "bbox": [ + 400, + 488, + 477, + 500 + ], + "score": 0.92, + "content": "J ( \\mu ) = W _ { 1 } ( \\mu , \\delta _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 487, + 505, + 501 + ], + "score": 1.0, + "content": "evalu-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 136, + 511 + ], + "score": 1.0, + "content": "ated at", + "type": "text" + }, + { + "bbox": [ + 136, + 498, + 205, + 512 + ], + "score": 0.93, + "content": "\\mu = \\textstyle { \\frac { 1 } { 2 } } \\delta _ { - 1 } + \\frac { 1 } { 2 } \\delta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 498, + 298, + 511 + ], + "score": 1.0, + "content": ", a mixture of spikes at", + "type": "text" + }, + { + "bbox": [ + 299, + 500, + 312, + 510 + ], + "score": 0.85, + "content": "\\pm 1", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 498, + 435, + 511 + ], + "score": 1.0, + "content": ". The optimal discriminator of", + "type": "text" + }, + { + "bbox": [ + 436, + 500, + 443, + 509 + ], + "score": 0.81, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "at this mixture", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 281, + 523 + ], + "score": 1.0, + "content": "is the Kantorovich potential that transfers", + "type": "text" + }, + { + "bbox": [ + 281, + 512, + 289, + 522 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 510, + 302, + 523 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 302, + 511, + 312, + 522 + ], + "score": 0.87, + "content": "\\delta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 510, + 356, + 523 + ], + "score": 1.0, + "content": ", which is", + "type": "text" + }, + { + "bbox": [ + 357, + 510, + 412, + 522 + ], + "score": 0.93, + "content": "\\Phi _ { \\mu } ( x ) = \\vert x \\vert", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 510, + 505, + 523 + ], + "score": 1.0, + "content": ". The gradient of this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 521, + 382, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 382, + 533 + ], + "score": 1.0, + "content": "Kantorovich potential is discontinuous at 0, and hence not Lipschitz.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 487, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 207, + 551 + ], + "score": 1.0, + "content": "For the minimax GAN", + "type": "text" + }, + { + "bbox": [ + 208, + 537, + 299, + 549 + ], + "score": 0.9, + "content": "J ( \\mu ) ~ = ~ { \\cal D } _ { \\mathrm { J S } } ( \\mu | | \\mu _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 536, + 320, + 551 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 320, + 540, + 328, + 549 + ], + "score": 0.79, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 536, + 349, + 551 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 349, + 540, + 361, + 549 + ], + "score": 0.83, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 536, + 495, + 551 + ], + "score": 1.0, + "content": "be probability distributions on", + "type": "text" + }, + { + "bbox": [ + 495, + 538, + 504, + 548 + ], + "score": 0.77, + "content": "\\mathbb { R }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 191, + 562 + ], + "score": 1.0, + "content": "whose densities are", + "type": "text" + }, + { + "bbox": [ + 192, + 550, + 283, + 561 + ], + "score": 0.9, + "content": "p ( x ) \\propto \\exp ( - | x | / 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 549, + 304, + 562 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 305, + 550, + 390, + 561 + ], + "score": 0.91, + "content": "p _ { 0 } ( x ) \\ \\propto \\ \\exp ( - | x | )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 549, + 506, + 562 + ], + "score": 1.0, + "content": ", respectively (i.e., Laplace", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 103, + 558, + 493, + 579 + ], + "spans": [ + { + "bbox": [ + 103, + 558, + 493, + 579 + ], + "score": 1.0, + "content": "distributions). Then, the optimal discriminator at µ is given as Φµ(x) = 12 log p(x)p(x)+p0(x)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 218, + 590 + ], + "score": 0.9, + "content": "- \\textstyle { \\frac { 1 } { 2 } } \\log ( 1 + 2 \\exp ( - | x | / 2 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 576, + 506, + 590 + ], + "score": 1.0, + "content": ". By elementary calculations, we can see that this function is not differ-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 149, + 600 + ], + "score": 1.0, + "content": "entiable at", + "type": "text" + }, + { + "bbox": [ + 149, + 589, + 174, + 598 + ], + "score": 0.86, + "content": "x = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 587, + 505, + 600 + ], + "score": 1.0, + "content": ", which implies that the minimax GAN does not satisfy (D2). For the non-saturating", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 597, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 333, + 613 + ], + "score": 1.0, + "content": "GAN, we obtain a similar result by swapping the role of", + "type": "text" + }, + { + "bbox": [ + 333, + 601, + 340, + 611 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 597, + 358, + 613 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 358, + 601, + 370, + 611 + ], + "score": 0.85, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 597, + 373, + 613 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 493, + 601, + 505, + 607 + ], + "score": 0.965, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 103, + 536, + 506, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 503, + 642 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 185, + 631 + ], + "score": 1.0, + "content": "Proposition 8. Let", + "type": "text" + }, + { + "bbox": [ + 186, + 617, + 253, + 630 + ], + "score": 0.91, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 615, + 505, + 631 + ], + "score": 1.0, + "content": "be a convex, proper, lower semicontinuous function, and define", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 628, + 371, + 643 + ], + "spans": [ + { + "bbox": [ + 107, + 629, + 160, + 642 + ], + "score": 0.92, + "content": "{ \\tilde { J } } : = J \\oplus R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 628, + 306, + 643 + ], + "score": 1.0, + "content": ". Then the optimal discriminator for", + "type": "text" + }, + { + "bbox": [ + 306, + 629, + 314, + 641 + ], + "score": 0.82, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 628, + 324, + 643 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 324, + 630, + 335, + 642 + ], + "score": 0.87, + "content": "\\beta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 628, + 371, + 643 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 615, + 505, + 643 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 339, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 340, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 136, + 671 + ], + "score": 1.0, + "content": "Proof.", + "type": "text" + }, + { + "bbox": [ + 136, + 655, + 144, + 667 + ], + "score": 0.83, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 654, + 340, + 671 + ], + "score": 1.0, + "content": "is convex, proper, lower semicontinuous. Hence", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 654, + 340, + 671 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 673, + 443, + 700 + ], + "lines": [ + { + "bbox": [ + 168, + 673, + 443, + 700 + ], + "spans": [ + { + "bbox": [ + 168, + 673, + 443, + 700 + ], + "score": 0.94, + "content": "\\tilde { J } ( \\mu ) = ( J \\oplus \\lambda | | \\cdot | | _ { \\mathcal { F } ^ { * } } ) ( \\mu ) = \\operatorname* { s u p } _ { \\varphi } \\int \\varphi d \\mu - J ^ { \\star } ( \\varphi ) - \\chi \\{ \\frac { 1 } { \\lambda } \\varphi \\in \\mathcal { F } \\} .", + "type": "interline_equation", + "image_path": "ddba0c050726f6bccaf74d400f7ff13bfc9a356ba8086af5187e4c36338269b1.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 168, + 673, + 443, + 700 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 706, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 705, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 705, + 235, + 723 + ], + "score": 1.0, + "content": "Then by the envelope theorem,", + "type": "text" + }, + { + "bbox": [ + 236, + 707, + 248, + 722 + ], + "score": 0.92, + "content": "\\frac { \\delta J } { \\delta \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 705, + 274, + 723 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 275, + 709, + 282, + 720 + ], + "score": 0.82, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 705, + 474, + 723 + ], + "score": 1.0, + "content": "that maximizes the right-hand side, and hence", + "type": "text" + }, + { + "bbox": [ + 474, + 707, + 505, + 722 + ], + "score": 0.89, + "content": "\\frac { 1 } { \\lambda } \\frac { \\delta J } { \\delta \\mu } \\in", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 719, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 116, + 730 + ], + "score": 0.79, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 719, + 120, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 720, + 505, + 731 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 705, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 400, + 96 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 400, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 253, + 96 + ], + "score": 1.0, + "content": "Lemma 4. The convex conjugate of", + "type": "text" + }, + { + "bbox": [ + 254, + 82, + 311, + 95 + ], + "score": 0.92, + "content": "\\mu \\mapsto \\beta _ { 1 } \\| \\mu \\| _ { S ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 81, + 322, + 96 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 323, + 82, + 397, + 95 + ], + "score": 0.9, + "content": "\\varphi \\mapsto \\chi \\{ \\varphi \\in \\beta _ { 1 } S \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 81, + 400, + 96 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 404, + 118 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 405, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 405, + 119 + ], + "score": 1.0, + "content": "Proof. The proof of Lemma 4 is nearly identical to the proof of Lemma 3.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 123, + 505, + 157 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 187, + 137 + ], + "score": 1.0, + "content": "Proposition 9. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 123, + 239, + 135 + ], + "score": 0.91, + "content": "f : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 122, + 378, + 137 + ], + "score": 1.0, + "content": "be a neural network consisting of", + "type": "text" + }, + { + "bbox": [ + 378, + 124, + 385, + 134 + ], + "score": 0.52, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 122, + 505, + 137 + ], + "score": 1.0, + "content": "layers whose linear transfor-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 135, + 504, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 496, + 147 + ], + "score": 1.0, + "content": "mations have spectral norm 1 and whose activation functions are 1-Lipschitz and 1-smooth. Then", + "type": "text" + }, + { + "bbox": [ + 497, + 136, + 504, + 147 + ], + "score": 0.79, + "content": "f", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 146, + 158, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 115, + 157 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 116, + 146, + 122, + 156 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 146, + 158, + 157 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 169, + 503, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 168, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 213, + 183 + ], + "score": 1.0, + "content": "Proof. 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d \\sigma ( y ) | | _ { 2 } = | | \\mathrm { d i a g } \\sigma ^ { \\prime } ( x ) - \\mathrm { d i a g } \\sigma ^ { \\prime } ( y ) | | _ { 2 } } \\\\ & { \\qquad = | | \\mathrm { d i a g } ( \\sigma ^ { \\prime } ( x ) - \\sigma ^ { \\prime } ( y ) ) | | _ { 2 } } \\\\ & { \\qquad = \\underset { i } { \\operatorname* { m a x } } | \\sigma ^ { \\prime } ( x _ { i } ) - \\sigma ^ { \\prime } ( y _ { i } ) | } \\\\ & { \\qquad \\le \\underset { i } { \\operatorname* { m a x } } s ( \\sigma ) | x _ { i } - y _ { i } | } \\\\ & { \\qquad = s ( \\sigma ) | | x - y | | _ { \\infty } } \\\\ & { \\qquad \\le s ( \\sigma ) | | x - y | | _ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "c7eccebc73791bf2828f2a43d66c89987b4124486976dde8e2e52c4376b0538d.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 201, + 384, + 408, + 398.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 201, + 398.0, + 408, + 412.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 201, + 412.0, + 408, + 426.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 201, + 426.0, + 408, + 440.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 201, + 440.0, + 408, + 454.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 201, + 454.0, + 408, + 468.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 201, + 468.0, + 408, + 482.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 489, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 307, + 503 + ], + "score": 1.0, + "content": "We apply the inequality (27) recursively for all", + "type": "text" + }, + { + "bbox": [ + 307, + 491, + 314, + 500 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 489, + 493, + 503 + ], + "score": 1.0, + "content": "layers to obtain that the entire network is", + "type": "text" + }, + { + "bbox": [ + 493, + 491, + 501, + 500 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 489, + 506, + 503 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 500, + 142, + 513 + ], + "score": 1.0, + "content": "smooth.", + "type": "text" + }, + { + "bbox": [ + 495, + 502, + 505, + 511 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 105, + 517, + 441, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 442, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 169, + 533 + ], + "score": 1.0, + "content": "Lemma 5. Let", + "type": "text" + }, + { + "bbox": [ + 169, + 519, + 179, + 529 + ], + "score": 0.56, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 516, + 360, + 533 + ], + "score": 1.0, + "content": "be an RKHS with the Gaussian kernel. Then", + "type": "text" + }, + { + "bbox": [ + 360, + 516, + 438, + 532 + ], + "score": 0.91, + "content": "\\begin{array} { r } { | | \\hat { \\mu } | | _ { \\mathcal { H } } \\leq \\frac { \\sqrt { 2 } } { \\sigma ^ { 2 } } | | \\mu | | _ { S ^ { * } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 516, + 442, + 533 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 350, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 351, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 199, + 561 + ], + "score": 1.0, + "content": "Proof. 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\\frac { \\partial ^ { 2 } K ( x , y ) } { \\partial x _ { i } \\partial \\tilde { x } _ { i } } \\right| .", + "type": "interline_equation", + "image_path": "7085940f73378390bbf831149966e8072c3436e32ef90f1d1beb8fc5119f5b91.jpg" + } + ] + } + ], + "index": 33.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 701, + 393, + 718.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 218, + 718.0, + 393, + 735.0 + ], + "spans": [], + "index": 34 + } + ] + } + ], + "page_idx": 22, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 106, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 495, + 108, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 495, + 108, + 505, + 118 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 400, + 96 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 400, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 253, + 96 + ], + "score": 1.0, + "content": "Lemma 4. The convex conjugate of", + "type": "text" + }, + { + "bbox": [ + 254, + 82, + 311, + 95 + ], + "score": 0.92, + "content": "\\mu \\mapsto \\beta _ { 1 } \\| \\mu \\| _ { S ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 81, + 322, + 96 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 323, + 82, + 397, + 95 + ], + "score": 0.9, + "content": "\\varphi \\mapsto \\chi \\{ \\varphi \\in \\beta _ { 1 } S \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 81, + 400, + 96 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 400, + 96 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 404, + 118 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 405, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 405, + 119 + ], + "score": 1.0, + "content": "Proof. The proof of Lemma 4 is nearly identical to the proof of Lemma 3.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 105, + 405, + 119 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 123, + 505, + 157 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 187, + 137 + ], + "score": 1.0, + "content": "Proposition 9. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 123, + 239, + 135 + ], + "score": 0.91, + "content": "f : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 122, + 378, + 137 + ], + "score": 1.0, + "content": "be a neural network consisting of", + "type": "text" + }, + { + "bbox": [ + 378, + 124, + 385, + 134 + ], + "score": 0.52, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 122, + 505, + 137 + ], + "score": 1.0, + "content": "layers whose linear transfor-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 135, + 504, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 496, + 147 + ], + "score": 1.0, + "content": "mations have spectral norm 1 and whose activation functions are 1-Lipschitz and 1-smooth. 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Then", + "type": "text" + }, + { + "bbox": [ + 198, + 311, + 236, + 323 + ], + "score": 0.93, + "content": "\\ell ( A ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 309, + 254, + 324 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 311, + 293, + 323 + ], + "score": 0.93, + "content": "s ( A ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 309, + 309, + 324 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 299, + 505, + 324 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 325, + 387, + 356 + ], + "lines": [ + { + "bbox": [ + 223, + 325, + 387, + 356 + ], + "spans": [ + { + "bbox": [ + 223, + 325, + 387, + 356 + ], + "score": 0.9, + "content": "\\begin{array} { l } { { \\ell ( \\sigma \\circ A ) \\leq \\ell ( \\sigma ) \\ell ( A ) = 1 } } \\\\ { { s ( \\sigma \\circ A ) \\leq s ( \\sigma ) \\ell ( A ) ^ { 2 } + \\ell ( \\sigma ) s ( A ) = 1 . } } \\end{array}", + "type": "interline_equation", + "image_path": "5d73fa6e99ea3bb19d9b563de57bc7654f349873c7f32d5ac5785d08be633db4.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 325, + 387, + 340.5 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 223, + 340.5, + 387, + 356.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 359, + 504, + 382 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 278, + 372 + ], + "score": 1.0, + "content": "We note that we can use the same value of", + "type": "text" + }, + { + "bbox": [ + 278, + 359, + 298, + 371 + ], + "score": 0.92, + "content": "s ( \\sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 358, + 384, + 372 + ], + "score": 1.0, + "content": "whether we consider", + "type": "text" + }, + { + "bbox": [ + 384, + 360, + 430, + 369 + ], + "score": 0.91, + "content": "\\sigma : \\mathbb { R } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 358, + 505, + 372 + ], + "score": 1.0, + "content": "or elementwise as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 368, + 313, + 383 + ], + "spans": [ + { + "bbox": [ + 107, + 370, + 161, + 380 + ], + "score": 0.91, + "content": "\\sigma : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 368, + 266, + 383 + ], + "score": 1.0, + "content": ", since for an elementwise", + "type": "text" + }, + { + "bbox": [ + 266, + 373, + 273, + 380 + ], + "score": 0.8, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 368, + 313, + 383 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 358, + 505, + 383 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 384, + 408, + 482 + ], + "lines": [ + { + "bbox": [ + 201, + 384, + 408, + 482 + ], + "spans": [ + { + "bbox": [ + 201, + 384, + 408, + 482 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { | | d \\sigma ( x ) - d \\sigma ( y ) | | _ { 2 } = | | \\mathrm { d i a g } \\sigma ^ { \\prime } ( x ) - \\mathrm { d i a g } \\sigma ^ { \\prime } ( y ) | | _ { 2 } } \\\\ & { \\qquad = | | \\mathrm { d i a g } ( \\sigma ^ { \\prime } ( x ) - \\sigma ^ { \\prime } ( y ) ) | | _ { 2 } } \\\\ & { \\qquad = \\underset { i } { \\operatorname* { m a x } } | \\sigma ^ { \\prime } ( x _ { i } ) - \\sigma ^ { \\prime } ( y _ { i } ) | } \\\\ & { \\qquad \\le \\underset { i } { \\operatorname* { m a x } } s ( \\sigma ) | x _ { i } - y _ { i } | } \\\\ & { \\qquad = s ( \\sigma ) | | x - y | | _ { \\infty } } \\\\ & { \\qquad \\le s ( \\sigma ) | | x - y | | _ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "c7eccebc73791bf2828f2a43d66c89987b4124486976dde8e2e52c4376b0538d.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 201, + 384, + 408, + 398.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 201, + 398.0, + 408, + 412.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 201, + 412.0, + 408, + 426.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 201, + 426.0, + 408, + 440.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 201, + 440.0, + 408, + 454.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 201, + 454.0, + 408, + 468.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 201, + 468.0, + 408, + 482.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 489, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 307, + 503 + ], + "score": 1.0, + "content": "We apply the inequality (27) recursively for all", + "type": "text" + }, + { + "bbox": [ + 307, + 491, + 314, + 500 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 489, + 493, + 503 + ], + "score": 1.0, + "content": "layers to obtain that the entire network is", + "type": "text" + }, + { + "bbox": [ + 493, + 491, + 501, + 500 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 489, + 506, + 503 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 500, + 142, + 513 + ], + "score": 1.0, + "content": "smooth.", + "type": "text" + }, + { + "bbox": [ + 495, + 502, + 505, + 511 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 489, + 506, + 513 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 517, + 441, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 442, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 169, + 533 + ], + "score": 1.0, + "content": "Lemma 5. Let", + "type": "text" + }, + { + "bbox": [ + 169, + 519, + 179, + 529 + ], + "score": 0.56, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 516, + 360, + 533 + ], + "score": 1.0, + "content": "be an RKHS with the Gaussian kernel. Then", + "type": "text" + }, + { + "bbox": [ + 360, + 516, + 438, + 532 + ], + "score": 0.91, + "content": "\\begin{array} { r } { | | \\hat { \\mu } | | _ { \\mathcal { H } } \\leq \\frac { \\sqrt { 2 } } { \\sigma ^ { 2 } } | | \\mu | | _ { S ^ { * } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 516, + 442, + 533 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 516, + 442, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 350, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 351, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 199, + 561 + ], + "score": 1.0, + "content": "Proof. It follows from", + "type": "text" + }, + { + "bbox": [ + 200, + 543, + 283, + 560 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { \\partial f ( { \\boldsymbol x } ) } { \\partial { \\boldsymbol x } _ { i } } = \\langle f , \\frac { \\partial K ( { \\boldsymbol x } , \\cdot ) } { \\partial x _ { i } } \\rangle } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 542, + 351, + 559 + ], + "score": 1.0, + "content": ", ∂K(x,·) i that, for f ∈ H,", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 542, + 351, + 561 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 564, + 465, + 684 + ], + "lines": [ + { + "bbox": [ + 145, + 564, + 465, + 684 + ], + "spans": [ + { + "bbox": [ + 145, + 564, + 465, + 684 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle \\| \\nabla f ( x ) - \\nabla f ( y ) \\| ^ { 2 } = \\sum _ { i = 1 } ^ { d } \\left| \\frac { \\partial f ( x ) } { \\partial x _ { i } } - \\frac { \\partial f ( y ) } { \\partial y _ { i } } \\right| ^ { 2 } } \\\\ { \\displaystyle \\leq \\sum _ { i = 1 } ^ { d } \\| f \\| _ { \\mathcal H } ^ { 2 } \\left\\| \\frac { \\partial K ( x , \\cdot ) } { \\partial x _ { i } } - \\frac { \\partial K ( y , \\cdot ) } { \\partial y _ { i } } \\right\\| _ { \\mathcal H } ^ { 2 } } \\\\ { \\displaystyle = \\| f \\| _ { \\mathcal H } ^ { 2 } \\sum _ { i = 1 } ^ { d } \\left\\{ \\frac { \\partial ^ { 2 } K ( x , \\tilde { x } ) } { \\partial x _ { i } \\partial \\tilde { x } _ { i } } - 2 \\frac { \\partial ^ { 2 } K ( x , y ) } { \\partial x _ { i } \\partial y _ { i } } + \\frac { \\partial ^ { 2 } K ( y , \\tilde { y } ) } { \\partial y _ { i } \\partial \\tilde { y } _ { i } } \\right\\} } \\\\ { \\displaystyle \\leq \\| f \\| _ { \\mathcal H } ^ { 2 } \\cdot ( \\Gamma ( x , y ) + \\Gamma ( y , x ) ) , } \\end{array}", + "type": "interline_equation", + "image_path": "f924ff17dd940a5e643bb380e6127e72d5fd536b4c31c94d92bef5d0cee23f29.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 145, + 564, + 465, + 604.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 145, + 604.0, + 465, + 644.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 145, + 644.0, + 465, + 684.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 686, + 285, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 685, + 286, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 133, + 699 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 687, + 140, + 696 + ], + "score": 0.81, + "content": "\\tilde { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 685, + 158, + 699 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 158, + 687, + 165, + 698 + ], + "score": 0.84, + "content": "\\tilde { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 685, + 234, + 699 + ], + "score": 1.0, + "content": "denote copies of", + "type": "text" + }, + { + "bbox": [ + 234, + 690, + 241, + 696 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 685, + 259, + 699 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 259, + 690, + 265, + 698 + ], + "score": 0.77, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 685, + 286, + 699 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 685, + 286, + 699 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 701, + 393, + 735 + ], + "lines": [ + { + "bbox": [ + 218, + 701, + 393, + 735 + ], + "spans": [ + { + "bbox": [ + 218, + 701, + 393, + 735 + ], + "score": 0.94, + "content": "\\Gamma ( x , y ) = \\sum _ { i = 1 } ^ { d } \\left| \\frac { \\partial ^ { 2 } K ( x , \\tilde { x } ) } { \\partial x _ { i } \\partial \\tilde { x } _ { i } } - \\frac { \\partial ^ { 2 } K ( x , y ) } { \\partial x _ { i } \\partial \\tilde { x } _ { i } } \\right| .", + "type": "interline_equation", + "image_path": "7085940f73378390bbf831149966e8072c3436e32ef90f1d1beb8fc5119f5b91.jpg" + } + ] + } + ], + "index": 33.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 701, + 393, + 718.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 218, + 718.0, + 393, + 735.0 + ], + "spans": [], + "index": 34 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 291, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 291, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 291, + 95 + ], + "score": 1.0, + "content": "From (30), we find that for a Gaussian kernel,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 100, + 393, + 126 + ], + "lines": [ + { + "bbox": [ + 217, + 100, + 393, + 126 + ], + "spans": [ + { + "bbox": [ + 217, + 100, + 393, + 126 + ], + "score": 0.93, + "content": "\\Gamma ( x , y ) = \\frac { 1 } { \\sigma ^ { 4 } } \\exp \\Bigl ( - \\frac { \\| x - y \\| ^ { 2 } } { 2 \\sigma ^ { 2 } } \\Bigr ) \\| x - y \\| ^ { 2 } ,", + "type": "interline_equation", + "image_path": "5b6febddd8961d29a19bc49927bd3b7a177d43c22e40ab0fe9f8cfec8d9edcaf.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 217, + 100, + 393, + 126 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 144, + 143 + ], + "lines": [ + { + "bbox": [ + 105, + 130, + 147, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 147, + 145 + ], + "score": 1.0, + "content": "and thus,", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 143, + 389, + 169 + ], + "lines": [ + { + "bbox": [ + 221, + 143, + 389, + 169 + ], + "spans": [ + { + "bbox": [ + 221, + 143, + 389, + 169 + ], + "score": 0.94, + "content": "\\| \\nabla f ( x ) - \\nabla f ( y ) \\| \\leq { \\frac { \\sqrt { 2 } } { \\sigma ^ { 2 } } } \\| f \\| _ { \\mathcal H } \\| x - y \\| .", + "type": "interline_equation", + "image_path": "4a6043292436b32c9ee054c39b5cfe7a5d9b11eeb61e9b78ca58045962738a6e.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 221, + 143, + 389, + 169 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 176, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 177, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 177, + 192 + ], + "score": 1.0, + "content": "This implies that", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 198, + 435, + 231 + ], + "lines": [ + { + "bbox": [ + 175, + 198, + 435, + 231 + ], + "spans": [ + { + "bbox": [ + 175, + 198, + 435, + 231 + ], + "score": 0.93, + "content": "\\| \\hat { \\mu } \\| _ { \\mathcal H } = \\operatorname* { s u p } _ { f \\in \\mathcal H , \\| f \\| _ { \\mathcal H } \\leq 1 } \\int f d \\mu \\leq \\operatorname* { s u p } _ { \\| f \\| _ { \\mathcal S } \\leq \\frac { \\sqrt { 2 } } { \\sigma ^ { 2 } } } \\int f d \\mu = \\frac { \\sqrt { 2 } } { \\sigma ^ { 2 } } \\| \\mu \\| _ { \\mathcal S ^ { * } } .", + "type": "interline_equation", + "image_path": "079b85eee7b6bdf455f45d7ec4064145674604d2acc607913d3637370d218e0b.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 175, + 198, + 435, + 209.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 175, + 209.0, + 435, + 220.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 175, + 220.0, + 435, + 231.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 267, + 252, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 253, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 253, + 282 + ], + "score": 1.0, + "content": "H PROOFS FOR SECTION 6", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 105, + 291, + 388, + 308 + ], + "lines": [ + { + "bbox": [ + 102, + 286, + 386, + 311 + ], + "spans": [ + { + "bbox": [ + 102, + 286, + 253, + 311 + ], + "score": 1.0, + "content": "Lemma 6. The convex conjugate of", + "type": "text" + }, + { + "bbox": [ + 253, + 292, + 312, + 307 + ], + "score": 0.82, + "content": "\\mu \\mapsto \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathrm { K R } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 286, + 322, + 311 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 323, + 292, + 386, + 307 + ], + "score": 0.45, + "content": "\\begin{array} { r } { f \\mapsto \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 103, + 321, + 504, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 321, + 504, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 504, + 334 + ], + "score": 1.0, + "content": "Proof. This proof is generalized from the finite-dimensional case (Boyd & Vandenberghe, 2004).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 332, + 312, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 312, + 347 + ], + "score": 1.0, + "content": "We can bound the convex conjugate from above by", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 351, + 435, + 460 + ], + "lines": [ + { + "bbox": [ + 176, + 351, + 435, + 460 + ], + "spans": [ + { + "bbox": [ + 176, + 351, + 435, + 460 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { { \\operatorname* { s u p } _ { \\mu } [ \\int f d \\mu - \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathrm { K R } } ^ { 2 } ] \\leq \\operatorname* { s u p } _ { \\mu } [ | | f | | _ { \\mathrm { L i p } } | | \\mu | | _ { \\mathrm { K R } } - \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathrm { K R } } ^ { 2 } ] } } \\\\ & { \\leq \\operatorname* { s u p } _ { z \\in \\mathbb { R } } [ | | f | | _ { \\mathrm { L i p } } z - \\frac { \\lambda } { 2 } z ^ { 2 } ] } \\\\ & { = | | f | | _ { \\mathrm { L i p } } \\cdot \\frac { | | f | | _ { \\mathrm { L i p } } } { \\lambda } - \\frac { \\lambda } { 2 } \\cdot \\frac { | | f | | _ { \\mathrm { L i p } } ^ { 2 } } { \\lambda ^ { 2 } } } \\\\ & { = \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "8823eec8192bdc74a2279c05288666ac79d22f70be16dcb0e96b5d0c2ca5221a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 176, + 351, + 435, + 387.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 176, + 387.3333333333333, + 435, + 423.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 176, + 423.66666666666663, + 435, + 459.99999999999994 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 325, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 325, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 115, + 483 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 116, + 470, + 123, + 482 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 469, + 255, + 483 + ], + "score": 1.0, + "content": "is constant, then we may choose", + "type": "text" + }, + { + "bbox": [ + 255, + 471, + 280, + 482 + ], + "score": 0.91, + "content": "\\mu = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 469, + 325, + 483 + ], + "score": 1.0, + "content": "to see that", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 487, + 399, + 515 + ], + "lines": [ + { + "bbox": [ + 211, + 487, + 399, + 515 + ], + "spans": [ + { + "bbox": [ + 211, + 487, + 399, + 515 + ], + "score": 0.93, + "content": "\\operatorname* { s u p } _ { \\mu } \\left[ \\int f d \\mu - 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f ( y ) ) } } { \\Big ( } f ( x ) - f ( y ) { \\Big ) } = { \\frac { 1 } { \\lambda } } | | f | | _ { \\mathrm { L i p } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "80254740d87d38e8bd2806afe8f63643d678f40adf431d46040ad5a47854aed6.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 185, + 707, + 425, + 735 + ], + "spans": [], + "index": 29 + } + ] + } + ], + "page_idx": 23, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 16, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 238, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 495, + 239, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 495, + 239, + 505, + 251 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 291, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 291, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 291, + 95 + ], + "score": 1.0, + "content": "From (30), we find that for a Gaussian kernel,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 291, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 100, + 393, + 126 + ], + "lines": [ + { + "bbox": [ + 217, + 100, + 393, + 126 + ], + "spans": [ + { + "bbox": [ + 217, + 100, + 393, + 126 + ], + "score": 0.93, + "content": "\\Gamma ( x , y ) = \\frac { 1 } { \\sigma ^ { 4 } } \\exp \\Bigl ( - 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y \\| .", + "type": "interline_equation", + "image_path": "4a6043292436b32c9ee054c39b5cfe7a5d9b11eeb61e9b78ca58045962738a6e.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 221, + 143, + 389, + 169 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 176, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 177, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 177, + 192 + ], + "score": 1.0, + "content": "This implies that", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 177, + 177, + 192 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 198, + 435, + 231 + ], + "lines": [ + { + "bbox": [ + 175, + 198, + 435, + 231 + ], + "spans": [ + { + "bbox": [ + 175, + 198, + 435, + 231 + ], + "score": 0.93, + "content": "\\| \\hat { \\mu } \\| _ { \\mathcal H } = \\operatorname* { s u p } _ { f \\in \\mathcal H , \\| f \\| _ { \\mathcal H } \\leq 1 } \\int f d \\mu \\leq \\operatorname* { s u p } _ { \\| f \\| _ { \\mathcal S } \\leq \\frac { \\sqrt { 2 } } { \\sigma ^ { 2 } } } \\int f d \\mu = \\frac { \\sqrt { 2 } } { \\sigma ^ { 2 } } \\| \\mu \\| _ { \\mathcal S ^ { * } } .", + "type": "interline_equation", + "image_path": "079b85eee7b6bdf455f45d7ec4064145674604d2acc607913d3637370d218e0b.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 175, + 198, + 435, + 209.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 175, + 209.0, + 435, + 220.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 175, + 220.0, + 435, + 231.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 267, + 252, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 253, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 253, + 282 + ], + "score": 1.0, + "content": "H PROOFS FOR SECTION 6", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 105, + 291, + 388, + 308 + ], + "lines": [ + { + "bbox": [ + 102, + 286, + 386, + 311 + ], + "spans": [ + { + "bbox": [ + 102, + 286, + 253, + 311 + ], + "score": 1.0, + "content": "Lemma 6. The convex conjugate of", + "type": "text" + }, + { + "bbox": [ + 253, + 292, + 312, + 307 + ], + "score": 0.82, + "content": "\\mu \\mapsto \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathrm { K R } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 286, + 322, + 311 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 323, + 292, + 386, + 307 + ], + "score": 0.45, + "content": "\\begin{array} { r } { f \\mapsto \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 102, + 286, + 386, + 311 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 321, + 504, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 321, + 504, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 504, + 334 + ], + "score": 1.0, + "content": "Proof. 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\\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathrm { K R } } ^ { 2 } \\right] \\geq \\underset { f ( x ) \\neq f ( y ) } { \\operatorname* { s u p } } \\left[ \\int f d \\mu _ { x , y } - \\frac { \\lambda } { 2 } | | \\mu _ { x , y } | | _ { \\mathrm { K R } } ^ { 2 } \\right] } & { } \\\\ { = \\underset { f ( x ) \\neq f ( y ) } { \\operatorname* { s u p } } \\left[ \\frac { 1 } { \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } - \\frac { \\lambda } { 2 } \\big ( \\frac { 1 } { \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } \\frac { d ( x , y ) } { f ( x ) - f ( y ) } \\big ) ^ { 2 } \\right] } & { } \\\\ { = \\frac { 1 } { \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } - \\frac { \\lambda } { 2 } \\big ( \\frac { 1 } { \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } \\frac { 1 } { | | f | | _ { \\mathrm { L i p } } } \\big ) ^ { 2 } } & { } \\\\ { = \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathrm { L i p } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "447bf7e9c569a13dad90756baa47e5ef2d5a5a72497fbac85f0ba282309ee55c.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 138, + 97, + 473, + 134.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 138, + 134.66666666666666, + 473, + 172.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 138, + 172.33333333333331, + 473, + 209.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 230, + 505, + 257 + ], + "lines": [ + { + "bbox": [ + 105, + 229, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 192, + 245 + ], + "score": 1.0, + "content": "Proposition 10. Let", + "type": "text" + }, + { + "bbox": [ + 193, + 231, + 266, + 243 + ], + "score": 0.91, + "content": "J : \\mathcal { M } ( X ) \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 229, + 383, + 245 + ], + "score": 1.0, + "content": "be a convex function. Then", + "type": "text" + }, + { + "bbox": [ + 383, + 232, + 391, + 241 + ], + "score": 0.74, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 229, + 427, + 245 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 428, + 232, + 447, + 243 + ], + "score": 0.53, + "content": "( D 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 229, + 506, + 245 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 241, + 306, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 215, + 257 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathfrak { D } _ { J } ( \\nu , \\mu ) \\le \\frac { \\beta _ { 2 } } { 2 } \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 241, + 243, + 258 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 244, + 244, + 301, + 256 + ], + "score": 0.91, + "content": "\\mu , \\nu \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 241, + 306, + 258 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 332, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 333, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 333, + 283 + ], + "score": 1.0, + "content": "Proof. 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Before proceeding, note that the Bregman divergence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 674, + 401, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 145, + 686 + ], + "score": 0.93, + "content": "{ \\mathfrak { D } } _ { J } ( \\nu , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 674, + 304, + 687 + ], + "score": 1.0, + "content": "is always non-negative. 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Let", + "type": "text" + }, + { + "bbox": [ + 193, + 231, + 266, + 243 + ], + "score": 0.91, + "content": "J : \\mathcal { M } ( X ) \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 229, + 383, + 245 + ], + "score": 1.0, + "content": "be a convex function. Then", + "type": "text" + }, + { + "bbox": [ + 383, + 232, + 391, + 241 + ], + "score": 0.74, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 229, + 427, + 245 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 428, + 232, + 447, + 243 + ], + "score": 0.53, + "content": "( D 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 229, + 506, + 245 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 241, + 306, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 215, + 257 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathfrak { D } _ { J } ( \\nu , \\mu ) \\le \\frac { \\beta _ { 2 } } { 2 } \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 241, + 243, + 258 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 244, + 244, + 301, + 256 + ], + "score": 0.91, + "content": "\\mu , \\nu \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 241, + 306, + 258 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 229, + 506, + 258 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 332, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 333, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 333, + 283 + ], + "score": 1.0, + "content": "Proof. 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\\mu _ { s } \\| _ { \\mathrm { K R } } = | t - s | \\cdot \\| \\mu - \\nu \\| _ { \\mathrm { K R } } .", + "type": "interline_equation", + "image_path": "073b099b59e22d148db45c2fe2542da63ffe209cdd6f9e5025157035bec0e46b.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 173, + 408, + 438, + 423 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 167, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 168, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 168, + 440 + ], + "score": 1.0, + "content": "Then, we have", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 426, + 168, + 440 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 445, + 421, + 644 + ], + "lines": [ + { + "bbox": [ + 190, + 445, + 421, + 644 + ], + "spans": [ + { + "bbox": [ + 190, + 445, + 421, + 644 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { \\| \\mathcal { D } _ { J } ( v , p ) \\| = \\bigg | \\int _ { 0 } ^ { 1 } \\frac { d } { d t } \\hat { { \\boldsymbol { H } } } ( \\mu _ { t } ) \\hat { { \\boldsymbol { d } } } \\boldsymbol { L } \\bigg | \\hat { \\psi } - \\int \\hat { \\mathbf { q } } _ { \\mu } \\hat { { \\boldsymbol { d } } } ( v - \\mu ) \\bigg | } \\\\ & { = \\bigg | \\int _ { 0 } ^ { 1 } \\int \\Phi _ { \\mu } , d ( v - p ) d \\hbar - \\int \\Phi _ { \\mu } d ( v - \\mu ) \\bigg | } \\\\ & { = \\bigg | \\int _ { 0 } ^ { 1 } \\int ( \\Phi _ { \\mu \\nu } - \\Phi _ { \\mu } ) d v - \\mu ) d v - \\mu \\big | } \\\\ & { \\leq \\int _ { 0 } ^ { 1 } \\Big | \\| \\Phi _ { \\mu \\nu } - \\Phi _ { \\mu } \\| _ { \\mathrm { L E } ^ { \\infty } } \\| \\| \\mu _ { t } \\| = \\nu | \\mathrm { k i n } d t } \\\\ & { \\leq \\int _ { 0 } ^ { 1 } \\hat { \\mathcal { S } } _ { 2 } \\| u _ { t } - \\mu | \\mathrm { s g n } \\| u \\| \\mu - \\nu \\| \\mathrm { k i n } d t } \\\\ & { \\leq \\int _ { 0 } ^ { 1 } \\hat { \\mathcal { S } } _ { 2 } \\| u _ { t } - \\nu \\| _ { \\mathrm { L E } ^ { \\infty } } d u } \\\\ & { \\leq \\int _ { 0 } ^ { 1 } \\hat { \\mathcal { S } } _ { 2 } \\| u _ { t } - \\nu \\| _ { \\mathrm { L E } ^ { \\infty } } ^ { 2 } d u } \\\\ & { = \\frac { \\mathcal { S } _ { 2 } } { \\mathcal { S } } \\| u _ { t } - \\nu \\| _ { \\mathrm { L E } ^ { \\infty } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "6ab74676b3e85cc18d0b0b0eccafa6af6324314e2fb84c81c4c3be1276cb94cb.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 190, + 445, + 421, + 459.2142857142857 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 190, + 459.2142857142857, + 421, + 473.42857142857144 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 190, + 473.42857142857144, + 421, + 487.64285714285717 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 190, + 487.64285714285717, + 421, + 501.8571428571429 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 190, + 501.8571428571429, + 421, + 516.0714285714286 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 190, + 516.0714285714286, + 421, + 530.2857142857142 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 190, + 530.2857142857142, + 421, + 544.4999999999999 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 190, + 544.4999999999999, + 421, + 558.7142857142856 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 190, + 558.7142857142856, + 421, + 572.9285714285712 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 190, + 572.9285714285712, + 421, + 587.1428571428569 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 190, + 587.1428571428569, + 421, + 601.3571428571425 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 190, + 601.3571428571425, + 421, + 615.5714285714282 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 190, + 615.5714285714282, + 421, + 629.7857142857139 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 190, + 629.7857142857139, + 421, + 643.9999999999995 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 646, + 426, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 426, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 185, + 660 + ], + "score": 1.0, + "content": "Since the choice of", + "type": "text" + }, + { + "bbox": [ + 185, + 648, + 192, + 658 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 644, + 210, + 660 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 210, + 649, + 217, + 656 + ], + "score": 0.76, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 644, + 426, + 660 + ], + "score": 1.0, + "content": "is arbitrary, we have proved (28) by assuming (D2).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 644, + 426, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 506, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "score": 1.0, + "content": "Next, we move on to prove the converse. Before proceeding, note that the Bregman divergence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 674, + 401, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 145, + 686 + ], + "score": 0.93, + "content": "{ \\mathfrak { D } } _ { J } ( \\nu , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 674, + 304, + 687 + ], + "score": 1.0, + "content": "is always non-negative. In fact, for any", + "type": "text" + }, + { + "bbox": [ + 304, + 674, + 361, + 686 + ], + "score": 0.93, + "content": "\\mu , \\nu \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 674, + 401, + 687 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 662, + 505, + 687 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 691, + 436, + 735 + ], + "lines": [ + { + "bbox": [ + 174, + 691, + 436, + 735 + ], + "spans": [ + { + "bbox": [ + 174, + 691, + 436, + 735 + ], + "score": 0.94, + "content": "\\int \\Phi _ { \\mu } d ( \\nu - \\mu ) = \\underbrace { \\int \\Phi _ { \\mu } d \\nu - J ^ { \\star } ( \\Phi _ { \\mu } ) } _ { \\leq J ( \\nu ) } - \\underbrace { \\left[ \\int \\Phi _ { \\mu } d \\mu - J ^ { \\star } ( \\Phi _ { \\mu } ) \\right] } _ { = J ( \\mu ) }", + "type": "interline_equation", + "image_path": "72d6c37f06f87e65748b814e032cfafeb225f3bff76a41d04a27ef753c964b46.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 174, + 691, + 436, + 705.6666666666666 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 174, + 705.6666666666666, + 436, + 720.3333333333333 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 174, + 720.3333333333333, + 436, + 734.9999999999999 + ], + "spans": [], + "index": 35 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 239, + 81, + 306, + 95 + ], + "lines": [ + { + "bbox": [ + 239, + 81, + 306, + 95 + ], + "spans": [ + { + "bbox": [ + 239, + 81, + 306, + 95 + ], + "score": 0.73, + "content": "\\leq J ( \\nu ) - J ( \\mu ) ,", + "type": "interline_equation", + "image_path": "abccbe16d4fe3ec153db55a1c980fdeab0488934f53e3fea6f5b9642be29dcf3.jpg" + } + ] + } + ], + "index": 0, + "virtual_lines": [ + { + "bbox": [ + 239, + 81, + 306, + 95 + ], + "spans": [], + "index": 0 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 99, + 225, + 112 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 225, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 165, + 113 + ], + "score": 1.0, + "content": "which implies", + "type": "text" + }, + { + "bbox": [ + 165, + 100, + 222, + 112 + ], + "score": 0.93, + "content": "\\mathfrak { D } _ { J } ( \\nu , \\mu ) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 99, + 225, + 113 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 116, + 505, + 140 + ], + "lines": [ + { + "bbox": [ + 106, + 116, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 139, + 130 + ], + "score": 1.0, + "content": "Choose", + "type": "text" + }, + { + "bbox": [ + 139, + 117, + 186, + 129 + ], + "score": 0.92, + "content": "\\xi \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 116, + 241, + 130 + ], + "score": 1.0, + "content": "arbitrarily. If", + "type": "text" + }, + { + "bbox": [ + 242, + 117, + 267, + 129 + ], + "score": 0.91, + "content": "\\xi = \\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 116, + 328, + 130 + ], + "score": 1.0, + "content": ", the inequality", + "type": "text" + }, + { + "bbox": [ + 329, + 117, + 466, + 129 + ], + "score": 0.93, + "content": "\\| \\nabla \\Phi _ { \\mu } - \\nabla \\Phi _ { \\xi } \\| _ { 2 } \\leq \\beta _ { 2 } \\| \\xi - \\mu \\| _ { \\mathrm { K R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 116, + 505, + 130 + ], + "score": 1.0, + "content": "is trivial,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 456, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 164, + 141 + ], + "score": 1.0, + "content": "so we assume", + "type": "text" + }, + { + "bbox": [ + 164, + 129, + 189, + 140 + ], + "score": 0.9, + "content": "\\xi \\neq \\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 128, + 456, + 141 + ], + "score": 1.0, + "content": "below. By the non-negativity of the Bregman divergence, we have", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 143, + 466, + 241 + ], + "lines": [ + { + "bbox": [ + 144, + 143, + 466, + 241 + ], + "spans": [ + { + "bbox": [ + 144, + 143, + 466, + 241 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { { 0 \\leq \\mathfrak { D } _ { J } ( \\nu , \\mu ) } } \\\\ & { = J ( \\nu ) - [ J ( \\mu ) + \\int \\Phi _ { \\mu } d ( \\nu - \\mu ) ] } \\\\ & { \\leq [ J ( \\xi ) + \\int \\Phi _ { \\xi } d ( \\nu - \\xi ) + \\frac { \\beta _ { 2 } } { 2 } | | \\xi - \\nu | | _ { \\mathrm { K R } } ^ { 2 } ] - [ J ( \\mu ) + \\int \\Phi _ { \\mu } d ( \\nu - \\mu ) ] } \\\\ & { = J ( \\xi ) - J ( \\mu ) - \\int \\Phi _ { \\mu } d ( \\xi - \\mu ) + \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\nu - \\xi ) + \\frac { \\beta _ { 2 } } { 2 } | | \\xi - \\nu | | _ { \\mathrm { K R } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "181dc6ada7b4ee912d5479f53cc558c5860d0e25fc92e3a09bbf11f96af7bb8e.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 144, + 143, + 466, + 175.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 144, + 175.66666666666666, + 466, + 208.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 144, + 208.33333333333331, + 466, + 240.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 245, + 504, + 268 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 268, + 258 + ], + "score": 1.0, + "content": "Since the above inequality holds for all", + "type": "text" + }, + { + "bbox": [ + 268, + 245, + 318, + 258 + ], + "score": 0.92, + "content": "\\nu \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 244, + 506, + 258 + ], + "score": 1.0, + "content": ", the last expression is still non-negative if we", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 256, + 273, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 209, + 268 + ], + "score": 1.0, + "content": "take the infimum over all", + "type": "text" + }, + { + "bbox": [ + 209, + 259, + 215, + 266 + ], + "score": 0.73, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 256, + 273, + 268 + ], + "score": 1.0, + "content": ". In particular,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 272, + 430, + 387 + ], + "lines": [ + { + "bbox": [ + 181, + 272, + 430, + 387 + ], + "spans": [ + { + "bbox": [ + 181, + 272, + 430, + 387 + ], + "score": 0.95, + "content": "\\begin{array} { l } { { \\displaystyle \\operatorname* { i n f } _ { \\nu \\in \\mathcal { M } ( X ) } \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\nu - \\xi ) + \\frac { \\beta _ { 2 } } { 2 } \\| \\xi - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } } } \\\\ { ~ = - \\displaystyle \\operatorname* { s u p } _ { \\nu \\in \\mathcal { M } ( X ) } \\left[ \\int [ \\Phi _ { \\mu } - \\Phi _ { \\xi } ] d ( \\nu - \\xi ) - \\frac { \\beta _ { 2 } } { 2 } \\| \\xi - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } \\right] } \\\\ { ~ = - \\displaystyle \\operatorname* { s u p } _ { \\zeta \\in \\mathcal { M } ( X ) } \\left[ \\int [ \\Phi _ { \\mu } - \\Phi _ { \\xi } ] d \\zeta - \\frac { \\beta _ { 2 } } { 2 } \\| \\zeta \\| _ { \\mathrm { K R } } ^ { 2 } \\right] } \\\\ { ~ = - \\displaystyle \\frac { 1 } { 2 \\beta _ { 2 } } \\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\| _ { \\mathrm { L i p } } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "df828e39fb614b94ffcca73c3d69067f65c43486ec0876458e0ee22851b6e449.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 181, + 272, + 430, + 310.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 181, + 310.3333333333333, + 430, + 348.66666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 181, + 348.66666666666663, + 430, + 386.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 252, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 253, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 253, + 404 + ], + "score": 1.0, + "content": "using Lemma 6 for the last equality.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 232, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 232, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 232, + 420 + ], + "score": 1.0, + "content": "Continuing from (29), we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 422, + 426, + 449 + ], + "lines": [ + { + "bbox": [ + 185, + 422, + 426, + 449 + ], + "spans": [ + { + "bbox": [ + 185, + 422, + 426, + 449 + ], + "score": 0.94, + "content": "0 \\leq J ( \\xi ) - J ( \\mu ) - \\int \\Phi _ { \\mu } d ( \\xi - \\mu ) - \\frac { 1 } { 2 \\beta _ { 2 } } \\left. \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right. _ { \\mathrm { L i p } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "61ee9e5826d63e35d7fb5078a547d921cb0ba068c7fc24c859a3d14240480b5e.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 185, + 422, + 426, + 449 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 503, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 198, + 466 + ], + "score": 1.0, + "content": "Swapping the roles of", + "type": "text" + }, + { + "bbox": [ + 198, + 455, + 205, + 464 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 451, + 224, + 466 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 224, + 453, + 230, + 464 + ], + "score": 0.83, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 451, + 505, + 466 + ], + "score": 1.0, + "content": ", we obtain a similar inequality. Adding both sides of thus obtained", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 464, + 217, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 217, + 476 + ], + "score": 1.0, + "content": "two inequalities, we obtain", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 480, + 405, + 506 + ], + "lines": [ + { + "bbox": [ + 205, + 480, + 405, + 506 + ], + "spans": [ + { + "bbox": [ + 205, + 480, + 405, + 506 + ], + "score": 0.92, + "content": "0 \\leq \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\xi - \\mu ) - \\frac { 1 } { \\beta _ { 2 } } \\left\\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right\\| _ { \\mathrm { L i p } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "7eca492bb5087690b5a524d9b13cb915b6dd4c8eff8bea400a7eaba1ec24ba64.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 205, + 480, + 405, + 506 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 173, + 521 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 174, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 174, + 522 + ], + "score": 1.0, + "content": "Finally, we have", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 526, + 459, + 552 + ], + "lines": [ + { + "bbox": [ + 151, + 526, + 459, + 552 + ], + "spans": [ + { + "bbox": [ + 151, + 526, + 459, + 552 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\left\\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right\\| _ { \\mathrm { L i p } } ^ { 2 } \\leq \\beta _ { 2 } \\displaystyle \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\xi - \\mu ) \\leq \\beta _ { 2 } \\left\\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right\\| _ { \\mathrm { L i p } } \\left\\| \\xi - \\mu \\right\\| _ { \\mathrm { K R } } , } \\end{array}", + "type": "interline_equation", + "image_path": "cce09b6860735101984ff334234c8433e78c6c97dab782ec7cd89a1b4c5df44d.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 151, + 526, + 459, + 552 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 257, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 257, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 257, + 567 + ], + "score": 1.0, + "content": "and hence we have the desired result:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 572, + 371, + 588 + ], + "lines": [ + { + "bbox": [ + 239, + 572, + 371, + 588 + ], + "spans": [ + { + "bbox": [ + 239, + 572, + 371, + 588 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\| _ { \\mathrm { L i p } } \\leq \\beta _ { 2 } \\| \\xi - \\mu \\| _ { \\mathrm { K R } } . } \\end{array}", + "type": "interline_equation", + "image_path": "8141b90d5613fb2a4449966f2b194c53b90495f0bf422a025dfa1e46e446d6b3.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 239, + 572, + 371, + 588 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 211, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 211, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 211, + 648 + ], + "score": 1.0, + "content": "Proof. First, we note that", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 651, + 483, + 737 + ], + "lines": [ + { + "bbox": [ + 127, + 651, + 483, + 737 + ], + "spans": [ + { + "bbox": [ + 127, + 651, + 483, + 737 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathrm { M M D } _ { K } ^ { 2 } ( \\mu , \\nu ) = \\displaystyle \\int K ( x , y ) \\left( \\mu - \\nu \\right) ( d x ) \\left( \\mu - \\nu \\right) ( d y ) } \\\\ & { \\qquad \\le \\left[ \\displaystyle \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } } \\frac { \\int \\left( K ( x , y ) - K ( x ^ { \\prime } , y ) \\right) \\left( \\mu - \\nu \\right) \\left( d y \\right) } { d ( x , x ^ { \\prime } ) } \\right] \\| \\mu - \\nu \\| _ { \\mathrm { K R } } } \\\\ & { \\qquad \\le \\left[ \\displaystyle \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } , y \\neq y ^ { \\prime } } \\frac { K ( x , y ) - K ( x , y ^ { \\prime } ) - K ( x ^ { \\prime } , y ) + K ( x ^ { \\prime } , y ^ { \\prime } ) } { d ( x , x ^ { \\prime } ) d ( y , y ^ { \\prime } ) } \\right] \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "0a09a5f92d9a9ff556ff1e4799710a4fcc5fb544a84cf2d414ac05beb4e7f6d3.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 127, + 651, + 483, + 679.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 127, + 679.6666666666666, + 483, + 708.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 127, + 708.3333333333333, + 483, + 736.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + } + ], + "page_idx": 25, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 592, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 496, + 594, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 496, + 594, + 505, + 603 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 239, + 81, + 306, + 95 + ], + "lines": [ + { + "bbox": [ + 239, + 81, + 306, + 95 + ], + "spans": [ + { + "bbox": [ + 239, + 81, + 306, + 95 + ], + "score": 0.73, + "content": "\\leq J ( \\nu ) - J ( \\mu ) ,", + "type": "interline_equation", + "image_path": "abccbe16d4fe3ec153db55a1c980fdeab0488934f53e3fea6f5b9642be29dcf3.jpg" + } + ] + } + ], + "index": 0, + "virtual_lines": [ + { + "bbox": [ + 239, + 81, + 306, + 95 + ], + "spans": [], + "index": 0 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 99, + 225, + 112 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 225, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 165, + 113 + ], + "score": 1.0, + "content": "which implies", + "type": "text" + }, + { + "bbox": [ + 165, + 100, + 222, + 112 + ], + "score": 0.93, + "content": "\\mathfrak { D } _ { J } ( \\nu , \\mu ) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 99, + 225, + 113 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 99, + 225, + 113 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 116, + 505, + 140 + ], + "lines": [ + { + "bbox": [ + 106, + 116, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 139, + 130 + ], + "score": 1.0, + "content": "Choose", + "type": "text" + }, + { + "bbox": [ + 139, + 117, + 186, + 129 + ], + "score": 0.92, + "content": "\\xi \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 116, + 241, + 130 + ], + "score": 1.0, + "content": "arbitrarily. If", + "type": "text" + }, + { + "bbox": [ + 242, + 117, + 267, + 129 + ], + "score": 0.91, + "content": "\\xi = \\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 116, + 328, + 130 + ], + "score": 1.0, + "content": ", the inequality", + "type": "text" + }, + { + "bbox": [ + 329, + 117, + 466, + 129 + ], + "score": 0.93, + "content": "\\| \\nabla \\Phi _ { \\mu } - \\nabla \\Phi _ { \\xi } \\| _ { 2 } \\leq \\beta _ { 2 } \\| \\xi - \\mu \\| _ { \\mathrm { K R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 116, + 505, + 130 + ], + "score": 1.0, + "content": "is trivial,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 456, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 164, + 141 + ], + "score": 1.0, + "content": "so we assume", + "type": "text" + }, + { + "bbox": [ + 164, + 129, + 189, + 140 + ], + "score": 0.9, + "content": "\\xi \\neq \\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 128, + 456, + 141 + ], + "score": 1.0, + "content": "below. By the non-negativity of the Bregman divergence, we have", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 116, + 505, + 141 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 143, + 466, + 241 + ], + "lines": [ + { + "bbox": [ + 144, + 143, + 466, + 241 + ], + "spans": [ + { + "bbox": [ + 144, + 143, + 466, + 241 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { { 0 \\leq \\mathfrak { D } _ { J } ( \\nu , \\mu ) } } \\\\ & { = J ( \\nu ) - [ J ( \\mu ) + \\int \\Phi _ { \\mu } d ( \\nu - \\mu ) ] } \\\\ & { \\leq [ J ( \\xi ) + \\int \\Phi _ { \\xi } d ( \\nu - \\xi ) + \\frac { \\beta _ { 2 } } { 2 } | | \\xi - \\nu | | _ { \\mathrm { K R } } ^ { 2 } ] - [ J ( \\mu ) + \\int \\Phi _ { \\mu } d ( \\nu - \\mu ) ] } \\\\ & { = J ( \\xi ) - J ( \\mu ) - \\int \\Phi _ { \\mu } d ( \\xi - \\mu ) + \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\nu - \\xi ) + \\frac { \\beta _ { 2 } } { 2 } | | \\xi - \\nu | | _ { \\mathrm { K R } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "181dc6ada7b4ee912d5479f53cc558c5860d0e25fc92e3a09bbf11f96af7bb8e.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 144, + 143, + 466, + 175.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 144, + 175.66666666666666, + 466, + 208.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 144, + 208.33333333333331, + 466, + 240.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 245, + 504, + 268 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 268, + 258 + ], + "score": 1.0, + "content": "Since the above inequality holds for all", + "type": "text" + }, + { + "bbox": [ + 268, + 245, + 318, + 258 + ], + "score": 0.92, + "content": "\\nu \\in \\mathcal { M } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 244, + 506, + 258 + ], + "score": 1.0, + "content": ", the last expression is still non-negative if we", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 256, + 273, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 209, + 268 + ], + "score": 1.0, + "content": "take the infimum over all", + "type": "text" + }, + { + "bbox": [ + 209, + 259, + 215, + 266 + ], + "score": 0.73, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 256, + 273, + 268 + ], + "score": 1.0, + "content": ". In particular,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 244, + 506, + 268 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 272, + 430, + 387 + ], + "lines": [ + { + "bbox": [ + 181, + 272, + 430, + 387 + ], + "spans": [ + { + "bbox": [ + 181, + 272, + 430, + 387 + ], + "score": 0.95, + "content": "\\begin{array} { l } { { \\displaystyle \\operatorname* { i n f } _ { \\nu \\in \\mathcal { M } ( X ) } \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\nu - \\xi ) + \\frac { \\beta _ { 2 } } { 2 } \\| \\xi - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } } } \\\\ { ~ = - \\displaystyle \\operatorname* { s u p } _ { \\nu \\in \\mathcal { M } ( X ) } \\left[ \\int [ \\Phi _ { \\mu } - \\Phi _ { \\xi } ] d ( \\nu - \\xi ) - \\frac { \\beta _ { 2 } } { 2 } \\| \\xi - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } \\right] } \\\\ { ~ = - \\displaystyle \\operatorname* { s u p } _ { \\zeta \\in \\mathcal { M } ( X ) } \\left[ \\int [ \\Phi _ { \\mu } - \\Phi _ { \\xi } ] d \\zeta - \\frac { \\beta _ { 2 } } { 2 } \\| \\zeta \\| _ { \\mathrm { K R } } ^ { 2 } \\right] } \\\\ { ~ = - \\displaystyle \\frac { 1 } { 2 \\beta _ { 2 } } \\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\| _ { \\mathrm { L i p } } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "df828e39fb614b94ffcca73c3d69067f65c43486ec0876458e0ee22851b6e449.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 181, + 272, + 430, + 310.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 181, + 310.3333333333333, + 430, + 348.66666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 181, + 348.66666666666663, + 430, + 386.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 252, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 253, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 253, + 404 + ], + "score": 1.0, + "content": "using Lemma 6 for the last equality.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 387, + 253, + 404 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 232, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 232, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 232, + 420 + ], + "score": 1.0, + "content": "Continuing from (29), we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 405, + 232, + 420 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 422, + 426, + 449 + ], + "lines": [ + { + "bbox": [ + 185, + 422, + 426, + 449 + ], + "spans": [ + { + "bbox": [ + 185, + 422, + 426, + 449 + ], + "score": 0.94, + "content": "0 \\leq J ( \\xi ) - J ( \\mu ) - \\int \\Phi _ { \\mu } d ( \\xi - \\mu ) - \\frac { 1 } { 2 \\beta _ { 2 } } \\left. \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right. _ { \\mathrm { L i p } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "61ee9e5826d63e35d7fb5078a547d921cb0ba068c7fc24c859a3d14240480b5e.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 185, + 422, + 426, + 449 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 503, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 198, + 466 + ], + "score": 1.0, + "content": "Swapping the roles of", + "type": "text" + }, + { + "bbox": [ + 198, + 455, + 205, + 464 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 451, + 224, + 466 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 224, + 453, + 230, + 464 + ], + "score": 0.83, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 451, + 505, + 466 + ], + "score": 1.0, + "content": ", we obtain a similar inequality. Adding both sides of thus obtained", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 464, + 217, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 217, + 476 + ], + "score": 1.0, + "content": "two inequalities, we obtain", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 106, + 451, + 505, + 476 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 480, + 405, + 506 + ], + "lines": [ + { + "bbox": [ + 205, + 480, + 405, + 506 + ], + "spans": [ + { + "bbox": [ + 205, + 480, + 405, + 506 + ], + "score": 0.92, + "content": "0 \\leq \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\xi - \\mu ) - \\frac { 1 } { \\beta _ { 2 } } \\left\\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right\\| _ { \\mathrm { L i p } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "7eca492bb5087690b5a524d9b13cb915b6dd4c8eff8bea400a7eaba1ec24ba64.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 205, + 480, + 405, + 506 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 173, + 521 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 174, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 174, + 522 + ], + "score": 1.0, + "content": "Finally, we have", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 509, + 174, + 522 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 526, + 459, + 552 + ], + "lines": [ + { + "bbox": [ + 151, + 526, + 459, + 552 + ], + "spans": [ + { + "bbox": [ + 151, + 526, + 459, + 552 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\left\\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right\\| _ { \\mathrm { L i p } } ^ { 2 } \\leq \\beta _ { 2 } \\displaystyle \\int [ \\Phi _ { \\xi } - \\Phi _ { \\mu } ] d ( \\xi - \\mu ) \\leq \\beta _ { 2 } \\left\\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\right\\| _ { \\mathrm { L i p } } \\left\\| \\xi - \\mu \\right\\| _ { \\mathrm { K R } } , } \\end{array}", + "type": "interline_equation", + "image_path": "cce09b6860735101984ff334234c8433e78c6c97dab782ec7cd89a1b4c5df44d.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 151, + 526, + 459, + 552 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 257, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 257, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 257, + 567 + ], + "score": 1.0, + "content": "and hence we have the desired result:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 555, + 257, + 567 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 572, + 371, + 588 + ], + "lines": [ + { + "bbox": [ + 239, + 572, + 371, + 588 + ], + "spans": [ + { + "bbox": [ + 239, + 572, + 371, + 588 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\| \\Phi _ { \\mu } - \\Phi _ { \\xi } \\| _ { \\mathrm { L i p } } \\leq \\beta _ { 2 } \\| \\xi - \\mu \\| _ { \\mathrm { K R } } . } \\end{array}", + "type": "interline_equation", + "image_path": "8141b90d5613fb2a4449966f2b194c53b90495f0bf422a025dfa1e46e446d6b3.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 239, + 572, + 371, + 588 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 211, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 211, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 211, + 648 + ], + "score": 1.0, + "content": "Proof. First, we note that", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 635, + 211, + 648 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 651, + 483, + 737 + ], + "lines": [ + { + "bbox": [ + 127, + 651, + 483, + 737 + ], + "spans": [ + { + "bbox": [ + 127, + 651, + 483, + 737 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathrm { M M D } _ { K } ^ { 2 } ( \\mu , \\nu ) = \\displaystyle \\int K ( x , y ) \\left( \\mu - \\nu \\right) ( d x ) \\left( \\mu - \\nu \\right) ( d y ) } \\\\ & { \\qquad \\le \\left[ \\displaystyle \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } } \\frac { \\int \\left( K ( x , y ) - K ( x ^ { \\prime } , y ) \\right) \\left( \\mu - \\nu \\right) \\left( d y \\right) } { d ( x , x ^ { \\prime } ) } \\right] \\| \\mu - \\nu \\| _ { \\mathrm { K R } } } \\\\ & { \\qquad \\le \\left[ \\displaystyle \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } , y \\neq y ^ { \\prime } } \\frac { K ( x , y ) - K ( x , y ^ { \\prime } ) - K ( x ^ { \\prime } , y ) + K ( x ^ { \\prime } , y ^ { \\prime } ) } { d ( x , x ^ { \\prime } ) d ( y , y ^ { \\prime } ) } \\right] \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "0a09a5f92d9a9ff556ff1e4799710a4fcc5fb544a84cf2d414ac05beb4e7f6d3.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 127, + 651, + 483, + 679.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 127, + 679.6666666666666, + 483, + 708.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 127, + 708.3333333333333, + 483, + 736.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 185, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 186, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 186, + 95 + ], + "score": 1.0, + "content": "Next, we show that", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 93, + 439, + 122 + ], + "lines": [ + { + "bbox": [ + 171, + 93, + 439, + 122 + ], + "spans": [ + { + "bbox": [ + 171, + 93, + 439, + 122 + ], + "score": 0.92, + "content": "c = \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } , y \\neq y ^ { \\prime } } \\frac { K ( x , y ) - K ( x , y ^ { \\prime } ) - K ( x ^ { \\prime } , y ) + K ( x ^ { \\prime } , y ^ { \\prime } ) } { d ( x , x ^ { \\prime } ) d ( y , y ^ { \\prime } ) } = \\frac { 1 } { \\sigma ^ { 2 } } .", + "type": "interline_equation", + "image_path": "716797c1ae632c17565dff4e7aeba7bb8a2079444acd04b9e422905103b06767.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 171, + 93, + 439, + 102.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 171, + 102.66666666666667, + 439, + 112.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 171, + 112.33333333333334, + 439, + 122.00000000000001 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 129, + 503, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 127, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 142, + 142 + ], + "score": 1.0, + "content": "Because", + "type": "text" + }, + { + "bbox": [ + 142, + 130, + 152, + 139 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 127, + 396, + 142 + ], + "score": 1.0, + "content": "is differentiable, it suffices to check that the operator norm of", + "type": "text" + }, + { + "bbox": [ + 396, + 129, + 456, + 142 + ], + "score": 0.94, + "content": "\\partial _ { x _ { i } } \\partial _ { y _ { j } } K ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 127, + 505, + 142 + ], + "score": 1.0, + "content": "is bounded.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 140, + 191, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 191, + 153 + ], + "score": 1.0, + "content": "To see this, note that", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 148, + 390, + 230 + ], + "lines": [ + { + "bbox": [ + 219, + 148, + 390, + 230 + ], + "spans": [ + { + "bbox": [ + 219, + 148, + 390, + 230 + ], + "score": 0.93, + "content": "\\begin{array} { l } { c = \\displaystyle \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } } \\frac { \\mathrm { L i p } \\left( K ( x , \\cdot ) - K ( x ^ { \\prime } , \\cdot ) \\right) } { | | x - x ^ { \\prime } | | } } \\\\ { = \\displaystyle \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } , y } \\frac { | | \\nabla _ { y } K ( x , y ) - \\nabla _ { y } K ( x ^ { \\prime } , y ) | | _ { 2 } } { | | x - x ^ { \\prime } | | } } \\\\ { \\leq \\displaystyle \\operatorname* { s u p } _ { x , y } | | \\nabla _ { x } \\nabla _ { y } K ( x , y ) | | _ { \\mathrm { o p } } , } \\end{array}", + "type": "interline_equation", + "image_path": "41e12e95c1cabe9cd78e4b9f7cae0757931efe39a87393c3fe63693f46721d20.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 219, + 148, + 390, + 164.4 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 219, + 164.4, + 390, + 180.8 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 219, + 180.8, + 390, + 197.20000000000002 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 219, + 197.20000000000002, + 390, + 213.60000000000002 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 219, + 213.60000000000002, + 390, + 230.00000000000003 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 229, + 504, + 252 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 243 + ], + "score": 1.0, + "content": "where we obtained by the last equality by applying the vector-valued mean value theorem (Rudin,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 239, + 337, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 142, + 255 + ], + "score": 1.0, + "content": "1964) to", + "type": "text" + }, + { + "bbox": [ + 143, + 240, + 261, + 253 + ], + "score": 0.89, + "content": "t \\mapsto \\nabla _ { y } K ( ( 1 - t ) x + t x ^ { \\prime } , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 239, + 337, + 255 + ], + "score": 1.0, + "content": ", thereby obtaining", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 253, + 461, + 300 + ], + "lines": [ + { + "bbox": [ + 149, + 253, + 461, + 300 + ], + "spans": [ + { + "bbox": [ + 149, + 253, + 461, + 300 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { | | \\nabla _ { y } K ( x , y ) - \\nabla _ { y } K ( x ^ { \\prime } , y ) | | _ { 2 } \\leq | | \\partial _ { t } \\nabla _ { y } K ( ( 1 - t ) x + t x ^ { \\prime } , y ) | | _ { 2 } } \\\\ & { \\qquad = | | ( x ^ { \\prime } - x ) \\cdot \\nabla _ { x } \\nabla _ { y } K ( ( 1 - t ) x + t x ^ { \\prime } , y ) | | _ { 2 } } \\\\ & { \\qquad \\leq | | x ^ { \\prime } - x | | _ { 2 } | | \\nabla _ { x } \\nabla _ { y } K ( ( 1 - t ) x + t x ^ { \\prime } , y ) | | _ { \\mathrm { { o p } } } . } \\end{array}", + "type": "interline_equation", + "image_path": "ee58ffae8098cb0a6b2f8b875cb9bf9b9054c84eb7988728fde285c4eaaaab11.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 149, + 253, + 461, + 268.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 149, + 268.6666666666667, + 461, + 284.33333333333337 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 149, + 284.33333333333337, + 461, + 300.00000000000006 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 128, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 129, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 129, + 317 + ], + "score": 1.0, + "content": "Now", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 314, + 470, + 405 + ], + "lines": [ + { + "bbox": [ + 140, + 314, + 470, + 405 + ], + "spans": [ + { + "bbox": [ + 140, + 314, + 470, + 405 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\partial _ { x _ { i } } \\partial _ { y _ { j } } K ( x , y ) = \\partial _ { x _ { i } } \\partial _ { y _ { j } } \\exp \\left( - 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\\frac { | | x - y | | ^ { 2 } } { 2 \\sigma ^ { 2 } } } \\Big ) \\Big [ \\frac { 1 } { \\sigma ^ { 2 } } ( x - y ) ( x - y ) ^ { T } - I \\Big ] .", + "type": "interline_equation", + "image_path": "e6d13c84d565034af6fb08a48cad498b26df72450f503e27875b43c0b4bb45c2.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 203, + 417, + 407, + 443 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 102, + 439, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 102, + 439, + 212, + 464 + ], + "score": 1.0, + "content": "By inspection, we see that", + "type": "text" + }, + { + "bbox": [ + 212, + 448, + 234, + 459 + ], + "score": 0.91, + "content": "x - y", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 439, + 368, + 464 + ], + "score": 1.0, + "content": "is an eigenvector with eigenvalue", + "type": "text" + }, + { + "bbox": [ + 369, + 443, + 506, + 460 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { \\sigma ^ { 2 } } \\exp ( { - \\frac { | | x - y | | ^ { 2 } } { 2 \\sigma ^ { 2 } } } ) ( \\frac { 1 } { \\sigma ^ { 2 } } | | x - y | | ^ { 2 } - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 101, + 453, + 484, + 482 + ], + "spans": [ + { + "bbox": [ + 101, + 453, + 484, + 482 + ], + "score": 1.0, + "content": "1), and the other eigenvectors are orthogonal to x − y with eigenvalues − 1σ2 exp(− ||x−y||22σ2 ).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 109, + 479, + 503, + 506 + ], + "lines": [ + { + "bbox": [ + 107, + 479, + 504, + 498 + ], + "spans": [ + { + "bbox": [ + 107, + 479, + 137, + 498 + ], + "score": 1.0, + "content": "Setting", + "type": "text" + }, + { + "bbox": [ + 137, + 480, + 187, + 496 + ], + "score": 0.93, + "content": "\\begin{array} { r } { z = \\frac { | | x - y | | ^ { 2 } } { 2 \\sigma ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 479, + 504, + 498 + ], + "score": 1.0, + "content": "and taking the absolute value of the eigenvalues, we see the maximum operator", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 494, + 224, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 162, + 506 + ], + "score": 1.0, + "content": "norm over all", + "type": "text" + }, + { + "bbox": [ + 163, + 497, + 179, + 506 + ], + "score": 0.87, + "content": "x , y", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 494, + 224, + 506 + ], + "score": 1.0, + "content": "is equal to", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 507, + 403, + 533 + ], + "lines": [ + { + "bbox": [ + 206, + 507, + 403, + 533 + ], + "spans": [ + { + "bbox": [ + 206, + 507, + 403, + 533 + ], + "score": 0.82, + "content": "\\operatorname* { m a x } \\bigg \\{ \\operatorname* { s u p } _ { z \\ge 0 } \\frac { 1 } { \\sigma ^ { 2 } } e ^ { - z } | 2 z - 1 | , \\operatorname* { s u p } _ { z \\ge 0 } \\frac { 1 } { \\sigma ^ { 2 } } e ^ { - z } \\bigg \\} = \\frac { 1 } { \\sigma ^ { 2 } } .", + "type": "interline_equation", + "image_path": "b0964649def180ef259ab1b3372b6c936b942bb04fd129d0e2e27af19125700d.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 206, + 507, + 403, + 533 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 148, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 148, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 148, + 546 + ], + "score": 1.0, + "content": "Therefore", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 288, + 542, + 323, + 566 + ], + "lines": [ + { + "bbox": [ + 288, + 542, + 323, + 566 + ], + "spans": [ + { + "bbox": [ + 288, + 542, + 323, + 566 + ], + "score": 0.75, + "content": "c \\leq { \\frac { 1 } { \\sigma ^ { 2 } } } .", + "type": "interline_equation", + "image_path": "71a174ec59ca6c4d5c117b7d40f9d71badc788cf2dd794d7ee75a4faf6e3fe44.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 288, + 542, + 323, + 566 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 570, + 504, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 459, + 583 + ], + "score": 1.0, + "content": "It is not strictly necessary for the proof to show equality. 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K ( x ^ { \\prime } , \\cdot ) \\right) } { | | x - x ^ { \\prime } | | } } \\\\ { = \\displaystyle \\operatorname* { s u p } _ { x \\neq x ^ { \\prime } , y } \\frac { | | \\nabla _ { y } K ( x , y ) - \\nabla _ { y } K ( x ^ { \\prime } , y ) | | _ { 2 } } { | | x - x ^ { \\prime } | | } } \\\\ { \\leq \\displaystyle \\operatorname* { s u p } _ { x , y } | | \\nabla _ { x } \\nabla _ { y } K ( x , y ) | | _ { \\mathrm { o p } } , } \\end{array}", + "type": "interline_equation", + "image_path": "41e12e95c1cabe9cd78e4b9f7cae0757931efe39a87393c3fe63693f46721d20.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 219, + 148, + 390, + 164.4 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 219, + 164.4, + 390, + 180.8 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 219, + 180.8, + 390, + 197.20000000000002 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 219, + 197.20000000000002, + 390, + 213.60000000000002 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 219, + 213.60000000000002, + 390, + 230.00000000000003 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 229, + 504, + 252 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 243 + ], + "score": 1.0, + "content": "where we obtained by the last equality by applying the vector-valued mean value theorem (Rudin,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 239, + 337, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 142, + 255 + ], + "score": 1.0, + "content": "1964) to", + "type": "text" + }, + { + "bbox": [ + 143, + 240, + 261, + 253 + ], + "score": 0.89, + "content": "t \\mapsto \\nabla _ { y } K ( ( 1 - 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z } | 2 z - 1 | , \\operatorname* { s u p } _ { z \\ge 0 } \\frac { 1 } { \\sigma ^ { 2 } } e ^ { - z } \\bigg \\} = \\frac { 1 } { \\sigma ^ { 2 } } .", + "type": "interline_equation", + "image_path": "b0964649def180ef259ab1b3372b6c936b942bb04fd129d0e2e27af19125700d.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 206, + 507, + 403, + 533 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 148, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 148, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 148, + 546 + ], + "score": 1.0, + "content": "Therefore", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 534, + 148, + 546 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 288, + 542, + 323, + 566 + ], + "lines": [ + { + "bbox": [ + 288, + 542, + 323, + 566 + ], + "spans": [ + { + "bbox": [ + 288, + 542, + 323, + 566 + ], + "score": 0.75, + "content": "c \\leq { \\frac { 1 } { \\sigma ^ { 2 } } } .", + "type": "interline_equation", + "image_path": "71a174ec59ca6c4d5c117b7d40f9d71badc788cf2dd794d7ee75a4faf6e3fe44.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 288, + 542, + 323, + 566 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 570, + 504, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 459, + 583 + ], + "score": 1.0, + "content": "It is not strictly necessary for the proof to show equality. 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Then", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 570, + 505, + 594 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 594, + 413, + 737 + ], + "lines": [ + { + "bbox": [ + 196, + 594, + 413, + 737 + ], + "spans": [ + { + "bbox": [ + 196, + 594, + 413, + 737 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { c \\geq \\underset { t \\neq 0 } { \\operatorname* { s u p } } \\frac { K ( t u , t u ) - K ( t u , 0 ) - K ( 0 , t u ) + K ( 0 , 0 ) } { d ( t u , 0 ) d ( t u , 0 ) } } \\\\ & { = \\underset { t \\neq 0 } { \\operatorname* { s u p } } \\frac { 2 - 2 \\exp ( - \\frac { t ^ { 2 } } { 2 \\sigma ^ { 2 } } ) } { { t ^ { 2 } } } } \\\\ & { \\geq \\underset { t \\neq 0 } { \\operatorname* { l i m } } \\frac { 2 - 2 \\exp ( - \\frac { t ^ { 2 } } { 2 \\sigma ^ { 2 } } ) } { { t ^ { 2 } } } } \\\\ & { = \\underset { t \\neq 0 } { \\operatorname* { l i m } } \\frac { - 2 \\exp ( - \\frac { t ^ { 2 } } { 2 \\sigma ^ { 2 } } ) \\cdot - \\frac { 2 t } { 2 \\sigma ^ { 2 } } } { 2 t } } \\\\ & { = \\frac { 1 } { t \\neq 0 } , } \\end{array}", + "type": "interline_equation", + "image_path": "a4134feb8370abe258458b2371f6fb5e993f6fbfac5fca607870342569d68ad3.jpg" + } + ] + } + ], + "index": 35.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 594, + 413, + 608.3 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 196, + 608.3, + 413, + 622.5999999999999 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 196, + 622.5999999999999, + 413, + 636.8999999999999 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 196, + 636.8999999999999, + 413, + 651.1999999999998 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 196, + 651.1999999999998, + 413, + 665.4999999999998 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 196, + 665.4999999999998, + 413, + 679.7999999999997 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 196, + 679.7999999999997, + 413, + 694.0999999999997 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 196, + 694.0999999999997, + 413, + 708.3999999999996 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 196, + 708.3999999999996, + 413, + 722.6999999999996 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 196, + 722.6999999999996, + 413, + 736.9999999999995 + ], + "spans": [], + "index": 40 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 315, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 315, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 315, + 95 + ], + "score": 1.0, + "content": "where we used l’Hopital’s rule to evaluate the limit. ˆ", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 486, + 113 + ], + "lines": [ + { + "bbox": [ + 105, + 98, + 483, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 413, + 115 + ], + "score": 1.0, + "content": "Proposition 11. The minimax and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 413, + 101, + 433, + 112 + ], + "score": 0.62, + "content": "( D 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 98, + 471, + 115 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 471, + 103, + 483, + 112 + ], + "score": 0.78, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 124, + 336, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 337, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 337, + 139 + ], + "score": 1.0, + "content": "Proof. For the minimax loss, the Bregman divergence is:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 140, + 501, + 304 + ], + "lines": [ + { + "bbox": [ + 113, + 140, + 501, + 304 + ], + "spans": [ + { + "bbox": [ + 113, + 140, + 501, + 304 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\mathfrak { D } _ { B \\ni \\{ \\cdot \\} \\parallel \\mu _ { 0 } } ( v , \\mu ) = } & { \\displaystyle \\int \\left[ \\frac { 1 } { 2 } \\log \\frac { \\mu _ { 0 } } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } } \\ d \\mu _ { 0 } + \\frac { 1 } { 2 } \\log \\frac { v } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } d \\nu \\right] } \\\\ & { \\quad - \\displaystyle \\int \\left[ \\frac { 1 } { 2 } \\log \\frac { \\mu _ { 0 } } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\mu } d \\mu _ { 0 } + \\frac { 1 } { 2 } \\log \\frac { \\mu } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\mu } d \\mu \\right] } \\\\ & { \\quad - \\displaystyle \\int \\frac { 1 } { 2 } \\log \\frac { \\mu } { \\mu _ { 0 } + \\mu } d ( \\nu - \\mu ) } \\\\ & { = \\frac { 1 } { 2 } \\int \\log \\frac { \\mu _ { 0 } + \\mu } { \\mu _ { 0 } + \\nu } d \\mu _ { 0 } + \\frac { 1 } { 2 } \\int \\left[ \\log \\frac { \\nu } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } - \\log \\frac { \\mu } { \\mu _ { 0 } + \\mu } \\right] d \\nu + \\frac { 1 } { 2 } \\log \\frac { 1 } { 2 } } \\\\ & { = \\frac { 1 } { 2 } \\int \\log \\frac { \\mu _ { 0 } + \\mu } { \\mu _ { 0 } + \\nu } d ( \\mu _ { 0 } + \\nu ) + \\frac { 1 } { 2 } \\int \\log \\frac { \\nu } { \\mu } d \\nu } \\\\ & { = D _ { \\mathrm { K L } } ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } ) \\parallel \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } ) + \\frac { 1 } { 2 } D _ { \\mathrm { K L } } ( \\nu | \\mu ) , } \\end{array}", + "type": "interline_equation", + "image_path": "ab011b3d7abc2c586431d4b326d7c8c5d5dba719d53a70a1f96be2e66fc544ea.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 113, + 140, + 501, + 194.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 113, + 194.66666666666666, + 501, + 249.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 113, + 249.33333333333331, + 501, + 304.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 306, + 342, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 341, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 341, + 320 + ], + "score": 1.0, + "content": "and for the non-saturating loss, the Bregman divergence is", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 323, + 479, + 452 + ], + "lines": [ + { + "bbox": [ + 132, + 323, + 479, + 452 + ], + "spans": [ + { + "bbox": [ + 132, + 323, + 479, + 452 + ], + "score": 0.95, + "content": "\\begin{array} { l } { { \\displaystyle \\mathfrak { D } _ { N \\mathrm { k } , ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\mu _ { 0 } ) \\| \\mu _ { 0 } ) } ( \\nu , \\mu ) = \\int \\log \\frac { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } { \\mu _ { 0 } } d ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } ) } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } - \\int \\log \\frac { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\mu } { \\mu _ { 0 } } d ( \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } ) } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } + \\int \\frac { 1 } { 2 } \\log \\frac { \\mu _ { 0 } } { \\mu _ { 0 } + \\mu } d ( \\nu - \\mu ) } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } = \\frac { 1 } { 2 } \\int \\log \\frac { \\mu _ { 0 } + \\nu } { \\mu _ { 0 } + \\mu } d \\mu _ { 0 } + \\frac { 1 } { 2 } \\int \\log \\frac { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } { \\mu _ { 0 } + \\mu } d \\nu - \\frac { 1 } { 2 } \\log \\frac { 1 } { 2 } } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } = D _ { \\mathrm { K L } } ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } | | \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "624234490f122d9c8c8a64f264c0cffd9317857f4240f8be3fbea7675223677c.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 132, + 323, + 479, + 366.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 132, + 366.0, + 479, + 409.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 132, + 409.0, + 479, + 452.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 148, + 471 + ], + "score": 1.0, + "content": "Choosing", + "type": "text" + }, + { + "bbox": [ + 148, + 460, + 155, + 467 + ], + "score": 0.72, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 456, + 312, + 471 + ], + "score": 1.0, + "content": "to be not absolutely continuous w.r.t.", + "type": "text" + }, + { + "bbox": [ + 313, + 456, + 357, + 470 + ], + "score": 0.93, + "content": "{ \\scriptstyle { \\frac { 1 } { 2 } } } \\mu + { \\scriptstyle { \\frac { 1 } { 2 } } } \\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 456, + 490, + 471 + ], + "score": 1.0, + "content": "makes the Bregman divergence", + "type": "text" + }, + { + "bbox": [ + 490, + 460, + 501, + 468 + ], + "score": 0.76, + "content": "\\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 456, + 506, + 471 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 465, + 503, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 406, + 485 + ], + "score": 1.0, + "content": "which is sufficient to show that the Bregman divergence is not bounded by", + "type": "text" + }, + { + "bbox": [ + 406, + 469, + 453, + 482 + ], + "score": 0.92, + "content": "| | \\mu - \\nu | | _ { \\mathrm { K R } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 465, + 458, + 485 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 474, + 503, + 478 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 478, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 479, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 358, + 502 + ], + "score": 1.0, + "content": "Proposition 12. The MMD loss with Gaussian kernel satisfies", + "type": "text" + }, + { + "bbox": [ + 359, + 488, + 378, + 499 + ], + "score": 0.66, + "content": "( D 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 485, + 399, + 502 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 399, + 488, + 435, + 499 + ], + "score": 0.89, + "content": "\\beta _ { 2 } = 2 \\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 485, + 463, + 502 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 464, + 488, + 474, + 499 + ], + "score": 0.84, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 485, + 479, + 502 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 460, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 459, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 276, + 530 + ], + "score": 1.0, + "content": "Proof. We work with the Gaussian kernel", + "type": "text" + }, + { + "bbox": [ + 276, + 511, + 362, + 529 + ], + "score": 0.93, + "content": "K ( x , y ) = e ^ { - { \\frac { | | x - y | | ^ { 2 } } { 2 \\sigma ^ { 2 } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 516, + 459, + 529 + ], + "score": 1.0, + "content": "and use Proposition 10:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 534, + 466, + 687 + ], + "lines": [ + { + "bbox": [ + 145, + 534, + 466, + 687 + ], + "spans": [ + { + "bbox": [ + 145, + 534, + 466, + 687 + ], + "score": 0.96, + "content": "\\begin{array} { r l } { \\mathfrak { D } _ { \\frac { 1 } { 2 } \\mathrm { M M D } ^ { 2 } ( \\cdot , \\cdot , \\mu _ { 0 } ) } ( \\nu , \\mu ) = \\displaystyle \\frac { 1 } { 2 } \\int K ( x , y ) \\big ( \\nu - \\mu _ { 0 } ) ( d x ) \\big ( \\nu - \\mu _ { 0 } ) ( d y ) } & { } \\\\ & { \\qquad - \\frac { 1 } { 2 } \\int K ( x , y ) ( \\mu - \\mu _ { 0 } ) ( d x ) \\big ( \\mu - \\mu _ { 0 } ) ( d y ) } \\\\ & { \\qquad - \\displaystyle \\int ( K ( x , y ) \\mu ( d y ) - K ( x , y ) \\mu _ { 0 } ( d y ) \\big ) \\big ( \\nu - \\mu ) ( d x ) } \\\\ & { = \\displaystyle \\frac { 1 } { 2 } \\int K ( x , y ) \\big ( \\nu - \\mu ) ( d x ) \\big ( \\nu - \\mu ) ( d y ) } \\\\ & { = \\frac { 1 } { 2 } \\mathrm { M M D } ^ { 2 } ( \\mu , \\nu ) } \\\\ & { \\leq \\frac { 1 } { 2 \\pi ^ { 2 } } \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "8c6d7841abea1bee82f67bd624e7ca88be97bd4bfa59a83774faaae4385ae7e5.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 145, + 534, + 466, + 585.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 145, + 585.0, + 466, + 636.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 145, + 636.0, + 466, + 687.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 691, + 413, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 412, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 351, + 706 + ], + "score": 1.0, + "content": "where the last line is from Lemma 7. The result follows with", + "type": "text" + }, + { + "bbox": [ + 352, + 691, + 409, + 705 + ], + "score": 0.93, + "content": "\\sigma = ( 2 \\pi ) ^ { - 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 691, + 412, + 706 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 105, + 709, + 507, + 734 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 475, + 722 + ], + "score": 1.0, + "content": "The result actually applies more generally for the MMD loss with a differentiable kernel", + "type": "text" + }, + { + "bbox": [ + 475, + 710, + 486, + 720 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 140, + 734 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 140, + 720, + 293, + 734 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\beta _ { 2 } : = \\operatorname* { s u p } _ { x , y } \\bar { \\| \\nabla _ { x } \\nabla _ { y } K ( \\bar { x } , y ) \\| _ { 2 } } < \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 719, + 297, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 721, + 506, + 731 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + } + ], + "page_idx": 27, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "28", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 82, + 505, + 93 + ], + "lines": [ + { + "bbox": [ + 496, + 84, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 496, + 84, + 505, + 93 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 315, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 315, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 315, + 95 + ], + "score": 1.0, + "content": "where we used l’Hopital’s rule to evaluate the limit. ˆ", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 315, + 95 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 486, + 113 + ], + "lines": [ + { + "bbox": [ + 105, + 98, + 483, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 413, + 115 + ], + "score": 1.0, + "content": "Proposition 11. The minimax and non-saturating GAN losses do not satisfy", + "type": "text" + }, + { + "bbox": [ + 413, + 101, + 433, + 112 + ], + "score": 0.62, + "content": "( D 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 98, + 471, + 115 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 471, + 103, + 483, + 112 + ], + "score": 0.78, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 98, + 483, + 115 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 124, + 336, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 337, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 337, + 139 + ], + "score": 1.0, + "content": "Proof. For the minimax loss, the Bregman divergence is:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 122, + 337, + 139 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 140, + 501, + 304 + ], + "lines": [ + { + "bbox": [ + 113, + 140, + 501, + 304 + ], + "spans": [ + { + "bbox": [ + 113, + 140, + 501, + 304 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\mathfrak { D } _ { B \\ni \\{ \\cdot \\} \\parallel \\mu _ { 0 } } ( v , \\mu ) = } & { \\displaystyle \\int \\left[ \\frac { 1 } { 2 } \\log \\frac { \\mu _ { 0 } } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } } \\ d \\mu _ { 0 } + \\frac { 1 } { 2 } \\log \\frac { v } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } d \\nu \\right] } \\\\ & { \\quad - \\displaystyle \\int \\left[ \\frac { 1 } { 2 } \\log \\frac { \\mu _ { 0 } } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\mu } d \\mu _ { 0 } + \\frac { 1 } { 2 } \\log \\frac { \\mu } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\mu } d \\mu \\right] } \\\\ & { \\quad - \\displaystyle \\int \\frac { 1 } { 2 } \\log \\frac { \\mu } { \\mu _ { 0 } + \\mu } d ( \\nu - \\mu ) } \\\\ & { = \\frac { 1 } { 2 } \\int \\log \\frac { \\mu _ { 0 } + \\mu } { \\mu _ { 0 } + \\nu } d \\mu _ { 0 } + \\frac { 1 } { 2 } \\int \\left[ \\log \\frac { \\nu } { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } - \\log \\frac { \\mu } { \\mu _ { 0 } + \\mu } \\right] d \\nu + \\frac { 1 } { 2 } \\log \\frac { 1 } { 2 } } \\\\ & { = \\frac { 1 } { 2 } \\int \\log \\frac { \\mu _ { 0 } + \\mu } { \\mu _ { 0 } + \\nu } d ( \\mu _ { 0 } + \\nu ) + \\frac { 1 } { 2 } \\int \\log \\frac { \\nu } { \\mu } d \\nu } \\\\ & { = D _ { \\mathrm { K L } } ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } ) \\parallel \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } ) + \\frac { 1 } { 2 } D _ { \\mathrm { K L } } ( \\nu | \\mu ) , } \\end{array}", + "type": "interline_equation", + "image_path": "ab011b3d7abc2c586431d4b326d7c8c5d5dba719d53a70a1f96be2e66fc544ea.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 113, + 140, + 501, + 194.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 113, + 194.66666666666666, + 501, + 249.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 113, + 249.33333333333331, + 501, + 304.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 306, + 342, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 341, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 341, + 320 + ], + "score": 1.0, + "content": "and for the non-saturating loss, the Bregman divergence is", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 306, + 341, + 320 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 323, + 479, + 452 + ], + "lines": [ + { + "bbox": [ + 132, + 323, + 479, + 452 + ], + "spans": [ + { + "bbox": [ + 132, + 323, + 479, + 452 + ], + "score": 0.95, + "content": "\\begin{array} { l } { { \\displaystyle \\mathfrak { D } _ { N \\mathrm { k } , ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\mu _ { 0 } ) \\| \\mu _ { 0 } ) } ( \\nu , \\mu ) = \\int \\log \\frac { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } { \\mu _ { 0 } } d ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } ) } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } - \\int \\log \\frac { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\mu } { \\mu _ { 0 } } d ( \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } ) } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } + \\int \\frac { 1 } { 2 } \\log \\frac { \\mu _ { 0 } } { \\mu _ { 0 } + \\mu } d ( \\nu - \\mu ) } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } = \\frac { 1 } { 2 } \\int \\log \\frac { \\mu _ { 0 } + \\nu } { \\mu _ { 0 } + \\mu } d \\mu _ { 0 } + \\frac { 1 } { 2 } \\int \\log \\frac { \\frac { 1 } { 2 } \\mu _ { 0 } + \\frac { 1 } { 2 } \\nu } { \\mu _ { 0 } + \\mu } d \\nu - \\frac { 1 } { 2 } \\log \\frac { 1 } { 2 } } } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 2 } } = D _ { \\mathrm { K L } } ( \\frac { 1 } { 2 } \\nu + \\frac { 1 } { 2 } \\mu _ { 0 } | | \\frac { 1 } { 2 } \\mu + \\frac { 1 } { 2 } \\mu _ { 0 } ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "624234490f122d9c8c8a64f264c0cffd9317857f4240f8be3fbea7675223677c.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 132, + 323, + 479, + 366.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 132, + 366.0, + 479, + 409.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 132, + 409.0, + 479, + 452.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 148, + 471 + ], + "score": 1.0, + "content": "Choosing", + "type": "text" + }, + { + "bbox": [ + 148, + 460, + 155, + 467 + ], + "score": 0.72, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 456, + 312, + 471 + ], + "score": 1.0, + "content": "to be not absolutely continuous w.r.t.", + "type": "text" + }, + { + "bbox": [ + 313, + 456, + 357, + 470 + ], + "score": 0.93, + "content": "{ \\scriptstyle { \\frac { 1 } { 2 } } } \\mu + { \\scriptstyle { \\frac { 1 } { 2 } } } \\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 456, + 490, + 471 + ], + "score": 1.0, + "content": "makes the Bregman divergence", + "type": "text" + }, + { + "bbox": [ + 490, + 460, + 501, + 468 + ], + "score": 0.76, + "content": "\\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 456, + 506, + 471 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 465, + 503, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 406, + 485 + ], + "score": 1.0, + "content": "which is sufficient to show that the Bregman divergence is not bounded by", + "type": "text" + }, + { + "bbox": [ + 406, + 469, + 453, + 482 + ], + "score": 0.92, + "content": "| | \\mu - \\nu | | _ { \\mathrm { K R } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 465, + 458, + 485 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 474, + 503, + 478 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 456, + 506, + 485 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 478, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 479, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 358, + 502 + ], + "score": 1.0, + "content": "Proposition 12. The MMD loss with Gaussian kernel satisfies", + "type": "text" + }, + { + "bbox": [ + 359, + 488, + 378, + 499 + ], + "score": 0.66, + "content": "( D 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 485, + 399, + 502 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 399, + 488, + 435, + 499 + ], + "score": 0.89, + "content": "\\beta _ { 2 } = 2 \\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 485, + 463, + 502 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 464, + 488, + 474, + 499 + ], + "score": 0.84, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 485, + 479, + 502 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 485, + 479, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 460, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 459, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 276, + 530 + ], + "score": 1.0, + "content": "Proof. We work with the Gaussian kernel", + "type": "text" + }, + { + "bbox": [ + 276, + 511, + 362, + 529 + ], + "score": 0.93, + "content": "K ( x , y ) = e ^ { - { \\frac { | | x - y | | ^ { 2 } } { 2 \\sigma ^ { 2 } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 516, + 459, + 529 + ], + "score": 1.0, + "content": "and use Proposition 10:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 510, + 459, + 530 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 534, + 466, + 687 + ], + "lines": [ + { + "bbox": [ + 145, + 534, + 466, + 687 + ], + "spans": [ + { + "bbox": [ + 145, + 534, + 466, + 687 + ], + "score": 0.96, + "content": "\\begin{array} { r l } { \\mathfrak { D } _ { \\frac { 1 } { 2 } \\mathrm { M M D } ^ { 2 } ( \\cdot , \\cdot , \\mu _ { 0 } ) } ( \\nu , \\mu ) = \\displaystyle \\frac { 1 } { 2 } \\int K ( x , y ) \\big ( \\nu - \\mu _ { 0 } ) ( d x ) \\big ( \\nu - \\mu _ { 0 } ) ( d y ) } & { } \\\\ & { \\qquad - \\frac { 1 } { 2 } \\int K ( x , y ) ( \\mu - \\mu _ { 0 } ) ( d x ) \\big ( \\mu - \\mu _ { 0 } ) ( d y ) } \\\\ & { \\qquad - \\displaystyle \\int ( K ( x , y ) \\mu ( d y ) - K ( x , y ) \\mu _ { 0 } ( d y ) \\big ) \\big ( \\nu - \\mu ) ( d x ) } \\\\ & { = \\displaystyle \\frac { 1 } { 2 } \\int K ( x , y ) \\big ( \\nu - \\mu ) ( d x ) \\big ( \\nu - \\mu ) ( d y ) } \\\\ & { = \\frac { 1 } { 2 } \\mathrm { M M D } ^ { 2 } ( \\mu , \\nu ) } \\\\ & { \\leq \\frac { 1 } { 2 \\pi ^ { 2 } } \\| \\mu - \\nu \\| _ { \\mathrm { K R } } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "8c6d7841abea1bee82f67bd624e7ca88be97bd4bfa59a83774faaae4385ae7e5.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 145, + 534, + 466, + 585.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 145, + 585.0, + 466, + 636.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 145, + 636.0, + 466, + 687.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 691, + 413, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 412, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 351, + 706 + ], + "score": 1.0, + "content": "where the last line is from Lemma 7. The result follows with", + "type": "text" + }, + { + "bbox": [ + 352, + 691, + 409, + 705 + ], + "score": 0.93, + "content": "\\sigma = ( 2 \\pi ) ^ { - 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 691, + 412, + 706 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 691, + 412, + 706 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 709, + 507, + 734 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 475, + 722 + ], + "score": 1.0, + "content": "The result actually applies more generally for the MMD loss with a differentiable kernel", + "type": "text" + }, + { + "bbox": [ + 475, + 710, + 486, + 720 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 140, + 734 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 140, + 720, + 293, + 734 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\beta _ { 2 } : = \\operatorname* { s u p } _ { x , y } \\bar { \\| \\nabla _ { x } \\nabla _ { y } K ( \\bar { x } , y ) \\| _ { 2 } } < \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 719, + 297, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 721, + 506, + 731 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 709, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 109 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 344, + 96 + ], + "score": 1.0, + "content": "Proposition 13 (Moreau–Yosida regularization). Suppose", + "type": "text" + }, + { + "bbox": [ + 344, + 82, + 416, + 95 + ], + "score": 0.92, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 80, + 505, + 96 + ], + "score": 1.0, + "content": "is convex, and define", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 90, + 369, + 111 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 161, + 106 + ], + "score": 0.9, + "content": "{ \\tilde { J } } : = J \\oplus R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 90, + 187, + 111 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 188, + 94, + 196, + 106 + ], + "score": 0.82, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 90, + 255, + 111 + ], + "score": 1.0, + "content": "is convex, and", + "type": "text" + }, + { + "bbox": [ + 255, + 93, + 365, + 109 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathfrak { D } _ { \\widetilde { J } } ( \\nu , \\mu ) \\le \\frac { \\beta _ { 2 } } { 2 } \\vert \\vert \\mu - \\nu \\vert \\vert _ { \\mathrm { K R } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 90, + 369, + 111 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 105, + 151, + 491, + 168 + ], + "lines": [ + { + "bbox": [ + 101, + 148, + 492, + 172 + ], + "spans": [ + { + "bbox": [ + 101, + 148, + 447, + 172 + ], + "score": 1.0, + "content": "Proof. We work on the more general case of K(x, y) = (2πσ2)−d/2 exp(− kx−yk22σ2 ),", + "type": "text" + }, + { + "bbox": [ + 443, + 154, + 492, + 167 + ], + "score": 1.0, + "content": "and define", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 182, + 382, + 207 + ], + "lines": [ + { + "bbox": [ + 228, + 182, + 382, + 207 + ], + "spans": [ + { + "bbox": [ + 228, + 182, + 382, + 207 + ], + "score": 0.93, + "content": "\\tilde { J } ( \\mu ) = \\operatorname * { i n f } _ { \\tilde { \\mu } } J ( \\tilde { \\mu } ) + \\frac { c L } { 2 } \\mathrm { M M D } _ { K } ^ { 2 } ( \\mu , \\tilde { \\mu } )", + "type": "interline_equation", + "image_path": "fa50d72d8c9073b097d83aa6240cf0912e7cb04b02653c86e316076706e38cac.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 228, + 182, + 382, + 207 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 223, + 195, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 196, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 120, + 237 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 223, + 192, + 236 + ], + "score": 0.91, + "content": "c = \\sigma ^ { 2 } ( 2 \\pi \\sigma ^ { 2 } ) ^ { d / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 222, + 196, + 237 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 105, + 241, + 504, + 264 + ], + "lines": [ + { + "bbox": [ + 104, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 123, + 255 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 242, + 135, + 253 + ], + "score": 0.88, + "content": "\\mu ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "be the unique minimizer of the infimum, which exists because the function is a strongly", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 252, + 314, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 314, + 264 + ], + "score": 1.0, + "content": "convex. By the envelope theorem, we compute that", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 277, + 459, + 370 + ], + "lines": [ + { + "bbox": [ + 151, + 277, + 459, + 370 + ], + "spans": [ + { + "bbox": [ + 151, + 277, + 459, + 370 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\frac { d } { d \\epsilon } \\tilde { J } ( \\mu + \\epsilon \\chi ) \\Big | _ { \\epsilon = 0 } = \\frac { d } { d \\epsilon } \\frac { c L } { 2 } | | \\mu + \\epsilon \\chi - \\mu ^ { * } | | _ { \\mathcal { H } } ^ { 2 } \\Big | _ { \\epsilon = 0 } } \\\\ & { \\qquad = \\frac { d } { d \\epsilon } \\frac { c L } { 2 } \\big ( | | \\mu - \\mu ^ { * } | | _ { \\mathcal { H } } ^ { 2 } + 2 \\epsilon \\langle \\mu - \\mu ^ { * } , \\chi \\rangle _ { \\mathcal { H } } + \\epsilon ^ { 2 } | | \\chi | | _ { \\mathcal { H } } ^ { 2 } \\big ) \\Big | _ { \\epsilon = 0 } } \\\\ & { \\qquad = c L \\langle \\mu - \\mu ^ { * } , \\chi \\rangle _ { \\mathcal { H } } } \\\\ & { \\qquad = c L \\int ( \\mu - \\mu ^ { * } ) d \\chi , } \\end{array}", + "type": "interline_equation", + "image_path": "7ffd00c12dd3d738b41e3f3e443cb91be716fec8f159ea9e266b8d26f611c8df.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 151, + 277, + 459, + 308.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 151, + 308.0, + 459, + 339.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 151, + 339.0, + 459, + 370.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 194, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 191, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 118, + 402 + ], + "score": 1.0, + "content": "so", + "type": "text" + }, + { + "bbox": [ + 118, + 383, + 191, + 401 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { \\delta \\tilde { J } } { \\delta \\mu } = c L ( \\mu - \\mu ^ { * } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 129, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 131, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 131, + 417 + ], + "score": 1.0, + "content": "Then", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 429, + 491, + 581 + ], + "lines": [ + { + "bbox": [ + 119, + 429, + 491, + 581 + ], + "spans": [ + { + "bbox": [ + 119, + 429, + 491, + 581 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathfrak { D } _ { \\mathcal { J } } ( \\nu , \\mu ) = \\mathcal { \\tilde { J } } ( \\nu ) - \\mathcal { \\tilde { J } } ( \\mu ) - \\displaystyle \\int \\frac { \\delta \\mathcal { \\tilde { J } } } { \\delta \\mu } d ( \\nu - \\mu ) } \\\\ & { \\qquad = \\operatorname* { i n f } J ( \\tilde { \\nu } ) + \\frac { c L } { 2 } \\| \\nu - \\tilde { \\nu } \\| _ { \\mathcal H } ^ { 2 } - \\left[ J ( \\mu ^ { * } ) + \\frac { c L } { 2 } \\| \\mu - \\mu ^ { * } \\| _ { \\mathcal H } ^ { 2 } \\right] - c L \\langle \\mu - \\mu ^ { * } , \\nu - \\mu \\rangle _ { \\mathcal H } } \\\\ & { \\qquad \\le J ( \\mu ^ { * } ) + \\frac { c L } { 2 } \\| \\nu - \\mu ^ { * } \\| _ { \\mathcal H } ^ { 2 } - \\left[ J ( \\mu ^ { * } ) + \\frac { c L } { 2 } \\| \\mu - \\mu ^ { * } \\| _ { \\mathcal H } ^ { 2 } \\right] - c L \\langle \\mu - \\mu ^ { * } , \\nu - \\mu \\rangle _ { \\mathcal H } } \\\\ & { \\qquad = \\frac { c L } { 2 } \\| \\mu - \\nu \\| _ { \\mathcal H } ^ { 2 } } \\\\ & { \\qquad \\le \\frac { c L } { 2 } \\cdot ( 2 \\pi \\sigma ^ { 2 } ) ^ { - d / 2 } \\cdot \\frac { 1 } { \\sigma ^ { 2 } } \\| \\mu - \\nu \\| _ { \\mathcal H } ^ { 2 } } \\\\ & { \\qquad = \\frac { L } { 2 } \\| \\mu - \\nu \\| _ { \\mathbb K ^ { 0 } } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "6c3a00581fb70cabe3bb42b8516aef3b8aa0a2270f5120d88531855fe188a19a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 119, + 429, + 491, + 479.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 119, + 479.6666666666667, + 491, + 530.3333333333334 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 119, + 530.3333333333334, + 491, + 581.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 595, + 489, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 486, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 428, + 609 + ], + "score": 1.0, + "content": "where we used Lemma 7 for the second-to-last line. The proposition follows for", + "type": "text" + }, + { + "bbox": [ + 429, + 594, + 486, + 608 + ], + "score": 0.93, + "content": "\\sigma = ( 2 \\pi ) ^ { - 1 / 2 }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 612, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 205, + 625 + ], + "score": 1.0, + "content": "Regarding this choice of", + "type": "text" + }, + { + "bbox": [ + 205, + 615, + 212, + 622 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 612, + 505, + 625 + ], + "score": 1.0, + "content": ", it will turn out that in the general case, the dual penalty includes a numer-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 622, + 507, + 638 + ], + "spans": [ + { + "bbox": [ + 104, + 622, + 219, + 638 + ], + "score": 1.0, + "content": "ically unfavorable factor of", + "type": "text" + }, + { + "bbox": [ + 220, + 623, + 268, + 636 + ], + "score": 0.93, + "content": "( 2 \\pi \\sigma ^ { 2 } ) ^ { - d / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 622, + 507, + 638 + ], + "score": 1.0, + "content": ", dependent on the dimension of the problem. In practical", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 636, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 269, + 648 + ], + "score": 1.0, + "content": "applications, such as image generation,", + "type": "text" + }, + { + "bbox": [ + 269, + 636, + 276, + 645 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 636, + 506, + 648 + ], + "score": 1.0, + "content": "can be quite large, making the accurate computation of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 646, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 107, + 646, + 156, + 659 + ], + "score": 0.91, + "content": "( \\bar { 2 } \\pi \\sigma ^ { 2 } ) ^ { - d / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 646, + 506, + 661 + ], + "score": 1.0, + "content": "completely infeasible. For numerical stability, we propose choosing the critical param-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 659, + 504, + 673 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 124, + 673 + ], + "score": 1.0, + "content": "eter", + "type": "text" + }, + { + "bbox": [ + 124, + 659, + 181, + 673 + ], + "score": 0.94, + "content": "\\sigma = ( 2 \\pi ) ^ { - 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 659, + 378, + 673 + ], + "score": 1.0, + "content": ", which corresponds to the dimension-free kernel", + "type": "text" + }, + { + "bbox": [ + 379, + 659, + 469, + 673 + ], + "score": 0.92, + "content": "K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 659, + 474, + 673 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 662, + 504, + 671 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 734 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 171, + 722 + ], + "score": 1.0, + "content": "Lemma 8. Let", + "type": "text" + }, + { + "bbox": [ + 171, + 710, + 180, + 720 + ], + "score": 0.66, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 708, + 338, + 722 + ], + "score": 1.0, + "content": "be an RKHS with a continuous kernel", + "type": "text" + }, + { + "bbox": [ + 339, + 710, + 349, + 720 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 708, + 441, + 722 + ], + "score": 1.0, + "content": "on a compact domain", + "type": "text" + }, + { + "bbox": [ + 441, + 710, + 451, + 720 + ], + "score": 0.65, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 708, + 506, + 722 + ], + "score": 1.0, + "content": ". The convex", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 718, + 340, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 159, + 735 + ], + "score": 1.0, + "content": "conjugate of", + "type": "text" + }, + { + "bbox": [ + 159, + 720, + 212, + 734 + ], + "score": 0.93, + "content": "\\mu \\mapsto \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathcal { H } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 718, + 223, + 735 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 224, + 720, + 336, + 734 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f \\mapsto \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathcal { H } } ^ { 2 } + \\chi \\{ f \\in \\mathcal { H } \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 718, + 340, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + } + ], + "page_idx": 28, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 109 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 344, + 96 + ], + "score": 1.0, + "content": "Proposition 13 (Moreau–Yosida regularization). Suppose", + "type": "text" + }, + { + "bbox": [ + 344, + 82, + 416, + 95 + ], + "score": 0.92, + "content": "J : \\mathcal { M } ( X ) \\bar { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 80, + 505, + 96 + ], + "score": 1.0, + "content": "is convex, and define", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 90, + 369, + 111 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 161, + 106 + ], + "score": 0.9, + "content": "{ \\tilde { J } } : = J \\oplus R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 90, + 187, + 111 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 188, + 94, + 196, + 106 + ], + "score": 0.82, + "content": "\\tilde { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 90, + 255, + 111 + ], + "score": 1.0, + "content": "is convex, and", + "type": "text" + }, + { + "bbox": [ + 255, + 93, + 365, + 109 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathfrak { D } _ { \\widetilde { J } } ( \\nu , \\mu ) \\le \\frac { \\beta _ { 2 } } { 2 } \\vert \\vert \\mu - \\nu \\vert \\vert _ { \\mathrm { K R } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 90, + 369, + 111 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 80, + 505, + 111 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 151, + 491, + 168 + ], + "lines": [ + { + "bbox": [ + 101, + 148, + 492, + 172 + ], + "spans": [ + { + "bbox": [ + 101, + 148, + 447, + 172 + ], + "score": 1.0, + "content": "Proof. We work on the more general case of K(x, y) = (2πσ2)−d/2 exp(− kx−yk22σ2 ),", + "type": "text" + }, + { + "bbox": [ + 443, + 154, + 492, + 167 + ], + "score": 1.0, + "content": "and define", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 101, + 148, + 492, + 172 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 182, + 382, + 207 + ], + "lines": [ + { + "bbox": [ + 228, + 182, + 382, + 207 + ], + "spans": [ + { + "bbox": [ + 228, + 182, + 382, + 207 + ], + "score": 0.93, + "content": "\\tilde { J } ( \\mu ) = \\operatorname * { i n f } _ { \\tilde { \\mu } } J ( \\tilde { \\mu } ) + \\frac { c L } { 2 } \\mathrm { M M D } _ { K } ^ { 2 } ( \\mu , \\tilde { \\mu } )", + "type": "interline_equation", + "image_path": "fa50d72d8c9073b097d83aa6240cf0912e7cb04b02653c86e316076706e38cac.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 228, + 182, + 382, + 207 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 223, + 195, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 196, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 120, + 237 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 223, + 192, + 236 + ], + "score": 0.91, + "content": "c = \\sigma ^ { 2 } ( 2 \\pi \\sigma ^ { 2 } ) ^ { d / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 222, + 196, + 237 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 222, + 196, + 237 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 241, + 504, + 264 + ], + "lines": [ + { + "bbox": [ + 104, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 123, + 255 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 242, + 135, + 253 + ], + "score": 0.88, + "content": "\\mu ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "be the unique minimizer of the infimum, which exists because the function is a strongly", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 252, + 314, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 314, + 264 + ], + "score": 1.0, + "content": "convex. By the envelope theorem, we compute that", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 240, + 505, + 264 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 277, + 459, + 370 + ], + "lines": [ + { + "bbox": [ + 151, + 277, + 459, + 370 + ], + "spans": [ + { + "bbox": [ + 151, + 277, + 459, + 370 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\frac { d } { d \\epsilon } \\tilde { J } ( \\mu + \\epsilon \\chi ) \\Big | _ { \\epsilon = 0 } = \\frac { d } { d \\epsilon } \\frac { c L } { 2 } | | \\mu + \\epsilon \\chi - \\mu ^ { * } | | _ { \\mathcal { H } } ^ { 2 } \\Big | _ { \\epsilon = 0 } } \\\\ & { \\qquad = \\frac { d } { d \\epsilon } \\frac { c L } { 2 } \\big ( | | \\mu - \\mu ^ { * } | | _ { \\mathcal { H } } ^ { 2 } + 2 \\epsilon \\langle \\mu - \\mu ^ { * } , \\chi \\rangle _ { \\mathcal { H } } + \\epsilon ^ { 2 } | | \\chi | | _ { \\mathcal { H } } ^ { 2 } \\big ) \\Big | _ { \\epsilon = 0 } } \\\\ & { \\qquad = c L \\langle \\mu - \\mu ^ { * } , \\chi \\rangle _ { \\mathcal { H } } } \\\\ & { \\qquad = c L \\int ( \\mu - \\mu ^ { * } ) d \\chi , } \\end{array}", + "type": "interline_equation", + "image_path": "7ffd00c12dd3d738b41e3f3e443cb91be716fec8f159ea9e266b8d26f611c8df.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 151, + 277, + 459, + 308.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 151, + 308.0, + 459, + 339.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 151, + 339.0, + 459, + 370.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 194, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 191, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 118, + 402 + ], + "score": 1.0, + "content": "so", + "type": "text" + }, + { + "bbox": [ + 118, + 383, + 191, + 401 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { \\delta \\tilde { J } } { \\delta \\mu } = c L ( \\mu - \\mu ^ { * } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 382, + 191, + 402 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 129, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 131, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 131, + 417 + ], + "score": 1.0, + "content": "Then", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 402, + 131, + 417 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 429, + 491, + 581 + ], + "lines": [ + { + "bbox": [ + 119, + 429, + 491, + 581 + ], + "spans": [ + { + "bbox": [ + 119, + 429, + 491, + 581 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathfrak { D } _ { \\mathcal { J } } ( \\nu , \\mu ) = \\mathcal { \\tilde { J } } ( \\nu ) - \\mathcal { \\tilde { J } } ( \\mu ) - \\displaystyle \\int \\frac { \\delta \\mathcal { \\tilde { J } } } { \\delta \\mu } d ( \\nu - \\mu ) } \\\\ & { \\qquad = \\operatorname* { i n f } J ( \\tilde { \\nu } ) + \\frac { c L } { 2 } \\| \\nu - \\tilde { \\nu } \\| _ { \\mathcal H } ^ { 2 } - \\left[ J ( \\mu ^ { * } ) + \\frac { c L } { 2 } \\| \\mu - \\mu ^ { * } \\| _ { \\mathcal H } ^ { 2 } \\right] - c L \\langle \\mu - \\mu ^ { * } , \\nu - \\mu \\rangle _ { \\mathcal H } } \\\\ & { \\qquad \\le J ( \\mu ^ { * } ) + \\frac { c L } { 2 } \\| \\nu - \\mu ^ { * } \\| _ { \\mathcal H } ^ { 2 } - \\left[ J ( \\mu ^ { * } ) + \\frac { c L } { 2 } \\| \\mu - \\mu ^ { * } \\| _ { \\mathcal H } ^ { 2 } \\right] - c L \\langle \\mu - \\mu ^ { * } , \\nu - \\mu \\rangle _ { \\mathcal H } } \\\\ & { \\qquad = \\frac { c L } { 2 } \\| \\mu - \\nu \\| _ { \\mathcal H } ^ { 2 } } \\\\ & { \\qquad \\le \\frac { c L } { 2 } \\cdot ( 2 \\pi \\sigma ^ { 2 } ) ^ { - d / 2 } \\cdot \\frac { 1 } { \\sigma ^ { 2 } } \\| \\mu - \\nu \\| _ { \\mathcal H } ^ { 2 } } \\\\ & { \\qquad = \\frac { L } { 2 } \\| \\mu - \\nu \\| _ { \\mathbb K ^ { 0 } } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "6c3a00581fb70cabe3bb42b8516aef3b8aa0a2270f5120d88531855fe188a19a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 119, + 429, + 491, + 479.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 119, + 479.6666666666667, + 491, + 530.3333333333334 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 119, + 530.3333333333334, + 491, + 581.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 595, + 489, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 486, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 428, + 609 + ], + "score": 1.0, + "content": "where we used Lemma 7 for the second-to-last line. The proposition follows for", + "type": "text" + }, + { + "bbox": [ + 429, + 594, + 486, + 608 + ], + "score": 0.93, + "content": "\\sigma = ( 2 \\pi ) ^ { - 1 / 2 }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 593, + 486, + 609 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 612, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 205, + 625 + ], + "score": 1.0, + "content": "Regarding this choice of", + "type": "text" + }, + { + "bbox": [ + 205, + 615, + 212, + 622 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 612, + 505, + 625 + ], + "score": 1.0, + "content": ", it will turn out that in the general case, the dual penalty includes a numer-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 622, + 507, + 638 + ], + "spans": [ + { + "bbox": [ + 104, + 622, + 219, + 638 + ], + "score": 1.0, + "content": "ically unfavorable factor of", + "type": "text" + }, + { + "bbox": [ + 220, + 623, + 268, + 636 + ], + "score": 0.93, + "content": "( 2 \\pi \\sigma ^ { 2 } ) ^ { - d / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 622, + 507, + 638 + ], + "score": 1.0, + "content": ", dependent on the dimension of the problem. In practical", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 636, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 269, + 648 + ], + "score": 1.0, + "content": "applications, such as image generation,", + "type": "text" + }, + { + "bbox": [ + 269, + 636, + 276, + 645 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 636, + 506, + 648 + ], + "score": 1.0, + "content": "can be quite large, making the accurate computation of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 646, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 107, + 646, + 156, + 659 + ], + "score": 0.91, + "content": "( \\bar { 2 } \\pi \\sigma ^ { 2 } ) ^ { - d / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 646, + 506, + 661 + ], + "score": 1.0, + "content": "completely infeasible. For numerical stability, we propose choosing the critical param-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 659, + 504, + 673 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 124, + 673 + ], + "score": 1.0, + "content": "eter", + "type": "text" + }, + { + "bbox": [ + 124, + 659, + 181, + 673 + ], + "score": 0.94, + "content": "\\sigma = ( 2 \\pi ) ^ { - 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 659, + 378, + 673 + ], + "score": 1.0, + "content": ", which corresponds to the dimension-free kernel", + "type": "text" + }, + { + "bbox": [ + 379, + 659, + 469, + 673 + ], + "score": 0.92, + "content": "K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 659, + 474, + 673 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 662, + 504, + 671 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 612, + 507, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 734 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 171, + 722 + ], + "score": 1.0, + "content": "Lemma 8. Let", + "type": "text" + }, + { + "bbox": [ + 171, + 710, + 180, + 720 + ], + "score": 0.66, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 708, + 338, + 722 + ], + "score": 1.0, + "content": "be an RKHS with a continuous kernel", + "type": "text" + }, + { + "bbox": [ + 339, + 710, + 349, + 720 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 708, + 441, + 722 + ], + "score": 1.0, + "content": "on a compact domain", + "type": "text" + }, + { + "bbox": [ + 441, + 710, + 451, + 720 + ], + "score": 0.65, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 708, + 506, + 722 + ], + "score": 1.0, + "content": ". The convex", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 718, + 340, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 159, + 735 + ], + "score": 1.0, + "content": "conjugate of", + "type": "text" + }, + { + "bbox": [ + 159, + 720, + 212, + 734 + ], + "score": 0.93, + "content": "\\mu \\mapsto \\frac { \\lambda } { 2 } | | \\mu | | _ { \\mathcal { H } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 718, + 223, + 735 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 224, + 720, + 336, + 734 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f \\mapsto \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathcal { H } } ^ { 2 } + \\chi \\{ f \\in \\mathcal { H } \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 718, + 340, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 708, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 304, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 304, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 273, + 96 + ], + "score": 1.0, + "content": "Proof. First we show an upper bound for", + "type": "text" + }, + { + "bbox": [ + 273, + 83, + 300, + 94 + ], + "score": 0.91, + "content": "f \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 82, + 304, + 96 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 99, + 424, + 235 + ], + "lines": [ + { + "bbox": [ + 186, + 99, + 424, + 235 + ], + "spans": [ + { + "bbox": [ + 186, + 99, + 424, + 235 + ], + "score": 0.96, + "content": "\\begin{array} { l } { \\displaystyle \\operatorname* { s u p } _ { \\mu } [ \\int f d \\mu - \\frac \\lambda 2 | | \\mu | | _ { \\mathcal H } ^ { 2 } ] = \\operatorname* { s u p } _ { \\mu } [ \\langle f , \\mu \\rangle _ { \\mathcal H } - \\frac \\lambda 2 | | \\mu | | _ { \\mathcal H } ^ { 2 } ] } \\\\ { \\displaystyle \\quad \\leq \\operatorname* { s u p } _ { \\mu } [ | | f | | \\mu | | \\mu | ] _ { \\mathcal H } - \\frac \\lambda 2 | | \\mu | | _ { \\mathcal H } ^ { 2 } ] } \\\\ { \\displaystyle \\quad \\leq \\operatorname* { s u p } _ { z \\in \\mathbb R } [ | | f | | \\mu z - \\frac \\lambda 2 z ^ { 2 } ] } \\\\ { \\displaystyle \\quad = | | f | | _ { \\mathcal H } \\cdot \\frac { | | f | | _ { \\mathcal H } } { \\lambda } - \\frac { \\lambda } { 2 } \\cdot \\frac { | | f | | _ { \\mathcal H } ^ { 2 } } { \\lambda ^ { 2 } } } \\\\ { \\displaystyle \\quad = \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathcal H } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "cfeb19d9fe1087df3a7d5a9f63b774727f1173339edc9e4dd644357d389a913d.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 186, + 99, + 424, + 114.11111111111111 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 186, + 114.11111111111111, + 424, + 129.22222222222223 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 186, + 129.22222222222223, + 424, + 144.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 186, + 144.33333333333334, + 424, + 159.44444444444446 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 186, + 159.44444444444446, + 424, + 174.55555555555557 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 186, + 174.55555555555557, + 424, + 189.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 186, + 189.66666666666669, + 424, + 204.7777777777778 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 186, + 204.7777777777778, + 424, + 219.8888888888889 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 186, + 219.8888888888889, + 424, + 235.00000000000003 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 244, + 503, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 256, + 258 + ], + "score": 1.0, + "content": "To derive a lower bound for general", + "type": "text" + }, + { + "bbox": [ + 256, + 244, + 300, + 257 + ], + "score": 0.93, + "content": "f \\in { \\mathcal { C } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 243, + 505, + 258 + ], + "score": 1.0, + "content": ", we use the Mercer decomposition of the positive", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 255, + 181, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 166, + 267 + ], + "score": 1.0, + "content": "definite kernel", + "type": "text" + }, + { + "bbox": [ + 167, + 256, + 176, + 266 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 255, + 181, + 267 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 265, + 365, + 298 + ], + "lines": [ + { + "bbox": [ + 246, + 265, + 365, + 298 + ], + "spans": [ + { + "bbox": [ + 246, + 265, + 365, + 298 + ], + "score": 0.95, + "content": "K ( x , y ) = \\sum _ { i = 1 } ^ { \\infty } \\gamma _ { i } \\phi _ { i } ( x ) \\phi _ { i } ( y )", + "type": "interline_equation", + "image_path": "f73c3dff23878cabfa9851d9dc9c885ccc20f9c15583059c39242e1bd2092147.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 265, + 365, + 281.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 246, + 281.5, + 365, + 298.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 133, + 317 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 303, + 162, + 314 + ], + "score": 0.91, + "content": "\\gamma _ { i } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 300, + 245, + 317 + ], + "score": 1.0, + "content": "are eigenvalues and", + "type": "text" + }, + { + "bbox": [ + 245, + 302, + 278, + 315 + ], + "score": 0.93, + "content": "\\{ \\phi _ { i } \\} _ { i = 1 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 300, + 438, + 317 + ], + "score": 1.0, + "content": "is a complete orthonormal sequence of", + "type": "text" + }, + { + "bbox": [ + 438, + 302, + 482, + 315 + ], + "score": 0.94, + "content": "L ^ { 2 } ( X ; d x )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 300, + 507, + 317 + ], + "score": 1.0, + "content": ". It is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 312, + 322, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 322, + 327 + ], + "score": 1.0, + "content": "well-known that the corresponding RKHS is given by", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 330, + 407, + 365 + ], + "lines": [ + { + "bbox": [ + 203, + 330, + 407, + 365 + ], + "spans": [ + { + "bbox": [ + 203, + 330, + 407, + 365 + ], + "score": 0.93, + "content": "\\mathcal { H } = \\left\\{ g \\in L ^ { 2 } ( X , d x ) \\mid \\sum _ { i = 1 } ^ { \\infty } \\frac { ( g , \\phi _ { i } ) _ { L ^ { 2 } ( X ) } ^ { 2 } } { \\gamma _ { i } } < \\infty \\right\\} ,", + "type": "interline_equation", + "image_path": "a0b43008f029132cf6e1f4818b4b8095c38d1f36ace7f206aada2c77666db033.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 330, + 407, + 347.5 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 203, + 347.5, + 407, + 365.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 213, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 368, + 213, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 172, + 383 + ], + "score": 1.0, + "content": "and the norm of", + "type": "text" + }, + { + "bbox": [ + 173, + 370, + 199, + 381 + ], + "score": 0.92, + "content": "g \\in \\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 368, + 213, + 383 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 381, + 360, + 415 + ], + "lines": [ + { + "bbox": [ + 250, + 381, + 360, + 415 + ], + "spans": [ + { + "bbox": [ + 250, + 381, + 360, + 415 + ], + "score": 0.94, + "content": "\\| g \\| _ { \\mathcal { H } } ^ { 2 } = \\sum _ { j = 1 } ^ { \\infty } \\frac { ( g , \\phi _ { i } ) _ { L ^ { 2 } ( X ) } ^ { 2 } } { \\gamma _ { i } } .", + "type": "interline_equation", + "image_path": "b400f2fc35ebd1d326e20b21252bf1468bdfbe1c7236951f737f3e3ce0b0afb3.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 381, + 360, + 398.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 250, + 398.0, + 360, + 415.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 104, + 423, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 104, + 423, + 123, + 439 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 424, + 225, + 437 + ], + "score": 0.92, + "content": "f \\in \\mathcal { C } ( X ) \\subset L ^ { 2 } ( X , d x )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 423, + 505, + 439 + ], + "score": 1.0, + "content": "be an arbitrary function. 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We use", + "type": "text" + }, + { + "bbox": [ + 322, + 487, + 343, + 499 + ], + "score": 0.92, + "content": "\\mu ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "for the Radon–Nykodim derivative with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 498, + 204, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 204, + 510 + ], + "score": 1.0, + "content": "slight abuse of notation.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 504, + 540 + ], + "lines": [ + { + "bbox": [ + 102, + 508, + 509, + 532 + ], + "spans": [ + { + "bbox": [ + 102, + 508, + 143, + 532 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 144, + 514, + 236, + 528 + ], + "score": 0.91, + "content": "\\textstyle f ( x ) = \\sum _ { j = 1 } ^ { \\infty } a _ { j } \\phi _ { j } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 508, + 255, + 532 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 255, + 514, + 348, + 529 + ], + "score": 0.92, + "content": "\\textstyle \\mu ( x ) = \\sum _ { j = 1 } ^ { \\infty } b _ { j } \\phi _ { j } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 508, + 509, + 532 + ], + "score": 1.0, + "content": "are the expansion. 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If", + "type": "text" + }, + { + "bbox": [ + 238, + 710, + 266, + 721 + ], + "score": 0.91, + "content": "f \\notin { \\mathcal { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", the value (32) is infinite, and the conjugate function takes", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 722, + 125, + 731 + ], + "score": 0.85, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 721, + 131, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 721, + 505, + 731 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + } + ], + "page_idx": 29, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "30", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 304, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 304, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 273, + 96 + ], + "score": 1.0, + "content": "Proof. First we show an upper bound for", + "type": "text" + }, + { + "bbox": [ + 273, + 83, + 300, + 94 + ], + "score": 0.91, + "content": "f \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 82, + 304, + 96 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 304, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 99, + 424, + 235 + ], + "lines": [ + { + "bbox": [ + 186, + 99, + 424, + 235 + ], + "spans": [ + { + "bbox": [ + 186, + 99, + 424, + 235 + ], + "score": 0.96, + "content": "\\begin{array} { l } { \\displaystyle \\operatorname* { s u p } _ { \\mu } [ \\int f d \\mu - \\frac \\lambda 2 | | \\mu | | _ { \\mathcal H } ^ { 2 } ] = \\operatorname* { s u p } _ { \\mu } [ \\langle f , \\mu \\rangle _ { \\mathcal H } - \\frac \\lambda 2 | | \\mu | | _ { \\mathcal H } ^ { 2 } ] } \\\\ { \\displaystyle \\quad \\leq \\operatorname* { s u p } _ { \\mu } [ | | f | | \\mu | | \\mu | ] _ { \\mathcal H } - \\frac \\lambda 2 | | \\mu | | _ { \\mathcal H } ^ { 2 } ] } \\\\ { \\displaystyle \\quad \\leq \\operatorname* { s u p } _ { z \\in \\mathbb R } [ | | f | | \\mu z - \\frac \\lambda 2 z ^ { 2 } ] } \\\\ { \\displaystyle \\quad = | | f | | _ { \\mathcal H } \\cdot \\frac { | | f | | _ { \\mathcal H } } { \\lambda } - \\frac { \\lambda } { 2 } \\cdot \\frac { | | f | | _ { \\mathcal H } ^ { 2 } } { \\lambda ^ { 2 } } } \\\\ { \\displaystyle \\quad = \\frac { 1 } { 2 \\lambda } | | f | | _ { \\mathcal H } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "cfeb19d9fe1087df3a7d5a9f63b774727f1173339edc9e4dd644357d389a913d.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 186, + 99, + 424, + 114.11111111111111 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 186, + 114.11111111111111, + 424, + 129.22222222222223 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 186, + 129.22222222222223, + 424, + 144.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 186, + 144.33333333333334, + 424, + 159.44444444444446 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 186, + 159.44444444444446, + 424, + 174.55555555555557 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 186, + 174.55555555555557, + 424, + 189.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 186, + 189.66666666666669, + 424, + 204.7777777777778 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 186, + 204.7777777777778, + 424, + 219.8888888888889 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 186, + 219.8888888888889, + 424, + 235.00000000000003 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 244, + 503, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 256, + 258 + ], + "score": 1.0, + "content": "To derive a lower bound for general", + "type": "text" + }, + { + "bbox": [ + 256, + 244, + 300, + 257 + ], + "score": 0.93, + "content": "f \\in { \\mathcal { C } } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 243, + 505, + 258 + ], + "score": 1.0, + "content": ", we use the Mercer decomposition of the positive", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 255, + 181, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 166, + 267 + ], + "score": 1.0, + "content": "definite kernel", + "type": "text" + }, + { + "bbox": [ + 167, + 256, + 176, + 266 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 255, + 181, + 267 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 243, + 505, + 267 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 265, + 365, + 298 + ], + "lines": [ + { + "bbox": [ + 246, + 265, + 365, + 298 + ], + "spans": [ + { + "bbox": [ + 246, + 265, + 365, + 298 + ], + "score": 0.95, + "content": "K ( x , y ) = \\sum _ { i = 1 } ^ { \\infty } \\gamma _ { i } \\phi _ { i } ( x ) \\phi _ { i } ( y )", + "type": "interline_equation", + "image_path": "f73c3dff23878cabfa9851d9dc9c885ccc20f9c15583059c39242e1bd2092147.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 265, + 365, + 281.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 246, + 281.5, + 365, + 298.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 133, + 317 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 303, + 162, + 314 + ], + "score": 0.91, + "content": "\\gamma _ { i } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 300, + 245, + 317 + ], + "score": 1.0, + "content": "are eigenvalues and", + "type": "text" + }, + { + "bbox": [ + 245, + 302, + 278, + 315 + ], + "score": 0.93, + "content": "\\{ \\phi _ { i } \\} _ { i = 1 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 300, + 438, + 317 + ], + "score": 1.0, + "content": "is a complete orthonormal sequence of", + "type": "text" + }, + { + "bbox": [ + 438, + 302, + 482, + 315 + ], + "score": 0.94, + "content": "L ^ { 2 } ( X ; d x )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 300, + 507, + 317 + ], + "score": 1.0, + "content": ". It is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 312, + 322, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 322, + 327 + ], + "score": 1.0, + "content": "well-known that the corresponding RKHS is given by", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 300, + 507, + 327 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 330, + 407, + 365 + ], + "lines": [ + { + "bbox": [ + 203, + 330, + 407, + 365 + ], + "spans": [ + { + "bbox": [ + 203, + 330, + 407, + 365 + ], + "score": 0.93, + "content": "\\mathcal { H } = \\left\\{ g \\in L ^ { 2 } ( X , d x ) \\mid \\sum _ { i = 1 } ^ { \\infty } \\frac { ( g , \\phi _ { i } ) _ { L ^ { 2 } ( X ) } ^ { 2 } } { \\gamma _ { i } } < \\infty \\right\\} ,", + "type": "interline_equation", + "image_path": "a0b43008f029132cf6e1f4818b4b8095c38d1f36ace7f206aada2c77666db033.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 330, + 407, + 347.5 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 203, + 347.5, + 407, + 365.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 213, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 368, + 213, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 172, + 383 + ], + "score": 1.0, + "content": "and the norm of", + "type": "text" + }, + { + "bbox": [ + 173, + 370, + 199, + 381 + ], + "score": 0.92, + "content": "g \\in \\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 368, + 213, + 383 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 368, + 213, + 383 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 381, + 360, + 415 + ], + "lines": [ + { + "bbox": [ + 250, + 381, + 360, + 415 + ], + "spans": [ + { + "bbox": [ + 250, + 381, + 360, + 415 + ], + "score": 0.94, + "content": "\\| g \\| _ { \\mathcal { H } } ^ { 2 } = \\sum _ { j = 1 } ^ { \\infty } \\frac { ( g , \\phi _ { i } ) _ { L ^ { 2 } ( X ) } ^ { 2 } } { \\gamma _ { i } } .", + "type": "interline_equation", + "image_path": "b400f2fc35ebd1d326e20b21252bf1468bdfbe1c7236951f737f3e3ce0b0afb3.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 381, + 360, + 398.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 250, + 398.0, + 360, + 415.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 104, + 423, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 104, + 423, + 123, + 439 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 424, + 225, + 437 + ], + "score": 0.92, + "content": "f \\in \\mathcal { C } ( X ) \\subset L ^ { 2 } ( X , d x )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 423, + 505, + 439 + ], + "score": 1.0, + "content": "be an arbitrary function. 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If", + "type": "text" + }, + { + "bbox": [ + 238, + 710, + 266, + 721 + ], + "score": 0.91, + "content": "f \\notin { \\mathcal { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", the value (32) is infinite, and the conjugate function takes", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 722, + 125, + 731 + ], + "score": 0.85, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 721, + 131, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 721, + 505, + 731 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 697, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 103, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 78, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 78, + 193, + 96 + ], + "score": 1.0, + "content": "Proposition 14. Let", + "type": "text" + }, + { + "bbox": [ + 193, + 83, + 203, + 93 + ], + "score": 0.66, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 78, + 365, + 96 + ], + "score": 1.0, + "content": "be an RKHS with the Gaussian kernel", + "type": "text" + }, + { + "bbox": [ + 366, + 81, + 460, + 95 + ], + "score": 0.92, + "content": "K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 78, + 506, + 96 + ], + "score": 1.0, + "content": ". Then for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 91, + 139, + 107 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 134, + 105 + ], + "score": 0.87, + "content": "f \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 91, + 139, + 107 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 109, + 457, + 163 + ], + "lines": [ + { + "bbox": [ + 153, + 109, + 457, + 163 + ], + "spans": [ + { + "bbox": [ + 153, + 109, + 457, + 163 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\displaystyle | | f | | _ { \\mathcal { H } } ^ { 2 } = \\sum _ { k = 0 } ^ { \\infty } ( 4 \\pi ) ^ { - k } \\sum _ { \\stackrel { k _ { 1 } + \\cdots + k _ { d } = k } { k _ { 1 } \\prod _ { i = 1 } ^ { d } k _ { i } ! } } | | \\partial _ { x _ { 1 } } ^ { k _ { 1 } } \\cdot \\cdot \\cdot \\partial _ { x _ { d } } ^ { k _ { d } } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } \\\\ & { \\quad \\quad = | | f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 4 \\pi } | | \\nabla f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 1 6 \\pi ^ { 2 } } | | \\nabla ^ { 2 } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + o t h e r t e r m s . } \\end{array}", + "type": "interline_equation", + "image_path": "4e5335bb7f6a79dc83592e1cc93b80593c6b89b9c6e21ce206fd1b2902fc1f58.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 153, + 109, + 457, + 127.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 153, + 127.0, + 457, + 145.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 153, + 145.0, + 457, + 163.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 172, + 504, + 187 + ], + "lines": [ + { + "bbox": [ + 102, + 169, + 508, + 191 + ], + "spans": [ + { + "bbox": [ + 102, + 169, + 288, + 191 + ], + "score": 1.0, + "content": "Proof. We consider the more general case of", + "type": "text" + }, + { + "bbox": [ + 288, + 172, + 439, + 187 + ], + "score": 0.91, + "content": "K ( x , y ) = ( 2 \\pi \\sigma ^ { 2 } ) ^ { - d / 2 } e ^ { - | | x - y | | ^ { 2 } / 2 \\sigma ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 169, + 508, + 191 + ], + "score": 1.0, + "content": ", where we have", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 192, + 459, + 245 + ], + "lines": [ + { + "bbox": [ + 151, + 192, + 459, + 245 + ], + "spans": [ + { + "bbox": [ + 151, + 192, + 459, + 245 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\displaystyle | | f | | _ { \\mathcal { H } } ^ { 2 } = \\sum _ { k = 0 } ^ { \\infty } ( \\frac { 1 } { 2 } \\sigma ^ { 2 } ) ^ { k } \\sum _ { { k _ { 1 } + \\cdots + k _ { d } = k } \\atop = | | f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } \\frac { 1 } { | | \\nabla f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } | | \\partial _ { x _ { 1 } } ^ { k _ { 1 } } \\cdot \\cdot \\cdot \\partial _ { x _ { d } } ^ { k _ { d } } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } \\\\ & { \\quad \\quad \\quad = | | f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 2 } \\sigma ^ { 2 } | | \\nabla f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 4 } \\sigma ^ { 4 } | | \\nabla ^ { 2 } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\mathrm { o t h e r ~ t e r m s } . } \\end{array}", + "type": "interline_equation", + "image_path": "e5b45c1eb9abda125f0ea4c739835f2877d44333f4eeae9e297115c0d5ae9312.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 151, + 192, + 459, + 209.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 151, + 209.66666666666666, + 459, + 227.33333333333331 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 151, + 227.33333333333331, + 459, + 244.99999999999997 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 253, + 506, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 269, + 267 + ], + "score": 1.0, + "content": "Novak et al. (2018) prove this result for", + "type": "text" + }, + { + "bbox": [ + 269, + 254, + 295, + 264 + ], + "score": 0.89, + "content": "\\sigma = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 253, + 505, + 267 + ], + "score": 1.0, + "content": ". We sketch the proof for the general case here. We", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 265, + 275, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 275, + 277 + ], + "score": 1.0, + "content": "use the Fourier transform convention that", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 279, + 459, + 306 + ], + "lines": [ + { + "bbox": [ + 150, + 279, + 459, + 306 + ], + "spans": [ + { + "bbox": [ + 150, + 279, + 459, + 306 + ], + "score": 0.91, + "content": "\\hat { f } ( \\boldsymbol { k } ) = ( 2 \\pi ) ^ { - d / 2 } \\int _ { \\mathbb { R } ^ { d } } f ( \\boldsymbol { x } ) e ^ { - i \\boldsymbol { k } \\cdot \\boldsymbol { x } } d \\boldsymbol { x } , \\quad f ( \\boldsymbol { x } ) = ( 2 \\pi ) ^ { - d / 2 } \\int _ { \\mathbb { R } ^ { d } } \\hat { f } ( \\boldsymbol { k } ) e ^ { i \\boldsymbol { k } \\cdot \\boldsymbol { x } } d \\boldsymbol { k } .", + "type": "interline_equation", + "image_path": "bdd526ae349d382f5be4edd161494406194d851fa0ee68dc437ee8d61cae6034.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 150, + 279, + 459, + 306 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 216, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 216, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 216, + 330 + ], + "score": 1.0, + "content": "Consider the inner product", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 333, + 380, + 359 + ], + "lines": [ + { + "bbox": [ + 230, + 333, + 380, + 359 + ], + "spans": [ + { + "bbox": [ + 230, + 333, + 380, + 359 + ], + "score": 0.94, + "content": "\\langle f , g \\rangle = \\int _ { \\mathbb { R } ^ { d } } e ^ { \\sigma ^ { 2 } | | k | | ^ { 2 } / 2 } \\hat { f } ( k ) \\overline { { \\hat { g } ( k ) } } d k ,", + "type": "interline_equation", + "image_path": "e120d4868f2caf4db8d26813f760524a9c33c46f04f4c1d1f320c3a067b7dff4.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 230, + 333, + 380, + 359 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 195, + 378 + ], + "score": 1.0, + "content": "defined for functions", + "type": "text" + }, + { + "bbox": [ + 195, + 364, + 240, + 376 + ], + "score": 0.92, + "content": "| | f | | < \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 363, + 506, + 378 + ], + "score": 1.0, + "content": ". Expanding the exponential in Taylor series, this inner product", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "gives the equation for the norm in the proposition. 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Let", + "type": "text" + }, + { + "bbox": [ + 193, + 83, + 203, + 93 + ], + "score": 0.66, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 78, + 365, + 96 + ], + "score": 1.0, + "content": "be an RKHS with the Gaussian kernel", + "type": "text" + }, + { + "bbox": [ + 366, + 81, + 460, + 95 + ], + "score": 0.92, + "content": "K ( x , y ) = e ^ { - \\pi | | x - y | | ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 78, + 506, + 96 + ], + "score": 1.0, + "content": ". Then for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 91, + 139, + 107 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 134, + 105 + ], + "score": 0.87, + "content": "f \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 91, + 139, + 107 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 78, + 506, + 107 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 109, + 457, + 163 + ], + "lines": [ + { + "bbox": [ + 153, + 109, + 457, + 163 + ], + "spans": [ + { + "bbox": [ + 153, + 109, + 457, + 163 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\displaystyle | | f | | _ { \\mathcal { H } } ^ { 2 } = \\sum _ { k = 0 } ^ { \\infty } ( 4 \\pi ) ^ { - k } \\sum _ { \\stackrel { k _ { 1 } + \\cdots + k _ { d } = k } { k _ { 1 } \\prod _ { i = 1 } ^ { d } k _ { i } ! } } | | \\partial _ { x _ { 1 } } ^ { k _ { 1 } } \\cdot \\cdot \\cdot \\partial _ { x _ { d } } ^ { k _ { d } } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } \\\\ & { \\quad \\quad = | | f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 4 \\pi } | | \\nabla f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 1 6 \\pi ^ { 2 } } | | \\nabla ^ { 2 } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + o t h e r t e r m s . } \\end{array}", + "type": "interline_equation", + "image_path": "4e5335bb7f6a79dc83592e1cc93b80593c6b89b9c6e21ce206fd1b2902fc1f58.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 153, + 109, + 457, + 127.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 153, + 127.0, + 457, + 145.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 153, + 145.0, + 457, + 163.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 172, + 504, + 187 + ], + "lines": [ + { + "bbox": [ + 102, + 169, + 508, + 191 + ], + "spans": [ + { + "bbox": [ + 102, + 169, + 288, + 191 + ], + "score": 1.0, + "content": "Proof. We consider the more general case of", + "type": "text" + }, + { + "bbox": [ + 288, + 172, + 439, + 187 + ], + "score": 0.91, + "content": "K ( x , y ) = ( 2 \\pi \\sigma ^ { 2 } ) ^ { - d / 2 } e ^ { - | | x - y | | ^ { 2 } / 2 \\sigma ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 169, + 508, + 191 + ], + "score": 1.0, + "content": ", where we have", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 102, + 169, + 508, + 191 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 192, + 459, + 245 + ], + "lines": [ + { + "bbox": [ + 151, + 192, + 459, + 245 + ], + "spans": [ + { + "bbox": [ + 151, + 192, + 459, + 245 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\displaystyle | | f | | _ { \\mathcal { H } } ^ { 2 } = \\sum _ { k = 0 } ^ { \\infty } ( \\frac { 1 } { 2 } \\sigma ^ { 2 } ) ^ { k } \\sum _ { { k _ { 1 } + \\cdots + k _ { d } = k } \\atop = | | f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } \\frac { 1 } { | | \\nabla f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } | | \\partial _ { x _ { 1 } } ^ { k _ { 1 } } \\cdot \\cdot \\cdot \\partial _ { x _ { d } } ^ { k _ { d } } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } } \\\\ & { \\quad \\quad \\quad = | | f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 2 } \\sigma ^ { 2 } | | \\nabla f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\frac { 1 } { 4 } \\sigma ^ { 4 } | | \\nabla ^ { 2 } f | | _ { L ^ { 2 } ( \\mathbb { R } ^ { d } ) } ^ { 2 } + \\mathrm { o t h e r ~ t e r m s } . } \\end{array}", + "type": "interline_equation", + "image_path": "e5b45c1eb9abda125f0ea4c739835f2877d44333f4eeae9e297115c0d5ae9312.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 151, + 192, + 459, + 209.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 151, + 209.66666666666666, + 459, + 227.33333333333331 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 151, + 227.33333333333331, + 459, + 244.99999999999997 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 253, + 506, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 269, + 267 + ], + "score": 1.0, + "content": "Novak et al. (2018) prove this result for", + "type": "text" + }, + { + "bbox": [ + 269, + 254, + 295, + 264 + ], + "score": 0.89, + "content": "\\sigma = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 253, + 505, + 267 + ], + "score": 1.0, + "content": ". We sketch the proof for the general case here. We", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 265, + 275, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 275, + 277 + ], + "score": 1.0, + "content": "use the Fourier transform convention that", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 253, + 505, + 277 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 279, + 459, + 306 + ], + "lines": [ + { + "bbox": [ + 150, + 279, + 459, + 306 + ], + "spans": [ + { + "bbox": [ + 150, + 279, + 459, + 306 + ], + "score": 0.91, + "content": "\\hat { f } ( \\boldsymbol { k } ) = ( 2 \\pi ) ^ { - d / 2 } \\int _ { \\mathbb { R } ^ { d } } f ( \\boldsymbol { x } ) e ^ { - i \\boldsymbol { k } \\cdot \\boldsymbol { x } } d \\boldsymbol { x } , \\quad f ( \\boldsymbol { x } ) = ( 2 \\pi ) ^ { - d / 2 } \\int _ { \\mathbb { R } ^ { d } } \\hat { f } ( \\boldsymbol { k } ) e ^ { i \\boldsymbol { k } \\cdot \\boldsymbol { x } } d \\boldsymbol { k } .", + "type": "interline_equation", + "image_path": "bdd526ae349d382f5be4edd161494406194d851fa0ee68dc437ee8d61cae6034.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 150, + 279, + 459, + 306 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 216, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 216, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 216, + 330 + ], + "score": 1.0, + "content": "Consider the inner product", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 315, + 216, + 330 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 333, + 380, + 359 + ], + "lines": [ + { + "bbox": [ + 230, + 333, + 380, + 359 + ], + "spans": [ + { + "bbox": [ + 230, + 333, + 380, + 359 + ], + "score": 0.94, + "content": "\\langle f , g \\rangle = \\int _ { \\mathbb { R } ^ { d } } e ^ { \\sigma ^ { 2 } | | k | | ^ { 2 } / 2 } \\hat { f } ( k ) \\overline { { \\hat { g } ( k ) } } d k ,", + "type": "interline_equation", + "image_path": "e120d4868f2caf4db8d26813f760524a9c33c46f04f4c1d1f320c3a067b7dff4.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 230, + 333, + 380, + 359 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 195, + 378 + ], + "score": 1.0, + "content": "defined for functions", + "type": "text" + }, + { + "bbox": [ + 195, + 364, + 240, + 376 + ], + "score": 0.92, + "content": "| | f | | < \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 363, + 506, + 378 + ], + "score": 1.0, + "content": ". Expanding the exponential in Taylor series, this inner product", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "gives the equation for the norm in the proposition. By use of the Fourier inversion formula, it can be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 386, + 248, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 153, + 399 + ], + "score": 1.0, + "content": "shown that", + "type": "text" + }, + { + "bbox": [ + 153, + 386, + 217, + 398 + ], + "score": 0.93, + "content": "\\langle f , K _ { x } \\rangle = f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 387, + 248, + 399 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 363, + 506, + 399 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 403, + 380, + 420 + ], + "lines": [ + { + "bbox": [ + 231, + 403, + 380, + 420 + ], + "spans": [ + { + "bbox": [ + 231, + 403, + 380, + 420 + ], + "score": 0.92, + "content": "K _ { x } ( y ) = ( 2 \\pi \\sigma ^ { 2 } ) ^ { - d / 2 } e ^ { - | | x - y | | ^ { 2 } / 2 \\sigma ^ { 2 } } ,", + "type": "interline_equation", + "image_path": "c838bf10a41a4d619b0750db846400961ae55377bec74063bd8fc927d9798371.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 231, + 403, + 380, + 420 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 288, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 288, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 288, + 437 + ], + "score": 1.0, + "content": "so this is an RKHS with the Gaussian kernel.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 425, + 288, + 437 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/HJeOekHKwr/HJeOekHKwr_model.json b/parse/train/HJeOekHKwr/HJeOekHKwr_model.json new file mode 100644 index 0000000000000000000000000000000000000000..4f5d50ff51b8ba08d24eae7fe130665dc882c936 --- /dev/null +++ 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Loss function J(μ)Optimal discriminator Φμ(x)
Minimax GANDjs(μ|l μo)μ(x）
Non-saturating GANDKL(¹μ+μo |μo)-110g（）+(@） μo(x）
Wasserstein GANW1(μ, μo)argmaXfeLip1 Ey~μ[f(y)] -Ey~μo[f(y)]
GMMN,Coulomb GAN1MMD²(μ, μo)Ey~μ[K(x,y)]-Ey~μo[K(x,y)]
IPM-GANIPMj(μ, μo)arg maxf∈F Ey~μ[f(y)] -Ey~μof(y)]
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(D1)(D2)(D3)
Minimax GAN×X×
Non-saturating GAN Wasserstein GANXX？
√X
×？
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PurposeSource lossR()R（）GAN techniques
(D1)W1|KRx{lφ|lLip ≤1}spectral norm
(D2)IPMs|/| s*x{∈s}smooth activations, spectral norm
(D3)MMD²gradient penalties
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D _ { \\theta } f _ { \\theta } ( z ) \\right. _ { \\mathrm { o p } } , \\quad \\mathrm { a n d } } \\\\ { ( \\mathrm { G } 2 ^ { \\prime } ) \\quad } & { \\mathbb { E } _ { z \\sim \\omega } \\vert \\vert D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } ( z ) - D _ { \\theta _ { 2 } } f _ { \\theta _ { 2 } } ( z ) \\vert \\vert _ { \\mathrm { o p } } \\leq B \\vert \\vert \\theta _ { 1 } - \\theta _ { 2 } \\vert \\vert . } \\end{array}" + }, + { + "category_id": 14, + "poly": [ + 360, + 307, + 1282, + 307, + 1282, + 676, + 360, + 676 + ], + "score": 0.94, + "latex": "\\begin{array} { r l } & { \\quad \\| \\nabla _ { \\theta _ { 1 } } J ( \\mu _ { \\theta _ { 1 } } ) - \\nabla _ { \\theta _ { 2 } } J ( \\mu _ { \\theta _ { 2 } } ) \\| } \\\\ & { = \\big \\| \\mathbb { E } _ { z \\sim \\omega } \\big [ \\nabla \\Phi _ { \\mu _ { \\theta _ { 1 } } } ( f _ { \\theta _ { 1 } } ( z ) ) \\cdot D _ { \\theta _ { 1 } } f _ { \\theta _ { 1 } } - \\nabla \\Phi _ { \\mu _ { \\theta _ { 2 } } } ( f 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/dev/null +++ b/parse/train/HyeVtoRqtQ/images/f4404e63ad0cb3d5dda5c4c3dff2a4fae689163b7205f30c6401d62c8f9b304b.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6f5a9f2a79587eeceae1d1220567ed6417d82c3bc11a6fe20a004cb4a45c42dc +size 20358 diff --git a/parse/train/HylzTiC5Km/HylzTiC5Km.md b/parse/train/HylzTiC5Km/HylzTiC5Km.md new file mode 100644 index 0000000000000000000000000000000000000000..2831ea56f73b92bcb218cfcb870d0bba779b5ae1 --- /dev/null +++ b/parse/train/HylzTiC5Km/HylzTiC5Km.md @@ -0,0 +1,227 @@ +# GENERATING HIGH FIDELITY IMAGES WITH SUBSCALE PIXEL NETWORKS AND MULTIDIMENSIONAL UPSCALING + +Jacob Menick∗ +DeepMind +jmenick@google.com + +Nal Kalchbrenner∗ Google Brain Amsterdam nalk@google.com + +# ABSTRACT + +The unconditional generation of high fidelity images is a longstanding benchmark for testing the performance of image decoders. Autoregressive image models have been able to generate small images unconditionally, but the extension of these methods to large images where fidelity can be more readily assessed has remained an open problem. Among the major challenges are the capacity to encode the vast previous context and the sheer difficulty of learning a distribution that preserves both global semantic coherence and exactness of detail. To address the former challenge, we propose the Subscale Pixel Network (SPN), a conditional decoder architecture that generates an image as a sequence of sub-images of equal size. The SPN compactly captures image-wide spatial dependencies and requires a fraction of the memory and the computation required by other fully autoregressive models. To address the latter challenge, we propose to use Multidimensional Upscaling to grow an image in both size and depth via intermediate stages utilising distinct SPNs. We evaluate SPNs on the unconditional generation of CelebAHQ of size 256 and of ImageNet from size 32 to 256. We achieve state-of-the-art likelihood results in multiple settings, set up new benchmark results in previously unexplored settings and are able to generate very high fidelity large scale samples on the basis of both datasets. + +# 1 INTRODUCTION + +A successful generative model has two core aspects: it produces targets that have high fidelity and it generalizes well on held-out data. Autoregressive (AR) models trained by conventional maximum likelihood estimation (MLE) have produced superior scores on held-out data across a wide range of domains such as text (Vaswani et al., 2017; Wu et al., 2016), audio (van den Oord et al., 2016a), images (Parmar et al., 2018) and videos (Kalchbrenner et al., 2016). These scores are a measure of the models’ ability to generalize in that setting. From the perspective of sample fidelity, the outputs generated by AR models have also achieved state-of-the-art fidelity in many of the aforementioned domains with one notable exception. In the domain of unconditional large-scale image generation, AR samples have yet to manifest long-range structure and semantic coherence. + +One source of difficulties impeding high-fidelity image generation is the multi-faceted relationship between the MLE scores achieved by a model and the model’s sample fidelity. On the one hand, MLE is a well-defined measure as improvements in held-out scores generally produce improvements in the visual fidelity of the samples. On the other hand, as opposed to for example adversarial methods (Arora & Zhang, 2017), MLE forces the model to support the entire empirical distribution. This guarantees the model’s ability to generalize at the cost of allotting capacity to parts of the distribution that are irrelevant to fidelity. A second source of difficulties arises from the high dimensionality of large images. A $2 5 6 \times 2 5 6 \times 3$ image has a total of 196,608 positions that need to be architecturally connected in order to learn dependencies among them; the representations at each position require sufficient capacity to express their respective surrounding contexts. These requirements translate to large amounts of memory and computation. + +![](images/95c2aef047f12f2740eba99f379e7a1c4e685396bb70dd2a36d7926b37c9409b.jpg) +Figure 1: A representation of Multidimensional Upscaling. Left: depth upscaling is applied to a generated 3-bit $2 5 6 \times 2 5 6$ RGB subimage from CelebAHQ to map it to a full 8-bit $2 5 6 \times 2 5 6$ RGB image. Right: size upscaling followed by depth upscaling are applied to a generated 3-bit $3 2 \times 3 2$ RGB subimage from ImageNet to map it to the target resolution of the 8-bit $1 2 8 \times 1 2 8$ RGB image. We stress that the rightmost column of both figures are true unconditional samples from our model at full 8bit depth. + +These difficulties notwithstanding, we aim to learn the full distribution over 8-bit RGB images of size up to $2 5 6 \times 2 5 6$ well enough so that the samples have high fidelity. We aim to guide the model to focus first on visually more salient bits of the distribution and later on the visually less salient bits. We identify two visually salient subsets of the distribution: first, the subset determined by sub-images (“slices”) of smaller size (e.g. $3 2 \times 3 2$ ) sub-sampled at all positions from the original image; and secondly, the subset determined by the few (e.g. 3) most significant bits of each RGB channel in the image. We use Multidimensional Upscaling to map from one subset of the distribution to the other one by upscaling images in size or in depth. For example, the generation of a $1 2 8 \times 1 2 8 :$ 8-bit RGB image proceeds by first upscaling it in size from a $3 2 \times 3 2$ 3-bit RGB image to a $1 2 8 \times 1 2 8$ 3-bit RGB image; we then upscale the resulting image in depth to the original resolution of the $1 2 8 \times 1 2 8$ 8-bit RGB image. We thus train three networks: (a) a decoder on the small size, low depth image slices subsampled at every $n$ pixels from the original image with the desired target resolution; (b) a size-upscaling decoder that generates the large size, low depth image conditioned on the small size, low depth image; and (c) a depth-upscaling decoder that generates the large size, high depth image conditioned on the large size, low depth image. Figure 1 illustrates this process. + +To address the latter difficulties that ensue in the training of decoders (b) and (c), we develop the Subscale Pixel Network (SPN) architecture. The SPN divides an image of size $N \times N$ into sub-images of size $\begin{array} { r } { \frac { N } { S _ { . } } \times \frac { N } { S } } \end{array}$ sliced out at interleaving positions (see Figure 2), which implicitly also captures a form of size upscaling. The $N \times N$ image is generated one slice at a time conditioned on previously generated slices in a way that encodes a rich spatial structure. SPN consists of two networks, a conditioning network that embeds previous slices and a decoder proper that predicts a single target slice given the context embedding. The decoding part of the SPN acts over image slices with the same spatial structure and it can share weights for all of them. The SPN is an independent image decoder with an implicit size upscaling mechanism, but it can also be used as an explicit size upscaling network by initializing the first slice of the SPN input at sampling time with one generated separately during step (a). + +We extensively evaluate the performance of SPN and the size and depth upscaling methods both quantitatively and from a fidelity perspective on two unconditional image generation benchmarks, CelebAHQ-256 and ImageNet of various sizes up to 256. From a MLE scores perspective, we compare with previous work to obtain state-of-the-art results on CelebAHQ-256, both at full 8-bit resolution and at the reduced 5-bit resolution (Kingma & Dhariwal, 2018), and on ImageNet-64. We also establish MLE baselines for ImageNet-128 and ImageNet-256. From a sample fidelity perspective, we show the strong benefits of multidimensional upscaling as well as the benefits of the SPN. We produce CelebAHQ-256 samples (at full 8-bit resolution) that are of similar visual fidelity to those produced with methods such as GANs that lack however an intrinsic measure of generalization (Mescheder, 2018; Karras et al., 2017). We also produce some of the first successful samples on unconditional ImageNet-128 (also at 8-bit) showing again the striking impact of the SPN and of multidimensional upscaling on sample quality and setting a fidelity baseline for future methods. + +![](images/7b2a844dbd9243a20e217da5e13212d83a158d5360aa6104f93056ee1c073ba0.jpg) +.Figure 2: The receptive field in a Subscale Pixel Networks (a) and the four image slices subsampled from the original image (b) + +# 2 MODEL + +# 2.1 CONVENTIONAL GENERATION ORDERING + +A standard AR image model such as the PixelCNN (van den Oord et al., 2016b) generates an $H x W$ colour image starting at the top-left position and ending at the bottom-right position, fully generating the three 8-bit channels of each pixel in a given position: + +$$ +P ( \mathbf { x } ) = \prod _ { h = 1 } ^ { H } \prod _ { w = 1 } ^ { W } \prod _ { c } ^ { \{ R , G , B \} } P ( x _ { h , w , c } | \mathbf { x } _ { < } ) +$$ + +where $\mathbf { x } _ { < }$ corresponds to all previously generated intensity values in the ordering and $h , w .$ , and $c$ are row, column, and colour channel indices. The raster scan ordering (Figure 3(a)) is conventionally used in AR models. Each conditional distribution $P ( x _ { h , w , c } | \mathbf { x } _ { < } )$ is parametrized by a deep neural network (van den Oord et al., 2016b). + +# 2.2 SUBSCALE ORDERING IN IMAGES + +We define an alternative ordering that divides a large image into a sequence of equally sized slices and has various core properties. First, it makes it easy to compactly encode long-range dependencies across the many pixels in the large images. It also induces a spatial structure over the original image by aligning the subsampled slices; this also has an implicit size upscaling side effect. From the perspective of the neural architecture, it makes it possible for the same decoder within the SPN to be consistently applied to all slices, since they are structurally similar; the smaller slices also allow for self-attention (Vaswani et al., 2017) in the SPN to be used without local contexts (Parmar et al., 2018). We think of the ordering as the two-dimensional analogue of the one-dimensional subscale ordering introduced in Kalchbrenner et al. (2018). + +The subscale ordering is defined as follows: + +$$ +P ( \mathbf { x } ) = \prod _ { i = 1 } ^ { S } \prod _ { j = 1 } ^ { S } \prod _ { h = 1 } ^ { H / S } \prod _ { w = 1 } ^ { W / S } \prod _ { c } ^ { \{ R , G , B \} } P ( x _ { i + S * h , j + S * w , c } | \mathbf { x } _ { < } ) +$$ + +where $\mathbf { x } _ { < }$ corresponds to all previously generated intensity values according to this ordering. Figure 3(d) illustrates the subscale ordering. A scaling factor $S$ is selected and each slice of size $H / S \times W / S$ is obtained by selecting a pixel every $S$ pixels in both height and width; there are thus $S ^ { 2 }$ interleaved slices in the original image, with each specified by its row and column offset $( i , j )$ . We sometimes refer to this offset as the “meta-position” of a slice. + +![](images/98559d46079f8d28599a7a5163c09abb9d3f0b63ff32e33727c28ac6d3f9a3f2.jpg) +Figure 3: Different generation ordering schemes, where the numbers indicate the step-by-step order. Distinct colors correspond to distinct neural networks. (a) and (b) are from (van den Oord et al., 2016b). (c) is from (Reed et al., 2017). The Subscale ordering alone, with size-only, depth-only and with multidimensional upscaling are, respectively, in blocks (d), (e), (f) and (g). + +# 2.3 SIZE UPSCALING IN SUBSCALE ORDERING + +The subscale ordering itself already captures size upscaling implicitly. Analogous to the multi-scale ordering (van den Oord et al., 2016b), and depicted in 3(b), we can perform size upscaling explicitly, by training a single slice decoder on subimages and generate the first slice of a subscale ordering from the single slice decoder itself. The rest of the image is then generated according to the subscale ordering by the main network (see 3(e)). The single-slice model can be trained on just the first slices of images, or on slices at all positions in all images given the shared spatial structure among the slices. For this reason, the same SPN that captures the subscale ordering can act simultaneously as a full-blown image model as well as a size upscaling model if initialized with the outputs of a single-slice decoder. A separate formulation of size upscaling is the Parallel Multi-Scale (Reed et al., 2017) ordering where the pixels in an image are doubled at every stage by distinct neural networks and are generated in parallel without sequentiality (3(c)). + +# 2.4 DEPTH UPSCALING + +Multidimensional upscaling applies upscaling not just in the height and width of the image, but also in the remaining dimension that is channel depth. This is performed in stages such that a network first generates the $d _ { 1 }$ most significant bits of an image using a conventional or subscale ordering; then a second network generates the next $d _ { 2 }$ most significant bits of the image conditioned on all the $d _ { 1 }$ bits of the image; and so on to further stages. Using the conventional ordering as basis, the first stage of depth upscaling looks as follows: + +$$ +P ( \mathbf { x } ^ { : \mathbf { d } _ { 1 } } ) = \prod _ { h = 1 } ^ { H } \prod _ { w = 1 } ^ { W } \prod _ { c } ^ { \{ R , G , B \} } P ( \mathbf { x } _ { \mathbf { h } , \mathbf { w } , \mathbf { c } } ^ { : \mathbf { d } _ { 1 } } | \mathbf { x } ^ { : \mathbf { d } _ { 1 } } < ) +$$ + +Next, a second stage of depth upscaling has the following form, conditioned on the first $d _ { 1 }$ bits of each channel: + +$$ +P ( \mathbf { x ^ { d _ { 1 } : d _ { 2 } } } ) = \prod _ { h = 1 } ^ { H } \prod _ { w = 1 } ^ { W } \prod _ { c } ^ { \{ R , G , B \} } P ( \mathbf { x _ { h , w , c } ^ { d _ { 1 } : d _ { 2 } } } | \mathbf { x ^ { d _ { 1 } : d _ { 2 } } } _ { < } , \mathbf { x } ^ { : d _ { 1 } } ) +$$ + +We do not share weights among the networks at different stages of depth upscaling. We note that in depth upscaling bits of lower significance are only generated when the more significants bits at all positions have been generated in a previous stage. Just like for size upscaling from the previous section, the goal of multidimensional upscaling is to let the model focus on visually salient bits of an image unaffected by less salient and less predictable bits of the image. Depth upscaling is related to the method underlying the Grayscale PixelCNN that models 4-bit greyscale images subsampled from colored images Kolesnikov & Lampert (2016a). + +![](images/fe7ca922f4649bae3603e8639b2d4ae3c036fa4269b86d943ed1c5ad94c6973b.jpg) +(a) (b)Figure 4: (a) The architecture of a Subscale Pixel Network, with a conditional and a decoding part. (b) Scheme of the parts in the decoder itself. + +# 3 ARCHITECTURE + +# 3.1 CHALLENGES IN TRAINING CONVENTIONAL DECODERS ON LARGE IMAGES + +Using a conventional generation ordering, models such as PixelRNN, PixelCNN (van den Oord et al., 2016b) and Image Transformer (Parmar et al., 2018) construct a representation of the generated context for each dimension of each pixel. Existing AR approaches inherently require an amount of computation and memory that is superlinear in the number of pixels. In particular, the quadratic memory requirements of self-attention become severely limiting for images larger than $3 2 \times 3 2$ and intractable in practice for the 196,608 distinct positions we consider in a $2 5 6 \times 2 5 6$ colour image. + +Mitigating the memory requirements and computational requirements of encoding the dependencies amongst so many variables often comes at the expense of global context. Modeling choices such as cropping images within the decoders (Kalchbrenner et al., 2016) or performing self-attention over local neighborhoods (Parmar et al., 2018) neglect global dependencies, while model parallelism, though technically feasible with the joint use of a very large number of accelerators, does not overcome the challenges in learning the global structure. + +# 3.2 SUBSCALE PIXEL NETWORK + +To address these challenges, we devise the Subscale Pixel Network (SPN), an architecture that embodies the subscale ordering from Section 2.2. For an image of size $H \times W \times 3 \times D$ , where $D$ is the number of bits used for the current generation stage, one first chooses a scaling factor $S$ and obtains the $S ^ { 2 }$ slices of the original image of size $H / S \times W / S \times 3 \times D$ . We use $H = W = 2 5 6$ and $S = 8$ as well as $H = W = 1 2 8$ and $S = 4$ , for the larger images we process, so that the slices have size $3 2 \times 3 2 \times 3 \times D$ . This scheme of choosing $S$ such that slices are always $3 2 \times 3 2$ renders the memory and computation requirements effectively constant as the true image size $H \times W$ changes. + +The SPN architecture is composed of two parts: an embedding part for slices at preceding metapositions that conditions the decoder for the current slice that is being generated (Figure 4 (a)). The embedding part is a convolutional neural network with residual blocks that takes as input preceding slices that are concatenated along the depth dimension. One detail is the way the slices are ordered along the channel dimension when concatenated. As illustrated in Figure 4 (a), empty padding slices are used to preserve relative meta-positions of each preceding slice with respect to the current target slice. For example, the slice above any target slice in the two-dimensional meta-grid is always aligned in the same position along the depth axis in the input. This achieves equivariance in the embedding architecture with respect to the $( i , j )$ offset of a slice. The padding slices also ensure that the depth of the input slice tensor remains the same for all target slices. In addition to the slice tensor, the embedding part also receives as input the meta-position of the target slice as an embedding of 8 units tiled spatially across the slice tensor. The pixel intensity values are also embedded as one-hot indices of size 8. The context embedding network passes its input through a series of self-attention layers and then a series of residual blocks, finally emitting a slice-sized feature map s that summarizes the context for the decoder. + +![](images/e6f31c8d7a0d48ad915bfecc5d546389faa9b99b3c07f683678a398e09811c80.jpg) +Figure 5: Left: 8-bit 128x128 RGB ImageNet samples from SPN with depth upscaling only. Right: 8-bit 128x128 RGB ImageNet samples from SPN with full-blown Multidimensional Upscaling. Temperature is 0.99 for the initial $3 2 \mathbf { x } 3 2$ sub-image and otherwise 1.0 + +The decoder takes as input the encoded slice tensor s in a position-preserving manner: each position in the target slice is given as input the encoded representations of pixels at that same position in the preceding slices. In addition it processes the target slice in the raster-scan order. The decoder that we use is a hybrid architecture combining masked convolution and self-attention (Chen et al., 2017). We employ an initial 1D self-attention network Vaswani et al. (2017) that is used to gather the entire available context in the slice (see Figure 4(b)). The slice is reshaped into a 1D tensor before it is given as input to masked 1D self-attention layers; the masking is performed over the previous pixels only (as opposed to over the current RGB channels) in the self-attention layers. Then the output of the layers is reshaped back into a 2D tensor, concatenated depth-wise with the output of the slice embedding network, and given as conditioning input to a Gated PixelCNN as in Equation 5 in van den Oord et al. (2016c). The PixelCNN network models the target slice with full masking over pixels and channel dimensions. We can see how memory requirements are significantly lower - up to $\bar { S } ^ { 2 } = 6 4 \times$ lower with $S = 8$ - due to the smaller spatial size of the slices and their compact concatenation along the channel dimension of the input tensor. Due to this structure, the entire previously generated context is captured at each position of the decoder. + +# 3.3 LEARNING + +The log-likelihood derived from equation 2 decomposes as a sum over slices. An unbiased estimator of the log-loss is obtained by uniformly sampling a choice of target slice and evaluating its logprobability conditioned upon previous slices as depicted in Figure 4 (a). We perform maximum likelihood learning by doing stochastic gradient descent on this Monte Carlo estimate, with all gradients computed by backpropagation. + +# 3.4 MULTIDIMENSIONAL UPSCALING WITH SPN + +As seen in Section 2.3, the SPN naturally serves as a size-upscaling network when the first slice of the input tensor is initialized with with an externally generated subimage. In our experiments, we ensure that the smaller subimages used for the initialization and those used in the training of the SPN decoder are identical to each other. + +Analogously, the SPN can be used to upscale the depth of the channels of an image. The image to be upscaled in depth is itself divided into slices by the subscale method (secion 2) and the slices are then concatenated along the channel dimension into a slice tensor for the conditioning image $\mathbf { x } ^ { \mathrm { { : d } _ { 1 } } }$ . The latter is then added as a fixed additional input to the embedding part of the SPN in order to model $P ( \mathbf { x _ { h , w , c } ^ { d _ { 1 } : d _ { 2 } } } | \mathbf { x ^ { d _ { 1 } : d _ { 2 } } } < , \mathbf { x ^ { : d _ { 1 } } } )$ . The model of $P ( \mathbf { x } ^ { \mathbf { : d _ { 1 } } } )$ is a normal SPN, but trained on data with low bit depth. + +Table 1: Negative Log-likelihood (NLL) scores for Downsampled Imagenet (van den Oord et al., 2016b) in bits/dim. The parenthesized numbers in Depth Upscaling rows indicate NLL for $P ( \mathbf { x } ^ { \mathbf { : d _ { 1 } } } )$ and P (xd1:d2h,w,c| $P ( \mathbf { x _ { h , w , c } ^ { d _ { 1 } : d _ { 2 } } } | \mathbf { x ^ { d _ { 1 } : d _ { 2 } } } < , \mathbf { x ^ { : d _ { 1 } } } )$ respectively. + +
ImageNet 32x32ImageNet 64x64
Gated PixelCNN (van den Oord et al.,2016c)Parallel Multiscale (Reed et al.,2017)PixelSNAIL (Chen et al., 2017)Image Transformer (Parmar et al.,2018)Glow (Kingma & Dhariwal,2018)3.833.953.803.774.093.573.70--3.81
Decoder baselineSPNSPN +Depth UpscalingSPN + 16x16 slicesSPN +8x8 slices3.79113.853.913.523.533.53 (0.63, 2.90)=
+ +Table 2: NLL scores for high-resUpscaling rows indicate NLL for $P ( \mathbf { x } ^ { \mathbf { : d _ { 1 } } } )$ mage and $P ( \mathbf { x _ { h , w , c } ^ { d _ { 1 } : d _ { 2 } } } | \mathbf { x ^ { d _ { 1 } : d _ { 2 } } } < , \mathbf { \bar { x } ^ { : d _ { 1 } } } )$ thesized numbers in Depth respectively. + +
ImageNet 128 x 128ImageNet256 x 256
Parallel Multiscale (Reed et al., 2017)3.551
SPN SPN + Depth-Upscaling3.08 3.08 (0.46, 2.62)2.97 3.01 (0.40,2.61)
+ +# 4 EXPERIMENTS + +We demonstrate experimentally that our model is capable of high fidelity samples at high resolution, producing unconditional CelebA-HQ samples of quality better than the Glow model (Kingma & Dhariwal, 2018) and improving the MLE scores. Furthermore, we show that these results extend to high-resolution ImageNet images, with state-of-the-art log-likelihoods at $1 2 8 \mathrm { x } 1 2 8$ by a large margin and the first benchmark on $2 5 6 \times 2 5 6$ ImageNet. Unconditional samples at these resolutions are characterized by unprecedented global coherence. + +Because our networks operate on small images $3 2 \times 3 2$ slices), we can train large networks both in terms of the number of hidden units and in terms of network depth (see Appendix C for details of sizes). The context-embedding network contains 5 convolutional layers and 6-8 self-attention layers depending on the dataset. The masked decoder consists of a PixelCNN with 15 layers in all experiments. The 1D Transformer in the decoder (Figure 4(b)) has between 8 and 10 layers depending on the dataset. See Table 4 for all dataset-specific hyperparameter details. + +# 4.1 DOWNSAMPLED IMAGENET AT $3 2 \times 3 2$ AND $6 4 \times 6 4$ + +We first benchmark the performance of our hybrid decoder alone (i.e. no subscaling, Figure 4(b)) and show that it compares favorably to state of the art models on $3 2 \times 3 2$ Downsampled ImageNet (see Table 1). We find that SPN hurts in this low-resolution setting with $S = 2$ and even further with $S = 4$ . This is likely because the size of the resulting image slices becomes very small and the image coarse grained. + +On $6 4 \times 6 4$ Downsampled ImageNet, we achieve a state of the art log-likelihood of 3.52 bits/dim. We hypothesize that PixelSNAIL would achieve a similar score, but results at this resolution were not reported in Chen et al. (2017). At this resolution, SPN scores similarly with 3.53 bits/dim. The improvement over Glow in the 5-bit setting is very significant (Table 3). + +Table 3: Negative Log-likelihood scores for 5-bit datasets in bits/dim. + +
ImageNet 64 x 64 (5bit)CelebA-HQ 256 x 256 (5bit)
Glow (Kingma &Dhariwal,2018)1.761.03
SPN1.410.61
+ +# 4.2 IMAGENET AT $1 2 8 \times 1 2 8$ + +For these experiments we use the standard ILSVRC Imagenet dataset (Kolesnikov & Lampert, 2016b) resized with Tensorflow’s resize area function. Parallel Multiscale PixelCNN (Reed et al., 2017) is the only model in the literature which reports log-likelihood on $1 2 8 \times 1 2 8$ ImageNet. SPN improves the log-likelihood over this model from 3.55 bits/dim to 3.08 bits/dim (see Table 2). Figure 5 gives $1 2 8 \times 1 2 8$ 8-bit ImageNet samples for both the setting of depth upscaling only and of complete multidimensional upscaling. These settings do not affect the NLL, but the samples with depth upscaling show significant semantic coherence that is usually lacking in samples without upscaling. In addition, multidimensional upscaling seems to increase the overall rate of success of the samples. Additional intermediate ImageNet samples can be seen in Figures 10, 11 and 12 in the Appendix. + +# 4.3 CELEBAHQ + +At $2 5 6 \times 2 5 6$ we can produce high-fidelity samples of celebrity faces from the CelebAHQ dataset. The quality compares favorably to the samples of other models such as Glow and GANs (Karras et al., 2017). We show in Table 3 that the achieved MLE scores are a significant improvement over previously reported scores. Figure 6 showcases some samples for 8-bit CelebAHQ-256. Figure 7 in the Appendix includes 5-bit samples, Figure 8 includes 3-bit samples while Figure 9 includes 3-bit samples with the temperature of the output distribution set to 0.95. + +# 5 CONCLUSION + +The problem of whether it is possible to learn the distribution of complex natural images and attain high sample fidelity has been a long-standing one in the tradition of generative models. The SPN and Multidimensional Upscaling model that we introduce accomplishes a large step towards solving this problem, by attaining both state-of-the-art MLE scores on large-scale images from complex domains such as CelebAHQ-256 and ImageNet-128 and by being able to generate high fidelity full 8-bit samples from the resulting learnt distributions without alterations to the sampling process (via e.g. heavy modifications of the temperature of the output distribution). The generated samples show an unprecedented amount of semantic coherence and exactness of details even at the large scale size of full 8-bit $1 2 8 \times 1 2 8$ and $2 5 6 \times 2 5 6$ images. + +# 6 ACKNOWLEDGEMENTS + +We would like to thank Alex Graves, Karen Simonyan, Aaron van den Oord, Tim Harley, Sander Dieleman, Tim Salimans, Lasse Espeholt, Ali Razavi, Jeffrey De Fauw and Andy Brock for insightful discussions. In particular we wish to thank Andriy Mnih for formative ideas about autoregressivity in the bit-depth of an image. + +# REFERENCES + +Sanjeev Arora and Yi Zhang. Do gans actually learn the distribution? an empirical study. CoRR, abs/1706.08224, 2017. URL http://arxiv.org/abs/1706.08224. + +Xi Chen, Nikhil Mishra, Mostafa Rohaninejad, and Pieter Abbeel. Pixelsnail: An improved autoregressive generative model. CoRR, abs/1712.09763, 2017. URL http://arxiv.org/abs/ 1712.09763. + +Norman P. Jouppi, Cliff Young, Nishant Patil, David A. Patterson, Gaurav Agrawal, Raminder Bajwa, Sarah Bates, Suresh Bhatia, Nan Boden, Al Borchers, Rick Boyle, Pierre-luc Cantin, + +Clifford Chao, Chris Clark, Jeremy Coriell, Mike Daley, Matt Dau, Jeffrey Dean, Ben Gelb, Tara Vazir Ghaemmaghami, Rajendra Gottipati, William Gulland, Robert Hagmann, Richard C. 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URL http://arxiv.org/abs/1803.07416. + +Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V. Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, Jeff Klingner, Apurva Shah, Melvin Johnson, Xiaobing Liu, Lukasz Kaiser, Stephan Gouws, Yoshikiyo Kato, Taku Kudo, Hideto Kazawa, Keith Stevens, George Kurian, Nishant Patil, Wei Wang, Cliff Young, Jason Smith, Jason Riesa, Alex Rudnick, Oriol Vinyals, Greg Corrado, Macduff Hughes, and Jeffrey Dean. Google’s neural machine translation system: Bridging the gap between human and machine translation. CoRR, abs/1609.08144, 2016. URL http://arxiv.org/abs/1609.08144. + +# APPENDIX A + +In some cases for purposes of analysis the entropy of the softmax output distributions has been artificially reduced via a “temperature” divisor on the predicted logits. When we say the temperature is 0.95, we mean that the logits of a trained model have been divided by this constant at sampling time. + +![](images/d9e179ba71ce959d7d1f3a8cee2639304460ecce18c6d99ecacc0bb2dd11b905.jpg) +Figure 6: 8-bit 256x256 RGB CelebA-HQ samples from SPN with Depth-Upscaling. Temperature is 0.99 for the low-bit-depth image and otherwise 1.0 + +![](images/504d0665a5b40d06191c7bd8627c201458e6521b40015b4d0b70db79bfe3b9b9.jpg) +Figure 7: 256x256 CelebA-HQ 5bit samples from SPN + +# APPENDIX B + +Please see Table 4 for all the detailed hyperparameters. + +# APPENDIX C + +Our experiments operate at a fairly large scale in terms of both the amount of compute used and the size of the networks. We proportionately increase the batch size so that the number of pixels in a batch is not affected by the subscaling. These large batch sizes (a maximum of 2048) are achieved by increasing the degree of data parallelism by running on Google Cloud TPU pods (Jouppi et al., 2017). For Imagenet 32 we used 64 tensorcores. For ImageNet 64, 128 and 256, we use 128 tensorcores. The fast interconnect between these devices affords much faster synchronous gradient computation than would be possible using the same number of GPUs. When overfitting is a problem, as in small datasets like CelebA-HQ, we rather decrease the batch size and use a lower number of 32 tensorcores. + +Our SPN architectures have between ${ \sim } 5 0 \mathbf { M }$ and $\sim 2 5 0 \mathrm { M }$ parameters depending on the dataset. See Table 4 for the number of parameters in the SPN architecture for each dataset. Depth-upscaling doubles the number of parameters due to using two separate networks with untied weights. Sizeupscaling adds more parameters still for the separate decoder-only network which models the first slice as seen in Figure 3 (g). Thus the maximal number of parameters used to generate a sample in the paper occurs in the multidimensional upscaling setting for ImageNet 128, where the total parameter count reaches ${ \sim } 6 5 0 \mathrm { M }$ (the decoder-only network used to model the first slice has $\mathord { \sim } 1 5 0 \mathrm { M }$ parameters). + +![](images/d848a209d6b7533bcfb3b34e9070b3c0d63348e6b8f6869d3f908d184dba7b69.jpg) +Figure 8: 256x256 CelebA-HQ 3bit samples from SPN + +![](images/697e394f5b5990e1e08a4707158de53d0bcaed683c4c1de9516612c7dd56baad.jpg) +Figure 9: 256x256 CelebA-HQ 3bit samples from SPN with temperature 0.95 + +![](images/112ca671283b8d1c9be472807544de4f71075d12f5a63d4abf83c2837648da53.jpg) +Figure 10: 128x128 ImageNet 3bit; upscaled $3 2 \mathrm { x } 3 2 $ slices + +![](images/f059f8be6aedb09ab4801a67ed6fa109022eb0a363f720ce1ea1d8f95bb67805.jpg) +Figure 11: 128x128 ImageNet 3bit samples from model trained on $3 2 \mathrm { x } 3 2 $ slices + +![](images/1ad9d7898f245e7d4d3450a51ccd621e11cd56b7342e3076a4759202e18bc3e1.jpg) +Figure 12: 128x128 ImageNet 3bit samples from SPN + +Table 4: SPN Hyperparameters. A learning rate marked “sched” utilizes a piecewise-constant schedule starting at 1e-4, and decreasing to 3e-5 and finally 1e-5 at training steps $5 0 \mathrm { k }$ and $1 0 0 \mathrm { k }$ respectively. The “attention” parameters listed are configurable hyperparameters of the open source Transformer implementation in Vaswani et al. (2018). The parameter “residual channels“ refers to the number of hidden units in residual convolution layers within the Slice Embedder or PixelCNN networks. + +
Optimization
batch size(1024,2048,2048,2048)256
learning rate(sched,sched, le-5,1e-5)5e-5
rmsprop momentum0.90.9
rmsprop decay0.950.95
rmsprop epsilon1e-81e-8
polyak decay0.99990.9999
Decoder
PixelCNN layers1515
PixelCNN conv channels(256,256, 384, 384)128
PixelCNN residual channels12801280
PixelCNN nonlinearitygatedgated
PixelCNN filter size33
masked self-attention layers85
attention heads105
attention channels128128
attention ffn layer"parameter_attention""parameter_attention"
SliceEmbedder
conv layers55
conv filter size33
conv channels256256
residual channels10241024
nonlinearityrelurelu
self-attention layers(6,6, 8, 8)6
attention heads(4,4,8,8)4
attention channels(64,64, 128, 128)64
attention ffn layer"parameter_attention""parameter_attention"
Number of parameters ~150M ~150M ~ 250M ~ 250M) ~ 50M
\ No newline at end of file diff --git a/parse/train/HylzTiC5Km/HylzTiC5Km_content_list.json b/parse/train/HylzTiC5Km/HylzTiC5Km_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..6817a576901bfa7f2b5af65c46691f6afdafd659 --- /dev/null +++ b/parse/train/HylzTiC5Km/HylzTiC5Km_content_list.json @@ -0,0 +1,1247 @@ +[ + { + "type": "text", + "text": "GENERATING HIGH FIDELITY IMAGES WITH SUBSCALE PIXEL NETWORKS AND MULTIDIMENSIONAL UPSCALING ", + "text_level": 1, + "bbox": [ + 174, + 98, + 635, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jacob Menick∗ \nDeepMind \njmenick@google.com ", + "bbox": [ + 184, + 195, + 362, + 237 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Nal Kalchbrenner∗ Google Brain Amsterdam nalk@google.com ", + "bbox": [ + 509, + 195, + 681, + 237 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 273, + 544, + 289 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The unconditional generation of high fidelity images is a longstanding benchmark for testing the performance of image decoders. Autoregressive image models have been able to generate small images unconditionally, but the extension of these methods to large images where fidelity can be more readily assessed has remained an open problem. Among the major challenges are the capacity to encode the vast previous context and the sheer difficulty of learning a distribution that preserves both global semantic coherence and exactness of detail. To address the former challenge, we propose the Subscale Pixel Network (SPN), a conditional decoder architecture that generates an image as a sequence of sub-images of equal size. The SPN compactly captures image-wide spatial dependencies and requires a fraction of the memory and the computation required by other fully autoregressive models. To address the latter challenge, we propose to use Multidimensional Upscaling to grow an image in both size and depth via intermediate stages utilising distinct SPNs. We evaluate SPNs on the unconditional generation of CelebAHQ of size 256 and of ImageNet from size 32 to 256. We achieve state-of-the-art likelihood results in multiple settings, set up new benchmark results in previously unexplored settings and are able to generate very high fidelity large scale samples on the basis of both datasets. ", + "bbox": [ + 232, + 306, + 766, + 555 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 583, + 336, + 599 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A successful generative model has two core aspects: it produces targets that have high fidelity and it generalizes well on held-out data. Autoregressive (AR) models trained by conventional maximum likelihood estimation (MLE) have produced superior scores on held-out data across a wide range of domains such as text (Vaswani et al., 2017; Wu et al., 2016), audio (van den Oord et al., 2016a), images (Parmar et al., 2018) and videos (Kalchbrenner et al., 2016). These scores are a measure of the models’ ability to generalize in that setting. From the perspective of sample fidelity, the outputs generated by AR models have also achieved state-of-the-art fidelity in many of the aforementioned domains with one notable exception. In the domain of unconditional large-scale image generation, AR samples have yet to manifest long-range structure and semantic coherence. ", + "bbox": [ + 174, + 614, + 825, + 741 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "One source of difficulties impeding high-fidelity image generation is the multi-faceted relationship between the MLE scores achieved by a model and the model’s sample fidelity. On the one hand, MLE is a well-defined measure as improvements in held-out scores generally produce improvements in the visual fidelity of the samples. On the other hand, as opposed to for example adversarial methods (Arora & Zhang, 2017), MLE forces the model to support the entire empirical distribution. This guarantees the model’s ability to generalize at the cost of allotting capacity to parts of the distribution that are irrelevant to fidelity. A second source of difficulties arises from the high dimensionality of large images. A $2 5 6 \\times 2 5 6 \\times 3$ image has a total of 196,608 positions that need to be architecturally connected in order to learn dependencies among them; the representations at each position require sufficient capacity to express their respective surrounding contexts. These requirements translate to large amounts of memory and computation. ", + "bbox": [ + 174, + 747, + 825, + 900 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/95c2aef047f12f2740eba99f379e7a1c4e685396bb70dd2a36d7926b37c9409b.jpg", + "image_caption": [ + "Figure 1: A representation of Multidimensional Upscaling. Left: depth upscaling is applied to a generated 3-bit $2 5 6 \\times 2 5 6$ RGB subimage from CelebAHQ to map it to a full 8-bit $2 5 6 \\times 2 5 6$ RGB image. Right: size upscaling followed by depth upscaling are applied to a generated 3-bit $3 2 \\times 3 2$ RGB subimage from ImageNet to map it to the target resolution of the 8-bit $1 2 8 \\times 1 2 8$ RGB image. We stress that the rightmost column of both figures are true unconditional samples from our model at full 8bit depth. " + ], + "image_footnote": [], + "bbox": [ + 199, + 98, + 799, + 323 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "These difficulties notwithstanding, we aim to learn the full distribution over 8-bit RGB images of size up to $2 5 6 \\times 2 5 6$ well enough so that the samples have high fidelity. We aim to guide the model to focus first on visually more salient bits of the distribution and later on the visually less salient bits. We identify two visually salient subsets of the distribution: first, the subset determined by sub-images (“slices”) of smaller size (e.g. $3 2 \\times 3 2$ ) sub-sampled at all positions from the original image; and secondly, the subset determined by the few (e.g. 3) most significant bits of each RGB channel in the image. We use Multidimensional Upscaling to map from one subset of the distribution to the other one by upscaling images in size or in depth. For example, the generation of a $1 2 8 \\times 1 2 8 :$ 8-bit RGB image proceeds by first upscaling it in size from a $3 2 \\times 3 2$ 3-bit RGB image to a $1 2 8 \\times 1 2 8$ 3-bit RGB image; we then upscale the resulting image in depth to the original resolution of the $1 2 8 \\times 1 2 8$ 8-bit RGB image. We thus train three networks: (a) a decoder on the small size, low depth image slices subsampled at every $n$ pixels from the original image with the desired target resolution; (b) a size-upscaling decoder that generates the large size, low depth image conditioned on the small size, low depth image; and (c) a depth-upscaling decoder that generates the large size, high depth image conditioned on the large size, low depth image. Figure 1 illustrates this process. ", + "bbox": [ + 174, + 465, + 825, + 674 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To address the latter difficulties that ensue in the training of decoders (b) and (c), we develop the Subscale Pixel Network (SPN) architecture. The SPN divides an image of size $N \\times N$ into sub-images of size $\\begin{array} { r } { \\frac { N } { S _ { . } } \\times \\frac { N } { S } } \\end{array}$ sliced out at interleaving positions (see Figure 2), which implicitly also captures a form of size upscaling. The $N \\times N$ image is generated one slice at a time conditioned on previously generated slices in a way that encodes a rich spatial structure. SPN consists of two networks, a conditioning network that embeds previous slices and a decoder proper that predicts a single target slice given the context embedding. The decoding part of the SPN acts over image slices with the same spatial structure and it can share weights for all of them. The SPN is an independent image decoder with an implicit size upscaling mechanism, but it can also be used as an explicit size upscaling network by initializing the first slice of the SPN input at sampling time with one generated separately during step (a). ", + "bbox": [ + 174, + 680, + 825, + 833 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We extensively evaluate the performance of SPN and the size and depth upscaling methods both quantitatively and from a fidelity perspective on two unconditional image generation benchmarks, CelebAHQ-256 and ImageNet of various sizes up to 256. From a MLE scores perspective, we compare with previous work to obtain state-of-the-art results on CelebAHQ-256, both at full 8-bit resolution and at the reduced 5-bit resolution (Kingma & Dhariwal, 2018), and on ImageNet-64. We also establish MLE baselines for ImageNet-128 and ImageNet-256. From a sample fidelity perspective, we show the strong benefits of multidimensional upscaling as well as the benefits of the SPN. We produce CelebAHQ-256 samples (at full 8-bit resolution) that are of similar visual fidelity to those produced with methods such as GANs that lack however an intrinsic measure of generalization (Mescheder, 2018; Karras et al., 2017). We also produce some of the first successful samples on unconditional ImageNet-128 (also at 8-bit) showing again the striking impact of the SPN and of multidimensional upscaling on sample quality and setting a fidelity baseline for future methods. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/7b2a844dbd9243a20e217da5e13212d83a158d5360aa6104f93056ee1c073ba0.jpg", + "image_caption": [ + ".Figure 2: The receptive field in a Subscale Pixel Networks (a) and the four image slices subsampled from the original image (b) " + ], + "image_footnote": [], + "bbox": [ + 191, + 99, + 812, + 237 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 296, + 826, + 382 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 MODEL ", + "text_level": 1, + "bbox": [ + 174, + 400, + 269, + 416 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 CONVENTIONAL GENERATION ORDERING ", + "text_level": 1, + "bbox": [ + 174, + 431, + 508, + 446 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A standard AR image model such as the PixelCNN (van den Oord et al., 2016b) generates an $H x W$ colour image starting at the top-left position and ending at the bottom-right position, fully generating the three 8-bit channels of each pixel in a given position: ", + "bbox": [ + 174, + 457, + 825, + 500 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/aed0d7c54a9e53f029e8bf827d7dd5e2df0d49ed2d355984974af3222f0be857.jpg", + "text": "$$\nP ( \\mathbf { x } ) = \\prod _ { h = 1 } ^ { H } \\prod _ { w = 1 } ^ { W } \\prod _ { c } ^ { \\{ R , G , B \\} } P ( x _ { h , w , c } | \\mathbf { x } _ { < } )\n$$", + "text_format": "latex", + "bbox": [ + 367, + 503, + 630, + 549 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbf { x } _ { < }$ corresponds to all previously generated intensity values in the ordering and $h , w .$ , and $c$ are row, column, and colour channel indices. The raster scan ordering (Figure 3(a)) is conventionally used in AR models. Each conditional distribution $P ( x _ { h , w , c } | \\mathbf { x } _ { < } )$ is parametrized by a deep neural network (van den Oord et al., 2016b). ", + "bbox": [ + 174, + 558, + 825, + 613 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 SUBSCALE ORDERING IN IMAGES ", + "text_level": 1, + "bbox": [ + 176, + 630, + 449, + 643 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We define an alternative ordering that divides a large image into a sequence of equally sized slices and has various core properties. First, it makes it easy to compactly encode long-range dependencies across the many pixels in the large images. It also induces a spatial structure over the original image by aligning the subsampled slices; this also has an implicit size upscaling side effect. From the perspective of the neural architecture, it makes it possible for the same decoder within the SPN to be consistently applied to all slices, since they are structurally similar; the smaller slices also allow for self-attention (Vaswani et al., 2017) in the SPN to be used without local contexts (Parmar et al., 2018). We think of the ordering as the two-dimensional analogue of the one-dimensional subscale ordering introduced in Kalchbrenner et al. (2018). ", + "bbox": [ + 173, + 655, + 826, + 781 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The subscale ordering is defined as follows: ", + "bbox": [ + 176, + 787, + 462, + 803 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/7719f0fd6d1f96bd360d15a09e562580430206baff6eaab1a0ebf4089a981037.jpg", + "text": "$$\nP ( \\mathbf { x } ) = \\prod _ { i = 1 } ^ { S } \\prod _ { j = 1 } ^ { S } \\prod _ { h = 1 } ^ { H / S } \\prod _ { w = 1 } ^ { W / S } \\prod _ { c } ^ { \\{ R , G , B \\} } P ( x _ { i + S * h , j + S * w , c } | \\mathbf { x } _ { < } )\n$$", + "text_format": "latex", + "bbox": [ + 310, + 805, + 687, + 852 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbf { x } _ { < }$ corresponds to all previously generated intensity values according to this ordering. Figure 3(d) illustrates the subscale ordering. A scaling factor $S$ is selected and each slice of size $H / S \\times W / S$ is obtained by selecting a pixel every $S$ pixels in both height and width; there are thus $S ^ { 2 }$ interleaved slices in the original image, with each specified by its row and column offset $( i , j )$ . We sometimes refer to this offset as the “meta-position” of a slice. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/98559d46079f8d28599a7a5163c09abb9d3f0b63ff32e33727c28ac6d3f9a3f2.jpg", + "image_caption": [ + "Figure 3: Different generation ordering schemes, where the numbers indicate the step-by-step order. Distinct colors correspond to distinct neural networks. (a) and (b) are from (van den Oord et al., 2016b). (c) is from (Reed et al., 2017). The Subscale ordering alone, with size-only, depth-only and with multidimensional upscaling are, respectively, in blocks (d), (e), (f) and (g). " + ], + "image_footnote": [], + "bbox": [ + 256, + 99, + 730, + 347 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 SIZE UPSCALING IN SUBSCALE ORDERING ", + "text_level": 1, + "bbox": [ + 178, + 441, + 511, + 455 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The subscale ordering itself already captures size upscaling implicitly. Analogous to the multi-scale ordering (van den Oord et al., 2016b), and depicted in 3(b), we can perform size upscaling explicitly, by training a single slice decoder on subimages and generate the first slice of a subscale ordering from the single slice decoder itself. The rest of the image is then generated according to the subscale ordering by the main network (see 3(e)). The single-slice model can be trained on just the first slices of images, or on slices at all positions in all images given the shared spatial structure among the slices. For this reason, the same SPN that captures the subscale ordering can act simultaneously as a full-blown image model as well as a size upscaling model if initialized with the outputs of a single-slice decoder. A separate formulation of size upscaling is the Parallel Multi-Scale (Reed et al., 2017) ordering where the pixels in an image are doubled at every stage by distinct neural networks and are generated in parallel without sequentiality (3(c)). ", + "bbox": [ + 173, + 468, + 826, + 621 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.4 DEPTH UPSCALING ", + "text_level": 1, + "bbox": [ + 176, + 638, + 349, + 652 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Multidimensional upscaling applies upscaling not just in the height and width of the image, but also in the remaining dimension that is channel depth. This is performed in stages such that a network first generates the $d _ { 1 }$ most significant bits of an image using a conventional or subscale ordering; then a second network generates the next $d _ { 2 }$ most significant bits of the image conditioned on all the $d _ { 1 }$ bits of the image; and so on to further stages. Using the conventional ordering as basis, the first stage of depth upscaling looks as follows: ", + "bbox": [ + 173, + 665, + 825, + 748 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/fee3bf4b2621aed419218210fbbefd00c7aa96a36e8b413b113bfc4c74439a5a.jpg", + "text": "$$\nP ( \\mathbf { x } ^ { : \\mathbf { d } _ { 1 } } ) = \\prod _ { h = 1 } ^ { H } \\prod _ { w = 1 } ^ { W } \\prod _ { c } ^ { \\{ R , G , B \\} } P ( \\mathbf { x } _ { \\mathbf { h } , \\mathbf { w } , \\mathbf { c } } ^ { : \\mathbf { d } _ { 1 } } | \\mathbf { x } ^ { : \\mathbf { d } _ { 1 } } < )\n$$", + "text_format": "latex", + "bbox": [ + 346, + 756, + 651, + 801 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Next, a second stage of depth upscaling has the following form, conditioned on the first $d _ { 1 }$ bits of each channel: ", + "bbox": [ + 171, + 808, + 821, + 835 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2d158c9d9d286611adb44019233c1bcead43afe0546d39b4cf8716bd54b4c33e.jpg", + "text": "$$\nP ( \\mathbf { x ^ { d _ { 1 } : d _ { 2 } } } ) = \\prod _ { h = 1 } ^ { H } \\prod _ { w = 1 } ^ { W } \\prod _ { c } ^ { \\{ R , G , B \\} } P ( \\mathbf { x _ { h , w , c } ^ { d _ { 1 } : d _ { 2 } } } | \\mathbf { x ^ { d _ { 1 } : d _ { 2 } } } _ { < } , \\mathbf { x } ^ { : d _ { 1 } } )\n$$", + "text_format": "latex", + "bbox": [ + 312, + 832, + 686, + 878 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We do not share weights among the networks at different stages of depth upscaling. We note that in depth upscaling bits of lower significance are only generated when the more significants bits at all positions have been generated in a previous stage. Just like for size upscaling from the previous section, the goal of multidimensional upscaling is to let the model focus on visually salient bits of an image unaffected by less salient and less predictable bits of the image. Depth upscaling is related to the method underlying the Grayscale PixelCNN that models 4-bit greyscale images subsampled from colored images Kolesnikov & Lampert (2016a). ", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/fe7ca922f4649bae3603e8639b2d4ae3c036fa4269b86d943ed1c5ad94c6973b.jpg", + "image_caption": [ + "(a) (b)Figure 4: (a) The architecture of a Subscale Pixel Network, with a conditional and a decoding part. (b) Scheme of the parts in the decoder itself. " + ], + "image_footnote": [], + "bbox": [ + 205, + 99, + 790, + 222 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 290, + 825, + 347 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 368, + 339, + 385 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 CHALLENGES IN TRAINING CONVENTIONAL DECODERS ON LARGE IMAGES ", + "text_level": 1, + "bbox": [ + 178, + 390, + 740, + 406 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Using a conventional generation ordering, models such as PixelRNN, PixelCNN (van den Oord et al., 2016b) and Image Transformer (Parmar et al., 2018) construct a representation of the generated context for each dimension of each pixel. Existing AR approaches inherently require an amount of computation and memory that is superlinear in the number of pixels. In particular, the quadratic memory requirements of self-attention become severely limiting for images larger than $3 2 \\times 3 2$ and intractable in practice for the 196,608 distinct positions we consider in a $2 5 6 \\times 2 5 6$ colour image. ", + "bbox": [ + 174, + 417, + 825, + 501 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Mitigating the memory requirements and computational requirements of encoding the dependencies amongst so many variables often comes at the expense of global context. Modeling choices such as cropping images within the decoders (Kalchbrenner et al., 2016) or performing self-attention over local neighborhoods (Parmar et al., 2018) neglect global dependencies, while model parallelism, though technically feasible with the joint use of a very large number of accelerators, does not overcome the challenges in learning the global structure. ", + "bbox": [ + 174, + 508, + 825, + 592 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 SUBSCALE PIXEL NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 612, + 413, + 626 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To address these challenges, we devise the Subscale Pixel Network (SPN), an architecture that embodies the subscale ordering from Section 2.2. For an image of size $H \\times W \\times 3 \\times D$ , where $D$ is the number of bits used for the current generation stage, one first chooses a scaling factor $S$ and obtains the $S ^ { 2 }$ slices of the original image of size $H / S \\times W / S \\times 3 \\times D$ . We use $H = W = 2 5 6$ and $S = 8$ as well as $H = W = 1 2 8$ and $S = 4$ , for the larger images we process, so that the slices have size $3 2 \\times 3 2 \\times 3 \\times D$ . This scheme of choosing $S$ such that slices are always $3 2 \\times 3 2$ renders the memory and computation requirements effectively constant as the true image size $H \\times W$ changes. ", + "bbox": [ + 174, + 638, + 825, + 736 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The SPN architecture is composed of two parts: an embedding part for slices at preceding metapositions that conditions the decoder for the current slice that is being generated (Figure 4 (a)). The embedding part is a convolutional neural network with residual blocks that takes as input preceding slices that are concatenated along the depth dimension. One detail is the way the slices are ordered along the channel dimension when concatenated. As illustrated in Figure 4 (a), empty padding slices are used to preserve relative meta-positions of each preceding slice with respect to the current target slice. For example, the slice above any target slice in the two-dimensional meta-grid is always aligned in the same position along the depth axis in the input. This achieves equivariance in the embedding architecture with respect to the $( i , j )$ offset of a slice. The padding slices also ensure that the depth of the input slice tensor remains the same for all target slices. In addition to the slice tensor, the embedding part also receives as input the meta-position of the target slice as an embedding of 8 units tiled spatially across the slice tensor. The pixel intensity values are also embedded as one-hot indices of size 8. The context embedding network passes its input through a series of self-attention layers and then a series of residual blocks, finally emitting a slice-sized feature map s that summarizes the context for the decoder. ", + "bbox": [ + 174, + 743, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/e6f31c8d7a0d48ad915bfecc5d546389faa9b99b3c07f683678a398e09811c80.jpg", + "image_caption": [ + "Figure 5: Left: 8-bit 128x128 RGB ImageNet samples from SPN with depth upscaling only. Right: 8-bit 128x128 RGB ImageNet samples from SPN with full-blown Multidimensional Upscaling. Temperature is 0.99 for the initial $3 2 \\mathbf { x } 3 2$ sub-image and otherwise 1.0 " + ], + "image_footnote": [], + "bbox": [ + 174, + 103, + 838, + 349 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 438, + 823, + 465 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The decoder takes as input the encoded slice tensor s in a position-preserving manner: each position in the target slice is given as input the encoded representations of pixels at that same position in the preceding slices. In addition it processes the target slice in the raster-scan order. The decoder that we use is a hybrid architecture combining masked convolution and self-attention (Chen et al., 2017). We employ an initial 1D self-attention network Vaswani et al. (2017) that is used to gather the entire available context in the slice (see Figure 4(b)). The slice is reshaped into a 1D tensor before it is given as input to masked 1D self-attention layers; the masking is performed over the previous pixels only (as opposed to over the current RGB channels) in the self-attention layers. Then the output of the layers is reshaped back into a 2D tensor, concatenated depth-wise with the output of the slice embedding network, and given as conditioning input to a Gated PixelCNN as in Equation 5 in van den Oord et al. (2016c). The PixelCNN network models the target slice with full masking over pixels and channel dimensions. We can see how memory requirements are significantly lower - up to $\\bar { S } ^ { 2 } = 6 4 \\times$ lower with $S = 8$ - due to the smaller spatial size of the slices and their compact concatenation along the channel dimension of the input tensor. Due to this structure, the entire previously generated context is captured at each position of the decoder. ", + "bbox": [ + 174, + 472, + 825, + 679 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 691, + 289, + 705 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The log-likelihood derived from equation 2 decomposes as a sum over slices. An unbiased estimator of the log-loss is obtained by uniformly sampling a choice of target slice and evaluating its logprobability conditioned upon previous slices as depicted in Figure 4 (a). We perform maximum likelihood learning by doing stochastic gradient descent on this Monte Carlo estimate, with all gradients computed by backpropagation. ", + "bbox": [ + 174, + 718, + 825, + 787 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4 MULTIDIMENSIONAL UPSCALING WITH SPN", + "text_level": 1, + "bbox": [ + 174, + 806, + 526, + 820 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As seen in Section 2.3, the SPN naturally serves as a size-upscaling network when the first slice of the input tensor is initialized with with an externally generated subimage. In our experiments, we ensure that the smaller subimages used for the initialization and those used in the training of the SPN decoder are identical to each other. ", + "bbox": [ + 174, + 833, + 825, + 888 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Analogously, the SPN can be used to upscale the depth of the channels of an image. The image to be upscaled in depth is itself divided into slices by the subscale method (secion 2) and the slices are then concatenated along the channel dimension into a slice tensor for the conditioning image $\\mathbf { x } ^ { \\mathrm { { : d } _ { 1 } } }$ . The latter is then added as a fixed additional input to the embedding part of the SPN in order to model $P ( \\mathbf { x _ { h , w , c } ^ { d _ { 1 } : d _ { 2 } } } | \\mathbf { x ^ { d _ { 1 } : d _ { 2 } } } < , \\mathbf { x ^ { : d _ { 1 } } } )$ . The model of $P ( \\mathbf { x } ^ { \\mathbf { : d _ { 1 } } } )$ is a normal SPN, but trained on data with low bit depth. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/fb8c9c354dac3417c972b3cab0ea60198e27d9392a087115aa45ccd6b38d1d82.jpg", + "table_caption": [ + "Table 1: Negative Log-likelihood (NLL) scores for Downsampled Imagenet (van den Oord et al., 2016b) in bits/dim. The parenthesized numbers in Depth Upscaling rows indicate NLL for $P ( \\mathbf { x } ^ { \\mathbf { : d _ { 1 } } } )$ and P (xd1:d2h,w,c| $P ( \\mathbf { x _ { h , w , c } ^ { d _ { 1 } : d _ { 2 } } } | \\mathbf { x ^ { d _ { 1 } : d _ { 2 } } } < , \\mathbf { x ^ { : d _ { 1 } } } )$ respectively. " + ], + "table_footnote": [], + "table_body": "
ImageNet 32x32ImageNet 64x64
Gated PixelCNN (van den Oord et al.,2016c)Parallel Multiscale (Reed et al.,2017)PixelSNAIL (Chen et al., 2017)Image Transformer (Parmar et al.,2018)Glow (Kingma & Dhariwal,2018)3.833.953.803.774.093.573.70--3.81
Decoder baselineSPNSPN +Depth UpscalingSPN + 16x16 slicesSPN +8x8 slices3.79113.853.913.523.533.53 (0.63, 2.90)=
", + "bbox": [ + 235, + 99, + 763, + 248 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/97540deaa9124ade285886a4f72d8b85e1912725731a0972ece9692b295e5fe6.jpg", + "table_caption": [ + "Table 2: NLL scores for high-resUpscaling rows indicate NLL for $P ( \\mathbf { x } ^ { \\mathbf { : d _ { 1 } } } )$ mage and $P ( \\mathbf { x _ { h , w , c } ^ { d _ { 1 } : d _ { 2 } } } | \\mathbf { x ^ { d _ { 1 } : d _ { 2 } } } < , \\mathbf { \\bar { x } ^ { : d _ { 1 } } } )$ thesized numbers in Depth respectively. " + ], + "table_footnote": [], + "table_body": "
ImageNet 128 x 128ImageNet256 x 256
Parallel Multiscale (Reed et al., 2017)3.551
SPN SPN + Depth-Upscaling3.08 3.08 (0.46, 2.62)2.97 3.01 (0.40,2.61)
", + "bbox": [ + 235, + 324, + 761, + 385 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 459, + 825, + 520 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 540, + 326, + 556 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We demonstrate experimentally that our model is capable of high fidelity samples at high resolution, producing unconditional CelebA-HQ samples of quality better than the Glow model (Kingma & Dhariwal, 2018) and improving the MLE scores. Furthermore, we show that these results extend to high-resolution ImageNet images, with state-of-the-art log-likelihoods at $1 2 8 \\mathrm { x } 1 2 8$ by a large margin and the first benchmark on $2 5 6 \\times 2 5 6$ ImageNet. Unconditional samples at these resolutions are characterized by unprecedented global coherence. ", + "bbox": [ + 174, + 571, + 825, + 656 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Because our networks operate on small images $3 2 \\times 3 2$ slices), we can train large networks both in terms of the number of hidden units and in terms of network depth (see Appendix C for details of sizes). The context-embedding network contains 5 convolutional layers and 6-8 self-attention layers depending on the dataset. The masked decoder consists of a PixelCNN with 15 layers in all experiments. The 1D Transformer in the decoder (Figure 4(b)) has between 8 and 10 layers depending on the dataset. See Table 4 for all dataset-specific hyperparameter details. ", + "bbox": [ + 174, + 662, + 825, + 747 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 DOWNSAMPLED IMAGENET AT $3 2 \\times 3 2$ AND $6 4 \\times 6 4$ ", + "text_level": 1, + "bbox": [ + 173, + 763, + 578, + 779 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We first benchmark the performance of our hybrid decoder alone (i.e. no subscaling, Figure 4(b)) and show that it compares favorably to state of the art models on $3 2 \\times 3 2$ Downsampled ImageNet (see Table 1). We find that SPN hurts in this low-resolution setting with $S = 2$ and even further with $S = 4$ . This is likely because the size of the resulting image slices becomes very small and the image coarse grained. ", + "bbox": [ + 173, + 790, + 825, + 861 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "On $6 4 \\times 6 4$ Downsampled ImageNet, we achieve a state of the art log-likelihood of 3.52 bits/dim. We hypothesize that PixelSNAIL would achieve a similar score, but results at this resolution were not reported in Chen et al. (2017). At this resolution, SPN scores similarly with 3.53 bits/dim. The improvement over Glow in the 5-bit setting is very significant (Table 3). ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/da9fe299b4603133810bba74557dec948b267f7d9bd96d1a0ea8233637161001.jpg", + "table_caption": [ + "Table 3: Negative Log-likelihood scores for 5-bit datasets in bits/dim. " + ], + "table_footnote": [], + "table_body": "
ImageNet 64 x 64 (5bit)CelebA-HQ 256 x 256 (5bit)
Glow (Kingma &Dhariwal,2018)1.761.03
SPN1.410.61
", + "bbox": [ + 210, + 101, + 787, + 148 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 IMAGENET AT $1 2 8 \\times 1 2 8$ ", + "text_level": 1, + "bbox": [ + 176, + 194, + 387, + 209 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For these experiments we use the standard ILSVRC Imagenet dataset (Kolesnikov & Lampert, 2016b) resized with Tensorflow’s resize area function. Parallel Multiscale PixelCNN (Reed et al., 2017) is the only model in the literature which reports log-likelihood on $1 2 8 \\times 1 2 8$ ImageNet. SPN improves the log-likelihood over this model from 3.55 bits/dim to 3.08 bits/dim (see Table 2). Figure 5 gives $1 2 8 \\times 1 2 8$ 8-bit ImageNet samples for both the setting of depth upscaling only and of complete multidimensional upscaling. These settings do not affect the NLL, but the samples with depth upscaling show significant semantic coherence that is usually lacking in samples without upscaling. In addition, multidimensional upscaling seems to increase the overall rate of success of the samples. Additional intermediate ImageNet samples can be seen in Figures 10, 11 and 12 in the Appendix. ", + "bbox": [ + 173, + 220, + 826, + 347 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 CELEBAHQ ", + "text_level": 1, + "bbox": [ + 176, + 363, + 299, + 377 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "At $2 5 6 \\times 2 5 6$ we can produce high-fidelity samples of celebrity faces from the CelebAHQ dataset. The quality compares favorably to the samples of other models such as Glow and GANs (Karras et al., 2017). We show in Table 3 that the achieved MLE scores are a significant improvement over previously reported scores. Figure 6 showcases some samples for 8-bit CelebAHQ-256. Figure 7 in the Appendix includes 5-bit samples, Figure 8 includes 3-bit samples while Figure 9 includes 3-bit samples with the temperature of the output distribution set to 0.95. ", + "bbox": [ + 174, + 390, + 825, + 473 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 493, + 318, + 510 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The problem of whether it is possible to learn the distribution of complex natural images and attain high sample fidelity has been a long-standing one in the tradition of generative models. The SPN and Multidimensional Upscaling model that we introduce accomplishes a large step towards solving this problem, by attaining both state-of-the-art MLE scores on large-scale images from complex domains such as CelebAHQ-256 and ImageNet-128 and by being able to generate high fidelity full 8-bit samples from the resulting learnt distributions without alterations to the sampling process (via e.g. heavy modifications of the temperature of the output distribution). The generated samples show an unprecedented amount of semantic coherence and exactness of details even at the large scale size of full 8-bit $1 2 8 \\times 1 2 8$ and $2 5 6 \\times 2 5 6$ images. ", + "bbox": [ + 173, + 525, + 825, + 651 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 672, + 398, + 688 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We would like to thank Alex Graves, Karen Simonyan, Aaron van den Oord, Tim Harley, Sander Dieleman, Tim Salimans, Lasse Espeholt, Ali Razavi, Jeffrey De Fauw and Andy Brock for insightful discussions. In particular we wish to thank Andriy Mnih for formative ideas about autoregressivity in the bit-depth of an image. 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Google’s neural machine translation system: Bridging the gap between human and machine translation. CoRR, abs/1609.08144, 2016. URL http://arxiv.org/abs/1609.08144. ", + "bbox": [ + 176, + 181, + 826, + 280 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "APPENDIX A ", + "text_level": 1, + "bbox": [ + 176, + 319, + 282, + 335 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In some cases for purposes of analysis the entropy of the softmax output distributions has been artificially reduced via a “temperature” divisor on the predicted logits. When we say the temperature is 0.95, we mean that the logits of a trained model have been divided by this constant at sampling time. ", + "bbox": [ + 174, + 354, + 825, + 410 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/d9e179ba71ce959d7d1f3a8cee2639304460ecce18c6d99ecacc0bb2dd11b905.jpg", + "image_caption": [ + "Figure 6: 8-bit 256x256 RGB CelebA-HQ samples from SPN with Depth-Upscaling. Temperature is 0.99 for the low-bit-depth image and otherwise 1.0 " + ], + "image_footnote": [], + "bbox": [ + 179, + 428, + 818, + 859 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/504d0665a5b40d06191c7bd8627c201458e6521b40015b4d0b70db79bfe3b9b9.jpg", + "image_caption": [ + "Figure 7: 256x256 CelebA-HQ 5bit samples from SPN " + ], + "image_footnote": [], + "bbox": [ + 174, + 102, + 846, + 623 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "APPENDIX B ", + "text_level": 1, + "bbox": [ + 176, + 679, + 282, + 695 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Please see Table 4 for all the detailed hyperparameters. ", + "bbox": [ + 174, + 710, + 534, + 724 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "APPENDIX C ", + "text_level": 1, + "bbox": [ + 176, + 746, + 282, + 761 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Our experiments operate at a fairly large scale in terms of both the amount of compute used and the size of the networks. We proportionately increase the batch size so that the number of pixels in a batch is not affected by the subscaling. These large batch sizes (a maximum of 2048) are achieved by increasing the degree of data parallelism by running on Google Cloud TPU pods (Jouppi et al., 2017). For Imagenet 32 we used 64 tensorcores. For ImageNet 64, 128 and 256, we use 128 tensorcores. The fast interconnect between these devices affords much faster synchronous gradient computation than would be possible using the same number of GPUs. When overfitting is a problem, as in small datasets like CelebA-HQ, we rather decrease the batch size and use a lower number of 32 tensorcores. ", + "bbox": [ + 174, + 776, + 826, + 888 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Our SPN architectures have between ${ \\sim } 5 0 \\mathbf { M }$ and $\\sim 2 5 0 \\mathrm { M }$ parameters depending on the dataset. See Table 4 for the number of parameters in the SPN architecture for each dataset. Depth-upscaling doubles the number of parameters due to using two separate networks with untied weights. Sizeupscaling adds more parameters still for the separate decoder-only network which models the first slice as seen in Figure 3 (g). Thus the maximal number of parameters used to generate a sample in the paper occurs in the multidimensional upscaling setting for ImageNet 128, where the total parameter count reaches ${ \\sim } 6 5 0 \\mathrm { M }$ (the decoder-only network used to model the first slice has $\\mathord { \\sim } 1 5 0 \\mathrm { M }$ parameters). ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/d848a209d6b7533bcfb3b34e9070b3c0d63348e6b8f6869d3f908d184dba7b69.jpg", + "image_caption": [ + "Figure 8: 256x256 CelebA-HQ 3bit samples from SPN " + ], + "image_footnote": [], + "bbox": [ + 174, + 101, + 844, + 623 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 680, + 826, + 763 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/697e394f5b5990e1e08a4707158de53d0bcaed683c4c1de9516612c7dd56baad.jpg", + "image_caption": [ + "Figure 9: 256x256 CelebA-HQ 3bit samples from SPN with temperature 0.95 " + ], + "image_footnote": [], + "bbox": [ + 174, + 231, + 846, + 760 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/112ca671283b8d1c9be472807544de4f71075d12f5a63d4abf83c2837648da53.jpg", + "image_caption": [ + "Figure 10: 128x128 ImageNet 3bit; upscaled $3 2 \\mathrm { x } 3 2 $ slices " + ], + "image_footnote": [], + "bbox": [ + 330, + 121, + 666, + 381 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/f059f8be6aedb09ab4801a67ed6fa109022eb0a363f720ce1ea1d8f95bb67805.jpg", + "image_caption": [ + "Figure 11: 128x128 ImageNet 3bit samples from model trained on $3 2 \\mathrm { x } 3 2 $ slices " + ], + "image_footnote": [], + "bbox": [ + 454, + 460, + 544, + 529 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/1ad9d7898f245e7d4d3450a51ccd621e11cd56b7342e3076a4759202e18bc3e1.jpg", + "image_caption": [ + "Figure 12: 128x128 ImageNet 3bit samples from SPN " + ], + "image_footnote": [], + "bbox": [ + 330, + 608, + 666, + 868 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/9501916cfd87c9ce505d8f73c868ceec25be43a82c6ccc08a526c51471d6f698.jpg", + "table_caption": [ + "Table 4: SPN Hyperparameters. A learning rate marked “sched” utilizes a piecewise-constant schedule starting at 1e-4, and decreasing to 3e-5 and finally 1e-5 at training steps $5 0 \\mathrm { k }$ and $1 0 0 \\mathrm { k }$ respectively. The “attention” parameters listed are configurable hyperparameters of the open source Transformer implementation in Vaswani et al. (2018). The parameter “residual channels“ refers to the number of hidden units in residual convolution layers within the Slice Embedder or PixelCNN networks. " + ], + "table_footnote": [ + "" + ], + "table_body": "
Optimization
batch size(1024,2048,2048,2048)256
learning rate(sched,sched, le-5,1e-5)5e-5
rmsprop momentum0.90.9
rmsprop decay0.950.95
rmsprop epsilon1e-81e-8
polyak decay0.99990.9999
Decoder
PixelCNN layers1515
PixelCNN conv channels(256,256, 384, 384)128
PixelCNN residual channels12801280
PixelCNN nonlinearitygatedgated
PixelCNN filter size33
masked self-attention layers85
attention heads105
attention channels128128
attention ffn layer"parameter_attention""parameter_attention"
SliceEmbedder
conv layers55
conv filter size33
conv channels256256
residual channels10241024
nonlinearityrelurelu
self-attention layers(6,6, 8, 8)6
attention heads(4,4,8,8)4
attention channels(64,64, 128, 128)64
attention ffn layer"parameter_attention""parameter_attention"
Number of parameters ~150M ~150M ~ 250M ~ 250M) ~ 50M
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Autoregressive image models have", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 265, + 470, + 278 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 470, + 278 + ], + "score": 1.0, + "content": "been able to generate small images unconditionally, but the extension of these", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "score": 1.0, + "content": "methods to large images where fidelity can be more readily assessed has remained", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 287, + 470, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 299 + ], + "score": 1.0, + "content": "an open problem. 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We evaluate SPNs on the unconditional generation of CelebAHQ of size", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 397, + 470, + 408 + ], + "spans": [ + { + "bbox": [ + 142, + 397, + 470, + 408 + ], + "score": 1.0, + "content": "256 and of ImageNet from size 32 to 256. We achieve state-of-the-art likelihood", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 408, + 469, + 420 + ], + "spans": [ + { + "bbox": [ + 141, + 408, + 469, + 420 + ], + "score": 1.0, + "content": "results in multiple settings, set up new benchmark results in previously unexplored", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 419, + 469, + 430 + ], + "spans": [ + { + "bbox": [ + 141, + 419, + 469, + 430 + ], + "score": 1.0, + "content": "settings and are able to generate very high fidelity large scale samples on the basis", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 429, + 210, + 441 + ], + "spans": [ + { + "bbox": [ + 141, + 429, + 210, + 441 + ], + "score": 1.0, + "content": "of both datasets.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 462, + 206, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 208, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 208, + 478 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "A successful generative model has two core aspects: it produces targets that have high fidelity and it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "generalizes well on held-out data. Autoregressive (AR) models trained by conventional maximum", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "likelihood estimation (MLE) have produced superior scores on held-out data across a wide range", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "of domains such as text (Vaswani et al., 2017; Wu et al., 2016), audio (van den Oord et al., 2016a),", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 532, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 506, + 544 + ], + "score": 1.0, + "content": "images (Parmar et al., 2018) and videos (Kalchbrenner et al., 2016). These scores are a measure of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "the models’ ability to generalize in that setting. From the perspective of sample fidelity, the outputs", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "score": 1.0, + "content": "generated by AR models have also achieved state-of-the-art fidelity in many of the aforementioned", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 563, + 507, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 507, + 578 + ], + "score": 1.0, + "content": "domains with one notable exception. 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Autoregressive image models have", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 265, + 470, + 278 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 470, + 278 + ], + "score": 1.0, + "content": "been able to generate small images unconditionally, but the extension of these", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "score": 1.0, + "content": "methods to large images where fidelity can be more readily assessed has remained", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 287, + 470, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 299 + ], + "score": 1.0, + "content": "an open problem. 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We aim to guide the model to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "focus first on visually more salient bits of the distribution and later on the visually less salient bits.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 400, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 416 + ], + "score": 1.0, + "content": "We identify two visually salient subsets of the distribution: first, the subset determined by sub-images", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 231, + 425 + ], + "score": 1.0, + "content": "(“slices”) of smaller size (e.g.", + "type": "text" + }, + { + "bbox": [ + 231, + 413, + 266, + 424 + ], + "score": 0.88, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 412, + 506, + 425 + ], + "score": 1.0, + "content": ") sub-sampled at all positions from the original image; and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "secondly, the subset determined by the few (e.g. 3) most significant bits of each RGB channel in the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "image. We use Multidimensional Upscaling to map from one subset of the distribution to the other", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 416, + 458 + ], + "score": 1.0, + "content": "one by upscaling images in size or in depth. For example, the generation of a", + "type": "text" + }, + { + "bbox": [ + 416, + 446, + 463, + 456 + ], + "score": 0.78, + "content": "1 2 8 \\times 1 2 8 :", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "8-bit RGB", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 457, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 312, + 468 + ], + "score": 1.0, + "content": "image proceeds by first upscaling it in size from a", + "type": "text" + }, + { + "bbox": [ + 312, + 457, + 347, + 467 + ], + "score": 0.82, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 457, + 438, + 468 + ], + "score": 1.0, + "content": "3-bit RGB image to a", + "type": "text" + }, + { + "bbox": [ + 439, + 457, + 483, + 467 + ], + "score": 0.84, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 457, + 506, + 468 + ], + "score": 1.0, + "content": "3-bit", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 467, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 461, + 480 + ], + "score": 1.0, + "content": "RGB image; we then upscale the resulting image in depth to the original resolution of the", + "type": "text" + }, + { + "bbox": [ + 461, + 468, + 504, + 478 + ], + "score": 0.89, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "score": 1.0, + "content": "8-bit RGB image. We thus train three networks: (a) a decoder on the small size, low depth image", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 215, + 502 + ], + "score": 1.0, + "content": "slices subsampled at every", + "type": "text" + }, + { + "bbox": [ + 215, + 491, + 222, + 500 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "pixels from the original image with the desired target resolution; (b) a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 501, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 506, + 513 + ], + "score": 1.0, + "content": "size-upscaling decoder that generates the large size, low depth image conditioned on the small size,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "low depth image; and (c) a depth-upscaling decoder that generates the large size, high depth image", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 523, + 427, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 427, + 535 + ], + "score": 1.0, + "content": "conditioned on the large size, low depth image. Figure 1 illustrates this process.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "To address the latter difficulties that ensue in the training of decoders (b) and (c), we develop the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 549, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 411, + 564 + ], + "score": 1.0, + "content": "Subscale Pixel Network (SPN) architecture. The SPN divides an image of size", + "type": "text" + }, + { + "bbox": [ + 412, + 551, + 440, + 560 + ], + "score": 0.9, + "content": "N \\times N", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 549, + 505, + 564 + ], + "score": 1.0, + "content": "into sub-images", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 136, + 573 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 137, + 560, + 168, + 574 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { N } { S _ { . } } \\times \\frac { N } { S } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "sliced out at interleaving positions (see Figure 2), which implicitly also captures a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 217, + 585 + ], + "score": 1.0, + "content": "form of size upscaling. The", + "type": "text" + }, + { + "bbox": [ + 218, + 573, + 249, + 582 + ], + "score": 0.89, + "content": "N \\times N", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "image is generated one slice at a time conditioned on previously", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "generated slices in a way that encodes a rich spatial structure. SPN consists of two networks, a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "conditioning network that embeds previous slices and a decoder proper that predicts a single target", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "slice given the context embedding. The decoding part of the SPN acts over image slices with the same", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "score": 1.0, + "content": "spatial structure and it can share weights for all of them. The SPN is an independent image decoder", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "with an implicit size upscaling mechanism, but it can also be used as an explicit size upscaling", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "network by initializing the first slice of the SPN input at sampling time with one generated separately", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 649, + 170, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 170, + 662 + ], + "score": 1.0, + "content": "during step (a).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We extensively evaluate the performance of SPN and the size and depth upscaling methods both", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "score": 1.0, + "content": "quantitatively and from a fidelity perspective on two unconditional image generation benchmarks,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "score": 1.0, + "content": "CelebAHQ-256 and ImageNet of various sizes up to 256. From a MLE scores perspective, we", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "compare with previous work to obtain state-of-the-art results on CelebAHQ-256, both at full 8-bit", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "resolution and at the reduced 5-bit resolution (Kingma & Dhariwal, 2018), and on ImageNet-64.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "We also establish MLE baselines for ImageNet-128 and ImageNet-256. 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Left: depth upscaling is applied to a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 169, + 288 + ], + "score": 1.0, + "content": "generated 3-bit", + "type": "text" + }, + { + "bbox": [ + 170, + 276, + 213, + 286 + ], + "score": 0.88, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 276, + 438, + 288 + ], + "score": 1.0, + "content": "RGB subimage from CelebAHQ to map it to a full 8-bit", + "type": "text" + }, + { + "bbox": [ + 438, + 276, + 481, + 286 + ], + "score": 0.87, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 276, + 506, + 288 + ], + "score": 1.0, + "content": "RGB", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 470, + 300 + ], + "score": 1.0, + "content": "image. Right: size upscaling followed by depth upscaling are applied to a generated 3-bit", + "type": "text" + }, + { + "bbox": [ + 471, + 287, + 505, + 298 + ], + "score": 0.89, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 296, + 507, + 312 + ], + "spans": [ + { + "bbox": [ + 104, + 296, + 410, + 312 + ], + "score": 1.0, + "content": "RGB subimage from ImageNet to map it to the target resolution of the 8-bit", + "type": "text" + }, + { + "bbox": [ + 410, + 298, + 454, + 308 + ], + "score": 0.89, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 296, + 507, + 312 + ], + "score": 1.0, + "content": "RGB image.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "We stress that the rightmost column of both figures are true unconditional samples from our model at", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 319, + 168, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 168, + 333 + ], + "score": 1.0, + "content": "full 8bit depth.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "These difficulties notwithstanding, we aim to learn the full distribution over 8-bit RGB images of size", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 129, + 393 + ], + "score": 1.0, + "content": "up to", + "type": "text" + }, + { + "bbox": [ + 130, + 380, + 173, + 390 + ], + "score": 0.89, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "well enough so that the samples have high fidelity. We aim to guide the model to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "focus first on visually more salient bits of the distribution and later on the visually less salient bits.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 400, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 416 + ], + "score": 1.0, + "content": "We identify two visually salient subsets of the distribution: first, the subset determined by sub-images", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 231, + 425 + ], + "score": 1.0, + "content": "(“slices”) of smaller size (e.g.", + "type": "text" + }, + { + "bbox": [ + 231, + 413, + 266, + 424 + ], + "score": 0.88, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 412, + 506, + 425 + ], + "score": 1.0, + "content": ") sub-sampled at all positions from the original image; and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "secondly, the subset determined by the few (e.g. 3) most significant bits of each RGB channel in the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "image. We use Multidimensional Upscaling to map from one subset of the distribution to the other", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 416, + 458 + ], + "score": 1.0, + "content": "one by upscaling images in size or in depth. For example, the generation of a", + "type": "text" + }, + { + "bbox": [ + 416, + 446, + 463, + 456 + ], + "score": 0.78, + "content": "1 2 8 \\times 1 2 8 :", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "8-bit RGB", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 457, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 312, + 468 + ], + "score": 1.0, + "content": "image proceeds by first upscaling it in size from a", + "type": "text" + }, + { + "bbox": [ + 312, + 457, + 347, + 467 + ], + "score": 0.82, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 457, + 438, + 468 + ], + "score": 1.0, + "content": "3-bit RGB image to a", + "type": "text" + }, + { + "bbox": [ + 439, + 457, + 483, + 467 + ], + "score": 0.84, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 457, + 506, + 468 + ], + "score": 1.0, + "content": "3-bit", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 467, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 461, + 480 + ], + "score": 1.0, + "content": "RGB image; we then upscale the resulting image in depth to the original resolution of the", + "type": "text" + }, + { + "bbox": [ + 461, + 468, + 504, + 478 + ], + "score": 0.89, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "score": 1.0, + "content": "8-bit RGB image. We thus train three networks: (a) a decoder on the small size, low depth image", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 215, + 502 + ], + "score": 1.0, + "content": "slices subsampled at every", + "type": "text" + }, + { + "bbox": [ + 215, + 491, + 222, + 500 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "pixels from the original image with the desired target resolution; (b) a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 501, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 506, + 513 + ], + "score": 1.0, + "content": "size-upscaling decoder that generates the large size, low depth image conditioned on the small size,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "low depth image; and (c) a depth-upscaling decoder that generates the large size, high depth image", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 523, + 427, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 427, + 535 + ], + "score": 1.0, + "content": "conditioned on the large size, low depth image. Figure 1 illustrates this process.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 369, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "To address the latter difficulties that ensue in the training of decoders (b) and (c), we develop the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 549, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 411, + 564 + ], + "score": 1.0, + "content": "Subscale Pixel Network (SPN) architecture. The SPN divides an image of size", + "type": "text" + }, + { + "bbox": [ + 412, + 551, + 440, + 560 + ], + "score": 0.9, + "content": "N \\times N", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 549, + 505, + 564 + ], + "score": 1.0, + "content": "into sub-images", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 136, + 573 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 137, + 560, + 168, + 574 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { N } { S _ { . } } \\times \\frac { N } { S } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "sliced out at interleaving positions (see Figure 2), which implicitly also captures a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 217, + 585 + ], + "score": 1.0, + "content": "form of size upscaling. The", + "type": "text" + }, + { + "bbox": [ + 218, + 573, + 249, + 582 + ], + "score": 0.89, + "content": "N \\times N", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "image is generated one slice at a time conditioned on previously", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "generated slices in a way that encodes a rich spatial structure. SPN consists of two networks, a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "conditioning network that embeds previous slices and a decoder proper that predicts a single target", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "slice given the context embedding. The decoding part of the SPN acts over image slices with the same", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "score": 1.0, + "content": "spatial structure and it can share weights for all of them. The SPN is an independent image decoder", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "with an implicit size upscaling mechanism, but it can also be used as an explicit size upscaling", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "network by initializing the first slice of the SPN input at sampling time with one generated separately", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 649, + 170, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 170, + 662 + ], + "score": 1.0, + "content": "during step (a).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 538, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We extensively evaluate the performance of SPN and the size and depth upscaling methods both", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "score": 1.0, + "content": "quantitatively and from a fidelity perspective on two unconditional image generation benchmarks,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "score": 1.0, + "content": "CelebAHQ-256 and ImageNet of various sizes up to 256. From a MLE scores perspective, we", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "compare with previous work to obtain state-of-the-art results on CelebAHQ-256, both at full 8-bit", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "resolution and at the reduced 5-bit resolution (Kingma & Dhariwal, 2018), and on ImageNet-64.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "We also establish MLE baselines for ImageNet-128 and ImageNet-256. 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We produce CelebAHQ-256 samples (at full 8-bit resolution) that are of similar visual fidelity to", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "those produced with methods such as GANs that lack however an intrinsic measure of generalization", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "(Mescheder, 2018; Karras et al., 2017). We also produce some of the first successful samples on", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "unconditional ImageNet-128 (also at 8-bit) showing again the striking impact of the SPN and of", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 291, + 491, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 491, + 304 + ], + "score": 1.0, + "content": "multidimensional upscaling on sample quality and setting a fidelity baseline for future methods.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 665, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 117, + 79, + 497, + 188 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 79, + 497, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 79, + 497, + 188 + ], + "spans": [ + { + "bbox": [ + 117, + 79, + 497, + 188 + ], + "score": 0.972, + "type": "image", + "image_path": "7b2a844dbd9243a20e217da5e13212d83a158d5360aa6104f93056ee1c073ba0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 79, + 497, + 115.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 115.33333333333334, + 497, + 151.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 151.66666666666669, + 497, + 188.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 194, + 505, + 218 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "score": 1.0, + "content": ".Figure 2: The receptive field in a Subscale Pixel Networks (a) and the four image slices subsampled", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 205, + 218, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 218, + 219 + ], + "score": 1.0, + "content": "from the original image (b)", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 235, + 506, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "perspective, we show the strong benefits of multidimensional upscaling as well as the benefits of the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "score": 1.0, + "content": "SPN. We produce CelebAHQ-256 samples (at full 8-bit resolution) that are of similar visual fidelity to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "those produced with methods such as GANs that lack however an intrinsic measure of generalization", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "(Mescheder, 2018; Karras et al., 2017). We also produce some of the first successful samples on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "unconditional ImageNet-128 (also at 8-bit) showing again the striking impact of the SPN and of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 291, + 491, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 491, + 304 + ], + "score": 1.0, + "content": "multidimensional upscaling on sample quality and setting a fidelity baseline for future methods.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 107, + 317, + 165, + 330 + ], + "lines": [ + { + "bbox": [ + 104, + 316, + 168, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 316, + 168, + 333 + ], + "score": 1.0, + "content": "2 MODEL", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 107, + 342, + 311, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 311, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 311, + 355 + ], + "score": 1.0, + "content": "2.1 CONVENTIONAL GENERATION ORDERING", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 477, + 376 + ], + "score": 1.0, + "content": "A standard AR image model such as the PixelCNN (van den Oord et al., 2016b) generates an", + "type": "text" + }, + { + "bbox": [ + 477, + 363, + 504, + 373 + ], + "score": 0.76, + "content": "H x W", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 373, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 388 + ], + "score": 1.0, + "content": "colour image starting at the top-left position and ending at the bottom-right position, fully generating", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 384, + 335, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 335, + 397 + ], + "score": 1.0, + "content": "the three 8-bit channels of each pixel in a given position:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 399, + 386, + 435 + ], + "lines": [ + { + "bbox": [ + 225, + 399, + 386, + 435 + ], + "spans": [ + { + "bbox": [ + 225, + 399, + 386, + 435 + ], + "score": 0.94, + "content": "P ( \\mathbf { x } ) = \\prod _ { h = 1 } ^ { H } \\prod _ { w = 1 } ^ { W } \\prod _ { c } ^ { \\{ R , G , B \\} } P ( x _ { h , w , c } | \\mathbf { x } _ { < } )", + "type": "interline_equation", + "image_path": "aed0d7c54a9e53f029e8bf827d7dd5e2df0d49ed2d355984974af3222f0be857.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 225, + 399, + 386, + 417.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 225, + 417.0, + 386, + 435.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 132, + 455 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 444, + 147, + 453 + ], + "score": 0.82, + "content": "\\mathbf { x } _ { < }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 442, + 444, + 455 + ], + "score": 1.0, + "content": "corresponds to all previously generated intensity values in the ordering and", + "type": "text" + }, + { + "bbox": [ + 444, + 443, + 464, + 452 + ], + "score": 0.54, + "content": "h , w .", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 442, + 483, + 455 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 484, + 445, + 489, + 452 + ], + "score": 0.73, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "row, column, and colour channel indices. 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A scaling factor", + "type": "text" + }, + { + "bbox": [ + 317, + 690, + 324, + 698 + ], + "score": 0.86, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 687, + 452, + 699 + ], + "score": 1.0, + "content": "is selected and each slice of size", + "type": "text" + }, + { + "bbox": [ + 452, + 688, + 504, + 700 + ], + "score": 0.9, + "content": "H / S \\times W / S", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 254, + 711 + ], + "score": 1.0, + "content": "is obtained by selecting a pixel every", + "type": "text" + }, + { + "bbox": [ + 254, + 701, + 262, + 709 + ], + "score": 0.86, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 698, + 446, + 711 + ], + "score": 1.0, + "content": "pixels in both height and width; there are thus", + "type": "text" + }, + { + "bbox": [ + 446, + 699, + 457, + 709 + ], + "score": 0.88, + "content": "S ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "interleaved", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 418, + 722 + ], + "score": 1.0, + "content": "slices in the original image, with each specified by its row and column offset", + "type": "text" + }, + { + "bbox": [ + 419, + 710, + 439, + 722 + ], + "score": 0.91, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 710, + 505, + 722 + ], + "score": 1.0, + "content": ". We sometimes", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 720, + 312, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 312, + 734 + ], + "score": 1.0, + "content": "refer to this offset as the “meta-position” of a slice.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 677, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 157, + 79, + 447, + 275 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 157, + 79, + 447, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 157, + 79, + 447, + 275 + ], + "spans": [ + { + "bbox": [ + 157, + 79, + 447, + 275 + ], + "score": 0.967, + "type": "image", + "image_path": "98559d46079f8d28599a7a5163c09abb9d3f0b63ff32e33727c28ac6d3f9a3f2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 157, + 79, + 447, + 144.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 157, + 144.33333333333331, + 447, + 209.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 157, + 209.66666666666663, + 447, + 274.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 284, + 506, + 328 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 284, + 507, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 507, + 297 + ], + "score": 1.0, + "content": "Figure 3: Different generation ordering schemes, where the numbers indicate the step-by-step order.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "Distinct colors correspond to distinct neural networks. (a) and (b) are from (van den Oord et al.,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "2016b). (c) is from (Reed et al., 2017). The Subscale ordering alone, with size-only, depth-only and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 316, + 426, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 316, + 426, + 330 + ], + "score": 1.0, + "content": "with multidimensional upscaling are, respectively, in blocks (d), (e), (f) and (g).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 109, + 350, + 313, + 361 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 315, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 315, + 363 + ], + "score": 1.0, + "content": "2.3 SIZE UPSCALING IN SUBSCALE ORDERING", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 506, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "The subscale ordering itself already captures size upscaling implicitly. Analogous to the multi-scale", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "ordering (van den Oord et al., 2016b), and depicted in 3(b), we can perform size upscaling explicitly,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 392, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 407 + ], + "score": 1.0, + "content": "by training a single slice decoder on subimages and generate the first slice of a subscale ordering", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "from the single slice decoder itself. 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We note that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "in depth upscaling bits of lower significance are only generated when the more significants bits at", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "all positions have been generated in a previous stage. Just like for size upscaling from the previous", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 157, + 79, + 447, + 275 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 157, + 79, + 447, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 157, + 79, + 447, + 275 + ], + "spans": [ + { + "bbox": [ + 157, + 79, + 447, + 275 + ], + "score": 0.967, + "type": "image", + "image_path": "98559d46079f8d28599a7a5163c09abb9d3f0b63ff32e33727c28ac6d3f9a3f2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 157, + 79, + 447, + 144.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 157, + 144.33333333333331, + 447, + 209.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 157, + 209.66666666666663, + 447, + 274.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 284, + 506, + 328 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 284, + 507, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 507, + 297 + ], + "score": 1.0, + "content": "Figure 3: Different generation ordering schemes, where the numbers indicate the step-by-step order.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "Distinct colors correspond to distinct neural networks. (a) and (b) are from (van den Oord et al.,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "2016b). (c) is from (Reed et al., 2017). 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For this reason, the same SPN that captures the subscale ordering can act simultaneously", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "as a full-blown image model as well as a size upscaling model if initialized with the outputs of a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "single-slice decoder. 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Existing AR approaches inherently require an amount of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "score": 1.0, + "content": "computation and memory that is superlinear in the number of pixels. 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Modeling choices such as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "cropping images within the decoders (Kalchbrenner et al., 2016) or performing self-attention over", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "local neighborhoods (Parmar et al., 2018) neglect global dependencies, while model parallelism,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "though technically feasible with the joint use of a very large number of accelerators, does not", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 458, + 335, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 335, + 471 + ], + "score": 1.0, + "content": "overcome the challenges in learning the global structure.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 403, + 506, + 471 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 485, + 253, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 255, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 255, + 498 + ], + "score": 1.0, + "content": "3.2 SUBSCALE PIXEL NETWORK", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "To address these challenges, we devise the Subscale Pixel Network (SPN), an architecture that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 517, + 504, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 392, + 529 + ], + "score": 1.0, + "content": "embodies the subscale ordering from Section 2.2. For an image of size", + "type": "text" + }, + { + "bbox": [ + 392, + 518, + 463, + 528 + ], + "score": 0.92, + "content": "H \\times W \\times 3 \\times D", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 517, + 494, + 529 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 494, + 518, + 504, + 527 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 479, + 541 + ], + "score": 1.0, + "content": "is the number of bits used for the current generation stage, one first chooses a scaling factor", + "type": "text" + }, + { + "bbox": [ + 479, + 529, + 487, + 538 + ], + "score": 0.84, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 151, + 552 + ], + "score": 1.0, + "content": "obtains the", + "type": "text" + }, + { + "bbox": [ + 151, + 539, + 164, + 549 + ], + "score": 0.89, + "content": "S ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 538, + 300, + 552 + ], + "score": 1.0, + "content": "slices of the original image of size", + "type": "text" + }, + { + "bbox": [ + 301, + 539, + 390, + 551 + ], + "score": 0.93, + "content": "H / S \\times W / S \\times 3 \\times D", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 538, + 424, + 552 + ], + "score": 1.0, + "content": ". We use", + "type": "text" + }, + { + "bbox": [ + 425, + 539, + 487, + 550 + ], + "score": 0.9, + "content": "H = W = 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 107, + 550, + 133, + 560 + ], + "score": 0.9, + "content": "S = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 549, + 174, + 563 + ], + "score": 1.0, + "content": "as well as", + "type": "text" + }, + { + "bbox": [ + 175, + 550, + 237, + 560 + ], + "score": 0.9, + "content": "H = W = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 549, + 255, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 255, + 550, + 281, + 560 + ], + "score": 0.9, + "content": "S = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 549, + 506, + 563 + ], + "score": 1.0, + "content": ", for the larger images we process, so that the slices have", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 125, + 574 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 125, + 561, + 196, + 572 + ], + "score": 0.92, + "content": "3 2 \\times 3 2 \\times 3 \\times D", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 560, + 304, + 574 + ], + "score": 1.0, + "content": ". This scheme of choosing", + "type": "text" + }, + { + "bbox": [ + 304, + 562, + 312, + 572 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 560, + 423, + 574 + ], + "score": 1.0, + "content": "such that slices are always", + "type": "text" + }, + { + "bbox": [ + 423, + 561, + 456, + 572 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "renders the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 434, + 585 + ], + "score": 1.0, + "content": "memory and computation requirements effectively constant as the true image size", + "type": "text" + }, + { + "bbox": [ + 434, + 572, + 468, + 583 + ], + "score": 0.91, + "content": "H \\times W", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "changes.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 506, + 506, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 507, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 507, + 602 + ], + "score": 1.0, + "content": "The SPN architecture is composed of two parts: an embedding part for slices at preceding meta-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 599, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 614 + ], + "score": 1.0, + "content": "positions that conditions the decoder for the current slice that is being generated (Figure 4 (a)). The", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "embedding part is a convolutional neural network with residual blocks that takes as input preceding", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "slices that are concatenated along the depth dimension. One detail is the way the slices are ordered", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "along the channel dimension when concatenated. As illustrated in Figure 4 (a), empty padding slices", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "are used to preserve relative meta-positions of each preceding slice with respect to the current target", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "slice. For example, the slice above any target slice in the two-dimensional meta-grid is always aligned", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 664, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 680 + ], + "score": 1.0, + "content": "in the same position along the depth axis in the input. This achieves equivariance in the embedding", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 233, + 689 + ], + "score": 1.0, + "content": "architecture with respect to the", + "type": "text" + }, + { + "bbox": [ + 234, + 677, + 254, + 689 + ], + "score": 0.94, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "offset of a slice. The padding slices also ensure that the depth", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "of the input slice tensor remains the same for all target slices. In addition to the slice tensor, the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "embedding part also receives as input the meta-position of the target slice as an embedding of 8 units", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "tiled spatially across the slice tensor. The pixel intensity values are also embedded as one-hot indices", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 720, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 504, + 733 + ], + "score": 1.0, + "content": "of size 8. The context embedding network passes its input through a series of self-attention layers", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 346, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 359 + ], + "score": 1.0, + "content": "and then a series of residual blocks, finally emitting a slice-sized feature map s that summarizes the", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 358, + 202, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 202, + 369 + ], + "score": 1.0, + "content": "context for the decoder.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 588, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 82, + 513, + 277 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 82, + 513, + 277 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 82, + 513, + 277 + ], + "spans": [ + { + "bbox": [ + 107, + 82, + 513, + 277 + ], + "score": 0.976, + "type": "image", + "image_path": "e6f31c8d7a0d48ad915bfecc5d546389faa9b99b3c07f683678a398e09811c80.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 82, + 513, + 147.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 147.0, + 513, + 212.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 212.0, + 513, + 277.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 286, + 506, + 319 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "Figure 5: Left: 8-bit 128x128 RGB ImageNet samples from SPN with depth upscaling only. Right:", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 295, + 507, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 507, + 311 + ], + "score": 1.0, + "content": "8-bit 128x128 RGB ImageNet samples from SPN with full-blown Multidimensional Upscaling.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 307, + 388, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 242, + 320 + ], + "score": 1.0, + "content": "Temperature is 0.99 for the initial", + "type": "text" + }, + { + "bbox": [ + 243, + 308, + 269, + 318 + ], + "score": 0.4, + "content": "3 2 \\mathbf { x } 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 307, + 388, + 320 + ], + "score": 1.0, + "content": "sub-image and otherwise 1.0", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 108, + 347, + 504, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 359 + ], + "score": 1.0, + "content": "and then a series of residual blocks, finally emitting a slice-sized feature map s that summarizes the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 358, + 202, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 202, + 369 + ], + "score": 1.0, + "content": "context for the decoder.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "The decoder takes as input the encoded slice tensor s in a position-preserving manner: each position", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "in the target slice is given as input the encoded representations of pixels at that same position in the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "score": 1.0, + "content": "preceding slices. In addition it processes the target slice in the raster-scan order. The decoder that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 408, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 420 + ], + "score": 1.0, + "content": "we use is a hybrid architecture combining masked convolution and self-attention (Chen et al., 2017).", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "We employ an initial 1D self-attention network Vaswani et al. (2017) that is used to gather the entire", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "available context in the slice (see Figure 4(b)). The slice is reshaped into a 1D tensor before it is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "given as input to masked 1D self-attention layers; the masking is performed over the previous pixels", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "only (as opposed to over the current RGB channels) in the self-attention layers. Then the output of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "the layers is reshaped back into a 2D tensor, concatenated depth-wise with the output of the slice", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "embedding network, and given as conditioning input to a Gated PixelCNN as in Equation 5 in van den", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "Oord et al. (2016c). The PixelCNN network models the target slice with full masking over pixels and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 494, + 504, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 460, + 508 + ], + "score": 1.0, + "content": "channel dimensions. We can see how memory requirements are significantly lower - up to", + "type": "text" + }, + { + "bbox": [ + 460, + 495, + 504, + 506 + ], + "score": 0.91, + "content": "\\bar { S } ^ { 2 } = 6 4 \\times", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 153, + 519 + ], + "score": 1.0, + "content": "lower with", + "type": "text" + }, + { + "bbox": [ + 154, + 506, + 182, + 516 + ], + "score": 0.88, + "content": "S = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "- due to the smaller spatial size of the slices and their compact concatenation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "along the channel dimension of the input tensor. 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ImageNet 32x32ImageNet 64x64
Gated PixelCNN (van den Oord et al.,2016c)Parallel Multiscale (Reed et al.,2017)PixelSNAIL (Chen et al., 2017)Image Transformer (Parmar et al.,2018)Glow (Kingma & Dhariwal,2018)3.833.953.803.774.093.573.70--3.81
Decoder baselineSPNSPN +Depth UpscalingSPN + 16x16 slicesSPN +8x8 slices3.79113.853.913.523.533.53 (0.63, 2.90)=
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ImageNet 128 x 128ImageNet256 x 256
Parallel Multiscale (Reed et al., 2017)3.551
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ImageNet 32x32ImageNet 64x64
Gated PixelCNN (van den Oord et al.,2016c)Parallel Multiscale (Reed et al.,2017)PixelSNAIL (Chen et al., 2017)Image Transformer (Parmar et al.,2018)Glow (Kingma & Dhariwal,2018)3.833.953.803.774.093.573.70--3.81
Decoder baselineSPNSPN +Depth UpscalingSPN + 16x16 slicesSPN +8x8 slices3.79113.853.913.523.533.53 (0.63, 2.90)=
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ImageNet 128 x 128ImageNet256 x 256
Parallel Multiscale (Reed et al., 2017)3.551
SPN SPN + Depth-Upscaling3.08 3.08 (0.46, 2.62)2.97 3.01 (0.40,2.61)
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Furthermore, we show that these results extend", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 486, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 422, + 501 + ], + "score": 1.0, + "content": "to high-resolution ImageNet images, with state-of-the-art log-likelihoods at", + "type": "text" + }, + { + "bbox": [ + 423, + 487, + 459, + 497 + ], + "score": 0.47, + "content": "1 2 8 \\mathrm { x } 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 486, + 505, + 501 + ], + "score": 1.0, + "content": "by a large", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 247, + 510 + ], + "score": 1.0, + "content": "margin and the first benchmark on", + "type": "text" + }, + { + "bbox": [ + 247, + 498, + 284, + 508 + ], + "score": 0.43, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "ImageNet. Unconditional samples at these resolutions", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 509, + 322, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 322, + 522 + ], + "score": 1.0, + "content": "are characterized by unprecedented global coherence.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 454, + 506, + 522 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 526, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 303, + 537 + ], + "score": 1.0, + "content": "Because our networks operate on small images", + "type": "text" + }, + { + "bbox": [ + 304, + 526, + 337, + 536 + ], + "score": 0.87, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 526, + 505, + 537 + ], + "score": 1.0, + "content": "slices), we can train large networks both", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "in terms of the number of hidden units and in terms of network depth (see Appendix C for details", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 548, + 504, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 504, + 559 + ], + "score": 1.0, + "content": "of sizes). The context-embedding network contains 5 convolutional layers and 6-8 self-attention", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "layers depending on the dataset. The masked decoder consists of a PixelCNN with 15 layers in all", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 568, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 505, + 584 + ], + "score": 1.0, + "content": "experiments. The 1D Transformer in the decoder (Figure 4(b)) has between 8 and 10 layers depending", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 579, + 402, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 402, + 593 + ], + "score": 1.0, + "content": "on the dataset. 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Figure 5 gives", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 218, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 152, + 230 + ], + "score": 0.76, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 218, + 506, + 232 + ], + "score": 1.0, + "content": "8-bit ImageNet samples for both the setting of depth upscaling only and of complete", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "multidimensional upscaling. 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We show in Table 3 that the achieved MLE scores are a significant improvement over", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "previously reported scores. Figure 6 showcases some samples for 8-bit CelebAHQ-256. 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In particular we wish to thank Andriy Mnih for formative ideas about autoregressivity in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 212, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 212, + 604 + ], + "score": 1.0, + "content": "the bit-depth of an image.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 619, + 175, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 619, + 176, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 176, + 632 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 637, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 506, + 650 + ], + "score": 1.0, + "content": "Sanjeev Arora and Yi Zhang. Do gans actually learn the distribution? an empirical study. 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URL http://arxiv.org/abs/", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 116, + 690, + 182, + 702 + ], + "spans": [ + { + "bbox": [ + 116, + 690, + 182, + 702 + ], + "score": 1.0, + "content": "1712.09763.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 721 + ], + "score": 1.0, + "content": "Norman P. Jouppi, Cliff Young, Nishant Patil, David A. 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The generated samples show", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "an unprecedented amount of semantic coherence and exactness of details even at the large scale size", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 503, + 294, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 155, + 518 + ], + "score": 1.0, + "content": "of full 8-bit", + "type": "text" + }, + { + "bbox": [ + 155, + 505, + 199, + 515 + ], + "score": 0.9, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 503, + 216, + 518 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 217, + 505, + 260, + 515 + ], + "score": 0.88, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 503, + 294, + 518 + ], + "score": 1.0, + "content": "images.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 416, + 506, + 518 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 533, + 244, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 245, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 245, + 547 + ], + "score": 1.0, + "content": "6 ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "We would like to thank Alex Graves, Karen Simonyan, Aaron van den Oord, Tim Harley, Sander", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "Dieleman, Tim Salimans, Lasse Espeholt, Ali Razavi, Jeffrey De Fauw and Andy Brock for insightful", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "discussions. In particular we wish to thank Andriy Mnih for formative ideas about autoregressivity in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 212, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 212, + 604 + ], + "score": 1.0, + "content": "the bit-depth of an image.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 558, + 505, + 604 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 619, + 175, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 619, + 176, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 176, + 632 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 637, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 506, + 650 + ], + "score": 1.0, + "content": "Sanjeev Arora and Yi Zhang. 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Optimization
batch size(1024,2048,2048,2048)256
learning rate(sched,sched, le-5,1e-5)5e-5
rmsprop momentum0.90.9
rmsprop decay0.950.95
rmsprop epsilon1e-81e-8
polyak decay0.99990.9999
Decoder
PixelCNN layers1515
PixelCNN conv channels(256,256, 384, 384)128
PixelCNN residual channels12801280
PixelCNN nonlinearitygatedgated
PixelCNN filter size33
masked self-attention layers85
attention heads105
attention channels128128
attention ffn layer"parameter_attention""parameter_attention"
SliceEmbedder
conv layers55
conv filter size33
conv channels256256
residual channels10241024
nonlinearityrelurelu
self-attention layers(6,6, 8, 8)6
attention heads(4,4,8,8)4
attention channels(64,64, 128, 128)64
attention ffn layer"parameter_attention""parameter_attention"
Number of parameters ~150M ~150M ~ 250M ~ 250M) ~ 50M
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Optimization
batch size(1024,2048,2048,2048)256
learning rate(sched,sched, le-5,1e-5)5e-5
rmsprop momentum0.90.9
rmsprop decay0.950.95
rmsprop epsilon1e-81e-8
polyak decay0.99990.9999
Decoder
PixelCNN layers1515
PixelCNN conv channels(256,256, 384, 384)128
PixelCNN residual channels12801280
PixelCNN nonlinearitygatedgated
PixelCNN filter size33
masked self-attention layers85
attention heads105
attention channels128128
attention ffn layer"parameter_attention""parameter_attention"
SliceEmbedder
conv layers55
conv filter size33
conv channels256256
residual channels10241024
nonlinearityrelurelu
self-attention layers(6,6, 8, 8)6
attention heads(4,4,8,8)4
attention channels(64,64, 128, 128)64
attention ffn layer"parameter_attention""parameter_attention"
Number of parameters ~150M ~150M ~ 250M ~ 250M) ~ 50M
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A learning rate marked “sched” utilizes a piecewise-constant", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 555, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 446, + 566 + ], + "score": 1.0, + "content": "schedule starting at 1e-4, and decreasing to 3e-5 and finally 1e-5 at training steps", + "type": "text" + }, + { + "bbox": [ + 446, + 555, + 464, + 565 + ], + "score": 0.41, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 555, + 483, + 566 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 483, + 555, + 505, + 565 + ], + "score": 0.54, + "content": "1 0 0 \\mathrm { k }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "respectively. The “attention” parameters listed are configurable hyperparameters of the open source", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 577, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 588 + ], + "score": 1.0, + "content": "Transformer implementation in Vaswani et al. (2018). The parameter “residual channels“ refers to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "the number of hidden units in residual convolution layers within the Slice Embedder or PixelCNN", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 599, + 147, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 147, + 610 + ], + "score": 1.0, + "content": "networks.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + } + ], + "index": 3 + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/HylzTiC5Km/HylzTiC5Km_model.json b/parse/train/HylzTiC5Km/HylzTiC5Km_model.json new file mode 100644 index 0000000000000000000000000000000000000000..b2f1ea59825d42a962db8ba41bc5731cd78ece12 --- /dev/null +++ b/parse/train/HylzTiC5Km/HylzTiC5Km_model.json @@ -0,0 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ImageNet 128 x 128ImageNet256 x 256
Parallel Multiscale (Reed et al., 2017)3.551
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Optimization
batch size(1024,2048,2048,2048)256
learning rate(sched,sched, le-5,1e-5)5e-5
rmsprop momentum0.90.9
rmsprop decay0.950.95
rmsprop epsilon1e-81e-8
polyak decay0.99990.9999
Decoder
PixelCNN layers1515
PixelCNN conv channels(256,256, 384, 384)128
PixelCNN residual channels12801280
PixelCNN nonlinearitygatedgated
PixelCNN filter size33
masked self-attention layers85
attention heads105
attention channels128128
attention ffn layer"parameter_attention""parameter_attention"
SliceEmbedder
conv layers55
conv filter size33
conv channels256256
residual channels10241024
nonlinearityrelurelu
self-attention layers(6,6, 8, 8)6
attention heads(4,4,8,8)4
attention channels(64,64, 128, 128)64
attention ffn layer"parameter_attention""parameter_attention"
Number of parameters ~150M ~150M ~ 250M ~ 250M) ~ 50M
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In contrast, standard classification benchmarks: 1) consider only a fixed set of category labels, 2) do not evaluate compositional concept learning and 3) do not explicitly capture a notion of reasoning under uncertainty. We introduce a new few-shot, meta-learning benchmark, Compositional Reasoning Under Uncertainty (CURI) to bridge this gap. CURI evaluates different aspects of productive and systematic generalization, including abstract understandings of disentangling, productive generalization, learning boolean operations, variable binding, etc. Importantly, it also defines a model-independent “compositionality gap” to evaluate difficulty of generalizing out-of-distribution along each of these axes. Extensive evaluations across a range of modeling choices spanning different modalities (image, schemas, and sounds), splits, privileged auxiliary concept information, and choices of negatives reveal substantial scope for modeling advances on the proposed task. All code and datasets will be available online. + +# 1 INTRODUCTION + +Human concept learning is more flexible than today’s AI systems. Human conceptual knowledge is productive: people can understand and generate novel concepts via compositions of existing concepts (“an apartment dog”) (Murphy, 2002), unlike standard machine classifiers that are limited to a fixed set of classes (“dog”, “cat”, etc.). Further, humans can induce goal-based, “ad hoc” categories such as “things to take from one’s apartment in a fire” (children, dogs, keepsakes, etc.) (Barsalou, 1983). Thus, unlike AI systems, humans reason seamlessly in large, essentially “unbounded” concept spaces. + +Beyond unboundedness, a natural challenge in such concept spaces is uncertainty – the right concept to be inferred is uncertain, as a plethora of candidate concepts could explain observations. For e.g. in Figure 1 (top, image panel), the “right” concept could be that “All objects are blue and have the same size”, but it could also be “There are less than four objects in the scene”, or “All objects have the same color”. Humans gracefully handle such uncertainty and underdetermination (Tenenbaum & Griffiths, 2001; Xu & Tenenbaum, 2007; Goodman et al., 2008; Piantadosi et al., 2016). Popular compositional reasoning benchmarks such as CLEVR (Johnson & Zhang, 2016) for visual question answering and Ravens Progressive Matrices (Santoro et al., 2017) for deductive, analogical reasoning are compositionally rich and challenging in nature, but do not tackle ambiguity and underdetermination. + +We address this gap in the literature, and propose the Compositional Reasoning Under Uncertainty (CURI) benchmark to study how modern machine learning systems can learn concepts spanning a large, productively defined space (Figure 1). In pursuit of this goal, we instantiate a meta learning task where a model must acquire a compositional concept from finite samples. A signature of productivity in human thought is our ability to handle novel combinations of known, atomic components. Thus, in CURI we instantiate different systematic train-test splits to analyze different forms of generalization in concept learning, involving novel combinations of intrinsic properties (e.g. color, shape) with boolean operators, counting, extrinsic object properties (e.g. object location), and a novel test of variable binding in context of compositional learning. + +While related systematic splits have been proposed in prior work in context of other tasks such as question answering and analogical reasoning (Barrett et al., 2018; Hill et al., 2019; Agrawal et al., + +![](images/97c8b1053b99d529776e044cb9e1891e2b2b0025d94aba8cabb7e42abef57b5b.jpg) +Figure 1: Concept Space. Three example concepts (rows) along with schematic positive examples. Actual scenes are rendered in multiple ways including the CLEVR renderer (Johnson et al., 2016) (see Figure 2). Right: The grammar of variables, quantifiers, functions and operators to induce compositional concepts. + +2017; Johnson et al., 2016; Vedantam et al., 2017; Higgins et al., 2017; Bakhtin et al., 2019; Lake & Baroni, 2018; Ruis et al., 2020), ours is the first benchmark which tests different qualitative aspects of reasoning about productive concepts under uncertainty. + +Compositional Reasoning Under Uncertainty (CURI) Task. Concretely, the CURI task tests few-shot learning of relational concepts in a large compositional conceptual space, with design inspiration from studies in cognitive modeling using a language of thought (LOT) approach (Fodor, 1975; Piantadosi, 2011; Kemp et al., 2005). CURI includes scene-based concepts such as “All objects have the same color” and “There exists a blue object while the rest are triangles” (Figure 1) but unlike CLEVR (Johnson et al., 2016) there are too few examples to deduce answers with certainty. Our benchmark is defined through a series of meta-learning episodes (see example in Figure 2): given positive and negative examples of a new concept $D _ { \mathrm { s u p p } }$ (known as the “support set”), the goal of an episode is to classify new examples $D _ { \mathrm { q u e r y } }$ (the “query set”). As in few-shot classification (Fei-Fei et al., 2006), meta-learning (Vinyals et al., 2016), and other open-set tasks (Lampert et al., 2014), models are evaluated on novel classes outside the (meta-)training set. Unlike previous work (Triantafillou et al., 2019; Lake et al., 2019) that focuses on atomic concepts, our benchmarks concerns more structured, relational concepts built compositionally from a set of atomic concepts, and involves reasoning under uncertainty – an ideal learner must marginalize over many hypotheses when making predictions (Gelman et al., 2004; Xu & Tenenbaum, 2007; Piantadosi et al., 2016). + +We also vary the modality in which scenes are presented—rendering them as images, symbolic schemas, and sounds— enabling future research on modality-specific representational choices for compositional reasoning under uncertainty. Finally, we vary the concepts learned by the model during meta-training and meta-testing to test different aspects of systematic generalization. + +Compositionality Gap. In addition to defining systematic splits, we also characterize (for the first time, in our knowledge), the difficulty of + +![](images/bcbd5ef86b8117329029374b7393e74ddac4a499b1f454de03a928ce74d8a8ce.jpg) +Figure 2: CURI Task. Given as input a support set $D _ { \mathrm { s u p p } }$ , with positive and negative examples corresponding to concept, the model has to infer the concept and produce accurate predictions on novel images (right). + +generalization entailed by each split by introducing the notion of a model-independent “compositionality gap”. Concretely, the compositionality gap is the difference in test performance between an ideal Bayesian learner with access to the full hypothesis space, and a Bayesian learner with access to only a (potentially large) list of the hypotheses examined during meta-training. A large gap indicates that any learner must extrapolate compositionally from the training hypotheses to solve the task; additionally, models can be compared to ideal learners that either do or do not engage in such extrapolation. We anticipate that this tool will be more broadly useful for analyzing other benchmarks with compositional splits. + +Models. We evaluate models around various dimensions which concern the difficulty of learning productive concepts under uncertainty, including: 1) the modality in which the input is rendered (image, schemas, sounds), 2) method used for reasoning across objects in a scene (transformer, relation-network, global average pooling, concatenation), 3) whether or not training provides groundtruth symbolic descriptions of concepts, and 4) how negative examples are sampled. Overall, our evaluations suggest that there is substantial room for improvement in compositional reasoning under uncertainty, w.r.t the compositionality gap, representing a novel challenge for compositional learning. + +![](images/abd840279a188f051f7bac99c975cad4d32983a558c17d915300fd79ca71bc69.jpg) +Figure 3: Language of thought. All valid (type-consistent) compositions of functions are potential complex concepts in our dataset. Note that the functions are illustrated for the case of images and schemas. Location, size, shape etc. correspond to different properties for sounds. + +Summary of contributions: 1) We introduce the Compositional Reasoning Under Uncertainty (CURI) benchmark for evaluating compositional, relational learning under uncertainty from observational data; 2) We introduce a ‘compositionality gap’ metric for measuring the difficulty of systematic generalization from train to test; 3) We provide various baseline models for benchmarking progess. + +# 2 RELATED WORK + +Compositional Learning. Related work has examined systematic generalization in pattern completion using Raven’s matrices (PGM) (Santoro et al., 2017; Hill et al., 2019) and visual question answering with CLEVR (Johnson et al., 2016; Bahdanau et al., 2019). CURI’s use of the CLEVR renderer further invites particular comparison with that benchmark. Compared to these more deductive reasoning tests, CURI examines few-shot concept learning under substantial inherent uncertainty. Unlike puzzle solving or question answering, an ideal inductive learner on CURI cannot know the right rule with certainty. In essence, unlike CLEVR the “question” to be answered is not given to the model as input, but must be inferred – making the task more challenging. While PGMs do involve such an inference, once the constraints of a puzzle are identified, it does not: 1) have any uncertainty in the reasoning (which is crucial) and 2) involve any “concept” learning – where a concept applies to multiple images – as much as it involves “instance” matching to complete a sequence. In contrast, a successful CURI model behaves as if marginalizing over many hypotheses consistent with the observations e.g., (Tenenbaum & Griffiths, 2001; Xu & Tenenbaum, 2007; Piantadosi et al., 2016), an ability which is rarely studied directly in deep learning models (although see (Grant et al., 2019)). + +Recently, Keysers et al. (2019) proposed a method to create “difficult” systematic splits based on the principle that they should share atoms but have maximally different compositions. This is complementary to our splits, which provide interpretable notions of what each split tests such as disentangling, complexity, variable binding etc. Moreover, our variable binding split is predicated on having different atoms between train and test, and thus cannot be recovered by their methodology. + +Language of Thought (LOT). Our choice of compositional concepts was most closely inspired by (Piantadosi et al., 2016) along with other studies of human concept learning in the Language of Thought (LOT) framework (Fodor, 1975; Goodman et al., 2008; Kemp & Jern, 2009; Piantadosi et al., 2012; Goodman et al., 2015; Overlan et al., 2017; Lake & Piantadosi, 2019). In typical LOT studies of human learning, the conceptual space $\mathcal { H }$ is defined through a probabilistic context-free grammar $\mathcal { G }$ , which specifies a set of conceptual primitives and their rules of combination. Here, we use a LOT-inspired grammar $\mathcal { G }$ to generate an unbounded set concepts $\mathcal { H }$ , while evaluating machine learning models trained without access to the underlying LOT. + +# 3 COMPOSITIONAL REASONING UNDER UNCERTAINTY (CURI) DATASET + +Concept space. The compositional concepts in CURI were inspired by the empirical and cognitive modeling work of Piantadosi et al. (2016). The space of concepts (LOT) is defined by a context free grammar $( { \mathcal { G } } )$ . Figure 3 shows the LOT and specifies how primitives and functions compose to produce a large unbounded concept space. The LOT has three variables: $\mathbf { x }$ , representing an object in a scene, $S = \mathbf { \bar { \{ x \} } } _ { i = 1 } ^ { N }$ representing the set of all objects in the scene, and $S _ { - \mathbf { x } } \overset { \cdot } { = } S / \{ \mathbf { x } \}$ , representing the set of all objects in the scene except x. Each concept describes a rule composed of object and scene properties, logical operators, and/or comparison operators, and can be evaluated on a given scene $S$ to determine whether the scene satisfies the rule. + +Object and scene properties are defined by functions which can be applied to objects or scenes: for example, $\mathsf { s i z e ? } ( \mathbf { x } )$ yields the size of an object $\mathbf { x }$ , while $\mathsf { s i z e ? } ( S )$ returns a set with the sizes of all the objects $\langle \lnot \mathrm { i } \lnot \mathbf { e } ? ( \mathbf { x } ) : \mathbf { x } \in S \}$ ). Comparison and logical operators can be used to compare and relate various properties of objects in scenes. In contrast to Piantadosi et al. (2016), we include a count operator, which determines how many times a condition is satisfied by a set, which allows us to check how well deep learning models are able to count (Chattopadhyay et al., 2016; Johnson et al., 2016; Agrawal et al., 2017). Finally, quantifiers such as exists and for-all enrich the LOT by specifying the number of objects which must satisfy a given condition. + +Consider the following example concept (Figure 1 bottom): “There exists a blue object in the scene and the rest of the objects are squares.” To access the color of a given object, we use color? $\mathbf { \Psi } ( \mathbf { x } )$ and to access the shape of a given object, we use shape? $\mathbf { \mu } ( \mathbf { x } )$ . To determine whether an object matches a specific property, we can combine this with equality: shape? $\begin{array} { r l r } { ( { \bf x } ) } & { { } = } & { { \bf w } _ { \sf S q u a r e ^ { \prime \prime } } } \end{array}$ . Finally, we can use exists to specify that at least one object must be blue, $S _ { - \mathbf { x } }$ to specify all the objects except for that blue object, and all to specify that all the objects in $S _ { - \bf { x } }$ must be squares. Putting it all together: exists $\mathbf { x } \in S$ $\begin{array} { r l } { ( \mathtt { c o l o r 2 } \mathtt { ( x ) } } & { { } = \begin{array} { l } { \mathtt { w } _ { \mathrm { b l u e } ^ { \prime \prime } } ; } \end{array} } \end{array}$ ) and all (shape? $( S _ { - { \bf x } } ) ~ = ~ \ " { \bf s } _ { \mathrm { q u a r e } ^ { \prime \prime } ) }$ . + +Structured Generalization Splits. A signature of productivity is the ability to handle novel combinations of known components (Fodor, 1975; Fodor & Pylyshyn, 1988). Thus, in CURI, we consider splits that require generalizing to novel combinations of known elements from our LOT (Figure 3), including combinations of constants, variables, and functions. We achieve this by creating disjoint splits of concepts $\mathcal { H } _ { t r a i n }$ and $\mathcal { H } _ { t e s t }$ for training and evaluating models. By varying the held out elements and their combinations, we obtain splits that evaluate different axes of generalization. In practice, we use our grammar $\mathcal { G }$ to sample and filter a large set of concepts (see Appendix B.2 for more details), which yields a set of 14,929 concepts $\mathcal { H }$ for training and evaluation. We next describe how each split divides $\mathcal { H }$ into $\mathcal { H } _ { t r a i n }$ and $\mathcal { H } _ { t e s t }$ , to test productive, out of distribution generalization: + +• Instance IID: Evaluates generalization to novel episodes from the same concept set. This is the standard setup in machine learning (Murphy, 2013), in which $\mathcal { H } _ { t r a i n } = \mathcal { H } _ { t e s t }$ . This is the only split where train and test concepts overlap. Concept IID: Evaluates generalization to novel concepts based on an arbitrary random split of the concepts into $\mathcal { H } _ { t r a i n }$ and $\mathcal { H } _ { t e s t }$ .1 Counting: Evaluates the ability to learn a new concept $h$ with novel property-count combinations, e.g, the training concepts never filter for exactly $^ { \bullet } 3$ squares’. Extrinsic properties: Evaluates the ability to learn a new concept $h$ , with novel combinations of extrinsic (e.g. location) and intrinsic (e.g. color) object properties. +Intrinsic properties: Evaluates the ability to learn a new concept $h$ with novel combinations of intrinsic properties, e.g., the training concepts never reference both ‘red’ and ‘rubber’. Boolean operations: Evaluates the ability to learn concepts which require application of a familiar boolean operation to a property to which the operation has never been applied previously. +• Complexity split: Evaluates generalization from simple concepts (those which have less than or equal to 10 symbols) to more complex concepts (longer than 10 symbols). This is indicative of the productivity (Fodor, 1975) exhibited by models, in generalizing from simpler concepts to more complex concepts. +Variable binding: Evaluates learning of entirely novel intrinsic properties, e.g. the training concepts involve only “red”, “blue”, and “green” but test concepts involve “yellow” (although ‘yellow’ objects can still appear in training scenes). This is indicative of inferential coherence (Fodor, 1975) in models, in generalizing rules of inference to novel atoms. + +A model that infers the underlying LOT during meta-training would be expected to perform well on any such systematic split. By comparing the performance of current models to to such ideal learners, this benchmark will allow us to evaluate progress on the systematic out-of-distribution generalization capabilities of our current models. Appendix C provides more details on the strucutred splits. + +From Concepts to Meta-learning Episodes. A single episode comprises a support set $( D _ { \mathrm { s u p p } } )$ and a query set $( D _ { \mathrm { q u e r y } } )$ , each of which is generated from a given concept, $h$ . Formally, a support or query set $D$ has input data $\mathbf { u }$ and corresponding label $y$ , i.e. $D = \{ \{ y _ { i } \} _ { i = 1 } ^ { N }$ , $\{ { \mathbf { u } } _ { i } \} _ { i = 1 } ^ { N } \}$ . Each support and query set contains 5 positive and 20 negative examples — negative examples are oversampled since the space of negatives is generally much larger than that for positives. The set of positive examples are sampled uniformly from a categorical distribution over all positives. However, we consider two types of negatives: 1) easy negatives, in which the negatives are also sampled at random, and 2) hard negatives, in which negatives are generated from a closely related concept which also evaluates true on the positive examples in $D _ { s u p p }$ , such that these negatives are maximally confusing. Altogether, for each split, our train, validation, and test sets contain 500000, 5000, and 20000 episodes, respectively. + +Compositionality Gap. A key aspect of our benchmark is to define the difficulty in learning that arises from the compositional structure of the concept space. Most of the splits above are structured in a way such that $\mathcal { H } _ { \mathrm { t e s t } } \cap \mathcal { H } _ { \mathrm { t r a i n } } = \emptyset .$ – forcing a learner to use the compositional structure of the concept space to generalize to ${ \mathcal { H } } _ { \mathrm { t e s t } }$ . We conceptualize the difficulty of this task through the notion of its compositionality gap. Intuitively, the compositionality gap captures the difference between the generalization performance of an ideal compositional learner (strong oracle) compared to an ideal non-compositional learner that is unable to extrapolate outside the training concepts (weak oracle). + +Formally, let $\Omega ~ \in ~ \{ \mathrm { s t r o n g } , \mathrm { w e a k } \}$ denote an oracle over a concept space $\mathcal { H } _ { \Omega }$ . The posterior predictive distribution of an oracle for query scene $\mathbf { u }$ and query label $y \in \{ 0 , 1 \}$ is then given as: $\begin{array} { r l r } { p _ { \Omega } ( y | \mathbf { u } , D _ { \mathrm { s u p p } } ) } & { = } & { \sum _ { h \in \mathcal { H } _ { \Omega } } \bar { p _ { \Omega } } ( y | h , \mathbf { u } ) p _ { \Omega } ( h | D _ { \mathrm { s u p p } } ) } \end{array}$ , where $p _ { \Omega } ( h | D _ { \mathrm { s u p p } } ) \ d t \propto$ $p _ { \Omega } ( h ) p ( \{ y _ { i } \} _ { i = 1 } ^ { N } | h ; \{ \mathbf { u } _ { i } \} _ { i = 1 } ^ { N } )$ and $p _ { \Omega } ( h )$ denote the posterior and prior, respectively. Given a metric of interest $M$ (e.g., mean average precision or accuracy), the compositionality gap of a learning task is then simply defined as the difference in performance of the strong and weak oracle when evaluating on concepts from $\mathcal { H } _ { \mathrm { t e s t } }$ , i.e., $M ( p _ { \mathrm { s t r o n g } } ) - M ( p _ { \mathrm { w e a k } } )$ . + +Using this notion of compositionality gap, we can then define ideal learners, i.e., the strong and weak oracle, simply via their priors. In particular, let $w ( h )$ denote a weight on importance of each hypothesis2 and let I denote the indicator function. We then define the prior of an oracle as $\begin{array} { r } { p _ { \Omega } ( h ) \stackrel { } { = } \sum _ { h ^ { \prime } \in \mathcal { H } _ { \Omega } } w ( h ^ { \prime } ) { \bf I } [ h ^ { \prime } = h ] , } \end{array}$ . The difference between strong and weak oracle lies in which concepts can be accessed in these priors. + +In this formalism, the strong oracle has access to the union of train and test concepts; that is $\mathcal { H } _ { \mathrm { s t r o n g } } = \mathcal { H } _ { \mathrm { t r a i n } } \cup \mathcal { H } _ { \mathrm { t e s t } }$ . The weak oracle, on the other hand only assumes access to $\mathcal { H } _ { \mathrm { w e a k } } = \mathcal { H } _ { \mathrm { t r a i n } }$ which means it is unable to consider any hypothesis outside what has been seen in training and assigning it zero probability mass. Given a support set $D _ { \mathrm { s u p p } }$ this difference in priors leads then to different inferences on posteriors and allows us to quantify how compositionally novel a learning task is relative to these ideal learners. + +# 4 METRICS AND BASELINES + +During meta-test, given $D _ { \mathrm { s u p p } }$ models are evaluated on their ability to learn novel concepts. We use two metrics for quantifying this: 1) Accuracy: evaluates the accuracy of model predictions across the query set $D _ { q u e r y }$ , as is standard practice in meta-learning (Lake et al., 2019; Snell et al., 2017). Since there are more negative than positive labels, we report class balanced accuracy for better interpretability, averaging accuracies for the positive and negative query examples; and 2) mean Average Precision (mAP): evaluates models on a much larger number of test scenes $\tau$ for each episode (comprising 44,787 scenes, 3 per each concept in $\mathcal { H }$ ). This resolves an issue that with a small query set, a strong model could achieve perfect accuracy without grasping the concept. Since episodes typically have many more negative than positive examples, Average Precision sweeps over different thresholds of a model’s score and reports the average of the precision values at different recall rates, e.g., (Everingham et al., 2010). mAP is then the mean across all of the meta-test episodes. + +![](images/b996d6f1a26afe350829ee2e314ba7a4972274ec9a6a0686e57215a46af6a944.jpg) +Figure 4: Baseline models (left). Different choices for the encoder $f ( \mathbf { u } )$ parameterization explored in the paper. We consider three modalities each of which is processed with a modality specific encoder, followed by four kinds of pooling architecture which take as input objects and their corresponding locations to provide an encoding for the datapoint. Training (right). The model is trained by processing the support images $D _ { \mathrm { s u p p } }$ with positive (green) and negative (red) images, using $f ( \mathbf { x } )$ to compute $\mathcal { L } _ { \mathrm { q u e r y } }$ which computes generalization error on queries and $\mathcal { L } _ { \mathrm { c o n c e p t } }$ which learns to decode the true concept as an auxiliary task. Losses are weighted by $\alpha \geq 0$ . + +# 4.1 TRAINING LOSS + +Denote by $\mathbf { u } \in \mathbb { R } ^ { M }$ the input to the model, which can be either in the form of image, sound or schema. We work in a binary classification setting with labels $y$ live in the space $\mathcal { V } \in \bar { \{ 0 , 1 \} }$ . Then, given a support set $D _ { \mathrm { s u p p } } = \{ \mathbf { u } _ { i } , y _ { i } \} _ { i = 1 } ^ { T }$ and a query set $D _ { \mathrm { q u e r y } } \stackrel { \cdot } { = } \{ \mathbf { u } _ { i } , y _ { i } \} _ { i = 1 } ^ { T }$ , sampled in accordance with a productive concept $h$ , our training objective for a single training instance can be written as $\mathcal { L } _ { \mathrm { q u e r y } } + \alpha \mathcal { L } _ { \mathrm { c o n c e p t } }$ . Here $\begin{array} { r } { \mathcal { L } _ { \mathrm { q u e r y } } = \sum _ { \mathbf { u } , y \in D _ { \mathrm { q u e r y } } } ^ { - } \log p ( Y = \bar { y } | \mathbf { u } , D _ { \mathrm { s u p p } } ) } \end{array}$ is a standard maximum likelihood meta-learning loss (Ravi & Larochelle, 2016; Snell et al., 2017; Finn et al., 2017), and $\mathcal { L } _ { \mathrm { c o n c e p t } } = \log p ( H = h | \bar { D } _ { \mathrm { s u p p } } )$ is an optional regularizer designed to encourage retaining information about the hypothesis of interest from the support set. + +# 4.2 BASELINE MODEL ARCHITECTURES + +Our baseline models (shown in Figure 4) parameterize the probability in the $\mathcal { L } _ { \mathrm { q u e r y } }$ term above using prototypical networks (Snell et al., 2017). The prototypical network consists of an embedding function $f = f _ { \theta }$ and uses it to compute prototypes $\mathbf { c } _ { p }$ and $\mathbf { c } _ { n }$ for positive and negative examples by averaging $f ( \mathbf { u } )$ for positive and negative examples in the support set respectively. In equations, given a query datapoint $\mathbf { u } ^ { \prime }$ , we compute + +$$ +p ( Y = y | \mathbf { u } ; D _ { \mathrm { s u p p } } ) = \frac { e x p ( | | f ( \mathbf { u } ^ { \prime } ) - \mathbf { c } _ { p } | | ^ { 2 } } { e x p ( | | f ( \mathbf { u } ^ { \prime } ) - \mathbf { c } _ { p } | | ^ { 2 } ) + e x p ( | | f ( \mathbf { u } ^ { \prime } ) - \mathbf { c } _ { n } | | ^ { 2 } } +$$ + +In this formalism, the models we study in this paper span different choices for $f$ . Roughly, in each modality, we start with an encoder that converts the raw input into a set of vectors, and then a pooling operation that converts that set of vectors into a single vector. In the case of images and sound (input as spectrograms), the encoder is a ResNet-18; and the set of vectors is a subsampling of spatial locations; and for schemas we vectorize components with a lookup table and combine them into a set via feed-forward networks. In the case of images and sounds, the output of the encoder is enriched with position vectors. For the pooling operation, we study global averaging, concatenation, relation networks (Santoro et al., 2017) and transformers (Vaswani et al., 2017) equipped with different pooling operations (max, mean, sum, min) for reasoning inspired by Wang et al. (2019) (Figure 4 middle panel, also see Appendix F for more details). + +For the probability in $\mathcal { L } _ { \mathrm { c o n c e p t } }$ , we represent the concept as a sequence by prefix serialization and then use an LSTM (Hochreiter & Schmidhuber, 1997) to parameterize $p ( h | D _ { \mathrm { s u p p } } ) =$ $\Pi _ { s = 1 } ^ { S } p ( h _ { s } | h _ { 1 \cdots t - 1 } ; D _ { s u p p } )$ . At each step of the LSTM we concatenate $[ \mathbf { c } _ { p } , \mathbf { c } _ { n } ]$ to the input. + +![](images/2ea5088d670d6fa4797dca4586c3097660e37e47c12eebe19396d0937331179b.jpg) +Figure 5: Compositionality Gap. Different splits ( $\mathbf { \hat { x } } \mathbf { \cdot }$ -axis) plotted w.r.t performance of the strong oracle (green line) and weak oracle (red line) on the mAP (top) and Accuracy (bottom) evaluated on respective test splits (using hard negatives in support and query sets). Difference between the two is the compositionality gap (comp gap). Yellow: shows the (best) relation-net model on schema inputs, purple: shows the model on image inputs, and gray: shows the model on sound inputs. Error bars are std across 3 independent model runs. + +# 5 EXPERIMENTAL RESULTS + +We first discuss the compositionality gap induced by the different generalization splits and then delve into the impact of modeling choices on performance on the generalization splits. All models are trained for 1 million steps, and are run with with 3 independent training runs to report standard deviations. We sweep over 3 modalities (image, schema, sound), 4 pooling schemes (avg-pool, concat, relation-net, transformer), 2 choices of negatives (hard negatives, random negatives) and choice of language $( \alpha = 0 . 0 , 1 . 0 )$ . Unless mentioned otherwise in the main paper we focus on results with hard negatives and $\alpha = 0 . 0$ . When instantiated for a given modality, we note that the encoders $f ( \mathbf { u } )$ (Figure 4) all have a similar number of parameters. The appendix contains details of the exact hyperparameters (Appendix E), and more comprehensive results for each split (Appendix G.7). + +# 5.1 DATASET DESIGN AND COMPOSITIONALITY + +How compositional are the structured splits?. Our main results are shown in Figure 5. Using our model-independent measure of the compositionality gap (Section 3), different splits present varying challenges for generalizing from train to test. The most difficult splits, with the largest compositionality gaps, are the Binding (color) and Binding (shape), which is reasonable since they require learning concepts with entirely new property-values. In contrast, the easiest split with the smallest compositionality gaps is the Instance IID split since it does not require compositionality. Finally, while the mAP metric exposes a larger value of comp gap, the ordering of splits in terms of comp gap is same for both metrics – suggesting similar coarse-grained notions of compositionality. + +Results for the best overall architecture, a relation network (relation-net), is shown in Figure 5. Network performance on the easiest data format (schema; yellow bars) is generally better than the weak oracle, but substantially worse than the strong oracle. Counting is a particularly challenging split where the models underperform even the weak oracle. Broadly, this suggests that the models capture some notion of compositionality – especially for images and schemas – relative to a weak oracle that rigidly considers only training hypotheses, but there is substantial room to improve (especially with respect to the more stringent mAP metric). These results demonstrate that CURI provides a challenging yet tractable setting for evaluating the compositional capabilities of models. + +Finally, we found that the performance on the Instance IID split is not equal to the weak (and strong) oracle—which are both equal in this case—indicating that the best model does not make ideal posterior predictions even when compositionality is not an issue. Ideal predictions in this case would require the network to behave as if marginalizing over the training hypotheses, as the strong oracle does. A similar plot to Figure 5 can be found in Appendix G.4 for random negatives. + +Influence of Negatives. Previous work (Hill et al., 2019) has shown that the choice of random v.s. hard negatives for training and evaluation impacts compositional generalization substantially in the case of a particular set of analogical reasoning models. However, we argue that such decisions on dataset design can be made more objectively if one can evaluate the model-independent comp gap. In our context, we find that the comp gap with mAP when using random negatives decreases on average by $5 . 5 \pm 1 . 4 \%$ compared to when we use hard negatives. This indicates that it is not only the choice of $\mathcal { H } _ { t r a i n }$ and $\mathcal { H } _ { t e s t }$ , which are identical for a given compositional split (say Counting), but also the choice of the negatives which “makes” the task compositionally novel. More generally, this indicates that the comp gap has utility as a more general diagnostic tool for making principled design decisions in compositional learning settings without the confound of specific model decisions. + +# 5.2 DIFFERENCES BETWEEN MODELS + +Best Models. In general, the best performing model is the relation-net applied to schema inputs, outperforming other combinations of models and input modalities on the Boolean, Concept IID, Complexity, and Instance IID splits on both the mAP as well as accuracy metrics (Figure 5); although as mentioned above, none of the models are close to the strong oracle. It is closely followed by the transformer model on schema inputs, which performs the best on Binding (color), Binding (shape), and Intrinsic splits (Appendix G.7). Utilizing schema inputs proves easier for abstraction except for the Extrinsic setting, where the task requires generalization to novel locations for objects in images, which is well supported by the inductive bias of the CNN encoder (Figure 4). In this case, the image-transformer gets an mAP of $6 2 . 1 \pm 0 . 7 \%$ , compared to the next best schema-transformer model at $6 0 . 9 \pm 0 . 7$ . Further, relational learning proves more crucial in the schema case than for images, with all image models (regardless of pooling) performing better than $5 9 . 4 \pm 1 . 3 \%$ mAP (achieved for image-avg-pool) while schema-avg-pool models get only get to $5 3 . 4 \pm 1 . 5 \%$ . + +When to use a transformer? Transformer models appear to outperform relation networks in splits concerning disentangling. For instance, for the Intrinsic split with schema-relation-net is at $5 5 . 1 \pm 0 . 8 \%$ v.s. $5 7 . 9 \pm 0 . 6 \%$ for schema-transformer. Similarly, for the Extrinsic split the imagetransformer is at $6 2 . 1 \pm 0 . 7 \%$ compared to the image-relation-net at $6 0 . 8 \pm 1 . 1 \%$ . We hypothesize that this is because the iterative message passing via. attention in transformers improves object representations for disentangling compared to relation networks that lack such a mechanism. + +What is the relative difficulty of abstraction from different modalities? One of the key contributions of our work is in providing multiple modalities (image, schema, sound) for productive concept learning. We next characterize the difficulty of abstraction based on modality for the various generalization settings. In the Intrinsic setting, we find that the schema models, which have access to a “perfect” disentangled representation significantly outperform image models—a schema-avg-pool model gets an mAP of $5 2 . 7 \pm 3 . 1 \%$ while an image-avg-pool model gets to $3 4 . 4 \pm 0 . 0 \%$ mAP. + +Similarly, for the Counting split where the total number of objects are exactly specified in the schema (Figure 4), schemas are substantially better than images. For example, schema-relation-nets get to $5 6 . 2 5 \pm 5 . 3 2 \%$ mAP while image-avg-pool is at $4 8 . 4 \pm 1 . 2 \%$ mAP. Interestingly, the next best model—image-relation-net—is substantially worse, at $3 9 . 4 5 \pm 1 . 6 \%$ . Curiously, for while transformer models perform well at disentangling, they seem to be quite poor for Counting, with image-transformer models getting to only $3 2 . 4 \pm 1 . 4 \%$ mAP, suggesting a potential weakness for transformers. Overall, there appears to be an intimate link between the generalization setting and the input modality, suggesting avenues where representation learning could be improved for a given modality (e.g. images), relative to the kind of reasoning one is interested in (e.g. counting). + +When does language help? On average, training models with explicit concept supervision using the concept loss (Section 4.1) improves performance by $2 . 8 \pm 0 . 6 \%$ mAP (SEM error). This is a small boost relative to the gap between the original model and the strong oracle, suggesting that this simple auxiliary loss is not sufficient to internalize the LOT in a neural network. Overall, image models benefit more from language than schema models which natively utilize symbols (Appendix G.3). + +# 6 CONCLUSION + +We introduced the compositional reasoning under uncertainty (CURI) benchmark for evaluating fewshot concept learning in a large compositional space, capturing the kinds of producitivity, unboundness and underdetermination that characterize human conceptual reasoning. We instantiate a series of meta-learning tasks, and evaluate numerous baseline models on various aspects of compositional reasoning under uncertainty, including inferential coherence, boolean operation learning, counting, disentangling, etc. Further, we introduce the notion of a compositionality gap to quantify the difficultly of each generalization type, and to estimate the degree of compositionality in current deep learning models. 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Psychological Review, 114:245–272, 2007. + +# A EXAMPLE EPISODES FROM THE DATASET + +We show examples from the Concept IID split test set comprising the ground truth productive concept (top), along with the support and query sets for meta learning (rendered as images), the alternate hypotheses which are consistent with the support set – that is, other hypotheses which could also have generated the positive and negative examples in the support set – and the concepts based on which we pick the hard negatives Figures 6 to 11. + +# B ADDITIONAL DATASET DETAILS + +We first provide more details of the concept space $\mathcal { G }$ , then explain how we obtain $\mathcal { H }$ , the space of concepts for training and evaluation, provide more details of the structured splits, and finally explain the weight $w ( h )$ based on which we sample concepts. + +# B.1 MORE DETAILS OF THE GRAMMAR + +We provide below the full grammar used to sample concepts, where $A \to B | C$ means that $A$ can expand to either $B$ or $C$ under the rules defined by the grammar. We always start expanding at the START token and then follow the rules of the grammar until we hit a terminal node (which does not have any expansions defined). As and where possible, we followed the insights from Piantadosi et al. (2016) in choosing the sampling probabilities for various completions based on how well humans seem to be able to learn the corresponding primitive. For example, we sample utterances with disjunctions (or) less frequently since they are known to be difficult for humans to learn. Based on Kemp & Jern (2009), we chose to represent location as a discrete entity, such that relative, and categorical notions of left or right simply become comparisons in the location space (location? $\textbf { x } >$ location? $S _ { - \mathbf { x } , }$ ), unlike the CLEVR dataset (Johnson et al., 2016) which defines categorical relational objects. + +Here is the full grammar $\mathcal { G }$ used for sampling the concepts (as explained in the main paper, $S _ { - { \bf x } } =$ $S / \{ \mathbf { x } \} )$ . Note that the grammar always generates strings in postfix notation and thus the operands in each expansion occur before the operation: + +![](images/ce51437271d11b3923b461a18ecfba57684729b9acfbc6d616027c265c143df5.jpg) + +# Valid Hypotheses + +\* for-all $\mathsf { x }$ in ${ \mathsf { S } } =$ (purple,color?( $\textsf { x } )$ ）{log-prob[-0.852]] +\*for-all $\times$ in $\gamma =$ (color?(x)，purple）{log-prob[-0.852]} +\*all(color?(S),purple）{log-prob[-1.950]} +\*exists $\times$ inSand $( =$ (metal,il? $\textsf { x } )$ ),al(color?(S),purple)）{log-prob[-8.523]} +\*for-all $\times$ in Sand $\scriptstyle =$ (color?(x),purple),any(material？(S)，metal)）{log-prob[-8.523]} + +Hypotheses for Hard Negatives + +$^ { \ast }$ for-all $\times$ inSor $( =$ (shape?( $\times$ ),cube ),all(color?(S),purple )) \*for-all $\times$ inSor $: =$ (purple,color？ $\textsf { x } )$ ), $=$ (sphere,shape?(×))) \*exists $\times$ inSand(any(color?(S_{-x}),purple),all(color?(S_{-x}),purple)) \*for-all xinSo $\mathbf { \partial } ^ { \cdot } ( =$ (shape？( $\times$ )，sphere), $=$ (purple,color?(x)）） \*for-all xinSor $( =$ (purple,color？ $\textsf { x } )$ ）， $=$ (shape?(x),cube)) + +![](images/9982c10744a33a41760dc776accfdcf610e9ea57fe0a93e16d9c69fed8a64d74.jpg) +Figure 6: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. + +# Valid Hypotheses + +\* any(locationY?(S),7）{log-prob[-1.621]} \*exists $\mathsf { x }$ in $\mathsf { S } >$ (locationY?(x),6）{log-prob[-1.621]} \*exists $\times$ inSany(locationY?(S_{-x})，7）{log-prob[-1.621]} \*exists $\times$ inSall(locationY?( $S _ { - } \{ - x \}$ ),7）{log-prob[-1.647]} \*exists $\mathsf { x }$ in ${ \mathsf { S } } =$ (locationY?(×)，7）{log-prob[-2.314]} + +Hypotheses for Hard Negatives + +\*exists $\times$ in S and(al(color?(S_{-x}),red),>(8,locationX?(x)))\*exists $\mathsf { x }$ inSand(all(color?(S_{-x})，red）， ${ \tt > } ( 7 _ { \tt { i } }$ locationx?(x)))\*exists $\times$ inSand(al(color？(S_{-x}）,red）,any(size(S_{-x}）,0.5））\*exists $\mathsf { x }$ inSand $( >$ (8,locationY? $, \times ) ~ ,$ ， $>$ (locationY?( $\times ~ 5 .$ 5））\*exists $\mathsf { x }$ inSor(al(shape?(S),sphere),all(color?(S_{-x}),red)) + +Figure 7: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. + +![](images/a01628a9c0491627343c835820bbc7a21990351f0912b975e7c67523f79be486.jpg) + +# Valid Hypotheses + +\* for-all $\times$ in $\mathsf { S } =$ (sphere,shape? $\textsf { x } )$ ）{log-prob[-0.855]} +\*for-all $\times$ in s $=$ (shape?(x)，sphere）{log-prob[-0.855]} +\*all(shape?(S）,sphere）{log-prob[-1.954]} +\*for-all $\times$ inSand $( = 1$ (sphere,shape？ $( \times )$ ）， ${ } = { \mathsf { I } }$ (shape?(×),sphere））{log-prob[-7.807]} +\*for-all $\mathsf { x }$ in Sand $\mathop { \left. \sum \right. }$ (shape?(x),sphere), $=$ (sphere,shape?(x))）{log-prob[-7.807]} + +Hypotheses for Hard Negatives + +$^ *$ for-all $\times$ inSor $( =$ (sphere,shape?( $\textsf { x }$ ）） $= ( \mathsf { c o l o r ? } ( \textsf { X } )$ ，green））\*for-all $\times$ inSor $: =$ (purple,color？(x））, $=$ (sphere,shape?(x)))\*for-all $\mathsf { x }$ inSor $: =$ (sphere,shape? $\times )$ ） $=$ (green,colo？ $\times ) )$ ）\*for-all $\times$ inSor(al(shape?(S)，sphere)，=(3,ocationx?(x)))\*for-allxin Sor $( =$ (shape？( $\times$ )，'sphere), $=$ (purple,color?(x))) + +Productive Concept:exists x in Sand(al(locationY?(S_{-x}),6),any(size?(S_{-x} ),0.7)) + +![](images/b105df7e0fb61953c0f0cbc38ab2c6bef9485d4b1af91e615a037b47da81e616.jpg) +Figure 8: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. + +# Valid Hypotheses + +\*exists $\times$ in Sal(locationY?(S_{-x}),6）{log-prob[-0.008]} \*exists $\mathsf { x }$ in $\mathsf { S } >$ (count=(locationY?(S_{-x}),6),count=(size?(S_{-x}),0.35)）{log-prob[-6.668]} \*for-all $\times$ in $\mathsf { S } = \mathsf { ( c o u n t = }$ (locationY?(S）,6）,o $: =$ (shape?(S),shape？(x))）{log-prob[-6.690]} \*for-all $\times$ in ${ \sf S } =$ (count=(shape?(S),sape？(x)）,ount=(location？(S）,6)){log-prob[-6.690]} \*exists $\times$ in Sand $\mathrm { . } >$ (locationY?(x),5), $>$ (size?(x),0.35)）{og-prob[-7.45] + +Hypotheses for Hard Negatives + +\*and(any(size?(S),O.7),any(locationY?(S),6)) \*exists $\times$ inSand(any(locationY?(S),6), $>$ (locationY?(×),locationX?(×))) \*exists $\times$ in Sand $: = 1$ (locationY?( $\times )$ 0,6),any(shape?(S)，sphere)) \*exists $\mathsf { x }$ in ${ \mathsf { S } } = { \mathsf { I } }$ (coun $: = 1$ (locationY?(S),6),count=(locationY?(S),locationY?(x))) \*exists $\mathsf { x }$ ins $>$ (count=(locationY?(S_{-x})，6）,count $=$ (locationY?(S),2)) + +Figure 9: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. + +Productive Concept: exists $\times$ in S and(all(material?( $\mathsf { S \_ f } \cdot \mathsf { X } \mathsf { \Sigma }$ ),rubber),all(color?(S_{-x}),cyan）) + +![](images/175fb5b3d8f181ce9318ca5b23355bd199dd16ba5b63902407ad08561f5fcb76.jpg) + +# Valid Hypotheses + +\*exists $\mathsf { x }$ in Sand(any(material?(S $\{ - \mathsf { x } \}$ ).rubber),al(color?(S_{-x}),cyan））{log-prob[-0.983]}\*exists $\times$ inSand(all(color？(S $\{ - { \boldsymbol { \times } } \bar \} \}$ ,cyan),a(material？(S_{-x}),rubber））{og-prob[-1.162]}\*exists $\times$ inSand(al(material？(S_{-x}),rubber),al(color？(S_{-x})，cyan））{log-prob[-1.162]} + +Hypotheses for Hard Negatives + +$* >$ (count=(color?(S),cyan),count=(material?(S),metal)) \*existsx in Sand(al(color?(S_{-x}),cyan),any(locationY?(S),3)) \*existsxinSall(color?(S_{-x}),cyan） \*existsxin Sand(all(color?(S $\{ - \mathsf { x } \}$ ),cyan),any(color?(S_{-x}),cyan)) \*exists $\mathsf { x }$ inSand(al(color？(S_{-x})，cyan),any(size?(S)，0.7)) + +Productive Concept: exists x in S and(al(locationY?( S_{-x} ),5), $>$ (locationY?(×),locationX?(×))) + +![](images/6eed4f451f8da33d850d98fba50a0bcb7bb12c18e81c7685beaa7d2369d8b85e.jpg) +Figure 10: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. + +Valid Hypotheses + +$^ *$ exists $\times$ in Sand(al(locationY?(S_{-x}),5), $>$ (locationY?(x),locationx?( $\times$ ）））{log-prob[0.000]} + +Hypotheses for Hard Negatives + +\* for-all $\times$ in S or(al(locationX?(S),1), $>$ (locationY?( $\textsf { x } )$ ,locationx?(×）)) \*for-all $\times$ inSand $\mathrm { . } >$ (locationY?( $\times$ ).locationx?( $\times 1$ )）, $>$ (8,locationY?(x))) \*for-allxin Sor(>(locationY? $\times 1$ ),locationx?(x)) $=$ (color?(x）,brown)) \*for-all $\times$ inSor $( =$ (yellow，color？ $\mathbf { \epsilon } ( \mathbf { \epsilon } )$ ）， $>$ (locationY?(x),locationx?(×))) \*for-all $\times$ inSor $( =$ (purple,color？(x)), $>$ (locationY?(x）,ocationx?(×))) + +Figure 11: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. + +START $- > \lambda$ S . BOOL e x i s $\mathrm { ~ ~ t ~ s ~ } = \mathrm { ~ ~ | ~ } \lambda \mathrm { ~ ~ S ~ }$ . BOOL f o r − a l l = BOOL $- >$ BOOL BOOL and | BOOL BOOL o r | BOOL n o t | C C = | SH SH = | M M = | S I $\mathrm { ~ S ~ I ~ } = \mathrm { ~ \Omega ~ } |$ L $\mathrm { ~ L ~ } = \mathrm { ~ \Omega ~ }$ NUM NUM $=$ | S I S I > | L L > | NUM NUM > | SETFC C a l l | SETFSH SH a l l | SETFM M a l l | SETFSI S I a l l | SETFL L a l l | SETFC C a n y | SETFSH SH any | SETFM M any | SETFSI S I any | SETFL L any + +NUM $- >$ SETFC C c o u n t $=$ | SETFSH SH c o u n t $=$ | SETFM M c o u n t $=$ | SETFSI S I c o u n t $=$ | SETFL L c o u n t = NUM > 1 | 2 | 3 + +SETFC $- >$ SET FC SETFSH $>$ SET FSH SETFM $- >$ SET FM SETFSI $- >$ SET F S I SETFL $- >$ SET FL + +FC > c o l o r ? +FSH −> s h a p e ? +FM $- >$ m a t e r i a l ? +F S I $- >$ s i z e ? +FL $- >$ l o c a t i o n X ? | l o c a t i o n Y ? + +OBJECT $- >$ x SET : S | $S _ { - \bf { x } }$ + +# B.2 SAMPLING + +We sample 2000000 initial hypotheses from the CFG $\mathcal { G }$ , and impose a maximum depth in the recursion tree of 6 when sampling. That is, no node has a depth larger than 6 in the recursion through which we generate concepts from the grammar $\mathcal { G }$ . We then reject and filter the hypotheses to obtain a set of “interesting” hypotheses $\mathcal { H }$ used in the main paper explained in more detail below: + +Rejection Sampling: We reject the following string combinations after sampling from the grammar $\mathcal { G }$ : + +• All programs which contain " $\lambda S$ . for-all $\mathbf { X } ^ { " }$ and " $S _ { - { \bf x } } "$ " in the same program. This is asking that for all objects in a scene, a certain property is satisfied by everything other than the object, which is the same as saying, for all objects in the scene. + +• All programs where we compare the properties of the same object to itself, e.g. color? $\mathrm { ~ ~ { ~ ( ~ x ~ ) ~ } ~ } = = \mathrm { ~ \cos ~ } \tt c o l o r ? \mathrm { ~ ~ { ~ ( ~ x ~ ) ~ } ~ }$ , where color? can be any function applied to the object. All programs where we have the following string: exists(color?(S) $= =$ color?(x)) or for-all(color?(S) $= =$ color?(x)) where color? can be any function applied to the object. +All programs which evaluate to true on schemas more than $10 \%$ of the time and less than 10 times. The former condition ensures that we work with concepts which are in some sense interesting and surprising (as opposed to concepts which are always trivially true), and the second condition ensures that we have unique schmeas or datapoints to place in the support and query sets, which both have 5 positive images each. + +We provide examples of concepts which get rejected for being true too often below: + +exist $_ { \mathrm { { S } } } = \mathrm { { x } }$ \in S or( $=$ (locationX?( x ), locationY?( x ) ), any(color?( S ), brown ) +) +exist $_ { \mathrm { { S } } } = \mathrm { { x } }$ \in S and( exists $=$ (locationY?( S ), locationX?( x ) ), any(color?( S ), brown ) +) +exist $_ { \mathrm { { S } } } = \mathrm { { x } }$ \in S or( all(color?( S ), gray ), all(color?( S ), brown ) +) + +See Appendix C for more details on the structured generalization splits which yeild train concepts $\mathcal { H } _ { \mathrm { t r a i n } }$ and test concepts $\mathcal { H } _ { \mathrm { t e s t } }$ . + +# B.3 CONCEPT PRIOR WEIGHT $w ( h )$ + +We next explain the form of the prior weight $w ( h )$ that we use for defining the prior over the concepts provided to the models (both oracles as well as deep learning models). Given $l ( h )$ , the number of tokens in the postfix serialization of the concept $h$ , the unnormalized weight $\tilde { w } ( h )$ is log-linear in the length, and is defined as follows: + +$$ +\tilde { w } ( h ) \propto \exp { - 0 . 2 \cdot l ( h ) } +$$ + +Given a split $\Omega \in \{ t r a i n , t e s t \}$ , the final, normalized weight is given as: + +$$ +w ( h ) = \frac { \tilde { w } ( h ) } { \sum _ { \mathcal { H } _ { \Omega } } \tilde { w } ( h ) } +$$ + +As explained in the main paper, the final prior for a hypothesis given a split $\Omega$ is $p ( h ) \ =$ $\begin{array} { r } { \sum _ { h ^ { \prime } \in \mathcal { H } _ { \Omega } } \mathcal { w } ( h ) \mathbf { I } [ h = h ^ { \prime } ] } \end{array}$ . + +Our choice of the log-linear weight is inspired by the observation in cognitive science that longer boolean concepts are harder for people to learn (Feldman, 2000). + +Here are some examples of hypotheses with a high weight (computed on $\Omega = \operatorname { t r a i n } \cup$ test): + +$$ +\begin{array} { r l } & { \mathrm { ~ e ~ x ~ i ~ s ~ t ~ s ~ \ x ~ \ x ~ \ i n ~ \ S ~ \ = ( 2 , ~ c ~ o ~ u n t = ( c ~ o ~ l o r ~ 2 ( { \it \Phi } ~ \mathbb { S } \mathbb { S } _ { - } \{ - \mathbb { x } \} \mathbb { S } _ { \mathrm { \Lambda } } ) , ~ { \it \Phi } ~ \mathrm { c y a n } ) ~ ) ~ } , } \\ & { \mathrm { ~ e ~ x ~ i ~ s ~ t ~ s ~ \ x ~ \ x ~ \ i n ~ \ S ~ > ( 1 ~ o ~ c ~ a ~ t ~ i ~ o ~ n ~ Y ~ 2 ( { \it \Lambda } ~ \mathrm { x } ~ \Lambda ~ ) ~ , ~ \mathbb { \Lambda } ~ 6 ~ ) ~ } , } \\ & { = ( \mathrm { ~ c ~ o ~ u n t = ( c ~ o ~ l o r ~ 2 ( { \it \Lambda } ~ \mathrm { S } ~ ) ~ , ~ b r o w n ~ \Lambda ~ ) ~ , ~ 3 ~ \Lambda ~ ) ~ } , } \\ & { > ( \mathrm { ~ c ~ o ~ u n t = ( ~ l o ~ c ~ a t i o n ~ { \cal X } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ ) ~ } , } \\ & { \mathrm { ~ a n y ~ ( 1 ~ o ~ c ~ a ~ t ~ i ~ o ~ n ~ Y ~ 2 ( { \it \Lambda } ~ \mathrm { S } ~ \Lambda ~ ) ~ , ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ ) ~ } , } \\ & { = ( 1 , ~ \mathrm { c ~ o ~ u n t = ( ~ l o ~ c ~ a t ~ i ~ o n ~ Y ~ 2 ( { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ ) ~ } ) ~ , } \\ & { = ( 3 , ~ \mathrm { c ~ o ~ u n t = ( ~ l o ~ c ~ a t ~ i ~ o n ~ Y ~ 2 ( { \it \Lambda } ~ { \it \Lambda } ~ { \it \Lambda } ~ ) ~ , ~ 3 ~ { \it \Lambda } ~ { \it \Lambda } ~ ) ~ } ) ~ , } \\ & \mathrm ~ a l ~ ( 1 ~ o ~ c ~ a ~ t ~ i ~ o n ~ { \cal X } ~ 2 \end{array} +$$ + +$= ( 2$ , c o u n t $=$ ( c o l o r ? ( S ) , b l u e ) ) , f o r − a l l x i n S n o t ( $>$ ( 6 , l o c a t i o n X ? ( x ) ) ) , $\mathbf { \Lambda } = ( \mathrm { ~ c ~ o ~ u ~ n ~ t ~ } = ( \mathrm { ~ c ~ o ~ l ~ o ~ r ~ } ? ( \mathrm { ~ \textbf ~ { ~ S ~ } ~ } )$ , g r a y ) , 2 ) , $= ( 2$ , c o u n t $=$ ( c o l o r ? ( S ) , g r a y ) ) + +# B.4 EXECUTION ON IMAGES. + +In order to create the perceptual inputs in the dataset $\mathcal { U }$ , we sample images using the renderer for CLEVR from Johnson et al. (2016), changing the range of objects to [2, 5], to reduce clutter and enable easier learning of predicates like any and all for models. 3 The CLEVR renderer produces scenes u with pixels as well as an associated schema file $\mathbf { u } ^ { s }$ detailing the properties of all the objects sampled in the scene, including their location, shape, size, material, and rotation. Based on this, we convert our sampled concepts into postfix notation and execute them on the schemas using an operator stack. Concretely, execution of the concept $h \in \mathcal H$ on $\mathbf { u } ^ { s }$ yields a boolean true or false value $\{ 0 , 1 \}$ . We execute each such hypothesis on a set of 990K images, yielding scores of how often a hypothesis is true for an image. We threshold this score to retain the subset of hypotheses which are true for no more than $10 \%$ of the images and are true at least for 10 images, to pick a subset of “interesting” hypotheses $\mathcal { H } ^ { \prime }$ for training models. + +Bias. The image dataset here sampled itself has a bias in terms of the location coordinates (in the pixel space). The CLEVR dataset generation process samples objects in the 3d (top-down x, y) coordinate space uniformly (from a grid of $^ { - 3 }$ , to $+ 3$ ). However, since the camera is always looking into the scene from outside, the image formation geometry implies that in the camera / image coordinates most of the objects appear to be away from the scene and very few are close to the camera. Thus, in terms of the y-coordinates we observe in the image coordinates a bias in terms of the distribution not being unifrom. This also makes sense in general, as even in the real world, objects are not found very close to the camera or very far away from the camera in general. See Figure 12 for all the biases in the observation space u computed over 990K sampled images. + +# B.5 AUDIO. + +To build the audio data, we use clips of orchestral instruments playing various pitches downloaded from https://philharmonia.co.uk/resources/sound-samples/. We make the following mappings of object properties: + +· $x$ location temporal location. larger $x$ bin means the note is played later. +• $y$ location pitch. All pitches between the instruments are the same (up to octaves). +color instrument – gray trumpet – red clarinet – blue violin – green flute – brown oboe – purple saxaphone – cyan french-horn – yellow guitar + +• shape amplitude profile; either getting louder, getting softer, or constant volume + +size total volume + +• material low-pass filtering or no filtering. + +All binned quantities use the same number of bins as in the image domain. + +![](images/d50281e9fb8b1c2569d993b39f1456f0a63a3891542c3a78614e4eca783867e3.jpg) +Figure 12: Histogram of properties found in inputs $\mathbf { u }$ in the dataset. We notice that the properties are all largely uniform with bias in x and y-coordinates towards the center of the image. + +# B.6 ANALYSIS OF SYNONOMY OF CONCEPTS. + +We next show an analysis of concepts which have the same evaluation signatures on a large set of 990K images, and are thus synonymous (in context of the dataset at hand). Note that while some of these concepts might be truly synonymous to each other (for example, $A > B$ is the same as $B < A$ ), others might be synonymous in context of the image distribution we work with. For example, size can never be greater than 0.7 in our dataset and location can never be greater than 8, and thus asking if location is greater than 8 or size is greater than 0.7 has the same semantics on our dataset. In Figure 13 we show each such “concept” or “meaning”, which is a cluster of hypotheses which all evaluate to the same truth value and plot a histogram of how many hypotheses each cluster tends to have. We notice that most of the concepts have 1 synonym (i.e. there is only one concept with the particular) evaluation signature, with a long tail going upto 80 synonyms in a concept. In the Concept IID split we ensure that none of the concepts which have the same signature are found across the train/val/test splits. + +# C DETAILED DISCUSSION OF THE STRUCTURED SPLITS + +We provide more details on how each of the structured splits described in Sec. 3 of the main paper are created. Assuming access to $\mathcal { H }$ , the space of concepts sampled and filtered from the grammar $\mathcal { G }$ , we use various heuristics to produce the generalization splits in the paper: + +• Instance IID: This split is trivial since $\mathcal { H } _ { \mathrm { t r a i n } } = \mathcal { H } _ { \mathrm { t e s t } } = \mathcal { H }$ +• Concept IID: This split divides concepts into train and test by picking concepts at random from $\mathcal { H }$ and assigning them to $\mathcal { H } _ { \mathrm { t r a i n } }$ or $\mathcal { H } _ { \mathrm { t e s t } }$ while ensuring that no two concepts which are synonyms Appendix B.6 are found in different splits. +• Boolean: This split forms cross product of all possible colors and $\{ \harpoonright , \ \mathrm { o r } \}$ boolean operators, and partitions a subset of such combinations which we want to only occur in test. We use the following tokens for test: + +![](images/44c0a4f91414e41f7139ec4f48443841663a8cffa1f32381163d8b0af7eee6eb.jpg) +igure 13: Histogram of number hypotheses with same evaluation signatures. + +‘green’, ‘or’ | ‘purple’, ‘and’ | ‘cyan’, ‘and’ | ‘red’, ‘or’ | ‘green’, ‘and’ + +We then create $\mathcal { H } _ { \mathrm { t e s t } }$ to contain all concepts which have any of the combinations above. For example, if a concept has both green and or we would place it in ${ \mathcal { H } } _ { \mathrm { t e s t } }$ . After every feasible candidate is placed in $\mathcal { H } _ { \mathrm { t e s t } }$ based on this heurisitc, the remaining concepts in $\mathcal { H }$ are assigned to ${ \mathcal { H } } _ { \operatorname { t r a i n } }$ . + +• Extrinsic: This split forms cross product of all possible colors and locations in the dataset, and partitions a subset of such combinations that we only want to occur in test. We use the following tokens for test (only a subset shown for illustration): + +$$ +\begin{array} { r l } & { \mathrm { ~ \gamma ~ } _ { \mathrm { J } } ^ { \prime \prime } , \quad \mathrm { ~ \gamma ~ } _ { \mathrm { J } } ^ { \mathrm { v } } \mathrm { a y } ^ { \prime } , \quad | \mathrm { ~ \gamma ~ } _ { \mathrm { L } } ^ { \mathrm { v } } { \mathrm { ~ \gamma ~ } } _ { \mathrm { ~ \gamma ~ } } ^ { \prime } , \quad \mathrm { ~ \gamma ~ } _ { \mathrm { c y a n } ^ { \prime } } ^ { \mathrm { v } } , \quad | \mathrm { ~ \gamma ~ } _ { \mathrm { S } } ^ { \mathrm { v } } { \mathrm { ~ \gamma ~ } } _ { \mathrm { ~ \gamma ~ } } ^ { \mathrm { v } } , \quad \mathrm { \ p u r p } { \mathrm { ~ \gamma ~ } } _ { \mathrm { J } } ^ { \mathrm { v } } } \\ & { \mathrm { ~ \gamma ~ } _ { \mathrm { J } } ^ { \mathrm { v } } , \quad \mathrm { ~ \gamma ~ } _ { \mathrm { J } } ^ { \mathrm { v } } \mathrm { e e } ^ { \prime } , \quad | \mathrm { ~ \gamma ~ } _ { \mathrm { ~ \gamma ~ } } ^ { \mathrm { v } } { \mathrm { ~ \gamma ~ } } _ { \mathrm { J } } ^ { \mathrm { v } } , \quad \mathrm { ~ \gamma ~ } _ { \mathrm { v y a n } ^ { \prime } } ^ { \mathrm { v } } , \quad | \mathrm { ~ \gamma ~ } _ { \mathrm { v } } ^ { \mathrm { v } } { \mathrm { ~ \gamma ~ } } _ { \mathrm { J } } ^ { \mathrm { v } } , \quad \mathrm { ~ \gamma ~ } _ { \mathrm { p u r p } { \mathrm { ~ \gamma ~ } } } ^ { \mathrm { v } } | \mathrm { e } ^ { \prime } , | } \\ & { \mathrm { ~ \gamma ~ } _ { \mathrm { J } } ^ { \mathrm { v } } , \quad \mathrm { ~ \gamma ~ } _ { \mathrm { b } } ^ { \mathrm { v } } \mathrm { l u e } ^ { \prime } , \quad | \mathrm { ~ \gamma ~ } _ { \mathrm { S } } ^ { \mathrm { v } } { \mathrm { ~ \gamma ~ } } _ { \mathrm { v } } ^ { \mathrm { v } } \mathrm { a n } ^ { \prime } } \end{array} +$$ + +We then create $\mathcal { H } _ { \mathrm { t e s t } }$ to contain all concepts which have any of the combinations above. For example, if a concept has both gray and 7, and is related to location, that is contains locationX? or locationY? keywords, we would place it in ${ \mathcal { H } } _ { \mathrm { t e s t } }$ . After every feasible candidate is placed in ${ \mathcal { H } } _ { \mathrm { t e s t } }$ based on this heurisitc, the remaining concepts in $\mathcal { H }$ are assigned to ${ \mathcal { H } } _ { \mathrm { t r a i n } }$ . + +• Intrinsic: This split forms cross product of all possible colors and materials in the dataset, and partitions a subset of such combinations that we only want to occur in test. We use the following tokens for test: + +‘green’, ‘metal’ | ‘purple’, ‘rubber’ | ‘cyan’, ‘rubber’ | ‘red’, ‘metal’ | ‘green’, ‘rubber’ + +We then create $\mathcal { H } _ { \mathrm { t e s t } }$ to contain all concepts which have any of the combinations above. For example, if a concept has both green and metal, and is related to material, that is contains material? keyword, we would place it in $\mathcal { H } _ { \mathrm { t e s t } }$ . After every feasible candidate is placed in $\mathcal { H } _ { \mathrm { t e s t } }$ based on this heurisitc, the remaining concepts in $\mathcal { H }$ are assigned to $\mathcal { H } _ { \mathrm { t r a i n } }$ • Binding (color): This split takes all possible colors in the dataset, and partitions a subset of colors that we only want to occur in test. We use the following tokens for test: + +We then create ${ \mathcal { H } } _ { \mathrm { t e s t } }$ to contain all concepts which have any of the tokens above. For example, if a concept has purple, we would place it in $\mathcal { H } _ { \mathrm { t e s t } }$ . After every feasible candidate is placed in $\mathcal { H } _ { \mathrm { t e s t } }$ based on this heurisitc, the remaining concepts in $\mathcal { H }$ are assigned to $\mathcal { H } _ { \mathrm { t r a i n } }$ . + +• Binding (shape): This split takes all possible shapes in the dataset, and partitions a subset of shapes that we only want to occur in test. We use the following tokens for test: + +‘cylinder’ + +We then create ${ \mathcal { H } } _ { \mathrm { t e s t } }$ to contain all concepts which have any of the tokens above. For example, if a concept has cylinder, we would place it in $\mathcal { H } _ { \mathrm { t e s t } }$ . After every feasible candidate is placed in $\mathcal { H } _ { \mathrm { t e s t } }$ based on this heurisitc, the remaining concepts in $\mathcal { H }$ are assigned to $\mathcal { H } _ { \mathrm { t r a i n } }$ . + +• Complexity: This split partitions into train and test based on length of the postfix serialization of the concept. Specifically, concepts shorter than 10 tokens are placed in $\mathcal { H } _ { \mathrm { t r a i n } }$ and longer concepts are placed in ${ \mathcal { H } } _ { \mathrm { t e s t } }$ . + +# D CREATING SUPPORT AND QUERY SETS + +We next explain how we go from the initial dataset $\mathcal { U } -$ which contains a large number of images, schema and sounds – and a concept space $\mathcal { H } _ { \mathrm { t r a i n } }$ and ${ \mathcal { H } } _ { \mathrm { t e s t } }$ , to a dataset for meta learning. To create the training/validation/test sets for models, we sample a series of episodes, each containing a support set and a query set. We illustrate the sampling procedure for a training episode below: + +# Support Set Sampling with Hard Negatives + +1. Pick a concept $h \sim p _ { \mathrm { t r a i n } } ( h )$ , with a preference for shorter hypotheses being more frequent based on the weights used to define the prior Appendix B.3 +2. Pick 5 images $( P )$ , uniformly at random from $\mathcal { U }$ such that $h ( \mathbf { u } ^ { s } ) = 1$ , where the concept is evaluated on the schema to determine the label Appendix B.4 +3. Identify other concepts $h ^ { \prime } \in \mathcal { H }$ s.t. $h ( u ( S ) ) = 1$ and $h ^ { \prime } \neq h$ +4. Pick images such that $h ^ { \prime } ( \mathbf { u } ^ { s } ) = 1$ and $h ( \mathbf { u } ^ { s } ) = 0$ as negatives $( N )$ . If no such images exist, pick random images from $\mathcal { U }$ as negatives until we have 20 negatives. +5. Return $D _ { \operatorname { s u p p } } = P \cup N$ . + +The sampling procedure for the Query set iterates all the steps above (except step 1, where we choose the concept $h$ ). Step 3 and 4 outline a procedure for identifying hard negatives for training the model, by looking at other hypotheses which also explain a chosen set of positives $P$ and using them to clarify what the concept of interest is. + +We give below an analogous procedure for easy negatives: + +# Support Set Sampling with Easy Negatives + +1. Pick a concept $h \sim p _ { \mathrm { t r a i n } } ( h )$ , with a preference for shorter hypotheses being more frequent based on the weights used to define the prior Appendix B.3 +2. Pick 5 images $( P )$ , uniformly at random from $\mathcal { U }$ such that $h ( \mathbf { u } ^ { s } ) = 1$ , where the concept is evaluated on the schema to determine the label Appendix B.4 +3. Pick 20 random images from $\mathcal { U }$ as negatives, $\mathbf { N }$ . +4. Return $D _ { \operatorname { s u p p } } = P \cup N$ . + +Similar to hard negatives, the sampling procedure for the Query set iterates all the steps above (except step 1, where we choose the concept $h$ ). + +# E REPRODUCIBILITY AND HYPERPARAMETERS + +For all the models in Figure. 4 in the main paper, we use the following hyperparameters. All the modalities are processed into a set of objects $\ \bar { \{ o _ { i } \} } _ { i = 1 } ^ { N }$ where each $o _ { i } \in \mathbb { R } ^ { { \overline { { 6 4 } } } }$ for image and sound models while for schema $o _ { i } \in \mathbb { R } ^ { 9 6 }$ . Further, the we use a learning rate of $\beth - 4$ for image models, $\beth \in - 3$ for schema models, and $5 \in - 5$ using the best learning rate for each modality across an initial sweep. The batch size for image and sound models is 8 episodes per batch, while for schema we use a batch size of 64. All models use the Adam optimizer. The overall scene representation across all the modalities is set to 256, that is, $ { \mathbf { u } } \in \mathbb { R } ^ { 2 5 6 }$ . All our models are initialized with the method proposed in (Glorot & Bengio, 2010), which we found to be crucial for training relation networks well. The initial representation from the first stage of the encoder (Fig. 4 in main paper) with the objects for images has a size of $\mathsf { 1 0 x 8 }$ i.e. there are 80 objects, while for sound representations have 38 objects. In the schema case the number of objects is the ground truth number of objects which is provided as input to the model. + +We trained all the models on the training set, and picked the best performing checkpoints on training – measured in terms of mAP– to report all the results in the main paper. Our image and schema models are trained for 1 million steps (16 epochs for images, 128 epochs for schemas) while the sound models are trained for 500K steps, and checkpoints are stored after every 30K steps. + +All our models fit on a GPU with 16GB capacity except the relation network trained with image inputs, which needs at 32GB GPU. We use the pytorch framework to implement all our models. + +# F MODEL ARCHITECTURES FOR POOLING + +In this section we detail the exact architectures used for the different pooling operations we consider in this paper as shown in Figure 4 center panel. We first establish some notation. Let $o _ { i } \in \mathbb { R } ^ { K }$ be the output object feature from the modality specific encoder (Figure 4, left panel), and let us denote by $O = \{ o _ { i } \} _ { i = 1 } ^ { | O | }$ the set of features for each of the objects in the scene, which includes optional position information indicating where the object is present in the scene (Figure 4). Let $N$ be the requested dimensionality of the feature space from the pooling operation. Given this, we can describe the pooling operations used as follows: + +• avg-pool: We first average the representations across all the objects $\{ o _ { i } \} _ { i = 1 } ^ { | O | }$ and then pass the averaged representation through an MLP with $2 5 6 \texttt { x } 5 1 2 \texttt { x } 3 8 4 \texttt { x N }$ units with batch normalization and rectified linear unit nonlinearity in the hidden layers. +concat: We first concatenate all the object representations in $O$ , followed by an MLP with $2 5 6 ~ \mathrm { ~ x ~ } ~ 5 1 2 ~ \mathrm { ~ x ~ } ~ 2 5 6 ~ \mathrm { ~ x ~ } ~ \mathrm { ~ N ~ }$ units with batch normalization and rectified linear units nonlinearity in the hidden layers. +relation-net: For relation networks, following (Santoro et al., 2017) we use relative position encoding that captures the relative positioning of the objects in a scene for image and sound modalities, and use the location information already present in the schema modality. Based on this, in the terminology of Santoro et al. (2017) our $g ( \ u )$ MLP has $2 5 6 ~ \times ~ 2 5 6 ~ \times ~ 2 5 6$ $\texttt { x } \ 2 5 6$ hidden units with rectified linear unit non linearity and batch normalization whereas our $f ( )$ MLP has $2 5 6 ~ \mathrm { ~ x ~ } ~ 2 5 6 ~ \mathrm { ~ x ~ N ~ }$ units with recitifed linear unit non linearlity and batch normalization in the middle (non-output) layers. Different from the original paper, we do not use dropout as we did not observe any overfitting in our experiments. +transformer: We use a 2-head multi-head attention layer stacked 4 times, with the feedforward network dimenstions set to 512. After forwarding through this module, we take the output vectors $o _ { i } ^ { \prime }$ for each object processed through these initial layers and pool across objects by doing max(), mean(), sum(), min() operations and concatenating their outputs, similar to previous work by Wang et al. (2019). The final representation then does a linear projection of this concatenated vector to $N$ , the dimensionality expected from the pooling module. + +# G ADDITIONAL RESULTS + +# G.1 HYPERPARAMETER SWEEPS – OBJECT FEATURE DIMENSIONS + +We next show the hyperparameter sweeps for image models in deterimining the choice of the dimensionality to represent each object $o _ { i }$ for our image models (Figure 14). The same choice of dimensionality was replicated for sound models. In our initial sweeps, on the Concept IID split, across different choices of the dimensionality of objects, we found relation networks to outperform concat and global average pooling models substantially, and thus we picked the object dimensions based on what performs best for relation networks since overall we are interested in the best possible choice of models for a given split and modality. Based on the results in Figure 14 we picked $\boldsymbol { \partial } _ { i } \in \mathbb { R } ^ { 6 4 }$ . + +![](images/bf6a77c2a6709f6ce6e6adbfe162da677142fe8242a7f9fdec34d5e5babb42d4.jpg) +Figure 14: mAP on validation for hard negatives (y-axis) vs number of training steps $\mathbf { \bar { x } }$ -axis) for relation network models on images with different dimensionality for the object embedding $o _ { i }$ . + +![](images/8792990cb4db47ca305e666971ad1dbd79120b4c495ea476a7e83107a22bfff9.jpg) +Figure 15: mAP on validation for hard negatives (y-axis) vs number of training steps $\mathbf { \check { x } }$ -axis) for relation network models on images with different learning rates. + +# G.2 IMAGE RELATION NETWORKS LEARNING RATE SWEEPS + +We picked the learning rate for image models based on the best performing image relation network model, which across an initial sweep we found to yeild the best class of models. Figure 15 shows the performance of the models across learning rates of $\{ 1 { \mathrm { e } } { - } 4 , \quad 5 { \mathrm { e } } { - } 4 , \quad 2 . 5 { \mathrm { e } } { - } 4 \}$ . + +# G.3 SWEEP ON USE OF LANGUAGE + +As explained in the main paper (Figure. 4), the parameter $\alpha$ controls the tradeoff between the query accuracy and the likelihood of the concept expressed as a prefix string. We generally found across a broad range of values in $\{ \boldsymbol { 0 } . \boldsymbol { 0 } , 0 . 0 1 , 0 . 1 \boldsymbol { 0 } , 1 . 0 \}$ the models generally performed the best at $\alpha = 1 . 0$ . Our initial experiments with $\alpha = 1 0 . 0$ suggested substantially worse performance so we discareded it from the sweep. See Figures 16 to 18 for the corresponding results. + +![](images/e9bf606123ff5f6f556c7e2091ebdef2828453c8304b3070a9b6a2cbee43b705.jpg) +Figure 16: mAP on validation for hard negatives (y-axis) vs number of training steps ( $\mathbf { \dot { x } }$ -axis) for relation network models on images with different amounts of language usage by varying the parameter $\alpha$ . + +![](images/c19322b9e649e0a16c2e6020f8b351c538174e76228d31d8fd5271b29f1d618f.jpg) +Figure 17: mAP on validation for hard negatives (y-axis) vs number of training steps $\mathbf { \bar { x } }$ -axis) for relation network models on schemas with different amounts of language usage by varying the parameter $\alpha$ . + +![](images/46b42656f77da39ba1de9d0cb18b1e13567d2412ee84e050fdb73f8279e66e4a.jpg) +Figure 18: mAP on validation for hard negatives (y-axis) vs number of training steps $\mathbf { \dot { x } }$ -axis) for concat pooling models on schemas with different amounts of language usage by varying the parameter $\alpha$ . + +![](images/926daed86f654d1c15bce3718d9a58607179ab3f387dd0059eb7185d13f96b32.jpg) +Figure 19: mAP (top) and accuracy (bottom) metrics for the different splits presented in the paper when easy negatives are used (ordered by their corresponding comp gap). Yellow: shows the schema relation-net model while purple: shows the image relation-net model. Notice that compared to Fig. 5 in the main paper, the comp gap is smaller and the models appear to be substantially closer to the strong oracle in this setting compared to when we use hard negatives. + +# G.4 RESULTS ON EASY NEGATIVES + +In Figure 19 we show results for the relation-net model on various splits, where easy negatives are used to populate the support and query sets during training and evaluation, unlike the case of hard negatives discussed in the main paper (Figure. 5). Notice that the compositionality gap (comp gap) is lower in general for easy negatives compared to the hard negatives as reported in the main paper. Further, we find that the best models are substantially closer to the strong oracle compared to Figure. 5 main paper, showing that on the easier, less compositional task it is easier for machine learning models to approach the strong oracle (especially in terms of accuracy). Finally, it is interesting to note that with easy negatives it appears that the best models outperform the weak oracle on the Counting split, while with the hard negatives one finds that the models are worse than the weak oracle, suggesting poor generalization for counting. + +# G.5 FINER $\alpha$ SWEEP FOR COUNTING + +Finally, we ran a finer alpha sweep for the Counting split since it appeared on our initial sweep that the counting split was not performing better with language. Concretely, we ran a new set of experiments sweeping over $\alpha$ values of $\{ \boldsymbol { 0 } , 0 1 , \quad 0 . 1 0 , \quad 1 . 0 , \quad 5 . 0 , \quad 1 0 . 0 , \quad 1 0 0 . 0 \}$ . Across this broader range of values, we found models still did not show any statistically significant gains from using language v.s. not for the Counting split. + +# G.6 CHOICE OF METRIC: MAP v.s. ACCURACY + +In general, the mAP metric opens up a larger comp gap for the various splits than indicated by CBA. For example, with hard negatives, while CBA indicates a gap of $1 4 . 2 \%$ for Counting compared to $0 \%$ for Instance IID, mAP suggests a gap of $3 4 . 4 \%$ for Counting relative to $0 \%$ for Instance IID. For the Binding (color) split its $8 6 . 5 \%$ comp gap (mAP) v.s. $3 4 . 0 \%$ for CBA. mAP, while being more expensive to compute evaluates more thoroughly to test if a concept $h$ is truly learnt by the model, by probing its performance on a large, representative set of negatives $\tau$ , providing a more stringent test of compositional generalization. + +Table 1: Performance on meta-test, sorted based on mAP $( \mathrm { i n } \% )$ on Binding (color) with hard negatives + +
modalitypoolingmAP (mean)mAP (std)
schematransformer15.90.9
schemarelation-net15.30.9
imageavg-pool15.10.4
imagerelation-net14.80.7
schemaavg-pool14.41.2
imagetransformer14.20.2
schemaconcat14.00.7
imageconcat13.50.4
soundavg-pool9.40.6
soundrelation-net9.20.3
soundconcat8.00.6
+ +Table 2: Performance on meta-test, sorted based on mAP (in $\%$ ) on Boolean with hard negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net51.11.4
schemaavg-pool48.02.1
schemaconcat47.12.1
schematransformer46.91.1
imagetransformer44.02.2
imagerelation-net38.41.0
imageavg-pool36.32.3
imageconcat29.62.4
soundrelation-net25.50.8
soundavg-pool24.11.6
soundconcat22.31.1
+ +G.7 DETAILED RESULTS ON ALL THE SPLITS IN HARD NEGATIVES SETTING + +In this section we provide the full results of all of the tested models on each of the splits considered in the paper, in the hard negatives setting. Tables 1-9 show the results of different models (sorted in a descending order based on mAP for each of the splits considered in the paper, in the case where models do not have access to language. + +# G.8 DETAILED RESULTS ON ALL THE SPLITS IN EASY NEGATIVES SETTING. + +In this section we provide the full results of all of the tested models on each of the splits considered in the paper, in the easy negatives setting. Tables 10-18 show the results of different models (sorted in a descending order based on mAP for each of the splits considered in the paper, in the case where models do not have access to language. Note that we did not evaluate transformer models or sound models in this setting as this is qualitatively less interesting than the hard negatives setting and is not the main focus of the paper. + +Table 3: Performance on meta-test, sorted based on mAP (in $\%$ ) on Intrinsic with hard negatives + +
modalitypoolingmAP (mean)mAP (std)
schematransformer57.90.6
schemarelation-net55.10.9
imagetransformer54.13.0
schemaavg-pool53.33.0
schemaconcat52.23.5
imagerelation-net47.67.5
imageavg-pool34.50.4
imageconcat34.01.0
soundconcat29.41.6
soundavg-pool26.52.3
soundrelation-net24.32.4
+ +Table 4: Performance on meta-test, sorted based on mAP (in $\%$ ) on Concept IID with hard negatives + +
modalitypoolingmAP (mean)mAP (std)
imagetransformer60.80.4
schemarelation-net60.70.3
schematransformer59.80.2
imagerelation-net58.50.7
imageavg-pool56.71.0
schemaavg-pool54.61.3
schemaconcat53.30.9
imageconcat53.00.6
soundconcat23.62.7
soundavg-pool22.72.3
soundrelation-net21.8NaN
+ +Table 5: Performance on meta-test, sorted based on mAP $( \mathrm { i n } \%$ ) on Instance IID with hard negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net63.90.4
imagetransformer61.50.9
schematransformer59.80.5
imagerelation-net58.61.3
imageavg-pool58.10.7
schemaavg-pool57.40.9
imageconcat57.10.9
schemaconcat57.11.4
soundavg-pool23.51.4
soundrelation-net22.32.3
soundconcat21.70.5
+ +Table 6: Performance on meta-test, sorted based on mAP (in $\%$ ) on Extrinsic with hard negatives + +
modalitypoolingmAP (mean)mAP (std)
imagetransformer62.10.8
schematransformer60.90.6
imagerelation-net60.81.1
imageconcat60.80.7
schemarelation-net59.80.4
imageavg-pool59.41.3
schemaconcat54.32.3
schemaavg-pool53.41.5
soundavg-pool25.7NaN
soundconcat23.21.4
soundrelation-net18.31.7
+ +Table 7: Performance on meta-test, sorted based on mAP (in $\%$ ) on Complexity with hard negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net52.80.7
schematransformer49.70.6
schemaavg-pool48.41.0
schemaconcat48.01.2
imageavg-pool45.80.4
imagetransformer45.60.2
imagerelation-net45.40.4
imageconcat42.63.4
soundconcat26.0NaN
soundrelation-net23.31.0
soundavg-pool23.01.7
+ +Table 8: Performance on meta-test, sorted based on mAP (in $\%$ ) on Binding (shape) with hard negatives + +
modalitypoolingmAP (mean)mAP (std)
schematransformer30.90.6
schemarelation-net30.41.5
imagerelation-net29.11.8
imageavg-pool28.71.0
imagetransformer28.70.1
schemaavg-pool28.31.9
imageconcat27.00.8
schemaconcat27.00.8
soundrelation-net17.50.4
soundconcat17.41.3
soundavg-pool16.21.6
+ +Table 9: Performance on meta-test, sorted based on mAP (in $\%$ ) on Counting with hard negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net57.44.8
schemaavg-pool55.06.4
schemaconcat50.03.2
imageavg-pool48.51.3
schematransformer40.70.4
imagerelation-net40.32.0
imageconcat38.52.8
imagetransformer32.41.4
soundconcat13.82.4
soundavg-pool13.41.0
soundrelation-net13.01.7
+ +Table 10: Performance on meta-test, sorted based on mAP (in $\%$ ) on Binding (color) with easy negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net18.61.6
imagerelation-net16.30.8
imageavg-pool16.30.5
imageconcat15.60.5
schemaavg-pool15.50.5
schemaconcat15.50.5
+ +Table 11: Performance on meta-test, sorted based on mAP (in $\%$ ) on Boolean with easy negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net58.12.0
schemaavg-pool55.02.1
schemaconcat54.72.2
imagerelation-net50.77.3
imageavg-pool42.91.6
imageconcat40.91.7
+ +Table 12: Performance on meta-test, sorted based on mAP (in $\%$ ) on Counting with easy negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net67.98.5
schemaavg-pool64.47.3
imageavg-pool62.26.6
imageconcat61.26.9
schemaconcat60.86.7
imagerelation-net58.85.7
+ +Table 13: Performance on meta-test, sorted based on mAP (in $\%$ ) on Extrinsic with easy negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net63.71.4
imagerelation-net62.91.1
imageconcat61.82.4
imageavg-pool61.51.4
schemaavg-pool57.51.2
schemaconcat57.21.5
+ +Table 14: Performance on meta-test, sorted based on mAP (in $\%$ ) on Intrinsic with easy negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net66.54.8
imagerelation-net64.95.3
imageavg-pool64.44.5
schemaavg-pool63.14.7
imageconcat62.74.5
schemaconcat60.84.6
+ +Table 15: Performance on meta-test, sorted based on mAP (in $\%$ ) on Concept IID with easy negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net68.21.0
imagerelation-net65.80.4
imageavg-pool65.10.4
imageconcat64.70.4
schemaavg-pool63.30.4
schemaconcat61.90.2
+ +Table 16: Performance on meta-test, sorted based on mAP (in $\%$ ) on Instance IID with easy negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net69.03.6
imageavg-pool66.83.3
imagerelation-net66.53.2
imageconcat65.43.4
schemaavg-pool63.53.1
schemaconcat63.33.1
+ +Table 17: Performance on meta-test, sorted based on mAP $( \mathrm { i n } \% )$ ) on Complexity with easy negatives + +
modalitypoolingmAP (mean)mAP (std)
schemarelation-net55.11.8
schemaavg-pool51.92.0
schemaconcat51.31.9
imageavg-pool49.01.6
imageconcat48.21.7
imagerelation-net46.61.5
+ +Table 18: Performance on meta-test, sorted based on mAP (in $\%$ ) on Binding (shape) with easy negati + +
modalitypoolingmAP (mean)mAP (std)
imagerelation-net33.92.4
schemaavg-pool32.11.7
imageavg-pool31.91.8
schemarelation-net31.71.6
imageconcat31.31.7
schemaconcat31.01.5
\ No newline at end of file diff --git a/parse/train/LuyryrCs6Ez/LuyryrCs6Ez_content_list.json b/parse/train/LuyryrCs6Ez/LuyryrCs6Ez_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..a879167541f3f362a234ff4c62a190e94169c915 --- /dev/null +++ b/parse/train/LuyryrCs6Ez/LuyryrCs6Ez_content_list.json @@ -0,0 +1,3151 @@ +[ + { + "type": "text", + "text": "CURI: A BENCHMARK FOR PRODUCTIVE CONCEPT LEARNING UNDER UNCERTAINTY ", + "text_level": 1, + "bbox": [ + 176, + 99, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Humans can learn and reason under substantial uncertainty in a space of infinitely many concepts, including structured relational concepts (“a scene with objects that have the same color”) and ad-hoc categories defined through goals (“objects that could fall on one’s head”). In contrast, standard classification benchmarks: 1) consider only a fixed set of category labels, 2) do not evaluate compositional concept learning and 3) do not explicitly capture a notion of reasoning under uncertainty. We introduce a new few-shot, meta-learning benchmark, Compositional Reasoning Under Uncertainty (CURI) to bridge this gap. CURI evaluates different aspects of productive and systematic generalization, including abstract understandings of disentangling, productive generalization, learning boolean operations, variable binding, etc. Importantly, it also defines a model-independent “compositionality gap” to evaluate difficulty of generalizing out-of-distribution along each of these axes. Extensive evaluations across a range of modeling choices spanning different modalities (image, schemas, and sounds), splits, privileged auxiliary concept information, and choices of negatives reveal substantial scope for modeling advances on the proposed task. All code and datasets will be available online. ", + "bbox": [ + 232, + 263, + 766, + 486 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 510, + 336, + 526 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Human concept learning is more flexible than today’s AI systems. Human conceptual knowledge is productive: people can understand and generate novel concepts via compositions of existing concepts (“an apartment dog”) (Murphy, 2002), unlike standard machine classifiers that are limited to a fixed set of classes (“dog”, “cat”, etc.). Further, humans can induce goal-based, “ad hoc” categories such as “things to take from one’s apartment in a fire” (children, dogs, keepsakes, etc.) (Barsalou, 1983). Thus, unlike AI systems, humans reason seamlessly in large, essentially “unbounded” concept spaces. ", + "bbox": [ + 173, + 540, + 825, + 625 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Beyond unboundedness, a natural challenge in such concept spaces is uncertainty – the right concept to be inferred is uncertain, as a plethora of candidate concepts could explain observations. For e.g. in Figure 1 (top, image panel), the “right” concept could be that “All objects are blue and have the same size”, but it could also be “There are less than four objects in the scene”, or “All objects have the same color”. Humans gracefully handle such uncertainty and underdetermination (Tenenbaum & Griffiths, 2001; Xu & Tenenbaum, 2007; Goodman et al., 2008; Piantadosi et al., 2016). Popular compositional reasoning benchmarks such as CLEVR (Johnson & Zhang, 2016) for visual question answering and Ravens Progressive Matrices (Santoro et al., 2017) for deductive, analogical reasoning are compositionally rich and challenging in nature, but do not tackle ambiguity and underdetermination. ", + "bbox": [ + 174, + 631, + 825, + 756 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We address this gap in the literature, and propose the Compositional Reasoning Under Uncertainty (CURI) benchmark to study how modern machine learning systems can learn concepts spanning a large, productively defined space (Figure 1). In pursuit of this goal, we instantiate a meta learning task where a model must acquire a compositional concept from finite samples. A signature of productivity in human thought is our ability to handle novel combinations of known, atomic components. Thus, in CURI we instantiate different systematic train-test splits to analyze different forms of generalization in concept learning, involving novel combinations of intrinsic properties (e.g. color, shape) with boolean operators, counting, extrinsic object properties (e.g. object location), and a novel test of variable binding in context of compositional learning. ", + "bbox": [ + 174, + 762, + 825, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "While related systematic splits have been proposed in prior work in context of other tasks such as question answering and analogical reasoning (Barrett et al., 2018; Hill et al., 2019; Agrawal et al., ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/97c8b1053b99d529776e044cb9e1891e2b2b0025d94aba8cabb7e42abef57b5b.jpg", + "image_caption": [ + "Figure 1: Concept Space. Three example concepts (rows) along with schematic positive examples. Actual scenes are rendered in multiple ways including the CLEVR renderer (Johnson et al., 2016) (see Figure 2). Right: The grammar of variables, quantifiers, functions and operators to induce compositional concepts. " + ], + "image_footnote": [], + "bbox": [ + 174, + 99, + 823, + 279 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2017; Johnson et al., 2016; Vedantam et al., 2017; Higgins et al., 2017; Bakhtin et al., 2019; Lake & Baroni, 2018; Ruis et al., 2020), ours is the first benchmark which tests different qualitative aspects of reasoning about productive concepts under uncertainty. ", + "bbox": [ + 174, + 325, + 826, + 367 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Compositional Reasoning Under Uncertainty (CURI) Task. Concretely, the CURI task tests few-shot learning of relational concepts in a large compositional conceptual space, with design inspiration from studies in cognitive modeling using a language of thought (LOT) approach (Fodor, 1975; Piantadosi, 2011; Kemp et al., 2005). CURI includes scene-based concepts such as “All objects have the same color” and “There exists a blue object while the rest are triangles” (Figure 1) but unlike CLEVR (Johnson et al., 2016) there are too few examples to deduce answers with certainty. Our benchmark is defined through a series of meta-learning episodes (see example in Figure 2): given positive and negative examples of a new concept $D _ { \\mathrm { s u p p } }$ (known as the “support set”), the goal of an episode is to classify new examples $D _ { \\mathrm { q u e r y } }$ (the “query set”). As in few-shot classification (Fei-Fei et al., 2006), meta-learning (Vinyals et al., 2016), and other open-set tasks (Lampert et al., 2014), models are evaluated on novel classes outside the (meta-)training set. Unlike previous work (Triantafillou et al., 2019; Lake et al., 2019) that focuses on atomic concepts, our benchmarks concerns more structured, relational concepts built compositionally from a set of atomic concepts, and involves reasoning under uncertainty – an ideal learner must marginalize over many hypotheses when making predictions (Gelman et al., 2004; Xu & Tenenbaum, 2007; Piantadosi et al., 2016). ", + "bbox": [ + 173, + 375, + 826, + 583 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We also vary the modality in which scenes are presented—rendering them as images, symbolic schemas, and sounds— enabling future research on modality-specific representational choices for compositional reasoning under uncertainty. Finally, we vary the concepts learned by the model during meta-training and meta-testing to test different aspects of systematic generalization. ", + "bbox": [ + 174, + 604, + 485, + 715 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Compositionality Gap. In addition to defining systematic splits, we also characterize (for the first time, in our knowledge), the difficulty of ", + "bbox": [ + 174, + 722, + 483, + 763 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/bcbd5ef86b8117329029374b7393e74ddac4a499b1f454de03a928ce74d8a8ce.jpg", + "image_caption": [ + "Figure 2: CURI Task. Given as input a support set $D _ { \\mathrm { s u p p } }$ , with positive and negative examples corresponding to concept, the model has to infer the concept and produce accurate predictions on novel images (right). " + ], + "image_footnote": [], + "bbox": [ + 503, + 595, + 815, + 684 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "generalization entailed by each split by introducing the notion of a model-independent “compositionality gap”. Concretely, the compositionality gap is the difference in test performance between an ideal Bayesian learner with access to the full hypothesis space, and a Bayesian learner with access to only a (potentially large) list of the hypotheses examined during meta-training. A large gap indicates that any learner must extrapolate compositionally from the training hypotheses to solve the task; additionally, models can be compared to ideal learners that either do or do not engage in such extrapolation. We anticipate that this tool will be more broadly useful for analyzing other benchmarks with compositional splits. ", + "bbox": [ + 174, + 763, + 825, + 875 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Models. We evaluate models around various dimensions which concern the difficulty of learning productive concepts under uncertainty, including: 1) the modality in which the input is rendered (image, schemas, sounds), 2) method used for reasoning across objects in a scene (transformer, relation-network, global average pooling, concatenation), 3) whether or not training provides groundtruth symbolic descriptions of concepts, and 4) how negative examples are sampled. Overall, our evaluations suggest that there is substantial room for improvement in compositional reasoning under uncertainty, w.r.t the compositionality gap, representing a novel challenge for compositional learning. ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/abd840279a188f051f7bac99c975cad4d32983a558c17d915300fd79ca71bc69.jpg", + "image_caption": [ + "Figure 3: Language of thought. All valid (type-consistent) compositions of functions are potential complex concepts in our dataset. Note that the functions are illustrated for the case of images and schemas. Location, size, shape etc. correspond to different properties for sounds. " + ], + "image_footnote": [], + "bbox": [ + 173, + 101, + 823, + 239 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 284, + 825, + 339 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Summary of contributions: 1) We introduce the Compositional Reasoning Under Uncertainty (CURI) benchmark for evaluating compositional, relational learning under uncertainty from observational data; 2) We introduce a ‘compositionality gap’ metric for measuring the difficulty of systematic generalization from train to test; 3) We provide various baseline models for benchmarking progess. ", + "bbox": [ + 174, + 347, + 825, + 402 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 422, + 344, + 439 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Compositional Learning. Related work has examined systematic generalization in pattern completion using Raven’s matrices (PGM) (Santoro et al., 2017; Hill et al., 2019) and visual question answering with CLEVR (Johnson et al., 2016; Bahdanau et al., 2019). CURI’s use of the CLEVR renderer further invites particular comparison with that benchmark. Compared to these more deductive reasoning tests, CURI examines few-shot concept learning under substantial inherent uncertainty. Unlike puzzle solving or question answering, an ideal inductive learner on CURI cannot know the right rule with certainty. In essence, unlike CLEVR the “question” to be answered is not given to the model as input, but must be inferred – making the task more challenging. While PGMs do involve such an inference, once the constraints of a puzzle are identified, it does not: 1) have any uncertainty in the reasoning (which is crucial) and 2) involve any “concept” learning – where a concept applies to multiple images – as much as it involves “instance” matching to complete a sequence. In contrast, a successful CURI model behaves as if marginalizing over many hypotheses consistent with the observations e.g., (Tenenbaum & Griffiths, 2001; Xu & Tenenbaum, 2007; Piantadosi et al., 2016), an ability which is rarely studied directly in deep learning models (although see (Grant et al., 2019)). ", + "bbox": [ + 174, + 454, + 825, + 648 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Recently, Keysers et al. (2019) proposed a method to create “difficult” systematic splits based on the principle that they should share atoms but have maximally different compositions. This is complementary to our splits, which provide interpretable notions of what each split tests such as disentangling, complexity, variable binding etc. Moreover, our variable binding split is predicated on having different atoms between train and test, and thus cannot be recovered by their methodology. ", + "bbox": [ + 174, + 655, + 825, + 724 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Language of Thought (LOT). Our choice of compositional concepts was most closely inspired by (Piantadosi et al., 2016) along with other studies of human concept learning in the Language of Thought (LOT) framework (Fodor, 1975; Goodman et al., 2008; Kemp & Jern, 2009; Piantadosi et al., 2012; Goodman et al., 2015; Overlan et al., 2017; Lake & Piantadosi, 2019). In typical LOT studies of human learning, the conceptual space $\\mathcal { H }$ is defined through a probabilistic context-free grammar $\\mathcal { G }$ , which specifies a set of conceptual primitives and their rules of combination. Here, we use a LOT-inspired grammar $\\mathcal { G }$ to generate an unbounded set concepts $\\mathcal { H }$ , while evaluating machine learning models trained without access to the underlying LOT. ", + "bbox": [ + 174, + 732, + 825, + 843 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 COMPOSITIONAL REASONING UNDER UNCERTAINTY (CURI) DATASET ", + "text_level": 1, + "bbox": [ + 173, + 863, + 799, + 880 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Concept space. The compositional concepts in CURI were inspired by the empirical and cognitive modeling work of Piantadosi et al. (2016). The space of concepts (LOT) is defined by a context free grammar $( { \\mathcal { G } } )$ . Figure 3 shows the LOT and specifies how primitives and functions compose to produce a large unbounded concept space. The LOT has three variables: $\\mathbf { x }$ , representing an object in a scene, $S = \\mathbf { \\bar { \\{ x \\} } } _ { i = 1 } ^ { N }$ representing the set of all objects in the scene, and $S _ { - \\mathbf { x } } \\overset { \\cdot } { = } S / \\{ \\mathbf { x } \\}$ , representing the set of all objects in the scene except x. Each concept describes a rule composed of object and scene properties, logical operators, and/or comparison operators, and can be evaluated on a given scene $S$ to determine whether the scene satisfies the rule. ", + "bbox": [ + 174, + 895, + 820, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 186 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Object and scene properties are defined by functions which can be applied to objects or scenes: for example, $\\mathsf { s i z e ? } ( \\mathbf { x } )$ yields the size of an object $\\mathbf { x }$ , while $\\mathsf { s i z e ? } ( S )$ returns a set with the sizes of all the objects $\\langle \\lnot \\mathrm { i } \\lnot \\mathbf { e } ? ( \\mathbf { x } ) : \\mathbf { x } \\in S \\}$ ). Comparison and logical operators can be used to compare and relate various properties of objects in scenes. In contrast to Piantadosi et al. (2016), we include a count operator, which determines how many times a condition is satisfied by a set, which allows us to check how well deep learning models are able to count (Chattopadhyay et al., 2016; Johnson et al., 2016; Agrawal et al., 2017). Finally, quantifiers such as exists and for-all enrich the LOT by specifying the number of objects which must satisfy a given condition. ", + "bbox": [ + 174, + 194, + 825, + 306 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Consider the following example concept (Figure 1 bottom): “There exists a blue object in the scene and the rest of the objects are squares.” To access the color of a given object, we use color? $\\mathbf { \\Psi } ( \\mathbf { x } )$ and to access the shape of a given object, we use shape? $\\mathbf { \\mu } ( \\mathbf { x } )$ . To determine whether an object matches a specific property, we can combine this with equality: shape? $\\begin{array} { r l r } { ( { \\bf x } ) } & { { } = } & { { \\bf w } _ { \\sf S q u a r e ^ { \\prime \\prime } } } \\end{array}$ . Finally, we can use exists to specify that at least one object must be blue, $S _ { - \\mathbf { x } }$ to specify all the objects except for that blue object, and all to specify that all the objects in $S _ { - \\bf { x } }$ must be squares. Putting it all together: exists $\\mathbf { x } \\in S$ $\\begin{array} { r l } { ( \\mathtt { c o l o r 2 } \\mathtt { ( x ) } } & { { } = \\begin{array} { l } { \\mathtt { w } _ { \\mathrm { b l u e } ^ { \\prime \\prime } } ; } \\end{array} } \\end{array}$ ) and all (shape? $( S _ { - { \\bf x } } ) ~ = ~ \\ \" { \\bf s } _ { \\mathrm { q u a r e } ^ { \\prime \\prime } ) }$ . ", + "bbox": [ + 174, + 311, + 825, + 410 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Structured Generalization Splits. A signature of productivity is the ability to handle novel combinations of known components (Fodor, 1975; Fodor & Pylyshyn, 1988). Thus, in CURI, we consider splits that require generalizing to novel combinations of known elements from our LOT (Figure 3), including combinations of constants, variables, and functions. We achieve this by creating disjoint splits of concepts $\\mathcal { H } _ { t r a i n }$ and $\\mathcal { H } _ { t e s t }$ for training and evaluating models. By varying the held out elements and their combinations, we obtain splits that evaluate different axes of generalization. In practice, we use our grammar $\\mathcal { G }$ to sample and filter a large set of concepts (see Appendix B.2 for more details), which yields a set of 14,929 concepts $\\mathcal { H }$ for training and evaluation. We next describe how each split divides $\\mathcal { H }$ into $\\mathcal { H } _ { t r a i n }$ and $\\mathcal { H } _ { t e s t }$ , to test productive, out of distribution generalization: ", + "bbox": [ + 173, + 416, + 826, + 542 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Instance IID: Evaluates generalization to novel episodes from the same concept set. This is the standard setup in machine learning (Murphy, 2013), in which $\\mathcal { H } _ { t r a i n } = \\mathcal { H } _ { t e s t }$ . This is the only split where train and test concepts overlap. Concept IID: Evaluates generalization to novel concepts based on an arbitrary random split of the concepts into $\\mathcal { H } _ { t r a i n }$ and $\\mathcal { H } _ { t e s t }$ .1 Counting: Evaluates the ability to learn a new concept $h$ with novel property-count combinations, e.g, the training concepts never filter for exactly $^ { \\bullet } 3$ squares’. Extrinsic properties: Evaluates the ability to learn a new concept $h$ , with novel combinations of extrinsic (e.g. location) and intrinsic (e.g. color) object properties. \nIntrinsic properties: Evaluates the ability to learn a new concept $h$ with novel combinations of intrinsic properties, e.g., the training concepts never reference both ‘red’ and ‘rubber’. Boolean operations: Evaluates the ability to learn concepts which require application of a familiar boolean operation to a property to which the operation has never been applied previously. \n• Complexity split: Evaluates generalization from simple concepts (those which have less than or equal to 10 symbols) to more complex concepts (longer than 10 symbols). This is indicative of the productivity (Fodor, 1975) exhibited by models, in generalizing from simpler concepts to more complex concepts. \nVariable binding: Evaluates learning of entirely novel intrinsic properties, e.g. the training concepts involve only “red”, “blue”, and “green” but test concepts involve “yellow” (although ‘yellow’ objects can still appear in training scenes). This is indicative of inferential coherence (Fodor, 1975) in models, in generalizing rules of inference to novel atoms. ", + "bbox": [ + 189, + 545, + 826, + 837 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A model that infers the underlying LOT during meta-training would be expected to perform well on any such systematic split. By comparing the performance of current models to to such ideal learners, this benchmark will allow us to evaluate progress on the systematic out-of-distribution generalization capabilities of our current models. Appendix C provides more details on the strucutred splits. ", + "bbox": [ + 178, + 848, + 823, + 877 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 821, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "From Concepts to Meta-learning Episodes. A single episode comprises a support set $( D _ { \\mathrm { s u p p } } )$ and a query set $( D _ { \\mathrm { q u e r y } } )$ , each of which is generated from a given concept, $h$ . Formally, a support or query set $D$ has input data $\\mathbf { u }$ and corresponding label $y$ , i.e. $D = \\{ \\{ y _ { i } \\} _ { i = 1 } ^ { N }$ , $\\{ { \\mathbf { u } } _ { i } \\} _ { i = 1 } ^ { N } \\}$ . Each support and query set contains 5 positive and 20 negative examples — negative examples are oversampled since the space of negatives is generally much larger than that for positives. The set of positive examples are sampled uniformly from a categorical distribution over all positives. However, we consider two types of negatives: 1) easy negatives, in which the negatives are also sampled at random, and 2) hard negatives, in which negatives are generated from a closely related concept which also evaluates true on the positive examples in $D _ { s u p p }$ , such that these negatives are maximally confusing. Altogether, for each split, our train, validation, and test sets contain 500000, 5000, and 20000 episodes, respectively. ", + "bbox": [ + 173, + 138, + 825, + 280 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Compositionality Gap. A key aspect of our benchmark is to define the difficulty in learning that arises from the compositional structure of the concept space. Most of the splits above are structured in a way such that $\\mathcal { H } _ { \\mathrm { t e s t } } \\cap \\mathcal { H } _ { \\mathrm { t r a i n } } = \\emptyset .$ – forcing a learner to use the compositional structure of the concept space to generalize to ${ \\mathcal { H } } _ { \\mathrm { t e s t } }$ . We conceptualize the difficulty of this task through the notion of its compositionality gap. Intuitively, the compositionality gap captures the difference between the generalization performance of an ideal compositional learner (strong oracle) compared to an ideal non-compositional learner that is unable to extrapolate outside the training concepts (weak oracle). ", + "bbox": [ + 174, + 286, + 825, + 383 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Formally, let $\\Omega ~ \\in ~ \\{ \\mathrm { s t r o n g } , \\mathrm { w e a k } \\}$ denote an oracle over a concept space $\\mathcal { H } _ { \\Omega }$ . The posterior predictive distribution of an oracle for query scene $\\mathbf { u }$ and query label $y \\in \\{ 0 , 1 \\}$ is then given as: $\\begin{array} { r l r } { p _ { \\Omega } ( y | \\mathbf { u } , D _ { \\mathrm { s u p p } } ) } & { = } & { \\sum _ { h \\in \\mathcal { H } _ { \\Omega } } \\bar { p _ { \\Omega } } ( y | h , \\mathbf { u } ) p _ { \\Omega } ( h | D _ { \\mathrm { s u p p } } ) } \\end{array}$ , where $p _ { \\Omega } ( h | D _ { \\mathrm { s u p p } } ) \\ d t \\propto$ $p _ { \\Omega } ( h ) p ( \\{ y _ { i } \\} _ { i = 1 } ^ { N } | h ; \\{ \\mathbf { u } _ { i } \\} _ { i = 1 } ^ { N } )$ and $p _ { \\Omega } ( h )$ denote the posterior and prior, respectively. Given a metric of interest $M$ (e.g., mean average precision or accuracy), the compositionality gap of a learning task is then simply defined as the difference in performance of the strong and weak oracle when evaluating on concepts from $\\mathcal { H } _ { \\mathrm { t e s t } }$ , i.e., $M ( p _ { \\mathrm { s t r o n g } } ) - M ( p _ { \\mathrm { w e a k } } )$ . ", + "bbox": [ + 174, + 391, + 825, + 493 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Using this notion of compositionality gap, we can then define ideal learners, i.e., the strong and weak oracle, simply via their priors. In particular, let $w ( h )$ denote a weight on importance of each hypothesis2 and let I denote the indicator function. We then define the prior of an oracle as $\\begin{array} { r } { p _ { \\Omega } ( h ) \\stackrel { } { = } \\sum _ { h ^ { \\prime } \\in \\mathcal { H } _ { \\Omega } } w ( h ^ { \\prime } ) { \\bf I } [ h ^ { \\prime } = h ] , } \\end{array}$ . The difference between strong and weak oracle lies in which concepts can be accessed in these priors. ", + "bbox": [ + 174, + 498, + 825, + 568 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this formalism, the strong oracle has access to the union of train and test concepts; that is $\\mathcal { H } _ { \\mathrm { s t r o n g } } = \\mathcal { H } _ { \\mathrm { t r a i n } } \\cup \\mathcal { H } _ { \\mathrm { t e s t } }$ . The weak oracle, on the other hand only assumes access to $\\mathcal { H } _ { \\mathrm { w e a k } } = \\mathcal { H } _ { \\mathrm { t r a i n } }$ which means it is unable to consider any hypothesis outside what has been seen in training and assigning it zero probability mass. Given a support set $D _ { \\mathrm { s u p p } }$ this difference in priors leads then to different inferences on posteriors and allows us to quantify how compositionally novel a learning task is relative to these ideal learners. ", + "bbox": [ + 174, + 575, + 825, + 659 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 METRICS AND BASELINES", + "text_level": 1, + "bbox": [ + 176, + 685, + 423, + 702 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "During meta-test, given $D _ { \\mathrm { s u p p } }$ models are evaluated on their ability to learn novel concepts. We use two metrics for quantifying this: 1) Accuracy: evaluates the accuracy of model predictions across the query set $D _ { q u e r y }$ , as is standard practice in meta-learning (Lake et al., 2019; Snell et al., 2017). Since there are more negative than positive labels, we report class balanced accuracy for better interpretability, averaging accuracies for the positive and negative query examples; and 2) mean Average Precision (mAP): evaluates models on a much larger number of test scenes $\\tau$ for each episode (comprising 44,787 scenes, 3 per each concept in $\\mathcal { H }$ ). This resolves an issue that with a small query set, a strong model could achieve perfect accuracy without grasping the concept. Since episodes typically have many more negative than positive examples, Average Precision sweeps over different thresholds of a model’s score and reports the average of the precision values at different recall rates, e.g., (Everingham et al., 2010). mAP is then the mean across all of the meta-test episodes. ", + "bbox": [ + 173, + 720, + 826, + 875 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/b996d6f1a26afe350829ee2e314ba7a4972274ec9a6a0686e57215a46af6a944.jpg", + "image_caption": [ + "Figure 4: Baseline models (left). Different choices for the encoder $f ( \\mathbf { u } )$ parameterization explored in the paper. We consider three modalities each of which is processed with a modality specific encoder, followed by four kinds of pooling architecture which take as input objects and their corresponding locations to provide an encoding for the datapoint. Training (right). The model is trained by processing the support images $D _ { \\mathrm { s u p p } }$ with positive (green) and negative (red) images, using $f ( \\mathbf { x } )$ to compute $\\mathcal { L } _ { \\mathrm { q u e r y } }$ which computes generalization error on queries and $\\mathcal { L } _ { \\mathrm { c o n c e p t } }$ which learns to decode the true concept as an auxiliary task. Losses are weighted by $\\alpha \\geq 0$ . " + ], + "image_footnote": [], + "bbox": [ + 178, + 99, + 823, + 300 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 TRAINING LOSS ", + "text_level": 1, + "bbox": [ + 174, + 393, + 328, + 407 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Denote by $\\mathbf { u } \\in \\mathbb { R } ^ { M }$ the input to the model, which can be either in the form of image, sound or schema. We work in a binary classification setting with labels $y$ live in the space $\\mathcal { V } \\in \\bar { \\{ 0 , 1 \\} }$ . Then, given a support set $D _ { \\mathrm { s u p p } } = \\{ \\mathbf { u } _ { i } , y _ { i } \\} _ { i = 1 } ^ { T }$ and a query set $D _ { \\mathrm { q u e r y } } \\stackrel { \\cdot } { = } \\{ \\mathbf { u } _ { i } , y _ { i } \\} _ { i = 1 } ^ { T }$ , sampled in accordance with a productive concept $h$ , our training objective for a single training instance can be written as $\\mathcal { L } _ { \\mathrm { q u e r y } } + \\alpha \\mathcal { L } _ { \\mathrm { c o n c e p t } }$ . Here $\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { q u e r y } } = \\sum _ { \\mathbf { u } , y \\in D _ { \\mathrm { q u e r y } } } ^ { - } \\log p ( Y = \\bar { y } | \\mathbf { u } , D _ { \\mathrm { s u p p } } ) } \\end{array}$ is a standard maximum likelihood meta-learning loss (Ravi & Larochelle, 2016; Snell et al., 2017; Finn et al., 2017), and $\\mathcal { L } _ { \\mathrm { c o n c e p t } } = \\log p ( H = h | \\bar { D } _ { \\mathrm { s u p p } } )$ is an optional regularizer designed to encourage retaining information about the hypothesis of interest from the support set. ", + "bbox": [ + 173, + 421, + 826, + 537 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 BASELINE MODEL ARCHITECTURES", + "text_level": 1, + "bbox": [ + 176, + 563, + 465, + 577 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our baseline models (shown in Figure 4) parameterize the probability in the $\\mathcal { L } _ { \\mathrm { q u e r y } }$ term above using prototypical networks (Snell et al., 2017). The prototypical network consists of an embedding function $f = f _ { \\theta }$ and uses it to compute prototypes $\\mathbf { c } _ { p }$ and $\\mathbf { c } _ { n }$ for positive and negative examples by averaging $f ( \\mathbf { u } )$ for positive and negative examples in the support set respectively. In equations, given a query datapoint $\\mathbf { u } ^ { \\prime }$ , we compute ", + "bbox": [ + 174, + 592, + 825, + 662 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/abb76188158222f0134b4f59e996d1af4987afe148e164f3359fde038f1efb45.jpg", + "text": "$$\np ( Y = y | \\mathbf { u } ; D _ { \\mathrm { s u p p } } ) = \\frac { e x p ( | | f ( \\mathbf { u } ^ { \\prime } ) - \\mathbf { c } _ { p } | | ^ { 2 } } { e x p ( | | f ( \\mathbf { u } ^ { \\prime } ) - \\mathbf { c } _ { p } | | ^ { 2 } ) + e x p ( | | f ( \\mathbf { u } ^ { \\prime } ) - \\mathbf { c } _ { n } | | ^ { 2 } }\n$$", + "text_format": "latex", + "bbox": [ + 277, + 676, + 718, + 712 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this formalism, the models we study in this paper span different choices for $f$ . Roughly, in each modality, we start with an encoder that converts the raw input into a set of vectors, and then a pooling operation that converts that set of vectors into a single vector. In the case of images and sound (input as spectrograms), the encoder is a ResNet-18; and the set of vectors is a subsampling of spatial locations; and for schemas we vectorize components with a lookup table and combine them into a set via feed-forward networks. In the case of images and sounds, the output of the encoder is enriched with position vectors. For the pooling operation, we study global averaging, concatenation, relation networks (Santoro et al., 2017) and transformers (Vaswani et al., 2017) equipped with different pooling operations (max, mean, sum, min) for reasoning inspired by Wang et al. (2019) (Figure 4 middle panel, also see Appendix F for more details). ", + "bbox": [ + 173, + 733, + 825, + 873 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For the probability in $\\mathcal { L } _ { \\mathrm { c o n c e p t } }$ , we represent the concept as a sequence by prefix serialization and then use an LSTM (Hochreiter & Schmidhuber, 1997) to parameterize $p ( h | D _ { \\mathrm { s u p p } } ) =$ $\\Pi _ { s = 1 } ^ { S } p ( h _ { s } | h _ { 1 \\cdots t - 1 } ; D _ { s u p p } )$ . At each step of the LSTM we concatenate $[ \\mathbf { c } _ { p } , \\mathbf { c } _ { n } ]$ to the input. ", + "bbox": [ + 174, + 880, + 825, + 925 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/2ea5088d670d6fa4797dca4586c3097660e37e47c12eebe19396d0937331179b.jpg", + "image_caption": [ + "Figure 5: Compositionality Gap. Different splits ( $\\mathbf { \\hat { x } } \\mathbf { \\cdot }$ -axis) plotted w.r.t performance of the strong oracle (green line) and weak oracle (red line) on the mAP (top) and Accuracy (bottom) evaluated on respective test splits (using hard negatives in support and query sets). Difference between the two is the compositionality gap (comp gap). Yellow: shows the (best) relation-net model on schema inputs, purple: shows the model on image inputs, and gray: shows the model on sound inputs. Error bars are std across 3 independent model runs. " + ], + "image_footnote": [], + "bbox": [ + 179, + 99, + 823, + 234 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 308, + 419, + 324 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We first discuss the compositionality gap induced by the different generalization splits and then delve into the impact of modeling choices on performance on the generalization splits. All models are trained for 1 million steps, and are run with with 3 independent training runs to report standard deviations. We sweep over 3 modalities (image, schema, sound), 4 pooling schemes (avg-pool, concat, relation-net, transformer), 2 choices of negatives (hard negatives, random negatives) and choice of language $( \\alpha = 0 . 0 , 1 . 0 )$ . Unless mentioned otherwise in the main paper we focus on results with hard negatives and $\\alpha = 0 . 0$ . When instantiated for a given modality, we note that the encoders $f ( \\mathbf { u } )$ (Figure 4) all have a similar number of parameters. The appendix contains details of the exact hyperparameters (Appendix E), and more comprehensive results for each split (Appendix G.7). ", + "bbox": [ + 173, + 340, + 825, + 467 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 DATASET DESIGN AND COMPOSITIONALITY ", + "text_level": 1, + "bbox": [ + 174, + 486, + 519, + 500 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "How compositional are the structured splits?. Our main results are shown in Figure 5. Using our model-independent measure of the compositionality gap (Section 3), different splits present varying challenges for generalizing from train to test. The most difficult splits, with the largest compositionality gaps, are the Binding (color) and Binding (shape), which is reasonable since they require learning concepts with entirely new property-values. In contrast, the easiest split with the smallest compositionality gaps is the Instance IID split since it does not require compositionality. Finally, while the mAP metric exposes a larger value of comp gap, the ordering of splits in terms of comp gap is same for both metrics – suggesting similar coarse-grained notions of compositionality. ", + "bbox": [ + 174, + 512, + 825, + 625 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results for the best overall architecture, a relation network (relation-net), is shown in Figure 5. Network performance on the easiest data format (schema; yellow bars) is generally better than the weak oracle, but substantially worse than the strong oracle. Counting is a particularly challenging split where the models underperform even the weak oracle. Broadly, this suggests that the models capture some notion of compositionality – especially for images and schemas – relative to a weak oracle that rigidly considers only training hypotheses, but there is substantial room to improve (especially with respect to the more stringent mAP metric). These results demonstrate that CURI provides a challenging yet tractable setting for evaluating the compositional capabilities of models. ", + "bbox": [ + 174, + 631, + 825, + 743 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Finally, we found that the performance on the Instance IID split is not equal to the weak (and strong) oracle—which are both equal in this case—indicating that the best model does not make ideal posterior predictions even when compositionality is not an issue. Ideal predictions in this case would require the network to behave as if marginalizing over the training hypotheses, as the strong oracle does. A similar plot to Figure 5 can be found in Appendix G.4 for random negatives. ", + "bbox": [ + 174, + 750, + 825, + 819 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Influence of Negatives. Previous work (Hill et al., 2019) has shown that the choice of random v.s. hard negatives for training and evaluation impacts compositional generalization substantially in the case of a particular set of analogical reasoning models. However, we argue that such decisions on dataset design can be made more objectively if one can evaluate the model-independent comp gap. In our context, we find that the comp gap with mAP when using random negatives decreases on average by $5 . 5 \\pm 1 . 4 \\%$ compared to when we use hard negatives. This indicates that it is not only the choice of $\\mathcal { H } _ { t r a i n }$ and $\\mathcal { H } _ { t e s t }$ , which are identical for a given compositional split (say Counting), but also the choice of the negatives which “makes” the task compositionally novel. More generally, this indicates that the comp gap has utility as a more general diagnostic tool for making principled design decisions in compositional learning settings without the confound of specific model decisions. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 DIFFERENCES BETWEEN MODELS ", + "text_level": 1, + "bbox": [ + 176, + 162, + 450, + 178 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Best Models. In general, the best performing model is the relation-net applied to schema inputs, outperforming other combinations of models and input modalities on the Boolean, Concept IID, Complexity, and Instance IID splits on both the mAP as well as accuracy metrics (Figure 5); although as mentioned above, none of the models are close to the strong oracle. It is closely followed by the transformer model on schema inputs, which performs the best on Binding (color), Binding (shape), and Intrinsic splits (Appendix G.7). Utilizing schema inputs proves easier for abstraction except for the Extrinsic setting, where the task requires generalization to novel locations for objects in images, which is well supported by the inductive bias of the CNN encoder (Figure 4). In this case, the image-transformer gets an mAP of $6 2 . 1 \\pm 0 . 7 \\%$ , compared to the next best schema-transformer model at $6 0 . 9 \\pm 0 . 7$ . Further, relational learning proves more crucial in the schema case than for images, with all image models (regardless of pooling) performing better than $5 9 . 4 \\pm 1 . 3 \\%$ mAP (achieved for image-avg-pool) while schema-avg-pool models get only get to $5 3 . 4 \\pm 1 . 5 \\%$ . ", + "bbox": [ + 174, + 189, + 825, + 356 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "When to use a transformer? Transformer models appear to outperform relation networks in splits concerning disentangling. For instance, for the Intrinsic split with schema-relation-net is at $5 5 . 1 \\pm 0 . 8 \\%$ v.s. $5 7 . 9 \\pm 0 . 6 \\%$ for schema-transformer. Similarly, for the Extrinsic split the imagetransformer is at $6 2 . 1 \\pm 0 . 7 \\%$ compared to the image-relation-net at $6 0 . 8 \\pm 1 . 1 \\%$ . We hypothesize that this is because the iterative message passing via. attention in transformers improves object representations for disentangling compared to relation networks that lack such a mechanism. ", + "bbox": [ + 174, + 363, + 825, + 446 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "What is the relative difficulty of abstraction from different modalities? One of the key contributions of our work is in providing multiple modalities (image, schema, sound) for productive concept learning. We next characterize the difficulty of abstraction based on modality for the various generalization settings. In the Intrinsic setting, we find that the schema models, which have access to a “perfect” disentangled representation significantly outperform image models—a schema-avg-pool model gets an mAP of $5 2 . 7 \\pm 3 . 1 \\%$ while an image-avg-pool model gets to $3 4 . 4 \\pm 0 . 0 \\%$ mAP. ", + "bbox": [ + 174, + 454, + 825, + 537 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Similarly, for the Counting split where the total number of objects are exactly specified in the schema (Figure 4), schemas are substantially better than images. For example, schema-relation-nets get to $5 6 . 2 5 \\pm 5 . 3 2 \\%$ mAP while image-avg-pool is at $4 8 . 4 \\pm 1 . 2 \\%$ mAP. Interestingly, the next best model—image-relation-net—is substantially worse, at $3 9 . 4 5 \\pm 1 . 6 \\%$ . Curiously, for while transformer models perform well at disentangling, they seem to be quite poor for Counting, with image-transformer models getting to only $3 2 . 4 \\pm 1 . 4 \\%$ mAP, suggesting a potential weakness for transformers. Overall, there appears to be an intimate link between the generalization setting and the input modality, suggesting avenues where representation learning could be improved for a given modality (e.g. images), relative to the kind of reasoning one is interested in (e.g. counting). ", + "bbox": [ + 174, + 544, + 825, + 670 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "When does language help? On average, training models with explicit concept supervision using the concept loss (Section 4.1) improves performance by $2 . 8 \\pm 0 . 6 \\%$ mAP (SEM error). This is a small boost relative to the gap between the original model and the strong oracle, suggesting that this simple auxiliary loss is not sufficient to internalize the LOT in a neural network. Overall, image models benefit more from language than schema models which natively utilize symbols (Appendix G.3). ", + "bbox": [ + 174, + 676, + 823, + 746 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 767, + 318, + 782 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We introduced the compositional reasoning under uncertainty (CURI) benchmark for evaluating fewshot concept learning in a large compositional space, capturing the kinds of producitivity, unboundness and underdetermination that characterize human conceptual reasoning. We instantiate a series of meta-learning tasks, and evaluate numerous baseline models on various aspects of compositional reasoning under uncertainty, including inferential coherence, boolean operation learning, counting, disentangling, etc. Further, we introduce the notion of a compositionality gap to quantify the difficultly of each generalization type, and to estimate the degree of compositionality in current deep learning models. 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", + "bbox": [ + 176, + 409, + 825, + 438 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A EXAMPLE EPISODES FROM THE DATASET ", + "text_level": 1, + "bbox": [ + 174, + 469, + 547, + 483 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We show examples from the Concept IID split test set comprising the ground truth productive concept (top), along with the support and query sets for meta learning (rendered as images), the alternate hypotheses which are consistent with the support set – that is, other hypotheses which could also have generated the positive and negative examples in the support set – and the concepts based on which we pick the hard negatives Figures 6 to 11. ", + "bbox": [ + 173, + 501, + 825, + 570 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "B ADDITIONAL DATASET DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 593, + 478, + 609 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We first provide more details of the concept space $\\mathcal { G }$ , then explain how we obtain $\\mathcal { H }$ , the space of concepts for training and evaluation, provide more details of the structured splits, and finally explain the weight $w ( h )$ based on which we sample concepts. ", + "bbox": [ + 174, + 626, + 825, + 667 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "B.1 MORE DETAILS OF THE GRAMMAR ", + "text_level": 1, + "bbox": [ + 176, + 688, + 455, + 702 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We provide below the full grammar used to sample concepts, where $A \\to B | C$ means that $A$ can expand to either $B$ or $C$ under the rules defined by the grammar. We always start expanding at the START token and then follow the rules of the grammar until we hit a terminal node (which does not have any expansions defined). As and where possible, we followed the insights from Piantadosi et al. (2016) in choosing the sampling probabilities for various completions based on how well humans seem to be able to learn the corresponding primitive. For example, we sample utterances with disjunctions (or) less frequently since they are known to be difficult for humans to learn. Based on Kemp & Jern (2009), we chose to represent location as a discrete entity, such that relative, and categorical notions of left or right simply become comparisons in the location space (location? $\\textbf { x } >$ location? $S _ { - \\mathbf { x } , }$ ), unlike the CLEVR dataset (Johnson et al., 2016) which defines categorical relational objects. ", + "bbox": [ + 173, + 713, + 825, + 853 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Here is the full grammar $\\mathcal { G }$ used for sampling the concepts (as explained in the main paper, $S _ { - { \\bf x } } =$ $S / \\{ \\mathbf { x } \\} )$ . Note that the grammar always generates strings in postfix notation and thus the operands in each expansion occur before the operation: ", + "bbox": [ + 176, + 859, + 823, + 902 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/ce51437271d11b3923b461a18ecfba57684729b9acfbc6d616027c265c143df5.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 254, + 135, + 759, + 262 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Valid Hypotheses ", + "text_level": 1, + "bbox": [ + 279, + 300, + 357, + 309 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "\\* for-all $\\mathsf { x }$ in ${ \\mathsf { S } } =$ (purple,color?( $\\textsf { x } )$ ）{log-prob[-0.852]] \n\\*for-all $\\times$ in $\\gamma =$ (color?(x)，purple）{log-prob[-0.852]} \n\\*all(color?(S),purple）{log-prob[-1.950]} \n\\*exists $\\times$ inSand $( =$ (metal,il? $\\textsf { x } )$ ),al(color?(S),purple)）{log-prob[-8.523]} \n\\*for-all $\\times$ in Sand $\\scriptstyle =$ (color?(x),purple),any(material？(S)，metal)）{log-prob[-8.523]}", + "bbox": [ + 277, + 315, + 683, + 356 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Hypotheses for Hard Negatives ", + "bbox": [ + 279, + 362, + 421, + 371 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "$^ { \\ast }$ for-all $\\times$ inSor $( =$ (shape?( $\\times$ ),cube ),all(color?(S),purple )) \\*for-all $\\times$ inSor $: =$ (purple,color？ $\\textsf { x } )$ ), $=$ (sphere,shape?(×))) \\*exists $\\times$ inSand(any(color?(S_{-x}),purple),all(color?(S_{-x}),purple)) \\*for-all xinSo $\\mathbf { \\partial } ^ { \\cdot } ( =$ (shape？( $\\times$ )，sphere), $=$ (purple,color?(x)）） \\*for-all xinSor $( =$ (purple,color？ $\\textsf { x } )$ ）， $=$ (shape?(x),cube)) ", + "bbox": [ + 277, + 377, + 632, + 417 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/9982c10744a33a41760dc776accfdcf610e9ea57fe0a93e16d9c69fed8a64d74.jpg", + "image_caption": [ + "Figure 6: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. " + ], + "image_footnote": [], + "bbox": [ + 254, + 526, + 761, + 676 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Valid Hypotheses ", + "text_level": 1, + "bbox": [ + 279, + 713, + 357, + 722 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "\\* any(locationY?(S),7）{log-prob[-1.621]} \\*exists $\\mathsf { x }$ in $\\mathsf { S } >$ (locationY?(x),6）{log-prob[-1.621]} \\*exists $\\times$ inSany(locationY?(S_{-x})，7）{log-prob[-1.621]} \\*exists $\\times$ inSall(locationY?( $S _ { - } \\{ - x \\}$ ),7）{log-prob[-1.647]} \\*exists $\\mathsf { x }$ in ${ \\mathsf { S } } =$ (locationY?(×)，7）{log-prob[-2.314]} ", + "bbox": [ + 277, + 728, + 562, + 768 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Hypotheses for Hard Negatives ", + "bbox": [ + 279, + 775, + 421, + 784 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "\\*exists $\\times$ in S and(al(color?(S_{-x}),red),>(8,locationX?(x)))\\*exists $\\mathsf { x }$ inSand(all(color?(S_{-x})，red）， ${ \\tt > } ( 7 _ { \\tt { i } }$ locationx?(x)))\\*exists $\\times$ inSand(al(color？(S_{-x}）,red）,any(size(S_{-x}）,0.5））\\*exists $\\mathsf { x }$ inSand $( >$ (8,locationY? $, \\times ) ~ ,$ ， $>$ (locationY?( $\\times ~ 5 .$ 5））\\*exists $\\mathsf { x }$ inSor(al(shape?(S),sphere),all(color?(S_{-x}),red))", + "bbox": [ + 277, + 790, + 606, + 830 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Figure 7: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. ", + "bbox": [ + 214, + 900, + 782, + 912 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/a01628a9c0491627343c835820bbc7a21990351f0912b975e7c67523f79be486.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 254, + 133, + 759, + 263 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Valid Hypotheses ", + "text_level": 1, + "bbox": [ + 277, + 300, + 357, + 310 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "\\* for-all $\\times$ in $\\mathsf { S } =$ (sphere,shape? $\\textsf { x } )$ ）{log-prob[-0.855]} \n\\*for-all $\\times$ in s $=$ (shape?(x)，sphere）{log-prob[-0.855]} \n\\*all(shape?(S）,sphere）{log-prob[-1.954]} \n\\*for-all $\\times$ inSand $( = 1$ (sphere,shape？ $( \\times )$ ）， ${ } = { \\mathsf { I } }$ (shape?(×),sphere））{log-prob[-7.807]} \n\\*for-all $\\mathsf { x }$ in Sand $\\mathop { \\left. \\sum \\right. }$ (shape?(x),sphere), $=$ (sphere,shape?(x))）{log-prob[-7.807]}", + "bbox": [ + 277, + 315, + 676, + 356 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Hypotheses for Hard Negatives ", + "bbox": [ + 279, + 362, + 421, + 371 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$^ *$ for-all $\\times$ inSor $( =$ (sphere,shape?( $\\textsf { x }$ ）） $= ( \\mathsf { c o l o r ? } ( \\textsf { X } )$ ，green））\\*for-all $\\times$ inSor $: =$ (purple,color？(x））, $=$ (sphere,shape?(x)))\\*for-all $\\mathsf { x }$ inSor $: =$ (sphere,shape? $\\times )$ ） $=$ (green,colo？ $\\times ) )$ ）\\*for-all $\\times$ inSor(al(shape?(S)，sphere)，=(3,ocationx?(x)))\\*for-allxin Sor $( =$ (shape？( $\\times$ )，'sphere), $=$ (purple,color?(x)))", + "bbox": [ + 277, + 377, + 571, + 417 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Productive Concept:exists x in Sand(al(locationY?(S_{-x}),6),any(size?(S_{-x} ),0.7)) ", + "bbox": [ + 289, + 529, + 709, + 539 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/b105df7e0fb61953c0f0cbc38ab2c6bef9485d4b1af91e615a037b47da81e616.jpg", + "image_caption": [ + "Figure 8: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. " + ], + "image_footnote": [], + "bbox": [ + 254, + 537, + 759, + 676 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Valid Hypotheses ", + "text_level": 1, + "bbox": [ + 277, + 713, + 357, + 723 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "\\*exists $\\times$ in Sal(locationY?(S_{-x}),6）{log-prob[-0.008]} \\*exists $\\mathsf { x }$ in $\\mathsf { S } >$ (count=(locationY?(S_{-x}),6),count=(size?(S_{-x}),0.35)）{log-prob[-6.668]} \\*for-all $\\times$ in $\\mathsf { S } = \\mathsf { ( c o u n t = }$ (locationY?(S）,6）,o $: =$ (shape?(S),shape？(x))）{log-prob[-6.690]} \\*for-all $\\times$ in ${ \\sf S } =$ (count=(shape?(S),sape？(x)）,ount=(location？(S）,6)){log-prob[-6.690]} \\*exists $\\times$ in Sand $\\mathrm { . } >$ (locationY?(x),5), $>$ (size?(x),0.35)）{og-prob[-7.45] ", + "bbox": [ + 277, + 728, + 735, + 770 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Hypotheses for Hard Negatives ", + "bbox": [ + 279, + 776, + 421, + 785 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "\\*and(any(size?(S),O.7),any(locationY?(S),6)) \\*exists $\\times$ inSand(any(locationY?(S),6), $>$ (locationY?(×),locationX?(×))) \\*exists $\\times$ in Sand $: = 1$ (locationY?( $\\times )$ 0,6),any(shape?(S)，sphere)) \\*exists $\\mathsf { x }$ in ${ \\mathsf { S } } = { \\mathsf { I } }$ (coun $: = 1$ (locationY?(S),6),count=(locationY?(S),locationY?(x))) \\*exists $\\mathsf { x }$ ins $>$ (count=(locationY?(S_{-x})，6）,count $=$ (locationY?(S),2)) ", + "bbox": [ + 277, + 791, + 663, + 830 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Figure 9: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. ", + "bbox": [ + 214, + 900, + 782, + 912 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Productive Concept: exists $\\times$ in S and(all(material?( $\\mathsf { S \\_ f } \\cdot \\mathsf { X } \\mathsf { \\Sigma }$ ),rubber),all(color?(S_{-x}),cyan）) ", + "bbox": [ + 274, + 114, + 722, + 125 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/175fb5b3d8f181ce9318ca5b23355bd199dd16ba5b63902407ad08561f5fcb76.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 254, + 133, + 759, + 263 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Valid Hypotheses ", + "text_level": 1, + "bbox": [ + 277, + 313, + 357, + 323 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "\\*exists $\\mathsf { x }$ in Sand(any(material?(S $\\{ - \\mathsf { x } \\}$ ).rubber),al(color?(S_{-x}),cyan））{log-prob[-0.983]}\\*exists $\\times$ inSand(all(color？(S $\\{ - { \\boldsymbol { \\times } } \\bar \\} \\}$ ,cyan),a(material？(S_{-x}),rubber））{og-prob[-1.162]}\\*exists $\\times$ inSand(al(material？(S_{-x}),rubber),al(color？(S_{-x})，cyan））{log-prob[-1.162]}", + "bbox": [ + 277, + 329, + 733, + 354 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Hypotheses for Hard Negatives ", + "bbox": [ + 279, + 361, + 421, + 369 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "$* >$ (count=(color?(S),cyan),count=(material?(S),metal)) \\*existsx in Sand(al(color?(S_{-x}),cyan),any(locationY?(S),3)) \\*existsxinSall(color?(S_{-x}),cyan） \\*existsxin Sand(all(color?(S $\\{ - \\mathsf { x } \\}$ ),cyan),any(color?(S_{-x}),cyan)) \\*exists $\\mathsf { x }$ inSand(al(color？(S_{-x})，cyan),any(size?(S)，0.7)) ", + "bbox": [ + 277, + 376, + 619, + 416 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Productive Concept: exists x in S and(al(locationY?( S_{-x} ),5), $>$ (locationY?(×),locationX?(×))) ", + "bbox": [ + 267, + 529, + 728, + 539 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/6eed4f451f8da33d850d98fba50a0bcb7bb12c18e81c7685beaa7d2369d8b85e.jpg", + "image_caption": [ + "Figure 10: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. " + ], + "image_footnote": [], + "bbox": [ + 254, + 547, + 761, + 678 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Valid Hypotheses ", + "bbox": [ + 277, + 744, + 357, + 753 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "$^ *$ exists $\\times$ in Sand(al(locationY?(S_{-x}),5), $>$ (locationY?(x),locationx?( $\\times$ ）））{log-prob[0.000]} ", + "bbox": [ + 276, + 760, + 736, + 770 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Hypotheses for Hard Negatives ", + "bbox": [ + 279, + 776, + 421, + 785 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "\\* for-all $\\times$ in S or(al(locationX?(S),1), $>$ (locationY?( $\\textsf { x } )$ ,locationx?(×）)) \\*for-all $\\times$ inSand $\\mathrm { . } >$ (locationY?( $\\times$ ).locationx?( $\\times 1$ )）, $>$ (8,locationY?(x))) \\*for-allxin Sor(>(locationY? $\\times 1$ ),locationx?(x)) $=$ (color?(x）,brown)) \\*for-all $\\times$ inSor $( =$ (yellow，color？ $\\mathbf { \\epsilon } ( \\mathbf { \\epsilon } )$ ）， $>$ (locationY?(x),locationx?(×))) \\*for-all $\\times$ inSor $( =$ (purple,color？(x)), $>$ (locationY?(x）,ocationx?(×))) ", + "bbox": [ + 277, + 791, + 620, + 832 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Figure 11: Qualitative Example of an Episode in CURI dataset. Best viewed zooming in, in color. ", + "bbox": [ + 210, + 900, + 785, + 914 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "START $- > \\lambda$ S . BOOL e x i s $\\mathrm { ~ ~ t ~ s ~ } = \\mathrm { ~ ~ | ~ } \\lambda \\mathrm { ~ ~ S ~ }$ . BOOL f o r − a l l = BOOL $- >$ BOOL BOOL and | BOOL BOOL o r | BOOL n o t | C C = | SH SH = | M M = | S I $\\mathrm { ~ S ~ I ~ } = \\mathrm { ~ \\Omega ~ } |$ L $\\mathrm { ~ L ~ } = \\mathrm { ~ \\Omega ~ }$ NUM NUM $=$ | S I S I > | L L > | NUM NUM > | SETFC C a l l | SETFSH SH a l l | SETFM M a l l | SETFSI S I a l l | SETFL L a l l | SETFC C a n y | SETFSH SH any | SETFM M any | SETFSI S I any | SETFL L any ", + "bbox": [ + 171, + 102, + 669, + 229 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "NUM $- >$ SETFC C c o u n t $=$ | SETFSH SH c o u n t $=$ | SETFM M c o u n t $=$ | SETFSI S I c o u n t $=$ | SETFL L c o u n t = NUM > 1 | 2 | 3 ", + "bbox": [ + 169, + 241, + 686, + 284 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "SETFC $- >$ SET FC SETFSH $>$ SET FSH SETFM $- >$ SET FM SETFSI $- >$ SET F S I SETFL $- >$ SET FL ", + "bbox": [ + 173, + 296, + 334, + 367 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "FC > c o l o r ? \nFSH −> s h a p e ? \nFM $- >$ m a t e r i a l ? \nF S I $- >$ s i z e ? \nFL $- >$ l o c a t i o n X ? | l o c a t i o n Y ? ", + "bbox": [ + 173, + 602, + 488, + 671 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "OBJECT $- >$ x SET : S | $S _ { - \\bf { x } }$ ", + "bbox": [ + 173, + 684, + 294, + 713 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.2 SAMPLING ", + "text_level": 1, + "bbox": [ + 174, + 747, + 294, + 762 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We sample 2000000 initial hypotheses from the CFG $\\mathcal { G }$ , and impose a maximum depth in the recursion tree of 6 when sampling. That is, no node has a depth larger than 6 in the recursion through which we generate concepts from the grammar $\\mathcal { G }$ . We then reject and filter the hypotheses to obtain a set of “interesting” hypotheses $\\mathcal { H }$ used in the main paper explained in more detail below: ", + "bbox": [ + 173, + 776, + 825, + 832 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Rejection Sampling: We reject the following string combinations after sampling from the grammar $\\mathcal { G }$ : ", + "bbox": [ + 166, + 838, + 823, + 867 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• All programs which contain \" $\\lambda S$ . for-all $\\mathbf { X } ^ { \" }$ and \" $S _ { - { \\bf x } } \"$ \" in the same program. This is asking that for all objects in a scene, a certain property is satisfied by everything other than the object, which is the same as saying, for all objects in the scene. ", + "bbox": [ + 217, + 881, + 823, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• All programs where we compare the properties of the same object to itself, e.g. color? $\\mathrm { ~ ~ { ~ ( ~ x ~ ) ~ } ~ } = = \\mathrm { ~ \\cos ~ } \\tt c o l o r ? \\mathrm { ~ ~ { ~ ( ~ x ~ ) ~ } ~ }$ , where color? can be any function applied to the object. All programs where we have the following string: exists(color?(S) $= =$ color?(x)) or for-all(color?(S) $= =$ color?(x)) where color? can be any function applied to the object. \nAll programs which evaluate to true on schemas more than $10 \\%$ of the time and less than 10 times. The former condition ensures that we work with concepts which are in some sense interesting and surprising (as opposed to concepts which are always trivially true), and the second condition ensures that we have unique schmeas or datapoints to place in the support and query sets, which both have 5 positive images each. ", + "bbox": [ + 215, + 103, + 825, + 253 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We provide examples of concepts which get rejected for being true too often below: ", + "bbox": [ + 173, + 265, + 720, + 280 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "exist $_ { \\mathrm { { S } } } = \\mathrm { { x } }$ \\in S or( $=$ (locationX?( x ), locationY?( x ) ), any(color?( S ), brown ) \n) \nexist $_ { \\mathrm { { S } } } = \\mathrm { { x } }$ \\in S and( exists $=$ (locationY?( S ), locationX?( x ) ), any(color?( S ), brown ) \n) \nexist $_ { \\mathrm { { S } } } = \\mathrm { { x } }$ \\in S or( all(color?( S ), gray ), all(color?( S ), brown ) \n) ", + "bbox": [ + 210, + 291, + 653, + 458 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "See Appendix C for more details on the structured generalization splits which yeild train concepts $\\mathcal { H } _ { \\mathrm { t r a i n } }$ and test concepts $\\mathcal { H } _ { \\mathrm { t e s t } }$ . ", + "bbox": [ + 176, + 469, + 823, + 500 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.3 CONCEPT PRIOR WEIGHT $w ( h )$ ", + "text_level": 1, + "bbox": [ + 176, + 515, + 429, + 530 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We next explain the form of the prior weight $w ( h )$ that we use for defining the prior over the concepts provided to the models (both oracles as well as deep learning models). Given $l ( h )$ , the number of tokens in the postfix serialization of the concept $h$ , the unnormalized weight $\\tilde { w } ( h )$ is log-linear in the length, and is defined as follows: ", + "bbox": [ + 173, + 540, + 825, + 597 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/62905cde2e2da9f11e9e36f0ed77017a7a39033ee907ae7f48c4efa9d8f8175a.jpg", + "text": "$$\n\\tilde { w } ( h ) \\propto \\exp { - 0 . 2 \\cdot l ( h ) }\n$$", + "text_format": "latex", + "bbox": [ + 418, + 603, + 580, + 621 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Given a split $\\Omega \\in \\{ t r a i n , t e s t \\}$ , the final, normalized weight is given as: ", + "bbox": [ + 173, + 633, + 650, + 650 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/7d69283655897150d808676592622221f344878a91d3cf386f8b6291ffe99fd0.jpg", + "text": "$$\nw ( h ) = \\frac { \\tilde { w } ( h ) } { \\sum _ { \\mathcal { H } _ { \\Omega } } \\tilde { w } ( h ) }\n$$", + "text_format": "latex", + "bbox": [ + 429, + 656, + 568, + 693 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "As explained in the main paper, the final prior for a hypothesis given a split $\\Omega$ is $p ( h ) \\ =$ $\\begin{array} { r } { \\sum _ { h ^ { \\prime } \\in \\mathcal { H } _ { \\Omega } } \\mathcal { w } ( h ) \\mathbf { I } [ h = h ^ { \\prime } ] } \\end{array}$ . ", + "bbox": [ + 173, + 707, + 825, + 736 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Our choice of the log-linear weight is inspired by the observation in cognitive science that longer boolean concepts are harder for people to learn (Feldman, 2000). ", + "bbox": [ + 174, + 741, + 823, + 771 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Here are some examples of hypotheses with a high weight (computed on $\\Omega = \\operatorname { t r a i n } \\cup$ test): ", + "bbox": [ + 171, + 776, + 767, + 792 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/10d5c556f2c547685ce1933c33ecfc253713e222a1aa2e77912c31322f6f5198.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { ~ e ~ x ~ i ~ s ~ t ~ s ~ \\ x ~ \\ x ~ \\ i n ~ \\ S ~ \\ = ( 2 , ~ c ~ o ~ u n t = ( c ~ o ~ l o r ~ 2 ( { \\it \\Phi } ~ \\mathbb { S } \\mathbb { S } _ { - } \\{ - \\mathbb { x } \\} \\mathbb { S } _ { \\mathrm { \\Lambda } } ) , ~ { \\it \\Phi } ~ \\mathrm { c y a n } ) ~ ) ~ } , } \\\\ & { \\mathrm { ~ e ~ x ~ i ~ s ~ t ~ s ~ \\ x ~ \\ x ~ \\ i n ~ \\ S ~ > ( 1 ~ o ~ c ~ a ~ t ~ i ~ o ~ n ~ Y ~ 2 ( { \\it \\Lambda } ~ \\mathrm { x } ~ \\Lambda ~ ) ~ , ~ \\mathbb { \\Lambda } ~ 6 ~ ) ~ } , } \\\\ & { = ( \\mathrm { ~ c ~ o ~ u n t = ( c ~ o ~ l o r ~ 2 ( { \\it \\Lambda } ~ \\mathrm { S } ~ ) ~ , ~ b r o w n ~ \\Lambda ~ ) ~ , ~ 3 ~ \\Lambda ~ ) ~ } , } \\\\ & { > ( \\mathrm { ~ c ~ o ~ u n t = ( ~ l o ~ c ~ a t i o n ~ { \\cal X } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ ) ~ } , } \\\\ & { \\mathrm { ~ a n y ~ ( 1 ~ o ~ c ~ a ~ t ~ i ~ o ~ n ~ Y ~ 2 ( { \\it \\Lambda } ~ \\mathrm { S } ~ \\Lambda ~ ) ~ , ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ ) ~ } , } \\\\ & { = ( 1 , ~ \\mathrm { c ~ o ~ u n t = ( ~ l o ~ c ~ a t ~ i ~ o n ~ Y ~ 2 ( { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ ) ~ } ) ~ , } \\\\ & { = ( 3 , ~ \\mathrm { c ~ o ~ u n t = ( ~ l o ~ c ~ a t ~ i ~ o n ~ Y ~ 2 ( { \\it \\Lambda } ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ ) ~ , ~ 3 ~ { \\it \\Lambda } ~ { \\it \\Lambda } ~ ) ~ } ) ~ , } \\\\ & \\mathrm ~ a l ~ ( 1 ~ o ~ c ~ a ~ t ~ i ~ o n ~ { \\cal X } ~ 2 \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 209, + 797, + 753, + 926 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "$= ( 2$ , c o u n t $=$ ( c o l o r ? ( S ) , b l u e ) ) , f o r − a l l x i n S n o t ( $>$ ( 6 , l o c a t i o n X ? ( x ) ) ) , $\\mathbf { \\Lambda } = ( \\mathrm { ~ c ~ o ~ u ~ n ~ t ~ } = ( \\mathrm { ~ c ~ o ~ l ~ o ~ r ~ } ? ( \\mathrm { ~ \\textbf ~ { ~ S ~ } ~ } )$ , g r a y ) , 2 ) , $= ( 2$ , c o u n t $=$ ( c o l o r ? ( S ) , g r a y ) ) ", + "bbox": [ + 212, + 103, + 655, + 160 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B.4 EXECUTION ON IMAGES. ", + "text_level": 1, + "bbox": [ + 174, + 188, + 388, + 202 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In order to create the perceptual inputs in the dataset $\\mathcal { U }$ , we sample images using the renderer for CLEVR from Johnson et al. (2016), changing the range of objects to [2, 5], to reduce clutter and enable easier learning of predicates like any and all for models. 3 The CLEVR renderer produces scenes u with pixels as well as an associated schema file $\\mathbf { u } ^ { s }$ detailing the properties of all the objects sampled in the scene, including their location, shape, size, material, and rotation. Based on this, we convert our sampled concepts into postfix notation and execute them on the schemas using an operator stack. Concretely, execution of the concept $h \\in \\mathcal H$ on $\\mathbf { u } ^ { s }$ yields a boolean true or false value $\\{ 0 , 1 \\}$ . We execute each such hypothesis on a set of 990K images, yielding scores of how often a hypothesis is true for an image. We threshold this score to retain the subset of hypotheses which are true for no more than $10 \\%$ of the images and are true at least for 10 images, to pick a subset of “interesting” hypotheses $\\mathcal { H } ^ { \\prime }$ for training models. ", + "bbox": [ + 173, + 214, + 825, + 367 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Bias. The image dataset here sampled itself has a bias in terms of the location coordinates (in the pixel space). The CLEVR dataset generation process samples objects in the 3d (top-down x, y) coordinate space uniformly (from a grid of $^ { - 3 }$ , to $+ 3$ ). However, since the camera is always looking into the scene from outside, the image formation geometry implies that in the camera / image coordinates most of the objects appear to be away from the scene and very few are close to the camera. Thus, in terms of the y-coordinates we observe in the image coordinates a bias in terms of the distribution not being unifrom. This also makes sense in general, as even in the real world, objects are not found very close to the camera or very far away from the camera in general. See Figure 12 for all the biases in the observation space u computed over 990K sampled images. ", + "bbox": [ + 173, + 373, + 825, + 500 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B.5 AUDIO. ", + "text_level": 1, + "bbox": [ + 174, + 518, + 269, + 532 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "To build the audio data, we use clips of orchestral instruments playing various pitches downloaded from https://philharmonia.co.uk/resources/sound-samples/. We make the following mappings of object properties: ", + "bbox": [ + 176, + 545, + 825, + 587 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "· $x$ location temporal location. larger $x$ bin means the note is played later. \n• $y$ location pitch. All pitches between the instruments are the same (up to octaves). \ncolor instrument – gray trumpet – red clarinet – blue violin – green flute – brown oboe – purple saxaphone – cyan french-horn – yellow guitar ", + "bbox": [ + 214, + 598, + 790, + 799 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "• shape amplitude profile; either getting louder, getting softer, or constant volume ", + "bbox": [ + 217, + 801, + 774, + 818 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "size total volume ", + "bbox": [ + 217, + 823, + 367, + 837 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "• material low-pass filtering or no filtering. ", + "bbox": [ + 217, + 843, + 522, + 857 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "All binned quantities use the same number of bins as in the image domain. ", + "bbox": [ + 176, + 869, + 661, + 885 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/d50281e9fb8b1c2569d993b39f1456f0a63a3891542c3a78614e4eca783867e3.jpg", + "image_caption": [ + "Figure 12: Histogram of properties found in inputs $\\mathbf { u }$ in the dataset. We notice that the properties are all largely uniform with bias in x and y-coordinates towards the center of the image. " + ], + "image_footnote": [], + "bbox": [ + 173, + 104, + 794, + 479 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.6 ANALYSIS OF SYNONOMY OF CONCEPTS. ", + "text_level": 1, + "bbox": [ + 176, + 515, + 500, + 529 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We next show an analysis of concepts which have the same evaluation signatures on a large set of 990K images, and are thus synonymous (in context of the dataset at hand). Note that while some of these concepts might be truly synonymous to each other (for example, $A > B$ is the same as $B < A$ ), others might be synonymous in context of the image distribution we work with. For example, size can never be greater than 0.7 in our dataset and location can never be greater than 8, and thus asking if location is greater than 8 or size is greater than 0.7 has the same semantics on our dataset. In Figure 13 we show each such “concept” or “meaning”, which is a cluster of hypotheses which all evaluate to the same truth value and plot a histogram of how many hypotheses each cluster tends to have. We notice that most of the concepts have 1 synonym (i.e. there is only one concept with the particular) evaluation signature, with a long tail going upto 80 synonyms in a concept. In the Concept IID split we ensure that none of the concepts which have the same signature are found across the train/val/test splits. ", + "bbox": [ + 173, + 540, + 825, + 707 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C DETAILED DISCUSSION OF THE STRUCTURED SPLITS ", + "text_level": 1, + "bbox": [ + 174, + 729, + 645, + 744 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We provide more details on how each of the structured splits described in Sec. 3 of the main paper are created. Assuming access to $\\mathcal { H }$ , the space of concepts sampled and filtered from the grammar $\\mathcal { G }$ , we use various heuristics to produce the generalization splits in the paper: ", + "bbox": [ + 176, + 760, + 826, + 803 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "• Instance IID: This split is trivial since $\\mathcal { H } _ { \\mathrm { t r a i n } } = \\mathcal { H } _ { \\mathrm { t e s t } } = \\mathcal { H }$ \n• Concept IID: This split divides concepts into train and test by picking concepts at random from $\\mathcal { H }$ and assigning them to $\\mathcal { H } _ { \\mathrm { t r a i n } }$ or $\\mathcal { H } _ { \\mathrm { t e s t } }$ while ensuring that no two concepts which are synonyms Appendix B.6 are found in different splits. \n• Boolean: This split forms cross product of all possible colors and $\\{ \\harpoonright , \\ \\mathrm { o r } \\}$ boolean operators, and partitions a subset of such combinations which we want to only occur in test. We use the following tokens for test: ", + "bbox": [ + 217, + 814, + 825, + 924 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/44c0a4f91414e41f7139ec4f48443841663a8cffa1f32381163d8b0af7eee6eb.jpg", + "image_caption": [ + "igure 13: Histogram of number hypotheses with same evaluation signatures. " + ], + "image_footnote": [], + "bbox": [ + 191, + 130, + 759, + 352 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "‘green’, ‘or’ | ‘purple’, ‘and’ | ‘cyan’, ‘and’ | ‘red’, ‘or’ | ‘green’, ‘and’ ", + "bbox": [ + 312, + 375, + 790, + 404 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We then create $\\mathcal { H } _ { \\mathrm { t e s t } }$ to contain all concepts which have any of the combinations above. For example, if a concept has both green and or we would place it in ${ \\mathcal { H } } _ { \\mathrm { t e s t } }$ . After every feasible candidate is placed in $\\mathcal { H } _ { \\mathrm { t e s t } }$ based on this heurisitc, the remaining concepts in $\\mathcal { H }$ are assigned to ${ \\mathcal { H } } _ { \\operatorname { t r a i n } }$ . ", + "bbox": [ + 230, + 425, + 825, + 482 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• Extrinsic: This split forms cross product of all possible colors and locations in the dataset, and partitions a subset of such combinations that we only want to occur in test. We use the following tokens for test (only a subset shown for illustration): ", + "bbox": [ + 218, + 488, + 825, + 530 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/9b3218553be57cb519424c2c4df8b0e8af96a7efe08014b6356623b14a56259d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { ~ \\gamma ~ } _ { \\mathrm { J } } ^ { \\prime \\prime } , \\quad \\mathrm { ~ \\gamma ~ } _ { \\mathrm { J } } ^ { \\mathrm { v } } \\mathrm { a y } ^ { \\prime } , \\quad | \\mathrm { ~ \\gamma ~ } _ { \\mathrm { L } } ^ { \\mathrm { v } } { \\mathrm { ~ \\gamma ~ } } _ { \\mathrm { ~ \\gamma ~ } } ^ { \\prime } , \\quad \\mathrm { ~ \\gamma ~ } _ { \\mathrm { c y a n } ^ { \\prime } } ^ { \\mathrm { v } } , \\quad | \\mathrm { ~ \\gamma ~ } _ { \\mathrm { S } } ^ { \\mathrm { v } } { \\mathrm { ~ \\gamma ~ } } _ { \\mathrm { ~ \\gamma ~ } } ^ { \\mathrm { v } } , \\quad \\mathrm { \\ p u r p } { \\mathrm { ~ \\gamma ~ } } _ { \\mathrm { J } } ^ { \\mathrm { v } } } \\\\ & { \\mathrm { ~ \\gamma ~ } _ { \\mathrm { J } } ^ { \\mathrm { v } } , \\quad \\mathrm { ~ \\gamma ~ } _ { \\mathrm { J } } ^ { \\mathrm { v } } \\mathrm { e e } ^ { \\prime } , \\quad | \\mathrm { ~ \\gamma ~ } _ { \\mathrm { ~ \\gamma ~ } } ^ { \\mathrm { v } } { \\mathrm { ~ \\gamma ~ } } _ { \\mathrm { J } } ^ { \\mathrm { v } } , \\quad \\mathrm { ~ \\gamma ~ } _ { \\mathrm { v y a n } ^ { \\prime } } ^ { \\mathrm { v } } , \\quad | \\mathrm { ~ \\gamma ~ } _ { \\mathrm { v } } ^ { \\mathrm { v } } { \\mathrm { ~ \\gamma ~ } } _ { \\mathrm { J } } ^ { \\mathrm { v } } , \\quad \\mathrm { ~ \\gamma ~ } _ { \\mathrm { p u r p } { \\mathrm { ~ \\gamma ~ } } } ^ { \\mathrm { v } } | \\mathrm { e } ^ { \\prime } , | } \\\\ & { \\mathrm { ~ \\gamma ~ } _ { \\mathrm { J } } ^ { \\mathrm { v } } , \\quad \\mathrm { ~ \\gamma ~ } _ { \\mathrm { b } } ^ { \\mathrm { v } } \\mathrm { l u e } ^ { \\prime } , \\quad | \\mathrm { ~ \\gamma ~ } _ { \\mathrm { S } } ^ { \\mathrm { v } } { \\mathrm { ~ \\gamma ~ } } _ { \\mathrm { v } } ^ { \\mathrm { v } } \\mathrm { a n } ^ { \\prime } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 307, + 537, + 767, + 598 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We then create $\\mathcal { H } _ { \\mathrm { t e s t } }$ to contain all concepts which have any of the combinations above. For example, if a concept has both gray and 7, and is related to location, that is contains locationX? or locationY? keywords, we would place it in ${ \\mathcal { H } } _ { \\mathrm { t e s t } }$ . After every feasible candidate is placed in ${ \\mathcal { H } } _ { \\mathrm { t e s t } }$ based on this heurisitc, the remaining concepts in $\\mathcal { H }$ are assigned to ${ \\mathcal { H } } _ { \\mathrm { t r a i n } }$ . ", + "bbox": [ + 230, + 617, + 826, + 688 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• Intrinsic: This split forms cross product of all possible colors and materials in the dataset, and partitions a subset of such combinations that we only want to occur in test. We use the following tokens for test: ", + "bbox": [ + 215, + 693, + 826, + 736 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "‘green’, ‘metal’ | ‘purple’, ‘rubber’ | ‘cyan’, ‘rubber’ | ‘red’, ‘metal’ | ‘green’, ‘rubber’ ", + "bbox": [ + 302, + 744, + 867, + 773 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We then create $\\mathcal { H } _ { \\mathrm { t e s t } }$ to contain all concepts which have any of the combinations above. For example, if a concept has both green and metal, and is related to material, that is contains material? keyword, we would place it in $\\mathcal { H } _ { \\mathrm { t e s t } }$ . After every feasible candidate is placed in $\\mathcal { H } _ { \\mathrm { t e s t } }$ based on this heurisitc, the remaining concepts in $\\mathcal { H }$ are assigned to $\\mathcal { H } _ { \\mathrm { t r a i n } }$ • Binding (color): This split takes all possible colors in the dataset, and partitions a subset of colors that we only want to occur in test. We use the following tokens for test: ", + "bbox": [ + 228, + 795, + 825, + 852 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 217, + 858, + 826, + 887 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We then create ${ \\mathcal { H } } _ { \\mathrm { t e s t } }$ to contain all concepts which have any of the tokens above. For example, if a concept has purple, we would place it in $\\mathcal { H } _ { \\mathrm { t e s t } }$ . After every feasible candidate is placed in $\\mathcal { H } _ { \\mathrm { t e s t } }$ based on this heurisitc, the remaining concepts in $\\mathcal { H }$ are assigned to $\\mathcal { H } _ { \\mathrm { t r a i n } }$ . ", + "bbox": [ + 230, + 103, + 825, + 146 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "• Binding (shape): This split takes all possible shapes in the dataset, and partitions a subset of shapes that we only want to occur in test. We use the following tokens for test: ", + "bbox": [ + 217, + 150, + 825, + 179 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "‘cylinder’ ", + "bbox": [ + 313, + 185, + 406, + 200 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We then create ${ \\mathcal { H } } _ { \\mathrm { t e s t } }$ to contain all concepts which have any of the tokens above. For example, if a concept has cylinder, we would place it in $\\mathcal { H } _ { \\mathrm { t e s t } }$ . After every feasible candidate is placed in $\\mathcal { H } _ { \\mathrm { t e s t } }$ based on this heurisitc, the remaining concepts in $\\mathcal { H }$ are assigned to $\\mathcal { H } _ { \\mathrm { t r a i n } }$ . ", + "bbox": [ + 227, + 219, + 825, + 263 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "• Complexity: This split partitions into train and test based on length of the postfix serialization of the concept. Specifically, concepts shorter than 10 tokens are placed in $\\mathcal { H } _ { \\mathrm { t r a i n } }$ and longer concepts are placed in ${ \\mathcal { H } } _ { \\mathrm { t e s t } }$ . ", + "bbox": [ + 215, + 267, + 823, + 310 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "D CREATING SUPPORT AND QUERY SETS ", + "text_level": 1, + "bbox": [ + 174, + 329, + 534, + 347 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We next explain how we go from the initial dataset $\\mathcal { U } -$ which contains a large number of images, schema and sounds – and a concept space $\\mathcal { H } _ { \\mathrm { t r a i n } }$ and ${ \\mathcal { H } } _ { \\mathrm { t e s t } }$ , to a dataset for meta learning. To create the training/validation/test sets for models, we sample a series of episodes, each containing a support set and a query set. We illustrate the sampling procedure for a training episode below: ", + "bbox": [ + 174, + 361, + 826, + 417 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Support Set Sampling with Hard Negatives ", + "text_level": 1, + "bbox": [ + 174, + 424, + 477, + 439 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "1. Pick a concept $h \\sim p _ { \\mathrm { t r a i n } } ( h )$ , with a preference for shorter hypotheses being more frequent based on the weights used to define the prior Appendix B.3 \n2. Pick 5 images $( P )$ , uniformly at random from $\\mathcal { U }$ such that $h ( \\mathbf { u } ^ { s } ) = 1$ , where the concept is evaluated on the schema to determine the label Appendix B.4 \n3. Identify other concepts $h ^ { \\prime } \\in \\mathcal { H }$ s.t. $h ( u ( S ) ) = 1$ and $h ^ { \\prime } \\neq h$ \n4. Pick images such that $h ^ { \\prime } ( \\mathbf { u } ^ { s } ) = 1$ and $h ( \\mathbf { u } ^ { s } ) = 0$ as negatives $( N )$ . If no such images exist, pick random images from $\\mathcal { U }$ as negatives until we have 20 negatives. \n5. Return $D _ { \\operatorname { s u p p } } = P \\cup N$ . ", + "bbox": [ + 209, + 450, + 826, + 583 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The sampling procedure for the Query set iterates all the steps above (except step 1, where we choose the concept $h$ ). Step 3 and 4 outline a procedure for identifying hard negatives for training the model, by looking at other hypotheses which also explain a chosen set of positives $P$ and using them to clarify what the concept of interest is. ", + "bbox": [ + 174, + 594, + 825, + 650 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We give below an analogous procedure for easy negatives: ", + "bbox": [ + 173, + 656, + 555, + 671 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Support Set Sampling with Easy Negatives ", + "text_level": 1, + "bbox": [ + 174, + 678, + 473, + 693 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "1. Pick a concept $h \\sim p _ { \\mathrm { t r a i n } } ( h )$ , with a preference for shorter hypotheses being more frequent based on the weights used to define the prior Appendix B.3 \n2. Pick 5 images $( P )$ , uniformly at random from $\\mathcal { U }$ such that $h ( \\mathbf { u } ^ { s } ) = 1$ , where the concept is evaluated on the schema to determine the label Appendix B.4 \n3. Pick 20 random images from $\\mathcal { U }$ as negatives, $\\mathbf { N }$ . \n4. Return $D _ { \\operatorname { s u p p } } = P \\cup N$ . ", + "bbox": [ + 210, + 704, + 825, + 805 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Similar to hard negatives, the sampling procedure for the Query set iterates all the steps above (except step 1, where we choose the concept $h$ ). ", + "bbox": [ + 174, + 815, + 823, + 844 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E REPRODUCIBILITY AND HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 173, + 863, + 581, + 881 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For all the models in Figure. 4 in the main paper, we use the following hyperparameters. All the modalities are processed into a set of objects $\\ \\bar { \\{ o _ { i } \\} } _ { i = 1 } ^ { N }$ where each $o _ { i } \\in \\mathbb { R } ^ { { \\overline { { 6 4 } } } }$ for image and sound models while for schema $o _ { i } \\in \\mathbb { R } ^ { 9 6 }$ . Further, the we use a learning rate of $\\beth - 4$ for image models, $\\beth \\in - 3$ for schema models, and $5 \\in - 5$ using the best learning rate for each modality across an initial sweep. The batch size for image and sound models is 8 episodes per batch, while for schema we use a batch size of 64. All models use the Adam optimizer. The overall scene representation across all the modalities is set to 256, that is, $ { \\mathbf { u } } \\in \\mathbb { R } ^ { 2 5 6 }$ . All our models are initialized with the method proposed in (Glorot & Bengio, 2010), which we found to be crucial for training relation networks well. The initial representation from the first stage of the encoder (Fig. 4 in main paper) with the objects for images has a size of $\\mathsf { 1 0 x 8 }$ i.e. there are 80 objects, while for sound representations have 38 objects. In the schema case the number of objects is the ground truth number of objects which is provided as input to the model. ", + "bbox": [ + 174, + 895, + 825, + 925 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 242 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We trained all the models on the training set, and picked the best performing checkpoints on training – measured in terms of mAP– to report all the results in the main paper. Our image and schema models are trained for 1 million steps (16 epochs for images, 128 epochs for schemas) while the sound models are trained for 500K steps, and checkpoints are stored after every 30K steps. ", + "bbox": [ + 173, + 250, + 825, + 305 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "All our models fit on a GPU with 16GB capacity except the relation network trained with image inputs, which needs at 32GB GPU. We use the pytorch framework to implement all our models. ", + "bbox": [ + 173, + 313, + 823, + 340 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "F MODEL ARCHITECTURES FOR POOLING ", + "text_level": 1, + "bbox": [ + 174, + 362, + 540, + 377 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "In this section we detail the exact architectures used for the different pooling operations we consider in this paper as shown in Figure 4 center panel. We first establish some notation. Let $o _ { i } \\in \\mathbb { R } ^ { K }$ be the output object feature from the modality specific encoder (Figure 4, left panel), and let us denote by $O = \\{ o _ { i } \\} _ { i = 1 } ^ { | O | }$ the set of features for each of the objects in the scene, which includes optional position information indicating where the object is present in the scene (Figure 4). Let $N$ be the requested dimensionality of the feature space from the pooling operation. Given this, we can describe the pooling operations used as follows: ", + "bbox": [ + 174, + 392, + 825, + 493 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• avg-pool: We first average the representations across all the objects $\\{ o _ { i } \\} _ { i = 1 } ^ { | O | }$ and then pass the averaged representation through an MLP with $2 5 6 \\texttt { x } 5 1 2 \\texttt { x } 3 8 4 \\texttt { x N }$ units with batch normalization and rectified linear unit nonlinearity in the hidden layers. \nconcat: We first concatenate all the object representations in $O$ , followed by an MLP with $2 5 6 ~ \\mathrm { ~ x ~ } ~ 5 1 2 ~ \\mathrm { ~ x ~ } ~ 2 5 6 ~ \\mathrm { ~ x ~ } ~ \\mathrm { ~ N ~ }$ units with batch normalization and rectified linear units nonlinearity in the hidden layers. \nrelation-net: For relation networks, following (Santoro et al., 2017) we use relative position encoding that captures the relative positioning of the objects in a scene for image and sound modalities, and use the location information already present in the schema modality. Based on this, in the terminology of Santoro et al. (2017) our $g ( \\ u )$ MLP has $2 5 6 ~ \\times ~ 2 5 6 ~ \\times ~ 2 5 6$ $\\texttt { x } \\ 2 5 6$ hidden units with rectified linear unit non linearity and batch normalization whereas our $f ( )$ MLP has $2 5 6 ~ \\mathrm { ~ x ~ } ~ 2 5 6 ~ \\mathrm { ~ x ~ N ~ }$ units with recitifed linear unit non linearlity and batch normalization in the middle (non-output) layers. Different from the original paper, we do not use dropout as we did not observe any overfitting in our experiments. \ntransformer: We use a 2-head multi-head attention layer stacked 4 times, with the feedforward network dimenstions set to 512. After forwarding through this module, we take the output vectors $o _ { i } ^ { \\prime }$ for each object processed through these initial layers and pool across objects by doing max(), mean(), sum(), min() operations and concatenating their outputs, similar to previous work by Wang et al. (2019). The final representation then does a linear projection of this concatenated vector to $N$ , the dimensionality expected from the pooling module. ", + "bbox": [ + 212, + 506, + 826, + 818 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "G ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 838, + 400, + 853 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "G.1 HYPERPARAMETER SWEEPS – OBJECT FEATURE DIMENSIONS ", + "text_level": 1, + "bbox": [ + 176, + 871, + 640, + 883 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We next show the hyperparameter sweeps for image models in deterimining the choice of the dimensionality to represent each object $o _ { i }$ for our image models (Figure 14). The same choice of dimensionality was replicated for sound models. In our initial sweeps, on the Concept IID split, across different choices of the dimensionality of objects, we found relation networks to outperform concat and global average pooling models substantially, and thus we picked the object dimensions based on what performs best for relation networks since overall we are interested in the best possible choice of models for a given split and modality. Based on the results in Figure 14 we picked $\\boldsymbol { \\partial } _ { i } \\in \\mathbb { R } ^ { 6 4 }$ . ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/bf6a77c2a6709f6ce6e6adbfe162da677142fe8242a7f9fdec34d5e5babb42d4.jpg", + "image_caption": [ + "Figure 14: mAP on validation for hard negatives (y-axis) vs number of training steps $\\mathbf { \\bar { x } }$ -axis) for relation network models on images with different dimensionality for the object embedding $o _ { i }$ . " + ], + "image_footnote": [], + "bbox": [ + 225, + 118, + 761, + 325 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/8792990cb4db47ca305e666971ad1dbd79120b4c495ea476a7e83107a22bfff9.jpg", + "image_caption": [ + "Figure 15: mAP on validation for hard negatives (y-axis) vs number of training steps $\\mathbf { \\check { x } }$ -axis) for relation network models on images with different learning rates. " + ], + "image_footnote": [], + "bbox": [ + 225, + 396, + 761, + 603 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 648, + 825, + 718 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "G.2 IMAGE RELATION NETWORKS LEARNING RATE SWEEPS ", + "text_level": 1, + "bbox": [ + 174, + 739, + 599, + 752 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We picked the learning rate for image models based on the best performing image relation network model, which across an initial sweep we found to yeild the best class of models. Figure 15 shows the performance of the models across learning rates of $\\{ 1 { \\mathrm { e } } { - } 4 , \\quad 5 { \\mathrm { e } } { - } 4 , \\quad 2 . 5 { \\mathrm { e } } { - } 4 \\}$ . ", + "bbox": [ + 176, + 765, + 823, + 808 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "G.3 SWEEP ON USE OF LANGUAGE ", + "text_level": 1, + "bbox": [ + 176, + 827, + 429, + 840 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "As explained in the main paper (Figure. 4), the parameter $\\alpha$ controls the tradeoff between the query accuracy and the likelihood of the concept expressed as a prefix string. We generally found across a broad range of values in $\\{ \\boldsymbol { 0 } . \\boldsymbol { 0 } , 0 . 0 1 , 0 . 1 \\boldsymbol { 0 } , 1 . 0 \\}$ the models generally performed the best at $\\alpha = 1 . 0$ . Our initial experiments with $\\alpha = 1 0 . 0$ suggested substantially worse performance so we discareded it from the sweep. See Figures 16 to 18 for the corresponding results. ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/e9bf606123ff5f6f556c7e2091ebdef2828453c8304b3070a9b6a2cbee43b705.jpg", + "image_caption": [ + "Figure 16: mAP on validation for hard negatives (y-axis) vs number of training steps ( $\\mathbf { \\dot { x } }$ -axis) for relation network models on images with different amounts of language usage by varying the parameter $\\alpha$ . " + ], + "image_footnote": [], + "bbox": [ + 225, + 122, + 764, + 328 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/c19322b9e649e0a16c2e6020f8b351c538174e76228d31d8fd5271b29f1d618f.jpg", + "image_caption": [ + "Figure 17: mAP on validation for hard negatives (y-axis) vs number of training steps $\\mathbf { \\bar { x } }$ -axis) for relation network models on schemas with different amounts of language usage by varying the parameter $\\alpha$ . " + ], + "image_footnote": [], + "bbox": [ + 225, + 396, + 761, + 604 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/46b42656f77da39ba1de9d0cb18b1e13567d2412ee84e050fdb73f8279e66e4a.jpg", + "image_caption": [ + "Figure 18: mAP on validation for hard negatives (y-axis) vs number of training steps $\\mathbf { \\dot { x } }$ -axis) for concat pooling models on schemas with different amounts of language usage by varying the parameter $\\alpha$ . " + ], + "image_footnote": [], + "bbox": [ + 240, + 672, + 764, + 880 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/926daed86f654d1c15bce3718d9a58607179ab3f387dd0059eb7185d13f96b32.jpg", + "image_caption": [ + "Figure 19: mAP (top) and accuracy (bottom) metrics for the different splits presented in the paper when easy negatives are used (ordered by their corresponding comp gap). Yellow: shows the schema relation-net model while purple: shows the image relation-net model. Notice that compared to Fig. 5 in the main paper, the comp gap is smaller and the models appear to be substantially closer to the strong oracle in this setting compared to when we use hard negatives. " + ], + "image_footnote": [], + "bbox": [ + 181, + 106, + 820, + 400 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "G.4 RESULTS ON EASY NEGATIVES ", + "text_level": 1, + "bbox": [ + 176, + 479, + 434, + 493 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In Figure 19 we show results for the relation-net model on various splits, where easy negatives are used to populate the support and query sets during training and evaluation, unlike the case of hard negatives discussed in the main paper (Figure. 5). Notice that the compositionality gap (comp gap) is lower in general for easy negatives compared to the hard negatives as reported in the main paper. Further, we find that the best models are substantially closer to the strong oracle compared to Figure. 5 main paper, showing that on the easier, less compositional task it is easier for machine learning models to approach the strong oracle (especially in terms of accuracy). Finally, it is interesting to note that with easy negatives it appears that the best models outperform the weak oracle on the Counting split, while with the hard negatives one finds that the models are worse than the weak oracle, suggesting poor generalization for counting. ", + "bbox": [ + 173, + 508, + 825, + 648 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "G.5 FINER $\\alpha$ SWEEP FOR COUNTING ", + "text_level": 1, + "bbox": [ + 176, + 672, + 437, + 688 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Finally, we ran a finer alpha sweep for the Counting split since it appeared on our initial sweep that the counting split was not performing better with language. Concretely, we ran a new set of experiments sweeping over $\\alpha$ values of $\\{ \\boldsymbol { 0 } , 0 1 , \\quad 0 . 1 0 , \\quad 1 . 0 , \\quad 5 . 0 , \\quad 1 0 . 0 , \\quad 1 0 0 . 0 \\}$ . Across this broader range of values, we found models still did not show any statistically significant gains from using language v.s. not for the Counting split. ", + "bbox": [ + 174, + 702, + 825, + 772 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "G.6 CHOICE OF METRIC: MAP v.s. ACCURACY ", + "text_level": 1, + "bbox": [ + 174, + 797, + 506, + 811 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In general, the mAP metric opens up a larger comp gap for the various splits than indicated by CBA. For example, with hard negatives, while CBA indicates a gap of $1 4 . 2 \\%$ for Counting compared to $0 \\%$ for Instance IID, mAP suggests a gap of $3 4 . 4 \\%$ for Counting relative to $0 \\%$ for Instance IID. For the Binding (color) split its $8 6 . 5 \\%$ comp gap (mAP) v.s. $3 4 . 0 \\%$ for CBA. mAP, while being more expensive to compute evaluates more thoroughly to test if a concept $h$ is truly learnt by the model, by probing its performance on a large, representative set of negatives $\\tau$ , providing a more stringent test of compositional generalization. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/644337f4c67e6963dc7a44600e16a64ddf30646fb43b574dc02e3eeccc51037d.jpg", + "table_caption": [ + "Table 1: Performance on meta-test, sorted based on mAP $( \\mathrm { i n } \\% )$ on Binding (color) with hard negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
schematransformer15.90.9
schemarelation-net15.30.9
imageavg-pool15.10.4
imagerelation-net14.80.7
schemaavg-pool14.41.2
imagetransformer14.20.2
schemaconcat14.00.7
imageconcat13.50.4
soundavg-pool9.40.6
soundrelation-net9.20.3
soundconcat8.00.6
", + "bbox": [ + 313, + 116, + 681, + 297 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/c662a08e77ae4907298777059fe512e6709943d9216cd97e8cb2519836b1c16e.jpg", + "table_caption": [ + "Table 2: Performance on meta-test, sorted based on mAP (in $\\%$ ) on Boolean with hard negatives " + ], + "table_footnote": [ + "G.7 DETAILED RESULTS ON ALL THE SPLITS IN HARD NEGATIVES SETTING " + ], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
schemarelation-net51.11.4
schemaavg-pool48.02.1
schemaconcat47.12.1
schematransformer46.91.1
imagetransformer44.02.2
imagerelation-net38.41.0
imageavg-pool36.32.3
imageconcat29.62.4
soundrelation-net25.50.8
soundavg-pool24.11.6
soundconcat22.31.1
", + "bbox": [ + 313, + 327, + 687, + 508 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "In this section we provide the full results of all of the tested models on each of the splits considered in the paper, in the hard negatives setting. Tables 1-9 show the results of different models (sorted in a descending order based on mAP for each of the splits considered in the paper, in the case where models do not have access to language. ", + "bbox": [ + 174, + 539, + 825, + 594 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "G.8 DETAILED RESULTS ON ALL THE SPLITS IN EASY NEGATIVES SETTING. ", + "text_level": 1, + "bbox": [ + 176, + 612, + 705, + 626 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "In this section we provide the full results of all of the tested models on each of the splits considered in the paper, in the easy negatives setting. Tables 10-18 show the results of different models (sorted in a descending order based on mAP for each of the splits considered in the paper, in the case where models do not have access to language. Note that we did not evaluate transformer models or sound models in this setting as this is qualitatively less interesting than the hard negatives setting and is not the main focus of the paper. ", + "bbox": [ + 174, + 637, + 826, + 720 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/8740cecc3963af23d4b46c7d86147d6845b10a9c9c9884f3cd1ad00e61aa5f11.jpg", + "table_caption": [ + "Table 3: Performance on meta-test, sorted based on mAP (in $\\%$ ) on Intrinsic with hard negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
schematransformer57.90.6
schemarelation-net55.10.9
imagetransformer54.13.0
schemaavg-pool53.33.0
schemaconcat52.23.5
imagerelation-net47.67.5
imageavg-pool34.50.4
imageconcat34.01.0
soundconcat29.41.6
soundavg-pool26.52.3
soundrelation-net24.32.4
", + "bbox": [ + 315, + 739, + 681, + 921 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/8b9141020d9fa6c3deb0f79d0e54df1aedc7b7b5d0874fce711334bb2250f4cb.jpg", + "table_caption": [ + "Table 4: Performance on meta-test, sorted based on mAP (in $\\%$ ) on Concept IID with hard negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
imagetransformer60.80.4
schemarelation-net60.70.3
schematransformer59.80.2
imagerelation-net58.50.7
imageavg-pool56.71.0
schemaavg-pool54.61.3
schemaconcat53.30.9
imageconcat53.00.6
soundconcat23.62.7
soundavg-pool22.72.3
soundrelation-net21.8NaN
", + "bbox": [ + 315, + 117, + 681, + 299 + ], + "page_idx": 25 + }, + { + "type": "table", + "img_path": "images/e3c9f4e8e0b4ac361855e9eafb6160c40841c26d2ab48e168107a77bf32330f3.jpg", + "table_caption": [ + "Table 5: Performance on meta-test, sorted based on mAP $( \\mathrm { i n } \\%$ ) on Instance IID with hard negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
schemarelation-net63.90.4
imagetransformer61.50.9
schematransformer59.80.5
imagerelation-net58.61.3
imageavg-pool58.10.7
schemaavg-pool57.40.9
imageconcat57.10.9
schemaconcat57.11.4
soundavg-pool23.51.4
soundrelation-net22.32.3
soundconcat21.70.5
", + "bbox": [ + 313, + 325, + 681, + 506 + ], + "page_idx": 25 + }, + { + "type": "table", + "img_path": "images/8f5bdafb93b699e92b04765246c7fcf3f95ff546b368a39fa7ea600a5c9fb66d.jpg", + "table_caption": [ + "Table 6: Performance on meta-test, sorted based on mAP (in $\\%$ ) on Extrinsic with hard negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
imagetransformer62.10.8
schematransformer60.90.6
imagerelation-net60.81.1
imageconcat60.80.7
schemarelation-net59.80.4
imageavg-pool59.41.3
schemaconcat54.32.3
schemaavg-pool53.41.5
soundavg-pool25.7NaN
soundconcat23.21.4
soundrelation-net18.31.7
", + "bbox": [ + 313, + 531, + 681, + 713 + ], + "page_idx": 25 + }, + { + "type": "table", + "img_path": "images/05c0e779630959daef838b9bc5252d18231c9715365ae8ba05b4011fc3a1fc99.jpg", + "table_caption": [ + "Table 7: Performance on meta-test, sorted based on mAP (in $\\%$ ) on Complexity with hard negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
schemarelation-net52.80.7
schematransformer49.70.6
schemaavg-pool48.41.0
schemaconcat48.01.2
imageavg-pool45.80.4
imagetransformer45.60.2
imagerelation-net45.40.4
imageconcat42.63.4
soundconcat26.0NaN
soundrelation-net23.31.0
soundavg-pool23.01.7
", + "bbox": [ + 315, + 739, + 681, + 920 + ], + "page_idx": 25 + }, + { + "type": "table", + "img_path": "images/c2691d2a11dfc37464944f5d9b9749a2d048aea59f756c2f9923c99c5be65113.jpg", + "table_caption": [ + "Table 8: Performance on meta-test, sorted based on mAP (in $\\%$ ) on Binding (shape) with hard negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
schematransformer30.90.6
schemarelation-net30.41.5
imagerelation-net29.11.8
imageavg-pool28.71.0
imagetransformer28.70.1
schemaavg-pool28.31.9
imageconcat27.00.8
schemaconcat27.00.8
soundrelation-net17.50.4
soundconcat17.41.3
soundavg-pool16.21.6
", + "bbox": [ + 313, + 117, + 681, + 299 + ], + "page_idx": 26 + }, + { + "type": "table", + "img_path": "images/402ba8e37e649ede5af304a7e1788284080ce52e158dbc3a54d49388f99ceff4.jpg", + "table_caption": [ + "Table 9: Performance on meta-test, sorted based on mAP (in $\\%$ ) on Counting with hard negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
schemarelation-net57.44.8
schemaavg-pool55.06.4
schemaconcat50.03.2
imageavg-pool48.51.3
schematransformer40.70.4
imagerelation-net40.32.0
imageconcat38.52.8
imagetransformer32.41.4
soundconcat13.82.4
soundavg-pool13.41.0
soundrelation-net13.01.7
", + "bbox": [ + 313, + 325, + 681, + 506 + ], + "page_idx": 26 + }, + { + "type": "table", + "img_path": "images/9555e919de8e8a6cf16fafbf2aa3c5b1adb724d606156d5aa00aecc6742fae12.jpg", + "table_caption": [ + "Table 10: Performance on meta-test, sorted based on mAP (in $\\%$ ) on Binding (color) with easy negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
schemarelation-net18.61.6
imagerelation-net16.30.8
imageavg-pool16.30.5
imageconcat15.60.5
schemaavg-pool15.50.5
schemaconcat15.50.5
", + "bbox": [ + 315, + 531, + 681, + 645 + ], + "page_idx": 26 + }, + { + "type": "table", + "img_path": "images/7bbb15137772f07a962a77da643f9d107289eab605edbb7a43908156ecc5df21.jpg", + "table_caption": [ + "Table 11: Performance on meta-test, sorted based on mAP (in $\\%$ ) on Boolean with easy negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
schemarelation-net58.12.0
schemaavg-pool55.02.1
schemaconcat54.72.2
imagerelation-net50.77.3
imageavg-pool42.91.6
imageconcat40.91.7
", + "bbox": [ + 315, + 670, + 681, + 782 + ], + "page_idx": 26 + }, + { + "type": "table", + "img_path": "images/6472b56d0ad7d936c05ad9da812abbe484557072026ce6a8dd678c48c34d30e6.jpg", + "table_caption": [ + "Table 12: Performance on meta-test, sorted based on mAP (in $\\%$ ) on Counting with easy negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
schemarelation-net67.98.5
schemaavg-pool64.47.3
imageavg-pool62.26.6
imageconcat61.26.9
schemaconcat60.86.7
imagerelation-net58.85.7
", + "bbox": [ + 315, + 808, + 681, + 921 + ], + "page_idx": 26 + }, + { + "type": "table", + "img_path": "images/4021164103016f73202251e8772a60bd309a4fda39a9d87ee779b687568255f9.jpg", + "table_caption": [ + "Table 13: Performance on meta-test, sorted based on mAP (in $\\%$ ) on Extrinsic with easy negatives " + ], + "table_footnote": [], + "table_body": "
modalitypoolingmAP (mean)mAP (std)
schemarelation-net63.71.4
imagerelation-net62.91.1
imageconcat61.82.4
imageavg-pool61.51.4
schemaavg-pool57.51.2
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Extensive evaluations across a range of modeling choices spanning different", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 352, + 470, + 365 + ], + "spans": [ + { + "bbox": [ + 142, + 352, + 470, + 365 + ], + "score": 1.0, + "content": "modalities (image, schemas, and sounds), splits, privileged auxiliary concept infor-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 363, + 469, + 375 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 469, + 375 + ], + "score": 1.0, + "content": "mation, and choices of negatives reveal substantial scope for modeling advances", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 374, + 416, + 386 + ], + "spans": [ + { + "bbox": [ + 141, + 374, + 416, + 386 + ], + "score": 1.0, + "content": "on the proposed task. All code and datasets will be available online.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5, + "bbox_fs": [ + 141, + 209, + 470, + 386 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 404, + 206, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 208, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 208, + 420 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "Human concept learning is more flexible than today’s AI systems. Human conceptual knowledge is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "productive: people can understand and generate novel concepts via compositions of existing concepts", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "score": 1.0, + "content": "(“an apartment dog”) (Murphy, 2002), unlike standard machine classifiers that are limited to a fixed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "set of classes (“dog”, “cat”, etc.). 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Three example concepts (rows) along with schematic positive examples. Actual", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 234, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 506, + 245 + ], + "score": 1.0, + "content": "scenes are rendered in multiple ways including the CLEVR renderer (Johnson et al., 2016) (see Figure 2). 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Concretely, the CURI task tests", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 307, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 322 + ], + "score": 1.0, + "content": "few-shot learning of relational concepts in a large compositional conceptual space, with design", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 318, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 333 + ], + "score": 1.0, + "content": "inspiration from studies in cognitive modeling using a language of thought (LOT) approach (Fodor,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 329, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 343 + ], + "score": 1.0, + "content": "1975; Piantadosi, 2011; Kemp et al., 2005). CURI includes scene-based concepts such as “All objects", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "have the same color” and “There exists a blue object while the rest are triangles” (Figure 1) but", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "unlike CLEVR (Johnson et al., 2016) there are too few examples to deduce answers with certainty.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "Our benchmark is defined through a series of meta-learning episodes (see example in Figure 2):", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 327, + 388 + ], + "score": 1.0, + "content": "given positive and negative examples of a new concept", + "type": "text" + }, + { + "bbox": [ + 327, + 374, + 351, + 386 + ], + "score": 0.89, + "content": "D _ { \\mathrm { s u p p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "(known as the “support set”), the goal", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 384, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 281, + 399 + ], + "score": 1.0, + "content": "of an episode is to classify new examples", + "type": "text" + }, + { + "bbox": [ + 281, + 385, + 307, + 397 + ], + "score": 0.91, + "content": "D _ { \\mathrm { q u e r y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 384, + 506, + 399 + ], + "score": 1.0, + "content": "(the “query set”). As in few-shot classification", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "score": 1.0, + "content": "(Fei-Fei et al., 2006), meta-learning (Vinyals et al., 2016), and other open-set tasks (Lampert et al.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "2014), models are evaluated on novel classes outside the (meta-)training set. Unlike previous", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "work (Triantafillou et al., 2019; Lake et al., 2019) that focuses on atomic concepts, our benchmarks", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "concerns more structured, relational concepts built compositionally from a set of atomic concepts,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "and involves reasoning under uncertainty – an ideal learner must marginalize over many hypotheses", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "when making predictions (Gelman et al., 2004; Xu & Tenenbaum, 2007; Piantadosi et al., 2016).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 297, + 506, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 297, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 297, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 297, + 491 + ], + "score": 1.0, + "content": "We also vary the modality in which scenes are", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 489, + 297, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 297, + 501 + ], + "score": 1.0, + "content": "presented—rendering them as images, symbolic", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 500, + 297, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 297, + 513 + ], + "score": 1.0, + "content": "schemas, and sounds— enabling future research", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 511, + 298, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 298, + 523 + ], + "score": 1.0, + "content": "on modality-specific representational choices for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 523, + 298, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 298, + 534 + ], + "score": 1.0, + "content": "compositional reasoning under uncertainty. Fi-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 533, + 298, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 298, + 545 + ], + "score": 1.0, + "content": "nally, we vary the concepts learned by the model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 544, + 299, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 299, + 556 + ], + "score": 1.0, + "content": "during meta-training and meta-testing to test dif-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 556, + 280, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 280, + 568 + ], + "score": 1.0, + "content": "ferent aspects of systematic generalization.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 479, + 299, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 296, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 571, + 298, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 298, + 585 + ], + "score": 1.0, + "content": "Compositionality Gap. In addition to defining", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 583, + 297, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 297, + 595 + ], + "score": 1.0, + "content": "systematic splits, we also characterize (for the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 594, + 298, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 298, + 606 + ], + "score": 1.0, + "content": "first time, in our knowledge), the difficulty of", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 571, + 298, + 606 + ] + }, + { + "type": "image", + "bbox": [ + 308, + 472, + 499, + 542 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 308, + 472, + 499, + 542 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 308, + 472, + 499, + 542 + ], + "spans": [ + { + "bbox": [ + 308, + 472, + 499, + 542 + ], + "score": 0.969, + "type": "image", + "image_path": "bcbd5ef86b8117329029374b7393e74ddac4a499b1f454de03a928ce74d8a8ce.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 308, + 472, + 499, + 486.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 308, + 486.0, + 499, + 500.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 308, + 500.0, + 499, + 514.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 308, + 514.0, + 499, + 528.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 308, + 528.0, + 499, + 542.0 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 546, + 505, + 586 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 303, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 303, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "Figure 2: CURI Task. Given as input a support set", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 305, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 305, + 556, + 325, + 567 + ], + "score": 0.89, + "content": "D _ { \\mathrm { s u p p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 556, + 506, + 568 + ], + "score": 1.0, + "content": ", with positive and negative examples correspond-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 303, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 303, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "ing to concept, the model has to infer the concept and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 303, + 576, + 498, + 587 + ], + "spans": [ + { + "bbox": [ + 303, + 576, + 498, + 587 + ], + "score": 1.0, + "content": "produce accurate predictions on novel images (right).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + } + ], + "index": 39.25 + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "generalization entailed by each split by introducing the notion of a model-independent “composi-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "tionality gap”. Concretely, the compositionality gap is the difference in test performance between", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "an ideal Bayesian learner with access to the full hypothesis space, and a Bayesian learner with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 637, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 652 + ], + "score": 1.0, + "content": "access to only a (potentially large) list of the hypotheses examined during meta-training. A large gap", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "indicates that any learner must extrapolate compositionally from the training hypotheses to solve the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "task; additionally, models can be compared to ideal learners that either do or do not engage in such", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "extrapolation. We anticipate that this tool will be more broadly useful for analyzing other benchmarks", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 681, + 211, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 211, + 695 + ], + "score": 1.0, + "content": "with compositional splits.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 605, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "score": 1.0, + "content": "Models. We evaluate models around various dimensions which concern the difficulty of learning", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "productive concepts under uncertainty, including: 1) the modality in which the input is rendered", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "(image, schemas, sounds), 2) method used for reasoning across objects in a scene (transformer,", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "relation-network, global average pooling, concatenation), 3) whether or not training provides ground-", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "truth symbolic descriptions of concepts, and 4) how negative examples are sampled. 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All valid (type-consistent) compositions of functions are potential complex", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "score": 1.0, + "content": "concepts in our dataset. Note that the functions are illustrated for the case of images and schemas. Location,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 211, + 327, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 327, + 221 + ], + "score": 1.0, + "content": "size, shape etc. correspond to different properties for sounds.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 505, + 269 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "relation-network, global average pooling, concatenation), 3) whether or not training provides ground-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "truth symbolic descriptions of concepts, and 4) how negative examples are sampled. Overall, our", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "evaluations suggest that there is substantial room for improvement in compositional reasoning under", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "uncertainty, w.r.t the compositionality gap, representing a novel challenge for compositional learning.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "Summary of contributions: 1) We introduce the Compositional Reasoning Under Uncertainty", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "(CURI) benchmark for evaluating compositional, relational learning under uncertainty from observa-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 296, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 310 + ], + "score": 1.0, + "content": "tional data; 2) We introduce a ‘compositionality gap’ metric for measuring the difficulty of systematic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 307, + 502, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 502, + 321 + ], + "score": 1.0, + "content": "generalization from train to test; 3) We provide various baseline models for benchmarking progess.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 335, + 211, + 348 + ], + "lines": [ + { + "bbox": [ + 104, + 334, + 213, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 334, + 213, + 351 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 360, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "Compositional Learning. Related work has examined systematic generalization in pattern com-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "pletion using Raven’s matrices (PGM) (Santoro et al., 2017; Hill et al., 2019) and visual question", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 381, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 381, + 505, + 395 + ], + "score": 1.0, + "content": "answering with CLEVR (Johnson et al., 2016; Bahdanau et al., 2019). CURI’s use of the CLEVR", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 394, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 506, + 405 + ], + "score": 1.0, + "content": "renderer further invites particular comparison with that benchmark. Compared to these more deduc-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 405, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 506, + 416 + ], + "score": 1.0, + "content": "tive reasoning tests, CURI examines few-shot concept learning under substantial inherent uncertainty.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 414, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 429 + ], + "score": 1.0, + "content": "Unlike puzzle solving or question answering, an ideal inductive learner on CURI cannot know the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "right rule with certainty. In essence, unlike CLEVR the “question” to be answered is not given to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "score": 1.0, + "content": "model as input, but must be inferred – making the task more challenging. While PGMs do involve", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "such an inference, once the constraints of a puzzle are identified, it does not: 1) have any uncertainty", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "in the reasoning (which is crucial) and 2) involve any “concept” learning – where a concept applies to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 470, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 506, + 482 + ], + "score": 1.0, + "content": "multiple images – as much as it involves “instance” matching to complete a sequence. In contrast,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "a successful CURI model behaves as if marginalizing over many hypotheses consistent with the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "observations e.g., (Tenenbaum & Griffiths, 2001; Xu & Tenenbaum, 2007; Piantadosi et al., 2016),", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "an ability which is rarely studied directly in deep learning models (although see (Grant et al., 2019)).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "Recently, Keysers et al. 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This is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "complementary to our splits, which provide interpretable notions of what each split tests such as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 567 + ], + "score": 1.0, + "content": "disentangling, complexity, variable binding etc. Moreover, our variable binding split is predicated on", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 562, + 500, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 500, + 578 + ], + "score": 1.0, + "content": "having different atoms between train and test, and thus cannot be recovered by their methodology.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "score": 1.0, + "content": "Language of Thought (LOT). Our choice of compositional concepts was most closely inspired by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "(Piantadosi et al., 2016) along with other studies of human concept learning in the Language of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "Thought (LOT) framework (Fodor, 1975; Goodman et al., 2008; Kemp & Jern, 2009; Piantadosi", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "et al., 2012; Goodman et al., 2015; Overlan et al., 2017; Lake & Piantadosi, 2019). In typical LOT", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 304, + 637 + ], + "score": 1.0, + "content": "studies of human learning, the conceptual space", + "type": "text" + }, + { + "bbox": [ + 305, + 625, + 315, + 635 + ], + "score": 0.78, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "is defined through a probabilistic context-free", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 145, + 648 + ], + "score": 1.0, + "content": "grammar", + "type": "text" + }, + { + "bbox": [ + 145, + 636, + 153, + 646 + ], + "score": 0.77, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 635, + 505, + 648 + ], + "score": 1.0, + "content": ", which specifies a set of conceptual primitives and their rules of combination. Here, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 223, + 659 + ], + "score": 1.0, + "content": "use a LOT-inspired grammar", + "type": "text" + }, + { + "bbox": [ + 223, + 647, + 231, + 657 + ], + "score": 0.83, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 646, + 387, + 659 + ], + "score": 1.0, + "content": "to generate an unbounded set concepts", + "type": "text" + }, + { + "bbox": [ + 388, + 647, + 397, + 657 + ], + "score": 0.68, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 646, + 505, + 659 + ], + "score": 1.0, + "content": ", while evaluating machine", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 658, + 358, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 358, + 669 + ], + "score": 1.0, + "content": "learning models trained without access to the underlying LOT.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 106, + 684, + 489, + 697 + ], + "lines": [ + { + "bbox": [ + 103, + 682, + 492, + 701 + ], + "spans": [ + { + "bbox": [ + 103, + 682, + 492, + 701 + ], + "score": 1.0, + "content": "3 COMPOSITIONAL REASONING UNDER UNCERTAINTY (CURI) DATASET", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "Concept space. The compositional concepts in CURI were inspired by the empirical and cognitive", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 721, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 733 + ], + "score": 1.0, + "content": "modeling work of Piantadosi et al. (2016). 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All valid (type-consistent) compositions of functions are potential complex", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "score": 1.0, + "content": "concepts in our dataset. Note that the functions are illustrated for the case of images and schemas. Location,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 211, + 327, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 327, + 221 + ], + "score": 1.0, + "content": "size, shape etc. correspond to different properties for sounds.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 505, + 269 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 105, + 225, + 506, + 271 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "Summary of contributions: 1) We introduce the Compositional Reasoning Under Uncertainty", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "(CURI) benchmark for evaluating compositional, relational learning under uncertainty from observa-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 296, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 310 + ], + "score": 1.0, + "content": "tional data; 2) We introduce a ‘compositionality gap’ metric for measuring the difficulty of systematic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 307, + 502, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 502, + 321 + ], + "score": 1.0, + "content": "generalization from train to test; 3) We provide various baseline models for benchmarking progess.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 274, + 506, + 321 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 335, + 211, + 348 + ], + "lines": [ + { + "bbox": [ + 104, + 334, + 213, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 334, + 213, + 351 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 360, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "Compositional Learning. Related work has examined systematic generalization in pattern com-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "pletion using Raven’s matrices (PGM) (Santoro et al., 2017; Hill et al., 2019) and visual question", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 381, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 381, + 505, + 395 + ], + "score": 1.0, + "content": "answering with CLEVR (Johnson et al., 2016; Bahdanau et al., 2019). CURI’s use of the CLEVR", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 394, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 506, + 405 + ], + "score": 1.0, + "content": "renderer further invites particular comparison with that benchmark. Compared to these more deduc-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 405, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 506, + 416 + ], + "score": 1.0, + "content": "tive reasoning tests, CURI examines few-shot concept learning under substantial inherent uncertainty.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 414, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 429 + ], + "score": 1.0, + "content": "Unlike puzzle solving or question answering, an ideal inductive learner on CURI cannot know the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "right rule with certainty. 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In contrast,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "a successful CURI model behaves as if marginalizing over many hypotheses consistent with the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "observations e.g., (Tenenbaum & Griffiths, 2001; Xu & Tenenbaum, 2007; Piantadosi et al., 2016),", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "an ability which is rarely studied directly in deep learning models (although see (Grant et al., 2019)).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 360, + 506, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "Recently, Keysers et al. 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This is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "complementary to our splits, which provide interpretable notions of what each split tests such as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 567 + ], + "score": 1.0, + "content": "disentangling, complexity, variable binding etc. Moreover, our variable binding split is predicated on", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 562, + 500, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 500, + 578 + ], + "score": 1.0, + "content": "having different atoms between train and test, and thus cannot be recovered by their methodology.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 520, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "score": 1.0, + "content": "Language of Thought (LOT). Our choice of compositional concepts was most closely inspired by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "(Piantadosi et al., 2016) along with other studies of human concept learning in the Language of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "Thought (LOT) framework (Fodor, 1975; Goodman et al., 2008; Kemp & Jern, 2009; Piantadosi", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "et al., 2012; Goodman et al., 2015; Overlan et al., 2017; Lake & Piantadosi, 2019). In typical LOT", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 304, + 637 + ], + "score": 1.0, + "content": "studies of human learning, the conceptual space", + "type": "text" + }, + { + "bbox": [ + 305, + 625, + 315, + 635 + ], + "score": 0.78, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "is defined through a probabilistic context-free", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 145, + 648 + ], + "score": 1.0, + "content": "grammar", + "type": "text" + }, + { + "bbox": [ + 145, + 636, + 153, + 646 + ], + "score": 0.77, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 635, + 505, + 648 + ], + "score": 1.0, + "content": ", which specifies a set of conceptual primitives and their rules of combination. Here, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 223, + 659 + ], + "score": 1.0, + "content": "use a LOT-inspired grammar", + "type": "text" + }, + { + "bbox": [ + 223, + 647, + 231, + 657 + ], + "score": 0.83, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 646, + 387, + 659 + ], + "score": 1.0, + "content": "to generate an unbounded set concepts", + "type": "text" + }, + { + "bbox": [ + 388, + 647, + 397, + 657 + ], + "score": 0.68, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 646, + 505, + 659 + ], + "score": 1.0, + "content": ", while evaluating machine", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 658, + 358, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 358, + 669 + ], + "score": 1.0, + "content": "learning models trained without access to the underlying LOT.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 580, + 505, + 669 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 684, + 489, + 697 + ], + "lines": [ + { + "bbox": [ + 103, + 682, + 492, + 701 + ], + "spans": [ + { + "bbox": [ + 103, + 682, + 492, + 701 + ], + "score": 1.0, + "content": "3 COMPOSITIONAL REASONING UNDER UNCERTAINTY (CURI) DATASET", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "Concept space. The compositional concepts in CURI were inspired by the empirical and cognitive", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 721, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 733 + ], + "score": 1.0, + "content": "modeling work of Piantadosi et al. (2016). The space of concepts (LOT) is defined by a context", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 164, + 95 + ], + "score": 1.0, + "content": "free grammar", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 164, + 83, + 177, + 94 + ], + "score": 0.77, + "content": "( { \\mathcal { G } } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 178, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ". Figure 3 shows the LOT and specifies how primitives and functions compose to", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 394, + 106 + ], + "score": 1.0, + "content": "produce a large unbounded concept space. The LOT has three variables:", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 394, + 96, + 402, + 104 + ], + "score": 0.33, + "content": "\\mathbf { x }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 402, + 93, + 505, + 106 + ], + "score": 1.0, + "content": ", representing an object in", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 103, + 507, + 120 + ], + "spans": [ + { + "bbox": [ + 104, + 103, + 140, + 120 + ], + "score": 1.0, + "content": "a scene,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 140, + 104, + 190, + 117 + ], + "score": 0.94, + "content": "S = \\mathbf { \\bar { \\{ x \\} } } _ { i = 1 } ^ { N }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 191, + 103, + 391, + 120 + ], + "score": 1.0, + "content": "representing the set of all objects in the scene, and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 392, + 104, + 450, + 117 + ], + "score": 0.93, + "content": "S _ { - \\mathbf { x } } \\overset { \\cdot } { = } S / \\{ \\mathbf { x } \\}", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 451, + 103, + 507, + 120 + ], + "score": 1.0, + "content": ", representing", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "the set of all objects in the scene except x. Each concept describes a rule composed of object and", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "scene properties, logical operators, and/or comparison operators, and can be evaluated on a given", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 336, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 131, + 149 + ], + "score": 1.0, + "content": "scene", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 131, + 138, + 139, + 147 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 140, + 137, + 336, + 149 + ], + "score": 1.0, + "content": "to determine whether the scene satisfies the rule.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 43.5, + "bbox_fs": [ + 106, + 709, + 504, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 164, + 95 + ], + "score": 1.0, + "content": "free grammar", + "type": "text" + }, + { + "bbox": [ + 164, + 83, + 177, + 94 + ], + "score": 0.77, + "content": "( { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ". Figure 3 shows the LOT and specifies how primitives and functions compose to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 394, + 106 + ], + "score": 1.0, + "content": "produce a large unbounded concept space. The LOT has three variables:", + "type": "text" + }, + { + "bbox": [ + 394, + 96, + 402, + 104 + ], + "score": 0.33, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 93, + 505, + 106 + ], + "score": 1.0, + "content": ", representing an object in", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 103, + 507, + 120 + ], + "spans": [ + { + "bbox": [ + 104, + 103, + 140, + 120 + ], + "score": 1.0, + "content": "a scene,", + "type": "text" + }, + { + "bbox": [ + 140, + 104, + 190, + 117 + ], + "score": 0.94, + "content": "S = \\mathbf { \\bar { \\{ x \\} } } _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 103, + 391, + 120 + ], + "score": 1.0, + "content": "representing the set of all objects in the scene, and", + "type": "text" + }, + { + "bbox": [ + 392, + 104, + 450, + 117 + ], + "score": 0.93, + "content": "S _ { - \\mathbf { x } } \\overset { \\cdot } { = } S / \\{ \\mathbf { x } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 103, + 507, + 120 + ], + "score": 1.0, + "content": ", representing", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "the set of all objects in the scene except x. Each concept describes a rule composed of object and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "scene properties, logical operators, and/or comparison operators, and can be evaluated on a given", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 336, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 131, + 149 + ], + "score": 1.0, + "content": "scene", + "type": "text" + }, + { + "bbox": [ + 131, + 138, + 139, + 147 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 137, + 336, + 149 + ], + "score": 1.0, + "content": "to determine whether the scene satisfies the rule.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Object and scene properties are defined by functions which can be applied to objects or scenes: for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 146, + 178 + ], + "score": 1.0, + "content": "example,", + "type": "text" + }, + { + "bbox": [ + 146, + 165, + 190, + 177 + ], + "score": 0.72, + "content": "\\mathsf { s i z e ? } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 165, + 302, + 178 + ], + "score": 1.0, + "content": "yields the size of an object", + "type": "text" + }, + { + "bbox": [ + 302, + 167, + 309, + 176 + ], + "score": 0.44, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 165, + 338, + 178 + ], + "score": 1.0, + "content": ", while", + "type": "text" + }, + { + "bbox": [ + 339, + 165, + 383, + 177 + ], + "score": 0.7, + "content": "\\mathsf { s i z e ? } ( S )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "returns a set with the sizes of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 168, + 189 + ], + "score": 1.0, + "content": "all the objects", + "type": "text" + }, + { + "bbox": [ + 168, + 176, + 258, + 188 + ], + "score": 0.89, + "content": "\\langle \\lnot \\mathrm { i } \\lnot \\mathbf { e } ? ( \\mathbf { x } ) : \\mathbf { x } \\in S \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "). Comparison and logical operators can be used to compare", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "and relate various properties of objects in scenes. In contrast to Piantadosi et al. (2016), we include a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "count operator, which determines how many times a condition is satisfied by a set, which allows us", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "to check how well deep learning models are able to count (Chattopadhyay et al., 2016; Johnson et al.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "2016; Agrawal et al., 2017). Finally, quantifiers such as exists and for-all enrich the LOT by", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 391, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 391, + 244 + ], + "score": 1.0, + "content": "specifying the number of objects which must satisfy a given condition.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 247, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "Consider the following example concept (Figure 1 bottom): “There exists a blue object in the scene", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 470, + 271 + ], + "score": 1.0, + "content": "and the rest of the objects are squares.” To access the color of a given object, we use color?", + "type": "text" + }, + { + "bbox": [ + 471, + 260, + 485, + 270 + ], + "score": 0.25, + "content": "\\mathbf { \\Psi } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 318, + 282 + ], + "score": 1.0, + "content": "to access the shape of a given object, we use shape?", + "type": "text" + }, + { + "bbox": [ + 318, + 271, + 333, + 281 + ], + "score": 0.52, + "content": "\\mathbf { \\mu } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 269, + 506, + 282 + ], + "score": 1.0, + "content": ". To determine whether an object matches a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 355, + 294 + ], + "score": 1.0, + "content": "specific property, we can combine this with equality: shape?", + "type": "text" + }, + { + "bbox": [ + 356, + 281, + 438, + 292 + ], + "score": 0.83, + "content": "\\begin{array} { r l r } { ( { \\bf x } ) } & { { } = } & { { \\bf w } _ { \\sf S q u a r e ^ { \\prime \\prime } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 280, + 506, + 294 + ], + "score": 1.0, + "content": ". Finally, we can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 345, + 304 + ], + "score": 1.0, + "content": "use exists to specify that at least one object must be blue,", + "type": "text" + }, + { + "bbox": [ + 346, + 292, + 365, + 303 + ], + "score": 0.9, + "content": "S _ { - \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "to specify all the objects except for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 331, + 315 + ], + "score": 1.0, + "content": "that blue object, and all to specify that all the objects in", + "type": "text" + }, + { + "bbox": [ + 331, + 303, + 350, + 314 + ], + "score": 0.91, + "content": "S _ { - \\bf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "must be squares. Putting it all together:", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 313, + 501, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 147, + 326 + ], + "score": 1.0, + "content": "exists", + "type": "text" + }, + { + "bbox": [ + 148, + 314, + 175, + 324 + ], + "score": 0.87, + "content": "\\mathbf { x } \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 314, + 296, + 325 + ], + "score": 0.67, + "content": "\\begin{array} { r l } { ( \\mathtt { c o l o r 2 } \\mathtt { ( x ) } } & { { } = \\begin{array} { l } { \\mathtt { w } _ { \\mathrm { b l u e } ^ { \\prime \\prime } } ; } \\end{array} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 313, + 397, + 326 + ], + "score": 1.0, + "content": ") and all (shape?", + "type": "text" + }, + { + "bbox": [ + 398, + 314, + 496, + 325 + ], + "score": 0.84, + "content": "( S _ { - { \\bf x } } ) ~ = ~ \\ \" { \\bf s } _ { \\mathrm { q u a r e } ^ { \\prime \\prime } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 313, + 501, + 326 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 506, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "Structured Generalization Splits. A signature of productivity is the ability to handle novel combi-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "nations of known components (Fodor, 1975; Fodor & Pylyshyn, 1988). Thus, in CURI, we consider", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 507, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 507, + 366 + ], + "score": 1.0, + "content": "splits that require generalizing to novel combinations of known elements from our LOT (Figure 3),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "including combinations of constants, variables, and functions. We achieve this by creating disjoint", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 181, + 389 + ], + "score": 1.0, + "content": "splits of concepts", + "type": "text" + }, + { + "bbox": [ + 181, + 375, + 210, + 385 + ], + "score": 0.91, + "content": "\\mathcal { H } _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 374, + 230, + 389 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 230, + 375, + 254, + 386 + ], + "score": 0.93, + "content": "\\mathcal { H } _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 374, + 506, + 389 + ], + "score": 1.0, + "content": "for training and evaluating models. By varying the held out", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "elements and their combinations, we obtain splits that evaluate different axes of generalization. In", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 230, + 410 + ], + "score": 1.0, + "content": "practice, we use our grammar", + "type": "text" + }, + { + "bbox": [ + 230, + 397, + 237, + 407 + ], + "score": 0.84, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "to sample and filter a large set of concepts (see Appendix B.2 for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 313, + 420 + ], + "score": 1.0, + "content": "more details), which yields a set of 14,929 concepts", + "type": "text" + }, + { + "bbox": [ + 314, + 408, + 324, + 417 + ], + "score": 0.77, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "for training and evaluation. We next describe", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 196, + 432 + ], + "score": 1.0, + "content": "how each split divides", + "type": "text" + }, + { + "bbox": [ + 196, + 419, + 206, + 429 + ], + "score": 0.81, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 418, + 225, + 432 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 225, + 419, + 254, + 430 + ], + "score": 0.92, + "content": "\\mathcal { H } _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 418, + 272, + 432 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 272, + 419, + 295, + 430 + ], + "score": 0.91, + "content": "\\mathcal { H } _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 418, + 506, + 432 + ], + "score": 1.0, + "content": ", to test productive, out of distribution generalization:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 116, + 432, + 506, + 663 + ], + "lines": [ + { + "bbox": [ + 117, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 117, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "• Instance IID: Evaluates generalization to novel episodes from the same concept set. This is the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 124, + 443, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 124, + 443, + 374, + 457 + ], + "score": 1.0, + "content": "standard setup in machine learning (Murphy, 2013), in which", + "type": "text" + }, + { + "bbox": [ + 375, + 444, + 436, + 455 + ], + "score": 0.92, + "content": "\\mathcal { H } _ { t r a i n } = \\mathcal { H } _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 443, + 505, + 457 + ], + "score": 1.0, + "content": ". This is the only", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 125, + 455, + 298, + 467 + ], + "spans": [ + { + "bbox": [ + 125, + 455, + 298, + 467 + ], + "score": 1.0, + "content": "split where train and test concepts overlap.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 123, + 464, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 123, + 464, + 506, + 479 + ], + "score": 1.0, + "content": "Concept IID: Evaluates generalization to novel concepts based on an arbitrary random split of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 124, + 474, + 273, + 491 + ], + "spans": [ + { + "bbox": [ + 124, + 474, + 196, + 491 + ], + "score": 1.0, + "content": "the concepts into", + "type": "text" + }, + { + "bbox": [ + 196, + 477, + 226, + 488 + ], + "score": 0.91, + "content": "\\mathcal { H } _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 474, + 243, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 476, + 267, + 488 + ], + "score": 0.9, + "content": "\\mathcal { H } _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 474, + 273, + 491 + ], + "score": 1.0, + "content": ".1", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 123, + 487, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 123, + 487, + 350, + 501 + ], + "score": 1.0, + "content": "Counting: Evaluates the ability to learn a new concept", + "type": "text" + }, + { + "bbox": [ + 350, + 488, + 357, + 497 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 487, + 506, + 501 + ], + "score": 1.0, + "content": "with novel property-count combina-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 124, + 498, + 393, + 511 + ], + "spans": [ + { + "bbox": [ + 124, + 498, + 344, + 511 + ], + "score": 1.0, + "content": "tions, e.g, the training concepts never filter for exactly", + "type": "text" + }, + { + "bbox": [ + 344, + 499, + 354, + 509 + ], + "score": 0.49, + "content": "^ { \\bullet } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 498, + 393, + 511 + ], + "score": 1.0, + "content": "squares’.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 123, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 123, + 509, + 393, + 522 + ], + "score": 1.0, + "content": "Extrinsic properties: Evaluates the ability to learn a new concept", + "type": "text" + }, + { + "bbox": [ + 393, + 510, + 400, + 519 + ], + "score": 0.8, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 509, + 505, + 522 + ], + "score": 1.0, + "content": ", with novel combinations", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 124, + 520, + 403, + 533 + ], + "spans": [ + { + "bbox": [ + 124, + 520, + 403, + 533 + ], + "score": 1.0, + "content": "of extrinsic (e.g. location) and intrinsic (e.g. color) object properties.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 119, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 119, + 531, + 387, + 543 + ], + "score": 1.0, + "content": "Intrinsic properties: Evaluates the ability to learn a new concept", + "type": "text" + }, + { + "bbox": [ + 387, + 532, + 394, + 541 + ], + "score": 0.78, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "with novel combinations of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 124, + 542, + 472, + 555 + ], + "spans": [ + { + "bbox": [ + 124, + 542, + 472, + 555 + ], + "score": 1.0, + "content": "intrinsic properties, e.g., the training concepts never reference both ‘red’ and ‘rubber’.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 123, + 553, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 123, + 553, + 506, + 565 + ], + "score": 1.0, + "content": "Boolean operations: Evaluates the ability to learn concepts which require application of a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 123, + 563, + 507, + 577 + ], + "spans": [ + { + "bbox": [ + 123, + 563, + 507, + 577 + ], + "score": 1.0, + "content": "familiar boolean operation to a property to which the operation has never been applied previously.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 120, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 120, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "• Complexity split: Evaluates generalization from simple concepts (those which have less than", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 125, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 125, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "or equal to 10 symbols) to more complex concepts (longer than 10 symbols). This is indicative", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 125, + 596, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 125, + 596, + 506, + 611 + ], + "score": 1.0, + "content": "of the productivity (Fodor, 1975) exhibited by models, in generalizing from simpler concepts to", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 125, + 608, + 226, + 621 + ], + "spans": [ + { + "bbox": [ + 125, + 608, + 226, + 621 + ], + "score": 1.0, + "content": "more complex concepts.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 122, + 618, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 122, + 618, + 505, + 632 + ], + "score": 1.0, + "content": "Variable binding: Evaluates learning of entirely novel intrinsic properties, e.g. the training", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 125, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 125, + 629, + 506, + 643 + ], + "score": 1.0, + "content": "concepts involve only “red”, “blue”, and “green” but test concepts involve “yellow” (although", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 125, + 641, + 507, + 654 + ], + "spans": [ + { + "bbox": [ + 125, + 641, + 507, + 654 + ], + "score": 1.0, + "content": "‘yellow’ objects can still appear in training scenes). This is indicative of inferential coher-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 126, + 651, + 446, + 664 + ], + "spans": [ + { + "bbox": [ + 126, + 651, + 446, + 664 + ], + "score": 1.0, + "content": "ence (Fodor, 1975) in models, in generalizing rules of inference to novel atoms.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 109, + 672, + 504, + 695 + ], + "lines": [ + { + "bbox": [ + 107, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 107, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "A model that infers the underlying LOT during meta-training would be expected to perform well on", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 683, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 506, + 696 + ], + "score": 1.0, + "content": "any such systematic split. By comparing the performance of current models to to such ideal learners,", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 701, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 194, + 714 + ], + "score": 1.0, + "content": "1While some strings", + "type": "text" + }, + { + "bbox": [ + 195, + 702, + 201, + 711 + ], + "score": 0.42, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "might be different in surface form, they may yeild the same results when applied to", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 712, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 505, + 722 + ], + "score": 1.0, + "content": "images. In this split we account for such synonomy, and ensure that no two concepts which are synonyms are in", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 293, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 293, + 732 + ], + "score": 1.0, + "content": "different splits. See Appendix B.6 for more details.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 104, + 82, + 507, + 149 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Object and scene properties are defined by functions which can be applied to objects or scenes: for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 146, + 178 + ], + "score": 1.0, + "content": "example,", + "type": "text" + }, + { + "bbox": [ + 146, + 165, + 190, + 177 + ], + "score": 0.72, + "content": "\\mathsf { s i z e ? } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 165, + 302, + 178 + ], + "score": 1.0, + "content": "yields the size of an object", + "type": "text" + }, + { + "bbox": [ + 302, + 167, + 309, + 176 + ], + "score": 0.44, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 165, + 338, + 178 + ], + "score": 1.0, + "content": ", while", + "type": "text" + }, + { + "bbox": [ + 339, + 165, + 383, + 177 + ], + "score": 0.7, + "content": "\\mathsf { s i z e ? } ( S )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "returns a set with the sizes of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 168, + 189 + ], + "score": 1.0, + "content": "all the objects", + "type": "text" + }, + { + "bbox": [ + 168, + 176, + 258, + 188 + ], + "score": 0.89, + "content": "\\langle \\lnot \\mathrm { i } \\lnot \\mathbf { e } ? ( \\mathbf { x } ) : \\mathbf { x } \\in S \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "). Comparison and logical operators can be used to compare", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "and relate various properties of objects in scenes. In contrast to Piantadosi et al. (2016), we include a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "count operator, which determines how many times a condition is satisfied by a set, which allows us", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "to check how well deep learning models are able to count (Chattopadhyay et al., 2016; Johnson et al.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "2016; Agrawal et al., 2017). Finally, quantifiers such as exists and for-all enrich the LOT by", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 391, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 391, + 244 + ], + "score": 1.0, + "content": "specifying the number of objects which must satisfy a given condition.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 154, + 506, + 244 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 247, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "Consider the following example concept (Figure 1 bottom): “There exists a blue object in the scene", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 470, + 271 + ], + "score": 1.0, + "content": "and the rest of the objects are squares.” To access the color of a given object, we use color?", + "type": "text" + }, + { + "bbox": [ + 471, + 260, + 485, + 270 + ], + "score": 0.25, + "content": "\\mathbf { \\Psi } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 318, + 282 + ], + "score": 1.0, + "content": "to access the shape of a given object, we use shape?", + "type": "text" + }, + { + "bbox": [ + 318, + 271, + 333, + 281 + ], + "score": 0.52, + "content": "\\mathbf { \\mu } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 269, + 506, + 282 + ], + "score": 1.0, + "content": ". To determine whether an object matches a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 355, + 294 + ], + "score": 1.0, + "content": "specific property, we can combine this with equality: shape?", + "type": "text" + }, + { + "bbox": [ + 356, + 281, + 438, + 292 + ], + "score": 0.83, + "content": "\\begin{array} { r l r } { ( { \\bf x } ) } & { { } = } & { { \\bf w } _ { \\sf S q u a r e ^ { \\prime \\prime } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 280, + 506, + 294 + ], + "score": 1.0, + "content": ". Finally, we can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 345, + 304 + ], + "score": 1.0, + "content": "use exists to specify that at least one object must be blue,", + "type": "text" + }, + { + "bbox": [ + 346, + 292, + 365, + 303 + ], + "score": 0.9, + "content": "S _ { - \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "to specify all the objects except for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 331, + 315 + ], + "score": 1.0, + "content": "that blue object, and all to specify that all the objects in", + "type": "text" + }, + { + "bbox": [ + 331, + 303, + 350, + 314 + ], + "score": 0.91, + "content": "S _ { - \\bf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "must be squares. Putting it all together:", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 313, + 501, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 147, + 326 + ], + "score": 1.0, + "content": "exists", + "type": "text" + }, + { + "bbox": [ + 148, + 314, + 175, + 324 + ], + "score": 0.87, + "content": "\\mathbf { x } \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 314, + 296, + 325 + ], + "score": 0.67, + "content": "\\begin{array} { r l } { ( \\mathtt { c o l o r 2 } \\mathtt { ( x ) } } & { { } = \\begin{array} { l } { \\mathtt { w } _ { \\mathrm { b l u e } ^ { \\prime \\prime } } ; } \\end{array} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 313, + 397, + 326 + ], + "score": 1.0, + "content": ") and all (shape?", + "type": "text" + }, + { + "bbox": [ + 398, + 314, + 496, + 325 + ], + "score": 0.84, + "content": "( S _ { - { \\bf x } } ) ~ = ~ \\ \" { \\bf s } _ { \\mathrm { q u a r e } ^ { \\prime \\prime } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 313, + 501, + 326 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 247, + 506, + 326 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 506, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "Structured Generalization Splits. A signature of productivity is the ability to handle novel combi-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "nations of known components (Fodor, 1975; Fodor & Pylyshyn, 1988). Thus, in CURI, we consider", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 507, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 507, + 366 + ], + "score": 1.0, + "content": "splits that require generalizing to novel combinations of known elements from our LOT (Figure 3),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "including combinations of constants, variables, and functions. 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By varying the held out", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "elements and their combinations, we obtain splits that evaluate different axes of generalization. In", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 230, + 410 + ], + "score": 1.0, + "content": "practice, we use our grammar", + "type": "text" + }, + { + "bbox": [ + 230, + 397, + 237, + 407 + ], + "score": 0.84, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "to sample and filter a large set of concepts (see Appendix B.2 for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 313, + 420 + ], + "score": 1.0, + "content": "more details), which yields a set of 14,929 concepts", + "type": "text" + }, + { + "bbox": [ + 314, + 408, + 324, + 417 + ], + "score": 0.77, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "for training and evaluation. We next describe", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 196, + 432 + ], + "score": 1.0, + "content": "how each split divides", + "type": "text" + }, + { + "bbox": [ + 196, + 419, + 206, + 429 + ], + "score": 0.81, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 418, + 225, + 432 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 225, + 419, + 254, + 430 + ], + "score": 0.92, + "content": "\\mathcal { H } _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 418, + 272, + 432 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 272, + 419, + 295, + 430 + ], + "score": 0.91, + "content": "\\mathcal { H } _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 418, + 506, + 432 + ], + "score": 1.0, + "content": ", to test productive, out of distribution generalization:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 330, + 507, + 432 + ] + }, + { + "type": "list", + "bbox": [ + 116, + 432, + 506, + 663 + ], + "lines": [ + { + "bbox": [ + 117, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 117, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "• Instance IID: Evaluates generalization to novel episodes from the same concept set. This is the", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 124, + 443, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 124, + 443, + 374, + 457 + ], + "score": 1.0, + "content": "standard setup in machine learning (Murphy, 2013), in which", + "type": "text" + }, + { + "bbox": [ + 375, + 444, + 436, + 455 + ], + "score": 0.92, + "content": "\\mathcal { H } _ { t r a i n } = \\mathcal { H } _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 443, + 505, + 457 + ], + "score": 1.0, + "content": ". 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This is indicative of inferential coher-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 126, + 651, + 446, + 664 + ], + "spans": [ + { + "bbox": [ + 126, + 651, + 446, + 664 + ], + "score": 1.0, + "content": "ence (Fodor, 1975) in models, in generalizing rules of inference to novel atoms.", + "type": "text" + } + ], + "index": 50, + "is_list_end_line": true + } + ], + "index": 40, + "bbox_fs": [ + 117, + 433, + 507, + 664 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 672, + 504, + 695 + ], + "lines": [ + { + "bbox": [ + 107, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 107, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "A model that infers the underlying LOT during meta-training would be expected to perform well on", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 683, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 506, + 696 + ], + "score": 1.0, + "content": "any such systematic split. 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Each support and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "score": 1.0, + "content": "query set contains 5 positive and 20 negative examples — negative examples are oversampled since", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "score": 1.0, + "content": "the space of negatives is generally much larger than that for positives. The set of positive examples", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 166, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 506, + 181 + ], + "score": 1.0, + "content": "are sampled uniformly from a categorical distribution over all positives. However, we consider two", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "types of negatives: 1) easy negatives, in which the negatives are also sampled at random, and 2) hard", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "negatives, in which negatives are generated from a closely related concept which also evaluates true", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 214, + 213 + ], + "score": 1.0, + "content": "on the positive examples in", + "type": "text" + }, + { + "bbox": [ + 215, + 200, + 241, + 212 + ], + "score": 0.93, + "content": "D _ { s u p p }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 199, + 506, + 213 + ], + "score": 1.0, + "content": ", such that these negatives are maximally confusing. Altogether, for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 507, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 507, + 225 + ], + "score": 1.0, + "content": "each split, our train, validation, and test sets contain 500000, 5000, and 20000 episodes, respectively.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 227, + 505, + 304 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "Compositionality Gap. A key aspect of our benchmark is to define the difficulty in learning that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 250 + ], + "score": 1.0, + "content": "arises from the compositional structure of the concept space. Most of the splits above are structured", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 184, + 263 + ], + "score": 1.0, + "content": "in a way such that", + "type": "text" + }, + { + "bbox": [ + 185, + 249, + 262, + 261 + ], + "score": 0.9, + "content": "\\mathcal { H } _ { \\mathrm { t e s t } } \\cap \\mathcal { H } _ { \\mathrm { t r a i n } } = \\emptyset .", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "– forcing a learner to use the compositional structure of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 260, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 226, + 273 + ], + "score": 1.0, + "content": "concept space to generalize to", + "type": "text" + }, + { + "bbox": [ + 226, + 261, + 246, + 271 + ], + "score": 0.9, + "content": "{ \\mathcal { H } } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 260, + 506, + 273 + ], + "score": 1.0, + "content": ". We conceptualize the difficulty of this task through the notion of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "its compositionality gap. Intuitively, the compositionality gap captures the difference between the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "generalization performance of an ideal compositional learner (strong oracle) compared to an ideal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 294, + 502, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 502, + 306 + ], + "score": 1.0, + "content": "non-compositional learner that is unable to extrapolate outside the training concepts (weak oracle).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 310, + 505, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 507, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 167, + 323 + ], + "score": 1.0, + "content": "Formally, let", + "type": "text" + }, + { + "bbox": [ + 167, + 310, + 258, + 322 + ], + "score": 0.88, + "content": "\\Omega ~ \\in ~ \\{ \\mathrm { s t r o n g } , \\mathrm { w e a k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 309, + 437, + 323 + ], + "score": 1.0, + "content": "denote an oracle over a concept space", + "type": "text" + }, + { + "bbox": [ + 437, + 310, + 452, + 321 + ], + "score": 0.89, + "content": "\\mathcal { H } _ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 309, + 507, + 323 + ], + "score": 1.0, + "content": ". The pos-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 504, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 364, + 334 + ], + "score": 1.0, + "content": "terior predictive distribution of an oracle for query scene", + "type": "text" + }, + { + "bbox": [ + 364, + 323, + 373, + 331 + ], + "score": 0.45, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 320, + 450, + 334 + ], + "score": 1.0, + "content": "and query label", + "type": "text" + }, + { + "bbox": [ + 450, + 321, + 504, + 333 + ], + "score": 0.89, + "content": "y \\in \\{ 0 , 1 \\}", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 330, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 104, + 330, + 187, + 348 + ], + "score": 1.0, + "content": "is then given as:", + "type": "text" + }, + { + "bbox": [ + 187, + 331, + 400, + 345 + ], + "score": 0.86, + "content": "\\begin{array} { r l r } { p _ { \\Omega } ( y | \\mathbf { u } , D _ { \\mathrm { s u p p } } ) } & { = } & { \\sum _ { h \\in \\mathcal { H } _ { \\Omega } } \\bar { p _ { \\Omega } } ( y | h , \\mathbf { u } ) p _ { \\Omega } ( h | D _ { \\mathrm { s u p p } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 330, + 437, + 348 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 437, + 333, + 505, + 344 + ], + "score": 0.86, + "content": "p _ { \\Omega } ( h | D _ { \\mathrm { s u p p } } ) \\ d t \\propto", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 343, + 507, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 225, + 357 + ], + "score": 0.89, + "content": "p _ { \\Omega } ( h ) p ( \\{ y _ { i } \\} _ { i = 1 } ^ { N } | h ; \\{ \\mathbf { u } _ { i } \\} _ { i = 1 } ^ { N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 343, + 243, + 360 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 243, + 345, + 269, + 357 + ], + "score": 0.92, + "content": "p _ { \\Omega } ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 343, + 507, + 360 + ], + "score": 1.0, + "content": "denote the posterior and prior, respectively. Given a metric", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 355, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 149, + 370 + ], + "score": 1.0, + "content": "of interest", + "type": "text" + }, + { + "bbox": [ + 149, + 358, + 161, + 366 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 355, + 506, + 370 + ], + "score": 1.0, + "content": "(e.g., mean average precision or accuracy), the compositionality gap of a learning task", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "is then simply defined as the difference in performance of the strong and weak oracle when evaluating", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 378, + 316, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 178, + 392 + ], + "score": 1.0, + "content": "on concepts from", + "type": "text" + }, + { + "bbox": [ + 178, + 379, + 198, + 390 + ], + "score": 0.9, + "content": "\\mathcal { H } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 378, + 219, + 392 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 219, + 378, + 312, + 391 + ], + "score": 0.91, + "content": "M ( p _ { \\mathrm { s t r o n g } } ) - M ( p _ { \\mathrm { w e a k } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 378, + 316, + 392 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "Using this notion of compositionality gap, we can then define ideal learners, i.e., the strong and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 336, + 418 + ], + "score": 1.0, + "content": "weak oracle, simply via their priors. In particular, let", + "type": "text" + }, + { + "bbox": [ + 336, + 406, + 358, + 418 + ], + "score": 0.91, + "content": "w ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "denote a weight on importance of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "each hypothesis2 and let I denote the indicator function. We then define the prior of an oracle as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 426, + 507, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 247, + 441 + ], + "score": 0.89, + "content": "\\begin{array} { r } { p _ { \\Omega } ( h ) \\stackrel { } { = } \\sum _ { h ^ { \\prime } \\in \\mathcal { H } _ { \\Omega } } w ( h ^ { \\prime } ) { \\bf I } [ h ^ { \\prime } = h ] , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 426, + 507, + 443 + ], + "score": 1.0, + "content": ". The difference between strong and weak oracle lies in which", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 439, + 271, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 271, + 451 + ], + "score": 1.0, + "content": "concepts can be accessed in these priors.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "In this formalism, the strong oracle has access to the union of train and test concepts; that is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 465, + 502, + 482 + ], + "spans": [ + { + "bbox": [ + 107, + 467, + 199, + 479 + ], + "score": 0.9, + "content": "\\mathcal { H } _ { \\mathrm { s t r o n g } } = \\mathcal { H } _ { \\mathrm { t r a i n } } \\cup \\mathcal { H } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 465, + 442, + 482 + ], + "score": 1.0, + "content": ". The weak oracle, on the other hand only assumes access to", + "type": "text" + }, + { + "bbox": [ + 443, + 467, + 502, + 478 + ], + "score": 0.92, + "content": "\\mathcal { H } _ { \\mathrm { w e a k } } = \\mathcal { H } _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "which means it is unable to consider any hypothesis outside what has been seen in training and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 329, + 501 + ], + "score": 1.0, + "content": "assigning it zero probability mass. Given a support set", + "type": "text" + }, + { + "bbox": [ + 330, + 489, + 353, + 501 + ], + "score": 0.9, + "content": "D _ { \\mathrm { s u p p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "this difference in priors leads then to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "score": 1.0, + "content": "different inferences on posteriors and allows us to quantify how compositionally novel a learning", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 510, + 258, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 258, + 523 + ], + "score": 1.0, + "content": "task is relative to these ideal learners.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 108, + 543, + 259, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 262, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 262, + 558 + ], + "score": 1.0, + "content": "4 METRICS AND BASELINES", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 506, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 207, + 586 + ], + "score": 1.0, + "content": "During meta-test, given", + "type": "text" + }, + { + "bbox": [ + 207, + 573, + 230, + 584 + ], + "score": 0.9, + "content": "D _ { \\mathrm { s u p p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "models are evaluated on their ability to learn novel concepts. We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "use two metrics for quantifying this: 1) Accuracy: evaluates the accuracy of model predictions", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 594, + 507, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 594, + 188, + 608 + ], + "score": 1.0, + "content": "across the query set", + "type": "text" + }, + { + "bbox": [ + 188, + 595, + 218, + 606 + ], + "score": 0.91, + "content": "D _ { q u e r y }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 594, + 507, + 608 + ], + "score": 1.0, + "content": ", as is standard practice in meta-learning (Lake et al., 2019; Snell et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "2017). Since there are more negative than positive labels, we report class balanced accuracy for better", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "interpretability, averaging accuracies for the positive and negative query examples; and 2) mean", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 457, + 639 + ], + "score": 1.0, + "content": "Average Precision (mAP): evaluates models on a much larger number of test scenes", + "type": "text" + }, + { + "bbox": [ + 458, + 627, + 468, + 637 + ], + "score": 0.82, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "for each", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 333, + 650 + ], + "score": 1.0, + "content": "episode (comprising 44,787 scenes, 3 per each concept in", + "type": "text" + }, + { + "bbox": [ + 333, + 639, + 343, + 648 + ], + "score": 0.73, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "). This resolves an issue that with a small", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 104, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "query set, a strong model could achieve perfect accuracy without grasping the concept. Since episodes", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "typically have many more negative than positive examples, Average Precision sweeps over different", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 671, + 507, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 507, + 684 + ], + "score": 1.0, + "content": "thresholds of a model’s score and reports the average of the precision values at different recall rates,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 682, + 470, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 470, + 695 + ], + "score": 1.0, + "content": "e.g., (Everingham et al., 2010). mAP is then the mean across all of the meta-test episodes.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 711, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 117, + 709, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 117, + 709, + 506, + 725 + ], + "score": 1.0, + "content": "2Set to log-linear in the prefix serialization length of hypothesis, inspired by the observation that longer", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 450, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 450, + 732 + ], + "score": 1.0, + "content": "hypotheses are more difficult for humans (Feldman, 2000). See Appendix B.3 for more details.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 503, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 106, + 81, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 457, + 124 + ], + "score": 1.0, + "content": "From Concepts to Meta-learning Episodes. A single episode comprises a support set", + "type": "text" + }, + { + "bbox": [ + 458, + 111, + 487, + 123 + ], + "score": 0.85, + "content": "( D _ { \\mathrm { s u p p } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 109, + 506, + 124 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 120, + 505, + 136 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 151, + 136 + ], + "score": 1.0, + "content": "a query set", + "type": "text" + }, + { + "bbox": [ + 152, + 122, + 183, + 134 + ], + "score": 0.87, + "content": "( D _ { \\mathrm { q u e r y } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 120, + 380, + 136 + ], + "score": 1.0, + "content": ", each of which is generated from a given concept,", + "type": "text" + }, + { + "bbox": [ + 381, + 122, + 387, + 131 + ], + "score": 0.7, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 120, + 505, + 136 + ], + "score": 1.0, + "content": ". Formally, a support or query", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 103, + 130, + 507, + 151 + ], + "spans": [ + { + "bbox": [ + 103, + 130, + 120, + 151 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 134, + 130, + 144 + ], + "score": 0.79, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 130, + 190, + 151 + ], + "score": 1.0, + "content": "has input data", + "type": "text" + }, + { + "bbox": [ + 190, + 136, + 198, + 144 + ], + "score": 0.34, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 130, + 299, + 151 + ], + "score": 1.0, + "content": "and corresponding label", + "type": "text" + }, + { + "bbox": [ + 299, + 135, + 306, + 145 + ], + "score": 0.71, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 130, + 325, + 151 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 325, + 133, + 385, + 146 + ], + "score": 0.74, + "content": "D = \\{ \\{ y _ { i } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 130, + 388, + 151 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 389, + 133, + 427, + 146 + ], + "score": 0.79, + "content": "\\{ { \\mathbf { u } } _ { i } \\} _ { i = 1 } ^ { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 130, + 507, + 151 + ], + "score": 1.0, + "content": ". Each support and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "score": 1.0, + "content": "query set contains 5 positive and 20 negative examples — negative examples are oversampled since", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "score": 1.0, + "content": "the space of negatives is generally much larger than that for positives. The set of positive examples", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 166, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 506, + 181 + ], + "score": 1.0, + "content": "are sampled uniformly from a categorical distribution over all positives. However, we consider two", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "types of negatives: 1) easy negatives, in which the negatives are also sampled at random, and 2) hard", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "negatives, in which negatives are generated from a closely related concept which also evaluates true", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 214, + 213 + ], + "score": 1.0, + "content": "on the positive examples in", + "type": "text" + }, + { + "bbox": [ + 215, + 200, + 241, + 212 + ], + "score": 0.93, + "content": "D _ { s u p p }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 199, + 506, + 213 + ], + "score": 1.0, + "content": ", such that these negatives are maximally confusing. Altogether, for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 507, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 507, + 225 + ], + "score": 1.0, + "content": "each split, our train, validation, and test sets contain 500000, 5000, and 20000 episodes, respectively.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5, + "bbox_fs": [ + 103, + 109, + 507, + 225 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 227, + 505, + 304 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "Compositionality Gap. A key aspect of our benchmark is to define the difficulty in learning that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 250 + ], + "score": 1.0, + "content": "arises from the compositional structure of the concept space. Most of the splits above are structured", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 184, + 263 + ], + "score": 1.0, + "content": "in a way such that", + "type": "text" + }, + { + "bbox": [ + 185, + 249, + 262, + 261 + ], + "score": 0.9, + "content": "\\mathcal { H } _ { \\mathrm { t e s t } } \\cap \\mathcal { H } _ { \\mathrm { t r a i n } } = \\emptyset .", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "– forcing a learner to use the compositional structure of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 260, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 226, + 273 + ], + "score": 1.0, + "content": "concept space to generalize to", + "type": "text" + }, + { + "bbox": [ + 226, + 261, + 246, + 271 + ], + "score": 0.9, + "content": "{ \\mathcal { H } } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 260, + 506, + 273 + ], + "score": 1.0, + "content": ". We conceptualize the difficulty of this task through the notion of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "its compositionality gap. Intuitively, the compositionality gap captures the difference between the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "generalization performance of an ideal compositional learner (strong oracle) compared to an ideal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 294, + 502, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 502, + 306 + ], + "score": 1.0, + "content": "non-compositional learner that is unable to extrapolate outside the training concepts (weak oracle).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 227, + 506, + 306 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 310, + 505, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 507, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 167, + 323 + ], + "score": 1.0, + "content": "Formally, let", + "type": "text" + }, + { + "bbox": [ + 167, + 310, + 258, + 322 + ], + "score": 0.88, + "content": "\\Omega ~ \\in ~ \\{ \\mathrm { s t r o n g } , \\mathrm { w e a k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 309, + 437, + 323 + ], + "score": 1.0, + "content": "denote an oracle over a concept space", + "type": "text" + }, + { + "bbox": [ + 437, + 310, + 452, + 321 + ], + "score": 0.89, + "content": "\\mathcal { H } _ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 309, + 507, + 323 + ], + "score": 1.0, + "content": ". The pos-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 504, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 364, + 334 + ], + "score": 1.0, + "content": "terior predictive distribution of an oracle for query scene", + "type": "text" + }, + { + "bbox": [ + 364, + 323, + 373, + 331 + ], + "score": 0.45, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 320, + 450, + 334 + ], + "score": 1.0, + "content": "and query label", + "type": "text" + }, + { + "bbox": [ + 450, + 321, + 504, + 333 + ], + "score": 0.89, + "content": "y \\in \\{ 0 , 1 \\}", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 330, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 104, + 330, + 187, + 348 + ], + "score": 1.0, + "content": "is then given as:", + "type": "text" + }, + { + "bbox": [ + 187, + 331, + 400, + 345 + ], + "score": 0.86, + "content": "\\begin{array} { r l r } { p _ { \\Omega } ( y | \\mathbf { u } , D _ { \\mathrm { s u p p } } ) } & { = } & { \\sum _ { h \\in \\mathcal { H } _ { \\Omega } } \\bar { p _ { \\Omega } } ( y | h , \\mathbf { u } ) p _ { \\Omega } ( h | D _ { \\mathrm { s u p p } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 330, + 437, + 348 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 437, + 333, + 505, + 344 + ], + "score": 0.86, + "content": "p _ { \\Omega } ( h | D _ { \\mathrm { s u p p } } ) \\ d t \\propto", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 343, + 507, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 225, + 357 + ], + "score": 0.89, + "content": "p _ { \\Omega } ( h ) p ( \\{ y _ { i } \\} _ { i = 1 } ^ { N } | h ; \\{ \\mathbf { u } _ { i } \\} _ { i = 1 } ^ { N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 343, + 243, + 360 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 243, + 345, + 269, + 357 + ], + "score": 0.92, + "content": "p _ { \\Omega } ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 343, + 507, + 360 + ], + "score": 1.0, + "content": "denote the posterior and prior, respectively. Given a metric", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 355, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 149, + 370 + ], + "score": 1.0, + "content": "of interest", + "type": "text" + }, + { + "bbox": [ + 149, + 358, + 161, + 366 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 355, + 506, + 370 + ], + "score": 1.0, + "content": "(e.g., mean average precision or accuracy), the compositionality gap of a learning task", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "is then simply defined as the difference in performance of the strong and weak oracle when evaluating", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 378, + 316, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 178, + 392 + ], + "score": 1.0, + "content": "on concepts from", + "type": "text" + }, + { + "bbox": [ + 178, + 379, + 198, + 390 + ], + "score": 0.9, + "content": "\\mathcal { H } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 378, + 219, + 392 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 219, + 378, + 312, + 391 + ], + "score": 0.91, + "content": "M ( p _ { \\mathrm { s t r o n g } } ) - M ( p _ { \\mathrm { w e a k } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 378, + 316, + 392 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 309, + 507, + 392 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "Using this notion of compositionality gap, we can then define ideal learners, i.e., the strong and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 336, + 418 + ], + "score": 1.0, + "content": "weak oracle, simply via their priors. In particular, let", + "type": "text" + }, + { + "bbox": [ + 336, + 406, + 358, + 418 + ], + "score": 0.91, + "content": "w ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "denote a weight on importance of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "each hypothesis2 and let I denote the indicator function. We then define the prior of an oracle as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 426, + 507, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 247, + 441 + ], + "score": 0.89, + "content": "\\begin{array} { r } { p _ { \\Omega } ( h ) \\stackrel { } { = } \\sum _ { h ^ { \\prime } \\in \\mathcal { H } _ { \\Omega } } w ( h ^ { \\prime } ) { \\bf I } [ h ^ { \\prime } = h ] , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 426, + 507, + 443 + ], + "score": 1.0, + "content": ". The difference between strong and weak oracle lies in which", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 439, + 271, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 271, + 451 + ], + "score": 1.0, + "content": "concepts can be accessed in these priors.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 395, + 507, + 451 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "In this formalism, the strong oracle has access to the union of train and test concepts; that is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 465, + 502, + 482 + ], + "spans": [ + { + "bbox": [ + 107, + 467, + 199, + 479 + ], + "score": 0.9, + "content": "\\mathcal { H } _ { \\mathrm { s t r o n g } } = \\mathcal { H } _ { \\mathrm { t r a i n } } \\cup \\mathcal { H } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 465, + 442, + 482 + ], + "score": 1.0, + "content": ". The weak oracle, on the other hand only assumes access to", + "type": "text" + }, + { + "bbox": [ + 443, + 467, + 502, + 478 + ], + "score": 0.92, + "content": "\\mathcal { H } _ { \\mathrm { w e a k } } = \\mathcal { H } _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "which means it is unable to consider any hypothesis outside what has been seen in training and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 329, + 501 + ], + "score": 1.0, + "content": "assigning it zero probability mass. Given a support set", + "type": "text" + }, + { + "bbox": [ + 330, + 489, + 353, + 501 + ], + "score": 0.9, + "content": "D _ { \\mathrm { s u p p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "this difference in priors leads then to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "score": 1.0, + "content": "different inferences on posteriors and allows us to quantify how compositionally novel a learning", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 510, + 258, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 258, + 523 + ], + "score": 1.0, + "content": "task is relative to these ideal learners.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 456, + 506, + 523 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 543, + 259, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 262, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 262, + 558 + ], + "score": 1.0, + "content": "4 METRICS AND BASELINES", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 506, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 207, + 586 + ], + "score": 1.0, + "content": "During meta-test, given", + "type": "text" + }, + { + "bbox": [ + 207, + 573, + 230, + 584 + ], + "score": 0.9, + "content": "D _ { \\mathrm { s u p p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "models are evaluated on their ability to learn novel concepts. We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "use two metrics for quantifying this: 1) Accuracy: evaluates the accuracy of model predictions", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 594, + 507, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 594, + 188, + 608 + ], + "score": 1.0, + "content": "across the query set", + "type": "text" + }, + { + "bbox": [ + 188, + 595, + 218, + 606 + ], + "score": 0.91, + "content": "D _ { q u e r y }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 594, + 507, + 608 + ], + "score": 1.0, + "content": ", as is standard practice in meta-learning (Lake et al., 2019; Snell et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "2017). 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Different choices for the encoder", + "type": "text" + }, + { + "bbox": [ + 346, + 241, + 366, + 252 + ], + "score": 0.89, + "content": "f ( \\mathbf { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 239, + 506, + 253 + ], + "score": 1.0, + "content": "parameterization explored in the paper.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "We consider three modalities each of which is processed with a modality specific encoder, followed by four kinds", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "score": 1.0, + "content": "of pooling architecture which take as input objects and their corresponding locations to provide an encoding", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 269, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 434, + 284 + ], + "score": 1.0, + "content": "for the datapoint. 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Different choices for the encoder", + "type": "text" + }, + { + "bbox": [ + 346, + 241, + 366, + 252 + ], + "score": 0.89, + "content": "f ( \\mathbf { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 239, + 506, + 253 + ], + "score": 1.0, + "content": "parameterization explored in the paper.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "We consider three modalities each of which is processed with a modality specific encoder, followed by four kinds", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "score": 1.0, + "content": "of pooling architecture which take as input objects and their corresponding locations to provide an encoding", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 269, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 434, + 284 + ], + "score": 1.0, + "content": "for the datapoint. 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In the case of images and sounds, the output of the encoder is enriched", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 648, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 659 + ], + "score": 1.0, + "content": "with position vectors. For the pooling operation, we study global averaging, concatenation, relation", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "networks (Santoro et al., 2017) and transformers (Vaswani et al., 2017) equipped with different", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "pooling operations (max, mean, sum, min) for reasoning inspired by Wang et al. (2019) (Figure 4", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 681, + 317, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 317, + 693 + ], + "score": 1.0, + "content": "middle panel, also see Appendix F for more details).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 581, + 506, + 693 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 697, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 507, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 205, + 711 + ], + "score": 1.0, + "content": "For the probability in", + "type": "text" + }, + { + "bbox": [ + 206, + 698, + 236, + 709 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { \\mathrm { c o n c e p t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 696, + 507, + 711 + ], + "score": 1.0, + "content": ", we represent the concept as a sequence by prefix serializa-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 707, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 446, + 722 + ], + "score": 1.0, + "content": "tion and then use an LSTM (Hochreiter & Schmidhuber, 1997) to parameterize", + "type": "text" + }, + { + "bbox": [ + 446, + 708, + 505, + 721 + ], + "score": 0.9, + "content": "p ( h | D _ { \\mathrm { s u p p } } ) =", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 718, + 473, + 735 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 215, + 733 + ], + "score": 0.9, + "content": "\\Pi _ { s = 1 } ^ { S } p ( h _ { s } | h _ { 1 \\cdots t - 1 } ; D _ { s u p p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 718, + 390, + 735 + ], + "score": 1.0, + "content": ". At each step of the LSTM we concatenate", + "type": "text" + }, + { + "bbox": [ + 390, + 720, + 421, + 733 + ], + "score": 0.93, + "content": "[ \\mathbf { c } _ { p } , \\mathbf { c } _ { n } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 718, + 473, + 735 + ], + "score": 1.0, + "content": "to the input.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 696, + 507, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 79, + 504, + 186 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 79, + 504, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 79, + 504, + 186 + ], + "spans": [ + { + "bbox": [ + 110, + 79, + 504, + 186 + ], + "score": 0.963, + "type": "image", + "image_path": "2ea5088d670d6fa4797dca4586c3097660e37e47c12eebe19396d0937331179b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 79, + 504, + 114.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 114.66666666666666, + 504, + 150.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 150.33333333333331, + 504, + 185.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 189, + 505, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 188, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 290, + 200 + ], + "score": 1.0, + "content": "Figure 5: Compositionality Gap. Different splits (", + "type": "text" + }, + { + "bbox": [ + 290, + 190, + 296, + 198 + ], + "score": 0.54, + "content": "\\mathbf { \\hat { x } } \\mathbf { \\cdot }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 188, + 505, + 200 + ], + "score": 1.0, + "content": "-axis) plotted w.r.t performance of the strong oracle (green", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "line) and weak oracle (red line) on the mAP (top) and Accuracy (bottom) evaluated on respective test splits", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "(using hard negatives in support and query sets). Difference between the two is the compositionality gap (comp", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 217, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 507, + 232 + ], + "score": 1.0, + "content": "gap). Yellow: shows the (best) relation-net model on schema inputs, purple: shows the model on image inputs,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 229, + 461, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 461, + 240 + ], + "score": 1.0, + "content": "and gray: shows the model on sound inputs. Error bars are std across 3 independent model runs.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "title", + "bbox": [ + 108, + 244, + 257, + 257 + ], + "lines": [ + { + "bbox": [ + 104, + 243, + 258, + 259 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 258, + 259 + ], + "score": 1.0, + "content": "5 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "We first discuss the compositionality gap induced by the different generalization splits and then", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "delve into the impact of modeling choices on performance on the generalization splits. All models", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "score": 1.0, + "content": "are trained for 1 million steps, and are run with with 3 independent training runs to report standard", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "deviations. We sweep over 3 modalities (image, schema, sound), 4 pooling schemes (avg-pool,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "concat, relation-net, transformer), 2 choices of negatives (hard negatives, random negatives) and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 185, + 338 + ], + "score": 1.0, + "content": "choice of language", + "type": "text" + }, + { + "bbox": [ + 185, + 326, + 238, + 336 + ], + "score": 0.75, + "content": "( \\alpha = 0 . 0 , 1 . 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 326, + 505, + 338 + ], + "score": 1.0, + "content": ". Unless mentioned otherwise in the main paper we focus on results", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 203, + 349 + ], + "score": 1.0, + "content": "with hard negatives and", + "type": "text" + }, + { + "bbox": [ + 203, + 337, + 237, + 347 + ], + "score": 0.87, + "content": "\\alpha = 0 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 337, + 505, + 349 + ], + "score": 1.0, + "content": ". When instantiated for a given modality, we note that the encoders", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 107, + 347, + 128, + 360 + ], + "score": 0.9, + "content": "f ( \\mathbf { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "(Figure 4) all have a similar number of parameters. The appendix contains details of the exact", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 358, + 488, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 488, + 371 + ], + "score": 1.0, + "content": "hyperparameters (Appendix E), and more comprehensive results for each split (Appendix G.7).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 385, + 318, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 319, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 319, + 398 + ], + "score": 1.0, + "content": "5.1 DATASET DESIGN AND COMPOSITIONALITY", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 421 + ], + "score": 1.0, + "content": "How compositional are the structured splits?. Our main results are shown in Figure 5. Using", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "score": 1.0, + "content": "our model-independent measure of the compositionality gap (Section 3), different splits present", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "varying challenges for generalizing from train to test. The most difficult splits, with the largest", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "compositionality gaps, are the Binding (color) and Binding (shape), which is reasonable since they", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 451, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 462 + ], + "score": 1.0, + "content": "require learning concepts with entirely new property-values. In contrast, the easiest split with the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "smallest compositionality gaps is the Instance IID split since it does not require compositionality.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "Finally, while the mAP metric exposes a larger value of comp gap, the ordering of splits in terms of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "score": 1.0, + "content": "comp gap is same for both metrics – suggesting similar coarse-grained notions of compositionality.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Results for the best overall architecture, a relation network (relation-net), is shown in Figure 5.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "Network performance on the easiest data format (schema; yellow bars) is generally better than the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "weak oracle, but substantially worse than the strong oracle. Counting is a particularly challenging split", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "where the models underperform even the weak oracle. Broadly, this suggests that the models capture", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "some notion of compositionality – especially for images and schemas – relative to a weak oracle", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "that rigidly considers only training hypotheses, but there is substantial room to improve (especially", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "with respect to the more stringent mAP metric). These results demonstrate that CURI provides a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 577, + 459, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 459, + 590 + ], + "score": 1.0, + "content": "challenging yet tractable setting for evaluating the compositional capabilities of models.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "Finally, we found that the performance on the Instance IID split is not equal to the weak (and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "strong) oracle—which are both equal in this case—indicating that the best model does not make ideal", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "posterior predictions even when compositionality is not an issue. Ideal predictions in this case would", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "require the network to behave as if marginalizing over the training hypotheses, as the strong oracle", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 446, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 446, + 651 + ], + "score": 1.0, + "content": "does. A similar plot to Figure 5 can be found in Appendix G.4 for random negatives.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "Influence of Negatives. Previous work (Hill et al., 2019) has shown that the choice of random v.s.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "hard negatives for training and evaluation impacts compositional generalization substantially in the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "case of a particular set of analogical reasoning models. However, we argue that such decisions on", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 686, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 507, + 702 + ], + "score": 1.0, + "content": "dataset design can be made more objectively if one can evaluate the model-independent comp gap.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "In our context, we find that the comp gap with mAP when using random negatives decreases on", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 153, + 722 + ], + "score": 1.0, + "content": "average by", + "type": "text" + }, + { + "bbox": [ + 153, + 710, + 201, + 720 + ], + "score": 0.9, + "content": "5 . 5 \\pm 1 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "compared to when we use hard negatives. This indicates that it is not only", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 161, + 734 + ], + "score": 1.0, + "content": "the choice of", + "type": "text" + }, + { + "bbox": [ + 161, + 721, + 190, + 732 + ], + "score": 0.91, + "content": "\\mathcal { H } _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 720, + 208, + 734 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 209, + 721, + 232, + 732 + ], + "score": 0.91, + "content": "\\mathcal { H } _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 720, + 506, + 734 + ], + "score": 1.0, + "content": ", which are identical for a given compositional split (say Counting),", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 79, + 504, + 186 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 79, + 504, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 79, + 504, + 186 + ], + "spans": [ + { + "bbox": [ + 110, + 79, + 504, + 186 + ], + "score": 0.963, + "type": "image", + "image_path": "2ea5088d670d6fa4797dca4586c3097660e37e47c12eebe19396d0937331179b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 79, + 504, + 114.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 114.66666666666666, + 504, + 150.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 150.33333333333331, + 504, + 185.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 189, + 505, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 188, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 290, + 200 + ], + "score": 1.0, + "content": "Figure 5: Compositionality Gap. Different splits (", + "type": "text" + }, + { + "bbox": [ + 290, + 190, + 296, + 198 + ], + "score": 0.54, + "content": "\\mathbf { \\hat { x } } \\mathbf { \\cdot }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 188, + 505, + 200 + ], + "score": 1.0, + "content": "-axis) plotted w.r.t performance of the strong oracle (green", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "line) and weak oracle (red line) on the mAP (top) and Accuracy (bottom) evaluated on respective test splits", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "(using hard negatives in support and query sets). Difference between the two is the compositionality gap (comp", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 217, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 507, + 232 + ], + "score": 1.0, + "content": "gap). Yellow: shows the (best) relation-net model on schema inputs, purple: shows the model on image inputs,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 229, + 461, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 461, + 240 + ], + "score": 1.0, + "content": "and gray: shows the model on sound inputs. Error bars are std across 3 independent model runs.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "title", + "bbox": [ + 108, + 244, + 257, + 257 + ], + "lines": [ + { + "bbox": [ + 104, + 243, + 258, + 259 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 258, + 259 + ], + "score": 1.0, + "content": "5 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "We first discuss the compositionality gap induced by the different generalization splits and then", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "delve into the impact of modeling choices on performance on the generalization splits. All models", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "score": 1.0, + "content": "are trained for 1 million steps, and are run with with 3 independent training runs to report standard", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "deviations. We sweep over 3 modalities (image, schema, sound), 4 pooling schemes (avg-pool,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "concat, relation-net, transformer), 2 choices of negatives (hard negatives, random negatives) and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 185, + 338 + ], + "score": 1.0, + "content": "choice of language", + "type": "text" + }, + { + "bbox": [ + 185, + 326, + 238, + 336 + ], + "score": 0.75, + "content": "( \\alpha = 0 . 0 , 1 . 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 326, + 505, + 338 + ], + "score": 1.0, + "content": ". Unless mentioned otherwise in the main paper we focus on results", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 203, + 349 + ], + "score": 1.0, + "content": "with hard negatives and", + "type": "text" + }, + { + "bbox": [ + 203, + 337, + 237, + 347 + ], + "score": 0.87, + "content": "\\alpha = 0 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 337, + 505, + 349 + ], + "score": 1.0, + "content": ". When instantiated for a given modality, we note that the encoders", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 107, + 347, + 128, + 360 + ], + "score": 0.9, + "content": "f ( \\mathbf { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "(Figure 4) all have a similar number of parameters. The appendix contains details of the exact", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 358, + 488, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 488, + 371 + ], + "score": 1.0, + "content": "hyperparameters (Appendix E), and more comprehensive results for each split (Appendix G.7).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 271, + 506, + 371 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 385, + 318, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 319, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 319, + 398 + ], + "score": 1.0, + "content": "5.1 DATASET DESIGN AND COMPOSITIONALITY", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 421 + ], + "score": 1.0, + "content": "How compositional are the structured splits?. Our main results are shown in Figure 5. Using", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "score": 1.0, + "content": "our model-independent measure of the compositionality gap (Section 3), different splits present", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "varying challenges for generalizing from train to test. The most difficult splits, with the largest", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "compositionality gaps, are the Binding (color) and Binding (shape), which is reasonable since they", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 451, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 462 + ], + "score": 1.0, + "content": "require learning concepts with entirely new property-values. In contrast, the easiest split with the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "smallest compositionality gaps is the Instance IID split since it does not require compositionality.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "Finally, while the mAP metric exposes a larger value of comp gap, the ordering of splits in terms of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "score": 1.0, + "content": "comp gap is same for both metrics – suggesting similar coarse-grained notions of compositionality.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 405, + 506, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Results for the best overall architecture, a relation network (relation-net), is shown in Figure 5.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "Network performance on the easiest data format (schema; yellow bars) is generally better than the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "weak oracle, but substantially worse than the strong oracle. Counting is a particularly challenging split", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "where the models underperform even the weak oracle. Broadly, this suggests that the models capture", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "some notion of compositionality – especially for images and schemas – relative to a weak oracle", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "that rigidly considers only training hypotheses, but there is substantial room to improve (especially", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "with respect to the more stringent mAP metric). These results demonstrate that CURI provides a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 577, + 459, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 459, + 590 + ], + "score": 1.0, + "content": "challenging yet tractable setting for evaluating the compositional capabilities of models.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 500, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "Finally, we found that the performance on the Instance IID split is not equal to the weak (and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "strong) oracle—which are both equal in this case—indicating that the best model does not make ideal", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "posterior predictions even when compositionality is not an issue. Ideal predictions in this case would", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "require the network to behave as if marginalizing over the training hypotheses, as the strong oracle", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 446, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 446, + 651 + ], + "score": 1.0, + "content": "does. A similar plot to Figure 5 can be found in Appendix G.4 for random negatives.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 594, + 505, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "Influence of Negatives. Previous work (Hill et al., 2019) has shown that the choice of random v.s.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "hard negatives for training and evaluation impacts compositional generalization substantially in the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "case of a particular set of analogical reasoning models. However, we argue that such decisions on", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 686, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 507, + 702 + ], + "score": 1.0, + "content": "dataset design can be made more objectively if one can evaluate the model-independent comp gap.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "In our context, we find that the comp gap with mAP when using random negatives decreases on", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 153, + 722 + ], + "score": 1.0, + "content": "average by", + "type": "text" + }, + { + "bbox": [ + 153, + 710, + 201, + 720 + ], + "score": 0.9, + "content": "5 . 5 \\pm 1 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "compared to when we use hard negatives. This indicates that it is not only", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 161, + 734 + ], + "score": 1.0, + "content": "the choice of", + "type": "text" + }, + { + "bbox": [ + 161, + 721, + 190, + 732 + ], + "score": 0.91, + "content": "\\mathcal { H } _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 720, + 208, + 734 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 209, + 721, + 232, + 732 + ], + "score": 0.91, + "content": "\\mathcal { H } _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 720, + 506, + 734 + ], + "score": 1.0, + "content": ", which are identical for a given compositional split (say Counting),", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "but also the choice of the negatives which “makes” the task compositionally novel. More generally,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "this indicates that the comp gap has utility as a more general diagnostic tool for making principled", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 507, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 507, + 118 + ], + "score": 1.0, + "content": "design decisions in compositional learning settings without the confound of specific model decisions.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 655, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "but also the choice of the negatives which “makes” the task compositionally novel. More generally,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "this indicates that the comp gap has utility as a more general diagnostic tool for making principled", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 507, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 507, + 118 + ], + "score": 1.0, + "content": "design decisions in compositional learning settings without the confound of specific model decisions.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 108, + 129, + 276, + 141 + ], + "lines": [ + { + "bbox": [ + 105, + 128, + 277, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 277, + 142 + ], + "score": 1.0, + "content": "5.2 DIFFERENCES BETWEEN MODELS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 507, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 507, + 164 + ], + "score": 1.0, + "content": "Best Models. In general, the best performing model is the relation-net applied to schema inputs,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 162, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 174 + ], + "score": 1.0, + "content": "outperforming other combinations of models and input modalities on the Boolean, Concept IID,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 173, + 504, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 504, + 185 + ], + "score": 1.0, + "content": "Complexity, and Instance IID splits on both the mAP as well as accuracy metrics (Figure 5); although", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "as mentioned above, none of the models are close to the strong oracle. It is closely followed by the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 507, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 507, + 208 + ], + "score": 1.0, + "content": "transformer model on schema inputs, which performs the best on Binding (color), Binding (shape),", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "and Intrinsic splits (Appendix G.7). Utilizing schema inputs proves easier for abstraction except", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "score": 1.0, + "content": "for the Extrinsic setting, where the task requires generalization to novel locations for objects in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 507, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 507, + 240 + ], + "score": 1.0, + "content": "images, which is well supported by the inductive bias of the CNN encoder (Figure 4). In this case,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 238, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 262, + 250 + ], + "score": 1.0, + "content": "the image-transformer gets an mAP of", + "type": "text" + }, + { + "bbox": [ + 262, + 238, + 314, + 249 + ], + "score": 0.91, + "content": "6 2 . 1 \\pm 0 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 238, + 506, + 250 + ], + "score": 1.0, + "content": ", compared to the next best schema-transformer", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 145, + 262 + ], + "score": 1.0, + "content": "model at", + "type": "text" + }, + { + "bbox": [ + 145, + 249, + 190, + 260 + ], + "score": 0.88, + "content": "6 0 . 9 \\pm 0 . 7", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 249, + 506, + 262 + ], + "score": 1.0, + "content": ". Further, relational learning proves more crucial in the schema case than for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 427, + 272 + ], + "score": 1.0, + "content": "images, with all image models (regardless of pooling) performing better than", + "type": "text" + }, + { + "bbox": [ + 427, + 260, + 480, + 271 + ], + "score": 0.89, + "content": "5 9 . 4 \\pm 1 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 260, + 506, + 272 + ], + "score": 1.0, + "content": "mAP", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 473, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 416, + 284 + ], + "score": 1.0, + "content": "(achieved for image-avg-pool) while schema-avg-pool models get only get to", + "type": "text" + }, + { + "bbox": [ + 417, + 271, + 468, + 282 + ], + "score": 0.9, + "content": "5 3 . 4 \\pm 1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 271, + 473, + 284 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "When to use a transformer? Transformer models appear to outperform relation networks in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 311 + ], + "score": 1.0, + "content": "splits concerning disentangling. For instance, for the Intrinsic split with schema-relation-net is at", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 308, + 507, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 159, + 321 + ], + "score": 0.85, + "content": "5 5 . 1 \\pm 0 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 308, + 175, + 323 + ], + "score": 1.0, + "content": "v.s.", + "type": "text" + }, + { + "bbox": [ + 175, + 310, + 228, + 320 + ], + "score": 0.86, + "content": "5 7 . 9 \\pm 0 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 308, + 507, + 323 + ], + "score": 1.0, + "content": "for schema-transformer. Similarly, for the Extrinsic split the image-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 174, + 333 + ], + "score": 1.0, + "content": "transformer is at", + "type": "text" + }, + { + "bbox": [ + 175, + 321, + 227, + 331 + ], + "score": 0.89, + "content": "6 2 . 1 \\pm 0 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 320, + 382, + 333 + ], + "score": 1.0, + "content": "compared to the image-relation-net at", + "type": "text" + }, + { + "bbox": [ + 382, + 321, + 434, + 331 + ], + "score": 0.9, + "content": "6 0 . 8 \\pm 1 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 320, + 505, + 333 + ], + "score": 1.0, + "content": ". We hypothesize", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "that this is because the iterative message passing via. attention in transformers improves object", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 343, + 478, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 478, + 355 + ], + "score": 1.0, + "content": "representations for disentangling compared to relation networks that lack such a mechanism.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 360, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "What is the relative difficulty of abstraction from different modalities? One of the key con-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "tributions of our work is in providing multiple modalities (image, schema, sound) for productive", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "concept learning. We next characterize the difficulty of abstraction based on modality for the various", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "score": 1.0, + "content": "generalization settings. In the Intrinsic setting, we find that the schema models, which have access to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "a “perfect” disentangled representation significantly outperform image models—a schema-avg-pool", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 415, + 489, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 198, + 426 + ], + "score": 1.0, + "content": "model gets an mAP of", + "type": "text" + }, + { + "bbox": [ + 198, + 415, + 250, + 425 + ], + "score": 0.9, + "content": "5 2 . 7 \\pm 3 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 415, + 410, + 426 + ], + "score": 1.0, + "content": "while an image-avg-pool model gets to", + "type": "text" + }, + { + "bbox": [ + 410, + 415, + 462, + 425 + ], + "score": 0.87, + "content": "3 4 . 4 \\pm 0 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 415, + 489, + 426 + ], + "score": 1.0, + "content": "mAP.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 431, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 443 + ], + "score": 1.0, + "content": "Similarly, for the Counting split where the total number of objects are exactly specified in the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "schema (Figure 4), schemas are substantially better than images. For example, schema-relation-nets", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 452, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 132, + 467 + ], + "score": 1.0, + "content": "get to", + "type": "text" + }, + { + "bbox": [ + 133, + 453, + 196, + 464 + ], + "score": 0.9, + "content": "5 6 . 2 5 \\pm 5 . 3 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 452, + 333, + 467 + ], + "score": 1.0, + "content": "mAP while image-avg-pool is at", + "type": "text" + }, + { + "bbox": [ + 333, + 453, + 386, + 464 + ], + "score": 0.89, + "content": "4 8 . 4 \\pm 1 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 452, + 506, + 467 + ], + "score": 1.0, + "content": "mAP. Interestingly, the next", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 353, + 476 + ], + "score": 1.0, + "content": "best model—image-relation-net—is substantially worse, at", + "type": "text" + }, + { + "bbox": [ + 353, + 464, + 412, + 475 + ], + "score": 0.91, + "content": "3 9 . 4 5 \\pm 1 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 464, + 505, + 476 + ], + "score": 1.0, + "content": ". Curiously, for while", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "transformer models perform well at disentangling, they seem to be quite poor for Counting, with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 278, + 499 + ], + "score": 1.0, + "content": "image-transformer models getting to only", + "type": "text" + }, + { + "bbox": [ + 279, + 486, + 331, + 497 + ], + "score": 0.89, + "content": "3 2 . 4 \\pm 1 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "mAP, suggesting a potential weakness for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "transformers. Overall, there appears to be an intimate link between the generalization setting and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "the input modality, suggesting avenues where representation learning could be improved for a given", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 518, + 472, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 472, + 533 + ], + "score": 1.0, + "content": "modality (e.g. images), relative to the kind of reasoning one is interested in (e.g. counting).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 504, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "When does language help? On average, training models with explicit concept supervision using the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 318, + 559 + ], + "score": 1.0, + "content": "concept loss (Section 4.1) improves performance by", + "type": "text" + }, + { + "bbox": [ + 318, + 547, + 365, + 558 + ], + "score": 0.9, + "content": "2 . 8 \\pm 0 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "mAP (SEM error). This is a small", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "boost relative to the gap between the original model and the strong oracle, suggesting that this simple", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "auxiliary loss is not sufficient to internalize the LOT in a neural network. Overall, image models", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 579, + 495, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 495, + 593 + ], + "score": 1.0, + "content": "benefit more from language than schema models which natively utilize symbols (Appendix G.3).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 108, + 608, + 195, + 620 + ], + "lines": [ + { + "bbox": [ + 104, + 605, + 197, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 605, + 197, + 623 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "We introduced the compositional reasoning under uncertainty (CURI) benchmark for evaluating few-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "shot concept learning in a large compositional space, capturing the kinds of producitivity, unboundness", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "and underdetermination that characterize human conceptual reasoning. We instantiate a series of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "meta-learning tasks, and evaluate numerous baseline models on various aspects of compositional", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "reasoning under uncertainty, including inferential coherence, boolean operation learning, counting,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "disentangling, etc. Further, we introduce the notion of a compositionality gap to quantify the difficultly", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 697, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 506, + 714 + ], + "score": 1.0, + "content": "of each generalization type, and to estimate the degree of compositionality in current deep learning", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "models. We hope our contributions of dataset, compositionality gaps, evaluation metrics and baseline", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 479, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 479, + 734 + ], + "score": 1.0, + "content": "models help spur progress in the important research direction of productive concept learning.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 116 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 81, + 507, + 118 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 129, + 276, + 141 + ], + "lines": [ + { + "bbox": [ + 105, + 128, + 277, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 277, + 142 + ], + "score": 1.0, + "content": "5.2 DIFFERENCES BETWEEN MODELS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 507, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 507, + 164 + ], + "score": 1.0, + "content": "Best Models. In general, the best performing model is the relation-net applied to schema inputs,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 162, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 174 + ], + "score": 1.0, + "content": "outperforming other combinations of models and input modalities on the Boolean, Concept IID,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 173, + 504, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 504, + 185 + ], + "score": 1.0, + "content": "Complexity, and Instance IID splits on both the mAP as well as accuracy metrics (Figure 5); although", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "as mentioned above, none of the models are close to the strong oracle. It is closely followed by the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 507, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 507, + 208 + ], + "score": 1.0, + "content": "transformer model on schema inputs, which performs the best on Binding (color), Binding (shape),", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "and Intrinsic splits (Appendix G.7). Utilizing schema inputs proves easier for abstraction except", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "score": 1.0, + "content": "for the Extrinsic setting, where the task requires generalization to novel locations for objects in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 507, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 507, + 240 + ], + "score": 1.0, + "content": "images, which is well supported by the inductive bias of the CNN encoder (Figure 4). In this case,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 238, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 262, + 250 + ], + "score": 1.0, + "content": "the image-transformer gets an mAP of", + "type": "text" + }, + { + "bbox": [ + 262, + 238, + 314, + 249 + ], + "score": 0.91, + "content": "6 2 . 1 \\pm 0 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 238, + 506, + 250 + ], + "score": 1.0, + "content": ", compared to the next best schema-transformer", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 145, + 262 + ], + "score": 1.0, + "content": "model at", + "type": "text" + }, + { + "bbox": [ + 145, + 249, + 190, + 260 + ], + "score": 0.88, + "content": "6 0 . 9 \\pm 0 . 7", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 249, + 506, + 262 + ], + "score": 1.0, + "content": ". Further, relational learning proves more crucial in the schema case than for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 427, + 272 + ], + "score": 1.0, + "content": "images, with all image models (regardless of pooling) performing better than", + "type": "text" + }, + { + "bbox": [ + 427, + 260, + 480, + 271 + ], + "score": 0.89, + "content": "5 9 . 4 \\pm 1 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 260, + 506, + 272 + ], + "score": 1.0, + "content": "mAP", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 473, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 416, + 284 + ], + "score": 1.0, + "content": "(achieved for image-avg-pool) while schema-avg-pool models get only get to", + "type": "text" + }, + { + "bbox": [ + 417, + 271, + 468, + 282 + ], + "score": 0.9, + "content": "5 3 . 4 \\pm 1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 271, + 473, + 284 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 150, + 507, + 284 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "When to use a transformer? Transformer models appear to outperform relation networks in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 311 + ], + "score": 1.0, + "content": "splits concerning disentangling. For instance, for the Intrinsic split with schema-relation-net is at", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 308, + 507, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 159, + 321 + ], + "score": 0.85, + "content": "5 5 . 1 \\pm 0 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 308, + 175, + 323 + ], + "score": 1.0, + "content": "v.s.", + "type": "text" + }, + { + "bbox": [ + 175, + 310, + 228, + 320 + ], + "score": 0.86, + "content": "5 7 . 9 \\pm 0 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 308, + 507, + 323 + ], + "score": 1.0, + "content": "for schema-transformer. Similarly, for the Extrinsic split the image-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 174, + 333 + ], + "score": 1.0, + "content": "transformer is at", + "type": "text" + }, + { + "bbox": [ + 175, + 321, + 227, + 331 + ], + "score": 0.89, + "content": "6 2 . 1 \\pm 0 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 320, + 382, + 333 + ], + "score": 1.0, + "content": "compared to the image-relation-net at", + "type": "text" + }, + { + "bbox": [ + 382, + 321, + 434, + 331 + ], + "score": 0.9, + "content": "6 0 . 8 \\pm 1 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 320, + 505, + 333 + ], + "score": 1.0, + "content": ". We hypothesize", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "that this is because the iterative message passing via. attention in transformers improves object", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 343, + 478, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 478, + 355 + ], + "score": 1.0, + "content": "representations for disentangling compared to relation networks that lack such a mechanism.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 287, + 507, + 355 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 360, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "What is the relative difficulty of abstraction from different modalities? One of the key con-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "tributions of our work is in providing multiple modalities (image, schema, sound) for productive", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "concept learning. We next characterize the difficulty of abstraction based on modality for the various", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "score": 1.0, + "content": "generalization settings. In the Intrinsic setting, we find that the schema models, which have access to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "a “perfect” disentangled representation significantly outperform image models—a schema-avg-pool", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 415, + 489, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 198, + 426 + ], + "score": 1.0, + "content": "model gets an mAP of", + "type": "text" + }, + { + "bbox": [ + 198, + 415, + 250, + 425 + ], + "score": 0.9, + "content": "5 2 . 7 \\pm 3 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 415, + 410, + 426 + ], + "score": 1.0, + "content": "while an image-avg-pool model gets to", + "type": "text" + }, + { + "bbox": [ + 410, + 415, + 462, + 425 + ], + "score": 0.87, + "content": "3 4 . 4 \\pm 0 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 415, + 489, + 426 + ], + "score": 1.0, + "content": "mAP.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 359, + 506, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 431, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 443 + ], + "score": 1.0, + "content": "Similarly, for the Counting split where the total number of objects are exactly specified in the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "schema (Figure 4), schemas are substantially better than images. For example, schema-relation-nets", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 452, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 132, + 467 + ], + "score": 1.0, + "content": "get to", + "type": "text" + }, + { + "bbox": [ + 133, + 453, + 196, + 464 + ], + "score": 0.9, + "content": "5 6 . 2 5 \\pm 5 . 3 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 452, + 333, + 467 + ], + "score": 1.0, + "content": "mAP while image-avg-pool is at", + "type": "text" + }, + { + "bbox": [ + 333, + 453, + 386, + 464 + ], + "score": 0.89, + "content": "4 8 . 4 \\pm 1 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 452, + 506, + 467 + ], + "score": 1.0, + "content": "mAP. Interestingly, the next", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 353, + 476 + ], + "score": 1.0, + "content": "best model—image-relation-net—is substantially worse, at", + "type": "text" + }, + { + "bbox": [ + 353, + 464, + 412, + 475 + ], + "score": 0.91, + "content": "3 9 . 4 5 \\pm 1 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 464, + 505, + 476 + ], + "score": 1.0, + "content": ". Curiously, for while", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "transformer models perform well at disentangling, they seem to be quite poor for Counting, with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 278, + 499 + ], + "score": 1.0, + "content": "image-transformer models getting to only", + "type": "text" + }, + { + "bbox": [ + 279, + 486, + 331, + 497 + ], + "score": 0.89, + "content": "3 2 . 4 \\pm 1 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "mAP, suggesting a potential weakness for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "transformers. Overall, there appears to be an intimate link between the generalization setting and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "the input modality, suggesting avenues where representation learning could be improved for a given", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 518, + 472, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 472, + 533 + ], + "score": 1.0, + "content": "modality (e.g. images), relative to the kind of reasoning one is interested in (e.g. counting).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 432, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 504, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "When does language help? On average, training models with explicit concept supervision using the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 318, + 559 + ], + "score": 1.0, + "content": "concept loss (Section 4.1) improves performance by", + "type": "text" + }, + { + "bbox": [ + 318, + 547, + 365, + 558 + ], + "score": 0.9, + "content": "2 . 8 \\pm 0 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "mAP (SEM error). This is a small", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "boost relative to the gap between the original model and the strong oracle, suggesting that this simple", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "auxiliary loss is not sufficient to internalize the LOT in a neural network. Overall, image models", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 579, + 495, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 495, + 593 + ], + "score": 1.0, + "content": "benefit more from language than schema models which natively utilize symbols (Appendix G.3).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 535, + 505, + 593 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 608, + 195, + 620 + ], + "lines": [ + { + "bbox": [ + 104, + 605, + 197, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 605, + 197, + 623 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "We introduced the compositional reasoning under uncertainty (CURI) benchmark for evaluating few-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "shot concept learning in a large compositional space, capturing the kinds of producitivity, unboundness", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "and underdetermination that characterize human conceptual reasoning. We instantiate a series of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "meta-learning tasks, and evaluate numerous baseline models on various aspects of compositional", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "reasoning under uncertainty, including inferential coherence, boolean operation learning, counting,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "disentangling, etc. Further, we introduce the notion of a compositionality gap to quantify the difficultly", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 697, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 506, + 714 + ], + "score": 1.0, + "content": "of each generalization type, and to estimate the degree of compositionality in current deep learning", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "models. 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March", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 115, + 92, + 143, + 106 + ], + "spans": [ + { + "bbox": [ + 115, + 92, + 143, + 106 + ], + "score": 1.0, + "content": "2017.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 115, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "J B Tenenbaum and T L Griffiths. Generalization, similarity, and Bayesian inference. Behavioral and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 115, + 126, + 253, + 138 + ], + "spans": [ + { + "bbox": [ + 115, + 126, + 253, + 138 + ], + "score": 1.0, + "content": "Brain Sciences, 24:629–40, 2001.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 147, + 506, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "score": 1.0, + "content": "Eleni Triantafillou, Tyler Zhu, Vincent Dumoulin, Pascal Lamblin, Utku Evci, Kelvin Xu, Ross", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 116, + 159, + 507, + 172 + ], + "spans": [ + { + "bbox": [ + 116, + 159, + 507, + 172 + ], + "score": 1.0, + "content": "Goroshin, Carles Gelada, Kevin Swersky, Pierre-Antoine Manzagol, and Hugo Larochelle. Meta-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 116, + 170, + 456, + 183 + ], + "spans": [ + { + "bbox": [ + 116, + 170, + 456, + 183 + ], + "score": 1.0, + "content": "Dataset: A dataset of datasets for learning to learn from few examples. March 2019.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 105, + 192, + 504, + 215 + ], + "lines": [ + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "score": 1.0, + "content": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 116, + 203, + 383, + 215 + ], + "spans": [ + { + "bbox": [ + 116, + 203, + 383, + 215 + ], + "score": 1.0, + "content": "Kaiser, and Illia Polosukhin. Attention is all you need. June 2017.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 504, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "Ramakrishna Vedantam, Ian Fischer, Jonathan Huang, and Kevin Murphy. Generative models of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 116, + 236, + 290, + 248 + ], + "spans": [ + { + "bbox": [ + 116, + 236, + 290, + 248 + ], + "score": 1.0, + "content": "visually grounded imagination. May 2017.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 507, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 507, + 272 + ], + "score": 1.0, + "content": "Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Koray Kavukcuoglu, and Daan Wierstra. Match-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 115, + 270, + 306, + 281 + ], + "spans": [ + { + "bbox": [ + 115, + 270, + 306, + 281 + ], + "score": 1.0, + "content": "ing networks for one shot learning. June 2016.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 504, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "score": 1.0, + "content": "Duo Wang, Mateja Jamnik, and Pietro Lio. Abstract diagrammatic reasoning with multiplex graph", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 115, + 303, + 229, + 315 + ], + "spans": [ + { + "bbox": [ + 115, + 303, + 229, + 315 + ], + "score": 1.0, + "content": "networks. September 2019.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 108, + 324, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 104, + 322, + 507, + 337 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 507, + 337 + ], + "score": 1.0, + "content": "F Xu and J B Tenenbaum. Word learning as Bayesian inference. Psychological Review, 114:245–272,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 114, + 334, + 143, + 348 + ], + "spans": [ + { + "bbox": [ + 114, + 334, + 143, + 348 + ], + "score": 1.0, + "content": "2007.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 107, + 372, + 335, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 337, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 337, + 385 + ], + "score": 1.0, + "content": "A EXAMPLE EPISODES FROM THE DATASET", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "We show examples from the Concept IID split test set comprising the ground truth productive concept", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "(top), along with the support and query sets for meta learning (rendered as images), the alternate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "hypotheses which are consistent with the support set – that is, other hypotheses which could also", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 430, + 504, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 504, + 442 + ], + "score": 1.0, + "content": "have generated the positive and negative examples in the support set – and the concepts based on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 441, + 306, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 306, + 453 + ], + "score": 1.0, + "content": "which we pick the hard negatives Figures 6 to 11.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 470, + 293, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 294, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 294, + 485 + ], + "score": 1.0, + "content": "B ADDITIONAL DATASET DETAILS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 312, + 509 + ], + "score": 1.0, + "content": "We first provide more details of the concept space", + "type": "text" + }, + { + "bbox": [ + 313, + 497, + 321, + 507 + ], + "score": 0.78, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 496, + 439, + 509 + ], + "score": 1.0, + "content": ", then explain how we obtain", + "type": "text" + }, + { + "bbox": [ + 440, 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"index": 25 + }, + { + "type": "title", + "bbox": [ + 108, + 545, + 279, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 544, + 281, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 281, + 557 + ], + "score": 1.0, + "content": "B.1 MORE DETAILS OF THE GRAMMAR", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 386, + 578 + ], + "score": 1.0, + "content": "We provide below the full grammar used to sample concepts, where", + "type": "text" + }, + { + "bbox": [ + 387, + 566, + 430, + 578 + ], + "score": 0.93, + "content": "A \\to B | C", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 566, + 478, + 578 + ], + "score": 1.0, + "content": "means that", + "type": "text" + }, + { + "bbox": [ + 478, + 566, + 487, + 576 + ], + "score": 0.73, + "content": "A", + "type": "inline_equation" + }, 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We always start expanding at the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 506, + 600 + ], + "score": 1.0, + "content": "START token and then follow the rules of the grammar until we hit a terminal node (which does not", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "have any expansions defined). As and where possible, we followed the insights from Piantadosi et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "(2016) in choosing the sampling probabilities for various completions based on how well humans seem", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "to be able to learn the corresponding primitive. For example, we sample utterances with disjunctions", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "(or) less frequently since they are known to be difficult for humans to learn. Based on Kemp & Jern", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 643, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 655 + ], + "score": 1.0, + "content": "(2009), we chose to represent location as a discrete entity, such that relative, and categorical notions of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 420, + 667 + ], + "score": 1.0, + "content": "left or right simply become comparisons in the location space (location?", + "type": "text" + }, + { + "bbox": [ + 421, + 654, + 440, + 664 + ], + "score": 0.8, + "content": "\\textbf { x } >", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 654, + 480, + 667 + ], + "score": 1.0, + "content": "location?", + "type": "text" + }, + { + "bbox": [ + 480, + 654, + 500, + 666 + ], + "score": 0.87, + "content": "S _ { - \\mathbf { x } , }", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "),", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 664, + 478, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 478, + 678 + ], + "score": 1.0, + "content": "unlike the CLEVR dataset (Johnson et al., 2016) which defines categorical relational objects.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 108, + 681, + 504, + 715 + ], + "lines": [ + { + "bbox": [ + 105, + 680, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 208, + 695 + ], + "score": 1.0, + "content": "Here is the full grammar", + "type": "text" + }, + { + "bbox": [ + 208, + 682, + 216, + 692 + ], + "score": 0.82, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 680, + 474, + 695 + ], + "score": 1.0, + "content": "used for sampling the concepts (as explained in the main paper,", + "type": "text" + }, + { + "bbox": [ + 474, + 681, + 505, + 693 + ], + "score": 0.89, + "content": "S _ { - { \\bf x } } =", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 691, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 137, + 705 + ], + "score": 0.9, + "content": "S / \\{ \\mathbf { x } \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 691, + 506, + 706 + ], + "score": 1.0, + "content": ". Note that the grammar always generates strings in postfix notation and thus the operands in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 703, + 281, + 717 + ], + "spans": [ + { + "bbox": [ + 106, + 703, + 281, + 717 + ], + "score": 1.0, + "content": "each expansion occur before the operation:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + } + ], + "page_idx": 10, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Jake Snell, Kevin Swersky, and Richard S Zemel. Prototypical networks for few-shot learning. March", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 115, + 92, + 143, + 106 + ], + "spans": [ + { + "bbox": [ + 115, + 92, + 143, + 106 + ], + "score": 1.0, + "content": "2017.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 115, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "J B Tenenbaum and T L Griffiths. Generalization, similarity, and Bayesian inference. Behavioral and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 115, + 126, + 253, + 138 + ], + "spans": [ + { + "bbox": [ + 115, + 126, + 253, + 138 + ], + "score": 1.0, + "content": "Brain Sciences, 24:629–40, 2001.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 115, + 505, + 138 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 147, + 506, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "score": 1.0, + "content": "Eleni Triantafillou, Tyler Zhu, Vincent Dumoulin, Pascal Lamblin, Utku Evci, Kelvin Xu, Ross", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 116, + 159, + 507, + 172 + ], + "spans": [ + { + "bbox": [ + 116, + 159, + 507, + 172 + ], + "score": 1.0, + "content": "Goroshin, Carles Gelada, Kevin Swersky, Pierre-Antoine Manzagol, and Hugo Larochelle. Meta-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 116, + 170, + 456, + 183 + ], + "spans": [ + { + "bbox": [ + 116, + 170, + 456, + 183 + ], + "score": 1.0, + "content": "Dataset: A dataset of datasets for learning to learn from few examples. March 2019.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 147, + 507, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 192, + 504, + 215 + ], + "lines": [ + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "score": 1.0, + "content": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 116, + 203, + 383, + 215 + ], + "spans": [ + { + "bbox": [ + 116, + 203, + 383, + 215 + ], + "score": 1.0, + "content": "Kaiser, and Illia Polosukhin. Attention is all you need. June 2017.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 106, + 193, + 505, + 215 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 504, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "Ramakrishna Vedantam, Ian Fischer, Jonathan Huang, and Kevin Murphy. Generative models of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 116, + 236, + 290, + 248 + ], + "spans": [ + { + "bbox": [ + 116, + 236, + 290, + 248 + ], + "score": 1.0, + "content": "visually grounded imagination. May 2017.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 106, + 225, + 505, + 248 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 507, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 507, + 272 + ], + "score": 1.0, + "content": "Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Koray Kavukcuoglu, and Daan Wierstra. Match-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 115, + 270, + 306, + 281 + ], + "spans": [ + { + "bbox": [ + 115, + 270, + 306, + 281 + ], + "score": 1.0, + "content": "ing networks for one shot learning. June 2016.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 257, + 507, + 281 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 504, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "score": 1.0, + "content": "Duo Wang, Mateja Jamnik, and Pietro Lio. Abstract diagrammatic reasoning with multiplex graph", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 115, + 303, + 229, + 315 + ], + "spans": [ + { + "bbox": [ + 115, + 303, + 229, + 315 + ], + "score": 1.0, + "content": "networks. September 2019.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 290, + 505, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 324, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 104, + 322, + 507, + 337 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 507, + 337 + ], + "score": 1.0, + "content": "F Xu and J B Tenenbaum. Word learning as Bayesian inference. Psychological Review, 114:245–272,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 114, + 334, + 143, + 348 + ], + "spans": [ + { + "bbox": [ + 114, + 334, + 143, + 348 + ], + "score": 1.0, + "content": "2007.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 322, + 507, + 348 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 372, + 335, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 337, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 337, + 385 + ], + "score": 1.0, + "content": "A EXAMPLE EPISODES FROM THE DATASET", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "We show examples from the Concept IID split test set comprising the ground truth productive concept", + "type": 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"score": 1.0, + "content": "which we pick the hard negatives Figures 6 to 11.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 396, + 505, + 453 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 470, + 293, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 294, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 294, + 485 + ], + "score": 1.0, + "content": "B ADDITIONAL DATASET DETAILS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 312, + 509 + ], + "score": 1.0, + "content": "We first provide more details of the concept space", + "type": "text" + }, + { + "bbox": [ + 313, + 497, + 321, + 507 + ], + "score": 0.78, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 496, + 439, + 509 + ], + "score": 1.0, 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We always start expanding at the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 506, + 600 + ], + "score": 1.0, + "content": "START token and then follow the rules of the grammar until we hit a terminal node (which does not", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "have any expansions defined). As and where possible, we followed the insights from Piantadosi et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "(2016) in choosing the sampling probabilities for various completions based on how well humans seem", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "to be able to learn the corresponding primitive. For example, we sample utterances with disjunctions", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "(or) less frequently since they are known to be difficult for humans to learn. 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We then reject and filter the hypotheses to obtain a set of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 649, + 437, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 205, + 660 + ], + "score": 1.0, + "content": "“interesting” hypotheses", + "type": "text" + }, + { + "bbox": [ + 205, + 649, + 215, + 658 + ], + "score": 0.72, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 649, + 437, + 660 + ], + "score": 1.0, + "content": "used in the main paper explained in more detail below:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 614, + 506, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 102, + 664, + 504, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "Rejection Sampling: We reject the following string combinations after sampling from the grammar", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 675, + 119, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 114, + 687 + ], + "score": 0.75, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 675, + 119, + 689 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 664, + 506, + 689 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 132, + 696, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 132, + 696, + 260, + 713 + ], + "score": 1.0, + "content": "• All programs which contain \"", + "type": "text" + }, + { + "bbox": [ + 261, + 700, + 273, + 709 + ], + "score": 0.58, + "content": "\\lambda S", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 696, + 304, + 713 + ], + "score": 1.0, + "content": ". for-all", + "type": "text" + }, + { + "bbox": [ + 304, + 699, + 315, + 709 + ], + "score": 0.35, + "content": "\\mathbf { X } ^ { \" }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 696, + 337, + 713 + ], + "score": 1.0, + "content": "and \"", + "type": "text" + }, + { + "bbox": [ + 337, + 699, + 357, + 710 + ], + "score": 0.79, + "content": "S _ { - { \\bf x } } \"", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 696, + 505, + 713 + ], + "score": 1.0, + "content": "\" in the same program. This is asking", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 142, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 142, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "that for all objects in a scene, a certain property is satisfied by everything other than the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 720, + 397, + 733 + ], + "spans": [ + { + "bbox": [ + 142, + 720, + 397, + 733 + ], + "score": 1.0, + "content": "object, which is the same as saying, for all objects in the scene.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 132, + 696, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 132, + 82, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 132, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 132, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "• All programs where we compare the properties of the same object to itself, e.g. color?", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 144, + 94, + 475, + 106 + ], + "spans": [ + { + "bbox": [ + 144, + 94, + 249, + 105 + ], + "score": 0.32, + "content": "\\mathrm { ~ ~ { ~ ( ~ x ~ ) ~ } ~ } = = \\mathrm { ~ \\cos ~ } \\tt c o l o r ? \\mathrm { ~ ~ { ~ ( ~ x ~ ) ~ } ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 94, + 475, + 106 + ], + "score": 1.0, + "content": ", where color? can be any function applied to the object.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 140, + 109, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 140, + 109, + 489, + 120 + ], + "score": 1.0, + "content": "All programs where we have the following string: exists(color?(S)", + "type": "text" + }, + { + "bbox": [ + 489, + 110, + 505, + 119 + ], + "score": 0.67, + "content": "= =", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 118, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 141, + 118, + 322, + 133 + ], + "score": 1.0, + "content": "color?(x)) or for-all(color?(S)", + "type": "text" + }, + { + "bbox": [ + 322, + 120, + 339, + 129 + ], + "score": 0.78, + "content": "= =", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 118, + 505, + 133 + ], + "score": 1.0, + "content": "color?(x)) where color? can be any", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 131, + 264, + 143 + ], + "spans": [ + { + "bbox": [ + 141, + 131, + 264, + 143 + ], + "score": 1.0, + "content": "function applied to the object.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 134, + 145, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 134, + 145, + 375, + 157 + ], + "score": 1.0, + "content": "All programs which evaluate to true on schemas more than", + "type": "text" + }, + { + "bbox": [ + 375, + 145, + 394, + 156 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 145, + 505, + 157 + ], + "score": 1.0, + "content": "of the time and less than 10", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 156, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 141, + 156, + 505, + 169 + ], + "score": 1.0, + "content": "times. 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Note that while some of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 382, + 463 + ], + "score": 1.0, + "content": "these concepts might be truly synonymous to each other (for example,", + "type": "text" + }, + { + "bbox": [ + 382, + 451, + 412, + 461 + ], + "score": 0.91, + "content": "A > B", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 451, + 469, + 463 + ], + "score": 1.0, + "content": "is the same as", + "type": "text" + }, + { + "bbox": [ + 469, + 451, + 500, + 461 + ], + "score": 0.89, + "content": "B < A", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "),", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "others might be synonymous in context of the image distribution we work with. 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We use the following tokens for test:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 192, + 147, + 249, + 159 + ], + "lines": [ + { + "bbox": [ + 191, + 146, + 251, + 160 + ], + "spans": [ + { + "bbox": [ + 191, + 146, + 251, + 160 + ], + "score": 1.0, + "content": "‘cylinder’", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 139, + 174, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 141, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 141, + 174, + 201, + 188 + ], + "score": 1.0, + "content": "We then create", + "type": "text" + }, + { + "bbox": [ + 201, + 176, + 221, + 186 + ], + "score": 0.91, + "content": "{ \\mathcal { H } } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "to contain all concepts which have any of the tokens above. For example,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 185, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 141, + 185, + 346, + 199 + ], + "score": 1.0, + "content": "if a concept has cylinder, we would place it in", + "type": "text" + }, + { + "bbox": [ + 347, + 186, + 366, + 198 + ], + "score": 0.91, + "content": "\\mathcal { H } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 185, + 505, + 199 + ], + "score": 1.0, + "content": ". After every feasible candidate is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 195, + 502, + 211 + ], + "spans": [ + { + "bbox": [ + 141, + 195, + 181, + 211 + ], + "score": 1.0, + "content": "placed in", + "type": "text" + }, + { + "bbox": [ + 181, + 198, + 201, + 208 + ], + "score": 0.9, + "content": "\\mathcal { H } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 195, + 401, + 211 + ], + "score": 1.0, + "content": "based on this heurisitc, the remaining concepts in", + "type": "text" + }, + { + "bbox": [ + 401, + 198, + 411, + 207 + ], + "score": 0.81, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 195, + 474, + 211 + ], + "score": 1.0, + "content": "are assigned to", + "type": "text" + }, + { + "bbox": [ + 474, + 197, + 496, + 208 + ], + "score": 0.9, + "content": "\\mathcal { H } _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 195, + 502, + 211 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 132, + 212, + 504, + 246 + ], + "lines": [ + { + "bbox": [ + 134, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 134, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "• Complexity: This split partitions into train and test based on length of the postfix serialization", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 222, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 222, + 436, + 236 + ], + "score": 1.0, + "content": "of the concept. Specifically, concepts shorter than 10 tokens are placed in", + "type": "text" + }, + { + "bbox": [ + 436, + 223, + 459, + 234 + ], + "score": 0.9, + "content": "\\mathcal { H } _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 222, + 505, + 236 + ], + "score": 1.0, + "content": "and longer", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 233, + 257, + 248 + ], + "spans": [ + { + "bbox": [ + 142, + 233, + 233, + 248 + ], + "score": 1.0, + "content": "concepts are placed in", + "type": "text" + }, + { + "bbox": [ + 233, + 235, + 253, + 245 + ], + "score": 0.9, + "content": "{ \\mathcal { H } } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 233, + 257, + 248 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 261, + 327, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 329, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 329, + 277 + ], + "score": 1.0, + "content": "D CREATING SUPPORT AND QUERY SETS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 506, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 316, + 300 + ], + "score": 1.0, + "content": "We next explain how we go from the initial dataset", + "type": "text" + }, + { + "bbox": [ + 316, + 287, + 333, + 297 + ], + "score": 0.34, + "content": "\\mathcal { U } -", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 285, + 506, + 300 + ], + "score": 1.0, + "content": "which contains a large number of images,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 277, + 310 + ], + "score": 1.0, + "content": "schema and sounds – and a concept space", + "type": "text" + }, + { + "bbox": [ + 277, + 298, + 300, + 309 + ], + "score": 0.91, + "content": "\\mathcal { H } _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 298, + 318, + 310 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 319, + 298, + 339, + 308 + ], + "score": 0.9, + "content": "{ \\mathcal { H } } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 298, + 505, + 310 + ], + "score": 1.0, + "content": ", to a dataset for meta learning. To create", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "the training/validation/test sets for models, we sample a series of episodes, each containing a support", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 320, + 451, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 451, + 332 + ], + "score": 1.0, + "content": "set and a query set. We illustrate the sampling procedure for a training episode below:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 107, + 336, + 292, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 293, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 293, + 350 + ], + "score": 1.0, + "content": "Support Set Sampling with Hard Negatives", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 128, + 357, + 506, + 462 + ], + "lines": [ + { + "bbox": [ + 129, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 129, + 357, + 203, + 370 + ], + "score": 1.0, + "content": "1. Pick a concept", + "type": "text" + }, + { + "bbox": [ + 203, + 357, + 255, + 369 + ], + "score": 0.93, + "content": "h \\sim p _ { \\mathrm { t r a i n } } ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 357, + 505, + 370 + ], + "score": 1.0, + "content": ", with a preference for shorter hypotheses being more frequent", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 368, + 381, + 381 + ], + "spans": [ + { + "bbox": [ + 141, + 368, + 381, + 381 + ], + "score": 1.0, + "content": "based on the weights used to define the prior Appendix B.3", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 128, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 128, + 383, + 200, + 396 + ], + "score": 1.0, + "content": "2. Pick 5 images", + "type": "text" + }, + { + "bbox": [ + 201, + 383, + 216, + 394 + ], + "score": 0.77, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 383, + 325, + 396 + ], + "score": 1.0, + "content": ", uniformly at random from", + "type": "text" + }, + { + "bbox": [ + 326, + 383, + 335, + 393 + ], + "score": 0.76, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 383, + 374, + 396 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 374, + 383, + 417, + 395 + ], + "score": 0.92, + "content": "h ( \\mathbf { u } ^ { s } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 383, + 505, + 396 + ], + "score": 1.0, + "content": ", where the concept is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 394, + 390, + 406 + ], + "spans": [ + { + "bbox": [ + 141, + 394, + 390, + 406 + ], + "score": 1.0, + "content": "evaluated on the schema to determine the label Appendix B.4", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 128, + 408, + 384, + 422 + ], + "spans": [ + { + "bbox": [ + 128, + 408, + 236, + 422 + ], + "score": 1.0, + "content": "3. Identify other concepts", + "type": "text" + }, + { + "bbox": [ + 237, + 408, + 267, + 420 + ], + "score": 0.9, + "content": "h ^ { \\prime } \\in \\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 408, + 283, + 422 + ], + "score": 1.0, + "content": "s.t.", + "type": "text" + }, + { + "bbox": [ + 284, + 408, + 336, + 421 + ], + "score": 0.92, + "content": "h ( u ( S ) ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 408, + 354, + 422 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 354, + 408, + 384, + 421 + ], + "score": 0.86, + "content": "h ^ { \\prime } \\neq h", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 128, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 128, + 423, + 230, + 437 + ], + "score": 1.0, + "content": "4. Pick images such that", + "type": "text" + }, + { + "bbox": [ + 231, + 423, + 277, + 435 + ], + "score": 0.92, + "content": "h ^ { \\prime } ( \\mathbf { u } ^ { s } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 423, + 295, + 437 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 295, + 423, + 339, + 436 + ], + "score": 0.93, + "content": "h ( \\mathbf { u } ^ { s } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 423, + 389, + 437 + ], + "score": 1.0, + "content": "as negatives", + "type": "text" + }, + { + "bbox": [ + 390, + 424, + 406, + 435 + ], + "score": 0.75, + "content": "( N )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 423, + 506, + 437 + ], + "score": 1.0, + "content": ". If no such images exist,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 435, + 418, + 448 + ], + "spans": [ + { + "bbox": [ + 141, + 435, + 247, + 448 + ], + "score": 1.0, + "content": "pick random images from", + "type": "text" + }, + { + "bbox": [ + 248, + 436, + 256, + 445 + ], + "score": 0.8, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 435, + 418, + 448 + ], + "score": 1.0, + "content": "as negatives until we have 20 negatives.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 128, + 449, + 241, + 463 + ], + "spans": [ + { + "bbox": [ + 128, + 449, + 172, + 463 + ], + "score": 1.0, + "content": "5. Return", + "type": "text" + }, + { + "bbox": [ + 172, + 450, + 236, + 462 + ], + "score": 0.91, + "content": "D _ { \\operatorname { s u p p } } = P \\cup N", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 449, + 241, + 463 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "The sampling procedure for the Query set iterates all the steps above (except step 1, where we choose", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 482, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 154, + 493 + ], + "score": 1.0, + "content": "the concept", + "type": "text" + }, + { + "bbox": [ + 154, + 482, + 161, + 492 + ], + "score": 0.72, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 482, + 506, + 493 + ], + "score": 1.0, + "content": "). Step 3 and 4 outline a procedure for identifying hard negatives for training the model,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 418, + 505 + ], + "score": 1.0, + "content": "by looking at other hypotheses which also explain a chosen set of positives", + "type": "text" + }, + { + "bbox": [ + 418, + 493, + 427, + 502 + ], + "score": 0.83, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "and using them to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 504, + 259, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 259, + 516 + ], + "score": 1.0, + "content": "clarify what the concept of interest is.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 340, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 342, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 342, + 535 + ], + "score": 1.0, + "content": "We give below an analogous procedure for easy negatives:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 107, + 537, + 290, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 290, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 290, + 552 + ], + "score": 1.0, + "content": "Support Set Sampling with Easy Negatives", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 129, + 558, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 129, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 129, + 558, + 203, + 572 + ], + "score": 1.0, + "content": "1. Pick a concept", + "type": "text" + }, + { + "bbox": [ + 203, + 559, + 255, + 570 + ], + "score": 0.93, + "content": "h \\sim p _ { \\mathrm { t r a i n } } ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 558, + 505, + 572 + ], + "score": 1.0, + "content": ", with a preference for shorter hypotheses being more frequent", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 569, + 381, + 582 + ], + "spans": [ + { + "bbox": [ + 141, + 569, + 381, + 582 + ], + "score": 1.0, + "content": "based on the weights used to define the prior Appendix B.3", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 129, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 129, + 584, + 200, + 597 + ], + "score": 1.0, + "content": "2. 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To create", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "the training/validation/test sets for models, we sample a series of episodes, each containing a support", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 320, + 451, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 451, + 332 + ], + "score": 1.0, + "content": "set and a query set. 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Identify other concepts", + "type": "text" + }, + { + "bbox": [ + 237, + 408, + 267, + 420 + ], + "score": 0.9, + "content": "h ^ { \\prime } \\in \\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 408, + 283, + 422 + ], + "score": 1.0, + "content": "s.t.", + "type": "text" + }, + { + "bbox": [ + 284, + 408, + 336, + 421 + ], + "score": 0.92, + "content": "h ( u ( S ) ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 408, + 354, + 422 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 354, + 408, + 384, + 421 + ], + "score": 0.86, + "content": "h ^ { \\prime } \\neq h", + "type": "inline_equation" + } + ], + "index": 22, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 128, + 423, + 230, + 437 + ], + "score": 1.0, + "content": "4. 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The", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "initial representation from the first stage of the encoder (Fig. 4 in main paper) with the objects for", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 188, + 171 + ], + "score": 1.0, + "content": "images has a size of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 189, + 160, + 213, + 170 + ], + "score": 0.76, + "content": "\\mathsf { 1 0 x 8 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 214, + 160, + 506, + 171 + ], + "score": 1.0, + "content": "i.e. there are 80 objects, while for sound representations have 38 objects.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "In the schema case the number of objects is the ground truth number of objects which is provided as", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 184, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 184, + 194 + ], + "score": 1.0, + "content": "input to the model.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 41.5, + "bbox_fs": [ + 103, + 709, + 508, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 210, + 96 + ], + "score": 1.0, + "content": "models while for schema", + "type": "text" + }, + { + "bbox": [ + 211, + 82, + 248, + 93 + ], + "score": 0.92, + "content": "o _ { i } \\in \\mathbb { R } ^ { 9 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 80, + 404, + 96 + ], + "score": 1.0, + "content": ". 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The", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "initial representation from the first stage of the encoder (Fig. 4 in main paper) with the objects for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 188, + 171 + ], + "score": 1.0, + "content": "images has a size of", + "type": "text" + }, + { + "bbox": [ + 189, + 160, + 213, + 170 + ], + "score": 0.76, + "content": "\\mathsf { 1 0 x 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 160, + 506, + 171 + ], + "score": 1.0, + "content": "i.e. there are 80 objects, while for sound representations have 38 objects.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "In the schema case the number of objects is the ground truth number of objects which is provided as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 184, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 184, + 194 + ], + "score": 1.0, + "content": "input to the model.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "score": 1.0, + "content": "We trained all the models on the training set, and picked the best performing checkpoints on training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "– measured in terms of mAP– to report all the results in the main paper. 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Let", + "type": "text" + }, + { + "bbox": [ + 455, + 322, + 491, + 334 + ], + "score": 0.92, + "content": "o _ { i } \\in \\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "the output object feature from the modality specific encoder (Figure 4, left panel), and let us denote", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 102, + 340, + 509, + 364 + ], + "spans": [ + { + "bbox": [ + 102, + 340, + 120, + 364 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 344, + 176, + 360 + ], + "score": 0.94, + "content": "O = \\{ o _ { i } \\} _ { i = 1 } ^ { | O | }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 340, + 509, + 364 + ], + "score": 1.0, + "content": "the set of features for each of the objects in the scene, which includes optional", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 465, + 370 + ], + "score": 1.0, + "content": "position information indicating where the object is present in the scene (Figure 4). 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modalitypoolingmAP (mean)mAP (std)
schematransformer15.90.9
schemarelation-net15.30.9
imageavg-pool15.10.4
imagerelation-net14.80.7
schemaavg-pool14.41.2
imagetransformer14.20.2
schemaconcat14.00.7
imageconcat13.50.4
soundavg-pool9.40.6
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0000000000000000000000000000000000000000..6581f122b03fbb08c234f4096e2fff9f985941eb --- /dev/null +++ b/parse/train/S1fduCl0b/S1fduCl0b.md @@ -0,0 +1,356 @@ +# LIFELONG GENERATIVE MODELING + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Lifelong learning is the problem of learning multiple consecutive tasks in a sequential manner where knowledge gained from previous tasks is retained and used for future learning. It is essential towards the development of intelligent machines that can adapt to their surroundings. In this work we focus on a lifelong learning approach to generative modeling where we continuously incorporate newly observed streaming distributions into our learnt model. We do so through a student-teacher architecture which allows us to learn and preserve all the distributions seen so far without the need to retain the past data nor the past models. Through the introduction of a novel cross-model regularizer, the student model leverages the information learnt by the teacher, which acts as a summary of everything seen till now. The regularizer has the additional benefit of reducing the effect of catastrophic interference that appears when we learn over streaming data. We demonstrate its efficacy on streaming distributions as well as its ability to learn a common latent representation across a complex transfer learning scenario. + +# 1 INTRODUCTION + +Deep unsupervised generative learning allows us to take advantage of the massive amount of unlabeled data available in order to build models that efficiently compress and learn an approximation of the true data distribution. It has numerous applications such as image denoising, inpainting, super-resolution, structured prediction, clustering, pre-training and many more. However, something that is lacking in the modern ML toolbox is an efficient way to learn these deep generative models in a sequential, lifelong setting. + +In a lot of real world scenarios we observe distributions sequentially. Examples of this include streaming data from sensors such as cameras and microphones or other similar time series data. A system can also be resource limited wherein all of the past data or learnt models cannot be stored. We are interested in the lifelong learning setting for generative models where data arrives sequentially in a stream and where the storage of all data is infeasible. Within the stream, instances are generated according to some non-observed distribution which changes at given time-points. We assume we know the time points at which the transitions occur and whether the latent distribution is a completely new one or one that has been observed before. We do not however know the underlying identity of the individual distributions. Our goal is to learn a generative model that can summarize all the distributions seen so far in the stream. We give an example of such a setting in figure 1(a) using MNIST LeCun & Cortes (2010), where we have three unique distributions and one that is repeated. + +Since we only observe one distribution at a time we need to develop a strategy of retaining the previously learnt knowledge (i.e. the previously learnt distributions) and integrate it into future learning. To accumulate additional distributions in the current generative model we utilize a student-teacher architecture similar to that in distillation methods Hinton et al. (2015); Furlanello et al. (2016). The teacher contains a summary of all past distributions and is used to augment the data used to train the student model. The student model thus receives data samples from the currently observable distribution as well as synthetic data samples from previous distributions. This allows the student model to learn a distribution that summarizes the current as well as all previously observed distributions. Once a new distribution shift occurs the existing teacher model is discarded, the student becomes the teacher and a new student is instantiated. + +We further leverage the generative model of the teacher by introducing a regularizer in the learning objective function of the student that brings the posterior distribution of the latter close to that of the former. This allows us to build upon and extend the teacher’s generative model in the student each time the latter is re-instantiated (rather than re-learning it from scratch). By coupling this regularizer with a weight transfer from the teacher to the student we also allow for faster convergence of the student model. We empirically show that the regularizer allows us to learn a much larger set of distributions without catastrophic interference McCloskey & Cohen (1989). + +![](images/e4a0de89f53f3da3ca7f5c7c076ac99f174d75abf3b0fa6674908d15676319ae.jpg) +Figure 1: (a) Our problem setting where we sequentially observe samples from multiple unknown distributions; (b)Visualization of a learnt two-dimensional posterior of MNIST, evaluated with samples from the full test set. + +We build our lifelong generative models over Variational Autoencoders (VAEs) Kingma & Welling (2014). VAEs learn the posterior distribution of a latent variable model using an encoder network; they generate data by sampling from a prior and decoding the sample through a conditional distribution learnt by a decoder network. + +Using a vanilla VAE as a teacher to generate synthetic data for the student is problematic due to a couple of limitations of the VAE generative process. 1) Sampling the prior can select a point in the latent space that is in between two separate distributions, causing generation of unrealistic synthetic data and eventually leading to loss of previously learnt distributions. 2) Additionally, data points mapped to the posterior that are further away from the prior mean will be sampled less frequently resulting in an unbalanced sampling of the constituent distributions. Both limitations can be understood by visually inspecting the learnt posterior distribution of a standard VAE evaluated on test images from MNIST as shown in figure 1(b). To address the VAE’s sampling limitations we decompose the latent variable vector into a continuous and a discrete component. The discrete component is used to summarize the discriminative information of the individual generative distributions while the continuous caters for the remaining sample variability. By independently sampling the discrete and continuous components we preserve the distributional boundaries and circumvent the two problems above. + +This sampling strategy, combined with the proposed regularizer allows us to learn and remember all the individual distributions observed in the past. In addition we are also able to generate samples from any of the past distributions at will; we call this property consistent sampling. + +# 2 RELATED WORK + +Past work in sequential learning of generative models has focused on learning Gaussian mixture models Singer & Warmuth (1999); Declercq & Piater (2008) or on variational methods such as Variational EM Ghahramani & Attias (2000). Work that is closer to ours is the online or sequential learning of generative models in a streaming setting. Variational methods have been adapted for a streaming setting, e.g: Streaming Variational Bayes Broderick et al. (2013), Streaming Variational Mixture models Tank et al. (2015), and the Population Posterior McInerney et al. (2015). However their learning objectives are very different from ours. The objective of these methods is to adjust the learnt model such that it reflects the current data distribution as accurately as possible, while forgetting the previously observed distributions. Instead we want to do lifelong learning and retain all previously observed distributions within our learnt model. As far as we know our work is the first one that tries to bring generative models, and in particular VAEs, into a lifelong setting where distributions are seen, learnt, and remembered sequentially. + +VAEs rely on an encoder and a decoder neural network in order to learn the parameters of the posterior and likelihood. One of the central problems that arise when training a neural network in an sequential manner is that it causes the model to run into the problem of catastrophic interference McCloskey & Cohen (1989). Catastrophic interference appears when we train neural networks in a sequential manner and model parameters start to become biased to the most recent samples observed, while forgetting what was learnt from older samples. This generally happens when we stop exposing the model to past data. There have been a number of attempts to solve the problem of catastrophic interference in neural networks. These range from distillation methods such as the original method Hinton et al. (2015) and ALTM Furlanello et al. (2016), to utilizing privileged information Lopez-Paz et al. (2016), as well as transfer learning approaches such as Learning Without Forgetting Li & Hoiem (2016) and methods that relay information from previously learnt hidden layers such as in Progressive Neural Networks Rusu et al. (2016) and Deep Block-Modular Neural Networks Terekhov et al. (2015). All of these methods necessitate the storage of previous models or data; our method does not. + +The recent work of elastic weight consolidation (EWC) Kirkpatrick et al. (2017) utilizes the Fisher Information matrix (FIM) to avoid the problem of catastrophic interference. The FIM captures the sensitivity of the log-likelihood with respect to the model parameters; EWC leverages this (via a linear approximation of the FIM) to control the change of model parameter values between varying distributions. Intuitively, important parameters should not have their values changed, while non-important parameters are left unconstrained. Since EWC assumes model parameters being distributed under an exponential family, it allows for the utilization of the FIM as a quadratic approximationJeffreys (1946) to the Kullback-Leibler (KL) divergence. Our model makes no such distributional assumptions about the model parameters. Instead of constraining the parameters of the model as in EWC, we restrict the posterior representation of the student model to be close to that of the teacher for the previous distributions accumulated by the teacher. This allows the model parameters to vary as necessary in order to best fit the data. + +# 3 BACKGROUND + +We consider an unsupervised setting where we observe a sample $\mathbf { X }$ of $K \geq 1$ realizations $\mathbf { X } =$ $\{ \mathbf { x } ^ { ( 0 ) } , \mathbf { x } ^ { ( 1 ) } , . . . , \mathbf { x } ^ { ( K ) } \}$ from an unknown true distribution $P ^ { * } ( \mathbf { x } )$ with $\mathbf { x } \in \mathcal { R } ^ { N }$ . We assume that the data is generated by a random process involving a non-observed random variable $z \in \mathcal { R } ^ { M }$ . In order to incorporate our prior knowledge we posit a prior $P ( z )$ over $_ { z }$ . Our objective is to approximate the true underlying data distribution by a model $P _ { \pmb { \theta } } ( \mathbf { x } )$ such that $P _ { \pmb \theta } ( \mathbf { x } ) \approx P ^ { * } ( \mathbf { x } )$ . + +Given a latent variable model $P _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } ) P ( \mathbf { z } )$ we obtain the marginal likelihood $P _ { \pmb { \theta } } ( \mathbf { x } )$ by integrating out the latent variable $\mathbf { z }$ from the joint distribution. The joint distribution can in turn be factorized using the conditional distribution $P _ { \theta } ( \mathbf { x } | z )$ or the posterior $P _ { \boldsymbol { \theta } } ( z | \mathbf { x } )$ . + +$$ +P _ { \theta } ( { \bf x } ) = \int P _ { \theta } ( { \bf x } , z ) \delta z = \int P _ { \theta } ( z | { \bf x } ) P _ { \theta } ( { \bf x } ) \delta z = \int P _ { \theta } ( { \bf x } | z ) P ( z ) \delta z +$$ + +We model the conditional distribution $P _ { \theta } ( \mathbf { x } | z )$ by a decoder, typically a neural network. Very often the marginal likelihood $P _ { \pmb { \theta } } ( \mathbf { x } )$ will be intractable because the integral in equation (1) does not have an analytical form nor an efficient estimator (Kingma (2017)). As a result the respective posterior distribution, $P _ { \pmb { \theta } } ( \pmb { z } | \mathbf { x } )$ , is also intractable. + +Variational inference side-steps the intractability of the posterior by approximating it with a tractable distribution $Q _ { \phi } ( z | \mathbf { x } ) \approx P _ { \theta } \bar { ( } z | \mathbf { x } )$ . VAEs use an encoder (generally a neural network) to model the approximate posterior $Q _ { \phi } ( z | \mathbf { x } )$ and optimize the parameters $\phi$ to minimize the reverse KL divergence $K L [ Q _ { \phi } ( z | \mathbf { x } ) | | P _ { \theta } ( z | \mathbf { x } ) ]$ between the approximate posterior distribution $Q _ { \phi } ( z | \mathbf { x } )$ and the true posterior $P _ { \pmb { \theta } } ( \pmb { z } | \mathbf { x } )$ . Given that $Q _ { \phi } ( z | \mathbf { x } )$ is a powerful model (such that the KL divergence against the true posterior will be close to zero) we maximize the tractable Evidence Lower BOund (ELBO) to the intractable marginal likelihood. $\mathcal { L } _ { \pmb { \theta } } ( \mathbf { x } ) \leq P _ { \pmb { \theta } } ( \mathbf { x } )$ (full derivation available in the appendix) + +$$ +\begin{array} { r l } { \mathrm { E L B O : } } & { { } \mathcal { L } _ { \pmb { \theta } } ( \mathbf { x } ) = \mathbb { E } _ { Q _ { \phi } ( z | \mathbf { x } ) } [ \log P _ { \pmb { \theta } } ( \mathbf { x } | z ) ] - K L [ Q _ { \phi } ( z | \mathbf { x } ) \ | | \ P ( z ) ] } \end{array} +$$ + +By sharing the variational parameters $\phi$ of the encoder across the data points (amortized inference Gershman & Goodman (2014)), variational autoencoders avoid per-data optimization loops typically needed by mean-field approaches. + +# 3.1 SEQUENTIAL GENERATIVE MODELING + +The standard setting in maximum-likelihood generative modeling is to estimate the set of parameters $\pmb \theta$ that will maximize the marginal likelihood $P _ { \pmb { \theta } } ( \mathbf { x } )$ for data sample $\mathbf { X }$ generated IID from a single true data distribution $P ^ { * } ( \mathbf { x } )$ . In our work we assume the data are generated from multiple distributions $P _ { i } ^ { * } ( \mathbf { x } )$ such that $\begin{array} { r } { P ^ { * } ( \mathbf { \bar { x } } ) = \sum _ { i } \pi _ { i } ^ { * } P _ { i } ^ { * } ( \mathbf { x } ) } \end{array}$ . In classical batch generative modelling, the individual data points are not associated with the specific generative distributions $P _ { i } ^ { * } ( \mathbf { x } )$ . Instead, the whole sample $\mathbf { X }$ is considered to be generated from the mixture distribution $P ^ { * } ( \mathbf { x } )$ . Latent variable models $P _ { \pmb { \theta } } ( \mathbf { x } , \mathbf { z } ) = P _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } ) P ( \mathbf { z } )$ (such as VAEs) capture the complex structures in $P ^ { * } ( \mathbf { x } )$ by conditioning the observed variables $\mathbf { x }$ on the latent variables $\mathbf { z }$ and combining these in (possibly infinite) mixtures $\begin{array} { r } { P _ { \pmb { \theta } } ( \mathbf { x } ) = \int P _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } ) P ( \mathbf { z } ) \delta z } \end{array}$ . + +Our sequential setting is vastly different from the batch approach described above. We receive a stream of (possibly infinite) data $\begin{array} { r c l } { \mathbf { X } } & { = } & { \{ \mathbf { X } _ { 1 } , \mathbf { X } _ { 2 } , \ldots \} } \end{array}$ where the data samples $\begin{array} { r l } { \mathbf { X } _ { i } } & { { } = } \end{array}$ $\{ \mathbf { x } _ { i } ^ { ( 1 ) } , \mathbf { x } _ { i } ^ { ( 2 ) } , \ldots , \mathbf { x } _ { i } ^ { ( K _ { i } ) } \}$ originate from the components $P _ { i } ^ { * } ( \mathbf { x } )$ of the generative distribution. At any given time we observe the latest sample $\mathbf { X } _ { i }$ generated from a single component $P _ { i } ^ { * } ( \mathbf { x } )$ without access to any of the previous samples generated by the other components of $P ^ { * } ( \mathbf { x } )$ . Our goal is to sequentially build an approximation $P _ { \pmb { \theta } } ( \mathbf { x } )$ of the true mixture $P ^ { * } ( \mathbf { x } )$ by only observing data from a single component $P _ { i } ^ { * } ( \mathbf { x } )$ at a time. + +# 4 MODEL + +![](images/3b6612bebe19aa888fb068ec1bfd1318a4431d12d36e772e46e52fb3f6dd2577.jpg) +Figure 2: Shown above is the relationship of the teacher and the student generative models. Data generated from the teacher model is used to augment the student model’s training data and consistency is applied between posteriors. Best viewed in color. + +To enable lifelong generative learning we propose a dual model architecture based on a student-teacher model. The teacher and the student have rather different roles throughout the learning process: the teacher’s role is to preserve the memory of the previously learned tasks and to pass this knowledge onto the student; the student’s role is to learn the distributions over the new incoming data while accommodating for the knowledge obtained from the teacher. The dual model architecture is summarized in figure 2. + +The top part represents the teacher model. At any given time the teacher contains a summary of all previous distributions within the learned parameters of the encoder $Q _ { \Phi } ( z | \mathbf { x } )$ and the decoder $P _ { \Theta } ( \mathbf { x } | z )$ . The teacher is used to generate synthetic samples $\hat { \bf x }$ from these past distributions by decoding samples from the prior $\hat { z } \sim P ( z )$ through the decoder $\hat { \mathbf { x } } \sim P _ { \Theta } ( \bar { \mathbf { x } } | \hat { z } )$ . The generated synthetic samples $\hat { \bf x }$ are passed onto the student model as a form of knowledge transfer about the past distributions. + +The bottom part of figure 2 represents the student, which is responsible for updating the parameters of the encoder $Q _ { \phi } ( z | \mathbf { x } )$ and decoder $P _ { \theta } ( \mathbf { x } | z )$ models over the newly observed data. The student is exposed to a mixture of learning instances $\mathbf { x }$ sampled from $\mathbf { x } \sim \mathcal { \dot { P } } ( \omega ) P ( \mathbf { x } | \omega )$ , $\omega \sim \mathrm { B e r } ( \pi )$ ; it sees synthetic instances generated by the teacher $P ( \mathbf { x } | \boldsymbol { \omega } = 0 ) = P _ { \Theta } ( \mathbf { x } | z )$ , and real ones sampled from the currently active training distribution $P ( { \bf x } | \omega = 1 ) = P ^ { * } ( { \bf x } )$ . The mean $\pi$ of the Bernouli distribution controls the sampling proportion of the previously learnt distributions to the current one. + +If we have seen k distinct distributions prior to the currently active one then π = $\textstyle \pi = { \frac { k } { k + 1 } }$ . In this way we ensure that all the past distributions and the current one are equally represented in the training set used by the student model. + +Once a new distribution is signalled, the old teacher is dropped, the student model is frozen and becomes the new teacher $( \phi \to \Phi , \theta \to \Theta )$ , and a new student is initiated with the latest weights $\phi$ and $\pmb { \theta }$ from the previous student (the new teacher). + +# 4.1 TEACHER-STUDENT CONSISTENCY + +Each new student instantiation uses the input data mix to learn a new approximate posterior $Q _ { \phi } ( z | \mathbf { x } )$ . In addition to being initiated by the new teacher’s weights and receiving information about the teacher’s knowledge via the synthetic samples $\hat { \bf x }$ , we further foster the lifelong learning idea by bringing the latent variable posterior induced by the student model closer to the respective posterior induced by the teacher model. We enforce the latter constraint only over the synthetic samples, ensuring that the previously learnt latent variable posteriors are preserved over the different models. In doing so, we alleviate the effect of catastrophic interference. + +To achieve this, we complement the classical VAE objective (equation (2)) with a term minimizing the KL divergence $K L [ \partial _ { \phi } ( z | \hat { \mathbf { x } } ) | | Q _ { \Phi } ( z | \hat { \mathbf { x } } ) ]$ between the student’s and the teacher’s posteriors over the synthetic data $\hat { \bf x }$ . The teacher’s encoder model, which already has the accumulated knowledge from the previous learning steps, is thus reused within the new student’s objective. Under certain mild assumptions, we show that this objective reparameterizes the student model’s posterior, while preserving the same learning objective as a standard VAE (appendix section 7.0.1). + +# 4.2 LATENT VARIABLE + +A critical component of our model is the synthetic data generation by the teacher’s decoder $\hat { \mathbf { x } } ~ \sim ~ P _ { \Theta } ( \mathbf { x } | z )$ . The synthetic samples need to be representative of all the previously observed distributions in order to provide the student with ample information about the learning history. The teacher generates these synthetic samples by first sampling the latent variable from the prior $\hat { z } ~ \sim ~ P ( z )$ followed by the decoding step $\begin{array} { r } { \hat { \mathbf { x } } \sim \mathcal { P } _ { \Theta } ( \mathbf { x } | \hat { z } ) } \end{array}$ . As we will describe shortly, the latent variable $\hat { z }$ has a categorical component which corresponds to all the past distributions. This categorical component allows us to uniformly sample synthetic instances from all past distributions. + +A simple unimodal prior distribution $P ( z )$ , such as the isotropic Gaussian typically used in classical VAEs, results in an undersampling of the data points that are mapped to a posterior mean that is further away from the prior mean. Visualizing the 2d latent posterior of MNIST in figure 1(b) allows us to get a better intuition of this problem. If for example the prior mean corresponds to a point in latent space between two disparate distributions, the sample generated will not correspond to a sample from the real distribution. Since we use synthetic samples from the teacher in the student model, this aliased sample corresponding to the prior mean, will be reused over and over again, causing corruption in the learning process. In addition, we would under represent the respective true distributions in the learning input mix of the student and eventually lead to distribution loss. + +We circumvent this in our model by decomposing the latent variable $_ z$ into a discrete component $\boldsymbol { z } _ { d } ~ \in ~ \mathcal { R } ^ { J }$ and a continuous component $\tilde { z _ { c } } ~ \in ~ \bar { \mathcal { R } } ^ { F }$ , $z ~ = ~ [ z _ { d } , z _ { c } ]$ . The discrete component $z _ { d }$ shall summarise the most discriminative information about each of the true generating distributions $P _ { i } ^ { * } ( \mathbf { x } )$ . We use the uniform multivariate categorical prior $\begin{array} { r } { z _ { d } \sim C a t ( \frac { 1 } { J } ) } \end{array}$ to represent it and the same parametric family for the approximate posterior $Q _ { \Phi } ( z | \mathbf { x } )$ . The continuous $z _ { c }$ component is the global representation of the distributional variability and we use the multivariate standard normal as the prior $z _ { c } \sim { N ( \mathbf { 0 } , \pmb { I } ) }$ and the isotropic multivariate normal $N ( \mu , \sigma ^ { 2 } I )$ for the approximate posterior. + +When generating synthetic data, the teacher now independently samples from the discrete and continuous priors $\hat { z } _ { d } \sim P ( z _ { d } ) , \hat { z } _ { c } \sim P ( z _ { c } )$ and uses the composition of these to condition the decoding step $\hat { \mathbf { x } } \sim P _ { \Theta } ( \mathbf { x } | \hat { z } _ { d } , \hat { z } _ { c } )$ . Since the discrete representation $\hat { z } _ { d }$ is associated with the true generative distribution components $P _ { i } ^ { * } ( \mathbf { x } )$ , uniformly sampling the discrete prior ensures that that the distributions are well represented in the synthetic mix that the student observes. + +In general, the capacity of a categorical distribution is less than that of a continuous normal distribution. To prevent the VAE’s encoder from using primarily the continuous representation while disregarding the discrete one we further complement the learning objective by a term maximising the mutual information between the discrete representation and the data $I ( z _ { d } ; \mathbf { x } ) = H ( z _ { d } ) - H ( z _ { d } | \mathbf { x } )$ . $H ( z _ { d } )$ is used to denote the marginal entropy of $z _ { d }$ and $H ( z _ { d } | \mathbf { x } )$ denotes the conditional entropy of $z _ { d }$ given $\mathbf { x }$ . 1 + +# 4.3 LEARNING OBJECTIVE + +The final learning objective for each of the student models is the maximization of the ELBO from equation (2), augmented by the negative of the cross-model consistency term introduced in section 4.1 and the mutual information term proposed in section 4.2. + +$$ +\begin{array} { r } { \mathbb { E } _ { Q _ { \phi } } [ l o g P _ { \theta } ( \mathbf { x } | z ) ] - K L [ Q _ { \phi } ( z | \mathbf { x } ) | | P ( z ) ] - \mathbb { 1 } ( \omega = 0 ) K L [ Q _ { \phi } ( z _ { d } | \mathbf { x } ) | | Q _ { \Phi } ( z _ { d } | \mathbf { x } ) ] + \lambda I ( z _ { d } ; \mathbf { x } ) \ , } \end{array} +$$ + +We sample the training instances $\mathbf { x }$ from $\mathbf { x } \sim P ( \omega ) P ( \mathbf { x } | \omega ) , \omega \sim B e r ( \pi )$ as described in section 4. Thus they can either be generated from the teacher model $\omega = 0$ ) or come from the training set of the currently active distribution $( \omega = 1 )$ ). $\mathbb { 1 } ( . )$ is the indicator function which evaluates to 1 if its argument is true and zero otherwise; it makes sure that the consistency regularizer is applied only over the synthetic samples generated by the teacher. The $\lambda$ hyper-parameter controls the importance of the mutual information regularizer. We present the analytical evaluation of the consistency regularizer in appendix section 7.0.1. + +# 5 EXPERIMENTS + +We conducted a set of experiments to explore the behaviour and properties of the method we propose. We specifically concentrate on the benefits our model brings in the lifelong learning setting which is the main motivation of our work. We explain the settings of the individual experiments and their focus in the following three sections. + +In all the experiments we use the notion of a distributional ‘interval’: the interval in which we observe samples from a single distribution $P _ { i } ^ { * } ( \mathbf { x } )$ before the transition to the next distribution $P _ { i + 1 } ^ { * } ( \mathbf { x } )$ occurs. The length of the intervals is in principle random and we developed a heuristic to generate these. We provide further details on this together with other technical details related to the network implementation and training common for all the experiments in the appendix. + +# 5.1 FASHION MNIST : SEQUENTIAL GENERATION + +In this experiment, we seek to establish the performance benefit that our augmented objective formulation in section 4.3 brings into the learning in contrast to the simple ELBO objective 2. We do so by training two models with identical student-teacher architectures as introduced in section 4, with one using the consistency and mutual information augmented objective (with consistency) and the other using the standard ELBO objective (without consistency). We also demonstrate the ability of our model to disambiguate distributional boundaries from the distributional variations. + +We use Fashion MNIST Xiao et al. (2017) 2 to simulate our sequential learning setting. We treat each object as a different distribution and present the model with samples drawn from a single distribution at a time. We sequentially progress over the ten available distributions. When a distribution transition occurs (new object) we signal the model, make the latest student the new teacher and instantiate a new student model. + +We quantify the performance of the generative models by computing the ELBO over the standard Fashion MNIST test set after every distributional transition. The test set contains objects from all of the individual distributions. We run this procedure ten times and report the average test ELBO over the ten repetitions in figure 3(c). We see that around the 3rd interval (the 3rd distributional transition), the negative ELBO of the with consistency model is systematically below $\sim 2 0$ nats ) that of the without consistency model. This confirms the benefits of our new objective formulation for reducing the effects of the catastrophic interference, a crucial property in our lifelong learning setting. In the same figure we also plot the ELBO of the baseline batch VAE. The batch VAE will always outperform our model because it has simultaneous access to all of the distributions during training. + +![](images/de756f9933d2c746151e9331f505d0ed4cb3706f7841c6aa359e35e7728162f2.jpg) +Figure 3: (a) Generation with consistency regularizer. (b) Generation without consistency regularizer. (c) Average negative ELBO over ten trials (each) for the ten distributions within Fashion MNIST. + +After observing and training over all ten distributions we generate samples from the final students of the two models. We do this by fixing the discrete distribution $z _ { d }$ to one-hot vectors over the whole categorical distribution, while randomly sampling the continuous prior $ { \boldsymbol { z } } _ { c } \sim \mathcal { N } ( 0 , I )$ . We contrast samples generated from the model with consistency (figure 3(a)) to the model without consistency (figure 3(b)). Our model learns to separate ’style’ from the distributional boundaries. For example, in the last row of our with consistency model, we observe the various styles of shoes. The without consistency model mixes the distributions randomly. This illustrates the benefits that our augmented objective has for achieving consistent sampling from the individual distributional components. + +# 5.2 ROTATED MNIST $:$ LONG TERM DISTRIBUTION ACCUMULATION + +In this experiment we dig deeper into the benefits our objective formulation brings for the lifelong learning setting. We expose the models to a much larger number of distributions and we explore how our augmented objective from 4.3 helps in preserving the previously learned knowledge. As in section 5.1, we compare models with and without consistency with identical teacher-student architectures. We measure the ability of the models to recall the previously learned information by looking at the consistency between the posterior of the student and the teacher models over the test data set + +$$ +\mathrm { \sf t e n c y : } \quad \# \{ k : Q _ { \Phi } ( z _ { d } | { \bf x } _ { k } ) = = Q _ { \phi } ( z _ { d } | { \bf x } _ { k } ) , { \bf x } _ { k } \in { \bf X } _ { t e s t } \} \mathrm { ~ . ~ } +$$ + +We use the MNIST dataset in which we rotate each of the original digit samples by angles $\nu =$ $[ 3 0 ^ { \circ }$ , $7 0 ^ { \circ }$ , $1 3 0 ^ { \circ }$ , $2 0 0 ^ { \circ }$ , $2 5 0 ^ { \circ }$ ]. We treat each rotation of a single digit family as an individual distribution $\{ P _ { i } ^ { * } ( \mathbf { x } ) \} _ { i = 1 } ^ { 7 0 }$ . Within each distributional interval, we sample the data by first sampling (uniformly with replacement) one of the 70 distributions and then sampling the data instances $\mathbf { x }$ from the selected distribution. + +Figure 4(b) compares the consistency results of the two tested models throughout the learning process. Our model with the augmented objective clearly outperforms the model that uses the simple ELBO objective. This confirms the usefulness of the additional terms in our objective for preserving the previously learned knowledge in accordance with the lifelong learning paradigms. In addition, similarly as in experiment 5.1, figure 4(a) documents that the model with the augmented objective (thanks to reducing the effects of the catastrophic interference) achieves lower negative test ELBO systematically over the much longer course of learning ( $\sim 3 0$ nats). + +We also visualise in figure 4(c) how the accumulation of knowledge speeds up the learning process. For each distributional interval we plot the norms of the model gradients across the learning iterations. We observe that for later distributional intervals the curves become steeper much quicker, reducing the gradients and reaching (lower) steady states much faster then in the early learning stages. This suggests that the latter models are able to learn quicker in our proposed architecture. + +![](images/7e68be6459eb4fe191ce9481bdea3906681f59cc2a823e02e61cb7449cfb295e.jpg) +Figure 4: (a) Negative test ELBO over the learning history; (b) consistency between the teacher and student posteriors across the test data samples normalized by the test data set size; (c) speed of learning convergence across distributional intervals. Best viewed in color. + +# 5.3 SVHN TO MNIST + +In this experiment we explore the ability of our model to retain and transfer knowledge across completely different datasets. We use MNIST and SVHN Netzer et al. (2011) to demonstrate this. We treat all samples from SVHN as being generated by one distribution $P _ { 1 } ^ { * } ( \mathbf { x } )$ and all the MNIST3 samples as generated by another distribution $P _ { 2 } ^ { * } ( \mathbf { x } )$ (irrespective of the specific digit). + +We first train a student model (standard VAE) over the entire SVHN data set. Once done, we freeze the parameters of the encoder and the decoder and transfer the model into the teacher state $( \phi \to \Phi , \theta \to \Theta )$ . We then use this teacher to aid the learning of the new student over the mix of the teacher-generated synthetic SVHN samples $\hat { \bf x }$ and the true MNIST data. + +![](images/5b74ad0e049511c04d3e0276bca9b4fd870fefca948ab30f0adbba4860afe937.jpg) +Figure 5: (a) Reconstructions of test samples from SVHN[left] and MNIST[right]; (b) Decoded samples $\hat { \mathbf { x } } \sim P _ { \pmb { \theta } } ( \mathbf { x } | z _ { d } , z _ { c } )$ based on linear interpolation of $z _ { c } \in \mathcal { R } ^ { 2 }$ with $z _ { d } = [ 0 , 1 ]$ ; (c) Same as (b) but with $z _ { d } = [ 1 , 0 ]$ . + +We use the final student model to reconstruct samples from the two datasets by passing them through the learned encoding/decoding flow: $\mathbf { x } \ \sim \ P _ { i } ^ { * } \ \mathrm { ( \mathbf { x } ) } \ \ z \ \sim \ Q _ { \phi } ( z | \mathbf { x } ) \ \ \hat { \mathbf { x } } \ \sim \ P _ { \theta } ( \mathbf { x } | z )$ . We visualise examples of the true inputs $\mathbf { x }$ and the respective reconstructions $\hat { \bf x }$ in figure 5(a). We see that even though the only true data the final model received for training were from MNIST, it can still reconstruct SVHN data. This confirms the ability of our architecture to transition between complex distributions while still preserving the knowledge learned from the previously observed distributions. + +Finally, in figure 5(b) and 5(c) we illustrate the data generated from an interpolation of a 2-dimensional continuous latent space. For this we specifically trained the models with the continuous latent variable $z _ { c } \in \mathcal { R } ^ { 2 }$ . To generate the data, we fix the discrete categorical $z _ { d }$ to one of the possible values $\{ [ 0 , 1 ] , [ 1 , 0 ] \}$ and linearly interpolate the continuous $z _ { c }$ over the range $[ - 3 , 3 ]$ . We then decode these to obtain the samples $\hat { \mathbf { x } } \sim P _ { \pmb { \theta } } ( \mathbf { x } | \pmb { z } _ { d } , \pmb { z } _ { c } )$ . The model learns a common continuous structure for the two distributions which can be followed by observing the development in the generated samples from top left to bottom right on both figure 5(b) and 5(c). + +# 6 CONCLUSION + +In this work we propose a novel method for learning generative models over streaming data following the lifelong learning principles. The principal assumption for the data is that they are generated by multiple distributions and presented to the learner in a sequential manner (a set of observations from a single distribution followed by a distributional transition). A key limitation for the learning is that the method can only access data generated by the current distribution and has no access to any of the data generated by any of the previous distributions. + +The proposed method is based on a dual student-teacher architecture where the teacher’s role is to preserve the past knowledge and aid the student in future learning. We argue for and augment the standard VAE’s ELBO objective by terms helping the teacher-student knowledge transfer. We demonstrate on a series of experiments the benefits this augmented objective brings in the lifelong learning settings by supporting the retention of previously learned knowledge (models) and limiting the usual effects of catastrophic interference. + +In our future work we will explore the possibilities to extend our architecture to GAN-like Goodfellow et al. (2014) learning with the prospect to further improve the generative abilities of our method. GANs, however, do not use a metric for measuring the quality of the learned distributions such as the marginal likelihood or the ELBO in their objective and therefore the transfer of our architecture to these is not straightforward. + +# REFERENCES + +Tamara Broderick, Nicholas Boyd, Andre Wibisono, Ashia C. Wilson, and Michael I. Jordan. Streaming variational bayes. In Burges et al. (2013), pp. 1727–1735. URL http://papers. nips.cc/paper/4980-streaming-variational-bayes. + +Christopher J. C. Burges, Leon Bottou, Zoubin Ghahramani, and Kilian Q. Weinberger (eds.).´ Advances in Neural Information Processing Systems 26: 27th Annual Conference on Neural Information Processing Systems 2013. Proceedings of a meeting held December 5-8, 2013, Lake Tahoe, Nevada, United States, 2013. URL http://papers.nips.cc/book/ advances-in-neural-information-processing-systems-26-2013. + +Xi Chen, Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. 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Streaming variational inference for bayesian nonparametric mixture models. In Artificial Intelligence and Statistics, pp. 968–976, 2015. + +Alexander V Terekhov, Guglielmo Montone, and J Kevin O’Regan. Knowledge transfer in deep block-modular neural networks. In Proceedings of the 4th International Conference on Biomimetic and Biohybrid Systems-Volume 9222, pp. 268–279. Springer-Verlag New York, Inc., 2015. + +Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, 2017. + +# 7 APPENDIX + +# 7.0.1 UNDERSTANDING THE CONSISTENCY REGULARIZER + +The analytical derivations of the consistency regularizer show that the regularizer can be interpreted as an a transformation of the standard VAE regularizer. In the case of an isotropic gaussian posterior, the proposed regularizer scales the mean and variance of the student posterior by the variance of the teacher 7.0.2 and adds an extra ’volume’ term. This interpretation of the consistency regularizer shows that the proposed regularizer preserves the same learning objective as that of the standard VAE. Below we present the analytical form of the consistency regularizer with categorical and isotropic gaussian posteriors: + +Corollary 7.0.1 We parameterize the learnt posterior of the teacher by $\begin{array} { r } { \Phi _ { i } \ = \ \frac { \exp ( p _ { i } ^ { E } ) } { \sum _ { i = 1 } ^ { J } \exp ( p _ { i } ^ { E } ) } } \end{array}$ and the posterior of the student by φi = $\begin{array} { r } { c ^ { E } = \sum _ { i = 1 } ^ { J } \exp ( p _ { i } ^ { E } ) } \end{array}$ and $\begin{array} { r } { c ^ { S } = \sum _ { i = 1 } ^ { J } \exp ( p _ { i } ^ { S } ) . } \end{array}$ $\begin{array} { r } { \phi _ { i } = \frac { \exp ( p _ { i } ^ { S } ) } { \sum _ { i = 1 } ^ { J } \exp ( p _ { i } ^ { S } ) } } \end{array}$ . We also redefine the normalizing constants as the teacher and student models respectively. The reverse $K L$ divergence in equation 8 can now be re-written as: + +$$ +\begin{array} { c } { { \displaystyle K L ( Q _ { \phi } ( z _ { d } | x ) | | Q _ { \Phi } ( z _ { d } | x ) ) = \sum _ { i = 1 } ^ { J } \frac { \exp ( p _ { i } ^ { S } ) } { c ^ { S } } l o g \left( \frac { \exp ( p _ { i } ^ { S } ) } { c ^ { S } } \frac { c ^ { E } } { \exp ( p _ { i } ^ { E } ) } \right) } } \\ { { = H ( p ^ { S } , p ^ { S } - p ^ { E } ) = - H ( p ^ { s } ) + H ( p ^ { S } , p ^ { E } ) } } \end{array} +$$ + +where $H ( \mathbf { \Sigma } _ { - } )$ is the entropy operator and $H ( \ l _ { - } , \ l _ { - } )$ is the cross-entropy operator. + +Corollary 7.0.2 We assume the learnt posterior of the teacher is parameterized by a centered, isotropic gaussian with $\Phi \ : = \ : [ \pmb { \mu } ^ { E } \ : = \ : \mathbf { 0 } , \pmb { \Sigma } _ { \sim } ^ { E } \ : = \ : \sigma ^ { E ^ { 2 } } \underline { { { I } } } ]$ and the posterior of our student by $a$ non-centered isotropic gaussian with $\phi = [ \mu ^ { S } , \Sigma ^ { S } = \sigma ^ { S 2 } I ]$ , then + +$$ +\begin{array} { l } { { \displaystyle \nabla L ( Q _ { \phi } ( z | x ) | | Q _ { \Phi } ( z | x ) ) = 0 . 5 \biggl [ t r ( \Sigma ^ { E ^ { - 1 } } \Sigma ^ { S } ) + ( \mu ^ { E } - \mu ^ { S } ) ^ { T } \Sigma ^ { E ^ { - 1 } } ( \mu ^ { E } - \mu ^ { S } ) - F + l o g \biggl ( \frac { | \Sigma ^ { E } | } { | \Sigma ^ { S } | } \biggr ) \biggr ] } } \\ { { \displaystyle = 0 . 5 \sum _ { j = 1 } ^ { F } \biggl [ \frac { 1 } { \sigma ^ { E ^ { 2 } } ( j ) } ( \sigma ^ { S 2 } ( j ) + \mu ^ { S 2 } ( j ) ) - 1 + l o g \sigma ^ { E 2 } ( j ) - l o g \sigma ^ { S 2 } ( j ) \biggr ] } } \\ { { \displaystyle = K L ( Q _ { \phi ^ { * } } ( z | x ) | | \mathcal { N } ( 0 , I ) ) - l o g | \Sigma ^ { E } | } } \end{array} +$$ + +Via a reparameterization of the student’s parameters: + +$$ +\begin{array} { c } { { \displaystyle \phi ^ { * } = [ \mu ^ { S * } , \sigma ^ { S * 2 } ] } } \\ { { \displaystyle \mu ^ { S * } = \frac { \mu ^ { S } ( j ) } { \sigma ^ { E 2 } ( j ) } ; \sigma ^ { S * 2 } = \frac { \sigma ^ { S 2 } ( j ) } { \sigma ^ { E 2 } ( j ) } } } \end{array} +$$ + +It is also interesting to note that our posterior regularizer becomes the prior if: + +$$ +l i m _ { \sigma ^ { E ^ { 2 } } \mapsto 1 } K L ( Q _ { \phi } ( z | x ) | | Q _ { \Phi } ( z | x ) ) = K L ( Q _ { \phi } ( z | x ) | | \mathcal { N } ( 0 , I ) ) +$$ + +# 7.1 ELBO DERIVATION + +Variational inference Hoffman et al. (2013) side-steps the intractability of the posterior distribution by approximating it with a tractable distribution $Q _ { \Phi } ( z | \mathbf { x } )$ ; we then optimize the parameters $\Phi$ in order to bring this distribution close to $P _ { \Phi } ( z | \mathbf { x } )$ . The form of this approximate distribution is fixed and is generally conjugate to the prior $P ( z )$ . Variational inference converts the problem of posterior inference into an optimization problem over $\Phi$ . This allows us to utilize stochastic gradient descent to solve our problem. To be more concrete, variational inference tries to minimize the reverse Kullback-Leibler (KL) divergence between the variational posterior distribution $Q _ { \Phi } ( z | \mathbf { x } )$ and the true posterior $P _ { \pmb { \theta } } ( \pmb { z } | \mathbf { x } )$ : + +$$ +K L [ Q _ { \Phi } ( z | \mathbf { x } ) | | P _ { \theta } ( z | \mathbf { x } ) ] = \log P _ { \theta } ( \mathbf { x } ) - \underbrace { \mathbb { E } _ { Q _ { \Phi } ( z | \mathbf { x } ) } \left[ \log \frac { P _ { \theta } ( x , z ) } { Q _ { \Phi } ( z | \mathbf { x } ) } \right] } _ { \mathcal { L } _ { \theta } } +$$ + +Rearranging the terms in equation 8 and utilizing the fact that the KL divergence is a measure, we can derive the evidence lower bound $\mathcal { L } _ { \theta }$ (ELBO) which is the objective function we directly optimize: + +$$ +l o g P _ { \pmb \theta } ( \mathbf { x } ) \geq \mathbb { E } _ { Q _ { \Phi } ( z | \mathbf { x } ) } [ l o g P _ { \pmb \theta } ( \mathbf { x } | z ) ] - K L ( Q _ { \Phi } ( z | \mathbf { x } ) | | P ( z ) ) = \mathcal { L } _ { \pmb \theta } +$$ + +In order to backpropagate it is necessary to remove the dependence on the stochastic variable $_ { z }$ . To achieve this, we push the sampling operation outside of the computational graph for the normal distribution via the reparameterization trick Kingma & Welling (2014) and the gumbel-softmax reparameterization Maddison et al. (2016); Jang et al. (2017) for the discrete distribution. In essence the reparameterization trick allows us to introduce a distribution $P ( \epsilon )$ that is not a function of the data or computational graph in order to move the gradient operator into the expectation: + +$$ +\nabla \mathbb { E } _ { Q _ { \Phi } ( z | \mathbf { x } ) } \bigg [ \mathrm { l o g } \frac { P _ { \theta } ( x , z ) } { Q _ { \Phi } ( z | \mathbf { x } ) } \bigg ] \mapsto \mathbb { E } _ { P ( \epsilon ) } \bigg [ \nabla \mathrm { l o g } \frac { P _ { \theta } ( x , z ) } { Q _ { \Phi } ( z | \mathbf { x } ) } \bigg ] +$$ + +# 7.2 MODEL RELATED + +In this section we provide extra details of our model architecture. + +# 7.2.1 MODEL ARCHITECTURE + +We utilized two different architectures for our experiments. The first two utilize a standard deep neural network with two layers of 512 to map to the latent representation and two layers of 512 to map back to the reconstruction for the decoder. We used batch norm Ioffe & Szegedy (2015) and ELU activations for all the layers barring the layer projecting into the latent representation and the output layer. + +The final experiment with the transfer from SVHN to MNIST utilizes a fully convolutional architecture with only strided convolutional layers in the encoder (where the number of filters are doubled at each layer). The final projection layer for the encoder maps the data to a $[ { \mathsf { C } } { = } | z _ { d } |$ , 1, 1] output which is then reparameterized in the standard way. The decoder utilizes fractional strides for the convolutional-transpose (de-convolution) layers where we reduce the number of filters in half at each layer. The full architecture can be examined in our code repository [which will be de-anonymized after the review process]. All layers used batch norm Ioffe & Szegedy (2015) and ELU activations. + +We utilized Adam Kingma & Ba (2015) to optimize all of our problems with a learning rate of 1e-4. When we utilized weight transfer we re-initialized the accumulated momentum vector of Adam as well as the aggregated mean and covariance of the Batch Norm layers. Our code is already available online under an MIT license at 4 + +# 7.2.2 GUMBEL REPARAMETERIZATION + +Since we model our latent variable as a combination of a discrete and a continuous distribution we also use the Gumbel-Softmax reparameterization Maddison et al. (2016); Jang et al. (2017). The Gumbel-Softmax reparameterization over logits [linear output of the last layer in the encoder] $\pmb { p } \in \mathcal { R } ^ { M }$ and an annealed temperature parameter $\tau \in \mathcal { R }$ is defined as: + +$$ +z = s o f t m a x ( \frac { l o g ( p ) + g } { \tau } ) ; g = - l o g ( - l o g ( { \pmb u } \sim U n i f ( 0 , 1 ) ) ) +$$ + +$\pmb { u } \in \mathcal { R } ^ { M } , \pmb { g } \in \mathcal { R } ^ { M }$ . As the temperature parameter $\tau \mapsto 0$ , $_ { z }$ converges to a categorical. + +# 7.2.3 EXPANDABLE MODEL CAPACITY AND REPRESENTATIONS + +Multilayer neural networks with sigmoidal activations have a VC dimension bounded between $O ( \rho ^ { 2 } )$ Sontag (1998) and $O ( \rho ^ { 4 } )$ Karpinski & Macintyre (1997) where $\rho$ are the number of parameters. A model that is able to consistently add new information should also be able to expand its VC dimension by adding new parameters over time. Our formulation imposes no restrictions on the model architecture: i.e. new layers can be added freely to the new student model. + +In addition we also allow the dimensionality of $\boldsymbol { z } _ { d } \in \mathcal { R } ^ { J }$ , our discrete latent representation to grow in order to accommodate new distributions. This is possible because the KL divergence between two categorical distributions of different sizes can be evaluated by simply zero padding the teacher’s smaller discrete distribution. Since we also transfer weights between the teacher and the student model, we need to handle the case of expanding latent representations appropriately. In the event that we add a new distribution we copy all the weights besides the ones immediately surrounding the projection into and out of the latent distribution. These surrounding weights are reinitialized to their standard Glorot initializations Glorot & Bengio (2010). + +# 7.3 FORWARD VS. REVERSE KL + +![](images/c20e180406d015f1a528955ddb39a6aeaad6b034df00717d8727af083540f5d4.jpg) +Figure 6: Reverse vs. forward KL on FashionMNIST 5.1 + +In our setting we have the ability to utilize the zero forcing (reverse or mode-seeking) KL or the zero avoiding (forward) KL divergence. In general, if the true underlying posterior is multi-modal, it is preferable to operate with the reverse KL divergence (Murphy (2012) 21.2.2). In addition, utilizing the mode-seeking KL divergence generates more realistic results when operating over image data. + +In order to validate this, we repeat the experiment in 5.1. We train two models: one with the forward KL posterior regularizer and one with the reverse. We evaluate the -ELBO mean and variance over ten trials. Empirically, we observed no difference between the different measures. This is demonstrated in figure 6. + +# 7.4 NUMBER OF REQUIRED SAMPLES + +Our method derives its sample complexity from standard VAEs. In practice we evaluate the number of required real and synthetic samples by utilizing early stopping. When the negative ELBO on the validation set stops decreasing for 50 steps we stop training the current model and transition to the next distribution interval. Using this and the fact that we keep equal proportions of all observed distributions in our minibatch, we can evaluate the number of synthetic and real samples used during the single distribution interval. We demonstrate this procedure on experiment 5.1 in figure 7. + +![](images/cc4dab3f21baf14bf3afeebd6d4c9bdb4ca56c427a0d6ee500748df09837e33e.jpg) +Figure 7: FashionMNIST experiment 5.1 data efficiency analysis. Left: Real samples used; Right: Synthetic samples used + +We observe a rapid decrease of the number of required real samples as we assimilate more distributions into our model. + +# 7.5 EXPERIMENTS RELATED + +In this section we provide an extra experiment run on MNIST as well as some extra images from the rotated MNIST experiment. + +# 7.5.1 MNIST : GENERATION AND ELBO + +![](images/15a1a1e12ba4d93f4e0643a470665c0942c237892d48141604db03027ffeeb15.jpg) +Figure 8: (a) Generation with consistency regularizer. (b) Without consistency regularizer. (c) Average log-likelihood over ten trials (each) for the ten separated distributions within MNIST. + +In this experiment, we seek to establish the performance benefit that the consistency regularizer brings into the learning process. We do so by evaluating the ELBO for a model with and without the consistency and mutual information regularizers. We also demonstrate the ability of the regularizers to disambiguate distributional boundaries and their inter-distributional variations. I.e. for MNIST this separates the MNIST digits from their inter-class variants (i.e drawing style). + +We use MNIST to simulate our sequential learning setting. We treat each digit as a different distribution and present the model with samples drawn from a single distribution at a time. For the purpose of this experiment we sequentially progress over the ten distributions (i.e. interval sampling involves linearly iterating over all the distributions ). + +When an interval transition occurs we signal the model, make the student the new teacher and instantiate a new student model. We contrast this to a model that utilizes the same graphical model, without our consistency and mutual information regularizers. We quantify the performance of the generative models by computing the ELBO over the standard MNIST test set at every interval. The test set contains digits from all of the individual distributions. We run this procedure ten times and report the average ELBO over the test set. + +After observing all ten distributions we evaluate samples generated from the final student model. We do this by fixing the discrete distribution $z _ { d }$ , while randomly sampling $z _ { c } \sim \mathcal { N } ( 0 , I )$ . We contrast samples generated from the model with both regularizers (left-most image in 8) to the model without the regularizers (center image in 8). Our model learns to separate ’style’ from distributional boundaries. This is demonstrated by observing the digit $" 2 "$ : i.e. different samples of $z _ { c }$ produce different styles of writing a $" 2 "$ . + +# 7.5.2 ROTATED MNIST EXPERIMENT + +We provide a larger sized image for the ELBO from experiment 5.2. We also visualize reconstructions from the rotated MNIST problem (visualized in figure 10). Finally in figure 11 we show the effects on the reconstructions when we do not use the mutual information regularizer. We believe this is due to the fact that the network utilizes the larger continuous representation to model the discriminative aspects of the observed distribution. + +![](images/f19a75a72a657bd9f71a621bf6deb35b993d8e2c51ff3349c27ba3f7cb3d3923.jpg) +Figure 9: Visualization of ELBO for rotated MNIST evaluated at the last model (the one at the 70th interval) + +![](images/52599e0d0ae24dc9212c75394825fdde9b6e11ed1bf838c77e4af811fe8676d3.jpg) +Figure 10: Visualization of reconstructions for rotated MNIST evaluated at the last model (the one at the 70th interval) + +![](images/e0fedbcb6e99ba86a31cb6d4751934b6f6afe937d08337dcb166f202aef3e30c.jpg) +Figure 11: Visualization of reconstructions when we do not use the mutual information regularizer \ No newline at end of file diff --git a/parse/train/S1fduCl0b/S1fduCl0b_content_list.json b/parse/train/S1fduCl0b/S1fduCl0b_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..a020fe0ffd85423afe2388e01b3205d29907ccb5 --- /dev/null +++ b/parse/train/S1fduCl0b/S1fduCl0b_content_list.json @@ -0,0 +1,1902 @@ +[ + { + "type": "text", + "text": "LIFELONG GENERATIVE MODELING ", + "text_level": 1, + "bbox": [ + 174, + 99, + 609, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 145, + 398, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 210, + 544, + 224 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Lifelong learning is the problem of learning multiple consecutive tasks in a sequential manner where knowledge gained from previous tasks is retained and used for future learning. It is essential towards the development of intelligent machines that can adapt to their surroundings. In this work we focus on a lifelong learning approach to generative modeling where we continuously incorporate newly observed streaming distributions into our learnt model. We do so through a student-teacher architecture which allows us to learn and preserve all the distributions seen so far without the need to retain the past data nor the past models. Through the introduction of a novel cross-model regularizer, the student model leverages the information learnt by the teacher, which acts as a summary of everything seen till now. The regularizer has the additional benefit of reducing the effect of catastrophic interference that appears when we learn over streaming data. We demonstrate its efficacy on streaming distributions as well as its ability to learn a common latent representation across a complex transfer learning scenario. ", + "bbox": [ + 233, + 242, + 764, + 436 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 467, + 334, + 482 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep unsupervised generative learning allows us to take advantage of the massive amount of unlabeled data available in order to build models that efficiently compress and learn an approximation of the true data distribution. It has numerous applications such as image denoising, inpainting, super-resolution, structured prediction, clustering, pre-training and many more. However, something that is lacking in the modern ML toolbox is an efficient way to learn these deep generative models in a sequential, lifelong setting. ", + "bbox": [ + 174, + 500, + 823, + 583 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In a lot of real world scenarios we observe distributions sequentially. Examples of this include streaming data from sensors such as cameras and microphones or other similar time series data. A system can also be resource limited wherein all of the past data or learnt models cannot be stored. We are interested in the lifelong learning setting for generative models where data arrives sequentially in a stream and where the storage of all data is infeasible. Within the stream, instances are generated according to some non-observed distribution which changes at given time-points. We assume we know the time points at which the transitions occur and whether the latent distribution is a completely new one or one that has been observed before. We do not however know the underlying identity of the individual distributions. Our goal is to learn a generative model that can summarize all the distributions seen so far in the stream. We give an example of such a setting in figure 1(a) using MNIST LeCun & Cortes (2010), where we have three unique distributions and one that is repeated. ", + "bbox": [ + 174, + 589, + 825, + 742 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Since we only observe one distribution at a time we need to develop a strategy of retaining the previously learnt knowledge (i.e. the previously learnt distributions) and integrate it into future learning. To accumulate additional distributions in the current generative model we utilize a student-teacher architecture similar to that in distillation methods Hinton et al. (2015); Furlanello et al. (2016). The teacher contains a summary of all past distributions and is used to augment the data used to train the student model. The student model thus receives data samples from the currently observable distribution as well as synthetic data samples from previous distributions. This allows the student model to learn a distribution that summarizes the current as well as all previously observed distributions. Once a new distribution shift occurs the existing teacher model is discarded, the student becomes the teacher and a new student is instantiated. ", + "bbox": [ + 174, + 750, + 823, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We further leverage the generative model of the teacher by introducing a regularizer in the learning objective function of the student that brings the posterior distribution of the latter close to that of the former. This allows us to build upon and extend the teacher’s generative model in the student each time the latter is re-instantiated (rather than re-learning it from scratch). By coupling this regularizer with a weight transfer from the teacher to the student we also allow for faster convergence of the student model. We empirically show that the regularizer allows us to learn a much larger set of distributions without catastrophic interference McCloskey & Cohen (1989). ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/e4a0de89f53f3da3ca7f5c7c076ac99f174d75abf3b0fa6674908d15676319ae.jpg", + "image_caption": [ + "Figure 1: (a) Our problem setting where we sequentially observe samples from multiple unknown distributions; (b)Visualization of a learnt two-dimensional posterior of MNIST, evaluated with samples from the full test set. " + ], + "image_footnote": [], + "bbox": [ + 155, + 121, + 841, + 262 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 329, + 825, + 400 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We build our lifelong generative models over Variational Autoencoders (VAEs) Kingma & Welling (2014). VAEs learn the posterior distribution of a latent variable model using an encoder network; they generate data by sampling from a prior and decoding the sample through a conditional distribution learnt by a decoder network. ", + "bbox": [ + 174, + 406, + 823, + 462 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Using a vanilla VAE as a teacher to generate synthetic data for the student is problematic due to a couple of limitations of the VAE generative process. 1) Sampling the prior can select a point in the latent space that is in between two separate distributions, causing generation of unrealistic synthetic data and eventually leading to loss of previously learnt distributions. 2) Additionally, data points mapped to the posterior that are further away from the prior mean will be sampled less frequently resulting in an unbalanced sampling of the constituent distributions. Both limitations can be understood by visually inspecting the learnt posterior distribution of a standard VAE evaluated on test images from MNIST as shown in figure 1(b). To address the VAE’s sampling limitations we decompose the latent variable vector into a continuous and a discrete component. The discrete component is used to summarize the discriminative information of the individual generative distributions while the continuous caters for the remaining sample variability. By independently sampling the discrete and continuous components we preserve the distributional boundaries and circumvent the two problems above. ", + "bbox": [ + 174, + 469, + 825, + 648 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This sampling strategy, combined with the proposed regularizer allows us to learn and remember all the individual distributions observed in the past. In addition we are also able to generate samples from any of the past distributions at will; we call this property consistent sampling. ", + "bbox": [ + 176, + 656, + 825, + 699 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 722, + 343, + 738 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Past work in sequential learning of generative models has focused on learning Gaussian mixture models Singer & Warmuth (1999); Declercq & Piater (2008) or on variational methods such as Variational EM Ghahramani & Attias (2000). Work that is closer to ours is the online or sequential learning of generative models in a streaming setting. Variational methods have been adapted for a streaming setting, e.g: Streaming Variational Bayes Broderick et al. (2013), Streaming Variational Mixture models Tank et al. (2015), and the Population Posterior McInerney et al. (2015). However their learning objectives are very different from ours. The objective of these methods is to adjust the learnt model such that it reflects the current data distribution as accurately as possible, while forgetting the previously observed distributions. Instead we want to do lifelong learning and retain all previously observed distributions within our learnt model. As far as we know our work is the first one that tries to bring generative models, and in particular VAEs, into a lifelong setting where distributions are seen, learnt, and remembered sequentially. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "VAEs rely on an encoder and a decoder neural network in order to learn the parameters of the posterior and likelihood. One of the central problems that arise when training a neural network in an sequential manner is that it causes the model to run into the problem of catastrophic interference McCloskey & Cohen (1989). Catastrophic interference appears when we train neural networks in a sequential manner and model parameters start to become biased to the most recent samples observed, while forgetting what was learnt from older samples. This generally happens when we stop exposing the model to past data. There have been a number of attempts to solve the problem of catastrophic interference in neural networks. These range from distillation methods such as the original method Hinton et al. (2015) and ALTM Furlanello et al. (2016), to utilizing privileged information Lopez-Paz et al. (2016), as well as transfer learning approaches such as Learning Without Forgetting Li & Hoiem (2016) and methods that relay information from previously learnt hidden layers such as in Progressive Neural Networks Rusu et al. (2016) and Deep Block-Modular Neural Networks Terekhov et al. (2015). All of these methods necessitate the storage of previous models or data; our method does not. ", + "bbox": [ + 174, + 103, + 825, + 297 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The recent work of elastic weight consolidation (EWC) Kirkpatrick et al. (2017) utilizes the Fisher Information matrix (FIM) to avoid the problem of catastrophic interference. The FIM captures the sensitivity of the log-likelihood with respect to the model parameters; EWC leverages this (via a linear approximation of the FIM) to control the change of model parameter values between varying distributions. Intuitively, important parameters should not have their values changed, while non-important parameters are left unconstrained. Since EWC assumes model parameters being distributed under an exponential family, it allows for the utilization of the FIM as a quadratic approximationJeffreys (1946) to the Kullback-Leibler (KL) divergence. Our model makes no such distributional assumptions about the model parameters. Instead of constraining the parameters of the model as in EWC, we restrict the posterior representation of the student model to be close to that of the teacher for the previous distributions accumulated by the teacher. This allows the model parameters to vary as necessary in order to best fit the data. ", + "bbox": [ + 174, + 304, + 825, + 472 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 491, + 326, + 507 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We consider an unsupervised setting where we observe a sample $\\mathbf { X }$ of $K \\geq 1$ realizations $\\mathbf { X } =$ $\\{ \\mathbf { x } ^ { ( 0 ) } , \\mathbf { x } ^ { ( 1 ) } , . . . , \\mathbf { x } ^ { ( K ) } \\}$ from an unknown true distribution $P ^ { * } ( \\mathbf { x } )$ with $\\mathbf { x } \\in \\mathcal { R } ^ { N }$ . We assume that the data is generated by a random process involving a non-observed random variable $z \\in \\mathcal { R } ^ { M }$ . In order to incorporate our prior knowledge we posit a prior $P ( z )$ over $_ { z }$ . Our objective is to approximate the true underlying data distribution by a model $P _ { \\pmb { \\theta } } ( \\mathbf { x } )$ such that $P _ { \\pmb \\theta } ( \\mathbf { x } ) \\approx P ^ { * } ( \\mathbf { x } )$ . ", + "bbox": [ + 174, + 521, + 825, + 594 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given a latent variable model $P _ { \\pmb { \\theta } } ( \\mathbf { x } | \\mathbf { z } ) P ( \\mathbf { z } )$ we obtain the marginal likelihood $P _ { \\pmb { \\theta } } ( \\mathbf { x } )$ by integrating out the latent variable $\\mathbf { z }$ from the joint distribution. The joint distribution can in turn be factorized using the conditional distribution $P _ { \\theta } ( \\mathbf { x } | z )$ or the posterior $P _ { \\boldsymbol { \\theta } } ( z | \\mathbf { x } )$ . ", + "bbox": [ + 174, + 599, + 825, + 642 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8a03c8228aa7008917fafb0710f6747c163f21099c3206f5a30bc5a9b8915def.jpg", + "text": "$$\nP _ { \\theta } ( { \\bf x } ) = \\int P _ { \\theta } ( { \\bf x } , z ) \\delta z = \\int P _ { \\theta } ( z | { \\bf x } ) P _ { \\theta } ( { \\bf x } ) \\delta z = \\int P _ { \\theta } ( { \\bf x } | z ) P ( z ) \\delta z\n$$", + "text_format": "latex", + "bbox": [ + 264, + 646, + 732, + 678 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We model the conditional distribution $P _ { \\theta } ( \\mathbf { x } | z )$ by a decoder, typically a neural network. Very often the marginal likelihood $P _ { \\pmb { \\theta } } ( \\mathbf { x } )$ will be intractable because the integral in equation (1) does not have an analytical form nor an efficient estimator (Kingma (2017)). As a result the respective posterior distribution, $P _ { \\pmb { \\theta } } ( \\pmb { z } | \\mathbf { x } )$ , is also intractable. ", + "bbox": [ + 174, + 680, + 825, + 737 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Variational inference side-steps the intractability of the posterior by approximating it with a tractable distribution $Q _ { \\phi } ( z | \\mathbf { x } ) \\approx P _ { \\theta } \\bar { ( } z | \\mathbf { x } )$ . VAEs use an encoder (generally a neural network) to model the approximate posterior $Q _ { \\phi } ( z | \\mathbf { x } )$ and optimize the parameters $\\phi$ to minimize the reverse KL divergence $K L [ Q _ { \\phi } ( z | \\mathbf { x } ) | | P _ { \\theta } ( z | \\mathbf { x } ) ]$ between the approximate posterior distribution $Q _ { \\phi } ( z | \\mathbf { x } )$ and the true posterior $P _ { \\pmb { \\theta } } ( \\pmb { z } | \\mathbf { x } )$ . Given that $Q _ { \\phi } ( z | \\mathbf { x } )$ is a powerful model (such that the KL divergence against the true posterior will be close to zero) we maximize the tractable Evidence Lower BOund (ELBO) to the intractable marginal likelihood. $\\mathcal { L } _ { \\pmb { \\theta } } ( \\mathbf { x } ) \\leq P _ { \\pmb { \\theta } } ( \\mathbf { x } )$ (full derivation available in the appendix) ", + "bbox": [ + 173, + 742, + 825, + 854 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2269a1dffe7a71dd3526bccbcf02f92150a052047ec38edb9bda1b34f9c61194.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathrm { E L B O : } } & { { } \\mathcal { L } _ { \\pmb { \\theta } } ( \\mathbf { x } ) = \\mathbb { E } _ { Q _ { \\phi } ( z | \\mathbf { x } ) } [ \\log P _ { \\pmb { \\theta } } ( \\mathbf { x } | z ) ] - K L [ Q _ { \\phi } ( z | \\mathbf { x } ) \\ | | \\ P ( z ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 271, + 853, + 725, + 872 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "By sharing the variational parameters $\\phi$ of the encoder across the data points (amortized inference Gershman & Goodman (2014)), variational autoencoders avoid per-data optimization loops typically needed by mean-field approaches. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 SEQUENTIAL GENERATIVE MODELING ", + "text_level": 1, + "bbox": [ + 176, + 103, + 483, + 117 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The standard setting in maximum-likelihood generative modeling is to estimate the set of parameters $\\pmb \\theta$ that will maximize the marginal likelihood $P _ { \\pmb { \\theta } } ( \\mathbf { x } )$ for data sample $\\mathbf { X }$ generated IID from a single true data distribution $P ^ { * } ( \\mathbf { x } )$ . In our work we assume the data are generated from multiple distributions $P _ { i } ^ { * } ( \\mathbf { x } )$ such that $\\begin{array} { r } { P ^ { * } ( \\mathbf { \\bar { x } } ) = \\sum _ { i } \\pi _ { i } ^ { * } P _ { i } ^ { * } ( \\mathbf { x } ) } \\end{array}$ . In classical batch generative modelling, the individual data points are not associated with the specific generative distributions $P _ { i } ^ { * } ( \\mathbf { x } )$ . Instead, the whole sample $\\mathbf { X }$ is considered to be generated from the mixture distribution $P ^ { * } ( \\mathbf { x } )$ . Latent variable models $P _ { \\pmb { \\theta } } ( \\mathbf { x } , \\mathbf { z } ) = P _ { \\pmb { \\theta } } ( \\mathbf { x } | \\mathbf { z } ) P ( \\mathbf { z } )$ (such as VAEs) capture the complex structures in $P ^ { * } ( \\mathbf { x } )$ by conditioning the observed variables $\\mathbf { x }$ on the latent variables $\\mathbf { z }$ and combining these in (possibly infinite) mixtures $\\begin{array} { r } { P _ { \\pmb { \\theta } } ( \\mathbf { x } ) = \\int P _ { \\pmb { \\theta } } ( \\mathbf { x } | \\mathbf { z } ) P ( \\mathbf { z } ) \\delta z } \\end{array}$ . ", + "bbox": [ + 173, + 128, + 825, + 256 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our sequential setting is vastly different from the batch approach described above. We receive a stream of (possibly infinite) data $\\begin{array} { r c l } { \\mathbf { X } } & { = } & { \\{ \\mathbf { X } _ { 1 } , \\mathbf { X } _ { 2 } , \\ldots \\} } \\end{array}$ where the data samples $\\begin{array} { r l } { \\mathbf { X } _ { i } } & { { } = } \\end{array}$ $\\{ \\mathbf { x } _ { i } ^ { ( 1 ) } , \\mathbf { x } _ { i } ^ { ( 2 ) } , \\ldots , \\mathbf { x } _ { i } ^ { ( K _ { i } ) } \\}$ originate from the components $P _ { i } ^ { * } ( \\mathbf { x } )$ of the generative distribution. At any given time we observe the latest sample $\\mathbf { X } _ { i }$ generated from a single component $P _ { i } ^ { * } ( \\mathbf { x } )$ without access to any of the previous samples generated by the other components of $P ^ { * } ( \\mathbf { x } )$ . Our goal is to sequentially build an approximation $P _ { \\pmb { \\theta } } ( \\mathbf { x } )$ of the true mixture $P ^ { * } ( \\mathbf { x } )$ by only observing data from a single component $P _ { i } ^ { * } ( \\mathbf { x } )$ at a time. ", + "bbox": [ + 173, + 262, + 825, + 363 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 MODEL ", + "text_level": 1, + "bbox": [ + 174, + 375, + 269, + 390 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/3b6612bebe19aa888fb068ec1bfd1318a4431d12d36e772e46e52fb3f6dd2577.jpg", + "image_caption": [ + "Figure 2: Shown above is the relationship of the teacher and the student generative models. Data generated from the teacher model is used to augment the student model’s training data and consistency is applied between posteriors. Best viewed in color. " + ], + "image_footnote": [], + "bbox": [ + 197, + 404, + 799, + 590 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To enable lifelong generative learning we propose a dual model architecture based on a student-teacher model. The teacher and the student have rather different roles throughout the learning process: the teacher’s role is to preserve the memory of the previously learned tasks and to pass this knowledge onto the student; the student’s role is to learn the distributions over the new incoming data while accommodating for the knowledge obtained from the teacher. The dual model architecture is summarized in figure 2. ", + "bbox": [ + 174, + 659, + 825, + 742 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The top part represents the teacher model. At any given time the teacher contains a summary of all previous distributions within the learned parameters of the encoder $Q _ { \\Phi } ( z | \\mathbf { x } )$ and the decoder $P _ { \\Theta } ( \\mathbf { x } | z )$ . The teacher is used to generate synthetic samples $\\hat { \\bf x }$ from these past distributions by decoding samples from the prior $\\hat { z } \\sim P ( z )$ through the decoder $\\hat { \\mathbf { x } } \\sim P _ { \\Theta } ( \\bar { \\mathbf { x } } | \\hat { z } )$ . The generated synthetic samples $\\hat { \\bf x }$ are passed onto the student model as a form of knowledge transfer about the past distributions. ", + "bbox": [ + 173, + 750, + 825, + 833 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The bottom part of figure 2 represents the student, which is responsible for updating the parameters of the encoder $Q _ { \\phi } ( z | \\mathbf { x } )$ and decoder $P _ { \\theta } ( \\mathbf { x } | z )$ models over the newly observed data. The student is exposed to a mixture of learning instances $\\mathbf { x }$ sampled from $\\mathbf { x } \\sim \\mathcal { \\dot { P } } ( \\omega ) P ( \\mathbf { x } | \\omega )$ , $\\omega \\sim \\mathrm { B e r } ( \\pi )$ ; it sees synthetic instances generated by the teacher $P ( \\mathbf { x } | \\boldsymbol { \\omega } = 0 ) = P _ { \\Theta } ( \\mathbf { x } | z )$ , and real ones sampled from the currently active training distribution $P ( { \\bf x } | \\omega = 1 ) = P ^ { * } ( { \\bf x } )$ . The mean $\\pi$ of the Bernouli distribution controls the sampling proportion of the previously learnt distributions to the current one. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "If we have seen k distinct distributions prior to the currently active one then π = $\\textstyle \\pi = { \\frac { k } { k + 1 } }$ . In this way we ensure that all the past distributions and the current one are equally represented in the training set used by the student model. ", + "bbox": [ + 174, + 103, + 825, + 147 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Once a new distribution is signalled, the old teacher is dropped, the student model is frozen and becomes the new teacher $( \\phi \\to \\Phi , \\theta \\to \\Theta )$ , and a new student is initiated with the latest weights $\\phi$ and $\\pmb { \\theta }$ from the previous student (the new teacher). ", + "bbox": [ + 176, + 154, + 823, + 196 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 TEACHER-STUDENT CONSISTENCY ", + "text_level": 1, + "bbox": [ + 176, + 223, + 454, + 238 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Each new student instantiation uses the input data mix to learn a new approximate posterior $Q _ { \\phi } ( z | \\mathbf { x } )$ . In addition to being initiated by the new teacher’s weights and receiving information about the teacher’s knowledge via the synthetic samples $\\hat { \\bf x }$ , we further foster the lifelong learning idea by bringing the latent variable posterior induced by the student model closer to the respective posterior induced by the teacher model. We enforce the latter constraint only over the synthetic samples, ensuring that the previously learnt latent variable posteriors are preserved over the different models. In doing so, we alleviate the effect of catastrophic interference. ", + "bbox": [ + 173, + 253, + 825, + 352 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To achieve this, we complement the classical VAE objective (equation (2)) with a term minimizing the KL divergence $K L [ \\partial _ { \\phi } ( z | \\hat { \\mathbf { x } } ) | | Q _ { \\Phi } ( z | \\hat { \\mathbf { x } } ) ]$ between the student’s and the teacher’s posteriors over the synthetic data $\\hat { \\bf x }$ . The teacher’s encoder model, which already has the accumulated knowledge from the previous learning steps, is thus reused within the new student’s objective. Under certain mild assumptions, we show that this objective reparameterizes the student model’s posterior, while preserving the same learning objective as a standard VAE (appendix section 7.0.1). ", + "bbox": [ + 174, + 358, + 825, + 441 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 LATENT VARIABLE ", + "text_level": 1, + "bbox": [ + 176, + 469, + 344, + 483 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "A critical component of our model is the synthetic data generation by the teacher’s decoder $\\hat { \\mathbf { x } } ~ \\sim ~ P _ { \\Theta } ( \\mathbf { x } | z )$ . The synthetic samples need to be representative of all the previously observed distributions in order to provide the student with ample information about the learning history. The teacher generates these synthetic samples by first sampling the latent variable from the prior $\\hat { z } ~ \\sim ~ P ( z )$ followed by the decoding step $\\begin{array} { r } { \\hat { \\mathbf { x } } \\sim \\mathcal { P } _ { \\Theta } ( \\mathbf { x } | \\hat { z } ) } \\end{array}$ . As we will describe shortly, the latent variable $\\hat { z }$ has a categorical component which corresponds to all the past distributions. This categorical component allows us to uniformly sample synthetic instances from all past distributions. ", + "bbox": [ + 173, + 498, + 825, + 597 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "A simple unimodal prior distribution $P ( z )$ , such as the isotropic Gaussian typically used in classical VAEs, results in an undersampling of the data points that are mapped to a posterior mean that is further away from the prior mean. Visualizing the 2d latent posterior of MNIST in figure 1(b) allows us to get a better intuition of this problem. If for example the prior mean corresponds to a point in latent space between two disparate distributions, the sample generated will not correspond to a sample from the real distribution. Since we use synthetic samples from the teacher in the student model, this aliased sample corresponding to the prior mean, will be reused over and over again, causing corruption in the learning process. In addition, we would under represent the respective true distributions in the learning input mix of the student and eventually lead to distribution loss. ", + "bbox": [ + 173, + 603, + 825, + 728 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We circumvent this in our model by decomposing the latent variable $_ z$ into a discrete component $\\boldsymbol { z } _ { d } ~ \\in ~ \\mathcal { R } ^ { J }$ and a continuous component $\\tilde { z _ { c } } ~ \\in ~ \\bar { \\mathcal { R } } ^ { F }$ , $z ~ = ~ [ z _ { d } , z _ { c } ]$ . The discrete component $z _ { d }$ shall summarise the most discriminative information about each of the true generating distributions $P _ { i } ^ { * } ( \\mathbf { x } )$ . We use the uniform multivariate categorical prior $\\begin{array} { r } { z _ { d } \\sim C a t ( \\frac { 1 } { J } ) } \\end{array}$ to represent it and the same parametric family for the approximate posterior $Q _ { \\Phi } ( z | \\mathbf { x } )$ . The continuous $z _ { c }$ component is the global representation of the distributional variability and we use the multivariate standard normal as the prior $z _ { c } \\sim { N ( \\mathbf { 0 } , \\pmb { I } ) }$ and the isotropic multivariate normal $N ( \\mu , \\sigma ^ { 2 } I )$ for the approximate posterior. ", + "bbox": [ + 174, + 734, + 825, + 847 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "When generating synthetic data, the teacher now independently samples from the discrete and continuous priors $\\hat { z } _ { d } \\sim P ( z _ { d } ) , \\hat { z } _ { c } \\sim P ( z _ { c } )$ and uses the composition of these to condition the decoding step $\\hat { \\mathbf { x } } \\sim P _ { \\Theta } ( \\mathbf { x } | \\hat { z } _ { d } , \\hat { z } _ { c } )$ . Since the discrete representation $\\hat { z } _ { d }$ is associated with the true generative distribution components $P _ { i } ^ { * } ( \\mathbf { x } )$ , uniformly sampling the discrete prior ensures that that the distributions are well represented in the synthetic mix that the student observes. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In general, the capacity of a categorical distribution is less than that of a continuous normal distribution. To prevent the VAE’s encoder from using primarily the continuous representation while disregarding the discrete one we further complement the learning objective by a term maximising the mutual information between the discrete representation and the data $I ( z _ { d } ; \\mathbf { x } ) = H ( z _ { d } ) - H ( z _ { d } | \\mathbf { x } )$ . $H ( z _ { d } )$ is used to denote the marginal entropy of $z _ { d }$ and $H ( z _ { d } | \\mathbf { x } )$ denotes the conditional entropy of $z _ { d }$ given $\\mathbf { x }$ . 1 ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 LEARNING OBJECTIVE ", + "text_level": 1, + "bbox": [ + 174, + 204, + 374, + 219 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The final learning objective for each of the student models is the maximization of the ELBO from equation (2), augmented by the negative of the cross-model consistency term introduced in section 4.1 and the mutual information term proposed in section 4.2. ", + "bbox": [ + 174, + 229, + 821, + 272 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/aa57063cf8176c5be781d36c0fb0d018fe4aa8377463028bf79abc20b67c8986.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { Q _ { \\phi } } [ l o g P _ { \\theta } ( \\mathbf { x } | z ) ] - K L [ Q _ { \\phi } ( z | \\mathbf { x } ) | | P ( z ) ] - \\mathbb { 1 } ( \\omega = 0 ) K L [ Q _ { \\phi } ( z _ { d } | \\mathbf { x } ) | | Q _ { \\Phi } ( z _ { d } | \\mathbf { x } ) ] + \\lambda I ( z _ { d } ; \\mathbf { x } ) \\ , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 176, + 279, + 815, + 297 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We sample the training instances $\\mathbf { x }$ from $\\mathbf { x } \\sim P ( \\omega ) P ( \\mathbf { x } | \\omega ) , \\omega \\sim B e r ( \\pi )$ as described in section 4. Thus they can either be generated from the teacher model $\\omega = 0$ ) or come from the training set of the currently active distribution $( \\omega = 1 )$ ). $\\mathbb { 1 } ( . )$ is the indicator function which evaluates to 1 if its argument is true and zero otherwise; it makes sure that the consistency regularizer is applied only over the synthetic samples generated by the teacher. The $\\lambda$ hyper-parameter controls the importance of the mutual information regularizer. We present the analytical evaluation of the consistency regularizer in appendix section 7.0.1. ", + "bbox": [ + 174, + 337, + 825, + 435 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 455, + 326, + 472 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We conducted a set of experiments to explore the behaviour and properties of the method we propose. We specifically concentrate on the benefits our model brings in the lifelong learning setting which is the main motivation of our work. We explain the settings of the individual experiments and their focus in the following three sections. ", + "bbox": [ + 174, + 487, + 825, + 542 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In all the experiments we use the notion of a distributional ‘interval’: the interval in which we observe samples from a single distribution $P _ { i } ^ { * } ( \\mathbf { x } )$ before the transition to the next distribution $P _ { i + 1 } ^ { * } ( \\mathbf { x } )$ occurs. The length of the intervals is in principle random and we developed a heuristic to generate these. We provide further details on this together with other technical details related to the network implementation and training common for all the experiments in the appendix. ", + "bbox": [ + 174, + 549, + 825, + 619 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 FASHION MNIST : SEQUENTIAL GENERATION ", + "text_level": 1, + "bbox": [ + 174, + 636, + 539, + 651 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this experiment, we seek to establish the performance benefit that our augmented objective formulation in section 4.3 brings into the learning in contrast to the simple ELBO objective 2. We do so by training two models with identical student-teacher architectures as introduced in section 4, with one using the consistency and mutual information augmented objective (with consistency) and the other using the standard ELBO objective (without consistency). We also demonstrate the ability of our model to disambiguate distributional boundaries from the distributional variations. ", + "bbox": [ + 174, + 662, + 825, + 746 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We use Fashion MNIST Xiao et al. (2017) 2 to simulate our sequential learning setting. We treat each object as a different distribution and present the model with samples drawn from a single distribution at a time. We sequentially progress over the ten available distributions. When a distribution transition occurs (new object) we signal the model, make the latest student the new teacher and instantiate a new student model. ", + "bbox": [ + 174, + 752, + 825, + 823 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We quantify the performance of the generative models by computing the ELBO over the standard Fashion MNIST test set after every distributional transition. The test set contains objects from all of the individual distributions. We run this procedure ten times and report the average test ELBO over the ten repetitions in figure 3(c). We see that around the 3rd interval (the 3rd distributional transition), the negative ELBO of the with consistency model is systematically below $\\sim 2 0$ nats ) that of the without consistency model. This confirms the benefits of our new objective formulation for reducing the effects of the catastrophic interference, a crucial property in our lifelong learning setting. In the same figure we also plot the ELBO of the baseline batch VAE. The batch VAE will always outperform our model because it has simultaneous access to all of the distributions during training. ", + "bbox": [ + 174, + 830, + 825, + 886 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/de756f9933d2c746151e9331f505d0ed4cb3706f7841c6aa359e35e7728162f2.jpg", + "image_caption": [ + "Figure 3: (a) Generation with consistency regularizer. (b) Generation without consistency regularizer. (c) Average negative ELBO over ten trials (each) for the ten distributions within Fashion MNIST. " + ], + "image_footnote": [], + "bbox": [ + 174, + 99, + 820, + 243 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 303, + 825, + 387 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "After observing and training over all ten distributions we generate samples from the final students of the two models. We do this by fixing the discrete distribution $z _ { d }$ to one-hot vectors over the whole categorical distribution, while randomly sampling the continuous prior $ { \\boldsymbol { z } } _ { c } \\sim \\mathcal { N } ( 0 , I )$ . We contrast samples generated from the model with consistency (figure 3(a)) to the model without consistency (figure 3(b)). Our model learns to separate ’style’ from the distributional boundaries. For example, in the last row of our with consistency model, we observe the various styles of shoes. The without consistency model mixes the distributions randomly. This illustrates the benefits that our augmented objective has for achieving consistent sampling from the individual distributional components. ", + "bbox": [ + 173, + 393, + 825, + 506 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 ROTATED MNIST $:$ LONG TERM DISTRIBUTION ACCUMULATION ", + "text_level": 1, + "bbox": [ + 176, + 521, + 665, + 536 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this experiment we dig deeper into the benefits our objective formulation brings for the lifelong learning setting. We expose the models to a much larger number of distributions and we explore how our augmented objective from 4.3 helps in preserving the previously learned knowledge. As in section 5.1, we compare models with and without consistency with identical teacher-student architectures. We measure the ability of the models to recall the previously learned information by looking at the consistency between the posterior of the student and the teacher models over the test data set ", + "bbox": [ + 173, + 546, + 825, + 645 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/e1dc5338bebb0752930464ee0f9a698300af1135479b10028325c92cedcdf9b2.jpg", + "text": "$$\n\\mathrm { \\sf t e n c y : } \\quad \\# \\{ k : Q _ { \\Phi } ( z _ { d } | { \\bf x } _ { k } ) = = Q _ { \\phi } ( z _ { d } | { \\bf x } _ { k } ) , { \\bf x } _ { k } \\in { \\bf X } _ { t e s t } \\} \\mathrm { ~ . ~ }\n$$", + "text_format": "latex", + "bbox": [ + 326, + 646, + 715, + 664 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We use the MNIST dataset in which we rotate each of the original digit samples by angles $\\nu =$ $[ 3 0 ^ { \\circ }$ , $7 0 ^ { \\circ }$ , $1 3 0 ^ { \\circ }$ , $2 0 0 ^ { \\circ }$ , $2 5 0 ^ { \\circ }$ ]. We treat each rotation of a single digit family as an individual distribution $\\{ P _ { i } ^ { * } ( \\mathbf { x } ) \\} _ { i = 1 } ^ { 7 0 }$ . Within each distributional interval, we sample the data by first sampling (uniformly with replacement) one of the 70 distributions and then sampling the data instances $\\mathbf { x }$ from the selected distribution. ", + "bbox": [ + 174, + 672, + 825, + 742 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 4(b) compares the consistency results of the two tested models throughout the learning process. Our model with the augmented objective clearly outperforms the model that uses the simple ELBO objective. This confirms the usefulness of the additional terms in our objective for preserving the previously learned knowledge in accordance with the lifelong learning paradigms. In addition, similarly as in experiment 5.1, figure 4(a) documents that the model with the augmented objective (thanks to reducing the effects of the catastrophic interference) achieves lower negative test ELBO systematically over the much longer course of learning ( $\\sim 3 0$ nats). ", + "bbox": [ + 173, + 750, + 825, + 848 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We also visualise in figure 4(c) how the accumulation of knowledge speeds up the learning process. For each distributional interval we plot the norms of the model gradients across the learning iterations. We observe that for later distributional intervals the curves become steeper much quicker, reducing the gradients and reaching (lower) steady states much faster then in the early learning stages. This suggests that the latter models are able to learn quicker in our proposed architecture. ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/7e68be6459eb4fe191ce9481bdea3906681f59cc2a823e02e61cb7449cfb295e.jpg", + "image_caption": [ + "Figure 4: (a) Negative test ELBO over the learning history; (b) consistency between the teacher and student posteriors across the test data samples normalized by the test data set size; (c) speed of learning convergence across distributional intervals. Best viewed in color. " + ], + "image_footnote": [], + "bbox": [ + 176, + 103, + 846, + 247 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 SVHN TO MNIST ", + "text_level": 1, + "bbox": [ + 176, + 311, + 343, + 325 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this experiment we explore the ability of our model to retain and transfer knowledge across completely different datasets. We use MNIST and SVHN Netzer et al. (2011) to demonstrate this. We treat all samples from SVHN as being generated by one distribution $P _ { 1 } ^ { * } ( \\mathbf { x } )$ and all the MNIST3 samples as generated by another distribution $P _ { 2 } ^ { * } ( \\mathbf { x } )$ (irrespective of the specific digit). ", + "bbox": [ + 174, + 339, + 825, + 395 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We first train a student model (standard VAE) over the entire SVHN data set. Once done, we freeze the parameters of the encoder and the decoder and transfer the model into the teacher state $( \\phi \\to \\Phi , \\theta \\to \\Theta )$ . We then use this teacher to aid the learning of the new student over the mix of the teacher-generated synthetic SVHN samples $\\hat { \\bf x }$ and the true MNIST data. ", + "bbox": [ + 174, + 401, + 825, + 458 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/5b74ad0e049511c04d3e0276bca9b4fd870fefca948ab30f0adbba4860afe937.jpg", + "image_caption": [ + "Figure 5: (a) Reconstructions of test samples from SVHN[left] and MNIST[right]; (b) Decoded samples $\\hat { \\mathbf { x } } \\sim P _ { \\pmb { \\theta } } ( \\mathbf { x } | z _ { d } , z _ { c } )$ based on linear interpolation of $z _ { c } \\in \\mathcal { R } ^ { 2 }$ with $z _ { d } = [ 0 , 1 ]$ ; (c) Same as (b) but with $z _ { d } = [ 1 , 0 ]$ . " + ], + "image_footnote": [], + "bbox": [ + 173, + 473, + 825, + 636 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We use the final student model to reconstruct samples from the two datasets by passing them through the learned encoding/decoding flow: $\\mathbf { x } \\ \\sim \\ P _ { i } ^ { * } \\ \\mathrm { ( \\mathbf { x } ) } \\ \\ z \\ \\sim \\ Q _ { \\phi } ( z | \\mathbf { x } ) \\ \\ \\hat { \\mathbf { x } } \\ \\sim \\ P _ { \\theta } ( \\mathbf { x } | z )$ . We visualise examples of the true inputs $\\mathbf { x }$ and the respective reconstructions $\\hat { \\bf x }$ in figure 5(a). We see that even though the only true data the final model received for training were from MNIST, it can still reconstruct SVHN data. This confirms the ability of our architecture to transition between complex distributions while still preserving the knowledge learned from the previously observed distributions. ", + "bbox": [ + 173, + 707, + 825, + 804 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Finally, in figure 5(b) and 5(c) we illustrate the data generated from an interpolation of a 2-dimensional continuous latent space. For this we specifically trained the models with the continuous latent variable $z _ { c } \\in \\mathcal { R } ^ { 2 }$ . To generate the data, we fix the discrete categorical $z _ { d }$ to one of the possible values $\\{ [ 0 , 1 ] , [ 1 , 0 ] \\}$ and linearly interpolate the continuous $z _ { c }$ over the range $[ - 3 , 3 ]$ . We then decode these to obtain the samples $\\hat { \\mathbf { x } } \\sim P _ { \\pmb { \\theta } } ( \\mathbf { x } | \\pmb { z } _ { d } , \\pmb { z } _ { c } )$ . The model learns a common continuous structure for the two distributions which can be followed by observing the development in the generated samples from top left to bottom right on both figure 5(b) and 5(c). ", + "bbox": [ + 174, + 810, + 825, + 881 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 151, + 318, + 167 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work we propose a novel method for learning generative models over streaming data following the lifelong learning principles. The principal assumption for the data is that they are generated by multiple distributions and presented to the learner in a sequential manner (a set of observations from a single distribution followed by a distributional transition). A key limitation for the learning is that the method can only access data generated by the current distribution and has no access to any of the data generated by any of the previous distributions. ", + "bbox": [ + 174, + 183, + 825, + 266 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The proposed method is based on a dual student-teacher architecture where the teacher’s role is to preserve the past knowledge and aid the student in future learning. We argue for and augment the standard VAE’s ELBO objective by terms helping the teacher-student knowledge transfer. We demonstrate on a series of experiments the benefits this augmented objective brings in the lifelong learning settings by supporting the retention of previously learned knowledge (models) and limiting the usual effects of catastrophic interference. ", + "bbox": [ + 174, + 273, + 825, + 357 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In our future work we will explore the possibilities to extend our architecture to GAN-like Goodfellow et al. (2014) learning with the prospect to further improve the generative abilities of our method. GANs, however, do not use a metric for measuring the quality of the learned distributions such as the marginal likelihood or the ELBO in their objective and therefore the transfer of our architecture to these is not straightforward. ", + "bbox": [ + 174, + 364, + 825, + 434 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 455, + 285, + 469 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Tamara Broderick, Nicholas Boyd, Andre Wibisono, Ashia C. Wilson, and Michael I. Jordan. Streaming variational bayes. In Burges et al. (2013), pp. 1727–1735. URL http://papers. nips.cc/paper/4980-streaming-variational-bayes. ", + "bbox": [ + 178, + 477, + 821, + 520 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Christopher J. C. Burges, Leon Bottou, Zoubin Ghahramani, and Kilian Q. 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", + "bbox": [ + 174, + 141, + 825, + 196 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, 2017. ", + "bbox": [ + 176, + 207, + 823, + 234 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "7 APPENDIX ", + "text_level": 1, + "bbox": [ + 174, + 102, + 294, + 117 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "7.0.1 UNDERSTANDING THE CONSISTENCY REGULARIZER ", + "text_level": 1, + "bbox": [ + 174, + 133, + 593, + 148 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The analytical derivations of the consistency regularizer show that the regularizer can be interpreted as an a transformation of the standard VAE regularizer. In the case of an isotropic gaussian posterior, the proposed regularizer scales the mean and variance of the student posterior by the variance of the teacher 7.0.2 and adds an extra ’volume’ term. This interpretation of the consistency regularizer shows that the proposed regularizer preserves the same learning objective as that of the standard VAE. Below we present the analytical form of the consistency regularizer with categorical and isotropic gaussian posteriors: ", + "bbox": [ + 173, + 157, + 826, + 257 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Corollary 7.0.1 We parameterize the learnt posterior of the teacher by $\\begin{array} { r } { \\Phi _ { i } \\ = \\ \\frac { \\exp ( p _ { i } ^ { E } ) } { \\sum _ { i = 1 } ^ { J } \\exp ( p _ { i } ^ { E } ) } } \\end{array}$ and the posterior of the student by φi = $\\begin{array} { r } { c ^ { E } = \\sum _ { i = 1 } ^ { J } \\exp ( p _ { i } ^ { E } ) } \\end{array}$ and $\\begin{array} { r } { c ^ { S } = \\sum _ { i = 1 } ^ { J } \\exp ( p _ { i } ^ { S } ) . } \\end{array}$ $\\begin{array} { r } { \\phi _ { i } = \\frac { \\exp ( p _ { i } ^ { S } ) } { \\sum _ { i = 1 } ^ { J } \\exp ( p _ { i } ^ { S } ) } } \\end{array}$ . We also redefine the normalizing constants as the teacher and student models respectively. The reverse $K L$ divergence in equation 8 can now be re-written as: ", + "bbox": [ + 173, + 270, + 826, + 348 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/c02315e469553ceaae815a6c6f53ea9a024a68c4349768c7981f12a1f3410d9f.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle K L ( Q _ { \\phi } ( z _ { d } | x ) | | Q _ { \\Phi } ( z _ { d } | x ) ) = \\sum _ { i = 1 } ^ { J } \\frac { \\exp ( p _ { i } ^ { S } ) } { c ^ { S } } l o g \\left( \\frac { \\exp ( p _ { i } ^ { S } ) } { c ^ { S } } \\frac { c ^ { E } } { \\exp ( p _ { i } ^ { E } ) } \\right) } } \\\\ { { = H ( p ^ { S } , p ^ { S } - p ^ { E } ) = - H ( p ^ { s } ) + H ( p ^ { S } , p ^ { E } ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 356, + 746, + 429 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "where $H ( \\mathbf { \\Sigma } _ { - } )$ is the entropy operator and $H ( \\ l _ { - } , \\ l _ { - } )$ is the cross-entropy operator. ", + "bbox": [ + 178, + 435, + 683, + 452 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Corollary 7.0.2 We assume the learnt posterior of the teacher is parameterized by a centered, isotropic gaussian with $\\Phi \\ : = \\ : [ \\pmb { \\mu } ^ { E } \\ : = \\ : \\mathbf { 0 } , \\pmb { \\Sigma } _ { \\sim } ^ { E } \\ : = \\ : \\sigma ^ { E ^ { 2 } } \\underline { { { I } } } ]$ and the posterior of our student by $a$ non-centered isotropic gaussian with $\\phi = [ \\mu ^ { S } , \\Sigma ^ { S } = \\sigma ^ { S 2 } I ]$ , then ", + "bbox": [ + 174, + 463, + 826, + 510 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/077631f76bc7d179308607709bfe30d5f49beeac5b9297bf55cf0a317126eafe.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\nabla L ( Q _ { \\phi } ( z | x ) | | Q _ { \\Phi } ( z | x ) ) = 0 . 5 \\biggl [ t r ( \\Sigma ^ { E ^ { - 1 } } \\Sigma ^ { S } ) + ( \\mu ^ { E } - \\mu ^ { S } ) ^ { T } \\Sigma ^ { E ^ { - 1 } } ( \\mu ^ { E } - \\mu ^ { S } ) - F + l o g \\biggl ( \\frac { | \\Sigma ^ { E } | } { | \\Sigma ^ { S } | } \\biggr ) \\biggr ] } } \\\\ { { \\displaystyle = 0 . 5 \\sum _ { j = 1 } ^ { F } \\biggl [ \\frac { 1 } { \\sigma ^ { E ^ { 2 } } ( j ) } ( \\sigma ^ { S 2 } ( j ) + \\mu ^ { S 2 } ( j ) ) - 1 + l o g \\sigma ^ { E 2 } ( j ) - l o g \\sigma ^ { S 2 } ( j ) \\biggr ] } } \\\\ { { \\displaystyle = K L ( Q _ { \\phi ^ { * } } ( z | x ) | | \\mathcal { N } ( 0 , I ) ) - l o g | \\Sigma ^ { E } | } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 516, + 839, + 621 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Via a reparameterization of the student’s parameters: ", + "bbox": [ + 174, + 640, + 524, + 655 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/e322135b510dd78bc5a21432d47dea889934964a4865930105d378db84c5a6b3.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\phi ^ { * } = [ \\mu ^ { S * } , \\sigma ^ { S * 2 } ] } } \\\\ { { \\displaystyle \\mu ^ { S * } = \\frac { \\mu ^ { S } ( j ) } { \\sigma ^ { E 2 } ( j ) } ; \\sigma ^ { S * 2 } = \\frac { \\sigma ^ { S 2 } ( j ) } { \\sigma ^ { E 2 } ( j ) } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 388, + 660, + 607, + 718 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "It is also interesting to note that our posterior regularizer becomes the prior if: ", + "bbox": [ + 173, + 731, + 683, + 746 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/2cb58ee9dda29b24f5254bc832586c8ff2c085087b2ecb6b689a7d5fcc73324d.jpg", + "text": "$$\nl i m _ { \\sigma ^ { E ^ { 2 } } \\mapsto 1 } K L ( Q _ { \\phi } ( z | x ) | | Q _ { \\Phi } ( z | x ) ) = K L ( Q _ { \\phi } ( z | x ) | | \\mathcal { N } ( 0 , I ) )\n$$", + "text_format": "latex", + "bbox": [ + 284, + 752, + 714, + 771 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "7.1 ELBO DERIVATION ", + "text_level": 1, + "bbox": [ + 176, + 785, + 352, + 801 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Variational inference Hoffman et al. (2013) side-steps the intractability of the posterior distribution by approximating it with a tractable distribution $Q _ { \\Phi } ( z | \\mathbf { x } )$ ; we then optimize the parameters $\\Phi$ in order to bring this distribution close to $P _ { \\Phi } ( z | \\mathbf { x } )$ . The form of this approximate distribution is fixed and is generally conjugate to the prior $P ( z )$ . Variational inference converts the problem of posterior inference into an optimization problem over $\\Phi$ . This allows us to utilize stochastic gradient descent to solve our problem. To be more concrete, variational inference tries to minimize the reverse Kullback-Leibler (KL) divergence between the variational posterior distribution $Q _ { \\Phi } ( z | \\mathbf { x } )$ and the true posterior $P _ { \\pmb { \\theta } } ( \\pmb { z } | \\mathbf { x } )$ : ", + "bbox": [ + 173, + 810, + 825, + 925 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/aa133bbcc02ccfbbdcdbd2e63334ee56b4dfbab03b832e62ee835c5d1e50817c.jpg", + "text": "$$\nK L [ Q _ { \\Phi } ( z | \\mathbf { x } ) | | P _ { \\theta } ( z | \\mathbf { x } ) ] = \\log P _ { \\theta } ( \\mathbf { x } ) - \\underbrace { \\mathbb { E } _ { Q _ { \\Phi } ( z | \\mathbf { x } ) } \\left[ \\log \\frac { P _ { \\theta } ( x , z ) } { Q _ { \\Phi } ( z | \\mathbf { x } ) } \\right] } _ { \\mathcal { L } _ { \\theta } }\n$$", + "text_format": "latex", + "bbox": [ + 277, + 122, + 720, + 178 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Rearranging the terms in equation 8 and utilizing the fact that the KL divergence is a measure, we can derive the evidence lower bound $\\mathcal { L } _ { \\theta }$ (ELBO) which is the objective function we directly optimize: ", + "bbox": [ + 173, + 178, + 825, + 220 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/1fd08028e0b897b0db9eb83c509dafded2b24f3f14955e580646139d00eac1b3.jpg", + "text": "$$\nl o g P _ { \\pmb \\theta } ( \\mathbf { x } ) \\geq \\mathbb { E } _ { Q _ { \\Phi } ( z | \\mathbf { x } ) } [ l o g P _ { \\pmb \\theta } ( \\mathbf { x } | z ) ] - K L ( Q _ { \\Phi } ( z | \\mathbf { x } ) | | P ( z ) ) = \\mathcal { L } _ { \\pmb \\theta }\n$$", + "text_format": "latex", + "bbox": [ + 269, + 228, + 727, + 247 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In order to backpropagate it is necessary to remove the dependence on the stochastic variable $_ { z }$ . To achieve this, we push the sampling operation outside of the computational graph for the normal distribution via the reparameterization trick Kingma & Welling (2014) and the gumbel-softmax reparameterization Maddison et al. (2016); Jang et al. (2017) for the discrete distribution. In essence the reparameterization trick allows us to introduce a distribution $P ( \\epsilon )$ that is not a function of the data or computational graph in order to move the gradient operator into the expectation: ", + "bbox": [ + 173, + 261, + 825, + 345 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/95589251212178f5ee5fa6e8701c9f481debd08fd4e47444f8907327aafbda26.jpg", + "text": "$$\n\\nabla \\mathbb { E } _ { Q _ { \\Phi } ( z | \\mathbf { x } ) } \\bigg [ \\mathrm { l o g } \\frac { P _ { \\theta } ( x , z ) } { Q _ { \\Phi } ( z | \\mathbf { x } ) } \\bigg ] \\mapsto \\mathbb { E } _ { P ( \\epsilon ) } \\bigg [ \\nabla \\mathrm { l o g } \\frac { P _ { \\theta } ( x , z ) } { Q _ { \\Phi } ( z | \\mathbf { x } ) } \\bigg ]\n$$", + "text_format": "latex", + "bbox": [ + 308, + 353, + 687, + 388 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "7.2 MODEL RELATED ", + "text_level": 1, + "bbox": [ + 174, + 404, + 338, + 417 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this section we provide extra details of our model architecture. ", + "bbox": [ + 173, + 430, + 599, + 445 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "7.2.1 MODEL ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 462, + 398, + 477 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We utilized two different architectures for our experiments. The first two utilize a standard deep neural network with two layers of 512 to map to the latent representation and two layers of 512 to map back to the reconstruction for the decoder. We used batch norm Ioffe & Szegedy (2015) and ELU activations for all the layers barring the layer projecting into the latent representation and the output layer. ", + "bbox": [ + 173, + 486, + 825, + 556 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The final experiment with the transfer from SVHN to MNIST utilizes a fully convolutional architecture with only strided convolutional layers in the encoder (where the number of filters are doubled at each layer). The final projection layer for the encoder maps the data to a $[ { \\mathsf { C } } { = } | z _ { d } |$ , 1, 1] output which is then reparameterized in the standard way. The decoder utilizes fractional strides for the convolutional-transpose (de-convolution) layers where we reduce the number of filters in half at each layer. The full architecture can be examined in our code repository [which will be de-anonymized after the review process]. All layers used batch norm Ioffe & Szegedy (2015) and ELU activations. ", + "bbox": [ + 173, + 563, + 825, + 674 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We utilized Adam Kingma & Ba (2015) to optimize all of our problems with a learning rate of 1e-4. When we utilized weight transfer we re-initialized the accumulated momentum vector of Adam as well as the aggregated mean and covariance of the Batch Norm layers. Our code is already available online under an MIT license at 4 ", + "bbox": [ + 174, + 681, + 825, + 737 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "7.2.2 GUMBEL REPARAMETERIZATION ", + "text_level": 1, + "bbox": [ + 176, + 753, + 457, + 768 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Since we model our latent variable as a combination of a discrete and a continuous distribution we also use the Gumbel-Softmax reparameterization Maddison et al. (2016); Jang et al. (2017). The Gumbel-Softmax reparameterization over logits [linear output of the last layer in the encoder] $\\pmb { p } \\in \\mathcal { R } ^ { M }$ and an annealed temperature parameter $\\tau \\in \\mathcal { R }$ is defined as: ", + "bbox": [ + 173, + 779, + 825, + 835 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/c800d0a0048e6db147a5ca7153eb48d53f13937f2e3b8f66933c7fdfc16ad105.jpg", + "text": "$$\nz = s o f t m a x ( \\frac { l o g ( p ) + g } { \\tau } ) ; g = - l o g ( - l o g ( { \\pmb u } \\sim U n i f ( 0 , 1 ) ) )\n$$", + "text_format": "latex", + "bbox": [ + 284, + 842, + 712, + 875 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$\\pmb { u } \\in \\mathcal { R } ^ { M } , \\pmb { g } \\in \\mathcal { R } ^ { M }$ . As the temperature parameter $\\tau \\mapsto 0$ , $_ { z }$ converges to a categorical. ", + "bbox": [ + 178, + 882, + 745, + 898 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "7.2.3 EXPANDABLE MODEL CAPACITY AND REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 103, + 629, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Multilayer neural networks with sigmoidal activations have a VC dimension bounded between $O ( \\rho ^ { 2 } )$ Sontag (1998) and $O ( \\rho ^ { 4 } )$ Karpinski & Macintyre (1997) where $\\rho$ are the number of parameters. A model that is able to consistently add new information should also be able to expand its VC dimension by adding new parameters over time. Our formulation imposes no restrictions on the model architecture: i.e. new layers can be added freely to the new student model. ", + "bbox": [ + 174, + 128, + 825, + 199 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In addition we also allow the dimensionality of $\\boldsymbol { z } _ { d } \\in \\mathcal { R } ^ { J }$ , our discrete latent representation to grow in order to accommodate new distributions. This is possible because the KL divergence between two categorical distributions of different sizes can be evaluated by simply zero padding the teacher’s smaller discrete distribution. Since we also transfer weights between the teacher and the student model, we need to handle the case of expanding latent representations appropriately. In the event that we add a new distribution we copy all the weights besides the ones immediately surrounding the projection into and out of the latent distribution. These surrounding weights are reinitialized to their standard Glorot initializations Glorot & Bengio (2010). ", + "bbox": [ + 174, + 205, + 825, + 318 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "7.3 FORWARD VS. REVERSE KL ", + "text_level": 1, + "bbox": [ + 176, + 338, + 410, + 353 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/c20e180406d015f1a528955ddb39a6aeaad6b034df00717d8727af083540f5d4.jpg", + "image_caption": [ + "Figure 6: Reverse vs. forward KL on FashionMNIST 5.1 " + ], + "image_footnote": [], + "bbox": [ + 281, + 386, + 686, + 622 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In our setting we have the ability to utilize the zero forcing (reverse or mode-seeking) KL or the zero avoiding (forward) KL divergence. In general, if the true underlying posterior is multi-modal, it is preferable to operate with the reverse KL divergence (Murphy (2012) 21.2.2). In addition, utilizing the mode-seeking KL divergence generates more realistic results when operating over image data. ", + "bbox": [ + 174, + 671, + 825, + 728 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In order to validate this, we repeat the experiment in 5.1. We train two models: one with the forward KL posterior regularizer and one with the reverse. We evaluate the -ELBO mean and variance over ten trials. Empirically, we observed no difference between the different measures. This is demonstrated in figure 6. ", + "bbox": [ + 174, + 734, + 825, + 791 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "7.4 NUMBER OF REQUIRED SAMPLES ", + "text_level": 1, + "bbox": [ + 178, + 813, + 444, + 827 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Our method derives its sample complexity from standard VAEs. In practice we evaluate the number of required real and synthetic samples by utilizing early stopping. When the negative ELBO on the validation set stops decreasing for 50 steps we stop training the current model and transition to the next distribution interval. Using this and the fact that we keep equal proportions of all observed distributions in our minibatch, we can evaluate the number of synthetic and real samples used during the single distribution interval. We demonstrate this procedure on experiment 5.1 in figure 7. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/cc4dab3f21baf14bf3afeebd6d4c9bdb4ca56c427a0d6ee500748df09837e33e.jpg", + "image_caption": [ + "Figure 7: FashionMNIST experiment 5.1 data efficiency analysis. Left: Real samples used; Right: Synthetic samples used " + ], + "image_footnote": [], + "bbox": [ + 176, + 111, + 794, + 286 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We observe a rapid decrease of the number of required real samples as we assimilate more distributions into our model. ", + "bbox": [ + 173, + 334, + 823, + 362 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "7.5 EXPERIMENTS RELATED ", + "text_level": 1, + "bbox": [ + 176, + 381, + 385, + 395 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this section we provide an extra experiment run on MNIST as well as some extra images from the rotated MNIST experiment. ", + "bbox": [ + 174, + 406, + 823, + 435 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "7.5.1 MNIST : GENERATION AND ELBO ", + "text_level": 1, + "bbox": [ + 174, + 452, + 473, + 467 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/15a1a1e12ba4d93f4e0643a470665c0942c237892d48141604db03027ffeeb15.jpg", + "image_caption": [ + "Figure 8: (a) Generation with consistency regularizer. (b) Without consistency regularizer. (c) Average log-likelihood over ten trials (each) for the ten separated distributions within MNIST. " + ], + "image_footnote": [], + "bbox": [ + 174, + 500, + 821, + 643 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this experiment, we seek to establish the performance benefit that the consistency regularizer brings into the learning process. We do so by evaluating the ELBO for a model with and without the consistency and mutual information regularizers. We also demonstrate the ability of the regularizers to disambiguate distributional boundaries and their inter-distributional variations. I.e. for MNIST this separates the MNIST digits from their inter-class variants (i.e drawing style). ", + "bbox": [ + 174, + 700, + 825, + 770 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We use MNIST to simulate our sequential learning setting. We treat each digit as a different distribution and present the model with samples drawn from a single distribution at a time. For the purpose of this experiment we sequentially progress over the ten distributions (i.e. interval sampling involves linearly iterating over all the distributions ). ", + "bbox": [ + 174, + 777, + 823, + 833 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "When an interval transition occurs we signal the model, make the student the new teacher and instantiate a new student model. We contrast this to a model that utilizes the same graphical model, without our consistency and mutual information regularizers. We quantify the performance of the generative models by computing the ELBO over the standard MNIST test set at every interval. The test set contains digits from all of the individual distributions. We run this procedure ten times and report the average ELBO over the test set. ", + "bbox": [ + 174, + 840, + 825, + 922 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "After observing all ten distributions we evaluate samples generated from the final student model. We do this by fixing the discrete distribution $z _ { d }$ , while randomly sampling $z _ { c } \\sim \\mathcal { N } ( 0 , I )$ . We contrast samples generated from the model with both regularizers (left-most image in 8) to the model without the regularizers (center image in 8). Our model learns to separate ’style’ from distributional boundaries. This is demonstrated by observing the digit $\" 2 \"$ : i.e. different samples of $z _ { c }$ produce different styles of writing a $\" 2 \"$ . ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "7.5.2 ROTATED MNIST EXPERIMENT ", + "text_level": 1, + "bbox": [ + 176, + 202, + 447, + 217 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We provide a larger sized image for the ELBO from experiment 5.2. We also visualize reconstructions from the rotated MNIST problem (visualized in figure 10). Finally in figure 11 we show the effects on the reconstructions when we do not use the mutual information regularizer. We believe this is due to the fact that the network utilizes the larger continuous representation to model the discriminative aspects of the observed distribution. ", + "bbox": [ + 173, + 227, + 825, + 296 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/f19a75a72a657bd9f71a621bf6deb35b993d8e2c51ff3349c27ba3f7cb3d3923.jpg", + "image_caption": [ + "Figure 9: Visualization of ELBO for rotated MNIST evaluated at the last model (the one at the 70th interval) " + ], + "image_footnote": [], + "bbox": [ + 315, + 330, + 678, + 563 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/52599e0d0ae24dc9212c75394825fdde9b6e11ed1bf838c77e4af811fe8676d3.jpg", + "image_caption": [ + "Figure 10: Visualization of reconstructions for rotated MNIST evaluated at the last model (the one at the 70th interval) " + ], + "image_footnote": [], + "bbox": [ + 176, + 643, + 820, + 829 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/e0fedbcb6e99ba86a31cb6d4751934b6f6afe937d08337dcb166f202aef3e30c.jpg", + "image_caption": [ + "Figure 11: Visualization of reconstructions when we do not use the mutual information regularizer " + ], + "image_footnote": [], + "bbox": [ + 178, + 406, + 820, + 609 + ], + "page_idx": 16 + } +] \ No newline at end of file diff --git a/parse/train/S1fduCl0b/S1fduCl0b_middle.json b/parse/train/S1fduCl0b/S1fduCl0b_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..6b8e3867bb92191cf0ebcbc62ed45e9a0e9fe60e --- /dev/null +++ b/parse/train/S1fduCl0b/S1fduCl0b_middle.json @@ -0,0 +1,41382 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 79, + 373, + 96 + ], + "lines": [ + { + "bbox": [ + 104, + 77, + 375, + 98 + ], + "spans": [ + { + "bbox": [ + 104, + 77, + 375, + 98 + ], + "score": 1.0, + "content": "LIFELONG GENERATIVE MODELING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 112, + 115, + 244, + 137 + ], + "lines": [ + { + "bbox": [ + 113, + 115, + 201, + 127 + ], + "spans": [ + { + "bbox": [ + 113, + 115, + 201, + 127 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 111, + 126, + 245, + 138 + ], + "spans": [ + { + "bbox": [ + 111, + 126, + 245, + 138 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 278, + 167, + 333, + 178 + ], + "lines": [ + { + "bbox": [ + 276, + 166, + 335, + 180 + ], + "spans": [ + { + "bbox": [ + 276, + 166, + 335, + 180 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 143, + 192, + 468, + 346 + ], + "lines": [ + { + "bbox": [ + 142, + 193, + 470, + 204 + ], + "spans": [ + { + "bbox": [ + 142, + 193, + 470, + 204 + ], + "score": 1.0, + "content": "Lifelong learning is the problem of learning multiple consecutive tasks in a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 204, + 470, + 215 + ], + "spans": [ + { + "bbox": [ + 141, + 204, + 470, + 215 + ], + "score": 1.0, + "content": "sequential manner where knowledge gained from previous tasks is retained and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 214, + 470, + 227 + ], + "spans": [ + { + "bbox": [ + 141, + 214, + 470, + 227 + ], + "score": 1.0, + "content": "used for future learning. 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It has numerous applications such as image", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "denoising, inpainting, super-resolution, structured prediction, clustering, pre-training and many", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "more. However, something that is lacking in the modern ML toolbox is an efficient way to learn", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 353, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 353, + 464 + ], + "score": 1.0, + "content": "these deep generative models in a sequential, lifelong setting.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "In a lot of real world scenarios we observe distributions sequentially. 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We give an example of such a setting in figure 1(a) using", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "MNIST LeCun & Cortes (2010), where we have three unique distributions and one that is repeated.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Since we only observe one distribution at a time we need to develop a strategy of retaining the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "previously learnt knowledge (i.e. the previously learnt distributions) and integrate it into future", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "learning. To accumulate additional distributions in the current generative model we utilize a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 638 + ], + "score": 1.0, + "content": "student-teacher architecture similar to that in distillation methods Hinton et al. (2015); Furlanello", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "et al. (2016). The teacher contains a summary of all past distributions and is used to augment", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "the data used to train the student model. 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It is essential towards the development of intelligent", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 224, + 470, + 239 + ], + "spans": [ + { + "bbox": [ + 141, + 224, + 470, + 239 + ], + "score": 1.0, + "content": "machines that can adapt to their surroundings. In this work we focus on a lifelong", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 237, + 470, + 249 + ], + "spans": [ + { + "bbox": [ + 141, + 237, + 470, + 249 + ], + "score": 1.0, + "content": "learning approach to generative modeling where we continuously incorporate", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 247, + 470, + 260 + ], + "spans": [ + { + "bbox": [ + 141, + 247, + 470, + 260 + ], + "score": 1.0, + "content": "newly observed streaming distributions into our learnt model. We do so through", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 259, + 470, + 271 + ], + "spans": [ + { + "bbox": [ + 141, + 259, + 470, + 271 + ], + "score": 1.0, + "content": "a student-teacher architecture which allows us to learn and preserve all the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 270, + 470, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 270, + 470, + 281 + ], + "score": 1.0, + "content": "distributions seen so far without the need to retain the past data nor the past", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 280, + 470, + 292 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 470, + 292 + ], + "score": 1.0, + "content": "models. Through the introduction of a novel cross-model regularizer, the student", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 291, + 469, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 469, + 304 + ], + "score": 1.0, + "content": "model leverages the information learnt by the teacher, which acts as a summary", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 302, + 469, + 315 + ], + "spans": [ + { + "bbox": [ + 141, + 302, + 469, + 315 + ], + "score": 1.0, + "content": "of everything seen till now. The regularizer has the additional benefit of reducing", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 313, + 470, + 326 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 470, + 326 + ], + "score": 1.0, + "content": "the effect of catastrophic interference that appears when we learn over streaming", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 324, + 469, + 336 + ], + "spans": [ + { + "bbox": [ + 141, + 324, + 469, + 336 + ], + "score": 1.0, + "content": "data. We demonstrate its efficacy on streaming distributions as well as its ability to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 335, + 469, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 469, + 348 + ], + "score": 1.0, + "content": "learn a common latent representation across a complex transfer learning scenario.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 10.5, + "bbox_fs": [ + 141, + 193, + 470, + 348 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 370, + 205, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 208, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 208, + 385 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "Deep unsupervised generative learning allows us to take advantage of the massive amount", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "of unlabeled data available in order to build models that efficiently compress and learn an", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 416, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 432 + ], + "score": 1.0, + "content": "approximation of the true data distribution. It has numerous applications such as image", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "denoising, inpainting, super-resolution, structured prediction, clustering, pre-training and many", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "more. However, something that is lacking in the modern ML toolbox is an efficient way to learn", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 353, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 353, + 464 + ], + "score": 1.0, + "content": "these deep generative models in a sequential, lifelong setting.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 395, + 506, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "In a lot of real world scenarios we observe distributions sequentially. Examples of this include", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 490 + ], + "score": 1.0, + "content": "streaming data from sensors such as cameras and microphones or other similar time series data. A", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "system can also be resource limited wherein all of the past data or learnt models cannot be stored. We", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "are interested in the lifelong learning setting for generative models where data arrives sequentially", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "in a stream and where the storage of all data is infeasible. Within the stream, instances are generated", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "according to some non-observed distribution which changes at given time-points. We assume we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "know the time points at which the transitions occur and whether the latent distribution is a completely", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "new one or one that has been observed before. We do not however know the underlying identity of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "the individual distributions. Our goal is to learn a generative model that can summarize all the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "distributions seen so far in the stream. We give an example of such a setting in figure 1(a) using", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "MNIST LeCun & Cortes (2010), where we have three unique distributions and one that is repeated.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 466, + 506, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Since we only observe one distribution at a time we need to develop a strategy of retaining the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "previously learnt knowledge (i.e. the previously learnt distributions) and integrate it into future", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "learning. 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The student model thus receives data samples from the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "currently observable distribution as well as synthetic data samples from previous distributions. This", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "allows the student model to learn a distribution that summarizes the current as well as all previously", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "observed distributions. Once a new distribution shift occurs the existing teacher model is discarded,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 693, + 369, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 369, + 704 + ], + "score": 1.0, + "content": "the student becomes the teacher and a new student is instantiated.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 594, + 506, + 704 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "We further leverage the generative model of the teacher by introducing a regularizer in the learning", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "score": 1.0, + "content": "objective function of the student that brings the posterior distribution of the latter close to that of the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 262, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 273 + ], + "score": 1.0, + "content": "former. This allows us to build upon and extend the teacher’s generative model in the student each", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "time the latter is re-instantiated (rather than re-learning it from scratch). By coupling this regularizer", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "with a weight transfer from the teacher to the student we also allow for faster convergence of the", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "student model. 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By independently", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "sampling the discrete and continuous components we preserve the distributional boundaries and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 504, + 253, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 253, + 516 + ], + "score": 1.0, + "content": "circumvent the two problems above.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 372, + 506, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 520, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "This sampling strategy, combined with the proposed regularizer allows us to learn and remember all", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "the individual distributions observed in the past. In addition we are also able to generate samples", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 542, + 439, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 439, + 556 + ], + "score": 1.0, + "content": "from any of the past distributions at will; we call this property consistent sampling.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 520, + 505, + 556 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 572, + 210, + 585 + ], + "lines": [ + { + "bbox": [ + 104, + 571, + 213, + 588 + ], + "spans": [ + { + "bbox": [ + 104, + 571, + 213, + 588 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Past work in sequential learning of generative models has focused on learning Gaussian mixture", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "models Singer & Warmuth (1999); Declercq & Piater (2008) or on variational methods such as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "Variational EM Ghahramani & Attias (2000). Work that is closer to ours is the online or sequential", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "learning of generative models in a streaming setting. Variational methods have been adapted for a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "streaming setting, e.g: Streaming Variational Bayes Broderick et al. (2013), Streaming Variational", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "Mixture models Tank et al. (2015), and the Population Posterior McInerney et al. (2015). However", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "their learning objectives are very different from ours. The objective of these methods is to adjust", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "the learnt model such that it reflects the current data distribution as accurately as possible, while", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "forgetting the previously observed distributions. Instead we want to do lifelong learning and retain", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "all previously observed distributions within our learnt model. As far as we know our work is the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "first one that tries to bring generative models, and in particular VAEs, into a lifelong setting where", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 719, + 345, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 345, + 734 + ], + "score": 1.0, + "content": "distributions are seen, learnt, and remembered sequentially.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 600, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "VAEs rely on an encoder and a decoder neural network in order to learn the parameters of the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "posterior and likelihood. One of the central problems that arise when training a neural network in", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "an sequential manner is that it causes the model to run into the problem of catastrophic interference", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "McCloskey & Cohen (1989). Catastrophic interference appears when we train neural networks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "in a sequential manner and model parameters start to become biased to the most recent samples", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "observed, while forgetting what was learnt from older samples. This generally happens when we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "stop exposing the model to past data. There have been a number of attempts to solve the problem", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "of catastrophic interference in neural networks. These range from distillation methods such as the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "original method Hinton et al. (2015) and ALTM Furlanello et al. (2016), to utilizing privileged", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 195 + ], + "score": 1.0, + "content": "information Lopez-Paz et al. (2016), as well as transfer learning approaches such as Learning", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Without Forgetting Li & Hoiem (2016) and methods that relay information from previously learnt", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "hidden layers such as in Progressive Neural Networks Rusu et al. (2016) and Deep Block-Modular", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "Neural Networks Terekhov et al. (2015). All of these methods necessitate the storage of previous", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 226, + 257, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 257, + 237 + ], + "score": 1.0, + "content": "models or data; our method does not.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 241, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "The recent work of elastic weight consolidation (EWC) Kirkpatrick et al. (2017) utilizes the Fisher", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "Information matrix (FIM) to avoid the problem of catastrophic interference. The FIM captures", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "the sensitivity of the log-likelihood with respect to the model parameters; EWC leverages this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "(via a linear approximation of the FIM) to control the change of model parameter values between", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "varying distributions. Intuitively, important parameters should not have their values changed, while", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 296, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 104, + 296, + 505, + 311 + ], + "score": 1.0, + "content": "non-important parameters are left unconstrained. Since EWC assumes model parameters being", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "distributed under an exponential family, it allows for the utilization of the FIM as a quadratic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "approximationJeffreys (1946) to the Kullback-Leibler (KL) divergence. Our model makes no such", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "distributional assumptions about the model parameters. Instead of constraining the parameters of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "the model as in EWC, we restrict the posterior representation of the student model to be close to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "that of the teacher for the previous distributions accumulated by the teacher. This allows the model", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 363, + 345, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 345, + 375 + ], + "score": 1.0, + "content": "parameters to vary as necessary in order to best fit the data.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 108, + 389, + 200, + 402 + ], + "lines": [ + { + "bbox": [ + 104, + 388, + 201, + 405 + ], + "spans": [ + { + "bbox": [ + 104, + 388, + 201, + 405 + ], + "score": 1.0, + "content": "3 BACKGROUND", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 374, + 427 + ], + "score": 1.0, + "content": "We consider an unsupervised setting where we observe a sample", + "type": "text" + }, + { + "bbox": [ + 375, + 414, + 385, + 424 + ], + "score": 0.6, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 413, + 398, + 427 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 399, + 414, + 430, + 425 + ], + "score": 0.9, + "content": "K \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 413, + 482, + 427 + ], + "score": 1.0, + "content": "realizations", + "type": "text" + }, + { + "bbox": [ + 482, + 414, + 505, + 425 + ], + "score": 0.84, + "content": "\\mathbf { X } =", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 422, + 507, + 440 + ], + "spans": [ + { + "bbox": [ + 107, + 425, + 192, + 438 + ], + "score": 0.92, + "content": "\\{ \\mathbf { x } ^ { ( 0 ) } , \\mathbf { x } ^ { ( 1 ) } , . . . , \\mathbf { x } ^ { ( K ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 422, + 334, + 440 + ], + "score": 1.0, + "content": "from an unknown true distribution", + "type": "text" + }, + { + "bbox": [ + 335, + 426, + 362, + 438 + ], + "score": 0.92, + "content": "P ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 422, + 383, + 440 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 384, + 425, + 420, + 437 + ], + "score": 0.92, + "content": "\\mathbf { x } \\in \\mathcal { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 422, + 507, + 440 + ], + "score": 1.0, + "content": ". We assume that the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 430, + 449 + ], + "score": 1.0, + "content": "data is generated by a random process involving a non-observed random variable", + "type": "text" + }, + { + "bbox": [ + 430, + 436, + 466, + 447 + ], + "score": 0.9, + "content": "z \\in \\mathcal { R } ^ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 435, + 506, + 449 + ], + "score": 1.0, + "content": ". In order", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 317, + 461 + ], + "score": 1.0, + "content": "to incorporate our prior knowledge we posit a prior", + "type": "text" + }, + { + "bbox": [ + 317, + 448, + 340, + 460 + ], + "score": 0.92, + "content": "P ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 447, + 362, + 461 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 362, + 450, + 369, + 458 + ], + "score": 0.73, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 447, + 505, + 461 + ], + "score": 1.0, + "content": ". Our objective is to approximate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 435, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 299, + 473 + ], + "score": 1.0, + "content": "the true underlying data distribution by a model", + "type": "text" + }, + { + "bbox": [ + 299, + 459, + 325, + 471 + ], + "score": 0.92, + "content": "P _ { \\pmb { \\theta } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 457, + 365, + 473 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 365, + 459, + 430, + 471 + ], + "score": 0.93, + "content": "P _ { \\pmb \\theta } ( \\mathbf { x } ) \\approx P ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 457, + 435, + 473 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 226, + 489 + ], + "score": 1.0, + "content": "Given a latent variable model", + "type": "text" + }, + { + "bbox": [ + 226, + 476, + 281, + 487 + ], + "score": 0.93, + "content": "P _ { \\pmb { \\theta } } ( \\mathbf { x } | \\mathbf { z } ) P ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 473, + 419, + 489 + ], + "score": 1.0, + "content": "we obtain the marginal likelihood", + "type": "text" + }, + { + "bbox": [ + 419, + 475, + 446, + 487 + ], + "score": 0.92, + "content": "P _ { \\pmb { \\theta } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 473, + 505, + 489 + ], + "score": 1.0, + "content": "by integrating", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 487, + 504, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 196, + 498 + ], + "score": 1.0, + "content": "out the latent variable", + "type": "text" + }, + { + "bbox": [ + 197, + 489, + 203, + 497 + ], + "score": 0.67, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 487, + 504, + 498 + ], + "score": 1.0, + "content": "from the joint distribution. The joint distribution can in turn be factorized", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 497, + 380, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 241, + 510 + ], + "score": 1.0, + "content": "using the conditional distribution", + "type": "text" + }, + { + "bbox": [ + 241, + 498, + 276, + 510 + ], + "score": 0.93, + "content": "P _ { \\theta } ( \\mathbf { x } | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 498, + 341, + 510 + ], + "score": 1.0, + "content": "or the posterior", + "type": "text" + }, + { + "bbox": [ + 341, + 497, + 376, + 510 + ], + "score": 0.93, + "content": "P _ { \\boldsymbol { \\theta } } ( z | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 498, + 380, + 510 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 512, + 448, + 537 + ], + "lines": [ + { + "bbox": [ + 162, + 512, + 448, + 537 + ], + "spans": [ + { + "bbox": [ + 162, + 512, + 448, + 537 + ], + "score": 0.92, + "content": "P _ { \\theta } ( { \\bf x } ) = \\int P _ { \\theta } ( { \\bf x } , z ) \\delta z = \\int P _ { \\theta } ( z | { \\bf x } ) P _ { \\theta } ( { \\bf x } ) \\delta z = \\int P _ { \\theta } ( { \\bf x } | z ) P ( z ) \\delta z", + "type": "interline_equation", + "image_path": "8a03c8228aa7008917fafb0710f6747c163f21099c3206f5a30bc5a9b8915def.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 162, + 512, + 448, + 537 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 259, + 552 + ], + "score": 1.0, + "content": "We model the conditional distribution", + "type": "text" + }, + { + "bbox": [ + 259, + 540, + 294, + 552 + ], + "score": 0.93, + "content": "P _ { \\theta } ( \\mathbf { x } | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "by a decoder, typically a neural network. Very often", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 202, + 563 + ], + "score": 1.0, + "content": "the marginal likelihood", + "type": "text" + }, + { + "bbox": [ + 202, + 551, + 228, + 563 + ], + "score": 0.92, + "content": "P _ { \\pmb { \\theta } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "will be intractable because the integral in equation (1) does not have", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "an analytical form nor an efficient estimator (Kingma (2017)). As a result the respective posterior", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 572, + 271, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 157, + 585 + ], + "score": 1.0, + "content": "distribution,", + "type": "text" + }, + { + "bbox": [ + 157, + 572, + 192, + 585 + ], + "score": 0.92, + "content": "P _ { \\pmb { \\theta } } ( \\pmb { z } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 572, + 271, + 585 + ], + "score": 1.0, + "content": ", is also intractable.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 588, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "score": 1.0, + "content": "Variational inference side-steps the intractability of the posterior by approximating it with a tractable", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 156, + 613 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 156, + 600, + 245, + 613 + ], + "score": 0.91, + "content": "Q _ { \\phi } ( z | \\mathbf { x } ) \\approx P _ { \\theta } \\bar { ( } z | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 599, + 506, + 613 + ], + "score": 1.0, + "content": ". VAEs use an encoder (generally a neural network) to model", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 610, + 507, + 626 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 216, + 626 + ], + "score": 1.0, + "content": "the approximate posterior", + "type": "text" + }, + { + "bbox": [ + 216, + 612, + 253, + 623 + ], + "score": 0.9, + "content": "Q _ { \\phi } ( z | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 610, + 376, + 626 + ], + "score": 1.0, + "content": "and optimize the parameters", + "type": "text" + }, + { + "bbox": [ + 377, + 612, + 385, + 623 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 610, + 507, + 626 + ], + "score": 1.0, + "content": "to minimize the reverse KL", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 153, + 635 + ], + "score": 1.0, + "content": "divergence", + "type": "text" + }, + { + "bbox": [ + 153, + 623, + 252, + 634 + ], + "score": 0.92, + "content": "K L [ Q _ { \\phi } ( z | \\mathbf { x } ) | | P _ { \\theta } ( z | \\mathbf { x } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 622, + 448, + 635 + ], + "score": 1.0, + "content": "between the approximate posterior distribution", + "type": "text" + }, + { + "bbox": [ + 448, + 622, + 486, + 635 + ], + "score": 0.92, + "content": "Q _ { \\phi } ( z | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 178, + 646 + ], + "score": 1.0, + "content": "the true posterior", + "type": "text" + }, + { + "bbox": [ + 178, + 634, + 213, + 645 + ], + "score": 0.91, + "content": "P _ { \\pmb { \\theta } } ( \\pmb { z } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 633, + 263, + 646 + ], + "score": 1.0, + "content": ". Given that", + "type": "text" + }, + { + "bbox": [ + 264, + 633, + 301, + 646 + ], + "score": 0.94, + "content": "Q _ { \\phi } ( z | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "is a powerful model (such that the KL divergence", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "against the true posterior will be close to zero) we maximize the tractable Evidence Lower BOund", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 303, + 668 + ], + "score": 1.0, + "content": "(ELBO) to the intractable marginal likelihood.", + "type": "text" + }, + { + "bbox": [ + 304, + 655, + 373, + 667 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { \\pmb { \\theta } } ( \\mathbf { x } ) \\leq P _ { \\pmb { \\theta } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "(full derivation available in the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 667, + 150, + 679 + ], + "spans": [ + { + "bbox": [ + 104, + 667, + 150, + 679 + ], + "score": 1.0, + "content": "appendix)", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43.5 + }, + { + "type": "interline_equation", + "bbox": [ + 166, + 676, + 444, + 691 + ], + "lines": [ + { + "bbox": [ + 166, + 676, + 444, + 691 + ], + "spans": [ + { + "bbox": [ + 166, + 676, + 444, + 691 + ], + "score": 0.88, + "content": "\\begin{array} { r l } { \\mathrm { E L B O : } } & { { } \\mathcal { L } _ { \\pmb { \\theta } } ( \\mathbf { x } ) = \\mathbb { E } _ { Q _ { \\phi } ( z | \\mathbf { x } ) } [ \\log P _ { \\pmb { \\theta } } ( \\mathbf { x } | z ) ] - K L [ Q _ { \\phi } ( z | \\mathbf { x } ) \\ | | \\ P ( z ) ] } \\end{array}", + "type": "interline_equation", + "image_path": "2269a1dffe7a71dd3526bccbcf02f92150a052047ec38edb9bda1b34f9c61194.jpg" + } + ] + } + ], + "index": 48, + "virtual_lines": [ + { + "bbox": [ + 166, + 676, + 444, + 691 + ], + "spans": [], + "index": 48 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 260, + 712 + ], + "score": 1.0, + "content": "By sharing the variational parameters", + "type": "text" + }, + { + "bbox": [ + 260, + 700, + 268, + 710 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "of the encoder across the data points (amortized inference", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 504, + 723 + ], + "score": 1.0, + "content": "Gershman & Goodman (2014)), variational autoencoders avoid per-data optimization loops typically", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 244, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 244, + 732 + ], + "score": 1.0, + "content": "needed by mean-field approaches.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "VAEs rely on an encoder and a decoder neural network in order to learn the parameters of the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "posterior and likelihood. One of the central problems that arise when training a neural network in", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "an sequential manner is that it causes the model to run into the problem of catastrophic interference", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "McCloskey & Cohen (1989). Catastrophic interference appears when we train neural networks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "in a sequential manner and model parameters start to become biased to the most recent samples", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "observed, while forgetting what was learnt from older samples. This generally happens when we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "stop exposing the model to past data. There have been a number of attempts to solve the problem", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "of catastrophic interference in neural networks. These range from distillation methods such as the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "original method Hinton et al. (2015) and ALTM Furlanello et al. (2016), to utilizing privileged", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 195 + ], + "score": 1.0, + "content": "information Lopez-Paz et al. (2016), as well as transfer learning approaches such as Learning", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Without Forgetting Li & Hoiem (2016) and methods that relay information from previously learnt", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "hidden layers such as in Progressive Neural Networks Rusu et al. (2016) and Deep Block-Modular", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "Neural Networks Terekhov et al. (2015). All of these methods necessitate the storage of previous", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 226, + 257, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 257, + 237 + ], + "score": 1.0, + "content": "models or data; our method does not.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 82, + 506, + 237 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 241, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "The recent work of elastic weight consolidation (EWC) Kirkpatrick et al. (2017) utilizes the Fisher", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "Information matrix (FIM) to avoid the problem of catastrophic interference. The FIM captures", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "the sensitivity of the log-likelihood with respect to the model parameters; EWC leverages this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "(via a linear approximation of the FIM) to control the change of model parameter values between", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "varying distributions. Intuitively, important parameters should not have their values changed, while", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 296, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 104, + 296, + 505, + 311 + ], + "score": 1.0, + "content": "non-important parameters are left unconstrained. Since EWC assumes model parameters being", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "distributed under an exponential family, it allows for the utilization of the FIM as a quadratic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "approximationJeffreys (1946) to the Kullback-Leibler (KL) divergence. Our model makes no such", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "distributional assumptions about the model parameters. Instead of constraining the parameters of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "the model as in EWC, we restrict the posterior representation of the student model to be close to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "that of the teacher for the previous distributions accumulated by the teacher. 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In addition to being initiated by the new teacher’s weights and receiving information", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 222, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 222, + 337, + 237 + ], + "score": 1.0, + "content": "about the teacher’s knowledge via the synthetic samples", + "type": "text" + }, + { + "bbox": [ + 337, + 224, + 345, + 234 + ], + "score": 0.49, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 222, + 505, + 237 + ], + "score": 1.0, + "content": ", we further foster the lifelong learning", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "idea by bringing the latent variable posterior induced by the student model closer to the respective", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "posterior induced by the teacher model. 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In doing so, we alleviate the effect of catastrophic interference.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 505, + 350 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 298 + ], + "score": 1.0, + "content": "To achieve this, we complement the classical VAE objective (equation (2)) with a term minimizing", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 182, + 308 + ], + "score": 1.0, + "content": "the KL divergence", + "type": "text" + }, + { + "bbox": [ + 183, + 295, + 284, + 307 + ], + "score": 0.92, + "content": "K L [ \\partial _ { \\phi } ( z | \\hat { \\mathbf { x } } ) | | Q _ { \\Phi } ( z | \\hat { \\mathbf { x } } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "between the student’s and the teacher’s posteriors over", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 180, + 319 + ], + "score": 1.0, + "content": "the synthetic data", + "type": "text" + }, + { + "bbox": [ + 180, + 307, + 187, + 316 + ], + "score": 0.46, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 306, + 505, + 319 + ], + "score": 1.0, + "content": ". The teacher’s encoder model, which already has the accumulated knowledge", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "from the previous learning steps, is thus reused within the new student’s objective. Under certain", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 328, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 341 + ], + "score": 1.0, + "content": "mild assumptions, we show that this objective reparameterizes the student model’s posterior, while", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 339, + 439, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 439, + 351 + ], + "score": 1.0, + "content": "preserving the same learning objective as a standard VAE (appendix section 7.0.1).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 108, + 372, + 211, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 213, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 213, + 384 + ], + "score": 1.0, + "content": "4.2 LATENT VARIABLE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "A critical component of our model is the synthetic data generation by the teacher’s decoder", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 168, + 419 + ], + "score": 0.92, + "content": "\\hat { \\mathbf { x } } ~ \\sim ~ P _ { \\Theta } ( \\mathbf { x } | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 407, + 505, + 419 + ], + "score": 1.0, + "content": ". The synthetic samples need to be representative of all the previously observed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 432 + ], + "score": 1.0, + "content": "distributions in order to provide the student with ample information about the learning history.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "The teacher generates these synthetic samples by first sampling the latent variable from the prior", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 156, + 451 + ], + "score": 0.91, + "content": "\\hat { z } ~ \\sim ~ P ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 439, + 291, + 452 + ], + "score": 1.0, + "content": "followed by the decoding step", + "type": "text" + }, + { + "bbox": [ + 291, + 439, + 356, + 452 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\hat { \\mathbf { x } } \\sim \\mathcal { P } _ { \\Theta } ( \\mathbf { x } | \\hat { z } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 439, + 506, + 452 + ], + "score": 1.0, + "content": ". 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The discrete component", + "type": "text" + }, + { + "bbox": [ + 492, + 596, + 504, + 605 + ], + "score": 0.83, + "content": "z _ { d }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 618 + ], + "score": 1.0, + "content": "shall summarise the most discriminative information about each of the true generating distributions", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 133, + 628 + ], + "score": 0.92, + "content": "P _ { i } ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 615, + 336, + 629 + ], + "score": 1.0, + "content": ". 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In addition to being initiated by the new teacher’s weights and receiving information", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 222, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 222, + 337, + 237 + ], + "score": 1.0, + "content": "about the teacher’s knowledge via the synthetic samples", + "type": "text" + }, + { + "bbox": [ + 337, + 224, + 345, + 234 + ], + "score": 0.49, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 222, + 505, + 237 + ], + "score": 1.0, + "content": ", we further foster the lifelong learning", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "idea by bringing the latent variable posterior induced by the student model closer to the respective", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "posterior induced by the teacher model. We enforce the latter constraint only over the synthetic", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "samples, ensuring that the previously learnt latent variable posteriors are preserved over the different", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 267, + 394, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 394, + 280 + ], + "score": 1.0, + "content": "models. In doing so, we alleviate the effect of catastrophic interference.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 200, + 506, + 280 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 505, + 350 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 298 + ], + "score": 1.0, + "content": "To achieve this, we complement the classical VAE objective (equation (2)) with a term minimizing", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 182, + 308 + ], + "score": 1.0, + "content": "the KL divergence", + "type": "text" + }, + { + "bbox": [ + 183, + 295, + 284, + 307 + ], + "score": 0.92, + "content": "K L [ \\partial _ { \\phi } ( z | \\hat { \\mathbf { x } } ) | | Q _ { \\Phi } ( z | \\hat { \\mathbf { x } } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "between the student’s and the teacher’s posteriors over", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 180, + 319 + ], + "score": 1.0, + "content": "the synthetic data", + "type": "text" + }, + { + "bbox": [ + 180, + 307, + 187, + 316 + ], + "score": 0.46, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 306, + 505, + 319 + ], + "score": 1.0, + "content": ". The teacher’s encoder model, which already has the accumulated knowledge", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "from the previous learning steps, is thus reused within the new student’s objective. Under certain", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 328, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 341 + ], + "score": 1.0, + "content": "mild assumptions, we show that this objective reparameterizes the student model’s posterior, while", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 339, + 439, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 439, + 351 + ], + "score": 1.0, + "content": "preserving the same learning objective as a standard VAE (appendix section 7.0.1).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 282, + 506, + 351 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 372, + 211, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 213, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 213, + 384 + ], + "score": 1.0, + "content": "4.2 LATENT VARIABLE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "A critical component of our model is the synthetic data generation by the teacher’s decoder", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 168, + 419 + ], + "score": 0.92, + "content": "\\hat { \\mathbf { x } } ~ \\sim ~ P _ { \\Theta } ( \\mathbf { x } | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 407, + 505, + 419 + ], + "score": 1.0, + "content": ". The synthetic samples need to be representative of all the previously observed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 432 + ], + "score": 1.0, + "content": "distributions in order to provide the student with ample information about the learning history.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "The teacher generates these synthetic samples by first sampling the latent variable from the prior", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 156, + 451 + ], + "score": 0.91, + "content": "\\hat { z } ~ \\sim ~ P ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 439, + 291, + 452 + ], + "score": 1.0, + "content": "followed by the decoding step", + "type": "text" + }, + { + "bbox": [ + 291, + 439, + 356, + 452 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\hat { \\mathbf { x } } \\sim \\mathcal { P } _ { \\Theta } ( \\mathbf { x } | \\hat { z } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 439, + 506, + 452 + ], + "score": 1.0, + "content": ". As we will describe shortly, the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 450, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 166, + 462 + ], + "score": 1.0, + "content": "latent variable", + "type": "text" + }, + { + "bbox": [ + 166, + 451, + 174, + 460 + ], + "score": 0.72, + "content": "\\hat { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 450, + 504, + 462 + ], + "score": 1.0, + "content": "has a categorical component which corresponds to all the past distributions. This", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "categorical component allows us to uniformly sample synthetic instances from all past distributions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 395, + 506, + 475 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 254, + 491 + ], + "score": 1.0, + "content": "A simple unimodal prior distribution", + "type": "text" + }, + { + "bbox": [ + 254, + 478, + 276, + 490 + ], + "score": 0.92, + "content": "P ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 478, + 505, + 491 + ], + "score": 1.0, + "content": ", such as the isotropic Gaussian typically used in classical", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "VAEs, results in an undersampling of the data points that are mapped to a posterior mean that is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 498, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 104, + 498, + 506, + 514 + ], + "score": 1.0, + "content": "further away from the prior mean. Visualizing the 2d latent posterior of MNIST in figure 1(b)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "allows us to get a better intuition of this problem. If for example the prior mean corresponds to a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "point in latent space between two disparate distributions, the sample generated will not correspond to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "a sample from the real distribution. 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The discrete component", + "type": "text" + }, + { + "bbox": [ + 492, + 596, + 504, + 605 + ], + "score": 0.83, + "content": "z _ { d }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 618 + ], + "score": 1.0, + "content": "shall summarise the most discriminative information about each of the true generating distributions", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 133, + 628 + ], + "score": 0.92, + "content": "P _ { i } ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 615, + 336, + 629 + ], + "score": 1.0, + "content": ". 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We specifically concentrate on the benefits our model brings in the lifelong learning setting", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "which is the main motivation of our work. We explain the settings of the individual experiments and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 419, + 277, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 277, + 432 + ], + "score": 1.0, + "content": "their focus in the following three sections.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "In all the experiments we use the notion of a distributional ‘interval’: the interval in which we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 288, + 461 + ], + "score": 1.0, + "content": "observe samples from a single distribution", + "type": "text" + }, + { + "bbox": [ + 289, + 447, + 316, + 459 + ], + "score": 0.92, + "content": "P _ { i } ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 446, + 506, + 461 + ], + "score": 1.0, + "content": "before the transition to the next distribution", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 455, + 507, + 474 + ], + "spans": [ + { + "bbox": [ + 107, + 458, + 141, + 471 + ], + "score": 0.93, + "content": "P _ { i + 1 } ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 455, + 507, + 474 + ], + "score": 1.0, + "content": "occurs. The length of the intervals is in principle random and we developed a heuristic", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "to generate these. We provide further details on this together with other technical details related to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 480, + 467, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 467, + 493 + ], + "score": 1.0, + "content": "the network implementation and training common for all the experiments in the appendix.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 504, + 330, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 331, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 331, + 518 + ], + "score": 1.0, + "content": "5.1 FASHION MNIST : SEQUENTIAL GENERATION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "In this experiment, we seek to establish the performance benefit that our augmented objective", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "formulation in section 4.3 brings into the learning in contrast to the simple ELBO objective 2. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "do so by training two models with identical student-teacher architectures as introduced in section 4,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "with one using the consistency and mutual information augmented objective (with consistency) and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "the other using the standard ELBO objective (without consistency). We also demonstrate the ability", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 581, + 464, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 464, + 592 + ], + "score": 1.0, + "content": "of our model to disambiguate distributional boundaries from the distributional variations.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 610 + ], + "score": 1.0, + "content": "We use Fashion MNIST Xiao et al. (2017) 2 to simulate our sequential learning setting. We treat", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "each object as a different distribution and present the model with samples drawn from a single", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "score": 1.0, + "content": "distribution at a time. We sequentially progress over the ten available distributions. When a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "distribution transition occurs (new object) we signal the model, make the latest student the new", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 641, + 284, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 284, + 652 + ], + "score": 1.0, + "content": "teacher and instantiate a new student model.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 658, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 670 + ], + "score": 1.0, + "content": "We quantify the performance of the generative models by computing the ELBO over the standard", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "Fashion MNIST test set after every distributional transition. The test set contains objects from all", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "score": 1.0, + "content": "of the individual distributions. We run this procedure ten times and report the average test ELBO", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "score": 1.0, + "content": "over the ten repetitions in figure 3(c). We see that around the 3rd interval (the 3rd distributional", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 710, + 335, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 709, + 336, + 722 + ], + "spans": [ + { + "bbox": [ + 119, + 709, + 336, + 722 + ], + "score": 1.0, + "content": "1A similar idea is leveraged in InfoGAN Chen et al. 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To prevent the VAE’s encoder from using primarily the continuous representation while", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "disregarding the discrete one we further complement the learning objective by a term maximising the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 379, + 128 + ], + "score": 1.0, + "content": "mutual information between the discrete representation and the data", + "type": "text" + }, + { + "bbox": [ + 379, + 115, + 501, + 127 + ], + "score": 0.91, + "content": "I ( z _ { d } ; \\mathbf { x } ) = H ( z _ { d } ) - H ( z _ { d } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 115, + 505, + 128 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 125, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 107, + 126, + 135, + 138 + ], + "score": 0.92, + "content": "H ( z _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 125, + 299, + 140 + ], + "score": 1.0, + "content": "is used to denote the marginal entropy of", + "type": "text" + }, + { + "bbox": [ + 300, + 127, + 311, + 137 + ], + "score": 0.87, + "content": "z _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 125, + 329, + 140 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 329, + 126, + 366, + 138 + ], + "score": 0.93, + "content": "H ( z _ { d } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 125, + 507, + 140 + ], + "score": 1.0, + "content": "denotes the conditional entropy of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 158, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 118, + 149 + ], + "score": 0.84, + "content": "z _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 137, + 143, + 149 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 143, + 138, + 151, + 147 + ], + "score": 0.57, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 137, + 158, + 149 + ], + "score": 1.0, + "content": ". 1", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 507, + 149 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 162, + 229, + 174 + ], + "lines": [ + { + "bbox": [ + 105, + 161, + 230, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 230, + 175 + ], + "score": 1.0, + "content": "4.3 LEARNING OBJECTIVE", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 503, + 216 + ], + "lines": [ + { + "bbox": [ + 106, + 183, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 505, + 195 + ], + "score": 1.0, + "content": "The final learning objective for each of the student models is the maximization of the ELBO from", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "equation (2), augmented by the negative of the cross-model consistency term introduced in section", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 351, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 351, + 217 + ], + "score": 1.0, + "content": "4.1 and the mutual information term proposed in section 4.2.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 183, + 505, + 217 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 108, + 221, + 499, + 236 + ], + "lines": [ + { + "bbox": [ + 108, + 221, + 499, + 236 + ], + "spans": [ + { + "bbox": [ + 108, + 221, + 499, + 236 + ], + "score": 0.44, + "content": "\\begin{array} { r } { \\mathbb { E } _ { Q _ { \\phi } } [ l o g P _ { \\theta } ( \\mathbf { x } | z ) ] - K L [ Q _ { \\phi } ( z | \\mathbf { x } ) | | P ( z ) ] - \\mathbb { 1 } ( \\omega = 0 ) K L [ Q _ { \\phi } ( z _ { d } | \\mathbf { x } ) | | Q _ { \\Phi } ( z _ { d } | \\mathbf { x } ) ] + \\lambda I ( z _ { d } ; 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it makes sure that the consistency regularizer is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 393, + 325 + ], + "score": 1.0, + "content": "applied only over the synthetic samples generated by the teacher. The", + "type": "text" + }, + { + "bbox": [ + 393, + 312, + 400, + 322 + ], + "score": 0.73, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "hyper-parameter controls", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "the importance of the mutual information regularizer. We present the analytical evaluation of the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 334, + 307, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 307, + 346 + ], + "score": 1.0, + "content": "consistency regularizer in appendix section 7.0.1.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 268, + 506, + 346 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 361, + 200, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 201, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 201, + 376 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "We conducted a set of experiments to explore the behaviour and properties of the method we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "score": 1.0, + "content": "propose. We specifically concentrate on the benefits our model brings in the lifelong learning setting", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "which is the main motivation of our work. We explain the settings of the individual experiments and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 419, + 277, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 277, + 432 + ], + "score": 1.0, + "content": "their focus in the following three sections.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 386, + 505, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "In all the experiments we use the notion of a distributional ‘interval’: the interval in which we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 288, + 461 + ], + "score": 1.0, + "content": "observe samples from a single distribution", + "type": "text" + }, + { + "bbox": [ + 289, + 447, + 316, + 459 + ], + "score": 0.92, + "content": "P _ { i } ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 446, + 506, + 461 + ], + "score": 1.0, + "content": "before the transition to the next distribution", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 455, + 507, + 474 + ], + "spans": [ + { + "bbox": [ + 107, + 458, + 141, + 471 + ], + "score": 0.93, + "content": "P _ { i + 1 } ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 455, + 507, + 474 + ], + "score": 1.0, + "content": "occurs. The length of the intervals is in principle random and we developed a heuristic", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "to generate these. We provide further details on this together with other technical details related to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 480, + 467, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 467, + 493 + ], + "score": 1.0, + "content": "the network implementation and training common for all the experiments in the appendix.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 436, + 507, + 493 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 504, + 330, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 331, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 331, + 518 + ], + "score": 1.0, + "content": "5.1 FASHION MNIST : SEQUENTIAL GENERATION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "In this experiment, we seek to establish the performance benefit that our augmented objective", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "formulation in section 4.3 brings into the learning in contrast to the simple ELBO objective 2. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "do so by training two models with identical student-teacher architectures as introduced in section 4,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "with one using the consistency and mutual information augmented objective (with consistency) and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "the other using the standard ELBO objective (without consistency). We also demonstrate the ability", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 581, + 464, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 464, + 592 + ], + "score": 1.0, + "content": "of our model to disambiguate distributional boundaries from the distributional variations.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 524, + 506, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 610 + ], + "score": 1.0, + "content": "We use Fashion MNIST Xiao et al. (2017) 2 to simulate our sequential learning setting. We treat", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "each object as a different distribution and present the model with samples drawn from a single", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "score": 1.0, + "content": "distribution at a time. We sequentially progress over the ten available distributions. When a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "distribution transition occurs (new object) we signal the model, make the latest student the new", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 641, + 284, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 284, + 652 + ], + "score": 1.0, + "content": "teacher and instantiate a new student model.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 596, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 658, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 670 + ], + "score": 1.0, + "content": "We quantify the performance of the generative models by computing the ELBO over the standard", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "Fashion MNIST test set after every distributional transition. The test set contains objects from all", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "score": 1.0, + "content": "of the individual distributions. We run this procedure ten times and report the average test ELBO", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "score": 1.0, + "content": "over the ten repetitions in figure 3(c). We see that around the 3rd interval (the 3rd distributional", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 240, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 456, + 254 + ], + "score": 1.0, + "content": "transition), the negative ELBO of the with consistency model is systematically below", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 456, + 241, + 479, + 252 + ], + "score": 0.72, + "content": "\\sim 2 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 479, + 240, + 506, + 254 + ], + "score": 1.0, + "content": "nats )", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "score": 1.0, + "content": "that of the without consistency model. This confirms the benefits of our new objective formulation", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 261, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 276 + ], + "score": 1.0, + "content": "for reducing the effects of the catastrophic interference, a crucial property in our lifelong learning", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 274, + 504, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 504, + 285 + ], + "score": 1.0, + "content": "setting. In the same figure we also plot the ELBO of the baseline batch VAE. The batch VAE will", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "always outperform our model because it has simultaneous access to all of the distributions during", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 295, + 143, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 143, + 310 + ], + "score": 1.0, + "content": "training.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 657, + 506, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 79, + 502, + 193 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 79, + 502, + 193 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 79, + 502, + 193 + ], + "spans": [ + { + "bbox": [ + 107, + 79, + 502, + 193 + ], + "score": 0.974, + "type": "image", + "image_path": "de756f9933d2c746151e9331f505d0ed4cb3706f7841c6aa359e35e7728162f2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 79, + 502, + 117.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 117.0, + 502, + 155.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 155.0, + 502, + 193.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 202, + 504, + 235 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "score": 1.0, + "content": "Figure 3: (a) Generation with consistency regularizer. (b) Generation without consistency", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "regularizer. (c) Average negative ELBO over ten trials (each) for the ten distributions within Fashion", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 224, + 142, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 142, + 236 + ], + "score": 1.0, + "content": "MNIST.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 240, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 456, + 254 + ], + "score": 1.0, + "content": "transition), the negative ELBO of the with consistency model is systematically below", + "type": "text" + }, + { + "bbox": [ + 456, + 241, + 479, + 252 + ], + "score": 0.72, + "content": "\\sim 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 240, + 506, + 254 + ], + "score": 1.0, + "content": "nats )", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "score": 1.0, + "content": "that of the without consistency model. This confirms the benefits of our new objective formulation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 261, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 276 + ], + "score": 1.0, + "content": "for reducing the effects of the catastrophic interference, a crucial property in our lifelong learning", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 274, + 504, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 504, + 285 + ], + "score": 1.0, + "content": "setting. In the same figure we also plot the ELBO of the baseline batch VAE. The batch VAE will", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "always outperform our model because it has simultaneous access to all of the distributions during", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 295, + 143, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 143, + 310 + ], + "score": 1.0, + "content": "training.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 506, + 325 + ], + "score": 1.0, + "content": "After observing and training over all ten distributions we generate samples from the final students of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 355, + 336 + ], + "score": 1.0, + "content": "the two models. We do this by fixing the discrete distribution", + "type": "text" + }, + { + "bbox": [ + 355, + 325, + 367, + 334 + ], + "score": 0.85, + "content": "z _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "to one-hot vectors over the whole", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 392, + 347 + ], + "score": 1.0, + "content": "categorical distribution, while randomly sampling the continuous prior", + "type": "text" + }, + { + "bbox": [ + 393, + 334, + 450, + 347 + ], + "score": 0.93, + "content": " { \\boldsymbol { z } } _ { c } \\sim \\mathcal { N } ( 0 , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 335, + 506, + 347 + ], + "score": 1.0, + "content": ". We contrast", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 345, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 505, + 359 + ], + "score": 1.0, + "content": "samples generated from the model with consistency (figure 3(a)) to the model without consistency", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "(figure 3(b)). Our model learns to separate ’style’ from the distributional boundaries. For example,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "in the last row of our with consistency model, we observe the various styles of shoes. The without", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 379, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 391 + ], + "score": 1.0, + "content": "consistency model mixes the distributions randomly. This illustrates the benefits that our augmented", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 389, + 485, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 485, + 402 + ], + "score": 1.0, + "content": "objective has for achieving consistent sampling from the individual distributional components.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 413, + 407, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 409, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 207, + 426 + ], + "score": 1.0, + "content": "5.2 ROTATED MNIST", + "type": "text" + }, + { + "bbox": [ + 208, + 415, + 214, + 423 + ], + "score": 0.3, + "content": ":", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 412, + 409, + 426 + ], + "score": 1.0, + "content": "LONG TERM DISTRIBUTION ACCUMULATION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 104, + 432, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 506, + 447 + ], + "score": 1.0, + "content": "In this experiment we dig deeper into the benefits our objective formulation brings for the lifelong", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "learning setting. We expose the models to a much larger number of distributions and we explore", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "how our augmented objective from 4.3 helps in preserving the previously learned knowledge. As", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "in section 5.1, we compare models with and without consistency with identical teacher-student", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "architectures. We measure the ability of the models to recall the previously learned information", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "by looking at the consistency between the posterior of the student and the teacher models over the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 501, + 156, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 156, + 511 + ], + "score": 1.0, + "content": "test data set", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 512, + 438, + 526 + ], + "lines": [ + { + "bbox": [ + 200, + 512, + 438, + 526 + ], + "spans": [ + { + "bbox": [ + 200, + 512, + 438, + 526 + ], + "score": 0.85, + "content": "\\mathrm { \\sf t e n c y : } \\quad \\# \\{ k : Q _ { \\Phi } ( z _ { d } | { \\bf x } _ { k } ) = = Q _ { \\phi } ( z _ { d } | { \\bf x } _ { k } ) , { \\bf x } _ { k } \\in { \\bf X } _ { t e s t } \\} \\mathrm { ~ . ~ }", + "type": "interline_equation", + "image_path": "e1dc5338bebb0752930464ee0f9a698300af1135479b10028325c92cedcdf9b2.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 200, + 512, + 438, + 526 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 487, + 546 + ], + "score": 1.0, + "content": "We use the MNIST dataset in which we rotate each of the original digit samples by angles", + "type": "text" + }, + { + "bbox": [ + 488, + 535, + 505, + 544 + ], + "score": 0.83, + "content": "\\nu =", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 107, + 545, + 125, + 555 + ], + "score": 0.78, + "content": "[ 3 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 544, + 130, + 557 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 131, + 545, + 147, + 555 + ], + "score": 0.79, + "content": "7 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 544, + 153, + 557 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 153, + 545, + 173, + 555 + ], + "score": 0.8, + "content": "1 3 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 544, + 179, + 557 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 179, + 545, + 200, + 555 + ], + "score": 0.78, + "content": "2 0 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 544, + 205, + 557 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 205, + 545, + 228, + 555 + ], + "score": 0.73, + "content": "2 5 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "]. 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Within each distributional interval, we sample the data by first sampling", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 565, + 504, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 496, + 579 + ], + "score": 1.0, + "content": "(uniformly with replacement) one of the 70 distributions and then sampling the data instances", + "type": "text" + }, + { + "bbox": [ + 496, + 568, + 504, + 576 + ], + "score": 0.39, + "content": "\\mathbf { x }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 577, + 228, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 228, + 590 + ], + "score": 1.0, + "content": "from the selected distribution.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "score": 1.0, + "content": "Figure 4(b) compares the consistency results of the two tested models throughout the learning", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "process. Our model with the augmented objective clearly outperforms the model that uses the simple", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "score": 1.0, + "content": "ELBO objective. This confirms the usefulness of the additional terms in our objective for preserving", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "the previously learned knowledge in accordance with the lifelong learning paradigms. In addition,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "similarly as in experiment 5.1, figure 4(a) documents that the model with the augmented objective", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "(thanks to reducing the effects of the catastrophic interference) achieves lower negative test ELBO", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 660, + 378, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 331, + 673 + ], + "score": 1.0, + "content": "systematically over the much longer course of learning (", + "type": "text" + }, + { + "bbox": [ + 331, + 660, + 353, + 671 + ], + "score": 0.63, + "content": "\\sim 3 0", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 660, + 378, + 673 + ], + "score": 1.0, + "content": "nats).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 675, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 690 + ], + "score": 1.0, + "content": "We also visualise in figure 4(c) how the accumulation of knowledge speeds up the learning process.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "score": 1.0, + "content": "For each distributional interval we plot the norms of the model gradients across the learning", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 697, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 506, + 713 + ], + "score": 1.0, + "content": "iterations. We observe that for later distributional intervals the curves become steeper much quicker,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 707, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 506, + 724 + ], + "score": 1.0, + "content": "reducing the gradients and reaching (lower) steady states much faster then in the early learning", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 720, + 495, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 495, + 734 + ], + "score": 1.0, + "content": "stages. This suggests that the latter models are able to learn quicker in our proposed architecture.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 79, + 502, + 193 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 79, + 502, + 193 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 79, + 502, + 193 + ], + "spans": [ + { + "bbox": [ + 107, + 79, + 502, + 193 + ], + "score": 0.974, + "type": "image", + "image_path": "de756f9933d2c746151e9331f505d0ed4cb3706f7841c6aa359e35e7728162f2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 79, + 502, + 117.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 117.0, + 502, + 155.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 155.0, + 502, + 193.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 202, + 504, + 235 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "score": 1.0, + "content": "Figure 3: (a) Generation with consistency regularizer. 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(c) Average negative ELBO over ten trials (each) for the ten distributions within Fashion", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 224, + 142, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 142, + 236 + ], + "score": 1.0, + "content": "MNIST.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 505, + 307 + ], + "lines": [], + "index": 8.5, + "bbox_fs": [ + 105, + 240, + 506, + 310 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 506, + 325 + ], + "score": 1.0, + "content": "After observing and training over all ten distributions we generate samples from the final students of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 355, + 336 + ], + "score": 1.0, + "content": "the two models. 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We contrast", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 345, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 505, + 359 + ], + "score": 1.0, + "content": "samples generated from the model with consistency (figure 3(a)) to the model without consistency", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "(figure 3(b)). Our model learns to separate ’style’ from the distributional boundaries. For example,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "in the last row of our with consistency model, we observe the various styles of shoes. The without", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 379, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 391 + ], + "score": 1.0, + "content": "consistency model mixes the distributions randomly. This illustrates the benefits that our augmented", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 389, + 485, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 485, + 402 + ], + "score": 1.0, + "content": "objective has for achieving consistent sampling from the individual distributional components.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 313, + 506, + 402 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 413, + 407, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 409, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 207, + 426 + ], + "score": 1.0, + "content": "5.2 ROTATED MNIST", + "type": "text" + }, + { + "bbox": [ + 208, + 415, + 214, + 423 + ], + "score": 0.3, + "content": ":", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 412, + 409, + 426 + ], + "score": 1.0, + "content": "LONG TERM DISTRIBUTION ACCUMULATION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 104, + 432, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 506, + 447 + ], + "score": 1.0, + "content": "In this experiment we dig deeper into the benefits our objective formulation brings for the lifelong", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "learning setting. We expose the models to a much larger number of distributions and we explore", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "how our augmented objective from 4.3 helps in preserving the previously learned knowledge. As", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "in section 5.1, we compare models with and without consistency with identical teacher-student", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "architectures. We measure the ability of the models to recall the previously learned information", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "by looking at the consistency between the posterior of the student and the teacher models over the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 501, + 156, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 156, + 511 + ], + "score": 1.0, + "content": "test data set", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 432, + 506, + 511 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 512, + 438, + 526 + ], + "lines": [ + { + "bbox": [ + 200, + 512, + 438, + 526 + ], + "spans": [ + { + "bbox": [ + 200, + 512, + 438, + 526 + ], + "score": 0.85, + "content": "\\mathrm { \\sf t e n c y : } \\quad \\# \\{ k : Q _ { \\Phi } ( z _ { d } | { \\bf x } _ { k } ) = = Q _ { \\phi } ( z _ { d } | { \\bf x } _ { k } ) , { \\bf x } _ { k } \\in { \\bf X } _ { t e s t } \\} \\mathrm { ~ . ~ }", + "type": "interline_equation", + "image_path": "e1dc5338bebb0752930464ee0f9a698300af1135479b10028325c92cedcdf9b2.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 200, + 512, + 438, + 526 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 487, + 546 + ], + "score": 1.0, + "content": "We use the MNIST dataset in which we rotate each of the original digit samples by angles", + "type": "text" + }, + { + "bbox": [ + 488, + 535, + 505, + 544 + ], + "score": 0.83, + "content": "\\nu =", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 107, + 545, + 125, + 555 + ], + "score": 0.78, + "content": "[ 3 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 544, + 130, + 557 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 131, + 545, + 147, + 555 + ], + "score": 0.79, + "content": "7 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 544, + 153, + 557 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 153, + 545, + 173, + 555 + ], + "score": 0.8, + "content": "1 3 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 544, + 179, + 557 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 179, + 545, + 200, + 555 + ], + "score": 0.78, + "content": "2 0 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 544, + 205, + 557 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 205, + 545, + 228, + 555 + ], + "score": 0.73, + "content": "2 5 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "]. 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Our model with the augmented objective clearly outperforms the model that uses the simple", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 630 + ], + "score": 1.0, + "content": "ELBO objective. This confirms the usefulness of the additional terms in our objective for preserving", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "the previously learned knowledge in accordance with the lifelong learning paradigms. In addition,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "similarly as in experiment 5.1, figure 4(a) documents that the model with the augmented objective", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "(thanks to reducing the effects of the catastrophic interference) achieves lower negative test ELBO", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 660, + 378, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 331, + 673 + ], + "score": 1.0, + "content": "systematically over the much longer course of learning (", + "type": "text" + }, + { + "bbox": [ + 331, + 660, + 353, + 671 + ], + "score": 0.63, + "content": "\\sim 3 0", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 660, + 378, + 673 + ], + "score": 1.0, + "content": "nats).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 594, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 675, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 690 + ], + "score": 1.0, + "content": "We also visualise in figure 4(c) how the accumulation of knowledge speeds up the learning process.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "score": 1.0, + "content": "For each distributional interval we plot the norms of the model gradients across the learning", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 697, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 506, + 713 + ], + "score": 1.0, + "content": "iterations. We observe that for later distributional intervals the curves become steeper much quicker,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 707, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 506, + 724 + ], + "score": 1.0, + "content": "reducing the gradients and reaching (lower) steady states much faster then in the early learning", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 720, + 495, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 495, + 734 + ], + "score": 1.0, + "content": "stages. This suggests that the latter models are able to learn quicker in our proposed architecture.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 675, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 82, + 518, + 196 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 82, + 518, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 82, + 518, + 196 + ], + "spans": [ + { + "bbox": [ + 108, + 82, + 518, + 196 + ], + "score": 0.76, + "type": "image", + "image_path": "7e68be6459eb4fe191ce9481bdea3906681f59cc2a823e02e61cb7449cfb295e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 82, + 518, + 120.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 120.0, + 518, + 158.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 158.0, + 518, + 196.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 207, + 506, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 206, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 220 + ], + "score": 1.0, + "content": "Figure 4: (a) Negative test ELBO over the learning history; (b) consistency between the teacher", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "and student posteriors across the test data samples normalized by the test data set size; (c) speed of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 229, + 402, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 402, + 241 + ], + "score": 1.0, + "content": "learning convergence across distributional intervals. Best viewed in color.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 247, + 210, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 212, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 212, + 260 + ], + "score": 1.0, + "content": "5.3 SVHN TO MNIST", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 313 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "In this experiment we explore the ability of our model to retain and transfer knowledge across", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "completely different datasets. We use MNIST and SVHN Netzer et al. (2011) to demonstrate this.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 289, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 395, + 304 + ], + "score": 1.0, + "content": "We treat all samples from SVHN as being generated by one distribution", + "type": "text" + }, + { + "bbox": [ + 395, + 291, + 422, + 303 + ], + "score": 0.92, + "content": "P _ { 1 } ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 289, + 506, + 304 + ], + "score": 1.0, + "content": "and all the MNIST3", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 302, + 451, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 286, + 315 + ], + "score": 1.0, + "content": "samples as generated by another distribution", + "type": "text" + }, + { + "bbox": [ + 286, + 302, + 313, + 314 + ], + "score": 0.93, + "content": "P _ { 2 } ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 302, + 451, + 315 + ], + "score": 1.0, + "content": "(irrespective of the specific digit).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "We first train a student model (standard VAE) over the entire SVHN data set. Once done, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "freeze the parameters of the encoder and the decoder and transfer the model into the teacher state", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 108, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 108, + 340, + 181, + 352 + ], + "score": 0.9, + "content": "( \\phi \\to \\Phi , \\theta \\to \\Theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 340, + 506, + 352 + ], + "score": 1.0, + "content": ". We then use this teacher to aid the learning of the new student over the mix of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 351, + 411, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 296, + 363 + ], + "score": 1.0, + "content": "the teacher-generated synthetic SVHN samples", + "type": "text" + }, + { + "bbox": [ + 297, + 352, + 304, + 361 + ], + "score": 0.41, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 351, + 411, + 363 + ], + "score": 1.0, + "content": "and the true MNIST data.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "image", + "bbox": [ + 106, + 375, + 505, + 504 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 375, + 505, + 504 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 504 + ], + "score": 0.97, + "type": "image", + "image_path": "5b74ad0e049511c04d3e0276bca9b4fd870fefca948ab30f0adbba4860afe937.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 106, + 375, + 505, + 418.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 106, + 418.0, + 505, + 461.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 106, + 461.0, + 505, + 504.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 513, + 506, + 547 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "Figure 5: (a) Reconstructions of test samples from SVHN[left] and MNIST[right]; (b) Decoded", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 141, + 538 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 141, + 524, + 215, + 536 + ], + "score": 0.9, + "content": "\\hat { \\mathbf { x } } \\sim P _ { \\pmb { \\theta } } ( \\mathbf { x } | z _ { d } , z _ { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 524, + 345, + 538 + ], + "score": 1.0, + "content": "based on linear interpolation of", + "type": "text" + }, + { + "bbox": [ + 346, + 524, + 383, + 536 + ], + "score": 0.92, + "content": "z _ { c } \\in \\mathcal { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 524, + 405, + 538 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 405, + 525, + 451, + 537 + ], + "score": 0.92, + "content": "z _ { d } = [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "; (c) Same as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 535, + 205, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 155, + 548 + ], + "score": 1.0, + "content": "(b) but with", + "type": "text" + }, + { + "bbox": [ + 156, + 536, + 200, + 547 + ], + "score": 0.92, + "content": "z _ { d } = [ 1 , 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 535, + 205, + 548 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 560, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "We use the final student model to reconstruct samples from the two datasets by passing them through", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 262, + 584 + ], + "score": 1.0, + "content": "the learned encoding/decoding flow:", + "type": "text" + }, + { + "bbox": [ + 262, + 571, + 479, + 583 + ], + "score": 0.54, + "content": "\\mathbf { x } \\ \\sim \\ P _ { i } ^ { * } \\ \\mathrm { ( \\mathbf { x } ) } \\ \\ z \\ \\sim \\ Q _ { \\phi } ( z | \\mathbf { x } ) \\ \\ \\hat { \\mathbf { x } } \\ \\sim \\ P _ { \\theta } ( \\mathbf { x } | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 570, + 506, + 584 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 582, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 262, + 594 + ], + "score": 1.0, + "content": "visualise examples of the true inputs", + "type": "text" + }, + { + "bbox": [ + 262, + 584, + 270, + 592 + ], + "score": 0.47, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 582, + 414, + 594 + ], + "score": 1.0, + "content": "and the respective reconstructions", + "type": "text" + }, + { + "bbox": [ + 415, + 583, + 423, + 592 + ], + "score": 0.56, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 582, + 506, + 594 + ], + "score": 1.0, + "content": "in figure 5(a). We", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 593, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 605 + ], + "score": 1.0, + "content": "see that even though the only true data the final model received for training were from MNIST, it", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 604, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 616 + ], + "score": 1.0, + "content": "can still reconstruct SVHN data. This confirms the ability of our architecture to transition between", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "score": 1.0, + "content": "complex distributions while still preserving the knowledge learned from the previously observed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 625, + 162, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 162, + 638 + ], + "score": 1.0, + "content": "distributions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 642, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "Finally, in figure 5(b) and 5(c) we illustrate the data generated from an interpolation of a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "2-dimensional continuous latent space. For this we specifically trained the models with the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 663, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 214, + 677 + ], + "score": 1.0, + "content": "continuous latent variable", + "type": "text" + }, + { + "bbox": [ + 215, + 664, + 255, + 676 + ], + "score": 0.91, + "content": "z _ { c } \\in \\mathcal { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 663, + 480, + 677 + ], + "score": 1.0, + "content": ". 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We then decode these to obtain the samples", + "type": "text" + }, + { + "bbox": [ + 313, + 686, + 386, + 699 + ], + "score": 0.93, + "content": "\\hat { \\mathbf { x } } \\sim P _ { \\pmb { \\theta } } ( \\mathbf { x } | \\pmb { z } _ { d } , \\pmb { z } _ { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 686, + 505, + 700 + ], + "score": 1.0, + "content": ". The model learns a common", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 711, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 441, + 723 + ], + "score": 1.0, + "content": "3In order to work over both of these datasets we convert MNIST to RGB and resize it to", + "type": "text" + }, + { + "bbox": [ + 441, + 712, + 465, + 721 + ], + "score": 0.67, + "content": "3 2 \\mathrm { x } 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "to make it", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 258, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 258, + 731 + ], + "score": 1.0, + "content": "consistent with the dimensions of SVHN.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 82, + 518, + 196 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 82, + 518, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 82, + 518, + 196 + ], + "spans": [ + { + "bbox": [ + 108, + 82, + 518, + 196 + ], + "score": 0.76, + "type": "image", + "image_path": "7e68be6459eb4fe191ce9481bdea3906681f59cc2a823e02e61cb7449cfb295e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 82, + 518, + 120.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 120.0, + 518, + 158.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 158.0, + 518, + 196.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 207, + 506, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 206, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 220 + ], + "score": 1.0, + "content": "Figure 4: (a) Negative test ELBO over the learning history; (b) consistency between the teacher", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "and student posteriors across the test data samples normalized by the test data set size; (c) speed of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 229, + 402, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 402, + 241 + ], + "score": 1.0, + "content": "learning convergence across distributional intervals. Best viewed in color.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 247, + 210, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 212, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 212, + 260 + ], + "score": 1.0, + "content": "5.3 SVHN TO MNIST", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 313 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "In this experiment we explore the ability of our model to retain and transfer knowledge across", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "completely different datasets. We use MNIST and SVHN Netzer et al. (2011) to demonstrate this.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 289, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 395, + 304 + ], + "score": 1.0, + "content": "We treat all samples from SVHN as being generated by one distribution", + "type": "text" + }, + { + "bbox": [ + 395, + 291, + 422, + 303 + ], + "score": 0.92, + "content": "P _ { 1 } ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 289, + 506, + 304 + ], + "score": 1.0, + "content": "and all the MNIST3", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 302, + 451, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 286, + 315 + ], + "score": 1.0, + "content": "samples as generated by another distribution", + "type": "text" + }, + { + "bbox": [ + 286, + 302, + 313, + 314 + ], + "score": 0.93, + "content": "P _ { 2 } ^ { * } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 302, + 451, + 315 + ], + "score": 1.0, + "content": "(irrespective of the specific digit).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 269, + 506, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "We first train a student model (standard VAE) over the entire SVHN data set. Once done, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "freeze the parameters of the encoder and the decoder and transfer the model into the teacher state", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 108, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 108, + 340, + 181, + 352 + ], + "score": 0.9, + "content": "( \\phi \\to \\Phi , \\theta \\to \\Theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 340, + 506, + 352 + ], + "score": 1.0, + "content": ". We then use this teacher to aid the learning of the new student over the mix of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 351, + 411, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 296, + 363 + ], + "score": 1.0, + "content": "the teacher-generated synthetic SVHN samples", + "type": "text" + }, + { + "bbox": [ + 297, + 352, + 304, + 361 + ], + "score": 0.41, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 351, + 411, + 363 + ], + "score": 1.0, + "content": "and the true MNIST data.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 318, + 506, + 363 + ] + }, + { + "type": "image", + "bbox": [ + 106, + 375, + 505, + 504 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 375, + 505, + 504 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 504 + ], + "score": 0.97, + "type": "image", + "image_path": "5b74ad0e049511c04d3e0276bca9b4fd870fefca948ab30f0adbba4860afe937.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 106, + 375, + 505, + 418.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 106, + 418.0, + 505, + 461.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 106, + 461.0, + 505, + 504.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 513, + 506, + 547 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "Figure 5: (a) Reconstructions of test samples from SVHN[left] and MNIST[right]; (b) Decoded", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 141, + 538 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 141, + 524, + 215, + 536 + ], + "score": 0.9, + "content": "\\hat { \\mathbf { x } } \\sim P _ { \\pmb { \\theta } } ( \\mathbf { x } | z _ { d } , z _ { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 524, + 345, + 538 + ], + "score": 1.0, + "content": "based on linear interpolation of", + "type": "text" + }, + { + "bbox": [ + 346, + 524, + 383, + 536 + ], + "score": 0.92, + "content": "z _ { c } \\in \\mathcal { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 524, + 405, + 538 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 405, + 525, + 451, + 537 + ], + "score": 0.92, + "content": "z _ { d } = [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "; (c) Same as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 535, + 205, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 155, + 548 + ], + "score": 1.0, + "content": "(b) but with", + "type": "text" + }, + { + "bbox": [ + 156, + 536, + 200, + 547 + ], + "score": 0.92, + "content": "z _ { d } = [ 1 , 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 535, + 205, + 548 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 560, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "We use the final student model to reconstruct samples from the two datasets by passing them through", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 262, + 584 + ], + "score": 1.0, + "content": "the learned encoding/decoding flow:", + "type": "text" + }, + { + "bbox": [ + 262, + 571, + 479, + 583 + ], + "score": 0.54, + "content": "\\mathbf { x } \\ \\sim \\ P _ { i } ^ { * } \\ \\mathrm { ( \\mathbf { x } ) } \\ \\ z \\ \\sim \\ Q _ { \\phi } ( z | \\mathbf { x } ) \\ \\ \\hat { \\mathbf { x } } \\ \\sim \\ P _ { \\theta } ( \\mathbf { x } | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 570, + 506, + 584 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 582, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 262, + 594 + ], + "score": 1.0, + "content": "visualise examples of the true inputs", + "type": "text" + }, + { + "bbox": [ + 262, + 584, + 270, + 592 + ], + "score": 0.47, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 582, + 414, + 594 + ], + "score": 1.0, + "content": "and the respective reconstructions", + "type": "text" + }, + { + "bbox": [ + 415, + 583, + 423, + 592 + ], + "score": 0.56, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 582, + 506, + 594 + ], + "score": 1.0, + "content": "in figure 5(a). We", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 593, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 605 + ], + "score": 1.0, + "content": "see that even though the only true data the final model received for training were from MNIST, it", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 604, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 616 + ], + "score": 1.0, + "content": "can still reconstruct SVHN data. This confirms the ability of our architecture to transition between", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "score": 1.0, + "content": "complex distributions while still preserving the knowledge learned from the previously observed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 625, + 162, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 162, + 638 + ], + "score": 1.0, + "content": "distributions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 560, + 506, + 638 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 642, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "Finally, in figure 5(b) and 5(c) we illustrate the data generated from an interpolation of a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "2-dimensional continuous latent space. For this we specifically trained the models with the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 663, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 214, + 677 + ], + "score": 1.0, + "content": "continuous latent variable", + "type": "text" + }, + { + "bbox": [ + 215, + 664, + 255, + 676 + ], + "score": 0.91, + "content": "z _ { c } \\in \\mathcal { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 663, + 480, + 677 + ], + "score": 1.0, + "content": ". To generate the data, we fix the discrete categorical", + "type": "text" + }, + { + "bbox": [ + 481, + 666, + 493, + 676 + ], + "score": 0.85, + "content": "z _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 663, + 506, + 677 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 674, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 215, + 690 + ], + "score": 1.0, + "content": "one of the possible values", + "type": "text" + }, + { + "bbox": [ + 215, + 676, + 271, + 688 + ], + "score": 0.91, + "content": "\\{ [ 0 , 1 ] , [ 1 , 0 ] \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 674, + 431, + 690 + ], + "score": 1.0, + "content": "and linearly interpolate the continuous", + "type": "text" + }, + { + "bbox": [ + 432, + 677, + 443, + 687 + ], + "score": 0.85, + "content": "z _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 674, + 506, + 690 + ], + "score": 1.0, + "content": "over the range", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 686, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 686, + 135, + 699 + ], + "score": 0.82, + "content": "[ - 3 , 3 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 686, + 313, + 700 + ], + "score": 1.0, + "content": ". We then decode these to obtain the samples", + "type": "text" + }, + { + "bbox": [ + 313, + 686, + 386, + 699 + ], + "score": 0.93, + "content": "\\hat { \\mathbf { x } } \\sim P _ { \\pmb { \\theta } } ( \\mathbf { x } | \\pmb { z } _ { d } , \\pmb { z } _ { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 686, + 505, + 700 + ], + "score": 1.0, + "content": ". The model learns a common", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "continuous structure for the two distributions which can be followed by observing the development", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 437, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 437, + 105 + ], + "score": 1.0, + "content": "in the generated samples from top left to bottom right on both figure 5(b) and 5(c).", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 642, + 506, + 700 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "continuous structure for the two distributions which can be followed by observing the development", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 437, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 437, + 105 + ], + "score": 1.0, + "content": "in the generated samples from top left to bottom right on both figure 5(b) and 5(c).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 107, + 120, + 195, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 197, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 197, + 136 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "score": 1.0, + "content": "In this work we propose a novel method for learning generative models over streaming data", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 156, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 506, + 169 + ], + "score": 1.0, + "content": "following the lifelong learning principles. The principal assumption for the data is that they are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 167, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 506, + 180 + ], + "score": 1.0, + "content": "generated by multiple distributions and presented to the learner in a sequential manner (a set of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "observations from a single distribution followed by a distributional transition). A key limitation for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "the learning is that the method can only access data generated by the current distribution and has no", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 200, + 393, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 393, + 213 + ], + "score": 1.0, + "content": "access to any of the data generated by any of the previous distributions.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 505, + 283 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "The proposed method is based on a dual student-teacher architecture where the teacher’s role is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 240 + ], + "score": 1.0, + "content": "to preserve the past knowledge and aid the student in future learning. We argue for and augment", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 237, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 252 + ], + "score": 1.0, + "content": "the standard VAE’s ELBO objective by terms helping the teacher-student knowledge transfer. We", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "score": 1.0, + "content": "demonstrate on a series of experiments the benefits this augmented objective brings in the lifelong", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 275 + ], + "score": 1.0, + "content": "learning settings by supporting the retention of previously learned knowledge (models) and limiting", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 288, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 288, + 285 + ], + "score": 1.0, + "content": "the usual effects of catastrophic interference.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 301 + ], + "score": 1.0, + "content": "In our future work we will explore the possibilities to extend our architecture to GAN-like", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "Goodfellow et al. (2014) learning with the prospect to further improve the generative abilities of our", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "method. GANs, however, do not use a metric for measuring the quality of the learned distributions", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "such as the marginal likelihood or the ELBO in their objective and therefore the transfer of our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 333, + 279, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 279, + 345 + ], + "score": 1.0, + "content": "architecture to these is not straightforward.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 361, + 175, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 176, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 176, + 374 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 109, + 378, + 503, + 412 + ], + "lines": [ + { + "bbox": [ + 107, + 378, + 504, + 390 + ], + "spans": [ + { + "bbox": [ + 107, + 378, + 504, + 390 + ], + "score": 1.0, + "content": "Tamara Broderick, Nicholas Boyd, Andre Wibisono, Ashia C. 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In the case of an isotropic gaussian posterior,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 147, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 160 + ], + "score": 1.0, + "content": "the proposed regularizer scales the mean and variance of the student posterior by the variance of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "teacher 7.0.2 and adds an extra ’volume’ term. This interpretation of the consistency regularizer", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "shows that the proposed regularizer preserves the same learning objective as that of the standard", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 506, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 193 + ], + "score": 1.0, + "content": "VAE. Below we present the analytical form of the consistency regularizer with categorical and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 225, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 225, + 204 + ], + "score": 1.0, + "content": "isotropic gaussian posteriors:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 124, + 506, + 204 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 214, + 506, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 403, + 235 + ], + "score": 1.0, + "content": "Corollary 7.0.1 We parameterize the learnt posterior of the teacher by", + "type": "text" + }, + { + "bbox": [ + 404, + 213, + 485, + 234 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Phi _ { i } \\ = \\ \\frac { \\exp ( p _ { i } ^ { E } ) } { \\sum _ { i = 1 } ^ { J } \\exp ( p _ { i } ^ { E } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 210, + 506, + 235 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 232, + 511, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 262, + 250 + ], + "score": 1.0, + "content": "the posterior of the student by φi =", + "type": "text" + }, + { + "bbox": [ + 107, + 251, + 192, + 266 + ], + "score": 0.88, + "content": "\\begin{array} { r } { c ^ { E } = \\sum _ { i = 1 } ^ { J } \\exp ( p _ { i } ^ { E } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 236, + 211, + 271 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 212, + 253, + 295, + 266 + ], + "score": 0.86, + "content": "\\begin{array} { r } { c ^ { S } = \\sum _ { i = 1 } ^ { J } \\exp ( p _ { i } ^ { S } ) . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 232, + 309, + 253 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\phi _ { i } = \\frac { \\exp ( p _ { i } ^ { S } ) } { \\sum _ { i = 1 } ^ { J } \\exp ( p _ { i } ^ { S } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 236, + 511, + 271 + ], + "score": 1.0, + "content": ". 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K L ( Q _ { \\Phi } ( z | \\mathbf { x } ) | | P ( z ) ) = \\mathcal { L } _ { \\pmb \\theta }", + "type": "interline_equation", + "image_path": "1fd08028e0b897b0db9eb83c509dafded2b24f3f14955e580646139d00eac1b3.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 165, + 181, + 445, + 196 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 207, + 505, + 274 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 494, + 220 + ], + "score": 1.0, + "content": "In order to backpropagate it is necessary to remove the dependence on the stochastic variable", + "type": "text" + }, + { + "bbox": [ + 494, + 210, + 501, + 218 + ], + "score": 0.71, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 207, + 505, + 220 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "To achieve this, we push the sampling operation outside of the computational graph for the normal", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "distribution via the reparameterization trick Kingma & Welling (2014) and the gumbel-softmax", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 240, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 506, + 254 + ], + "score": 1.0, + "content": "reparameterization Maddison et al. (2016); Jang et al. (2017) for the discrete distribution. In essence", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 369, + 263 + ], + "score": 1.0, + "content": "the reparameterization trick allows us to introduce a distribution", + "type": "text" + }, + { + "bbox": [ + 369, + 251, + 390, + 263 + ], + "score": 0.91, + "content": "P ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 252, + 505, + 263 + ], + "score": 1.0, + "content": "that is not a function of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 262, + 459, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 459, + 276 + ], + "score": 1.0, + "content": "data or computational graph in order to move the gradient operator into the expectation:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 280, + 421, + 308 + ], + "lines": [ + { + "bbox": [ + 189, + 280, + 421, + 308 + ], + "spans": [ + { + "bbox": [ + 189, + 280, + 421, + 308 + ], + "score": 0.93, + "content": "\\nabla \\mathbb { E } _ { Q _ { \\Phi } ( z | \\mathbf { x } ) } \\bigg [ \\mathrm { l o g } \\frac { P _ { \\theta } ( x , z ) } { Q _ { \\Phi } ( z | \\mathbf { x } ) } \\bigg ] \\mapsto \\mathbb { E } _ { P ( \\epsilon ) } \\bigg [ \\nabla \\mathrm { l o g } \\frac { P _ { \\theta } ( x , z ) } { Q _ { \\Phi } ( z | \\mathbf { x } ) } \\bigg ]", + "type": "interline_equation", + "image_path": "95589251212178f5ee5fa6e8701c9f481debd08fd4e47444f8907327aafbda26.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 189, + 280, + 421, + 294.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 189, + 294.0, + 421, + 308.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 320, + 207, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 208, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 208, + 333 + ], + "score": 1.0, + "content": "7.2 MODEL RELATED", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 341, + 367, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 368, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 368, + 354 + ], + "score": 1.0, + "content": "In this section we provide extra details of our model architecture.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 366, + 244, + 378 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 245, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 245, + 378 + ], + "score": 1.0, + "content": "7.2.1 MODEL ARCHITECTURE", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 399 + ], + "score": 1.0, + "content": "We utilized two different architectures for our experiments. 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These surrounding weights are reinitialized to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 240, + 350, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 350, + 253 + ], + "score": 1.0, + "content": "their standard Glorot initializations Glorot & Bengio (2010).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 162, + 506, + 253 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 268, + 251, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 267, + 253, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 253, + 282 + ], + "score": 1.0, + "content": "7.3 FORWARD VS. REVERSE KL", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "image", + "bbox": [ + 172, + 306, + 420, + 493 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 172, + 306, + 420, + 493 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 172, + 306, + 420, + 493 + ], + "spans": [ + { + "bbox": [ + 172, + 306, + 420, + 493 + ], + "score": 0.972, + "type": "image", + "image_path": "c20e180406d015f1a528955ddb39a6aeaad6b034df00717d8727af083540f5d4.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 172, + 306, + 420, + 368.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 172, + 368.3333333333333, + 420, + 430.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 172, + 430.66666666666663, + 420, + 492.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 190, + 505, + 419, + 516 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 190, + 504, + 421, + 518 + ], + "spans": [ + { + "bbox": [ + 190, + 504, + 421, + 518 + ], + "score": 1.0, + "content": "Figure 6: Reverse vs. forward KL on FashionMNIST 5.1", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + } + ], + "index": 17.0 + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "In our setting we have the ability to utilize the zero forcing (reverse or mode-seeking) KL or the zero", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "avoiding (forward) KL divergence. In general, if the true underlying posterior is multi-modal, it is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "preferable to operate with the reverse KL divergence (Murphy (2012) 21.2.2). In addition, utilizing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 565, + 499, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 499, + 578 + ], + "score": 1.0, + "content": "the mode-seeking KL divergence generates more realistic results when operating over image data.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 533, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "In order to validate this, we repeat the experiment in 5.1. We train two models: one with the forward", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 594, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 605 + ], + "score": 1.0, + "content": "KL posterior regularizer and one with the reverse. We evaluate the -ELBO mean and variance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "over ten trials. Empirically, we observed no difference between the different measures. This is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 615, + 208, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 208, + 627 + ], + "score": 1.0, + "content": "demonstrated in figure 6.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 582, + 506, + 627 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 644, + 272, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 273, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 273, + 657 + ], + "score": 1.0, + "content": "7.4 NUMBER OF REQUIRED SAMPLES", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Our method derives its sample complexity from standard VAEs. In practice we evaluate the number", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "of required real and synthetic samples by utilizing early stopping. When the negative ELBO on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "the validation set stops decreasing for 50 steps we stop training the current model and transition to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "the next distribution interval. Using this and the fact that we keep equal proportions of all observed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "distributions in our minibatch, we can evaluate the number of synthetic and real samples used during", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 720, + 478, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 478, + 732 + ], + "score": 1.0, + "content": "the single distribution interval. We demonstrate this procedure on experiment 5.1 in figure 7.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 665, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 88, + 486, + 227 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 88, + 486, + 227 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 88, + 486, + 227 + ], + "spans": [ + { + "bbox": [ + 108, + 88, + 486, + 227 + ], + "score": 0.97, + "type": "image", + "image_path": "cc4dab3f21baf14bf3afeebd6d4c9bdb4ca56c427a0d6ee500748df09837e33e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 88, + 486, + 134.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 134.33333333333334, + 486, + 180.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 180.66666666666669, + 486, + 227.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 235, + 504, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 235, + 504, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 504, + 249 + ], + "score": 1.0, + "content": "Figure 7: FashionMNIST experiment 5.1 data efficiency analysis. Left: Real samples used; Right:", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 246, + 203, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 203, + 259 + ], + "score": 1.0, + "content": "Synthetic samples used", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 504, + 287 + ], + "lines": [ + { + "bbox": [ + 106, + 264, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 278 + ], + "score": 1.0, + "content": "We observe a rapid decrease of the number of required real samples as we assimilate more", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 275, + 222, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 222, + 287 + ], + "score": 1.0, + "content": "distributions into our model.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 302, + 236, + 313 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 237, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 237, + 314 + ], + "score": 1.0, + "content": "7.5 EXPERIMENTS RELATED", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 322, + 504, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "In this section we provide an extra experiment run on MNIST as well as some extra images from the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 333, + 219, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 219, + 346 + ], + "score": 1.0, + "content": "rotated MNIST experiment.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 107, + 358, + 290, + 370 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 291, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 291, + 371 + ], + "score": 1.0, + "content": "7.5.1 MNIST : GENERATION AND ELBO", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "image", + "bbox": [ + 107, + 396, + 503, + 510 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 396, + 503, + 510 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 396, + 503, + 510 + ], + "spans": [ + { + "bbox": [ + 107, + 396, + 503, + 510 + ], + "score": 0.971, + "type": "image", + "image_path": "15a1a1e12ba4d93f4e0643a470665c0942c237892d48141604db03027ffeeb15.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 107, + 396, + 503, + 434.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 107, + 434.0, + 503, + 472.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 107, + 472.0, + 503, + 510.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 518, + 505, + 541 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "Figure 8: (a) Generation with consistency regularizer. (b) Without consistency regularizer. (c)", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 529, + 484, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 484, + 542 + ], + "score": 1.0, + "content": "Average log-likelihood over ten trials (each) for the ten separated distributions within MNIST.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + } + ], + "index": 13.25 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "In this experiment, we seek to establish the performance benefit that the consistency regularizer", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "brings into the learning process. We do so by evaluating the ELBO for a model with and without the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "consistency and mutual information regularizers. We also demonstrate the ability of the regularizers", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "to disambiguate distributional boundaries and their inter-distributional variations. I.e. for MNIST", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 600, + 432, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 432, + 612 + ], + "score": 1.0, + "content": "this separates the MNIST digits from their inter-class variants (i.e drawing style).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "We use MNIST to simulate our sequential learning setting. We treat each digit as a different", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "distribution and present the model with samples drawn from a single distribution at a time. For the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 637, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 652 + ], + "score": 1.0, + "content": "purpose of this experiment we sequentially progress over the ten distributions (i.e. interval sampling", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 649, + 318, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 318, + 660 + ], + "score": 1.0, + "content": "involves linearly iterating over all the distributions ).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "When an interval transition occurs we signal the model, make the student the new teacher and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "instantiate a new student model. We contrast this to a model that utilizes the same graphical model,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "without our consistency and mutual information regularizers. We quantify the performance of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "generative models by computing the ELBO over the standard MNIST test set at every interval. The", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "test set contains digits from all of the individual distributions. We run this procedure ten times and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 720, + 276, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 276, + 733 + ], + "score": 1.0, + "content": "report the average ELBO over the test set.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 88, + 486, + 227 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 88, + 486, + 227 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 88, + 486, + 227 + ], + "spans": [ + { + "bbox": [ + 108, + 88, + 486, + 227 + ], + "score": 0.97, + "type": "image", + "image_path": "cc4dab3f21baf14bf3afeebd6d4c9bdb4ca56c427a0d6ee500748df09837e33e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 88, + 486, + 134.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 134.33333333333334, + 486, + 180.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 180.66666666666669, + 486, + 227.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 235, + 504, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 235, + 504, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 504, + 249 + ], + "score": 1.0, + "content": "Figure 7: FashionMNIST experiment 5.1 data efficiency analysis. Left: Real samples used; Right:", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 246, + 203, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 203, + 259 + ], + "score": 1.0, + "content": "Synthetic samples used", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 504, + 287 + ], + "lines": [ + { + "bbox": [ + 106, + 264, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 278 + ], + "score": 1.0, + "content": "We observe a rapid decrease of the number of required real samples as we assimilate more", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 275, + 222, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 222, + 287 + ], + "score": 1.0, + "content": "distributions into our model.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 106, + 264, + 505, + 287 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 302, + 236, + 313 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 237, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 237, + 314 + ], + "score": 1.0, + "content": "7.5 EXPERIMENTS RELATED", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 322, + 504, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "In this section we provide an extra experiment run on MNIST as well as some extra images from the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 333, + 219, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 219, + 346 + ], + "score": 1.0, + "content": "rotated MNIST experiment.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 322, + 505, + 346 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 358, + 290, + 370 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 291, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 291, + 371 + ], + "score": 1.0, + "content": "7.5.1 MNIST : GENERATION AND ELBO", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "image", + "bbox": [ + 107, + 396, + 503, + 510 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 396, + 503, + 510 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 396, + 503, + 510 + ], + "spans": [ + { + "bbox": [ + 107, + 396, + 503, + 510 + ], + "score": 0.971, + "type": "image", + "image_path": "15a1a1e12ba4d93f4e0643a470665c0942c237892d48141604db03027ffeeb15.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 107, + 396, + 503, + 434.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 107, + 434.0, + 503, + 472.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 107, + 472.0, + 503, + 510.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 518, + 505, + 541 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "Figure 8: (a) Generation with consistency regularizer. (b) Without consistency regularizer. (c)", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 529, + 484, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 484, + 542 + ], + "score": 1.0, + "content": "Average log-likelihood over ten trials (each) for the ten separated distributions within MNIST.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + } + ], + "index": 13.25 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "In this experiment, we seek to establish the performance benefit that the consistency regularizer", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "brings into the learning process. We do so by evaluating the ELBO for a model with and without the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "consistency and mutual information regularizers. We also demonstrate the ability of the regularizers", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "to disambiguate distributional boundaries and their inter-distributional variations. I.e. for MNIST", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 600, + 432, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 432, + 612 + ], + "score": 1.0, + "content": "this separates the MNIST digits from their inter-class variants (i.e drawing style).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 554, + 505, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "We use MNIST to simulate our sequential learning setting. We treat each digit as a different", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "distribution and present the model with samples drawn from a single distribution at a time. 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0000000000000000000000000000000000000000..b453c5279ed0b48f5539abb817485703da85d7b2 --- /dev/null +++ b/parse/train/SJg7KhVKPH/SJg7KhVKPH.md @@ -0,0 +1,410 @@ +# DEPTH-ADAPTIVE TRANSFORMER + +Maha Elbayad∗ Univ. Grenoble Alpes + +Jiatao Gu, Edouard Grave, Michael Auli Facebook AI Research + +# ABSTRACT + +State of the art sequence-to-sequence models for large scale tasks perform a fixed number of computations for each input sequence regardless of whether it is easy or hard to process. In this paper, we train Transformer models which can make output predictions at different stages of the network and we investigate different ways to predict how much computation is required for a particular sequence. Unlike dynamic computation in Universal Transformers, which applies the same set of layers iteratively, we apply different layers at every step to adjust both the amount of computation as well as the model capacity. On IWSLT German-English translation our approach matches the accuracy of a well tuned baseline Transformer while using less than a quarter of the decoder layers. + +# 1 INTRODUCTION + +The size of modern neural sequence models (Gehring et al., 2017; Vaswani et al., 2017; Devlin et al., 2019) can amount to billions of parameters (Radford et al., 2019). For example, the winning entry of the WMT’19 news machine translation task in English-German used an ensemble totaling two billion parameters $\mathrm { N g }$ et al., 2019). While large models are required to do better on hard examples, small models are likely to perform as well on easy ones, e.g., the aforementioned ensemble is probably not required to translate a short phrase such as "Thank you". However, current models apply the same amount of computation regardless of whether the input is easy or hard. + +In this paper, we propose Transformers which adapt the number of layers to each input in order to achieve a good speed-accuracy trade off at inference time. We extend Graves (2016; ACT) who introduced dynamic computation to recurrent neural networks in several ways: we apply different layers at each stage, we investigate a range of designs and training targets for the halting module and we explicitly supervise through simple oracles to achieve good performance on large-scale tasks. + +Universal Transformers (UT) rely on ACT for dynamic computation and repeatedly apply the same layer (Dehghani et al., 2018). Our work considers a variety of mechanisms to estimate the network depth and applies a different layer at each step. Moreover, Dehghani et al. (2018) fix the number of steps for large-scale machine translation whereas we vary the number of steps to demonstrate substantial improvements in speed at no loss in accuracy. UT uses a layer which contains as many weights as an entire standard Transformer and this layer is applied several times which impacts speed. Our approach does not increase the size of individual layers. We also extend the resource efficient object classification work of Huang et al. (2017) and Bolukbasi et al. (2017) to structured prediction where dynamic computation decisions impact future computation. Related work from computer vision includes Teerapittayanon et al. (2016); Figurnov et al. (2017) and Wang et al. (2018) who explored the idea of dynamic routing either by exiting early or by skipping layers. + +We encode the input sequence using a standard Transformer encoder to generate the output sequence with a varying amount of computation in the decoder network. Dynamic computation poses a challenge for self-attention because omitted layers in prior time-steps may be required in the future. We experiment with two approaches to address this and show that a simple approach works well (§2). Next, we investigate different mechanisms to control the amount of computation in the decoder network, either for the entire sequence or on a per-token basis. This includes multinomial and binomial classifiers supervised by the model likelihood or whether the argmax is already correct as well as simply thresholding the model score (§3). Experiments on IWSLT14 German-English translation (Cettolo et al., 2014) as well as WMT’14 English-French translation show that we can match the performance of well tuned baseline models at up to $76 \%$ less computation (§4). + +![](images/c6b3f9ef1172307eaa082df73a5bf06366df36c55192e6d65cd4faba2c5300b2.jpg) +Figure 1: Training regimes for decoder networks able to emit outputs at any layer. Aligned training optimizes all output classifiers $\mathcal { C } _ { n }$ simultaneously assuming all previous hidden states for the current layer are available. Mixed training samples $M$ paths of random exits at which the model is assumed to have exited; missing previous hidden states are copied from below. + +# 2 ANYTIME STRUCTURED PREDICTION + +We first present a model that can make predictions at different layers. This is known as anytime prediction for computer vision models (Huang et al., 2017) and we extend it to structured prediction. + +# 2.1 TRANSFORMER WITH MULTIPLE OUTPUT CLASSIFIERS + +We base our approach on the Transformer sequence-to-sequence model (Vaswani et al., 2017). Both encoder and decoder networks contain $N$ stacked blocks where each has several sub-blocks surrounded by residual skip-connections. The first sub-block is a multi-head dot-product self-attention and the second a position-wise fully connected feed-forward network. For the decoder, there is an additional sub-block after the self-attention to add source context via another multi-head attention. + +Given a pair of source-target sequences $( { \pmb x } , { \pmb y } )$ , $_ { \textbf { \em x } }$ is processed with the encoder to give representations $\pmb { s } = ( s _ { 1 } , \ldots , s _ { | \pmb { x } | } )$ . Next, the decoder generates $\textbf { { y } }$ step-by-step. For every new token $\pmb { y } _ { t }$ input |to the decoder at time $t$ , the $N$ decoder blocks process it to yield hidden states $\left( h _ { t } ^ { n } \right) _ { 1 \leq n \leq N }$ : + +$$ +h _ { t } ^ { 0 } = \mathrm { e m b e d } ( y _ { t } ) , \quad h _ { t } ^ { n } = \mathrm { b l o c k } _ { n } ( h _ { \leq t } ^ { n - 1 } , \pmb { s } ) , +$$ + +where $\mathrm { b l o c k } _ { n }$ is the mapping associated with the $n ^ { \mathrm { t h } }$ block and embed is a lookup table. + +The output distribution for predicting the next token is computed by feeding the activations of the last decoder layer $h _ { t } ^ { N }$ into a softmax normalized output classifier $W$ : + +$$ +p ( y _ { t + 1 } | h _ { t } ^ { N } ) = \mathrm { s o f t m a x } ( W h _ { t } ^ { N } ) +$$ + +Standard Transformers have a single output classifier attached to the top of the decoder network. However, for dynamic computation we need to be able to make predictions at different stages of the network. To achieve this, we attach output classifiers $\mathcal { C } _ { n }$ parameterized by $W _ { n }$ to the output $h _ { t } ^ { n }$ of each of the $N$ decoder blocks: + +$$ +\forall n , \ p ( y _ { t + 1 } | h _ { t } ^ { n } ) = \operatorname { s o f t m a x } ( W _ { n } h _ { t } ^ { n } ) +$$ + +The classifiers can be parameterized independently or we can share the weights across the $N$ blocks. + +# 2.2 TRAINING MULTIPLE OUTPUT CLASSIFIERS + +Dynamic computation enables the model to use any of the $N$ exit classifiers instead of just the final one. Some of our models can choose a different output classifier at each time-step which results in an exponential number of possible output classifier combinations in the sequence length. + +We consider two possible ways to train the decoder network (Figure 1). Aligned training optimizes all classifiers simultaneously and assumes all previous hidden states required by the self-attention are available. However, at test time this is often not the case when we choose a different exit for every token which leads to misaligned states. Instead, mixed training samples several sequences of exits for a given sentence and exposes the model to hidden states from different layers. + +Generally, for a given output sequence $\textbf { { y } }$ , we have a sequence of chosen exits $( n _ { 1 } , \ldots , n _ { | { \pmb y } | } )$ and we denote the block at which we exit at time $t$ as $n _ { t }$ . + +# 2.2.1 ALIGNED TRAINING + +Aligned training assumes all hidden states $h _ { 1 } ^ { n - 1 } , \ldots , h _ { t } ^ { n - 1 }$ are available in order to compute selfattention and it optimizes $N$ loss terms, one for each exit (Figure 1a): + +$$ +\mathrm { L L } _ { t } ^ { n } = \log p ( y _ { t } | h _ { t - 1 } ^ { n } ) , \quad \mathrm { L L } ^ { n } = \sum _ { t = 1 } ^ { | y | } \mathrm { L L } _ { t } ^ { n } , \quad \mathcal { L } _ { d e c } ( \pmb { x } , \pmb { y } ) = - \frac { 1 } { \sum _ { n } \omega _ { n } } \sum _ { n = 1 } ^ { N } \omega _ { n } \mathrm { L L } ^ { n } . +$$ + +The compound loss $\mathcal { L } _ { d e c } ( \pmb { x } , \pmb { y } )$ is a weighted average of $N$ terms w.r.t. to $\left( \omega _ { 1 } , \hdots \omega _ { N } \right)$ . We found that uniform weights achieve better BLEU compared to other weighing schemes $_ { . c . f }$ . Appendix A). At inference time, not all time-steps will have hidden states for the current layer since the model exited early. In this case, we simply copy the last computed state to all upper layers, similar to mixed training $( \ S 2 . 2 . 2 )$ . However, we do apply layer-specific key and value projections to the copied state. + +# 2.2.2 MIXED TRAINING + +Aligned training assumes that all hidden states of the previous time-steps are available but this assumption is unrealistic since an early exit may have been chosen previously. This creates a mismatch between training and testing. Mixed training reduces the mismatch by training the model to use hidden states from different blocks of previous time-steps for self-attention. We sample $M$ different exit sequences (n(m)1 , . $( n _ { 1 } ^ { ( m ) } , \dots n _ { | \pmb { y } | } ^ { ( m ) } ) _ { 1 \leq m \leq M }$ and evaluate the following loss: + +$$ +\mathrm { L L } ( n _ { 1 } , \dots , n _ { | \mathbf { y } | } ) = \sum _ { t = 1 } ^ { | \mathbf { y } | } \log p ( y _ { t } | h _ { t - 1 } ^ { n _ { t } } ) , \quad \mathscr { L } _ { d e c } ( \mathbf { x } , \mathbf { y } ) = - \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \mathrm { L L } ( n _ { 1 } ^ { ( m ) } , \dots , n _ { | \mathbf { y } | } ^ { ( m ) } ) . +$$ + +When $n _ { t } ~ < ~ N$ , we copy the last evaluated hidden state $h _ { t } ^ { n }$ to the subsequent layers so that the self-attention of future time steps can function as usual (see Figure 1b). + +# 3 ADAPTIVE DEPTH ESTIMATION + +We present a variety of mechanisms to predict the decoder block at which the model will stop and output the next token, or when it should exit to achieve a good speed-accuracy trade-off. We consider two approaches: sequence-specific depth decodes all output tokens using the same block (§3.1) while token-specific depth determines a separate exit for each individual token (§3.2). + +We model the distribution of exiting at time-step $t$ with a parametric distribution $q _ { t }$ where $q _ { t } ( n )$ is the probability of computing ${ \mathrm { b l o c k } } _ { 1 } , \ldots , { \mathrm { b l o c k } } _ { n }$ and then emitting a prediction with $\mathcal { C } _ { n }$ . The parameters of $q _ { t }$ are optimized to match an oracle distribution $q _ { t } ^ { * }$ with cross-entropy: + +$$ +\mathcal { L } _ { \mathrm { e x i t } } ( { \pmb x } , { \pmb y } ) = \sum _ { t } H ( q _ { t } ^ { * } ( { \pmb x } , { \pmb y } ) , q _ { t } ( { \pmb x } ) ) +$$ + +The exit loss $( \mathcal { L } _ { \mathrm { e x i t } } )$ is back-propagated to the encoder-decoder parameters. We simultaneously optimize the decoding loss (Eq. (4)) and the exit loss (Eq. (6)) balanced by a hyper-parameter $\alpha$ to ensure that the model maintains good generation accuracy. The final loss takes the form: + +$$ +\begin{array} { r } { \mathcal { L } ( \pmb { x } , \pmb { y } ) = \mathcal { L } _ { d e c } ( \pmb { x } , \pmb { y } ) + \alpha \mathcal { L } _ { \mathrm { e x i t } } ( \pmb { x } , \pmb { y } ) , } \end{array} +$$ + +In the following we describe for each approach how the exit distribution $q _ { t }$ is modeled (illustrated in Figure 2) and how the oracle distribution $q _ { t } ^ { * }$ is inferred. + +![](images/9ffbc9ee07d1225e2d2a69f3d20fd736e999c534160e2004ae183049c51fafd3.jpg) +Figure 2: Variants of the adaptive depth prediction classifiers. Sequence-specific depth uses a multinomial classifier to choose an exit for the entire output sequence based on the encoder output $s$ (2a). It then outputs a token at this depth with classifier $\mathcal { C } _ { n }$ . The token-specific multinomial classifier determines the exit after the first block and proceeds up to the predicted depth before outputting the next token (2b). The token geometric-like classifier (2c) makes a binary decision after every block to dictate whether to continue (C) to the next block or to stop (S) and emit an output distribution. + +# 3.1 SEQUENCE-SPECIFIC DEPTH: + +For sequence-specific depth, the exit distribution $q$ and the oracle distribution $q ^ { * }$ are independent of the time-step so we drop subscript $t$ . We condition the exit on the source sequence by feeding the average $s$ of the encoder outputs to a multinomial classifier: + +$$ +s = \frac { 1 } { | \pmb { x } | } \sum _ { t } s _ { t } , \quad q ( n | \pmb { x } ) = \mathrm { s o f t m a x } ( W _ { h } s + b _ { h } ) \in \mathbb { R } ^ { N } , +$$ + +where $W _ { h }$ and $b _ { h }$ are the weights and biases of the halting mechanism. We consider two oracles to determine which of the $N$ blocks should be chosen. The first is based on the sequence likelihood and the second looks at an aggregate of the correctly predicted tokens at each block. + +Likelihood-based: This oracle is based on the likelihood of the entire sequence after each block and we optimize it with the Dirac delta centered around the exit with the highest sequence likelihood. + +$$ +q ^ { * } ( { \pmb x } , { \pmb y } ) = \delta ( \arg \operatorname* { m a x } _ { n } { \mathrm { L L } ^ { n } } ) . +$$ + +We add a regularization term to encourage lower exits that achieve good likelihood: + +$$ +q ^ { * } ( { \pmb x } , { \pmb y } ) = \delta ( \arg \operatorname* { m a x } _ { n } { \mathrm { L L } ^ { n } } - \lambda n ) . +$$ + +Correctness-based: Likelihood ignores whether the model already assigns the highest score to the correct target. Instead, this oracle chooses the lowest block that assigns the largest score to the correct prediction. For each block, we count the number of correctly predicted tokens over the sequence and choose the block with the most number of correct tokens. A regularization term controls the trade-off between speed and accuracy. + +$$ +C ^ { n } = \# \{ t | y _ { t } = \arg \operatorname* { m a x } _ { y } p ( y | h _ { t - 1 } ^ { n } ) \} , \quad q ^ { * } ( x , y ) = \delta ( \arg \operatorname* { m a x } _ { n } C ^ { n } - \lambda n ) . +$$ + +Oracles based on test metrics such as BLEU are feasible but expensive to compute since we would need to decode every training sentence $N$ times. We leave this for future work. + +# 3.2 TOKEN-SPECIFIC DEPTH: + +The token-specific approach can choose a different exit at every time-step. We consider two options for the exit distribution $q _ { t }$ at time-step t: a multinomial with a classifier conditioned on the first decoder hidden state $h _ { t } ^ { 1 }$ and a geometric-like where an exit probability $\chi _ { t } ^ { n }$ is estimated after each block based on the activations of the current block $h _ { t } ^ { n }$ . + +# Multinomial $q _ { t }$ + +$$ +q _ { t } ( n | \pmb { x } , \pmb { y } _ { < t } ) = \mathrm { s o f t m a x } ( { W _ { h } } h _ { t } ^ { 1 } + b _ { h } ) , +$$ + +The most probable exit arg max $q _ { t } ( n | \pmb { x } , \pmb { y } _ { < t } )$ is selected at inference. + +Geometric-like $q _ { t }$ + +$$ +\begin{array} { l } { \forall n \in [ 1 . . N - 1 ] , ~ \chi _ { t } ^ { n } = \mathrm { s i g m o i d } ( w _ { h } ^ { \top } h _ { t } ^ { n } + b _ { h } ) , } \\ { q _ { t } ( n | \pmb { x } , \pmb { y } _ { < t } ) = \left\{ \begin{array} { l l } { \chi _ { t } ^ { n } \displaystyle \prod _ { n ^ { \prime } < n } ( 1 - \chi _ { t } ^ { n ^ { \prime } } ) , \mathrm { i f } n < N } \\ { \displaystyle \prod _ { n ^ { \prime } < N } ( 1 - \chi _ { t } ^ { n ^ { \prime } } ) , \mathrm { o t h e r w i s e } } \end{array} \right. } \end{array} +$$ + +where, $d$ is the dimension of the decoder states, $W _ { h } \ \in \ \mathbb { R } ^ { N \times d }$ and $w _ { h } \in \mathbb { R } ^ { d }$ are the weights of the halting mechanisms, and $b _ { h }$ their biases. During inference the decoder exits when the halting signal $\chi _ { t } ^ { n }$ exceeds a threshold $\tau _ { n }$ which we tune on the valid set to achieve a better accuracy-speed trade-off. If thresholds $\left( \tau _ { n } \right) _ { 1 \leq n < N }$ have not been exceeded, then we default to exiting at block $N$ . + +The two classifiers are trained to minimize the cross-entropy with respect to either one the following oracle distributions: + +Likelihood-based: At each time-step $t$ , we choose the block whose exit classifier has the highest likelihood plus a regularization term weighted by $\lambda$ to encourage lower exits. + +$$ +q _ { t } ^ { * } ( \pmb { x } , \pmb { y } ) = \delta ( \arg \operatorname* { m a x } _ { n } \mathrm { L L } _ { t } ^ { n } - \lambda n ) +$$ + +This oracle ignores the impact of the current decision on the future time-steps and we therefore consider smoothing the likelihoods with an RBF kernel. + +$$ +\kappa ( t , t ^ { \prime } ) = e ^ { - \frac { \lvert t - t ^ { \prime } \rvert ^ { 2 } } { \sigma } } , \quad \widetilde { \mathrm { L L } _ { t } ^ { n } } = \sum _ { t ^ { \prime } = 1 } ^ { \lfloor y \rfloor } \kappa ( t , t ^ { \prime } ) \mathrm { L L } _ { t ^ { \prime } } ^ { n } , \quad q _ { t } ^ { * } ( { \pmb x } , { \pmb y } ) = \delta ( \arg \operatorname* { m a x } _ { n } \widetilde { \mathrm { L L } _ { t } ^ { n } } - \lambda n ) , +$$ + +where we control the size of the surrounding context with $\sigma$ the kernel width. We refer to this oracle as $\operatorname { L L } ( \sigma , \lambda )$ including the case where we only look at the likelihood of the current token with $\sigma \to 0$ . + +Correctness-based: Similar to the likelihood-based oracle we can look at the correctness of the prediction at time-step $t$ as well as surrounding positions. We define the target $q _ { t } ^ { * }$ as follows: + +$$ +\begin{array} { l } { { C _ { t } ^ { n } = \mathbb { 1 } \big [ y _ { t } = \arg \operatorname* { m a x } _ { y } p ( y | h _ { t - 1 } ^ { n } ) \big ] , \quad \widetilde { C _ { t } ^ { n } } = \displaystyle \sum _ { t ^ { \prime } = 1 } ^ { | y | } \kappa ( t , t ^ { \prime } ) C _ { t } ^ { n } , } } \\ { { q _ { t } ^ { * } ( { \pmb x } , { \pmb y } ) = \delta ( \arg \operatorname* { m a x } _ { n } \widetilde { C _ { t } ^ { n } } - \lambda n ) . } } \end{array} +$$ + +Confidence thresholding Finally, we consider thresholding the model predictions (§2), i.e., exit when the maximum score of the current output classifier $p \bar { ( } y _ { t + 1 } | h _ { t } ^ { n } )$ exceeds a hyper-parameter threshold $\tau _ { n }$ . This does not require training and the thresholds $\pmb { \tau } = ( \tau _ { 1 } , \dots , \tau _ { N - 1 } )$ are simply tuned on the valid set to maximize BLEU. Concretely, for 10k iterations, we sample a sequence of thresholds $\tau \sim \mathcal { U } ( 0 , 1 ) ^ { N - 1 }$ , decode the valid set with the sampled thresholds and then evaluate the BLEU score and computational cost achieved with this choice of $\tau$ . After $1 0 \mathrm { k }$ evaluations we pick the best performing thresholds, that is $\tau$ with the highest BLEU in each cost segment. + +# 4 EXPERIMENTS + +# 4.1 EXPERIMENTAL SETUP + +We evaluate on several benchmarks and measure tokenized BLEU (Papineni et al., 2002): + +IWSLT’14 German to English (De-En). We use the setup of Edunov et al. (2018) and train on 160K sentence pairs. We use $N = 6$ blocks, a feed-forward network (ffn) of intermediate-dimension + +Table 1: Aligned vs. mixed training on IWSLT De-En. We report valid BLEU for a uniformly sampled exit $\overset { \cdot } { n } \sim \mathcal { U } ( [ 1 . . 6 ] )$ at each token, a fixed exit $n \in [ 1 . . 6 ]$ for all tokens, as well as the average BLEU over the fixed exits. As baseline we show six standard Transformer models with 1-6 blocks. + +
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline134.235.335.635.735.635.935.4
Aligned (ωn = 1)35.534.135.535.836.136.136.235.6
Mixed M =134.132.934.334.534.534.634.534.2
Mixed M = 335.133.935.235.435.535.535.535.2
Mixed M = 635.334.235.435.835.935.835.935.5
+ +1024, 4 heads, dropout 0.3, embedding dimension $d _ { \mathrm { e n c } } = 5 1 2$ for the encoder and $d _ { \mathrm { d e c } } = 2 5 6$ for the decoder. Embeddings are untied with 6 different output classifiers. We evaluate with a single checkpoint and a beam of width 5. + +WMT’14 English to French $( \mathbf { E n } \mathbf { - } \mathbf { F r } )$ . We also experiment on the much larger WMT’14 EnglishFrench task comprising $3 5 . 5 \mathrm { m }$ training sentence pairs. We develop on 26k held out pairs and test on newstest14. The vocabulary consists of 44k joint BPE types (Sennrich et al., 2016). We use a Transformer big architecture and tie the embeddings of the encoder, the decoder and the output classifiers $( ( W _ { n } ) _ { 1 \leq n \leq 6 } ; \ S 2 . 1 )$ . We average the last ten checkpoints and use a beam of width 4. + +Models are implemented in fairseq (Ott et al., 2019) and are trained with Adam (Kingma & Ba, 2015). We train for 50k updates on 128 GPUs with a batch size of 460k tokens for WMT’14 En-Fr and on 2 GPUs with $^ \mathrm { 8 k }$ tokens per batch for IWSLT’14 De-En. To stabilize training, we re-normalize the gradients if the norm exceeds $g _ { \mathrm { c l i p } } = 3$ . + +For models with adaptive exits, we first train without exit prediction ( $\alpha = 0$ in Eq. (7)) using the aligned mode $( c . f . \ S 2 . 2 . 1 )$ for $5 0 \mathrm { k }$ updates and then continue training with $\alpha \neq 0$ until convergence. The exit prediction classifiers are parameterized by a single linear layer (Eq. (8)) with the same input dimension as the embedding dimension, e.g., 1024 for a big Transformer; the output dimension is $N$ for a multinomial classifier or one for geometric-like. We exit when $\chi _ { t , n } > 0 . 5$ for geometric-like classifiers. + +# 4.2 TRAINING MULTIPLE OUTPUT CLASSIFIERS + +We first compare the two training regimes for our model $( \ S 2 . 2 )$ . Aligned training performs selfattention on aligned states (§2.2.1) and mixed training exposes self-attention to hidden states from different blocks $( \ S 2 . 2 . 2 )$ . + +We compare the two training modes when choosing either a uniformly sampled exit or a fixed exit $n = 1 , \ldots , 6$ at inference time for every time-step. The sampled exit experiment tests the robustness to mixed hidden states and the fixed exit setup simulates an ideal setting where all previous states are available. As baselines we show six separate standard Transformers with $N \in [ 1 . . 6 ]$ decoder blocks. All models are trained with an equal number of updates and mixed training with $M { = } 6$ paths is most comparable to aligned training since the number of losses per sample is identical. + +Table 1 shows that aligned training outperforms mixed training both for fixed exits as well as for randomly sampled exits. The latter is surprising since aligned training never exposes the self-attention mechanism to hidden states from other blocks. We suspect that this is due to the residual connections which copy features from lower blocks to subsequent layers and which are ubiquitous in Transformer models (§2). Aligned training also performs very competitively to the individual baseline models. + +Aligned training is conceptually simple and fast. We can process a training example with $N$ exits in a single forward/backward pass while $M$ passes are needed for mixed training. In the remaining paper, we use the aligned mode to train our models. Appendix A reports experiments with weighing the various output classifiers differently but we found that a uniform weighting scheme worked well. On our largest setup, WMT’14 English-French, the training time of an aligned model with six output classifiers increases only marginally by about $1 \%$ compared to a baseline with a single output classifier keeping everything else equal. + +![](images/08aa6865d38f0ae73b75393499522f9a002e3778b5c4d0a3402639a98af647f1.jpg) +Figure 3: Trade-off between speed (average exit or AE) and accuracy (BLEU) for depth-adaptive methods on the IWSLT14 De-En test set. + +![](images/8208d9e007a1f6ef7e4f203a9c4bf94a1261af5e96cf9c95b86e08c15a27b5bb.jpg) +Figure 4: Effect of the hyper-parameters $\sigma$ and $\lambda$ on the average exit (AE) measured on the valid set of IWSLT’14 De-En. + +# 4.3 ADAPTIVE DEPTH ESTIMATION + +Next, we train models with aligned states and compare adaptive depth classifiers in terms of BLEU as well as computational effort. We measure the latter as the average exit per output token (AE). + +As baselines we use again six separate standard Transformers with $N \in [ 1 . . 6 ]$ with a single output classifier. We also measure the performance of the aligned mode trained model for fixed exits $n \in$ [1..6]. For the adaptive depth token-specific models (Tok), we train four combinations: likelihoodbased oracle (LL) $^ +$ geometric-like, likelihood-based oracle (LL) $^ +$ multinomial, correctness based oracle $\left( \mathbf { C } \right) +$ geometric-like and correctness-based oracle $\left( \mathbf { C } \right) +$ multinomial. Sequence-specific models (Seq) are trained with the correctness oracle (C) and the likelihood oracle (LL) with different values for the regularization weight $\lambda$ . All parameters are tuned on the valid set and we report results on the test set for a range of average exits. + +Figure 3 shows that the aligned model (blue line) can match the accuracy of a standard 6-block Transformer (black line) at half the number of layers ${ \mathrm { ~ \ : ~ } } _ { n } = 3 { \mathrm { ~ \ : ~ } }$ ) by always exiting at the third block. The aligned model outperforms the baseline for $n = 2 , \ldots , 6$ . + +For token specific halting mechanisms (Figure 3a) the geometric-like classifiers achieves a better speed-accuracy trade-off than the multinomial classifiers (filled vs. empty triangles). For geometriclike classifiers, the correctness oracle outperforms the likelihood oracle (Tok-C geometric-like vs. Tok-LL geometric-like) but the trend is less clear for multinomial classifiers. At the sequence-level, likelihood is the better oracle (Figure 3b). + +The rightmost Tok-C geometric-like point $( \sigma = 0$ , $\lambda = 0 . 1$ ) achieves 34.73 BLEU at $\mathrm { A E } = 1 . 4 2$ which corresponds to similar accuracy as the $N \ = \ 6$ baseline at $76 \%$ fewer decoding blocks. + +![](images/72eb06ffdaeb2b8f53f187689e2b19f54d633c162199e0bfd63806ba35e8e616.jpg) +Figure 5: Speed and accuracy on the WMT’14 English-French benchmark $( c . f .$ . Figure 3). + +The best accuracy of the aligned model is 34.95 BLEU at exit 5 and the best comparable Tok-C geometric-like configuration achieves 34.99 BLEU at $\mathrm { A E } = 1 . 9 7$ , or $61 \%$ fewer decoding blocks. When fixing the budget to two decoder blocks, Tok-C geometric-like with $\mathrm { A E } = 1 . 9 7$ achieves BLEU 35, a 0.64 BLEU improvement over the baseline ( $N = 2$ ) and aligned which both achieve BLEU 34.35. + +Confidence thresholding (Figure 3c) performs very well but cannot outperform Tok-C geometriclike. + +Ablation of hyper-parameters In this section, we look at the effect of the two main hyperparameters on IWSLT’14 De-En: $\lambda$ the regularization scale ( $_ { . c . f }$ . Eq. (9)), and the RBF kernel width $\sigma$ used to smooth the scores $( c . f$ . Eq. (15)). We train Tok-LL Geometric-like models and evaluate them with their default thresholds (exit if $\chi _ { t } ^ { n } > 0 . 5 )$ . Figure 4a shows that higher values of $\lambda$ lead to lower exits. Figure 4b shows the effect of $\sigma$ for two values of $\lambda$ . In both curves, we see that wider kernels favor higher exits. + +# 4.4 SCALING THE ADAPTIVE-DEPTH MODELS + +Finally, we take the best performing models form the IWSLT benchmark and test them on the large WMT’14 English-French benchmark. Results on the test set (Figure 5a) show that adaptive depth still shows improvements but that they are diminished in this very large-scale setup. Confidence thresholding works very well and sequence-specific depth approaches improve only marginally over the baseline. Tok-LL geometric-like can match the best baseline result of BLEU 43.4 $N = 6$ ) by using only $\mathrm { A E } = 2 . 4 0$ which corresponds to $40 \%$ of the decoder blocks; the best aligned result of BLEU 43.6 can be matched with $\mathrm { A E } = 3 . 2 5$ . In this setup, Tok-LL geometric-like slightly outperforms the Tok-C counterpart. + +Confidence thresholding matches the accuracy of the $N { = } 6$ baseline with AE 2.5 or $59 \%$ fewer decoding blocks. However, confidence thresholding requires computing the output classifier at each block to determine whether to halt or continue. This is a large overhead since output classifiers predict 44k types for this benchmark (§4.1). To better account for this, we measure the average number of FLOPs per output token (details in Appendix B). Figure 5b shows that the Tok-LL geometric-like approach provides a better trade-off when the overhead of the output classifiers is considered. + +# 4.5 QUALITATIVE RESULTS + +The exit distribution for a given sample can give insights into what a Depth-Adaptive Transformer decoder considers to be a difficult task. In this section, for each hypothesis $\widetilde { \boldsymbol { y } }$ , we will look at the sequence of selected exits $( n _ { 1 } , \ldots , n _ { | \tilde { \pmb { y } } | } )$ and the probability scores $( p _ { 1 } , \hdots p _ { | \widetilde { \pmb { y } } | } )$ with $\begin{array} { r l } { p _ { t } } & { { } = } \end{array}$ $p ( \widetilde { y } _ { t } | h _ { t - 1 } ^ { n _ { t } } )$ e i.e. the confidence of the model in the sampled token at the selected exit. + +Figures 6 and 7 show hypotheses from the WMT’14 En-Fr and IWSLT’14 De-En test sets, respectively. For each hypothesis we state the exits and the probability scores. In Figure 6a, predicting + +![](images/73aee9ecfc92f773d75e7f5027eab3b98129a636e72fa4ceb61bb68a12f85cc2.jpg) + +![](images/684d9684562ed9f33faf762808bedff084b7388602287045fea2467431609d51.jpg) + +(a) Src: Chi@@rac , the Prime Minister , was there . Ref: Chi@@rac , Premier ministre , est là . + +(b) Src: But passengers shoul@@dn’t expect changes to happen immediately . Ref: Mais les passagers ne devraient pas ${ \mathrm { ~ s ~ } } ^ { \prime }$ attendre à des changements immédiats . + +Figure 6: Examples from the WMT’14 En-Fr test set (newstest14) with Tok-LL geometric-like depth estimation. Token exits are in blue and confidence scores are in gray. The $\bullet _ { @ \mathbb { Q } } ,$ are due to BPE or subword tokenization. For each example the source (Src) and the reference (Ref) are provided in the caption. + +![](images/c09a06b4990673a86c9b78ffc1d7e7a2e7d8b550d430e028ee6b5dbf8b98eb27.jpg) +Figure 7: Example from the IWSLT’14 De-En test set with Tok-LL geometric-like depth estimation. See Figure 6 for more details. + +(a) Src: diesen trick können sie ihren freunden und nachbarn vor@@führen . danke . Ref: there is a trick you can do for your friends and neighb@@ors . thanks . + +‘présent’ (meaning ‘present’) is hard. A straightforward translation is ‘était là’ but the model chooses ‘present’ which is also appropriate. In Figure 6b, the model uses more computation to predict the definite article ‘les’ since the source has omitted the article for ‘passengers’. + +A clear trend in both benchmarks is that the model requires less computation near the end of decoding to generate the end of sequence marker $< / \mathrm { s } >$ and the preceding full-stop when relevant. In Figure 8, we show the distribution of the exits at the beginning and near the end of test set hypotheses. We consider the beginning of a sequence to be the first $10 \%$ of tokens and the end as the last $10 \%$ of tokens. The exit distributions are shown for three models on WMT’14 En-Fr: Model1 has an average exit of $\mathrm { A E = 2 . 5 3 }$ , Model2 exits at $\mathrm { A E } = 3 . 7 9$ on average and $\mathbf { M o d e l } _ { 3 }$ with $\mathrm { A E } = 4 . 6 8$ . Within the same models, deep exits late are used at the beginning of the sequence and early exits are selected near the end. For heavily regularized models such as $\mathbf { M o d e l } _ { 1 }$ with $\mathrm { A E = 2 . 5 3 }$ , the disparity between beginning and end is less severe as the model exits early most of the time. $\mathbf { M o d e l } _ { 2 }$ and Model3 are less regularized (higher AE) and tend to use late exits at the beginning of the sequence and early exits near the end. On the other hand, the more regularized $\mathbf { M o d e l } _ { 1 }$ with $\mathrm { A E = 2 . 5 3 }$ exits early most of the time. There is also a correlation between the model probability and the amount of computation, particularly in models with low AE . Figure 9 shows the joint histogram of the scores and the selected exit. For both $\mathbf { M o d e l } _ { 1 }$ and $\mathbf { M o d e l } _ { 2 }$ , low exits $( n \leq 2 )$ are used in the high confidence range $[ 0 . 8 - 1 ]$ and high exits $( n \geq 4 )$ ) are used in the low-confidence range $[ 0 - 0 . { \bar { 5 } } ]$ . $\mathbf { M o d e l } _ { 3 }$ has a high average exit $\mathrm { ( A E = 4 . 6 8 ) }$ ) so most tokens exit late, however, in low confidence ranges the model does not exit earlier than $n = 5$ . + +![](images/2f53098f789a934627fba81e59ffbd43421d69fdd9ce9be0f5f4c72460559af7.jpg) +Figure 8: WMT’14 En-Fr test set: exit distributions in the beginning (relative-position: $\mathrm { r p o s } { < } 0 . 1 $ ) and near the end $\mathrm { ( r p o s > 0 . 9 ) }$ ) of the hypotheses of three models. + +![](images/dd8c2c9e3222d3d928628a33e80512b5c1b5a33ff8368bceafa11540393d5485.jpg) +Figure 9: Joint histogram of the exits and the confidence scores for 3 Tok-LL geometric-like models on newstest14. + +# 5 CONCLUSION + +We extended anytime prediction to the structured prediction setting and introduced simple but effective methods to equip sequence models to make predictions at different points in the network. We compared a number of different mechanisms to predict the required network depth and find that a simple correctness based geometric-like classifier obtains the best trade-off between speed and accuracy. Results show that the number of decoder layers can be reduced by more than three quarters at no loss in accuracy compared to a well tuned Transformer baseline. + +# ACKNOWLEDGMENTS + +We thank Laurens van der Maaten for fruitful comments and suggestions. + +# REFERENCES + +Tolga Bolukbasi, Joseph Wang, Ofer Dekel, and Venkatesh Saligrama. Adaptive neural networks for efficient inference. In Proc. of ICML, 2017. +M. Cettolo, J. Niehues, S. Stüker, L. Bentivogli, and M. Federico. Report on the 11th iwslt evaluation campaign. In IWSLT, 2014. +Mostafa Dehghani, Stephan Gouws, Oriol Vinyals, Jakob Uszkoreit, and Lukasz Kaiser. Universal transformers. In Proc. of ICLR, 2018. +Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In Proc. of NAACL, 2019. +Sergey Edunov, Myle Ott, Michael Auli, David Grangier, and Marc’Aurelio Ranzato. Classical structured prediction losses for sequence to sequence learning. In Proc. of NAACL, 2018. +Michael Figurnov, Artem Sobolev, and Dmitry P. Vetrov. Probabilistic adaptive computation time. In ArXiv preprint, 2017. +Jonas Gehring, Michael Auli, David Grangier, Denis Yarats, and Yann N Dauphin. Convolutional sequence to sequence learning. In Proc. of ICML, 2017. +Alex Graves. Adaptive computation time for recurrent neural networks. In ArXiv preprint, 2016. +Gao Huang, Danlu Chen, Tianhong Li, Felix Wu, Laurens van der Maaten, and Kilian Q Weinberger. Multi-scale dense networks for resource efficient image classification. In Proc. of ICLR, 2017. +D. Kingma and J. Ba. Adam: A method for stochastic optimization. In Proc. of ICLR, 2015. +Nathan Ng, Kyra Yee, Alexei Baevski, Myle Ott, Michael Auli, and Sergey Edunov. Facebook fair’s wmt19 news translation task submission. In Proc. of WMT, 2019. +Myle Ott, Sergey Edunov, Alexei Baevski, Angela Fan, Sam Gross, Nathan Ng, David Grangier, and Michael Auli. Fairseq: A fast, extensible toolkit for sequence modeling. In Proc. of NAACL, 2019. +K. Papineni, S. Roukos, T. Ward, and W.-J. Zhu. BLEU: a method for automatic evaluation of machine translation. In Proc. of ACL, 2002. +Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. In Technical report, OpenAI., 2019. +R. Sennrich, B. Haddow, and A. Birch. Neural machine translation of rare words with subword units. In Proc. of ACL, 2016. +Surat Teerapittayanon, Bradley McDanel, and Hsiang-Tsung Kung. Branchynet: Fast inference via early exiting from deep neural networks. In ICPR, 2016. +A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need. In Proc. of NeurIPS, 2017. +Xin Wang, Fisher Yu, Zi-Yi Dou, Trevor Darrell, and Joseph E Gonzalez. Skipnet: Learning dynamic routing in convolutional networks. In Proc. of ECCV, 2018. + +# APPENDIX A LOSS SCALING + +In this section we experiment with different weights for scaling the output classifier losses. Instead of uniform weighting, we bias towards specific output classifiers by assigning higher weights to their losses. Table 2 shows that weighing the classifiers equally provides good results. + +(b) IWSLT De-En - Test + +
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline-34.235.335.635.735.635.935.4
Wn=135.534.135.535.836.136.136.235.6
Wn=n35.332.235.035.836.036.236.335.2
Wn=√n35.433.335.235.835.936.136.135.4
Wn=1/√n35.634.535.435.735.835.835.935.5
Wn=1/n35.334.735.335.535.735.835.835.5
(a) IWSLT De-En - Valid
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline133.734.634.634.634.634.834.5
Wn=134.433.234.434.834.935.034.934.5
Wn=n34.231.433.834.734.834.834.934.1
Wn=√n34.432.534.134.834.935.035.134.4
Wn=1/√n34.633.734.334.634.834.834.934.5
wn=1/n34.233.834.334.534.634.734.734.4
+ +Table 2: Aligned training with different weights $\left( \omega _ { n } \right)$ on IWSLT De-En. For each model we report BLEU on the dev set evaluated with a uniformly sampled exit $n \sim \mathcal { U } ( [ 1 . . 6 ] )$ for each token and a fixed exit $n \in [ 1 . . 6 ]$ throughout the sequence. The average corresponds to the average BLEU over the fixed exits. + +Gradient scaling Adding intermediate supervision at different levels of the decoder results in richer gradients for lower blocks compared to upper blocks. This is because earlier layers affect more loss terms in the compound loss of Eq. (4). To balance the gradients of each block in the decoder, we scale up the gradients of each loss term $\left( - \operatorname { L L } _ { n } \right)$ when it is updating the parameters of its associated block $\mathrm { ( b l o c k } _ { n }$ with parameters $\theta _ { n }$ ) and revert it back to its normal scale before back-propagating it to the previous blocks. Figure 10 and Algorithm 1 illustrate this gradient scaling procedure. The $\theta _ { n }$ are updated with $\gamma _ { n }$ -amplified gradients from the block’s supervision and $( N { - } n )$ gradients from the subsequent blocks. We choose $\gamma _ { n } = \gamma ( N - n )$ to control the ratio $\gamma { : } 1$ as the ratio of the block supervision to the subsequent blocks’ supervisions. + +Table 3 shows that gradient scaling can benefit the lowest layer at the expense of higher layers. +However, no scaling generally works very well. + +![](images/ea1e31e6c741caa889c0b1175b0bed7b176d252c5f35d54ca54e3389a0dd0932.jpg) +Figure 10: Illustration of gradient scaling. + +# Algorithm 1 Pseudo-code for gradient scaling (illustrated for a single step t) + +1: for $n \in 1 . . N$ do +2: $h _ { t } ^ { n } = \mathrm { b l o c k } _ { n } ( h _ { t } ^ { n - 1 } )$ +3: $p ( y _ { t + 1 } | h _ { t } ^ { n } ) = \mathrm { s o f t m a x } ( W _ { n } h _ { t } ^ { n } )$ +4: p(yt+1|hnt ) = SCALE_GRADIENT(p(yt+1|hnt ), γn) +5: if $n < N$ then hnt = SCALE_GRADIENT(hnt , 1 ) +γn+1 +6: end for +7: function SCALE_GRADIENT(Tensor $x$ , scale $\gamma$ ) +8: return $\gamma x + ( 1 - \gamma )$ STOP_GRADIENT $( x )$ +9: $\triangleright$ STOP_GRADIENT in PyTorch with $x$ .detach(). + +# 10: end function + +(b) IWSLT De-En - Test + +
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline134.235.335.635.735.635.935.4
@35.534.135.535.836.136.136.235.6
γ = 0.335.133.734.735.335.735.836.035.2
γ = 0.535.434.835.435.635.635.735.635.4
γ = 0.734.934.635.135.135.235.435.335.1
γ = 0.934.934.835.335.335.335.435.535.3
γ = 1.135.134.935.235.335.335.335.335.2
(a) IWSLT De-En - Valid
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline33.734.634.634.634.634.834.5
034.433.234.434.834.935.034.934.5
γ = 0.334.232.833.934.334.634.835.034.2
γ = 0.534.533.834.234.634.534.734.734.6
γ = 0.734.033.734.234.334.334.334.334.2
γ = 0.934.134.034.234.334.434.434.434.3
γ = 1.134.234.034.334.334.334.334.234.2
+ +Table 3: Aligned training with different gradient scaling ratios $\gamma : 1$ on IWSLT’14 De-En. For each model we report the BLEU4 score evaluated with a uniformly sampled exit $n \sim \mathcal { U } ( [ 1 . . 6 ] )$ for each token and a fixed exit $n \in [ 1 . . 6 ]$ . The average corresponds to the average BLEU4 of all fixed exits. + +# APPENDIX B FLOPS APPROXIMATION + +This section details the computation of the FLOPS we report. The per token FLOPS are for the decoder network only since we use an encoder of the same size for all models. We breakdown the FLOPS of every operation in Algorithm 2 (blue front of the algorithmic statement). We omit non-linearities, normalizations and residual connections. The main operations we account for are dot-products and by extension matrix-vector products since those represent the vast majority of FLOPS (we assume batch size one to simplify the calculation). + +Parameters + +Table 4: FLOPS of basic operations, key parameters and variables for the FLOPS estimation. + +
Parameters decoder embedding dimension.FLOPS
dd de df x t V output vocabulary size.encoder embedding dimension.Operation
The feed-forward network dimension.Dot-product (d)2d-1
source length. Current time-estep (t ≥ 1).Linear din →dout2dindout
+ +With this breakdown, the total computational cost at time-step $t$ of a decoder block that we actually go through, denoted with FC, is: + +$$ +\mathrm { F C } ( \pmb { x } , t ) = 1 2 d _ { d } ^ { 2 } + 4 d _ { f } d _ { d } + 4 t d _ { d } + 4 | \pmb { x } | d _ { d } + 4 [ \mathrm { F i r s t C a l l } ] | \pmb { x } | d _ { d } d _ { e } , +$$ + +where the cost of mapping the source’ keys and values is incurred the first time the block is called (flagged with FirstCall). This occurs at $t = 1$ for the baseline model but it is input-dependent with depth adaptive estimation and may never occur if all tokens exit early. + +If skipped, a block still has to compute the keys and value of the self-attention block so the selfattention of future time-steps can function. We will denote this cost with FS and we have $\mathrm { F S } = 4 d _ { d } ^ { 2 }$ . + +Depending on the halting mechanism, an exit prediction cost, denoted wit FP, is added: + +$$ +\begin{array} { r l } & { \mathrm { F P } ( t , q ( t ) ) = 2 [ [ t = 1 ] ] N d _ { d } } \\ & { \mathrm { F P } ( t , q ( t ) ) = 2 N d _ { d } } \\ & { \mathrm { F P } ( t , q ( t ) ) = 2 d _ { d } q ( t ) } \\ & { \mathrm { F P } ( t , q ( t ) ) = 2 q ( t ) V d _ { d } } \end{array} +$$ + +For a set of source sequences $\{ \pmb { x } ^ { ( i ) } \} _ { i \in \mathcal { I } }$ and generated hypotheses $\{ \pmb { y } ^ { ( i ) } \} _ { i \in \mathcal { I } }$ , the average flops per token is: + +$$ +\begin{array} { r l } & { \mathrm { s e l i n e ~ } ( N \mathrm { ~ b l o c k s } ) ; \quad \frac { 1 } { \sum _ { i } | y ^ { ( i ) } | } \sum _ { i } \sum _ { t = 1 } ^ { | y ^ { ( i ) } | } \left( N \mathrm { ~ F C } ( { \pmb x } ^ { ( i ) } , t ) + 2 V d _ { d } \right) } \\ & { \quad \mathrm { A d a p t i v e ~ d e p t h } ; \quad \frac { 1 } { \sum _ { i } | y ^ { ( i ) } | } \sum _ { i } \sum _ { t = 1 } ^ { | y ^ { ( i ) } | } \left( q ( t ) \mathrm { F C } ( { \pmb x } ^ { ( i ) } , t ) + ( N - q ( t ) ) \mathrm { F S } + \mathrm { F P } ( t , q ( t ) ) + 2 V d _ { d } \right) } \end{array} +$$ + +In the case of confidence thresholding the final output prediction cost $( 2 V d _ { d } )$ is already accounted for in the exit prediction cost FP. + +# Algorithm 2 Adaptive decoding with Tok-geometric-like + +1: Input: source codes $\pmb { s }$ , incremental state +2: Initialization: $t = 1$ , $y _ { 1 } = < s >$ +3: for $n \in 1 \ldots N$ do +4: FirstCall[n] $=$ True. . A flag signaling if the source’ keys and values should be evalua +5: end for +6: while $y _ { t } \neq < / \mathrm { s } >$ do +7: Embed the last output token $y _ { t }$ . +8: for $n \in 1 \ldots N$ do +9: $\triangleright$ Self-attention. +10: - Map the input into a key $( k )$ and value $( v )$ . FLOPS $\scriptstyle \dot { = } 4 d _ { d } ^ { 2 }$ +11: - Map the input into a query $q$ $\cdot \mathrm { F L O P S } { = } 2 d _ { d } ^ { 2 }$ +12: - Score the memory keys with $q$ to get the attention weights $\alpha$ . FLOPS $\scriptstyle \sum 4 t d _ { d }$ +13: - Map the attention output. $\mathrm { F L O P S } { = } 2 d _ { d } ^ { 2 }$ +14: $\triangleright$ Encoder-Decoder interaction. +15: if FirstCall $[ n ]$ then +16: Map the source states into keys and values for the nth block. FLOPS=4|x|dedd +17: FirstCall[n] $=$ False +18: end if +19: - Map the input into a query $q$ . $\mathrm { F L O P S } { = } 2 d _ { d } ^ { 2 }$ +20: - Score the memory keys with $q$ to get the attention weights $\alpha$ . FLOPS=4|x|dd +21: - Map the attention output. $\mathrm { F L O P S } { = } 2 d _ { d } ^ { 2 }$ +22: Feed-forward network. FLO $\scriptstyle { \sum } = 4 d _ { d } d _ { f }$ +23: Estimate the halting probability $\chi _ { t , n }$ . FLOP ${ \mathrm { S } } { = } 2 d _ { d }$ +24: if $\chi _ { t , n } > 0 . 5$ then +25: Exit the loop (Line 8) +26: end if +27: end for +28: if $n < N$ then +29: $\triangleright$ Skipped blocks. +30: for $n _ { s } \in n + 1 \ldots N$ do +31: Copy and map the copied state into a key $( k )$ and value (v). FLOPS $\scriptstyle \cdot = 4 d _ { d } ^ { 2 }$ +32: end for +33: end if +34: Project the final state and sample a new output token. FLOP $\ S { = } 2 V d _ { d }$ +35: $t { + + }$ +36: end while \ No newline at end of file diff --git a/parse/train/SJg7KhVKPH/SJg7KhVKPH_content_list.json b/parse/train/SJg7KhVKPH/SJg7KhVKPH_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..7ed04c26cfa08ba50dbd0bef8f49bfc036843213 --- /dev/null +++ b/parse/train/SJg7KhVKPH/SJg7KhVKPH_content_list.json @@ -0,0 +1,1776 @@ +[ + { + "type": "text", + "text": "DEPTH-ADAPTIVE TRANSFORMER ", + "text_level": 1, + "bbox": [ + 176, + 99, + 588, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Maha Elbayad∗ Univ. Grenoble Alpes ", + "bbox": [ + 184, + 145, + 328, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jiatao Gu, Edouard Grave, Michael Auli Facebook AI Research ", + "bbox": [ + 436, + 145, + 722, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 210, + 544, + 224 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "State of the art sequence-to-sequence models for large scale tasks perform a fixed number of computations for each input sequence regardless of whether it is easy or hard to process. In this paper, we train Transformer models which can make output predictions at different stages of the network and we investigate different ways to predict how much computation is required for a particular sequence. Unlike dynamic computation in Universal Transformers, which applies the same set of layers iteratively, we apply different layers at every step to adjust both the amount of computation as well as the model capacity. On IWSLT German-English translation our approach matches the accuracy of a well tuned baseline Transformer while using less than a quarter of the decoder layers. ", + "bbox": [ + 233, + 242, + 764, + 382 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 412, + 336, + 428 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The size of modern neural sequence models (Gehring et al., 2017; Vaswani et al., 2017; Devlin et al., 2019) can amount to billions of parameters (Radford et al., 2019). For example, the winning entry of the WMT’19 news machine translation task in English-German used an ensemble totaling two billion parameters $\\mathrm { N g }$ et al., 2019). While large models are required to do better on hard examples, small models are likely to perform as well on easy ones, e.g., the aforementioned ensemble is probably not required to translate a short phrase such as \"Thank you\". However, current models apply the same amount of computation regardless of whether the input is easy or hard. ", + "bbox": [ + 174, + 445, + 823, + 542 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we propose Transformers which adapt the number of layers to each input in order to achieve a good speed-accuracy trade off at inference time. We extend Graves (2016; ACT) who introduced dynamic computation to recurrent neural networks in several ways: we apply different layers at each stage, we investigate a range of designs and training targets for the halting module and we explicitly supervise through simple oracles to achieve good performance on large-scale tasks. ", + "bbox": [ + 174, + 549, + 823, + 619 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Universal Transformers (UT) rely on ACT for dynamic computation and repeatedly apply the same layer (Dehghani et al., 2018). Our work considers a variety of mechanisms to estimate the network depth and applies a different layer at each step. Moreover, Dehghani et al. (2018) fix the number of steps for large-scale machine translation whereas we vary the number of steps to demonstrate substantial improvements in speed at no loss in accuracy. UT uses a layer which contains as many weights as an entire standard Transformer and this layer is applied several times which impacts speed. Our approach does not increase the size of individual layers. We also extend the resource efficient object classification work of Huang et al. (2017) and Bolukbasi et al. (2017) to structured prediction where dynamic computation decisions impact future computation. Related work from computer vision includes Teerapittayanon et al. (2016); Figurnov et al. (2017) and Wang et al. (2018) who explored the idea of dynamic routing either by exiting early or by skipping layers. ", + "bbox": [ + 174, + 626, + 825, + 779 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We encode the input sequence using a standard Transformer encoder to generate the output sequence with a varying amount of computation in the decoder network. Dynamic computation poses a challenge for self-attention because omitted layers in prior time-steps may be required in the future. We experiment with two approaches to address this and show that a simple approach works well (§2). Next, we investigate different mechanisms to control the amount of computation in the decoder network, either for the entire sequence or on a per-token basis. This includes multinomial and binomial classifiers supervised by the model likelihood or whether the argmax is already correct as well as simply thresholding the model score (§3). Experiments on IWSLT14 German-English translation (Cettolo et al., 2014) as well as WMT’14 English-French translation show that we can match the performance of well tuned baseline models at up to $76 \\%$ less computation (§4). ", + "bbox": [ + 174, + 786, + 825, + 897 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/c6b3f9ef1172307eaa082df73a5bf06366df36c55192e6d65cd4faba2c5300b2.jpg", + "image_caption": [ + "Figure 1: Training regimes for decoder networks able to emit outputs at any layer. Aligned training optimizes all output classifiers $\\mathcal { C } _ { n }$ simultaneously assuming all previous hidden states for the current layer are available. Mixed training samples $M$ paths of random exits at which the model is assumed to have exited; missing previous hidden states are copied from below. " + ], + "image_footnote": [], + "bbox": [ + 232, + 99, + 779, + 284 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 373, + 823, + 402 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 ANYTIME STRUCTURED PREDICTION ", + "text_level": 1, + "bbox": [ + 174, + 422, + 513, + 438 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We first present a model that can make predictions at different layers. This is known as anytime prediction for computer vision models (Huang et al., 2017) and we extend it to structured prediction. ", + "bbox": [ + 173, + 452, + 825, + 481 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 TRANSFORMER WITH MULTIPLE OUTPUT CLASSIFIERS ", + "text_level": 1, + "bbox": [ + 174, + 497, + 589, + 511 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We base our approach on the Transformer sequence-to-sequence model (Vaswani et al., 2017). Both encoder and decoder networks contain $N$ stacked blocks where each has several sub-blocks surrounded by residual skip-connections. The first sub-block is a multi-head dot-product self-attention and the second a position-wise fully connected feed-forward network. For the decoder, there is an additional sub-block after the self-attention to add source context via another multi-head attention. ", + "bbox": [ + 174, + 522, + 825, + 593 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Given a pair of source-target sequences $( { \\pmb x } , { \\pmb y } )$ , $_ { \\textbf { \\em x } }$ is processed with the encoder to give representations $\\pmb { s } = ( s _ { 1 } , \\ldots , s _ { | \\pmb { x } | } )$ . Next, the decoder generates $\\textbf { { y } }$ step-by-step. For every new token $\\pmb { y } _ { t }$ input |to the decoder at time $t$ , the $N$ decoder blocks process it to yield hidden states $\\left( h _ { t } ^ { n } \\right) _ { 1 \\leq n \\leq N }$ : ", + "bbox": [ + 173, + 599, + 825, + 643 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/173e6671d5a92e8a753034fc631b60d8c2bdc6736989c2617137defad93ccea6.jpg", + "text": "$$\nh _ { t } ^ { 0 } = \\mathrm { e m b e d } ( y _ { t } ) , \\quad h _ { t } ^ { n } = \\mathrm { b l o c k } _ { n } ( h _ { \\leq t } ^ { n - 1 } , \\pmb { s } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 647, + 643, + 667 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\mathrm { b l o c k } _ { n }$ is the mapping associated with the $n ^ { \\mathrm { t h } }$ block and embed is a lookup table. ", + "bbox": [ + 171, + 670, + 750, + 685 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The output distribution for predicting the next token is computed by feeding the activations of the last decoder layer $h _ { t } ^ { N }$ into a softmax normalized output classifier $W$ : ", + "bbox": [ + 173, + 690, + 823, + 719 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/987f4b801efa8ca2862a3a1e648e5812689ffb77afdcca743932f63a867db52e.jpg", + "text": "$$\np ( y _ { t + 1 } | h _ { t } ^ { N } ) = \\mathrm { s o f t m a x } ( W h _ { t } ^ { N } )\n$$", + "text_format": "latex", + "bbox": [ + 392, + 722, + 604, + 741 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Standard Transformers have a single output classifier attached to the top of the decoder network. However, for dynamic computation we need to be able to make predictions at different stages of the network. To achieve this, we attach output classifiers $\\mathcal { C } _ { n }$ parameterized by $W _ { n }$ to the output $h _ { t } ^ { n }$ of each of the $N$ decoder blocks: ", + "bbox": [ + 173, + 748, + 825, + 804 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/71d7d0b35c4f7ef40121d8ff357a8c783c54b9916cf82a99943b21c6bdd25dbd.jpg", + "text": "$$\n\\forall n , \\ p ( y _ { t + 1 } | h _ { t } ^ { n } ) = \\operatorname { s o f t m a x } ( W _ { n } h _ { t } ^ { n } )\n$$", + "text_format": "latex", + "bbox": [ + 375, + 806, + 619, + 824 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The classifiers can be parameterized independently or we can share the weights across the $N$ blocks. ", + "bbox": [ + 178, + 825, + 821, + 840 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 TRAINING MULTIPLE OUTPUT CLASSIFIERS ", + "text_level": 1, + "bbox": [ + 174, + 856, + 514, + 871 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Dynamic computation enables the model to use any of the $N$ exit classifiers instead of just the final one. Some of our models can choose a different output classifier at each time-step which results in an exponential number of possible output classifier combinations in the sequence length. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We consider two possible ways to train the decoder network (Figure 1). Aligned training optimizes all classifiers simultaneously and assumes all previous hidden states required by the self-attention are available. However, at test time this is often not the case when we choose a different exit for every token which leads to misaligned states. Instead, mixed training samples several sequences of exits for a given sentence and exposes the model to hidden states from different layers. ", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Generally, for a given output sequence $\\textbf { { y } }$ , we have a sequence of chosen exits $( n _ { 1 } , \\ldots , n _ { | { \\pmb y } | } )$ and we denote the block at which we exit at time $t$ as $n _ { t }$ . ", + "bbox": [ + 173, + 180, + 823, + 209 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2.1 ALIGNED TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 224, + 369, + 238 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Aligned training assumes all hidden states $h _ { 1 } ^ { n - 1 } , \\ldots , h _ { t } ^ { n - 1 }$ are available in order to compute selfattention and it optimizes $N$ loss terms, one for each exit (Figure 1a): ", + "bbox": [ + 174, + 247, + 823, + 277 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c4d129c2c1599741dcd4467f0fe324e5495ff8416146f1b49790801fd970f8dd.jpg", + "text": "$$\n\\mathrm { L L } _ { t } ^ { n } = \\log p ( y _ { t } | h _ { t - 1 } ^ { n } ) , \\quad \\mathrm { L L } ^ { n } = \\sum _ { t = 1 } ^ { | y | } \\mathrm { L L } _ { t } ^ { n } , \\quad \\mathcal { L } _ { d e c } ( \\pmb { x } , \\pmb { y } ) = - \\frac { 1 } { \\sum _ { n } \\omega _ { n } } \\sum _ { n = 1 } ^ { N } \\omega _ { n } \\mathrm { L L } ^ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 225, + 284, + 776, + 329 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The compound loss $\\mathcal { L } _ { d e c } ( \\pmb { x } , \\pmb { y } )$ is a weighted average of $N$ terms w.r.t. to $\\left( \\omega _ { 1 } , \\hdots \\omega _ { N } \\right)$ . We found that uniform weights achieve better BLEU compared to other weighing schemes $_ { . c . f }$ . Appendix A). At inference time, not all time-steps will have hidden states for the current layer since the model exited early. In this case, we simply copy the last computed state to all upper layers, similar to mixed training $( \\ S 2 . 2 . 2 )$ . However, we do apply layer-specific key and value projections to the copied state. ", + "bbox": [ + 173, + 334, + 825, + 406 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2.2 MIXED TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 420, + 352, + 435 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Aligned training assumes that all hidden states of the previous time-steps are available but this assumption is unrealistic since an early exit may have been chosen previously. This creates a mismatch between training and testing. Mixed training reduces the mismatch by training the model to use hidden states from different blocks of previous time-steps for self-attention. We sample $M$ different exit sequences (n(m)1 , . $( n _ { 1 } ^ { ( m ) } , \\dots n _ { | \\pmb { y } | } ^ { ( m ) } ) _ { 1 \\leq m \\leq M }$ and evaluate the following loss: ", + "bbox": [ + 173, + 444, + 826, + 525 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/72797010deb0549235109abb626da946579d717a38de4c8e5951c1db2a56aa7c.jpg", + "text": "$$\n\\mathrm { L L } ( n _ { 1 } , \\dots , n _ { | \\mathbf { y } | } ) = \\sum _ { t = 1 } ^ { | \\mathbf { y } | } \\log p ( y _ { t } | h _ { t - 1 } ^ { n _ { t } } ) , \\quad \\mathscr { L } _ { d e c } ( \\mathbf { x } , \\mathbf { y } ) = - \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\mathrm { L L } ( n _ { 1 } ^ { ( m ) } , \\dots , n _ { | \\mathbf { y } | } ^ { ( m ) } ) .\n$$", + "text_format": "latex", + "bbox": [ + 196, + 531, + 781, + 577 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "When $n _ { t } ~ < ~ N$ , we copy the last evaluated hidden state $h _ { t } ^ { n }$ to the subsequent layers so that the self-attention of future time steps can function as usual (see Figure 1b). ", + "bbox": [ + 174, + 582, + 825, + 611 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 ADAPTIVE DEPTH ESTIMATION ", + "text_level": 1, + "bbox": [ + 176, + 632, + 462, + 647 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We present a variety of mechanisms to predict the decoder block at which the model will stop and output the next token, or when it should exit to achieve a good speed-accuracy trade-off. We consider two approaches: sequence-specific depth decodes all output tokens using the same block (§3.1) while token-specific depth determines a separate exit for each individual token (§3.2). ", + "bbox": [ + 174, + 662, + 825, + 719 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We model the distribution of exiting at time-step $t$ with a parametric distribution $q _ { t }$ where $q _ { t } ( n )$ is the probability of computing ${ \\mathrm { b l o c k } } _ { 1 } , \\ldots , { \\mathrm { b l o c k } } _ { n }$ and then emitting a prediction with $\\mathcal { C } _ { n }$ . The parameters of $q _ { t }$ are optimized to match an oracle distribution $q _ { t } ^ { * }$ with cross-entropy: ", + "bbox": [ + 173, + 726, + 825, + 768 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/23a8696a2f5ad07d1733da4481e66333edcd93669204595414f50193eee75a9a.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { e x i t } } ( { \\pmb x } , { \\pmb y } ) = \\sum _ { t } H ( q _ { t } ^ { * } ( { \\pmb x } , { \\pmb y } ) , q _ { t } ( { \\pmb x } ) )\n$$", + "text_format": "latex", + "bbox": [ + 374, + 775, + 624, + 809 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The exit loss $( \\mathcal { L } _ { \\mathrm { e x i t } } )$ is back-propagated to the encoder-decoder parameters. We simultaneously optimize the decoding loss (Eq. (4)) and the exit loss (Eq. (6)) balanced by a hyper-parameter $\\alpha$ to ensure that the model maintains good generation accuracy. The final loss takes the form: ", + "bbox": [ + 173, + 814, + 826, + 857 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/932ae1a6796b7e3939d672c670a69a5d33b381a1c69de2129bc624b937848d7c.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( \\pmb { x } , \\pmb { y } ) = \\mathcal { L } _ { d e c } ( \\pmb { x } , \\pmb { y } ) + \\alpha \\mathcal { L } _ { \\mathrm { e x i t } } ( \\pmb { x } , \\pmb { y } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 370, + 863, + 625, + 881 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the following we describe for each approach how the exit distribution $q _ { t }$ is modeled (illustrated in Figure 2) and how the oracle distribution $q _ { t } ^ { * }$ is inferred. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/9ffbc9ee07d1225e2d2a69f3d20fd736e999c534160e2004ae183049c51fafd3.jpg", + "image_caption": [ + "Figure 2: Variants of the adaptive depth prediction classifiers. Sequence-specific depth uses a multinomial classifier to choose an exit for the entire output sequence based on the encoder output $s$ (2a). It then outputs a token at this depth with classifier $\\mathcal { C } _ { n }$ . The token-specific multinomial classifier determines the exit after the first block and proceeds up to the predicted depth before outputting the next token (2b). The token geometric-like classifier (2c) makes a binary decision after every block to dictate whether to continue (C) to the next block or to stop (S) and emit an output distribution. " + ], + "image_footnote": [], + "bbox": [ + 173, + 98, + 816, + 267 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 SEQUENCE-SPECIFIC DEPTH: ", + "text_level": 1, + "bbox": [ + 176, + 388, + 413, + 404 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For sequence-specific depth, the exit distribution $q$ and the oracle distribution $q ^ { * }$ are independent of the time-step so we drop subscript $t$ . We condition the exit on the source sequence by feeding the average $s$ of the encoder outputs to a multinomial classifier: ", + "bbox": [ + 173, + 415, + 825, + 457 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f4bc96fd7bbe813173f5ef7cc88e7087261864689df8aea127d97e0e420537a0.jpg", + "text": "$$\ns = \\frac { 1 } { | \\pmb { x } | } \\sum _ { t } s _ { t } , \\quad q ( n | \\pmb { x } ) = \\mathrm { s o f t m a x } ( W _ { h } s + b _ { h } ) \\in \\mathbb { R } ^ { N } ,\n$$", + "text_format": "latex", + "bbox": [ + 312, + 463, + 684, + 500 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $W _ { h }$ and $b _ { h }$ are the weights and biases of the halting mechanism. We consider two oracles to determine which of the $N$ blocks should be chosen. The first is based on the sequence likelihood and the second looks at an aggregate of the correctly predicted tokens at each block. ", + "bbox": [ + 174, + 505, + 825, + 547 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Likelihood-based: This oracle is based on the likelihood of the entire sequence after each block and we optimize it with the Dirac delta centered around the exit with the highest sequence likelihood. ", + "bbox": [ + 173, + 563, + 825, + 592 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b1c9b63a9273a6fe3e74a982881b171debd5a5bfb29d25bcd78cd55c30168f21.jpg", + "text": "$$\nq ^ { * } ( { \\pmb x } , { \\pmb y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } { \\mathrm { L L } ^ { n } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 597, + 596, + 619 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We add a regularization term to encourage lower exits that achieve good likelihood: ", + "bbox": [ + 174, + 626, + 720, + 641 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a8cb505c8f388593729aa3b677775078daa07e53f8e34aeebbb6042f05054c72.jpg", + "text": "$$\nq ^ { * } ( { \\pmb x } , { \\pmb y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } { \\mathrm { L L } ^ { n } } - \\lambda n ) .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 646, + 612, + 670 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Correctness-based: Likelihood ignores whether the model already assigns the highest score to the correct target. Instead, this oracle chooses the lowest block that assigns the largest score to the correct prediction. For each block, we count the number of correctly predicted tokens over the sequence and choose the block with the most number of correct tokens. A regularization term controls the trade-off between speed and accuracy. ", + "bbox": [ + 173, + 681, + 825, + 753 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2a1f65cdafa7f4af34fddaa01de5644d9b4343db42c4cf60e9a6d80f7eacf9cd.jpg", + "text": "$$\nC ^ { n } = \\# \\{ t | y _ { t } = \\arg \\operatorname* { m a x } _ { y } p ( y | h _ { t - 1 } ^ { n } ) \\} , \\quad q ^ { * } ( x , y ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } C ^ { n } - \\lambda n ) .\n$$", + "text_format": "latex", + "bbox": [ + 250, + 758, + 748, + 782 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Oracles based on test metrics such as BLEU are feasible but expensive to compute since we would need to decode every training sentence $N$ times. We leave this for future work. ", + "bbox": [ + 174, + 796, + 825, + 825 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 TOKEN-SPECIFIC DEPTH: ", + "text_level": 1, + "bbox": [ + 174, + 842, + 387, + 856 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The token-specific approach can choose a different exit at every time-step. We consider two options for the exit distribution $q _ { t }$ at time-step t: a multinomial with a classifier conditioned on the first decoder hidden state $h _ { t } ^ { 1 }$ and a geometric-like where an exit probability $\\chi _ { t } ^ { n }$ is estimated after each block based on the activations of the current block $h _ { t } ^ { n }$ . ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Multinomial $q _ { t }$ ", + "text_level": 1, + "bbox": [ + 174, + 103, + 285, + 118 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/52e827a70be346a7ac2d729f516e014c50964b46500aab28a94300feccb38731.jpg", + "text": "$$\nq _ { t } ( n | \\pmb { x } , \\pmb { y } _ { < t } ) = \\mathrm { s o f t m a x } ( { W _ { h } } h _ { t } ^ { 1 } + b _ { h } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 369, + 123, + 627, + 142 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The most probable exit arg max $q _ { t } ( n | \\pmb { x } , \\pmb { y } _ { < t } )$ is selected at inference. ", + "bbox": [ + 174, + 147, + 629, + 164 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Geometric-like $q _ { t }$ ", + "bbox": [ + 174, + 178, + 302, + 193 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/37a64ea2c3490986c8851f9d201ea6b6c287cff9e6addc31c3a435dd73b3c3d5.jpg", + "text": "$$\n\\begin{array} { l } { \\forall n \\in [ 1 . . N - 1 ] , ~ \\chi _ { t } ^ { n } = \\mathrm { s i g m o i d } ( w _ { h } ^ { \\top } h _ { t } ^ { n } + b _ { h } ) , } \\\\ { q _ { t } ( n | \\pmb { x } , \\pmb { y } _ { < t } ) = \\left\\{ \\begin{array} { l l } { \\chi _ { t } ^ { n } \\displaystyle \\prod _ { n ^ { \\prime } < n } ( 1 - \\chi _ { t } ^ { n ^ { \\prime } } ) , \\mathrm { i f } n < N } \\\\ { \\displaystyle \\prod _ { n ^ { \\prime } < N } ( 1 - \\chi _ { t } ^ { n ^ { \\prime } } ) , \\mathrm { o t h e r w i s e } } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 195, + 642, + 272 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where, $d$ is the dimension of the decoder states, $W _ { h } \\ \\in \\ \\mathbb { R } ^ { N \\times d }$ and $w _ { h } \\in \\mathbb { R } ^ { d }$ are the weights of the halting mechanisms, and $b _ { h }$ their biases. During inference the decoder exits when the halting signal $\\chi _ { t } ^ { n }$ exceeds a threshold $\\tau _ { n }$ which we tune on the valid set to achieve a better accuracy-speed trade-off. If thresholds $\\left( \\tau _ { n } \\right) _ { 1 \\leq n < N }$ have not been exceeded, then we default to exiting at block $N$ . ", + "bbox": [ + 173, + 279, + 825, + 337 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The two classifiers are trained to minimize the cross-entropy with respect to either one the following oracle distributions: ", + "bbox": [ + 173, + 342, + 823, + 371 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Likelihood-based: At each time-step $t$ , we choose the block whose exit classifier has the highest likelihood plus a regularization term weighted by $\\lambda$ to encourage lower exits. ", + "bbox": [ + 173, + 386, + 821, + 415 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/14352d99319d701f91c816a211d885a60b4119c1196408ed92c89f9fd5fd2834.jpg", + "text": "$$\nq _ { t } ^ { * } ( \\pmb { x } , \\pmb { y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } \\mathrm { L L } _ { t } ^ { n } - \\lambda n )\n$$", + "text_format": "latex", + "bbox": [ + 385, + 421, + 612, + 444 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This oracle ignores the impact of the current decision on the future time-steps and we therefore consider smoothing the likelihoods with an RBF kernel. ", + "bbox": [ + 173, + 449, + 826, + 478 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/bcccd6b00d0cea955cbf50d76b119d50b34bd42e0a2c080e2ae2fb030deb3672.jpg", + "text": "$$\n\\kappa ( t , t ^ { \\prime } ) = e ^ { - \\frac { \\lvert t - t ^ { \\prime } \\rvert ^ { 2 } } { \\sigma } } , \\quad \\widetilde { \\mathrm { L L } _ { t } ^ { n } } = \\sum _ { t ^ { \\prime } = 1 } ^ { \\lfloor y \\rfloor } \\kappa ( t , t ^ { \\prime } ) \\mathrm { L L } _ { t ^ { \\prime } } ^ { n } , \\quad q _ { t } ^ { * } ( { \\pmb x } , { \\pmb y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } \\widetilde { \\mathrm { L L } _ { t } ^ { n } } - \\lambda n ) ,\n$$", + "text_format": "latex", + "bbox": [ + 202, + 484, + 767, + 530 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where we control the size of the surrounding context with $\\sigma$ the kernel width. We refer to this oracle as $\\operatorname { L L } ( \\sigma , \\lambda )$ including the case where we only look at the likelihood of the current token with $\\sigma \\to 0$ . ", + "bbox": [ + 173, + 535, + 826, + 564 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Correctness-based: Similar to the likelihood-based oracle we can look at the correctness of the prediction at time-step $t$ as well as surrounding positions. We define the target $q _ { t } ^ { * }$ as follows: ", + "bbox": [ + 171, + 578, + 823, + 608 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b259a9bae7ebbd7cf6cb842668fb15f4b34352ed9f421e2dc514cff7db3c615b.jpg", + "text": "$$\n\\begin{array} { l } { { C _ { t } ^ { n } = \\mathbb { 1 } \\big [ y _ { t } = \\arg \\operatorname* { m a x } _ { y } p ( y | h _ { t - 1 } ^ { n } ) \\big ] , \\quad \\widetilde { C _ { t } ^ { n } } = \\displaystyle \\sum _ { t ^ { \\prime } = 1 } ^ { | y | } \\kappa ( t , t ^ { \\prime } ) C _ { t } ^ { n } , } } \\\\ { { q _ { t } ^ { * } ( { \\pmb x } , { \\pmb y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } \\widetilde { C _ { t } ^ { n } } - \\lambda n ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 614, + 696, + 686 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Confidence thresholding Finally, we consider thresholding the model predictions (§2), i.e., exit when the maximum score of the current output classifier $p \\bar { ( } y _ { t + 1 } | h _ { t } ^ { n } )$ exceeds a hyper-parameter threshold $\\tau _ { n }$ . This does not require training and the thresholds $\\pmb { \\tau } = ( \\tau _ { 1 } , \\dots , \\tau _ { N - 1 } )$ are simply tuned on the valid set to maximize BLEU. Concretely, for 10k iterations, we sample a sequence of thresholds $\\tau \\sim \\mathcal { U } ( 0 , 1 ) ^ { N - 1 }$ , decode the valid set with the sampled thresholds and then evaluate the BLEU score and computational cost achieved with this choice of $\\tau$ . After $1 0 \\mathrm { k }$ evaluations we pick the best performing thresholds, that is $\\tau$ with the highest BLEU in each cost segment. ", + "bbox": [ + 173, + 698, + 825, + 797 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 816, + 326, + 833 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 174, + 848, + 374, + 863 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We evaluate on several benchmarks and measure tokenized BLEU (Papineni et al., 2002): ", + "bbox": [ + 169, + 873, + 759, + 888 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "IWSLT’14 German to English (De-En). We use the setup of Edunov et al. (2018) and train on 160K sentence pairs. We use $N = 6$ blocks, a feed-forward network (ffn) of intermediate-dimension ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/40ca2a722a6fbe8403a4a3e403efaeeb90e7d9b542c5a79a161332a8d6633134.jpg", + "table_caption": [ + "Table 1: Aligned vs. mixed training on IWSLT De-En. We report valid BLEU for a uniformly sampled exit $\\overset { \\cdot } { n } \\sim \\mathcal { U } ( [ 1 . . 6 ] )$ at each token, a fixed exit $n \\in [ 1 . . 6 ]$ for all tokens, as well as the average BLEU over the fixed exits. As baseline we show six standard Transformer models with 1-6 blocks. " + ], + "table_footnote": [], + "table_body": "
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline134.235.335.635.735.635.935.4
Aligned (ωn = 1)35.534.135.535.836.136.136.235.6
Mixed M =134.132.934.334.534.534.634.534.2
Mixed M = 335.133.935.235.435.535.535.535.2
Mixed M = 635.334.235.435.835.935.835.935.5
", + "bbox": [ + 176, + 101, + 823, + 207 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1024, 4 heads, dropout 0.3, embedding dimension $d _ { \\mathrm { e n c } } = 5 1 2$ for the encoder and $d _ { \\mathrm { d e c } } = 2 5 6$ for the decoder. Embeddings are untied with 6 different output classifiers. We evaluate with a single checkpoint and a beam of width 5. ", + "bbox": [ + 176, + 289, + 821, + 332 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "WMT’14 English to French $( \\mathbf { E n } \\mathbf { - } \\mathbf { F r } )$ . We also experiment on the much larger WMT’14 EnglishFrench task comprising $3 5 . 5 \\mathrm { m }$ training sentence pairs. We develop on 26k held out pairs and test on newstest14. The vocabulary consists of 44k joint BPE types (Sennrich et al., 2016). We use a Transformer big architecture and tie the embeddings of the encoder, the decoder and the output classifiers $( ( W _ { n } ) _ { 1 \\leq n \\leq 6 } ; \\ S 2 . 1 )$ . We average the last ten checkpoints and use a beam of width 4. ", + "bbox": [ + 174, + 338, + 823, + 409 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Models are implemented in fairseq (Ott et al., 2019) and are trained with Adam (Kingma & Ba, 2015). We train for 50k updates on 128 GPUs with a batch size of 460k tokens for WMT’14 En-Fr and on 2 GPUs with $^ \\mathrm { 8 k }$ tokens per batch for IWSLT’14 De-En. To stabilize training, we re-normalize the gradients if the norm exceeds $g _ { \\mathrm { c l i p } } = 3$ . ", + "bbox": [ + 174, + 415, + 825, + 472 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For models with adaptive exits, we first train without exit prediction ( $\\alpha = 0$ in Eq. (7)) using the aligned mode $( c . f . \\ S 2 . 2 . 1 )$ for $5 0 \\mathrm { k }$ updates and then continue training with $\\alpha \\neq 0$ until convergence. The exit prediction classifiers are parameterized by a single linear layer (Eq. (8)) with the same input dimension as the embedding dimension, e.g., 1024 for a big Transformer; the output dimension is $N$ for a multinomial classifier or one for geometric-like. We exit when $\\chi _ { t , n } > 0 . 5$ for geometric-like classifiers. ", + "bbox": [ + 174, + 478, + 825, + 561 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 TRAINING MULTIPLE OUTPUT CLASSIFIERS ", + "text_level": 1, + "bbox": [ + 174, + 583, + 513, + 597 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We first compare the two training regimes for our model $( \\ S 2 . 2 )$ . Aligned training performs selfattention on aligned states (§2.2.1) and mixed training exposes self-attention to hidden states from different blocks $( \\ S 2 . 2 . 2 )$ . ", + "bbox": [ + 174, + 609, + 825, + 652 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We compare the two training modes when choosing either a uniformly sampled exit or a fixed exit $n = 1 , \\ldots , 6$ at inference time for every time-step. The sampled exit experiment tests the robustness to mixed hidden states and the fixed exit setup simulates an ideal setting where all previous states are available. As baselines we show six separate standard Transformers with $N \\in [ 1 . . 6 ]$ decoder blocks. All models are trained with an equal number of updates and mixed training with $M { = } 6$ paths is most comparable to aligned training since the number of losses per sample is identical. ", + "bbox": [ + 174, + 659, + 823, + 742 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Table 1 shows that aligned training outperforms mixed training both for fixed exits as well as for randomly sampled exits. The latter is surprising since aligned training never exposes the self-attention mechanism to hidden states from other blocks. We suspect that this is due to the residual connections which copy features from lower blocks to subsequent layers and which are ubiquitous in Transformer models (§2). Aligned training also performs very competitively to the individual baseline models. ", + "bbox": [ + 174, + 750, + 825, + 819 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Aligned training is conceptually simple and fast. We can process a training example with $N$ exits in a single forward/backward pass while $M$ passes are needed for mixed training. In the remaining paper, we use the aligned mode to train our models. Appendix A reports experiments with weighing the various output classifiers differently but we found that a uniform weighting scheme worked well. On our largest setup, WMT’14 English-French, the training time of an aligned model with six output classifiers increases only marginally by about $1 \\%$ compared to a baseline with a single output classifier keeping everything else equal. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/08aa6865d38f0ae73b75393499522f9a002e3778b5c4d0a3402639a98af647f1.jpg", + "image_caption": [ + "Figure 3: Trade-off between speed (average exit or AE) and accuracy (BLEU) for depth-adaptive methods on the IWSLT14 De-En test set. " + ], + "image_footnote": [], + "bbox": [ + 176, + 98, + 818, + 276 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/8208d9e007a1f6ef7e4f203a9c4bf94a1261af5e96cf9c95b86e08c15a27b5bb.jpg", + "image_caption": [ + "Figure 4: Effect of the hyper-parameters $\\sigma$ and $\\lambda$ on the average exit (AE) measured on the valid set of IWSLT’14 De-En. " + ], + "image_footnote": [], + "bbox": [ + 307, + 347, + 691, + 515 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 ADAPTIVE DEPTH ESTIMATION ", + "text_level": 1, + "bbox": [ + 176, + 588, + 428, + 602 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Next, we train models with aligned states and compare adaptive depth classifiers in terms of BLEU as well as computational effort. We measure the latter as the average exit per output token (AE). ", + "bbox": [ + 173, + 616, + 823, + 645 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As baselines we use again six separate standard Transformers with $N \\in [ 1 . . 6 ]$ with a single output classifier. We also measure the performance of the aligned mode trained model for fixed exits $n \\in$ [1..6]. For the adaptive depth token-specific models (Tok), we train four combinations: likelihoodbased oracle (LL) $^ +$ geometric-like, likelihood-based oracle (LL) $^ +$ multinomial, correctness based oracle $\\left( \\mathbf { C } \\right) +$ geometric-like and correctness-based oracle $\\left( \\mathbf { C } \\right) +$ multinomial. Sequence-specific models (Seq) are trained with the correctness oracle (C) and the likelihood oracle (LL) with different values for the regularization weight $\\lambda$ . All parameters are tuned on the valid set and we report results on the test set for a range of average exits. ", + "bbox": [ + 174, + 651, + 825, + 762 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 3 shows that the aligned model (blue line) can match the accuracy of a standard 6-block Transformer (black line) at half the number of layers ${ \\mathrm { ~ \\ : ~ } } _ { n } = 3 { \\mathrm { ~ \\ : ~ } }$ ) by always exiting at the third block. The aligned model outperforms the baseline for $n = 2 , \\ldots , 6$ . ", + "bbox": [ + 174, + 770, + 825, + 811 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For token specific halting mechanisms (Figure 3a) the geometric-like classifiers achieves a better speed-accuracy trade-off than the multinomial classifiers (filled vs. empty triangles). For geometriclike classifiers, the correctness oracle outperforms the likelihood oracle (Tok-C geometric-like vs. Tok-LL geometric-like) but the trend is less clear for multinomial classifiers. At the sequence-level, likelihood is the better oracle (Figure 3b). ", + "bbox": [ + 174, + 818, + 825, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The rightmost Tok-C geometric-like point $( \\sigma = 0$ , $\\lambda = 0 . 1$ ) achieves 34.73 BLEU at $\\mathrm { A E } = 1 . 4 2$ which corresponds to similar accuracy as the $N \\ = \\ 6$ baseline at $76 \\%$ fewer decoding blocks. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/72eb06ffdaeb2b8f53f187689e2b19f54d633c162199e0bfd63806ba35e8e616.jpg", + "image_caption": [ + "Figure 5: Speed and accuracy on the WMT’14 English-French benchmark $( c . f .$ . Figure 3). " + ], + "image_footnote": [], + "bbox": [ + 285, + 99, + 712, + 276 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The best accuracy of the aligned model is 34.95 BLEU at exit 5 and the best comparable Tok-C geometric-like configuration achieves 34.99 BLEU at $\\mathrm { A E } = 1 . 9 7$ , or $61 \\%$ fewer decoding blocks. When fixing the budget to two decoder blocks, Tok-C geometric-like with $\\mathrm { A E } = 1 . 9 7$ achieves BLEU 35, a 0.64 BLEU improvement over the baseline ( $N = 2$ ) and aligned which both achieve BLEU 34.35. ", + "bbox": [ + 174, + 339, + 825, + 410 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Confidence thresholding (Figure 3c) performs very well but cannot outperform Tok-C geometriclike. ", + "bbox": [ + 173, + 416, + 823, + 445 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Ablation of hyper-parameters In this section, we look at the effect of the two main hyperparameters on IWSLT’14 De-En: $\\lambda$ the regularization scale ( $_ { . c . f }$ . Eq. (9)), and the RBF kernel width $\\sigma$ used to smooth the scores $( c . f$ . Eq. (15)). We train Tok-LL Geometric-like models and evaluate them with their default thresholds (exit if $\\chi _ { t } ^ { n } > 0 . 5 )$ . Figure 4a shows that higher values of $\\lambda$ lead to lower exits. Figure 4b shows the effect of $\\sigma$ for two values of $\\lambda$ . In both curves, we see that wider kernels favor higher exits. ", + "bbox": [ + 174, + 460, + 825, + 544 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 SCALING THE ADAPTIVE-DEPTH MODELS ", + "text_level": 1, + "bbox": [ + 176, + 560, + 500, + 574 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Finally, we take the best performing models form the IWSLT benchmark and test them on the large WMT’14 English-French benchmark. Results on the test set (Figure 5a) show that adaptive depth still shows improvements but that they are diminished in this very large-scale setup. Confidence thresholding works very well and sequence-specific depth approaches improve only marginally over the baseline. Tok-LL geometric-like can match the best baseline result of BLEU 43.4 $N = 6$ ) by using only $\\mathrm { A E } = 2 . 4 0$ which corresponds to $40 \\%$ of the decoder blocks; the best aligned result of BLEU 43.6 can be matched with $\\mathrm { A E } = 3 . 2 5$ . In this setup, Tok-LL geometric-like slightly outperforms the Tok-C counterpart. ", + "bbox": [ + 174, + 585, + 825, + 698 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Confidence thresholding matches the accuracy of the $N { = } 6$ baseline with AE 2.5 or $59 \\%$ fewer decoding blocks. However, confidence thresholding requires computing the output classifier at each block to determine whether to halt or continue. This is a large overhead since output classifiers predict 44k types for this benchmark (§4.1). To better account for this, we measure the average number of FLOPs per output token (details in Appendix B). Figure 5b shows that the Tok-LL geometric-like approach provides a better trade-off when the overhead of the output classifiers is considered. ", + "bbox": [ + 174, + 704, + 825, + 789 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.5 QUALITATIVE RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 805, + 375, + 819 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The exit distribution for a given sample can give insights into what a Depth-Adaptive Transformer decoder considers to be a difficult task. In this section, for each hypothesis $\\widetilde { \\boldsymbol { y } }$ , we will look at the sequence of selected exits $( n _ { 1 } , \\ldots , n _ { | \\tilde { \\pmb { y } } | } )$ and the probability scores $( p _ { 1 } , \\hdots p _ { | \\widetilde { \\pmb { y } } | } )$ with $\\begin{array} { r l } { p _ { t } } & { { } = } \\end{array}$ $p ( \\widetilde { y } _ { t } | h _ { t - 1 } ^ { n _ { t } } )$ e i.e. the confidence of the model in the sampled token at the selected exit. ", + "bbox": [ + 174, + 832, + 825, + 888 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figures 6 and 7 show hypotheses from the WMT’14 En-Fr and IWSLT’14 De-En test sets, respectively. For each hypothesis we state the exits and the probability scores. In Figure 6a, predicting ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/73aee9ecfc92f773d75e7f5027eab3b98129a636e72fa4ceb61bb68a12f85cc2.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 178, + 104, + 478, + 180 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/684d9684562ed9f33faf762808bedff084b7388602287045fea2467431609d51.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 495, + 104, + 857, + 180 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) Src: Chi@@rac , the Prime Minister , was there . Ref: Chi@@rac , Premier ministre , est là . ", + "bbox": [ + 173, + 195, + 485, + 220 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(b) Src: But passengers shoul@@dn’t expect changes to happen immediately . Ref: Mais les passagers ne devraient pas ${ \\mathrm { ~ s ~ } } ^ { \\prime }$ attendre à des changements immédiats . ", + "bbox": [ + 485, + 195, + 821, + 246 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Figure 6: Examples from the WMT’14 En-Fr test set (newstest14) with Tok-LL geometric-like depth estimation. Token exits are in blue and confidence scores are in gray. The $\\bullet _ { @ \\mathbb { Q } } ,$ are due to BPE or subword tokenization. For each example the source (Src) and the reference (Ref) are provided in the caption. ", + "bbox": [ + 173, + 258, + 826, + 314 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/c09a06b4990673a86c9b78ffc1d7e7a2e7d8b550d430e028ee6b5dbf8b98eb27.jpg", + "image_caption": [ + "Figure 7: Example from the IWSLT’14 De-En test set with Tok-LL geometric-like depth estimation. See Figure 6 for more details. " + ], + "image_footnote": [], + "bbox": [ + 274, + 333, + 718, + 411 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) Src: diesen trick können sie ihren freunden und nachbarn vor@@führen . danke . Ref: there is a trick you can do for your friends and neighb@@ors . thanks . ", + "bbox": [ + 236, + 428, + 735, + 454 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "‘présent’ (meaning ‘present’) is hard. A straightforward translation is ‘était là’ but the model chooses ‘present’ which is also appropriate. In Figure 6b, the model uses more computation to predict the definite article ‘les’ since the source has omitted the article for ‘passengers’. ", + "bbox": [ + 174, + 522, + 825, + 565 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "A clear trend in both benchmarks is that the model requires less computation near the end of decoding to generate the end of sequence marker $< / \\mathrm { s } >$ and the preceding full-stop when relevant. In Figure 8, we show the distribution of the exits at the beginning and near the end of test set hypotheses. We consider the beginning of a sequence to be the first $10 \\%$ of tokens and the end as the last $10 \\%$ of tokens. The exit distributions are shown for three models on WMT’14 En-Fr: Model1 has an average exit of $\\mathrm { A E = 2 . 5 3 }$ , Model2 exits at $\\mathrm { A E } = 3 . 7 9$ on average and $\\mathbf { M o d e l } _ { 3 }$ with $\\mathrm { A E } = 4 . 6 8$ . Within the same models, deep exits late are used at the beginning of the sequence and early exits are selected near the end. For heavily regularized models such as $\\mathbf { M o d e l } _ { 1 }$ with $\\mathrm { A E = 2 . 5 3 }$ , the disparity between beginning and end is less severe as the model exits early most of the time. $\\mathbf { M o d e l } _ { 2 }$ and Model3 are less regularized (higher AE) and tend to use late exits at the beginning of the sequence and early exits near the end. On the other hand, the more regularized $\\mathbf { M o d e l } _ { 1 }$ with $\\mathrm { A E = 2 . 5 3 }$ exits early most of the time. There is also a correlation between the model probability and the amount of computation, particularly in models with low AE . Figure 9 shows the joint histogram of the scores and the selected exit. For both $\\mathbf { M o d e l } _ { 1 }$ and $\\mathbf { M o d e l } _ { 2 }$ , low exits $( n \\leq 2 )$ are used in the high confidence range $[ 0 . 8 - 1 ]$ and high exits $( n \\geq 4 )$ ) are used in the low-confidence range $[ 0 - 0 . { \\bar { 5 } } ]$ . $\\mathbf { M o d e l } _ { 3 }$ has a high average exit $\\mathrm { ( A E = 4 . 6 8 ) }$ ) so most tokens exit late, however, in low confidence ranges the model does not exit earlier than $n = 5$ . ", + "bbox": [ + 173, + 571, + 826, + 726 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/2f53098f789a934627fba81e59ffbd43421d69fdd9ce9be0f5f4c72460559af7.jpg", + "image_caption": [ + "Figure 8: WMT’14 En-Fr test set: exit distributions in the beginning (relative-position: $\\mathrm { r p o s } { < } 0 . 1 $ ) and near the end $\\mathrm { ( r p o s > 0 . 9 ) }$ ) of the hypotheses of three models. " + ], + "image_footnote": [], + "bbox": [ + 261, + 748, + 743, + 878 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/dd8c2c9e3222d3d928628a33e80512b5c1b5a33ff8368bceafa11540393d5485.jpg", + "image_caption": [ + "Figure 9: Joint histogram of the exits and the confidence scores for 3 Tok-LL geometric-like models on newstest14. " + ], + "image_footnote": [], + "bbox": [ + 321, + 98, + 679, + 261 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 328, + 825, + 411 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 431, + 318, + 448 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We extended anytime prediction to the structured prediction setting and introduced simple but effective methods to equip sequence models to make predictions at different points in the network. We compared a number of different mechanisms to predict the required network depth and find that a simple correctness based geometric-like classifier obtains the best trade-off between speed and accuracy. Results show that the number of decoder layers can be reduced by more than three quarters at no loss in accuracy compared to a well tuned Transformer baseline. ", + "bbox": [ + 174, + 463, + 825, + 546 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 564, + 326, + 577 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank Laurens van der Maaten for fruitful comments and suggestions. ", + "bbox": [ + 176, + 587, + 655, + 602 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 622, + 285, + 637 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Tolga Bolukbasi, Joseph Wang, Ofer Dekel, and Venkatesh Saligrama. Adaptive neural networks for efficient inference. In Proc. of ICML, 2017. \nM. Cettolo, J. Niehues, S. Stüker, L. Bentivogli, and M. Federico. Report on the 11th iwslt evaluation campaign. In IWSLT, 2014. \nMostafa Dehghani, Stephan Gouws, Oriol Vinyals, Jakob Uszkoreit, and Lukasz Kaiser. Universal transformers. In Proc. of ICLR, 2018. \nJacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In Proc. of NAACL, 2019. \nSergey Edunov, Myle Ott, Michael Auli, David Grangier, and Marc’Aurelio Ranzato. Classical structured prediction losses for sequence to sequence learning. In Proc. of NAACL, 2018. \nMichael Figurnov, Artem Sobolev, and Dmitry P. Vetrov. Probabilistic adaptive computation time. In ArXiv preprint, 2017. \nJonas Gehring, Michael Auli, David Grangier, Denis Yarats, and Yann N Dauphin. Convolutional sequence to sequence learning. In Proc. of ICML, 2017. \nAlex Graves. Adaptive computation time for recurrent neural networks. In ArXiv preprint, 2016. \nGao Huang, Danlu Chen, Tianhong Li, Felix Wu, Laurens van der Maaten, and Kilian Q Weinberger. Multi-scale dense networks for resource efficient image classification. In Proc. of ICLR, 2017. \nD. Kingma and J. Ba. Adam: A method for stochastic optimization. In Proc. of ICLR, 2015. \nNathan Ng, Kyra Yee, Alexei Baevski, Myle Ott, Michael Auli, and Sergey Edunov. Facebook fair’s wmt19 news translation task submission. In Proc. of WMT, 2019. \nMyle Ott, Sergey Edunov, Alexei Baevski, Angela Fan, Sam Gross, Nathan Ng, David Grangier, and Michael Auli. Fairseq: A fast, extensible toolkit for sequence modeling. In Proc. of NAACL, 2019. \nK. Papineni, S. Roukos, T. Ward, and W.-J. Zhu. BLEU: a method for automatic evaluation of machine translation. In Proc. of ACL, 2002. \nAlec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. In Technical report, OpenAI., 2019. \nR. Sennrich, B. Haddow, and A. Birch. Neural machine translation of rare words with subword units. In Proc. of ACL, 2016. \nSurat Teerapittayanon, Bradley McDanel, and Hsiang-Tsung Kung. Branchynet: Fast inference via early exiting from deep neural networks. In ICPR, 2016. \nA. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need. In Proc. of NeurIPS, 2017. \nXin Wang, Fisher Yu, Zi-Yi Dou, Trevor Darrell, and Joseph E Gonzalez. Skipnet: Learning dynamic routing in convolutional networks. In Proc. of ECCV, 2018. ", + "bbox": [ + 171, + 645, + 826, + 926 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 103, + 828, + 474 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "APPENDIX A LOSS SCALING ", + "text_level": 1, + "bbox": [ + 176, + 102, + 426, + 118 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In this section we experiment with different weights for scaling the output classifier losses. Instead of uniform weighting, we bias towards specific output classifiers by assigning higher weights to their losses. Table 2 shows that weighing the classifiers equally provides good results. ", + "bbox": [ + 174, + 142, + 825, + 185 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/9e4e5f5e2e677c89738e7d95736f5fad65aaa7475c149ea55c47ec0be62d3ea0.jpg", + "table_caption": [ + "(b) IWSLT De-En - Test " + ], + "table_footnote": [], + "table_body": "
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline-34.235.335.635.735.635.935.4
Wn=135.534.135.535.836.136.136.235.6
Wn=n35.332.235.035.836.036.236.335.2
Wn=√n35.433.335.235.835.936.136.135.4
Wn=1/√n35.634.535.435.735.835.835.935.5
Wn=1/n35.334.735.335.535.735.835.835.5
(a) IWSLT De-En - Valid
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline133.734.634.634.634.634.834.5
Wn=134.433.234.434.834.935.034.934.5
Wn=n34.231.433.834.734.834.834.934.1
Wn=√n34.432.534.134.834.935.035.134.4
Wn=1/√n34.633.734.334.634.834.834.934.5
wn=1/n34.233.834.334.534.634.734.734.4
", + "bbox": [ + 187, + 210, + 799, + 472 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Table 2: Aligned training with different weights $\\left( \\omega _ { n } \\right)$ on IWSLT De-En. For each model we report BLEU on the dev set evaluated with a uniformly sampled exit $n \\sim \\mathcal { U } ( [ 1 . . 6 ] )$ for each token and a fixed exit $n \\in [ 1 . . 6 ]$ throughout the sequence. The average corresponds to the average BLEU over the fixed exits. ", + "bbox": [ + 173, + 502, + 826, + 559 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Gradient scaling Adding intermediate supervision at different levels of the decoder results in richer gradients for lower blocks compared to upper blocks. This is because earlier layers affect more loss terms in the compound loss of Eq. (4). To balance the gradients of each block in the decoder, we scale up the gradients of each loss term $\\left( - \\operatorname { L L } _ { n } \\right)$ when it is updating the parameters of its associated block $\\mathrm { ( b l o c k } _ { n }$ with parameters $\\theta _ { n }$ ) and revert it back to its normal scale before back-propagating it to the previous blocks. Figure 10 and Algorithm 1 illustrate this gradient scaling procedure. The $\\theta _ { n }$ are updated with $\\gamma _ { n }$ -amplified gradients from the block’s supervision and $( N { - } n )$ gradients from the subsequent blocks. We choose $\\gamma _ { n } = \\gamma ( N - n )$ to control the ratio $\\gamma { : } 1$ as the ratio of the block supervision to the subsequent blocks’ supervisions. ", + "bbox": [ + 173, + 598, + 825, + 724 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Table 3 shows that gradient scaling can benefit the lowest layer at the expense of higher layers. \nHowever, no scaling generally works very well. ", + "bbox": [ + 173, + 729, + 821, + 760 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/ea1e31e6c741caa889c0b1175b0bed7b176d252c5f35d54ca54e3389a0dd0932.jpg", + "image_caption": [ + "Figure 10: Illustration of gradient scaling. " + ], + "image_footnote": [], + "bbox": [ + 222, + 785, + 779, + 892 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Algorithm 1 Pseudo-code for gradient scaling (illustrated for a single step t) ", + "text_level": 1, + "bbox": [ + 174, + 103, + 676, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1: for $n \\in 1 . . N$ do \n2: $h _ { t } ^ { n } = \\mathrm { b l o c k } _ { n } ( h _ { t } ^ { n - 1 } )$ \n3: $p ( y _ { t + 1 } | h _ { t } ^ { n } ) = \\mathrm { s o f t m a x } ( W _ { n } h _ { t } ^ { n } )$ \n4: p(yt+1|hnt ) = SCALE_GRADIENT(p(yt+1|hnt ), γn) \n5: if $n < N$ then hnt = SCALE_GRADIENT(hnt , 1 ) \nγn+1 \n6: end for \n7: function SCALE_GRADIENT(Tensor $x$ , scale $\\gamma$ ) \n8: return $\\gamma x + ( 1 - \\gamma )$ STOP_GRADIENT $( x )$ \n9: $\\triangleright$ STOP_GRADIENT in PyTorch with $x$ .detach(). ", + "bbox": [ + 179, + 122, + 570, + 263 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "10: end function ", + "text_level": 1, + "bbox": [ + 176, + 263, + 290, + 276 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/d877e08e623fcb1067f94f5dd7ef9e48950a92bf89da1633eeb60d40a0d9c942.jpg", + "table_caption": [ + "(b) IWSLT De-En - Test " + ], + "table_footnote": [], + "table_body": "
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline134.235.335.635.735.635.935.4
@35.534.135.535.836.136.136.235.6
γ = 0.335.133.734.735.335.735.836.035.2
γ = 0.535.434.835.435.635.635.735.635.4
γ = 0.734.934.635.135.135.235.435.335.1
γ = 0.934.934.835.335.335.335.435.535.3
γ = 1.135.134.935.235.335.335.335.335.2
(a) IWSLT De-En - Valid
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline33.734.634.634.634.634.834.5
034.433.234.434.834.935.034.934.5
γ = 0.334.232.833.934.334.634.835.034.2
γ = 0.534.533.834.234.634.534.734.734.6
γ = 0.734.033.734.234.334.334.334.334.2
γ = 0.934.134.034.234.334.434.434.434.3
γ = 1.134.234.034.334.334.334.334.234.2
", + "bbox": [ + 204, + 308, + 792, + 594 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Table 3: Aligned training with different gradient scaling ratios $\\gamma : 1$ on IWSLT’14 De-En. For each model we report the BLEU4 score evaluated with a uniformly sampled exit $n \\sim \\mathcal { U } ( [ 1 . . 6 ] )$ for each token and a fixed exit $n \\in [ 1 . . 6 ]$ . The average corresponds to the average BLEU4 of all fixed exits. ", + "bbox": [ + 174, + 625, + 826, + 667 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "APPENDIX B FLOPS APPROXIMATION ", + "text_level": 1, + "bbox": [ + 176, + 101, + 509, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This section details the computation of the FLOPS we report. The per token FLOPS are for the decoder network only since we use an encoder of the same size for all models. We breakdown the FLOPS of every operation in Algorithm 2 (blue front of the algorithmic statement). We omit non-linearities, normalizations and residual connections. The main operations we account for are dot-products and by extension matrix-vector products since those represent the vast majority of FLOPS (we assume batch size one to simplify the calculation). ", + "bbox": [ + 173, + 132, + 826, + 218 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Parameters ", + "bbox": [ + 222, + 232, + 297, + 246 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/6a950deecc6b2e3e57631277973835524c2aefeef6110aec7d29474d44184145.jpg", + "table_caption": [ + "Table 4: FLOPS of basic operations, key parameters and variables for the FLOPS estimation. " + ], + "table_footnote": [], + "table_body": "
Parameters decoder embedding dimension.FLOPS
dd de df x t V output vocabulary size.encoder embedding dimension.Operation
The feed-forward network dimension.Dot-product (d)2d-1
source length. Current time-estep (t ≥ 1).Linear din →dout2dindout
", + "bbox": [ + 178, + 238, + 728, + 337 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "With this breakdown, the total computational cost at time-step $t$ of a decoder block that we actually go through, denoted with FC, is: ", + "bbox": [ + 174, + 387, + 823, + 416 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/a289d410d68cd225a95f30b912ad76826d398c5a61c6bb49615810a38687ac3e.jpg", + "text": "$$\n\\mathrm { F C } ( \\pmb { x } , t ) = 1 2 d _ { d } ^ { 2 } + 4 d _ { f } d _ { d } + 4 t d _ { d } + 4 | \\pmb { x } | d _ { d } + 4 [ \\mathrm { F i r s t C a l l } ] | \\pmb { x } | d _ { d } d _ { e } ,\n$$", + "text_format": "latex", + "bbox": [ + 272, + 421, + 722, + 440 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where the cost of mapping the source’ keys and values is incurred the first time the block is called (flagged with FirstCall). This occurs at $t = 1$ for the baseline model but it is input-dependent with depth adaptive estimation and may never occur if all tokens exit early. ", + "bbox": [ + 174, + 446, + 825, + 489 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "If skipped, a block still has to compute the keys and value of the self-attention block so the selfattention of future time-steps can function. We will denote this cost with FS and we have $\\mathrm { F S } = 4 d _ { d } ^ { 2 }$ . ", + "bbox": [ + 173, + 496, + 825, + 525 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Depending on the halting mechanism, an exit prediction cost, denoted wit FP, is added: ", + "bbox": [ + 171, + 531, + 745, + 546 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/965fa907f1264b90ed34db9755b0338d5575212ce17081a37652455d69838b97.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { F P } ( t , q ( t ) ) = 2 [ [ t = 1 ] ] N d _ { d } } \\\\ & { \\mathrm { F P } ( t , q ( t ) ) = 2 N d _ { d } } \\\\ & { \\mathrm { F P } ( t , q ( t ) ) = 2 d _ { d } q ( t ) } \\\\ & { \\mathrm { F P } ( t , q ( t ) ) = 2 q ( t ) V d _ { d } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 504, + 556, + 704, + 617 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For a set of source sequences $\\{ \\pmb { x } ^ { ( i ) } \\} _ { i \\in \\mathcal { I } }$ and generated hypotheses $\\{ \\pmb { y } ^ { ( i ) } \\} _ { i \\in \\mathcal { I } }$ , the average flops per token is: ", + "bbox": [ + 171, + 626, + 825, + 657 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/33ba85117e194e298a12b73818c7b4463d345cabdf03212cd5dda42da22bc68c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { s e l i n e ~ } ( N \\mathrm { ~ b l o c k s } ) ; \\quad \\frac { 1 } { \\sum _ { i } | y ^ { ( i ) } | } \\sum _ { i } \\sum _ { t = 1 } ^ { | y ^ { ( i ) } | } \\left( N \\mathrm { ~ F C } ( { \\pmb x } ^ { ( i ) } , t ) + 2 V d _ { d } \\right) } \\\\ & { \\quad \\mathrm { A d a p t i v e ~ d e p t h } ; \\quad \\frac { 1 } { \\sum _ { i } | y ^ { ( i ) } | } \\sum _ { i } \\sum _ { t = 1 } ^ { | y ^ { ( i ) } | } \\left( q ( t ) \\mathrm { F C } ( { \\pmb x } ^ { ( i ) } , t ) + ( N - q ( t ) ) \\mathrm { F S } + \\mathrm { F P } ( t , q ( t ) ) + 2 V d _ { d } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 204, + 672, + 854, + 723 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In the case of confidence thresholding the final output prediction cost $( 2 V d _ { d } )$ is already accounted for in the exit prediction cost FP. ", + "bbox": [ + 171, + 727, + 825, + 756 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Algorithm 2 Adaptive decoding with Tok-geometric-like ", + "text_level": 1, + "bbox": [ + 173, + 103, + 549, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "1: Input: source codes $\\pmb { s }$ , incremental state \n2: Initialization: $t = 1$ , $y _ { 1 } = < s >$ \n3: for $n \\in 1 \\ldots N$ do \n4: FirstCall[n] $=$ True. . A flag signaling if the source’ keys and values should be evalua \n5: end for \n6: while $y _ { t } \\neq < / \\mathrm { s } >$ do \n7: Embed the last output token $y _ { t }$ . \n8: for $n \\in 1 \\ldots N$ do \n9: $\\triangleright$ Self-attention. \n10: - Map the input into a key $( k )$ and value $( v )$ . FLOPS $\\scriptstyle \\dot { = } 4 d _ { d } ^ { 2 }$ \n11: - Map the input into a query $q$ $\\cdot \\mathrm { F L O P S } { = } 2 d _ { d } ^ { 2 }$ \n12: - Score the memory keys with $q$ to get the attention weights $\\alpha$ . FLOPS $\\scriptstyle \\sum 4 t d _ { d }$ \n13: - Map the attention output. $\\mathrm { F L O P S } { = } 2 d _ { d } ^ { 2 }$ \n14: $\\triangleright$ Encoder-Decoder interaction. \n15: if FirstCall $[ n ]$ then \n16: Map the source states into keys and values for the nth block. FLOPS=4|x|dedd \n17: FirstCall[n] $=$ False \n18: end if \n19: - Map the input into a query $q$ . $\\mathrm { F L O P S } { = } 2 d _ { d } ^ { 2 }$ \n20: - Score the memory keys with $q$ to get the attention weights $\\alpha$ . FLOPS=4|x|dd \n21: - Map the attention output. $\\mathrm { F L O P S } { = } 2 d _ { d } ^ { 2 }$ \n22: Feed-forward network. FLO $\\scriptstyle { \\sum } = 4 d _ { d } d _ { f }$ \n23: Estimate the halting probability $\\chi _ { t , n }$ . FLOP ${ \\mathrm { S } } { = } 2 d _ { d }$ \n24: if $\\chi _ { t , n } > 0 . 5$ then \n25: Exit the loop (Line 8) \n26: end if \n27: end for \n28: if $n < N$ then \n29: $\\triangleright$ Skipped blocks. \n30: for $n _ { s } \\in n + 1 \\ldots N$ do \n31: Copy and map the copied state into a key $( k )$ and value (v). FLOPS $\\scriptstyle \\cdot = 4 d _ { d } ^ { 2 }$ \n32: end for \n33: end if \n34: Project the final state and sample a new output token. FLOP $\\ S { = } 2 V d _ { d }$ \n35: $t { + + }$ \n36: end while ", + "bbox": [ + 171, + 123, + 795, + 623 + ], + "page_idx": 14 + } +] \ No newline at end of file diff --git a/parse/train/SJg7KhVKPH/SJg7KhVKPH_middle.json b/parse/train/SJg7KhVKPH/SJg7KhVKPH_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..dfb0ba905dce5fc6a17eba9908706272704ef9fa --- /dev/null +++ b/parse/train/SJg7KhVKPH/SJg7KhVKPH_middle.json @@ -0,0 +1,40479 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 79, + 360, + 96 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 362, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 362, + 99 + ], + "score": 1.0, + "content": "DEPTH-ADAPTIVE TRANSFORMER", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 113, + 115, + 201, + 137 + ], + "lines": [ + { + "bbox": [ + 111, + 114, + 181, + 127 + ], + "spans": [ + { + "bbox": [ + 111, + 114, + 181, + 127 + ], + "score": 1.0, + "content": "Maha Elbayad∗", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 112, + 124, + 202, + 139 + ], + "spans": [ + { + "bbox": [ + 112, + 124, + 202, + 139 + ], + "score": 1.0, + "content": "Univ. 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Our work considers a variety of mechanisms to estimate the network", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "score": 1.0, + "content": "depth and applies a different layer at each step. Moreover, Dehghani et al. (2018) fix the number", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "of steps for large-scale machine translation whereas we vary the number of steps to demonstrate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 539, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 554 + ], + "score": 1.0, + "content": "substantial improvements in speed at no loss in accuracy. 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(2017) to structured", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "prediction where dynamic computation decisions impact future computation. Related work from", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "computer vision includes Teerapittayanon et al. (2016); Figurnov et al. (2017) and Wang et al. (2018)", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 606, + 454, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 454, + 618 + ], + "score": 1.0, + "content": "who explored the idea of dynamic routing either by exiting early or by skipping layers.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "We encode the input sequence using a standard Transformer encoder to generate the output sequence", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "with a varying amount of computation in the decoder network. Dynamic computation poses a chal-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 645, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 504, + 657 + ], + "score": 1.0, + "content": "lenge for self-attention because omitted layers in prior time-steps may be required in the future.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "We experiment with two approaches to address this and show that a simple approach works well", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "(§2). 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Unlike", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 248, + 470, + 260 + ], + "spans": [ + { + "bbox": [ + 142, + 248, + 470, + 260 + ], + "score": 1.0, + "content": "dynamic computation in Universal Transformers, which applies the same set of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 259, + 470, + 272 + ], + "spans": [ + { + "bbox": [ + 141, + 259, + 470, + 272 + ], + "score": 1.0, + "content": "layers iteratively, we apply different layers at every step to adjust both the amount", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 269, + 470, + 282 + ], + "spans": [ + { + "bbox": [ + 141, + 269, + 470, + 282 + ], + "score": 1.0, + "content": "of computation as well as the model capacity. On IWSLT German-English trans-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 281, + 469, + 293 + ], + "spans": [ + { + "bbox": [ + 141, + 281, + 469, + 293 + ], + "score": 1.0, + "content": "lation our approach matches the accuracy of a well tuned baseline Transformer", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 291, + 354, + 305 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 354, + 305 + ], + "score": 1.0, + "content": "while using less than a quarter of the decoder layers.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5, + "bbox_fs": [ + 141, + 193, + 470, + 305 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 327, + 206, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 208, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 208, + 343 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 504, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "The size of modern neural sequence models (Gehring et al., 2017; Vaswani et al., 2017; Devlin et al.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 362, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 378 + ], + "score": 1.0, + "content": "2019) can amount to billions of parameters (Radford et al., 2019). For example, the winning entry", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "of the WMT’19 news machine translation task in English-German used an ensemble totaling two", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 184, + 398 + ], + "score": 1.0, + "content": "billion parameters", + "type": "text" + }, + { + "bbox": [ + 184, + 387, + 199, + 397 + ], + "score": 0.35, + "content": "\\mathrm { N g }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "et al., 2019). While large models are required to do better on hard examples,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "small models are likely to perform as well on easy ones, e.g., the aforementioned ensemble is prob-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 406, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 421 + ], + "score": 1.0, + "content": "ably not required to translate a short phrase such as \"Thank you\". However, current models apply", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 419, + 428, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 428, + 432 + ], + "score": 1.0, + "content": "the same amount of computation regardless of whether the input is easy or hard.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 352, + 506, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 504, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "In this paper, we propose Transformers which adapt the number of layers to each input in order to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "achieve a good speed-accuracy trade off at inference time. We extend Graves (2016; ACT) who", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "introduced dynamic computation to recurrent neural networks in several ways: we apply different", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "layers at each stage, we investigate a range of designs and training targets for the halting module and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 479, + 495, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 495, + 492 + ], + "score": 1.0, + "content": "we explicitly supervise through simple oracles to achieve good performance on large-scale tasks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 435, + 506, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "Universal Transformers (UT) rely on ACT for dynamic computation and repeatedly apply the same", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "layer (Dehghani et al., 2018). Our work considers a variety of mechanisms to estimate the network", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "score": 1.0, + "content": "depth and applies a different layer at each step. Moreover, Dehghani et al. (2018) fix the number", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "of steps for large-scale machine translation whereas we vary the number of steps to demonstrate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 539, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 554 + ], + "score": 1.0, + "content": "substantial improvements in speed at no loss in accuracy. 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1 } ^ { n } ) , \\quad \\mathrm { L L } ^ { n } = \\sum _ { t = 1 } ^ { | y | } \\mathrm { L L } _ { t } ^ { n } , \\quad \\mathcal { L } _ { d e c } ( \\pmb { x } , \\pmb { y } ) = - \\frac { 1 } { \\sum _ { n } \\omega _ { n } } \\sum _ { n = 1 } ^ { N } \\omega _ { n } \\mathrm { L L } ^ { n } .", + "type": "interline_equation", + "image_path": "c4d129c2c1599741dcd4467f0fe324e5495ff8416146f1b49790801fd970f8dd.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 138, + 225, + 475, + 237.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 138, + 237.0, + 475, + 249.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 138, + 249.0, + 475, + 261.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 188, + 279 + ], + "score": 1.0, + "content": "The compound loss", + "type": "text" + }, + { + "bbox": [ + 189, + 266, + 233, + 278 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { d e c } ( \\pmb { x } , \\pmb { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 266, + 336, + 279 + ], + "score": 1.0, + "content": "is a weighted average of", + "type": "text" + }, + { + "bbox": [ + 336, + 267, + 347, + 276 + ], + "score": 0.85, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 266, + 407, + 279 + ], + "score": 1.0, + "content": "terms w.r.t. to", + "type": "text" + }, + { + "bbox": [ + 407, + 266, + 457, + 278 + ], + "score": 0.89, + "content": "\\left( \\omega _ { 1 } , \\hdots \\omega _ { N } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 266, + 505, + 279 + ], + "score": 1.0, + "content": ". We found", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 431, + 290 + ], + "score": 1.0, + "content": "that uniform weights achieve better BLEU compared to other weighing schemes", + "type": "text" + }, + { + "bbox": [ + 432, + 278, + 444, + 289 + ], + "score": 0.8, + "content": "_ { . c . f }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 276, + 505, + 290 + ], + "score": 1.0, + "content": ". Appendix A).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "score": 1.0, + "content": "At inference time, not all time-steps will have hidden states for the current layer since the model", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "exited early. 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1 } ^ { n } ) , \\quad \\mathrm { L L } ^ { n } = \\sum _ { t = 1 } ^ { | y | } \\mathrm { L L } _ { t } ^ { n } , \\quad \\mathcal { L } _ { d e c } ( \\pmb { x } , \\pmb { y } ) = - \\frac { 1 } { \\sum _ { n } \\omega _ { n } } \\sum _ { n = 1 } ^ { N } \\omega _ { n } \\mathrm { L L } ^ { n } .", + "type": "interline_equation", + "image_path": "c4d129c2c1599741dcd4467f0fe324e5495ff8416146f1b49790801fd970f8dd.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 138, + 225, + 475, + 237.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 138, + 237.0, + 475, + 249.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 138, + 249.0, + 475, + 261.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 188, + 279 + ], + "score": 1.0, + "content": "The compound loss", + "type": "text" + }, + { + "bbox": [ + 189, + 266, + 233, + 278 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { d e c } ( \\pmb { x } , \\pmb { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 266, + 336, + 279 + ], + "score": 1.0, + "content": "is a weighted average of", + "type": "text" + }, + { + "bbox": [ + 336, + 267, + 347, + 276 + ], + "score": 0.85, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 266, + 407, + 279 + ], + "score": 1.0, + "content": "terms w.r.t. to", + "type": "text" + }, + { + "bbox": [ + 407, + 266, + 457, + 278 + ], + "score": 0.89, + "content": "\\left( \\omega _ { 1 } , \\hdots \\omega _ { N } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 266, + 505, + 279 + ], + "score": 1.0, + "content": ". 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Appendix A).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "score": 1.0, + "content": "At inference time, not all time-steps will have hidden states for the current layer since the model", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "exited early. In this case, we simply copy the last computed state to all upper layers, similar to mixed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 309, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 141, + 323 + ], + "score": 1.0, + "content": "training", + "type": "text" + }, + { + "bbox": [ + 141, + 311, + 172, + 321 + ], + "score": 0.37, + "content": "( \\ S 2 . 2 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 309, + 505, + 323 + ], + "score": 1.0, + "content": ". However, we do apply layer-specific key and value projections to the copied state.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 266, + 506, + 323 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 333, + 216, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 332, + 217, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 217, + 346 + ], + "score": 1.0, + "content": "2.2.2 MIXED TRAINING", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 506, + 416 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "Aligned training assumes that all hidden states of the previous time-steps are available but this as-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 364, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 376 + ], + "score": 1.0, + "content": "sumption is unrealistic since an early exit may have been chosen previously. This creates a mismatch", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "between training and testing. Mixed training reduces the mismatch by training the model to use hid-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 455, + 398 + ], + "score": 1.0, + "content": "den states from different blocks of previous time-steps for self-attention. We sample", + "type": "text" + }, + { + "bbox": [ + 455, + 388, + 466, + 396 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "different", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 102, + 395, + 394, + 417 + ], + "spans": [ + { + "bbox": [ + 102, + 395, + 197, + 417 + ], + "score": 1.0, + "content": "exit sequences (n(m)1 , .", + "type": "text" + }, + { + "bbox": [ + 167, + 397, + 264, + 416 + ], + "score": 0.92, + "content": "( n _ { 1 } ^ { ( m ) } , \\dots n _ { | \\pmb { y } | } ^ { ( m ) } ) _ { 1 \\leq m \\leq M }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 401, + 394, + 411 + ], + "score": 1.0, + "content": "and evaluate the following loss:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 102, + 353, + 505, + 417 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 421, + 478, + 457 + ], + "lines": [ + { + "bbox": [ + 120, + 421, + 478, + 457 + ], + "spans": [ + { + "bbox": [ + 120, + 421, + 478, + 457 + ], + "score": 0.95, + "content": "\\mathrm { L L } ( n _ { 1 } , \\dots , n _ { | \\mathbf { y } | } ) = \\sum _ { t = 1 } ^ { | \\mathbf { y } | } \\log p ( y _ { t } | h _ { t - 1 } ^ { n _ { t } } ) , \\quad \\mathscr { L } _ { d e c } ( \\mathbf { x } , \\mathbf { y } ) = - \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\mathrm { L L } ( n _ { 1 } ^ { ( m ) } , \\dots , n _ { | \\mathbf { y } | } ^ { ( m ) } ) .", + "type": "interline_equation", + "image_path": "72797010deb0549235109abb626da946579d717a38de4c8e5951c1db2a56aa7c.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 120, + 421, + 478, + 433.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 120, + 433.0, + 478, + 445.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 120, + 445.0, + 478, + 457.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 133, + 475 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 134, + 462, + 171, + 473 + ], + "score": 0.91, + "content": "n _ { t } ~ < ~ N", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 460, + 342, + 475 + ], + "score": 1.0, + "content": ", we copy the last evaluated hidden state", + "type": "text" + }, + { + "bbox": [ + 342, + 462, + 354, + 474 + ], + "score": 0.89, + "content": "h _ { t } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 460, + 505, + 475 + ], + "score": 1.0, + "content": "to the subsequent layers so that the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 472, + 392, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 392, + 485 + ], + "score": 1.0, + "content": "self-attention of future time steps can function as usual (see Figure 1b).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 106, + 460, + 505, + 485 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 501, + 283, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 285, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 285, + 515 + ], + "score": 1.0, + "content": "3 ADAPTIVE DEPTH ESTIMATION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 539 + ], + "score": 1.0, + "content": "We present a variety of mechanisms to predict the decoder block at which the model will stop", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "and output the next token, or when it should exit to achieve a good speed-accuracy trade-off. We", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 548, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 559 + ], + "score": 1.0, + "content": "consider two approaches: sequence-specific depth decodes all output tokens using the same block", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 559, + 476, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 476, + 570 + ], + "score": 1.0, + "content": "(§3.1) while token-specific depth determines a separate exit for each individual token (§3.2).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 524, + 506, + 570 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 575, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 504, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 310, + 588 + ], + "score": 1.0, + "content": "We model the distribution of exiting at time-step", + "type": "text" + }, + { + "bbox": [ + 310, + 577, + 315, + 585 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 574, + 441, + 588 + ], + "score": 1.0, + "content": "with a parametric distribution", + "type": "text" + }, + { + "bbox": [ + 442, + 577, + 451, + 587 + ], + "score": 0.83, + "content": "q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 574, + 480, + 588 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 481, + 575, + 504, + 587 + ], + "score": 0.91, + "content": "q _ { t } ( n )", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 237, + 599 + ], + "score": 1.0, + "content": "is the probability of computing", + "type": "text" + }, + { + "bbox": [ + 237, + 586, + 316, + 597 + ], + "score": 0.84, + "content": "{ \\mathrm { b l o c k } } _ { 1 } , \\ldots , { \\mathrm { b l o c k } } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 586, + 466, + 599 + ], + "score": 1.0, + "content": "and then emitting a prediction with", + "type": "text" + }, + { + "bbox": [ + 466, + 587, + 479, + 597 + ], + "score": 0.89, + "content": "\\mathcal { C } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 586, + 505, + 599 + ], + "score": 1.0, + "content": ". 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We simultaneously", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 486, + 669 + ], + "score": 1.0, + "content": "optimize the decoding loss (Eq. (4)) and the exit loss (Eq. (6)) balanced by a hyper-parameter", + "type": "text" + }, + { + "bbox": [ + 486, + 659, + 493, + 667 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 668, + 461, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 461, + 680 + ], + "score": 1.0, + "content": "ensure that the model maintains good generation accuracy. 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The token geometric-like classifier (2c) makes a binary decision after every block", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 275, + 495, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 495, + 288 + ], + "score": 1.0, + "content": "to dictate whether to continue (C) to the next block or to stop (S) and emit an output distribution.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "title", + "bbox": [ + 108, + 308, + 253, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 255, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 255, + 321 + ], + "score": 1.0, + "content": "3.1 SEQUENCE-SPECIFIC DEPTH:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 302, + 342 + ], + "score": 1.0, + "content": "For sequence-specific depth, the exit distribution", + "type": "text" + }, + { + "bbox": [ + 303, + 331, + 309, + 340 + ], + "score": 0.8, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 329, + 416, + 342 + ], + "score": 1.0, + "content": "and the oracle distribution", + "type": "text" + }, + { + "bbox": [ + 417, + 330, + 427, + 341 + ], + "score": 0.88, + "content": "q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "are independent of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 248, + 352 + ], + "score": 1.0, + "content": "the time-step so we drop subscript", + "type": "text" + }, + { + "bbox": [ + 248, + 341, + 253, + 350 + ], + "score": 0.68, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 340, + 505, + 352 + ], + "score": 1.0, + "content": ". We condition the exit on the source sequence by feeding the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 352, + 347, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 140, + 363 + ], + "score": 1.0, + "content": "average", + "type": "text" + }, + { + "bbox": [ + 140, + 354, + 145, + 361 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 352, + 347, + 363 + ], + "score": 1.0, + "content": "of the encoder outputs to a multinomial classifier:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 367, + 419, + 396 + ], + "lines": [ + { + "bbox": [ + 191, + 367, + 419, + 396 + ], + "spans": [ + { + "bbox": [ + 191, + 367, + 419, + 396 + ], + "score": 0.94, + "content": "s = \\frac { 1 } { | \\pmb { x } | } \\sum _ { t } s _ { t } , \\quad q ( n | \\pmb { x } ) = \\mathrm { s o f t m a x } ( W _ { h } s + b _ { h } ) \\in \\mathbb { R } ^ { N } ,", + "type": "interline_equation", + "image_path": "f4bc96fd7bbe813173f5ef7cc88e7087261864689df8aea127d97e0e420537a0.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 367, + 419, + 381.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 191, + 381.5, + 419, + 396.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 400, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 133, + 413 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 401, + 149, + 412 + ], + "score": 0.9, + "content": "W _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 400, + 167, + 413 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 168, + 401, + 178, + 412 + ], + "score": 0.88, + "content": "b _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "are the weights and biases of the halting mechanism. We consider two oracles to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 203, + 424 + ], + "score": 1.0, + "content": "determine which of the", + "type": "text" + }, + { + "bbox": [ + 203, + 412, + 214, + 422 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "blocks should be chosen. The first is based on the sequence likelihood", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 422, + 444, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 444, + 436 + ], + "score": 1.0, + "content": "and the second looks at an aggregate of the correctly predicted tokens at each block.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 458 + ], + "score": 1.0, + "content": "Likelihood-based: This oracle is based on the likelihood of the entire sequence after each block", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "and we optimize it with the Dirac delta centered around the exit with the highest sequence likelihood.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 473, + 365, + 491 + ], + "lines": [ + { + "bbox": [ + 245, + 473, + 365, + 491 + ], + "spans": [ + { + "bbox": [ + 245, + 473, + 365, + 491 + ], + "score": 0.93, + "content": "q ^ { * } ( { \\pmb x } , { \\pmb y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } { \\mathrm { L L } ^ { n } } ) .", + "type": "interline_equation", + "image_path": "b1c9b63a9273a6fe3e74a982881b171debd5a5bfb29d25bcd78cd55c30168f21.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 245, + 473, + 365, + 491 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 441, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 442, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 442, + 509 + ], + "score": 1.0, + "content": "We add a regularization term to encourage lower exits that achieve good likelihood:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 512, + 375, + 531 + ], + "lines": [ + { + "bbox": [ + 235, + 512, + 375, + 531 + ], + "spans": [ + { + "bbox": [ + 235, + 512, + 375, + 531 + ], + "score": 0.93, + "content": "q ^ { * } ( { \\pmb x } , { \\pmb y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } { \\mathrm { L L } ^ { n } } - \\lambda n ) .", + "type": "interline_equation", + "image_path": "a8cb505c8f388593729aa3b677775078daa07e53f8e34aeebbb6042f05054c72.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 235, + 512, + 375, + 531 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "Correctness-based: Likelihood ignores whether the model already assigns the highest score to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "the correct target. Instead, this oracle chooses the lowest block that assigns the largest score to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 562, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 577 + ], + "score": 1.0, + "content": "the correct prediction. For each block, we count the number of correctly predicted tokens over", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 572, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 506, + 588 + ], + "score": 1.0, + "content": "the sequence and choose the block with the most number of correct tokens. A regularization term", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 585, + 311, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 311, + 599 + ], + "score": 1.0, + "content": "controls the trade-off between speed and accuracy.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 601, + 458, + 620 + ], + "lines": [ + { + "bbox": [ + 153, + 601, + 458, + 620 + ], + "spans": [ + { + "bbox": [ + 153, + 601, + 458, + 620 + ], + "score": 0.9, + "content": "C ^ { n } = \\# \\{ t | y _ { t } = \\arg \\operatorname* { m a x } _ { y } p ( y | h _ { t - 1 } ^ { n } ) \\} , \\quad q ^ { * } ( x , y ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } C ^ { n } - \\lambda n ) .", + "type": "interline_equation", + "image_path": "2a1f65cdafa7f4af34fddaa01de5644d9b4343db42c4cf60e9a6d80f7eacf9cd.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 153, + 601, + 458, + 620 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 630, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 644 + ], + "score": 1.0, + "content": "Oracles based on test metrics such as BLEU are feasible but expensive to compute since we would", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 642, + 423, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 263, + 654 + ], + "score": 1.0, + "content": "need to decode every training sentence", + "type": "text" + }, + { + "bbox": [ + 263, + 643, + 273, + 652 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 642, + 423, + 654 + ], + "score": 1.0, + "content": "times. 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Sequence-specific depth uses a multi-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 477, + 244 + ], + "score": 1.0, + "content": "nomial classifier to choose an exit for the entire output sequence based on the encoder output", + "type": "text" + }, + { + "bbox": [ + 477, + 234, + 483, + 242 + ], + "score": 0.64, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "(2a).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 316, + 255 + ], + "score": 1.0, + "content": "It then outputs a token at this depth with classifier", + "type": "text" + }, + { + "bbox": [ + 316, + 243, + 330, + 254 + ], + "score": 0.89, + "content": "\\mathcal { C } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 242, + 506, + 255 + ], + "score": 1.0, + "content": ". 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The token geometric-like classifier (2c) makes a binary decision after every block", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 275, + 495, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 495, + 288 + ], + "score": 1.0, + "content": "to dictate whether to continue (C) to the next block or to stop (S) and emit an output distribution.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "title", + "bbox": [ + 108, + 308, + 253, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 255, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 255, + 321 + ], + "score": 1.0, + "content": "3.1 SEQUENCE-SPECIFIC DEPTH:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 302, + 342 + ], + "score": 1.0, + "content": "For sequence-specific depth, the exit distribution", + "type": "text" + }, + { + "bbox": [ + 303, + 331, + 309, + 340 + ], + "score": 0.8, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 329, + 416, + 342 + ], + "score": 1.0, + "content": "and the oracle distribution", + "type": "text" + }, + { + "bbox": [ + 417, + 330, + 427, + 341 + ], + "score": 0.88, + "content": "q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "are independent of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 248, + 352 + ], + "score": 1.0, + "content": "the time-step so we drop subscript", + "type": "text" + }, + { + "bbox": [ + 248, + 341, + 253, + 350 + ], + "score": 0.68, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 340, + 505, + 352 + ], + "score": 1.0, + "content": ". We condition the exit on the source sequence by feeding the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 352, + 347, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 140, + 363 + ], + "score": 1.0, + "content": "average", + "type": "text" + }, + { + "bbox": [ + 140, + 354, + 145, + 361 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 352, + 347, + 363 + ], + "score": 1.0, + "content": "of the encoder outputs to a multinomial classifier:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 329, + 506, + 363 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 367, + 419, + 396 + ], + "lines": [ + { + "bbox": [ + 191, + 367, + 419, + 396 + ], + "spans": [ + { + "bbox": [ + 191, + 367, + 419, + 396 + ], + "score": 0.94, + "content": "s = \\frac { 1 } { | \\pmb { x } | } \\sum _ { t } s _ { t } , \\quad q ( n | \\pmb { x } ) = \\mathrm { s o f t m a x } ( W _ { h } s + b _ { h } ) \\in \\mathbb { R } ^ { N } ,", + "type": "interline_equation", + "image_path": "f4bc96fd7bbe813173f5ef7cc88e7087261864689df8aea127d97e0e420537a0.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 367, + 419, + 381.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 191, + 381.5, + 419, + 396.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 400, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 133, + 413 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 401, + 149, + 412 + ], + "score": 0.9, + "content": "W _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 400, + 167, + 413 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 168, + 401, + 178, + 412 + ], + "score": 0.88, + "content": "b _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "are the weights and biases of the halting mechanism. We consider two oracles to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 203, + 424 + ], + "score": 1.0, + "content": "determine which of the", + "type": "text" + }, + { + "bbox": [ + 203, + 412, + 214, + 422 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "blocks should be chosen. The first is based on the sequence likelihood", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 422, + 444, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 444, + 436 + ], + "score": 1.0, + "content": "and the second looks at an aggregate of the correctly predicted tokens at each block.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 400, + 505, + 436 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 458 + ], + "score": 1.0, + "content": "Likelihood-based: This oracle is based on the likelihood of the entire sequence after each block", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "and we optimize it with the Dirac delta centered around the exit with the highest sequence likelihood.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 106, + 447, + 505, + 470 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 473, + 365, + 491 + ], + "lines": [ + { + "bbox": [ + 245, + 473, + 365, + 491 + ], + "spans": [ + { + "bbox": [ + 245, + 473, + 365, + 491 + ], + "score": 0.93, + "content": "q ^ { * } ( { \\pmb x } , { \\pmb y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } { \\mathrm { L L } ^ { n } } ) .", + "type": "interline_equation", + "image_path": "b1c9b63a9273a6fe3e74a982881b171debd5a5bfb29d25bcd78cd55c30168f21.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 245, + 473, + 365, + 491 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 441, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 442, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 442, + 509 + ], + "score": 1.0, + "content": "We add a regularization term to encourage lower exits that achieve good likelihood:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 495, + 442, + 509 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 512, + 375, + 531 + ], + "lines": [ + { + "bbox": [ + 235, + 512, + 375, + 531 + ], + "spans": [ + { + "bbox": [ + 235, + 512, + 375, + 531 + ], + "score": 0.93, + "content": "q ^ { * } ( { \\pmb x } , { \\pmb y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } { \\mathrm { L L } ^ { n } } - \\lambda n ) .", + "type": "interline_equation", + "image_path": "a8cb505c8f388593729aa3b677775078daa07e53f8e34aeebbb6042f05054c72.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 235, + 512, + 375, + 531 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "Correctness-based: Likelihood ignores whether the model already assigns the highest score to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "the correct target. Instead, this oracle chooses the lowest block that assigns the largest score to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 562, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 577 + ], + "score": 1.0, + "content": "the correct prediction. For each block, we count the number of correctly predicted tokens over", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 572, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 506, + 588 + ], + "score": 1.0, + "content": "the sequence and choose the block with the most number of correct tokens. A regularization term", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 585, + 311, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 311, + 599 + ], + "score": 1.0, + "content": "controls the trade-off between speed and accuracy.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 541, + 506, + 599 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 601, + 458, + 620 + ], + "lines": [ + { + "bbox": [ + 153, + 601, + 458, + 620 + ], + "spans": [ + { + "bbox": [ + 153, + 601, + 458, + 620 + ], + "score": 0.9, + "content": "C ^ { n } = \\# \\{ t | y _ { t } = \\arg \\operatorname* { m a x } _ { y } p ( y | h _ { t - 1 } ^ { n } ) \\} , \\quad q ^ { * } ( x , y ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } C ^ { n } - \\lambda n ) .", + "type": "interline_equation", + "image_path": "2a1f65cdafa7f4af34fddaa01de5644d9b4343db42c4cf60e9a6d80f7eacf9cd.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 153, + 601, + 458, + 620 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 630, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 644 + ], + "score": 1.0, + "content": "Oracles based on test metrics such as BLEU are feasible but expensive to compute since we would", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 642, + 423, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 263, + 654 + ], + "score": 1.0, + "content": "need to decode every training sentence", + "type": "text" + }, + { + "bbox": [ + 263, + 643, + 273, + 652 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 642, + 423, + 654 + ], + "score": 1.0, + "content": "times. We leave this for future work.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 630, + 505, + 654 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 667, + 237, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 239, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 239, + 680 + ], + "score": 1.0, + "content": "3.2 TOKEN-SPECIFIC DEPTH:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 700 + ], + "score": 1.0, + "content": "The token-specific approach can choose a different exit at every time-step. 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N - 1 ] , ~ \\chi _ { t } ^ { n } = \\mathrm { s i g m o i d } ( w _ { h } ^ { \\top } h _ { t } ^ { n } + b _ { h } ) , } \\\\ { q _ { t } ( n | \\pmb { x } , \\pmb { y } _ { < t } ) = \\left\\{ \\begin{array} { l l } { \\chi _ { t } ^ { n } \\displaystyle \\prod _ { n ^ { \\prime } < n } ( 1 - \\chi _ { t } ^ { n ^ { \\prime } } ) , \\mathrm { i f } n < N } \\\\ { \\displaystyle \\prod _ { n ^ { \\prime } < N } ( 1 - \\chi _ { t } ^ { n ^ { \\prime } } ) , \\mathrm { o t h e r w i s e } } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "37a64ea2c3490986c8851f9d201ea6b6c287cff9e6addc31c3a435dd73b3c3d5.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 210, + 155, + 393, + 170.25 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 210, + 170.25, + 393, + 185.5 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 210, + 185.5, + 393, + 200.75 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 210, + 200.75, + 393, + 216.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 221, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 507, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 137, + 235 + ], + "score": 1.0, + "content": "where,", + "type": "text" + }, + { + "bbox": [ + 137, + 223, + 144, + 232 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 219, + 308, + 235 + ], + "score": 1.0, + "content": "is the dimension of the decoder states,", + "type": "text" + }, + { + "bbox": [ + 308, + 221, + 365, + 233 + ], + "score": 0.93, + "content": "W _ { h } \\ \\in \\ \\mathbb { R } ^ { N \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 219, + 384, + 235 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 221, + 426, + 233 + ], + "score": 0.92, + "content": "w _ { h } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 219, + 507, + 235 + ], + "score": 1.0, + "content": "are the weights of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 225, + 246 + ], + "score": 1.0, + "content": "the halting mechanisms, and", + "type": "text" + }, + { + "bbox": [ + 225, + 234, + 236, + 244 + ], + "score": 0.88, + "content": "b _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 232, + 505, + 246 + ], + "score": 1.0, + "content": "their biases. 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\\lambda n )", + "type": "interline_equation", + "image_path": "14352d99319d701f91c816a211d885a60b4119c1196408ed92c89f9fd5fd2834.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 236, + 334, + 375, + 352 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 506, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "This oracle ignores the impact of the current decision on the future time-steps and we therefore", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 368, + 333, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 333, + 379 + ], + "score": 1.0, + "content": "consider smoothing the likelihoods with an RBF kernel.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 384, + 470, + 420 + ], + "lines": [ + { + "bbox": [ + 124, + 384, + 470, + 420 + ], + "spans": [ + { + "bbox": [ + 124, + 384, + 470, + 420 + ], + "score": 0.94, + "content": "\\kappa ( t , t ^ { \\prime } ) = e ^ { - \\frac { \\lvert t - t ^ { \\prime } \\rvert ^ { 2 } } { \\sigma } } , \\quad \\widetilde { \\mathrm { L L } _ { t } ^ { n } } = \\sum _ { t ^ { \\prime } = 1 } ^ { \\lfloor y \\rfloor } \\kappa ( t , t ^ { \\prime } ) \\mathrm { L L } _ { t ^ { \\prime } } ^ { n } , \\quad q _ { t } ^ { * } ( { \\pmb x } , { \\pmb y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } \\widetilde { \\mathrm { L L } _ { t } ^ { n } } - \\lambda n ) ,", + "type": "interline_equation", + "image_path": "bcccd6b00d0cea955cbf50d76b119d50b34bd42e0a2c080e2ae2fb030deb3672.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 124, + 384, + 470, + 396.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 124, + 396.0, + 470, + 408.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 124, + 408.0, + 470, + 420.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 506, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 336, + 437 + ], + "score": 1.0, + "content": "where we control the size of the surrounding context with", + "type": "text" + }, + { + "bbox": [ + 337, + 426, + 344, + 434 + ], + "score": 0.8, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "the kernel width. 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We define the target", + "type": "text" + }, + { + "bbox": [ + 420, + 470, + 430, + 482 + ], + "score": 0.89, + "content": "q _ { t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 470, + 478, + 483 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 487, + 426, + 544 + ], + "lines": [ + { + "bbox": [ + 183, + 487, + 426, + 544 + ], + "spans": [ + { + "bbox": [ + 183, + 487, + 426, + 544 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { C _ { t } ^ { n } = \\mathbb { 1 } \\big [ y _ { t } = \\arg \\operatorname* { m a x } _ { y } p ( y | h _ { t - 1 } ^ { n } ) \\big ] , \\quad \\widetilde { C _ { t } ^ { n } } = \\displaystyle \\sum _ { t ^ { \\prime } = 1 } ^ { | y | } \\kappa ( t , t ^ { \\prime } ) C _ { t } ^ { n } , } } \\\\ { { q _ { t } ^ { * } ( { \\pmb x } , { \\pmb y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } \\widetilde { C _ { t } ^ { n } } - \\lambda n ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "b259a9bae7ebbd7cf6cb842668fb15f4b34352ed9f421e2dc514cff7db3c615b.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 183, + 487, + 426, + 501.25 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 183, + 501.25, + 426, + 515.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 183, + 515.5, + 426, + 529.75 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 183, + 529.75, + 426, + 544.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 553, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 566 + ], + "score": 1.0, + "content": "Confidence thresholding Finally, we consider thresholding the model predictions (§2), i.e., exit", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 345, + 578 + ], + "score": 1.0, + "content": "when the maximum score of the current output classifier", + "type": "text" + }, + { + "bbox": [ + 345, + 565, + 391, + 577 + ], + "score": 0.92, + "content": "p \\bar { ( } y _ { t + 1 } | h _ { t } ^ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 564, + 506, + 578 + ], + "score": 1.0, + "content": "exceeds a hyper-parameter", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 574, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 147, + 589 + ], + "score": 1.0, + "content": "threshold", + "type": "text" + }, + { + "bbox": [ + 147, + 577, + 158, + 587 + ], + "score": 0.85, + "content": "\\tau _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 574, + 371, + 589 + ], + "score": 1.0, + "content": ". This does not require training and the thresholds", + "type": "text" + }, + { + "bbox": [ + 371, + 576, + 457, + 588 + ], + "score": 0.92, + "content": "\\pmb { \\tau } = ( \\tau _ { 1 } , \\dots , \\tau _ { N - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 574, + 505, + 589 + ], + "score": 1.0, + "content": "are simply", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "tuned on the valid set to maximize BLEU. 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\\lambda n )", + "type": "interline_equation", + "image_path": "14352d99319d701f91c816a211d885a60b4119c1196408ed92c89f9fd5fd2834.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 236, + 334, + 375, + 352 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 506, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "This oracle ignores the impact of the current decision on the future time-steps and we therefore", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 368, + 333, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 333, + 379 + ], + "score": 1.0, + "content": "consider smoothing the likelihoods with an RBF kernel.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 356, + 505, + 379 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 384, + 470, + 420 + ], + "lines": [ + { + "bbox": [ + 124, + 384, + 470, + 420 + ], + "spans": [ + { + "bbox": [ + 124, + 384, + 470, + 420 + ], + "score": 0.94, + "content": "\\kappa ( t , t ^ { \\prime } ) = e ^ { - \\frac { \\lvert t - t ^ { \\prime } \\rvert ^ { 2 } } { \\sigma } } , \\quad \\widetilde { \\mathrm { L L } _ { t } ^ { n } } = \\sum _ { t ^ { \\prime } = 1 } ^ { \\lfloor y \\rfloor } \\kappa ( t , t ^ { \\prime } ) \\mathrm { L L } _ { t ^ { \\prime } } ^ { n } , \\quad q _ { t } ^ { * } ( { \\pmb x } , { \\pmb y } ) = \\delta ( \\arg \\operatorname* { m a x } _ { n } \\widetilde { \\mathrm { L L } _ { t } ^ { n } } - \\lambda n ) ,", + "type": "interline_equation", + "image_path": "bcccd6b00d0cea955cbf50d76b119d50b34bd42e0a2c080e2ae2fb030deb3672.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 124, + 384, + 470, + 396.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 124, + 396.0, + 470, + 408.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 124, + 408.0, + 470, + 420.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 506, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 336, + 437 + ], + "score": 1.0, + "content": "where we control the size of the surrounding context with", + "type": "text" + }, + { + "bbox": [ + 337, + 426, + 344, + 434 + ], + "score": 0.8, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "the kernel width. We refer to this oracle", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 117, + 448 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 117, + 435, + 154, + 447 + ], + "score": 0.87, + "content": "\\operatorname { L L } ( \\sigma , \\lambda )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 435, + 473, + 448 + ], + "score": 1.0, + "content": "including the case where we only look at the likelihood of the current token with", + "type": "text" + }, + { + "bbox": [ + 474, + 436, + 501, + 445 + ], + "score": 0.88, + "content": "\\sigma \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 435, + 505, + 448 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 106, + 424, + 505, + 448 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 458, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "Correctness-based: Similar to the likelihood-based oracle we can look at the correctness of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 470, + 478, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 198, + 483 + ], + "score": 1.0, + "content": "prediction at time-step", + "type": "text" + }, + { + "bbox": [ + 199, + 471, + 204, + 480 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 470, + 420, + 483 + ], + "score": 1.0, + "content": "as well as surrounding positions. 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This does not require training and the thresholds", + "type": "text" + }, + { + "bbox": [ + 371, + 576, + 457, + 588 + ], + "score": 0.92, + "content": "\\pmb { \\tau } = ( \\tau _ { 1 } , \\dots , \\tau _ { N - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 574, + 505, + 589 + ], + "score": 1.0, + "content": "are simply", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "tuned on the valid set to maximize BLEU. 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We use the setup of Edunov et al. (2018) and train on", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 222, + 733 + ], + "score": 1.0, + "content": "160K sentence pairs. 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Uniformn=1n=2n=3n=4n=5n=6Average
Baseline134.235.335.635.735.635.935.4
Aligned (ωn = 1)35.534.135.535.836.136.136.235.6
Mixed M =134.132.934.334.534.534.634.534.2
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Mixed M = 635.334.235.435.835.935.835.935.5
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We exit when", + "type": "text" + }, + { + "bbox": [ + 383, + 424, + 429, + 435 + ], + "score": 0.91, + "content": "\\chi _ { t , n } > 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "for geometric-like", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 433, + 151, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 151, + 445 + ], + "score": 1.0, + "content": "classifiers.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 462, + 314, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 315, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 315, + 474 + ], + "score": 1.0, + "content": "4.2 TRAINING MULTIPLE OUTPUT CLASSIFIERS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 343, + 496 + ], + "score": 1.0, + "content": "We first compare the two training regimes for our model", + "type": "text" + }, + { + "bbox": [ + 343, + 484, + 368, + 495 + ], + "score": 0.29, + "content": "( \\ S 2 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 483, + 505, + 496 + ], + "score": 1.0, + "content": ". Aligned training performs self-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "attention on aligned states (§2.2.1) and mixed training exposes self-attention to hidden states from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 208, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 173, + 518 + ], + "score": 1.0, + "content": "different blocks", + "type": "text" + }, + { + "bbox": [ + 173, + 506, + 204, + 517 + ], + "score": 0.46, + "content": "( \\ S 2 . 2 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 504, + 208, + 518 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "We compare the two training modes when choosing either a uniformly sampled exit or a fixed exit", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 107, + 534, + 159, + 545 + ], + "score": 0.91, + "content": "n = 1 , \\ldots , 6", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "at inference time for every time-step. The sampled exit experiment tests the robustness", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "to mixed hidden states and the fixed exit setup simulates an ideal setting where all previous states", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 423, + 568 + ], + "score": 1.0, + "content": "are available. As baselines we show six separate standard Transformers with", + "type": "text" + }, + { + "bbox": [ + 423, + 555, + 469, + 567 + ], + "score": 0.88, + "content": "N \\in [ 1 . . 6 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "decoder", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 456, + 578 + ], + "score": 1.0, + "content": "blocks. All models are trained with an equal number of updates and mixed training with", + "type": "text" + }, + { + "bbox": [ + 456, + 567, + 481, + 576 + ], + "score": 0.87, + "content": "M { = } 6", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "paths", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 578, + 464, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 464, + 589 + ], + "score": 1.0, + "content": "is most comparable to aligned training since the number of losses per sample is identical.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Table 1 shows that aligned training outperforms mixed training both for fixed exits as well as for ran-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "domly sampled exits. The latter is surprising since aligned training never exposes the self-attention", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "mechanism to hidden states from other blocks. We suspect that this is due to the residual connections", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "which copy features from lower blocks to subsequent layers and which are ubiquitous in Transformer", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 637, + 498, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 498, + 651 + ], + "score": 1.0, + "content": "models (§2). Aligned training also performs very competitively to the individual baseline models.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 471, + 667 + ], + "score": 1.0, + "content": "Aligned training is conceptually simple and fast. We can process a training example with", + "type": "text" + }, + { + "bbox": [ + 472, + 655, + 482, + 665 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "exits", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 665, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 270, + 680 + ], + "score": 1.0, + "content": "in a single forward/backward pass while", + "type": "text" + }, + { + "bbox": [ + 271, + 666, + 282, + 676 + ], + "score": 0.7, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 665, + 505, + 680 + ], + "score": 1.0, + "content": "passes are needed for mixed training. In the remaining", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "paper, we use the aligned mode to train our models. Appendix A reports experiments with weighing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "the various output classifiers differently but we found that a uniform weighting scheme worked", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "well. 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Uniformn=1n=2n=3n=4n=5n=6Average
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We report valid BLEU for a uniformly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 159, + 195 + ], + "score": 1.0, + "content": "sampled exit", + "type": "text" + }, + { + "bbox": [ + 159, + 183, + 215, + 195 + ], + "score": 0.93, + "content": "\\overset { \\cdot } { n } \\sim \\mathcal { U } ( [ 1 . . 6 ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 182, + 318, + 195 + ], + "score": 1.0, + "content": "at each token, a fixed exit", + "type": "text" + }, + { + "bbox": [ + 319, + 183, + 359, + 195 + ], + "score": 0.81, + "content": "n \\in [ 1 . . 6 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "for all tokens, as well as the average", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 193, + 504, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 504, + 206 + ], + "score": 1.0, + "content": "BLEU over the fixed exits. 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We use", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 300, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 315 + ], + "score": 1.0, + "content": "a Transformer big architecture and tie the embeddings of the encoder, the decoder and the output", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 312, + 486, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 149, + 325 + ], + "score": 1.0, + "content": "classifiers", + "type": "text" + }, + { + "bbox": [ + 149, + 313, + 227, + 325 + ], + "score": 0.81, + "content": "( ( W _ { n } ) _ { 1 \\leq n \\leq 6 } ; \\ S 2 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 312, + 486, + 325 + ], + "score": 1.0, + "content": ". We average the last ten checkpoints and use a beam of width 4.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 267, + 506, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "Models are implemented in fairseq (Ott et al., 2019) and are trained with Adam (Kingma & Ba,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "2015). 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(7)) using the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 164, + 403 + ], + "score": 1.0, + "content": "aligned mode", + "type": "text" + }, + { + "bbox": [ + 164, + 390, + 211, + 401 + ], + "score": 0.59, + "content": "( c . f . \\ S 2 . 2 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 389, + 226, + 403 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 226, + 390, + 243, + 401 + ], + "score": 0.39, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 389, + 402, + 403 + ], + "score": 1.0, + "content": "updates and then continue training with", + "type": "text" + }, + { + "bbox": [ + 403, + 390, + 429, + 401 + ], + "score": 0.91, + "content": "\\alpha \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "until convergence.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "The exit prediction classifiers are parameterized by a single linear layer (Eq. (8)) with the same input", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 411, + 504, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 493, + 424 + ], + "score": 1.0, + "content": "dimension as the embedding dimension, e.g., 1024 for a big Transformer; the output dimension is", + "type": "text" + }, + { + "bbox": [ + 494, + 412, + 504, + 422 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 383, + 437 + ], + "score": 1.0, + "content": "for a multinomial classifier or one for geometric-like. We exit when", + "type": "text" + }, + { + "bbox": [ + 383, + 424, + 429, + 435 + ], + "score": 0.91, + "content": "\\chi _ { t , n } > 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "for geometric-like", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 433, + 151, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 151, + 445 + ], + "score": 1.0, + "content": "classifiers.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 378, + 506, + 445 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 462, + 314, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 315, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 315, + 474 + ], + "score": 1.0, + "content": "4.2 TRAINING MULTIPLE OUTPUT CLASSIFIERS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 343, + 496 + ], + "score": 1.0, + "content": "We first compare the two training regimes for our model", + "type": "text" + }, + { + "bbox": [ + 343, + 484, + 368, + 495 + ], + "score": 0.29, + "content": "( \\ S 2 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 483, + 505, + 496 + ], + "score": 1.0, + "content": ". Aligned training performs self-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "attention on aligned states (§2.2.1) and mixed training exposes self-attention to hidden states from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 208, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 173, + 518 + ], + "score": 1.0, + "content": "different blocks", + "type": "text" + }, + { + "bbox": [ + 173, + 506, + 204, + 517 + ], + "score": 0.46, + "content": "( \\ S 2 . 2 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 504, + 208, + 518 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 483, + 505, + 518 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "We compare the two training modes when choosing either a uniformly sampled exit or a fixed exit", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 107, + 534, + 159, + 545 + ], + "score": 0.91, + "content": "n = 1 , \\ldots , 6", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "at inference time for every time-step. The sampled exit experiment tests the robustness", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "to mixed hidden states and the fixed exit setup simulates an ideal setting where all previous states", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 423, + 568 + ], + "score": 1.0, + "content": "are available. As baselines we show six separate standard Transformers with", + "type": "text" + }, + { + "bbox": [ + 423, + 555, + 469, + 567 + ], + "score": 0.88, + "content": "N \\in [ 1 . . 6 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "decoder", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 456, + 578 + ], + "score": 1.0, + "content": "blocks. All models are trained with an equal number of updates and mixed training with", + "type": "text" + }, + { + "bbox": [ + 456, + 567, + 481, + 576 + ], + "score": 0.87, + "content": "M { = } 6", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "paths", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 578, + 464, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 464, + 589 + ], + "score": 1.0, + "content": "is most comparable to aligned training since the number of losses per sample is identical.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 523, + 506, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Table 1 shows that aligned training outperforms mixed training both for fixed exits as well as for ran-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "domly sampled exits. The latter is surprising since aligned training never exposes the self-attention", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "mechanism to hidden states from other blocks. We suspect that this is due to the residual connections", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "which copy features from lower blocks to subsequent layers and which are ubiquitous in Transformer", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 637, + 498, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 498, + 651 + ], + "score": 1.0, + "content": "models (§2). Aligned training also performs very competitively to the individual baseline models.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 594, + 505, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 471, + 667 + ], + "score": 1.0, + "content": "Aligned training is conceptually simple and fast. We can process a training example with", + "type": "text" + }, + { + "bbox": [ + 472, + 655, + 482, + 665 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "exits", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 665, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 270, + 680 + ], + "score": 1.0, + "content": "in a single forward/backward pass while", + "type": "text" + }, + { + "bbox": [ + 271, + 666, + 282, + 676 + ], + "score": 0.7, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 665, + 505, + 680 + ], + "score": 1.0, + "content": "passes are needed for mixed training. In the remaining", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "paper, we use the aligned mode to train our models. Appendix A reports experiments with weighing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "the various output classifiers differently but we found that a uniform weighting scheme worked", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "well. 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We measure the latter as the average exit per output token (AE).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 377, + 528 + ], + "score": 1.0, + "content": "As baselines we use again six separate standard Transformers with", + "type": "text" + }, + { + "bbox": [ + 377, + 516, + 421, + 528 + ], + "score": 0.8, + "content": "N \\in [ 1 . . 6 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "with a single output", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 486, + 540 + ], + "score": 1.0, + "content": "classifier. We also measure the performance of the aligned mode trained model for fixed exits", + "type": "text" + }, + { + "bbox": [ + 487, + 528, + 505, + 538 + ], + "score": 0.83, + "content": "n \\in", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "[1..6]. 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}, + { + "type": "text", + "bbox": [ + 106, + 488, + 504, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "Next, we train models with aligned states and compare adaptive depth classifiers in terms of BLEU", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 498, + 493, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 493, + 512 + ], + "score": 1.0, + "content": "as well as computational effort. We measure the latter as the average exit per output token (AE).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 488, + 505, + 512 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 377, + 528 + ], + "score": 1.0, + "content": "As baselines we use again six separate standard Transformers with", + "type": "text" + }, + { + "bbox": [ + 377, + 516, + 421, + 528 + ], + "score": 0.8, + "content": "N \\in [ 1 . . 6 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "with a single output", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 486, + 540 + ], + "score": 1.0, + "content": "classifier. We also measure the performance of the aligned mode trained model for fixed exits", + "type": "text" + }, + { + "bbox": [ + 487, + 528, + 505, + 538 + ], + "score": 0.83, + "content": "n \\in", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "[1..6]. 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All parameters are tuned on the valid set and we report results", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 594, + 276, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 276, + 605 + ], + "score": 1.0, + "content": "on the test set for a range of average exits.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 516, + 505, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "Figure 3 shows that the aligned model (blue line) can match the accuracy of a standard 6-block", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 325, + 633 + ], + "score": 1.0, + "content": "Transformer (black line) at half the number of layers", + "type": "text" + }, + { + "bbox": [ + 325, + 621, + 353, + 632 + ], + "score": 0.85, + "content": "{ \\mathrm { ~ \\ : ~ } } _ { n } = 3 { \\mathrm { ~ \\ : ~ } }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 621, + 505, + 633 + ], + "score": 1.0, + "content": ") by always exiting at the third block.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 631, + 356, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 299, + 644 + ], + "score": 1.0, + "content": "The aligned model outperforms the baseline for", + "type": "text" + }, + { + "bbox": [ + 299, + 633, + 351, + 643 + ], + "score": 0.91, + "content": "n = 2 , \\ldots , 6", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 631, + 356, + 644 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 609, + 505, + 644 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "For token specific halting mechanisms (Figure 3a) the geometric-like classifiers achieves a better", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "speed-accuracy trade-off than the multinomial classifiers (filled vs. empty triangles). For geometric-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "like classifiers, the correctness oracle outperforms the likelihood oracle (Tok-C geometric-like vs.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "Tok-LL geometric-like) but the trend is less clear for multinomial classifiers. At the sequence-level,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 692, + 274, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 274, + 705 + ], + "score": 1.0, + "content": "likelihood is the better oracle (Figure 3b).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 649, + 505, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 281, + 722 + ], + "score": 1.0, + "content": "The rightmost Tok-C geometric-like point", + "type": "text" + }, + { + "bbox": [ + 282, + 710, + 309, + 720 + ], + "score": 0.84, + "content": "( \\sigma = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 709, + 313, + 722 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 314, + 710, + 349, + 720 + ], + "score": 0.76, + "content": "\\lambda = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 709, + 455, + 722 + ], + "score": 1.0, + "content": ") achieves 34.73 BLEU at", + "type": "text" + }, + { + "bbox": [ + 456, + 709, + 505, + 720 + ], + "score": 0.86, + "content": "\\mathrm { A E } = 1 . 4 2", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 299, + 732 + ], + "score": 1.0, + "content": "which corresponds to similar accuracy as the", + "type": "text" + }, + { + "bbox": [ + 299, + 721, + 334, + 731 + ], + "score": 0.89, + "content": "N \\ = \\ 6", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 720, + 384, + 732 + ], + "score": 1.0, + "content": "baseline at", + "type": "text" + }, + { + "bbox": [ + 384, + 721, + 404, + 731 + ], + "score": 0.87, + "content": "76 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 720, + 504, + 732 + ], + "score": 1.0, + "content": "fewer decoding blocks.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 106, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 175, + 79, + 436, + 219 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 175, + 79, + 436, + 219 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 175, + 79, + 436, + 219 + ], + "spans": [ + { + "bbox": [ + 175, + 79, + 436, + 219 + ], + "score": 0.971, + "type": "image", + "image_path": "72eb06ffdaeb2b8f53f187689e2b19f54d633c162199e0bfd63806ba35e8e616.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 175, + 79, + 436, + 125.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 175, + 125.66666666666666, + 436, + 172.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 175, + 172.33333333333331, + 436, + 218.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 124, + 229, + 484, + 242 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 228, + 486, + 244 + ], + "spans": [ + { + "bbox": [ + 124, + 228, + 427, + 244 + ], + "score": 1.0, + "content": "Figure 5: Speed and accuracy on the WMT’14 English-French benchmark", + "type": "text" + }, + { + "bbox": [ + 427, + 230, + 441, + 241 + ], + "score": 0.68, + "content": "( c . f .", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 228, + 486, + 244 + ], + "score": 1.0, + "content": ". Figure 3).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "The best accuracy of the aligned model is 34.95 BLEU at exit 5 and the best comparable Tok-C", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 324, + 294 + ], + "score": 1.0, + "content": "geometric-like configuration achieves 34.99 BLEU at", + "type": "text" + }, + { + "bbox": [ + 325, + 281, + 372, + 291 + ], + "score": 0.88, + "content": "\\mathrm { A E } = 1 . 9 7", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 279, + 388, + 294 + ], + "score": 1.0, + "content": ", or", + "type": "text" + }, + { + "bbox": [ + 388, + 281, + 408, + 291 + ], + "score": 0.87, + "content": "61 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 279, + 505, + 294 + ], + "score": 1.0, + "content": "fewer decoding blocks.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 415, + 304 + ], + "score": 1.0, + "content": "When fixing the budget to two decoder blocks, Tok-C geometric-like with", + "type": "text" + }, + { + "bbox": [ + 416, + 291, + 467, + 302 + ], + "score": 0.88, + "content": "\\mathrm { A E } = 1 . 9 7", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "achieves", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 301, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 339, + 316 + ], + "score": 1.0, + "content": "BLEU 35, a 0.64 BLEU improvement over the baseline (", + "type": "text" + }, + { + "bbox": [ + 339, + 303, + 370, + 313 + ], + "score": 0.86, + "content": "N = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 301, + 505, + 316 + ], + "score": 1.0, + "content": ") and aligned which both achieve", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 313, + 165, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 165, + 325 + ], + "score": 1.0, + "content": "BLEU 34.35.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 504, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 344 + ], + "score": 1.0, + "content": "Confidence thresholding (Figure 3c) performs very well but cannot outperform Tok-C geometric-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 339, + 128, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 128, + 354 + ], + "score": 1.0, + "content": "like.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "Ablation of hyper-parameters In this section, we look at the effect of the two main hyper-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 241, + 388 + ], + "score": 1.0, + "content": "parameters on IWSLT’14 De-En:", + "type": "text" + }, + { + "bbox": [ + 242, + 376, + 249, + 386 + ], + "score": 0.69, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 375, + 348, + 388 + ], + "score": 1.0, + "content": "the regularization scale (", + "type": "text" + }, + { + "bbox": [ + 348, + 376, + 361, + 387 + ], + "score": 0.74, + "content": "_ { . c . f }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 375, + 506, + 388 + ], + "score": 1.0, + "content": ". Eq. (9)), and the RBF kernel width", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 107, + 389, + 114, + 397 + ], + "score": 0.67, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 386, + 225, + 399 + ], + "score": 1.0, + "content": "used to smooth the scores", + "type": "text" + }, + { + "bbox": [ + 226, + 387, + 239, + 398 + ], + "score": 0.72, + "content": "( c . f", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 386, + 506, + 399 + ], + "score": 1.0, + "content": ". Eq. (15)). We train Tok-LL Geometric-like models and evaluate", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 275, + 411 + ], + "score": 1.0, + "content": "them with their default thresholds (exit if", + "type": "text" + }, + { + "bbox": [ + 275, + 398, + 317, + 410 + ], + "score": 0.9, + "content": "\\chi _ { t } ^ { n } > 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 398, + 477, + 411 + ], + "score": 1.0, + "content": ". Figure 4a shows that higher values of", + "type": "text" + }, + { + "bbox": [ + 477, + 398, + 484, + 408 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "lead", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 409, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 283, + 420 + ], + "score": 1.0, + "content": "to lower exits. Figure 4b shows the effect of", + "type": "text" + }, + { + "bbox": [ + 283, + 411, + 290, + 419 + ], + "score": 0.81, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 409, + 361, + 420 + ], + "score": 1.0, + "content": "for two values of", + "type": "text" + }, + { + "bbox": [ + 361, + 409, + 368, + 419 + ], + "score": 0.7, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 409, + 505, + 420 + ], + "score": 1.0, + "content": ". In both curves, we see that wider", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 419, + 212, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 212, + 432 + ], + "score": 1.0, + "content": "kernels favor higher exits.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 444, + 306, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 308, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 308, + 457 + ], + "score": 1.0, + "content": "4.4 SCALING THE ADAPTIVE-DEPTH MODELS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 464, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 478 + ], + "score": 1.0, + "content": "Finally, we take the best performing models form the IWSLT benchmark and test them on the large", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "WMT’14 English-French benchmark. Results on the test set (Figure 5a) show that adaptive depth", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "still shows improvements but that they are diminished in this very large-scale setup. Confidence", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "thresholding works very well and sequence-specific depth approaches improve only marginally over", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 507, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 457, + 522 + ], + "score": 1.0, + "content": "the baseline. Tok-LL geometric-like can match the best baseline result of BLEU 43.4", + "type": "text" + }, + { + "bbox": [ + 457, + 509, + 488, + 519 + ], + "score": 0.85, + "content": "N = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 507, + 505, + 522 + ], + "score": 1.0, + "content": ") by", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 152, + 532 + ], + "score": 1.0, + "content": "using only", + "type": "text" + }, + { + "bbox": [ + 153, + 520, + 203, + 531 + ], + "score": 0.87, + "content": "\\mathrm { A E } = 2 . 4 0", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 520, + 295, + 532 + ], + "score": 1.0, + "content": "which corresponds to", + "type": "text" + }, + { + "bbox": [ + 295, + 520, + 315, + 530 + ], + "score": 0.86, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "of the decoder blocks; the best aligned result", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 529, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 259, + 544 + ], + "score": 1.0, + "content": "of BLEU 43.6 can be matched with", + "type": "text" + }, + { + "bbox": [ + 259, + 531, + 310, + 541 + ], + "score": 0.87, + "content": "\\mathrm { A E } = 3 . 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 529, + 505, + 544 + ], + "score": 1.0, + "content": ". In this setup, Tok-LL geometric-like slightly", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 542, + 250, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 250, + 554 + ], + "score": 1.0, + "content": "outperforms the Tok-C counterpart.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 504, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 320, + 570 + ], + "score": 1.0, + "content": "Confidence thresholding matches the accuracy of the", + "type": "text" + }, + { + "bbox": [ + 321, + 559, + 344, + 569 + ], + "score": 0.86, + "content": "N { = } 6", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 559, + 443, + 570 + ], + "score": 1.0, + "content": "baseline with AE 2.5 or", + "type": "text" + }, + { + "bbox": [ + 444, + 559, + 464, + 569 + ], + "score": 0.86, + "content": "59 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 559, + 504, + 570 + ], + "score": 1.0, + "content": "fewer de-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "coding blocks. However, confidence thresholding requires computing the output classifier at each", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "score": 1.0, + "content": "block to determine whether to halt or continue. This is a large overhead since output classifiers pre-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "dict 44k types for this benchmark (§4.1). To better account for this, we measure the average number", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "of FLOPs per output token (details in Appendix B). Figure 5b shows that the Tok-LL geometric-like", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 613, + 480, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 480, + 626 + ], + "score": 1.0, + "content": "approach provides a better trade-off when the overhead of the output classifiers is considered.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 107, + 638, + 230, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 231, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 231, + 651 + ], + "score": 1.0, + "content": "4.5 QUALITATIVE RESULTS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "The exit distribution for a given sample can give insights into what a Depth-Adaptive Transformer", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 426, + 682 + ], + "score": 1.0, + "content": "decoder considers to be a difficult task. In this section, for each hypothesis", + "type": "text" + }, + { + "bbox": [ + 426, + 670, + 434, + 681 + ], + "score": 0.8, + "content": "\\widetilde { \\boldsymbol { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 669, + 505, + 682 + ], + "score": 1.0, + "content": ", we will look at", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 234, + 694 + ], + "score": 1.0, + "content": "the sequence of selected exits", + "type": "text" + }, + { + "bbox": [ + 235, + 681, + 291, + 694 + ], + "score": 0.92, + "content": "( n _ { 1 } , \\ldots , n _ { | \\tilde { \\pmb { y } } | } )", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 680, + 405, + 694 + ], + "score": 1.0, + "content": "and the probability scores", + "type": "text" + }, + { + "bbox": [ + 406, + 681, + 456, + 694 + ], + "score": 0.91, + "content": "( p _ { 1 } , \\hdots p _ { | \\widetilde { \\pmb { y } } | } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 680, + 480, + 694 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 481, + 682, + 505, + 693 + ], + "score": 0.85, + "content": "\\begin{array} { r l } { p _ { t } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 691, + 445, + 706 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 151, + 705 + ], + "score": 0.91, + "content": "p ( \\widetilde { y } _ { t } | h _ { t - 1 } ^ { n _ { t } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 691, + 445, + 706 + ], + "score": 1.0, + "content": "e i.e. the confidence of the model in the sampled token at the selected exit.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Figures 6 and 7 show hypotheses from the WMT’14 En-Fr and IWSLT’14 De-En test sets, respec-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "tively. For each hypothesis we state the exits and the probability scores. 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Figure 3).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "The best accuracy of the aligned model is 34.95 BLEU at exit 5 and the best comparable Tok-C", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 324, + 294 + ], + "score": 1.0, + "content": "geometric-like configuration achieves 34.99 BLEU at", + "type": "text" + }, + { + "bbox": [ + 325, + 281, + 372, + 291 + ], + "score": 0.88, + "content": "\\mathrm { A E } = 1 . 9 7", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 279, + 388, + 294 + ], + "score": 1.0, + "content": ", or", + "type": "text" + }, + { + "bbox": [ + 388, + 281, + 408, + 291 + ], + "score": 0.87, + "content": "61 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 279, + 505, + 294 + ], + "score": 1.0, + "content": "fewer decoding blocks.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 415, + 304 + ], + "score": 1.0, + "content": "When fixing the budget to two decoder blocks, Tok-C geometric-like with", + "type": "text" + }, + { + "bbox": [ + 416, + 291, + 467, + 302 + ], + "score": 0.88, + "content": "\\mathrm { A E } = 1 . 9 7", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "achieves", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 301, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 339, + 316 + ], + "score": 1.0, + "content": "BLEU 35, a 0.64 BLEU improvement over the baseline (", + "type": "text" + }, + { + "bbox": [ + 339, + 303, + 370, + 313 + ], + "score": 0.86, + "content": "N = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 301, + 505, + 316 + ], + "score": 1.0, + "content": ") and aligned which both achieve", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 313, + 165, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 165, + 325 + ], + "score": 1.0, + "content": "BLEU 34.35.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 269, + 505, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 504, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 344 + ], + "score": 1.0, + "content": "Confidence thresholding (Figure 3c) performs very well but cannot outperform Tok-C geometric-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 339, + 128, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 128, + 354 + ], + "score": 1.0, + "content": "like.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 329, + 505, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "Ablation of hyper-parameters In this section, we look at the effect of the two main hyper-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 241, + 388 + ], + "score": 1.0, + "content": "parameters on IWSLT’14 De-En:", + "type": "text" + }, + { + "bbox": [ + 242, + 376, + 249, + 386 + ], + "score": 0.69, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 375, + 348, + 388 + ], + "score": 1.0, + "content": "the regularization scale (", + "type": "text" + }, + { + "bbox": [ + 348, + 376, + 361, + 387 + ], + "score": 0.74, + "content": "_ { . c . f }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 375, + 506, + 388 + ], + "score": 1.0, + "content": ". Eq. (9)), and the RBF kernel width", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 107, + 389, + 114, + 397 + ], + "score": 0.67, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 386, + 225, + 399 + ], + "score": 1.0, + "content": "used to smooth the scores", + "type": "text" + }, + { + "bbox": [ + 226, + 387, + 239, + 398 + ], + "score": 0.72, + "content": "( c . f", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 386, + 506, + 399 + ], + "score": 1.0, + "content": ". Eq. (15)). We train Tok-LL Geometric-like models and evaluate", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 275, + 411 + ], + "score": 1.0, + "content": "them with their default thresholds (exit if", + "type": "text" + }, + { + "bbox": [ + 275, + 398, + 317, + 410 + ], + "score": 0.9, + "content": "\\chi _ { t } ^ { n } > 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 398, + 477, + 411 + ], + "score": 1.0, + "content": ". Figure 4a shows that higher values of", + "type": "text" + }, + { + "bbox": [ + 477, + 398, + 484, + 408 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "lead", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 409, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 283, + 420 + ], + "score": 1.0, + "content": "to lower exits. Figure 4b shows the effect of", + "type": "text" + }, + { + "bbox": [ + 283, + 411, + 290, + 419 + ], + "score": 0.81, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 409, + 361, + 420 + ], + "score": 1.0, + "content": "for two values of", + "type": "text" + }, + { + "bbox": [ + 361, + 409, + 368, + 419 + ], + "score": 0.7, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 409, + 505, + 420 + ], + "score": 1.0, + "content": ". In both curves, we see that wider", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 419, + 212, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 212, + 432 + ], + "score": 1.0, + "content": "kernels favor higher exits.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 364, + 506, + 432 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 444, + 306, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 308, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 308, + 457 + ], + "score": 1.0, + "content": "4.4 SCALING THE ADAPTIVE-DEPTH MODELS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 464, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 478 + ], + "score": 1.0, + "content": "Finally, we take the best performing models form the IWSLT benchmark and test them on the large", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "WMT’14 English-French benchmark. Results on the test set (Figure 5a) show that adaptive depth", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "still shows improvements but that they are diminished in this very large-scale setup. Confidence", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "thresholding works very well and sequence-specific depth approaches improve only marginally over", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 507, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 457, + 522 + ], + "score": 1.0, + "content": "the baseline. Tok-LL geometric-like can match the best baseline result of BLEU 43.4", + "type": "text" + }, + { + "bbox": [ + 457, + 509, + 488, + 519 + ], + "score": 0.85, + "content": "N = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 507, + 505, + 522 + ], + "score": 1.0, + "content": ") by", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 152, + 532 + ], + "score": 1.0, + "content": "using only", + "type": "text" + }, + { + "bbox": [ + 153, + 520, + 203, + 531 + ], + "score": 0.87, + "content": "\\mathrm { A E } = 2 . 4 0", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 520, + 295, + 532 + ], + "score": 1.0, + "content": "which corresponds to", + "type": "text" + }, + { + "bbox": [ + 295, + 520, + 315, + 530 + ], + "score": 0.86, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "of the decoder blocks; the best aligned result", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 529, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 259, + 544 + ], + "score": 1.0, + "content": "of BLEU 43.6 can be matched with", + "type": "text" + }, + { + "bbox": [ + 259, + 531, + 310, + 541 + ], + "score": 0.87, + "content": "\\mathrm { A E } = 3 . 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 529, + 505, + 544 + ], + "score": 1.0, + "content": ". In this setup, Tok-LL geometric-like slightly", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 542, + 250, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 250, + 554 + ], + "score": 1.0, + "content": "outperforms the Tok-C counterpart.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 463, + 506, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 504, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 320, + 570 + ], + "score": 1.0, + "content": "Confidence thresholding matches the accuracy of the", + "type": "text" + }, + { + "bbox": [ + 321, + 559, + 344, + 569 + ], + "score": 0.86, + "content": "N { = } 6", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 559, + 443, + 570 + ], + "score": 1.0, + "content": "baseline with AE 2.5 or", + "type": "text" + }, + { + "bbox": [ + 444, + 559, + 464, + 569 + ], + "score": 0.86, + "content": "59 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 559, + 504, + 570 + ], + "score": 1.0, + "content": "fewer de-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "coding blocks. However, confidence thresholding requires computing the output classifier at each", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "score": 1.0, + "content": "block to determine whether to halt or continue. This is a large overhead since output classifiers pre-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "dict 44k types for this benchmark (§4.1). To better account for this, we measure the average number", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "of FLOPs per output token (details in Appendix B). Figure 5b shows that the Tok-LL geometric-like", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 613, + 480, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 480, + 626 + ], + "score": 1.0, + "content": "approach provides a better trade-off when the overhead of the output classifiers is considered.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 559, + 505, + 626 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 638, + 230, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 231, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 231, + 651 + ], + "score": 1.0, + "content": "4.5 QUALITATIVE RESULTS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "The exit distribution for a given sample can give insights into what a Depth-Adaptive Transformer", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 426, + 682 + ], + "score": 1.0, + "content": "decoder considers to be a difficult task. In this section, for each hypothesis", + "type": "text" + }, + { + "bbox": [ + 426, + 670, + 434, + 681 + ], + "score": 0.8, + "content": "\\widetilde { \\boldsymbol { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 669, + 505, + 682 + ], + "score": 1.0, + "content": ", we will look at", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 234, + 694 + ], + "score": 1.0, + "content": "the sequence of selected exits", + "type": "text" + }, + { + "bbox": [ + 235, + 681, + 291, + 694 + ], + "score": 0.92, + "content": "( n _ { 1 } , \\ldots , n _ { | \\tilde { \\pmb { y } } | } )", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 680, + 405, + 694 + ], + "score": 1.0, + "content": "and the probability scores", + "type": "text" + }, + { + "bbox": [ + 406, + 681, + 456, + 694 + ], + "score": 0.91, + "content": "( p _ { 1 } , \\hdots p _ { | \\widetilde { \\pmb { y } } | } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 680, + 480, + 694 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 481, + 682, + 505, + 693 + ], + "score": 0.85, + "content": "\\begin{array} { r l } { p _ { t } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 691, + 445, + 706 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 151, + 705 + ], + "score": 0.91, + "content": "p ( \\widetilde { y } _ { t } | h _ { t - 1 } ^ { n _ { t } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 691, + 445, + 706 + ], + "score": 1.0, + "content": "e i.e. the confidence of the model in the sampled token at the selected exit.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 658, + 506, + 706 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Figures 6 and 7 show hypotheses from the WMT’14 En-Fr and IWSLT’14 De-En test sets, respec-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "tively. For each hypothesis we state the exits and the probability scores. 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Token exits are in blue and confidence scores are in gray. The", + "type": "text" + }, + { + "bbox": [ + 408, + 216, + 429, + 226 + ], + "score": 0.76, + "content": "\\bullet _ { @ \\mathbb { Q } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "are due to BPE or", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "subword tokenization. For each example the source (Src) and the reference (Ref) are provided in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 237, + 156, + 250 + ], + "spans": [ + { + "bbox": [ + 104, + 237, + 156, + 250 + ], + "score": 1.0, + "content": "the caption.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "image", + "bbox": [ + 168, + 264, + 440, + 326 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 168, + 264, + 440, + 326 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 168, + 264, + 440, + 326 + ], + "spans": [ + { + "bbox": [ + 168, + 264, + 440, + 326 + ], + "score": 0.964, + "type": "image", + "image_path": "c09a06b4990673a86c9b78ffc1d7e7a2e7d8b550d430e028ee6b5dbf8b98eb27.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 168, + 264, + 440, + 284.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 168, + 284.6666666666667, + 440, + 305.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 168, + 305.33333333333337, + 440, + 326.00000000000006 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 368, + 504, + 391 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "Figure 7: Example from the IWSLT’14 De-En test set with Tok-LL geometric-like depth estimation.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 380, + 227, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 227, + 392 + ], + "score": 1.0, + "content": "See Figure 6 for more details.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + } + ], + "index": 23.25 + }, + { + "type": "text", + "bbox": [ + 145, + 339, + 450, + 360 + ], + "lines": [ + { + "bbox": [ + 145, + 339, + 451, + 350 + ], + "spans": [ + { + "bbox": [ + 145, + 339, + 451, + 350 + ], + "score": 1.0, + "content": "(a) Src: diesen trick können sie ihren freunden und nachbarn vor@@führen . danke .", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 158, + 348, + 431, + 361 + ], + "spans": [ + { + "bbox": [ + 158, + 348, + 431, + 361 + ], + "score": 1.0, + "content": "Ref: there is a trick you can do for your friends and neighb@@ors . thanks .", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 107, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 107, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "‘présent’ (meaning ‘present’) is hard. A straightforward translation is ‘était là’ but the model chooses", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 108, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 108, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "‘present’ which is also appropriate. In Figure 6b, the model uses more computation to predict the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 436, + 412, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 412, + 449 + ], + "score": 1.0, + "content": "definite article ‘les’ since the source has omitted the article for ‘passengers’.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 506, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "A clear trend in both benchmarks is that the model requires less computation near the end of de-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 295, + 477 + ], + "score": 1.0, + "content": "coding to generate the end of sequence marker", + "type": "text" + }, + { + "bbox": [ + 296, + 465, + 321, + 477 + ], + "score": 0.94, + "content": "< / \\mathrm { s } >", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "and the preceding full-stop when relevant. In", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "Figure 8, we show the distribution of the exits at the beginning and near the end of test set hypothe-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 487, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 351, + 498 + ], + "score": 1.0, + "content": "ses. We consider the beginning of a sequence to be the first", + "type": "text" + }, + { + "bbox": [ + 351, + 487, + 371, + 497 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 487, + 505, + 498 + ], + "score": 1.0, + "content": "of tokens and the end as the last", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 126, + 508 + ], + "score": 0.85, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "of tokens. The exit distributions are shown for three models on WMT’14 En-Fr: Model1 has", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 179, + 520 + ], + "score": 1.0, + "content": "an average exit of", + "type": "text" + }, + { + "bbox": [ + 179, + 509, + 225, + 519 + ], + "score": 0.76, + "content": "\\mathrm { A E = 2 . 5 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 509, + 292, + 520 + ], + "score": 1.0, + "content": ", Model2 exits at", + "type": "text" + }, + { + "bbox": [ + 293, + 509, + 339, + 519 + ], + "score": 0.89, + "content": "\\mathrm { A E } = 3 . 7 9", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 509, + 402, + 520 + ], + "score": 1.0, + "content": "on average and", + "type": "text" + }, + { + "bbox": [ + 402, + 509, + 434, + 520 + ], + "score": 0.37, + "content": "\\mathbf { M o d e l } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 509, + 455, + 520 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 455, + 509, + 501, + 519 + ], + "score": 0.88, + "content": "\\mathrm { A E } = 4 . 6 8", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 509, + 505, + 520 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "Within the same models, deep exits late are used at the beginning of the sequence and early exits are", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 350, + 543 + ], + "score": 1.0, + "content": "selected near the end. For heavily regularized models such as", + "type": "text" + }, + { + "bbox": [ + 351, + 531, + 382, + 541 + ], + "score": 0.66, + "content": "\\mathbf { M o d e l } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 530, + 403, + 543 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 403, + 531, + 450, + 541 + ], + "score": 0.89, + "content": "\\mathrm { A E = 2 . 5 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 530, + 505, + 543 + ], + "score": 1.0, + "content": ", the disparity", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 453, + 553 + ], + "score": 1.0, + "content": "between beginning and end is less severe as the model exits early most of the time.", + "type": "text" + }, + { + "bbox": [ + 454, + 542, + 486, + 553 + ], + "score": 0.7, + "content": "\\mathbf { M o d e l } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 542, + 505, + 553 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "Model3 are less regularized (higher AE) and tend to use late exits at the beginning of the sequence", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 383, + 575 + ], + "score": 1.0, + "content": "and early exits near the end. 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A straightforward translation is ‘était là’ but the model chooses", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 108, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 108, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "‘present’ which is also appropriate. In Figure 6b, the model uses more computation to predict the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 436, + 412, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 412, + 449 + ], + "score": 1.0, + "content": "definite article ‘les’ since the source has omitted the article for ‘passengers’.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 414, + 505, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 506, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "A clear trend in both benchmarks is that the model requires less computation near the end of de-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 295, + 477 + ], + "score": 1.0, + "content": "coding to generate the end of sequence marker", + "type": "text" + }, + { + "bbox": [ + 296, + 465, + 321, + 477 + ], + "score": 0.94, + "content": "< / \\mathrm { s } >", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "and the preceding full-stop when relevant. In", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "Figure 8, we show the distribution of the exits at the beginning and near the end of test set hypothe-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 487, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 351, + 498 + ], + "score": 1.0, + "content": "ses. We consider the beginning of a sequence to be the first", + "type": "text" + }, + { + "bbox": [ + 351, + 487, + 371, + 497 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 487, + 505, + 498 + ], + "score": 1.0, + "content": "of tokens and the end as the last", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 126, + 508 + ], + "score": 0.85, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "of tokens. The exit distributions are shown for three models on WMT’14 En-Fr: Model1 has", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 179, + 520 + ], + "score": 1.0, + "content": "an average exit of", + "type": "text" + }, + { + "bbox": [ + 179, + 509, + 225, + 519 + ], + "score": 0.76, + "content": "\\mathrm { A E = 2 . 5 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 509, + 292, + 520 + ], + "score": 1.0, + "content": ", Model2 exits at", + "type": "text" + }, + { + "bbox": [ + 293, + 509, + 339, + 519 + ], + "score": 0.89, + "content": "\\mathrm { A E } = 3 . 7 9", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 509, + 402, + 520 + ], + "score": 1.0, + "content": "on average and", + "type": "text" + }, + { + "bbox": [ + 402, + 509, + 434, + 520 + ], + "score": 0.37, + "content": "\\mathbf { M o d e l } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 509, + 455, + 520 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 455, + 509, + 501, + 519 + ], + "score": 0.88, + "content": "\\mathrm { A E } = 4 . 6 8", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 509, + 505, + 520 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "Within the same models, deep exits late are used at the beginning of the sequence and early exits are", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 350, + 543 + ], + "score": 1.0, + "content": "selected near the end. For heavily regularized models such as", + "type": "text" + }, + { + "bbox": [ + 351, + 531, + 382, + 541 + ], + "score": 0.66, + "content": "\\mathbf { M o d e l } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 530, + 403, + 543 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 403, + 531, + 450, + 541 + ], + "score": 0.89, + "content": "\\mathrm { A E = 2 . 5 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 530, + 505, + 543 + ], + "score": 1.0, + "content": ", the disparity", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 453, + 553 + ], + "score": 1.0, + "content": "between beginning and end is less severe as the model exits early most of the time.", + "type": "text" + }, + { + "bbox": [ + 454, + 542, + 486, + 553 + ], + "score": 0.7, + "content": "\\mathbf { M o d e l } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 542, + 505, + 553 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "Model3 are less regularized (higher AE) and tend to use late exits at the beginning of the sequence", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 383, + 575 + ], + "score": 1.0, + "content": "and early exits near the end. On the other hand, the more regularized", + "type": "text" + }, + { + "bbox": [ + 383, + 563, + 414, + 574 + ], + "score": 0.37, + "content": "\\mathbf { M o d e l } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 564, + 436, + 575 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 436, + 563, + 483, + 574 + ], + "score": 0.89, + "content": "\\mathrm { A E = 2 . 5 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 564, + 505, + 575 + ], + "score": 1.0, + "content": "exits", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "score": 1.0, + "content": "early most of the time. There is also a correlation between the model probability and the amount", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "of computation, particularly in models with low AE . Figure 9 shows the joint histogram of the", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 260, + 295 + ], + "score": 1.0, + "content": "scores and the selected exit. 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Figure 10 and Algorithm 1 illustrate this gradient scaling", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 539, + 504, + 554 + ], + "spans": [ + { + "bbox": [ + 104, + 539, + 169, + 554 + ], + "score": 1.0, + "content": "procedure. 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We choose", + "type": "text" + }, + { + "bbox": [ + 312, + 551, + 381, + 563 + ], + "score": 0.93, + "content": "\\gamma _ { n } = \\gamma ( N - n )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 551, + 462, + 564 + ], + "score": 1.0, + "content": "to control the ratio", + "type": "text" + }, + { + "bbox": [ + 462, + 552, + 477, + 563 + ], + "score": 0.43, + "content": "\\gamma { : } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "as the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 562, + 384, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 384, + 575 + ], + "score": 1.0, + "content": "ratio of the block supervision to the subsequent blocks’ supervisions.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 503, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 504, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 504, + 592 + ], + "score": 1.0, + "content": "Table 3 shows that gradient scaling can benefit the lowest layer at the expense of higher layers.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 590, + 298, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 298, + 603 + ], + "score": 1.0, + "content": "However, no scaling generally works very well.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "image", + "bbox": [ + 136, + 622, + 477, + 707 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 136, + 622, + 477, + 707 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 136, + 622, + 477, + 707 + ], + "spans": [ + { + "bbox": [ + 136, + 622, + 477, + 707 + ], + "score": 0.967, + "type": "image", + "image_path": "ea1e31e6c741caa889c0b1175b0bed7b176d252c5f35d54ca54e3389a0dd0932.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 136, + 622, + 477, + 650.3333333333334 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 136, + 650.3333333333334, + 477, + 678.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 136, + 678.6666666666667, + 477, + 707.0000000000001 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 221, + 717, + 390, + 729 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 219, + 714, + 391, + 732 + ], + "spans": [ + { + "bbox": [ + 219, + 714, + 391, + 732 + ], + "score": 1.0, + "content": "Figure 10: Illustration of gradient scaling.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + } + ], + "index": 25.0 + } + ], + "page_idx": 11, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "12", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 261, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 263, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 263, + 96 + ], + "score": 1.0, + "content": "APPENDIX A LOSS SCALING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 113, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 106, + 113, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 505, + 126 + ], + "score": 1.0, + "content": "In this section we experiment with different weights for scaling the output classifier losses. 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(a) IWSLT De-En - Valid
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Uniformn=1n=2n=3n=4n=5n=6Average
Baseline134.235.335.635.735.635.935.4
@35.534.135.535.836.136.136.235.6
γ = 0.335.133.734.735.335.735.836.035.2
γ = 0.535.434.835.435.635.635.735.635.4
γ = 0.734.934.635.135.135.235.435.335.1
γ = 0.934.934.835.335.335.335.435.535.3
γ = 1.135.134.935.235.335.335.335.335.2
(a) IWSLT De-En - Valid
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline33.734.634.634.634.634.834.5
034.433.234.434.834.935.034.934.5
γ = 0.334.232.833.934.334.634.835.034.2
γ = 0.534.533.834.234.634.534.734.734.6
γ = 0.734.033.734.234.334.334.334.334.2
γ = 0.934.134.034.234.334.434.434.434.3
γ = 1.134.234.034.334.334.334.334.234.2
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Uniformn=1n=2n=3n=4n=5n=6Average
Baseline134.235.335.635.735.635.935.4
@35.534.135.535.836.136.136.235.6
γ = 0.335.133.734.735.335.735.836.035.2
γ = 0.535.434.835.435.635.635.735.635.4
γ = 0.734.934.635.135.135.235.435.335.1
γ = 0.934.934.835.335.335.335.435.535.3
γ = 1.135.134.935.235.335.335.335.335.2
(a) IWSLT De-En - Valid
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline33.734.634.634.634.634.834.5
034.433.234.434.834.935.034.934.5
γ = 0.334.232.833.934.334.634.835.034.2
γ = 0.534.533.834.234.634.534.734.734.6
γ = 0.734.033.734.234.334.334.334.334.2
γ = 0.934.134.034.234.334.434.434.434.3
γ = 1.134.234.034.334.334.334.334.234.2
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The main operations we account for are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "dot-products and by extension matrix-vector products since those represent the vast majority of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 359, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 359, + 173 + ], + "score": 1.0, + "content": "FLOPS (we assume batch size one to simplify the calculation).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 136, + 184, + 182, + 195 + ], + "lines": [ + { + "bbox": [ + 136, + 183, + 183, + 195 + ], + "spans": [ + { + "bbox": [ + 136, + 183, + 183, + 195 + ], + "score": 1.0, + "content": "Parameters", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "table", + "bbox": [ + 109, + 189, + 446, + 267 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 189, + 446, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 189, + 446, + 267 + ], + "spans": [ + { + "bbox": [ + 109, + 189, + 446, + 267 + ], + "score": 0.566, + "html": "
Parameters decoder embedding dimension.FLOPS
dd de df x t V output vocabulary size.encoder embedding dimension.Operation
The feed-forward network dimension.Dot-product (d)2d-1
source length. Current time-estep (t ≥ 1).Linear din →dout2dindout
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\\quad \\frac { 1 } { \\sum _ { i } | y ^ { ( i ) } | } \\sum _ { i } \\sum _ { t = 1 } ^ { | y ^ { ( i ) } | } \\left( N \\mathrm { ~ F C } ( { \\pmb x } ^ { ( i ) } , t ) + 2 V d _ { d } \\right) } \\\\ & { \\quad \\mathrm { A d a p t i v e ~ d e p t h } ; \\quad \\frac { 1 } { \\sum _ { i } | y ^ { ( i ) } | } \\sum _ { i } \\sum _ { t = 1 } ^ { | y ^ { ( i ) } | } \\left( q ( t ) \\mathrm { F C } ( { \\pmb x } ^ { ( i ) } , t ) + ( N - q ( t ) ) \\mathrm { F S } + \\mathrm { F P } ( t , q ( t ) ) + 2 V d _ { d } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "33ba85117e194e298a12b73818c7b4463d345cabdf03212cd5dda42da22bc68c.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 125, + 533, + 523, + 546.3333333333334 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 125, + 546.3333333333334, + 523, + 559.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 125, + 559.6666666666667, + 523, + 573.0000000000001 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 576, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 389, + 589 + ], + "score": 1.0, + "content": "In the case of confidence thresholding the final output prediction cost", + "type": "text" + }, + { + "bbox": [ + 389, + 577, + 419, + 588 + ], + "score": 0.89, + "content": "( 2 V d _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "is already accounted", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 587, + 242, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 242, + 599 + ], + "score": 1.0, + "content": "for in the exit prediction cost FP.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 80, + 312, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 314, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 314, + 96 + ], + "score": 1.0, + "content": "APPENDIX B FLOPS APPROXIMATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 506, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "This section details the computation of the FLOPS we report. The per token FLOPS are for the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "decoder network only since we use an encoder of the same size for all models. We breakdown", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 140 + ], + "score": 1.0, + "content": "the FLOPS of every operation in Algorithm 2 (blue front of the algorithmic statement). We omit", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 505, + 151 + ], + "score": 1.0, + "content": "non-linearities, normalizations and residual connections. The main operations we account for are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "dot-products and by extension matrix-vector products since those represent the vast majority of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 359, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 359, + 173 + ], + "score": 1.0, + "content": "FLOPS (we assume batch size one to simplify the calculation).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 106, + 506, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 136, + 184, + 182, + 195 + ], + "lines": [ + { + "bbox": [ + 136, + 183, + 183, + 195 + ], + "spans": [ + { + "bbox": [ + 136, + 183, + 183, + 195 + ], + "score": 1.0, + "content": "Parameters", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 136, + 183, + 183, + 195 + ] + }, + { + "type": "table", + "bbox": [ + 109, + 189, + 446, + 267 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 189, + 446, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 189, + 446, + 267 + ], + "spans": [ + { + "bbox": [ + 109, + 189, + 446, + 267 + ], + "score": 0.566, + "html": "
Parameters decoder embedding dimension.FLOPS
dd de df x t V output vocabulary size.encoder embedding dimension.Operation
The feed-forward network dimension.Dot-product (d)2d-1
source length. Current time-estep (t ≥ 1).Linear din →dout2dindout
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Baseline134.235.335.635.735.635.935.4
Aligned (ωn = 1)35.534.135.535.836.136.136.235.6
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(a) IWSLT De-En - Valid
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline133.734.634.634.634.634.834.5
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Uniformn=1n=2n=3n=4n=5n=6Average
Baseline134.235.335.635.735.635.935.4
@35.534.135.535.836.136.136.235.6
γ = 0.335.133.734.735.335.735.836.035.2
γ = 0.535.434.835.435.635.635.735.635.4
γ = 0.734.934.635.135.135.235.435.335.1
γ = 0.934.934.835.335.335.335.435.535.3
γ = 1.135.134.935.235.335.335.335.335.2
(a) IWSLT De-En - Valid
Uniformn=1n=2n=3n=4n=5n=6Average
Baseline33.734.634.634.634.634.834.5
034.433.234.434.834.935.034.934.5
γ = 0.334.232.833.934.334.634.835.034.2
γ = 0.534.533.834.234.634.534.734.734.6
γ = 0.734.033.734.234.334.334.334.334.2
γ = 0.934.134.034.234.334.434.434.434.3
γ = 1.134.234.034.334.334.334.334.234.2
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Parameters decoder embedding dimension.FLOPS
dd de df x t V output vocabulary size.encoder embedding dimension.Operation
The feed-forward network dimension.Dot-product (d)2d-1
source length. Current time-estep (t ≥ 1).Linear din →dout2dindout
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0000000000000000000000000000000000000000..7e0cf6d2498c2e5857cf54cbd406092dc900617f --- /dev/null +++ b/parse/train/SJx9GQb0-/SJx9GQb0-.md @@ -0,0 +1,382 @@ +# IMPROVING THE IMPROVED TRAINING OF WASSERSTEIN GANS: A CONSISTENCY TERM AND ITS DUAL EFFECT + +Xiang $\mathbf { W e i } ^ { 1 , 2 * }$ , Boqing $\mathbf { G o n g ^ { 3 * } }$ , Zixia $\mathbf { L i u ^ { 1 } }$ , Wei $\mathbf { L } \mathbf { u } ^ { 2 }$ , Liqiang Wang1 1Department of Computer Science, University of Central Florida, Orlando, FL, USA 32816 2School of Software Engineering, Beijing Jiaotong University, Beijing, China 100044 3Tencent AI Lab, Bellevue, WA, USA 98004 yqweixiang@knights.ucf.edu, boqinggo@outlook.com zixia@knights.ucf.edu, luwei@bjtu.edu.cn, lwang@cs.ucf.edu + +# ABSTRACT + +Despite being impactful on a variety of problems and applications, the generative adversarial nets (GANs) are remarkably difficult to train. This issue is formally analyzed by Arjovsky & Bottou (2017), who also propose an alternative direction to avoid the caveats in the minmax two-player training of GANs. The corresponding algorithm, called Wasserstein GAN (WGAN), hinges on the 1-Lipschitz continuity of the discriminator. In this paper, we propose a novel approach to enforcing the Lipschitz continuity in the training procedure of WGANs. Our approach seamlessly connects WGAN with one of the recent semi-supervised learning methods. As a result, it gives rise to not only better photo-realistic samples than the previous methods but also state-of-the-art semi-supervised learning results. In particular, our approach gives rise to the inception score of more than 5.0 with only 1,000 CIFAR-10 images and is the first that exceeds the accuracy of $90 \%$ on the CIFAR-10 dataset using only 4,000 labeled images, to the best of our knowledge. + +# 1 INTRODUCTION + +We have witnessed a great surge of interests in deep generative networks in recent years (Kingma & Welling, 2013; Goodfellow et al., 2014; Li et al., 2015). The central idea therein is to feed a random vector to a (e.g., feedforward) neural network and then take the output as the desired sample. This sampling procedure is very efficient without the need of any Markov chains. + +In order to train such a deep generative network, two broad categories of methods are proposed. The first is to use stochastic variational inference (Kingma & Welling, 2013; Rezende et al., 2014; Kingma et al., 2014) to optimize the lower bound of the data likelihood. The other is to use the samples as a proxy to minimize the distribution divergence between the model and the real through a two-player game (Goodfellow et al., 2014; Salimans et al., 2016), maximum mean discrepancy (Li et al., 2015; Dziugaite et al., 2015; Li et al., 2017b), f-divergence (Nowozin et al., 2016; Nock et al., 2017), and the most recent Wasserstein distance (Arjovsky et al., 2017; Gulrajani et al., 2017). + +With no doubt, the generative adversarial networks (GANs) among them (Goodfellow et al., 2014) have the biggest impact thus far on a variety of problems and applications (Radford et al., 2015; Denton et al., 2015; Im et al., 2016; Isola et al., 2016; Springenberg, 2015; Sutskever et al., 2015; Odena, 2016; Zhu et al., 2017). GANs learn the generative network (generator) by playing a twoplayer game between the generator and an auxiliary discriminator network. While the generator has no difference from other deep generative models in the sense that it translates a random vector into a desired sample, it is impossible to calculate the sample likelihood from it. Instead, the discriminator serves to evaluate the quality of the generated samples by checking how difficult it is to differentiate them from real data points. + +However, it is remarkably difficult to train GANs without good heuristics (Goodfellow et al., 2014; Salimans et al., 2016; Radford et al., 2015) which may not generalize across different network architectures or application domains. The training dynamics are often unstable and the generated samples could collapse to limited modes. These issues are formally analyzed by Arjovsky & Bottou (2017), who also propose an alternative direction (Arjovsky et al., 2017) to avoid the caveats in the minmax two-player training of GANs. The corresponding algorithm, namely, Wasserstein GAN (WGAN), shows not only superior performance over GANs but also a nice correlation between the sample quality and the value function that GANs lack. + +# 1.1 BACKGROUND: WGAN AND THE IMPROVED TRAINING OF WGAN + +WGAN (Arjovsky et al., 2017) aims to learn the generator network $G ( z )$ , for any random vector $z \sim \mathbb { P } _ { z }$ , such that the Wasserstein distance is minimized between the resulting distribution $\mathbb { P } _ { G }$ of the generated samples $\{ G ( z ) \}$ and the real distribution $\mathbb { P } _ { r }$ underlying the observed data points $\{ { \pmb x } \}$ ; in other words, $\mathrm { m i n } _ { G } W ( \mathbb { P } _ { r } , \mathbb { P } _ { G } )$ . The Wasserstein distance $W ( \mathbb { P } _ { r } , \mathbb { P } _ { G } )$ is shown a more sensible cost function for learning the distributions supported by low-dimensional manifolds than the other popular distribution divergences and distances — for example, the Jensen-Shannon (JS) divergence implicitly employed in GANs (Goodfellow et al., 2014). + +Due to the Kantorovich-Rubinstein duality (Villani, 2008) for calculating the Wasserstein distance, the value function of WGAN is then written as + +$$ +\operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D \in \mathcal { D } } \mathbb { E } _ { { \pmb x } \sim \mathbb { P } _ { r } } \big [ D ( { \pmb x } ) \big ] - \mathbb { E } _ { { \pmb z } \sim \mathbb { P } _ { z } } \big [ D ( G ( { \pmb z } ) ) \big ] , +$$ + +where $\mathcal { D }$ is the set of 1-Lipschitz functions. Analogous to GANs, we still call $D$ the “discriminator” although it is actually a real-valued function and not a classifier at all. Arjovsky et al. (2017) specify this family of functions $\mathcal { D }$ by neural networks and then use weight clipping to enforce the Lipschitz continuity. However, as the authors note, the networks’ capacities become limited due to the weight clipping and there could be gradient vanishing problems in the training. + +Improved training of WGAN. Gulrajani et al. (2017) give more concrete examples to illustrate the perils of the weight clipping and propose an alternative way of imposing the Lipschitz continuity. In particular, they introduce a gradient penalty term by noting that the differentiable discriminator $D ( \cdot )$ is 1-Lipschitz if and only if the norm of its gradients is at most 1 everywhere, + +$$ +G P | _ { \widehat { \pmb { x } } } : = \mathbb { E } _ { \widehat { \pmb { x } } } \Big [ \big ( \| \nabla _ { \widehat { \pmb { x } } } D ( \widehat { \pmb { x } } ) \| _ { 2 } - 1 \big ) ^ { 2 } \Big ] +$$ + +where $\widehat { \mathbf { x } }$ is uniformly sampled from the straight line between a pair of data points sampled from the model $\mathbb { P } _ { G }$ and the real $\mathbb { P } _ { r }$ , respectively. A similar regularization is used by Kodali et al. (2017). + +Potential caveats. Unlike the weight clipping, however, by no means one can penalize everywhere using this term through a finite number of training iterations. As a result, the gradient penalty term $G P$ takes effect only upon the sampled points $\widehat { \mathbf { x } }$ , leaving significant parts of the support domain not bexamined at all. In particular, consider the observed data points and their underlying manifold that supports the real distribution $\mathbb { P } _ { r }$ . At the beginning of the training stage, the generated sample $G ( z )$ and hence $\widehat { \mathbf { x } }$ could be distant from the manifold. The Lipschitz continuity over the manifold is not benforced until the generative model $\mathbb { P } _ { G }$ becomes close enough to the real one $\mathbb { P } _ { r }$ , if it can. + +# 1.2 OUR APPROACH AND CONTRIBUTIONS + +In light of the above pros and cons, we propose to improve the improved training of WGAN by additionally laying the Lipschitz continuity condition over the manifold of the real data $\mathbf { \boldsymbol { x } } \sim \mathbb { P } _ { \boldsymbol { r } }$ . Moreover, instead of focusing on one particular data point at a time, we devise a regularization over a pair of data points drawn near the manifold following the most basic definition of the 1-Lipschitz continuity. In particular, we perturb each real data point $_ { \textbf { \em x } }$ twice and use a Lipschitz constant to bound the difference between the discriminator’s responses to the perturbed data points ${ \mathbf { } } x ^ { \prime } , x ^ { \prime \prime }$ . + +Figure 1 illustrates our main idea. The gradient penalty $G P | _ { \widehat { \pmb { x } } }$ often fails to check the continuity of region near the real data $_ { \textbf { \em x } }$ b, around which the discriminator function can freely violate the 1- Lipschitz continuity. We alleviate this issue by explicitly checking the continuity condition using the two perturbed version ${ \mathbf { } } x ^ { \prime } , x ^ { \prime \prime }$ near any observed real data point $_ { \textbf { \em x } }$ . + +In this paper, we make the following contributions. (1) We propose an alternative mechanism for enforcing the Lipschitz continuity over the family of discriminators by resorting to the basic definition of the Lipschitz continuity. It effectively improves the gradient penalty method (Gulrajani et al., 2017) and gives rise to generators with more photo-realistic samples and higher inception scores (Salimans et al., 2016). (2) Our approach is very data efficient in terms of being less prone to overfitting even for very small training sets. We do not observe obvious overfitting phenomena even when the model is trained on only 1000 images of CIFAR-10 (Krizhevsky & Hinton, 2009). (3) Our approach can be seamlessly integrated with GANs to be a competitive semi-supervised training technique (Chapelle et al., 2009) thanks to that both inject noise to the real data points. + +As results, we are able to report the state-of-the-art results on the generative model with a inception score of $8 . 8 1 \pm 0 . 1 3$ on CIFAR-10, and the semi-supervised learning results of $9 . 9 8 \pm 0 . 2 1$ on CIFAR-10 using only 4,000 labeled images — especially, they are significantly better than all the existing GAN-based semi-supervised learning results, to the best of our knowledge. + +![](images/2db90a73234f20ca5dd9bd5010bdddf5b59275cab4cc2969bba65b1bfa731f59.jpg) +Figure 1: Illustration of our main idea. In addition to the gradient penalty over $\widehat { \mathbf { x } }$ , we also examine $\mathbf { x } ^ { \prime }$ and $\mathbf { x } ^ { \prime \prime }$ baround the real data point $_ { \textbf { \em x } }$ in each iteration. + +# 2 APPROACH + +We firstly review the definition of the Lipschitz continuity and then discuss how to use it to regularize the training of WGAN. We then arrive at an approach that can be seamlessly integrated with the semi-supervised learning method (Laine & Aila, 2016). By bringing the best of the two worlds, we report better semi-supervised learning results than both (Laine & Aila, 2016) and existing GANbased methods. + +# 2.1 IMPROVING THE IMPROVED TRAINING OF WGAN + +Let $d$ denote the $\ell _ { 2 }$ metric on an input space used in this paper. A discriminator $D : \mathcal { X } \mapsto \mathcal { Y }$ is Lipschitz continuous if there exists a real constant $M \geqslant 0$ such that, for all $\pmb { x } _ { 1 } , \pmb { x } _ { 2 } \in \mathcal { X }$ , + +$$ +d ( D ( \pmb { x } _ { 1 } ) , D ( \pmb { x } _ { 2 } ) ) \leqslant M \cdot d ( \pmb { x } _ { 1 } , \pmb { x } _ { 2 } ) . +$$ + +Immediately, we can add the following soft consistency term $( C T )$ to the value function of WGAN in order to penalize the violations to the inequality in eq. (3), + +$$ +C T | _ { x _ { 1 } , x _ { 2 } } = \mathbb { E } _ { x _ { 1 } , x _ { 2 } } \left[ \operatorname* { m a x } \left( 0 , \frac { d ( D ( x _ { 1 } ) , D ( x _ { 2 } ) ) } { d ( x _ { 1 } , x _ { 2 } ) } - M ^ { \prime } \right) \right] +$$ + +Remarks. Here we face the same snag as in (Gulrajani et al., 2017), i.e., it is impractical to substitute all the possibilities of $( \pmb { x } _ { 1 } , \pmb { x } _ { 2 } )$ pairs into eq. (4). What pairs and which regions of the input set $\mathcal { X }$ should we check for eq. (4)? Arguably, it is fairly safe to limit our scope to the manifold that supports the real data distribution $\mathbb { P } _ { r }$ and its surrounding regions mainly for two reasons. First, we keep the gradient penalty term and improve it by the proposed consistency term in our overall approach. While the former enforces the continuity over the points sampled between the real and generated points, the latter complement the former by focusing on the region around the real data manifold instead. Second, the distribution of the generative model $\mathbb { P } _ { G }$ is virtually desired to be as close as possible to $\mathbb { P } _ { r }$ . We use the notation $M$ in eq. (3) and a different $M ^ { \prime }$ in eq. (4) to reflect the fact that the continuity will be checked only sparsely at finite data points in practice. + +Perturbing the real data. To this end, the very first version we tried was to directly add Gaussian noise $\pmb { \delta }$ to each real data point, resulting in a pair of ${ \pmb x } + \delta _ { 1 } , { \pmb x } + \delta _ { 2 }$ , where $\mathbf { \boldsymbol { x } } \sim \mathbb { P } _ { \boldsymbol { r } }$ . However, as noted by Arjovsky et al. (2017) and Wu et al. (2016), we found that the samples from the generator become blurry due to the Gaussian noise used in the training. We have also tested the dropout noise that is applied to the input and found that the resulting MNIST samples are cut off here and there. + +Algorithm 1 our proposed CT-GAN for training a generative neural net. Most of our experiments are conducted with the default values $\lambda _ { 1 } = 1 0 , \lambda _ { 2 } = 2 .$ , $N = 5$ , $\gamma = 0 . 0 0 0 2$ , $b = 6 4$ , $i t e r = 1 0 ^ { 6 }$ + +Require: $b$ , the batch size. $\lambda _ { 1 } , \lambda _ { 2 }$ , weights. $\gamma$ , the Learning rate. iter, number of iterations. +1: for iter of training iterations do +2: for $N$ of iterations do +3: for $i = 1 , . . . , b$ do +4: Sample real data $\mathbf { \boldsymbol { x } } \sim \mathbb { P } _ { \boldsymbol { r } }$ , random vector $z \sim \mathbb { P } _ { z }$ , and a random number $\epsilon \sim U [ 0 , 1 ]$ . +5: $\hat { x } \overset { } { \epsilon } { \pmb x } + ( 1 - \epsilon ) G ( z )$ +6: $\begin{array} { r l } & { ~ L ^ { ( i ) } \gets \dot { D ( G ( z ) ) ^ { ' } } \dot { D ( } x ) + \lambda _ { 1 } G P | _ { \widehat { \mathbf { x } } } + \lambda _ { 2 } C T | _ { \mathbf { x } ^ { \prime } , \mathbf { x } ^ { \prime \prime } } } \\ & { \mathbf { e n d } \mathbf { f o r } } \\ & { \theta _ { d } \gets A d a m ( \nabla _ { \theta _ { d } } \frac { 1 } { b } \sum _ { i = 1 } ^ { b } L ^ { ( i ) } , \theta _ { d } , \gamma ) } \end{array}$ +7: +8: +9: end for +10: Sample $b$ random vectors $\big \{ z ^ { ( i ) } \big \}$ +11: $\begin{array} { r } { \underrightarrow { \theta } _ { g } A d a m ( \nabla _ { \theta _ { g } } \frac { 1 } { b } \sum _ { i = 1 } ^ { b } - D ( G ( z ^ { ( i ) } ) ) , \theta _ { g } , \gamma ) } \end{array}$ +12: end for + +The success comes after we perturb the hidden layers of the discriminator using dropout, as opposed to the input $_ { \textbf { \em x } }$ . When the dropout rate is small, the perturbed discriminator’s output can be considered as the output of the clean discriminator in response to a “virtual” data point $\mathbf { x } ^ { \prime }$ that is not far from $_ { \textbf { \em x } }$ . Thus, we denote by $D ( \pmb { x } ^ { \prime } )$ the discriminator output after applying dropout to its hidden layers. In the same manner, we find the second virtual point $\scriptstyle { \pmb x } ^ { \prime \prime }$ around $_ { \textbf { \em x } }$ by applying the (stochastic) dropout again to the hidden layers of the discriminator, and denote by $D ( \pmb { x } ^ { \prime \prime } )$ the corresponding output. + +Note that, however, it becomes impossible to compute the distance $d ( { \pmb x } ^ { \prime } , { \pmb x } ^ { \prime \prime } )$ between the two virtual data points. In this work, we assume it is bounded by a constant and absorb the constant to $M ^ { \prime }$ . Accordingly, we tune $M ^ { \prime }$ in our experiments to take account of this unknown constant; the best results are obtained between $M ^ { \prime } = 0$ and $M ^ { \prime } = 0 . 2$ . For consistency, we use $M ^ { \prime } = 0$ to report all the results in this paper. + +A consistency regularization. Our final consistency regularization takes the following form, + +$$ +C T | _ { \mathbf { x } ^ { \prime } , \mathbf { x } ^ { \prime \prime } } = \mathbb { E } _ { \mathbf { x } \sim \mathbb { P } _ { r } } \left[ \operatorname* { m a x } \left( 0 , d ( D ( \mathbf { x } ^ { \prime } ) , D ( \mathbf { x } ^ { \prime \prime } ) ) + 0 . 1 \cdot d ( D _ { - } ( \mathbf { x } ^ { \prime } ) , D _ { - } ( \mathbf { x } ^ { \prime \prime } ) ) - M ^ { \prime } \right) \right] +$$ + +where $D ( { \pmb x } ^ { \prime } )$ is the output of the discriminator given the input $_ { \textbf { \em x } }$ and after we apply dropout to the hidden layers of the discriminator. We envision this is equivalent to passing a “virtual” data point $\mathbf { x } ^ { \prime }$ through the clean discriminator. We find that it slightly improves the performance by further controlling the second-to-last layer $D _ { - } ( \cdot )$ of the discriminator, $i . e .$ , the $d ( D _ { - } ^ { \mathbf { \hat { \rho } } } ( { \pmb x } ^ { \prime } ) , D _ { - } ( { \pmb x } ^ { \prime \prime } ) )$ above. + +This new consistent regularization $C T | _ { \pmb { x } ^ { \prime } , \pmb { x } ^ { \prime \prime } }$ enforces the Lipschitz continuity over the data manifold and its surrounding regions, effectively complementing and improving the gradient penalty $G P | _ { \widehat { \pmb { x } } }$ bused in the improved training of WGAN. Putting them together, our new objective function for updating the weigts of the discriminator is + +$$ +L = \mathbb { E } _ { z \sim \mathbb { P } _ { z } } \left[ D ( G ( z ) ) \right] - \mathbb { E } _ { \pmb { x } \sim \mathbb { P } _ { r } } \left[ D ( \pmb { x } ) \right] + \lambda _ { 1 } G P | _ { \hat { \pmb { x } } } + \lambda _ { 2 } C T | _ { \pmb { x } ^ { \prime } , \pmb { x } ^ { \prime \prime } } . +$$ + +Algorithm 1 shows the complete algorithm for learning a WGAN in this paper. For the hyperparameters, we borrow $\lambda _ { 1 } = 1 0$ from Gulrajani et al. (2017) and use $\lambda _ { 2 } = 2$ for all our experiments no matter on which dataset. Another hyper-parameter is $M ^ { \prime }$ in eq. (5). As stated previously, $M ^ { \prime }$ taking values between 0 and 0.2 gives rise to about the same results in our experiments. + +# 2.2 A SEAMLESS CONNECTION WITH A SEMI-SUPERVISED LEARNING METHOD + +In this section, we extend the WGAN to a semi-supervised learning approach by drawing insights from two related works. + +• Following (Salimans et al., 2016), we modify the output layer of the discriminator such that it has $K + 1$ output neurons, where $K$ is the number of classes of interest and the $( K + 1 )$ -th neuron is reserved for contrasting the generated samples with the real data using the Wasserstein distance in the WGAN. We use a $( K + 1 )$ -way softmax as the activation function of the last layer. + +![](images/b65ca53fe23c10d4de9a1eee5825e7f0f587db7d9a1f3a42ade0843100429e27.jpg) +Figure 2: Framework for the semi-supervised training. For clarity, we have omitted the generator. + +• Following (Laine & Aila, 2016), we add our consistency regularization $C T | _ { x ^ { \prime } , x ^ { \prime \prime } }$ to the objective function of the semi-supervised training, in order to take advantage of the effect of temporal ensembling. + +Figure 2 shows the framework for training the discriminator, with the objective function below, + +$$ +\begin{array} { r l } & { L _ { \mathrm { s e m i . d i s } } = - \operatorname { \mathbb { E } } _ { \pmb { x } , \pmb { y } \sim \mathbb { P } _ { \mathrm { x y } } } \left[ \log D ( \pmb { y } | \pmb { x } ) \right] - \operatorname { \mathbb { E } } _ { \pmb { z } \sim \mathbb { P } _ { z } } \left[ \log D ( K + 1 | G ( \pmb { z } ) ) \right] } \\ & { \qquad - \operatorname { \mathbb { E } } _ { \pmb { x } \sim \mathbb { P } _ { r } } \left[ \log ( 1 - D ( K + 1 | \pmb { x } ) ) \right] + \lambda C T | _ { \pmb { x } ^ { \prime } , \pmb { x } ^ { \prime \prime } } , } \end{array} +$$ + +where the first three terms are the same as in (Salimans et al., 2016), while the last consistency regularization is calculated after we apply dropout to the discriminator. The last term essentially leads to a temporal self-ensembling scheme to benefit the semi-supervised learning. Please see (Laine & Aila, 2016) for more insightful discussions about it. + +The generator loss matches the expected features of the generated sample and the real data points, + +$$ +L _ { \mathrm { s e m i . g e n } } = \| \mathbb { E } _ { z \sim \mathbb { P } _ { z } } ( D _ { - } ( G ( z ) ) - \mathbb { E } _ { { \pmb x } \sim \mathbb { P } _ { r } } ( D _ { - } ( { \pmb x } ) ) \| _ { 2 } ^ { 2 } . +$$ + +# 3 EXPERIMENTAL RESULTS + +We conduct experiments on the prevalent MNIST (LeCun et al., 1998), and CIFAR-10 (Krizhevsky & Hinton, 2009) datasets. The code is available on https://github.com/biuyq/CT-GAN to facilitate the reproducibility of our results. + +# 3.1 MNIST + +The MNIST dataset provides 70,000 handwritten digits in total and 10,000 of them are often left out for the testing purpose. Following (Gulrajani et al., 2017), we use only 1,000 of them to train the WGAN for a fair comparison with it. We use all the 60,000 training examples in the semi-supervised learning experiments and reveal the labels of 100 of them (10 per class). This setup is the same as in (Rasmus et al., 2015). No data augmentation is used. Please see Appendix A for the network architectures of the generator and discriminator, respectively. + +Qualitative results. Figure 3 shows the generated samples with improved training of WGAN by the gradient penalty (GP-WGAN) and ours with the consistency regularization (CT-GAN), respectively, after 50,000 generator iterations. It is clear that our approach gives rise to more realistic samples than GP-WGAN. The contrasts of our samples between the foreground and the background are in general sharper than those of GP-WGAN. + +Overfitting. We find that our approach is less prone to overfitting. To demonstrate this point, we show the convergence curves of the discriminator’s value functions by GP-WGAN and our CT-GAN in Figure 4. The red curves are evaluated on the training set and the blue ones are on the test set. We can see that the results on the test set become saturated pretty early in GP-WGAN, while ours can consistently decrease the costs on both the training and the test sets. This observation also holds for the CIFAR-10 dataset (cf. Appendix E). + +![](images/8681645893db2d6f35b2e9b969262d80ba64bd7b995fdd6924754e6edc4bf18f.jpg) +Figure 3: Images generated by (a) GP-WGAN (Gulrajani et al., 2017) and (b) Our CT-GAN, respectively. + +![](images/b3fedcd75934f5d21cb397b2c5678493c20d58bf6f361902bcd6d953a888d842.jpg) +Figure 4: Convergence curves of the discriminator cost: (a) GP-WGAN and (b) Our CT-GAN. + +Semi-supervised learning results. We compare our semi-supervised learning results with those of several competitive methods in Table 1. Our approach is among the best on this MNIST dataset. + +# 3.2 CIFAR-10 + +CIFAR-10 (Krizhevsky & Hinton, 2009) contains 50,000 natural images of size $3 2 \times 3 2$ . We use it to test two networks for the generative model: a small CNN and a ResNet (cf. Appendix A for the network structures). For the former we use only 1,000 images to train the model and we use the whole training set to learn the ResNet. + +Qualitative results. Figure 5 contrasts the samples we generated to those by GP-WGAN when the generator is the small CNN. Figure 7 shows the results by a larger-scale ResNet. Our results are more photo-realistic. + +Additionally, we also draw the histograms of the discriminator’s weights in Figure 6 after we train it using GP-WGAN and CT-GAN, respectively. It is interesting to see that ours controls the weights within a smaller and more symmetric range $[ - 0 . 6 7 , 0 . 9 6 ]$ than the $[ - 2 . 0 0 , 1 0 . 1 2 ]$ by GP-WGAN, partially explaining why our approach is less prone to overfitting. + +Table 1: Comparing our semi-supervised learning approach with state-of-the-art ones on MNIST. + +
MethodTest error (%)
Ladder (Rasmus et al., 2015)1.06±0.37
VAT (Miyato et al., 2017)1.36
CatGAN (Springenberg,2015)1.39±0.28
Improved GAN (Salimans et al., 2016)0.93 ± 0.065
Triple GAN (Li et al., 2017a)0.91 ± 0.58
Our CT-GAN0.89 ± 0.13
+ +![](images/aa12002406502061af7c698f3350a49fb5750013e7322ac6cab2d6c4287e124b.jpg) +Figure 5: Generated samples by (a) GP-WGAN and (b) our CT-GAN. Here the generator is a small CNN. See Figure 7 for the samples by a ResNet. + +![](images/249fc7181e82a988faf3d3e8057cd4dfd28a1b23595766db419b398a8d1059d8.jpg) +Figure 6: Histograms of the weights of the discriminators trained by (a) GP-WGAN and (b) CTGAN, respectively. + +Table 2: Inception score and accuracy of different models on CIFAR-10 + +
MethodSupervised ISUnsupervised ISAccuracy(%)
SteinGANs (Wang&Liu,2016)6.35
DCGANs (Radford et al., 2015)6.586.16 ± 0.07
Improved GANs (Salimans et al., 2016)8.09 ± 0.071
AC-GANs (Odena et al., 2016)8.25 ± 0.07
GP-WGAN (Gulrajani et al., 2017)8.42 ± 0.107.86± 0.0791.85
SGANs (Huang et al., 2016)8.59 ±0.12
ALI (Warde-Farley & Bengio,2016)15.34 ± 0.05
BEGAN (Berthelot et al., 2017)一5.62
EGAN-Ent-VI (Dai et al., 2017)7.07 ± 0.10
DFM(Warde-Farley & Bengio,2016)17.72 ± 0.071
Our CT-GAN8.81±0.138.12±0.1295.91
+ +Comparison of the inception scores. Finally, we compare our approach with GP-WGAN on the whole training set for both unsupervised and supervised generative-purpose task using ResNet. For model selection, we use the first 50,000 samples to compute the inception scores (Salimans et al., 2016), then choose the best model, and finally report the “test” score on another 50,000 samples. The experiment follows the previous setup in (Odena et al., 2016). From the comparison results in Tables 2, we conclude that our proposed model achieves the highest inception score on the CIFAR10 dataset, to the best of our knowledge. Some generated samples are shown in Figure 7. + +For the small CNN based generator, the inception scores of GP-WGAN and our CT-GAN are $2 . 9 8 \pm$ 0.11 and $5 . 1 3 \pm 0 . 1 2$ , respectively. + +![](images/3c70b1dead917ecf56456f0a87519cf0cad0fc5a71fe26bb94f6752bce509e04.jpg) +Figure 7: Generated samples by our ResNet model: (a) Generated samples by unsupervised model and (b) Generated samples by supervised model. Each column corresponds to one class in the CIFAR-10 dataset. + +Table 3: Comparing our semi-supervised learning approach with state-of-the-art ones on CIFAR-10 + +
MethodTest error(%)
Ladder (Rasmus et al., 2015)20.40±0.47
VAT (Miyato et al., 2017)10.55
TE (Laine & Aila,2016)12.16 ±0.24
Teacher-Student (Tarvainen & Valpola,2017)12.31 ± 0.28
CatGANs (Springenberg,2015)19.58 ± 0.58
Improved GANs (Salimans et al., 2016)18.63 ± 2.32
ALI (Dumoulin et al., 2016)17.99 ±1.62
CLS-GAN (Qi, 2017)17.30 ± 0.50
Triple GAN (Li et al., 2017a)16.99 ± 0.36
Improved semi-GAN (Kumar et al., 2017)16.78 ± 1.80
Our CT-GAN9.98± 0.21
+ +Semi-supervised learning. For the semi-supervised learning approach, we follow the standard training/test split of the dataset but use only 4,000 labels in the training. A regular data augmentation with flipping the images horizontally and randomly translating the images within [-2,2] pixels is used in our paper (No ZCA whitening). We report the semi-supervised learning results in Table 3. The mean and standard errors are obtained by running the experiments 5 rounds. Comparing to several very competitive methods, ours is able to achieve state-of-the-art results. Notably, our CT-GAN outperfroms all the GAN based methods by a large margin. Please see Appendix A for the network architectures and Appendix C for the ablation study of our algorithm. + +# 4 CONCLUSION + +In this paper, we present a consistency term derived from Lipschitz inequality, which boosts the performance of GANs model. The proposed term has been demonstrated to be an efficient manner to ease the over-fitting problem when data amount is limited. Experiments show that our model obtains the state-of-the-art accuracy and Inception score on CIFAR-10 dataset for both the semisupervised learning task and the learning of generative models. + +Acknowledgements. This work is partially supported by NSF IIS-1741431, IIS-1566511, and ONR N00014-18-1-2121. B.G. and L.W. would also like to thank Adobe Research and NVIDIA, and Amazon, respectively, for their gift donations. + +# REFERENCES + +Martin Arjovsky and Leon Bottou. Towards principled methods for training generative adversarial´ networks. arXiv preprint arXiv:1701.04862, 2017. + +Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein gan. ´ arXiv preprint arXiv:1701.07875, 2017. + +David Berthelot, Tom Schumm, and Luke Metz. 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On the quantitative analysis of decoder-based generative models. arXiv preprint arXiv:1611.04273, 2016. + +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. + +Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. arXiv preprint arXiv:1703.10593, 2017. + +# Appendices + +APPENDIX A NETWORK ARCHITECTURES + +Table 4 (MNIST) and Table 5 (CIFAR-10) detail the network architectures used in our classificationpurpose CT-GAN, where the classifiers are the same as the widely used ones in most semi-supervised networks (Laine & Aila, 2016) except that we apply weight-norm rather than batch-norm. For the generators, we follow the network structures in GP-WGAN (Salimans et al., 2016), but we use lower dimensional noise (50D) as the input to the generator for CIFAR-10 in order not to reproduce the complicated images and instead shift the focus of the training to the classifier. + +Table 4: Networks for semi-supervised learning on MNIST + +
Classifier CGenerator G
Input: Labels y,28*28 Images x,Input: Noise 100 z
Gaussiannoise0.3,MLP1000,ReLU Gaussian noise 0.5,MLP 500,ReLUMLP 500, Softplus,Batch norm
Gaussian noise O.5,MLP 250,ReLUMLP 500, Softplus,Batch norm
Gaussian noise 0.5,MLP 250,ReLUMLP 784, Sigmoid,Weight norm
Gaussian noise 0.5,MLP 250,ReLU
Gaussian noise O.5,MLP 10, Softmax
+ +Table 5: Networks for semi-supervised learning on CIFAR-10 + +
Classifier CGenerator G
Input: Labels y, 32*32*3 Colored Image x,Input:Noise 50 z
0.2 Dropout 3*3 conv.128,Pad=1, Stride =1,lReLU,Weight norm 3*3 conv. 128, Pad =1, Stride =1, lReLU,Weight normMLP 8192, ReLU, Batch norm Reshape 512*4*4 5*5 deconv. 256*8*8,
3*3 conv. 128,Pad =1, Stride =2, lReLU, Weight norm 0.5 Dropout 3*3 conv. 256,Pad =1, Stride =1, lReLU, Weight norm 3*3 conv. 256,Pad=1, Stride =1, lReLU, Weight norm 3*3 conv. 256,Pad =1, Stride =2, lReLU, Weight normReLU,Batch norm 5*5 deconv. 128*16*16, ReLU,Batch norm
0.5 Dropout 3*3 conv. 512, Pad =O, Stride =1, lReLU, Weight norm 3*3 conv. 256,Pad =O, Stride =1,lReLU, Weight norm 3*3 conv. 128,Pad =O, Stride =1, lReLU, Weight norm5*5 deconv. 3*32*32,
Global pool MLP 10, Weight norm, SoftmaxTanh,Weight norm
+ +# APPENDIX B HYPER-PARAMETERS AND OTHER TRAINING DETAILS + +For the semi-supervised learning experiments, we set $\lambda = 1 . 0$ in Eq.(7) in all our experiments. For CIFAR-10, the number of training epochs is set to 1,000 with a constant learning rate of 0.0003. For + +Table 6: Generative model for MNIST + +
DiscriminatorGenerator
Input: 1*28*28 Image xInput:Noise z 128
5*5 conv. 64,Pad = same, Stride= 2, IReLU 0.5 DropoutMLP4096,ReLU Reshape 256*4*4
5*5 conv. 128,Pad = same, Stride= 2,lReLU 0.5 Dropout5*5 deconv. 128*8*8 ReLU, Cut 128*7*7
5*5 conv. 256, Pad = same, Stride = 2, IReLU 0.5 Dropout5*5 deconv. 64*14*14
Reshape 256*4*4 (D-)ReLU 5*5 deconv. 1*28*28
MLP 1 (D)Sigmoid
+ +Table 7: Generative model for CIFAR-10 + +
DiscriminatorGenerator
Input: 3*32*32 Image x,Input:Noise z 128
5*5 conv. 128, Pad = same, Stride =2, IReLU 0.5 DropoutMLP 8192, ReLU, Batch norm Reshape 512*4*4
5*5 conv. 256, Pad = same, Stride = 2, IReLU 0.5 Dropout5*5 deconv. 256*8*8 ReLU,Bach norm
5*5 conv. 512, Pad = same Stride= 2, IReLU5*5 deconv. 128*16*16
0.5 Dropout Reshape 512*4*4 (D-)ReLU, Batch norm 5*5 deconv. 3*32*32
MLP 1 (D)Tanh
+ +MNIST, the number of training epochs is set to 300 with a constant learning rate of 0.003. The other hyper-parameters are exactly the same as in the improved GAN (Salimans et al., 2016). + +For the experiments on the generative models, to have a fair comparison for our results with the existing ones, we keep the network structure and hyper-parameters the same as in the improved training of WGAN (Gulrajani et al., 2017) except that we add three dropout layers to some of the hidden layers as shown in tables 6 to 8. + +# APPENDIX C ABLATION STUDY OF OUR APPROACH TO SSL + +We run an ablation study of our approach to the semi-supervised learning (SSL). Thanks to the dual effect of the proposed consistency (CT) term, we are able to connect GAN with the temporal ensembling (TE) Laine & Aila (2016) method for SSL. Our superior results thus benefit from both of them, verified by the ablation study detailed in Table 9. + +Table 8: ResNet for CIFAR-10 + +
DiscriminatorGenerator
Input: 3*32*32 Image xInput: Noise bmz 128
[3*3]*2 Residual Block, Resample = DOWN 128*16*16MLP2048 Reshape 128*4*4
[3*3]*2 Residual Block, Resample = DOWN 128*8*8 0.2 Dropout[3*3]*2 Residual Block, Resample = UP
[3*3]*2 Residual Block, Resample = None 128*8*8 0.5 Dropout128*8*8 [3*3]*2 Residual Block, Resample = UP
[3*3]*2 Residual Block, Resample = None128*16*16 [3*3]*2 Residual Block, Resample = UP
128*8*8 0.5 Dropout128*32*32
ReLU, Global mean pool (D-) MLP 1 (D)3*3 conv. 3*32*32
Tanh
+ +If we remove the CT term, the test error goes up to 14.98, signifying the effectiveness of the CT regularization. + +If we remove GAN from our approach, it almost reduces to TE; in fact, all the settings here are the same as TE except that we use an extra regularization ${ \bf \it D _ { - } ( \dots ) }$ in CT) over the second-to-last layer. We can see that the error is still significantly larger than our overall method. + +We use the weight normalization as in Salimans et al. (2016), which becomes a core constituent of our approach. The batch normalization would actually invalidate the feature matching in Salimans et al. (2016). + +Finally, we also test the version without the regularization to the second-to-last layer and observe a little drop in the performance. + +Table 9: Ablation study of our semi-supervised learning method + +
MethodTest Error
w/o CT14.98±0.43
w/o GAN11.98±0.32
w batch norm
w/o D-(.,.) over the second-to-last layer10.70±0.24
Ours9.98±0.21
+ +# APPENDIX D EXAMINING THE 1-LIPSCHITZ CONTINUITY + +Norm of gradient. In our experiments, we find that although the GP-WGAN (Gulrajani et al., 2017) has applied a Lipschitz constraint in the form of the gradient penalty over the input sampled between a real data point and a generated one, the actual effect on the $\ell _ { 2 }$ norm of the gradient is not as good as our CT-GAN model in the real data points, as illustrated in Figure 1. We empirically verify this fact by Figure 8, which shows the $\ell _ { 2 }$ norms of the gradients of the discriminator with respect to the real data points. The closer to 1 the norms are, the better the 1-Lipschitz continuity is preserved. Figure 8 further demonstrates that our consistency (CT) regularization is able to improve GP-WGAN (Gulrajani et al., 2017). + +![](images/c6d2218ee59678ad1515a1a0032f1d261c7a9421305f20dc50ea49e964a774d5.jpg) +Figure 8: The maximum $\ell _ { 2 }$ norm of the gradients of the discriminator with respect to the input on CIFAR-10 testing set in each iteration of the training using 1000 CIFAR-10 training images. + +Definition of the Lipschitz continuity. Additionally, we check how much the 1-Lipschitz continuity is satisfied according to its vanilla definition (cf. Eq. (3)). Figures 9 and 10 plot the CTs of Eq. (4) and Eq. (5), respectively, over different iterations of the training using 1000 CIFAR-10 images. Figure 10 is the actual CTs used to train the generative model. Figure 9 is drawn as follows. + +For every 100 iterations, we randomly pick up 64 real examples and split them into two subsets of the same size. We compute $d ( D ( x _ { 1 } ) - D ( x _ { 2 } ) ) / d ( x _ { 1 } - x _ { 2 } )$ for all the $( x _ { 1 } , x _ { 2 } )$ pairs, where $x _ { 1 }$ is from the first subset and $x _ { 2 }$ is from the second. The maximum of $d ( D ( x _ { 1 } ) - D ( x _ { 2 } ) ) / d ( x _ { 1 } - x _ { 2 } )$ is plotted in Figure 9. We can see that the CT-GAN curve converges under a certain value much faster than GP-WGAN. The 1-Lipschitz continuity is better maintained by CT-GAN than GP-WGAN over the whole course of the training procedure on the manifold of the real data. + +![](images/d187e15086c5f6a9cb6de72fa06b6e70dfcfd085991fab599baf7f77e568762b.jpg) +Figure 9: CT (cf. Eq. (4)) over different iterations of the training using 1000 CIFAR-10 images. In each iteration, we randomly pick up two real data points to compute the CT. + +![](images/78fb5bbee816494a12aed3f7636ed7d567abaa59689f378bed1c9131bedf1d60.jpg) +Figure 10: CT (cf. Eq. (5)) over different iterations of the training using 1000 CIFAR-10 images. + +# APPENDIX E GP-WGAN WITH DROPOUT + +We run another experiment of GP-WGAN by adding the same dropout layers used in our method to the networks of GP-WGAN, denoted by GP-WGAN+dropout. This may help understand the contribution of our approach in preventing the overfitting problem — is it due to the CT regularization, or merely due to the dropout? This experiment is done by removing the CT term out of our value function for training WGAN using 1,000 CIFAR-10 images and yet keep the dropout layers. + +The Inception score of GP-WGAN+dropout is $4 . 2 9 \pm 0 . 1 2$ , and the generated samples are shown in Figure 11(a). We also plot the curve of CT in the training procedure of GP-WGAN $^ +$ dropout and contrast it to the curve of our CT-GAN in Figure 11(b). It is clear that GP-WGAN+dropout outperforms GP-WGAN $( 2 . 9 8 \pm 0 . 1 1 ) $ , and our method $( 5 . 1 3 \pm 0 . 1 2 )$ outperforms GP-WGAN with a large margin. + +![](images/95c8e3d138185e0fef7249995e192bbfa47adcdf3cf05b143734d134d5846b3a.jpg) +Figure 11: The results of GP-WGAN+dropout. + +Finally, we plot the convergence curves of the discriminators’ negative cost function learned by GPWGAN, GP-WGAN $^ +$ Dropout, and our CT-GAN in Figure 12. We can see that dropout is able to reduce the overfitting of GP-WGAN, but it is not as effective as our CT-GAN. + +![](images/42b83c1468918cfe11735af7355568c9222ce3d2ece267e863f5e088144006fe.jpg) +Figure 12: Convergence curves of the discriminator cost: (a) GP-WGAN, (b) GP-WGAN $^ +$ Dropout, and (c) CT-GAN (ours). + +# APPENDIX F EXPERIMENTS ON LARGE DATASET + +In this section, we further present experimental results on the large-scale ImageNet (Deng et al., 2009) and LSUN bedroom (Yu et al., 2015) datasets. The experiment setup (e.g., network architecture, learning rates, etc.) is exactly the same as in the GP-WGAN work (Gulrajani et al., 2017). We refer readers to (Gulrajani et al., 2017) or our code on GitHub (https://github.com/biuyq/CT-GAN) for the details of the setup. + +Our experiment on ImageNet is conducted using images of the size $6 4 \times 6 4$ . After 200, 000 generator iterations, the inception score of the proposed CT-GAN is ${ \bf 1 0 . 2 7 } \pm { \bf 0 . 1 5 }$ , whereas GP-WGAN’s is ${ \bf 9 . 8 5 \pm 0 . 1 7 }$ . In addition, the inception score comparison of GP-WGAN and our CT-GAN in each generator iteration is shown in Figure 13. We can observe that the inception score of our proposed CT-GAN becomes higher than GP-WGAN’s after early generator iterations. Finally, Figure 14 shows the samples generated by GP-WGAN and CT-GAN, respectively. + +![](images/8dad9e4369b46ff49ce7ce9c759e02deb243f3bc9efa497400438a821313cb50.jpg) +Figure 13: Inception score of GP-WGAN and our CT-GAN with respect to iterations + +![](images/072a288c32709d4684b8f5c996ac80ac24e2308673461cbdd52e9e49cc89c917.jpg) +Figure 14: Samples generated by GP-WGAN (a) and by CT-GAN (b), respectively. + +For the LSUN bedroom dataset, we show some results in Figure 15. They are the generated samples by our CT-GAN generator after $2 0 \mathrm { k }$ training iterations. + +![](images/c22c5ba777606a67430cb6fc56b70ba287ba6b7735bdf11341a64160d55db6f6.jpg) +Figure 15: Image samples generated by CT-GAN trained model using LSUN bedroom images. \ No newline at end of file diff --git a/parse/train/SJx9GQb0-/SJx9GQb0-_content_list.json b/parse/train/SJx9GQb0-/SJx9GQb0-_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..a7493b319928db74e48edd4f57fd9702d9de4c46 --- /dev/null +++ b/parse/train/SJx9GQb0-/SJx9GQb0-_content_list.json @@ -0,0 +1,1935 @@ +[ + { + "type": "text", + "text": "IMPROVING THE IMPROVED TRAINING OF WASSERSTEIN GANS: A CONSISTENCY TERM AND ITS DUAL EFFECT ", + "text_level": 1, + "bbox": [ + 176, + 98, + 741, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Xiang $\\mathbf { W e i } ^ { 1 , 2 * }$ , Boqing $\\mathbf { G o n g ^ { 3 * } }$ , Zixia $\\mathbf { L i u ^ { 1 } }$ , Wei $\\mathbf { L } \\mathbf { u } ^ { 2 }$ , Liqiang Wang1 1Department of Computer Science, University of Central Florida, Orlando, FL, USA 32816 2School of Software Engineering, Beijing Jiaotong University, Beijing, China 100044 3Tencent AI Lab, Bellevue, WA, USA 98004 yqweixiang@knights.ucf.edu, boqinggo@outlook.com zixia@knights.ucf.edu, luwei@bjtu.edu.cn, lwang@cs.ucf.edu ", + "bbox": [ + 184, + 193, + 784, + 280 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 318, + 544, + 332 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Despite being impactful on a variety of problems and applications, the generative adversarial nets (GANs) are remarkably difficult to train. This issue is formally analyzed by Arjovsky & Bottou (2017), who also propose an alternative direction to avoid the caveats in the minmax two-player training of GANs. The corresponding algorithm, called Wasserstein GAN (WGAN), hinges on the 1-Lipschitz continuity of the discriminator. In this paper, we propose a novel approach to enforcing the Lipschitz continuity in the training procedure of WGANs. Our approach seamlessly connects WGAN with one of the recent semi-supervised learning methods. As a result, it gives rise to not only better photo-realistic samples than the previous methods but also state-of-the-art semi-supervised learning results. In particular, our approach gives rise to the inception score of more than 5.0 with only 1,000 CIFAR-10 images and is the first that exceeds the accuracy of $90 \\%$ on the CIFAR-10 dataset using only 4,000 labeled images, to the best of our knowledge. ", + "bbox": [ + 233, + 352, + 764, + 531 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 565, + 336, + 582 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We have witnessed a great surge of interests in deep generative networks in recent years (Kingma & Welling, 2013; Goodfellow et al., 2014; Li et al., 2015). The central idea therein is to feed a random vector to a (e.g., feedforward) neural network and then take the output as the desired sample. This sampling procedure is very efficient without the need of any Markov chains. ", + "bbox": [ + 176, + 601, + 823, + 656 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In order to train such a deep generative network, two broad categories of methods are proposed. The first is to use stochastic variational inference (Kingma & Welling, 2013; Rezende et al., 2014; Kingma et al., 2014) to optimize the lower bound of the data likelihood. The other is to use the samples as a proxy to minimize the distribution divergence between the model and the real through a two-player game (Goodfellow et al., 2014; Salimans et al., 2016), maximum mean discrepancy (Li et al., 2015; Dziugaite et al., 2015; Li et al., 2017b), f-divergence (Nowozin et al., 2016; Nock et al., 2017), and the most recent Wasserstein distance (Arjovsky et al., 2017; Gulrajani et al., 2017). ", + "bbox": [ + 174, + 664, + 823, + 761 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "With no doubt, the generative adversarial networks (GANs) among them (Goodfellow et al., 2014) have the biggest impact thus far on a variety of problems and applications (Radford et al., 2015; Denton et al., 2015; Im et al., 2016; Isola et al., 2016; Springenberg, 2015; Sutskever et al., 2015; Odena, 2016; Zhu et al., 2017). GANs learn the generative network (generator) by playing a twoplayer game between the generator and an auxiliary discriminator network. While the generator has no difference from other deep generative models in the sense that it translates a random vector into a desired sample, it is impossible to calculate the sample likelihood from it. Instead, the discriminator serves to evaluate the quality of the generated samples by checking how difficult it is to differentiate them from real data points. ", + "bbox": [ + 174, + 768, + 825, + 892 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, it is remarkably difficult to train GANs without good heuristics (Goodfellow et al., 2014; Salimans et al., 2016; Radford et al., 2015) which may not generalize across different network architectures or application domains. The training dynamics are often unstable and the generated samples could collapse to limited modes. These issues are formally analyzed by Arjovsky & Bottou (2017), who also propose an alternative direction (Arjovsky et al., 2017) to avoid the caveats in the minmax two-player training of GANs. The corresponding algorithm, namely, Wasserstein GAN (WGAN), shows not only superior performance over GANs but also a nice correlation between the sample quality and the value function that GANs lack. ", + "bbox": [ + 173, + 103, + 825, + 215 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 BACKGROUND: WGAN AND THE IMPROVED TRAINING OF WGAN", + "text_level": 1, + "bbox": [ + 176, + 231, + 679, + 246 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "WGAN (Arjovsky et al., 2017) aims to learn the generator network $G ( z )$ , for any random vector $z \\sim \\mathbb { P } _ { z }$ , such that the Wasserstein distance is minimized between the resulting distribution $\\mathbb { P } _ { G }$ of the generated samples $\\{ G ( z ) \\}$ and the real distribution $\\mathbb { P } _ { r }$ underlying the observed data points $\\{ { \\pmb x } \\}$ ; in other words, $\\mathrm { m i n } _ { G } W ( \\mathbb { P } _ { r } , \\mathbb { P } _ { G } )$ . The Wasserstein distance $W ( \\mathbb { P } _ { r } , \\mathbb { P } _ { G } )$ is shown a more sensible cost function for learning the distributions supported by low-dimensional manifolds than the other popular distribution divergences and distances — for example, the Jensen-Shannon (JS) divergence implicitly employed in GANs (Goodfellow et al., 2014). ", + "bbox": [ + 174, + 256, + 825, + 356 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Due to the Kantorovich-Rubinstein duality (Villani, 2008) for calculating the Wasserstein distance, the value function of WGAN is then written as ", + "bbox": [ + 174, + 361, + 821, + 388 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/0619b4b527195bd840cce06292feaf3867008f918d7334c8ab0c01a426d81bc0.jpg", + "text": "$$\n\\operatorname* { m i n } _ { G } \\operatorname* { m a x } _ { D \\in \\mathcal { D } } \\mathbb { E } _ { { \\pmb x } \\sim \\mathbb { P } _ { r } } \\big [ D ( { \\pmb x } ) \\big ] - \\mathbb { E } _ { { \\pmb z } \\sim \\mathbb { P } _ { z } } \\big [ D ( G ( { \\pmb z } ) ) \\big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 344, + 392, + 650, + 416 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\mathcal { D }$ is the set of 1-Lipschitz functions. Analogous to GANs, we still call $D$ the “discriminator” although it is actually a real-valued function and not a classifier at all. Arjovsky et al. (2017) specify this family of functions $\\mathcal { D }$ by neural networks and then use weight clipping to enforce the Lipschitz continuity. However, as the authors note, the networks’ capacities become limited due to the weight clipping and there could be gradient vanishing problems in the training. ", + "bbox": [ + 174, + 417, + 823, + 489 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Improved training of WGAN. Gulrajani et al. (2017) give more concrete examples to illustrate the perils of the weight clipping and propose an alternative way of imposing the Lipschitz continuity. In particular, they introduce a gradient penalty term by noting that the differentiable discriminator $D ( \\cdot )$ is 1-Lipschitz if and only if the norm of its gradients is at most 1 everywhere, ", + "bbox": [ + 174, + 502, + 823, + 559 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/ee4a1c95f4df60014d8a15b90e0d091667c27b8c32dc0f0e8d3ecf7e85e631a2.jpg", + "text": "$$\nG P | _ { \\widehat { \\pmb { x } } } : = \\mathbb { E } _ { \\widehat { \\pmb { x } } } \\Big [ \\big ( \\| \\nabla _ { \\widehat { \\pmb { x } } } D ( \\widehat { \\pmb { x } } ) \\| _ { 2 } - 1 \\big ) ^ { 2 } \\Big ]\n$$", + "text_format": "latex", + "bbox": [ + 379, + 563, + 619, + 589 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\widehat { \\mathbf { x } }$ is uniformly sampled from the straight line between a pair of data points sampled from the model $\\mathbb { P } _ { G }$ and the real $\\mathbb { P } _ { r }$ , respectively. A similar regularization is used by Kodali et al. (2017). ", + "bbox": [ + 174, + 592, + 825, + 621 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Potential caveats. Unlike the weight clipping, however, by no means one can penalize everywhere using this term through a finite number of training iterations. As a result, the gradient penalty term $G P$ takes effect only upon the sampled points $\\widehat { \\mathbf { x } }$ , leaving significant parts of the support domain not bexamined at all. In particular, consider the observed data points and their underlying manifold that supports the real distribution $\\mathbb { P } _ { r }$ . At the beginning of the training stage, the generated sample $G ( z )$ and hence $\\widehat { \\mathbf { x } }$ could be distant from the manifold. The Lipschitz continuity over the manifold is not benforced until the generative model $\\mathbb { P } _ { G }$ becomes close enough to the real one $\\mathbb { P } _ { r }$ , if it can. ", + "bbox": [ + 173, + 633, + 825, + 733 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.2 OUR APPROACH AND CONTRIBUTIONS ", + "text_level": 1, + "bbox": [ + 176, + 748, + 485, + 762 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In light of the above pros and cons, we propose to improve the improved training of WGAN by additionally laying the Lipschitz continuity condition over the manifold of the real data $\\mathbf { \\boldsymbol { x } } \\sim \\mathbb { P } _ { \\boldsymbol { r } }$ . Moreover, instead of focusing on one particular data point at a time, we devise a regularization over a pair of data points drawn near the manifold following the most basic definition of the 1-Lipschitz continuity. In particular, we perturb each real data point $_ { \\textbf { \\em x } }$ twice and use a Lipschitz constant to bound the difference between the discriminator’s responses to the perturbed data points ${ \\mathbf { } } x ^ { \\prime } , x ^ { \\prime \\prime }$ . ", + "bbox": [ + 173, + 773, + 825, + 858 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Figure 1 illustrates our main idea. The gradient penalty $G P | _ { \\widehat { \\pmb { x } } }$ often fails to check the continuity of region near the real data $_ { \\textbf { \\em x } }$ b, around which the discriminator function can freely violate the 1- Lipschitz continuity. We alleviate this issue by explicitly checking the continuity condition using the two perturbed version ${ \\mathbf { } } x ^ { \\prime } , x ^ { \\prime \\prime }$ near any observed real data point $_ { \\textbf { \\em x } }$ . ", + "bbox": [ + 174, + 864, + 823, + 921 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we make the following contributions. (1) We propose an alternative mechanism for enforcing the Lipschitz continuity over the family of discriminators by resorting to the basic definition of the Lipschitz continuity. It effectively improves the gradient penalty method (Gulrajani et al., 2017) and gives rise to generators with more photo-realistic samples and higher inception scores (Salimans et al., 2016). (2) Our approach is very data efficient in terms of being less prone to overfitting even for very small training sets. We do not observe obvious overfitting phenomena even when the model is trained on only 1000 images of CIFAR-10 (Krizhevsky & Hinton, 2009). (3) Our approach can be seamlessly integrated with GANs to be a competitive semi-supervised training technique (Chapelle et al., 2009) thanks to that both inject noise to the real data points. ", + "bbox": [ + 174, + 104, + 568, + 311 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As results, we are able to report the state-of-the-art results on the generative model with a inception score of $8 . 8 1 \\pm 0 . 1 3$ on CIFAR-10, and the semi-supervised learning results of $9 . 9 8 \\pm 0 . 2 1$ on CIFAR-10 using only 4,000 labeled images — especially, they are significantly better than all the existing GAN-based semi-supervised learning results, to the best of our knowledge. ", + "bbox": [ + 173, + 319, + 568, + 347 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/2db90a73234f20ca5dd9bd5010bdddf5b59275cab4cc2969bba65b1bfa731f59.jpg", + "image_caption": [ + "Figure 1: Illustration of our main idea. In addition to the gradient penalty over $\\widehat { \\mathbf { x } }$ , we also examine $\\mathbf { x } ^ { \\prime }$ and $\\mathbf { x } ^ { \\prime \\prime }$ baround the real data point $_ { \\textbf { \\em x } }$ in each iteration. " + ], + "image_footnote": [], + "bbox": [ + 583, + 106, + 821, + 256 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 347, + 825, + 388 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 APPROACH", + "text_level": 1, + "bbox": [ + 174, + 410, + 297, + 425 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We firstly review the definition of the Lipschitz continuity and then discuss how to use it to regularize the training of WGAN. We then arrive at an approach that can be seamlessly integrated with the semi-supervised learning method (Laine & Aila, 2016). By bringing the best of the two worlds, we report better semi-supervised learning results than both (Laine & Aila, 2016) and existing GANbased methods. ", + "bbox": [ + 174, + 440, + 825, + 511 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 IMPROVING THE IMPROVED TRAINING OF WGAN ", + "text_level": 1, + "bbox": [ + 173, + 527, + 560, + 542 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $d$ denote the $\\ell _ { 2 }$ metric on an input space used in this paper. A discriminator $D : \\mathcal { X } \\mapsto \\mathcal { Y }$ is Lipschitz continuous if there exists a real constant $M \\geqslant 0$ such that, for all $\\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } \\in \\mathcal { X }$ , ", + "bbox": [ + 171, + 554, + 823, + 583 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1c9372f8c931466416e545c51b27303626650d7e933f6bf467d8444231fbec64.jpg", + "text": "$$\nd ( D ( \\pmb { x } _ { 1 } ) , D ( \\pmb { x } _ { 2 } ) ) \\leqslant M \\cdot d ( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 590, + 619, + 608 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Immediately, we can add the following soft consistency term $( C T )$ to the value function of WGAN in order to penalize the violations to the inequality in eq. (3), ", + "bbox": [ + 171, + 614, + 823, + 643 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6de6c29929ad322a7060ae54c3087f5b86aed194277dbebc66e8935ea0978b99.jpg", + "text": "$$\nC T | _ { x _ { 1 } , x _ { 2 } } = \\mathbb { E } _ { x _ { 1 } , x _ { 2 } } \\left[ \\operatorname* { m a x } \\left( 0 , \\frac { d ( D ( x _ { 1 } ) , D ( x _ { 2 } ) ) } { d ( x _ { 1 } , x _ { 2 } ) } - M ^ { \\prime } \\right) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 303, + 650, + 694, + 685 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Remarks. Here we face the same snag as in (Gulrajani et al., 2017), i.e., it is impractical to substitute all the possibilities of $( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } )$ pairs into eq. (4). What pairs and which regions of the input set $\\mathcal { X }$ should we check for eq. (4)? Arguably, it is fairly safe to limit our scope to the manifold that supports the real data distribution $\\mathbb { P } _ { r }$ and its surrounding regions mainly for two reasons. First, we keep the gradient penalty term and improve it by the proposed consistency term in our overall approach. While the former enforces the continuity over the points sampled between the real and generated points, the latter complement the former by focusing on the region around the real data manifold instead. Second, the distribution of the generative model $\\mathbb { P } _ { G }$ is virtually desired to be as close as possible to $\\mathbb { P } _ { r }$ . We use the notation $M$ in eq. (3) and a different $M ^ { \\prime }$ in eq. (4) to reflect the fact that the continuity will be checked only sparsely at finite data points in practice. ", + "bbox": [ + 174, + 698, + 825, + 838 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Perturbing the real data. To this end, the very first version we tried was to directly add Gaussian noise $\\pmb { \\delta }$ to each real data point, resulting in a pair of ${ \\pmb x } + \\delta _ { 1 } , { \\pmb x } + \\delta _ { 2 }$ , where $\\mathbf { \\boldsymbol { x } } \\sim \\mathbb { P } _ { \\boldsymbol { r } }$ . However, as noted by Arjovsky et al. (2017) and Wu et al. (2016), we found that the samples from the generator become blurry due to the Gaussian noise used in the training. We have also tested the dropout noise that is applied to the input and found that the resulting MNIST samples are cut off here and there. ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Algorithm 1 our proposed CT-GAN for training a generative neural net. Most of our experiments are conducted with the default values $\\lambda _ { 1 } = 1 0 , \\lambda _ { 2 } = 2 .$ , $N = 5$ , $\\gamma = 0 . 0 0 0 2$ , $b = 6 4$ , $i t e r = 1 0 ^ { 6 }$ ", + "bbox": [ + 171, + 102, + 825, + 132 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Require: $b$ , the batch size. $\\lambda _ { 1 } , \\lambda _ { 2 }$ , weights. $\\gamma$ , the Learning rate. iter, number of iterations. \n1: for iter of training iterations do \n2: for $N$ of iterations do \n3: for $i = 1 , . . . , b$ do \n4: Sample real data $\\mathbf { \\boldsymbol { x } } \\sim \\mathbb { P } _ { \\boldsymbol { r } }$ , random vector $z \\sim \\mathbb { P } _ { z }$ , and a random number $\\epsilon \\sim U [ 0 , 1 ]$ . \n5: $\\hat { x } \\overset { } { \\epsilon } { \\pmb x } + ( 1 - \\epsilon ) G ( z )$ \n6: $\\begin{array} { r l } & { ~ L ^ { ( i ) } \\gets \\dot { D ( G ( z ) ) ^ { ' } } \\dot { D ( } x ) + \\lambda _ { 1 } G P | _ { \\widehat { \\mathbf { x } } } + \\lambda _ { 2 } C T | _ { \\mathbf { x } ^ { \\prime } , \\mathbf { x } ^ { \\prime \\prime } } } \\\\ & { \\mathbf { e n d } \\mathbf { f o r } } \\\\ & { \\theta _ { d } \\gets A d a m ( \\nabla _ { \\theta _ { d } } \\frac { 1 } { b } \\sum _ { i = 1 } ^ { b } L ^ { ( i ) } , \\theta _ { d } , \\gamma ) } \\end{array}$ \n7: \n8: \n9: end for \n10: Sample $b$ random vectors $\\big \\{ z ^ { ( i ) } \\big \\}$ \n11: $\\begin{array} { r } { \\underrightarrow { \\theta } _ { g } A d a m ( \\nabla _ { \\theta _ { g } } \\frac { 1 } { b } \\sum _ { i = 1 } ^ { b } - D ( G ( z ^ { ( i ) } ) ) , \\theta _ { g } , \\gamma ) } \\end{array}$ \n12: end for ", + "bbox": [ + 178, + 142, + 825, + 324 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The success comes after we perturb the hidden layers of the discriminator using dropout, as opposed to the input $_ { \\textbf { \\em x } }$ . When the dropout rate is small, the perturbed discriminator’s output can be considered as the output of the clean discriminator in response to a “virtual” data point $\\mathbf { x } ^ { \\prime }$ that is not far from $_ { \\textbf { \\em x } }$ . Thus, we denote by $D ( \\pmb { x } ^ { \\prime } )$ the discriminator output after applying dropout to its hidden layers. In the same manner, we find the second virtual point $\\scriptstyle { \\pmb x } ^ { \\prime \\prime }$ around $_ { \\textbf { \\em x } }$ by applying the (stochastic) dropout again to the hidden layers of the discriminator, and denote by $D ( \\pmb { x } ^ { \\prime \\prime } )$ the corresponding output. ", + "bbox": [ + 173, + 349, + 825, + 434 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Note that, however, it becomes impossible to compute the distance $d ( { \\pmb x } ^ { \\prime } , { \\pmb x } ^ { \\prime \\prime } )$ between the two virtual data points. In this work, we assume it is bounded by a constant and absorb the constant to $M ^ { \\prime }$ . Accordingly, we tune $M ^ { \\prime }$ in our experiments to take account of this unknown constant; the best results are obtained between $M ^ { \\prime } = 0$ and $M ^ { \\prime } = 0 . 2$ . For consistency, we use $M ^ { \\prime } = 0$ to report all the results in this paper. ", + "bbox": [ + 173, + 439, + 825, + 511 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A consistency regularization. Our final consistency regularization takes the following form, ", + "bbox": [ + 171, + 525, + 794, + 540 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/de9358b9a5e621172b7a2b0803e3e72746c531dd76d358a89a79ac952fe34c07.jpg", + "text": "$$\nC T | _ { \\mathbf { x } ^ { \\prime } , \\mathbf { x } ^ { \\prime \\prime } } = \\mathbb { E } _ { \\mathbf { x } \\sim \\mathbb { P } _ { r } } \\left[ \\operatorname* { m a x } \\left( 0 , d ( D ( \\mathbf { x } ^ { \\prime } ) , D ( \\mathbf { x } ^ { \\prime \\prime } ) ) + 0 . 1 \\cdot d ( D _ { - } ( \\mathbf { x } ^ { \\prime } ) , D _ { - } ( \\mathbf { x } ^ { \\prime \\prime } ) ) - M ^ { \\prime } \\right) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 218, + 542, + 779, + 560 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $D ( { \\pmb x } ^ { \\prime } )$ is the output of the discriminator given the input $_ { \\textbf { \\em x } }$ and after we apply dropout to the hidden layers of the discriminator. We envision this is equivalent to passing a “virtual” data point $\\mathbf { x } ^ { \\prime }$ through the clean discriminator. We find that it slightly improves the performance by further controlling the second-to-last layer $D _ { - } ( \\cdot )$ of the discriminator, $i . e .$ , the $d ( D _ { - } ^ { \\mathbf { \\hat { \\rho } } } ( { \\pmb x } ^ { \\prime } ) , D _ { - } ( { \\pmb x } ^ { \\prime \\prime } ) )$ above. ", + "bbox": [ + 174, + 564, + 825, + 621 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This new consistent regularization $C T | _ { \\pmb { x } ^ { \\prime } , \\pmb { x } ^ { \\prime \\prime } }$ enforces the Lipschitz continuity over the data manifold and its surrounding regions, effectively complementing and improving the gradient penalty $G P | _ { \\widehat { \\pmb { x } } }$ bused in the improved training of WGAN. Putting them together, our new objective function for updating the weigts of the discriminator is ", + "bbox": [ + 174, + 627, + 823, + 683 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4f97d91d203da90d0ba216162ddfcb9e5c938bd407f94599af319a9d05a6ef2d.jpg", + "text": "$$\nL = \\mathbb { E } _ { z \\sim \\mathbb { P } _ { z } } \\left[ D ( G ( z ) ) \\right] - \\mathbb { E } _ { \\pmb { x } \\sim \\mathbb { P } _ { r } } \\left[ D ( \\pmb { x } ) \\right] + \\lambda _ { 1 } G P | _ { \\hat { \\pmb { x } } } + \\lambda _ { 2 } C T | _ { \\pmb { x } ^ { \\prime } , \\pmb { x } ^ { \\prime \\prime } } .\n$$", + "text_format": "latex", + "bbox": [ + 274, + 688, + 722, + 705 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1 shows the complete algorithm for learning a WGAN in this paper. For the hyperparameters, we borrow $\\lambda _ { 1 } = 1 0$ from Gulrajani et al. (2017) and use $\\lambda _ { 2 } = 2$ for all our experiments no matter on which dataset. Another hyper-parameter is $M ^ { \\prime }$ in eq. (5). As stated previously, $M ^ { \\prime }$ taking values between 0 and 0.2 gives rise to about the same results in our experiments. ", + "bbox": [ + 174, + 715, + 825, + 772 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 A SEAMLESS CONNECTION WITH A SEMI-SUPERVISED LEARNING METHOD", + "text_level": 1, + "bbox": [ + 174, + 789, + 728, + 804 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we extend the WGAN to a semi-supervised learning approach by drawing insights from two related works. ", + "bbox": [ + 171, + 814, + 823, + 843 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Following (Salimans et al., 2016), we modify the output layer of the discriminator such that it has $K + 1$ output neurons, where $K$ is the number of classes of interest and the $( K + 1 )$ -th neuron is reserved for contrasting the generated samples with the real data using the Wasserstein distance in the WGAN. We use a $( K + 1 )$ -way softmax as the activation function of the last layer. ", + "bbox": [ + 217, + 853, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/b65ca53fe23c10d4de9a1eee5825e7f0f587db7d9a1f3a42ade0843100429e27.jpg", + "image_caption": [ + "Figure 2: Framework for the semi-supervised training. For clarity, we have omitted the generator. " + ], + "image_footnote": [], + "bbox": [ + 259, + 104, + 738, + 231 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• Following (Laine & Aila, 2016), we add our consistency regularization $C T | _ { x ^ { \\prime } , x ^ { \\prime \\prime } }$ to the objective function of the semi-supervised training, in order to take advantage of the effect of temporal ensembling. ", + "bbox": [ + 215, + 291, + 823, + 335 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Figure 2 shows the framework for training the discriminator, with the objective function below, ", + "bbox": [ + 171, + 347, + 795, + 362 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6dc70829fb2e2d87b3229bd169f5a19bcbc3cb91c4d78141e9150a7a35641468.jpg", + "text": "$$\n\\begin{array} { r l } & { L _ { \\mathrm { s e m i . d i s } } = - \\operatorname { \\mathbb { E } } _ { \\pmb { x } , \\pmb { y } \\sim \\mathbb { P } _ { \\mathrm { x y } } } \\left[ \\log D ( \\pmb { y } | \\pmb { x } ) \\right] - \\operatorname { \\mathbb { E } } _ { \\pmb { z } \\sim \\mathbb { P } _ { z } } \\left[ \\log D ( K + 1 | G ( \\pmb { z } ) ) \\right] } \\\\ & { \\qquad - \\operatorname { \\mathbb { E } } _ { \\pmb { x } \\sim \\mathbb { P } _ { r } } \\left[ \\log ( 1 - D ( K + 1 | \\pmb { x } ) ) \\right] + \\lambda C T | _ { \\pmb { x } ^ { \\prime } , \\pmb { x } ^ { \\prime \\prime } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 276, + 366, + 722, + 406 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where the first three terms are the same as in (Salimans et al., 2016), while the last consistency regularization is calculated after we apply dropout to the discriminator. The last term essentially leads to a temporal self-ensembling scheme to benefit the semi-supervised learning. Please see (Laine & Aila, 2016) for more insightful discussions about it. ", + "bbox": [ + 176, + 410, + 823, + 467 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The generator loss matches the expected features of the generated sample and the real data points, ", + "bbox": [ + 171, + 473, + 813, + 488 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/77108c41b5e52d2c1aac1338bcded349b4461fc457aede680798cadb9edc14b5.jpg", + "text": "$$\nL _ { \\mathrm { s e m i . g e n } } = \\| \\mathbb { E } _ { z \\sim \\mathbb { P } _ { z } } ( D _ { - } ( G ( z ) ) - \\mathbb { E } _ { { \\pmb x } \\sim \\mathbb { P } _ { r } } ( D _ { - } ( { \\pmb x } ) ) \\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 493, + 671, + 512 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 539, + 416, + 555 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We conduct experiments on the prevalent MNIST (LeCun et al., 1998), and CIFAR-10 (Krizhevsky & Hinton, 2009) datasets. The code is available on https://github.com/biuyq/CT-GAN to facilitate the reproducibility of our results. ", + "bbox": [ + 174, + 570, + 825, + 613 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 MNIST ", + "text_level": 1, + "bbox": [ + 174, + 630, + 269, + 643 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The MNIST dataset provides 70,000 handwritten digits in total and 10,000 of them are often left out for the testing purpose. Following (Gulrajani et al., 2017), we use only 1,000 of them to train the WGAN for a fair comparison with it. We use all the 60,000 training examples in the semi-supervised learning experiments and reveal the labels of 100 of them (10 per class). This setup is the same as in (Rasmus et al., 2015). No data augmentation is used. Please see Appendix A for the network architectures of the generator and discriminator, respectively. ", + "bbox": [ + 174, + 655, + 825, + 739 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Qualitative results. Figure 3 shows the generated samples with improved training of WGAN by the gradient penalty (GP-WGAN) and ours with the consistency regularization (CT-GAN), respectively, after 50,000 generator iterations. It is clear that our approach gives rise to more realistic samples than GP-WGAN. The contrasts of our samples between the foreground and the background are in general sharper than those of GP-WGAN. ", + "bbox": [ + 173, + 755, + 825, + 825 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Overfitting. We find that our approach is less prone to overfitting. To demonstrate this point, we show the convergence curves of the discriminator’s value functions by GP-WGAN and our CT-GAN in Figure 4. The red curves are evaluated on the training set and the blue ones are on the test set. We can see that the results on the test set become saturated pretty early in GP-WGAN, while ours can consistently decrease the costs on both the training and the test sets. This observation also holds for the CIFAR-10 dataset (cf. Appendix E). ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/8681645893db2d6f35b2e9b969262d80ba64bd7b995fdd6924754e6edc4bf18f.jpg", + "image_caption": [ + "Figure 3: Images generated by (a) GP-WGAN (Gulrajani et al., 2017) and (b) Our CT-GAN, respectively. " + ], + "image_footnote": [], + "bbox": [ + 200, + 106, + 802, + 238 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/b3fedcd75934f5d21cb397b2c5678493c20d58bf6f361902bcd6d953a888d842.jpg", + "image_caption": [ + "Figure 4: Convergence curves of the discriminator cost: (a) GP-WGAN and (b) Our CT-GAN. " + ], + "image_footnote": [], + "bbox": [ + 214, + 290, + 777, + 468 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Semi-supervised learning results. We compare our semi-supervised learning results with those of several competitive methods in Table 1. Our approach is among the best on this MNIST dataset. ", + "bbox": [ + 174, + 507, + 820, + 536 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 CIFAR-10 ", + "text_level": 1, + "bbox": [ + 174, + 556, + 290, + 571 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "CIFAR-10 (Krizhevsky & Hinton, 2009) contains 50,000 natural images of size $3 2 \\times 3 2$ . We use it to test two networks for the generative model: a small CNN and a ResNet (cf. Appendix A for the network structures). For the former we use only 1,000 images to train the model and we use the whole training set to learn the ResNet. ", + "bbox": [ + 174, + 585, + 825, + 641 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Qualitative results. Figure 5 contrasts the samples we generated to those by GP-WGAN when the generator is the small CNN. Figure 7 shows the results by a larger-scale ResNet. Our results are more photo-realistic. ", + "bbox": [ + 174, + 661, + 825, + 704 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Additionally, we also draw the histograms of the discriminator’s weights in Figure 6 after we train it using GP-WGAN and CT-GAN, respectively. It is interesting to see that ours controls the weights within a smaller and more symmetric range $[ - 0 . 6 7 , 0 . 9 6 ]$ than the $[ - 2 . 0 0 , 1 0 . 1 2 ]$ by GP-WGAN, partially explaining why our approach is less prone to overfitting. ", + "bbox": [ + 174, + 710, + 825, + 767 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/50f81a96e31c0becfc7ad671a138cf8b083bf35206a3657a9fea0fa2e217ff0a.jpg", + "table_caption": [ + "Table 1: Comparing our semi-supervised learning approach with state-of-the-art ones on MNIST. " + ], + "table_footnote": [], + "table_body": "
MethodTest error (%)
Ladder (Rasmus et al., 2015)1.06±0.37
VAT (Miyato et al., 2017)1.36
CatGAN (Springenberg,2015)1.39±0.28
Improved GAN (Salimans et al., 2016)0.93 ± 0.065
Triple GAN (Li et al., 2017a)0.91 ± 0.58
Our CT-GAN0.89 ± 0.13
", + "bbox": [ + 299, + 819, + 697, + 920 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/aa12002406502061af7c698f3350a49fb5750013e7322ac6cab2d6c4287e124b.jpg", + "image_caption": [ + "Figure 5: Generated samples by (a) GP-WGAN and (b) our CT-GAN. Here the generator is a small CNN. See Figure 7 for the samples by a ResNet. " + ], + "image_footnote": [], + "bbox": [ + 204, + 107, + 797, + 239 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/249fc7181e82a988faf3d3e8057cd4dfd28a1b23595766db419b398a8d1059d8.jpg", + "image_caption": [ + "Figure 6: Histograms of the weights of the discriminators trained by (a) GP-WGAN and (b) CTGAN, respectively. " + ], + "image_footnote": [], + "bbox": [ + 210, + 321, + 772, + 478 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/713197a208107afa04c900cf7b7753e54bf6c9669d56616453b30f493f5192f8.jpg", + "table_caption": [ + "Table 2: Inception score and accuracy of different models on CIFAR-10 " + ], + "table_footnote": [], + "table_body": "
MethodSupervised ISUnsupervised ISAccuracy(%)
SteinGANs (Wang&Liu,2016)6.35
DCGANs (Radford et al., 2015)6.586.16 ± 0.07
Improved GANs (Salimans et al., 2016)8.09 ± 0.071
AC-GANs (Odena et al., 2016)8.25 ± 0.07
GP-WGAN (Gulrajani et al., 2017)8.42 ± 0.107.86± 0.0791.85
SGANs (Huang et al., 2016)8.59 ±0.12
ALI (Warde-Farley & Bengio,2016)15.34 ± 0.05
BEGAN (Berthelot et al., 2017)一5.62
EGAN-Ent-VI (Dai et al., 2017)7.07 ± 0.10
DFM(Warde-Farley & Bengio,2016)17.72 ± 0.071
Our CT-GAN8.81±0.138.12±0.1295.91
", + "bbox": [ + 176, + 585, + 820, + 757 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparison of the inception scores. Finally, we compare our approach with GP-WGAN on the whole training set for both unsupervised and supervised generative-purpose task using ResNet. For model selection, we use the first 50,000 samples to compute the inception scores (Salimans et al., 2016), then choose the best model, and finally report the “test” score on another 50,000 samples. The experiment follows the previous setup in (Odena et al., 2016). From the comparison results in Tables 2, we conclude that our proposed model achieves the highest inception score on the CIFAR10 dataset, to the best of our knowledge. Some generated samples are shown in Figure 7. ", + "bbox": [ + 173, + 790, + 825, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For the small CNN based generator, the inception scores of GP-WGAN and our CT-GAN are $2 . 9 8 \\pm$ 0.11 and $5 . 1 3 \\pm 0 . 1 2$ , respectively. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/3c70b1dead917ecf56456f0a87519cf0cad0fc5a71fe26bb94f6752bce509e04.jpg", + "image_caption": [ + "Figure 7: Generated samples by our ResNet model: (a) Generated samples by unsupervised model and (b) Generated samples by supervised model. Each column corresponds to one class in the CIFAR-10 dataset. " + ], + "image_footnote": [], + "bbox": [ + 202, + 108, + 795, + 351 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/073ec37d2e92c3393fa75d47650a227904b37fc9ece6bc142fd2ef7c1f65d0fd.jpg", + "table_caption": [ + "Table 3: Comparing our semi-supervised learning approach with state-of-the-art ones on CIFAR-10 " + ], + "table_footnote": [], + "table_body": "
MethodTest error(%)
Ladder (Rasmus et al., 2015)20.40±0.47
VAT (Miyato et al., 2017)10.55
TE (Laine & Aila,2016)12.16 ±0.24
Teacher-Student (Tarvainen & Valpola,2017)12.31 ± 0.28
CatGANs (Springenberg,2015)19.58 ± 0.58
Improved GANs (Salimans et al., 2016)18.63 ± 2.32
ALI (Dumoulin et al., 2016)17.99 ±1.62
CLS-GAN (Qi, 2017)17.30 ± 0.50
Triple GAN (Li et al., 2017a)16.99 ± 0.36
Improved semi-GAN (Kumar et al., 2017)16.78 ± 1.80
Our CT-GAN9.98± 0.21
", + "bbox": [ + 276, + 441, + 717, + 612 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Semi-supervised learning. For the semi-supervised learning approach, we follow the standard training/test split of the dataset but use only 4,000 labels in the training. A regular data augmentation with flipping the images horizontally and randomly translating the images within [-2,2] pixels is used in our paper (No ZCA whitening). We report the semi-supervised learning results in Table 3. The mean and standard errors are obtained by running the experiments 5 rounds. Comparing to several very competitive methods, ours is able to achieve state-of-the-art results. Notably, our CT-GAN outperfroms all the GAN based methods by a large margin. Please see Appendix A for the network architectures and Appendix C for the ablation study of our algorithm. ", + "bbox": [ + 174, + 636, + 825, + 747 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 766, + 318, + 782 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this paper, we present a consistency term derived from Lipschitz inequality, which boosts the performance of GANs model. The proposed term has been demonstrated to be an efficient manner to ease the over-fitting problem when data amount is limited. Experiments show that our model obtains the state-of-the-art accuracy and Inception score on CIFAR-10 dataset for both the semisupervised learning task and the learning of generative models. ", + "bbox": [ + 174, + 797, + 825, + 867 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Acknowledgements. This work is partially supported by NSF IIS-1741431, IIS-1566511, and ONR N00014-18-1-2121. 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", + "bbox": [ + 173, + 188, + 825, + 217 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Appendices ", + "text_level": 1, + "bbox": [ + 176, + 253, + 343, + 282 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "APPENDIX A NETWORK ARCHITECTURES ", + "bbox": [ + 176, + 305, + 535, + 321 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Table 4 (MNIST) and Table 5 (CIFAR-10) detail the network architectures used in our classificationpurpose CT-GAN, where the classifiers are the same as the widely used ones in most semi-supervised networks (Laine & Aila, 2016) except that we apply weight-norm rather than batch-norm. For the generators, we follow the network structures in GP-WGAN (Salimans et al., 2016), but we use lower dimensional noise (50D) as the input to the generator for CIFAR-10 in order not to reproduce the complicated images and instead shift the focus of the training to the classifier. ", + "bbox": [ + 174, + 335, + 825, + 420 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/b2e29437f5fcbed5346984e6f906f3b2bfca4f4173c20470db81ab1a6db2663a.jpg", + "table_caption": [ + "Table 4: Networks for semi-supervised learning on MNIST " + ], + "table_footnote": [], + "table_body": "
Classifier CGenerator G
Input: Labels y,28*28 Images x,Input: Noise 100 z
Gaussiannoise0.3,MLP1000,ReLU Gaussian noise 0.5,MLP 500,ReLUMLP 500, Softplus,Batch norm
Gaussian noise O.5,MLP 250,ReLUMLP 500, Softplus,Batch norm
Gaussian noise 0.5,MLP 250,ReLUMLP 784, Sigmoid,Weight norm
Gaussian noise 0.5,MLP 250,ReLU
Gaussian noise O.5,MLP 10, Softmax
", + "bbox": [ + 240, + 454, + 753, + 571 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/186d5c538f0050bffc97b03fc49ed138b6d7db3ea95e5f370ca0e38edb877b17.jpg", + "table_caption": [ + "Table 5: Networks for semi-supervised learning on CIFAR-10 " + ], + "table_footnote": [], + "table_body": "
Classifier CGenerator G
Input: Labels y, 32*32*3 Colored Image x,Input:Noise 50 z
0.2 Dropout 3*3 conv.128,Pad=1, Stride =1,lReLU,Weight norm 3*3 conv. 128, Pad =1, Stride =1, lReLU,Weight normMLP 8192, ReLU, Batch norm Reshape 512*4*4 5*5 deconv. 256*8*8,
3*3 conv. 128,Pad =1, Stride =2, lReLU, Weight norm 0.5 Dropout 3*3 conv. 256,Pad =1, Stride =1, lReLU, Weight norm 3*3 conv. 256,Pad=1, Stride =1, lReLU, Weight norm 3*3 conv. 256,Pad =1, Stride =2, lReLU, Weight normReLU,Batch norm 5*5 deconv. 128*16*16, ReLU,Batch norm
0.5 Dropout 3*3 conv. 512, Pad =O, Stride =1, lReLU, Weight norm 3*3 conv. 256,Pad =O, Stride =1,lReLU, Weight norm 3*3 conv. 128,Pad =O, Stride =1, lReLU, Weight norm5*5 deconv. 3*32*32,
Global pool MLP 10, Weight norm, SoftmaxTanh,Weight norm
", + "bbox": [ + 196, + 612, + 802, + 842 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "APPENDIX B HYPER-PARAMETERS AND OTHER TRAINING DETAILS ", + "text_level": 1, + "bbox": [ + 178, + 866, + 745, + 881 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "For the semi-supervised learning experiments, we set $\\lambda = 1 . 0$ in Eq.(7) in all our experiments. For CIFAR-10, the number of training epochs is set to 1,000 with a constant learning rate of 0.0003. For ", + "bbox": [ + 176, + 895, + 823, + 924 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/dbed09409edf0d50037f24549a8d86af413f80f9e0e56726ceca689aa12b98ff.jpg", + "table_caption": [ + "Table 6: Generative model for MNIST " + ], + "table_footnote": [], + "table_body": "
DiscriminatorGenerator
Input: 1*28*28 Image xInput:Noise z 128
5*5 conv. 64,Pad = same, Stride= 2, IReLU 0.5 DropoutMLP4096,ReLU Reshape 256*4*4
5*5 conv. 128,Pad = same, Stride= 2,lReLU 0.5 Dropout5*5 deconv. 128*8*8 ReLU, Cut 128*7*7
5*5 conv. 256, Pad = same, Stride = 2, IReLU 0.5 Dropout5*5 deconv. 64*14*14
Reshape 256*4*4 (D-)ReLU 5*5 deconv. 1*28*28
MLP 1 (D)Sigmoid
", + "bbox": [ + 253, + 125, + 745, + 272 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/25f2ce4d379ff56136c03b6f383c0e7884aaadd285ff0dc2112a14952ee1f905.jpg", + "table_caption": [ + "Table 7: Generative model for CIFAR-10 " + ], + "table_footnote": [], + "table_body": "
DiscriminatorGenerator
Input: 3*32*32 Image x,Input:Noise z 128
5*5 conv. 128, Pad = same, Stride =2, IReLU 0.5 DropoutMLP 8192, ReLU, Batch norm Reshape 512*4*4
5*5 conv. 256, Pad = same, Stride = 2, IReLU 0.5 Dropout5*5 deconv. 256*8*8 ReLU,Bach norm
5*5 conv. 512, Pad = same Stride= 2, IReLU5*5 deconv. 128*16*16
0.5 Dropout Reshape 512*4*4 (D-)ReLU, Batch norm 5*5 deconv. 3*32*32
MLP 1 (D)Tanh
", + "bbox": [ + 225, + 314, + 772, + 460 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "MNIST, the number of training epochs is set to 300 with a constant learning rate of 0.003. The other hyper-parameters are exactly the same as in the improved GAN (Salimans et al., 2016). ", + "bbox": [ + 174, + 489, + 823, + 517 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For the experiments on the generative models, to have a fair comparison for our results with the existing ones, we keep the network structure and hyper-parameters the same as in the improved training of WGAN (Gulrajani et al., 2017) except that we add three dropout layers to some of the hidden layers as shown in tables 6 to 8. ", + "bbox": [ + 174, + 525, + 825, + 580 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "APPENDIX C ABLATION STUDY OF OUR APPROACH TO SSL", + "text_level": 1, + "bbox": [ + 178, + 606, + 681, + 622 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We run an ablation study of our approach to the semi-supervised learning (SSL). Thanks to the dual effect of the proposed consistency (CT) term, we are able to connect GAN with the temporal ensembling (TE) Laine & Aila (2016) method for SSL. Our superior results thus benefit from both of them, verified by the ablation study detailed in Table 9. ", + "bbox": [ + 174, + 640, + 825, + 695 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/5e19f60486c0165686ef8346a3b41043e17bf4acd963e27f4b9d37b0fc9ccaca.jpg", + "table_caption": [ + "Table 8: ResNet for CIFAR-10 " + ], + "table_footnote": [], + "table_body": "
DiscriminatorGenerator
Input: 3*32*32 Image xInput: Noise bmz 128
[3*3]*2 Residual Block, Resample = DOWN 128*16*16MLP2048 Reshape 128*4*4
[3*3]*2 Residual Block, Resample = DOWN 128*8*8 0.2 Dropout[3*3]*2 Residual Block, Resample = UP
[3*3]*2 Residual Block, Resample = None 128*8*8 0.5 Dropout128*8*8 [3*3]*2 Residual Block, Resample = UP
[3*3]*2 Residual Block, Resample = None128*16*16 [3*3]*2 Residual Block, Resample = UP
128*8*8 0.5 Dropout128*32*32
ReLU, Global mean pool (D-) MLP 1 (D)3*3 conv. 3*32*32
Tanh
", + "bbox": [ + 196, + 748, + 802, + 922 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "If we remove the CT term, the test error goes up to 14.98, signifying the effectiveness of the CT regularization. ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "If we remove GAN from our approach, it almost reduces to TE; in fact, all the settings here are the same as TE except that we use an extra regularization ${ \\bf \\it D _ { - } ( \\dots ) }$ in CT) over the second-to-last layer. We can see that the error is still significantly larger than our overall method. ", + "bbox": [ + 176, + 138, + 823, + 181 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We use the weight normalization as in Salimans et al. (2016), which becomes a core constituent of our approach. The batch normalization would actually invalidate the feature matching in Salimans et al. (2016). ", + "bbox": [ + 174, + 188, + 825, + 229 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Finally, we also test the version without the regularization to the second-to-last layer and observe a little drop in the performance. ", + "bbox": [ + 173, + 237, + 823, + 265 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/64a8c6ceb0a5efaafe74ba227364fdaff251973c9ac441a175999a8580e5c9e9.jpg", + "table_caption": [ + "Table 9: Ablation study of our semi-supervised learning method " + ], + "table_footnote": [], + "table_body": "
MethodTest Error
w/o CT14.98±0.43
w/o GAN11.98±0.32
w batch norm
w/o D-(.,.) over the second-to-last layer10.70±0.24
Ours9.98±0.21
", + "bbox": [ + 305, + 304, + 692, + 393 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "APPENDIX D EXAMINING THE 1-LIPSCHITZ CONTINUITY ", + "text_level": 1, + "bbox": [ + 176, + 422, + 665, + 439 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Norm of gradient. In our experiments, we find that although the GP-WGAN (Gulrajani et al., 2017) has applied a Lipschitz constraint in the form of the gradient penalty over the input sampled between a real data point and a generated one, the actual effect on the $\\ell _ { 2 }$ norm of the gradient is not as good as our CT-GAN model in the real data points, as illustrated in Figure 1. We empirically verify this fact by Figure 8, which shows the $\\ell _ { 2 }$ norms of the gradients of the discriminator with respect to the real data points. The closer to 1 the norms are, the better the 1-Lipschitz continuity is preserved. Figure 8 further demonstrates that our consistency (CT) regularization is able to improve GP-WGAN (Gulrajani et al., 2017). ", + "bbox": [ + 173, + 454, + 825, + 566 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/c6d2218ee59678ad1515a1a0032f1d261c7a9421305f20dc50ea49e964a774d5.jpg", + "image_caption": [ + "Figure 8: The maximum $\\ell _ { 2 }$ norm of the gradients of the discriminator with respect to the input on CIFAR-10 testing set in each iteration of the training using 1000 CIFAR-10 training images. " + ], + "image_footnote": [], + "bbox": [ + 331, + 598, + 653, + 795 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Definition of the Lipschitz continuity. Additionally, we check how much the 1-Lipschitz continuity is satisfied according to its vanilla definition (cf. Eq. (3)). Figures 9 and 10 plot the CTs of Eq. (4) and Eq. (5), respectively, over different iterations of the training using 1000 CIFAR-10 images. Figure 10 is the actual CTs used to train the generative model. Figure 9 is drawn as follows. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For every 100 iterations, we randomly pick up 64 real examples and split them into two subsets of the same size. We compute $d ( D ( x _ { 1 } ) - D ( x _ { 2 } ) ) / d ( x _ { 1 } - x _ { 2 } )$ for all the $( x _ { 1 } , x _ { 2 } )$ pairs, where $x _ { 1 }$ is from the first subset and $x _ { 2 }$ is from the second. The maximum of $d ( D ( x _ { 1 } ) - D ( x _ { 2 } ) ) / d ( x _ { 1 } - x _ { 2 } )$ is plotted in Figure 9. We can see that the CT-GAN curve converges under a certain value much faster than GP-WGAN. The 1-Lipschitz continuity is better maintained by CT-GAN than GP-WGAN over the whole course of the training procedure on the manifold of the real data. ", + "bbox": [ + 173, + 103, + 825, + 188 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/d187e15086c5f6a9cb6de72fa06b6e70dfcfd085991fab599baf7f77e568762b.jpg", + "image_caption": [ + "Figure 9: CT (cf. Eq. (4)) over different iterations of the training using 1000 CIFAR-10 images. In each iteration, we randomly pick up two real data points to compute the CT. " + ], + "image_footnote": [], + "bbox": [ + 326, + 212, + 655, + 407 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/78fb5bbee816494a12aed3f7636ed7d567abaa59689f378bed1c9131bedf1d60.jpg", + "image_caption": [ + "Figure 10: CT (cf. Eq. (5)) over different iterations of the training using 1000 CIFAR-10 images. " + ], + "image_footnote": [], + "bbox": [ + 338, + 492, + 655, + 688 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "APPENDIX E GP-WGAN WITH DROPOUT ", + "text_level": 1, + "bbox": [ + 176, + 746, + 534, + 762 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We run another experiment of GP-WGAN by adding the same dropout layers used in our method to the networks of GP-WGAN, denoted by GP-WGAN+dropout. This may help understand the contribution of our approach in preventing the overfitting problem — is it due to the CT regularization, or merely due to the dropout? This experiment is done by removing the CT term out of our value function for training WGAN using 1,000 CIFAR-10 images and yet keep the dropout layers. ", + "bbox": [ + 174, + 776, + 825, + 847 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The Inception score of GP-WGAN+dropout is $4 . 2 9 \\pm 0 . 1 2$ , and the generated samples are shown in Figure 11(a). We also plot the curve of CT in the training procedure of GP-WGAN $^ +$ dropout and contrast it to the curve of our CT-GAN in Figure 11(b). It is clear that GP-WGAN+dropout outperforms GP-WGAN $( 2 . 9 8 \\pm 0 . 1 1 ) $ , and our method $( 5 . 1 3 \\pm 0 . 1 2 )$ outperforms GP-WGAN with a large margin. ", + "bbox": [ + 174, + 853, + 823, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/95c8e3d138185e0fef7249995e192bbfa47adcdf3cf05b143734d134d5846b3a.jpg", + "image_caption": [ + "Figure 11: The results of GP-WGAN+dropout. " + ], + "image_footnote": [], + "bbox": [ + 179, + 113, + 803, + 276 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Finally, we plot the convergence curves of the discriminators’ negative cost function learned by GPWGAN, GP-WGAN $^ +$ Dropout, and our CT-GAN in Figure 12. We can see that dropout is able to reduce the overfitting of GP-WGAN, but it is not as effective as our CT-GAN. ", + "bbox": [ + 174, + 330, + 825, + 372 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/42b83c1468918cfe11735af7355568c9222ce3d2ece267e863f5e088144006fe.jpg", + "image_caption": [ + "Figure 12: Convergence curves of the discriminator cost: (a) GP-WGAN, (b) GP-WGAN $^ +$ Dropout, and (c) CT-GAN (ours). " + ], + "image_footnote": [], + "bbox": [ + 207, + 401, + 776, + 777 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "APPENDIX F EXPERIMENTS ON LARGE DATASET ", + "text_level": 1, + "bbox": [ + 174, + 848, + 591, + 863 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this section, we further present experimental results on the large-scale ImageNet (Deng et al., 2009) and LSUN bedroom (Yu et al., 2015) datasets. The experiment setup (e.g., network architecture, learning rates, etc.) is exactly the same as in the GP-WGAN work (Gulrajani et al., 2017). We refer readers to (Gulrajani et al., 2017) or our code on GitHub (https://github.com/biuyq/CT-GAN) for the details of the setup. ", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 132 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Our experiment on ImageNet is conducted using images of the size $6 4 \\times 6 4$ . After 200, 000 generator iterations, the inception score of the proposed CT-GAN is ${ \\bf 1 0 . 2 7 } \\pm { \\bf 0 . 1 5 }$ , whereas GP-WGAN’s is ${ \\bf 9 . 8 5 \\pm 0 . 1 7 }$ . In addition, the inception score comparison of GP-WGAN and our CT-GAN in each generator iteration is shown in Figure 13. We can observe that the inception score of our proposed CT-GAN becomes higher than GP-WGAN’s after early generator iterations. Finally, Figure 14 shows the samples generated by GP-WGAN and CT-GAN, respectively. ", + "bbox": [ + 173, + 138, + 825, + 223 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/8dad9e4369b46ff49ce7ce9c759e02deb243f3bc9efa497400438a821313cb50.jpg", + "image_caption": [ + "Figure 13: Inception score of GP-WGAN and our CT-GAN with respect to iterations " + ], + "image_footnote": [], + "bbox": [ + 313, + 265, + 666, + 481 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/072a288c32709d4684b8f5c996ac80ac24e2308673461cbdd52e9e49cc89c917.jpg", + "image_caption": [ + "Figure 14: Samples generated by GP-WGAN (a) and by CT-GAN (b), respectively. " + ], + "image_footnote": [], + "bbox": [ + 202, + 583, + 795, + 827 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For the LSUN bedroom dataset, we show some results in Figure 15. They are the generated samples by our CT-GAN generator after $2 0 \\mathrm { k }$ training iterations. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/c22c5ba777606a67430cb6fc56b70ba287ba6b7735bdf11341a64160d55db6f6.jpg", + "image_caption": [ + "Figure 15: Image samples generated by CT-GAN trained model using LSUN bedroom images. 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The corresponding algorithm, namely, Wasserstein GAN", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "(WGAN), shows not only superior performance over GANs but also a nice correlation between the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 326, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 326, + 171 + ], + "score": 1.0, + "content": "sample quality and the value function that GANs lack.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 183, + 416, + 195 + ], + "lines": [ + { + "bbox": [ + 106, + 182, + 417, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 417, + 196 + ], + "score": 1.0, + "content": "1.1 BACKGROUND: WGAN AND THE IMPROVED TRAINING OF WGAN", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 282 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 384, + 217 + ], + "score": 1.0, + "content": "WGAN (Arjovsky et al., 2017) aims to learn the generator network", + "type": "text" + }, + { + "bbox": [ + 385, + 204, + 407, + 216 + ], + "score": 0.92, + "content": "G ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 204, + 506, + 217 + ], + "score": 1.0, + "content": ", for any random vector", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 215, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 139, + 226 + ], + "score": 0.9, + "content": "z \\sim \\mathbb { P } _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 215, + 478, + 227 + ], + "score": 1.0, + "content": ", such that the Wasserstein distance is minimized between the resulting distribution", + "type": "text" + }, + { + "bbox": [ + 478, + 215, + 492, + 226 + ], + "score": 0.9, + "content": "\\mathbb { P } _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 215, + 506, + 227 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 196, + 239 + ], + "score": 1.0, + "content": "the generated samples", + "type": "text" + }, + { + "bbox": [ + 197, + 226, + 229, + 238 + ], + "score": 0.92, + "content": "\\{ G ( z ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 226, + 326, + 239 + ], + "score": 1.0, + "content": "and the real distribution", + "type": "text" + }, + { + "bbox": [ + 327, + 226, + 338, + 237 + ], + "score": 0.89, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 226, + 483, + 239 + ], + "score": 1.0, + "content": "underlying the observed data points", + "type": "text" + }, + { + "bbox": [ + 484, + 226, + 501, + 238 + ], + "score": 0.91, + "content": "\\{ { \\pmb x } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 226, + 505, + 239 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 170, + 249 + ], + "score": 1.0, + "content": "in other words,", + "type": "text" + }, + { + "bbox": [ + 171, + 238, + 242, + 249 + ], + "score": 0.89, + "content": "\\mathrm { m i n } _ { G } W ( \\mathbb { P } _ { r } , \\mathbb { P } _ { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 237, + 352, + 249 + ], + "score": 1.0, + "content": ". The Wasserstein distance", + "type": "text" + }, + { + "bbox": [ + 353, + 237, + 400, + 249 + ], + "score": 0.93, + "content": "W ( \\mathbb { P } _ { r } , \\mathbb { P } _ { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "is shown a more sensible", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "cost function for learning the distributions supported by low-dimensional manifolds than the other", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "popular distribution divergences and distances — for example, the Jensen-Shannon (JS) divergence", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 270, + 333, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 333, + 283 + ], + "score": 1.0, + "content": "implicitly employed in GANs (Goodfellow et al., 2014).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 503, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 300 + ], + "score": 1.0, + "content": "Due to the Kantorovich-Rubinstein duality (Villani, 2008) for calculating the Wasserstein distance,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 297, + 296, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 296, + 309 + ], + "score": 1.0, + "content": "the value function of WGAN is then written as", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 311, + 398, + 330 + ], + "lines": [ + { + "bbox": [ + 211, + 311, + 398, + 330 + ], + "spans": [ + { + "bbox": [ + 211, + 311, + 398, + 330 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { G } \\operatorname* { m a x } _ { D \\in \\mathcal { D } } \\mathbb { E } _ { { \\pmb x } \\sim \\mathbb { P } _ { r } } \\big [ D ( { \\pmb x } ) \\big ] - \\mathbb { E } _ { { \\pmb z } \\sim \\mathbb { P } _ { z } } \\big [ D ( G ( { \\pmb z } ) ) \\big ] ,", + "type": "interline_equation", + "image_path": "0619b4b527195bd840cce06292feaf3867008f918d7334c8ab0c01a426d81bc0.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 211, + 311, + 398, + 330 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 133, + 344 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 333, + 142, + 342 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 331, + 415, + 344 + ], + "score": 1.0, + "content": "is the set of 1-Lipschitz functions. Analogous to GANs, we still call", + "type": "text" + }, + { + "bbox": [ + 415, + 333, + 425, + 342 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "the “discriminator”", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "although it is actually a real-valued function and not a classifier at all. Arjovsky et al. (2017) specify", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 202, + 367 + ], + "score": 1.0, + "content": "this family of functions", + "type": "text" + }, + { + "bbox": [ + 202, + 354, + 212, + 364 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "by neural networks and then use weight clipping to enforce the Lipschitz", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "continuity. However, as the authors note, the networks’ capacities become limited due to the weight", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 375, + 394, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 394, + 390 + ], + "score": 1.0, + "content": "clipping and there could be gradient vanishing problems in the training.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 504, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "Improved training of WGAN. Gulrajani et al. (2017) give more concrete examples to illustrate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "score": 1.0, + "content": "the perils of the weight clipping and propose an alternative way of imposing the Lipschitz continuity.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "In particular, they introduce a gradient penalty term by noting that the differentiable discriminator", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 432, + 439, + 444 + ], + "spans": [ + { + "bbox": [ + 107, + 432, + 127, + 444 + ], + "score": 0.91, + "content": "D ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 432, + 439, + 444 + ], + "score": 1.0, + "content": "is 1-Lipschitz if and only if the norm of its gradients is at most 1 everywhere,", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 446, + 379, + 467 + ], + "lines": [ + { + "bbox": [ + 232, + 446, + 379, + 467 + ], + "spans": [ + { + "bbox": [ + 232, + 446, + 379, + 467 + ], + "score": 0.95, + "content": "G P | _ { \\widehat { \\pmb { x } } } : = \\mathbb { E } _ { \\widehat { \\pmb { x } } } \\Big [ \\big ( \\| \\nabla _ { \\widehat { \\pmb { x } } } D ( \\widehat { \\pmb { x } } ) \\| _ { 2 } - 1 \\big ) ^ { 2 } \\Big ]", + "type": "interline_equation", + "image_path": "ee4a1c95f4df60014d8a15b90e0d091667c27b8c32dc0f0e8d3ecf7e85e631a2.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 232, + 446, + 379, + 467 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 133, + 482 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 469, + 141, + 479 + ], + "score": 0.83, + "content": "\\widehat { \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "is uniformly sampled from the straight line between a pair of data points sampled from the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 480, + 488, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 133, + 492 + ], + "score": 1.0, + "content": "model", + "type": "text" + }, + { + "bbox": [ + 134, + 480, + 148, + 491 + ], + "score": 0.88, + "content": "\\mathbb { P } _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 480, + 198, + 492 + ], + "score": 1.0, + "content": "and the real", + "type": "text" + }, + { + "bbox": [ + 198, + 480, + 209, + 491 + ], + "score": 0.89, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 480, + 488, + 492 + ], + "score": 1.0, + "content": ", respectively. A similar regularization is used by Kodali et al. (2017).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "Potential caveats. Unlike the weight clipping, however, by no means one can penalize everywhere", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "using this term through a finite number of training iterations. As a result, the gradient penalty term", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 525, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 107, + 525, + 123, + 535 + ], + "score": 0.82, + "content": "G P", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 525, + 291, + 537 + ], + "score": 1.0, + "content": "takes effect only upon the sampled points", + "type": "text" + }, + { + "bbox": [ + 291, + 525, + 299, + 535 + ], + "score": 0.83, + "content": "\\widehat { \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 525, + 506, + 537 + ], + "score": 1.0, + "content": ", leaving significant parts of the support domain not", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "bexamined at all. In particular, consider the observed data points and their underlying manifold that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 546, + 504, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 223, + 560 + ], + "score": 1.0, + "content": "supports the real distribution", + "type": "text" + }, + { + "bbox": [ + 223, + 547, + 235, + 558 + ], + "score": 0.88, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 546, + 481, + 560 + ], + "score": 1.0, + "content": ". At the beginning of the training stage, the generated sample", + "type": "text" + }, + { + "bbox": [ + 481, + 547, + 504, + 559 + ], + "score": 0.9, + "content": "G ( z )", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 149, + 570 + ], + "score": 1.0, + "content": "and hence", + "type": "text" + }, + { + "bbox": [ + 150, + 558, + 158, + 568 + ], + "score": 0.81, + "content": "\\widehat { \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "could be distant from the manifold. The Lipschitz continuity over the manifold is not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 569, + 468, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 250, + 582 + ], + "score": 1.0, + "content": "benforced until the generative model", + "type": "text" + }, + { + "bbox": [ + 250, + 569, + 264, + 580 + ], + "score": 0.89, + "content": "\\mathbb { P } _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 569, + 416, + 582 + ], + "score": 1.0, + "content": "becomes close enough to the real one", + "type": "text" + }, + { + "bbox": [ + 417, + 569, + 428, + 580 + ], + "score": 0.88, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 569, + 468, + 582 + ], + "score": 1.0, + "content": ", if it can.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 108, + 593, + 297, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 593, + 298, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 298, + 606 + ], + "score": 1.0, + "content": "1.2 OUR APPROACH AND CONTRIBUTIONS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "In light of the above pros and cons, we propose to improve the improved training of WGAN by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 623, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 466, + 638 + ], + "score": 1.0, + "content": "additionally laying the Lipschitz continuity condition over the manifold of the real data", + "type": "text" + }, + { + "bbox": [ + 467, + 625, + 501, + 636 + ], + "score": 0.9, + "content": "\\mathbf { \\boldsymbol { x } } \\sim \\mathbb { P } _ { \\boldsymbol { r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 623, + 506, + 638 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "score": 1.0, + "content": "Moreover, instead of focusing on one particular data point at a time, we devise a regularization over", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "a pair of data points drawn near the manifold following the most basic definition of the 1-Lipschitz", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 340, + 671 + ], + "score": 1.0, + "content": "continuity. In particular, we perturb each real data point", + "type": "text" + }, + { + "bbox": [ + 341, + 660, + 348, + 668 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "twice and use a Lipschitz constant to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 668, + 487, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 456, + 681 + ], + "score": 1.0, + "content": "bound the difference between the discriminator’s responses to the perturbed data points", + "type": "text" + }, + { + "bbox": [ + 457, + 668, + 483, + 680 + ], + "score": 0.92, + "content": "{ \\mathbf { } } x ^ { \\prime } , x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 668, + 487, + 681 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 685, + 504, + 730 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 338, + 698 + ], + "score": 1.0, + "content": "Figure 1 illustrates our main idea. The gradient penalty", + "type": "text" + }, + { + "bbox": [ + 338, + 686, + 363, + 698 + ], + "score": 0.91, + "content": "G P | _ { \\widehat { \\pmb { x } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "often fails to check the continuity", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 697, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 222, + 709 + ], + "score": 1.0, + "content": "of region near the real data", + "type": "text" + }, + { + "bbox": [ + 222, + 699, + 230, + 707 + ], + "score": 0.74, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 697, + 505, + 709 + ], + "score": 1.0, + "content": "b, around which the discriminator function can freely violate the 1-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 707, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 506, + 721 + ], + "score": 1.0, + "content": "Lipschitz continuity. We alleviate this issue by explicitly checking the continuity condition using", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 718, + 384, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 210, + 731 + ], + "score": 1.0, + "content": "the two perturbed version", + "type": "text" + }, + { + "bbox": [ + 211, + 718, + 237, + 730 + ], + "score": 0.91, + "content": "{ \\mathbf { } } x ^ { \\prime } , x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 718, + 373, + 731 + ], + "score": 1.0, + "content": "near any observed real data point", + "type": "text" + }, + { + "bbox": [ + 374, + 722, + 380, + 728 + ], + "score": 0.7, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 718, + 384, + 731 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "However, it is remarkably difficult to train GANs without good heuristics (Goodfellow et al., 2014;", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "Salimans et al., 2016; Radford et al., 2015) which may not generalize across different network", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "architectures or application domains. The training dynamics are often unstable and the generated", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "samples could collapse to limited modes. These issues are formally analyzed by Arjovsky & Bottou", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "(2017), who also propose an alternative direction (Arjovsky et al., 2017) to avoid the caveats in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "minmax two-player training of GANs. The corresponding algorithm, namely, Wasserstein GAN", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "(WGAN), shows not only superior performance over GANs but also a nice correlation between the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 326, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 326, + 171 + ], + "score": 1.0, + "content": "sample quality and the value function that GANs lack.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 82, + 506, + 171 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 183, + 416, + 195 + ], + "lines": [ + { + "bbox": [ + 106, + 182, + 417, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 417, + 196 + ], + "score": 1.0, + "content": "1.1 BACKGROUND: WGAN AND THE IMPROVED TRAINING OF WGAN", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 282 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 384, + 217 + ], + "score": 1.0, + "content": "WGAN (Arjovsky et al., 2017) aims to learn the generator network", + "type": "text" + }, + { + "bbox": [ + 385, + 204, + 407, + 216 + ], + "score": 0.92, + "content": "G ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 204, + 506, + 217 + ], + "score": 1.0, + "content": ", for any random vector", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 215, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 139, + 226 + ], + "score": 0.9, + "content": "z \\sim \\mathbb { P } _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 215, + 478, + 227 + ], + "score": 1.0, + "content": ", such that the Wasserstein distance is minimized between the resulting distribution", + "type": "text" + }, + { + "bbox": [ + 478, + 215, + 492, + 226 + ], + "score": 0.9, + "content": "\\mathbb { P } _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 215, + 506, + 227 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 196, + 239 + ], + "score": 1.0, + "content": "the generated samples", + "type": "text" + }, + { + "bbox": [ + 197, + 226, + 229, + 238 + ], + "score": 0.92, + "content": "\\{ G ( z ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 226, + 326, + 239 + ], + "score": 1.0, + "content": "and the real distribution", + "type": "text" + }, + { + "bbox": [ + 327, + 226, + 338, + 237 + ], + "score": 0.89, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 226, + 483, + 239 + ], + "score": 1.0, + "content": "underlying the observed data points", + "type": "text" + }, + { + "bbox": [ + 484, + 226, + 501, + 238 + ], + "score": 0.91, + "content": "\\{ { \\pmb x } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 226, + 505, + 239 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 170, + 249 + ], + "score": 1.0, + "content": "in other words,", + "type": "text" + }, + { + "bbox": [ + 171, + 238, + 242, + 249 + ], + "score": 0.89, + "content": "\\mathrm { m i n } _ { G } W ( \\mathbb { P } _ { r } , \\mathbb { P } _ { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 237, + 352, + 249 + ], + "score": 1.0, + "content": ". The Wasserstein distance", + "type": "text" + }, + { + "bbox": [ + 353, + 237, + 400, + 249 + ], + "score": 0.93, + "content": "W ( \\mathbb { P } _ { r } , \\mathbb { P } _ { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "is shown a more sensible", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "cost function for learning the distributions supported by low-dimensional manifolds than the other", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "popular distribution divergences and distances — for example, the Jensen-Shannon (JS) divergence", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 270, + 333, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 333, + 283 + ], + "score": 1.0, + "content": "implicitly employed in GANs (Goodfellow et al., 2014).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 204, + 506, + 283 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 503, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 300 + ], + "score": 1.0, + "content": "Due to the Kantorovich-Rubinstein duality (Villani, 2008) for calculating the Wasserstein distance,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 297, + 296, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 296, + 309 + ], + "score": 1.0, + "content": "the value function of WGAN is then written as", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 285, + 505, + 309 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 311, + 398, + 330 + ], + "lines": [ + { + "bbox": [ + 211, + 311, + 398, + 330 + ], + "spans": [ + { + "bbox": [ + 211, + 311, + 398, + 330 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { G } \\operatorname* { m a x } _ { D \\in \\mathcal { D } } \\mathbb { E } _ { { \\pmb x } \\sim \\mathbb { P } _ { r } } \\big [ D ( { \\pmb x } ) \\big ] - \\mathbb { E } _ { { \\pmb z } \\sim \\mathbb { P } _ { z } } \\big [ D ( G ( { \\pmb z } ) ) \\big ] ,", + "type": "interline_equation", + "image_path": "0619b4b527195bd840cce06292feaf3867008f918d7334c8ab0c01a426d81bc0.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 211, + 311, + 398, + 330 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 133, + 344 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 333, + 142, + 342 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 331, + 415, + 344 + ], + "score": 1.0, + "content": "is the set of 1-Lipschitz functions. Analogous to GANs, we still call", + "type": "text" + }, + { + "bbox": [ + 415, + 333, + 425, + 342 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "the “discriminator”", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "although it is actually a real-valued function and not a classifier at all. Arjovsky et al. (2017) specify", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 202, + 367 + ], + "score": 1.0, + "content": "this family of functions", + "type": "text" + }, + { + "bbox": [ + 202, + 354, + 212, + 364 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "by neural networks and then use weight clipping to enforce the Lipschitz", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "continuity. 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(2017) give more concrete examples to illustrate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "score": 1.0, + "content": "the perils of the weight clipping and propose an alternative way of imposing the Lipschitz continuity.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "In particular, they introduce a gradient penalty term by noting that the differentiable discriminator", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 432, + 439, + 444 + ], + "spans": [ + { + "bbox": [ + 107, + 432, + 127, + 444 + ], + "score": 0.91, + "content": "D ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 432, + 439, + 444 + ], + "score": 1.0, + "content": "is 1-Lipschitz if and only if the norm of its gradients is at most 1 everywhere,", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 398, + 506, + 444 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 446, + 379, + 467 + ], + "lines": [ + { + "bbox": [ + 232, + 446, + 379, + 467 + ], + "spans": [ + { + "bbox": [ + 232, + 446, + 379, + 467 + ], + "score": 0.95, + "content": "G P | _ { \\widehat { \\pmb { x } } } : = \\mathbb { E } _ { \\widehat { \\pmb { x } } } \\Big [ \\big ( \\| \\nabla _ { \\widehat { \\pmb { x } } } D ( \\widehat { \\pmb { x } } ) \\| _ { 2 } - 1 \\big ) ^ { 2 } \\Big ]", + "type": "interline_equation", + "image_path": "ee4a1c95f4df60014d8a15b90e0d091667c27b8c32dc0f0e8d3ecf7e85e631a2.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 232, + 446, + 379, + 467 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 133, + 482 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 469, + 141, + 479 + ], + "score": 0.83, + "content": "\\widehat { \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "is uniformly sampled from the straight line between a pair of data points sampled from the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 480, + 488, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 133, + 492 + ], + "score": 1.0, + "content": "model", + "type": "text" + }, + { + "bbox": [ + 134, + 480, + 148, + 491 + ], + "score": 0.88, + "content": "\\mathbb { P } _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 480, + 198, + 492 + ], + "score": 1.0, + "content": "and the real", + "type": "text" + }, + { + "bbox": [ + 198, + 480, + 209, + 491 + ], + "score": 0.89, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 480, + 488, + 492 + ], + "score": 1.0, + "content": ", respectively. A similar regularization is used by Kodali et al. (2017).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 106, + 469, + 505, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "Potential caveats. Unlike the weight clipping, however, by no means one can penalize everywhere", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "using this term through a finite number of training iterations. 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In particular, consider the observed data points and their underlying manifold that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 546, + 504, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 223, + 560 + ], + "score": 1.0, + "content": "supports the real distribution", + "type": "text" + }, + { + "bbox": [ + 223, + 547, + 235, + 558 + ], + "score": 0.88, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 546, + 481, + 560 + ], + "score": 1.0, + "content": ". At the beginning of the training stage, the generated sample", + "type": "text" + }, + { + "bbox": [ + 481, + 547, + 504, + 559 + ], + "score": 0.9, + "content": "G ( z )", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 149, + 570 + ], + "score": 1.0, + "content": "and hence", + "type": "text" + }, + { + "bbox": [ + 150, + 558, + 158, + 568 + ], + "score": 0.81, + "content": "\\widehat { \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "could be distant from the manifold. The Lipschitz continuity over the manifold is not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 569, + 468, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 250, + 582 + ], + "score": 1.0, + "content": "benforced until the generative model", + "type": "text" + }, + { + "bbox": [ + 250, + 569, + 264, + 580 + ], + "score": 0.89, + "content": "\\mathbb { P } _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 569, + 416, + 582 + ], + "score": 1.0, + "content": "becomes close enough to the real one", + "type": "text" + }, + { + "bbox": [ + 417, + 569, + 428, + 580 + ], + "score": 0.88, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 569, + 468, + 582 + ], + "score": 1.0, + "content": ", if it can.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 502, + 506, + 582 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 593, + 297, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 593, + 298, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 298, + 606 + ], + "score": 1.0, + "content": "1.2 OUR APPROACH AND CONTRIBUTIONS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "In light of the above pros and cons, we propose to improve the improved training of WGAN by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 623, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 466, + 638 + ], + "score": 1.0, + "content": "additionally laying the Lipschitz continuity condition over the manifold of the real data", + "type": "text" + }, + { + "bbox": [ + 467, + 625, + 501, + 636 + ], + "score": 0.9, + "content": "\\mathbf { \\boldsymbol { x } } \\sim \\mathbb { P } _ { \\boldsymbol { r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 623, + 506, + 638 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "score": 1.0, + "content": "Moreover, instead of focusing on one particular data point at a time, we devise a regularization over", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "a pair of data points drawn near the manifold following the most basic definition of the 1-Lipschitz", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 340, + 671 + ], + "score": 1.0, + "content": "continuity. In particular, we perturb each real data point", + "type": "text" + }, + { + "bbox": [ + 341, + 660, + 348, + 668 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "twice and use a Lipschitz constant to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 668, + 487, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 456, + 681 + ], + "score": 1.0, + "content": "bound the difference between the discriminator’s responses to the perturbed data points", + "type": "text" + }, + { + "bbox": [ + 457, + 668, + 483, + 680 + ], + "score": 0.92, + "content": "{ \\mathbf { } } x ^ { \\prime } , x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 668, + 487, + 681 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 614, + 506, + 681 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 685, + 504, + 730 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 338, + 698 + ], + "score": 1.0, + "content": "Figure 1 illustrates our main idea. The gradient penalty", + "type": "text" + }, + { + "bbox": [ + 338, + 686, + 363, + 698 + ], + "score": 0.91, + "content": "G P | _ { \\widehat { \\pmb { x } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "often fails to check the continuity", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 697, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 222, + 709 + ], + "score": 1.0, + "content": "of region near the real data", + "type": "text" + }, + { + "bbox": [ + 222, + 699, + 230, + 707 + ], + "score": 0.74, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 697, + 505, + 709 + ], + "score": 1.0, + "content": "b, around which the discriminator function can freely violate the 1-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 707, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 506, + 721 + ], + "score": 1.0, + "content": "Lipschitz continuity. We alleviate this issue by explicitly checking the continuity condition using", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 718, + 384, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 210, + 731 + ], + "score": 1.0, + "content": "the two perturbed version", + "type": "text" + }, + { + "bbox": [ + 211, + 718, + 237, + 730 + ], + "score": 0.91, + "content": "{ \\mathbf { } } x ^ { \\prime } , x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 718, + 373, + 731 + ], + "score": 1.0, + "content": "near any observed real data point", + "type": "text" + }, + { + "bbox": [ + 374, + 722, + 380, + 728 + ], + "score": 0.7, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 718, + 384, + 731 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 685, + 506, + 731 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 348, + 247 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 348, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 348, + 95 + ], + "score": 1.0, + "content": "In this paper, we make the following contributions. (1) We", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 349, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 349, + 108 + ], + "score": 1.0, + "content": "propose an alternative mechanism for enforcing the Lip-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 348, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 348, + 117 + ], + "score": 1.0, + "content": "schitz continuity over the family of discriminators by re-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 348, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 348, + 128 + ], + "score": 1.0, + "content": "sorting to the basic definition of the Lipschitz continuity.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 349, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 349, + 139 + ], + "score": 1.0, + "content": "It effectively improves the gradient penalty method (Gul-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 349, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 349, + 150 + ], + "score": 1.0, + "content": "rajani et al., 2017) and gives rise to generators with more", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 349, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 349, + 160 + ], + "score": 1.0, + "content": "photo-realistic samples and higher inception scores (Sali-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 348, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 348, + 171 + ], + "score": 1.0, + "content": "mans et al., 2016). (2) Our approach is very data efficient in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 349, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 349, + 183 + ], + "score": 1.0, + "content": "terms of being less prone to overfitting even for very small", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 349, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 349, + 194 + ], + "score": 1.0, + "content": "training sets. 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By bringing the best of the two worlds, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 383, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 395 + ], + "score": 1.0, + "content": "report better semi-supervised learning results than both (Laine & Aila, 2016) and existing GAN-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 393, + 169, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 169, + 405 + ], + "score": 1.0, + "content": "based methods.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 106, + 418, + 343, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 343, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 343, + 432 + ], + "score": 1.0, + "content": "2.1 IMPROVING THE IMPROVED TRAINING OF WGAN", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 105, + 439, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 123, + 452 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 441, + 130, + 450 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 439, + 176, + 452 + ], + "score": 1.0, + "content": "denote the", + "type": "text" + }, + { + "bbox": [ + 177, + 441, + 186, + 451 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 439, + 438, + 452 + ], + "score": 1.0, + "content": "metric on an input space used in this paper. A discriminator", + "type": "text" + }, + { + "bbox": [ + 439, + 440, + 494, + 451 + ], + "score": 0.91, + "content": "D : \\mathcal { X } \\mapsto \\mathcal { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 450, + 460, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 308, + 463 + ], + "score": 1.0, + "content": "Lipschitz continuous if there exists a real constant", + "type": "text" + }, + { + "bbox": [ + 309, + 451, + 339, + 462 + ], + "score": 0.93, + "content": "M \\geqslant 0", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 450, + 407, + 463 + ], + "score": 1.0, + "content": "such that, for all", + "type": "text" + }, + { + "bbox": [ + 407, + 451, + 455, + 462 + ], + "score": 0.91, + "content": "\\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 450, + 460, + 463 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 468, + 379, + 482 + ], + "lines": [ + { + "bbox": [ + 232, + 468, + 379, + 482 + ], + "spans": [ + { + "bbox": [ + 232, + 468, + 379, + 482 + ], + "score": 0.91, + "content": "d ( D ( \\pmb { x } _ { 1 } ) , D ( \\pmb { x } _ { 2 } ) ) \\leqslant M \\cdot d ( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } ) .", + "type": "interline_equation", + "image_path": "1c9372f8c931466416e545c51b27303626650d7e933f6bf467d8444231fbec64.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 232, + 468, + 379, + 482 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 487, + 504, + 510 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 352, + 500 + ], + "score": 1.0, + "content": "Immediately, we can add the following soft consistency term", + "type": "text" + }, + { + "bbox": [ + 353, + 488, + 375, + 498 + ], + "score": 0.82, + "content": "( C T )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "to the value function of WGAN", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 498, + 352, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 352, + 511 + ], + "score": 1.0, + "content": "in order to penalize the violations to the inequality in eq. (3),", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 515, + 425, + 543 + ], + "lines": [ + { + "bbox": [ + 186, + 515, + 425, + 543 + ], + "spans": [ + { + "bbox": [ + 186, + 515, + 425, + 543 + ], + "score": 0.94, + "content": "C T | _ { x _ { 1 } , x _ { 2 } } = \\mathbb { E } _ { x _ { 1 } , x _ { 2 } } \\left[ \\operatorname* { m a x } \\left( 0 , \\frac { d ( D ( x _ { 1 } ) , D ( x _ { 2 } ) ) } { d ( x _ { 1 } , x _ { 2 } ) } - M ^ { \\prime } \\right) \\right]", + "type": "interline_equation", + "image_path": "6de6c29929ad322a7060ae54c3087f5b86aed194277dbebc66e8935ea0978b99.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 515, + 425, + 529.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 186, + 529.0, + 425, + 543.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 553, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 504, + 567 + ], + "score": 1.0, + "content": "Remarks. Here we face the same snag as in (Gulrajani et al., 2017), i.e., it is impractical to sub-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 223, + 578 + ], + "score": 1.0, + "content": "stitute all the possibilities of", + "type": "text" + }, + { + "bbox": [ + 223, + 565, + 258, + 577 + ], + "score": 0.92, + "content": "( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 564, + 506, + 578 + ], + "score": 1.0, + "content": "pairs into eq. (4). What pairs and which regions of the input", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 120, + 588 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 121, + 577, + 131, + 586 + ], + "score": 0.81, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "should we check for eq. (4)? 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(2) Our approach is very data efficient in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 349, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 349, + 183 + ], + "score": 1.0, + "content": "terms of being less prone to overfitting even for very small", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 349, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 349, + 194 + ], + "score": 1.0, + "content": "training sets. 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In addition to the gradient", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 354, + 235, + 504, + 246 + ], + "spans": [ + { + "bbox": [ + 354, + 235, + 409, + 246 + ], + "score": 1.0, + "content": "penalty over", + "type": "text" + }, + { + "bbox": [ + 409, + 235, + 417, + 245 + ], + "score": 0.79, + "content": "\\widehat { \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 235, + 493, + 246 + ], + "score": 1.0, + "content": ", we also examine", + "type": "text" + }, + { + "bbox": [ + 493, + 235, + 504, + 245 + ], + "score": 0.85, + "content": "\\mathbf { x } ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 354, + 245, + 504, + 257 + ], + "spans": [ + { + "bbox": [ + 354, + 245, + 373, + 257 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 374, + 246, + 387, + 256 + ], + "score": 0.87, + "content": "\\mathbf { x } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 245, + 496, + 257 + ], + "score": 1.0, + "content": "baround the real data point", + "type": "text" + }, + { + "bbox": [ + 496, + 248, + 504, + 256 + ], + "score": 0.69, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 355, + 257, + 425, + 268 + ], + "spans": [ + { + "bbox": [ + 355, + 257, + 425, + 268 + ], + "score": 1.0, + "content": "in each iteration.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + } + ], + "index": 19.25 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 308 + ], + "lines": [], + "index": 25, + "bbox_fs": [ + 105, + 274, + 505, + 309 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 325, + 182, + 337 + ], + "lines": [ + { + "bbox": [ + 104, + 323, + 185, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 323, + 185, + 340 + ], + "score": 1.0, + "content": "2 APPROACH", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "We firstly review the definition of the Lipschitz continuity and then discuss how to use it to regularize", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "the training of WGAN. 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By bringing the best of the two worlds, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 383, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 395 + ], + "score": 1.0, + "content": "report better semi-supervised learning results than both (Laine & Aila, 2016) and existing GAN-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 393, + 169, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 169, + 405 + ], + "score": 1.0, + "content": "based methods.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 350, + 506, + 405 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 418, + 343, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 343, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 343, + 432 + ], + "score": 1.0, + "content": "2.1 IMPROVING THE IMPROVED TRAINING OF WGAN", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 105, + 439, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 123, + 452 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 441, + 130, + 450 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 439, + 176, + 452 + ], + "score": 1.0, + "content": "denote the", + "type": "text" + }, + { + "bbox": [ + 177, + 441, + 186, + 451 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 439, + 438, + 452 + ], + "score": 1.0, + "content": "metric on an input space used in this paper. A discriminator", + "type": "text" + }, + { + "bbox": [ + 439, + 440, + 494, + 451 + ], + "score": 0.91, + "content": "D : \\mathcal { X } \\mapsto \\mathcal { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 450, + 460, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 308, + 463 + ], + "score": 1.0, + "content": "Lipschitz continuous if there exists a real constant", + "type": "text" + }, + { + "bbox": [ + 309, + 451, + 339, + 462 + ], + "score": 0.93, + "content": "M \\geqslant 0", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 450, + 407, + 463 + ], + "score": 1.0, + "content": "such that, for all", + "type": "text" + }, + { + "bbox": [ + 407, + 451, + 455, + 462 + ], + "score": 0.91, + "content": "\\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 450, + 460, + 463 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 439, + 505, + 463 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 468, + 379, + 482 + ], + "lines": [ + { + "bbox": [ + 232, + 468, + 379, + 482 + ], + "spans": [ + { + "bbox": [ + 232, + 468, + 379, + 482 + ], + "score": 0.91, + "content": "d ( D ( \\pmb { x } _ { 1 } ) , D ( \\pmb { x } _ { 2 } ) ) \\leqslant M \\cdot d ( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } ) .", + "type": "interline_equation", + "image_path": "1c9372f8c931466416e545c51b27303626650d7e933f6bf467d8444231fbec64.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 232, + 468, + 379, + 482 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 487, + 504, + 510 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 352, + 500 + ], + "score": 1.0, + "content": "Immediately, we can add the following soft consistency term", + "type": "text" + }, + { + "bbox": [ + 353, + 488, + 375, + 498 + ], + "score": 0.82, + "content": "( C T )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "to the value function of WGAN", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 498, + 352, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 352, + 511 + ], + "score": 1.0, + "content": "in order to penalize the violations to the inequality in eq. (3),", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 487, + 505, + 511 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 515, + 425, + 543 + ], + "lines": [ + { + "bbox": [ + 186, + 515, + 425, + 543 + ], + "spans": [ + { + "bbox": [ + 186, + 515, + 425, + 543 + ], + "score": 0.94, + "content": "C T | _ { x _ { 1 } , x _ { 2 } } = \\mathbb { E } _ { x _ { 1 } , x _ { 2 } } \\left[ \\operatorname* { m a x } \\left( 0 , \\frac { d ( D ( x _ { 1 } ) , D ( x _ { 2 } ) ) } { d ( x _ { 1 } , x _ { 2 } ) } - M ^ { \\prime } \\right) \\right]", + "type": "interline_equation", + "image_path": "6de6c29929ad322a7060ae54c3087f5b86aed194277dbebc66e8935ea0978b99.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 515, + 425, + 529.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 186, + 529.0, + 425, + 543.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 553, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 504, + 567 + ], + "score": 1.0, + "content": "Remarks. 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In", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 306, + 334 + ], + "score": 1.0, + "content": "the same manner, we find the second virtual point", + "type": "text" + }, + { + "bbox": [ + 306, + 321, + 319, + 331 + ], + "score": 0.89, + "content": "\\scriptstyle { \\pmb x } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 321, + 350, + 334 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 350, + 323, + 358, + 331 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "by applying the (stochastic) dropout", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 331, + 488, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 353, + 345 + ], + "score": 1.0, + "content": "again to the hidden layers of the discriminator, and denote by", + "type": "text" + }, + { + "bbox": [ + 353, + 332, + 382, + 344 + ], + "score": 0.92, + "content": "D ( \\pmb { x } ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 331, + 488, + 345 + ], + "score": 1.0, + "content": "the corresponding output.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 370, + 361 + ], + "score": 1.0, + "content": "Note that, however, it becomes impossible to compute the distance", + "type": "text" + }, + { + "bbox": [ + 370, + 349, + 409, + 361 + ], + "score": 0.93, + "content": "d ( { \\pmb x } ^ { \\prime } , { \\pmb x } ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "between the two virtual", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 359, + 504, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 486, + 371 + ], + "score": 1.0, + "content": "data points. 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For consistency, we use", + "type": "text" + }, + { + "bbox": [ + 420, + 382, + 454, + 392 + ], + "score": 0.91, + "content": "M ^ { \\prime } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 381, + 505, + 395 + ], + "score": 1.0, + "content": "to report all", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 203, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 203, + 406 + ], + "score": 1.0, + "content": "the results in this paper.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 105, + 416, + 486, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 488, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 488, + 430 + ], + "score": 1.0, + "content": "A consistency regularization. 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In", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 306, + 334 + ], + "score": 1.0, + "content": "the same manner, we find the second virtual point", + "type": "text" + }, + { + "bbox": [ + 306, + 321, + 319, + 331 + ], + "score": 0.89, + "content": "\\scriptstyle { \\pmb x } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 321, + 350, + 334 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 350, + 323, + 358, + 331 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "by applying the (stochastic) dropout", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 331, + 488, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 353, + 345 + ], + "score": 1.0, + "content": "again to the hidden layers of the discriminator, and denote by", + "type": "text" + }, + { + "bbox": [ + 353, + 332, + 382, + 344 + ], + "score": 0.92, + "content": "D ( \\pmb { x } ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 331, + 488, + 345 + ], + "score": 1.0, + "content": "the corresponding output.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 276, + 506, + 345 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 370, + 361 + ], + "score": 1.0, + "content": "Note that, however, it becomes impossible to compute the distance", + "type": "text" + }, + { + "bbox": [ + 370, + 349, + 409, + 361 + ], + "score": 0.93, + "content": "d ( { \\pmb x } ^ { \\prime } , { \\pmb x } ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "between the two virtual", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 359, + 504, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 486, + 371 + ], + "score": 1.0, + "content": "data points. 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As stated previously,", + "type": "text" + }, + { + "bbox": [ + 489, + 591, + 504, + 600 + ], + "score": 0.83, + "content": "M ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 601, + 456, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 456, + 613 + ], + "score": 1.0, + "content": "taking values between 0 and 0.2 gives rise to about the same results in our experiments.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 567, + 506, + 613 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 625, + 446, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 625, + 448, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 448, + 638 + ], + "score": 1.0, + "content": "2.2 A SEAMLESS CONNECTION WITH A SEMI-SUPERVISED LEARNING METHOD", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 105, + 645, + 504, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "score": 1.0, + "content": "In this section, we extend the WGAN to a semi-supervised learning approach by drawing insights", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 656, + 204, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 204, + 668 + ], + "score": 1.0, + "content": "from two related works.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 645, + 505, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 132, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 132, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "• Following (Salimans et al., 2016), we modify the output layer of the discriminator such", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 141, + 687, + 186, + 700 + ], + "score": 1.0, + "content": "that it has", + "type": "text" + }, + { + "bbox": [ + 187, + 688, + 216, + 698 + ], + "score": 0.91, + "content": "K + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 687, + 312, + 700 + ], + "score": 1.0, + "content": "output neurons, where", + "type": "text" + }, + { + "bbox": [ + 313, + 688, + 323, + 698 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "is the number of classes of interest and the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 142, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 142, + 699, + 174, + 710 + ], + "score": 0.9, + "content": "( K + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "-th neuron is reserved for contrasting the generated samples with the real data using", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 142, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 142, + 710, + 345, + 721 + ], + "score": 1.0, + "content": "the Wasserstein distance in the WGAN. 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The last term essentially leads", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "to a temporal self-ensembling scheme to benefit the semi-supervised learning. Please see (Laine &", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 359, + 315, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 315, + 371 + ], + "score": 1.0, + "content": "Aila, 2016) for more insightful discussions about it.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 105, + 375, + 498, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 500, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 500, + 389 + ], + "score": 1.0, + "content": "The generator loss matches the expected features of the generated sample and the real data points,", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 391, + 411, + 406 + ], + "lines": [ + { + "bbox": [ + 199, + 391, + 411, + 406 + ], + "spans": [ + { + "bbox": [ + 199, + 391, + 411, + 406 + ], + "score": 0.89, + "content": "L _ { \\mathrm { s e m i . g e n } } = \\| \\mathbb { E } _ { z \\sim \\mathbb { P } _ { z } } ( D _ { - } ( G ( z ) ) - \\mathbb { E } _ { { \\pmb x } \\sim \\mathbb { P } _ { r } } ( D _ { - } ( { \\pmb x } ) ) \\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "77108c41b5e52d2c1aac1338bcded349b4461fc457aede680798cadb9edc14b5.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 199, + 391, + 411, + 406 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 427, + 255, + 440 + ], + "lines": [ + { + "bbox": [ + 104, + 426, + 257, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 426, + 257, + 442 + ], + "score": 1.0, + "content": "3 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "We conduct experiments on the prevalent MNIST (LeCun et al., 1998), and CIFAR-10 (Krizhevsky", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "& Hinton, 2009) datasets. The code is available on https://github.com/biuyq/CT-GAN", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 475, + 288, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 288, + 487 + ], + "score": 1.0, + "content": "to facilitate the reproducibility of our results.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 107, + 499, + 165, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 166, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 166, + 512 + ], + "score": 1.0, + "content": "3.1 MNIST", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "The MNIST dataset provides 70,000 handwritten digits in total and 10,000 of them are often left out", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "for the testing purpose. Following (Gulrajani et al., 2017), we use only 1,000 of them to train the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "WGAN for a fair comparison with it. We use all the 60,000 training examples in the semi-supervised", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "learning experiments and reveal the labels of 100 of them (10 per class). This setup is the same as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "in (Rasmus et al., 2015). No data augmentation is used. Please see Appendix A for the network", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 575, + 352, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 352, + 588 + ], + "score": 1.0, + "content": "architectures of the generator and discriminator, respectively.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "Qualitative results. Figure 3 shows the generated samples with improved training of WGAN by", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "the gradient penalty (GP-WGAN) and ours with the consistency regularization (CT-GAN), respec-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 633 + ], + "score": 1.0, + "content": "tively, after 50,000 generator iterations. It is clear that our approach gives rise to more realistic", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 630, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 104, + 630, + 506, + 645 + ], + "score": 1.0, + "content": "samples than GP-WGAN. The contrasts of our samples between the foreground and the background", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 642, + 301, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 301, + 655 + ], + "score": 1.0, + "content": "are in general sharper than those of GP-WGAN.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "Overfitting. We find that our approach is less prone to overfitting. 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The last term essentially leads", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "to a temporal self-ensembling scheme to benefit the semi-supervised learning. Please see (Laine &", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 359, + 315, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 315, + 371 + ], + "score": 1.0, + "content": "Aila, 2016) for more insightful discussions about it.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 323, + 505, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 375, + 498, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 500, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 500, + 389 + ], + "score": 1.0, + "content": "The generator loss matches the expected features of the generated sample and the real data points,", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 374, + 500, + 389 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 391, + 411, + 406 + ], + "lines": [ + { + "bbox": [ + 199, + 391, + 411, + 406 + ], + "spans": [ + { + "bbox": [ + 199, + 391, + 411, + 406 + ], + "score": 0.89, + "content": "L _ { \\mathrm { s e m i . g e n } } = \\| \\mathbb { E } _ { z \\sim \\mathbb { P } _ { z } } ( D _ { - } ( G ( z ) ) - \\mathbb { E } _ { { \\pmb x } \\sim \\mathbb { P } _ { r } } ( D _ { - } ( { \\pmb x } ) ) \\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "77108c41b5e52d2c1aac1338bcded349b4461fc457aede680798cadb9edc14b5.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 199, + 391, + 411, + 406 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 427, + 255, + 440 + ], + "lines": [ + { + "bbox": [ + 104, + 426, + 257, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 426, + 257, + 442 + ], + "score": 1.0, + "content": "3 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "We conduct experiments on the prevalent MNIST (LeCun et al., 1998), and CIFAR-10 (Krizhevsky", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "& Hinton, 2009) datasets. The code is available on https://github.com/biuyq/CT-GAN", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 475, + 288, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 288, + 487 + ], + "score": 1.0, + "content": "to facilitate the reproducibility of our results.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 451, + 505, + 487 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 499, + 165, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 166, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 166, + 512 + ], + "score": 1.0, + "content": "3.1 MNIST", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "The MNIST dataset provides 70,000 handwritten digits in total and 10,000 of them are often left out", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "for the testing purpose. Following (Gulrajani et al., 2017), we use only 1,000 of them to train the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "WGAN for a fair comparison with it. We use all the 60,000 training examples in the semi-supervised", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "learning experiments and reveal the labels of 100 of them (10 per class). This setup is the same as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "in (Rasmus et al., 2015). No data augmentation is used. Please see Appendix A for the network", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 575, + 352, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 352, + 588 + ], + "score": 1.0, + "content": "architectures of the generator and discriminator, respectively.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 520, + 506, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "Qualitative results. Figure 3 shows the generated samples with improved training of WGAN by", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "the gradient penalty (GP-WGAN) and ours with the consistency regularization (CT-GAN), respec-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 633 + ], + "score": 1.0, + "content": "tively, after 50,000 generator iterations. It is clear that our approach gives rise to more realistic", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 630, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 104, + 630, + 506, + 645 + ], + "score": 1.0, + "content": "samples than GP-WGAN. 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We compare our semi-supervised learning results with those", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 413, + 503, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 503, + 425 + ], + "score": 1.0, + "content": "of several competitive methods in Table 1. 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MethodTest error (%)
Ladder (Rasmus et al., 2015)1.06±0.37
VAT (Miyato et al., 2017)1.36
CatGAN (Springenberg,2015)1.39±0.28
Improved GAN (Salimans et al., 2016)0.93 ± 0.065
Triple GAN (Li et al., 2017a)0.91 ± 0.58
Our CT-GAN0.89 ± 0.13
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MethodTest error (%)
Ladder (Rasmus et al., 2015)1.06±0.37
VAT (Miyato et al., 2017)1.36
CatGAN (Springenberg,2015)1.39±0.28
Improved GAN (Salimans et al., 2016)0.93 ± 0.065
Triple GAN (Li et al., 2017a)0.91 ± 0.58
Our CT-GAN0.89 ± 0.13
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MethodSupervised ISUnsupervised ISAccuracy(%)
SteinGANs (Wang&Liu,2016)6.35
DCGANs (Radford et al., 2015)6.586.16 ± 0.07
Improved GANs (Salimans et al., 2016)8.09 ± 0.071
AC-GANs (Odena et al., 2016)8.25 ± 0.07
GP-WGAN (Gulrajani et al., 2017)8.42 ± 0.107.86± 0.0791.85
SGANs (Huang et al., 2016)8.59 ±0.12
ALI (Warde-Farley & Bengio,2016)15.34 ± 0.05
BEGAN (Berthelot et al., 2017)一5.62
EGAN-Ent-VI (Dai et al., 2017)7.07 ± 0.10
DFM(Warde-Farley & Bengio,2016)17.72 ± 0.071
Our CT-GAN8.81±0.138.12±0.1295.91
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MethodSupervised ISUnsupervised ISAccuracy(%)
SteinGANs (Wang&Liu,2016)6.35
DCGANs (Radford et al., 2015)6.586.16 ± 0.07
Improved GANs (Salimans et al., 2016)8.09 ± 0.071
AC-GANs (Odena et al., 2016)8.25 ± 0.07
GP-WGAN (Gulrajani et al., 2017)8.42 ± 0.107.86± 0.0791.85
SGANs (Huang et al., 2016)8.59 ±0.12
ALI (Warde-Farley & Bengio,2016)15.34 ± 0.05
BEGAN (Berthelot et al., 2017)一5.62
EGAN-Ent-VI (Dai et al., 2017)7.07 ± 0.10
DFM(Warde-Farley & Bengio,2016)17.72 ± 0.071
Our CT-GAN8.81±0.138.12±0.1295.91
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Each column corresponds to one class in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 307, + 184, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 184, + 319 + ], + "score": 1.0, + "content": "CIFAR-10 dataset.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 169, + 350, + 439, + 485 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 330, + 505, + 343 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 343 + ], + "score": 1.0, + "content": "Table 3: Comparing our semi-supervised learning approach with state-of-the-art ones on CIFAR-10", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "table_body", + "bbox": [ + 169, + 350, + 439, + 485 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 169, + 350, + 439, + 485 + ], + "spans": [ + { + "bbox": [ + 169, + 350, + 439, + 485 + ], + "score": 0.979, + "html": "
MethodTest error(%)
Ladder (Rasmus et al., 2015)20.40±0.47
VAT (Miyato et al., 2017)10.55
TE (Laine & Aila,2016)12.16 ±0.24
Teacher-Student (Tarvainen & Valpola,2017)12.31 ± 0.28
CatGANs (Springenberg,2015)19.58 ± 0.58
Improved GANs (Salimans et al., 2016)18.63 ± 2.32
ALI (Dumoulin et al., 2016)17.99 ±1.62
CLS-GAN (Qi, 2017)17.30 ± 0.50
Triple GAN (Li et al., 2017a)16.99 ± 0.36
Improved semi-GAN (Kumar et al., 2017)16.78 ± 1.80
Our CT-GAN9.98± 0.21
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The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 549, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 560 + ], + "score": 1.0, + "content": "mean and standard errors are obtained by running the experiments 5 rounds. Comparing to several", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "very competitive methods, ours is able to achieve state-of-the-art results. Notably, our CT-GAN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 570, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 506, + 582 + ], + "score": 1.0, + "content": "outperfroms all the GAN based methods by a large margin. Please see Appendix A for the network", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 581, + 385, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 385, + 593 + ], + "score": 1.0, + "content": "architectures and Appendix C for the ablation study of our algorithm.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 607, + 195, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 198, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 198, + 623 + ], + "score": 1.0, + "content": "4 CONCLUSION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "In this paper, we present a consistency term derived from Lipschitz inequality, which boosts the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "performance of GANs model. The proposed term has been demonstrated to be an efficient manner", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "to ease the over-fitting problem when data amount is limited. Experiments show that our model", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "obtains the state-of-the-art accuracy and Inception score on CIFAR-10 dataset for both the semi-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 676, + 360, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 360, + 689 + ], + "score": 1.0, + "content": "supervised learning task and the learning of generative models.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "Acknowledgements. This work is partially supported by NSF IIS-1741431, IIS-1566511, and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "ONR N00014-18-1-2121. B.G. and L.W. would also like to thank Adobe Research and NVIDIA,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 721, + 309, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 309, + 733 + ], + "score": 1.0, + "content": "and Amazon, respectively, for their gift donations.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 124, + 86, + 487, + 278 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 124, + 86, + 487, + 278 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 86, + 487, + 278 + ], + "spans": [ + { + "bbox": [ + 124, + 86, + 487, + 278 + ], + "score": 0.977, + "type": "image", + "image_path": "3c70b1dead917ecf56456f0a87519cf0cad0fc5a71fe26bb94f6752bce509e04.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 124, + 86, + 487, + 150.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 124, + 150.0, + 487, + 214.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 124, + 214.0, + 487, + 278.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 285, + 505, + 319 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "Figure 7: Generated samples by our ResNet model: (a) Generated samples by unsupervised model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "score": 1.0, + "content": "and (b) Generated samples by supervised model. 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MethodTest error(%)
Ladder (Rasmus et al., 2015)20.40±0.47
VAT (Miyato et al., 2017)10.55
TE (Laine & Aila,2016)12.16 ±0.24
Teacher-Student (Tarvainen & Valpola,2017)12.31 ± 0.28
CatGANs (Springenberg,2015)19.58 ± 0.58
Improved GANs (Salimans et al., 2016)18.63 ± 2.32
ALI (Dumoulin et al., 2016)17.99 ±1.62
CLS-GAN (Qi, 2017)17.30 ± 0.50
Triple GAN (Li et al., 2017a)16.99 ± 0.36
Improved semi-GAN (Kumar et al., 2017)16.78 ± 1.80
Our CT-GAN9.98± 0.21
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The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 549, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 560 + ], + "score": 1.0, + "content": "mean and standard errors are obtained by running the experiments 5 rounds. Comparing to several", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "very competitive methods, ours is able to achieve state-of-the-art results. Notably, our CT-GAN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 570, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 506, + 582 + ], + "score": 1.0, + "content": "outperfroms all the GAN based methods by a large margin. Please see Appendix A for the network", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 581, + 385, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 385, + 593 + ], + "score": 1.0, + "content": "architectures and Appendix C for the ablation study of our algorithm.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 504, + 506, + 593 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 607, + 195, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 198, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 198, + 623 + ], + "score": 1.0, + "content": "4 CONCLUSION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "In this paper, we present a consistency term derived from Lipschitz inequality, which boosts the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "performance of GANs model. 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Experiments show that our model", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "obtains the state-of-the-art accuracy and Inception score on CIFAR-10 dataset for both the semi-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 676, + 360, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 360, + 689 + ], + "score": 1.0, + "content": "supervised learning task and the learning of generative models.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 631, + 506, + 689 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "Acknowledgements. This work is partially supported by NSF IIS-1741431, IIS-1566511, and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "ONR N00014-18-1-2121. 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Classifier CGenerator G
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Classifier CGenerator G
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Classifier CGenerator G
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DiscriminatorGenerator
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5*5 conv. 128,Pad = same, Stride= 2,lReLU 0.5 Dropout5*5 deconv. 128*8*8 ReLU, Cut 128*7*7
5*5 conv. 256, Pad = same, Stride = 2, IReLU 0.5 Dropout5*5 deconv. 64*14*14
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DiscriminatorGenerator
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5*5 conv. 512, Pad = same Stride= 2, IReLU5*5 deconv. 128*16*16
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DiscriminatorGenerator
Input: 3*32*32 Image xInput: Noise bmz 128
[3*3]*2 Residual Block, Resample = DOWN 128*16*16MLP2048 Reshape 128*4*4
[3*3]*2 Residual Block, Resample = DOWN 128*8*8 0.2 Dropout[3*3]*2 Residual Block, Resample = UP
[3*3]*2 Residual Block, Resample = None 128*8*8 0.5 Dropout128*8*8 [3*3]*2 Residual Block, Resample = UP
[3*3]*2 Residual Block, Resample = None128*16*16 [3*3]*2 Residual Block, Resample = UP
128*8*8 0.5 Dropout128*32*32
ReLU, Global mean pool (D-) MLP 1 (D)3*3 conv. 3*32*32
Tanh
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DiscriminatorGenerator
Input: 1*28*28 Image xInput:Noise z 128
5*5 conv. 64,Pad = same, Stride= 2, IReLU 0.5 DropoutMLP4096,ReLU Reshape 256*4*4
5*5 conv. 128,Pad = same, Stride= 2,lReLU 0.5 Dropout5*5 deconv. 128*8*8 ReLU, Cut 128*7*7
5*5 conv. 256, Pad = same, Stride = 2, IReLU 0.5 Dropout5*5 deconv. 64*14*14
Reshape 256*4*4 (D-)ReLU 5*5 deconv. 1*28*28
MLP 1 (D)Sigmoid
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DiscriminatorGenerator
Input: 3*32*32 Image x,Input:Noise z 128
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5*5 conv. 256, Pad = same, Stride = 2, IReLU 0.5 Dropout5*5 deconv. 256*8*8 ReLU,Bach norm
5*5 conv. 512, Pad = same Stride= 2, IReLU5*5 deconv. 128*16*16
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DiscriminatorGenerator
Input: 3*32*32 Image xInput: Noise bmz 128
[3*3]*2 Residual Block, Resample = DOWN 128*16*16MLP2048 Reshape 128*4*4
[3*3]*2 Residual Block, Resample = DOWN 128*8*8 0.2 Dropout[3*3]*2 Residual Block, Resample = UP
[3*3]*2 Residual Block, Resample = None 128*8*8 0.5 Dropout128*8*8 [3*3]*2 Residual Block, Resample = UP
[3*3]*2 Residual Block, Resample = None128*16*16 [3*3]*2 Residual Block, Resample = UP
128*8*8 0.5 Dropout128*32*32
ReLU, Global mean pool (D-) MLP 1 (D)3*3 conv. 3*32*32
Tanh
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MethodTest Error
w/o CT14.98±0.43
w/o GAN11.98±0.32
w batch norm
w/o D-(.,.) over the second-to-last layer10.70±0.24
Ours9.98±0.21
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In our experiments, we find that although the GP-WGAN (Gulrajani et al.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "2017) has applied a Lipschitz constraint in the form of the gradient penalty over the input sampled", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 396, + 395 + ], + "score": 1.0, + "content": "between a real data point and a generated one, the actual effect on the", + "type": "text" + }, + { + "bbox": [ + 397, + 383, + 407, + 394 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 383, + 505, + 395 + ], + "score": 1.0, + "content": "norm of the gradient is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "not as good as our CT-GAN model in the real data points, as illustrated in Figure 1. 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Figure 9 is drawn as follows.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "If we remove the CT term, the test error goes up to 14.98, signifying the effectiveness of the CT", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 167, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 167, + 106 + ], + "score": 1.0, + "content": "regularization.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 110, + 504, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "If we remove GAN from our approach, it almost reduces to TE; in fact, all the settings here are the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 325, + 134 + ], + "score": 1.0, + "content": "same as TE except that we use an extra regularization", + "type": "text" + }, + { + "bbox": [ + 326, + 121, + 357, + 133 + ], + "score": 0.9, + "content": "{ \\bf \\it D _ { - } ( \\dots ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 121, + 505, + 134 + ], + "score": 1.0, + "content": "in CT) over the second-to-last layer.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 411, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 411, + 144 + ], + "score": 1.0, + "content": "We can see that the error is still significantly larger than our overall method.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 110, + 505, + 144 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 149, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "We use the weight normalization as in Salimans et al. 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MethodTest Error
w/o CT14.98±0.43
w/o GAN11.98±0.32
w batch norm
w/o D-(.,.) over the second-to-last layer10.70±0.24
Ours9.98±0.21
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We empirically", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 294, + 416 + ], + "score": 1.0, + "content": "verify this fact by Figure 8, which shows the", + "type": "text" + }, + { + "bbox": [ + 294, + 405, + 304, + 416 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "norms of the gradients of the discriminator with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "respect to the real data points. 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This may help understand the con-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "tribution of our approach in preventing the overfitting problem — is it due to the CT regularization,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "or merely due to the dropout? This experiment is done by removing the CT term out of our value", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 659, + 477, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 477, + 673 + ], + "score": 1.0, + "content": "function for training WGAN using 1,000 CIFAR-10 images and yet keep the dropout layers.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 297, + 690 + ], + "score": 1.0, + "content": "The Inception score of GP-WGAN+dropout is", + "type": "text" + }, + { + "bbox": [ + 298, + 677, + 347, + 687 + ], + "score": 0.88, + "content": "4 . 2 9 \\pm 0 . 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 676, + 506, + 690 + ], + "score": 1.0, + "content": ", and the generated samples are shown", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 466, + 700 + ], + "score": 1.0, + "content": "in Figure 11(a). 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(5)) over different iterations of the training using 1000 CIFAR-10 images.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + } + ], + "index": 27.0 + }, + { + "type": "title", + "bbox": [ + 108, + 591, + 327, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 329, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 329, + 606 + ], + "score": 1.0, + "content": "APPENDIX E GP-WGAN WITH DROPOUT", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "We run another experiment of GP-WGAN by adding the same dropout layers used in our method to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "the networks of GP-WGAN, denoted by GP-WGAN+dropout. This may help understand the con-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "tribution of our approach in preventing the overfitting problem — is it due to the CT regularization,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "or merely due to the dropout? This experiment is done by removing the CT term out of our value", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 659, + 477, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 477, + 673 + ], + "score": 1.0, + "content": "function for training WGAN using 1,000 CIFAR-10 images and yet keep the dropout layers.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 616, + 505, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 297, + 690 + ], + "score": 1.0, + "content": "The Inception score of GP-WGAN+dropout is", + "type": "text" + }, + { + "bbox": [ + 298, + 677, + 347, + 687 + ], + "score": 0.88, + "content": "4 . 2 9 \\pm 0 . 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 676, + 506, + 690 + ], + "score": 1.0, + "content": ", and the generated samples are shown", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 466, + 700 + ], + "score": 1.0, + "content": "in Figure 11(a). We also plot the curve of CT in the training procedure of GP-WGAN", + "type": "text" + }, + { + "bbox": [ + 466, + 689, + 473, + 697 + ], + "score": 0.44, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "dropout", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "and contrast it to the curve of our CT-GAN in Figure 11(b). It is clear that GP-WGAN+dropout", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 208, + 723 + ], + "score": 1.0, + "content": "outperforms GP-WGAN", + "type": "text" + }, + { + "bbox": [ + 208, + 710, + 260, + 721 + ], + "score": 0.86, + "content": "( 2 . 9 8 \\pm 0 . 1 1 ) ", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 709, + 331, + 723 + ], + "score": 1.0, + "content": ", and our method", + "type": "text" + }, + { + "bbox": [ + 331, + 710, + 383, + 721 + ], + "score": 0.84, + "content": "( 5 . 1 3 \\pm 0 . 1 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "outperforms GP-WGAN with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 721, + 169, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 169, + 735 + ], + "score": 1.0, + "content": "a large margin.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 676, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 90, + 492, + 219 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 90, + 492, + 219 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 90, + 492, + 219 + ], + "spans": [ + { + "bbox": [ + 110, + 90, + 492, + 219 + ], + "score": 0.964, + "type": "image", + "image_path": "95c8e3d138185e0fef7249995e192bbfa47adcdf3cf05b143734d134d5846b3a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 90, + 492, + 133.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 133.0, + 492, + 176.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 176.0, + 492, + 219.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 212, + 226, + 399, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 209, + 225, + 401, + 241 + ], + "spans": [ + { + "bbox": [ + 209, + 225, + 401, + 241 + ], + "score": 1.0, + "content": "Figure 11: The results of GP-WGAN+dropout.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "Finally, we plot the convergence curves of the discriminators’ negative cost function learned by GP-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 190, + 286 + ], + "score": 1.0, + "content": "WGAN, GP-WGAN", + "type": "text" + }, + { + "bbox": [ + 190, + 274, + 197, + 282 + ], + "score": 0.34, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 272, + 506, + 286 + ], + "score": 1.0, + "content": "Dropout, and our CT-GAN in Figure 12. 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Algorithm 1 our proposed CT-GAN for training a generative neural net. Most of our experiments are conducted with the default values X1 = 10,\\2 = 2,N = 5, γ = 0.0002,b = 64,iter = 106
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MethodTest error (%)
Ladder (Rasmus et al., 2015)1.06±0.37
VAT (Miyato et al., 2017)1.36
CatGAN (Springenberg,2015)1.39±0.28
Improved GAN (Salimans et al., 2016)0.93 ± 0.065
Triple GAN (Li et al., 2017a)0.91 ± 0.58
Our CT-GAN0.89 ± 0.13
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MethodSupervised ISUnsupervised ISAccuracy(%)
SteinGANs (Wang&Liu,2016)6.35
DCGANs (Radford et al., 2015)6.586.16 ± 0.07
Improved GANs (Salimans et al., 2016)8.09 ± 0.071
AC-GANs (Odena et al., 2016)8.25 ± 0.07
GP-WGAN (Gulrajani et al., 2017)8.42 ± 0.107.86± 0.0791.85
SGANs (Huang et al., 2016)8.59 ±0.12
ALI (Warde-Farley & Bengio,2016)15.34 ± 0.05
BEGAN (Berthelot et al., 2017)一5.62
EGAN-Ent-VI (Dai et al., 2017)7.07 ± 0.10
DFM(Warde-Farley & Bengio,2016)17.72 ± 0.071
Our CT-GAN8.81±0.138.12±0.1295.91
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MethodTest error(%)
Ladder (Rasmus et al., 2015)20.40±0.47
VAT (Miyato et al., 2017)10.55
TE (Laine & Aila,2016)12.16 ±0.24
Teacher-Student (Tarvainen & Valpola,2017)12.31 ± 0.28
CatGANs (Springenberg,2015)19.58 ± 0.58
Improved GANs (Salimans et al., 2016)18.63 ± 2.32
ALI (Dumoulin et al., 2016)17.99 ±1.62
CLS-GAN (Qi, 2017)17.30 ± 0.50
Triple GAN (Li et al., 2017a)16.99 ± 0.36
Improved semi-GAN (Kumar et al., 2017)16.78 ± 1.80
Our CT-GAN9.98± 0.21
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Classifier CGenerator G
Input: Labels y, 32*32*3 Colored Image x,Input:Noise 50 z
0.2 Dropout 3*3 conv.128,Pad=1, Stride =1,lReLU,Weight norm 3*3 conv. 128, Pad =1, Stride =1, lReLU,Weight normMLP 8192, ReLU, Batch norm Reshape 512*4*4 5*5 deconv. 256*8*8,
3*3 conv. 128,Pad =1, Stride =2, lReLU, Weight norm 0.5 Dropout 3*3 conv. 256,Pad =1, Stride =1, lReLU, Weight norm 3*3 conv. 256,Pad=1, Stride =1, lReLU, Weight norm 3*3 conv. 256,Pad =1, Stride =2, lReLU, Weight normReLU,Batch norm 5*5 deconv. 128*16*16, ReLU,Batch norm
0.5 Dropout 3*3 conv. 512, Pad =O, Stride =1, lReLU, Weight norm 3*3 conv. 256,Pad =O, Stride =1,lReLU, Weight norm 3*3 conv. 128,Pad =O, Stride =1, lReLU, Weight norm5*5 deconv. 3*32*32,
Global pool MLP 10, Weight norm, SoftmaxTanh,Weight norm
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Classifier CGenerator G
Input: Labels y,28*28 Images x,Input: Noise 100 z
Gaussiannoise0.3,MLP1000,ReLU Gaussian noise 0.5,MLP 500,ReLUMLP 500, Softplus,Batch norm
Gaussian noise O.5,MLP 250,ReLUMLP 500, Softplus,Batch norm
Gaussian noise 0.5,MLP 250,ReLUMLP 784, Sigmoid,Weight norm
Gaussian noise 0.5,MLP 250,ReLU
Gaussian noise O.5,MLP 10, Softmax
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DiscriminatorGenerator
Input: 3*32*32 Image xInput: Noise bmz 128
[3*3]*2 Residual Block, Resample = DOWN 128*16*16MLP2048 Reshape 128*4*4
[3*3]*2 Residual Block, Resample = DOWN 128*8*8 0.2 Dropout[3*3]*2 Residual Block, Resample = UP
[3*3]*2 Residual Block, Resample = None 128*8*8 0.5 Dropout128*8*8 [3*3]*2 Residual Block, Resample = UP
[3*3]*2 Residual Block, Resample = None128*16*16 [3*3]*2 Residual Block, Resample = UP
128*8*8 0.5 Dropout128*32*32
ReLU, Global mean pool (D-) MLP 1 (D)3*3 conv. 3*32*32
Tanh
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DiscriminatorGenerator
Input: 3*32*32 Image x,Input:Noise z 128
5*5 conv. 128, Pad = same, Stride =2, IReLU 0.5 DropoutMLP 8192, ReLU, Batch norm Reshape 512*4*4
5*5 conv. 256, Pad = same, Stride = 2, IReLU 0.5 Dropout5*5 deconv. 256*8*8 ReLU,Bach norm
5*5 conv. 512, Pad = same Stride= 2, IReLU5*5 deconv. 128*16*16
0.5 Dropout Reshape 512*4*4 (D-)ReLU, Batch norm 5*5 deconv. 3*32*32
MLP 1 (D)Tanh
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DiscriminatorGenerator
Input: 1*28*28 Image xInput:Noise z 128
5*5 conv. 64,Pad = same, Stride= 2, IReLU 0.5 DropoutMLP4096,ReLU Reshape 256*4*4
5*5 conv. 128,Pad = same, Stride= 2,lReLU 0.5 Dropout5*5 deconv. 128*8*8 ReLU, Cut 128*7*7
5*5 conv. 256, Pad = same, Stride = 2, IReLU 0.5 Dropout5*5 deconv. 64*14*14
Reshape 256*4*4 (D-)ReLU 5*5 deconv. 1*28*28
MLP 1 (D)Sigmoid
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w/o GAN11.98±0.32
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Structure LevelDimensionRepresentationExamples
ProteinsSmall Molecules
primary1Dlinear sequenceKVKALPDACC(C)CC(C)NO
secondary2Dchemical bond graph00001 O00-01 。 0 Q Oo00 oQ00Q obo O ○ o O80 ○
tertiary3Datomistic geometry)
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Name (Task Code)SchematicObjectiveSource
Small Molecule Properties (SMP)SMPropertiesQM9 (Ruddigkeit et al., 2012)
Protein Interface Prediction (PIP)P2 P1 G 与Amino Acid InteractionDIPS(Townshend et al.,2019) DB5 (Vreven et al., 2015)
Residue Identity (RES)PAmino Acid IdentityNew, created from PDB (Berman et al., 2000)
Mutation Stability Prediction (MSP)P1 白 P2 C vs. P1 P2 £ 白Effect of MutationNew, created from SKEMPI (Jankauskaité et al., 2019)
Ligand Binding Affinity (LBA)P SMBinding StrengthPDBBind (Wang et al., 2004)
Ligand Efficacy Prediction (LEP) Protein StructureP SM VS. SMDrug EfficacyNew, created from PDB (Berman et al., 2000)
PRankingCASP-QA (Kryshtafovych et al.,2019)
Ranking (PSR) RNA Structure
Ranking (RSR)RRankingFARFAR2-Puzzles (Watkins & Das,2019)
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TaskMetric3DNon-3D
3DCNNGNNENNOztirk et al. (2018)Karimi et al. (2019)
LBARMSD1.5201.936*1.4291.5651.893
glob.Rp0.5580.581*0.5410.5730.415
glob.Rs0.5560.647*0.5320.5740.426
LEPAUROC0.8240.6780.5690.6961
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TaskTarget3DNon-3D
3DCNNGNNENNTsubaki et al. (2019)Liu et al. (2019)
SMPμ[D]0.5720.0680.0460.4960.520
gap[eV]0.5890.0910.0650.1540.184
Ut[ev]1.6150.0700.0230.1820.218
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TaskMetric3DNon-3D
3DCNNGNNENNSanchez-Garcia et al. (2018)
PIPAUROC0.844*0.669一0.841
Rao et al. (2019)
RESaccuracy0.4510.082*0.072*0.30
MSPAUROC0.5200.6370.6780.554
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TaskMetric3D
3DCNNGNNSotA
PSRmean Rs0.1770.3270.432 (Pages et al., 2019)
glob. Rs0.8370.7160.796 (Pages et al., 2019)
RSRmean Rs0.4140.1950.173 (Alford et al.,2017)
glob. Rs0.6560.3090.304 (Alford et al., 2017)
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--- /dev/null +++ b/parse/train/rJxRmlStDB/images/f58d5815286747aef2747f7fbe122ea943cb2beaa19b0bebe0d6740e570cfdd7.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1bb0a607ae3b683bb60b03ab19b31390518d3a1f187570b1a2468b1d20c365e4 +size 46420 diff --git a/parse/train/ryiAv2xAZ/ryiAv2xAZ.md b/parse/train/ryiAv2xAZ/ryiAv2xAZ.md new file mode 100644 index 0000000000000000000000000000000000000000..14507a7a60c13c970ec567ed0269ee699714ce19 --- /dev/null +++ b/parse/train/ryiAv2xAZ/ryiAv2xAZ.md @@ -0,0 +1,337 @@ +# TRAINING CONFIDENCE-CALIBRATED CLASSIFIERS FOR DETECTING OUT-OF-DISTRIBUTION SAMPLES + +Kimin Lee∗ Honglak Lee§ ,† Kibok Lee† Jinwoo Shin∗ ∗Korea Advanced Institute of Science and Technology, Daejeon, Korea †University of Michigan, Ann Arbor, MI 48109 §Google Brain, Mountain View, CA 94043 + +# ABSTRACT + +The problem of detecting whether a test sample is from in-distribution (i.e., training distribution by a classifier) or out-of-distribution sufficiently different from it arises in many real-world machine learning applications. However, the state-of-art deep neural networks are known to be highly overconfident in their predictions, i.e., do not distinguish in- and out-of-distributions. Recently, to handle this issue, several threshold-based detectors have been proposed given pre-trained neural classifiers. However, the performance of prior works highly depends on how to train the classifiers since they only focus on improving inference procedures. In this paper, we develop a novel training method for classifiers so that such inference algorithms can work better. In particular, we suggest two additional terms added to the original loss (e.g., cross entropy). The first one forces samples from out-of-distribution less confident by the classifier and the second one is for (implicitly) generating most effective training samples for the first one. In essence, our method jointly trains both classification and generative neural networks for out-of-distribution. We demonstrate its effectiveness using deep convolutional neural networks on various popular image datasets. + +# 1 INTRODUCTION + +Deep neural networks (DNNs) have demonstrated state-of-the-art performance on many classification tasks, e.g., speech recognition (Hannun et al., 2014), image classification (Girshick, 2015), video prediction (Villegas et al., 2017) and medical diagnosis (Caruana et al., 2015). Even though DNNs achieve high accuracy, it has been addressed (Lakshminarayanan et al., 2017; Guo et al., 2017) that they are typically overconfident in their predictions. For example, DNNs trained to classify MNIST images often produce high confident probability $9 1 \%$ even for random noise (see the work of (Hendrycks & Gimpel, 2016)). Since evaluating the quality of their predictive uncertainty is hard, deploying them in real-world systems raises serious concerns in AI Safety (Amodei et al., 2016), e.g., one can easily break a secure authentication system that can be unlocked by detecting the gaze and iris of eyes using DNNs (Shrivastava et al., 2017). + +The overconfidence issue of DNNs is highly related to the problem of detecting out-of-distribution: detect whether a test sample is from in-distribution (i.e., training distribution by a classifier) or outof-distribution sufficiently different from it. Formally, it can be formulated as a binary classification problem. Let an input $\mathbf { x } \in \mathcal { X }$ and a label $y \in { \mathcal { Y } } = \mathbf { \bar { \rho } } \qquad $ be random variables that follow a joint data distribution $P _ { \mathrm { i n } } \left( \mathbf { x } , y \right) = P _ { \mathrm { i n } } \left( y | \mathbf { x } \right) P _ { \mathrm { i n } } \left( \mathbf { x } \right)$ . We assume that a classifier $P _ { \boldsymbol { \theta } } \left( y | \mathbf { x } \right)$ is trained on a dataset drawn from $P _ { \mathrm { i n } } \left( \mathbf { x } , y \right)$ , where $\theta$ denotes the model parameter. We let $P _ { \mathrm { o u t } }$ $\mathbf { \tau } ( \mathbf { x } )$ denote an out-of-distribution which is ‘far away’ from in-distribution $P _ { \mathrm { i n } } \left( \mathbf { x } \right)$ . Our problem of interest is determining if input $\mathbf { x }$ is from $P _ { \mathrm { i n } }$ or $P _ { \mathrm { o u t } }$ , possibly utilizing a well calibrated classifier $P _ { \boldsymbol { \theta } } \left( y | \mathbf { x } \right)$ . In other words, we aim to build a detector, $g ( \mathbf { x } ) : \mathcal { X } \{ 0 , 1 \}$ , which assigns label 1 if data is from in-distribution, and label 0 otherwise. + +There have been recent efforts toward developing efficient detection methods where they mostly have studied simple threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017) utilizing a pre-trained classifier. For each input $\mathbf { x }$ , it measures some confidence score $q ( \mathbf { x } )$ based on a pre-trained classifier, and compares the score to some threshold $\delta > 0$ . Then, the detector assigns label 1 if the confidence score $q ( \mathbf { x } )$ is above $\delta$ , and label 0, otherwise. Specifically, (Hendrycks & Gimpel, 2016) defined the confidence score as a maximum value of the predictive distribution, and (Liang et al., 2017) further improved the performance by using temperature scaling (Guo et al., 2017) and adding small controlled perturbations to the input data. Although such inference methods are computationally simple, their performances highly depend on the pre-trained classifier. Namely, they fail to work if the classifier does not separate the maximum value of predictive distribution well enough with respect to $P _ { \mathrm { i n } }$ and $P _ { \mathrm { o u t } }$ . Ideally, a classifier should be trained to separate all class-dependent in-distributions as well as out-of-distribution in the output space. As another line of research, Bayesian probabilistic models (Li & Gal, 2017; Louizos & Welling, 2017) and ensembles of classifiers (Lakshminarayanan et al., 2017) were also investigated. However, training or inferring those models are computationally more expensive. This motivates our approach of developing a new training method for the more plausible simple classifiers. Our direction is orthogonal to the Bayesian and ensemble approaches, where one can also combine them for even better performance. + +Contribution. In this paper, we develop such a training method for detecting out-of-distribution $P _ { \mathrm { o u t } }$ better without losing its original classification accuracy. First, we consider a new loss function, called confidence loss. Our key idea on the proposed loss is to additionally minimize the KullbackLeibler (KL) divergence from the predictive distribution on out-of-distribution samples to the uniform one in order to give less confident predictions on them. Then, in- and out-of-distributions are expected to be more separable. However, optimizing the confidence loss requires training samples from out-of-distribution, which are often hard to sample: a priori knowledge on out-of-distribution is not available or its underlying space is too huge to cover. To handle the issue, we consider a new generative adversarial network (GAN) (Goodfellow et al., 2014) for generating most effective samples from $P _ { \mathrm { o u t } }$ . Unlike the original GAN, the proposed GAN generates ‘boundary’ samples in the low-density area of $P _ { \mathrm { i n } }$ . Finally, we design a joint training scheme minimizing the classifier’s loss and new GAN loss alternatively, i.e., the confident classifier improves the GAN, and vice versa, as training proceeds. Here, we emphasize that the proposed GAN does not need to generate explicit samples under our scheme, and instead it implicitly encourages training a more confident classifier. + +We demonstrate the effectiveness of the proposed method using deep convolutional neural networks such as AlexNet (Krizhevsky, 2014) and VGGNet (Szegedy et al., 2015) for image classification tasks on CIFAR (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), ImageNet (Deng et al., 2009), and LSUN (Yu et al., 2015) datasets. The classifier trained by our proposed method drastically improves the detection performance of all threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017) in all experiments. In particular, VGGNet with 13 layers trained by our method improves the true negative rate (TNR), i.e., the fraction of detected out-of-distribution (LSUN) samples, compared to the baseline: $1 4 . 0 \% 3 9 . 1 \%$ and $4 6 . 3 \% 9 8 . 9 \%$ on CIFAR-10 and SVHN, respectively, when $9 5 \%$ of in-distribution samples are correctly detected. We also provide visual understandings on the proposed method using the image datasets. We believe that our method can be a strong guideline when other researchers will pursue these tasks in the future. + +# 2 TRAINING CONFIDENT NEURAL CLASSIFIERS + +In this section, we propose a novel training method for classifiers in order to improve the performance of prior threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017) (see Appendix A for more details). Our motivation is that such inference algorithms can work better if the classifiers are trained so that they map the samples from in- and out-of-distributions into the output space separately. Namely, we primarily focus on training an improved classifier, and then use prior detectors under the trained model to measure its performance. + +# 2.1 CONFIDENT CLASSIFIER FOR OUT-OF-DISTRIBUTION + +Without loss of generality, suppose that the cross entropy loss is used for training. Then, we propose the following new loss function, termed confidence loss: + +$$ +\operatorname* { m i n } _ { \theta } \mathbb { E } _ { P _ { \mathrm { i n } } ( \widehat { \mathbf { x } } , \widehat { y } ) } \big [ - \log P _ { \theta } \left( y = \widehat { y } | \widehat { \mathbf { x } } \right) \big ] + \beta \mathbb { E } _ { P _ { \mathrm { o u t } } ( \mathbf { x } ) } \big [ K L \left( \mathcal { U } \left( y \right) \mid \mid P _ { \theta } \left( y | \mathbf { x } \right) \right) \big ] , +$$ + +where $K L$ denotes the Kullback-Leibler (KL) divergence, $\mathcal { U } \left( y \right)$ is the uniform distribution and $\beta > 0$ is a penalty parameter. It is highly intuitive as the new loss forces the predictive distribution on out-of-distribution samples to be closer to the uniform one, i.e., zero confidence, while that for samples from in-distribution still follows the label-dependent probability. In other words, the proposed loss is designed for assigning higher maximum prediction values, i.e., $\operatorname* { m a x } _ { y } P _ { \theta } \left( y | \mathbf { x } \right)$ , to in-distribution samples than out-of-distribution ones. Here, a caveat is that adding the KL divergence term might degrade the classification performance. However, we found that it is not the case due to the high expressive power of deep neural networks, while in- and out-of-distributions become more separable with respect to the maximum prediction value by optimizing the confidence loss (see Section 3.1 for supporting experimental results). + +![](images/30ab0a6c3ac8a37d7cbd9bd3d2b7465e59b564853612866bbddd064ad62f4900.jpg) +Figure 1: Illustrating the behavior of classifier under different out-of-distribution training datasets. We generate the out-of-distribution samples from (a) 2D box $[ - 5 0 , 5 0 ] ^ { 2 }$ , and show (b) the corresponding decision boundary of classifier. We also generate the out-of-distribution samples from (c) 2D box $[ - 2 0 , 2 0 ] ^ { 2 }$ , and show (d) the corresponding decision boundary of classifier. + +We remark that minimizing a similar KL loss was studied recently for different purposes (Lee et al., 2017; Pereyra et al., 2017). Training samples for minimizing the KL divergence term is explicitly given in their settings while we might not. Ideally, one has to sample all (almost infinite) types of outof-distribution to minimize the KL term in (1), or require some prior information on testing out-ofdistribution for efficient sampling. However, this is often infeasible and fragile. To address the issue, we suggest to sample out-of-distribution close to in-distribution, which could be more effective in improving the detection performance, without any assumption on testing out-of-distribution. + +In order to explain our intuition in details, we consider a binary classification task on a simple example, where each class data is drawn from a Gaussian distribution and entire data space is bounded by 2D box $[ - 5 0 , 5 0 ] ^ { 2 }$ for visualization. We apply the confidence loss to simple fully-connected neural networks (2 hidden layers and 500 hidden units for each layer) using different types of outof-distribution training samples. First, as shown in Figure 1(a), we construct an out-of-distribution training dataset of 100 (green) points using rejection sampling on the entire data space $[ - 5 0 , 5 0 ] ^ { 2 }$ . Figure 1(b) shows the decision boundary of classifier optimizing the confidence loss on the corresponding dataset. One can observe that a classifier still shows overconfident predictions (red and blue regions) near the labeled in-distribution region. On the other hand, if we construct a training out-of-distribution dataset of 100 points from $[ - 2 0 , 2 0 ] ^ { 2 }$ , i.e., closer to target, in-distribution space (see Figure 1(c)), a classifier produces confident predictions only on the labeled region and zero confidence on the remaining in the entire data space $[ - 5 0 , 5 0 ] ^ { 2 }$ as shown in Figure 1(d). If one increases the number of training out-of-distribution samples which are generated from the entire space, i.e., $[ - 5 0 , 5 0 ] ^ { 2 }$ , Figure 1(b) is expected to be similar to Figure 1(d). In other words, one need more samples in order to train a confident classifier if samples are generated from the entire space. However, this might be impossible and not efficient since the number of out-of-distribution training samples might be almost infinite to cover its entire, huge actual data space. This implies that training out-of-distribution samples nearby the in-distribution region could be more effective in improving the detection performance. Our underlying intuition is that the effect of boundary of in-distribution region might propagate to the entire out-of-distribution space. Our experimental results in Section 3.1 also support this: realistic images are more useful as training out-of-distribution than synthetic datasets (e.g., Gaussian noise) for improving the detection performance when we consider an image classification task. This motivates us to develop a new generative adversarial network (GAN) for generating such effective out-of-distribution samples. + +# 2.2 ADVERSARIAL GENERATOR FOR OUT-OF-DISTRIBUTION + +In this section, we introduce a new training method for learning a generator of out-of-distribution inspired by generative adversarial network (GAN) (Goodfellow et al., 2014). We will first assume that the classifier for in-distribution is fixed, and also describe the joint learning framework in the next section. + +The GAN framework consists of two main components: discriminator $D$ and generator $G$ . The generator maps a latent variable $\mathbf { z }$ from a prior distribution $P _ { \mathrm { p r i } } \left( \mathbf { z } \right)$ to generated outputs $G \left( \mathbf { z } \right)$ , and discriminator $D : \mathcal { X } [ 0 , 1 ]$ represents a probability that sample $\mathbf { x }$ is from a target distribution. Suppose that we want to recover the in-distribution $P _ { \mathrm { i n } } ( x )$ using the generator $G$ . Then, one can optimize the following min-max objective for forcing $P _ { G } \approx P _ { \mathrm { i n } }$ : + +$$ +\operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } ~ \mathbb { E } _ { P _ { \mathrm { i n } } ( \mathbf { x } ) } \big [ \log D \left( \mathbf { x } \right) \big ] + \mathbb { E } _ { P _ { \mathrm { p r i } } ( \mathbf { z } ) } \big [ \log \left( 1 - D \left( G \left( \mathbf { z } \right) \right) \right) \big ] . +$$ + +However, unlike the original GAN, we want to make the generator recover an effective out-ofdistribution $P _ { \mathrm { o u t } }$ instead of $P _ { \mathrm { i n } }$ . To this end, we propose the following new GAN loss: + +$$ +\begin{array} { r l } { \underset { G } { \mathop { \operatorname* { m i n } } } \underset { D } { \mathop { \operatorname* { m a x } } } } & { \beta \underbrace { { \mathbb { E } } _ { P _ { G } ( \mathbf { x } ) } \left[ K L \left( \mathcal { U } \left( y \right) \mid P _ { \theta } \left( y | \mathbf { x } \right) \right) \right] } _ { \mathrm { ( a ) } } } \\ & { + \underbrace { { \mathbb { E } } _ { P _ { \mathrm { i n } } \left( \mathbf { x } \right) } \left[ \log D \left( \mathbf { x } \right) \right] + { \mathbb { E } } _ { P _ { G } ( \mathbf { x } ) } \left[ \log \left( 1 - D \left( \mathbf { x } \right) \right) \right] } _ { \mathrm { ( b ) } } , } \end{array} +$$ + +where $\theta$ is the model parameter of a classifier trained on in-distribution. The above objective can be interpreted as follows: the first term (a) corresponds to a replacement of the out-of-distribution $P _ { \mathrm { o u t } }$ in (1)’s KL loss with the generator distribution $P _ { G }$ . One can note that this forces the generator to generate low-density samples since it can be interpreted as minimizing the log negative likelihood of in-distribution using the classifier, i.e., $P _ { \mathrm { i n } } ( \mathbf { x } ) \approx \exp \left( K L \left( \mathcal { U } \left( y \right) \parallel P _ { \theta } \left( y | \mathbf { x } \right) \right) \right)$ . We remark that this approximation is also closely related to the inception score (Salimans et al., 2016) which is popularly used as a quantitative measure of visual fidelity of the samples. The second term (b) corresponds to the original GAN loss since we would like to have out-of-distribution samples close to in-distribution, as mentioned in Section 2.1. Suppose that the model parameter of classifier $\theta$ is set appropriately such that the classifier produces the uniform distribution for out of distribution samples. Then, the KL divergence term (a) in (3) is approximately 0 no matter what out-of-distribution samples are generated. However, if the samples are far away from boundary, the GAN loss (b) in (3) should be high, i.e., the GAN loss forces having samples being not too far from the in-distribution space. Therefore, one can expect that proposed loss can encourage the generator to produce the samples which are on the low-density boundary of the in-distribution space. We also provide its experimental evidences in Section 3.2. + +We also remark that (Dai et al., 2017) consider a similar GAN generating samples from out-ofdistribution for the purpose of semi-supervised learning. The authors assume the existence of a pretrained density estimation model such as $\mathrm { P i x e l C N N + + }$ (Salimans et al., 2017) for in-distribution, but such a model might not exist and be expensive to train in general. Instead, we use much simpler confident classifiers for approximating the density. Hence, under our fully-supervised setting, our GAN is much easier to train and more suitable. + +# 2.3 JOINT TRAINING METHOD OF CONFIDENT CLASSIFIER AND ADVERSARIAL GENERATOR + +In the previous section, we suggest training the proposed GAN using a pre-trained confident classifier. We remind that the converse is also possible, i.e., the motivation of having such a GAN is for training a better classifier. Hence, two models can be used for improving each other. This naturally suggests a joint training scheme where the confident classifier improves the proposed GAN, and vice versa, as training proceeds. Specifically, we suggest the following joint objective function: + +$$ +\begin{array} { r l } { \underset { G } { \mathop { \operatorname* { m i n } } } \underset { D } { \mathop { \operatorname* { m a x } } } \underset { \theta } { \mathop { \operatorname* { m i n } } } } & { \underbrace { { \mathbb { E } } _ { P _ { \mathrm { i n } } ( \widehat { \mathbf { x } } , \widehat { y } ) } { [ - \log P _ { \theta } ( y = \widehat { y } | \widehat { \mathbf { x } } ) ] } } _ { \mathrm { ( c ) } } + \beta \underbrace { { \mathbb { E } } _ { P _ { G } ( \mathbf { x } ) } { [ K L ( \mathcal { U } ( y ) \mid | P _ { \theta } ( y | \mathbf { x } ) ) ] } } _ { \mathrm { ( d ) } } } \\ & { \underbrace { + { \mathbb { E } } _ { P _ { \mathrm { i n } } ( \widehat { \mathbf { x } } ) } { [ \log D ( \widehat { \mathbf { x } } ) ] } + { \mathbb { E } } _ { P _ { G } ( \mathbf { x } ) } { [ \log ( 1 - D ( \mathbf { x } ) ) ] } . } _ { } } \end{array} +$$ + +The classifier’s confidence loss corresponds to ( ${ \vartriangle } \cosh + ( \mathrm { d } )$ , and the proposed GAN loss corresponds to $\mathrm { ( d ) + }$ (e), i.e., they share the KL divergence term (d) under joint training. To optimize the above objective efficiently, we propose an alternating algorithm, which optimizes model parameters $\{ \theta \}$ of classifier and GAN models $\{ G , D \}$ alternatively as shown in Algorithm 1. Since the algorithm monotonically decreases the objective function, it is guaranteed to converge. + +Algorithm 1 Alternating minimization for detecting and generating out-of-distribution. + +# repeat + +/∗ Update proposed $\mathrm { G A N * } /$ + +Sample $\left\{ \mathbf { z } _ { 1 } , \hdots , \mathbf { z } _ { M } \right\}$ and $\left\{ \mathbf { x } _ { 1 } , \hdots , \mathbf { x } _ { M } \right\}$ from prior $P _ { \mathrm { p r i } } \left( \mathbf { z } \right)$ and and in-distribution $P _ { \mathrm { i n } } \left( \mathbf { x } \right)$ respectively, and update the discriminator $D$ by ascending its stochastic gradient of + +$$ +\frac { 1 } { M } \sum _ { i = 1 } ^ { M } \left[ \log D \left( \mathbf { x } _ { i } \right) + \log \left( 1 - D \left( G \left( \mathbf { z } _ { i } \right) \right) \right) \right] . +$$ + +Sample $\left\{ \mathbf { z } _ { 1 } , \hdots , \mathbf { z } _ { M } \right\}$ from prior $P _ { \mathrm { p r i } } \left( \mathbf { z } \right)$ , and update the generator $G$ by descending its stochastic gradient of + +$$ +\frac { 1 } { M } \sum _ { i = 1 } ^ { M } \Big [ \log \big ( 1 - D \left( G \left( \mathbf { z } _ { i } \right) \right) \big ) \Big ] + \frac { \beta } { M } \sum _ { i = 1 } ^ { M } \Big [ K L \left( \mathcal { U } \left( y \right) \| P _ { \theta } \left( y | G \left( \mathbf { z } _ { i } \right) \right) \right) \Big ] . +$$ + +$/ *$ Update confident classifier $^ { * / }$ + +Sample $\left\{ \mathbf { z } _ { 1 } , \hdots , \mathbf { z } _ { M } \right\}$ and $\left\{ \left( \mathbf { x } _ { 1 } , y _ { 1 } \right) , \ldots , \left( \mathbf { x } _ { M } , y _ { M } \right) \right\}$ from prior $P _ { \mathrm { p r i } } \left( \mathbf { z } \right)$ and in-distribution $P _ { \mathrm { i n } } \left( \mathbf { x } , y \right)$ , respectively, and update the classifier $\theta$ by descending its stochastic gradient of + +$$ +\frac { 1 } { M } \sum _ { i = 1 } ^ { M } \Big [ - \log P _ { \theta } \left( y = y _ { i } | \mathbf { x } _ { i } \right) + \beta K L \left( \mathcal { U } \left( y \right) \parallel P _ { \theta } \left( y | G \left( \mathbf { z } _ { i } \right) \right) \right) \Big ] . +$$ + +until convergence + +# 3 EXPERIMENTAL RESULTS + +We demonstrate the effectiveness of our proposed method using various datasets: CIFAR (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), ImageNet (Deng et al., 2009), LSUN (Yu et al., 2015) and synthetic (Gaussian) noise distribution. We train convolutional neural networks (CNNs) including VGGNet (Szegedy et al., 2015) and AlexNet (Krizhevsky, 2014) for classifying CIFAR-10 and SVHN datasets. The corresponding test dataset is used as the in-distribution (positive) samples to measure the performance. We use realistic images and synthetic noises as the out-of-distribution (negative) samples. For evaluation, we measure the following metrics using the threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017): the true negative rate (TNR) at $9 5 \%$ true positive rate (TPR), the area under the receiver operating characteristic curve (AUROC), the area under the precision-recall curve (AUPR), and the detection accuracy, where larger values of all metrics indicate better detection performances. Due to the space limitation, more explanations about datasets, metrics and network architectures are given in Appendix B.1 + +
In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
Cross entropy loss /Confidence loss
SVHNCIFAR-10 (seen) TinyImageNet (unseen)93.82 /94.2347.4 /99.962.6 /99.978.6 /99.971.6 /99.991.2/99.4
49.0 /100.064.6 /100.079.6 /100.072.7 /100.091.6 /99.4
LSUN (unseen) Gaussian (unseen)46.3/100.0 56.1/100.061.8/100.0 72.0 /100.078.2 /100.0 83.4/100.071.1/100.0 77.2/100.090.8 /99.4 92.8 /99.4
CIFAR-10SVHN (seen)80.14 /80.56
13.7 / 99.8 13.6 /9.946.6 /99.966.6 /99.8 62.6 /58.661.4 /99.9 58.3 /55.373.5 /99.8 71.0 / 66.1
TinyImageNet (unseen)39.6 /31.8
LSUN (unseen) Gaussian (unseen)14.0 / 10.540.7 /34.863.2 / 60.258.7 /56.471.5 /68.0
2.8 / 3.310.2 /14.150.0/50.048.1 /49.439.9 /47.0
+ +Table 1: Performance of the baseline detector (Hendrycks & Gimpel, 2016) using VGGNet. All values are percentages and boldface values indicate relative the better results. For each in-distribution, we minimize the KL divergence term in (1) using training samples from an out-of-distribution dataset denoted by “seen”, where other “unseen” out-of-distributions were only used for testing. + +![](images/82eec49fdb49b288decb05d89feef7881d949c90c40d7146f2041eba0c997d44.jpg) +Figure 2: For all experiments in (a), (b) and (c), we commonly use the SVHN dataset for indistribution. Fraction of the maximum prediction value in softmax scores trained by (a) cross entropy loss and (b) confidence loss: the $\mathbf { X }$ -axis and y-axis represent the maximum prediction value and the fraction of images receiving the corresponding score, respectively. The receiver operating characteristic (ROC) curves under different losses are reported in (c): the red curve corresponds to the ROC curve of a model trained by optimizing the naive cross entropy loss, whereas other ones correspond to the ROC curves of models trained by optimizing the confidence loss. The KL divergence term in the confidence loss is optimized using explicit out-of-distribution datasets indicated in the parentheses, e.g., Confident loss (LSUN) means that we use the LSUN dataset for optimizing the KL divergence term. + +# 3.1 EFFECTS OF CONFIDENCE LOSS + +We first verify the effect of confidence loss in (1) trained by some explicit, say seen, out-ofdistribution datasets. First, we compare the quality of confidence level by applying various training losses. Specifically, the softmax classifier is used and simple CNNs (two convolutional layers followed by three fully-connected layers) are trained by minimizing the standard cross entropy loss on SVHN dataset. We also apply the confidence loss to the models by additionally optimizing the KL divergence term using CIFAR-10 dataset (as training out-of-distribution). In Figure 2(a) and 2(b), we report distributions of the maximum prediction value in softmax scores to evaluate the separation quality between in-distribution (i.e., SVHN) and out-of-distributions. It is clear that there exists a better separation between the SVHN test set (red bar) and other ones when the model is trained by the confidence loss. Here, we emphasize that the maximum prediction value is also low on even untrained (unseen) out-of-distributions, e.g., TinyImageNet, LSUN and synthetic datasets. Therefore, it is expected that one can distinguish in- and out-of-distributions more easily when a classifier is trained by optimizing the confidence loss. To verify that, we obtain the ROC curve using the baseline detector (Hendrycks & Gimpel, 2016) that computes the maximum value of predictive distribution on a test sample and classifies it as positive (i.e., in-distribution) if the confidence score is above some threshold. Figure 2(c) shows the ROC curves when we optimize the KL divergence term on various datasets. One can observe that realistic images such as TinyImageNet (aqua line) and LSUN (green line) are more useful than synthetic datasets (orange line) for improving the detection performance. This supports our intuition that out-of-distribution samples close to in-distribution could be more effective in improving the detection performance as we discussed in Section 2.1. + +We indeed evaluate the performance of the baseline detector for out-of-distribution using largescale CNNs, i.e., VGGNets with 13 layers, under various training scenarios, where more results on AlexNet and ODIN detector (Liang et al., 2017) can be found in Appendix C (the overall trends of results are similar). For optimizing the confidence loss in (1), SVHN and CIFAR-10 training datasets are used for optimizing the KL divergence term for the cases when the in-distribution is CIFAR-10 and SVHN, respectively. Table 1 shows the detection performance for each in- and out-of-distribution pair. When the in-distribution is SVHN, the classifier trained by our method drastically improves the detection performance across all out-of-distributions without hurting its original classification performance. However, when the in-distribution is CIFAR-10, the confidence loss does not improve the detection performance in overall, where we expect that this is because the trained/seen SVHN out-of-distribution does not effectively cover all tested out-of-distributions. Our joint confidence loss in (4), which was designed under the intuition, resolves the issue of the CIFAR-10 (in-distribution) classification case in Table 1 (see Figure 4(b)). + +![](images/e9f90921b82f9cee01a46da1fa1cfd34180f47111c99ee992c262ad48b241d9c.jpg) +Figure 3: The generated samples from original GAN (a)/(c) and proposed GAN (b)/(d). In (a)/(b), the grey area is the 2D histogram of training in-distribution samples drawn from a mixture of two Gaussian distributions and red points indicate generated samples by GANs. + +![](images/7c77a46a2cf9c6a821a4ce5593a77d3e597cc6179d1deaee26e3d473df7e41a5.jpg) +Figure 4: Performances of the baseline detector (Hendrycks & Gimpel, 2016) under various training losses. For training models by the confidence loss, the KL divergence term is optimized using samples indicated in the parentheses. For fair comparisons, we only plot the performances for unseen out-of-distributions, where those for seen out-of-distributions (used for minimizing the KL divergence term in (1)) can be found in Table 1. + +# 3.2 EFFECTS OF ADVERSARIAL GENERATOR AND JOINT CONFIDENCE LOSS + +In this section, we verify the effect of the proposed GAN in Section 2.2 and evaluate the detection performance of the joint confidence loss in (4). To verify that the proposed GAN can produce the samples nearby the low-density boundary of the in-distribution space, we first compare the generated samples by original GAN and proposed GAN on a simple example where the target distribution is a mixture of two Gaussian distributions. For both the generator and discriminator, we use fullyconnected neural networks with 2 hidden layers. For our method, we use a pre-trained classifier which minimizes the cross entropy on target distribution samples and the KL divergence on out-ofdistribution samples generated by rejection sampling on a bounded 2D box. As shown in Figure 3(a), the samples of original GAN cover the high-density area of the target distribution while those of proposed GAN does its boundary one (see Figure 3(b)). We also compare the generated samples of original and proposed GANs on MNIST dataset (LeCun et al., 1998), which consists of handwritten digits. For this experiment, we use deep convolutional GANs (DCGANs) (Radford et al., 2015). In this case, we use a pre-trained classifier which minimizes the cross entropy on MNIST training samples and the KL divergence on synthetic Gaussian noises. As shown in Figure 3(c) and 3(d), samples of original GAN looks more like digits than those of proposed GAN. Somewhat interestingly, the proposed GAN still generates some new digit-like images. + +![](images/d41db38747ca1efcc752cd0ff7ae1de2ed89e22acec41b351615403610c3ccf6.jpg) +Figure 5: Guided gradient (sensitivity) maps of the top-1 predicted class with respect to the input image under various training losses. + +We indeed evaluate the performance of our joint confidence loss in (4) utilizing the proposed GAN. To this end, we use VGGNets (as classifiers) and DCGANs (as GANs). We also test a variant of confidence loss which optimizes the KL divergence term on samples from a pre-trained original GAN (implicitly) modeling the in-distribution. One can expect that samples from the original GAN can be also useful for improving the detection performance since it may have bad generalization properties (Arora et al., 2017) and generate a few samples on the low-density boundary as like the proposed GAN. Figure 4 shows the performance of the baseline detector for each in- and out-of-distribution pair. First, observe that the joint confidence loss (blue bar) outperforms the confidence loss with some explicit out-of-distribution datasets (green bar). This is quite remarkable since the former is trained only using in-distribution datasets, while the latter utilizes additional out-of-distribution datasets. We also remark that our methods significantly outperform the baseline cross entropy loss (red bar) in all cases without harming its original classification performances (see Table 2 in Appendix C). Interestingly, the confidence loss with the original GAN (orange bar) is often (but not always) useful for improving the detection performance, whereas that with the proposed GAN (blue bar) still outperforms it in all cases. + +Finally, we also provide visual interpretations of models using the guided gradient maps (Springenberg et al., 2014). Here, the gradient can be interpreted as an importance value of each pixel which influences on the classification decision. As shown in Figure 5, the model trained by the cross entropy loss shows sharp gradient maps for both samples from in- and out-of-distributions, whereas models trained by the confidence losses do only on samples from in-distribution. For the case of SVHN in-distribution, all confidence losses gave almost zero gradients, which matches to the results in Figure 4(a): their detection performances are almost perfect. For the case of CIFAR10 distribution, one can now observe that there exists some connection between gradient maps and detection performances. This is intuitive because for detecting samples from out-of-distributions better, the classifier should look at more pixels as similar importance and the KL divergence term forces it. We think that our visualization results might give some ideas in future works for developing better inference methods for detecting out-of-distribution under our models. + +# 4 CONCLUSION + +In this paper, we aim to develop a training method for neural classification networks for detecting out-of-distribution better without losing its original classification accuracy. In essence, our method jointly trains two models for detecting and generating out-of-distribution by minimizing their losses alternatively. Although we primarily focus on image classification in our experiments, our method can be used for any classification tasks using deep neural networks. It is also interesting future directions applying our methods for other related tasks: regression (Malinin et al., 2017), network calibration (Guo et al., 2017), Bayesian probabilistic models (Li & Gal, 2017; Louizos & Welling, 2017), ensemble (Lakshminarayanan et al., 2017) and semi-supervised learning (Dai et al., 2017). + +# ACKNOWLEDGEMENTS + +This work was supported in part by the Institute for Information & communications Technology Promotion(IITP) grant funded by the Korea government(MSIT) (No.2017-0-01778, Development of Explainable Human-level Deep Machine Learning Inference Framework), the ICT R&D program of MSIP/IITP [R-20161130-004520, Research on Adaptive Machine Learning Technology Development for Intelligent Autonomous Digital Companion], DARPA Explainable AI (XAI) program #313498 and Sloan Research Fellowship. + +# REFERENCES + +Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mane. 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In International Conference on Learning Representations (ICLR), 2017. + +# A THRESHOLD-BASED DETECTORS + +In this section, we formally describe the detection procedure of threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017). For each data $\mathbf { x }$ , it measures some confidence score $q ( \mathbf { x } )$ by feeding the data into a pre-trained classifier. Here, (Hendrycks & Gimpel, 2016) defined the confidence score as a maximum value of the predictive distribution, and (Liang et al., 2017) further improved the performance by processing the predictive distribution (see Appendix C.3 for more details). Then, the detector, $g ( \mathbf { x } ) : \mathcal { X } \{ 0 , 1 \}$ , assigns label 1 if the confidence score $q ( \mathbf { x } )$ is above some threshold $\delta$ , and label 0, otherwise: + +$$ +g \left( \mathbf { x } \right) = \left\{ \begin{array} { l l } { \begin{array} { r l } { 1 } & { \mathrm { i f } q ( \mathbf { x } ) \geq \delta , } \\ { 0 } & { \mathrm { o t h e r w i s e } . } \end{array} } \end{array} \right. +$$ + +For this detector, we have to find a score threshold so that some positive examples are classified correctly, but this depends upon the trade-off between false negatives and false positives. To handle this issue, we use threshold-independent evaluation metrics such as area under the receiver operating characteristic curve (AUORC) and detection accuracy (see Appendix B). + +# B EXPERIMENTAL SETUPS IN SECTION 3 + +Datasets. We train deep models such as VGGNet (Szegedy et al., 2015) and AlexNet (Krizhevsky, 2014) for classifying CIFAR-10 and SVHN datasets: the former consists of 50,000 training and 10,000 test images with 10 image classes, and the latter consists of 73,257 training and 26,032 test images with 10 digits.2 The corresponding test dataset are used as the in-distribution (positive) samples to measure the performance. We use realistic images and synthetic noises as the out-ofdistribution (negative) samples: the TinyImageNet consists of 10,000 test images with 200 image classes from a subset of ImageNet images. The LSUN consists of 10,000 test images of 10 different scenes. We downsample each image of TinyImageNet and LSUN to size $3 2 \times 3 2$ . The Gaussian noise is independently and identically sampled from a Gaussian distribution with mean 0.5 and variance 1. We clip each pixel value into the range $[ 0 , 1 ]$ . + +Detailed CNN structure and training. The simple CNN that we use for evaluation shown in Figure 2 consists of two convolutional layers followed by three fully-connected layers. Convolutional layers have 128 and 256 filters, respectively. Each convolutional layer has a $5 \times 5$ receptive field applied with a stride of 1 pixel each followed by max pooling layer which pools $2 \times 2$ regions at strides of 2 pixels. AlexNet (Krizhevsky, 2014) consists of five convolutitonal layers followed by three fullyconnected layers. Convolutional layers have 64, 192, 384, 256 and 256 filters, respectively. First and second convolutional layers have a $5 \times 5$ receptive field applied with a stride of 1 pixel each followed by max pooling layer which pools $3 \times 3$ regions at strides of 2 pixels. Other convolutional layers have a $3 \times 3$ receptive field applied with a stride of 1 pixel followed by max pooling layer which pools $2 \times 2$ regions at strides of 2 pixels. Fully-connected layers have 2048, 1024 and 10 hidden units, respectively. Dropout (Hinton et al., 2012) was applied to only fully-connected layers of the network with the probability of retaining the unit being 0.5. All hidden units are ReLUs. Figure 6 shows the detailed structure of VGGNet (Szegedy et al., 2015) with three fully-connected layers and 10 convolutional layers. Each ConvReLU box in the figure indicates a $3 \times 3$ convolutional layer followed by ReLU activation. Also, all max pooling layers have $2 \times 2$ receptive fields with stride 2. Dropout was applied to only fully-connected layers of the network with the probability of retaining the unit being 0.5. For all experiments, the softmax classifier is used, and each model is trained by optimizing the objective function using Adam learning rule (Kingma & Ba, 2014). For each out-ofdistribution dataset, we randomly select 1,000 images for tuning the penalty parameter $\beta$ , mini-batch size and learning rate. The penalty parameter is chosen from $\beta \in \{ 0 , 0 . 1 , . . . 1 . 9 , 2 \}$ , the mini-batch size is chosen from $\{ 6 4 , 1 \bar { 2 } 8 \}$ and the learning rate is chosen from $\{ 0 . 0 0 1 , 0 . 0 0 0 5 , 0 . 0 0 0 2 \}$ . The optimal parameters are chosen to minimize the detection error on the validation set. We drop the learning rate by 0.1 at 60 epoch and models are trained for total 100 epochs. The best test result is reported for each method. + +Performance metrics. We measure the following metrics using threshold-based detectors: + +• True negative rate (TNR) at $95 \%$ true positive rate (TPR). Let TP, TN, FP, and FN denote true positive, true negative, false positive and false negative, respectively. We measure $\mathrm { T N R } = \mathrm { T N } / \left( \mathrm { F P + T N } \right)$ , when $\mathrm { T P R } = \mathrm { T P } .$ / $( \mathrm { T P + F N } )$ is $9 5 \%$ . + +• Area under the receiver operating characteristic curve (AUROC). The ROC curve is a graph plotting TPR against the false positive rate $= \mathrm { F P }$ / $\mathrm { \ F P { + } T N ) }$ by varying a threshold. + +• Area under the precision-recall curve (AUPR). The PR curve is a graph plotting the precision $=$ TP / $( \mathrm { T P + F P } )$ against recal $\mathrm { \Lambda } = \mathrm { T P } / \left( \mathrm { T P + F N } \right)$ by varying a threshold. AUPR-IN (or -OUT) is AUPR where in- (or out-of-) distribution samples are specified as positive. + +• Detection accuracy. This metric corresponds to the maximum classification probability over all possible thresholds $\delta$ : $1 \ \dot { - } \ \operatorname* { m i n } _ { \delta } \left\{ P _ { \mathrm { i n } } \left( q \left( \mathbf x \right) \leq \delta \right) P \left( \mathbf x \right) \right.$ is from $P _ { \mathrm { i n } }$ ) $^ +$ $P _ { \mathrm { o u t } }$ $\dot { \mathbf { \eta } } _ { q } \left( \mathbf { x } \right) > \delta ) \mathbf { \eta } _ { P }$ ( $\mathbf { x }$ is from $P _ { \mathrm { o u t } }$ ) $\}$ , where $q ( \mathbf { x } )$ is a confident score such as a maximum value of softmax. We assume that both positive and negative examples have equal probability of appearing in the test set, i.e., $P$ (x is from $P _ { \mathrm { i n } } ) \stackrel { - } { = } P$ ( $\mathbf { x }$ is from $P _ { \mathrm { o u t } } ) = \bar { 0 } . 5$ . + +Note that AUROC, AUPR and detection accuracy are threshold-independent evaluation metrics. + +![](images/cefa79cfd18994ae4d493f52482b99594b0c93029794a4fe28f238c72ccca960.jpg) +Figure 6: Detailed structure of VGGNet with 13 layers. + +Generating samples on a simple example. As shown in Figure 3(a) and Figure 3(b), we compare the generated samples by original GAN and proposed GAN on a simple example where the target distribution is a mixture of two Gaussian distributions. For both the generator and discriminator, we use fully-connected neural networks with 2 hidden layers and 500 hidden units for each layer. For all layers, we use ReLU activation function. We use a 100-dimensional Gaussian prior for the latent variable z. For our method, we pre-train the simple fully-connected neural networks (2 hidden layers and 500 ReLU units for each layer) by minimizing the cross entropy on target distribution samples and the KL divergence on out-of-distribution samples generated by rejection sampling on bounded 2D box $[ - 1 0 , 1 0 ] ^ { \frac { \ d } { 2 } }$ . The penalty parameter $\beta$ is set to 1. We use ADAM learning rule (Kingma & Ba, 2014) with a mini-batch size of 400. The initial learning rate is set to 0.002, and we train for total 100 epochs. + +Generating samples on MNIST. As shown in Figure 3(c) and Figure 3(d), we compare the generated samples of original and proposed GANs on MNIST dataset, which consists of greyscale images, each containing a digit 0 to 9 with 60,000 training and 10,000 test images. We expand each image to size $3 \times 3 2 \times 3 2$ . For both the generator and discriminator, we use deep convolutional GANs (DCGANs) (Radford et al., 2015). The discriminator and generator consist of four convolutional and deconvolutional layers, respectively. Convolutional layers have 128, 256, 512 and 1 filters, respectively. Each convolutional layer has a $4 \times 4$ receptive field applied with a stride of 2 pixel. The second and third convolutional layers are followed by batch normalization (Ioffe & Szegedy, 2015). For all layers, we use LeakyReLU activation function. Deconvolutional layers have 512, 256, 128 and 1 filters, respectively. Each deconvolutional layer has a $4 \times 4$ receptive field applied with a stride of 2 pixel followed by batch normalization (Ioffe & Szegedy, 2015) and ReLU activation. For our method, we use a pre-trained simple CNNs (two convolutional layers followed by three fully-connected layers) by minimizing the cross entropy on MNIST training samples and the KL divergence on synthetic Gaussian noise. Convolutional layers have 128 and 256 filters, respectively. Each convolutional layer has a $5 \times 5$ receptive field applied with a stride of 1 pixel each followed by max pooling layer which pools $2 \times 2$ regions at strides of 2 pixels. The penalty parameter $\beta$ is set to 1. We use ADAM learning rule (Kingma & Ba, 2014) with a mini-batch size of 128. The initial learning rate is set to 0.0002, and we train for total 50 epochs. + +# C MORE EXPERIMENTAL RESULTS + +# C.1 CLASSIFICATION PERFORMANCES + +Table 2 reports the classification accuracy of VGGNets on CIFAR-10 and SVHN datasets under various training losses shown in Figure 4. One can note that all methods do not degrade the original classification performance, where the differences in classification errors of across all tested single models are at most $1 \%$ in our experiments. +Table 2: Classification test set accuracy of VGGNets on CIFAR-10 and SVHN datasets under various training losses. + +
In-distributionCross entropyConfidence loss with original GANJoint confidence lossConfidence loss with explicit out-of-distribution samples
CIFAR-1080.1480.2781.3980.56
SVHN93.8294.0893.8194.23
+ +# C.2 CALIBRATION EFFECTS OF CONFIDENCE LOSS + +We also verify the calibration effects (Guo et al., 2017) of our methods: whether a classifier trained by our method can indicate when they are likely to be incorrect for test samples from the indistribution. In order to evaluate the calibration effects, we measure the expected calibration error (ECE) (Naeini et al., 2015). Given test data $\left\{ \left( x _ { 1 } , y _ { 1 } \right) , \ldots , \left( x _ { n } , y _ { n } \right) \right\}$ , we group the predictions into $M$ interval bins (each of size nfidence falls into the interv $1 / M )$ $B _ { m }$ be the set of indien, the accuracy of of samples whose predictionis defined as follows: $\textstyle { \bigl ( } { \frac { m - 1 } { M } } , { \frac { m } { M } } { \bigr ] }$ $B _ { m }$ + +$$ +\operatorname { a c c } ( B _ { m } ) = \frac { 1 } { | B _ { m } | } \sum _ { i \in B _ { m } } 1 _ { \{ y _ { i } = \arg \operatorname* { m a x } _ { y } P _ { \theta } ( y | \mathbf { x } _ { i } ) \} } , +$$ + +where $\theta$ is the model parameters of a classifier and $1 _ { A } \in \{ 0 , 1 \}$ is the indicator function for event $A$ . We also define the confidence of $B _ { m }$ as follows: + +$$ +\operatorname { c o n f } ( B _ { m } ) = { \frac { 1 } { \left| B _ { m } \right| } } \sum _ { i \in B _ { m } } q ( \mathbf { x } _ { i } ) , +$$ + +where $q ( \mathbf { x } _ { i } )$ is the confidence of data $i$ . Using these notations, we measure the expected calibration error (ECE) as follows: + +$$ +\mathrm { E C E } = \sum _ { m = 1 } ^ { M } \frac { | B _ { m } | } { n } | \mathrm { a c c } ( B _ { m } ) - \mathrm { c o n f } ( B _ { m } ) | . +$$ + +One can note that ECE is zero if the confidence score can represent the true distribution. Table 3 shows the calibration effects of confidence loss when we define the confidence score $q$ as the maximum predictive distribution of a classifier. We found that the ECE of a classifier trained by our methods is lower than that of a classifier trained by the standard cross entropy loss. This implies that our proposed method is effective at calibrating predictions. We also remark that the temperature scaling (Guo et al., 2017) provides further improvements under a classifier trained by our joint confidence loss. + +Table 3: Expected calibration error (ECE) of VGGNets on CIFAR-10 and SVHN datasets under various training losses. The number of bins $M$ is set to 20. All values are percentages and boldface values indicate relative the better results. + +
In-distributionWithout temperature scaling Cross entropy loss Joint confidence lossWith temperature scaling Cross entropy loss Joint confidence loss
CIFAR-1018.4514.627.076.19
SVHN5.305.132.801.39
+ +# C.3 EXPERIMENTAL RESULTS USING ODIN DETECTOR + +In this section, we verify the effects of confidence loss using ODIN detector (Liang et al., 2017) which is an advanced threshold-based detector using temperature scaling (Guo et al., 2017) and input perturbation. The key idea of ODIN is the temperature scaling which is defined as follows: + +$$ +P _ { \theta } ( y = \widehat { y } | \mathbf { x } ; T ) = \frac { \exp { ( f _ { \widehat { y } } ( \mathbf { x } ) / T ) } } { \sum _ { y } \exp { ( f _ { y } ( \mathbf { x } ) / T ) } } , +$$ + +where $T > 0$ is the temperature scaling parameter and $\mathbf { f } = \left( f _ { 1 } , \ldots , f _ { K } \right)$ is final feature vector of neural networks. For each data $\mathbf { x }$ , ODIN first calculates the pre-processed image $\widehat { \mathbf { x } }$ by adding the small perturbations as follows: + +$$ +\mathbf { x } ^ { \prime } = \mathbf { x } - \varepsilon \mathrm { s i g n } \left( - \bigtriangledown _ { \mathbf { x } } \log P _ { \theta } ( y = \widehat { y } | \mathbf { x } ; T ) \right) , +$$ + +where $\varepsilon$ is a magnitude of noise and $\widehat { y }$ is the predicted label. Next, ODIN feeds the prebprocessed data into the classifier, computes the maximum value of scaled predictive distribution, i.e., $\begin{array} { r } { \operatorname* { m a x } _ { y } P _ { \theta } ( y | \mathbf { x } ^ { \prime } ; T ) } \end{array}$ , and classifies it as positive (i.e., in-distribution) if the confidence score is above some threshold $\delta$ . + +For ODIN detector, the perturbation noise $\varepsilon$ is chosen from $\{ 0 , 0 . 0 0 0 1 , 0 . 0 0 1 , 0 . 0 1 \}$ , and the temperature $T$ is chosen from $\{ 1 , 1 0 , 1 0 0 , 5 0 0 , 1 0 0 0 \}$ . The optimal parameters are chosen to minimize the detection error on the validation set. Figure 7 shows the performance of the OIDN and baseline detector for each in- and out-of-distribution pair. First, we remark that the baseline detector using classifiers trained by our joint confidence loss (blue bar) typically outperforms the ODIN detector using classifiers trained by the cross entropy loss (orange bar). This means that our classifier can map in- and out-of-distributions more separately without pre-processing methods such as temperature scaling. The ODIN detector provides further improvements if one uses it with our joint confidence loss (green bar). In other words, our proposed training method can improve all prior detection methods. + +![](images/1380ade19cbcf946e7ad048c487ea59780f88d50e2289a37c6e92a096d1dc661.jpg) +Figure 7: Performances of the baseline detector (Hendrycks & Gimpel, 2016) and ODIN detector (Liang et al., 2017) under various training losses. + +# C.4 EXPERIMENTAL RESULTS ON ALEXNET + +Table 4 shows the detection performance for each in- and out-of-distribution pair when the classifier is AlexNet (Krizhevsky, 2014), which is one of popular CNN architectures. We remark that they show similar trends. + +Table 4: Performance of the baseline detector (Hendrycks & Gimpel, 2016) and ODIN detector (Liang et al., 2017) using AlexNet. All values are percentages and boldface values indicate relative the better results. For each in-distribution, we minimize the KL divergence term in (1) using training samples from an out-of-distribution dataset denoted by “seen”, where other “unseen” outof-distributions were also used for testing. + +
In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
Cross entropy loss /Confidence loss
Baseline (SVHN)CIFAR-10 (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)92.14 / 93.7742.0 / 99.9 45.6 /99.9 44.6/100.0 58.6/100.088.0/100.0 89.4/100.0 89.8/100.0 94.2/100.083.4/99.8 84.3/99.9 84.5 /99.9 88.8/100.088.7 /99.9 90.2/100.0 90.8/100.0 95.5/100.087.3/99.3 88.6/99.3 88.4/99.3 92.5 / 99.3
Baseline (CIFAR-10)SVHN (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)76.58 /76.1812.8 / 99.6 10.3/10.1 10.7 /8.1 6.7 /1.071.0 /99.9 59.2/52.1 56.3 / 51.5 49.6/13.573.2/99.6 64.2/62.0 64.3/61.8 61.3 / 50.074.3 /99.9 63.6/59.8 62.3/ 59.5 58.5 /43.770.7 /99.6 64.4/62.3 65.3/ 61.6 59.5/32.0
ODIN (SVHN)CIFAR-10 (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)92.14 /93.7755.5/99.9 59.5 /99.9 61.5 /100.0 82.6/100.089.1/99.2 90.5 /99.3 91.8 /99.382.4/99.8 83.8 /99.9 84.8 /99.985.9 /100.0 87.5 /100.0 90.5/100.089.0 /99.2 90.4 /99.3 91.3/99.3
37.1/99.697.0 /99.3 86.7 /99.691.6/100.0 79.3 /99.6 64.4/ 61.897.4/100.0 88.1/99.996.4 /99.3 84.2/99.6
ODIN (CIFAR-10)SVHN (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)76.58 / 76.1811.4 /8.4 13.3 / 7.1 3.8/ 0.069.1/ 65.6 71.9 /67.1 70.9/57.275.3/63.7 69.3 /40.471.4 /68.6 67.2/72.0 78.1/56.164.6/60.7 65.3/60.5 60.7 /40.7
+ +# D MAXIMIZING ENTROPY + +One might expect that the entropy of out-of-distribution is expected to be much higher compared to that of in-distribution since the out-of-distribution is typically on a much larger space than the indistribution. Therefore, one can add maximizing the entropy of generator distribution to new GAN loss in (3) and joint confidence loss in (4). However, maximizing the entropy of generator distribution is technically challenging since a GAN does not model the generator distribution explicitly. To handle the issue, one can leverage the pull-away term (PT) (Zhao et al., 2017): + +$$ +- \mathcal { H } \left( P _ { G } \left( \mathbf { x } \right) \right) \backsimeq \mathcal { P T } \left( P _ { G } \left( \mathbf { x } \right) \right) = \frac { 1 } { M ( M - 1 ) } \sum _ { i = 1 } ^ { M } \sum _ { j \neq i } \left( \frac { G \left( \mathbf { z } _ { i } \right) ^ { \top } G \left( \mathbf { z } _ { j } \right) } { \| G \left( \mathbf { z } _ { i } \right) \| \| G \left( \mathbf { z } _ { j } \right) \| } \right) ^ { 2 } , +$$ + +where $\mathcal { H } \left( \cdot \right)$ denotes the entropy, $\mathbf { z } _ { i } , \mathbf { z } _ { j } \sim P _ { \mathrm { p r i } } \left( \mathbf { z } \right)$ and $M$ is the number of samples. Intuitively, one can expect the effect of increasing the entropy by minimizing PT since it corresponds to the squared cosine similarity of generated samples. We note that (Dai et al., 2017) also used PT to maximize the entropy. Similarly as in Section 3.2, we verify the effects of PT using VGGNet. Table 5 shows the performance of the baseline detector for each in- and out-of-distribution pair. We found that joint confidence loss with PT tends to (but not always) improve the detection performance. However, since PT increases the training complexity and the gains from PT are relatively marginal (or controversial), we leave it as an auxiliary option for improving the performance. + +Table 5: Performance of the baseline detector (Hendrycks & Gimpel, 2016) using VGGNets trained by joint confidence loss with and without pull-away term (PT). All values are percentages and boldface values indicate relative the better results. + +
In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
CIFAR-10Joint confidence loss without PT/with PT
SVHNTinyImageNet LSUN93.81 /94.0590.1 /92.3 99.0 / 99.9 98.9 /100.097.6 /98.1 99.6 /100.0 99.6 /100.093.6 /94.6 97.6 /99.7 97.5 / 99.997.7 /98.2 99.7 /100.0 99.7 /100.097.9 /98.7 94.5 /100.0 95.5 /100.0
CIFAR-10SVHN TinyImageNet LSUN81.39 / 80.6025.4 / 13.2 35.0 /44.8 39.1 /49.166.8 / 69.5 72.0 / 78.4 75.1 /80.774.2 / 75.1 76.4 / 77.6 77.8 / 78.771.3 / 73.5 74.7 / 79.4 77.1 / 81.378.3 / 72.0 82.2 /84.4 83.6 /85.8
+ +# E ADDING OUT-OF-DISTRIBUTION CLASS + +Instead of forcing the predictive distribution on out-of-distribution samples to be closer to the uniform one, one can simply add an additional “out-of-distribution” class to a classifier as follows: + +$$ +\operatorname* { m i n } _ { \theta } \mathbb { E } _ { P _ { \mathrm { i n } } ( \widehat { \mathbf { x } } , \widehat { y } ) } \big [ - \log P _ { \theta } \left( y = \widehat { y } | \widehat { \mathbf { x } } \right) \big ] + \mathbb { E } _ { P _ { \mathrm { o u t } } ( \mathbf { x } ) } \big [ - \log P _ { \theta } \left( y = K + 1 | \mathbf { x } \right) \big ] , +$$ + +where $\theta$ is a model parameter. Similarly as in Section 3.1, we compare the performance of the confidence loss with that of above loss in (5) using VGGNets with for image classification on SVHN dataset. To optimize the KL divergence term in confidence loss and the second term of (5), CIFAR10 training datasets are used. In order to compare the detection performance, we define the the confidence score of input $\mathbf { x }$ as $1 - P _ { \theta }$ $\begin{array} { r } { \mathbf { \boldsymbol { y } } = K + 1 | \mathbf { \boldsymbol { x } } ) } \end{array}$ in case of (5). Table 6 shows the detection performance for out-of-distribution. First, the classifier trained by our method often significantly outperforms the alternative adding the new class label. This is because modeling explicitly out-ofdistribution can incur overfitting to trained out-of-distribution dataset. + +Table 6: Performance of the baseline detector (Hendrycks & Gimpel, 2016) and ODIN detector (Liang et al., 2017) using VGGNet. All values are percentages and boldface values indicate relative the better results. + +
DetectorOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
SVHN (seen)99.6 /99.8K+1 class loss in (5)/Confidence loss99.8 /99.999.7 /99.899.8 /99.999.9 /99.8
Baseline detectorTinyImageNet (unseen)79.61 /80.560.0 /9.95.2 /31.851.3 /58.650.7 /55.364.3 / 66.1
LSUN (unseen) Gaussian (unseen)0.0 /10.55.6 /34.851.5 /60.250.8/56.471.5 / 68.0
ODINSVHN (seen)0.0 /3.30.1/ 14.150.0 /50.0 99.1 /99.849.3 /49.412.2 /47.0
TinyImageNet (unseen)99.6 /99.899.9 /99.855.12/65.799.9 /99.999.9 /99.8
detectorLSUN (unseen)79.61 /80.560.3 /12.247.3 /70.656.6 /72.744.3 / 65.6
Gaussian (unseen)0.1 /13.7 0.0 /8.248.3 /73.1 28.3 /68.355.9 / 67.9 54.4 / 65.457.5 /75.2 47.8 /74.144.7 / 67.8 36.8 / 61.5
\ No newline at end of file diff --git a/parse/train/ryiAv2xAZ/ryiAv2xAZ_content_list.json b/parse/train/ryiAv2xAZ/ryiAv2xAZ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..0ef55ab0e8370ded53db73c8bd75c668bb569db2 --- /dev/null +++ b/parse/train/ryiAv2xAZ/ryiAv2xAZ_content_list.json @@ -0,0 +1,1751 @@ +[ + { + "type": "text", + "text": "TRAINING CONFIDENCE-CALIBRATED CLASSIFIERS FOR DETECTING OUT-OF-DISTRIBUTION SAMPLES ", + "text_level": 1, + "bbox": [ + 176, + 99, + 790, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Kimin Lee∗ Honglak Lee§ ,† Kibok Lee† Jinwoo Shin∗ ∗Korea Advanced Institute of Science and Technology, Daejeon, Korea †University of Michigan, Ann Arbor, MI 48109 §Google Brain, Mountain View, CA 94043 ", + "bbox": [ + 186, + 169, + 645, + 228 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 265, + 544, + 280 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The problem of detecting whether a test sample is from in-distribution (i.e., training distribution by a classifier) or out-of-distribution sufficiently different from it arises in many real-world machine learning applications. However, the state-of-art deep neural networks are known to be highly overconfident in their predictions, i.e., do not distinguish in- and out-of-distributions. Recently, to handle this issue, several threshold-based detectors have been proposed given pre-trained neural classifiers. However, the performance of prior works highly depends on how to train the classifiers since they only focus on improving inference procedures. In this paper, we develop a novel training method for classifiers so that such inference algorithms can work better. In particular, we suggest two additional terms added to the original loss (e.g., cross entropy). The first one forces samples from out-of-distribution less confident by the classifier and the second one is for (implicitly) generating most effective training samples for the first one. In essence, our method jointly trains both classification and generative neural networks for out-of-distribution. We demonstrate its effectiveness using deep convolutional neural networks on various popular image datasets. ", + "bbox": [ + 233, + 295, + 764, + 518 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 544, + 336, + 560 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep neural networks (DNNs) have demonstrated state-of-the-art performance on many classification tasks, e.g., speech recognition (Hannun et al., 2014), image classification (Girshick, 2015), video prediction (Villegas et al., 2017) and medical diagnosis (Caruana et al., 2015). Even though DNNs achieve high accuracy, it has been addressed (Lakshminarayanan et al., 2017; Guo et al., 2017) that they are typically overconfident in their predictions. For example, DNNs trained to classify MNIST images often produce high confident probability $9 1 \\%$ even for random noise (see the work of (Hendrycks & Gimpel, 2016)). Since evaluating the quality of their predictive uncertainty is hard, deploying them in real-world systems raises serious concerns in AI Safety (Amodei et al., 2016), e.g., one can easily break a secure authentication system that can be unlocked by detecting the gaze and iris of eyes using DNNs (Shrivastava et al., 2017). ", + "bbox": [ + 174, + 575, + 825, + 715 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The overconfidence issue of DNNs is highly related to the problem of detecting out-of-distribution: detect whether a test sample is from in-distribution (i.e., training distribution by a classifier) or outof-distribution sufficiently different from it. Formally, it can be formulated as a binary classification problem. Let an input $\\mathbf { x } \\in \\mathcal { X }$ and a label $y \\in { \\mathcal { Y } } = \\mathbf { \\bar { \\rho } } \\qquad $ be random variables that follow a joint data distribution $P _ { \\mathrm { i n } } \\left( \\mathbf { x } , y \\right) = P _ { \\mathrm { i n } } \\left( y | \\mathbf { x } \\right) P _ { \\mathrm { i n } } \\left( \\mathbf { x } \\right)$ . We assume that a classifier $P _ { \\boldsymbol { \\theta } } \\left( y | \\mathbf { x } \\right)$ is trained on a dataset drawn from $P _ { \\mathrm { i n } } \\left( \\mathbf { x } , y \\right)$ , where $\\theta$ denotes the model parameter. We let $P _ { \\mathrm { o u t } }$ $\\mathbf { \\tau } ( \\mathbf { x } )$ denote an out-of-distribution which is ‘far away’ from in-distribution $P _ { \\mathrm { i n } } \\left( \\mathbf { x } \\right)$ . Our problem of interest is determining if input $\\mathbf { x }$ is from $P _ { \\mathrm { i n } }$ or $P _ { \\mathrm { o u t } }$ , possibly utilizing a well calibrated classifier $P _ { \\boldsymbol { \\theta } } \\left( y | \\mathbf { x } \\right)$ . In other words, we aim to build a detector, $g ( \\mathbf { x } ) : \\mathcal { X } \\{ 0 , 1 \\}$ , which assigns label 1 if data is from in-distribution, and label 0 otherwise. ", + "bbox": [ + 174, + 722, + 825, + 861 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There have been recent efforts toward developing efficient detection methods where they mostly have studied simple threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017) utilizing a pre-trained classifier. For each input $\\mathbf { x }$ , it measures some confidence score $q ( \\mathbf { x } )$ based on a pre-trained classifier, and compares the score to some threshold $\\delta > 0$ . Then, the detector assigns label 1 if the confidence score $q ( \\mathbf { x } )$ is above $\\delta$ , and label 0, otherwise. Specifically, (Hendrycks & Gimpel, 2016) defined the confidence score as a maximum value of the predictive distribution, and (Liang et al., 2017) further improved the performance by using temperature scaling (Guo et al., 2017) and adding small controlled perturbations to the input data. Although such inference methods are computationally simple, their performances highly depend on the pre-trained classifier. Namely, they fail to work if the classifier does not separate the maximum value of predictive distribution well enough with respect to $P _ { \\mathrm { i n } }$ and $P _ { \\mathrm { o u t } }$ . Ideally, a classifier should be trained to separate all class-dependent in-distributions as well as out-of-distribution in the output space. As another line of research, Bayesian probabilistic models (Li & Gal, 2017; Louizos & Welling, 2017) and ensembles of classifiers (Lakshminarayanan et al., 2017) were also investigated. However, training or inferring those models are computationally more expensive. This motivates our approach of developing a new training method for the more plausible simple classifiers. Our direction is orthogonal to the Bayesian and ensemble approaches, where one can also combine them for even better performance. ", + "bbox": [ + 174, + 867, + 823, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 284 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contribution. In this paper, we develop such a training method for detecting out-of-distribution $P _ { \\mathrm { o u t } }$ better without losing its original classification accuracy. First, we consider a new loss function, called confidence loss. Our key idea on the proposed loss is to additionally minimize the KullbackLeibler (KL) divergence from the predictive distribution on out-of-distribution samples to the uniform one in order to give less confident predictions on them. Then, in- and out-of-distributions are expected to be more separable. However, optimizing the confidence loss requires training samples from out-of-distribution, which are often hard to sample: a priori knowledge on out-of-distribution is not available or its underlying space is too huge to cover. To handle the issue, we consider a new generative adversarial network (GAN) (Goodfellow et al., 2014) for generating most effective samples from $P _ { \\mathrm { o u t } }$ . Unlike the original GAN, the proposed GAN generates ‘boundary’ samples in the low-density area of $P _ { \\mathrm { i n } }$ . Finally, we design a joint training scheme minimizing the classifier’s loss and new GAN loss alternatively, i.e., the confident classifier improves the GAN, and vice versa, as training proceeds. Here, we emphasize that the proposed GAN does not need to generate explicit samples under our scheme, and instead it implicitly encourages training a more confident classifier. ", + "bbox": [ + 174, + 291, + 825, + 486 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We demonstrate the effectiveness of the proposed method using deep convolutional neural networks such as AlexNet (Krizhevsky, 2014) and VGGNet (Szegedy et al., 2015) for image classification tasks on CIFAR (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), ImageNet (Deng et al., 2009), and LSUN (Yu et al., 2015) datasets. The classifier trained by our proposed method drastically improves the detection performance of all threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017) in all experiments. In particular, VGGNet with 13 layers trained by our method improves the true negative rate (TNR), i.e., the fraction of detected out-of-distribution (LSUN) samples, compared to the baseline: $1 4 . 0 \\% 3 9 . 1 \\%$ and $4 6 . 3 \\% 9 8 . 9 \\%$ on CIFAR-10 and SVHN, respectively, when $9 5 \\%$ of in-distribution samples are correctly detected. We also provide visual understandings on the proposed method using the image datasets. We believe that our method can be a strong guideline when other researchers will pursue these tasks in the future. ", + "bbox": [ + 173, + 492, + 825, + 645 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 TRAINING CONFIDENT NEURAL CLASSIFIERS ", + "text_level": 1, + "bbox": [ + 174, + 667, + 578, + 683 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this section, we propose a novel training method for classifiers in order to improve the performance of prior threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017) (see Appendix A for more details). Our motivation is that such inference algorithms can work better if the classifiers are trained so that they map the samples from in- and out-of-distributions into the output space separately. Namely, we primarily focus on training an improved classifier, and then use prior detectors under the trained model to measure its performance. ", + "bbox": [ + 173, + 699, + 825, + 784 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 CONFIDENT CLASSIFIER FOR OUT-OF-DISTRIBUTION ", + "text_level": 1, + "bbox": [ + 173, + 801, + 576, + 815 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Without loss of generality, suppose that the cross entropy loss is used for training. Then, we propose the following new loss function, termed confidence loss: ", + "bbox": [ + 171, + 827, + 823, + 856 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/b88b52aa99ec6e4e628705cc11505897fdac6b170e23b5003e6e7cb2ec04707d.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\mathbb { E } _ { P _ { \\mathrm { i n } } ( \\widehat { \\mathbf { x } } , \\widehat { y } ) } \\big [ - \\log P _ { \\theta } \\left( y = \\widehat { y } | \\widehat { \\mathbf { x } } \\right) \\big ] + \\beta \\mathbb { E } _ { P _ { \\mathrm { o u t } } ( \\mathbf { x } ) } \\big [ K L \\left( \\mathcal { U } \\left( y \\right) \\mid \\mid P _ { \\theta } \\left( y | \\mathbf { x } \\right) \\right) \\big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 246, + 862, + 746, + 887 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $K L$ denotes the Kullback-Leibler (KL) divergence, $\\mathcal { U } \\left( y \\right)$ is the uniform distribution and $\\beta > 0$ is a penalty parameter. It is highly intuitive as the new loss forces the predictive distribution on out-of-distribution samples to be closer to the uniform one, i.e., zero confidence, while that for samples from in-distribution still follows the label-dependent probability. In other words, the proposed loss is designed for assigning higher maximum prediction values, i.e., $\\operatorname* { m a x } _ { y } P _ { \\theta } \\left( y | \\mathbf { x } \\right)$ , to in-distribution samples than out-of-distribution ones. Here, a caveat is that adding the KL divergence term might degrade the classification performance. However, we found that it is not the case due to the high expressive power of deep neural networks, while in- and out-of-distributions become more separable with respect to the maximum prediction value by optimizing the confidence loss (see Section 3.1 for supporting experimental results). ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/30ab0a6c3ac8a37d7cbd9bd3d2b7465e59b564853612866bbddd064ad62f4900.jpg", + "image_caption": [ + "Figure 1: Illustrating the behavior of classifier under different out-of-distribution training datasets. We generate the out-of-distribution samples from (a) 2D box $[ - 5 0 , 5 0 ] ^ { 2 }$ , and show (b) the corresponding decision boundary of classifier. We also generate the out-of-distribution samples from (c) 2D box $[ - 2 0 , 2 0 ] ^ { 2 }$ , and show (d) the corresponding decision boundary of classifier. " + ], + "image_footnote": [], + "bbox": [ + 205, + 111, + 779, + 217 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 296, + 825, + 409 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We remark that minimizing a similar KL loss was studied recently for different purposes (Lee et al., 2017; Pereyra et al., 2017). Training samples for minimizing the KL divergence term is explicitly given in their settings while we might not. Ideally, one has to sample all (almost infinite) types of outof-distribution to minimize the KL term in (1), or require some prior information on testing out-ofdistribution for efficient sampling. However, this is often infeasible and fragile. To address the issue, we suggest to sample out-of-distribution close to in-distribution, which could be more effective in improving the detection performance, without any assumption on testing out-of-distribution. ", + "bbox": [ + 174, + 415, + 823, + 512 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In order to explain our intuition in details, we consider a binary classification task on a simple example, where each class data is drawn from a Gaussian distribution and entire data space is bounded by 2D box $[ - 5 0 , 5 0 ] ^ { 2 }$ for visualization. We apply the confidence loss to simple fully-connected neural networks (2 hidden layers and 500 hidden units for each layer) using different types of outof-distribution training samples. First, as shown in Figure 1(a), we construct an out-of-distribution training dataset of 100 (green) points using rejection sampling on the entire data space $[ - 5 0 , 5 0 ] ^ { 2 }$ . Figure 1(b) shows the decision boundary of classifier optimizing the confidence loss on the corresponding dataset. One can observe that a classifier still shows overconfident predictions (red and blue regions) near the labeled in-distribution region. On the other hand, if we construct a training out-of-distribution dataset of 100 points from $[ - 2 0 , 2 0 ] ^ { 2 }$ , i.e., closer to target, in-distribution space (see Figure 1(c)), a classifier produces confident predictions only on the labeled region and zero confidence on the remaining in the entire data space $[ - 5 0 , 5 0 ] ^ { 2 }$ as shown in Figure 1(d). If one increases the number of training out-of-distribution samples which are generated from the entire space, i.e., $[ - 5 0 , 5 0 ] ^ { 2 }$ , Figure 1(b) is expected to be similar to Figure 1(d). In other words, one need more samples in order to train a confident classifier if samples are generated from the entire space. However, this might be impossible and not efficient since the number of out-of-distribution training samples might be almost infinite to cover its entire, huge actual data space. This implies that training out-of-distribution samples nearby the in-distribution region could be more effective in improving the detection performance. Our underlying intuition is that the effect of boundary of in-distribution region might propagate to the entire out-of-distribution space. Our experimental results in Section 3.1 also support this: realistic images are more useful as training out-of-distribution than synthetic datasets (e.g., Gaussian noise) for improving the detection performance when we consider an image classification task. This motivates us to develop a new generative adversarial network (GAN) for generating such effective out-of-distribution samples. ", + "bbox": [ + 173, + 520, + 825, + 852 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 ADVERSARIAL GENERATOR FOR OUT-OF-DISTRIBUTION ", + "text_level": 1, + "bbox": [ + 174, + 871, + 601, + 883 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we introduce a new training method for learning a generator of out-of-distribution inspired by generative adversarial network (GAN) (Goodfellow et al., 2014). We will first assume that the classifier for in-distribution is fixed, and also describe the joint learning framework in the next section. ", + "bbox": [ + 174, + 895, + 821, + 922 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The GAN framework consists of two main components: discriminator $D$ and generator $G$ . The generator maps a latent variable $\\mathbf { z }$ from a prior distribution $P _ { \\mathrm { p r i } } \\left( \\mathbf { z } \\right)$ to generated outputs $G \\left( \\mathbf { z } \\right)$ , and discriminator $D : \\mathcal { X } [ 0 , 1 ]$ represents a probability that sample $\\mathbf { x }$ is from a target distribution. Suppose that we want to recover the in-distribution $P _ { \\mathrm { i n } } ( x )$ using the generator $G$ . Then, one can optimize the following min-max objective for forcing $P _ { G } \\approx P _ { \\mathrm { i n } }$ : ", + "bbox": [ + 173, + 138, + 825, + 209 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6f4c8d8517375f94af4e0f1b7f1258dc4161d64907c250742398349ab46ad988.jpg", + "text": "$$\n\\operatorname* { m i n } _ { G } \\operatorname* { m a x } _ { D } ~ \\mathbb { E } _ { P _ { \\mathrm { i n } } ( \\mathbf { x } ) } \\big [ \\log D \\left( \\mathbf { x } \\right) \\big ] + \\mathbb { E } _ { P _ { \\mathrm { p r i } } ( \\mathbf { z } ) } \\big [ \\log \\left( 1 - D \\left( G \\left( \\mathbf { z } \\right) \\right) \\right) \\big ] .\n$$", + "text_format": "latex", + "bbox": [ + 281, + 209, + 715, + 234 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "However, unlike the original GAN, we want to make the generator recover an effective out-ofdistribution $P _ { \\mathrm { o u t } }$ instead of $P _ { \\mathrm { i n } }$ . To this end, we propose the following new GAN loss: ", + "bbox": [ + 173, + 236, + 828, + 263 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/3445bffdd36296f9eff224c6ffd4e5fcacd267cfc52dfd43a16a7136f34e9992.jpg", + "text": "$$\n\\begin{array} { r l } { \\underset { G } { \\mathop { \\operatorname* { m i n } } } \\underset { D } { \\mathop { \\operatorname* { m a x } } } } & { \\beta \\underbrace { { \\mathbb { E } } _ { P _ { G } ( \\mathbf { x } ) } \\left[ K L \\left( \\mathcal { U } \\left( y \\right) \\mid P _ { \\theta } \\left( y | \\mathbf { x } \\right) \\right) \\right] } _ { \\mathrm { ( a ) } } } \\\\ & { + \\underbrace { { \\mathbb { E } } _ { P _ { \\mathrm { i n } } \\left( \\mathbf { x } \\right) } \\left[ \\log D \\left( \\mathbf { x } \\right) \\right] + { \\mathbb { E } } _ { P _ { G } ( \\mathbf { x } ) } \\left[ \\log \\left( 1 - D \\left( \\mathbf { x } \\right) \\right) \\right] } _ { \\mathrm { ( b ) } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 265, + 709, + 342 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\theta$ is the model parameter of a classifier trained on in-distribution. The above objective can be interpreted as follows: the first term (a) corresponds to a replacement of the out-of-distribution $P _ { \\mathrm { o u t } }$ in (1)’s KL loss with the generator distribution $P _ { G }$ . One can note that this forces the generator to generate low-density samples since it can be interpreted as minimizing the log negative likelihood of in-distribution using the classifier, i.e., $P _ { \\mathrm { i n } } ( \\mathbf { x } ) \\approx \\exp \\left( K L \\left( \\mathcal { U } \\left( y \\right) \\parallel P _ { \\theta } \\left( y | \\mathbf { x } \\right) \\right) \\right)$ . We remark that this approximation is also closely related to the inception score (Salimans et al., 2016) which is popularly used as a quantitative measure of visual fidelity of the samples. The second term (b) corresponds to the original GAN loss since we would like to have out-of-distribution samples close to in-distribution, as mentioned in Section 2.1. Suppose that the model parameter of classifier $\\theta$ is set appropriately such that the classifier produces the uniform distribution for out of distribution samples. Then, the KL divergence term (a) in (3) is approximately 0 no matter what out-of-distribution samples are generated. However, if the samples are far away from boundary, the GAN loss (b) in (3) should be high, i.e., the GAN loss forces having samples being not too far from the in-distribution space. Therefore, one can expect that proposed loss can encourage the generator to produce the samples which are on the low-density boundary of the in-distribution space. We also provide its experimental evidences in Section 3.2. ", + "bbox": [ + 173, + 345, + 825, + 569 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We also remark that (Dai et al., 2017) consider a similar GAN generating samples from out-ofdistribution for the purpose of semi-supervised learning. The authors assume the existence of a pretrained density estimation model such as $\\mathrm { P i x e l C N N + + }$ (Salimans et al., 2017) for in-distribution, but such a model might not exist and be expensive to train in general. Instead, we use much simpler confident classifiers for approximating the density. Hence, under our fully-supervised setting, our GAN is much easier to train and more suitable. ", + "bbox": [ + 173, + 575, + 825, + 660 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 JOINT TRAINING METHOD OF CONFIDENT CLASSIFIER AND ADVERSARIAL GENERATOR ", + "text_level": 1, + "bbox": [ + 173, + 676, + 812, + 690 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the previous section, we suggest training the proposed GAN using a pre-trained confident classifier. We remind that the converse is also possible, i.e., the motivation of having such a GAN is for training a better classifier. Hence, two models can be used for improving each other. This naturally suggests a joint training scheme where the confident classifier improves the proposed GAN, and vice versa, as training proceeds. Specifically, we suggest the following joint objective function: ", + "bbox": [ + 174, + 700, + 823, + 771 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4599f52db8202c22e1c071f53bd9c20c46211a747226e4dac68c123e73ee7c8a.jpg", + "text": "$$\n\\begin{array} { r l } { \\underset { G } { \\mathop { \\operatorname* { m i n } } } \\underset { D } { \\mathop { \\operatorname* { m a x } } } \\underset { \\theta } { \\mathop { \\operatorname* { m i n } } } } & { \\underbrace { { \\mathbb { E } } _ { P _ { \\mathrm { i n } } ( \\widehat { \\mathbf { x } } , \\widehat { y } ) } { [ - \\log P _ { \\theta } ( y = \\widehat { y } | \\widehat { \\mathbf { x } } ) ] } } _ { \\mathrm { ( c ) } } + \\beta \\underbrace { { \\mathbb { E } } _ { P _ { G } ( \\mathbf { x } ) } { [ K L ( \\mathcal { U } ( y ) \\mid | P _ { \\theta } ( y | \\mathbf { x } ) ) ] } } _ { \\mathrm { ( d ) } } } \\\\ & { \\underbrace { + { \\mathbb { E } } _ { P _ { \\mathrm { i n } } ( \\widehat { \\mathbf { x } } ) } { [ \\log D ( \\widehat { \\mathbf { x } } ) ] } + { \\mathbb { E } } _ { P _ { G } ( \\mathbf { x } ) } { [ \\log ( 1 - D ( \\mathbf { x } ) ) ] } . } _ { } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 218, + 772, + 779, + 840 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The classifier’s confidence loss corresponds to ( ${ \\vartriangle } \\cosh + ( \\mathrm { d } )$ , and the proposed GAN loss corresponds to $\\mathrm { ( d ) + }$ (e), i.e., they share the KL divergence term (d) under joint training. To optimize the above objective efficiently, we propose an alternating algorithm, which optimizes model parameters $\\{ \\theta \\}$ of classifier and GAN models $\\{ G , D \\}$ alternatively as shown in Algorithm 1. Since the algorithm monotonically decreases the objective function, it is guaranteed to converge. ", + "bbox": [ + 173, + 854, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1 Alternating minimization for detecting and generating out-of-distribution. ", + "bbox": [ + 174, + 103, + 743, + 118 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "repeat ", + "text_level": 1, + "bbox": [ + 191, + 122, + 236, + 133 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "/∗ Update proposed $\\mathrm { G A N * } /$ ", + "bbox": [ + 215, + 133, + 408, + 147 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Sample $\\left\\{ \\mathbf { z } _ { 1 } , \\hdots , \\mathbf { z } _ { M } \\right\\}$ and $\\left\\{ \\mathbf { x } _ { 1 } , \\hdots , \\mathbf { x } _ { M } \\right\\}$ from prior $P _ { \\mathrm { p r i } } \\left( \\mathbf { z } \\right)$ and and in-distribution $P _ { \\mathrm { i n } } \\left( \\mathbf { x } \\right)$ respectively, and update the discriminator $D$ by ascending its stochastic gradient of ", + "bbox": [ + 189, + 147, + 821, + 176 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7b75326c8f8012406ba7c945db224b8c936d1502033cb0c3c03fc1801fd2e45f.jpg", + "text": "$$\n\\frac { 1 } { M } \\sum _ { i = 1 } ^ { M } \\left[ \\log D \\left( \\mathbf { x } _ { i } \\right) + \\log \\left( 1 - D \\left( G \\left( \\mathbf { z } _ { i } \\right) \\right) \\right) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 183, + 658, + 226 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Sample $\\left\\{ \\mathbf { z } _ { 1 } , \\hdots , \\mathbf { z } _ { M } \\right\\}$ from prior $P _ { \\mathrm { p r i } } \\left( \\mathbf { z } \\right)$ , and update the generator $G$ by descending its stochastic gradient of ", + "bbox": [ + 187, + 232, + 823, + 261 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0a4e275571bbddeab76d357f50aa2a23e4b9c0d892b0ef5598f08a482d19a1e8.jpg", + "text": "$$\n\\frac { 1 } { M } \\sum _ { i = 1 } ^ { M } \\Big [ \\log \\big ( 1 - D \\left( G \\left( \\mathbf { z } _ { i } \\right) \\right) \\big ) \\Big ] + \\frac { \\beta } { M } \\sum _ { i = 1 } ^ { M } \\Big [ K L \\left( \\mathcal { U } \\left( y \\right) \\| P _ { \\theta } \\left( y | G \\left( \\mathbf { z } _ { i } \\right) \\right) \\right) \\Big ] .\n$$", + "text_format": "latex", + "bbox": [ + 264, + 267, + 750, + 311 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$/ *$ Update confident classifier $^ { * / }$ ", + "bbox": [ + 215, + 318, + 431, + 332 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Sample $\\left\\{ \\mathbf { z } _ { 1 } , \\hdots , \\mathbf { z } _ { M } \\right\\}$ and $\\left\\{ \\left( \\mathbf { x } _ { 1 } , y _ { 1 } \\right) , \\ldots , \\left( \\mathbf { x } _ { M } , y _ { M } \\right) \\right\\}$ from prior $P _ { \\mathrm { p r i } } \\left( \\mathbf { z } \\right)$ and in-distribution $P _ { \\mathrm { i n } } \\left( \\mathbf { x } , y \\right)$ , respectively, and update the classifier $\\theta$ by descending its stochastic gradient of ", + "bbox": [ + 187, + 332, + 823, + 361 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/58f59868786d3bb587460b6210a35c813601f8a829cc5da5842ef0dbfac64d17.jpg", + "text": "$$\n\\frac { 1 } { M } \\sum _ { i = 1 } ^ { M } \\Big [ - \\log P _ { \\theta } \\left( y = y _ { i } | \\mathbf { x } _ { i } \\right) + \\beta K L \\left( \\mathcal { U } \\left( y \\right) \\parallel P _ { \\theta } \\left( y | G \\left( \\mathbf { z } _ { i } \\right) \\right) \\right) \\Big ] .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 367, + 722, + 411 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "until convergence ", + "bbox": [ + 189, + 416, + 318, + 431 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 463, + 419, + 479 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We demonstrate the effectiveness of our proposed method using various datasets: CIFAR (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), ImageNet (Deng et al., 2009), LSUN (Yu et al., 2015) and synthetic (Gaussian) noise distribution. We train convolutional neural networks (CNNs) including VGGNet (Szegedy et al., 2015) and AlexNet (Krizhevsky, 2014) for classifying CIFAR-10 and SVHN datasets. The corresponding test dataset is used as the in-distribution (positive) samples to measure the performance. We use realistic images and synthetic noises as the out-of-distribution (negative) samples. For evaluation, we measure the following metrics using the threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017): the true negative rate (TNR) at $9 5 \\%$ true positive rate (TPR), the area under the receiver operating characteristic curve (AUROC), the area under the precision-recall curve (AUPR), and the detection accuracy, where larger values of all metrics indicate better detection performances. Due to the space limitation, more explanations about datasets, metrics and network architectures are given in Appendix B.1 ", + "bbox": [ + 173, + 497, + 825, + 665 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/36eda71184e1688afb2df58e05bd429e4626c1b56f1b12e1a1e9895075b5d96b.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
Cross entropy loss /Confidence loss
SVHNCIFAR-10 (seen) TinyImageNet (unseen)93.82 /94.2347.4 /99.962.6 /99.978.6 /99.971.6 /99.991.2/99.4
49.0 /100.064.6 /100.079.6 /100.072.7 /100.091.6 /99.4
LSUN (unseen) Gaussian (unseen)46.3/100.0 56.1/100.061.8/100.0 72.0 /100.078.2 /100.0 83.4/100.071.1/100.0 77.2/100.090.8 /99.4 92.8 /99.4
CIFAR-10SVHN (seen)80.14 /80.56
13.7 / 99.8 13.6 /9.946.6 /99.966.6 /99.8 62.6 /58.661.4 /99.9 58.3 /55.373.5 /99.8 71.0 / 66.1
TinyImageNet (unseen)39.6 /31.8
LSUN (unseen) Gaussian (unseen)14.0 / 10.540.7 /34.863.2 / 60.258.7 /56.471.5 /68.0
2.8 / 3.310.2 /14.150.0/50.048.1 /49.439.9 /47.0
", + "bbox": [ + 174, + 680, + 825, + 815 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Table 1: Performance of the baseline detector (Hendrycks & Gimpel, 2016) using VGGNet. All values are percentages and boldface values indicate relative the better results. For each in-distribution, we minimize the KL divergence term in (1) using training samples from an out-of-distribution dataset denoted by “seen”, where other “unseen” out-of-distributions were only used for testing. ", + "bbox": [ + 173, + 825, + 825, + 882 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/82eec49fdb49b288decb05d89feef7881d949c90c40d7146f2041eba0c997d44.jpg", + "image_caption": [ + "Figure 2: For all experiments in (a), (b) and (c), we commonly use the SVHN dataset for indistribution. Fraction of the maximum prediction value in softmax scores trained by (a) cross entropy loss and (b) confidence loss: the $\\mathbf { X }$ -axis and y-axis represent the maximum prediction value and the fraction of images receiving the corresponding score, respectively. The receiver operating characteristic (ROC) curves under different losses are reported in (c): the red curve corresponds to the ROC curve of a model trained by optimizing the naive cross entropy loss, whereas other ones correspond to the ROC curves of models trained by optimizing the confidence loss. The KL divergence term in the confidence loss is optimized using explicit out-of-distribution datasets indicated in the parentheses, e.g., Confident loss (LSUN) means that we use the LSUN dataset for optimizing the KL divergence term. " + ], + "image_footnote": [], + "bbox": [ + 169, + 99, + 813, + 250 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 EFFECTS OF CONFIDENCE LOSS ", + "text_level": 1, + "bbox": [ + 176, + 433, + 429, + 445 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We first verify the effect of confidence loss in (1) trained by some explicit, say seen, out-ofdistribution datasets. First, we compare the quality of confidence level by applying various training losses. Specifically, the softmax classifier is used and simple CNNs (two convolutional layers followed by three fully-connected layers) are trained by minimizing the standard cross entropy loss on SVHN dataset. We also apply the confidence loss to the models by additionally optimizing the KL divergence term using CIFAR-10 dataset (as training out-of-distribution). In Figure 2(a) and 2(b), we report distributions of the maximum prediction value in softmax scores to evaluate the separation quality between in-distribution (i.e., SVHN) and out-of-distributions. It is clear that there exists a better separation between the SVHN test set (red bar) and other ones when the model is trained by the confidence loss. Here, we emphasize that the maximum prediction value is also low on even untrained (unseen) out-of-distributions, e.g., TinyImageNet, LSUN and synthetic datasets. Therefore, it is expected that one can distinguish in- and out-of-distributions more easily when a classifier is trained by optimizing the confidence loss. To verify that, we obtain the ROC curve using the baseline detector (Hendrycks & Gimpel, 2016) that computes the maximum value of predictive distribution on a test sample and classifies it as positive (i.e., in-distribution) if the confidence score is above some threshold. Figure 2(c) shows the ROC curves when we optimize the KL divergence term on various datasets. One can observe that realistic images such as TinyImageNet (aqua line) and LSUN (green line) are more useful than synthetic datasets (orange line) for improving the detection performance. This supports our intuition that out-of-distribution samples close to in-distribution could be more effective in improving the detection performance as we discussed in Section 2.1. ", + "bbox": [ + 174, + 458, + 825, + 736 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We indeed evaluate the performance of the baseline detector for out-of-distribution using largescale CNNs, i.e., VGGNets with 13 layers, under various training scenarios, where more results on AlexNet and ODIN detector (Liang et al., 2017) can be found in Appendix C (the overall trends of results are similar). For optimizing the confidence loss in (1), SVHN and CIFAR-10 training datasets are used for optimizing the KL divergence term for the cases when the in-distribution is CIFAR-10 and SVHN, respectively. Table 1 shows the detection performance for each in- and out-of-distribution pair. When the in-distribution is SVHN, the classifier trained by our method drastically improves the detection performance across all out-of-distributions without hurting its original classification performance. However, when the in-distribution is CIFAR-10, the confidence loss does not improve the detection performance in overall, where we expect that this is because the trained/seen SVHN out-of-distribution does not effectively cover all tested out-of-distributions. Our joint confidence loss in (4), which was designed under the intuition, resolves the issue of the CIFAR-10 (in-distribution) classification case in Table 1 (see Figure 4(b)). ", + "bbox": [ + 174, + 743, + 825, + 922 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/e9f90921b82f9cee01a46da1fa1cfd34180f47111c99ee992c262ad48b241d9c.jpg", + "image_caption": [ + "Figure 3: The generated samples from original GAN (a)/(c) and proposed GAN (b)/(d). In (a)/(b), the grey area is the 2D histogram of training in-distribution samples drawn from a mixture of two Gaussian distributions and red points indicate generated samples by GANs. " + ], + "image_footnote": [], + "bbox": [ + 184, + 102, + 818, + 219 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/7c77a46a2cf9c6a821a4ce5593a77d3e597cc6179d1deaee26e3d473df7e41a5.jpg", + "image_caption": [ + "Figure 4: Performances of the baseline detector (Hendrycks & Gimpel, 2016) under various training losses. For training models by the confidence loss, the KL divergence term is optimized using samples indicated in the parentheses. For fair comparisons, we only plot the performances for unseen out-of-distributions, where those for seen out-of-distributions (used for minimizing the KL divergence term in (1)) can be found in Table 1. " + ], + "image_footnote": [], + "bbox": [ + 176, + 299, + 802, + 606 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 EFFECTS OF ADVERSARIAL GENERATOR AND JOINT CONFIDENCE LOSS ", + "text_level": 1, + "bbox": [ + 176, + 715, + 702, + 729 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we verify the effect of the proposed GAN in Section 2.2 and evaluate the detection performance of the joint confidence loss in (4). To verify that the proposed GAN can produce the samples nearby the low-density boundary of the in-distribution space, we first compare the generated samples by original GAN and proposed GAN on a simple example where the target distribution is a mixture of two Gaussian distributions. For both the generator and discriminator, we use fullyconnected neural networks with 2 hidden layers. For our method, we use a pre-trained classifier which minimizes the cross entropy on target distribution samples and the KL divergence on out-ofdistribution samples generated by rejection sampling on a bounded 2D box. As shown in Figure 3(a), the samples of original GAN cover the high-density area of the target distribution while those of proposed GAN does its boundary one (see Figure 3(b)). We also compare the generated samples of original and proposed GANs on MNIST dataset (LeCun et al., 1998), which consists of handwritten digits. For this experiment, we use deep convolutional GANs (DCGANs) (Radford et al., 2015). In this case, we use a pre-trained classifier which minimizes the cross entropy on MNIST training samples and the KL divergence on synthetic Gaussian noises. As shown in Figure 3(c) and 3(d), samples of original GAN looks more like digits than those of proposed GAN. Somewhat interestingly, the proposed GAN still generates some new digit-like images. ", + "bbox": [ + 173, + 743, + 825, + 922 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/d41db38747ca1efcc752cd0ff7ae1de2ed89e22acec41b351615403610c3ccf6.jpg", + "image_caption": [ + "Figure 5: Guided gradient (sensitivity) maps of the top-1 predicted class with respect to the input image under various training losses. " + ], + "image_footnote": [], + "bbox": [ + 196, + 97, + 795, + 266 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 320, + 825, + 362 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We indeed evaluate the performance of our joint confidence loss in (4) utilizing the proposed GAN. To this end, we use VGGNets (as classifiers) and DCGANs (as GANs). We also test a variant of confidence loss which optimizes the KL divergence term on samples from a pre-trained original GAN (implicitly) modeling the in-distribution. One can expect that samples from the original GAN can be also useful for improving the detection performance since it may have bad generalization properties (Arora et al., 2017) and generate a few samples on the low-density boundary as like the proposed GAN. Figure 4 shows the performance of the baseline detector for each in- and out-of-distribution pair. First, observe that the joint confidence loss (blue bar) outperforms the confidence loss with some explicit out-of-distribution datasets (green bar). This is quite remarkable since the former is trained only using in-distribution datasets, while the latter utilizes additional out-of-distribution datasets. We also remark that our methods significantly outperform the baseline cross entropy loss (red bar) in all cases without harming its original classification performances (see Table 2 in Appendix C). Interestingly, the confidence loss with the original GAN (orange bar) is often (but not always) useful for improving the detection performance, whereas that with the proposed GAN (blue bar) still outperforms it in all cases. ", + "bbox": [ + 174, + 369, + 825, + 577 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Finally, we also provide visual interpretations of models using the guided gradient maps (Springenberg et al., 2014). Here, the gradient can be interpreted as an importance value of each pixel which influences on the classification decision. As shown in Figure 5, the model trained by the cross entropy loss shows sharp gradient maps for both samples from in- and out-of-distributions, whereas models trained by the confidence losses do only on samples from in-distribution. For the case of SVHN in-distribution, all confidence losses gave almost zero gradients, which matches to the results in Figure 4(a): their detection performances are almost perfect. For the case of CIFAR10 distribution, one can now observe that there exists some connection between gradient maps and detection performances. This is intuitive because for detecting samples from out-of-distributions better, the classifier should look at more pixels as similar importance and the KL divergence term forces it. We think that our visualization results might give some ideas in future works for developing better inference methods for detecting out-of-distribution under our models. ", + "bbox": [ + 174, + 584, + 825, + 751 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 777, + 318, + 792 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this paper, we aim to develop a training method for neural classification networks for detecting out-of-distribution better without losing its original classification accuracy. In essence, our method jointly trains two models for detecting and generating out-of-distribution by minimizing their losses alternatively. Although we primarily focus on image classification in our experiments, our method can be used for any classification tasks using deep neural networks. It is also interesting future directions applying our methods for other related tasks: regression (Malinin et al., 2017), network calibration (Guo et al., 2017), Bayesian probabilistic models (Li & Gal, 2017; Louizos & Welling, 2017), ensemble (Lakshminarayanan et al., 2017) and semi-supervised learning (Dai et al., 2017). ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 367, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This work was supported in part by the Institute for Information & communications Technology Promotion(IITP) grant funded by the Korea government(MSIT) (No.2017-0-01778, Development of Explainable Human-level Deep Machine Learning Inference Framework), the ICT R&D program of MSIP/IITP [R-20161130-004520, Research on Adaptive Machine Learning Technology Development for Intelligent Autonomous Digital Companion], DARPA Explainable AI (XAI) program #313498 and Sloan Research Fellowship. 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", + "bbox": [ + 174, + 484, + 825, + 527 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In Advances in neural information processing systems (NIPS), 2016. ", + "bbox": [ + 173, + 535, + 825, + 579 + ], + "page_idx": 9 + }, + { + "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. In International Conference on Learning Representations (ICLR), 2017. ", + "bbox": [ + 171, + 587, + 823, + 631 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Ashish Shrivastava, Tomas Pfister, Oncel Tuzel, Josh Susskind, Wenda Wang, and Russ Webb. Learning from simulated and unsupervised images through adversarial training. In Computer Vision and Pattern Recognition (CVPR), 2017. 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", + "bbox": [ + 173, + 780, + 823, + 823 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 832, + 821, + 875 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Junbo Zhao, Michael Mathieu, and Yann LeCun. Energy-based generative adversarial network. In International Conference on Learning Representations (ICLR), 2017. ", + "bbox": [ + 173, + 883, + 821, + 912 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A THRESHOLD-BASED DETECTORS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 480, + 118 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In this section, we formally describe the detection procedure of threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017). For each data $\\mathbf { x }$ , it measures some confidence score $q ( \\mathbf { x } )$ by feeding the data into a pre-trained classifier. Here, (Hendrycks & Gimpel, 2016) defined the confidence score as a maximum value of the predictive distribution, and (Liang et al., 2017) further improved the performance by processing the predictive distribution (see Appendix C.3 for more details). Then, the detector, $g ( \\mathbf { x } ) : \\mathcal { X } \\{ 0 , 1 \\}$ , assigns label 1 if the confidence score $q ( \\mathbf { x } )$ is above some threshold $\\delta$ , and label 0, otherwise: ", + "bbox": [ + 173, + 133, + 825, + 233 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/acce017d602a5d371b5e582ade033557dec4656e2e013103a6689702187aab67.jpg", + "text": "$$\ng \\left( \\mathbf { x } \\right) = \\left\\{ \\begin{array} { l l } { \\begin{array} { r l } { 1 } & { \\mathrm { i f } q ( \\mathbf { x } ) \\geq \\delta , } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } . } \\end{array} } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 393, + 241, + 599, + 276 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "For this detector, we have to find a score threshold so that some positive examples are classified correctly, but this depends upon the trade-off between false negatives and false positives. To handle this issue, we use threshold-independent evaluation metrics such as area under the receiver operating characteristic curve (AUORC) and detection accuracy (see Appendix B). ", + "bbox": [ + 173, + 285, + 825, + 340 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "B EXPERIMENTAL SETUPS IN SECTION 3 ", + "text_level": 1, + "bbox": [ + 174, + 363, + 529, + 380 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Datasets. We train deep models such as VGGNet (Szegedy et al., 2015) and AlexNet (Krizhevsky, 2014) for classifying CIFAR-10 and SVHN datasets: the former consists of 50,000 training and 10,000 test images with 10 image classes, and the latter consists of 73,257 training and 26,032 test images with 10 digits.2 The corresponding test dataset are used as the in-distribution (positive) samples to measure the performance. We use realistic images and synthetic noises as the out-ofdistribution (negative) samples: the TinyImageNet consists of 10,000 test images with 200 image classes from a subset of ImageNet images. The LSUN consists of 10,000 test images of 10 different scenes. We downsample each image of TinyImageNet and LSUN to size $3 2 \\times 3 2$ . The Gaussian noise is independently and identically sampled from a Gaussian distribution with mean 0.5 and variance 1. We clip each pixel value into the range $[ 0 , 1 ]$ . ", + "bbox": [ + 173, + 395, + 825, + 536 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Detailed CNN structure and training. The simple CNN that we use for evaluation shown in Figure 2 consists of two convolutional layers followed by three fully-connected layers. Convolutional layers have 128 and 256 filters, respectively. Each convolutional layer has a $5 \\times 5$ receptive field applied with a stride of 1 pixel each followed by max pooling layer which pools $2 \\times 2$ regions at strides of 2 pixels. AlexNet (Krizhevsky, 2014) consists of five convolutitonal layers followed by three fullyconnected layers. Convolutional layers have 64, 192, 384, 256 and 256 filters, respectively. First and second convolutional layers have a $5 \\times 5$ receptive field applied with a stride of 1 pixel each followed by max pooling layer which pools $3 \\times 3$ regions at strides of 2 pixels. Other convolutional layers have a $3 \\times 3$ receptive field applied with a stride of 1 pixel followed by max pooling layer which pools $2 \\times 2$ regions at strides of 2 pixels. Fully-connected layers have 2048, 1024 and 10 hidden units, respectively. Dropout (Hinton et al., 2012) was applied to only fully-connected layers of the network with the probability of retaining the unit being 0.5. All hidden units are ReLUs. Figure 6 shows the detailed structure of VGGNet (Szegedy et al., 2015) with three fully-connected layers and 10 convolutional layers. Each ConvReLU box in the figure indicates a $3 \\times 3$ convolutional layer followed by ReLU activation. Also, all max pooling layers have $2 \\times 2$ receptive fields with stride 2. Dropout was applied to only fully-connected layers of the network with the probability of retaining the unit being 0.5. For all experiments, the softmax classifier is used, and each model is trained by optimizing the objective function using Adam learning rule (Kingma & Ba, 2014). For each out-ofdistribution dataset, we randomly select 1,000 images for tuning the penalty parameter $\\beta$ , mini-batch size and learning rate. The penalty parameter is chosen from $\\beta \\in \\{ 0 , 0 . 1 , . . . 1 . 9 , 2 \\}$ , the mini-batch size is chosen from $\\{ 6 4 , 1 \\bar { 2 } 8 \\}$ and the learning rate is chosen from $\\{ 0 . 0 0 1 , 0 . 0 0 0 5 , 0 . 0 0 0 2 \\}$ . The optimal parameters are chosen to minimize the detection error on the validation set. We drop the learning rate by 0.1 at 60 epoch and models are trained for total 100 epochs. The best test result is reported for each method. ", + "bbox": [ + 173, + 541, + 825, + 875 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Performance metrics. We measure the following metrics using threshold-based detectors: ", + "bbox": [ + 173, + 882, + 766, + 896 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "• True negative rate (TNR) at $95 \\%$ true positive rate (TPR). Let TP, TN, FP, and FN denote true positive, true negative, false positive and false negative, respectively. We measure $\\mathrm { T N R } = \\mathrm { T N } / \\left( \\mathrm { F P + T N } \\right)$ , when $\\mathrm { T P R } = \\mathrm { T P } .$ / $( \\mathrm { T P + F N } )$ is $9 5 \\%$ . ", + "bbox": [ + 215, + 103, + 823, + 146 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "• Area under the receiver operating characteristic curve (AUROC). The ROC curve is a graph plotting TPR against the false positive rate $= \\mathrm { F P }$ / $\\mathrm { \\ F P { + } T N ) }$ by varying a threshold. ", + "bbox": [ + 215, + 150, + 823, + 179 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "• Area under the precision-recall curve (AUPR). The PR curve is a graph plotting the precision $=$ TP / $( \\mathrm { T P + F P } )$ against recal $\\mathrm { \\Lambda } = \\mathrm { T P } / \\left( \\mathrm { T P + F N } \\right)$ by varying a threshold. AUPR-IN (or -OUT) is AUPR where in- (or out-of-) distribution samples are specified as positive. ", + "bbox": [ + 217, + 183, + 823, + 224 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "• Detection accuracy. This metric corresponds to the maximum classification probability over all possible thresholds $\\delta$ : $1 \\ \\dot { - } \\ \\operatorname* { m i n } _ { \\delta } \\left\\{ P _ { \\mathrm { i n } } \\left( q \\left( \\mathbf x \\right) \\leq \\delta \\right) P \\left( \\mathbf x \\right) \\right.$ is from $P _ { \\mathrm { i n } }$ ) $^ +$ $P _ { \\mathrm { o u t } }$ $\\dot { \\mathbf { \\eta } } _ { q } \\left( \\mathbf { x } \\right) > \\delta ) \\mathbf { \\eta } _ { P }$ ( $\\mathbf { x }$ is from $P _ { \\mathrm { o u t } }$ ) $\\}$ , where $q ( \\mathbf { x } )$ is a confident score such as a maximum value of softmax. We assume that both positive and negative examples have equal probability of appearing in the test set, i.e., $P$ (x is from $P _ { \\mathrm { i n } } ) \\stackrel { - } { = } P$ ( $\\mathbf { x }$ is from $P _ { \\mathrm { o u t } } ) = \\bar { 0 } . 5$ . ", + "bbox": [ + 217, + 229, + 825, + 303 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Note that AUROC, AUPR and detection accuracy are threshold-independent evaluation metrics. ", + "bbox": [ + 173, + 313, + 799, + 328 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/cefa79cfd18994ae4d493f52482b99594b0c93029794a4fe28f238c72ccca960.jpg", + "image_caption": [ + "Figure 6: Detailed structure of VGGNet with 13 layers. " + ], + "image_footnote": [], + "bbox": [ + 326, + 344, + 704, + 462 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Generating samples on a simple example. As shown in Figure 3(a) and Figure 3(b), we compare the generated samples by original GAN and proposed GAN on a simple example where the target distribution is a mixture of two Gaussian distributions. For both the generator and discriminator, we use fully-connected neural networks with 2 hidden layers and 500 hidden units for each layer. For all layers, we use ReLU activation function. We use a 100-dimensional Gaussian prior for the latent variable z. For our method, we pre-train the simple fully-connected neural networks (2 hidden layers and 500 ReLU units for each layer) by minimizing the cross entropy on target distribution samples and the KL divergence on out-of-distribution samples generated by rejection sampling on bounded 2D box $[ - 1 0 , 1 0 ] ^ { \\frac { \\ d } { 2 } }$ . The penalty parameter $\\beta$ is set to 1. We use ADAM learning rule (Kingma & Ba, 2014) with a mini-batch size of 400. The initial learning rate is set to 0.002, and we train for total 100 epochs. ", + "bbox": [ + 173, + 508, + 825, + 661 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Generating samples on MNIST. As shown in Figure 3(c) and Figure 3(d), we compare the generated samples of original and proposed GANs on MNIST dataset, which consists of greyscale images, each containing a digit 0 to 9 with 60,000 training and 10,000 test images. We expand each image to size $3 \\times 3 2 \\times 3 2$ . For both the generator and discriminator, we use deep convolutional GANs (DCGANs) (Radford et al., 2015). The discriminator and generator consist of four convolutional and deconvolutional layers, respectively. Convolutional layers have 128, 256, 512 and 1 filters, respectively. Each convolutional layer has a $4 \\times 4$ receptive field applied with a stride of 2 pixel. The second and third convolutional layers are followed by batch normalization (Ioffe & Szegedy, 2015). For all layers, we use LeakyReLU activation function. Deconvolutional layers have 512, 256, 128 and 1 filters, respectively. Each deconvolutional layer has a $4 \\times 4$ receptive field applied with a stride of 2 pixel followed by batch normalization (Ioffe & Szegedy, 2015) and ReLU activation. For our method, we use a pre-trained simple CNNs (two convolutional layers followed by three fully-connected layers) by minimizing the cross entropy on MNIST training samples and the KL divergence on synthetic Gaussian noise. Convolutional layers have 128 and 256 filters, respectively. Each convolutional layer has a $5 \\times 5$ receptive field applied with a stride of 1 pixel each followed by max pooling layer which pools $2 \\times 2$ regions at strides of 2 pixels. The penalty parameter $\\beta$ is set to 1. We use ADAM learning rule (Kingma & Ba, 2014) with a mini-batch size of 128. The initial learning rate is set to 0.0002, and we train for total 50 epochs. ", + "bbox": [ + 173, + 667, + 825, + 919 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C MORE EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 473, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C.1 CLASSIFICATION PERFORMANCES ", + "text_level": 1, + "bbox": [ + 176, + 133, + 450, + 148 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/c042c11de4a455a7608c5fae748862559e56f37ce99cc889b69c6f689072c8e2.jpg", + "table_caption": [ + "Table 2 reports the classification accuracy of VGGNets on CIFAR-10 and SVHN datasets under various training losses shown in Figure 4. One can note that all methods do not degrade the original classification performance, where the differences in classification errors of across all tested single models are at most $1 \\%$ in our experiments. ", + "Table 2: Classification test set accuracy of VGGNets on CIFAR-10 and SVHN datasets under various training losses. " + ], + "table_footnote": [], + "table_body": "
In-distributionCross entropyConfidence loss with original GANJoint confidence lossConfidence loss with explicit out-of-distribution samples
CIFAR-1080.1480.2781.3980.56
SVHN93.8294.0893.8194.23
", + "bbox": [ + 176, + 228, + 825, + 292 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C.2 CALIBRATION EFFECTS OF CONFIDENCE LOSS ", + "text_level": 1, + "bbox": [ + 173, + 356, + 534, + 369 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We also verify the calibration effects (Guo et al., 2017) of our methods: whether a classifier trained by our method can indicate when they are likely to be incorrect for test samples from the indistribution. In order to evaluate the calibration effects, we measure the expected calibration error (ECE) (Naeini et al., 2015). Given test data $\\left\\{ \\left( x _ { 1 } , y _ { 1 } \\right) , \\ldots , \\left( x _ { n } , y _ { n } \\right) \\right\\}$ , we group the predictions into $M$ interval bins (each of size nfidence falls into the interv $1 / M )$ $B _ { m }$ be the set of indien, the accuracy of of samples whose predictionis defined as follows: $\\textstyle { \\bigl ( } { \\frac { m - 1 } { M } } , { \\frac { m } { M } } { \\bigr ] }$ $B _ { m }$ ", + "bbox": [ + 173, + 381, + 825, + 468 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/9bcd1d1200febd0642f987f7c590863d581525b1afdac42339683716d5eed03c.jpg", + "text": "$$\n\\operatorname { a c c } ( B _ { m } ) = \\frac { 1 } { | B _ { m } | } \\sum _ { i \\in B _ { m } } 1 _ { \\{ y _ { i } = \\arg \\operatorname* { m a x } _ { y } P _ { \\theta } ( y | \\mathbf { x } _ { i } ) \\} } ,\n$$", + "text_format": "latex", + "bbox": [ + 343, + 473, + 650, + 512 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $\\theta$ is the model parameters of a classifier and $1 _ { A } \\in \\{ 0 , 1 \\}$ is the indicator function for event $A$ . We also define the confidence of $B _ { m }$ as follows: ", + "bbox": [ + 173, + 518, + 823, + 547 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/827b0dbbe9ad1938d9b67ba42213b8a7a957e3aa235f72b6e33cf9f53cd1aefa.jpg", + "text": "$$\n\\operatorname { c o n f } ( B _ { m } ) = { \\frac { 1 } { \\left| B _ { m } \\right| } } \\sum _ { i \\in B _ { m } } q ( \\mathbf { x } _ { i } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 393, + 553, + 602, + 592 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $q ( \\mathbf { x } _ { i } )$ is the confidence of data $i$ . Using these notations, we measure the expected calibration error (ECE) as follows: ", + "bbox": [ + 174, + 599, + 823, + 627 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/c22f82fe3a9f5bc222c2f2c0e1ba5a0ad74945a7742198578fe81b13f41e7f65.jpg", + "text": "$$\n\\mathrm { E C E } = \\sum _ { m = 1 } ^ { M } \\frac { | B _ { m } | } { n } | \\mathrm { a c c } ( B _ { m } ) - \\mathrm { c o n f } ( B _ { m } ) | .\n$$", + "text_format": "latex", + "bbox": [ + 356, + 633, + 640, + 678 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "One can note that ECE is zero if the confidence score can represent the true distribution. Table 3 shows the calibration effects of confidence loss when we define the confidence score $q$ as the maximum predictive distribution of a classifier. We found that the ECE of a classifier trained by our methods is lower than that of a classifier trained by the standard cross entropy loss. This implies that our proposed method is effective at calibrating predictions. We also remark that the temperature scaling (Guo et al., 2017) provides further improvements under a classifier trained by our joint confidence loss. ", + "bbox": [ + 173, + 683, + 825, + 781 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/3f495c43d3eb637ba8600dec41abbe37448720531c5aeaeeb1a03bc6c0ac05f2.jpg", + "table_caption": [ + "Table 3: Expected calibration error (ECE) of VGGNets on CIFAR-10 and SVHN datasets under various training losses. The number of bins $M$ is set to 20. All values are percentages and boldface values indicate relative the better results. " + ], + "table_footnote": [], + "table_body": "
In-distributionWithout temperature scaling Cross entropy loss Joint confidence lossWith temperature scaling Cross entropy loss Joint confidence loss
CIFAR-1018.4514.627.076.19
SVHN5.305.132.801.39
", + "bbox": [ + 176, + 791, + 825, + 864 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C.3 EXPERIMENTAL RESULTS USING ODIN DETECTOR ", + "text_level": 1, + "bbox": [ + 173, + 103, + 568, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we verify the effects of confidence loss using ODIN detector (Liang et al., 2017) which is an advanced threshold-based detector using temperature scaling (Guo et al., 2017) and input perturbation. The key idea of ODIN is the temperature scaling which is defined as follows: ", + "bbox": [ + 174, + 128, + 825, + 171 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/2b8e95a21d71fc3b8e74d66755ce15ec4fa1f48ddd9fcdc377e83e6c85e0825c.jpg", + "text": "$$\nP _ { \\theta } ( y = \\widehat { y } | \\mathbf { x } ; T ) = \\frac { \\exp { ( f _ { \\widehat { y } } ( \\mathbf { x } ) / T ) } } { \\sum _ { y } \\exp { ( f _ { y } ( \\mathbf { x } ) / T ) } } ,\n$$", + "text_format": "latex", + "bbox": [ + 366, + 175, + 629, + 213 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $T > 0$ is the temperature scaling parameter and $\\mathbf { f } = \\left( f _ { 1 } , \\ldots , f _ { K } \\right)$ is final feature vector of neural networks. For each data $\\mathbf { x }$ , ODIN first calculates the pre-processed image $\\widehat { \\mathbf { x } }$ by adding the small perturbations as follows: ", + "bbox": [ + 174, + 215, + 825, + 257 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/a0d5476f34e6b5a0c7344c34935aefd1098da909b8041eb7e56a6a339f003646.jpg", + "text": "$$\n\\mathbf { x } ^ { \\prime } = \\mathbf { x } - \\varepsilon \\mathrm { s i g n } \\left( - \\bigtriangledown _ { \\mathbf { x } } \\log P _ { \\theta } ( y = \\widehat { y } | \\mathbf { x } ; T ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 349, + 262, + 647, + 280 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\varepsilon$ is a magnitude of noise and $\\widehat { y }$ is the predicted label. Next, ODIN feeds the prebprocessed data into the classifier, computes the maximum value of scaled predictive distribution, i.e., $\\begin{array} { r } { \\operatorname* { m a x } _ { y } P _ { \\theta } ( y | \\mathbf { x } ^ { \\prime } ; T ) } \\end{array}$ , and classifies it as positive (i.e., in-distribution) if the confidence score is above some threshold $\\delta$ . ", + "bbox": [ + 174, + 284, + 825, + 339 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For ODIN detector, the perturbation noise $\\varepsilon$ is chosen from $\\{ 0 , 0 . 0 0 0 1 , 0 . 0 0 1 , 0 . 0 1 \\}$ , and the temperature $T$ is chosen from $\\{ 1 , 1 0 , 1 0 0 , 5 0 0 , 1 0 0 0 \\}$ . The optimal parameters are chosen to minimize the detection error on the validation set. Figure 7 shows the performance of the OIDN and baseline detector for each in- and out-of-distribution pair. First, we remark that the baseline detector using classifiers trained by our joint confidence loss (blue bar) typically outperforms the ODIN detector using classifiers trained by the cross entropy loss (orange bar). This means that our classifier can map in- and out-of-distributions more separately without pre-processing methods such as temperature scaling. The ODIN detector provides further improvements if one uses it with our joint confidence loss (green bar). In other words, our proposed training method can improve all prior detection methods. ", + "bbox": [ + 173, + 345, + 825, + 484 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/1380ade19cbcf946e7ad048c487ea59780f88d50e2289a37c6e92a096d1dc661.jpg", + "image_caption": [ + "Figure 7: Performances of the baseline detector (Hendrycks & Gimpel, 2016) and ODIN detector (Liang et al., 2017) under various training losses. " + ], + "image_footnote": [], + "bbox": [ + 176, + 502, + 802, + 794 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.4 EXPERIMENTAL RESULTS ON ALEXNET ", + "text_level": 1, + "bbox": [ + 176, + 856, + 491, + 869 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Table 4 shows the detection performance for each in- and out-of-distribution pair when the classifier is AlexNet (Krizhevsky, 2014), which is one of popular CNN architectures. We remark that they show similar trends. ", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/20b4a80f454ba766eee4ab280d4bd4db286f17e2e6a1aaa3963996a9e958d931.jpg", + "table_caption": [ + "Table 4: Performance of the baseline detector (Hendrycks & Gimpel, 2016) and ODIN detector (Liang et al., 2017) using AlexNet. All values are percentages and boldface values indicate relative the better results. For each in-distribution, we minimize the KL divergence term in (1) using training samples from an out-of-distribution dataset denoted by “seen”, where other “unseen” outof-distributions were also used for testing. " + ], + "table_footnote": [], + "table_body": "
In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
Cross entropy loss /Confidence loss
Baseline (SVHN)CIFAR-10 (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)92.14 / 93.7742.0 / 99.9 45.6 /99.9 44.6/100.0 58.6/100.088.0/100.0 89.4/100.0 89.8/100.0 94.2/100.083.4/99.8 84.3/99.9 84.5 /99.9 88.8/100.088.7 /99.9 90.2/100.0 90.8/100.0 95.5/100.087.3/99.3 88.6/99.3 88.4/99.3 92.5 / 99.3
Baseline (CIFAR-10)SVHN (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)76.58 /76.1812.8 / 99.6 10.3/10.1 10.7 /8.1 6.7 /1.071.0 /99.9 59.2/52.1 56.3 / 51.5 49.6/13.573.2/99.6 64.2/62.0 64.3/61.8 61.3 / 50.074.3 /99.9 63.6/59.8 62.3/ 59.5 58.5 /43.770.7 /99.6 64.4/62.3 65.3/ 61.6 59.5/32.0
ODIN (SVHN)CIFAR-10 (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)92.14 /93.7755.5/99.9 59.5 /99.9 61.5 /100.0 82.6/100.089.1/99.2 90.5 /99.3 91.8 /99.382.4/99.8 83.8 /99.9 84.8 /99.985.9 /100.0 87.5 /100.0 90.5/100.089.0 /99.2 90.4 /99.3 91.3/99.3
37.1/99.697.0 /99.3 86.7 /99.691.6/100.0 79.3 /99.6 64.4/ 61.897.4/100.0 88.1/99.996.4 /99.3 84.2/99.6
ODIN (CIFAR-10)SVHN (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)76.58 / 76.1811.4 /8.4 13.3 / 7.1 3.8/ 0.069.1/ 65.6 71.9 /67.1 70.9/57.275.3/63.7 69.3 /40.471.4 /68.6 67.2/72.0 78.1/56.164.6/60.7 65.3/60.5 60.7 /40.7
", + "bbox": [ + 174, + 101, + 825, + 327 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D MAXIMIZING ENTROPY ", + "text_level": 1, + "bbox": [ + 176, + 433, + 403, + 449 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "One might expect that the entropy of out-of-distribution is expected to be much higher compared to that of in-distribution since the out-of-distribution is typically on a much larger space than the indistribution. Therefore, one can add maximizing the entropy of generator distribution to new GAN loss in (3) and joint confidence loss in (4). However, maximizing the entropy of generator distribution is technically challenging since a GAN does not model the generator distribution explicitly. To handle the issue, one can leverage the pull-away term (PT) (Zhao et al., 2017): ", + "bbox": [ + 173, + 465, + 826, + 550 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/392755eedc3b89d2a2f62e713f641f76d585f9831e1d675d4fc2cacecc3eb5f3.jpg", + "text": "$$\n- \\mathcal { H } \\left( P _ { G } \\left( \\mathbf { x } \\right) \\right) \\backsimeq \\mathcal { P T } \\left( P _ { G } \\left( \\mathbf { x } \\right) \\right) = \\frac { 1 } { M ( M - 1 ) } \\sum _ { i = 1 } ^ { M } \\sum _ { j \\neq i } \\left( \\frac { G \\left( \\mathbf { z } _ { i } \\right) ^ { \\top } G \\left( \\mathbf { z } _ { j } \\right) } { \\| G \\left( \\mathbf { z } _ { i } \\right) \\| \\| G \\left( \\mathbf { z } _ { j } \\right) \\| } \\right) ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 235, + 559, + 758, + 606 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $\\mathcal { H } \\left( \\cdot \\right)$ denotes the entropy, $\\mathbf { z } _ { i } , \\mathbf { z } _ { j } \\sim P _ { \\mathrm { p r i } } \\left( \\mathbf { z } \\right)$ and $M$ is the number of samples. Intuitively, one can expect the effect of increasing the entropy by minimizing PT since it corresponds to the squared cosine similarity of generated samples. We note that (Dai et al., 2017) also used PT to maximize the entropy. Similarly as in Section 3.2, we verify the effects of PT using VGGNet. Table 5 shows the performance of the baseline detector for each in- and out-of-distribution pair. We found that joint confidence loss with PT tends to (but not always) improve the detection performance. However, since PT increases the training complexity and the gains from PT are relatively marginal (or controversial), we leave it as an auxiliary option for improving the performance. ", + "bbox": [ + 173, + 612, + 825, + 726 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/4c4913cb128da0298b5b85b8f3bbd5a51ac938525257a11eb997068a05f95b0b.jpg", + "table_caption": [ + "Table 5: Performance of the baseline detector (Hendrycks & Gimpel, 2016) using VGGNets trained by joint confidence loss with and without pull-away term (PT). All values are percentages and boldface values indicate relative the better results. " + ], + "table_footnote": [], + "table_body": "
In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
CIFAR-10Joint confidence loss without PT/with PT
SVHNTinyImageNet LSUN93.81 /94.0590.1 /92.3 99.0 / 99.9 98.9 /100.097.6 /98.1 99.6 /100.0 99.6 /100.093.6 /94.6 97.6 /99.7 97.5 / 99.997.7 /98.2 99.7 /100.0 99.7 /100.097.9 /98.7 94.5 /100.0 95.5 /100.0
CIFAR-10SVHN TinyImageNet LSUN81.39 / 80.6025.4 / 13.2 35.0 /44.8 39.1 /49.166.8 / 69.5 72.0 / 78.4 75.1 /80.774.2 / 75.1 76.4 / 77.6 77.8 / 78.771.3 / 73.5 74.7 / 79.4 77.1 / 81.378.3 / 72.0 82.2 /84.4 83.6 /85.8
", + "bbox": [ + 176, + 739, + 825, + 862 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "E ADDING OUT-OF-DISTRIBUTION CLASS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 534, + 118 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Instead of forcing the predictive distribution on out-of-distribution samples to be closer to the uniform one, one can simply add an additional “out-of-distribution” class to a classifier as follows: ", + "bbox": [ + 173, + 133, + 823, + 161 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/7ccc949e04abd22deabc19c1ba301720db5b8b68d86bee89f432a7d195f472a2.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\mathbb { E } _ { P _ { \\mathrm { i n } } ( \\widehat { \\mathbf { x } } , \\widehat { y } ) } \\big [ - \\log P _ { \\theta } \\left( y = \\widehat { y } | \\widehat { \\mathbf { x } } \\right) \\big ] + \\mathbb { E } _ { P _ { \\mathrm { o u t } } ( \\mathbf { x } ) } \\big [ - \\log P _ { \\theta } \\left( y = K + 1 | \\mathbf { x } \\right) \\big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 245, + 167, + 748, + 191 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\theta$ is a model parameter. Similarly as in Section 3.1, we compare the performance of the confidence loss with that of above loss in (5) using VGGNets with for image classification on SVHN dataset. To optimize the KL divergence term in confidence loss and the second term of (5), CIFAR10 training datasets are used. In order to compare the detection performance, we define the the confidence score of input $\\mathbf { x }$ as $1 - P _ { \\theta }$ $\\begin{array} { r } { \\mathbf { \\boldsymbol { y } } = K + 1 | \\mathbf { \\boldsymbol { x } } ) } \\end{array}$ in case of (5). Table 6 shows the detection performance for out-of-distribution. First, the classifier trained by our method often significantly outperforms the alternative adding the new class label. This is because modeling explicitly out-ofdistribution can incur overfitting to trained out-of-distribution dataset. ", + "bbox": [ + 173, + 196, + 825, + 310 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/e7d2acb22e4ad021a5e07cc5d2c5a8893089c7e3340017ccd5fb173ff90bca92.jpg", + "table_caption": [ + "Table 6: Performance of the baseline detector (Hendrycks & Gimpel, 2016) and ODIN detector (Liang et al., 2017) using VGGNet. All values are percentages and boldface values indicate relative the better results. " + ], + "table_footnote": [], + "table_body": "
DetectorOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
SVHN (seen)99.6 /99.8K+1 class loss in (5)/Confidence loss99.8 /99.999.7 /99.899.8 /99.999.9 /99.8
Baseline detectorTinyImageNet (unseen)79.61 /80.560.0 /9.95.2 /31.851.3 /58.650.7 /55.364.3 / 66.1
LSUN (unseen) Gaussian (unseen)0.0 /10.55.6 /34.851.5 /60.250.8/56.471.5 / 68.0
ODINSVHN (seen)0.0 /3.30.1/ 14.150.0 /50.0 99.1 /99.849.3 /49.412.2 /47.0
TinyImageNet (unseen)99.6 /99.899.9 /99.855.12/65.799.9 /99.999.9 /99.8
detectorLSUN (unseen)79.61 /80.560.3 /12.247.3 /70.656.6 /72.744.3 / 65.6
Gaussian (unseen)0.1 /13.7 0.0 /8.248.3 /73.1 28.3 /68.355.9 / 67.9 54.4 / 65.457.5 /75.2 47.8 /74.144.7 / 67.8 36.8 / 61.5
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Let an input", + "type": "text" + }, + { + "bbox": [ + 199, + 605, + 228, + 615 + ], + "score": 0.9, + "content": "\\mathbf { x } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 605, + 276, + 617 + ], + "score": 1.0, + "content": "and a label", + "type": "text" + }, + { + "bbox": [ + 277, + 605, + 365, + 617 + ], + "score": 0.91, + "content": "y \\in { \\mathcal { Y } } = \\mathbf { \\bar { \\rho } } \\qquad ", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "be random variables that follow a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 192, + 630 + ], + "score": 1.0, + "content": "joint data distribution", + "type": "text" + }, + { + "bbox": [ + 193, + 616, + 318, + 628 + ], + "score": 0.89, + "content": "P _ { \\mathrm { i n } } \\left( \\mathbf { x } , y \\right) = P _ { \\mathrm { i n } } \\left( y | \\mathbf { x } \\right) P _ { \\mathrm { i n } } \\left( \\mathbf { x } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 615, + 430, + 630 + ], + "score": 1.0, + "content": ". We assume that a classifier", + "type": "text" + }, + { + "bbox": [ + 430, + 616, + 465, + 628 + ], + "score": 0.73, + "content": "P _ { \\boldsymbol { \\theta } } \\left( y | \\mathbf { x } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 615, + 506, + 630 + ], + "score": 1.0, + "content": "is trained", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 207, + 640 + ], + "score": 1.0, + "content": "on a dataset drawn from", + "type": "text" + }, + { + "bbox": [ + 207, + 628, + 248, + 639 + ], + "score": 0.91, + "content": "P _ { \\mathrm { i n } } \\left( \\mathbf { x } , y \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 627, + 280, + 640 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 280, + 628, + 286, + 637 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 627, + 438, + 640 + ], + "score": 1.0, + "content": "denotes the model parameter. We let", + "type": "text" + }, + { + "bbox": [ + 438, + 628, + 459, + 638 + ], + "score": 0.72, + "content": "P _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 627, + 474, + 638 + ], + "score": 0.32, + "content": "\\mathbf { \\tau } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "denote", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 359, + 650 + ], + "score": 1.0, + "content": "an out-of-distribution which is ‘far away’ from in-distribution", + "type": "text" + }, + { + "bbox": [ + 360, + 638, + 391, + 650 + ], + "score": 0.84, + "content": "P _ { \\mathrm { i n } } \\left( \\mathbf { x } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 637, + 505, + 650 + ], + "score": 1.0, + "content": ". Our problem of interest is", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 190, + 662 + ], + "score": 1.0, + "content": "determining if input", + "type": "text" + }, + { + "bbox": [ + 190, + 650, + 198, + 659 + ], + "score": 0.61, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 648, + 231, + 662 + ], + "score": 1.0, + "content": "is from", + "type": "text" + }, + { + "bbox": [ + 231, + 649, + 247, + 660 + ], + "score": 0.9, + "content": "P _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 648, + 260, + 662 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 260, + 649, + 280, + 660 + ], + "score": 0.93, + "content": "P _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 648, + 465, + 662 + ], + "score": 1.0, + "content": ", possibly utilizing a well calibrated classifier", + "type": "text" + }, + { + "bbox": [ + 466, + 649, + 501, + 661 + ], + "score": 0.9, + "content": "P _ { \\boldsymbol { \\theta } } \\left( y | \\mathbf { x } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 648, + 505, + 662 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 277, + 672 + ], + "score": 1.0, + "content": "In other words, we aim to build a detector,", + "type": "text" + }, + { + "bbox": [ + 277, + 660, + 356, + 672 + ], + "score": 0.92, + "content": "g ( \\mathbf { x } ) : \\mathcal { X } \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 659, + 505, + 672 + ], + "score": 1.0, + "content": ", which assigns label 1 if data is from", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 670, + 258, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 258, + 683 + ], + "score": 1.0, + "content": "in-distribution, and label 0 otherwise.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "There have been recent efforts toward developing efficient detection methods where they mostly", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 710 + ], + "score": 1.0, + "content": "have studied simple threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017) uti-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 286, + 722 + ], + "score": 1.0, + "content": "lizing a pre-trained classifier. 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However, the state-of-art", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 469, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 469, + 281 + ], + "score": 1.0, + "content": "deep neural networks are known to be highly overconfident in their predictions,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 278, + 469, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 469, + 291 + ], + "score": 1.0, + "content": "i.e., do not distinguish in- and out-of-distributions. Recently, to handle this is-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "score": 1.0, + "content": "sue, several threshold-based detectors have been proposed given pre-trained neu-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 301, + 469, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 469, + 313 + ], + "score": 1.0, + "content": "ral classifiers. However, the performance of prior works highly depends on how", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 470, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 470, + 325 + ], + "score": 1.0, + "content": "to train the classifiers since they only focus on improving inference procedures.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 470, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 470, + 335 + ], + "score": 1.0, + "content": "In this paper, we develop a novel training method for classifiers so that such in-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 469, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 346 + ], + "score": 1.0, + "content": "ference algorithms can work better. In particular, we suggest two additional terms", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 345, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 142, + 345, + 469, + 357 + ], + "score": 1.0, + "content": "added to the original loss (e.g., cross entropy). The first one forces samples from", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 356, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 356, + 469, + 367 + ], + "score": 1.0, + "content": "out-of-distribution less confident by the classifier and the second one is for (im-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 367, + 469, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 469, + 379 + ], + "score": 1.0, + "content": "plicitly) generating most effective training samples for the first one. In essence,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 378, + 470, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 378, + 470, + 389 + ], + "score": 1.0, + "content": "our method jointly trains both classification and generative neural networks for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 389, + 469, + 400 + ], + "spans": [ + { + "bbox": [ + 141, + 389, + 469, + 400 + ], + "score": 1.0, + "content": "out-of-distribution. We demonstrate its effectiveness using deep convolutional", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 399, + 349, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 349, + 412 + ], + "score": 1.0, + "content": "neural networks on various popular image datasets.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 14.5, + "bbox_fs": [ + 141, + 235, + 470, + 412 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 431, + 206, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "Deep neural networks (DNNs) have demonstrated state-of-the-art performance on many classifi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "cation tasks, e.g., speech recognition (Hannun et al., 2014), image classification (Girshick, 2015),", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "video prediction (Villegas et al., 2017) and medical diagnosis (Caruana et al., 2015). 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For example, DNNs trained to clas-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 356, + 524 + ], + "score": 1.0, + "content": "sify MNIST images often produce high confident probability", + "type": "text" + }, + { + "bbox": [ + 356, + 511, + 376, + 522 + ], + "score": 0.88, + "content": "9 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "even for random noise (see the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "work of (Hendrycks & Gimpel, 2016)). Since evaluating the quality of their predictive uncertainty", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "is hard, deploying them in real-world systems raises serious concerns in AI Safety (Amodei et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 559 + ], + "score": 1.0, + "content": "2016), e.g., one can easily break a secure authentication system that can be unlocked by detecting", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 362, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 362, + 568 + ], + "score": 1.0, + "content": "the gaze and iris of eyes using DNNs (Shrivastava et al., 2017).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 456, + 506, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "score": 1.0, + "content": "The overconfidence issue of DNNs is highly related to the problem of detecting out-of-distribution:", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "detect whether a test sample is from in-distribution (i.e., training distribution by a classifier) or out-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "of-distribution sufficiently different from it. 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Let an input", + "type": "text" + }, + { + "bbox": [ + 199, + 605, + 228, + 615 + ], + "score": 0.9, + "content": "\\mathbf { x } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 605, + 276, + 617 + ], + "score": 1.0, + "content": "and a label", + "type": "text" + }, + { + "bbox": [ + 277, + 605, + 365, + 617 + ], + "score": 0.91, + "content": "y \\in { \\mathcal { Y } } = \\mathbf { \\bar { \\rho } } \\qquad ", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "be random variables that follow a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 192, + 630 + ], + "score": 1.0, + "content": "joint data distribution", + "type": "text" + }, + { + "bbox": [ + 193, + 616, + 318, + 628 + ], + "score": 0.89, + "content": "P _ { \\mathrm { i n } } \\left( \\mathbf { x } , y \\right) = P _ { \\mathrm { i n } } \\left( y | \\mathbf { x } \\right) P _ { \\mathrm { i n } } \\left( \\mathbf { x } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 615, + 430, + 630 + ], + "score": 1.0, + "content": ". We assume that a classifier", + "type": "text" + }, + { + "bbox": [ + 430, + 616, + 465, + 628 + ], + "score": 0.73, + "content": "P _ { \\boldsymbol { \\theta } } \\left( y | \\mathbf { x } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 615, + 506, + 630 + ], + "score": 1.0, + "content": "is trained", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 207, + 640 + ], + "score": 1.0, + "content": "on a dataset drawn from", + "type": "text" + }, + { + "bbox": [ + 207, + 628, + 248, + 639 + ], + "score": 0.91, + "content": "P _ { \\mathrm { i n } } \\left( \\mathbf { x } , y \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 627, + 280, + 640 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 280, + 628, + 286, + 637 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 627, + 438, + 640 + ], + "score": 1.0, + "content": "denotes the model parameter. We let", + "type": "text" + }, + { + "bbox": [ + 438, + 628, + 459, + 638 + ], + "score": 0.72, + "content": "P _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 627, + 474, + 638 + ], + "score": 0.32, + "content": "\\mathbf { \\tau } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "denote", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 359, + 650 + ], + "score": 1.0, + "content": "an out-of-distribution which is ‘far away’ from in-distribution", + "type": "text" + }, + { + "bbox": [ + 360, + 638, + 391, + 650 + ], + "score": 0.84, + "content": "P _ { \\mathrm { i n } } \\left( \\mathbf { x } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 637, + 505, + 650 + ], + "score": 1.0, + "content": ". Our problem of interest is", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 190, + 662 + ], + "score": 1.0, + "content": "determining if input", + "type": "text" + }, + { + "bbox": [ + 190, + 650, + 198, + 659 + ], + "score": 0.61, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 648, + 231, + 662 + ], + "score": 1.0, + "content": "is from", + "type": "text" + }, + { + "bbox": [ + 231, + 649, + 247, + 660 + ], + "score": 0.9, + "content": "P _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 648, + 260, + 662 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 260, + 649, + 280, + 660 + ], + "score": 0.93, + "content": "P _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 648, + 465, + 662 + ], + "score": 1.0, + "content": ", possibly utilizing a well calibrated classifier", + "type": "text" + }, + { + "bbox": [ + 466, + 649, + 501, + 661 + ], + "score": 0.9, + "content": "P _ { \\boldsymbol { \\theta } } \\left( y | \\mathbf { x } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 648, + 505, + 662 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 277, + 672 + ], + "score": 1.0, + "content": "In other words, we aim to build a detector,", + "type": "text" + }, + { + "bbox": [ + 277, + 660, + 356, + 672 + ], + "score": 0.92, + "content": "g ( \\mathbf { x } ) : \\mathcal { X } \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 659, + 505, + 672 + ], + "score": 1.0, + "content": ", which assigns label 1 if data is from", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 670, + 258, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 258, + 683 + ], + "score": 1.0, + "content": "in-distribution, and label 0 otherwise.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 572, + 506, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "There have been recent efforts toward developing efficient detection methods where they mostly", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 710 + ], + "score": 1.0, + "content": "have studied simple threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017) uti-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 286, + 722 + ], + "score": 1.0, + "content": "lizing a pre-trained classifier. 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Then, the detector assigns", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 234, + 95 + ], + "score": 1.0, + "content": "label 1 if the confidence score", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 234, + 83, + 254, + 95 + ], + "score": 0.92, + "content": "q ( \\mathbf { x } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 254, + 82, + 293, + 95 + ], + "score": 1.0, + "content": "is above", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 293, + 83, + 299, + 93 + ], + "score": 0.71, + "content": "\\delta", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 300, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ", and label 0, otherwise. 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Namely,", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "they fail to work if the classifier does not separate the maximum value of predictive distribution", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 225, + 162 + ], + "score": 1.0, + "content": "well enough with respect to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 225, + 149, + 241, + 160 + ], + "score": 0.9, + "content": "P _ { \\mathrm { i n } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 242, + 148, + 261, + 162 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 262, + 149, + 282, + 160 + ], + "score": 0.91, + "content": "P _ { \\mathrm { o u t } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 282, + 148, + 506, + 162 + ], + "score": 1.0, + "content": ". Ideally, a classifier should be trained to separate all", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "class-dependent in-distributions as well as out-of-distribution in the output space. As another line of", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "research, Bayesian probabilistic models (Li & Gal, 2017; Louizos & Welling, 2017) and ensembles", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "score": 1.0, + "content": "of classifiers (Lakshminarayanan et al., 2017) were also investigated. However, training or inferring", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 507, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 507, + 206 + ], + "score": 1.0, + "content": "those models are computationally more expensive. This motivates our approach of developing a", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "new training method for the more plausible simple classifiers. Our direction is orthogonal to the", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "Bayesian and ensemble approaches, where one can also combine them for even better performance.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 688, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 234, + 95 + ], + "score": 1.0, + "content": "label 1 if the confidence score", + "type": "text" + }, + { + "bbox": [ + 234, + 83, + 254, + 95 + ], + "score": 0.92, + "content": "q ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 82, + 293, + 95 + ], + "score": 1.0, + "content": "is above", + "type": "text" + }, + { + "bbox": [ + 293, + 83, + 299, + 93 + ], + "score": 0.71, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ", and label 0, otherwise. 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Although such inference methods", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 504, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 504, + 139 + ], + "score": 1.0, + "content": "are computationally simple, their performances highly depend on the pre-trained classifier. Namely,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "they fail to work if the classifier does not separate the maximum value of predictive distribution", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 225, + 162 + ], + "score": 1.0, + "content": "well enough with respect to", + "type": "text" + }, + { + "bbox": [ + 225, + 149, + 241, + 160 + ], + "score": 0.9, + "content": "P _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 148, + 261, + 162 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 262, + 149, + 282, + 160 + ], + "score": 0.91, + "content": "P _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 148, + 506, + 162 + ], + "score": 1.0, + "content": ". Ideally, a classifier should be trained to separate all", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "class-dependent in-distributions as well as out-of-distribution in the output space. As another line of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "research, Bayesian probabilistic models (Li & Gal, 2017; Louizos & Welling, 2017) and ensembles", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "score": 1.0, + "content": "of classifiers (Lakshminarayanan et al., 2017) were also investigated. However, training or inferring", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 507, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 507, + 206 + ], + "score": 1.0, + "content": "those models are computationally more expensive. This motivates our approach of developing a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "new training method for the more plausible simple classifiers. 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In this paper, we develop such a training method for detecting out-of-distribution", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 240, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 127, + 253 + ], + "score": 0.89, + "content": "P _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 240, + 506, + 257 + ], + "score": 1.0, + "content": "better without losing its original classification accuracy. First, we consider a new loss function,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "called confidence loss. Our key idea on the proposed loss is to additionally minimize the Kullback-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "Leibler (KL) divergence from the predictive distribution on out-of-distribution samples to the uni-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "form one in order to give less confident predictions on them. Then, in- and out-of-distributions are", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "expected to be more separable. However, optimizing the confidence loss requires training samples", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "from out-of-distribution, which are often hard to sample: a priori knowledge on out-of-distribution", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "is not available or its underlying space is too huge to cover. 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Unlike the original GAN, the proposed GAN generates ‘boundary’ samples in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 202, + 353 + ], + "score": 1.0, + "content": "the low-density area of", + "type": "text" + }, + { + "bbox": [ + 202, + 341, + 218, + 352 + ], + "score": 0.9, + "content": "P _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 340, + 506, + 353 + ], + "score": 1.0, + "content": ". Finally, we design a joint training scheme minimizing the classifier’s", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "loss and new GAN loss alternatively, i.e., the confident classifier improves the GAN, and vice versa,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "as training proceeds. Here, we emphasize that the proposed GAN does not need to generate explicit", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 374, + 504, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 504, + 386 + ], + "score": 1.0, + "content": "samples under our scheme, and instead it implicitly encourages training a more confident classifier.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "We demonstrate the effectiveness of the proposed method using deep convolutional neural networks", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "such as AlexNet (Krizhevsky, 2014) and VGGNet (Szegedy et al., 2015) for image classification", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "tasks on CIFAR (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), ImageNet (Deng et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "2009), and LSUN (Yu et al., 2015) datasets. The classifier trained by our proposed method dras-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "tically improves the detection performance of all threshold-based detectors (Hendrycks & Gim-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "pel, 2016; Liang et al., 2017) in all experiments. In particular, VGGNet with 13 layers trained by", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "our method improves the true negative rate (TNR), i.e., the fraction of detected out-of-distribution", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 287, + 479 + ], + "score": 1.0, + "content": "(LSUN) samples, compared to the baseline:", + "type": "text" + }, + { + "bbox": [ + 287, + 467, + 357, + 478 + ], + "score": 0.92, + "content": "1 4 . 0 \\% 3 9 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 467, + 376, + 479 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 376, + 467, + 447, + 478 + ], + "score": 0.91, + "content": "4 6 . 3 \\% 9 8 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "on CIFAR-10", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 232, + 491 + ], + "score": 1.0, + "content": "and SVHN, respectively, when", + "type": "text" + }, + { + "bbox": [ + 232, + 478, + 252, + 489 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "of in-distribution samples are correctly detected. We also pro-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "vide visual understandings on the proposed method using the image datasets. We believe that our", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 500, + 482, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 482, + 513 + ], + "score": 1.0, + "content": "method can be a strong guideline when other researchers will pursue these tasks in the future.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 529, + 354, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 355, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 355, + 542 + ], + "score": 1.0, + "content": "2 TRAINING CONFIDENT NEURAL CLASSIFIERS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 566 + ], + "score": 1.0, + "content": "In this section, we propose a novel training method for classifiers in order to improve the perfor-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 564, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 104, + 564, + 505, + 579 + ], + "score": 1.0, + "content": "mance of prior threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017) (see Ap-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "pendix A for more details). Our motivation is that such inference algorithms can work better if the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "classifiers are trained so that they map the samples from in- and out-of-distributions into the output", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "space separately. Namely, we primarily focus on training an improved classifier, and then use prior", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 609, + 355, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 355, + 621 + ], + "score": 1.0, + "content": "detectors under the trained model to measure its performance.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 106, + 635, + 353, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 354, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 354, + 648 + ], + "score": 1.0, + "content": "2.1 CONFIDENT CLASSIFIER FOR OUT-OF-DISTRIBUTION", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 105, + 655, + 504, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 670 + ], + "score": 1.0, + "content": "Without loss of generality, suppose that the cross entropy loss is used for training. Then, we propose", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 667, + 333, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 333, + 678 + ], + "score": 1.0, + "content": "the following new loss function, termed confidence loss:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 683, + 457, + 703 + ], + "lines": [ + { + "bbox": [ + 151, + 683, + 457, + 703 + ], + "spans": [ + { + "bbox": [ + 151, + 683, + 457, + 703 + ], + "score": 0.88, + "content": "\\operatorname* { m i n } _ { \\theta } \\mathbb { E } _ { P _ { \\mathrm { i n } } ( \\widehat { \\mathbf { x } } , \\widehat { y } ) } \\big [ - \\log P _ { \\theta } \\left( y = \\widehat { y } | \\widehat { \\mathbf { x } } \\right) \\big ] + \\beta \\mathbb { E } _ { P _ { \\mathrm { o u t } } ( \\mathbf { x } ) } \\big [ K L \\left( \\mathcal { U } \\left( y \\right) \\mid \\mid P _ { \\theta } \\left( y | \\mathbf { x } \\right) \\right) \\big ] ,", + "type": "interline_equation", + "image_path": "b88b52aa99ec6e4e628705cc11505897fdac6b170e23b5003e6e7cb2ec04707d.jpg" + } + ] + } + ], + "index": 48, + "virtual_lines": [ + { + "bbox": [ + 151, + 683, + 457, + 703 + ], + "spans": [], + "index": 48 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 134, + 723 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 710, + 152, + 720 + ], + "score": 0.84, + "content": "K L", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 709, + 350, + 723 + ], + "score": 1.0, + "content": "denotes the Kullback-Leibler (KL) divergence,", + "type": "text" + }, + { + "bbox": [ + 350, + 709, + 374, + 722 + ], + "score": 0.91, + "content": "\\mathcal { U } \\left( y \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "is the uniform distribution and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 107, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 133, + 732 + ], + "score": 0.91, + "content": "\\beta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "is a penalty parameter. 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Finally, we design a joint training scheme minimizing the classifier’s", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "loss and new GAN loss alternatively, i.e., the confident classifier improves the GAN, and vice versa,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "as training proceeds. Here, we emphasize that the proposed GAN does not need to generate explicit", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 374, + 504, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 504, + 386 + ], + "score": 1.0, + "content": "samples under our scheme, and instead it implicitly encourages training a more confident classifier.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 231, + 506, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "We demonstrate the effectiveness of the proposed method using deep convolutional neural networks", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "such as AlexNet (Krizhevsky, 2014) and VGGNet (Szegedy et al., 2015) for image classification", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "tasks on CIFAR (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), ImageNet (Deng et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "2009), and LSUN (Yu et al., 2015) datasets. The classifier trained by our proposed method dras-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "tically improves the detection performance of all threshold-based detectors (Hendrycks & Gim-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "pel, 2016; Liang et al., 2017) in all experiments. In particular, VGGNet with 13 layers trained by", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "our method improves the true negative rate (TNR), i.e., the fraction of detected out-of-distribution", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 287, + 479 + ], + "score": 1.0, + "content": "(LSUN) samples, compared to the baseline:", + "type": "text" + }, + { + "bbox": [ + 287, + 467, + 357, + 478 + ], + "score": 0.92, + "content": "1 4 . 0 \\% 3 9 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 467, + 376, + 479 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 376, + 467, + 447, + 478 + ], + "score": 0.91, + "content": "4 6 . 3 \\% 9 8 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "on CIFAR-10", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 232, + 491 + ], + "score": 1.0, + "content": "and SVHN, respectively, when", + "type": "text" + }, + { + "bbox": [ + 232, + 478, + 252, + 489 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "of in-distribution samples are correctly detected. We also pro-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "vide visual understandings on the proposed method using the image datasets. We believe that our", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 500, + 482, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 482, + 513 + ], + "score": 1.0, + "content": "method can be a strong guideline when other researchers will pursue these tasks in the future.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 390, + 506, + 513 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 529, + 354, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 355, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 355, + 542 + ], + "score": 1.0, + "content": "2 TRAINING CONFIDENT NEURAL CLASSIFIERS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 566 + ], + "score": 1.0, + "content": "In this section, we propose a novel training method for classifiers in order to improve the perfor-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 564, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 104, + 564, + 505, + 579 + ], + "score": 1.0, + "content": "mance of prior threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017) (see Ap-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "pendix A for more details). Our motivation is that such inference algorithms can work better if the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "classifiers are trained so that they map the samples from in- and out-of-distributions into the output", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "space separately. Namely, we primarily focus on training an improved classifier, and then use prior", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 609, + 355, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 355, + 621 + ], + "score": 1.0, + "content": "detectors under the trained model to measure its performance.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 554, + 506, + 621 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 635, + 353, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 354, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 354, + 648 + ], + "score": 1.0, + "content": "2.1 CONFIDENT CLASSIFIER FOR OUT-OF-DISTRIBUTION", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 105, + 655, + 504, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 670 + ], + "score": 1.0, + "content": "Without loss of generality, suppose that the cross entropy loss is used for training. Then, we propose", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 667, + 333, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 333, + 678 + ], + "score": 1.0, + "content": "the following new loss function, termed confidence loss:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 654, + 505, + 678 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 683, + 457, + 703 + ], + "lines": [ + { + "bbox": [ + 151, + 683, + 457, + 703 + ], + "spans": [ + { + "bbox": [ + 151, + 683, + 457, + 703 + ], + "score": 0.88, + "content": "\\operatorname* { m i n } _ { \\theta } \\mathbb { E } _ { P _ { \\mathrm { i n } } ( \\widehat { \\mathbf { x } } , \\widehat { y } ) } \\big [ - \\log P _ { \\theta } \\left( y = \\widehat { y } | \\widehat { \\mathbf { x } } \\right) \\big ] + \\beta \\mathbb { E } _ { P _ { \\mathrm { o u t } } ( \\mathbf { x } ) } \\big [ K L \\left( \\mathcal { U } \\left( y \\right) \\mid \\mid P _ { \\theta } \\left( y | \\mathbf { x } \\right) \\right) \\big ] ,", + "type": "interline_equation", + "image_path": "b88b52aa99ec6e4e628705cc11505897fdac6b170e23b5003e6e7cb2ec04707d.jpg" + } + ] + } + ], + "index": 48, + "virtual_lines": [ + { + "bbox": [ + 151, + 683, + 457, + 703 + ], + "spans": [], + "index": 48 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 134, + 723 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 710, + 152, + 720 + ], + "score": 0.84, + "content": "K L", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 709, + 350, + 723 + ], + "score": 1.0, + "content": "denotes the Kullback-Leibler (KL) divergence,", + "type": "text" + }, + { + "bbox": [ + 350, + 709, + 374, + 722 + ], + "score": 0.91, + "content": "\\mathcal { U } \\left( y \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "is the uniform distribution and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 107, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 133, + 732 + ], + "score": 0.91, + "content": "\\beta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "is a penalty parameter. It is highly intuitive as the new loss forces the predictive distribution", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "on out-of-distribution samples to be closer to the uniform one, i.e., zero confidence, while that", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 247, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 258 + ], + "score": 1.0, + "content": "for samples from in-distribution still follows the label-dependent probability. In other words, the", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 430, + 270 + ], + "score": 1.0, + "content": "proposed loss is designed for assigning higher maximum prediction values, i.e.,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 430, + 257, + 490, + 269 + ], + "score": 0.91, + "content": "\\operatorname* { m a x } _ { y } P _ { \\theta } \\left( y | \\mathbf { x } \\right)", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 491, + 257, + 506, + 270 + ], + "score": 1.0, + "content": ", to", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "in-distribution samples than out-of-distribution ones. Here, a caveat is that adding the KL divergence", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 280, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "term might degrade the classification performance. However, we found that it is not the case due", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "score": 1.0, + "content": "to the high expressive power of deep neural networks, while in- and out-of-distributions become", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "more separable with respect to the maximum prediction value by optimizing the confidence loss", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 313, + 320, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 320, + 325 + ], + "score": 1.0, + "content": "(see Section 3.1 for supporting experimental results).", + "type": "text", + "cross_page": true + } + ], + "index": 14 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 709, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 126, + 88, + 477, + 172 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 126, + 88, + 477, + 172 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 88, + 477, + 172 + ], + "spans": [ + { + "bbox": [ + 126, + 88, + 477, + 172 + ], + "score": 0.974, + "type": "image", + "image_path": "30ab0a6c3ac8a37d7cbd9bd3d2b7465e59b564853612866bbddd064ad62f4900.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 126, + 88, + 477, + 116.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 126, + 116.0, + 477, + 144.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 126, + 144.0, + 477, + 172.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 178, + 505, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "Figure 1: Illustrating the behavior of classifier under different out-of-distribution training datasets.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 359, + 201 + ], + "score": 1.0, + "content": "We generate the out-of-distribution samples from (a) 2D box", + "type": "text" + }, + { + "bbox": [ + 359, + 189, + 401, + 201 + ], + "score": 0.93, + "content": "[ - 5 0 , 5 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 190, + 505, + 201 + ], + "score": 1.0, + "content": ", and show (b) the corre-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "sponding decision boundary of classifier. We also generate the out-of-distribution samples from (c)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 210, + 441, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 139, + 224 + ], + "score": 1.0, + "content": "2D box", + "type": "text" + }, + { + "bbox": [ + 139, + 211, + 181, + 223 + ], + "score": 0.92, + "content": "[ - 2 0 , 2 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 210, + 441, + 224 + ], + "score": 1.0, + "content": ", and show (d) the corresponding decision boundary of classifier.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "on out-of-distribution samples to be closer to the uniform one, i.e., zero confidence, while that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 247, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 258 + ], + "score": 1.0, + "content": "for samples from in-distribution still follows the label-dependent probability. In other words, the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 430, + 270 + ], + "score": 1.0, + "content": "proposed loss is designed for assigning higher maximum prediction values, i.e.,", + "type": "text" + }, + { + "bbox": [ + 430, + 257, + 490, + 269 + ], + "score": 0.91, + "content": "\\operatorname* { m a x } _ { y } P _ { \\theta } \\left( y | \\mathbf { x } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 257, + 506, + 270 + ], + "score": 1.0, + "content": ", to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "in-distribution samples than out-of-distribution ones. Here, a caveat is that adding the KL divergence", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 280, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "term might degrade the classification performance. However, we found that it is not the case due", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "score": 1.0, + "content": "to the high expressive power of deep neural networks, while in- and out-of-distributions become", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "more separable with respect to the maximum prediction value by optimizing the confidence loss", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 313, + 320, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 320, + 325 + ], + "score": 1.0, + "content": "(see Section 3.1 for supporting experimental results).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 504, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "We remark that minimizing a similar KL loss was studied recently for different purposes (Lee et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 339, + 504, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 504, + 353 + ], + "score": 1.0, + "content": "2017; Pereyra et al., 2017). Training samples for minimizing the KL divergence term is explicitly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "given in their settings while we might not. Ideally, one has to sample all (almost infinite) types of out-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 363, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 374 + ], + "score": 1.0, + "content": "of-distribution to minimize the KL term in (1), or require some prior information on testing out-of-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "distribution for efficient sampling. However, this is often infeasible and fragile. To address the issue,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "we suggest to sample out-of-distribution close to in-distribution, which could be more effective in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 395, + 476, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 476, + 408 + ], + "score": 1.0, + "content": "improving the detection performance, without any assumption on testing out-of-distribution.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "score": 1.0, + "content": "In order to explain our intuition in details, we consider a binary classification task on a simple ex-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "ample, where each class data is drawn from a Gaussian distribution and entire data space is bounded", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 154, + 446 + ], + "score": 1.0, + "content": "by 2D box", + "type": "text" + }, + { + "bbox": [ + 155, + 433, + 198, + 446 + ], + "score": 0.93, + "content": "[ - 5 0 , 5 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "for visualization. We apply the confidence loss to simple fully-connected", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 444, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 505, + 458 + ], + "score": 1.0, + "content": "neural networks (2 hidden layers and 500 hidden units for each layer) using different types of out-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "of-distribution training samples. First, as shown in Figure 1(a), we construct an out-of-distribution", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 458, + 480 + ], + "score": 1.0, + "content": "training dataset of 100 (green) points using rejection sampling on the entire data space", + "type": "text" + }, + { + "bbox": [ + 459, + 466, + 501, + 479 + ], + "score": 0.91, + "content": "[ - 5 0 , 5 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 466, + 505, + 480 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "Figure 1(b) shows the decision boundary of classifier optimizing the confidence loss on the corre-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "sponding dataset. One can observe that a classifier still shows overconfident predictions (red and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "score": 1.0, + "content": "blue regions) near the labeled in-distribution region. On the other hand, if we construct a training", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 291, + 523 + ], + "score": 1.0, + "content": "out-of-distribution dataset of 100 points from", + "type": "text" + }, + { + "bbox": [ + 291, + 510, + 334, + 522 + ], + "score": 0.92, + "content": "[ - 2 0 , 2 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 510, + 506, + 523 + ], + "score": 1.0, + "content": ", i.e., closer to target, in-distribution space", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "(see Figure 1(c)), a classifier produces confident predictions only on the labeled region and zero", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 532, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 325, + 545 + ], + "score": 1.0, + "content": "confidence on the remaining in the entire data space", + "type": "text" + }, + { + "bbox": [ + 326, + 532, + 369, + 544 + ], + "score": 0.93, + "content": "[ - 5 0 , 5 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "as shown in Figure 1(d). If one", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "increases the number of training out-of-distribution samples which are generated from the entire", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 151, + 567 + ], + "score": 1.0, + "content": "space, i.e.,", + "type": "text" + }, + { + "bbox": [ + 151, + 554, + 194, + 566 + ], + "score": 0.92, + "content": "[ - 5 0 , 5 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 554, + 506, + 567 + ], + "score": 1.0, + "content": ", Figure 1(b) is expected to be similar to Figure 1(d). In other words, one need", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "more samples in order to train a confident classifier if samples are generated from the entire space.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 573, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 506, + 591 + ], + "score": 1.0, + "content": "However, this might be impossible and not efficient since the number of out-of-distribution training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 601 + ], + "score": 1.0, + "content": "samples might be almost infinite to cover its entire, huge actual data space. This implies that training", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 596, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 104, + 596, + 506, + 613 + ], + "score": 1.0, + "content": "out-of-distribution samples nearby the in-distribution region could be more effective in improving", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "the detection performance. Our underlying intuition is that the effect of boundary of in-distribution", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "region might propagate to the entire out-of-distribution space. Our experimental results in Section", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "3.1 also support this: realistic images are more useful as training out-of-distribution than synthetic", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 641, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 655 + ], + "score": 1.0, + "content": "datasets (e.g., Gaussian noise) for improving the detection performance when we consider an image", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 652, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 506, + 666 + ], + "score": 1.0, + "content": "classification task. This motivates us to develop a new generative adversarial network (GAN) for", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 664, + 321, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 321, + 676 + ], + "score": 1.0, + "content": "generating such effective out-of-distribution samples.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 690, + 368, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 369, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 369, + 701 + ], + "score": 1.0, + "content": "2.2 ADVERSARIAL GENERATOR FOR OUT-OF-DISTRIBUTION", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In this section, we introduce a new training method for learning a generator of out-of-distribution", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "inspired by generative adversarial network (GAN) (Goodfellow et al., 2014). We will first assume", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 126, + 88, + 477, + 172 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 126, + 88, + 477, + 172 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 88, + 477, + 172 + ], + "spans": [ + { + "bbox": [ + 126, + 88, + 477, + 172 + ], + "score": 0.974, + "type": "image", + "image_path": "30ab0a6c3ac8a37d7cbd9bd3d2b7465e59b564853612866bbddd064ad62f4900.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 126, + 88, + 477, + 116.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 126, + 116.0, + 477, + 144.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 126, + 144.0, + 477, + 172.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 178, + 505, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "Figure 1: Illustrating the behavior of classifier under different out-of-distribution training datasets.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 359, + 201 + ], + "score": 1.0, + "content": "We generate the out-of-distribution samples from (a) 2D box", + "type": "text" + }, + { + "bbox": [ + 359, + 189, + 401, + 201 + ], + "score": 0.93, + "content": "[ - 5 0 , 5 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 190, + 505, + 201 + ], + "score": 1.0, + "content": ", and show (b) the corre-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "sponding decision boundary of classifier. We also generate the out-of-distribution samples from (c)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 210, + 441, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 139, + 224 + ], + "score": 1.0, + "content": "2D box", + "type": "text" + }, + { + "bbox": [ + 139, + 211, + 181, + 223 + ], + "score": 0.92, + "content": "[ - 2 0 , 2 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 210, + 441, + 224 + ], + "score": 1.0, + "content": ", and show (d) the corresponding decision boundary of classifier.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 505, + 324 + ], + "lines": [], + "index": 10.5, + "bbox_fs": [ + 105, + 235, + 506, + 325 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 504, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "We remark that minimizing a similar KL loss was studied recently for different purposes (Lee et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 339, + 504, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 504, + 353 + ], + "score": 1.0, + "content": "2017; Pereyra et al., 2017). Training samples for minimizing the KL divergence term is explicitly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "given in their settings while we might not. Ideally, one has to sample all (almost infinite) types of out-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 363, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 374 + ], + "score": 1.0, + "content": "of-distribution to minimize the KL term in (1), or require some prior information on testing out-of-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "distribution for efficient sampling. However, this is often infeasible and fragile. To address the issue,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "we suggest to sample out-of-distribution close to in-distribution, which could be more effective in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 395, + 476, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 476, + 408 + ], + "score": 1.0, + "content": "improving the detection performance, without any assumption on testing out-of-distribution.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 329, + 506, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "score": 1.0, + "content": "In order to explain our intuition in details, we consider a binary classification task on a simple ex-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "ample, where each class data is drawn from a Gaussian distribution and entire data space is bounded", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 154, + 446 + ], + "score": 1.0, + "content": "by 2D box", + "type": "text" + }, + { + "bbox": [ + 155, + 433, + 198, + 446 + ], + "score": 0.93, + "content": "[ - 5 0 , 5 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "for visualization. We apply the confidence loss to simple fully-connected", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 444, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 505, + 458 + ], + "score": 1.0, + "content": "neural networks (2 hidden layers and 500 hidden units for each layer) using different types of out-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "of-distribution training samples. First, as shown in Figure 1(a), we construct an out-of-distribution", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 458, + 480 + ], + "score": 1.0, + "content": "training dataset of 100 (green) points using rejection sampling on the entire data space", + "type": "text" + }, + { + "bbox": [ + 459, + 466, + 501, + 479 + ], + "score": 0.91, + "content": "[ - 5 0 , 5 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 466, + 505, + 480 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "Figure 1(b) shows the decision boundary of classifier optimizing the confidence loss on the corre-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "sponding dataset. One can observe that a classifier still shows overconfident predictions (red and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "score": 1.0, + "content": "blue regions) near the labeled in-distribution region. On the other hand, if we construct a training", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 291, + 523 + ], + "score": 1.0, + "content": "out-of-distribution dataset of 100 points from", + "type": "text" + }, + { + "bbox": [ + 291, + 510, + 334, + 522 + ], + "score": 0.92, + "content": "[ - 2 0 , 2 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 510, + 506, + 523 + ], + "score": 1.0, + "content": ", i.e., closer to target, in-distribution space", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "(see Figure 1(c)), a classifier produces confident predictions only on the labeled region and zero", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 532, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 325, + 545 + ], + "score": 1.0, + "content": "confidence on the remaining in the entire data space", + "type": "text" + }, + { + "bbox": [ + 326, + 532, + 369, + 544 + ], + "score": 0.93, + "content": "[ - 5 0 , 5 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "as shown in Figure 1(d). If one", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "increases the number of training out-of-distribution samples which are generated from the entire", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 151, + 567 + ], + "score": 1.0, + "content": "space, i.e.,", + "type": "text" + }, + { + "bbox": [ + 151, + 554, + 194, + 566 + ], + "score": 0.92, + "content": "[ - 5 0 , 5 0 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 554, + 506, + 567 + ], + "score": 1.0, + "content": ", Figure 1(b) is expected to be similar to Figure 1(d). In other words, one need", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "more samples in order to train a confident classifier if samples are generated from the entire space.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 573, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 506, + 591 + ], + "score": 1.0, + "content": "However, this might be impossible and not efficient since the number of out-of-distribution training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 601 + ], + "score": 1.0, + "content": "samples might be almost infinite to cover its entire, huge actual data space. This implies that training", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 596, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 104, + 596, + 506, + 613 + ], + "score": 1.0, + "content": "out-of-distribution samples nearby the in-distribution region could be more effective in improving", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "the detection performance. Our underlying intuition is that the effect of boundary of in-distribution", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "region might propagate to the entire out-of-distribution space. Our experimental results in Section", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "3.1 also support this: realistic images are more useful as training out-of-distribution than synthetic", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 641, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 655 + ], + "score": 1.0, + "content": "datasets (e.g., Gaussian noise) for improving the detection performance when we consider an image", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 652, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 506, + 666 + ], + "score": 1.0, + "content": "classification task. This motivates us to develop a new generative adversarial network (GAN) for", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 664, + 321, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 321, + 676 + ], + "score": 1.0, + "content": "generating such effective out-of-distribution samples.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 412, + 506, + 676 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 690, + 368, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 369, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 369, + 701 + ], + "score": 1.0, + "content": "2.2 ADVERSARIAL GENERATOR FOR OUT-OF-DISTRIBUTION", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In this section, we introduce a new training method for learning a generator of out-of-distribution", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "inspired by generative adversarial network (GAN) (Goodfellow et al., 2014). We will first assume", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "that the classifier for in-distribution is fixed, and also describe the joint learning framework in the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 158, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 158, + 105 + ], + "score": 1.0, + "content": "next section.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "that the classifier for in-distribution is fixed, and also describe the joint learning framework in the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 158, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 158, + 105 + ], + "score": 1.0, + "content": "next section.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 399, + 123 + ], + "score": 1.0, + "content": "The GAN framework consists of two main components: discriminator", + "type": "text" + }, + { + "bbox": [ + 399, + 111, + 409, + 120 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 110, + 470, + 123 + ], + "score": 1.0, + "content": "and generator", + "type": "text" + }, + { + "bbox": [ + 471, + 111, + 479, + 120 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 110, + 506, + 123 + ], + "score": 1.0, + "content": ". 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Then, one can", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 369, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 322, + 168 + ], + "score": 1.0, + "content": "optimize the following min-max objective for forcing", + "type": "text" + }, + { + "bbox": [ + 322, + 155, + 365, + 165 + ], + "score": 0.9, + "content": "P _ { G } \\approx P _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 154, + 369, + 168 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 166, + 438, + 186 + ], + "lines": [ + { + "bbox": [ + 172, + 166, + 438, + 186 + ], + "spans": [ + { + "bbox": [ + 172, + 166, + 438, + 186 + ], + "score": 0.89, + "content": "\\operatorname* { m i n } _ { G } \\operatorname* { m a x } _ { D } ~ \\mathbb { E } _ { P _ { \\mathrm { i n } } ( \\mathbf { x } ) } \\big [ \\log D \\left( \\mathbf { x } \\right) \\big ] + \\mathbb { E } _ { P _ { \\mathrm { p r i } } ( \\mathbf { z } ) } \\big [ \\log \\left( 1 - D \\left( G \\left( \\mathbf { z } \\right) \\right) \\right) \\big ] .", + "type": "interline_equation", + "image_path": "6f4c8d8517375f94af4e0f1b7f1258dc4161d64907c250742398349ab46ad988.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 172, + 166, + 438, + 186 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 187, + 507, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "However, unlike the original GAN, we want to make the generator recover an effective out-of-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 197, + 456, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 155, + 212 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 155, + 199, + 175, + 210 + ], + "score": 0.91, + "content": "P _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 197, + 218, + 212 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 218, + 199, + 234, + 210 + ], + "score": 0.9, + "content": "P _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 197, + 456, + 212 + ], + "score": 1.0, + "content": ". To this end, we propose the following new GAN loss:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 210, + 434, + 271 + ], + "lines": [ + { + "bbox": [ + 175, + 210, + 434, + 271 + ], + "spans": [ + { + "bbox": [ + 175, + 210, + 434, + 271 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\underset { G } { \\mathop { \\operatorname* { m i n } } } \\underset { D } { \\mathop { \\operatorname* { m a x } } } } & { \\beta \\underbrace { { \\mathbb { E } } _ { P _ { G } ( \\mathbf { x } ) } \\left[ K L \\left( \\mathcal { U } \\left( y \\right) \\mid P _ { \\theta } \\left( y | \\mathbf { x } \\right) \\right) \\right] } _ { \\mathrm { ( a ) } } } \\\\ & { + \\underbrace { { \\mathbb { E } } _ { P _ { \\mathrm { i n } } \\left( \\mathbf { x } \\right) } \\left[ \\log D \\left( \\mathbf { x } \\right) \\right] + { \\mathbb { E } } _ { P _ { G } ( \\mathbf { x } ) } \\left[ \\log \\left( 1 - D \\left( \\mathbf { x } \\right) \\right) \\right] } _ { \\mathrm { ( b ) } } , } \\end{array}", + "type": "interline_equation", + "image_path": "3445bffdd36296f9eff224c6ffd4e5fcacd267cfc52dfd43a16a7136f34e9992.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 175, + 210, + 434, + 230.33333333333334 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 175, + 230.33333333333334, + 434, + 250.66666666666669 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 175, + 250.66666666666669, + 434, + 271.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 274, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 133, + 287 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 276, + 140, + 285 + ], + "score": 0.82, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "is the model parameter of a classifier trained on in-distribution. The above objective can be", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 283, + 504, + 300 + ], + "spans": [ + { + "bbox": [ + 104, + 283, + 483, + 300 + ], + "score": 1.0, + "content": "interpreted as follows: the first term (a) corresponds to a replacement of the out-of-distribution", + "type": "text" + }, + { + "bbox": [ + 483, + 286, + 504, + 298 + ], + "score": 0.9, + "content": "P _ { \\mathrm { o u t } }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 296, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 104, + 296, + 299, + 310 + ], + "score": 1.0, + "content": "in (1)’s KL loss with the generator distribution", + "type": "text" + }, + { + "bbox": [ + 299, + 297, + 313, + 308 + ], + "score": 0.88, + "content": "P _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 296, + 506, + 310 + ], + "score": 1.0, + "content": ". One can note that this forces the generator to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "generate low-density samples since it can be interpreted as minimizing the log negative likelihood", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 273, + 333 + ], + "score": 1.0, + "content": "of in-distribution using the classifier, i.e.,", + "type": "text" + }, + { + "bbox": [ + 274, + 319, + 434, + 331 + ], + "score": 0.87, + "content": "P _ { \\mathrm { i n } } ( \\mathbf { x } ) \\approx \\exp \\left( K L \\left( \\mathcal { U } \\left( y \\right) \\parallel P _ { \\theta } \\left( y | \\mathbf { x } \\right) \\right) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 319, + 506, + 333 + ], + "score": 1.0, + "content": ". We remark that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "this approximation is also closely related to the inception score (Salimans et al., 2016) which is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "popularly used as a quantitative measure of visual fidelity of the samples. The second term (b) cor-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "responds to the original GAN loss since we would like to have out-of-distribution samples close to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 474, + 375 + ], + "score": 1.0, + "content": "in-distribution, as mentioned in Section 2.1. Suppose that the model parameter of classifier", + "type": "text" + }, + { + "bbox": [ + 474, + 363, + 481, + 373 + ], + "score": 0.74, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "is set", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 374, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 386 + ], + "score": 1.0, + "content": "appropriately such that the classifier produces the uniform distribution for out of distribution sam-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "ples. Then, the KL divergence term (a) in (3) is approximately 0 no matter what out-of-distribution", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "score": 1.0, + "content": "samples are generated. However, if the samples are far away from boundary, the GAN loss (b) in (3)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "should be high, i.e., the GAN loss forces having samples being not too far from the in-distribution", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "space. Therefore, one can expect that proposed loss can encourage the generator to produce the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "samples which are on the low-density boundary of the in-distribution space. We also provide its", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 440, + 263, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 263, + 452 + ], + "score": 1.0, + "content": "experimental evidences in Section 3.2.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 504, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 504, + 468 + ], + "score": 1.0, + "content": "We also remark that (Dai et al., 2017) consider a similar GAN generating samples from out-of-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 504, + 480 + ], + "score": 1.0, + "content": "distribution for the purpose of semi-supervised learning. The authors assume the existence of a pre-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 276, + 491 + ], + "score": 1.0, + "content": "trained density estimation model such as", + "type": "text" + }, + { + "bbox": [ + 276, + 479, + 331, + 489 + ], + "score": 0.44, + "content": "\\mathrm { P i x e l C N N + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "(Salimans et al., 2017) for in-distribution,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "but such a model might not exist and be expensive to train in general. Instead, we use much simpler", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "score": 1.0, + "content": "confident classifiers for approximating the density. Hence, under our fully-supervised setting, our", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 512, + 296, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 296, + 522 + ], + "score": 1.0, + "content": "GAN is much easier to train and more suitable.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 106, + 536, + 497, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 500, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 500, + 549 + ], + "score": 1.0, + "content": "2.3 JOINT TRAINING METHOD OF CONFIDENT CLASSIFIER AND ADVERSARIAL GENERATOR", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 504, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "In the previous section, we suggest training the proposed GAN using a pre-trained confident classi-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "score": 1.0, + "content": "fier. We remind that the converse is also possible, i.e., the motivation of having such a GAN is for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "training a better classifier. Hence, two models can be used for improving each other. This naturally", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "suggests a joint training scheme where the confident classifier improves the proposed GAN, and vice", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 600, + 469, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 469, + 613 + ], + "score": 1.0, + "content": "versa, as training proceeds. Specifically, we suggest the following joint objective function:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 134, + 612, + 477, + 666 + ], + "lines": [ + { + "bbox": [ + 134, + 612, + 477, + 666 + ], + "spans": [ + { + "bbox": [ + 134, + 612, + 477, + 666 + ], + "score": 0.88, + "content": "\\begin{array} { r l } { \\underset { G } { \\mathop { \\operatorname* { m i n } } } \\underset { D } { \\mathop { \\operatorname* { m a x } } } \\underset { \\theta } { \\mathop { \\operatorname* { m i n } } } } & { \\underbrace { { \\mathbb { E } } _ { P _ { \\mathrm { i n } } ( \\widehat { \\mathbf { x } } , \\widehat { y } ) } { [ - \\log P _ { \\theta } ( y = \\widehat { y } | \\widehat { \\mathbf { x } } ) ] } } _ { \\mathrm { ( c ) } } + \\beta \\underbrace { { \\mathbb { E } } _ { P _ { G } ( \\mathbf { x } ) } { [ K L ( \\mathcal { U } ( y ) \\mid | P _ { \\theta } ( y | \\mathbf { x } ) ) ] } } _ { \\mathrm { ( d ) } } } \\\\ & { \\underbrace { + { \\mathbb { E } } _ { P _ { \\mathrm { i n } } ( \\widehat { \\mathbf { x } } ) } { [ \\log D ( \\widehat { \\mathbf { x } } ) ] } + { \\mathbb { E } } _ { P _ { G } ( \\mathbf { x } ) } { [ \\log ( 1 - D ( \\mathbf { x } ) ) ] } . } _ { } } \\end{array}", + "type": "interline_equation", + "image_path": "4599f52db8202c22e1c071f53bd9c20c46211a747226e4dac68c123e73ee7c8a.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 134, + 612, + 477, + 630.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 134, + 630.0, + 477, + 648.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 134, + 648.0, + 477, + 666.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 303, + 690 + ], + "score": 1.0, + "content": "The classifier’s confidence loss corresponds to (", + "type": "text" + }, + { + "bbox": [ + 303, + 677, + 334, + 688 + ], + "score": 0.27, + "content": "{ \\vartriangle } \\cosh + ( \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 676, + 505, + 690 + ], + "score": 1.0, + "content": ", and the proposed GAN loss corresponds", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 117, + 700 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 688, + 141, + 699 + ], + "score": 0.29, + "content": "\\mathrm { ( d ) + }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "(e), i.e., they share the KL divergence term (d) under joint training. To optimize the above", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 699, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 488, + 712 + ], + "score": 1.0, + "content": "objective efficiently, we propose an alternating algorithm, which optimizes model parameters", + "type": "text" + }, + { + "bbox": [ + 488, + 699, + 504, + 711 + ], + "score": 0.9, + "content": "\\{ \\theta \\}", + "type": "inline_equation" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 230, + 722 + ], + "score": 1.0, + "content": "of classifier and GAN models", + "type": "text" + }, + { + "bbox": [ + 230, + 710, + 262, + 722 + ], + "score": 0.93, + "content": "\\{ G , D \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "alternatively as shown in Algorithm 1. Since the algorithm", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 413, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 413, + 734 + ], + "score": 1.0, + "content": "monotonically decreases the objective function, it is guaranteed to converge.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 105 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 399, + 123 + ], + "score": 1.0, + "content": "The GAN framework consists of two main components: discriminator", + "type": "text" + }, + { + "bbox": [ + 399, + 111, + 409, + 120 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 110, + 470, + 123 + ], + "score": 1.0, + "content": "and generator", + "type": "text" + }, + { + "bbox": [ + 471, + 111, + 479, + 120 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 110, + 506, + 123 + ], + "score": 1.0, + "content": ". 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Then, one can", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 369, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 322, + 168 + ], + "score": 1.0, + "content": "optimize the following min-max objective for forcing", + "type": "text" + }, + { + "bbox": [ + 322, + 155, + 365, + 165 + ], + "score": 0.9, + "content": "P _ { G } \\approx P _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 154, + 369, + 168 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 110, + 506, + 168 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 166, + 438, + 186 + ], + "lines": [ + { + "bbox": [ + 172, + 166, + 438, + 186 + ], + "spans": [ + { + "bbox": [ + 172, + 166, + 438, + 186 + ], + "score": 0.89, + "content": "\\operatorname* { m i n } _ { G } \\operatorname* { m a x } _ { D } ~ \\mathbb { E } _ { P _ { \\mathrm { i n } } ( \\mathbf { x } ) } \\big [ \\log D \\left( \\mathbf { x } \\right) \\big ] + \\mathbb { E } _ { P _ { \\mathrm { p r i } } ( \\mathbf { z } ) } \\big [ \\log \\left( 1 - D \\left( G \\left( \\mathbf { z } \\right) \\right) \\right) \\big ] .", + "type": "interline_equation", + "image_path": "6f4c8d8517375f94af4e0f1b7f1258dc4161d64907c250742398349ab46ad988.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 172, + 166, + 438, + 186 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 187, + 507, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "However, unlike the original GAN, we want to make the generator recover an effective out-of-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 197, + 456, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 155, + 212 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 155, + 199, + 175, + 210 + ], + "score": 0.91, + "content": "P _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 197, + 218, + 212 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 218, + 199, + 234, + 210 + ], + "score": 0.9, + "content": "P _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 197, + 456, + 212 + ], + "score": 1.0, + "content": ". To this end, we propose the following new GAN loss:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 187, + 505, + 212 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 210, + 434, + 271 + ], + "lines": [ + { + "bbox": [ + 175, + 210, + 434, + 271 + ], + "spans": [ + { + "bbox": [ + 175, + 210, + 434, + 271 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\underset { G } { \\mathop { \\operatorname* { m i n } } } \\underset { D } { \\mathop { \\operatorname* { m a x } } } } & { \\beta \\underbrace { { \\mathbb { E } } _ { P _ { G } ( \\mathbf { x } ) } \\left[ K L \\left( \\mathcal { U } \\left( y \\right) \\mid P _ { \\theta } \\left( y | \\mathbf { x } \\right) \\right) \\right] } _ { \\mathrm { ( a ) } } } \\\\ & { + \\underbrace { { \\mathbb { E } } _ { P _ { \\mathrm { i n } } \\left( \\mathbf { x } \\right) } \\left[ \\log D \\left( \\mathbf { x } \\right) \\right] + { \\mathbb { E } } _ { P _ { G } ( \\mathbf { x } ) } \\left[ \\log \\left( 1 - D \\left( \\mathbf { x } \\right) \\right) \\right] } _ { \\mathrm { ( b ) } } , } \\end{array}", + "type": "interline_equation", + "image_path": "3445bffdd36296f9eff224c6ffd4e5fcacd267cfc52dfd43a16a7136f34e9992.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 175, + 210, + 434, + 230.33333333333334 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 175, + 230.33333333333334, + 434, + 250.66666666666669 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 175, + 250.66666666666669, + 434, + 271.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 274, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 133, + 287 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 276, + 140, + 285 + ], + "score": 0.82, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "is the model parameter of a classifier trained on in-distribution. The above objective can be", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 283, + 504, + 300 + ], + "spans": [ + { + "bbox": [ + 104, + 283, + 483, + 300 + ], + "score": 1.0, + "content": "interpreted as follows: the first term (a) corresponds to a replacement of the out-of-distribution", + "type": "text" + }, + { + "bbox": [ + 483, + 286, + 504, + 298 + ], + "score": 0.9, + "content": "P _ { \\mathrm { o u t } }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 296, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 104, + 296, + 299, + 310 + ], + "score": 1.0, + "content": "in (1)’s KL loss with the generator distribution", + "type": "text" + }, + { + "bbox": [ + 299, + 297, + 313, + 308 + ], + "score": 0.88, + "content": "P _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 296, + 506, + 310 + ], + "score": 1.0, + "content": ". One can note that this forces the generator to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "generate low-density samples since it can be interpreted as minimizing the log negative likelihood", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 273, + 333 + ], + "score": 1.0, + "content": "of in-distribution using the classifier, i.e.,", + "type": "text" + }, + { + "bbox": [ + 274, + 319, + 434, + 331 + ], + "score": 0.87, + "content": "P _ { \\mathrm { i n } } ( \\mathbf { x } ) \\approx \\exp \\left( K L \\left( \\mathcal { U } \\left( y \\right) \\parallel P _ { \\theta } \\left( y | \\mathbf { x } \\right) \\right) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 319, + 506, + 333 + ], + "score": 1.0, + "content": ". We remark that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "this approximation is also closely related to the inception score (Salimans et al., 2016) which is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "popularly used as a quantitative measure of visual fidelity of the samples. The second term (b) cor-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "responds to the original GAN loss since we would like to have out-of-distribution samples close to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 474, + 375 + ], + "score": 1.0, + "content": "in-distribution, as mentioned in Section 2.1. Suppose that the model parameter of classifier", + "type": "text" + }, + { + "bbox": [ + 474, + 363, + 481, + 373 + ], + "score": 0.74, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "is set", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 374, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 386 + ], + "score": 1.0, + "content": "appropriately such that the classifier produces the uniform distribution for out of distribution sam-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "ples. Then, the KL divergence term (a) in (3) is approximately 0 no matter what out-of-distribution", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "score": 1.0, + "content": "samples are generated. However, if the samples are far away from boundary, the GAN loss (b) in (3)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "should be high, i.e., the GAN loss forces having samples being not too far from the in-distribution", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "space. Therefore, one can expect that proposed loss can encourage the generator to produce the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "samples which are on the low-density boundary of the in-distribution space. We also provide its", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 440, + 263, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 263, + 452 + ], + "score": 1.0, + "content": "experimental evidences in Section 3.2.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 275, + 506, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 504, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 504, + 468 + ], + "score": 1.0, + "content": "We also remark that (Dai et al., 2017) consider a similar GAN generating samples from out-of-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 504, + 480 + ], + "score": 1.0, + "content": "distribution for the purpose of semi-supervised learning. The authors assume the existence of a pre-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 276, + 491 + ], + "score": 1.0, + "content": "trained density estimation model such as", + "type": "text" + }, + { + "bbox": [ + 276, + 479, + 331, + 489 + ], + "score": 0.44, + "content": "\\mathrm { P i x e l C N N + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "(Salimans et al., 2017) for in-distribution,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "but such a model might not exist and be expensive to train in general. Instead, we use much simpler", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "score": 1.0, + "content": "confident classifiers for approximating the density. Hence, under our fully-supervised setting, our", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 512, + 296, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 296, + 522 + ], + "score": 1.0, + "content": "GAN is much easier to train and more suitable.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 456, + 506, + 522 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 536, + 497, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 500, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 500, + 549 + ], + "score": 1.0, + "content": "2.3 JOINT TRAINING METHOD OF CONFIDENT CLASSIFIER AND ADVERSARIAL GENERATOR", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 504, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "In the previous section, we suggest training the proposed GAN using a pre-trained confident classi-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "score": 1.0, + "content": "fier. We remind that the converse is also possible, i.e., the motivation of having such a GAN is for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "training a better classifier. Hence, two models can be used for improving each other. This naturally", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "suggests a joint training scheme where the confident classifier improves the proposed GAN, and vice", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 600, + 469, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 469, + 613 + ], + "score": 1.0, + "content": "versa, as training proceeds. 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In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
Cross entropy loss /Confidence loss
SVHNCIFAR-10 (seen) TinyImageNet (unseen)93.82 /94.2347.4 /99.962.6 /99.978.6 /99.971.6 /99.991.2/99.4
49.0 /100.064.6 /100.079.6 /100.072.7 /100.091.6 /99.4
LSUN (unseen) Gaussian (unseen)46.3/100.0 56.1/100.061.8/100.0 72.0 /100.078.2 /100.0 83.4/100.071.1/100.0 77.2/100.090.8 /99.4 92.8 /99.4
CIFAR-10SVHN (seen)80.14 /80.56
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TinyImageNet (unseen)39.6 /31.8
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2.8 / 3.310.2 /14.150.0/50.048.1 /49.439.9 /47.0
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We train convolutional neural networks", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "(CNNs) including VGGNet (Szegedy et al., 2015) and AlexNet (Krizhevsky, 2014) for classifying", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "CIFAR-10 and SVHN datasets. The corresponding test dataset is used as the in-distribution (pos-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "score": 1.0, + "content": "itive) samples to measure the performance. We use realistic images and synthetic noises as the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "out-of-distribution (negative) samples. For evaluation, we measure the following metrics using the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 470, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 485 + ], + "score": 1.0, + "content": "threshold-based detectors (Hendrycks & Gimpel, 2016; Liang et al., 2017): the true negative rate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 146, + 495 + ], + "score": 1.0, + "content": "(TNR) at", + "type": "text" + }, + { + "bbox": [ + 147, + 483, + 167, + 493 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "true positive rate (TPR), the area under the receiver operating characteristic curve", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "(AUROC), the area under the precision-recall curve (AUPR), and the detection accuracy, where", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 505, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 506, + 517 + ], + "score": 1.0, + "content": "larger values of all metrics indicate better detection performances. Due to the space limitation, more", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 515, + 463, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 463, + 528 + ], + "score": 1.0, + "content": "explanations about datasets, metrics and network architectures are given in Appendix B.1", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 394, + 506, + 528 + ] + }, + { + "type": "table", + "bbox": [ + 107, + 539, + 505, + 646 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 539, + 505, + 646 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 539, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 107, + 539, + 505, + 646 + ], + "score": 0.981, + "html": "
In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
Cross entropy loss /Confidence loss
SVHNCIFAR-10 (seen) TinyImageNet (unseen)93.82 /94.2347.4 /99.962.6 /99.978.6 /99.971.6 /99.991.2/99.4
49.0 /100.064.6 /100.079.6 /100.072.7 /100.091.6 /99.4
LSUN (unseen) Gaussian (unseen)46.3/100.0 56.1/100.061.8/100.0 72.0 /100.078.2 /100.0 83.4/100.071.1/100.0 77.2/100.090.8 /99.4 92.8 /99.4
CIFAR-10SVHN (seen)80.14 /80.56
13.7 / 99.8 13.6 /9.946.6 /99.966.6 /99.8 62.6 /58.661.4 /99.9 58.3 /55.373.5 /99.8 71.0 / 66.1
TinyImageNet (unseen)39.6 /31.8
LSUN (unseen) Gaussian (unseen)14.0 / 10.540.7 /34.863.2 / 60.258.7 /56.471.5 /68.0
2.8 / 3.310.2 /14.150.0/50.048.1 /49.439.9 /47.0
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All val-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "score": 1.0, + "content": "ues are percentages and boldface values indicate relative the better results. For each in-distribution,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "we minimize the KL divergence term in (1) using training samples from an out-of-distribution", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 492, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 492, + 700 + ], + "score": 1.0, + "content": "dataset denoted by “seen”, where other “unseen” out-of-distributions were only used for testing.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 654, + 505, + 700 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 104, + 79, + 498, + 198 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 79, + 498, + 198 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 79, + 498, + 198 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 498, + 198 + ], + "score": 0.972, + "type": "image", + "image_path": "82eec49fdb49b288decb05d89feef7881d949c90c40d7146f2041eba0c997d44.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 104, + 79, + 498, + 118.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 104, + 118.66666666666666, + 498, + 158.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 104, + 158.33333333333331, + 498, + 197.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 210, + 505, + 320 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "Figure 2: For all experiments in (a), (b) and (c), we commonly use the SVHN dataset for in-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "distribution. Fraction of the maximum prediction value in softmax scores trained by (a) cross en-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 266, + 245 + ], + "score": 1.0, + "content": "tropy loss and (b) confidence loss: the", + "type": "text" + }, + { + "bbox": [ + 266, + 234, + 272, + 243 + ], + "score": 0.38, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 232, + 505, + 245 + ], + "score": 1.0, + "content": "-axis and y-axis represent the maximum prediction value", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 243, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 257 + ], + "score": 1.0, + "content": "and the fraction of images receiving the corresponding score, respectively. The receiver operating", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 255, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 505, + 266 + ], + "score": 1.0, + "content": "characteristic (ROC) curves under different losses are reported in (c): the red curve corresponds to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "score": 1.0, + "content": "the ROC curve of a model trained by optimizing the naive cross entropy loss, whereas other ones", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 277, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 288 + ], + "score": 1.0, + "content": "correspond to the ROC curves of models trained by optimizing the confidence loss. The KL diver-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "gence term in the confidence loss is optimized using explicit out-of-distribution datasets indicated", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 297, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 104, + 297, + 506, + 312 + ], + "score": 1.0, + "content": "in the parentheses, e.g., Confident loss (LSUN) means that we use the LSUN dataset for optimizing", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 308, + 205, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 205, + 323 + ], + "score": 1.0, + "content": "the KL divergence term.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7.5 + } + ], + "index": 4.25 + }, + { + "type": "title", + "bbox": [ + 108, + 343, + 263, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 266, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 266, + 357 + ], + "score": 1.0, + "content": "3.1 EFFECTS OF CONFIDENCE LOSS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "We first verify the effect of confidence loss in (1) trained by some explicit, say seen, out-of-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 374, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 389 + ], + "score": 1.0, + "content": "distribution datasets. First, we compare the quality of confidence level by applying various training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 387, + 504, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 504, + 398 + ], + "score": 1.0, + "content": "losses. Specifically, the softmax classifier is used and simple CNNs (two convolutional layers fol-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "lowed by three fully-connected layers) are trained by minimizing the standard cross entropy loss on", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "SVHN dataset. We also apply the confidence loss to the models by additionally optimizing the KL", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "divergence term using CIFAR-10 dataset (as training out-of-distribution). In Figure 2(a) and 2(b),", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "we report distributions of the maximum prediction value in softmax scores to evaluate the separation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "quality between in-distribution (i.e., SVHN) and out-of-distributions. It is clear that there exists a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "better separation between the SVHN test set (red bar) and other ones when the model is trained by", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "the confidence loss. Here, we emphasize that the maximum prediction value is also low on even un-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "trained (unseen) out-of-distributions, e.g., TinyImageNet, LSUN and synthetic datasets. Therefore,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "it is expected that one can distinguish in- and out-of-distributions more easily when a classifier is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 496, + 504, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 504, + 507 + ], + "score": 1.0, + "content": "trained by optimizing the confidence loss. To verify that, we obtain the ROC curve using the baseline", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "detector (Hendrycks & Gimpel, 2016) that computes the maximum value of predictive distribution", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "score": 1.0, + "content": "on a test sample and classifies it as positive (i.e., in-distribution) if the confidence score is above", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "some threshold. Figure 2(c) shows the ROC curves when we optimize the KL divergence term on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "various datasets. One can observe that realistic images such as TinyImageNet (aqua line) and LSUN", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "(green line) are more useful than synthetic datasets (orange line) for improving the detection perfor-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "mance. This supports our intuition that out-of-distribution samples close to in-distribution could be", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 573, + 453, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 453, + 584 + ], + "score": 1.0, + "content": "more effective in improving the detection performance as we discussed in Section 2.1.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "We indeed evaluate the performance of the baseline detector for out-of-distribution using large-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "scale CNNs, i.e., VGGNets with 13 layers, under various training scenarios, where more results on", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "AlexNet and ODIN detector (Liang et al., 2017) can be found in Appendix C (the overall trends", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 620, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 636 + ], + "score": 1.0, + "content": "of results are similar). For optimizing the confidence loss in (1), SVHN and CIFAR-10 training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "datasets are used for optimizing the KL divergence term for the cases when the in-distribution is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "CIFAR-10 and SVHN, respectively. Table 1 shows the detection performance for each in- and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "out-of-distribution pair. When the in-distribution is SVHN, the classifier trained by our method", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "drastically improves the detection performance across all out-of-distributions without hurting its", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "original classification performance. However, when the in-distribution is CIFAR-10, the confidence", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "loss does not improve the detection performance in overall, where we expect that this is because", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "the trained/seen SVHN out-of-distribution does not effectively cover all tested out-of-distributions.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "Our joint confidence loss in (4), which was designed under the intuition, resolves the issue of the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 720, + 404, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 404, + 733 + ], + "score": 1.0, + "content": "CIFAR-10 (in-distribution) classification case in Table 1 (see Figure 4(b)).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 40 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 104, + 79, + 498, + 198 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 79, + 498, + 198 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 79, + 498, + 198 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 498, + 198 + ], + "score": 0.972, + "type": "image", + "image_path": "82eec49fdb49b288decb05d89feef7881d949c90c40d7146f2041eba0c997d44.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 104, + 79, + 498, + 118.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 104, + 118.66666666666666, + 498, + 158.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 104, + 158.33333333333331, + 498, + 197.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 210, + 505, + 320 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "Figure 2: For all experiments in (a), (b) and (c), we commonly use the SVHN dataset for in-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "distribution. Fraction of the maximum prediction value in softmax scores trained by (a) cross en-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 266, + 245 + ], + "score": 1.0, + "content": "tropy loss and (b) confidence loss: the", + "type": "text" + }, + { + "bbox": [ + 266, + 234, + 272, + 243 + ], + "score": 0.38, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 232, + 505, + 245 + ], + "score": 1.0, + "content": "-axis and y-axis represent the maximum prediction value", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 243, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 257 + ], + "score": 1.0, + "content": "and the fraction of images receiving the corresponding score, respectively. The receiver operating", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 255, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 505, + 266 + ], + "score": 1.0, + "content": "characteristic (ROC) curves under different losses are reported in (c): the red curve corresponds to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "score": 1.0, + "content": "the ROC curve of a model trained by optimizing the naive cross entropy loss, whereas other ones", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 277, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 288 + ], + "score": 1.0, + "content": "correspond to the ROC curves of models trained by optimizing the confidence loss. The KL diver-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "gence term in the confidence loss is optimized using explicit out-of-distribution datasets indicated", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 297, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 104, + 297, + 506, + 312 + ], + "score": 1.0, + "content": "in the parentheses, e.g., Confident loss (LSUN) means that we use the LSUN dataset for optimizing", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 308, + 205, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 205, + 323 + ], + "score": 1.0, + "content": "the KL divergence term.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7.5 + } + ], + "index": 4.25 + }, + { + "type": "title", + "bbox": [ + 108, + 343, + 263, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 266, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 266, + 357 + ], + "score": 1.0, + "content": "3.1 EFFECTS OF CONFIDENCE LOSS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "We first verify the effect of confidence loss in (1) trained by some explicit, say seen, out-of-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 374, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 389 + ], + "score": 1.0, + "content": "distribution datasets. First, we compare the quality of confidence level by applying various training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 387, + 504, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 504, + 398 + ], + "score": 1.0, + "content": "losses. Specifically, the softmax classifier is used and simple CNNs (two convolutional layers fol-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "lowed by three fully-connected layers) are trained by minimizing the standard cross entropy loss on", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "SVHN dataset. We also apply the confidence loss to the models by additionally optimizing the KL", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "divergence term using CIFAR-10 dataset (as training out-of-distribution). In Figure 2(a) and 2(b),", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "we report distributions of the maximum prediction value in softmax scores to evaluate the separation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "quality between in-distribution (i.e., SVHN) and out-of-distributions. It is clear that there exists a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "better separation between the SVHN test set (red bar) and other ones when the model is trained by", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "the confidence loss. Here, we emphasize that the maximum prediction value is also low on even un-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "trained (unseen) out-of-distributions, e.g., TinyImageNet, LSUN and synthetic datasets. Therefore,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "it is expected that one can distinguish in- and out-of-distributions more easily when a classifier is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 496, + 504, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 504, + 507 + ], + "score": 1.0, + "content": "trained by optimizing the confidence loss. To verify that, we obtain the ROC curve using the baseline", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "detector (Hendrycks & Gimpel, 2016) that computes the maximum value of predictive distribution", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "score": 1.0, + "content": "on a test sample and classifies it as positive (i.e., in-distribution) if the confidence score is above", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "some threshold. Figure 2(c) shows the ROC curves when we optimize the KL divergence term on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "various datasets. One can observe that realistic images such as TinyImageNet (aqua line) and LSUN", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "(green line) are more useful than synthetic datasets (orange line) for improving the detection perfor-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "mance. This supports our intuition that out-of-distribution samples close to in-distribution could be", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 573, + 453, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 453, + 584 + ], + "score": 1.0, + "content": "more effective in improving the detection performance as we discussed in Section 2.1.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 363, + 506, + 584 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "We indeed evaluate the performance of the baseline detector for out-of-distribution using large-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "scale CNNs, i.e., VGGNets with 13 layers, under various training scenarios, where more results on", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "AlexNet and ODIN detector (Liang et al., 2017) can be found in Appendix C (the overall trends", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 620, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 636 + ], + "score": 1.0, + "content": "of results are similar). For optimizing the confidence loss in (1), SVHN and CIFAR-10 training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "datasets are used for optimizing the KL divergence term for the cases when the in-distribution is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "CIFAR-10 and SVHN, respectively. Table 1 shows the detection performance for each in- and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "out-of-distribution pair. When the in-distribution is SVHN, the classifier trained by our method", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "drastically improves the detection performance across all out-of-distributions without hurting its", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "original classification performance. 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We also compare the generated samples", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "of original and proposed GANs on MNIST dataset (LeCun et al., 1998), which consists of hand-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "written digits. For this experiment, we use deep convolutional GANs (DCGANs) (Radford et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "2015). In this case, we use a pre-trained classifier which minimizes the cross entropy on MNIST", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "training samples and the KL divergence on synthetic Gaussian noises. As shown in Figure 3(c)", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "and 3(d), samples of original GAN looks more like digits than those of proposed GAN. Somewhat", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 276, + 410, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 410, + 290 + ], + "score": 1.0, + "content": "interestingly, the proposed GAN still generates some new digit-like images.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 590, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 120, + 77, + 487, + 211 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 120, + 77, + 487, + 211 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 77, + 487, + 211 + ], + "spans": [ + { + "bbox": [ + 120, + 77, + 487, + 211 + ], + "score": 0.974, + "type": "image", + "image_path": "d41db38747ca1efcc752cd0ff7ae1de2ed89e22acec41b351615403610c3ccf6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 120, + 77, + 487, + 121.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 120, + 121.66666666666666, + 487, + 166.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 120, + 166.33333333333331, + 487, + 210.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 217, + 505, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "Figure 5: Guided gradient (sensitivity) maps of the top-1 predicted class with respect to the input", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 229, + 252, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 252, + 240 + ], + "score": 1.0, + "content": "image under various training losses.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 287 + ], + "lines": [ + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "training samples and the KL divergence on synthetic Gaussian noises. As shown in Figure 3(c)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "and 3(d), samples of original GAN looks more like digits than those of proposed GAN. Somewhat", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 276, + 410, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 410, + 290 + ], + "score": 1.0, + "content": "interestingly, the proposed GAN still generates some new digit-like images.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 306 + ], + "score": 1.0, + "content": "We indeed evaluate the performance of our joint confidence loss in (4) utilizing the proposed GAN.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "To this end, we use VGGNets (as classifiers) and DCGANs (as GANs). We also test a variant of con-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "fidence loss which optimizes the KL divergence term on samples from a pre-trained original GAN", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "(implicitly) modeling the in-distribution. One can expect that samples from the original GAN can be", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "also useful for improving the detection performance since it may have bad generalization properties", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "(Arora et al., 2017) and generate a few samples on the low-density boundary as like the proposed", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "GAN. Figure 4 shows the performance of the baseline detector for each in- and out-of-distribution", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "pair. First, observe that the joint confidence loss (blue bar) outperforms the confidence loss with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "some explicit out-of-distribution datasets (green bar). This is quite remarkable since the former", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "is trained only using in-distribution datasets, while the latter utilizes additional out-of-distribution", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "datasets. We also remark that our methods significantly outperform the baseline cross entropy loss", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 412, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 427 + ], + "score": 1.0, + "content": "(red bar) in all cases without harming its original classification performances (see Table 2 in Ap-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "pendix C). Interestingly, the confidence loss with the original GAN (orange bar) is often (but not", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "always) useful for improving the detection performance, whereas that with the proposed GAN (blue", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 446, + 250, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 250, + 458 + ], + "score": 1.0, + "content": "bar) still outperforms it in all cases.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "Finally, we also provide visual interpretations of models using the guided gradient maps (Sprin-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "genberg et al., 2014). Here, the gradient can be interpreted as an importance value of each pixel", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "which influences on the classification decision. As shown in Figure 5, the model trained by the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "cross entropy loss shows sharp gradient maps for both samples from in- and out-of-distributions,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "whereas models trained by the confidence losses do only on samples from in-distribution. For the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "case of SVHN in-distribution, all confidence losses gave almost zero gradients, which matches to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "the results in Figure 4(a): their detection performances are almost perfect. For the case of CIFAR-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "10 distribution, one can now observe that there exists some connection between gradient maps and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "detection performances. This is intuitive because for detecting samples from out-of-distributions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 562, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 504, + 573 + ], + "score": 1.0, + "content": "better, the classifier should look at more pixels as similar importance and the KL divergence term", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 570, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 586 + ], + "score": 1.0, + "content": "forces it. We think that our visualization results might give some ideas in future works for developing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 584, + 411, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 411, + 596 + ], + "score": 1.0, + "content": "better inference methods for detecting out-of-distribution under our models.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 616, + 195, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 197, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 197, + 632 + ], + "score": 1.0, + "content": "4 CONCLUSION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "In this paper, we aim to develop a training method for neural classification networks for detecting", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "out-of-distribution better without losing its original classification accuracy. In essence, our method", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "jointly trains two models for detecting and generating out-of-distribution by minimizing their losses", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "alternatively. Although we primarily focus on image classification in our experiments, our method", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "can be used for any classification tasks using deep neural networks. It is also interesting future", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "directions applying our methods for other related tasks: regression (Malinin et al., 2017), network", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "calibration (Guo et al., 2017), Bayesian probabilistic models (Li & Gal, 2017; Louizos & Welling,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 499, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 499, + 733 + ], + "score": 1.0, + "content": "2017), ensemble (Lakshminarayanan et al., 2017) and semi-supervised learning (Dai et al., 2017).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 120, + 77, + 487, + 211 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 120, + 77, + 487, + 211 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 77, + 487, + 211 + ], + "spans": [ + { + "bbox": [ + 120, + 77, + 487, + 211 + ], + "score": 0.974, + "type": "image", + "image_path": "d41db38747ca1efcc752cd0ff7ae1de2ed89e22acec41b351615403610c3ccf6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 120, + 77, + 487, + 121.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 120, + 121.66666666666666, + 487, + 166.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 120, + 166.33333333333331, + 487, + 210.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 217, + 505, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "Figure 5: Guided gradient (sensitivity) maps of the top-1 predicted class with respect to the input", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 229, + 252, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 252, + 240 + ], + "score": 1.0, + "content": "image under various training losses.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 287 + ], + "lines": [], + "index": 6, + "bbox_fs": [ + 105, + 254, + 505, + 290 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 306 + ], + "score": 1.0, + "content": "We indeed evaluate the performance of our joint confidence loss in (4) utilizing the proposed GAN.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "To this end, we use VGGNets (as classifiers) and DCGANs (as GANs). We also test a variant of con-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "fidence loss which optimizes the KL divergence term on samples from a pre-trained original GAN", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "(implicitly) modeling the in-distribution. One can expect that samples from the original GAN can be", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "also useful for improving the detection performance since it may have bad generalization properties", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "(Arora et al., 2017) and generate a few samples on the low-density boundary as like the proposed", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "GAN. Figure 4 shows the performance of the baseline detector for each in- and out-of-distribution", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "pair. First, observe that the joint confidence loss (blue bar) outperforms the confidence loss with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "some explicit out-of-distribution datasets (green bar). This is quite remarkable since the former", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "is trained only using in-distribution datasets, while the latter utilizes additional out-of-distribution", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "datasets. We also remark that our methods significantly outperform the baseline cross entropy loss", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 412, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 427 + ], + "score": 1.0, + "content": "(red bar) in all cases without harming its original classification performances (see Table 2 in Ap-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "pendix C). Interestingly, the confidence loss with the original GAN (orange bar) is often (but not", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "always) useful for improving the detection performance, whereas that with the proposed GAN (blue", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 446, + 250, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 250, + 458 + ], + "score": 1.0, + "content": "bar) still outperforms it in all cases.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 292, + 506, + 458 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "Finally, we also provide visual interpretations of models using the guided gradient maps (Sprin-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "genberg et al., 2014). Here, the gradient can be interpreted as an importance value of each pixel", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "which influences on the classification decision. As shown in Figure 5, the model trained by the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "cross entropy loss shows sharp gradient maps for both samples from in- and out-of-distributions,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "whereas models trained by the confidence losses do only on samples from in-distribution. For the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "case of SVHN in-distribution, all confidence losses gave almost zero gradients, which matches to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "the results in Figure 4(a): their detection performances are almost perfect. For the case of CIFAR-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "10 distribution, one can now observe that there exists some connection between gradient maps and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "detection performances. This is intuitive because for detecting samples from out-of-distributions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 562, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 504, + 573 + ], + "score": 1.0, + "content": "better, the classifier should look at more pixels as similar importance and the KL divergence term", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 570, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 586 + ], + "score": 1.0, + "content": "forces it. We think that our visualization results might give some ideas in future works for developing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 584, + 411, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 411, + 596 + ], + "score": 1.0, + "content": "better inference methods for detecting out-of-distribution under our models.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 462, + 506, + 596 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 616, + 195, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 197, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 197, + 632 + ], + "score": 1.0, + "content": "4 CONCLUSION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "In this paper, we aim to develop a training method for neural classification networks for detecting", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "out-of-distribution better without losing its original classification accuracy. In essence, our method", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "jointly trains two models for detecting and generating out-of-distribution by minimizing their losses", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "alternatively. Although we primarily focus on image classification in our experiments, our method", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "can be used for any classification tasks using deep neural networks. It is also interesting future", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "directions applying our methods for other related tasks: regression (Malinin et al., 2017), network", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "calibration (Guo et al., 2017), Bayesian probabilistic models (Li & Gal, 2017; Louizos & Welling,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 499, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 499, + 733 + ], + "score": 1.0, + "content": "2017), ensemble (Lakshminarayanan et al., 2017) and semi-supervised learning (Dai et al., 2017).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 642, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 225, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 80, + 226, + 96 + ], + "spans": [ + { + "bbox": [ + 107, + 80, + 226, + 96 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "This work was supported in part by the Institute for Information & communications Technology", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "Promotion(IITP) grant funded by the Korea government(MSIT) (No.2017-0-01778, Development", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "score": 1.0, + "content": "of Explainable Human-level Deep Machine Learning Inference Framework), the ICT R&D program", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 504, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 504, + 151 + ], + "score": 1.0, + "content": "of MSIP/IITP [R-20161130-004520, Research on Adaptive Machine Learning Technology Devel-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 164 + ], + "score": 1.0, + "content": "opment for Intelligent Autonomous Digital Companion], DARPA Explainable AI (XAI) program", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 273, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 273, + 174 + ], + "score": 1.0, + "content": "#313498 and Sloan Research Fellowship.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 188, + 175, + 200 + ], + "lines": [ + { + "bbox": [ + 106, + 188, + 176, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 176, + 201 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 206, + 503, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 219 + ], + "score": 1.0, + "content": "Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mane. 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Convolutional layers", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 388, + 464 + ], + "score": 1.0, + "content": "have 128 and 256 filters, respectively. 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Fully-connected layers have 2048, 1024 and 10 hidden", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "units, respectively. Dropout (Hinton et al., 2012) was applied to only fully-connected layers of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "network with the probability of retaining the unit being 0.5. All hidden units are ReLUs. Figure", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "6 shows the detailed structure of VGGNet (Szegedy et al., 2015) with three fully-connected layers", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 403, + 585 + ], + "score": 1.0, + "content": "and 10 convolutional layers. Each ConvReLU box in the figure indicates a", + "type": "text" + }, + { + "bbox": [ + 403, + 573, + 425, + 583 + ], + "score": 0.9, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "convolutional layer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 363, + 596 + ], + "score": 1.0, + "content": "followed by ReLU activation. Also, all max pooling layers have", + "type": "text" + }, + { + "bbox": [ + 363, + 583, + 387, + 594 + ], + "score": 0.9, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "receptive fields with stride 2.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 592, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 506, + 609 + ], + "score": 1.0, + "content": "Dropout was applied to only fully-connected layers of the network with the probability of retaining", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "the unit being 0.5. For all experiments, the softmax classifier is used, and each model is trained by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "optimizing the objective function using Adam learning rule (Kingma & Ba, 2014). For each out-of-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 448, + 640 + ], + "score": 1.0, + "content": "distribution dataset, we randomly select 1,000 images for tuning the penalty parameter", + "type": "text" + }, + { + "bbox": [ + 448, + 627, + 456, + 638 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 626, + 506, + 640 + ], + "score": 1.0, + "content": ", mini-batch", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 349, + 650 + ], + "score": 1.0, + "content": "size and learning rate. The penalty parameter is chosen from", + "type": "text" + }, + { + "bbox": [ + 350, + 638, + 441, + 650 + ], + "score": 0.89, + "content": "\\beta \\in \\{ 0 , 0 . 1 , . . . 1 . 9 , 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 638, + 505, + 650 + ], + "score": 1.0, + "content": ", the mini-batch", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 189, + 662 + ], + "score": 1.0, + "content": "size is chosen from", + "type": "text" + }, + { + "bbox": [ + 189, + 649, + 229, + 661 + ], + "score": 0.93, + "content": "\\{ 6 4 , 1 \\bar { 2 } 8 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 648, + 382, + 662 + ], + "score": 1.0, + "content": "and the learning rate is chosen from", + "type": "text" + }, + { + "bbox": [ + 383, + 649, + 480, + 661 + ], + "score": 0.86, + "content": "\\{ 0 . 0 0 1 , 0 . 0 0 0 5 , 0 . 0 0 0 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 648, + 506, + 662 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "optimal parameters are chosen to minimize the detection error on the validation set. We drop the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "learning rate by 0.1 at 60 epoch and models are trained for total 100 epochs. The best test result is", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 682, + 212, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 212, + 694 + ], + "score": 1.0, + "content": "reported for each method.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 469, + 710 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 471, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 471, + 712 + ], + "score": 1.0, + "content": "Performance metrics. We measure the following metrics using threshold-based detectors:", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + } + ], + "page_idx": 10, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 721, + 307, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 308, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 308, + 734 + ], + "score": 1.0, + "content": "2We do not use the extra SVHN dataset for training.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 294, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 296, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 296, + 96 + ], + "score": 1.0, + "content": "A THRESHOLD-BASED DETECTORS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 505, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 107, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 505, + 119 + ], + "score": 1.0, + "content": "In this section, we formally describe the detection procedure of threshold-based detectors", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 119, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 372, + 130 + ], + "score": 1.0, + "content": "(Hendrycks & Gimpel, 2016; Liang et al., 2017). For each data", + "type": "text" + }, + { + "bbox": [ + 372, + 120, + 380, + 128 + ], + "score": 0.37, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 119, + 505, + 130 + ], + "score": 1.0, + "content": ", it measures some confidence", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 130, + 141 + ], + "score": 1.0, + "content": "score", + "type": "text" + }, + { + "bbox": [ + 130, + 129, + 150, + 141 + ], + "score": 0.91, + "content": "q ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 129, + 505, + 141 + ], + "score": 1.0, + "content": "by feeding the data into a pre-trained classifier. Here, (Hendrycks & Gimpel, 2016) de-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 505, + 153 + ], + "score": 1.0, + "content": "fined the confidence score as a maximum value of the predictive distribution, and (Liang et al., 2017)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 164 + ], + "score": 1.0, + "content": "further improved the performance by processing the predictive distribution (see Appendix C.3 for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 241, + 175 + ], + "score": 1.0, + "content": "more details). Then, the detector,", + "type": "text" + }, + { + "bbox": [ + 241, + 162, + 319, + 174 + ], + "score": 0.92, + "content": "g ( \\mathbf { x } ) : \\mathcal { X } \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 162, + 475, + 175 + ], + "score": 1.0, + "content": ", assigns label 1 if the confidence score", + "type": "text" + }, + { + "bbox": [ + 475, + 162, + 495, + 174 + ], + "score": 0.92, + "content": "q ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 162, + 506, + 175 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 173, + 298, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 195, + 185 + ], + "score": 1.0, + "content": "above some threshold", + "type": "text" + }, + { + "bbox": [ + 196, + 173, + 201, + 183 + ], + "score": 0.79, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 173, + 298, + 185 + ], + "score": 1.0, + "content": ", and label 0, otherwise:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 107, + 506, + 185 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 191, + 367, + 219 + ], + "lines": [ + { + "bbox": [ + 241, + 191, + 367, + 219 + ], + "spans": [ + { + "bbox": [ + 241, + 191, + 367, + 219 + ], + "score": 0.94, + "content": "g \\left( \\mathbf { x } \\right) = \\left\\{ \\begin{array} { l l } { \\begin{array} { r l } { 1 } & { \\mathrm { i f } q ( \\mathbf { x } ) \\geq \\delta , } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } . } \\end{array} } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "acce017d602a5d371b5e582ade033557dec4656e2e013103a6689702187aab67.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 241, + 191, + 367, + 205.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 241, + 205.0, + 367, + 219.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "For this detector, we have to find a score threshold so that some positive examples are classified", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "correctly, but this depends upon the trade-off between false negatives and false positives. To handle", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "this issue, we use threshold-independent evaluation metrics such as area under the receiver operating", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 399, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 399, + 272 + ], + "score": 1.0, + "content": "characteristic curve (AUORC) and detection accuracy (see Appendix B).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 226, + 505, + 272 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 288, + 324, + 301 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 324, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 324, + 302 + ], + "score": 1.0, + "content": "B EXPERIMENTAL SETUPS IN SECTION 3", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "score": 1.0, + "content": "Datasets. We train deep models such as VGGNet (Szegedy et al., 2015) and AlexNet (Krizhevsky,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 323, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 338 + ], + "score": 1.0, + "content": "2014) for classifying CIFAR-10 and SVHN datasets: the former consists of 50,000 training and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "10,000 test images with 10 image classes, and the latter consists of 73,257 training and 26,032 test", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "score": 1.0, + "content": "images with 10 digits.2 The corresponding test dataset are used as the in-distribution (positive)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "samples to measure the performance. We use realistic images and synthetic noises as the out-of-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 367, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 383 + ], + "score": 1.0, + "content": "distribution (negative) samples: the TinyImageNet consists of 10,000 test images with 200 image", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "classes from a subset of ImageNet images. The LSUN consists of 10,000 test images of 10 different", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 406, + 403 + ], + "score": 1.0, + "content": "scenes. We downsample each image of TinyImageNet and LSUN to size", + "type": "text" + }, + { + "bbox": [ + 407, + 391, + 441, + 402 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 390, + 505, + 403 + ], + "score": 1.0, + "content": ". The Gaussian", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 402, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 414 + ], + "score": 1.0, + "content": "noise is independently and identically sampled from a Gaussian distribution with mean 0.5 and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 412, + 335, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 310, + 425 + ], + "score": 1.0, + "content": "variance 1. We clip each pixel value into the range", + "type": "text" + }, + { + "bbox": [ + 311, + 413, + 331, + 425 + ], + "score": 0.55, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 412, + 335, + 425 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 313, + 506, + 425 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Detailed CNN structure and training. The simple CNN that we use for evaluation shown in Figure", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "score": 1.0, + "content": "2 consists of two convolutional layers followed by three fully-connected layers. Convolutional layers", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 388, + 464 + ], + "score": 1.0, + "content": "have 128 and 256 filters, respectively. Each convolutional layer has a", + "type": "text" + }, + { + "bbox": [ + 388, + 452, + 412, + 462 + ], + "score": 0.9, + "content": "5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "receptive field applied", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 398, + 476 + ], + "score": 1.0, + "content": "with a stride of 1 pixel each followed by max pooling layer which pools", + "type": "text" + }, + { + "bbox": [ + 399, + 463, + 423, + 473 + ], + "score": 0.89, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "regions at strides of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 472, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 104, + 472, + 505, + 487 + ], + "score": 1.0, + "content": "2 pixels. AlexNet (Krizhevsky, 2014) consists of five convolutitonal layers followed by three fully-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "connected layers. Convolutional layers have 64, 192, 384, 256 and 256 filters, respectively. First and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 245, + 508 + ], + "score": 1.0, + "content": "second convolutional layers have a", + "type": "text" + }, + { + "bbox": [ + 246, + 496, + 267, + 506 + ], + "score": 0.89, + "content": "5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "receptive field applied with a stride of 1 pixel each followed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 248, + 520 + ], + "score": 1.0, + "content": "by max pooling layer which pools", + "type": "text" + }, + { + "bbox": [ + 248, + 507, + 272, + 517 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "regions at strides of 2 pixels. 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Fully-connected layers have 2048, 1024 and 10 hidden", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "units, respectively. Dropout (Hinton et al., 2012) was applied to only fully-connected layers of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "network with the probability of retaining the unit being 0.5. All hidden units are ReLUs. Figure", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "6 shows the detailed structure of VGGNet (Szegedy et al., 2015) with three fully-connected layers", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 403, + 585 + ], + "score": 1.0, + "content": "and 10 convolutional layers. Each ConvReLU box in the figure indicates a", + "type": "text" + }, + { + "bbox": [ + 403, + 573, + 425, + 583 + ], + "score": 0.9, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "convolutional layer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 363, + 596 + ], + "score": 1.0, + "content": "followed by ReLU activation. Also, all max pooling layers have", + "type": "text" + }, + { + "bbox": [ + 363, + 583, + 387, + 594 + ], + "score": 0.9, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "receptive fields with stride 2.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 592, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 506, + 609 + ], + "score": 1.0, + "content": "Dropout was applied to only fully-connected layers of the network with the probability of retaining", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "the unit being 0.5. For all experiments, the softmax classifier is used, and each model is trained by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "optimizing the objective function using Adam learning rule (Kingma & Ba, 2014). For each out-of-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 448, + 640 + ], + "score": 1.0, + "content": "distribution dataset, we randomly select 1,000 images for tuning the penalty parameter", + "type": "text" + }, + { + "bbox": [ + 448, + 627, + 456, + 638 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 626, + 506, + 640 + ], + "score": 1.0, + "content": ", mini-batch", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 349, + 650 + ], + "score": 1.0, + "content": "size and learning rate. The penalty parameter is chosen from", + "type": "text" + }, + { + "bbox": [ + 350, + 638, + 441, + 650 + ], + "score": 0.89, + "content": "\\beta \\in \\{ 0 , 0 . 1 , . . . 1 . 9 , 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 638, + 505, + 650 + ], + "score": 1.0, + "content": ", the mini-batch", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 189, + 662 + ], + "score": 1.0, + "content": "size is chosen from", + "type": "text" + }, + { + "bbox": [ + 189, + 649, + 229, + 661 + ], + "score": 0.93, + "content": "\\{ 6 4 , 1 \\bar { 2 } 8 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 648, + 382, + 662 + ], + "score": 1.0, + "content": "and the learning rate is chosen from", + "type": "text" + }, + { + "bbox": [ + 383, + 649, + 480, + 661 + ], + "score": 0.86, + "content": "\\{ 0 . 0 0 1 , 0 . 0 0 0 5 , 0 . 0 0 0 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 648, + 506, + 662 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "optimal parameters are chosen to minimize the detection error on the validation set. We drop the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "learning rate by 0.1 at 60 epoch and models are trained for total 100 epochs. The best test result is", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 682, + 212, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 212, + 694 + ], + "score": 1.0, + "content": "reported for each method.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 429, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 469, + 710 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 471, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 471, + 712 + ], + "score": 1.0, + "content": "Performance metrics. We measure the following metrics using threshold-based detectors:", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 698, + 471, + 712 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 132, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 132, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 132, + 82, + 263, + 95 + ], + "score": 1.0, + "content": "• True negative rate (TNR) at", + "type": "text" + }, + { + "bbox": [ + 263, + 83, + 285, + 93 + ], + "score": 0.87, + "content": "95 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "true positive rate (TPR). Let TP, TN, FP, and FN de-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 142, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "note true positive, true negative, false positive and false negative, respectively. We measure", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 142, + 105, + 379, + 116 + ], + "spans": [ + { + "bbox": [ + 142, + 105, + 231, + 116 + ], + "score": 0.53, + "content": "\\mathrm { T N R } = \\mathrm { T N } / \\left( \\mathrm { F P + T N } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 105, + 259, + 116 + ], + "score": 1.0, + "content": ", when", + "type": "text" + }, + { + "bbox": [ + 259, + 105, + 304, + 115 + ], + "score": 0.51, + "content": "\\mathrm { T P R } = \\mathrm { T P } .", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 105, + 309, + 116 + ], + "score": 1.0, + "content": "/", + "type": "text" + }, + { + "bbox": [ + 310, + 105, + 345, + 116 + ], + "score": 0.56, + "content": "( \\mathrm { T P + F N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 105, + 356, + 116 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 356, + 105, + 375, + 115 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 105, + 379, + 116 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 132, + 119, + 504, + 142 + ], + "lines": [ + { + "bbox": [ + 132, + 118, + 506, + 132 + ], + "spans": [ + { + "bbox": [ + 132, + 118, + 506, + 132 + ], + "score": 1.0, + "content": "• Area under the receiver operating characteristic curve (AUROC). The ROC curve is a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 130, + 500, + 143 + ], + "spans": [ + { + "bbox": [ + 141, + 130, + 338, + 143 + ], + "score": 1.0, + "content": "graph plotting TPR against the false positive rate", + "type": "text" + }, + { + "bbox": [ + 339, + 131, + 361, + 141 + ], + "score": 0.27, + "content": "= \\mathrm { F P }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 130, + 369, + 143 + ], + "score": 1.0, + "content": "/", + "type": "text" + }, + { + "bbox": [ + 369, + 131, + 403, + 141 + ], + "score": 0.61, + "content": "\\mathrm { \\ F P { + } T N ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 130, + 500, + 143 + ], + "score": 1.0, + "content": "by varying a threshold.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 133, + 145, + 504, + 178 + ], + "lines": [ + { + "bbox": [ + 132, + 145, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 132, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "• Area under the precision-recall curve (AUPR). The PR curve is a graph plotting the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 156, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 141, + 156, + 180, + 169 + ], + "score": 1.0, + "content": "precision", + "type": "text" + }, + { + "bbox": [ + 180, + 157, + 189, + 166 + ], + "score": 0.47, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 156, + 210, + 169 + ], + "score": 1.0, + "content": "TP /", + "type": "text" + }, + { + "bbox": [ + 210, + 156, + 244, + 167 + ], + "score": 0.45, + "content": "( \\mathrm { T P + F P } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 156, + 299, + 169 + ], + "score": 1.0, + "content": "against recal", + "type": "text" + }, + { + "bbox": [ + 299, + 156, + 366, + 168 + ], + "score": 0.39, + "content": "\\mathrm { \\Lambda } = \\mathrm { T P } / \\left( \\mathrm { T P + F N } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 156, + 505, + 169 + ], + "score": 1.0, + "content": "by varying a threshold. 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As shown in Figure 3(a) and Figure 3(b), we compare", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "the generated samples by original GAN and proposed GAN on a simple example where the target", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "distribution is a mixture of two Gaussian distributions. For both the generator and discriminator, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "use fully-connected neural networks with 2 hidden layers and 500 hidden units for each layer. For", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "all layers, we use ReLU activation function. We use a 100-dimensional Gaussian prior for the latent", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "score": 1.0, + "content": "variable z. For our method, we pre-train the simple fully-connected neural networks (2 hidden layers", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "and 500 ReLU units for each layer) by minimizing the cross entropy on target distribution samples", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "and the KL divergence on out-of-distribution samples generated by rejection sampling on bounded", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 490, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 140, + 505 + ], + "score": 1.0, + "content": "2D box", + "type": "text" + }, + { + "bbox": [ + 140, + 491, + 182, + 504 + ], + "score": 0.92, + "content": "[ - 1 0 , 1 0 ] ^ { \\frac { \\ d } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 490, + 281, + 505 + ], + "score": 1.0, + "content": ". The penalty parameter", + "type": "text" + }, + { + "bbox": [ + 282, + 492, + 290, + 503 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 490, + 506, + 505 + ], + "score": 1.0, + "content": "is set to 1. We use ADAM learning rule (Kingma &", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "Ba, 2014) with a mini-batch size of 400. The initial learning rate is set to 0.002, and we train for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 513, + 177, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 177, + 526 + ], + "score": 1.0, + "content": "total 100 epochs.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 505, + 728 + ], + "lines": [ + { + "bbox": [ + 106, + 530, + 504, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 504, + 543 + ], + "score": 1.0, + "content": "Generating samples on MNIST. As shown in Figure 3(c) and Figure 3(d), we compare the gener-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 555 + ], + "score": 1.0, + "content": "ated samples of original and proposed GANs on MNIST dataset, which consists of greyscale images,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 551, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 566 + ], + "score": 1.0, + "content": "each containing a digit 0 to 9 with 60,000 training and 10,000 test images. We expand each image", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 136, + 576 + ], + "score": 1.0, + "content": "to size", + "type": "text" + }, + { + "bbox": [ + 136, + 563, + 189, + 574 + ], + "score": 0.91, + "content": "3 \\times 3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 563, + 505, + 576 + ], + "score": 1.0, + "content": ". For both the generator and discriminator, we use deep convolutional GANs", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "(DCGANs) (Radford et al., 2015). The discriminator and generator consist of four convolutional", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "score": 1.0, + "content": "and deconvolutional layers, respectively. Convolutional layers have 128, 256, 512 and 1 filters, re-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 276, + 609 + ], + "score": 1.0, + "content": "spectively. Each convolutional layer has a", + "type": "text" + }, + { + "bbox": [ + 276, + 596, + 300, + 607 + ], + "score": 0.89, + "content": "4 \\times 4", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "receptive field applied with a stride of 2 pixel. The", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "second and third convolutional layers are followed by batch normalization (Ioffe & Szegedy, 2015).", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "For all layers, we use LeakyReLU activation function. Deconvolutional layers have 512, 256, 128", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 356, + 642 + ], + "score": 1.0, + "content": "and 1 filters, respectively. 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The initial", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 717, + 355, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 717, + 355, + 730 + ], + "score": 1.0, + "content": "learning rate is set to 0.0002, and we train for total 50 epochs.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 41.5 + } + ], + "page_idx": 11, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "12", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 132, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 132, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 132, + 82, + 263, + 95 + ], + "score": 1.0, + "content": "• True negative rate (TNR) at", + "type": "text" + }, + { + "bbox": [ + 263, + 83, + 285, + 93 + ], + "score": 0.87, + "content": "95 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "true positive rate (TPR). 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We measure", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 142, + 105, + 379, + 116 + ], + "spans": [ + { + "bbox": [ + 142, + 105, + 231, + 116 + ], + "score": 0.53, + "content": "\\mathrm { T N R } = \\mathrm { T N } / \\left( \\mathrm { F P + T N } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 105, + 259, + 116 + ], + "score": 1.0, + "content": ", when", + "type": "text" + }, + { + "bbox": [ + 259, + 105, + 304, + 115 + ], + "score": 0.51, + "content": "\\mathrm { T P R } = \\mathrm { T P } .", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 105, + 309, + 116 + ], + "score": 1.0, + "content": "/", + "type": "text" + }, + { + "bbox": [ + 310, + 105, + 345, + 116 + ], + "score": 0.56, + "content": "( \\mathrm { T P + F N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 105, + 356, + 116 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 356, + 105, + 375, + 115 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 105, + 379, + 116 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 132, + 82, + 505, + 116 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 119, + 504, + 142 + ], + "lines": [ + { + "bbox": [ + 132, + 118, + 506, + 132 + ], + "spans": [ + { + "bbox": [ + 132, + 118, + 506, + 132 + ], + "score": 1.0, + "content": "• Area under the receiver operating characteristic curve (AUROC). 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AUPR-IN", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 167, + 493, + 180 + ], + "spans": [ + { + "bbox": [ + 142, + 167, + 493, + 180 + ], + "score": 1.0, + "content": "(or -OUT) is AUPR where in- (or out-of-) distribution samples are specified as positive.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 132, + 145, + 505, + 180 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 182, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 133, + 182, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 133, + 182, + 504, + 194 + ], + "score": 1.0, + "content": "• Detection accuracy. 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As shown in Figure 3(a) and Figure 3(b), we compare", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "the generated samples by original GAN and proposed GAN on a simple example where the target", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "distribution is a mixture of two Gaussian distributions. For both the generator and discriminator, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "use fully-connected neural networks with 2 hidden layers and 500 hidden units for each layer. For", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "all layers, we use ReLU activation function. We use a 100-dimensional Gaussian prior for the latent", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "score": 1.0, + "content": "variable z. For our method, we pre-train the simple fully-connected neural networks (2 hidden layers", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "and 500 ReLU units for each layer) by minimizing the cross entropy on target distribution samples", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "and the KL divergence on out-of-distribution samples generated by rejection sampling on bounded", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 490, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 140, + 505 + ], + "score": 1.0, + "content": "2D box", + "type": "text" + }, + { + "bbox": [ + 140, + 491, + 182, + 504 + ], + "score": 0.92, + "content": "[ - 1 0 , 1 0 ] ^ { \\frac { \\ d } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 490, + 281, + 505 + ], + "score": 1.0, + "content": ". The penalty parameter", + "type": "text" + }, + { + "bbox": [ + 282, + 492, + 290, + 503 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 490, + 506, + 505 + ], + "score": 1.0, + "content": "is set to 1. We use ADAM learning rule (Kingma &", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "Ba, 2014) with a mini-batch size of 400. The initial learning rate is set to 0.002, and we train for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 513, + 177, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 177, + 526 + ], + "score": 1.0, + "content": "total 100 epochs.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 404, + 506, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 505, + 728 + ], + "lines": [ + { + "bbox": [ + 106, + 530, + 504, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 504, + 543 + ], + "score": 1.0, + "content": "Generating samples on MNIST. As shown in Figure 3(c) and Figure 3(d), we compare the gener-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 555 + ], + "score": 1.0, + "content": "ated samples of original and proposed GANs on MNIST dataset, which consists of greyscale images,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 551, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 566 + ], + "score": 1.0, + "content": "each containing a digit 0 to 9 with 60,000 training and 10,000 test images. We expand each image", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 136, + 576 + ], + "score": 1.0, + "content": "to size", + "type": "text" + }, + { + "bbox": [ + 136, + 563, + 189, + 574 + ], + "score": 0.91, + "content": "3 \\times 3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 563, + 505, + 576 + ], + "score": 1.0, + "content": ". For both the generator and discriminator, we use deep convolutional GANs", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "(DCGANs) (Radford et al., 2015). The discriminator and generator consist of four convolutional", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "score": 1.0, + "content": "and deconvolutional layers, respectively. Convolutional layers have 128, 256, 512 and 1 filters, re-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 276, + 609 + ], + "score": 1.0, + "content": "spectively. Each convolutional layer has a", + "type": "text" + }, + { + "bbox": [ + 276, + 596, + 300, + 607 + ], + "score": 0.89, + "content": "4 \\times 4", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "receptive field applied with a stride of 2 pixel. The", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "second and third convolutional layers are followed by batch normalization (Ioffe & Szegedy, 2015).", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "For all layers, we use LeakyReLU activation function. Deconvolutional layers have 512, 256, 128", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 356, + 642 + ], + "score": 1.0, + "content": "and 1 filters, respectively. Each deconvolutional layer has a", + "type": "text" + }, + { + "bbox": [ + 356, + 630, + 381, + 640 + ], + "score": 0.9, + "content": "4 \\times 4", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "receptive field applied with a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "stride of 2 pixel followed by batch normalization (Ioffe & Szegedy, 2015) and ReLU activation.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 650, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 664 + ], + "score": 1.0, + "content": "For our method, we use a pre-trained simple CNNs (two convolutional layers followed by three", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 660, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 104, + 660, + 506, + 675 + ], + "score": 1.0, + "content": "fully-connected layers) by minimizing the cross entropy on MNIST training samples and the KL", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 673, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 505, + 686 + ], + "score": 1.0, + "content": "divergence on synthetic Gaussian noise. 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The initial", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 717, + 355, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 717, + 355, + 730 + ], + "score": 1.0, + "content": "learning rate is set to 0.0002, and we train for total 50 epochs.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 530, + 506, + 730 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 290, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 292, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 292, + 95 + ], + "score": 1.0, + "content": "C MORE EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 106, + 276, + 118 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 278, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 278, + 119 + ], + "score": 1.0, + "content": "C.1 CLASSIFICATION PERFORMANCES", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "table", + "bbox": [ + 108, + 181, + 505, + 232 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 126, + 505, + 172 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "Table 2 reports the classification accuracy of VGGNets on CIFAR-10 and SVHN datasets under", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "various training losses shown in Figure 4. One can note that all methods do not degrade the original", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "classification performance, where the differences in classification errors of across all tested single", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 279, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 183, + 172 + ], + "score": 1.0, + "content": "models are at most", + "type": "text" + }, + { + "bbox": [ + 184, + 160, + 199, + 171 + ], + "score": 0.87, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 160, + 279, + 172 + ], + "score": 1.0, + "content": "in our experiments.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 181, + 505, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 181, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 108, + 181, + 505, + 232 + ], + "score": 0.978, + "html": "
In-distributionCross entropyConfidence loss with original GANJoint confidence lossConfidence loss with explicit out-of-distribution samples
CIFAR-1080.1480.2781.3980.56
SVHN93.8294.0893.8194.23
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Table", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 469, + 564 + ], + "score": 1.0, + "content": "3 shows the calibration effects of confidence loss when we define the confidence score", + "type": "text" + }, + { + "bbox": [ + 470, + 555, + 476, + 564 + ], + "score": 0.79, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "as the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "maximum predictive distribution of a classifier. We found that the ECE of a classifier trained by our", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "methods is lower than that of a classifier trained by the standard cross entropy loss. This implies", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "that our proposed method is effective at calibrating predictions. We also remark that the temperature", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "scaling (Guo et al., 2017) provides further improvements under a classifier trained by our joint", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 608, + 173, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 173, + 619 + ], + "score": 1.0, + "content": "confidence loss.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31 + }, + { + "type": "table", + "bbox": [ + 108, + 627, + 505, + 685 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 627, + 505, + 685 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 627, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 108, + 627, + 505, + 685 + ], + "score": 0.975, + "html": "
In-distributionWithout temperature scaling Cross entropy loss Joint confidence lossWith temperature scaling Cross entropy loss Joint confidence loss
CIFAR-1018.4514.627.076.19
SVHN5.305.132.801.39
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In-distributionCross entropyConfidence loss with original GANJoint confidence lossConfidence loss with explicit out-of-distribution samples
CIFAR-1080.1480.2781.3980.56
SVHN93.8294.0893.8194.23
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In order to evaluate the calibration effects, we measure the expected calibration error", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 282, + 349 + ], + "score": 1.0, + "content": "(ECE) (Naeini et al., 2015). Given test data", + "type": "text" + }, + { + "bbox": [ + 282, + 336, + 382, + 348 + ], + "score": 0.91, + "content": "\\left\\{ \\left( x _ { 1 } , y _ { 1 } \\right) , \\ldots , \\left( x _ { n } , y _ { n } \\right) \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 335, + 505, + 349 + ], + "score": 1.0, + "content": ", we group the predictions into", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 346, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 107, + 347, + 119, + 357 + ], + "score": 0.79, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 346, + 230, + 376 + ], + "score": 1.0, + "content": "interval bins (each of size nfidence falls into the interv", + "type": "text" + }, + { + "bbox": [ + 231, + 347, + 255, + 358 + ], + "score": 0.86, + "content": "1 / M )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 348, + 296, + 358 + ], + "score": 0.89, + "content": "B _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 346, + 372, + 376 + ], + "score": 1.0, + "content": "be the set of indien, the accuracy of", + "type": "text" + }, + { + "bbox": [ + 389, + 346, + 506, + 376 + ], + "score": 1.0, + "content": "of samples whose predictionis defined as follows:", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 238, + 357, + 389, + 371 + ], + "spans": [ + { + "bbox": [ + 238, + 357, + 279, + 371 + ], + "score": 0.92, + "content": "\\textstyle { \\bigl ( } { \\frac { m - 1 } { M } } , { \\frac { m } { M } } { \\bigr ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 358, + 389, + 369 + ], + "score": 0.9, + "content": "B _ { m }", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 302, + 506, + 376 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 375, + 398, + 406 + ], + "lines": [ + { + "bbox": [ + 210, + 375, + 398, + 406 + ], + "spans": [ + { + "bbox": [ + 210, + 375, + 398, + 406 + ], + "score": 0.93, + "content": "\\operatorname { a c c } ( B _ { m } ) = \\frac { 1 } { | B _ { m } | } \\sum _ { i \\in B _ { m } } 1 _ { \\{ y _ { i } = \\arg \\operatorname* { m a x } _ { y } P _ { \\theta } ( y | \\mathbf { x } _ { i } ) \\} } ,", + "type": "interline_equation", + "image_path": "9bcd1d1200febd0642f987f7c590863d581525b1afdac42339683716d5eed03c.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 210, + 375, + 398, + 390.5 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 210, + 390.5, + 398, + 406.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 504, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 133, + 424 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 412, + 140, + 422 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 411, + 315, + 424 + ], + "score": 1.0, + "content": "is the model parameters of a classifier and", + "type": "text" + }, + { + "bbox": [ + 315, + 412, + 366, + 424 + ], + "score": 0.93, + "content": "1 _ { A } \\in \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "is the indicator function for event", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 421, + 315, + 435 + ], + "spans": [ + { + "bbox": [ + 107, + 423, + 115, + 433 + ], + "score": 0.73, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 421, + 252, + 435 + ], + "score": 1.0, + "content": ". 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Table", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 469, + 564 + ], + "score": 1.0, + "content": "3 shows the calibration effects of confidence loss when we define the confidence score", + "type": "text" + }, + { + "bbox": [ + 470, + 555, + 476, + 564 + ], + "score": 0.79, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "as the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "maximum predictive distribution of a classifier. 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In-distributionWithout temperature scaling Cross entropy loss Joint confidence lossWith temperature scaling Cross entropy loss Joint confidence loss
CIFAR-1018.4514.627.076.19
SVHN5.305.132.801.39
", + "type": "table", + "image_path": "3f495c43d3eb637ba8600dec41abbe37448720531c5aeaeeb1a03bc6c0ac05f2.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 108, + 627, + 505, + 646.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 108, + 646.3333333333334, + 505, + 665.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 108, + 665.6666666666667, + 505, + 685.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 106, + 692, + 506, + 726 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "score": 1.0, + "content": "Table 3: Expected calibration error (ECE) of VGGNets on CIFAR-10 and SVHN datasets under", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 703, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 282, + 716 + ], + "score": 1.0, + "content": "various training losses. 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First, we remark that the baseline detector using", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "classifiers trained by our joint confidence loss (blue bar) typically outperforms the ODIN detec-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "tor using classifiers trained by the cross entropy loss (orange bar). 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The key idea of ODIN is the temperature scaling which is defined as follows:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 103, + 505, + 137 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 139, + 385, + 169 + ], + "lines": [ + { + "bbox": [ + 224, + 139, + 385, + 169 + ], + "spans": [ + { + "bbox": [ + 224, + 139, + 385, + 169 + ], + "score": 0.95, + "content": "P _ { \\theta } ( y = \\widehat { y } | \\mathbf { x } ; T ) = \\frac { \\exp { ( f _ { \\widehat { y } } ( \\mathbf { x } ) / T ) } } { \\sum _ { y } \\exp { ( f _ { y } ( \\mathbf { x } ) / T ) } } ,", + "type": "interline_equation", + "image_path": "2b8e95a21d71fc3b8e74d66755ce15ec4fa1f48ddd9fcdc377e83e6c85e0825c.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 224, + 139, + 385, + 154.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 224, + 154.0, + 385, + 169.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 171, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 133, + 184 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 172, + 163, + 182 + ], + "score": 0.9, + "content": "T > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 171, + 331, + 184 + ], + "score": 1.0, + "content": "is the temperature scaling parameter and", + "type": "text" + }, + { + "bbox": [ + 331, + 171, + 403, + 183 + ], + "score": 0.93, + "content": "\\mathbf { f } = \\left( f _ { 1 } , \\ldots , f _ { K } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 171, + 506, + 184 + ], + "score": 1.0, + "content": "is final feature vector of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 236, + 195 + ], + "score": 1.0, + "content": "neural networks. For each data", + "type": "text" + }, + { + "bbox": [ + 236, + 185, + 243, + 192 + ], + "score": 0.64, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 182, + 437, + 195 + ], + "score": 1.0, + "content": ", ODIN first calculates the pre-processed image", + "type": "text" + }, + { + "bbox": [ + 438, + 183, + 446, + 192 + ], + "score": 0.81, + "content": "\\widehat { \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "by adding the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 193, + 232, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 232, + 205 + ], + "score": 1.0, + "content": "small perturbations as follows:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 171, + 506, + 205 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 208, + 396, + 222 + ], + "lines": [ + { + "bbox": [ + 214, + 208, + 396, + 222 + ], + "spans": [ + { + "bbox": [ + 214, + 208, + 396, + 222 + ], + "score": 0.9, + "content": "\\mathbf { x } ^ { \\prime } = \\mathbf { x } - \\varepsilon \\mathrm { s i g n } \\left( - \\bigtriangledown _ { \\mathbf { x } } \\log P _ { \\theta } ( y = \\widehat { y } | \\mathbf { x } ; T ) \\right) ,", + "type": "interline_equation", + "image_path": "a0d5476f34e6b5a0c7344c34935aefd1098da909b8041eb7e56a6a339f003646.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 214, + 208, + 396, + 222 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 505, + 269 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 135, + 237 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 136, + 227, + 142, + 235 + ], + "score": 0.74, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 225, + 272, + 237 + ], + "score": 1.0, + "content": "is a magnitude of noise and", + "type": "text" + }, + { + "bbox": [ + 272, + 225, + 280, + 237 + ], + "score": 0.85, + "content": "\\widehat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "is the predicted label. Next, ODIN feeds the pre-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "bprocessed data into the classifier, computes the maximum value of scaled predictive distribution,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 124, + 259 + ], + "score": 1.0, + "content": "i.e.,", + "type": "text" + }, + { + "bbox": [ + 125, + 247, + 198, + 259 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { y } P _ { \\theta } ( y | \\mathbf { x } ^ { \\prime } ; T ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 248, + 505, + 259 + ], + "score": 1.0, + "content": ", and classifies it as positive (i.e., in-distribution) if the confidence score is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 206, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 195, + 269 + ], + "score": 1.0, + "content": "above some threshold", + "type": "text" + }, + { + "bbox": [ + 196, + 259, + 201, + 268 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 258, + 206, + 269 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 225, + 505, + 269 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 274, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 279, + 288 + ], + "score": 1.0, + "content": "For ODIN detector, the perturbation noise", + "type": "text" + }, + { + "bbox": [ + 279, + 277, + 285, + 285 + ], + "score": 0.75, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 274, + 350, + 288 + ], + "score": 1.0, + "content": "is chosen from", + "type": "text" + }, + { + "bbox": [ + 350, + 275, + 447, + 287 + ], + "score": 0.9, + "content": "\\{ 0 , 0 . 0 0 0 1 , 0 . 0 0 1 , 0 . 0 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 274, + 505, + 288 + ], + "score": 1.0, + "content": ", and the tem-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 104, + 285, + 141, + 298 + ], + "score": 1.0, + "content": "perature", + "type": "text" + }, + { + "bbox": [ + 142, + 286, + 150, + 295 + ], + "score": 0.83, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 285, + 213, + 298 + ], + "score": 1.0, + "content": "is chosen from", + "type": "text" + }, + { + "bbox": [ + 214, + 286, + 307, + 298 + ], + "score": 0.88, + "content": "\\{ 1 , 1 0 , 1 0 0 , 5 0 0 , 1 0 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 285, + 506, + 298 + ], + "score": 1.0, + "content": ". The optimal parameters are chosen to minimize", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "score": 1.0, + "content": "the detection error on the validation set. Figure 7 shows the performance of the OIDN and baseline", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 306, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 321 + ], + "score": 1.0, + "content": "detector for each in- and out-of-distribution pair. First, we remark that the baseline detector using", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "classifiers trained by our joint confidence loss (blue bar) typically outperforms the ODIN detec-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "tor using classifiers trained by the cross entropy loss (orange bar). This means that our classifier", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "can map in- and out-of-distributions more separately without pre-processing methods such as tem-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "perature scaling. The ODIN detector provides further improvements if one uses it with our joint", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "confidence loss (green bar). In other words, our proposed training method can improve all prior", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 374, + 184, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 184, + 385 + ], + "score": 1.0, + "content": "detection methods.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 274, + 506, + 385 + ] + }, + { + "type": "image", + "bbox": [ + 108, + 398, + 491, + 629 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 398, + 491, + 629 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 398, + 491, + 629 + ], + "spans": [ + { + "bbox": [ + 108, + 398, + 491, + 629 + ], + "score": 0.975, + "type": "image", + "image_path": "1380ade19cbcf946e7ad048c487ea59780f88d50e2289a37c6e92a096d1dc661.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 108, + 398, + 491, + 475.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 108, + 475.0, + 491, + 552.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 108, + 552.0, + 491, + 629.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 635, + 504, + 658 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 634, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 648 + ], + "score": 1.0, + "content": "Figure 7: Performances of the baseline detector (Hendrycks & Gimpel, 2016) and ODIN detector", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 646, + 305, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 305, + 659 + ], + "score": 1.0, + "content": "(Liang et al., 2017) under various training losses.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + } + ], + "index": 26.25 + }, + { + "type": "title", + "bbox": [ + 108, + 678, + 301, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 302, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 302, + 690 + ], + "score": 1.0, + "content": "C.4 EXPERIMENTAL RESULTS ON ALEXNET", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Table 4 shows the detection performance for each in- and out-of-distribution pair when the classifier", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "is AlexNet (Krizhevsky, 2014), which is one of popular CNN architectures. We remark that they", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 721, + 189, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 189, + 732 + ], + "score": 1.0, + "content": "show similar trends.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 698, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 80, + 505, + 259 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 80, + 505, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 80, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 107, + 80, + 505, + 259 + ], + "score": 0.985, + "html": "
In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
Cross entropy loss /Confidence loss
Baseline (SVHN)CIFAR-10 (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)92.14 / 93.7742.0 / 99.9 45.6 /99.9 44.6/100.0 58.6/100.088.0/100.0 89.4/100.0 89.8/100.0 94.2/100.083.4/99.8 84.3/99.9 84.5 /99.9 88.8/100.088.7 /99.9 90.2/100.0 90.8/100.0 95.5/100.087.3/99.3 88.6/99.3 88.4/99.3 92.5 / 99.3
Baseline (CIFAR-10)SVHN (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)76.58 /76.1812.8 / 99.6 10.3/10.1 10.7 /8.1 6.7 /1.071.0 /99.9 59.2/52.1 56.3 / 51.5 49.6/13.573.2/99.6 64.2/62.0 64.3/61.8 61.3 / 50.074.3 /99.9 63.6/59.8 62.3/ 59.5 58.5 /43.770.7 /99.6 64.4/62.3 65.3/ 61.6 59.5/32.0
ODIN (SVHN)CIFAR-10 (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)92.14 /93.7755.5/99.9 59.5 /99.9 61.5 /100.0 82.6/100.089.1/99.2 90.5 /99.3 91.8 /99.382.4/99.8 83.8 /99.9 84.8 /99.985.9 /100.0 87.5 /100.0 90.5/100.089.0 /99.2 90.4 /99.3 91.3/99.3
37.1/99.697.0 /99.3 86.7 /99.691.6/100.0 79.3 /99.6 64.4/ 61.897.4/100.0 88.1/99.996.4 /99.3 84.2/99.6
ODIN (CIFAR-10)SVHN (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)76.58 / 76.1811.4 /8.4 13.3 / 7.1 3.8/ 0.069.1/ 65.6 71.9 /67.1 70.9/57.275.3/63.7 69.3 /40.471.4 /68.6 67.2/72.0 78.1/56.164.6/60.7 65.3/60.5 60.7 /40.7
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All values are percentages and boldface values indicate rela-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 287, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 304 + ], + "score": 1.0, + "content": "tive the better results. For each in-distribution, we minimize the KL divergence term in (1) using", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "training samples from an out-of-distribution dataset denoted by “seen”, where other “unseen” out-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 309, + 277, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 277, + 325 + ], + "score": 1.0, + "content": "of-distributions were also used for testing.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "title", + "bbox": [ + 108, + 343, + 247, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 249, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 249, + 357 + ], + "score": 1.0, + "content": "D MAXIMIZING ENTROPY", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 369, + 506, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "One might expect that the entropy of out-of-distribution is expected to be much higher compared to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "that of in-distribution since the out-of-distribution is typically on a much larger space than the in-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "distribution. Therefore, one can add maximizing the entropy of generator distribution to new GAN", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "loss in (3) and joint confidence loss in (4). However, maximizing the entropy of generator distribu-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "tion is technically challenging since a GAN does not model the generator distribution explicitly. To", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 424, + 421, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 421, + 437 + ], + "score": 1.0, + "content": "handle the issue, one can leverage the pull-away term (PT) (Zhao et al., 2017):", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 443, + 464, + 480 + ], + "lines": [ + { + "bbox": [ + 144, + 443, + 464, + 480 + ], + "spans": [ + { + "bbox": [ + 144, + 443, + 464, + 480 + ], + "score": 0.93, + "content": "- \\mathcal { H } \\left( P _ { G } \\left( \\mathbf { x } \\right) \\right) \\backsimeq \\mathcal { P T } \\left( P _ { G } \\left( \\mathbf { x } \\right) \\right) = \\frac { 1 } { M ( M - 1 ) } \\sum _ { i = 1 } ^ { M } \\sum _ { j \\neq i } \\left( \\frac { G \\left( \\mathbf { z } _ { i } \\right) ^ { \\top } G \\left( \\mathbf { z } _ { j } \\right) } { \\| G \\left( \\mathbf { z } _ { i } \\right) \\| \\| G \\left( \\mathbf { z } _ { j } \\right) \\| } \\right) ^ { 2 } ,", + "type": "interline_equation", + "image_path": "392755eedc3b89d2a2f62e713f641f76d585f9831e1d675d4fc2cacecc3eb5f3.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 144, + 443, + 464, + 455.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 144, + 455.3333333333333, + 464, + 467.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 144, + 467.66666666666663, + 464, + 479.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 133, + 500 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 486, + 156, + 498 + ], + "score": 0.9, + "content": "\\mathcal { H } \\left( \\cdot \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 486, + 242, + 500 + ], + "score": 1.0, + "content": "denotes the entropy,", + "type": "text" + }, + { + "bbox": [ + 243, + 486, + 316, + 499 + ], + "score": 0.92, + "content": "\\mathbf { z } _ { i } , \\mathbf { z } _ { j } \\sim P _ { \\mathrm { p r i } } \\left( \\mathbf { z } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 486, + 335, + 500 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 335, + 487, + 348, + 497 + ], + "score": 0.82, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "is the number of samples. Intuitively,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "one can expect the effect of increasing the entropy by minimizing PT since it corresponds to the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 520 + ], + "score": 1.0, + "content": "squared cosine similarity of generated samples. We note that (Dai et al., 2017) also used PT to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "maximize the entropy. Similarly as in Section 3.2, we verify the effects of PT using VGGNet. Table", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 531, + 504, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 504, + 542 + ], + "score": 1.0, + "content": "5 shows the performance of the baseline detector for each in- and out-of-distribution pair. We found", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "that joint confidence loss with PT tends to (but not always) improve the detection performance.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "However, since PT increases the training complexity and the gains from PT are relatively marginal", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 563, + 442, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 442, + 576 + ], + "score": 1.0, + "content": "(or controversial), we leave it as an auxiliary option for improving the performance.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "table", + "bbox": [ + 108, + 586, + 505, + 683 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 586, + 505, + 683 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 586, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 108, + 586, + 505, + 683 + ], + "score": 0.981, + "html": "
In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
CIFAR-10Joint confidence loss without PT/with PT
SVHNTinyImageNet LSUN93.81 /94.0590.1 /92.3 99.0 / 99.9 98.9 /100.097.6 /98.1 99.6 /100.0 99.6 /100.093.6 /94.6 97.6 /99.7 97.5 / 99.997.7 /98.2 99.7 /100.0 99.7 /100.097.9 /98.7 94.5 /100.0 95.5 /100.0
CIFAR-10SVHN TinyImageNet LSUN81.39 / 80.6025.4 / 13.2 35.0 /44.8 39.1 /49.166.8 / 69.5 72.0 / 78.4 75.1 /80.774.2 / 75.1 76.4 / 77.6 77.8 / 78.771.3 / 73.5 74.7 / 79.4 77.1 / 81.378.3 / 72.0 82.2 /84.4 83.6 /85.8
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In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
Cross entropy loss /Confidence loss
Baseline (SVHN)CIFAR-10 (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)92.14 / 93.7742.0 / 99.9 45.6 /99.9 44.6/100.0 58.6/100.088.0/100.0 89.4/100.0 89.8/100.0 94.2/100.083.4/99.8 84.3/99.9 84.5 /99.9 88.8/100.088.7 /99.9 90.2/100.0 90.8/100.0 95.5/100.087.3/99.3 88.6/99.3 88.4/99.3 92.5 / 99.3
Baseline (CIFAR-10)SVHN (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)76.58 /76.1812.8 / 99.6 10.3/10.1 10.7 /8.1 6.7 /1.071.0 /99.9 59.2/52.1 56.3 / 51.5 49.6/13.573.2/99.6 64.2/62.0 64.3/61.8 61.3 / 50.074.3 /99.9 63.6/59.8 62.3/ 59.5 58.5 /43.770.7 /99.6 64.4/62.3 65.3/ 61.6 59.5/32.0
ODIN (SVHN)CIFAR-10 (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)92.14 /93.7755.5/99.9 59.5 /99.9 61.5 /100.0 82.6/100.089.1/99.2 90.5 /99.3 91.8 /99.382.4/99.8 83.8 /99.9 84.8 /99.985.9 /100.0 87.5 /100.0 90.5/100.089.0 /99.2 90.4 /99.3 91.3/99.3
37.1/99.697.0 /99.3 86.7 /99.691.6/100.0 79.3 /99.6 64.4/ 61.897.4/100.0 88.1/99.996.4 /99.3 84.2/99.6
ODIN (CIFAR-10)SVHN (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)76.58 / 76.1811.4 /8.4 13.3 / 7.1 3.8/ 0.069.1/ 65.6 71.9 /67.1 70.9/57.275.3/63.7 69.3 /40.471.4 /68.6 67.2/72.0 78.1/56.164.6/60.7 65.3/60.5 60.7 /40.7
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All values are percentages and boldface values indicate rela-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 287, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 304 + ], + "score": 1.0, + "content": "tive the better results. For each in-distribution, we minimize the KL divergence term in (1) using", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "training samples from an out-of-distribution dataset denoted by “seen”, where other “unseen” out-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 309, + 277, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 277, + 325 + ], + "score": 1.0, + "content": "of-distributions were also used for testing.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "title", + "bbox": [ + 108, + 343, + 247, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 249, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 249, + 357 + ], + "score": 1.0, + "content": "D MAXIMIZING ENTROPY", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 369, + 506, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "One might expect that the entropy of out-of-distribution is expected to be much higher compared to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "that of in-distribution since the out-of-distribution is typically on a much larger space than the in-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "distribution. Therefore, one can add maximizing the entropy of generator distribution to new GAN", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "loss in (3) and joint confidence loss in (4). However, maximizing the entropy of generator distribu-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "tion is technically challenging since a GAN does not model the generator distribution explicitly. To", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 424, + 421, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 421, + 437 + ], + "score": 1.0, + "content": "handle the issue, one can leverage the pull-away term (PT) (Zhao et al., 2017):", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 369, + 506, + 437 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 443, + 464, + 480 + ], + "lines": [ + { + "bbox": [ + 144, + 443, + 464, + 480 + ], + "spans": [ + { + "bbox": [ + 144, + 443, + 464, + 480 + ], + "score": 0.93, + "content": "- \\mathcal { H } \\left( P _ { G } \\left( \\mathbf { x } \\right) \\right) \\backsimeq \\mathcal { P T } \\left( P _ { G } \\left( \\mathbf { x } \\right) \\right) = \\frac { 1 } { M ( M - 1 ) } \\sum _ { i = 1 } ^ { M } \\sum _ { j \\neq i } \\left( \\frac { G \\left( \\mathbf { z } _ { i } \\right) ^ { \\top } G \\left( \\mathbf { z } _ { j } \\right) } { \\| G \\left( \\mathbf { z } _ { i } \\right) \\| \\| G \\left( \\mathbf { z } _ { j } \\right) \\| } \\right) ^ { 2 } ,", + "type": "interline_equation", + "image_path": "392755eedc3b89d2a2f62e713f641f76d585f9831e1d675d4fc2cacecc3eb5f3.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 144, + 443, + 464, + 455.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 144, + 455.3333333333333, + 464, + 467.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 144, + 467.66666666666663, + 464, + 479.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 133, + 500 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 486, + 156, + 498 + ], + "score": 0.9, + "content": "\\mathcal { H } \\left( \\cdot \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 486, + 242, + 500 + ], + "score": 1.0, + "content": "denotes the entropy,", + "type": "text" + }, + { + "bbox": [ + 243, + 486, + 316, + 499 + ], + "score": 0.92, + "content": "\\mathbf { z } _ { i } , \\mathbf { z } _ { j } \\sim P _ { \\mathrm { p r i } } \\left( \\mathbf { z } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 486, + 335, + 500 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 335, + 487, + 348, + 497 + ], + "score": 0.82, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "is the number of samples. Intuitively,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "one can expect the effect of increasing the entropy by minimizing PT since it corresponds to the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 520 + ], + "score": 1.0, + "content": "squared cosine similarity of generated samples. We note that (Dai et al., 2017) also used PT to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "maximize the entropy. Similarly as in Section 3.2, we verify the effects of PT using VGGNet. Table", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 531, + 504, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 504, + 542 + ], + "score": 1.0, + "content": "5 shows the performance of the baseline detector for each in- and out-of-distribution pair. We found", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "that joint confidence loss with PT tends to (but not always) improve the detection performance.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "However, since PT increases the training complexity and the gains from PT are relatively marginal", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 563, + 442, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 442, + 576 + ], + "score": 1.0, + "content": "(or controversial), we leave it as an auxiliary option for improving the performance.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 486, + 506, + 576 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 586, + 505, + 683 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 586, + 505, + 683 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 586, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 108, + 586, + 505, + 683 + ], + "score": 0.981, + "html": "
In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
CIFAR-10Joint confidence loss without PT/with PT
SVHNTinyImageNet LSUN93.81 /94.0590.1 /92.3 99.0 / 99.9 98.9 /100.097.6 /98.1 99.6 /100.0 99.6 /100.093.6 /94.6 97.6 /99.7 97.5 / 99.997.7 /98.2 99.7 /100.0 99.7 /100.097.9 /98.7 94.5 /100.0 95.5 /100.0
CIFAR-10SVHN TinyImageNet LSUN81.39 / 80.6025.4 / 13.2 35.0 /44.8 39.1 /49.166.8 / 69.5 72.0 / 78.4 75.1 /80.774.2 / 75.1 76.4 / 77.6 77.8 / 78.771.3 / 73.5 74.7 / 79.4 77.1 / 81.378.3 / 72.0 82.2 /84.4 83.6 /85.8
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DetectorOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
SVHN (seen)99.6 /99.8K+1 class loss in (5)/Confidence loss99.8 /99.999.7 /99.899.8 /99.999.9 /99.8
Baseline detectorTinyImageNet (unseen)79.61 /80.560.0 /9.95.2 /31.851.3 /58.650.7 /55.364.3 / 66.1
LSUN (unseen) Gaussian (unseen)0.0 /10.55.6 /34.851.5 /60.250.8/56.471.5 / 68.0
ODINSVHN (seen)0.0 /3.30.1/ 14.150.0 /50.0 99.1 /99.849.3 /49.412.2 /47.0
TinyImageNet (unseen)99.6 /99.899.9 /99.855.12/65.799.9 /99.999.9 /99.8
detectorLSUN (unseen)79.61 /80.560.3 /12.247.3 /70.656.6 /72.744.3 / 65.6
Gaussian (unseen)0.1 /13.7 0.0 /8.248.3 /73.1 28.3 /68.355.9 / 67.9 54.4 / 65.457.5 /75.2 47.8 /74.144.7 / 67.8 36.8 / 61.5
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Table 6 shows the detection", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 104, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "performance for out-of-distribution. First, the classifier trained by our method often significantly", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "outperforms the alternative adding the new class label. 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DetectorOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
SVHN (seen)99.6 /99.8K+1 class loss in (5)/Confidence loss99.8 /99.999.7 /99.899.8 /99.999.9 /99.8
Baseline detectorTinyImageNet (unseen)79.61 /80.560.0 /9.95.2 /31.851.3 /58.650.7 /55.364.3 / 66.1
LSUN (unseen) Gaussian (unseen)0.0 /10.55.6 /34.851.5 /60.250.8/56.471.5 / 68.0
ODINSVHN (seen)0.0 /3.30.1/ 14.150.0 /50.0 99.1 /99.849.3 /49.412.2 /47.0
TinyImageNet (unseen)99.6 /99.899.9 /99.855.12/65.799.9 /99.999.9 /99.8
detectorLSUN (unseen)79.61 /80.560.3 /12.247.3 /70.656.6 /72.744.3 / 65.6
Gaussian (unseen)0.1 /13.7 0.0 /8.248.3 /73.1 28.3 /68.355.9 / 67.9 54.4 / 65.457.5 /75.2 47.8 /74.144.7 / 67.8 36.8 / 61.5
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In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
Cross entropy loss /Confidence loss
SVHNCIFAR-10 (seen) TinyImageNet (unseen)93.82 /94.2347.4 /99.962.6 /99.978.6 /99.971.6 /99.991.2/99.4
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LSUN (unseen) Gaussian (unseen)46.3/100.0 56.1/100.061.8/100.0 72.0 /100.078.2 /100.0 83.4/100.071.1/100.0 77.2/100.090.8 /99.4 92.8 /99.4
CIFAR-10SVHN (seen)80.14 /80.56
13.7 / 99.8 13.6 /9.946.6 /99.966.6 /99.8 62.6 /58.661.4 /99.9 58.3 /55.373.5 /99.8 71.0 / 66.1
TinyImageNet (unseen)39.6 /31.8
LSUN (unseen) Gaussian (unseen)14.0 / 10.540.7 /34.863.2 / 60.258.7 /56.471.5 /68.0
2.8 / 3.310.2 /14.150.0/50.048.1 /49.439.9 /47.0
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In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
Cross entropy loss /Confidence loss
Baseline (SVHN)CIFAR-10 (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)92.14 / 93.7742.0 / 99.9 45.6 /99.9 44.6/100.0 58.6/100.088.0/100.0 89.4/100.0 89.8/100.0 94.2/100.083.4/99.8 84.3/99.9 84.5 /99.9 88.8/100.088.7 /99.9 90.2/100.0 90.8/100.0 95.5/100.087.3/99.3 88.6/99.3 88.4/99.3 92.5 / 99.3
Baseline (CIFAR-10)SVHN (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)76.58 /76.1812.8 / 99.6 10.3/10.1 10.7 /8.1 6.7 /1.071.0 /99.9 59.2/52.1 56.3 / 51.5 49.6/13.573.2/99.6 64.2/62.0 64.3/61.8 61.3 / 50.074.3 /99.9 63.6/59.8 62.3/ 59.5 58.5 /43.770.7 /99.6 64.4/62.3 65.3/ 61.6 59.5/32.0
ODIN (SVHN)CIFAR-10 (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)92.14 /93.7755.5/99.9 59.5 /99.9 61.5 /100.0 82.6/100.089.1/99.2 90.5 /99.3 91.8 /99.382.4/99.8 83.8 /99.9 84.8 /99.985.9 /100.0 87.5 /100.0 90.5/100.089.0 /99.2 90.4 /99.3 91.3/99.3
37.1/99.697.0 /99.3 86.7 /99.691.6/100.0 79.3 /99.6 64.4/ 61.897.4/100.0 88.1/99.996.4 /99.3 84.2/99.6
ODIN (CIFAR-10)SVHN (seen) TinyImageNet (unseen) LSUN (unseen) Gaussian (unseen)76.58 / 76.1811.4 /8.4 13.3 / 7.1 3.8/ 0.069.1/ 65.6 71.9 /67.1 70.9/57.275.3/63.7 69.3 /40.471.4 /68.6 67.2/72.0 78.1/56.164.6/60.7 65.3/60.5 60.7 /40.7
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In-distOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
CIFAR-10Joint confidence loss without PT/with PT
SVHNTinyImageNet LSUN93.81 /94.0590.1 /92.3 99.0 / 99.9 98.9 /100.097.6 /98.1 99.6 /100.0 99.6 /100.093.6 /94.6 97.6 /99.7 97.5 / 99.997.7 /98.2 99.7 /100.0 99.7 /100.097.9 /98.7 94.5 /100.0 95.5 /100.0
CIFAR-10SVHN TinyImageNet LSUN81.39 / 80.6025.4 / 13.2 35.0 /44.8 39.1 /49.166.8 / 69.5 72.0 / 78.4 75.1 /80.774.2 / 75.1 76.4 / 77.6 77.8 / 78.771.3 / 73.5 74.7 / 79.4 77.1 / 81.378.3 / 72.0 82.2 /84.4 83.6 /85.8
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DetectorOut-of-distClassification accuracyTNR at TPR 95%AUROCDetection accuracyAUPR inAUPR out
SVHN (seen)99.6 /99.8K+1 class loss in (5)/Confidence loss99.8 /99.999.7 /99.899.8 /99.999.9 /99.8
Baseline detectorTinyImageNet (unseen)79.61 /80.560.0 /9.95.2 /31.851.3 /58.650.7 /55.364.3 / 66.1
LSUN (unseen) Gaussian (unseen)0.0 /10.55.6 /34.851.5 /60.250.8/56.471.5 / 68.0
ODINSVHN (seen)0.0 /3.30.1/ 14.150.0 /50.0 99.1 /99.849.3 /49.412.2 /47.0
TinyImageNet (unseen)99.6 /99.899.9 /99.855.12/65.799.9 /99.999.9 /99.8
detectorLSUN (unseen)79.61 /80.560.3 /12.247.3 /70.656.6 /72.744.3 / 65.6
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