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+# DIRECTED-INFO GAIL: LEARNING HIERARCHICALPOLICIES FROM UNSEGMENTED DEMONSTRATIONSUSING DIRECTED INFORMATION
+
+Mohit Sharma∗, Arjun Sharma∗, Nick Rhinehart, Kris M. Kitani   
+Robotics Institute   
+Carnegie Mellon University   
+Pittsburgh, PA 15213, USA   
+{mohits1,arjuns2,nrhineha,kkitani}@cs.cmu.edu
+
+# ABSTRACT
+
+The use of imitation learning to learn a single policy for a complex task that has multiple modes or hierarchical structure can be challenging. In fact, previous work has shown that when the modes are known, learning separate policies for each mode or sub-task can greatly improve the performance of imitation learning. In this work, we discover the interaction between sub-tasks from their resulting stateaction trajectory sequences using a directed graphical model. We propose a new algorithm based on the generative adversarial imitation learning framework which automatically learns sub-task policies from unsegmented demonstrations. Our approach maximizes the directed information flow in the graphical model between sub-task latent variables and their generated trajectories. We also show how our approach connects with the existing Options framework, which is commonly used to learn hierarchical policies.
+
+# 1 INTRODUCTION
+
+Complex human activities can often be broken down into various simpler sub-activities or sub-tasks that can serve as the basic building blocks for completing a variety of complicated tasks. For instance, when driving a car, a driver may perform several simpler sub-tasks such as driving straight in a lane, changing lanes, executing a turn and braking, in different orders and for varying times depending on the source, destination, traffic conditions etc. Using imitation learning to learn a single monolithic policy to represent a structured activity can be challenging as it does not make explicit the sub-structure between the parts within the activity. In this work, we develop an imitation learning framework that can learn a policy for each of these sub-tasks given unsegmented activity demonstrations and also learn a macro-policy which dictates switching from one sub-task policy to another. Learning sub-task specific policies has the benefit of shared learning. Each such sub-task policy also needs to specialize over a restricted state space, thus making the learning problem easier.
+
+Previous works in imitation learning (Li et al., 2017; Hausman et al., 2017) focus on learning each sub-task specific policy using segmented expert demonstrations by modeling the variability in each sub-task policy using a latent variable. This latent variable is inferred by enforcing high mutual information between the latent variable and expert demonstrations. This information theoretic perspective is equivalent to the graphical model shown in Figure 1 (Left), where the node $c$ represents the latent variable. However, since learning sub-task policies requires isolated demonstrations for each sub-task, this setup is difficult to scale to many real world scenarios where providing such segmented trajectories is cumbersome. Further, this setup does not learn a macro-policy to combine the learned sub-task policies in meaningful ways to achieve different tasks.
+
+In our work, we aim to learn each sub-task policy directly from unsegmented activity demonstrations. For example, given a task consisting of three sub-tasks — A, B and C, we wish to learn a policy to complete sub-task A, learn when to transition from A to B, finish sub-task B and so on. To achieve this we use a causal graphical model, which can be represented as a Dynamic Bayesian Network as shown in Figure 1 (Right). The nodes $c _ { t }$ denote latent variables which indicate the currently active sub-task and the nodes $\tau _ { t }$ denote the state-action pair at time $t$ . We consider as given, a set of expert demonstrations, each of which is represented by $\tau = \{ \tau _ { 1 } , \cdot \cdot \cdot , \tau _ { T } \}$ and has a corresponding sequence of latent factors $\pmb { c } = \{ c _ { 1 } , \cdots , c _ { T - 1 } \}$ . The sub-activity at time $t$ dictates what state-action pair was generated at time $t$ . The previous sub-task and the current state together cause the selection of the next sub-task.
+
+![](images/8775ac58f1c362d5030b1164fb5d245993b2840ab38985e87b517f3881f858f3.jpg)  
+Figure 1: Left: Graphical model used in Info-GAIL Li et al. (2017). Right: Causal model in this work. The latent code causes the policy to produce a trajectory. The current trajectory, and latent code produce the next latent code
+
+As we will discuss in Section 3, extending the use of mutual information to learn sub-task policies from unsegmented demonstrations is problematic, as it requires learning the macro-policy as a conditional probability distribution which depends on the unobserved future. This unobserved future is unknown during earlier points of interaction (Figure 1). To alleviate this, in our work we aim to force the policy to generate trajectories that maximize the directed information or causal information (Massey, 1990) flow from trajectories to latent factors of variation within the trajectories instead of mutual information. Using directed information requires us to learn a causally conditioned probability distribution (Kramer, 1998) which depends only on the observed past while allowing the unobserved future to be sequentially revealed. Further, since there exists feedback in our causal graphical model i.e., information flows from the latent variables to trajectories and vice versa, directed information also provides a better upper bound on this information flow between the latent variables and expert trajectories than does the conventional mutual information (Massey, 1990; Kramer, 1998).
+
+We also draw connections with existing work on learning sub-task policies using imitation learning with the options framework (Sutton et al., 1998; Daniel et al., 2016). We show that our work, while derived using the information theoretic perspective of maximizing directed information, bears a close resemblance to applying the options framework in a generative adversarial imitation setting. Thus, our approach combines the benefits of learning hierarchical policies using the options framework with the robustness of generative adversarial imitation learning, helping overcome problems such as compounding errors that plague behaviour cloning.
+
+In summary, the main contributions of our work include:
+
+• We extend existing generative adversarial imitation learning frameworks to allow for learning   
+of sub-task specific policies by maximizing directed information in a causal graph of subactivity latent variables and observed trajectory variables.   
+• We draw connections between previous works on imitation learning with sub-task policies   
+using options and show that our proposed approach can also be seen as option learning in a   
+generative adversarial setting.   
+We show through experiments on both discrete and continuous state-action spaces, the   
+ability of our approach to segment expert demonstrations into meaningful sub-tasks and combine sub-task specific policies to perform the desired task.
+
+# 2 RELATED WORK
+
+# 2.1 IMITATION LEARNING
+
+Imitation Learning (Pomerleau, 1989) aims at learning policies that can mimic expert behaviours from demonstrations. Modeling the problem as a Markov Decision Process (MDP), the goal in imitation learning is to learn a policy $\pi ( a | s )$ , which defines the conditional distribution over actions $a \in { \mathcal { A } }$ given the state $s \in { \mathcal { S } }$ , from state-action trajectories $\tau = ( s _ { 0 } , a _ { 0 } , \cdot \cdot \cdot , s _ { T } )$ of expert behaviour. Recently,
+
+Ho & Ermon (2016) introduced an imitation learning framework called Generative Adversarial Imitation Learning (GAIL) that is able to learn policies for complex high-dimensional physics-based control tasks. They reduce the imitation learning problem into an adversarial learning framework, for which they utilize Generative Adversarial Networks (GAN) (Goodfellow et al., 2014). The generator network of the GAN represents the agent’s policy $\pi$ while the discriminator network serves as a local reward function and learns to differentiate between state-action pairs from the expert policy $\pi _ { \mathbb { E } }$ and from the agent’s policy $\pi$ . Mathematically, it is equivalent to optimizing the following,
+
+$$
+\displaystyle { \operatorname* { m i n } _ { \pi } } \operatorname* { m a x } _ { D } \mathbb { E } _ { \pi } [ \log D ( s , a ) ] + \mathbb { E } _ { \pi _ { E } } [ 1 - \log D ( s , a ) ] - \lambda H ( \pi )
+$$
+
+InfoGAIL (Li et al., 2017) and Hausman et al. (2017) solve the problem of learning from policies generated by a mixture of experts. They introduce a latent variable $c$ into the policy function $\pi ( a | s , c )$ to separate different type of behaviours present in the demonstration. To incentivize the network to use the latent variable, they utilize an information-theoretic regularization enforcing that there should be high mutual information between $c$ and the state-action pairs in the generated trajectory, a concept that was first introduced in InfoGAN (Chen et al., 2016). They introduce a variational lower bound $L _ { 1 } ( \pi , Q )$ of the mutual information $I ( c ; \tau )$ to the loss function in GAIL.
+
+$$
+L _ { 1 } ( \pi , Q ) = \mathbb { E } _ { c \sim p ( c ) , a \sim \pi ( \cdot | s , c ) } \log Q ( c | \tau ) + H ( c ) \leq I ( c ; \tau )
+$$
+
+The modified objective can then be given as,
+
+$$
+\operatorname* { m i n } _ { \pi , q } \operatorname* { m a x } _ { D } \mathbb { E } _ { \pi } [ \log D ( s , a ) ] + \mathbb { E } _ { \pi _ { E } } [ 1 - \log D ( s , a ) ] - \lambda _ { 1 } L _ { 1 } ( \pi , q ) - \lambda _ { 2 } H ( \pi )
+$$
+
+InfoGAIL models variations between different trajectories as the latent codes correspond to trajectories coming from different demonstrators. In contrast, we aim to model intra-trajectory variations and latent codes in our work correspond to sub-tasks (variations) within a demonstration. In Section 3, we discuss why using a mutual information based loss is infeasible in our problem setting and describe our proposed approach.
+
+# 2.2 OPTIONS
+
+Consider an MDP with states $s \in S$ and actions $a \in { \mathcal { A } }$ . Under the options framework (Sutton et al., 1998), an option, indexed by $o \in \mathcal { O }$ consists of a sub-policy $\pi ( a | s , o )$ , a termination policy $\pi ( b | s , \bar { o } )$ and an option activation policy $\pi ( o | s )$ . After an option is initiated, actions are generated by the sub-policy until the option is terminated and a new option is selected.
+
+Options framework has been studied widely in RL literature. A challenging problem related to the options framework is to automatically infer options without supervision. Option discovery approaches often aim to find bottleneck states, i.e., states that the agent has to pass through to reach the goal. Many different approaches such as multiple-instance learning (McGovern & Barto, 2001), graph based algorithms (Menache et al., 2002; S¸ ims¸ek et al., 2005) have been used to find such bottleneck states. Once the bottleneck states are discovered, the above approaches find options policies to reach each such state. In contrast, we propose a unified framework using a information-theoretic approach to automatically discover relevant option policies without the need to discover bottleneck states.
+
+Daniel et al. (2016) formulate the options framework as a probabilistic graphical model where options are treated as latent variables which are then learned from expert data. The option policies $( \pi ( a | s , o ) )$ are analogous to sub-task policies in our work. These option policies are then learned by maximizing a lower bound using the Expectation-Maximization algorithm (Moon, 1996). We show how this lower bound is closely related to the objective derived in our work. We further show how this connection allows our method to be seen as a generative adversarial variant of their approach. Fox et al. (2017) propose to extend the EM based approach to multiple levels of option hierarchies. Further work on discovery of deep continuous options (Krishnan et al., 2017) allows the option policy to also select a continuous action in states where none of the options are applicable. Our proposed approach can also be extended to multi-level hierarchies (e.g. by learning VAEs introduced in section 3 with multiple sampling layers) or hybrid categorical-continuous macro-policies (e.g. using both categorical and continuous hidden units in the sampling layer in VAE).
+
+Shiarlis et al. (2018) learn options by assuming knowledge of task sketches (Andreas et al., 2017) along with the demonstrations. The work proposes a behavior cloning based approach using connectionist temporal classification (Graves et al., 2006) to simultaneously maximize the joint likelihood of the sketch sequences and the sub-policies. Our proposed approach does not expect task sketches as input, making it more amenable to problems where labeling demonstrations with sketch labels is difficult.
+
+Prior work in robot learning has also looked at learning motion primitives from unsegmented demonstrations. These primitives usually correspond to a particular skill and are analogous to options. Niekum & Barto (2011) used the Beta-Process Autoregressive Hidden Markov Model (BP-AR-HMM) to segment expert demonstrations and post-process these segments to learn motion primitives which provide the ability to use reinforcement learning for policy improvement. Alternately, Krishnan et al. (2018) use Dirichlet Process Gaussian Mixture Model (DP-GMM) to segment the expert demonstrations by finding transition states between linear dynamical segments. Similarly, Ranchod et al. (2015) use the BP-AR-HMM framework to initially segment the expert demonstrations and then use an inverse reinforcement learning step to infer the reward function for each segment. The use of appropriate priors allows these methods to discover options without a priori knowledge of the total number of skills. Kroemer et al. (2014) model the task of manipulation as an autoregressive Hidden Markov Model where the hidden phases of manipulation are learned from data using EM. However, unlike the above methods, in our proposed approach we also learn an appropriate policy over the extracted options. We show how this allows us to compose the individual option policies to induce novel behaviours which were not present in the expert demonstrations.
+
+# 3 PROPOSED APPROACH
+
+As mentioned in the previous section, while prior approaches can learn to disambiguate the multiple modalities in the demonstration of a sub-task and learn to imitate them, they cannot learn to imitate demonstrations of unsegmented long tasks that are formed by a combination of many small sub-tasks. To learn such sub-task policies from unsegmented deomonstrations we use the graphical model in Figure 1 (Right), i.e., consider a set of expert demonstrations, each of which is represented by $\tau = \{ \tau _ { 1 } , \cdot \cdot \cdot , \tau _ { T } \}$ where $\tau _ { t }$ is the state-action pair observed at time $t$ . Each such demonstration has a corresponding sequence of latent variables $\bar { \pmb { c } } = \{ c _ { 1 } , \cdots , c _ { T - 1 } \}$ which denote the sub-activity in the demonstration at any given time step.
+
+As noted before, previous approaches (Li et al., 2017; Hausman et al., 2017) model the expert sub-task demonstrations using only a single latent variable. To enforce the model to use this latent variable, these approaches propose to maximize the mutual information between the demonstrated sequence of state-action pairs and the latent embedding of the nature of the sub-activity. This is achieved by adding a lower bound to the mutual information between the latent variables and expert demonstrations. This variational lower bound of the mutual information is then combined with the the adversarial loss for imitation learning proposed in Ho & Ermon (2016). Extending this to our setting, where we have a sequence of latent variables $^ c$ , yields the following lower bound on the mutual information,
+
+$$
+L ( \pi , q ) = \sum _ { t } \mathbb { E } _ { c ^ { 1 : t } \sim p ( c ^ { 1 : t } ) , a ^ { t - 1 } \sim \pi ( \cdot \vert s ^ { t - 1 } , c ^ { 1 : t - 1 } ) } \Big [ \log q ( c ^ { t } \vert c ^ { 1 : t - 1 } , \tau ) \Big ] + H ( c ) \le I ( \tau ; c )
+$$
+
+Observe that the dependence of $q$ on the entire trajectory $\tau$ precludes the use of such a distribution at test time, where only the trajectory up to the current time is known. To overcome this limitation, in this work we propose to force the policy to generate trajectories that maximize the directed or causal information flow from trajectories to the sequence of latent sub-activity variables instead. As we show below, by using directed information instead of mutual information, we can replace the dependence on $\tau$ with a dependence on the trajectory generated up to current time $t$ .
+
+The directed information flow from a sequence $\boldsymbol { X }$ to $\mathbf { Y }$ is given by,
+
+$$
+I ( \boldsymbol { X } \to \boldsymbol { Y } ) = H ( \boldsymbol { Y } ) - H ( \boldsymbol { Y } | | \boldsymbol { X } )
+$$
+
+where $H ( Y \| X )$ is the causally-conditioned entropy. Replacing $\boldsymbol { X }$ and $\mathbf { Y }$ with sequences $\tau$ and $^ c$ ,
+
+$$
+\begin{array} { l } { { \displaystyle I ( \tau \to c ) = H ( c ) - H ( c \| \tau ) } } \\ { { \mathrm { ~ } = H ( c ) - \displaystyle \sum _ { t } H ( c ^ { t } | c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) } } \\ { { \mathrm { ~ } = H ( c ) + \displaystyle \sum _ { t } \displaystyle \sum _ { c ^ { 1 : t - 1 } , \tau ^ { 1 : t } } \left[ p ( c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) \right. } } \\ { { \displaystyle \left. \sum _ { c ^ { t } } p ( c ^ { t } | c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) \log p ( c ^ { t } | c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) \right] } } \end{array}
+$$
+
+Here $\tau ^ { 1 : t } = ( s _ { 1 } , \cdots , a _ { t - 1 } , s _ { t } )$ . A variational lower bound, $L _ { 1 } ( \pi , q )$ of the directed information, $I ( \tau \to c )$ which uses an approximate posterior $q ( c ^ { t } | c ^ { 1 : t - 1 } , \tau ^ { 1 : t } )$ instead of the true posterior $p ( c ^ { t } | c ^ { 1 : t - 1 } , \tau ^ { 1 : t } )$ can then be derived to get (See Appendix A.1 for the complete derivation),
+
+$$
+L _ { 1 } ( \pi , q ) = \sum _ { t } \mathbb { E } _ { c ^ { 1 : t } \sim p ( c ^ { 1 : t } ) , a ^ { t - 1 } \sim \pi ( \cdot | s ^ { t - 1 } , c ^ { 1 : t - 1 } ) } \left[ \log q ( c ^ { t } | c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) \right] + H ( c ) \le I ( \tau \to c )
+$$
+
+Thus, by maximizing directed information instead of mutual information, we can learn a posterior distribution over the next latent factor $c$ given the latent factors discovered up to now and the trajectory followed up to now, thereby removing the dependence on the future trajectory. In practice, we do not consider the $H ( c )$ term. This gives us the following objective,
+
+$$
+\operatorname* { m i n } _ { \pi , q } \operatorname* { m a x } _ { D } \mathbb { E } _ { \pi } \left[ \log D ( s , a ) \right] + \mathbb { E } _ { \pi _ { E } } \left[ 1 - \log D ( s , a ) \right] - \lambda _ { 1 } L _ { 1 } ( \pi , q ) - \lambda _ { 2 } H ( \pi )
+$$
+
+We call this approach Directed-Info GAIL. Notice that, to compute the loss in equation 3, we need to sample from the prior distribution $p ( c ^ { 1 : t } )$ . In order to estimate this distribution, we first pre-train a variational auto-encoder (VAE) (Kingma & Welling, 2013) on the expert trajectories, the details of which are described in the next sub-section.
+
+# 3.1 VAE PRE-TRAINING
+
+Figure 2 (left) shows the design of the VAE pictorially. The VAE consists of two multi-layer perceptrons that serve as the encoder and the decoder. The encoder uses the current state $s _ { t }$ and the previous latent variable $c _ { t - 1 }$ to produce the current latent variable $c _ { t }$ . We used the Gumbel-softmax trick (Jang et al., 2016) to obtain samples of latent variables from a categorical distribution. The decoder then takes $s _ { t }$ and $c _ { t }$ as input and outputs the action $a _ { t }$ . We use the following objective, which maximizes the lower bound of the probability of the trajectories $p ( \tau )$ , to train our VAE,
+
+$$
+L _ { \mathrm { V A E } } ( \pi , q ; \tau _ { i } ) = - \sum _ { t } \mathbb { E } _ { c ^ { t } \sim q } \Big [ \log \pi \big ( a ^ { t } | s ^ { t } , c ^ { 1 : t } \big ) \Big ] + \sum _ { t } D _ { \mathrm { K L } } \big ( q ( c ^ { t } | c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) \| p ( c ^ { t } | c ^ { 1 : t - 1 } ) \big )
+$$
+
+Figure 2 (right) gives an overview of the complete method. The VAE pre-training step allows us to get approximate samples from the distribution $\bar { \boldsymbol { p } } ( c ^ { 1 : t } )$ to optimize equation 4. This is done by using $q$ to obtain samples of latent variable sequence $^ c$ by using its output on the expert demonstrations. In practice, we fix the weights of the network $q$ to those obtained from the VAE pre-training step when optimizing the Directed-Info GAIL loss in equation 4.
+
+# 3.2 CONNECTION WITH OPTIONS FRAMEWORK
+
+In Daniel et al. (2016) the authors provide a probabilistic perspective of the options framework. Although, Daniel et al. (2016) consider separate termination and option latent variables ( $\mathit { b } ^ { t }$ and $o ^ { t }$ ), for the purpose of comparison, we collapse them into a single latent variable $c ^ { t }$ , similar to our framework with a distribution $p ( c ^ { t } | s ^ { t } , c ^ { t - 1 } )$ . The lower-bound derived in Daniel et al. (2016) which is maximized using Expectation-Maximization (EM) algorithm can then be written as (suppressing dependence on parameters),
+
+![](images/2e1e4ad06e734632536ea39085f2740dd03f9cdf84d3b49afd95b03b787b04e3.jpg)  
+Figure 2: Left: VAE pre-training step. The VAE encoder uses the current state $\left( { { s } _ { t } } \right)$ , and previous latent variable $\left( c _ { t - 1 } \right)$ to produce the current latent variable $\left( c _ { t } \right)$ . The decoder reconstructs the action $\left( \boldsymbol { a } _ { t } \right)$ using $s _ { t }$ and $c _ { t }$ . Right: An overview of the proposed approach. We use the VAE pre-training step to learn an approximate prior over the latent variables and use this to learn sub-task policies in the proposed Directed-Info GAIL step.
+
+![](images/fd0a563c5a474de6c02d09243c0acc3e863ac0de96a18bde945fd8fa905152d3.jpg)  
+Figure 3: Results on the Four Rooms environment. (a) and (b) show results for two different latent variables. The arrows in each cell indicate the direction (action) with highest probability in that state and using the given latent variable. (c) and (d) show expert and generated trajectories in this environment. Star $( ^ { * } )$ represents the start state. The expert trajectory is shown in red. The color of the generated trajectory represents the latent code used by the policy at each time step.
+
+$$
+p ( \tau ) \geq \sum _ { t } \sum _ { c ^ { t - 1 : t } } p ( c ^ { t - 1 : t } | \tau ) \log p ( c ^ { t } | s ^ { t } , c ^ { t - 1 } ) ) + \sum _ { t } \sum _ { c ^ { t } } p ( c ^ { t } | \tau ) \log \pi ( a ^ { t } | s ^ { t } , c ^ { t } )
+$$
+
+Note that the first term in equation $6 i . e .$ ., the expectation over the distribution $\log p ( c ^ { t } | s ^ { t } , c ^ { t - 1 } )$ is the same as equation 3 of our proposed approach with a one-step Markov assumption and a conditional expectation with given expert trajectories instead of an expectation with generated trajectories. The second term in equation 6 i.e., the expectation over $\log { \bar { \pi } } ( a ^ { t } | s ^ { t } , c ^ { t } )$ is replaced by the GAIL loss in equation 4. Our proposed Directed-Info GAIL can be therefore be considered as the generative adversarial variant of imitation learning using the options framework. The VAE behaviour cloning pretraining step in equation 5 is exactly equivalent to equation 6, where we use approximate variational inference using VAEs instead of EM. Thus, our approach combines the benefits of both behavior cloning and generative adversarial imitation learning. Using GAIL enables learning of robust policies that do not suffer from the problem of compounding errors. At the same time, conditioning GAIL on latent codes learned from the behavior cloning step prevents the issue of mode collapse in GANs.
+
+# 4 EXPERIMENTS
+
+We present results on both discrete and continuous state-action environments. In both of these settings we show that (1) our method is able to segment out sub-tasks from given expert trajectories, (2) learn sub-task conditioned policies, and (3) learn to combine these sub-task policies in order to achieve the task objective.
+
+<table><tr><td>Environment</td><td>GAIL (Ho &amp; Ermon,2016)</td><td>VAE</td><td>Directed-Info GAIL</td></tr><tr><td>Pendulum-v0</td><td>-121.42 ± 94.13</td><td>-142.89 ± 95.57</td><td>-125.39 ± 103.75</td></tr><tr><td>InvertedPendulum-v2</td><td>1000.0 ± 15.23</td><td>218.8 ± 7.95</td><td>1000.0±14.97</td></tr><tr><td>Hopper-v2</td><td>3623.4 ± 51.0</td><td>499.1 ± 86.2</td><td>3662.1 ± 21.7</td></tr><tr><td>Walker2d-v2</td><td>4858.0 ± 301.7</td><td>1549.5 ± 793.7</td><td>5083.9 ± 356.3</td></tr></table>
+
+Table 1: A comparison of returns for continuous environments. The returns were computed using 300 episodes. Our approach gives comparable returns to using GAIL but also segments expert demonstrations into sub-tasks. The proposed Directed-Info GAIL approach improves over the policy learned from the VAE pre-training step.
+
+# 4.1 DISCRETE ENVIRONMENT
+
+For the discrete setting, we choose a grid world environment which consists of a $1 5 \times 1 1$ grid with four rooms connected via corridors as shown in Figure 3. The agent spawns at a random location in the grid and its goal is to reach an apple, which spawns in one of the four rooms randomly, using the shortest possible path. Through this experiment we aim to see whether our proposed approach is able to infer sub-tasks which correspond to meaningful navigation strategies and combine them to plan paths to different goal states.
+
+Figure 3 shows sub-task policies learned by our approach in this task. The two plots on the left correspond to two of the four different values of the latent variable. The arrow at every state in the grid shows the agent action (direction) with the highest probability in that state for that latent variable. In the discussion that follows, we label the rooms from 1 to 4 starting from the room at the top left and moving in the clockwise direction. We observe that the sub-tasks extracted by our approach represent semantically meaningful navigation plans. Also, each latent variable is utilized for a different sub-task. For instance, the agent uses the latent code in Figure 3(a), to perform the sub-task of moving from room 1 to room 3 and from room 2 to room 4 and the code in Figure 3(b) to move in the opposite direction. Further, our approach learns to successfully combine these navigation strategies to achieve the given objectives. For example, Figure 3(c, d) show examples of how the macro-policy switches between various latent codes to achieve the desired goals of reaching the apples in rooms 1 and 2 respectively.
+
+# 4.2 CONTINUOUS ENVIRONMENTS
+
+To validate our proposed approach on continuous control tasks we experiment with 5 continuous state-action environments. The first environment involves learning to draw circles on a 2D plane and is called Circle-World. In this experiment, the agent must learn to draw a circle in both clockwise and counter-clockwise direction. The agent always starts at (0,0), completes a circle in clockwise direction and then retraces its path in the counter-clockwise direction. The trajectories differ in the radii of the circles. The state $\bar { s } \in \mathbb { R } ^ { 2 }$ is the $\mathbf { \Phi } ( \mathbf { x } , \mathbf { y } )$ co-ordinate and the actions $a \in \mathbb { R } ^ { 2 }$ is a unit vector representing the direction of motion. Notice that in Circle-World, the expert trajectories include two different actions (for clockwise and anti-clockwise direction) for every state $( x , y )$ in the trajectory, thus making the problem multi-modal in nature. This requires the agent to appropriately disambiguate between the two different phases of the trajectory.
+
+Further, to show the scalability of our approach to higher dimensional continuous control tasks we also show experiments on Pendulum, Inverted Pendulum, Hopper and Walker environments, provided in OpenAI Gym (Brockman et al., 2016). Each task is progressively more challenging, with a larger state and action space. Our aim with these experiments is to see whether our approach can identify certain action primitives which helps the agent to complete the given task successfully. To verify the effectiveness of our proposed approach we do a comparative analysis of our results with both GAIL (Ho & Ermon, 2016) and the supervised behavior cloning approaching using a VAE. To generate expert trajectories we train an agent using Proximal Policy Optimization (Schulman et al., 2017). We used 25 expert trajectories for the Pendulum and Inverted Pendulum tasks and 50 expert trajectories for experiments with the Hopper and Walker environments.
+
+Figures 4(a, b, c) show results on the Circle-World environment. As can be seen in Figure 4(a, b), when using two sub-task latent variables, our method learns to segment the demonstrations into two intuitive sub-tasks of drawing circles in clockwise and counterclockwise directions. Hence, our method is able to identify the underlying modes and thus find meaningful sub-task segmentations from unsegmented data. We also illustrate how the learned sub-task policies can be composed to perform new types of behavior that were unobserved in the expert data. In Figure 4(c) we show how the sub-task policies can be combined to draw the circles in inverted order of direction by swapping the learned macro-policy with a different desired policy. Thus, the sub-task policies can be utilized as a library of primitive actions which is a significant benefit over methods learning monolithic policies.
+
+![](images/8c3c3fe0f68cf95be6d1b031a3a50de574f6fe13b7d50a2c93125cfbc4e6c2d0.jpg)  
+Figure 4: Results for Directed-Info GAIL on continuous environments. (a) Our method learns to break down the Circle-World task into two different sub-activities, shown in green and blue. (b) Trajectory generated using our approach. Color denotes time step. (c) Trajectory generated in opposite direction. Color denotes time step. (d) Sub-activity latent variables as inferred by Directed-Info GAIL on Pendulum-v0. Different colors represent different context.
+
+![](images/04db31d0960cc22f6b6dbb3e1c42265fee143353435bf5e1c17084cfb2d5656c.jpg)  
+Figure 5: (a) shows the plot of the sub-task latent variable vs time on the Hopper and Walker tasks. (b) shows discovered sub-tasks using Directed-Info GAIL on these environments.
+
+We now discuss the results on the classical Pendulum environment. Figure 4(d) shows the sub-task latent variables assigned by our approach to the various states. As can be seen in the figure, the network is able to associate different latent variables to different sub-tasks. For instance, states that have a high velocity are assigned a particular latent variable (shown in blue). Similarly, states that lie close to position 0 and have low velocity (i.e. the desired target position) get assigned another latent variable (shown in green). The remaining states get classified as a separate sub-task.
+
+Figure 5 shows the results on the higher dimensional continuous control, Hopper and Walker, environments. Figure 5(a) shows a plots for sub-task latent variable assignment obtained on these environments. Our proposed method identifies basic action primitives which are then chained together to effectively perform the two locomotion tasks. Figure 5(b) shows that our approach learns to assign separate latent variable values for different action primitives such as, jumping, mid-air and landing phases of these tasks, with the latent variable changing approximately periodically as the agent performs the periodic hopping/walking motion.
+
+Finally, in Table 1 we also show the quantitative evaluation on the above continuous control environments. We report the mean and standard deviations of the returns over 300 episodes. As can be seen, our approach improves the performance over the VAE pre-training step, overcoming the issue of compounding errors. The performance of our approach is comparable to the state-of-the-art GAIL (Ho & Ermon, 2016). Our method moreover, has the added advantage of segmenting the demonstrations into sub-tasks and also providing composable sub-task policies.
+
+![](images/5c0ccdf53439b8388b8cd38d7baa1460824b68f92a7018a2a2c771a9f7aa9152.jpg)  
+Figure 6: Segmentations obtained using our proposed Directed-Info GAIL method on FetchPickandPlace-v1.
+
+Table 2: Mean returns over 100 episodes on FetchPickandPlace-v1 environment, calculated using the ‘dense’ reward setting.   
+
+<table><tr><td>Method</td><td>Returns</td></tr><tr><td>VAE</td><td>-14.07 ± 5.57</td></tr><tr><td>GAIL</td><td>-13.29 ± 5.84</td></tr><tr><td>Directed-Info GAIL</td><td>-11.74 ± 5.87</td></tr><tr><td>GAIL + L2 loss Directed-Info GAIL + L2 loss</td><td>-12.05 ± 4.94 -9.47 ± 4.84</td></tr></table>
+
+We further analyze our proposed approach in more detail in the Appendix. In Appendix A.4 we visualize the sub-tasks in a low-dimensional sub-space. Also, in Appendix A.5 we show results when using a larger dimensional sub-task latent variable. A video of our results on Hopper and Walker environments can be seen at https://sites.google.com/view/directedinfo-gail.
+
+# 4.3 OPENAI ROBOTICS ENVIRONMENT
+
+We further performed experiments on the FetchPickandPlace-v1 task in OpenAI Gym. In each episode of this task, the object and goal locations are selected randomly. The robot then must first reach and pick the object, and then move it to the goal location.
+
+We trained agents using both our proposed Directed-Info GAIL and the baseline GAIL approaches. We used 500 expert demonstrations. While our method was able to learn to segment the expert demonstrations into the Pick and Place sub-tasks correctly, as can be seen in Figure 6 and the videos at https://sites.google.com/view/directedinfo-gail/home#h.p_ 4dsbuC5expkZ, neither our approach, nor GAIL was able to successfully complete the task. In our preliminary results, we found that the robot, in both our proposed approach and GAIL, would reach the object but fail to grasp it despite repeated attempts. To the best of our knowledge, no other work has successfully trained GAIL on this task either. Our preliminary experiments suggested that stronger supervision may be necessary to teach the agent the subtle action of grasping.
+
+In order to provide this supervision, we additionally trained the policy to minimize the L2 distance between the policy action and the expert action on states in the expert demonstrations. At every training step, we compute the discriminator and policy (generator) gradient using the Directed-Info GAIL (or in the baseline, GAIL) loss using states and actions generated by the policy. Along with this gradient, we also sample a batch of states from the expert demonstrations and compute the policy gradient that minimizes the L2 loss between actions that the policy takes at these states and the actions taken by the expert. We weigh these two gradients to train the policy.
+
+Table 2 shows the returns computed over 100 episodes. Adding the L2 measure as an additional loss led to significant improvement. Our proposed approach Directed-Info $\mathrm { G A I L } + \mathrm { L } 2$ loss outperforms the baselines. Moreover, we believe that this quantitative improvement does not reflect the true performance gain obtained using our method. The reward function is such that a correct grasp but incorrect movement (e.g. motion in the opposite direction or dropping of the object) is penalized more than a failed grasp. Thus, the reward function does not capture the extent to which the task was completed.
+
+Qualitatively, we observed a much more significant difference in performance between the proposed approach and the baseline. This can be seen in the sample videos of the success and failure cases for our and the baseline method at https://sites.google.com/view/ directedinfo-gail/home#h.p_qM39qD8xQhJQ. Our proposed method succeeds much more often than the baseline method. The most common failure cases for our method include the agent picking up the object, but not reaching the goal state before the end of the episode, moving the object to an incorrect location or dropping the object while moving it to the goal. Agents trained using GAIL $+ \mathrm { L } 2$ loss on the other hand often fail to grasp the object, either not closing the gripper or closing the gripper prematurely. We believe that our approach helps the agent alleviate this issue by providing it with the sub-task code, helping it disambiguate between the very similar states the agent observes just before and just after grasping.
+
+# 5 CONCLUSION
+
+Learning separate sub-task policies can help improve the performance of imitation learning when the demonstrated task is complex and has a hierarchical structure. In this work, we present an algorithm that infers these latent sub-task policies directly from given unstructured and unlabelled expert demonstrations. We model the problem of imitation learning as a directed graph with sub-task latent variables and observed trajectory variables. We use the notion of directed information in a generative adversarial imitation learning framework to learn sub-task and macro policies. We further show theoretical connections with the options literature as used in hierarchical reinforcement and imitation learning. We evaluate our method on both discrete and continuous environments. Our experiments show that our method is able to segment the expert demonstrations into different sub-tasks, learn sub-task specific policies and also learn a macro-policy that can combines these sub-task.
+
+# REFERENCES
+
+Jacob Andreas, Dan Klein, and Sergey Levine. Modular multitask reinforcement learning with policy sketches. In International Conference on Machine Learning, pp. 166–175. JMLR. org, 2017.   
+Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym, 2016.   
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+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672–2680, 2014.   
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+Jonathan Ho and Stefano Ermon. Generative adversarial imitation learning. In Advances in Neural Information Processing Systems, pp. 4565–4573, 2016.   
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+Sanjay Krishnan, Roy Fox, Ion Stoica, and Ken Goldberg. DDCO: Discovery of deep continuous options for robot learning from demonstrations. In 1st Conference on Robot Learning (CoRL), 2017.
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+O. Kroemer, H. van Hoof, G. Neumann, and J. Peters. Learning to predict phases of manipulation tasks as hidden states. In Proceedings of 2014 IEEE International Conference on Robotics and Automation, pp. 4009–4014. IEEE, 2014.
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+James Massey. Causality, feedback and directed information. In Proc. Int. Symp. Inf. Theory Applic.(ISITA-90), pp. 303–305. Citeseer, 1990.
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+Amy McGovern and Andrew G Barto. Accelerating reinforcement learning through the discovery of useful subgoals. 2001.
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+Ishai Menache, Shie Mannor, and Nahum Shimkin. Q-cutdynamic discovery of sub-goals in reinforcement learning. In European Conference on Machine Learning, pp. 295–306. Springer, 2002.
+
+Todd K Moon. The expectation-maximization algorithm. IEEE Signal processing magazine, 13(6): 47–60, 1996.
+
+Scott Niekum and Andrew G Barto. Clustering via dirichlet process mixture models for portable skill discovery. In Advances in neural information processing systems, pp. 1818–1826, 2011.
+
+Dean A Pomerleau. Alvinn: An autonomous land vehicle in a neural network. In Advances in neural information processing systems, pp. 305–313, 1989.
+
+Pravesh Ranchod, Benjamin Rosman, and George Konidaris. Nonparametric bayesian reward segmentation for skill discovery using inverse reinforcement learning. In Intelligent Robots and Systems (IROS), 2015 IEEE/RSJ International Conference on, pp. 471–477. IEEE, 2015.
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+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.
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+Kyriacos Shiarlis, Markus Wulfmeier, Sasha Salter, Shimon Whiteson, and Ingmar Posner. Taco: Learning task decomposition via temporal alignment for control. In International Conference on Machine Learning, 2018.
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+
+Richard S Sutton, Doina Precup, and Satinder P Singh. Intra-option learning about temporally abstract actions. In ICML, volume 98, pp. 556–564, 1998.
+
+# A APPENDIX
+
+A.1 DERIVATION FOR DIRECTED-INFO LOSS
+
+The directed information flow from a sequence $\boldsymbol { X }$ to $\mathbf { Y }$ is given by:
+
+$$
+I ( \boldsymbol { X } \to \boldsymbol { Y } ) = H ( \boldsymbol { Y } ) - H ( \boldsymbol { Y } | | \boldsymbol { X } )
+$$
+
+where $H ( Y \| X )$ is the causally-conditioned entropy. Replacing $\boldsymbol { X }$ and $\mathbf { Y }$ with the sequences $\tau$ and $^ c$ give,
+
+$$
+\begin{array} { r l } & { I ( \tau  c ) = H ( c ) - H ( c ) \mid \tau } \\ & { \qquad = H ( c ) - \displaystyle \sum _ { \tau } H ( c ^ { i } \mid c ^ { i - 1 } , \tau ^ { 1 : t } ) } \\ & { \qquad = H ( c ) + \displaystyle \sum _ { \tau } \displaystyle \sum _ { c ^ { i + 1 } = 1 , \tau ^ { 1 : t } ) } [ p ( c ^ { i + t - 1 } , \tau ^ { 1 : t } ) \displaystyle \sum _ { c ^ { i } } p ( c ^ { i } \mid c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) \log p ( c ^ { i } \mid c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) ] } \\ & { \qquad = H ( c ) + \displaystyle \sum _ { \tau } \displaystyle \sum _ { c ^ { i + 1 } = 1 , \tau ^ { 1 : t } ) } [ p ( c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) [ D _ { K L } ( p ( \cdot \mid c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) | q ( \cdot \mid c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) )  } \\ & { \qquad \quad \qquad + \displaystyle \sum _ { c ^ { i } } p ( c ^ { i } \mid c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) \log q ( c ^ { i } \mid c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) ] ] } \\ & { \qquad \geq H ( c ) + \displaystyle \sum _ { \tau } \displaystyle \sum _ { c ^ { i + 1 } = 1 , \tau ^ { 1 : t } ) } [ p ( c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) \displaystyle \sum _ { c ^ { i } } p ( c ^ { i } \mid c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) \log q ( c ^ { i } \mid c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) ] . } \end{array}
+$$
+
+Here $\tau ^ { 1 : t } = ( s _ { 1 } , \cdots , a _ { t - 1 } , s _ { t } )$ . The lower bound in equation 7 requires us to know the true posterior distribution to compute the expectation. To avoid sampling from $p ( c ^ { t } | c ^ { 1 : t - 1 } , \tau ^ { 1 : t } )$ , we use the following,
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { s = 1 } \sum _ { \tau = 1 } \left[ p ( c ^ { \lfloor k - 1 \rfloor } , \tau ^ { \lfloor k \rfloor } ) \sum _ { \sigma ^ { \prime } } \overline { { p } } ( c ^ { \lfloor k \rfloor } c ^ { \lfloor k - 1 \rfloor } , \tau ^ { \lfloor k \rfloor } ) \log q ( c ^ { \lfloor k \rfloor - 1 } , \tau ^ { \lfloor k \rfloor } ) \right] } \\ & { \quad = \displaystyle \sum _ { s = 1 - 1 } \sum _ { \tau = 1 } \sum _ { \tau ^ { \lfloor k \rfloor } } \sum _ { \sigma ^ { \prime } } \left[ p ( c ^ { \lfloor k \rfloor - 1 } , \tau ^ { \lfloor k \rfloor } ) p ( c ^ { \lfloor k \rfloor - 1 } , \tau ^ { \lfloor k \rfloor } ) \log q ( c ^ { \lfloor k \rfloor - 1 } , \tau ^ { \lfloor k \rfloor } ) \right] } \\ & { \quad = \displaystyle \sum _ { s = 1 - 1 } \sum _ { \tau = 1 } \sum _ { \tau ^ { \lfloor k \rfloor } } \sum _ { \epsilon ^ { \ell } } \left[ p ( c ^ { \eta } , c ^ { \lfloor k \rfloor - 1 } , \tau ^ { \lfloor k \rfloor } ) \log q ( c ^ { \lfloor k \rfloor - 1 } , \tau ^ { \lfloor k \rfloor } ) \right] } \\ & { \quad = \displaystyle \sum _ { s = 1 - 1 } \sum _ { \tau \in \tau ^ { \lfloor k \rfloor } } \sum _ { \epsilon ^ { \ell } } \left[ p ( \tau ^ { \lfloor k \rfloor } | c , c ^ { \lfloor k - 1 \rfloor } ) p ( c ^ { \ell } , c ^ { \lfloor k - 1 \rfloor - 1 } ) \log q ( c ^ { \lfloor k \rfloor - 1 } , \tau ^ { \lfloor k \rfloor } ) \right] } \\ & { \quad = \displaystyle \sum _ { s = 1 } ^ { \lfloor \eta } p ( c ^ { \lfloor k \rfloor } ) \sum _ { \tau ^ { \lfloor k \rfloor } } \left[ p ( \tau ^ { \lfloor k \rfloor } | c ^ { \ell } , c ^ { \lfloor k - 1 \rfloor } ) \log q ( c ^ { \lfloor k \rfloor - 1 } , \tau ^ { \lfloor k \rfloor } ) \right] } \\ & { \quad = \displaystyle \sum _ { s = 1 } ^ { \lfloor \eta } p ( c ^ { \lfloor k \rfloor } ) \sum _ { \tau ^ { \lfloor k \rfloor } } \left[ p ( \tau ^ { \lfloor k \rfloor } | c ^ { \lfloor k - 1 \rfloor - 1 } ) \log q ( c ^ { \lfloor k \rfloor - 1 } , \tau ^ { \lfloor k \rfloor } ) \right] } \\ &  \quad = \displaystyle \sum _ { s = 1 } ^ { \lfloor \eta \rfloor } p ( c ^ { \lfloor k \rfloor } ) \sum _ { \tau ^ { \lfloor k \rfloor } } \end{array}
+$$
+
+where the last step follows from the causal restriction that future provided variables $( c ^ { t } )$ do not influence earlier predicted variables ( $\tau ^ { 1 : t }$ consists of states up to time $t$ . $c _ { t }$ does not effect state $s _ { t }$ ). Putting the result in equation 8 in equation 7 gives,
+
+$$
+L _ { 1 } ( \pi , q ) = \sum _ { t } \mathbb { E } _ { c ^ { 1 : t } \sim p ( c ^ { 1 : t } ) , a ^ { t - 1 } \sim \pi ( \cdot | s ^ { t - 1 } , c ^ { 1 : t - 1 } ) } \left[ \log q ( c ^ { t } | c ^ { 1 : t - 1 } , \tau ^ { 1 : t } ) \right] + H ( c ) \le I ( \tau \to c )
+$$
+
+Table 3: Experiment settings for all the different environments for both DirectedInfo-GAIL and VAE-pretraining step respectively.   
+
+<table><tr><td colspan="4">Directed Info-GAIL</td><td colspan="2">VAE pre-training</td></tr><tr><td>Environment</td><td>Epochs</td><td>Batch Size</td><td>posterior 入</td><td>Epochs</td><td>Batch Size</td></tr><tr><td>Discrete</td><td>1000</td><td>256</td><td>0.1</td><td>500</td><td>32</td></tr><tr><td>Circle-World</td><td>1000</td><td>512</td><td>0.01</td><td>1000</td><td>16</td></tr><tr><td>Pendulum (both)</td><td>2000</td><td>1024</td><td>0.01</td><td>1000</td><td>16</td></tr><tr><td>Hopper-v2</td><td>5000</td><td>4096</td><td>0.01</td><td>2000</td><td>32</td></tr><tr><td>Walker2d-v2</td><td>5000</td><td>8192</td><td>0.001</td><td>2000</td><td>32</td></tr></table>
+
+![](images/3957e6012ac4a833cae942728b11328572916d9caf0777a06c73a3337e91a170.jpg)  
+Figure 7: Latent variable assignment on the expert trajectories in Circle-World (a) with and (b) without smoothing penalty $L _ { s }$ . Blue and green colors represent the two different values of the context variable. The centres of the two circles are shifted for clarity.
+
+Thus, by maximizing directed information instead of mutual information, we can learn a posterior distribution over the next latent factor $c$ given the latent factors discovered up to now and the trajectory followed up to now, thereby removing the dependence on the future trajectory. In practice, we do not consider the $H ( c )$ term. This gives us the objective,
+
+$$
+\operatorname* { m i n } _ { \pi , q } \operatorname* { m a x } _ { D } \mathbb { E } _ { \pi } [ \log D ( s , a ) ] + \mathbb { E } _ { \pi _ { E } } [ 1 - \log D ( s , a ) ] - \lambda _ { 1 } L _ { 1 } ( \pi , q ) - \lambda _ { 2 } H ( \pi ) .
+$$
+
+In practice, we fix $q$ from the VAE pre-training and only minimize over the policy $\pi$ in equation 4.
+
+# A.2 IMPLEMENTATION DETAILS
+
+Table 3 lists the experiment settings for all of the different environments. We use multi-layer perceptrons for our policy (generator), value, reward (discriminator) and posterior function representations. Each network consisted of 2 hidden layers with 64 units in each layer and ReLU as our non-linearity function. We used Adam (Kingma & Ba, 2014) as our optimizer setting an initial learning rate of $3 e ^ { - 4 }$ . Further, we used the Proximal Policy Optimization algorithm (Schulman et al., 2017) to train our policy network with $\epsilon = 0 . 2$ . For the VAE pre-training step we set the VAE learning rate also to $3 e ^ { - \hat { 4 } }$ . For the Gumbel-Softmax distribution we set an initial temperature $\tau = 5 . 0$ . The temperature is annealed using using an exponential decay with the following schedule $\tau = \operatorname* { m a x } ( 0 . 1 , \stackrel { - } { \exp } ^ { - k t } )$ , where $k = 3 e - 3$ and $t$ is the current epoch.
+
+# A.3 CIRCLE-WORLD SMOOTHING
+
+In the Circle-World experiment, we added another loss term $L _ { s }$ to VAE pre-training loss $L _ { V A E }$ , which penalizes the number of times the latent variable switches from one value to another.
+
+$$
+L _ { s } = \sum _ { t } \left[ 1 - \frac { c _ { t - 1 } \cdot c _ { t } } { \operatorname* { m a x } ( | | c _ { t - 1 } | | _ { 2 } , | | c _ { t } | | _ { 2 } ) } \right]
+$$
+
+![](images/9c18955cc6e77a8396a8df44a443297bbdc4e9708adbc455ae37e457ceeec0a1.jpg)  
+Figure 8: PCA Visualization for Hopper and Walker environment with sub-task latent variable of size 4.
+
+![](images/e7fe68c4adc890e0944ac707b88d823abb69039c77f287f089be933b1cac256e.jpg)  
+Figure 9: Results on Hopper environment with sub-task latent variable of size 8.
+
+Figure 7 shows the segmentation of expert trajectories with and without the $L _ { s }$ term. We observed that without adding the smoothing penalty, the VAE learns to segment the expert trajectories into semi-circles as shown in Figure 7(a). While a valid solution, this does not match with the intuitive segmentation of the task into two sub-tasks of drawing circles in clockwise and counter-clockwise directions. The smoothing term can be thought of as a prior, forcing the network to change the latent variable as few times as possible. This helps reach a solution where the network switches between latent variables only when required. Figure 7(b) shows an example of segmentation obtained on expert trajectories after smoothing. Thus, adding more terms to the VAE pre-training loss can be a good way to introduce priors and bias solutions towards those that match with human notion of sub-tasks.
+
+# A.4 PCA VISUALIZATION OF SUB-TASKS
+
+In Figure 8, we show the plots expert states, reduced in dimensionality using Principal Component Analysis (PCA), in Hopper and Walker environments. States are color coded by the latent code assigned at these states. We reduced the dimension of states in Hopper from 11 to 2 and in Walker from 17 to 3. These low dimensional representations are able to cover $\sim 9 0 \%$ of variance in the states. As can be seen in the figure, states in different parts of the space get assigned different latent variables. This further shows that our proposed approach is able to segment trajectories in such a way so that states that are similar to each other get assigned to the same segment (latent variable).
+
+# A.5 USING LARGER CONTEXT
+
+For the following discussion we will represent a $k$ -dimensional categorical variable as belonging to $\Delta ^ { k - 1 }$ simplex. To observe how the dimensionality of the sub-task latent variable affects our proposed approach we show results with larger dimensionality for the categorical latent variable $c _ { t }$ . Since DirectedInfo-GAIL infers the sub-tasks in an unsupervised manner, we expect our approach to output meaningful sub-tasks irrespective of the dimensionality of $c _ { t }$ . Figure 9 shows results for using a higher dimensional sub-task latent variable. Precisely, we assume $c _ { t }$ to be a 8-dimensional one hot vector, i.e., $c _ { t } \in \Delta ^ { 7 }$ .
+
+As seen in the above figure, even with a larger context our approach identifies similar basic action primitives as done previously when $c _ { t } \in \bar { \Delta } ^ { 3 }$ . This shows that despite larger dimensionality our approach is able to reuse appropriate context inferred previously. We also visualize the context values for the low-dimensional state-space embedding obtained by PCA. Although not perfectly identical, these context values are similar to the visualizations observed previously for $\bar { c _ { t } } \in \Delta ^ { 3 }$ . Thus our proposed approach is able, to some extent, infer appropriate sub-task representations independent of the dimensionality of the context variable.

parse/train/BJeWUs05KQ/BJeWUs05KQ_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "DIRECTED-INFO GAIL: LEARNING HIERARCHICALPOLICIES FROM UNSEGMENTED DEMONSTRATIONSUSING DIRECTED INFORMATION",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mohit Sharma∗, Arjun Sharma∗, Nick Rhinehart, Kris M. Kitani   \nRobotics Institute   \nCarnegie Mellon University   \nPittsburgh, PA 15213, USA   \n{mohits1,arjuns2,nrhineha,kkitani}@cs.cmu.edu ",
+        "bbox": [
+            183,
+            194,
+            632,
+            265
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            301,
+            544,
+            316
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The use of imitation learning to learn a single policy for a complex task that has multiple modes or hierarchical structure can be challenging. In fact, previous work has shown that when the modes are known, learning separate policies for each mode or sub-task can greatly improve the performance of imitation learning. In this work, we discover the interaction between sub-tasks from their resulting stateaction trajectory sequences using a directed graphical model. We propose a new algorithm based on the generative adversarial imitation learning framework which automatically learns sub-task policies from unsegmented demonstrations. Our approach maximizes the directed information flow in the graphical model between sub-task latent variables and their generated trajectories. We also show how our approach connects with the existing Options framework, which is commonly used to learn hierarchical policies. ",
+        "bbox": [
+            233,
+            332,
+            766,
+            498
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+            176,
+            523,
+            336,
+            539
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Complex human activities can often be broken down into various simpler sub-activities or sub-tasks that can serve as the basic building blocks for completing a variety of complicated tasks. For instance, when driving a car, a driver may perform several simpler sub-tasks such as driving straight in a lane, changing lanes, executing a turn and braking, in different orders and for varying times depending on the source, destination, traffic conditions etc. Using imitation learning to learn a single monolithic policy to represent a structured activity can be challenging as it does not make explicit the sub-structure between the parts within the activity. In this work, we develop an imitation learning framework that can learn a policy for each of these sub-tasks given unsegmented activity demonstrations and also learn a macro-policy which dictates switching from one sub-task policy to another. Learning sub-task specific policies has the benefit of shared learning. Each such sub-task policy also needs to specialize over a restricted state space, thus making the learning problem easier. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Previous works in imitation learning (Li et al., 2017; Hausman et al., 2017) focus on learning each sub-task specific policy using segmented expert demonstrations by modeling the variability in each sub-task policy using a latent variable. This latent variable is inferred by enforcing high mutual information between the latent variable and expert demonstrations. This information theoretic perspective is equivalent to the graphical model shown in Figure 1 (Left), where the node $c$ represents the latent variable. However, since learning sub-task policies requires isolated demonstrations for each sub-task, this setup is difficult to scale to many real world scenarios where providing such segmented trajectories is cumbersome. Further, this setup does not learn a macro-policy to combine the learned sub-task policies in meaningful ways to achieve different tasks. ",
+        "bbox": [
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+        "type": "text",
+        "text": "In our work, we aim to learn each sub-task policy directly from unsegmented activity demonstrations. For example, given a task consisting of three sub-tasks — A, B and C, we wish to learn a policy to complete sub-task A, learn when to transition from A to B, finish sub-task B and so on. To achieve this we use a causal graphical model, which can be represented as a Dynamic Bayesian Network as shown in Figure 1 (Right). The nodes $c _ { t }$ denote latent variables which indicate the currently active sub-task and the nodes $\\tau _ { t }$ denote the state-action pair at time $t$ . We consider as given, a set of expert demonstrations, each of which is represented by $\\tau = \\{ \\tau _ { 1 } , \\cdot \\cdot \\cdot , \\tau _ { T } \\}$ and has a corresponding sequence of latent factors $\\pmb { c } = \\{ c _ { 1 } , \\cdots , c _ { T - 1 } \\}$ . The sub-activity at time $t$ dictates what state-action pair was generated at time $t$ . The previous sub-task and the current state together cause the selection of the next sub-task. ",
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+        "img_path": "images/8775ac58f1c362d5030b1164fb5d245993b2840ab38985e87b517f3881f858f3.jpg",
+        "image_caption": [
+            "Figure 1: Left: Graphical model used in Info-GAIL Li et al. (2017). Right: Causal model in this work. The latent code causes the policy to produce a trajectory. The current trajectory, and latent code produce the next latent code "
+        ],
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As we will discuss in Section 3, extending the use of mutual information to learn sub-task policies from unsegmented demonstrations is problematic, as it requires learning the macro-policy as a conditional probability distribution which depends on the unobserved future. This unobserved future is unknown during earlier points of interaction (Figure 1). To alleviate this, in our work we aim to force the policy to generate trajectories that maximize the directed information or causal information (Massey, 1990) flow from trajectories to latent factors of variation within the trajectories instead of mutual information. Using directed information requires us to learn a causally conditioned probability distribution (Kramer, 1998) which depends only on the observed past while allowing the unobserved future to be sequentially revealed. Further, since there exists feedback in our causal graphical model i.e., information flows from the latent variables to trajectories and vice versa, directed information also provides a better upper bound on this information flow between the latent variables and expert trajectories than does the conventional mutual information (Massey, 1990; Kramer, 1998). ",
+        "bbox": [
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+        "type": "text",
+        "text": "We also draw connections with existing work on learning sub-task policies using imitation learning with the options framework (Sutton et al., 1998; Daniel et al., 2016). We show that our work, while derived using the information theoretic perspective of maximizing directed information, bears a close resemblance to applying the options framework in a generative adversarial imitation setting. Thus, our approach combines the benefits of learning hierarchical policies using the options framework with the robustness of generative adversarial imitation learning, helping overcome problems such as compounding errors that plague behaviour cloning. ",
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+    {
+        "type": "text",
+        "text": "In summary, the main contributions of our work include: ",
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+        "type": "text",
+        "text": "• We extend existing generative adversarial imitation learning frameworks to allow for learning   \nof sub-task specific policies by maximizing directed information in a causal graph of subactivity latent variables and observed trajectory variables.   \n• We draw connections between previous works on imitation learning with sub-task policies   \nusing options and show that our proposed approach can also be seen as option learning in a   \ngenerative adversarial setting.   \nWe show through experiments on both discrete and continuous state-action spaces, the   \nability of our approach to segment expert demonstrations into meaningful sub-tasks and combine sub-task specific policies to perform the desired task. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "2.1 IMITATION LEARNING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Imitation Learning (Pomerleau, 1989) aims at learning policies that can mimic expert behaviours from demonstrations. Modeling the problem as a Markov Decision Process (MDP), the goal in imitation learning is to learn a policy $\\pi ( a | s )$ , which defines the conditional distribution over actions $a \\in { \\mathcal { A } }$ given the state $s \\in { \\mathcal { S } }$ , from state-action trajectories $\\tau = ( s _ { 0 } , a _ { 0 } , \\cdot \\cdot \\cdot , s _ { T } )$ of expert behaviour. Recently, ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Ho & Ermon (2016) introduced an imitation learning framework called Generative Adversarial Imitation Learning (GAIL) that is able to learn policies for complex high-dimensional physics-based control tasks. They reduce the imitation learning problem into an adversarial learning framework, for which they utilize Generative Adversarial Networks (GAN) (Goodfellow et al., 2014). The generator network of the GAN represents the agent’s policy $\\pi$ while the discriminator network serves as a local reward function and learns to differentiate between state-action pairs from the expert policy $\\pi _ { \\mathbb { E } }$ and from the agent’s policy $\\pi$ . Mathematically, it is equivalent to optimizing the following, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2df4c63bd8977f9931a162d75e1b466eee9b2c2ab8d78eb5af3e4ff765e24ce5.jpg",
+        "text": "$$\n\\displaystyle { \\operatorname* { m i n } _ { \\pi } } \\operatorname* { m a x } _ { D } \\mathbb { E } _ { \\pi } [ \\log D ( s , a ) ] + \\mathbb { E } _ { \\pi _ { E } } [ 1 - \\log D ( s , a ) ] - \\lambda H ( \\pi )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "InfoGAIL (Li et al., 2017) and Hausman et al. (2017) solve the problem of learning from policies generated by a mixture of experts. They introduce a latent variable $c$ into the policy function $\\pi ( a | s , c )$ to separate different type of behaviours present in the demonstration. To incentivize the network to use the latent variable, they utilize an information-theoretic regularization enforcing that there should be high mutual information between $c$ and the state-action pairs in the generated trajectory, a concept that was first introduced in InfoGAN (Chen et al., 2016). They introduce a variational lower bound $L _ { 1 } ( \\pi , Q )$ of the mutual information $I ( c ; \\tau )$ to the loss function in GAIL. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dd4bb0f6ec29cadf8107400dafb83c9aca6b5f37cc6f80928e0d9c2090f4d7cf.jpg",
+        "text": "$$\nL _ { 1 } ( \\pi , Q ) = \\mathbb { E } _ { c \\sim p ( c ) , a \\sim \\pi ( \\cdot | s , c ) } \\log Q ( c | \\tau ) + H ( c ) \\leq I ( c ; \\tau )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The modified objective can then be given as, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/132e9479778ad836f37dc7287e5c98b99614f9b30bdd329def9189617cc93cc4.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\pi , q } \\operatorname* { m a x } _ { D } \\mathbb { E } _ { \\pi } [ \\log D ( s , a ) ] + \\mathbb { E } _ { \\pi _ { E } } [ 1 - \\log D ( s , a ) ] - \\lambda _ { 1 } L _ { 1 } ( \\pi , q ) - \\lambda _ { 2 } H ( \\pi )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "InfoGAIL models variations between different trajectories as the latent codes correspond to trajectories coming from different demonstrators. In contrast, we aim to model intra-trajectory variations and latent codes in our work correspond to sub-tasks (variations) within a demonstration. In Section 3, we discuss why using a mutual information based loss is infeasible in our problem setting and describe our proposed approach. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "2.2 OPTIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Consider an MDP with states $s \\in S$ and actions $a \\in { \\mathcal { A } }$ . Under the options framework (Sutton et al., 1998), an option, indexed by $o \\in \\mathcal { O }$ consists of a sub-policy $\\pi ( a | s , o )$ , a termination policy $\\pi ( b | s , \\bar { o } )$ and an option activation policy $\\pi ( o | s )$ . After an option is initiated, actions are generated by the sub-policy until the option is terminated and a new option is selected. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Options framework has been studied widely in RL literature. A challenging problem related to the options framework is to automatically infer options without supervision. Option discovery approaches often aim to find bottleneck states, i.e., states that the agent has to pass through to reach the goal. Many different approaches such as multiple-instance learning (McGovern & Barto, 2001), graph based algorithms (Menache et al., 2002; S¸ ims¸ek et al., 2005) have been used to find such bottleneck states. Once the bottleneck states are discovered, the above approaches find options policies to reach each such state. In contrast, we propose a unified framework using a information-theoretic approach to automatically discover relevant option policies without the need to discover bottleneck states. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Daniel et al. (2016) formulate the options framework as a probabilistic graphical model where options are treated as latent variables which are then learned from expert data. The option policies $( \\pi ( a | s , o ) )$ are analogous to sub-task policies in our work. These option policies are then learned by maximizing a lower bound using the Expectation-Maximization algorithm (Moon, 1996). We show how this lower bound is closely related to the objective derived in our work. We further show how this connection allows our method to be seen as a generative adversarial variant of their approach. Fox et al. (2017) propose to extend the EM based approach to multiple levels of option hierarchies. Further work on discovery of deep continuous options (Krishnan et al., 2017) allows the option policy to also select a continuous action in states where none of the options are applicable. Our proposed approach can also be extended to multi-level hierarchies (e.g. by learning VAEs introduced in section 3 with multiple sampling layers) or hybrid categorical-continuous macro-policies (e.g. using both categorical and continuous hidden units in the sampling layer in VAE). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Shiarlis et al. (2018) learn options by assuming knowledge of task sketches (Andreas et al., 2017) along with the demonstrations. The work proposes a behavior cloning based approach using connectionist temporal classification (Graves et al., 2006) to simultaneously maximize the joint likelihood of the sketch sequences and the sub-policies. Our proposed approach does not expect task sketches as input, making it more amenable to problems where labeling demonstrations with sketch labels is difficult. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Prior work in robot learning has also looked at learning motion primitives from unsegmented demonstrations. These primitives usually correspond to a particular skill and are analogous to options. Niekum & Barto (2011) used the Beta-Process Autoregressive Hidden Markov Model (BP-AR-HMM) to segment expert demonstrations and post-process these segments to learn motion primitives which provide the ability to use reinforcement learning for policy improvement. Alternately, Krishnan et al. (2018) use Dirichlet Process Gaussian Mixture Model (DP-GMM) to segment the expert demonstrations by finding transition states between linear dynamical segments. Similarly, Ranchod et al. (2015) use the BP-AR-HMM framework to initially segment the expert demonstrations and then use an inverse reinforcement learning step to infer the reward function for each segment. The use of appropriate priors allows these methods to discover options without a priori knowledge of the total number of skills. Kroemer et al. (2014) model the task of manipulation as an autoregressive Hidden Markov Model where the hidden phases of manipulation are learned from data using EM. However, unlike the above methods, in our proposed approach we also learn an appropriate policy over the extracted options. We show how this allows us to compose the individual option policies to induce novel behaviours which were not present in the expert demonstrations. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 PROPOSED APPROACH",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As mentioned in the previous section, while prior approaches can learn to disambiguate the multiple modalities in the demonstration of a sub-task and learn to imitate them, they cannot learn to imitate demonstrations of unsegmented long tasks that are formed by a combination of many small sub-tasks. To learn such sub-task policies from unsegmented deomonstrations we use the graphical model in Figure 1 (Right), i.e., consider a set of expert demonstrations, each of which is represented by $\\tau = \\{ \\tau _ { 1 } , \\cdot \\cdot \\cdot , \\tau _ { T } \\}$ where $\\tau _ { t }$ is the state-action pair observed at time $t$ . Each such demonstration has a corresponding sequence of latent variables $\\bar { \\pmb { c } } = \\{ c _ { 1 } , \\cdots , c _ { T - 1 } \\}$ which denote the sub-activity in the demonstration at any given time step. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As noted before, previous approaches (Li et al., 2017; Hausman et al., 2017) model the expert sub-task demonstrations using only a single latent variable. To enforce the model to use this latent variable, these approaches propose to maximize the mutual information between the demonstrated sequence of state-action pairs and the latent embedding of the nature of the sub-activity. This is achieved by adding a lower bound to the mutual information between the latent variables and expert demonstrations. This variational lower bound of the mutual information is then combined with the the adversarial loss for imitation learning proposed in Ho & Ermon (2016). Extending this to our setting, where we have a sequence of latent variables $^ c$ , yields the following lower bound on the mutual information, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/431ff009928496358bf2417ad5859a105c8a1dc4db60a68decc969ebd04f8102.jpg",
+        "text": "$$\nL ( \\pi , q ) = \\sum _ { t } \\mathbb { E } _ { c ^ { 1 : t } \\sim p ( c ^ { 1 : t } ) , a ^ { t - 1 } \\sim \\pi ( \\cdot \\vert s ^ { t - 1 } , c ^ { 1 : t - 1 } ) } \\Big [ \\log q ( c ^ { t } \\vert c ^ { 1 : t - 1 } , \\tau ) \\Big ] + H ( c ) \\le I ( \\tau ; c )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Observe that the dependence of $q$ on the entire trajectory $\\tau$ precludes the use of such a distribution at test time, where only the trajectory up to the current time is known. To overcome this limitation, in this work we propose to force the policy to generate trajectories that maximize the directed or causal information flow from trajectories to the sequence of latent sub-activity variables instead. As we show below, by using directed information instead of mutual information, we can replace the dependence on $\\tau$ with a dependence on the trajectory generated up to current time $t$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The directed information flow from a sequence $\\boldsymbol { X }$ to $\\mathbf { Y }$ is given by, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/714641b9565eb9de682e92888e20a9e462dc6d721089049c3988af4714147be9.jpg",
+        "text": "$$\nI ( \\boldsymbol { X } \\to \\boldsymbol { Y } ) = H ( \\boldsymbol { Y } ) - H ( \\boldsymbol { Y } | | \\boldsymbol { X } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $H ( Y \\| X )$ is the causally-conditioned entropy. Replacing $\\boldsymbol { X }$ and $\\mathbf { Y }$ with sequences $\\tau$ and $^ c$ , ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/58e9f35e6be298156b6d16dffa4388d72cbdf2b6b76c5394248365bd4bfdba28.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle I ( \\tau \\to c ) = H ( c ) - H ( c \\| \\tau ) } } \\\\ { { \\mathrm { ~ } = H ( c ) - \\displaystyle \\sum _ { t } H ( c ^ { t } | c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) } } \\\\ { { \\mathrm { ~ } = H ( c ) + \\displaystyle \\sum _ { t } \\displaystyle \\sum _ { c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } } \\left[ p ( c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) \\right. } } \\\\ { { \\displaystyle \\left. \\sum _ { c ^ { t } } p ( c ^ { t } | c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) \\log p ( c ^ { t } | c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) \\right] } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here $\\tau ^ { 1 : t } = ( s _ { 1 } , \\cdots , a _ { t - 1 } , s _ { t } )$ . A variational lower bound, $L _ { 1 } ( \\pi , q )$ of the directed information, $I ( \\tau \\to c )$ which uses an approximate posterior $q ( c ^ { t } | c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } )$ instead of the true posterior $p ( c ^ { t } | c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } )$ can then be derived to get (See Appendix A.1 for the complete derivation), ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0e6153d4efe539c0592e34bbc9a60436715a09052ee9c57c850513c10aff9a14.jpg",
+        "text": "$$\nL _ { 1 } ( \\pi , q ) = \\sum _ { t } \\mathbb { E } _ { c ^ { 1 : t } \\sim p ( c ^ { 1 : t } ) , a ^ { t - 1 } \\sim \\pi ( \\cdot | s ^ { t - 1 } , c ^ { 1 : t - 1 } ) } \\left[ \\log q ( c ^ { t } | c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) \\right] + H ( c ) \\le I ( \\tau \\to c )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            183,
+            315,
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+            348
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Thus, by maximizing directed information instead of mutual information, we can learn a posterior distribution over the next latent factor $c$ given the latent factors discovered up to now and the trajectory followed up to now, thereby removing the dependence on the future trajectory. In practice, we do not consider the $H ( c )$ term. This gives us the following objective, ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/adf540dceea9f566442223b990d5643e52e2bcf03dfe46cca3f104f02c8acffb.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\pi , q } \\operatorname* { m a x } _ { D } \\mathbb { E } _ { \\pi } \\left[ \\log D ( s , a ) \\right] + \\mathbb { E } _ { \\pi _ { E } } \\left[ 1 - \\log D ( s , a ) \\right] - \\lambda _ { 1 } L _ { 1 } ( \\pi , q ) - \\lambda _ { 2 } H ( \\pi )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            436,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We call this approach Directed-Info GAIL. Notice that, to compute the loss in equation 3, we need to sample from the prior distribution $p ( c ^ { 1 : t } )$ . In order to estimate this distribution, we first pre-train a variational auto-encoder (VAE) (Kingma & Welling, 2013) on the expert trajectories, the details of which are described in the next sub-section. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 VAE PRE-TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            544,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 (left) shows the design of the VAE pictorially. The VAE consists of two multi-layer perceptrons that serve as the encoder and the decoder. The encoder uses the current state $s _ { t }$ and the previous latent variable $c _ { t - 1 }$ to produce the current latent variable $c _ { t }$ . We used the Gumbel-softmax trick (Jang et al., 2016) to obtain samples of latent variables from a categorical distribution. The decoder then takes $s _ { t }$ and $c _ { t }$ as input and outputs the action $a _ { t }$ . We use the following objective, which maximizes the lower bound of the probability of the trajectories $p ( \\tau )$ , to train our VAE, ",
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+    {
+        "type": "equation",
+        "img_path": "images/f6759532b1c9e5aec4416bc8a791b6e36ad9e8dac55921e24ef4c332972c53fb.jpg",
+        "text": "$$\nL _ { \\mathrm { V A E } } ( \\pi , q ; \\tau _ { i } ) = - \\sum _ { t } \\mathbb { E } _ { c ^ { t } \\sim q } \\Big [ \\log \\pi \\big ( a ^ { t } | s ^ { t } , c ^ { 1 : t } \\big ) \\Big ] + \\sum _ { t } D _ { \\mathrm { K L } } \\big ( q ( c ^ { t } | c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) \\| p ( c ^ { t } | c ^ { 1 : t - 1 } ) \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 (right) gives an overview of the complete method. The VAE pre-training step allows us to get approximate samples from the distribution $\\bar { \\boldsymbol { p } } ( c ^ { 1 : t } )$ to optimize equation 4. This is done by using $q$ to obtain samples of latent variable sequence $^ c$ by using its output on the expert demonstrations. In practice, we fix the weights of the network $q$ to those obtained from the VAE pre-training step when optimizing the Directed-Info GAIL loss in equation 4. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 CONNECTION WITH OPTIONS FRAMEWORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Daniel et al. (2016) the authors provide a probabilistic perspective of the options framework. Although, Daniel et al. (2016) consider separate termination and option latent variables ( $\\mathit { b } ^ { t }$ and $o ^ { t }$ ), for the purpose of comparison, we collapse them into a single latent variable $c ^ { t }$ , similar to our framework with a distribution $p ( c ^ { t } | s ^ { t } , c ^ { t - 1 } )$ . The lower-bound derived in Daniel et al. (2016) which is maximized using Expectation-Maximization (EM) algorithm can then be written as (suppressing dependence on parameters), ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/2e1e4ad06e734632536ea39085f2740dd03f9cdf84d3b49afd95b03b787b04e3.jpg",
+        "image_caption": [
+            "Figure 2: Left: VAE pre-training step. The VAE encoder uses the current state $\\left( { { s } _ { t } } \\right)$ , and previous latent variable $\\left( c _ { t - 1 } \\right)$ to produce the current latent variable $\\left( c _ { t } \\right)$ . The decoder reconstructs the action $\\left( \\boldsymbol { a } _ { t } \\right)$ using $s _ { t }$ and $c _ { t }$ . Right: An overview of the proposed approach. We use the VAE pre-training step to learn an approximate prior over the latent variables and use this to learn sub-task policies in the proposed Directed-Info GAIL step. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/fd0a563c5a474de6c02d09243c0acc3e863ac0de96a18bde945fd8fa905152d3.jpg",
+        "image_caption": [
+            "Figure 3: Results on the Four Rooms environment. (a) and (b) show results for two different latent variables. The arrows in each cell indicate the direction (action) with highest probability in that state and using the given latent variable. (c) and (d) show expert and generated trajectories in this environment. Star $( ^ { * } )$ represents the start state. The expert trajectory is shown in red. The color of the generated trajectory represents the latent code used by the policy at each time step. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/43ad81f9a226b3bb3b925f604524a6d1d648c117fb70f094fd9b5f3d97d6c5b8.jpg",
+        "text": "$$\np ( \\tau ) \\geq \\sum _ { t } \\sum _ { c ^ { t - 1 : t } } p ( c ^ { t - 1 : t } | \\tau ) \\log p ( c ^ { t } | s ^ { t } , c ^ { t - 1 } ) ) + \\sum _ { t } \\sum _ { c ^ { t } } p ( c ^ { t } | \\tau ) \\log \\pi ( a ^ { t } | s ^ { t } , c ^ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that the first term in equation $6 i . e .$ ., the expectation over the distribution $\\log p ( c ^ { t } | s ^ { t } , c ^ { t - 1 } )$ is the same as equation 3 of our proposed approach with a one-step Markov assumption and a conditional expectation with given expert trajectories instead of an expectation with generated trajectories. The second term in equation 6 i.e., the expectation over $\\log { \\bar { \\pi } } ( a ^ { t } | s ^ { t } , c ^ { t } )$ is replaced by the GAIL loss in equation 4. Our proposed Directed-Info GAIL can be therefore be considered as the generative adversarial variant of imitation learning using the options framework. The VAE behaviour cloning pretraining step in equation 5 is exactly equivalent to equation 6, where we use approximate variational inference using VAEs instead of EM. Thus, our approach combines the benefits of both behavior cloning and generative adversarial imitation learning. Using GAIL enables learning of robust policies that do not suffer from the problem of compounding errors. At the same time, conditioning GAIL on latent codes learned from the behavior cloning step prevents the issue of mode collapse in GANs. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We present results on both discrete and continuous state-action environments. In both of these settings we show that (1) our method is able to segment out sub-tasks from given expert trajectories, (2) learn sub-task conditioned policies, and (3) learn to combine these sub-task policies in order to achieve the task objective. ",
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+    {
+        "type": "table",
+        "img_path": "images/40dc5c188f4a7d5c960d0b2940c923474d0698cc4f59d67dcf3baed0c5b72ae3.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: A comparison of returns for continuous environments. The returns were computed using 300 episodes. Our approach gives comparable returns to using GAIL but also segments expert demonstrations into sub-tasks. The proposed Directed-Info GAIL approach improves over the policy learned from the VAE pre-training step. "
+        ],
+        "table_body": "<table><tr><td>Environment</td><td>GAIL (Ho &amp; Ermon,2016)</td><td>VAE</td><td>Directed-Info GAIL</td></tr><tr><td>Pendulum-v0</td><td>-121.42 ± 94.13</td><td>-142.89 ± 95.57</td><td>-125.39 ± 103.75</td></tr><tr><td>InvertedPendulum-v2</td><td>1000.0 ± 15.23</td><td>218.8 ± 7.95</td><td>1000.0±14.97</td></tr><tr><td>Hopper-v2</td><td>3623.4 ± 51.0</td><td>499.1 ± 86.2</td><td>3662.1 ± 21.7</td></tr><tr><td>Walker2d-v2</td><td>4858.0 ± 301.7</td><td>1549.5 ± 793.7</td><td>5083.9 ± 356.3</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "4.1 DISCRETE ENVIRONMENT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "For the discrete setting, we choose a grid world environment which consists of a $1 5 \\times 1 1$ grid with four rooms connected via corridors as shown in Figure 3. The agent spawns at a random location in the grid and its goal is to reach an apple, which spawns in one of the four rooms randomly, using the shortest possible path. Through this experiment we aim to see whether our proposed approach is able to infer sub-tasks which correspond to meaningful navigation strategies and combine them to plan paths to different goal states. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 3 shows sub-task policies learned by our approach in this task. The two plots on the left correspond to two of the four different values of the latent variable. The arrow at every state in the grid shows the agent action (direction) with the highest probability in that state for that latent variable. In the discussion that follows, we label the rooms from 1 to 4 starting from the room at the top left and moving in the clockwise direction. We observe that the sub-tasks extracted by our approach represent semantically meaningful navigation plans. Also, each latent variable is utilized for a different sub-task. For instance, the agent uses the latent code in Figure 3(a), to perform the sub-task of moving from room 1 to room 3 and from room 2 to room 4 and the code in Figure 3(b) to move in the opposite direction. Further, our approach learns to successfully combine these navigation strategies to achieve the given objectives. For example, Figure 3(c, d) show examples of how the macro-policy switches between various latent codes to achieve the desired goals of reaching the apples in rooms 1 and 2 respectively. ",
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+    {
+        "type": "text",
+        "text": "4.2 CONTINUOUS ENVIRONMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To validate our proposed approach on continuous control tasks we experiment with 5 continuous state-action environments. The first environment involves learning to draw circles on a 2D plane and is called Circle-World. In this experiment, the agent must learn to draw a circle in both clockwise and counter-clockwise direction. The agent always starts at (0,0), completes a circle in clockwise direction and then retraces its path in the counter-clockwise direction. The trajectories differ in the radii of the circles. The state $\\bar { s } \\in \\mathbb { R } ^ { 2 }$ is the $\\mathbf { \\Phi } ( \\mathbf { x } , \\mathbf { y } )$ co-ordinate and the actions $a \\in \\mathbb { R } ^ { 2 }$ is a unit vector representing the direction of motion. Notice that in Circle-World, the expert trajectories include two different actions (for clockwise and anti-clockwise direction) for every state $( x , y )$ in the trajectory, thus making the problem multi-modal in nature. This requires the agent to appropriately disambiguate between the two different phases of the trajectory. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Further, to show the scalability of our approach to higher dimensional continuous control tasks we also show experiments on Pendulum, Inverted Pendulum, Hopper and Walker environments, provided in OpenAI Gym (Brockman et al., 2016). Each task is progressively more challenging, with a larger state and action space. Our aim with these experiments is to see whether our approach can identify certain action primitives which helps the agent to complete the given task successfully. To verify the effectiveness of our proposed approach we do a comparative analysis of our results with both GAIL (Ho & Ermon, 2016) and the supervised behavior cloning approaching using a VAE. To generate expert trajectories we train an agent using Proximal Policy Optimization (Schulman et al., 2017). We used 25 expert trajectories for the Pendulum and Inverted Pendulum tasks and 50 expert trajectories for experiments with the Hopper and Walker environments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figures 4(a, b, c) show results on the Circle-World environment. As can be seen in Figure 4(a, b), when using two sub-task latent variables, our method learns to segment the demonstrations into two intuitive sub-tasks of drawing circles in clockwise and counterclockwise directions. Hence, our method is able to identify the underlying modes and thus find meaningful sub-task segmentations from unsegmented data. We also illustrate how the learned sub-task policies can be composed to perform new types of behavior that were unobserved in the expert data. In Figure 4(c) we show how the sub-task policies can be combined to draw the circles in inverted order of direction by swapping the learned macro-policy with a different desired policy. Thus, the sub-task policies can be utilized as a library of primitive actions which is a significant benefit over methods learning monolithic policies. ",
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+    {
+        "type": "image",
+        "img_path": "images/8c3c3fe0f68cf95be6d1b031a3a50de574f6fe13b7d50a2c93125cfbc4e6c2d0.jpg",
+        "image_caption": [
+            "Figure 4: Results for Directed-Info GAIL on continuous environments. (a) Our method learns to break down the Circle-World task into two different sub-activities, shown in green and blue. (b) Trajectory generated using our approach. Color denotes time step. (c) Trajectory generated in opposite direction. Color denotes time step. (d) Sub-activity latent variables as inferred by Directed-Info GAIL on Pendulum-v0. Different colors represent different context. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/04db31d0960cc22f6b6dbb3e1c42265fee143353435bf5e1c17084cfb2d5656c.jpg",
+        "image_caption": [
+            "Figure 5: (a) shows the plot of the sub-task latent variable vs time on the Hopper and Walker tasks. (b) shows discovered sub-tasks using Directed-Info GAIL on these environments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now discuss the results on the classical Pendulum environment. Figure 4(d) shows the sub-task latent variables assigned by our approach to the various states. As can be seen in the figure, the network is able to associate different latent variables to different sub-tasks. For instance, states that have a high velocity are assigned a particular latent variable (shown in blue). Similarly, states that lie close to position 0 and have low velocity (i.e. the desired target position) get assigned another latent variable (shown in green). The remaining states get classified as a separate sub-task. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 shows the results on the higher dimensional continuous control, Hopper and Walker, environments. Figure 5(a) shows a plots for sub-task latent variable assignment obtained on these environments. Our proposed method identifies basic action primitives which are then chained together to effectively perform the two locomotion tasks. Figure 5(b) shows that our approach learns to assign separate latent variable values for different action primitives such as, jumping, mid-air and landing phases of these tasks, with the latent variable changing approximately periodically as the agent performs the periodic hopping/walking motion. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Finally, in Table 1 we also show the quantitative evaluation on the above continuous control environments. We report the mean and standard deviations of the returns over 300 episodes. As can be seen, our approach improves the performance over the VAE pre-training step, overcoming the issue of compounding errors. The performance of our approach is comparable to the state-of-the-art GAIL (Ho & Ermon, 2016). Our method moreover, has the added advantage of segmenting the demonstrations into sub-tasks and also providing composable sub-task policies. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/5c0ccdf53439b8388b8cd38d7baa1460824b68f92a7018a2a2c771a9f7aa9152.jpg",
+        "image_caption": [
+            "Figure 6: Segmentations obtained using our proposed Directed-Info GAIL method on FetchPickandPlace-v1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/7f7f245c5522450d33ccaa512cf0f902bcb974ec88ec8a08692c589330d33132.jpg",
+        "table_caption": [
+            "Table 2: Mean returns over 100 episodes on FetchPickandPlace-v1 environment, calculated using the ‘dense’ reward setting. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Returns</td></tr><tr><td>VAE</td><td>-14.07 ± 5.57</td></tr><tr><td>GAIL</td><td>-13.29 ± 5.84</td></tr><tr><td>Directed-Info GAIL</td><td>-11.74 ± 5.87</td></tr><tr><td>GAIL + L2 loss Directed-Info GAIL + L2 loss</td><td>-12.05 ± 4.94 -9.47 ± 4.84</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We further analyze our proposed approach in more detail in the Appendix. In Appendix A.4 we visualize the sub-tasks in a low-dimensional sub-space. Also, in Appendix A.5 we show results when using a larger dimensional sub-task latent variable. A video of our results on Hopper and Walker environments can be seen at https://sites.google.com/view/directedinfo-gail. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 OPENAI ROBOTICS ENVIRONMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We further performed experiments on the FetchPickandPlace-v1 task in OpenAI Gym. In each episode of this task, the object and goal locations are selected randomly. The robot then must first reach and pick the object, and then move it to the goal location. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We trained agents using both our proposed Directed-Info GAIL and the baseline GAIL approaches. We used 500 expert demonstrations. While our method was able to learn to segment the expert demonstrations into the Pick and Place sub-tasks correctly, as can be seen in Figure 6 and the videos at https://sites.google.com/view/directedinfo-gail/home#h.p_ 4dsbuC5expkZ, neither our approach, nor GAIL was able to successfully complete the task. In our preliminary results, we found that the robot, in both our proposed approach and GAIL, would reach the object but fail to grasp it despite repeated attempts. To the best of our knowledge, no other work has successfully trained GAIL on this task either. Our preliminary experiments suggested that stronger supervision may be necessary to teach the agent the subtle action of grasping. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In order to provide this supervision, we additionally trained the policy to minimize the L2 distance between the policy action and the expert action on states in the expert demonstrations. At every training step, we compute the discriminator and policy (generator) gradient using the Directed-Info GAIL (or in the baseline, GAIL) loss using states and actions generated by the policy. Along with this gradient, we also sample a batch of states from the expert demonstrations and compute the policy gradient that minimizes the L2 loss between actions that the policy takes at these states and the actions taken by the expert. We weigh these two gradients to train the policy. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 2 shows the returns computed over 100 episodes. Adding the L2 measure as an additional loss led to significant improvement. Our proposed approach Directed-Info $\\mathrm { G A I L } + \\mathrm { L } 2$ loss outperforms the baselines. Moreover, we believe that this quantitative improvement does not reflect the true performance gain obtained using our method. The reward function is such that a correct grasp but incorrect movement (e.g. motion in the opposite direction or dropping of the object) is penalized more than a failed grasp. Thus, the reward function does not capture the extent to which the task was completed. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Qualitatively, we observed a much more significant difference in performance between the proposed approach and the baseline. This can be seen in the sample videos of the success and failure cases for our and the baseline method at https://sites.google.com/view/ directedinfo-gail/home#h.p_qM39qD8xQhJQ. Our proposed method succeeds much more often than the baseline method. The most common failure cases for our method include the agent picking up the object, but not reaching the goal state before the end of the episode, moving the object to an incorrect location or dropping the object while moving it to the goal. Agents trained using GAIL $+ \\mathrm { L } 2$ loss on the other hand often fail to grasp the object, either not closing the gripper or closing the gripper prematurely. We believe that our approach helps the agent alleviate this issue by providing it with the sub-task code, helping it disambiguate between the very similar states the agent observes just before and just after grasping. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Learning separate sub-task policies can help improve the performance of imitation learning when the demonstrated task is complex and has a hierarchical structure. In this work, we present an algorithm that infers these latent sub-task policies directly from given unstructured and unlabelled expert demonstrations. We model the problem of imitation learning as a directed graph with sub-task latent variables and observed trajectory variables. We use the notion of directed information in a generative adversarial imitation learning framework to learn sub-task and macro policies. We further show theoretical connections with the options literature as used in hierarchical reinforcement and imitation learning. We evaluate our method on both discrete and continuous environments. Our experiments show that our method is able to segment the expert demonstrations into different sub-tasks, learn sub-task specific policies and also learn a macro-policy that can combines these sub-task. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            347,
+            285,
+            361
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            299,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 DERIVATION FOR DIRECTED-INFO LOSS",
+        "bbox": [
+            174,
+            133,
+            500,
+            148
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The directed information flow from a sequence $\\boldsymbol { X }$ to $\\mathbf { Y }$ is given by: ",
+        "bbox": [
+            173,
+            159,
+            619,
+            176
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe3f7e2591de4dd93482e0e20a1c5b4a20172283890367479ba0bbc28455ae20.jpg",
+        "text": "$$\nI ( \\boldsymbol { X } \\to \\boldsymbol { Y } ) = H ( \\boldsymbol { Y } ) - H ( \\boldsymbol { Y } | | \\boldsymbol { X } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            195,
+            616,
+            213
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $H ( Y \\| X )$ is the causally-conditioned entropy. Replacing $\\boldsymbol { X }$ and $\\mathbf { Y }$ with the sequences $\\tau$ and $^ c$ give, ",
+        "bbox": [
+            173,
+            224,
+            828,
+            255
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b5139e4bfb7518c8d854de9b6c30be722f57c79550aaa22394474811bd8cc92.jpg",
+        "text": "$$\n\\begin{array} { r l } & { I ( \\tau  c ) = H ( c ) - H ( c ) \\mid \\tau } \\\\ & { \\qquad = H ( c ) - \\displaystyle \\sum _ { \\tau } H ( c ^ { i } \\mid c ^ { i - 1 } , \\tau ^ { 1 : t } ) } \\\\ & { \\qquad = H ( c ) + \\displaystyle \\sum _ { \\tau } \\displaystyle \\sum _ { c ^ { i + 1 } = 1 , \\tau ^ { 1 : t } ) } [ p ( c ^ { i + t - 1 } , \\tau ^ { 1 : t } ) \\displaystyle \\sum _ { c ^ { i } } p ( c ^ { i } \\mid c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) \\log p ( c ^ { i } \\mid c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) ] } \\\\ & { \\qquad = H ( c ) + \\displaystyle \\sum _ { \\tau } \\displaystyle \\sum _ { c ^ { i + 1 } = 1 , \\tau ^ { 1 : t } ) } [ p ( c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) [ D _ { K L } ( p ( \\cdot \\mid c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) | q ( \\cdot \\mid c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) )  } \\\\ & { \\qquad \\quad \\qquad + \\displaystyle \\sum _ { c ^ { i } } p ( c ^ { i } \\mid c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) \\log q ( c ^ { i } \\mid c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) ] ] } \\\\ & { \\qquad \\geq H ( c ) + \\displaystyle \\sum _ { \\tau } \\displaystyle \\sum _ { c ^ { i + 1 } = 1 , \\tau ^ { 1 : t } ) } [ p ( c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) \\displaystyle \\sum _ { c ^ { i } } p ( c ^ { i } \\mid c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) \\log q ( c ^ { i } \\mid c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            178,
+            280,
+            803,
+            487
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here $\\tau ^ { 1 : t } = ( s _ { 1 } , \\cdots , a _ { t - 1 } , s _ { t } )$ . The lower bound in equation 7 requires us to know the true posterior distribution to compute the expectation. To avoid sampling from $p ( c ^ { t } | c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } )$ , we use the following, ",
+        "bbox": [
+            173,
+            515,
+            825,
+            560
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8f7ded81bf8162752acaddab963c453ce71d1e1141fb230af1bf54e78e26ab23.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\sum _ { s = 1 } \\sum _ { \\tau = 1 } \\left[ p ( c ^ { \\lfloor k - 1 \\rfloor } , \\tau ^ { \\lfloor k \\rfloor } ) \\sum _ { \\sigma ^ { \\prime } } \\overline { { p } } ( c ^ { \\lfloor k \\rfloor } c ^ { \\lfloor k - 1 \\rfloor } , \\tau ^ { \\lfloor k \\rfloor } ) \\log q ( c ^ { \\lfloor k \\rfloor - 1 } , \\tau ^ { \\lfloor k \\rfloor } ) \\right] } \\\\ & { \\quad = \\displaystyle \\sum _ { s = 1 - 1 } \\sum _ { \\tau = 1 } \\sum _ { \\tau ^ { \\lfloor k \\rfloor } } \\sum _ { \\sigma ^ { \\prime } } \\left[ p ( c ^ { \\lfloor k \\rfloor - 1 } , \\tau ^ { \\lfloor k \\rfloor } ) p ( c ^ { \\lfloor k \\rfloor - 1 } , \\tau ^ { \\lfloor k \\rfloor } ) \\log q ( c ^ { \\lfloor k \\rfloor - 1 } , \\tau ^ { \\lfloor k \\rfloor } ) \\right] } \\\\ & { \\quad = \\displaystyle \\sum _ { s = 1 - 1 } \\sum _ { \\tau = 1 } \\sum _ { \\tau ^ { \\lfloor k \\rfloor } } \\sum _ { \\epsilon ^ { \\ell } } \\left[ p ( c ^ { \\eta } , c ^ { \\lfloor k \\rfloor - 1 } , \\tau ^ { \\lfloor k \\rfloor } ) \\log q ( c ^ { \\lfloor k \\rfloor - 1 } , \\tau ^ { \\lfloor k \\rfloor } ) \\right] } \\\\ & { \\quad = \\displaystyle \\sum _ { s = 1 - 1 } \\sum _ { \\tau \\in \\tau ^ { \\lfloor k \\rfloor } } \\sum _ { \\epsilon ^ { \\ell } } \\left[ p ( \\tau ^ { \\lfloor k \\rfloor } | c , c ^ { \\lfloor k - 1 \\rfloor } ) p ( c ^ { \\ell } , c ^ { \\lfloor k - 1 \\rfloor - 1 } ) \\log q ( c ^ { \\lfloor k \\rfloor - 1 } , \\tau ^ { \\lfloor k \\rfloor } ) \\right] } \\\\ & { \\quad = \\displaystyle \\sum _ { s = 1 } ^ { \\lfloor \\eta } p ( c ^ { \\lfloor k \\rfloor } ) \\sum _ { \\tau ^ { \\lfloor k \\rfloor } } \\left[ p ( \\tau ^ { \\lfloor k \\rfloor } | c ^ { \\ell } , c ^ { \\lfloor k - 1 \\rfloor } ) \\log q ( c ^ { \\lfloor k \\rfloor - 1 } , \\tau ^ { \\lfloor k \\rfloor } ) \\right] } \\\\ & { \\quad = \\displaystyle \\sum _ { s = 1 } ^ { \\lfloor \\eta } p ( c ^ { \\lfloor k \\rfloor } ) \\sum _ { \\tau ^ { \\lfloor k \\rfloor } } \\left[ p ( \\tau ^ { \\lfloor k \\rfloor } | c ^ { \\lfloor k - 1 \\rfloor - 1 } ) \\log q ( c ^ { \\lfloor k \\rfloor - 1 } , \\tau ^ { \\lfloor k \\rfloor } ) \\right] } \\\\ &  \\quad = \\displaystyle \\sum _ { s = 1 } ^ { \\lfloor \\eta \\rfloor } p ( c ^ { \\lfloor k \\rfloor } ) \\sum _ { \\tau ^ { \\lfloor k \\rfloor } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            583,
+            764,
+            805
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where the last step follows from the causal restriction that future provided variables $( c ^ { t } )$ do not influence earlier predicted variables ( $\\tau ^ { 1 : t }$ consists of states up to time $t$ . $c _ { t }$ does not effect state $s _ { t }$ ). Putting the result in equation 8 in equation 7 gives, ",
+        "bbox": [
+            173,
+            819,
+            826,
+            862
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aee178f521be447d3043e3871472a72507437fcb03ee65554048b9091b5af1ca.jpg",
+        "text": "$$\nL _ { 1 } ( \\pi , q ) = \\sum _ { t } \\mathbb { E } _ { c ^ { 1 : t } \\sim p ( c ^ { 1 : t } ) , a ^ { t - 1 } \\sim \\pi ( \\cdot | s ^ { t - 1 } , c ^ { 1 : t - 1 } ) } \\left[ \\log q ( c ^ { t } | c ^ { 1 : t - 1 } , \\tau ^ { 1 : t } ) \\right] + H ( c ) \\le I ( \\tau \\to c )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            887,
+            795,
+            921
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/24e2f69b21e7545044064bd605daff3d3e638829c8c2808bd839113eb1fc28c3.jpg",
+        "table_caption": [
+            "Table 3: Experiment settings for all the different environments for both DirectedInfo-GAIL and VAE-pretraining step respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"4\">Directed Info-GAIL</td><td colspan=\"2\">VAE pre-training</td></tr><tr><td>Environment</td><td>Epochs</td><td>Batch Size</td><td>posterior 入</td><td>Epochs</td><td>Batch Size</td></tr><tr><td>Discrete</td><td>1000</td><td>256</td><td>0.1</td><td>500</td><td>32</td></tr><tr><td>Circle-World</td><td>1000</td><td>512</td><td>0.01</td><td>1000</td><td>16</td></tr><tr><td>Pendulum (both)</td><td>2000</td><td>1024</td><td>0.01</td><td>1000</td><td>16</td></tr><tr><td>Hopper-v2</td><td>5000</td><td>4096</td><td>0.01</td><td>2000</td><td>32</td></tr><tr><td>Walker2d-v2</td><td>5000</td><td>8192</td><td>0.001</td><td>2000</td><td>32</td></tr></table>",
+        "bbox": [
+            173,
+            99,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/3957e6012ac4a833cae942728b11328572916d9caf0777a06c73a3337e91a170.jpg",
+        "image_caption": [
+            "Figure 7: Latent variable assignment on the expert trajectories in Circle-World (a) with and (b) without smoothing penalty $L _ { s }$ . Blue and green colors represent the two different values of the context variable. The centres of the two circles are shifted for clarity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            301,
+            787,
+            429
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Thus, by maximizing directed information instead of mutual information, we can learn a posterior distribution over the next latent factor $c$ given the latent factors discovered up to now and the trajectory followed up to now, thereby removing the dependence on the future trajectory. In practice, we do not consider the $H ( c )$ term. This gives us the objective, ",
+        "bbox": [
+            174,
+            507,
+            825,
+            564
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9df61962ee5ccd8f324f49def81ef75026d5d117e1ccbc253ccaafe7077aeb68.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\pi , q } \\operatorname* { m a x } _ { D } \\mathbb { E } _ { \\pi } [ \\log D ( s , a ) ] + \\mathbb { E } _ { \\pi _ { E } } [ 1 - \\log D ( s , a ) ] - \\lambda _ { 1 } L _ { 1 } ( \\pi , q ) - \\lambda _ { 2 } H ( \\pi ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            588,
+            751,
+            613
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In practice, we fix $q$ from the VAE pre-training and only minimize over the policy $\\pi$ in equation 4. ",
+        "bbox": [
+            173,
+            625,
+            813,
+            640
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            655,
+            410,
+            670
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 3 lists the experiment settings for all of the different environments. We use multi-layer perceptrons for our policy (generator), value, reward (discriminator) and posterior function representations. Each network consisted of 2 hidden layers with 64 units in each layer and ReLU as our non-linearity function. We used Adam (Kingma & Ba, 2014) as our optimizer setting an initial learning rate of $3 e ^ { - 4 }$ . Further, we used the Proximal Policy Optimization algorithm (Schulman et al., 2017) to train our policy network with $\\epsilon = 0 . 2$ . For the VAE pre-training step we set the VAE learning rate also to $3 e ^ { - \\hat { 4 } }$ . For the Gumbel-Softmax distribution we set an initial temperature $\\tau = 5 . 0$ . The temperature is annealed using using an exponential decay with the following schedule $\\tau = \\operatorname* { m a x } ( 0 . 1 , \\stackrel { - } { \\exp } ^ { - k t } )$ , where $k = 3 e - 3$ and $t$ is the current epoch. ",
+        "bbox": [
+            173,
+            680,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 CIRCLE-WORLD SMOOTHING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            823,
+            421,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the Circle-World experiment, we added another loss term $L _ { s }$ to VAE pre-training loss $L _ { V A E }$ , which penalizes the number of times the latent variable switches from one value to another. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bedcfe75f0f21f54a5397ccd30d4cc99546000ca4a2f8120ed7d1ac404b7062f.jpg",
+        "text": "$$\nL _ { s } = \\sum _ { t } \\left[ 1 - \\frac { c _ { t - 1 } \\cdot c _ { t } } { \\operatorname* { m a x } ( | | c _ { t - 1 } | | _ { 2 } , | | c _ { t } | | _ { 2 } ) } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/9c18955cc6e77a8396a8df44a443297bbdc4e9708adbc455ae37e457ceeec0a1.jpg",
+        "image_caption": [
+            "Figure 8: PCA Visualization for Hopper and Walker environment with sub-task latent variable of size 4. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            99,
+            802,
+            255
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/e7fe68c4adc890e0944ac707b88d823abb69039c77f287f089be933b1cac256e.jpg",
+        "image_caption": [
+            "Figure 9: Results on Hopper environment with sub-task latent variable of size 8. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            230,
+            314,
+            792,
+            516
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 shows the segmentation of expert trajectories with and without the $L _ { s }$ term. We observed that without adding the smoothing penalty, the VAE learns to segment the expert trajectories into semi-circles as shown in Figure 7(a). While a valid solution, this does not match with the intuitive segmentation of the task into two sub-tasks of drawing circles in clockwise and counter-clockwise directions. The smoothing term can be thought of as a prior, forcing the network to change the latent variable as few times as possible. This helps reach a solution where the network switches between latent variables only when required. Figure 7(b) shows an example of segmentation obtained on expert trajectories after smoothing. Thus, adding more terms to the VAE pre-training loss can be a good way to introduce priors and bias solutions towards those that match with human notion of sub-tasks. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.4 PCA VISUALIZATION OF SUB-TASKS ",
+        "text_level": 1,
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+        "text": "In Figure 8, we show the plots expert states, reduced in dimensionality using Principal Component Analysis (PCA), in Hopper and Walker environments. States are color coded by the latent code assigned at these states. We reduced the dimension of states in Hopper from 11 to 2 and in Walker from 17 to 3. These low dimensional representations are able to cover $\\sim 9 0 \\%$ of variance in the states. As can be seen in the figure, states in different parts of the space get assigned different latent variables. This further shows that our proposed approach is able to segment trajectories in such a way so that states that are similar to each other get assigned to the same segment (latent variable). ",
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+    {
+        "type": "text",
+        "text": "A.5 USING LARGER CONTEXT ",
+        "text_level": 1,
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+        "text": "For the following discussion we will represent a $k$ -dimensional categorical variable as belonging to $\\Delta ^ { k - 1 }$ simplex. To observe how the dimensionality of the sub-task latent variable affects our proposed approach we show results with larger dimensionality for the categorical latent variable $c _ { t }$ . Since DirectedInfo-GAIL infers the sub-tasks in an unsupervised manner, we expect our approach to output meaningful sub-tasks irrespective of the dimensionality of $c _ { t }$ . Figure 9 shows results for using a higher dimensional sub-task latent variable. Precisely, we assume $c _ { t }$ to be a 8-dimensional one hot vector, i.e., $c _ { t } \\in \\Delta ^ { 7 }$ . ",
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+    {
+        "type": "text",
+        "text": "As seen in the above figure, even with a larger context our approach identifies similar basic action primitives as done previously when $c _ { t } \\in \\bar { \\Delta } ^ { 3 }$ . This shows that despite larger dimensionality our approach is able to reuse appropriate context inferred previously. We also visualize the context values for the low-dimensional state-space embedding obtained by PCA. Although not perfectly identical, these context values are similar to the visualizations observed previously for $\\bar { c _ { t } } \\in \\Delta ^ { 3 }$ . Thus our proposed approach is able, to some extent, infer appropriate sub-task representations independent of the dimensionality of the context variable. ",
+        "bbox": [
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+]

parse/train/H1ltQ3R9KQ/H1ltQ3R9KQ.md

@@ -0,0 +1,300 @@
+# CAUSAL REASONING FROM META REINFORCEMENT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Discovering and exploiting the causal structure in the environment is a crucial challenge for intelligent agents. Here we explore whether modern deep reinforcement learning can be used to train agents to perform causal reasoning. We adopt a meta-learning approach, where the agent learns a policy for conducting experiments via causal interventions, in order to support a subsequent task which rewards making accurate causal inferences. We also found the agent could make sophisticated counterfactual predictions, as well as learn to draw causal inferences from purely observational data. Though powerful formalisms for causal reasoning have been developed, applying them in real-world domains can be difficult because fitting to large amounts of high dimensional data often requires making idealized assumptions. Our results suggest that causal reasoning in complex settings may benefit from powerful learning-based approaches. More generally, this work may offer new strategies for structured exploration in reinforcement learning, by providing agents with the ability to perform—and interpret—experiments.
+
+# 1 INTRODUCTION
+
+Many machine learning algorithms are rooted in discovering patterns of correlation in data. While this has been sufficient to excel in several areas (Krizhevsky et al., 2012; Cho et al., 2014), sometimes the problems we are interested in are fundamentally causal. Answering questions such as “Does smoking cause cancer?” or “Was this person denied a job due to racial discrimination?” or “Did this marketing campaign cause sales to go up?” all require an ability to reason about causes and effects and cannot be achieved by purely associative inference. Even for problems that are not obviously causal, like image classification, it has been suggested that some failure modes emerge from lack of causal understanding. Causal reasoning may be an essential component of natural intelligence and is present in human babies, rats and even birds (Leslie, 1982; Gopnik et al., 2001; 2004; Blaisdell et al., 2006; Lagnado et al., 2013). There is a rich literature on formal approaches for defining and performing causal reasoning (Pearl, 2000; Spirtes et al., 2000; Dawid, 2007; Pearl et al., 2016).
+
+Here we investigate whether procedures for learning and using causal structure can be produced by meta-learning. The approach of meta-learning is to learn the learning (or inference) procedure itself, directly from data. We adopt the specific method of Duan et al. (2016) and Wang et al. (2016), training a recurrent neural network (RNN) through model-free reinforcement learning. We train on a large family of tasks, each underpinned by a different causal structure.
+
+The use of meta-learning avoids the need to manually implement explicit causal reasoning methods in an algorithm, offers advantages of scalability by amortizing computations, and allows automatic incorporation of complex prior knowledge (Andrychowicz et al., 2016; Wang et al., 2016; Finn et al., 2017). Additionally, by learning end-to-end, the algorithm has the potential to find the internal representations of causal structure best suited for the types of causal inference required.
+
+# 2 PROBLEM SPECIFICATION AND APPROACH
+
+This work probed how an agent could learn to perform causal reasoning in three distinct settings – observational, interventional, and counterfactual – corresponding to different types of data available to the agent during the first phase of an episode.
+
+In the observational setting (Experiment 1), the agent could only obtain passive observations from the environment. This type of data allows an agent to infer associations (associative reasoning) and, when the structure of the underlying causal model permits it, to estimate the effect that changing a variable in the environment has on another variable, namely to estimate causal effects (cause-effect reasoning).
+
+In the interventional setting (Experiment 2), the agent could directly set the values of some variables in the environment. This type of data in principle allows an agent to estimate causal effects for any underlying causal model.
+
+In the counterfactual setting (Experiment 3), the agent first had an opportunity to learn about the causal graph through interventions. At the last step of the episode, it was asked a counterfactual question of the form “What would have happened if a different intervention had been made in the previous time-step?”.
+
+Next we will formalize these three settings and patterns of reasoning possible in each, using the graphical model framework (Pearl, 2000; Spirtes et al., 2000; Dawid, 2007)1, and introduce the meta-learning methods that we will use to train agents that are capable of such reasoning.
+
+# 2.1 CAUSALITY
+
+Causal relationships among random variables can be expressed using causal directed acyclic graphs (DAGs) (see Appendix). A causal DAG is a graphical model that captures both independence and causal relations. Each node $X _ { i }$ corresponds to a random variable, and the joint distribution $p ( X _ { 1 } , \ldots , X _ { N } )$ is given by the product of conditional distributions of each node $X _ { i }$ given its parent nodes $\operatorname { p a } ( X _ { i } )$ , i.e. $\begin{array} { r } { p ( X _ { 1 : N } \equiv X _ { 1 } , . . . , X _ { N } ) = \prod _ { i = 1 } ^ { N } p ( X _ { i } | \mathsf { p a } ( X _ { i } ) ) . } \end{array}$ .
+
+Edges carry causal semantics: if there exists a directed path from $X _ { i }$ to $X _ { j }$ , then $X _ { i }$ is a potential cause of $X _ { j }$ . Directed paths are also called causal paths. The causal effect of $X _ { i }$ on $X _ { j }$ is the conditional distribution of $X _ { j }$ given $X _ { i }$ restricted to only causal paths.
+
+![](images/f7edff51309caba959c03a89222e364720a09693b84ae7fa4c194a9ddbb97247.jpg)
+
+An example causal DAG $\mathcal { G }$ is given in the figure on the left, where $E$ represents hours of exercise in a week, $H$ cardiac health, and $A$ age. The causal effect of $E$ on $H$ is the conditional distribution restricted to the path $E \to H$ i.e. excluding the path $E \left. A \right. H$ . The variable $A$ is called a confounder, as it confounds the causal effect with non-causal statistical influence.
+
+Simply observing cardiac health conditioning on exercise level from $p ( H | E )$ (associative reasoning) cannot answer if change in exercise levels cause changes in cardiac health (cause-effect reasoning), since there is always the possibility that correlation between the two is because of the common confounder of age.
+
+Cause-effect Reasoning. The causal effect can be seen as the conditional distribution $p _ {  E = e } ( H | E =$ $e ) ^ { 2 }$ on the graph $\mathscr { G } _ {  E = e }$ above (right), resulting from intervening on $E$ by replacing $p ( E | A )$ with a delta distribution $\delta _ { E = e }$ (thereby removing the link from $A$ to $E$ ) and leaving the remaining conditional distributions $p ( H | E , A )$ and $p ( A )$ unaltered. The rules of do-calculus (Pearl, 2000; Pearl et al., 2016) tell us how to compute $\scriptstyle p \to E = e \left( H | E = e \right)$ using observations from $\mathcal { G }$ . In this case $\begin{array} { r } { p _ {  E = e } ( H | E = e ) = } \end{array}$ $\textstyle \sum _ { A } p ( H | E { = } e , A ) { \bar { p } } ( A ) ^ { 3 }$ . Therefore, do-calculus enables us to reason in the intervened graph $\mathscr { G } _ {  E = e }$ even if our observations are from $\mathcal { G }$ . This is the scenario captured by our observational setting outlined above.
+
+Such inferences are always possible if the confounders are observed, but in the presence of unobserved confounders, for many DAG structures the only way to compute causal effects is by collecting observations directly from $\mathcal { G } _ {  E }$ , i.e. by actively intervening on the world to fix the value of the variable $E = e$ and observing the remaining variables. In our interventional setting, outlined above, the agent has access to such interventions.
+
+Counterfactual Reasoning. Cause-effect reasoning can be used to correctly answer predictive questions of the type “Does exercising improve cardiac health?” by accounting for causal structure and confounding. However, it cannot answer retrospective questions about what would have happened. For example, given an individual $i$ who has died of a heart attack, this method would not be able to answer questions of the type “What would the cardiac health of this individual have been had they done more exercise?”. This type of question requires estimating unobserved sources of noise and then reasoning about the effects of this noise under a graph conditioned on a different intervention.
+
+# 2.2 META-LEARNING
+
+Meta-learning refers to a broad range of approaches in which aspects of the learning algorithm itself are learned from the data. Many individual components of deep learning algorithms have been successfully meta-learned, including the optimizer (Andrychowicz et al., 2016), initial parameter settings (Finn et al., 2017), a metric space (Vinyals et al., 2016), and use of external memory (Santoro et al., 2016).
+
+Following the approach of (Duan et al., 2016; Wang et al., 2016), we parameterize the entire learning algorithm as a recurrent neural network (RNN), and we train the weights of the RNN with model-free reinforcement learning (RL). The RNN is trained on a broad distribution of problems which each require learning. When trained in this way, the RNN is able to implement a learning algorithm capable of efficiently solving novel learning problems in or near the training distribution.
+
+Learning the weights of the RNN by model-free RL can be thought of as the “outer loop” of learning. The outer loop shapes the weights of the RNN into an “inner loop” learning algorithm. This inner loop algorithm plays out in the activation dynamics of the RNN and can continue learning even when the weights of the network are frozen. The inner loop algorithm can also have very different properties from the outer loop algorithm used to train it. For example, in previous work this approach was used to negotiate the exploration-exploitation tradeoff in multi-armed bandits (Duan et al., 2016) and learn algorithms which dynamically adjust their own learning rates (Wang et al., 2016; 2018). In the present work we explore the possibility of obtaining a causally-aware inner-loop learning algorithm. See the Appendix for a more formal approach to meta-learning.
+
+# 3 TASK SETUP AND AGENT ARCHITECTURE
+
+In the experiments, in each episode the agent interacted with a different causal DAG $\mathcal { G }$ . $\mathcal { G }$ was drawn randomly from the space of possible DAGs under the constraints given in the next paragraph. Each episode consisted of $T$ steps, and was divided into two phases: information and quiz. The information phase, corresponding to the first $T - 1$ steps, allowed the agent to collect information by interacting with or passively observing samples from $\mathcal { G }$ . The agent could potentially use this information to infer the connectivity and weights of $\mathcal { G }$ . The quiz phase, corresponding to the final step $T$ , required the agent to exploit the causal knowledge it collected in the information phase, to select the node with the highest value under a random external intervention.
+
+Causal graphs, observations, and actions. We generated all graphs on $N { = } 5$ nodes, with edges only in the upper triangular of the adjacency matrix (this guarantees that all the graphs obtained are DAGs), with edge weights, $w _ { j i } \in \{ - 1 , 0 , 1 \}$ (uniformly sampled), and removed 300 for held-out testing. The remaining 58749 (or $3 ^ { N ( N - 1 ) / 2 } - 3 0 0 )$ were used as the training set. Each node’s value, $X _ { i } \in \mathbb { R }$ , was Gaussiandistributed. The values of parentless nodes were drawn from $\mathcal { N } ( \mu = 0 . 0 , \sigma = 0 . 1 )$ . The conditional probability of a node with parents was $\begin{array} { r } { p ( X _ { i } | \mathsf { p a } ( X _ { i } ) ) = \mathcal { N } ( \mu = \sum _ { j } w _ { j i } X _ { j } , \sigma = 0 . 1 ) } \end{array}$ , where $\operatorname { p a } ( X _ { i } )$ represents the parents of node $X _ { i }$ in $\mathcal { G }$ . The values of the 4 observable nodes (the root node, was always hidden), were concatenated to create $v _ { t }$ and provided to the agent in its observation vector, $O _ { t } = [ v _ { t } , m _ { t } ]$ where $m _ { t }$ is a one-hot vector indicating external intervention during the quiz phase (explained below).4
+
+In both phases, on each step, $t$ , the agent’s action, $a _ { t }$ , was a discrete choice from the range $\left\{ 1 . . . 2 ( N { - } 1 ) \right\}$ . Action choices in $\{ 1 . . . N - 1 \}$ corresponded to information actions, and choices in $\{ N \ldots 2 ( N - 1 ) \}$ corresponded to quiz actions.
+
+Information phase. In the information phase, an information action, $a _ { t }$ , caused an intervention on the $a _ { t }$ -th node, setting its value to $X _ { a _ { t } } = 5$ . We choose an intervention value outside the likely range of sampled observations, to facilitate learning of the causal graph. The observation from the intervened graph, $\mathscr { G } _ {  X _ { a _ { t } } = 5 }$ , was sampled similarly to $\mathcal { G }$ , except the incoming edges to $X _ { a _ { t } }$ were severed, and its intervened value was used for conditioning its children’s values. The node values in $\mathscr { G } _ {  X _ { a _ { t } } = 5 }$ were distributed as $p _ {  X _ { i } = 5 } ( X _ { 1 : N \backslash i } | X _ { i } = 5 )$ . If a quiz action was chosen during the information phase, it was ignored, the $\mathcal { G }$ values were sampled as if no intervention had been made, and the agent was given a penalty of $r _ { t } = - 5$ in order to encourage it to take quiz actions at only during quiz phase. After the action was selected, an observation was provided to the agent. The default length of this phase was fixed to $T = N = 5$ since in the noise-free limit, a minimum of $T - 1 = 4$ interventions are required in general to resolve the causal structure, and score perfectly on the test phase.
+
+Quiz phase. In the quiz phase, one non-hidden node was selected at random to be intervened on externally, $X _ { j }$ , and its value was set to $- 5$ . We chose an intervention value of $- 5$ never previously observed by the agent in that episode, thus disallowing the agent from memorizing the results of interventions in the information phase to perform well on the quiz phase. The agent was informed of this by the observed $m _ { T - 1 }$ (a one-hot vector which indicated which node would be intervened on), from the final pre-quiz phase time-step, $T - 1$ . Note, $m _ { t }$ was set to a zero-vector for steps $t < T - 1$ . A quiz action, $a _ { T }$ , chosen by the agent indicated the node whose value would be given to the agent as a reward. In other words, the agent would receive reward, $r _ { T } = X _ { a _ { T } - ( N - 1 ) }$ . Again, if a quiz action was chosen during the information phase, the node values were not sampled and the agent was simply given a penalty of $r _ { T } = - 5$ .
+
+Active vs passive agents. Our agents had to perform two distinct tasks during the information phase: a) actively choose which nodes to set values on, and b) infer the causal DAG from its observations. We refer to this setup as the “active” condition. To control for (a), we created the “passive” condition, where the agent’s information phase actions are not learned. To provide a benchmark for how well the active agent can perform task (a), we fixed the passive agent’s intervention policy to be an exhaustive sweep through all observable nodes. This is close to optimal for this domain – in fact it is the optimal policy for noise-free conditional node values. We also compared the active agent’s performance to a baseline agent whose policy is to intervene randomly on the observable nodes in the information phase, in the Appendix.
+
+Two kinds of learning The “inner loop” of learning (see Section 2.2) occurs within each episode where the agent is learning from the evidence it gathers during the information phase in order to perform well in the quiz phase. The same agent then enters a new episode, where it has to repeat the task on a different DAG. Test performance is reported on DAGs that the agent has never previously seen, after all the weights of the RNN have been fixed. Hence, the only transfer from training to test (or the “outer loop” of learning) is the ability to discover causal dependencies based on observations in the information phase, and to perform causal inference in the quiz phase.
+
+# Agent Architecture and Training
+
+We used a long short-term memory (LSTM) network (Hochreiter & Schmidhuber, 1997) (with 96 hidden units) that, at each time-step $t$ , receives a concatenated vector containing $\left[ o _ { t } , a _ { t - 1 } , r _ { t - 1 } \right]$ as input, where $o _ { t }$ is the observation5, $a _ { t - 1 }$ is the previous action (as a one-hot vector) and $r _ { t - 1 }$ the reward (as a single real-value)6. The outputs, calculated as linear projections of the LSTM’s hidden state, are a set of policy logits (with dimensionality equal to the number of available actions), plus a scalar baseline. The policy logits are transformed by a softmax function, and then sampled to give a selected action.
+
+Learning was by asynchronous advantage actor-critic (Mnih et al., 2016). In this framework, the loss function consists of three terms – the policy gradient, the baseline cost and an entropy cost. The baseline cost was weighted by 0.05 relative to the policy gradient cost. The weighting of the entropy cost was annealed over the course of training from 0.05 to 0. Optimization was done by RMSProp with $\epsilon = 1 0 ^ { - 5 }$ , momentum $= 0 . 9$ and decay $= 0 . 9 5$ . Learning rate was annealed from $3 \times 1 0 ^ { - 6 }$ to 0. For all experiments, after training, the agent was tested with the learning rate set to zero, on a held-out test set.
+
+# 4 EXPERIMENTS
+
+Our three experiments (observational, interventional, and counterfactual) differed in the properties of the $v _ { t }$ that was observed by the agent during the information phase, and thereby limited the extent of causal reasoning possible within each data setting. Our measure of performance is the reward earned in the quiz phase for held-out DAGs. Choosing a random node node in the quiz phase results in a reward of $- 5 / 4 = - 1 . 2 5$ , since one node (the externally intervened node) always has value $- 5$ and the others have on average 0 value. By learning to simply avoid the externally intervened node, the agent can earn on average 0 reward. Consistently picking the node with the highest value in the quiz phase requires the agent to perform causal reasoning. For each agent, we take the average reward earned across 1200 episodes (300 held-out test DAGs, with 4 possible external interventions). We train 12 copies of each agent and report the average reward earned by these, with error bars showing $9 5 \%$ confidence intervals.
+
+# 4.1 EXPERIMENT 1: OBSERVATIONAL SETTING
+
+In Experiment 1, the agent could neither intervene to set the value of variables in the environment, nor observe any external interventions. In other words, it only received observations from $\mathcal { G }$ , not $\mathscr { G } _ {  X _ { j } }$ (where $X _ { j }$ is a node that has been intervened on). This limits the extent of causal inference possible. In this experiment, we tested six agents, four of which were learned: “Observational”, “Long Observational”, “Active Conditional”, “Passive Conditional”, “Observational MAP Baseline”(not learned) and the “Optimal Associative Baseline” (not learned). We also ran two other standard RL baselines—see the Appendix for details.
+
+Observational Agents: In the information phase, the actions of the agent were ignored7, and the observational agent always received the values of the observable nodes as sampled from the joint distribution associated with $\mathcal { G }$ . In addition to the default $T = 5$ episode length, we also trained this agent with $4 \times$ longer episode length (Long Observational Agent), to measure performance increase with more observational data.
+
+Conditional Agents: The information phase actions corresponded to observing a world in which the selected node $X _ { j }$ is equal to $X _ { j } = 5$ , and the remaining nodes are sampled from the conditional distribution $p ( X _ { 1 : N \backslash j } | X _ { j } = \mathsf { \bar { 5 } } )$ , where $X _ { 1 : N \backslash j }$ indicates the set of all nodes except $X _ { j }$ . This differs from intervening on the variable $X _ { j }$ by setting it to the value $X _ { j } = 5$ , since here we take a conditional sample from $\mathcal { G }$ rather than from $\mathscr { G } _ {  X _ { j } = 5 }$ (i.e. from $p _ {  X _ { j } = 5 } ( X _ { 1 : N \backslash j } | X _ { j } = 5 ) \rangle$ ), and inference about the corresponding node’s parents is possible. Therefore, this agent still has access to only observational data, as with the observational agents. However, on average it receives more diagnostic information about the relation between the random variables in $\mathcal { G }$ , since it can observe samples where a node takes a value far outside the likely range of sampled observations. We run active and passive versions of this agent as described in Section 3
+
+Optimal Associative Baseline: This baseline receives the true joint distribution $p ( X _ { 1 : N } )$ implied by the DAG in that episode, therefore it has full knowledge of the correlation structure of the environment8. It can therefore do exact associative reasoning of the form $p ( X _ { j } | X _ { i } = x )$ , but cannot do any cause-effect reasoning of the form $p _ {  X _ { i } = x } ( X _ { j } | X _ { i } = x ) \bar $ . In the quiz phase, this baseline chooses the node that has the maximum value according to the true $p ( X _ { j } | X _ { i } = x )$ in that episode, where $X _ { i }$ is the node externally intervened upon, and $x = - 5$ .
+
+Observational MAP Baseline: This baseline follows the traditional method of separating causal induction and causal inference. We first carry out exact maximum a posteriori (MAP) inference over the space of DAGs in each episode (i.e. causal induction) by selecting the DAG $( \mathcal { G } ^ { \mathrm { M A P } } )$ of the 59049 unique possibilities that maximizes the likelihood of the data observed, $v _ { 1 : T }$ , by the Observational Agent in that episode. This is equivalent to maximizing the posterior probability since the prior over graphs is uniform.
+
+# RESULTS
+
+We focus on three key questions in this experiment: (i) Can our agents learn to do associative reasoning with observational data?, (ii) Can they learn to do cause-effect reasoning from observational data?, and (iii) In addition to making causal inferences, can our agent also choose good actions in the information phase to generate the data it observes?
+
+![](images/749c52221b7aa659f857213bed214509bd030aeccb9f6e21dca673ef2ac5b823.jpg)  
+Figure 2: Experiment 1. Agents do associative and cause-effect reasoning from observational data. a) Average reward earned by the agents tested in this experiment. See main text for details. b) Performance split by the presence or absence of at least one parent (Parent and Orphan respectively) on the externally intervened node. c) Quiz phase for a test DAG. Green (red) edges indicate a weight of $+ 1$ $( - 1 )$ . Black represents the intervened node, green (red) nodes indicate a positive (negative) value at that node, white indicates a zero value. The blue circles indicate the agent’s choice. Left panel: $\mathcal { G }$ and the nodes taking the mean values prescribed by $p ( X _ { 1 : N \backslash j } | X _ { j } = - 5 )$ , including backward inference to the intervened node’s parent. The Optimal Associative Baseline’s choice is consistent with maximizing these (incorrect) node values. Right panel: $\mathscr { G } _ {  X _ { j } = - 5 }$ and the nodes taking the mean values prescribed by $p _ {  X _ { j } = - 5 } ( X _ { 1 : N \backslash j } | X _ { j } = - 5 )$ . We see that the Passive-Conditional Agent’s choice is consistent with maximizing these (correct) node values.
+
+For (i), we see that the Observational Agents achieve reward above the random baseline (see the Appendix), and that more observations (Long Observational Agent) lead to better performance (Fig. 2a), indicating that the agent is indeed learning the statistical dependencies between the nodes. We see that the performance of the Passive-Conditional Agent is better than either of the Observational Agents, since the data it observes is very informative about the statistical dependencies in the environment. Finally, we see that the PassiveConditional Agent’s performance is comparable (in fact surpasses as discussed below) the performance of the Optimal Associative Baseline, indicating that it is able to do perfect associative inference.
+
+![](images/69ae734c251061f5e5dae75f25ad12c00a96ccee89fecc8e16875f68d57b2a77.jpg)  
+Figure 1: Active and Passive Conditional Agents
+
+For (ii), we see the crucial result that the Passive-Conditional Agent’s performance is significantly above the Optimal Associative Baseline, i.e. it performs better than what is possible using only correlations. We compare their performances, split by whether or the node that was intervened on in the quiz phase of the episode has a parent (Fig. 2b). If the intervened node $X _ { j }$ has no parents, then $\mathscr { G } { = } \mathscr { G } _ {  X _ { j } }$ , and there is no advantage to being able to do cause-effect reasoning. We see indeed that the Passive-Conditional agent performs better than the Optimal Associative Baseline only when the intervened node has parents (denoted by hatched bars in Fig. 2b), indicating that this agent is able to carry out some cause-effect reasoning, despite access to only observational data – i.e. it learns some form of do-calculus. We show the quiz phase for an example test DAG in Fig. 2c, seeing that the Optimal Associative Baseline chooses according to the node values predicted by $\mathcal { G }$ whereas the Passive-Conditional Agent chooses according the node values predicted by $\mathscr { G } _ {  X _ { j } }$ .
+
+For (iii), we see (Fig. 2) that the Active-Conditional Agent’s performance is only marginally below the performance of the Passive-Conditional Agent, indicating that when the agent is allowed to choose its actions, it makes reasonable choices that allow good performance.
+
+# 4.2 EXPERIMENT 2: INTERVENTIONAL SETTING
+
+In Experiment 2, the agent receives interventional data in the information phase – it can choose to intervene on any observable node, $X _ { j }$ , and observe a sample from the resulting graph $\mathscr { G } _ {  X _ { j } }$ . As discussed in Section 2.1, access to intervention data permits cause-effect reasoning even in the presence of unobserved confounders, a feat which is in general impossible with access only to observational data. In this experiment, we test four new agents, two of which were learned: “Active Interventional”, “Passive Interventional”, “Interventional MAP Baseline”(not learned), and “Optimal Cause-Effect Baseline” (not learned).
+
+Interventional Agents: The information phase actions correspond to performing an intervention on the selected node $X _ { j }$ and sampling from $\mathscr { G } _ {  X _ { j } }$ (see Section 3 for details). We run active and passive versions of this agent as described in Section 3.
+
+Interventional MAP Baseline: This baseline infers a DAG by maximizing the likelihood of the data observed by the Passive Interventional Agent in that episode. In the quiz phase, we predict the values of each node according to ${ \mathcal { G } } _ {  X _ { j } } ^ { \mathrm { M A P } }$ where $X _ { j }$ is the node externally intervened upon (i.e. causal inference), and choose the node with the highest value.
+
+![](images/9a32279239e503f54b6b8c6d57fdabae8547106bb59b266884e801dbebc1216b.jpg)  
+Figure 4: Experiment 2. Agents do cause-effect reasoning from interventional data. a) Average reward earned by the agents tested in this experiment. See main text for details. b) Performance split by the presence or absence of unobserved confounders (abbreviated as Conf. and Unconf. respectively) on the externally intervened node. c) Quiz phase for a test DAG. See Fig. 2 for a legend. Here, the left panel shows the full $\mathcal { G }$ and the nodes taking the mean values prescribed by $p ( X _ { 1 : N \backslash j } | \bar { X } _ { j } = - 5 )$ . We see that the Passive-Cond Agent’s choice is consistent with choosing based on these (incorrect) node values. The right panel shows $\mathscr { G } _ {  X _ { j } = - 5 }$ and the nodes taking the mean values prescribed by $p _ {  X _ { j } = - 5 } ( X _ { 1 : N \backslash j } | X _ { j } = - \mathrm { \bar { 5 } } )$ We see that the Passive-Int. Agent’s choice is consistent with maximizing on these (correct) node value.
+
+Optimal Cause-Effect Baseline: This baseline receives the true DAG, $\mathcal { G }$ . In the quiz phase, it chooses the node that has the maximum value according to $\mathscr { G } _ {  X _ { j } }$ , where $X _ { j }$ is the node externally intervened upon.
+
+# RESULTS
+
+![](images/6b92a598cec55c4ca90267608194fd078c978d7e0d53f3d8083a35b1be04eb10.jpg)  
+Figure 3: Active and Passive Interventional Agents
+
+We focus on three key questions in this experiment: (i) Can our agents learn to do cause-effect reasoning from interventional data?, (ii) How does the cause-effect reasoning in our agents which have access to interventional data differ from the cause-effect reasoning measured in Experiment 1 (in agents that have access only to observational data)? (iii) In addition to making causal inferences, can our agent also choose good actions in the information phase to generate the data it observes?
+
+For (i) we see in Fig. 4a that the Passive-Interventional Agent’s performance is comparable to the Optimal Cause-Effect Baseline, indicating that it is able to do close to perfect cause-effect reasoning in this domain.
+
+For (ii) we see in Fig. 4a the crucial result that the Passive-Interventional Agent’s performance is significantly better than the Passive-Conditional Agent. We compare the performances of these two agents, split by whether the node that was intervened on in the quiz phase of the episode had unobserved confounders with other variables in the graph (Fig. 4b). In confounded cases, as described in Section 2.1, cause-effect reasoning is impossible with only observational data. We see that the performance of the Passive-Interventional Agent does not vary significantly with confoundedness, whereas the performance of the Passive-Conditional Agent is significantly lower in the confounded cases. This indicates that the improvement in the performance of the agent that has access to interventional data (as compared to the agents that had access to only observational data) is largely driven by its ability to also do cause-effect reasoning in the presence of confounders. This is highlighted by Fig. 4c, which shows the quiz phase for an example DAG, where the Passive-Conditional agent is unable to resolve the confounder, but the Passive-Interventional agent can.
+
+For (iii), we see in Fig. 3 that the Active-Interventional Agent’s performance is only marginally below the performance of the near optimal Passive-Interventional Agent, indicating that when the agent is allowed to choose its actions, it makes reasonable choices that allow good performance.
+
+# 4.3 EXPERIMENT 3: COUNTERFACTUAL SETTING
+
+In Experiment 3, the agent was again allowed to make interventions as in Experiment 2, but in this case the quiz phase task entailed answering a counterfactual question. We explain here what a counterfactual question in this domain looks like. Consider the conditional distribution $\scriptstyle { p ( \bar { X } _ { i } | \mathsf { p a } ( X _ { i } ) ) = N ( \sum _ { j } w _ { j i } X _ { j } , 0 . 1 ) }$ as described in Section 3 as $\begin{array} { r } { X _ { i } = \sum _ { j } w _ { j i } X _ { j } + \epsilon } \end{array}$ where $\epsilon$ is distributed as $\mathcal { N } ( 0 . 0 , 0 . 1 )$ , and represents the specific randomness introduced when taking one sample from the DAG. After observing the nodes $X _ { 1 : N }$ in the DAG in one sample, we can infer this specific randomness $\epsilon _ { i }$ for each node $X _ { i }$ (i.e. abduction as described in the Appendix) and answer counterfactual questions like “What would the values of the nodes be, had $X _ { j }$ in that particular sample taken on a different value than what we observed?”, for any of the nodes $X _ { j }$ . We test 2 new learned agents: “Active Counterfactual” and “Passive Counterfactual”.
+
+![](images/e971a44dda37b1eab99b010fc18fc5982ff913faa55ed88e63f6e3c8d79ae111.jpg)  
+Figure 5: Experiment 3. Agents do counterfactual reasoning. a) Average reward earned by the agents tested in this experiment. See main text for details. b) Performance split by if the maximum node value in the quiz phase is degenerate (Deg.) or distinct (Dist.). c) Quiz phase for an example test-DAG. See Fig. 2 for a legend. Here, the left panel shows $\mathscr { G } _ {  X _ { j } = - 5 }$ and the nodes taking the mean values prescribed by $p _ {  X _ { j } = - 5 } ( X _ { 1 : N \backslash j } | X _ { j } = - \bar { 5 } )$ . We see that the Passive-Int. Agent’s choice is consistent with maximizing on these node values, where it makes a random choice between two nodes with the same value. The right panel panel shows $\mathscr { G } _ {  X _ { j } = - 5 }$ and the nodes taking the exact values prescribed by the means of $p _ {  X _ { j } = - 5 } ( X _ { 1 : N \backslash j } | X _ { j } = - 5 )$ , combined with the specific randomness inferred from the previous time step. As a result of accounting for the randomness, the two previously degenerate maximum values are now distinct. We see that the Passive-CF. agent’s choice is consistent with maximizing on these node values.
+
+Counterfactual Agents: This agent is exactly analogous to the Interventional agent, with the addition that the exogenous noise in the last information phase step $t = T - 1$ (where say $X _ { p } = + 5$ ), is stored and the same noise is used in the quiz phase step $t = T$ (where say $X _ { f } = - 5$ ). While the question our agents have had to answer correctly so far in order to maximize their reward in the quiz phase was “Which of the nodes $X _ { 1 : N \backslash j }$ will have the highest value when $X _ { f }$ is set to $- 5 ? ^ { \prime }$ , in this setting, we ask “Which of the nodes $X _ { 1 : N \backslash j }$ would have had the highest value in the last step of the information phase, if instead of having $X _ { p } = + 5$ , we had $X _ { f } = - 5 ? ^ { \prime }$ . We run active and passive versions of this agent as described in Section 3.
+
+Optimal Counterfactual Baseline: This baseline receives the true DAG and does exact abduction based on the exogenous noise observed in the penultimate step of the information phase, and combines this correctly with the appropriate interventional inference on the true DAG in the quiz phase.
+
+# RESULTS
+
+We focus on two key questions in this experiment: (i) Can our agents learn to do counterfactual reasoning?, (ii) In addition to making causal inferences, can our agent also choose good actions in the information phase to generate the data it observes?
+
+For (i), we see that the Passive-Counterfactual Agent achieves higher reward than the Passive-Interventional Agent and the Optimal Cause-Effect Baseline. To evaluate whether this difference results from the agent’s use of abduction (see the Appendix for details), we split the test set into two groups, depending on whether or not the decision for which node will have the highest value in the quiz phase is affected by exogenous noise, i.e. whether or not the node with the maximum value in the quiz phase changes if the noise is resampled. This is most prevalent in cases where the maximum expected reward is degenerate, i.e. where several nodes give the same maximum reward (denoted by hatched bars in Figure 5b). Here, agents with no access to the noise have no basis for choosing one over the other, but different noise samples can give rise to significant differences in the actual values that these degenerate nodes have. We see indeed that there is no difference in the rewards received by the Passive-Counterfactual and Passive-Interventional Agents in the cases where the maximum values are distinct, however the Passive-Counterfactual Agent significantly outperforms the Passive-Interventional Agent in cases where there are degenerate maximum values.
+
+![](images/0102ff655040ab3e14e80ab25504d2a3a6eb52f8de6340142a4fc072189108a0.jpg)  
+Figure 6: Active and Passive Counterfactual Agents
+
+For (ii), we see in Fig. 6 that the Active-Counterfactual Agent’s performance is only marginally below the performance of the Passive-Counterfactual agent, indicating that when the agent is allowed to choose its actions, it makes reasonable choices that allow good performance.
+
+# 5 SUMMARY OF RESULTS
+
+We introduced and tested a framework for learning causal reasoning in various data settings—observational, interventional, and counterfactual—using deep meta-RL. Crucially, our approach did not require explicit encoding of formal principles of causal inference. Rather, by optimizing an agent to perform a task that depended on causal structure, the agent learned implicit strategies to use the available data for causal reasoning, including drawing inferences from passive observation, actively intervening, and making counterfactual predictions. Below, we summarize the keys results from each of the three experiments.
+
+In Section 4.1 and Fig. 2, we show that the agent learns to perform do-calculus. In Fig. 2(a) we see that, compared to the highest possible reward achievable without causal knowledge, the trained agent received more reward. This observation is corroborated by Fig. 2(b) which shows that performance increased selectively in cases where do-calculus made a prediction distinguishable from the predictions based on correlations. These are situations where the externally intervened node had a parent – meaning that the intervention resulted in a different graph.
+
+In Section 4.2 and Fig. 4, we show that the agent learns to resolve unobserved confounders using interventions (a feat impossible with only observational data). In Fig. 4(a) we see that the agent with access to interventional data performs better than an agent with access to only observational data. Fig. 4(b) shows that the performance increase is greater in cases where the intervened node shared an unobserved parent (a confounder) with other variables in the graph. In this section we also compare the agent’s performance to a MAP estimate of the causal structure and find that the agent’s performance matches it, indicating that the agent is indeed doing close to optimal causal inference.
+
+In Section 4.3 and Fig. 5, we show that the agent learns to use counterfactuals. In Fig. 5(a) we see that the agent with additional access to the specific randomness in the test phase performs better than an agent with access to only interventional data. In Fig. 5(b), we find that the increased performance is observed only in cases where the maximum mean value in the graph is degenerate, and optimal choice is affected by the exogenous noise – i.e. where multiple nodes have the same value on average and the specific randomness can be used to distinguish their actual values in that specific case.
+
+# 6 DISCUSSION AND FUTURE WORK
+
+This work is the first demonstration that causal reasoning can arise out of model-free reinforcement learning. This opens up the possibility of leveraging powerful learning-based methods for causal inference in complex settings. Traditional formal approaches usually decouple the two problems of causal induction (i.e. inferring the structure of the underlying model) and causal inference (i.e. estimating causal effects and answering counterfactual questions), and despite advances in both (Ortega & Stocker, 2015; Bramley et al., 2017; Parida et al., 2018; Sen et al., 2017; Forney et al., 2017; Lattimore et al., 2016), inducing models often requires assumptions that are difficult to fit to complex real-world conditions. By learning these end-to-end, our method can potentially find representations of causal structure best tuned to the specific causal inferences required. Another key advantage of our meta-RL approach is that it allows the agent to learn to interact with the environment in order to acquire necessary observations in the service of its task—i.e. to perform active learning. In our experimental domain, our agents’ active intervention policy was close to optimal, which demonstrates the promise of agents that can learn to experiment on their environment and perform rich causal reasoning on the observations.
+
+Future work should explore agents that perform experiments to support structured exploration in RL, and optimal experiment design in complex domains where large numbers of blind interventions are prohibitive. To this end, follow-up work should focus on scaling up our approach to larger environments, with more complex causal structure and a more diverse range of tasks. Though the results here are a first step in this direction which use relatively standard deep RL components, our approach will likely benefit from more advanced architectures (e.g. Espeholt et al., 2018; Hessel et al., 2018; Hester et al., 2017) that allow longer more complex episodes, as well as models which are more explicitly compositional (e.g. Battaglia et al., 2018; Andreas et al., 2016) or have richer semantics (e.g. Ganin et al., 2018), that more explicitly leverage symmetries like equivalance classes in the environment.
+
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+
+![](images/950a7be8955d94a6a74726f71f16e895a8d30fbb962aecd7c8593cedbd513a9b.jpg)  
+A ADDITIONAL BASELINES  
+Figure 7: Reward distribution
+
+We can also compare the performance of these agents to two standard model-free RL baselines. The Q-total agent learns a Q-value for each action across all steps for all the episodes. The Q-episode agent learns a Q-value for each action conditioned on the input at each time step $\left[ o _ { t } , a _ { t - 1 } , r _ { t - 1 } \right]$ , but with no LSTM memory to store previous actions and observations. Since the relationship between action and reward is random between episodes, Q-total was equivalent to selecting actions randomly, resulting in a considerably negative reward. The Q-episode agent essentially makes sure to not choose the arm that is indicated by $m _ { t }$ to be the external intervention (which is assured to be equal to $- 5 )$ , and essentially chooses randomly otherwise, giving an average reward of 0.
+
+# B FORMAL DESCRIPTION OF META-LEARNING
+
+Consider a distribution $\mathcal { D }$ over Markov Decision Processes (MDPs). We train an agent with memory (in our case an RNN-based agent) on this distribution. In each episode, we sample a task $m \sim \mathcal { D }$ . At each step $t$ within an episode, the agent sees an observation $o _ { t }$ , executes an action $a _ { t }$ , and receives a reward $r _ { t }$ . Both $a _ { t - 1 }$ and $r _ { t - 1 }$ are given as additional inputs to the network. Thus, via the recurrence of the network, each action is a function of the entire trajectory $\mathcal { H } _ { t } = \left\{ o _ { 0 } , a _ { 0 } , r _ { 0 } , . . . , o _ { t - 1 } , a _ { t - 1 } , r _ { t - 1 } , o _ { t } \right\}$ of the episode. Because this function is parameterized by the neural network, its complexity is limited only by the size of the network.
+
+# C ABDUCTION-ACTION-PREDICTION METHOD FOR COUNTERFACTUAL REASONING
+
+Pearl et al. (2016)’s “abduction-action-prediction” method prescribes one method for answering counterfactual queries, by estimating the specific unobserved makeup of individual $i$ and by transferring it to the counterfactual world. Assume, for example, the following model for $\mathcal { G }$ of Section 2.1: $E = w _ { A E } A + \eta$ , $H = w _ { A H } A + w _ { E H } E + \epsilon$ , where the weights $w _ { i j }$ represent the known causal effects in $\mathcal { G }$ and $\epsilon$ and $\eta$ are terms of (e.g.) Gaussian noise that represent the unobserved randomness in the makeup of each individual9. Suppose that for individual $i$ we observe: $A = a ^ { i }$ , $E = e ^ { i }$ , $H = h ^ { i }$ . We can answer the counterfactual question of “What if individual $i$ had done more exercise, i.e. $E { = } e ^ { \prime }$ , instead?” by: a) Abduction: estimate the individual’s specific makeup with $\epsilon ^ { i } = h ^ { i } - w _ { A H } a ^ { i } - w _ { E H } e ^ { i }$ , b) Action: set $E$ to more exercise $e ^ { \prime }$ , c) Prediction: predict a new value for cardiac health as $h ^ { \prime } { = } w _ { A H } a ^ { i } { + } w _ { E H } e ^ { \prime } { + } { \epsilon } ^ { i }$ .
+
+# D EXPERIMENT 4: NON-LINEAR CAUSAL GRAPHS
+
+![](images/4635b97ada0f7061f268d413085612f83173204d56deebcf8f2c4e789abdb1fe.jpg)  
+Figure 8: Experiment 4 results
+
+The purview of the previous experiments was to show a proof of concept on a simple tractable system, demonstrating that causal induction and inference can be learned and implemented via a meta-learned agent. In this experiment, we generalize some of the results to nonlinear, non-Gaussian causal graphs which are more typical of real-world causal graphs and to demonstrate that our results hold without loss of generality on such systems.
+
+Here we investigate causal DAGs with a quadratic dependence on the parents by changing the conditional distribution to $\begin{array} { r } { \overline { { p } } ( X _ { i } | \mathfrak { p a } ( X _ { i } ) ) = \mathcal { N } ( \frac { 1 } { N _ { i } } \overset { \cdot } { \sum _ { j } } w _ { j i } ( X _ { j } \overset { \cdot } { + } X _ { j } ^ { 2 } ) , \sigma ) } \end{array}$ . Here, although each node is normally distributed given its parents, the joint distribution is not multivariate Gaussian due to the non-linearity in how the means are determined. We find that the Long-Observational achieves more reward than the Observational agent indicating that the agent is in fact learning the statistical dependencies between the nodes, within an episode. We also find that although the Active-Interventional agent is not far behind the performance of the MAP baseline, and achieves reward well above the Long-Observational10 The fact that the MAP baseline gets so close to the Optimal Cause-Effect baseline indicates that the Active agent is choosing close to optimal actions.
+
+![](images/515d9cd2e9a7f0c794fecfd02afc3496625d8ffc4f9aefbbf5d09e6157700d51.jpg)  
+E EXPERIMENT 5: LARGER CAUSAL GRAPHS WITH GENERALIZATION TO NEW EQUIVALENCE CLASSES   
+Figure 9: (a) Comparing agent performances with different data. (b) Comparing information phase intervention policies.
+
+In the experiments reported in the main paper, the test set was a random subset of all graphs, and training examples were generated randomly subject to the constraint that they not be in the test set. However, this raised the possibility that any test graph might have an equivalent graph in the training set, which could result in a type of overfitting. We therefore ran a new set of experiments where
+
+the entire equivalence class of each test graph was held out from the training set11. Performance on the test set therefore indicates generalization of the inference procedures learned to previously unseen equivalence classes of causal DAGs. For these experiments, we used graphs with $N { = } 6$ nodes, because 5-node graphs have too few equivalence classes to partition in this way. All other details were the same as in the main paper.
+
+We see in Fig. 9a that the agents learn to generalize well to these held out examples, and we find the same pattern of behavior noted in the main text where the rewards earned are ordered such that Observational agent $<$ Passive-Conditional agent $<$ Passive-Interventional agent $<$ Passive-Counterfactual agent. We see additionally in Fig. 9b that the Active-Interventional agent performs at par with the Passive-Interventional agent (which is allowed to see the results of interventions on all nodes) and significantly better than an additional baseline we use here of the Random-Interventional agent whose information phase policy is to intervene on nodes at random, indicating that the intervention policy learned by the Active agent is good.
+
+# F GRAPHICAL MODELS AND BELIEF NETWORKS
+
+Graphical models (Pearl, 1988; Bishop, 2006; Koller & Friedman, 2009; Barber, 2012; Murphy, 2012) are a marriage between graph and probability theory that allows to graphically represent and assess statistical dependence. In the following sections, we give some basic definitions and describe a method $d \cdot$ -separation) for graphically assessing statistical independence in belief networks.
+
+# BASIC DEFINITIONS
+
+![](images/eb3d75dbb75e3bdfc96c2fcd4c13809197bbde2bcc0595748c0205078e4dbf78.jpg)  
+Figure 10: (a): Directed acyclic graph. The node $X _ { 3 }$ is a collider on the path $X _ { 1 } \right. X _ { 3 } \left. X _ { 2 }$ and a non-collider on the path $X _ { 2 }  X _ { 3 }  X _ { 4 }$ . (b): Cyclic graph obtained from (a) by adding a link from $X _ { 4 }$ to $X _ { 1 }$ .
+
+A graph is a collection of nodes and links connecting pairs of nodes. The links may be directed or undirected, giving rise to directed or undirected graphs respectively.
+
+A path from node $X _ { i }$ to node $X _ { j }$ is a sequence of linked nodes starting at $X _ { i }$ and ending at $X _ { j }$ . A directed path is a path whose links are directed and pointing from preceding towards following nodes in the sequence.
+
+A directed acyclic graph (DAG) is a directed graph with no directed paths starting and ending at the same node. For example, the directed graph in Fig. 10(a) is acyclic. The addition of a link from $X _ { 4 }$ to $X _ { 1 }$ gives rise to a cyclic graph (Fig. 10(b)).
+
+A node $X _ { i }$ with a directed link to $X _ { j }$ is called parent of $X _ { j }$ . In this case, $X _ { j }$ is called child of $X _ { i }$ .
+
+A node is a collider on a specified path if it has (at least) two parents on that path. Notice that a node can be a collider on a path and a non-collider on another path. For example, in Fig. 10(a) $X _ { 3 }$ is a collider on the path $X _ { 1 } \right. X _ { 3 } \left. X _ { 2 }$ and a non-collider on the path $X _ { 2 }  X _ { 3 }  X _ { 4 }$ .
+
+A node $X _ { i }$ is an ancestor of a node $X _ { j }$ if there exists a directed path from $X _ { i }$ to $X _ { j }$ . In this case, $X _ { j }$ is a descendant of $X _ { i }$ .
+
+A graphical model is a graph in which nodes represent random variables and links express statistical relationships between the variables.
+
+A belief network is a directed acyclic graphical model in which each node $X _ { i }$ is associated with the conditional distribution $p ( X _ { i } | \mathfrak { p a } ( X _ { i } ) )$ , where $\mathsf { p a } ( X _ { i } )$ indicates the parents of $X _ { i }$ . The joint distribution of all nodes in the graph, $p ( X _ { 1 : N } )$ , is given by the product of all conditional distributions, i.e.
+
+$$
+p ( X _ { 1 : N } ) { = } \prod _ { i = 1 } ^ { N } p ( X _ { i } | \mathsf { p a } ( X _ { i } ) ) .
+$$
+
+# ASSESSING STATISTICAL INDEPENDENCE IN BELIEF NETWORKS
+
+Given the sets of random variables $x , y$ and $\mathcal { Z }$ , $\mathcal { X }$ and $\mathcal { V }$ are statistically independent given $\mathcal { Z } \left( \mathcal { X } \perp \perp \mathcal { Y } | \mathcal { Z } \right)$ if all paths from any element of $\mathcal { X }$ to any element of $\mathcal { V }$ are closed (or blocked). A path is closed if at least one of the following conditions is satisfied:
+
+(Ia) There is a non-collider on the path which belongs to the conditioning set $\mathcal { Z }$ . (Ib) There is a collider on the path such that neither the collider nor any of its descendants belong to the conditioning set $\mathcal { Z }$ .

parse/train/H1ltQ3R9KQ/H1ltQ3R9KQ_content_list.json

@@ -0,0 +1,1634 @@
+[
+    {
+        "type": "text",
+        "text": "CAUSAL REASONING FROM META REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            823,
+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            171,
+            387,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            455,
+            236,
+            542,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Discovering and exploiting the causal structure in the environment is a crucial challenge for intelligent agents. Here we explore whether modern deep reinforcement learning can be used to train agents to perform causal reasoning. We adopt a meta-learning approach, where the agent learns a policy for conducting experiments via causal interventions, in order to support a subsequent task which rewards making accurate causal inferences. We also found the agent could make sophisticated counterfactual predictions, as well as learn to draw causal inferences from purely observational data. Though powerful formalisms for causal reasoning have been developed, applying them in real-world domains can be difficult because fitting to large amounts of high dimensional data often requires making idealized assumptions. Our results suggest that causal reasoning in complex settings may benefit from powerful learning-based approaches. More generally, this work may offer new strategies for structured exploration in reinforcement learning, by providing agents with the ability to perform—and interpret—experiments. ",
+        "bbox": [
+            233,
+            270,
+            766,
+            450
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            483,
+            330,
+            500
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many machine learning algorithms are rooted in discovering patterns of correlation in data. While this has been sufficient to excel in several areas (Krizhevsky et al., 2012; Cho et al., 2014), sometimes the problems we are interested in are fundamentally causal. Answering questions such as “Does smoking cause cancer?” or “Was this person denied a job due to racial discrimination?” or “Did this marketing campaign cause sales to go up?” all require an ability to reason about causes and effects and cannot be achieved by purely associative inference. Even for problems that are not obviously causal, like image classification, it has been suggested that some failure modes emerge from lack of causal understanding. Causal reasoning may be an essential component of natural intelligence and is present in human babies, rats and even birds (Leslie, 1982; Gopnik et al., 2001; 2004; Blaisdell et al., 2006; Lagnado et al., 2013). There is a rich literature on formal approaches for defining and performing causal reasoning (Pearl, 2000; Spirtes et al., 2000; Dawid, 2007; Pearl et al., 2016). ",
+        "bbox": [
+            174,
+            516,
+            825,
+            670
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Here we investigate whether procedures for learning and using causal structure can be produced by meta-learning. The approach of meta-learning is to learn the learning (or inference) procedure itself, directly from data. We adopt the specific method of Duan et al. (2016) and Wang et al. (2016), training a recurrent neural network (RNN) through model-free reinforcement learning. We train on a large family of tasks, each underpinned by a different causal structure. ",
+        "bbox": [
+            174,
+            676,
+            823,
+            746
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The use of meta-learning avoids the need to manually implement explicit causal reasoning methods in an algorithm, offers advantages of scalability by amortizing computations, and allows automatic incorporation of complex prior knowledge (Andrychowicz et al., 2016; Wang et al., 2016; Finn et al., 2017). Additionally, by learning end-to-end, the algorithm has the potential to find the internal representations of causal structure best suited for the types of causal inference required. ",
+        "bbox": [
+            174,
+            753,
+            825,
+            824
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2 PROBLEM SPECIFICATION AND APPROACH",
+        "text_level": 1,
+        "bbox": [
+            174,
+            848,
+            539,
+            863
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This work probed how an agent could learn to perform causal reasoning in three distinct settings – observational, interventional, and counterfactual – corresponding to different types of data available to the agent during the first phase of an episode. ",
+        "bbox": [
+            176,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the observational setting (Experiment 1), the agent could only obtain passive observations from the environment. This type of data allows an agent to infer associations (associative reasoning) and, when the structure of the underlying causal model permits it, to estimate the effect that changing a variable in the environment has on another variable, namely to estimate causal effects (cause-effect reasoning). ",
+        "bbox": [
+            174,
+            103,
+            823,
+            160
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the interventional setting (Experiment 2), the agent could directly set the values of some variables in the environment. This type of data in principle allows an agent to estimate causal effects for any underlying causal model. ",
+        "bbox": [
+            176,
+            166,
+            823,
+            208
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the counterfactual setting (Experiment 3), the agent first had an opportunity to learn about the causal graph through interventions. At the last step of the episode, it was asked a counterfactual question of the form “What would have happened if a different intervention had been made in the previous time-step?”. ",
+        "bbox": [
+            174,
+            215,
+            823,
+            257
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Next we will formalize these three settings and patterns of reasoning possible in each, using the graphical model framework (Pearl, 2000; Spirtes et al., 2000; Dawid, 2007)1, and introduce the meta-learning methods that we will use to train agents that are capable of such reasoning. ",
+        "bbox": [
+            176,
+            265,
+            823,
+            306
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 CAUSALITY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            330,
+            292,
+            344
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Causal relationships among random variables can be expressed using causal directed acyclic graphs (DAGs) (see Appendix). A causal DAG is a graphical model that captures both independence and causal relations. Each node $X _ { i }$ corresponds to a random variable, and the joint distribution $p ( X _ { 1 } , \\ldots , X _ { N } )$ is given by the product of conditional distributions of each node $X _ { i }$ given its parent nodes $\\operatorname { p a } ( X _ { i } )$ , i.e. $\\begin{array} { r } { p ( X _ { 1 : N } \\equiv X _ { 1 } , . . . , X _ { N } ) = \\prod _ { i = 1 } ^ { N } p ( X _ { i } | \\mathsf { p a } ( X _ { i } ) ) . } \\end{array}$ . ",
+        "bbox": [
+            173,
+            358,
+            825,
+            433
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Edges carry causal semantics: if there exists a directed path from $X _ { i }$ to $X _ { j }$ , then $X _ { i }$ is a potential cause of $X _ { j }$ . Directed paths are also called causal paths. The causal effect of $X _ { i }$ on $X _ { j }$ is the conditional distribution of $X _ { j }$ given $X _ { i }$ restricted to only causal paths. ",
+        "bbox": [
+            174,
+            438,
+            825,
+            481
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/f7edff51309caba959c03a89222e364720a09693b84ae7fa4c194a9ddbb97247.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            483,
+            442,
+            578
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "An example causal DAG $\\mathcal { G }$ is given in the figure on the left, where $E$ represents hours of exercise in a week, $H$ cardiac health, and $A$ age. The causal effect of $E$ on $H$ is the conditional distribution restricted to the path $E \\to H$ i.e. excluding the path $E \\left. A \\right. H$ . The variable $A$ is called a confounder, as it confounds the causal effect with non-causal statistical influence. ",
+        "bbox": [
+            467,
+            487,
+            826,
+            584
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Simply observing cardiac health conditioning on exercise level from $p ( H | E )$ (associative reasoning) cannot answer if change in exercise levels cause changes in cardiac health (cause-effect reasoning), since there is always the possibility that correlation between the two is because of the common confounder of age. ",
+        "bbox": [
+            174,
+            590,
+            825,
+            632
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Cause-effect Reasoning. The causal effect can be seen as the conditional distribution $p _ {  E = e } ( H | E =$ $e ) ^ { 2 }$ on the graph $\\mathscr { G } _ {  E = e }$ above (right), resulting from intervening on $E$ by replacing $p ( E | A )$ with a delta distribution $\\delta _ { E = e }$ (thereby removing the link from $A$ to $E$ ) and leaving the remaining conditional distributions $p ( H | E , A )$ and $p ( A )$ unaltered. The rules of do-calculus (Pearl, 2000; Pearl et al., 2016) tell us how to compute $\\scriptstyle p \\to E = e \\left( H | E = e \\right)$ using observations from $\\mathcal { G }$ . In this case $\\begin{array} { r } { p _ {  E = e } ( H | E = e ) = } \\end{array}$ $\\textstyle \\sum _ { A } p ( H | E { = } e , A ) { \\bar { p } } ( A ) ^ { 3 }$ . Therefore, do-calculus enables us to reason in the intervened graph $\\mathscr { G } _ {  E = e }$ even if our observations are from $\\mathcal { G }$ . This is the scenario captured by our observational setting outlined above. ",
+        "bbox": [
+            173,
+            638,
+            825,
+            737
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Such inferences are always possible if the confounders are observed, but in the presence of unobserved confounders, for many DAG structures the only way to compute causal effects is by collecting observations directly from $\\mathcal { G } _ {  E }$ , i.e. by actively intervening on the world to fix the value of the variable $E = e$ and observing the remaining variables. In our interventional setting, outlined above, the agent has access to such interventions. ",
+        "bbox": [
+            174,
+            743,
+            825,
+            813
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Counterfactual Reasoning. Cause-effect reasoning can be used to correctly answer predictive questions of the type “Does exercising improve cardiac health?” by accounting for causal structure and confounding. However, it cannot answer retrospective questions about what would have happened. For example, given an individual $i$ who has died of a heart attack, this method would not be able to answer questions of the type “What would the cardiac health of this individual have been had they done more exercise?”. This type of question requires estimating unobserved sources of noise and then reasoning about the effects of this noise under a graph conditioned on a different intervention. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            200
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 META-LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            219,
+            330,
+            233
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Meta-learning refers to a broad range of approaches in which aspects of the learning algorithm itself are learned from the data. Many individual components of deep learning algorithms have been successfully meta-learned, including the optimizer (Andrychowicz et al., 2016), initial parameter settings (Finn et al., 2017), a metric space (Vinyals et al., 2016), and use of external memory (Santoro et al., 2016). ",
+        "bbox": [
+            174,
+            246,
+            825,
+            301
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Following the approach of (Duan et al., 2016; Wang et al., 2016), we parameterize the entire learning algorithm as a recurrent neural network (RNN), and we train the weights of the RNN with model-free reinforcement learning (RL). The RNN is trained on a broad distribution of problems which each require learning. When trained in this way, the RNN is able to implement a learning algorithm capable of efficiently solving novel learning problems in or near the training distribution. ",
+        "bbox": [
+            174,
+            308,
+            825,
+            378
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Learning the weights of the RNN by model-free RL can be thought of as the “outer loop” of learning. The outer loop shapes the weights of the RNN into an “inner loop” learning algorithm. This inner loop algorithm plays out in the activation dynamics of the RNN and can continue learning even when the weights of the network are frozen. The inner loop algorithm can also have very different properties from the outer loop algorithm used to train it. For example, in previous work this approach was used to negotiate the exploration-exploitation tradeoff in multi-armed bandits (Duan et al., 2016) and learn algorithms which dynamically adjust their own learning rates (Wang et al., 2016; 2018). In the present work we explore the possibility of obtaining a causally-aware inner-loop learning algorithm. See the Appendix for a more formal approach to meta-learning. ",
+        "bbox": [
+            174,
+            385,
+            825,
+            511
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 TASK SETUP AND AGENT ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            534,
+            532,
+            547
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the experiments, in each episode the agent interacted with a different causal DAG $\\mathcal { G }$ . $\\mathcal { G }$ was drawn randomly from the space of possible DAGs under the constraints given in the next paragraph. Each episode consisted of $T$ steps, and was divided into two phases: information and quiz. The information phase, corresponding to the first $T - 1$ steps, allowed the agent to collect information by interacting with or passively observing samples from $\\mathcal { G }$ . The agent could potentially use this information to infer the connectivity and weights of $\\mathcal { G }$ . The quiz phase, corresponding to the final step $T$ , required the agent to exploit the causal knowledge it collected in the information phase, to select the node with the highest value under a random external intervention. ",
+        "bbox": [
+            174,
+            564,
+            825,
+            675
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Causal graphs, observations, and actions. We generated all graphs on $N { = } 5$ nodes, with edges only in the upper triangular of the adjacency matrix (this guarantees that all the graphs obtained are DAGs), with edge weights, $w _ { j i } \\in \\{ - 1 , 0 , 1 \\}$ (uniformly sampled), and removed 300 for held-out testing. The remaining 58749 (or $3 ^ { N ( N - 1 ) / 2 } - 3 0 0 )$ were used as the training set. Each node’s value, $X _ { i } \\in \\mathbb { R }$ , was Gaussiandistributed. The values of parentless nodes were drawn from $\\mathcal { N } ( \\mu = 0 . 0 , \\sigma = 0 . 1 )$ . The conditional probability of a node with parents was $\\begin{array} { r } { p ( X _ { i } | \\mathsf { p a } ( X _ { i } ) ) = \\mathcal { N } ( \\mu = \\sum _ { j } w _ { j i } X _ { j } , \\sigma = 0 . 1 ) } \\end{array}$ , where $\\operatorname { p a } ( X _ { i } )$ represents the parents of node $X _ { i }$ in $\\mathcal { G }$ . The values of the 4 observable nodes (the root node, was always hidden), were concatenated to create $v _ { t }$ and provided to the agent in its observation vector, $O _ { t } = [ v _ { t } , m _ { t } ]$ where $m _ { t }$ is a one-hot vector indicating external intervention during the quiz phase (explained below).4 ",
+        "bbox": [
+            174,
+            683,
+            825,
+            811
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In both phases, on each step, $t$ , the agent’s action, $a _ { t }$ , was a discrete choice from the range $\\left\\{ 1 . . . 2 ( N { - } 1 ) \\right\\}$ . Action choices in $\\{ 1 . . . N - 1 \\}$ corresponded to information actions, and choices in $\\{ N \\ldots 2 ( N - 1 ) \\}$ corresponded to quiz actions. ",
+        "bbox": [
+            174,
+            818,
+            826,
+            859
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Information phase. In the information phase, an information action, $a _ { t }$ , caused an intervention on the $a _ { t }$ -th node, setting its value to $X _ { a _ { t } } = 5$ . We choose an intervention value outside the likely range of sampled observations, to facilitate learning of the causal graph. The observation from the intervened graph, $\\mathscr { G } _ {  X _ { a _ { t } } = 5 }$ , was sampled similarly to $\\mathcal { G }$ , except the incoming edges to $X _ { a _ { t } }$ were severed, and its intervened value was used for conditioning its children’s values. The node values in $\\mathscr { G } _ {  X _ { a _ { t } } = 5 }$ were distributed as $p _ {  X _ { i } = 5 } ( X _ { 1 : N \\backslash i } | X _ { i } = 5 )$ . If a quiz action was chosen during the information phase, it was ignored, the $\\mathcal { G }$ values were sampled as if no intervention had been made, and the agent was given a penalty of $r _ { t } = - 5$ in order to encourage it to take quiz actions at only during quiz phase. After the action was selected, an observation was provided to the agent. The default length of this phase was fixed to $T = N = 5$ since in the noise-free limit, a minimum of $T - 1 = 4$ interventions are required in general to resolve the causal structure, and score perfectly on the test phase. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            258
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Quiz phase. In the quiz phase, one non-hidden node was selected at random to be intervened on externally, $X _ { j }$ , and its value was set to $- 5$ . We chose an intervention value of $- 5$ never previously observed by the agent in that episode, thus disallowing the agent from memorizing the results of interventions in the information phase to perform well on the quiz phase. The agent was informed of this by the observed $m _ { T - 1 }$ (a one-hot vector which indicated which node would be intervened on), from the final pre-quiz phase time-step, $T - 1$ . Note, $m _ { t }$ was set to a zero-vector for steps $t < T - 1$ . A quiz action, $a _ { T }$ , chosen by the agent indicated the node whose value would be given to the agent as a reward. In other words, the agent would receive reward, $r _ { T } = X _ { a _ { T } - ( N - 1 ) }$ . Again, if a quiz action was chosen during the information phase, the node values were not sampled and the agent was simply given a penalty of $r _ { T } = - 5$ . ",
+        "bbox": [
+            173,
+            265,
+            825,
+            391
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Active vs passive agents. Our agents had to perform two distinct tasks during the information phase: a) actively choose which nodes to set values on, and b) infer the causal DAG from its observations. We refer to this setup as the “active” condition. To control for (a), we created the “passive” condition, where the agent’s information phase actions are not learned. To provide a benchmark for how well the active agent can perform task (a), we fixed the passive agent’s intervention policy to be an exhaustive sweep through all observable nodes. This is close to optimal for this domain – in fact it is the optimal policy for noise-free conditional node values. We also compared the active agent’s performance to a baseline agent whose policy is to intervene randomly on the observable nodes in the information phase, in the Appendix. ",
+        "bbox": [
+            173,
+            397,
+            825,
+            508
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Two kinds of learning The “inner loop” of learning (see Section 2.2) occurs within each episode where the agent is learning from the evidence it gathers during the information phase in order to perform well in the quiz phase. The same agent then enters a new episode, where it has to repeat the task on a different DAG. Test performance is reported on DAGs that the agent has never previously seen, after all the weights of the RNN have been fixed. Hence, the only transfer from training to test (or the “outer loop” of learning) is the ability to discover causal dependencies based on observations in the information phase, and to perform causal inference in the quiz phase. ",
+        "bbox": [
+            173,
+            515,
+            825,
+            613
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Agent Architecture and Training ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            619,
+            393,
+            635
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We used a long short-term memory (LSTM) network (Hochreiter & Schmidhuber, 1997) (with 96 hidden units) that, at each time-step $t$ , receives a concatenated vector containing $\\left[ o _ { t } , a _ { t - 1 } , r _ { t - 1 } \\right]$ as input, where $o _ { t }$ is the observation5, $a _ { t - 1 }$ is the previous action (as a one-hot vector) and $r _ { t - 1 }$ the reward (as a single real-value)6. The outputs, calculated as linear projections of the LSTM’s hidden state, are a set of policy logits (with dimensionality equal to the number of available actions), plus a scalar baseline. The policy logits are transformed by a softmax function, and then sampled to give a selected action. ",
+        "bbox": [
+            173,
+            641,
+            825,
+            724
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Learning was by asynchronous advantage actor-critic (Mnih et al., 2016). In this framework, the loss function consists of three terms – the policy gradient, the baseline cost and an entropy cost. The baseline cost was weighted by 0.05 relative to the policy gradient cost. The weighting of the entropy cost was annealed over the course of training from 0.05 to 0. Optimization was done by RMSProp with $\\epsilon = 1 0 ^ { - 5 }$ , momentum $= 0 . 9$ and decay $= 0 . 9 5$ . Learning rate was annealed from $3 \\times 1 0 ^ { - 6 }$ to 0. For all experiments, after training, the agent was tested with the learning rate set to zero, on a held-out test set. ",
+        "bbox": [
+            174,
+            731,
+            825,
+            815
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            320,
+            117
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our three experiments (observational, interventional, and counterfactual) differed in the properties of the $v _ { t }$ that was observed by the agent during the information phase, and thereby limited the extent of causal reasoning possible within each data setting. Our measure of performance is the reward earned in the quiz phase for held-out DAGs. Choosing a random node node in the quiz phase results in a reward of $- 5 / 4 = - 1 . 2 5$ , since one node (the externally intervened node) always has value $- 5$ and the others have on average 0 value. By learning to simply avoid the externally intervened node, the agent can earn on average 0 reward. Consistently picking the node with the highest value in the quiz phase requires the agent to perform causal reasoning. For each agent, we take the average reward earned across 1200 episodes (300 held-out test DAGs, with 4 possible external interventions). We train 12 copies of each agent and report the average reward earned by these, with error bars showing $9 5 \\%$ confidence intervals. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENT 1: OBSERVATIONAL SETTING ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "In Experiment 1, the agent could neither intervene to set the value of variables in the environment, nor observe any external interventions. In other words, it only received observations from $\\mathcal { G }$ , not $\\mathscr { G } _ {  X _ { j } }$ (where $X _ { j }$ is a node that has been intervened on). This limits the extent of causal inference possible. In this experiment, we tested six agents, four of which were learned: “Observational”, “Long Observational”, “Active Conditional”, “Passive Conditional”, “Observational MAP Baseline”(not learned) and the “Optimal Associative Baseline” (not learned). We also ran two other standard RL baselines—see the Appendix for details. ",
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+        "text": "Observational Agents: In the information phase, the actions of the agent were ignored7, and the observational agent always received the values of the observable nodes as sampled from the joint distribution associated with $\\mathcal { G }$ . In addition to the default $T = 5$ episode length, we also trained this agent with $4 \\times$ longer episode length (Long Observational Agent), to measure performance increase with more observational data. ",
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+    {
+        "type": "text",
+        "text": "Conditional Agents: The information phase actions corresponded to observing a world in which the selected node $X _ { j }$ is equal to $X _ { j } = 5$ , and the remaining nodes are sampled from the conditional distribution $p ( X _ { 1 : N \\backslash j } | X _ { j } = \\mathsf { \\bar { 5 } } )$ , where $X _ { 1 : N \\backslash j }$ indicates the set of all nodes except $X _ { j }$ . This differs from intervening on the variable $X _ { j }$ by setting it to the value $X _ { j } = 5$ , since here we take a conditional sample from $\\mathcal { G }$ rather than from $\\mathscr { G } _ {  X _ { j } = 5 }$ (i.e. from $p _ {  X _ { j } = 5 } ( X _ { 1 : N \\backslash j } | X _ { j } = 5 ) \\rangle$ ), and inference about the corresponding node’s parents is possible. Therefore, this agent still has access to only observational data, as with the observational agents. However, on average it receives more diagnostic information about the relation between the random variables in $\\mathcal { G }$ , since it can observe samples where a node takes a value far outside the likely range of sampled observations. We run active and passive versions of this agent as described in Section 3 ",
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+    {
+        "type": "text",
+        "text": "Optimal Associative Baseline: This baseline receives the true joint distribution $p ( X _ { 1 : N } )$ implied by the DAG in that episode, therefore it has full knowledge of the correlation structure of the environment8. It can therefore do exact associative reasoning of the form $p ( X _ { j } | X _ { i } = x )$ , but cannot do any cause-effect reasoning of the form $p _ {  X _ { i } = x } ( X _ { j } | X _ { i } = x ) \\bar $ . In the quiz phase, this baseline chooses the node that has the maximum value according to the true $p ( X _ { j } | X _ { i } = x )$ in that episode, where $X _ { i }$ is the node externally intervened upon, and $x = - 5$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Observational MAP Baseline: This baseline follows the traditional method of separating causal induction and causal inference. We first carry out exact maximum a posteriori (MAP) inference over the space of DAGs in each episode (i.e. causal induction) by selecting the DAG $( \\mathcal { G } ^ { \\mathrm { M A P } } )$ of the 59049 unique possibilities that maximizes the likelihood of the data observed, $v _ { 1 : T }$ , by the Observational Agent in that episode. This is equivalent to maximizing the posterior probability since the prior over graphs is uniform. ",
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+    {
+        "type": "text",
+        "text": "RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We focus on three key questions in this experiment: (i) Can our agents learn to do associative reasoning with observational data?, (ii) Can they learn to do cause-effect reasoning from observational data?, and (iii) In addition to making causal inferences, can our agent also choose good actions in the information phase to generate the data it observes? ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/749c52221b7aa659f857213bed214509bd030aeccb9f6e21dca673ef2ac5b823.jpg",
+        "image_caption": [
+            "Figure 2: Experiment 1. Agents do associative and cause-effect reasoning from observational data. a) Average reward earned by the agents tested in this experiment. See main text for details. b) Performance split by the presence or absence of at least one parent (Parent and Orphan respectively) on the externally intervened node. c) Quiz phase for a test DAG. Green (red) edges indicate a weight of $+ 1$ $( - 1 )$ . Black represents the intervened node, green (red) nodes indicate a positive (negative) value at that node, white indicates a zero value. The blue circles indicate the agent’s choice. Left panel: $\\mathcal { G }$ and the nodes taking the mean values prescribed by $p ( X _ { 1 : N \\backslash j } | X _ { j } = - 5 )$ , including backward inference to the intervened node’s parent. The Optimal Associative Baseline’s choice is consistent with maximizing these (incorrect) node values. Right panel: $\\mathscr { G } _ {  X _ { j } = - 5 }$ and the nodes taking the mean values prescribed by $p _ {  X _ { j } = - 5 } ( X _ { 1 : N \\backslash j } | X _ { j } = - 5 )$ . We see that the Passive-Conditional Agent’s choice is consistent with maximizing these (correct) node values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For (i), we see that the Observational Agents achieve reward above the random baseline (see the Appendix), and that more observations (Long Observational Agent) lead to better performance (Fig. 2a), indicating that the agent is indeed learning the statistical dependencies between the nodes. We see that the performance of the Passive-Conditional Agent is better than either of the Observational Agents, since the data it observes is very informative about the statistical dependencies in the environment. Finally, we see that the PassiveConditional Agent’s performance is comparable (in fact surpasses as discussed below) the performance of the Optimal Associative Baseline, indicating that it is able to do perfect associative inference. ",
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+        "page_idx": 5
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+    {
+        "type": "image",
+        "img_path": "images/69ae734c251061f5e5dae75f25ad12c00a96ccee89fecc8e16875f68d57b2a77.jpg",
+        "image_caption": [
+            "Figure 1: Active and Passive Conditional Agents "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For (ii), we see the crucial result that the Passive-Conditional Agent’s performance is significantly above the Optimal Associative Baseline, i.e. it performs better than what is possible using only correlations. We compare their performances, split by whether or the node that was intervened on in the quiz phase of the episode has a parent (Fig. 2b). If the intervened node $X _ { j }$ has no parents, then $\\mathscr { G } { = } \\mathscr { G } _ {  X _ { j } }$ , and there is no advantage to being able to do cause-effect reasoning. We see indeed that the Passive-Conditional agent performs better than the Optimal Associative Baseline only when the intervened node has parents (denoted by hatched bars in Fig. 2b), indicating that this agent is able to carry out some cause-effect reasoning, despite access to only observational data – i.e. it learns some form of do-calculus. We show the quiz phase for an example test DAG in Fig. 2c, seeing that the Optimal Associative Baseline chooses according to the node values predicted by $\\mathcal { G }$ whereas the Passive-Conditional Agent chooses according the node values predicted by $\\mathscr { G } _ {  X _ { j } }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "For (iii), we see (Fig. 2) that the Active-Conditional Agent’s performance is only marginally below the performance of the Passive-Conditional Agent, indicating that when the agent is allowed to choose its actions, it makes reasonable choices that allow good performance. ",
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+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENT 2: INTERVENTIONAL SETTING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Experiment 2, the agent receives interventional data in the information phase – it can choose to intervene on any observable node, $X _ { j }$ , and observe a sample from the resulting graph $\\mathscr { G } _ {  X _ { j } }$ . As discussed in Section 2.1, access to intervention data permits cause-effect reasoning even in the presence of unobserved confounders, a feat which is in general impossible with access only to observational data. In this experiment, we test four new agents, two of which were learned: “Active Interventional”, “Passive Interventional”, “Interventional MAP Baseline”(not learned), and “Optimal Cause-Effect Baseline” (not learned). ",
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+    {
+        "type": "text",
+        "text": "Interventional Agents: The information phase actions correspond to performing an intervention on the selected node $X _ { j }$ and sampling from $\\mathscr { G } _ {  X _ { j } }$ (see Section 3 for details). We run active and passive versions of this agent as described in Section 3. ",
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+    },
+    {
+        "type": "text",
+        "text": "Interventional MAP Baseline: This baseline infers a DAG by maximizing the likelihood of the data observed by the Passive Interventional Agent in that episode. In the quiz phase, we predict the values of each node according to ${ \\mathcal { G } } _ {  X _ { j } } ^ { \\mathrm { M A P } }$ where $X _ { j }$ is the node externally intervened upon (i.e. causal inference), and choose the node with the highest value. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/9a32279239e503f54b6b8c6d57fdabae8547106bb59b266884e801dbebc1216b.jpg",
+        "image_caption": [
+            "Figure 4: Experiment 2. Agents do cause-effect reasoning from interventional data. a) Average reward earned by the agents tested in this experiment. See main text for details. b) Performance split by the presence or absence of unobserved confounders (abbreviated as Conf. and Unconf. respectively) on the externally intervened node. c) Quiz phase for a test DAG. See Fig. 2 for a legend. Here, the left panel shows the full $\\mathcal { G }$ and the nodes taking the mean values prescribed by $p ( X _ { 1 : N \\backslash j } | \\bar { X } _ { j } = - 5 )$ . We see that the Passive-Cond Agent’s choice is consistent with choosing based on these (incorrect) node values. The right panel shows $\\mathscr { G } _ {  X _ { j } = - 5 }$ and the nodes taking the mean values prescribed by $p _ {  X _ { j } = - 5 } ( X _ { 1 : N \\backslash j } | X _ { j } = - \\mathrm { \\bar { 5 } } )$ We see that the Passive-Int. Agent’s choice is consistent with maximizing on these (correct) node value. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Optimal Cause-Effect Baseline: This baseline receives the true DAG, $\\mathcal { G }$ . In the quiz phase, it chooses the node that has the maximum value according to $\\mathscr { G } _ {  X _ { j } }$ , where $X _ { j }$ is the node externally intervened upon. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/6b92a598cec55c4ca90267608194fd078c978d7e0d53f3d8083a35b1be04eb10.jpg",
+        "image_caption": [
+            "Figure 3: Active and Passive Interventional Agents "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We focus on three key questions in this experiment: (i) Can our agents learn to do cause-effect reasoning from interventional data?, (ii) How does the cause-effect reasoning in our agents which have access to interventional data differ from the cause-effect reasoning measured in Experiment 1 (in agents that have access only to observational data)? (iii) In addition to making causal inferences, can our agent also choose good actions in the information phase to generate the data it observes? ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "For (i) we see in Fig. 4a that the Passive-Interventional Agent’s performance is comparable to the Optimal Cause-Effect Baseline, indicating that it is able to do close to perfect cause-effect reasoning in this domain. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "For (ii) we see in Fig. 4a the crucial result that the Passive-Interventional Agent’s performance is significantly better than the Passive-Conditional Agent. We compare the performances of these two agents, split by whether the node that was intervened on in the quiz phase of the episode had unobserved confounders with other variables in the graph (Fig. 4b). In confounded cases, as described in Section 2.1, cause-effect reasoning is impossible with only observational data. We see that the performance of the Passive-Interventional Agent does not vary significantly with confoundedness, whereas the performance of the Passive-Conditional Agent is significantly lower in the confounded cases. This indicates that the improvement in the performance of the agent that has access to interventional data (as compared to the agents that had access to only observational data) is largely driven by its ability to also do cause-effect reasoning in the presence of confounders. This is highlighted by Fig. 4c, which shows the quiz phase for an example DAG, where the Passive-Conditional agent is unable to resolve the confounder, but the Passive-Interventional agent can. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "For (iii), we see in Fig. 3 that the Active-Interventional Agent’s performance is only marginally below the performance of the near optimal Passive-Interventional Agent, indicating that when the agent is allowed to choose its actions, it makes reasonable choices that allow good performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3 EXPERIMENT 3: COUNTERFACTUAL SETTING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Experiment 3, the agent was again allowed to make interventions as in Experiment 2, but in this case the quiz phase task entailed answering a counterfactual question. We explain here what a counterfactual question in this domain looks like. Consider the conditional distribution $\\scriptstyle { p ( \\bar { X } _ { i } | \\mathsf { p a } ( X _ { i } ) ) = N ( \\sum _ { j } w _ { j i } X _ { j } , 0 . 1 ) }$ as described in Section 3 as $\\begin{array} { r } { X _ { i } = \\sum _ { j } w _ { j i } X _ { j } + \\epsilon } \\end{array}$ where $\\epsilon$ is distributed as $\\mathcal { N } ( 0 . 0 , 0 . 1 )$ , and represents the specific randomness introduced when taking one sample from the DAG. After observing the nodes $X _ { 1 : N }$ in the DAG in one sample, we can infer this specific randomness $\\epsilon _ { i }$ for each node $X _ { i }$ (i.e. abduction as described in the Appendix) and answer counterfactual questions like “What would the values of the nodes be, had $X _ { j }$ in that particular sample taken on a different value than what we observed?”, for any of the nodes $X _ { j }$ . We test 2 new learned agents: “Active Counterfactual” and “Passive Counterfactual”. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/e971a44dda37b1eab99b010fc18fc5982ff913faa55ed88e63f6e3c8d79ae111.jpg",
+        "image_caption": [
+            "Figure 5: Experiment 3. Agents do counterfactual reasoning. a) Average reward earned by the agents tested in this experiment. See main text for details. b) Performance split by if the maximum node value in the quiz phase is degenerate (Deg.) or distinct (Dist.). c) Quiz phase for an example test-DAG. See Fig. 2 for a legend. Here, the left panel shows $\\mathscr { G } _ {  X _ { j } = - 5 }$ and the nodes taking the mean values prescribed by $p _ {  X _ { j } = - 5 } ( X _ { 1 : N \\backslash j } | X _ { j } = - \\bar { 5 } )$ . We see that the Passive-Int. Agent’s choice is consistent with maximizing on these node values, where it makes a random choice between two nodes with the same value. The right panel panel shows $\\mathscr { G } _ {  X _ { j } = - 5 }$ and the nodes taking the exact values prescribed by the means of $p _ {  X _ { j } = - 5 } ( X _ { 1 : N \\backslash j } | X _ { j } = - 5 )$ , combined with the specific randomness inferred from the previous time step. As a result of accounting for the randomness, the two previously degenerate maximum values are now distinct. We see that the Passive-CF. agent’s choice is consistent with maximizing on these node values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Counterfactual Agents: This agent is exactly analogous to the Interventional agent, with the addition that the exogenous noise in the last information phase step $t = T - 1$ (where say $X _ { p } = + 5$ ), is stored and the same noise is used in the quiz phase step $t = T$ (where say $X _ { f } = - 5$ ). While the question our agents have had to answer correctly so far in order to maximize their reward in the quiz phase was “Which of the nodes $X _ { 1 : N \\backslash j }$ will have the highest value when $X _ { f }$ is set to $- 5 ? ^ { \\prime }$ , in this setting, we ask “Which of the nodes $X _ { 1 : N \\backslash j }$ would have had the highest value in the last step of the information phase, if instead of having $X _ { p } = + 5$ , we had $X _ { f } = - 5 ? ^ { \\prime }$ . We run active and passive versions of this agent as described in Section 3. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Optimal Counterfactual Baseline: This baseline receives the true DAG and does exact abduction based on the exogenous noise observed in the penultimate step of the information phase, and combines this correctly with the appropriate interventional inference on the true DAG in the quiz phase. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            590,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We focus on two key questions in this experiment: (i) Can our agents learn to do counterfactual reasoning?, (ii) In addition to making causal inferences, can our agent also choose good actions in the information phase to generate the data it observes? ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For (i), we see that the Passive-Counterfactual Agent achieves higher reward than the Passive-Interventional Agent and the Optimal Cause-Effect Baseline. To evaluate whether this difference results from the agent’s use of abduction (see the Appendix for details), we split the test set into two groups, depending on whether or not the decision for which node will have the highest value in the quiz phase is affected by exogenous noise, i.e. whether or not the node with the maximum value in the quiz phase changes if the noise is resampled. This is most prevalent in cases where the maximum expected reward is degenerate, i.e. where several nodes give the same maximum reward (denoted by hatched bars in Figure 5b). Here, agents with no access to the noise have no basis for choosing one over the other, but different noise samples can give rise to significant differences in the actual values that these degenerate nodes have. We see indeed that there is no difference in the rewards received by the Passive-Counterfactual and Passive-Interventional Agents in the cases where the maximum values are distinct, however the Passive-Counterfactual Agent significantly outperforms the Passive-Interventional Agent in cases where there are degenerate maximum values. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/0102ff655040ab3e14e80ab25504d2a3a6eb52f8de6340142a4fc072189108a0.jpg",
+        "image_caption": [
+            "Figure 6: Active and Passive Counterfactual Agents "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "For (ii), we see in Fig. 6 that the Active-Counterfactual Agent’s performance is only marginally below the performance of the Passive-Counterfactual agent, indicating that when the agent is allowed to choose its actions, it makes reasonable choices that allow good performance. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "5 SUMMARY OF RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We introduced and tested a framework for learning causal reasoning in various data settings—observational, interventional, and counterfactual—using deep meta-RL. Crucially, our approach did not require explicit encoding of formal principles of causal inference. Rather, by optimizing an agent to perform a task that depended on causal structure, the agent learned implicit strategies to use the available data for causal reasoning, including drawing inferences from passive observation, actively intervening, and making counterfactual predictions. Below, we summarize the keys results from each of the three experiments. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "In Section 4.1 and Fig. 2, we show that the agent learns to perform do-calculus. In Fig. 2(a) we see that, compared to the highest possible reward achievable without causal knowledge, the trained agent received more reward. This observation is corroborated by Fig. 2(b) which shows that performance increased selectively in cases where do-calculus made a prediction distinguishable from the predictions based on correlations. These are situations where the externally intervened node had a parent – meaning that the intervention resulted in a different graph. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Section 4.2 and Fig. 4, we show that the agent learns to resolve unobserved confounders using interventions (a feat impossible with only observational data). In Fig. 4(a) we see that the agent with access to interventional data performs better than an agent with access to only observational data. Fig. 4(b) shows that the performance increase is greater in cases where the intervened node shared an unobserved parent (a confounder) with other variables in the graph. In this section we also compare the agent’s performance to a MAP estimate of the causal structure and find that the agent’s performance matches it, indicating that the agent is indeed doing close to optimal causal inference. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Section 4.3 and Fig. 5, we show that the agent learns to use counterfactuals. In Fig. 5(a) we see that the agent with additional access to the specific randomness in the test phase performs better than an agent with access to only interventional data. In Fig. 5(b), we find that the increased performance is observed only in cases where the maximum mean value in the graph is degenerate, and optimal choice is affected by the exogenous noise – i.e. where multiple nodes have the same value on average and the specific randomness can be used to distinguish their actual values in that specific case. ",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND FUTURE WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This work is the first demonstration that causal reasoning can arise out of model-free reinforcement learning. This opens up the possibility of leveraging powerful learning-based methods for causal inference in complex settings. Traditional formal approaches usually decouple the two problems of causal induction (i.e. inferring the structure of the underlying model) and causal inference (i.e. estimating causal effects and answering counterfactual questions), and despite advances in both (Ortega & Stocker, 2015; Bramley et al., 2017; Parida et al., 2018; Sen et al., 2017; Forney et al., 2017; Lattimore et al., 2016), inducing models often requires assumptions that are difficult to fit to complex real-world conditions. By learning these end-to-end, our method can potentially find representations of causal structure best tuned to the specific causal inferences required. Another key advantage of our meta-RL approach is that it allows the agent to learn to interact with the environment in order to acquire necessary observations in the service of its task—i.e. to perform active learning. In our experimental domain, our agents’ active intervention policy was close to optimal, which demonstrates the promise of agents that can learn to experiment on their environment and perform rich causal reasoning on the observations. ",
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+    },
+    {
+        "type": "text",
+        "text": "Future work should explore agents that perform experiments to support structured exploration in RL, and optimal experiment design in complex domains where large numbers of blind interventions are prohibitive. To this end, follow-up work should focus on scaling up our approach to larger environments, with more complex causal structure and a more diverse range of tasks. Though the results here are a first step in this direction which use relatively standard deep RL components, our approach will likely benefit from more advanced architectures (e.g. Espeholt et al., 2018; Hessel et al., 2018; Hester et al., 2017) that allow longer more complex episodes, as well as models which are more explicitly compositional (e.g. Battaglia et al., 2018; Andreas et al., 2016) or have richer semantics (e.g. Ganin et al., 2018), that more explicitly leverage symmetries like equivalance classes in the environment. ",
+        "bbox": [
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+    {
+        "type": "text",
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+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/950a7be8955d94a6a74726f71f16e895a8d30fbb962aecd7c8593cedbd513a9b.jpg",
+        "image_caption": [
+            "A ADDITIONAL BASELINES",
+            "Figure 7: Reward distribution "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We can also compare the performance of these agents to two standard model-free RL baselines. The Q-total agent learns a Q-value for each action across all steps for all the episodes. The Q-episode agent learns a Q-value for each action conditioned on the input at each time step $\\left[ o _ { t } , a _ { t - 1 } , r _ { t - 1 } \\right]$ , but with no LSTM memory to store previous actions and observations. Since the relationship between action and reward is random between episodes, Q-total was equivalent to selecting actions randomly, resulting in a considerably negative reward. The Q-episode agent essentially makes sure to not choose the arm that is indicated by $m _ { t }$ to be the external intervention (which is assured to be equal to $- 5 )$ , and essentially chooses randomly otherwise, giving an average reward of 0. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B FORMAL DESCRIPTION OF META-LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Consider a distribution $\\mathcal { D }$ over Markov Decision Processes (MDPs). We train an agent with memory (in our case an RNN-based agent) on this distribution. In each episode, we sample a task $m \\sim \\mathcal { D }$ . At each step $t$ within an episode, the agent sees an observation $o _ { t }$ , executes an action $a _ { t }$ , and receives a reward $r _ { t }$ . Both $a _ { t - 1 }$ and $r _ { t - 1 }$ are given as additional inputs to the network. Thus, via the recurrence of the network, each action is a function of the entire trajectory $\\mathcal { H } _ { t } = \\left\\{ o _ { 0 } , a _ { 0 } , r _ { 0 } , . . . , o _ { t - 1 } , a _ { t - 1 } , r _ { t - 1 } , o _ { t } \\right\\}$ of the episode. Because this function is parameterized by the neural network, its complexity is limited only by the size of the network. ",
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C ABDUCTION-ACTION-PREDICTION METHOD FOR COUNTERFACTUAL REASONING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Pearl et al. (2016)’s “abduction-action-prediction” method prescribes one method for answering counterfactual queries, by estimating the specific unobserved makeup of individual $i$ and by transferring it to the counterfactual world. Assume, for example, the following model for $\\mathcal { G }$ of Section 2.1: $E = w _ { A E } A + \\eta$ , $H = w _ { A H } A + w _ { E H } E + \\epsilon$ , where the weights $w _ { i j }$ represent the known causal effects in $\\mathcal { G }$ and $\\epsilon$ and $\\eta$ are terms of (e.g.) Gaussian noise that represent the unobserved randomness in the makeup of each individual9. Suppose that for individual $i$ we observe: $A = a ^ { i }$ , $E = e ^ { i }$ , $H = h ^ { i }$ . We can answer the counterfactual question of “What if individual $i$ had done more exercise, i.e. $E { = } e ^ { \\prime }$ , instead?” by: a) Abduction: estimate the individual’s specific makeup with $\\epsilon ^ { i } = h ^ { i } - w _ { A H } a ^ { i } - w _ { E H } e ^ { i }$ , b) Action: set $E$ to more exercise $e ^ { \\prime }$ , c) Prediction: predict a new value for cardiac health as $h ^ { \\prime } { = } w _ { A H } a ^ { i } { + } w _ { E H } e ^ { \\prime } { + } { \\epsilon } ^ { i }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "D EXPERIMENT 4: NON-LINEAR CAUSAL GRAPHS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/4635b97ada0f7061f268d413085612f83173204d56deebcf8f2c4e789abdb1fe.jpg",
+        "image_caption": [
+            "Figure 8: Experiment 4 results "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The purview of the previous experiments was to show a proof of concept on a simple tractable system, demonstrating that causal induction and inference can be learned and implemented via a meta-learned agent. In this experiment, we generalize some of the results to nonlinear, non-Gaussian causal graphs which are more typical of real-world causal graphs and to demonstrate that our results hold without loss of generality on such systems. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here we investigate causal DAGs with a quadratic dependence on the parents by changing the conditional distribution to $\\begin{array} { r } { \\overline { { p } } ( X _ { i } | \\mathfrak { p a } ( X _ { i } ) ) = \\mathcal { N } ( \\frac { 1 } { N _ { i } } \\overset { \\cdot } { \\sum _ { j } } w _ { j i } ( X _ { j } \\overset { \\cdot } { + } X _ { j } ^ { 2 } ) , \\sigma ) } \\end{array}$ . Here, although each node is normally distributed given its parents, the joint distribution is not multivariate Gaussian due to the non-linearity in how the means are determined. We find that the Long-Observational achieves more reward than the Observational agent indicating that the agent is in fact learning the statistical dependencies between the nodes, within an episode. We also find that although the Active-Interventional agent is not far behind the performance of the MAP baseline, and achieves reward well above the Long-Observational10 The fact that the MAP baseline gets so close to the Optimal Cause-Effect baseline indicates that the Active agent is choosing close to optimal actions. ",
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/515d9cd2e9a7f0c794fecfd02afc3496625d8ffc4f9aefbbf5d09e6157700d51.jpg",
+        "image_caption": [
+            "E EXPERIMENT 5: LARGER CAUSAL GRAPHS WITH GENERALIZATION TO NEW EQUIVALENCE CLASSES ",
+            "Figure 9: (a) Comparing agent performances with different data. (b) Comparing information phase intervention policies. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            273,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the experiments reported in the main paper, the test set was a random subset of all graphs, and training examples were generated randomly subject to the constraint that they not be in the test set. However, this raised the possibility that any test graph might have an equivalent graph in the training set, which could result in a type of overfitting. We therefore ran a new set of experiments where ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "the entire equivalence class of each test graph was held out from the training set11. Performance on the test set therefore indicates generalization of the inference procedures learned to previously unseen equivalence classes of causal DAGs. For these experiments, we used graphs with $N { = } 6$ nodes, because 5-node graphs have too few equivalence classes to partition in this way. All other details were the same as in the main paper. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We see in Fig. 9a that the agents learn to generalize well to these held out examples, and we find the same pattern of behavior noted in the main text where the rewards earned are ordered such that Observational agent $<$ Passive-Conditional agent $<$ Passive-Interventional agent $<$ Passive-Counterfactual agent. We see additionally in Fig. 9b that the Active-Interventional agent performs at par with the Passive-Interventional agent (which is allowed to see the results of interventions on all nodes) and significantly better than an additional baseline we use here of the Random-Interventional agent whose information phase policy is to intervene on nodes at random, indicating that the intervention policy learned by the Active agent is good. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "F GRAPHICAL MODELS AND BELIEF NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Graphical models (Pearl, 1988; Bishop, 2006; Koller & Friedman, 2009; Barber, 2012; Murphy, 2012) are a marriage between graph and probability theory that allows to graphically represent and assess statistical dependence. In the following sections, we give some basic definitions and describe a method $d \\cdot$ -separation) for graphically assessing statistical independence in belief networks. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "BASIC DEFINITIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "image_caption": [
+            "Figure 10: (a): Directed acyclic graph. The node $X _ { 3 }$ is a collider on the path $X _ { 1 } \\right. X _ { 3 } \\left. X _ { 2 }$ and a non-collider on the path $X _ { 2 }  X _ { 3 }  X _ { 4 }$ . (b): Cyclic graph obtained from (a) by adding a link from $X _ { 4 }$ to $X _ { 1 }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            220,
+            728,
+            488,
+            808
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A graph is a collection of nodes and links connecting pairs of nodes. The links may be directed or undirected, giving rise to directed or undirected graphs respectively. ",
+        "bbox": [
+            547,
+            729,
+            823,
+            785
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A path from node $X _ { i }$ to node $X _ { j }$ is a sequence of linked nodes starting at $X _ { i }$ and ending at $X _ { j }$ . A directed path is a path whose links are directed and pointing from preceding towards following nodes in the sequence. ",
+        "bbox": [
+            545,
+            791,
+            825,
+            873
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A directed acyclic graph (DAG) is a directed graph with no directed paths starting and ending at the same node. For example, the directed graph in Fig. 10(a) is acyclic. The addition of a link from $X _ { 4 }$ to $X _ { 1 }$ gives rise to a cyclic graph (Fig. 10(b)). ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A node $X _ { i }$ with a directed link to $X _ { j }$ is called parent of $X _ { j }$ . In this case, $X _ { j }$ is called child of $X _ { i }$ . ",
+        "bbox": [
+            173,
+            150,
+            785,
+            166
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A node is a collider on a specified path if it has (at least) two parents on that path. Notice that a node can be a collider on a path and a non-collider on another path. For example, in Fig. 10(a) $X _ { 3 }$ is a collider on the path $X _ { 1 } \\right. X _ { 3 } \\left. X _ { 2 }$ and a non-collider on the path $X _ { 2 }  X _ { 3 }  X _ { 4 }$ . ",
+        "bbox": [
+            173,
+            171,
+            825,
+            214
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A node $X _ { i }$ is an ancestor of a node $X _ { j }$ if there exists a directed path from $X _ { i }$ to $X _ { j }$ . In this case, $X _ { j }$ is a descendant of $X _ { i }$ . ",
+        "bbox": [
+            173,
+            219,
+            823,
+            248
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A graphical model is a graph in which nodes represent random variables and links express statistical relationships between the variables. ",
+        "bbox": [
+            173,
+            252,
+            823,
+            281
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A belief network is a directed acyclic graphical model in which each node $X _ { i }$ is associated with the conditional distribution $p ( X _ { i } | \\mathfrak { p a } ( X _ { i } ) )$ , where $\\mathsf { p a } ( X _ { i } )$ indicates the parents of $X _ { i }$ . The joint distribution of all nodes in the graph, $p ( X _ { 1 : N } )$ , is given by the product of all conditional distributions, i.e. ",
+        "bbox": [
+            174,
+            286,
+            825,
+            330
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aa6ce2b15fc19b138aa7e8c529d21609a2735f9276d9ff340e2f186731d1be77.jpg",
+        "text": "$$\np ( X _ { 1 : N } ) { = } \\prod _ { i = 1 } ^ { N } p ( X _ { i } | \\mathsf { p a } ( X _ { i } ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            337,
+            591,
+            380
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "ASSESSING STATISTICAL INDEPENDENCE IN BELIEF NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            393,
+            596,
+            409
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Given the sets of random variables $x , y$ and $\\mathcal { Z }$ , $\\mathcal { X }$ and $\\mathcal { V }$ are statistically independent given $\\mathcal { Z } \\left( \\mathcal { X } \\perp \\perp \\mathcal { Y } | \\mathcal { Z } \\right)$ if all paths from any element of $\\mathcal { X }$ to any element of $\\mathcal { V }$ are closed (or blocked). A path is closed if at least one of the following conditions is satisfied: ",
+        "bbox": [
+            176,
+            416,
+            826,
+            460
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(Ia) There is a non-collider on the path which belongs to the conditioning set $\\mathcal { Z }$ . (Ib) There is a collider on the path such that neither the collider nor any of its descendants belong to the conditioning set $\\mathcal { Z }$ . ",
+        "bbox": [
+            196,
+            470,
+            825,
+            520
+        ],
+        "page_idx": 13
+    }
+]

parse/train/HkzRQhR9YX/HkzRQhR9YX.md

@@ -0,0 +1,497 @@
+# TREE-STRUCTURED RECURRENT SWITCHING LINEARDYNAMICAL SYSTEMS FOR MULTI-SCALE MODELING
+
+Josue Nassar   
+Department of Electrical & Computer Engineering   
+Stony Brook University   
+Stony Brook, NY 11794   
+josue.nassar@stonybrook.edu   
+Scott W. Linderman   
+Department of Statistics   
+Columbia University   
+New York, NY 10027   
+scott.linderman@columbia.edu
+
+# Il Memming Park
+
+Mónica F. Bugallo   
+Department of Electrical & Computer Engineering   
+Stony Brook University   
+Stony Brook, NY, 11794   
+monica.bugallo@stonybrook.edu Department of Neurobiology and Behavior Stony Brook University   
+Stony Brook, NY, 11794   
+memming.park@stonybrook.edu
+
+# ABSTRACT
+
+Many real-world systems studied are governed by complex, nonlinear dynamics. By modeling these dynamics, we can gain insight into how these systems work, make predictions about how they will behave, and develop strategies for controlling them. While there are many methods for modeling nonlinear dynamical systems, existing techniques face a trade off between offering interpretable descriptions and making accurate predictions. Here, we develop a class of models that aims to achieve both simultaneously, smoothly interpolating between simple descriptions and more complex, yet also more accurate models1. Our probabilistic model achieves this multi-scale property through a hierarchy of locally linear dynamics that jointly approximate global nonlinear dynamics. We call it the tree-structured recurrent switching linear dynamical system. To fit this model, we present a fully-Bayesian sampling procedure using Pólya-Gamma data augmentation to allow for fast and conjugate Gibbs sampling. Through a variety of synthetic and real examples, we show how these models outperform existing methods in both interpretability and predictive capability.
+
+# 1 INTRODUCTION
+
+Complex systems can often be described at multiple levels of abstraction. A computer program can be characterized by the list of functions it calls, the sequence of statements it executes, or the assembly instructions it sends to the microprocessor. As we zoom in, we gain an increasingly nuanced view of the system and its dynamics. The same is true of many natural systems. For example, brain activity can be described in terms of high-level psychological states or via detailed ion channel activations; different tasks demand different levels of granularity. One of our principal aims as scientists is to identify appropriate levels of abstraction for complex natural phenomena and to discover the dynamics that govern how these systems behave at each level of resolution.
+
+Modern machine learning offers a powerful toolkit to aid in modeling the dynamics of complex systems. Bayesian state space models and inference algorithms enable posterior inference of the latent states of a system and the parameters that govern their dynamics (Särkkä, 2013; Barber et al., 2011; Doucet et al., 2001). In recent years, this toolkit has been expanded to incorporate increasingly flexible components like Gaussian processes (Frigola et al., 2014) and neural networks (Chung et al., 2015; Johnson et al., 2016; Gao et al., 2016; Krishnan et al., 2017) into probabilistic time series models. In neuroscience, sequential autoencoders offer highly accurate models of brain activity (Pandarinath et al., 2018). However, while these methods offer state of the art predictive models, their dynamics are specified at only the most granular resolution, leaving the practitioner to tease out higher level structure post hoc.
+
+Here we propose a probabilistic generative model that provides a multi-scale view of the dynamics through a hierarchical architecture. We call it the tree-structured recurrent switching linear dynamical system, or TrSLDS. The model builds on the recurrent SLDS (Linderman et al., 2017) to approximate latent nonlinear dynamics through a hierarchy of locally linear dynamics. Once fit, the TrSLDS can be queried at different levels of the hierarchy to obtain dynamical descriptions at multiple levels of resolution. As we proceed down the tree, we obtain higher fidelity, yet increasingly complex, descriptions. Thus, depth offers a simple knob for trading off interpretability and flexibility. The key contributions are two-fold2: first, we introduce a new form of tree-structured stick breaking for multinomial models that strictly generalizes the sequential stick breaking of the original rSLDS, while still permitting Pólya-gamma data augmentation (Polson et al., 2013) for efficient posterior inference; second, we develop a hierarchical prior that links dynamics parameters across levels of the tree, thereby providing descriptions that vary smoothly with depth. The paper is organized as follows. Section 2 provides background material on switching linear dynamical systems and their recurrent variants. Section 3 presents our tree-structured model and Section 4 derives an efficient fullyBayesian inference algorithm for the latent states and dynamics parameters. Finally, in Section 5 we show how our model yields multi-scale dynamics descriptions for synthetic data from two standard nonlinear dynamical systems—the Lorenz attractor and the FitzHugh-Nagumo model of nonlinear oscillation—as well as for a real dataset of neural responses to visual stimuli in a macaque monkey.
+
+# 2 BACKGROUND
+
+Let $x _ { t } \ \in \mathbb { R } ^ { d _ { x } }$ and $y _ { t } \in \mathbb { R } ^ { d _ { y } }$ denote the latent state and the observation of the system at time $t$ respectively. The system can be described using a state-space model:
+
+$$
+\begin{array} { r } { \begin{array} { l c r } { x _ { t } = f ( x _ { t - 1 } , w _ { t } ; \Theta ) , } & { w _ { t } \sim \mathrm { F } _ { w } } & { ( s t a t e d y n a m i c s ) } \\ { y _ { t } = g ( x _ { t } , v _ { t } ; \Psi ) , } & { v _ { t } \sim \mathrm { F } _ { v } } & { ( o b s e r \nu a t i o n ) } \end{array} } \end{array}
+$$
+
+where $\Theta$ denotes the dynamics parameters, $\Psi$ denotes the emission (observation) parameters, and $w _ { t }$ and $v _ { t }$ are the state and observation noises respectively. For simplicity, we restrict ourselves to systems of the form:
+
+$$
+\begin{array} { r } { x _ { t } = f ( x _ { t - 1 } ; \Theta ) + w _ { t } , \quad w _ { t } \sim \mathcal { N } ( 0 , Q ) , } \end{array}
+$$
+
+If the state space model is completely specified then recursive Bayesian inference can be applied to obtain an estimate of the latent states using the posterior $p \left( { x _ { 0 : T } | y _ { 1 : T } } \right)$ (Doucet et al., 2001). However in many applications, the parametric form of the state space model is unknown. While there exist methods that perform smoothing to obtain an estimate of $x _ { 0 : T }$ (Barber, 2006; Fox et al., 2009; Djuric & Bugallo, 2006), we are often interested in not only obtaining an estimate of the continuous latent states but also in learning the dynamics $f ( \cdot ; \Theta )$ that govern the dynamics of the system.
+
+In the simplest case, we can take a parametric approach to solving this joint state-parameter estimation problem. When $f ( \cdot ; \Theta )$ and $g ( \cdot ; \Psi )$ are assumed to be linear functions, the posterior distribution over latent states is available in closed-form and the parameters can be learned via expectationmaximization. On the other hand, we have nonparametric methods that use Gaussian processes and neural networks to learn highly nonlinear dynamics and observations where the joint estimation is untractable and approximations are necessarily imployed (Zhao & Park, 2016; 2018; Frigola et al., 2014; Sussillo et al., 2016). Switching linear dynamical systems (SLDS) (Ackerson & Fu, 1970; Chang & Athans, 1978; Hamilton, 1990; Ghahramani & Hinton, 1996; Murphy, 1998) balance between these two extremes, approximating the dynamics by stochastically transitioning between a small number of linear regimes.
+
+![](images/ce7777a334e74f3cbccb9f637258ca035d0f58ca91583b364912a9eeb2fff9e0.jpg)  
+Figure 1: State probability allocation through stick-breaking in standard rSLDS and the TrSLDS.
+
+# 2.1 SWITCHING LINEAR DYNAMICAL SYSTEMS
+
+SLDS approximate nonlinear dynamics by switching between a discrete set of linear regimes. An additional discrete latent state $z _ { t } \in \{ 1 , \ldots , K \}$ determines the linear dynamics at time $t$ ,
+
+$$
+x _ { t } = x _ { t - 1 } + A _ { z _ { t } } x _ { t - 1 } + b _ { z _ { t } } + w _ { t } , \quad w _ { t } \sim \mathcal { N } ( 0 , Q _ { z _ { t } } )
+$$
+
+where $A _ { k } , Q _ { k } \in \mathbb { R } ^ { d _ { x } \times d _ { x } }$ and $b _ { k } \in \mathbb { R } ^ { d _ { x } }$ for $k = 1 , \ldots , K$ . Typically, $z _ { t }$ is endowed with Markovian dynamics, $\operatorname* { P r } ( z _ { t } | z _ { t - 1 } = k ) = \pi _ { k }$ . The conditionally linear dynamics allow for fast and efficient learning of the model and can utilize the learning tools developed for linear systems (Haykin, 2001). While SLDS can estimate the continuous latent states $x _ { 0 : T }$ , the assumption of Markovian dynamics for the discrete latent states severely limits their generative capacity.
+
+# 2.2 RECURRENT SWITCHING LINEAR DYNAMICAL SYSTEMS
+
+Recurrent switching linear dynamical systems (rSLDS) (Linderman et al., 2017), also known as augmented SLDS (Barber, 2006), are an extension of SLDS where the transition density of the discrete latent state depends on the previous location in the continuous latent space
+
+$$
+\begin{array} { r } { z _ { t } | x _ { t - 1 } , \{ R , r \} \sim \pi _ { S B } \left( \nu _ { t } \right) , } \\ { \nu _ { t } = R x _ { t - 1 } + r , } \end{array}
+$$
+
+where $R \in \mathbb { R } ^ { K - 1 \times d _ { x } }$ and $r \in \mathbb { R } ^ { K - 1 }$ represents hyperplanes. $\pi _ { S B } : \mathbb { R } ^ { K - 1 } \to [ 0 , 1 ] ^ { K }$ maps from the reals to the probability simplex via stick-breaking:
+
+$$
+\pi _ { S B } ( \nu ) = \left( \pi _ { S B } ^ { ( 1 ) } ( \nu ) , \cdots , \pi _ { S B } ^ { ( K ) } ( \nu ) \right) , \quad \pi _ { S B } ^ { ( k ) } = \sigma ( \nu _ { k } ) \prod _ { j < k } \sigma \left( - \nu _ { j } \right) ,
+$$
+
+for k = 1, . . . , K − 1 and π(K)SB $\begin{array} { r } { \pi _ { S B } ^ { ( K ) } = \prod _ { k = 1 } ^ { K - 1 } \sigma \left( - \nu _ { k } \right) } \end{array}$ where $\nu _ { k }$ is the $k$ th component of of $\nu$ and $\sigma ( \nu ) = ( 1 + e ^ { - \nu } ) ^ { - 1 }$ is the logistic function (Fig. 1). By including this recurrence in the transition density of $z _ { t }$ , the rSLDS partitions the latent space into $K$ sections, where each section follows its own linear dynamics. It is through this combination of locally linear dynamical systems that the rSLDS approximates eq. (3); the partitioning of the space allows for a more interpretable visualization of the underlying dynamics.
+
+Recurrent SLDS can be learned efficiently and in a fully Bayesian manner, and experiments empirically show that they are adept in modeling the underlying generative process in many cases. However, the stick breaking process used to partition the space poses problems for inference due to its dependence on the permutation of the discrete states $\{ 1 , \cdots , K \}$ (Linderman et al., 2017).
+
+# 3 TREE-STRUCUTRED RECURRENT SWITCHING LINEAR DYNAMICAL SYSTEMS
+
+Building upon the rSLDS, we propose the tree-structured recurrent switching linear dynamical system (TrSLDS). Rather than sequentially partitioning the latent space using stick breaking, we use a treestructured stick breaking procedure (Adams et al., 2010) to partition the space.
+
+Let $\tau$ denote a tree structure with a finite set of nodes $\{ \epsilon , 1 , \cdots , N \}$ . Each node $n$ has a parent node denoted by $\operatorname { p a r } ( n )$ with the exception of the root node, $\epsilon$ , which has no parent. For simplicity, we initially restrict our scope to balanced binary trees where every internal node $n$ is the parent of two children, $\operatorname { l e f t } ( n )$ and right $( n )$ . Let $\mathrm { c h i l d } ( \bar { n } ) = \{ \mathrm { l e f t } ( n ) , \mathrm { r i g h t } ( n ) \}$ denote the set of children for internal node $n$ . Let $\mathcal { Z } \subseteq \mathcal { T }$ denote the set of leaf nodes, which have no children. Let depth $( n )$ denote the depth of a node $n$ in the tree, with $\mathrm { d e p t h } ( \epsilon ) = 0$ .
+
+At time instant $t$ , the discrete latent state $z _ { t }$ is chosen by starting at the root node and traversing down the tree until one of the $K$ leaf nodes are reached. The traversal is done through a sequence of left/right choices by the internal nodes. Unlike in standard regression trees where the choices are deterministic (Lakshminarayanan, 2016), we model the choices as random variables. The traversal through the tree can be described as a stick breaking process. We start at the root node with a unit-length stick $\pi _ { \epsilon } = 1$ , which we divide between its two children. The left child receives a fraction $\pi _ { \mathrm { l e f t } ( \epsilon ) } = \sigma ( \nu _ { \epsilon } )$ and the right child receives the remainder $\pi _ { \mathrm { r i g h t } ( \epsilon ) } = 1 - \sigma ( \nu _ { \epsilon } )$ such that $\nu _ { \epsilon } \in \mathbb { R }$ specifies the left/right balance. This process is repeated recursively, subdividing $\pi _ { n }$ into two pieces at each internal node until we reach the leaves of the tree (Fig. 1). The stick assigned to each node is thus,
+
+$$
+\pi _ { n } = \left\{ \begin{array} { l l } { \sigma ( \nu _ { \mathrm { p a r } ( n ) } ) ^ { \mathrm { I } [ n = \mathrm { l e f t } ( \mathrm { p a r } ( n ) ) ] } \left( 1 - \sigma ( \nu _ { \mathrm { p a r } ( n ) } ) \right) ^ { \mathrm { I } [ n = \mathrm { r i g h t } ( \mathrm { p a r } ( n ) ) ] } \pi _ { \mathrm { p a r } ( n ) } } & { n \neq \epsilon , } \\ { 1 } & { n = \epsilon . } \end{array} \right.
+$$
+
+We incorporate this into the TrSLDS by allowing $\nu _ { n }$ to be a function of the continuous latent state
+
+$$
+\nu _ { n } ( x _ { t - 1 } , R _ { n } , r _ { n } ) = R _ { n } ^ { T } x _ { t - 1 } + r _ { n } ,
+$$
+
+where the parameters $R _ { n }$ and $r _ { n }$ specify a linear hyperplane in the continuous latent state space. As the continuous latent state $x _ { t - 1 }$ evolves, the left/right choices become more or less probable. This in turn changes the probability distribution $\pi _ { k } ( x _ { t - 1 } , \Gamma , \mathcal { T } )$ over the $K$ leaf nodes, where $\Gamma = \{ R _ { n } , r _ { n } \} _ { n \in \mathcal { T } }$ In the TrSLDS, these leaf nodes correspond to the discrete latent states of the model, such that for each leaf node $k$ ,
+
+$$
+p \left( z _ { t } = k \mid x _ { t - 1 } , \Gamma , \mathcal { T } \right) = \pi _ { k } ( x _ { t - 1 } , \Gamma , \mathcal { T } ) .
+$$
+
+In general, the tree-structured stick-breaking is not restricted to balanced binary trees. We can allow more than two children through an ordered sequential stick-breaking at each level. In this sense, tree-structured stick-breaking is a strict generalization of stick-breaking. We also note that similar to rSLDS, the model can be made more flexible by introducing a dependence on the previous discrete latent in eq. (9) but for the rest of the paper, we stick to eq. (8).
+
+# 3.1 A HIERARCHICAL DYNAMICS PRIOR THAT RESPECTS THE TREE STRUCTURE
+
+Similar to standard rSLDS, the dynamics are conditionally linear given a leaf node $z _ { t }$ . A priori, it is natural to expect that locally linear dynamics of nearby regions in the latent space are similar. Thus, in the context of tree-structured stick breaking, we impose that partitions that share a common parent should have similar dynamics. We explicitly model this by enforcing a hierarchical prior on the dynamics that respects the tree structure.
+
+Let $\left\{ A _ { n } , b _ { n } \right\}$ be the dynamics parameters associated with node $n$ . Although the locally linear dynamics of a discrete state are specified by the leaf nodes, we introduce dynamics at the internal nodes as well. These internal dynamics serve as a link between the leaf node dynamics via a hierarchical prior,
+
+$$
+\operatorname { v e c } ( [ A _ { n } , b _ { n } ] ) | \operatorname { v e c } ( [ A _ { \mathrm { p a r } ( n ) } , b _ { \mathrm { p a r } ( n ) } ] ) \sim { \mathcal { N } } ( \operatorname { v e c } ( [ A _ { \mathrm { p a r } ( n ) } , b _ { \mathrm { p a r } ( n ) } ] ) , \Sigma _ { n } ) ,
+$$
+
+where $\mathrm { v e c } ( \cdot )$ is the vectorization operator. The prior on the root node is
+
+$$
+\mathrm { v e c } \left( \left[ A _ { \epsilon } , b _ { \epsilon } \right] \right) \sim { \mathcal N } \left( 0 , \Sigma _ { \epsilon } \right) .
+$$
+
+We impose the following constraint on the covariance matrix of the prior
+
+$$
+\begin{array} { r } { \Sigma _ { n } = \lambda ^ { \mathrm { d e p t h } ( n ) } \Sigma _ { \epsilon } , } \end{array}
+$$
+
+where $\lambda \in ( 0 , 1 )$ is a hyper parameter that dictates how "close" a parent and child are to one another. The prior over the parameters can be written as, where the affine term and the $\mathrm { v e c } ( \cdot )$ operator are dropped for compactness,
+
+$$
+p ( \{ A _ { n } \} _ { n \in \mathcal { T } } ) = p ( A _ { \epsilon } ) \prod _ { i \in \operatorname { c h i l d } ( \epsilon ) } p ( A _ { i } | A _ { \epsilon } ) \prod _ { j \in \operatorname { c h i l d } ( i ) } p ( A _ { j } | A _ { i } ) \ . . . \prod _ { z \in \mathcal { Z } } p ( A _ { z } | A _ { \operatorname { p a r } ( z ) } ) .
+$$
+
+It is through this hierarchical tree-structured prior that TrSLDS obtains a multi-scale view of the system. Parents are given the task of learning a higher level description of the dynamics over a larger region while children are tasked with learning the nuances of the dynamics. The use of hierarchical priors also allows for neighboring sections of latent space to share common underlying dynamics inherited from their parent. TrSLDS can be queried at different levels, where levels deeper in the tree provide more resolution.
+
+TrSLDS shares some features with regression trees (Lakshminarayanan, 2016), even though regression trees are primarily used for standard, static regression problems. The biggest differences are that our tree-structured model has stochastic choices and the internal nodes contribute to smoothing across partitions through the corresponding hierarchical prior.
+
+There are other hierarchical extensions of SLDS that have been proposed in the literature. In Stanculescu et al. (2014), they propose adding a layer to factorized SLDS where the top-level discrete latent variables determine the conditional distribution of $z _ { t }$ , with no dependence on $x _ { t - 1 }$ . While the tree-structured stick-breaking used in TrSLDS is also a hierarchy of discrete latent variables, the model proposed in Stanculescu et al. (2014) has no hierarchy of dynamics, preventing it from obtaining a multi-scale view of the dynamics. In Zoeter & Heskes (2003), the authors construct a tree of SLDSs where an SLDS with $K$ possible discrete states is first fit. An SLDS with $M$ discrete states is then fit to each of the $K$ clusters of points. This process continues iteratively, building a hierarchical collection of SLDSs that allow for a multi-scale, low-dimensional representation of the observed data. While similar in spirit to TrSLDS, there are key differences between the two models. First, it is through the tree-structured prior that TrSLDS obtains a multi-scale view of the dynamics, thus we only need to fit one instantiation of TrSLDS; in contrast, they fit a separate SLDS for each node in the tree, which is computationally expensive. There is also no explicit probabilistic connection between the dynamics of a parent and child in Zoeter & Heskes (2003). We also note that TrSLDS aims to learn a multiscale view of the dynamics while Zoeter & Heskes (2003) focuses on smoothing, that is, they aim to learn a multi-scale view of the latent states corresponding to data but not suitable for forecasting.
+
+In the next section we show an alternate view of TrSLDS which we will refer to as the residual model in which internal nodes do contribute to the dynamics. Nevertheless, this residual model will turn out to be equivalent to the TrSLDS.
+
+# 3.2 RESIDUAL MODEL
+
+Let $\{ \tilde { A } _ { n } , \tilde { b } _ { n } \}$ be the linear dynamics of node $n$ and let $\operatorname { p a t h } ( n ) = ( \epsilon , \dots , n )$ be the sequence of nodes visited to arrive at node $n$ . In contrast to TrSLDS, the dynamics for a leaf node are now determined by all the nodes in the tree:
+
+$$
+\begin{array} { r l } & { p ( x _ { t } | x _ { t - 1 } , \tilde { \Theta } , z _ { t } ) = \mathcal { N } ( x _ { t } | x _ { t - 1 } + \bar { A } _ { z _ { t } } x _ { t - 1 } + \bar { b } _ { z _ { t } } , \tilde { Q } _ { z _ { t } } ) , } \\ & { \bar { A } _ { z _ { t } } = \displaystyle \sum _ { j \in \mathrm { p a t h } ( z _ { t } ) } \tilde { A } _ { j } , \quad \bar { b } _ { z _ { t } } = \displaystyle \sum _ { j \in \mathrm { p a t h } ( z _ { t } ) } \tilde { b } _ { j } , } \end{array}
+$$
+
+We model the dynamics to be independent a priori, where once again the $\mathrm { v e c } ( \cdot )$ operator and the affine term aren’t shown for compactness,
+
+$$
+p ( \{ \tilde { A } _ { n } \} _ { n \in \mathcal { T } } ) = \prod _ { n \in \mathcal { T } } p ( \tilde { A } _ { n } ) , \quad p ( \tilde { A } _ { n } ) = \mathcal { N } ( 0 , \tilde { \Sigma } _ { n } ) ,
+$$
+
+where $\tilde { \Sigma } _ { n } = \tilde { \lambda } ^ { \mathrm { d e p t h } ( n ) } \tilde { \Sigma } _ { \epsilon }$ and $\tilde { \lambda } \in ( 0 , 1 )$ .
+
+The residual model offers a different perspective of TrSLDS. The covariance matrix can be seen as representing how much of the dynamics a node is tasked with learning. The root node is given the broadest prior because it is present in eq. (16) for all leaf nodes; thus it is given the task of learning the global dynamics. The children then have to learn to explain the residuals of the root node. Nodes deeper in the tree become more associated with certain regions of the space, so they are tasked with learning more localized dynamics which is represented by the prior being more sharply centered on 0. The model ultimately learns a multi-scale view of the dynamics where the root node captures a coarse estimate of the system while lower nodes learn a much finer grained picture. We show that TrSLDS and residual model yield the same joint distribution (See A for the proof).
+
+Theorem 1. TrSLDS and the residual model are equivalent if the following conditions are true: $A _ { \epsilon } = \tilde { A } _ { \epsilon } ,$ , $\begin{array} { r } { A _ { n } = \sum _ { j \in \mathrm { p a t h } ( n ) } \tilde { A } _ { j } } \end{array}$ , $Q _ { z } = \tilde { Q } _ { z } \forall z \in \mathrm { l e a v e s } ( \mathcal { T } )$ , $\Sigma _ { \epsilon } = \tilde { \Sigma } _ { \epsilon }$ and $\lambda = \tilde { \lambda }$
+
+# 4 BAYESIAN INFERENCE
+
+The linear dynamic matrices $\Theta$ , the hyperplanes $\Gamma = \{ R _ { n } , r _ { n } \} _ { n \in \mathcal { T } \backslash \mathcal { Z } }$ , the emission parameters $\Psi$ , the continuous latent states $x _ { 0 : T }$ and the discrete latent states $z _ { 1 : T }$ must be inferred from the data. Under the Bayesian framework, this implies computing the posterior,
+
+$$
+p \left( { { x } _ { 0 : T } } , { { z } _ { 0 : T } } , \Theta , \Psi , \Gamma \vert { { y } _ { 1 : T } } \right) = \frac { p \left( { { x } _ { 0 : T } } , { { z } _ { 1 : T } } , \Theta , \Psi , \Gamma , y _ { 1 : T } \right) } { p \left( { { y } _ { 1 : T } } \right) } .
+$$
+
+We perform fully Bayesian inference via Gibbs sampling (Brooks et al., 2011) to obtain samples from the posterior distribution described in eq. (18). To allow for fast and closed form conditional posteriors, we augment the model with Pólya-gamma auxiliary variables Polson et al. (2013).
+
+# 4.1 PÓLYA-GAMMA AUGMENTATION
+
+Consider a logistic regression from regressor $x _ { n } \in \mathbb { R } ^ { d _ { x } }$ to categorical distribution $z _ { n } \in \{ 0 , 1 \}$ ; the likelihood is
+
+$$
+p ( z _ { 1 : N } ) = \prod _ { n = 1 } ^ { N } \frac { \Big ( e ^ { x _ { n } ^ { T } \beta } \Big ) ^ { z _ { n } } } { 1 + e ^ { x _ { n } ^ { T } \beta } } .
+$$
+
+If a Gaussian prior is placed on $\beta$ then the model is non-conjugate and the posterior can’t be obtained in closed form. To circumvent this problem Polson et al. (2013) introduced a Pólya-Gamma (PG) augmentation scheme. This augmentation scheme is based on the following integral identity
+
+$$
+\frac { \left( e ^ { \psi } \right) ^ { a } } { \left( 1 + e ^ { \psi } \right) ^ { b } } = 2 ^ { - b } e ^ { \kappa \psi } \int _ { 0 } ^ { \infty } e ^ { - \frac { 1 } { 2 } \omega \psi ^ { 2 } } p ( \omega ) \mathrm { d } \omega
+$$
+
+where $\kappa = a - b / 2$ and $\omega \sim \mathrm { P G } ( b , 0 )$ . Setting $\psi = x ^ { T } \beta$ , it is evident that the integrand is a kernel for a Gaussian. Augmenting the model with PG axillary r.v.s $\{ \omega _ { n } \} _ { n = 1 } ^ { N }$ , eq. (19) can be expressed as
+
+$$
+p ( z _ { 1 : N } ) = \prod _ { n = 1 } ^ { N } \frac { \Big ( e ^ { x _ { n } ^ { T } \beta } \Big ) ^ { z _ { n } } } { 1 + e ^ { x _ { n } ^ { T } \beta } } \propto \prod _ { n = 1 } ^ { N } e ^ { \kappa _ { n } \psi _ { n } } \int _ { 0 } ^ { \infty } e ^ { - \frac { 1 } { 2 } \omega _ { n } \psi _ { n } ^ { 2 } } p ( \omega _ { n } ) \mathrm { d } \omega _ { n } = \prod _ { n = 1 } ^ { N } \mathbb { E } _ { \omega _ { n } } \big [ e ^ { - \frac { 1 } { 2 } ( \omega _ { n } \psi _ { n } ^ { 2 } - 2 \kappa _ { n } \psi _ { n } ) } \big ] .
+$$
+
+Conditioning on $\omega _ { n }$ , the posterior of $\beta$ is
+
+$$
+p ( \beta | \omega _ { 1 : N } , z _ { 1 : N } , x _ { 1 : N } ) \propto p ( \beta ) \prod _ { n = 1 } ^ { N } e ^ { - \frac { 1 } { 2 } \left( \omega _ { n } \psi _ { n } ^ { 2 } - 2 \kappa _ { n } \psi _ { n } \right) }
+$$
+
+where $\psi _ { n } = x _ { n } ^ { T } \beta$ and $\begin{array} { r } { \kappa _ { n } = z _ { n } - \frac { 1 } { 2 } } \end{array}$ . It can be shown that the conditional posterior of $\omega _ { n }$ is also PG where $\omega _ { n } | \beta , x _ { n } , z _ { n } \sim \mathrm { P G } ( 1 , \psi _ { n } )$ (Polson et al., 2013).
+
+# 4.2 CONDITIONAL POSTERIORS
+
+The structure of the model allows for closed form conditional posterior distributions that are easy to sample from. For clarity, the conditional posterior distributions for the TrSLDS are given below:
+
+1. The linear dynamic parameters $( A _ { k } , b _ { k } )$ and state variance $Q _ { k }$ of a leaf node $k$ are conjugate with a Matrix Normal Inverse Wishart (MNIW) prior
+
+$$
+p ( ( A _ { k } , b _ { k } ) , Q _ { k } | x _ { 0 : T } , z _ { 1 : T } ) \propto p ( ( A _ { k } , b _ { k } ) , Q _ { k } ) \prod _ { t = 1 } ^ { T } N ( x _ { t } | x _ { t - 1 } + A _ { z _ { t } } x _ { t - 1 } + b _ { z _ { t } } , Q _ { z _ { t } } ) ^ { \mathbb { 1 } [ z _ { t } = k ] } .
+$$
+
+2. The linear dynamic parameters of an internal node $n$ are conditionally Gaussian given a Gaussian prior on $\left( A _ { n } , b _ { n } \right)$
+
+$$
+p ( ( A _ { n } , b _ { n } ) | \Theta _ { - n } ) \propto p ( ( A _ { n } , b _ { n } ) | ( A _ { \mathrm { p a r } ( n ) } , b _ { \mathrm { p a r } ( n ) } ) ) \prod _ { j \in \coth \mathbb { 1 } \mathbb { d } ( n ) } p ( ( A _ { j } , b _ { j } ) | ( A _ { n } , b _ { n } ) ) .
+$$
+
+3. If we assume the observation model is linear and with additive white Gaussian noise then the emission parameters $\Psi = \{ ( C , d ) , S \}$ are also conjugate with a MNIW prior
+
+$$
+p ( ( C , d ) , S | x _ { 1 : T } , y _ { 1 : T } ) \propto p ( ( C , d ) , S ) \prod _ { t = 1 } ^ { T } \mathcal { N } ( y _ { t } | C x _ { t } + d , S ) .
+$$
+
+We can also handle Bernoulli observations through the use of Pólya-gamma augmentation.   
+In the interest of space, the details are explained in Section B.1 in the Appendix.
+
+4. The choice parameters are logistic regressions which follow from the conditional posterior
+
+$$
+p \left( \Gamma \middle | x _ { 0 : T } , z _ { 1 : T } \right) \propto p \left( \Gamma \right) \prod _ { t = 1 } ^ { T } p \left( z _ { t } \middle | x _ { t - 1 } , \Gamma \right) = p \left( \Gamma \right) \prod _ { t = 1 } ^ { T } \prod _ { n \in \mathrm { p a t h } \left( z _ { t } \right) \backslash z } \frac { \left( e ^ { \nu _ { n , t } } \right) ^ { \mathrm { 1 } \left( \mathrm { l e f t } \left( n \right) \in \mathrm { p a t h } \left( z _ { t } \right) \right) } } { 1 + e ^ { \nu _ { n , t } } } ,
+$$
+
+where $\nu _ { n , t } = R _ { n } ^ { T } x _ { t - 1 } + r _ { n }$ . The likelihood is of the same form as the left hand side of eq. (20), thus it is amenable to the PG augmentation. Let $\omega _ { n , t }$ be the auxiliary Pólya-gamma random variable introduced at time $t$ for an internal node $n$ . We can express the posterior over the hyperplane of an internal node $n$ as:
+
+$$
+p ( ( R _ { n } , r _ { n } ) | x _ { 0 : T } , z _ { 1 : T } , \omega _ { n , 1 : T } ) \propto p ( ( R _ { n } , r _ { n } ) ) \prod _ { t = 1 } ^ { T } \mathcal { N } ( \nu _ { n , t } | \kappa _ { n , t } / \omega _ { n , t } , 1 / \omega _ { n , t } ) ^ { 1 ( n \in \mathrm { p a t h } ( z _ { t } ) ) } ,
+$$
+
+where $\begin{array} { r } { \kappa _ { n , t } = \frac 1 2 \mathbb { 1 } [ j = \mathrm { l e f t } ( n ) ] - \frac 1 2 \mathbb { 1 } [ j = \mathrm { r i g h t } ( n ) ] } \end{array}$ , $j \in \mathrm { c h i l d } ( n )$ . Augmenting the model with Pólya-gamma random variables allows for the posterior to be conditionally Gaussian under a Gaussian prior.
+
+5. Conditioned on the discrete latent states, the continuous latent states are Gaussian. However, the presence of the tree-structured recurrence potentials $\psi ( x _ { t - 1 } , z _ { t } )$ introduced by eq. (10) destroys the Gaussinity of the conditional. When the model is augmented with PG random variables $\omega _ { n , t }$ , the augmented recurrence potential, $\psi ( x _ { t - 1 } , \boldsymbol { z } _ { t } , \omega _ { n , t } )$ , becomes effectively Gaussian, allowing for the use of message passing for efficient sampling. Linderman et al. (2017) shows how to perform message-passing using the Pólya-gamma augmented recurrence potentials $\psi ( x _ { t } , z _ { t } , w _ { n , t } )$ . In the interest of space, the details are explained in Section B.2 in the Appendix.
+
+6. The discrete latent variables $z _ { 1 : T }$ are conditionally independent given $x _ { 1 : T }$ thus
+
+$$
+p \left( \boldsymbol { z } _ { t } = k | \boldsymbol { x } _ { 1 : T } , \Theta , \Gamma \right) = \frac { p \left( x _ { t } | \boldsymbol { x } _ { t - 1 } , \theta _ { k } \right) p \left( \boldsymbol { z } _ { t } = k | \boldsymbol { x } _ { t - 1 } , \Gamma \right) } { \sum _ { l \in \mathrm { l e a v e s } ( T ) } p \left( \boldsymbol { x } _ { t } | \boldsymbol { x } _ { t - 1 } , \theta _ { l } \right) p \left( \boldsymbol { z } _ { t } = l | \boldsymbol { x } _ { t - 1 } , \Gamma \right) } , k \in \mathrm { l e a v e s } ( T ) .
+$$
+
+7. The conditional posterior of the Pólya-Gamma random variables are also Pólya-Gamma: $\omega _ { n , t } | z _ { t } , ( R _ { n } , r _ { n } ) , x _ { t - 1 } \sim \mathrm { P G } ( 1 , \nu _ { n , t } )$ .
+
+Due to the complexity of the model, good initialization is critical for the Gibbs sampler to converge to a mode in a reasonable number of iterations. Details of the initialization procedure are contained in Section C in the Appendix.
+
+# 5 EXPERIMENTS
+
+We demonstrate the potential of the proposed model by testing it on a number of non-linear dynamical systems. The first, FitzHugh-Nagumo, is a common nonlinear system utilized throughout neuroscience to describe an action potential. We show that the proposed method can offer different angles of the system. We also compare our model with other approaches and show that we can achieve state of the art performance. We then move on to the Lorenz attractor, a chaotic nonlinear dynamical system, and show that the proposed model can once again break down the dynamics and offer an interesting perspective. Finally, we apply the proposed method on the data from Graf et al. (2011).
+
+![](images/3c036b909ffe2d53f67b812be3bcd5e964c243cfab9e4743f08217a0662ee3a8.jpg)  
+Figure 2: TrSLDS applied to model the FitzHugh-Nagumo nonlinear oscillator. (a) The model was trained on 100 trajectories with random starting points. (b) The model can infer the latent trajectories. (c) The true vector field of FHN is shown where color of the arrow represents log-speed. The two nullclines are plotted in yellow and green. (d-f) The vector fields display the multi-scale view learned from the model where color of the arrows dictate log-speed The background color showcases the hierarchical partitioning learned by the model where the darker the color is, the higher the probability of ending up in that discrete state. As we go deeper in the tree, the resolution increases which is evident from the vector fields. (g) A deterministic trajectory from the leaf nodes (colored by most likely leaf node) with affine transformation onto a trajectory FHN (gray). (h) Plotting $w$ and $v$ over time, we see that the second level captures some of the oscillations but ultimately converges to a fixed point. The model learned by the leaf nodes captures the limit cycle accurately. (i) Performances compared for multi-step prediction. We see that TrSLDS outperforms rSLDS.
+
+# 5.1 FITZHUGH-NAGUMO
+
+The FitzHugh-Nagumo (FHN) model is a 2-dimensional reduction of the Hodgkin-Huxley model which is completely described by the following system of differential equations (Izhikevich, 2007):
+
+$$
+\dot { v } = v - \frac { v ^ { 3 } } { 3 } - w + I _ { e x t } , \qquad \tau \dot { w } = v + a - b w .
+$$
+
+We set the parameters to $a = 0 . 7$ , $b = 0 . 8$ , $\tau = 1 2 . 5$ , and $I _ { e x t } \sim \mathcal { N } ( 0 . 7 , 0 . 0 4 )$ . We trained our model with 100 trajectories where the starting points were sampled uniformly from $[ - 3 , 3 ] ^ { 2 }$ . Each of the trajectories consisted of 430 time points, where the last 30 time points of the trajectories were used for testing. The observation model is linear and Gaussian where $C = { \binom { 2 } { 0 } } \quad { \overset { 0 } { - } } { \overset { - } { 2 } } { \overset { - } { ) } } , d = [ 0 . 5 , 0 . 5$ ] and $S = 0 . 0 1 \mathbb { I } _ { 2 }$ where $\mathbb { I } _ { n }$ is an identity matrix of dimension n. We set the number of leaf nodes to be 4 and ran Gibbs for 1,000 samples; the last 50 samples were kept and we choose the sample that produced the highest log likelihood to produce Fig. 2 where the vector fields were produced using the mode of the conditional posteriors of the dynamics.
+
+To quantitatively measure the predictive power of TrSLDS, we compute the $k$ -step predictive mean squared error, $\mathbf { M S E } _ { k }$ , and its normalized version, $R _ { k } ^ { 2 }$ , on a test set where $\mathrm { M S E } _ { k }$ and $\bar { R } _ { k } ^ { 2 }$ are defined as
+
+$$
+\mathbf { M S E } _ { k } = \frac { 1 } { T - k } \sum _ { t = 0 } ^ { T - k } \left\| y _ { t + k } - \hat { y } _ { t + k } \right\| _ { 2 } ^ { 2 } , \qquad R _ { k } ^ { 2 } = 1 - \frac { ( T - k ) \mathbf { M S E } _ { k } } { \sum _ { t = 0 } ^ { T - k } \left\| y _ { t + k } - \bar { y } \right\| _ { 2 } ^ { 2 } } ,
+$$
+
+where $\bar { y }$ is the average of a trial and $\hat { y } _ { t + k }$ is the prediction at time $t + k$ which is obtained by (i) using the the samples produced by the sampler to obtain an estimate of $\hat { x } _ { T }$ given $y _ { 1 : T }$ , (ii) propagate $\hat { x } _ { T }$ for $k$ time steps forward to obtain $\hat { x } _ { t + k }$ and then (iii) obtain $\hat { y } _ { t + k }$ . We compare the model to LDS, SLDS and rSLDS for $k = 1 , \ldots , 3 0$ over the last 30 time steps for all 100 trajectories (Fig. 2I).
+
+# 5.2 LORENZ ATTRACTOR
+
+![](images/8ebda55e6b6b55c81c929fe7ad8029ddefad9375e1ce476a4bc4bbc2f246478d.jpg)  
+Figure 3: (a) The 50 trajectories used to train the model are plotted where the red "x" displays the starting point of the trajectory. (b) The inferred latent states are shown, colored by their discrete latent state. (c) We see that the second layer approximates the Lorenz attractor with 2 ellipsoids. A trajectory from the Lorenz attractor starting at the same initial point is shown for comparison. (d) Going one level lower in the tree, we see that in order to capture the nuances of the dynamics, each of the ellipsoids must be split in half. A trajectory from the Lorenz attractor is shown for comparison. (e) Plotting the dynamics, it is evident that the leaf nodes improve on it’s parent’s approximation. (f) The $R _ { k } ^ { 2 }$ demonstrates the predictive power of TrSLDS.
+
+Lorenz attractors are chaotic systems whose nonlinear dynamics are defined by,
+
+$$
+\begin{array} { r } { \dot { x _ { 1 } } = \sigma \left( x _ { 2 } - x _ { 1 } \right) , \quad \dot { x _ { 2 } } = x _ { 1 } ( \rho - x _ { 3 } ) - x _ { 2 } , \quad \dot { x _ { 3 } } = x _ { 1 } x _ { 2 } - \beta x _ { 3 } . } \end{array}
+$$
+
+The parameters were set to $\sigma = 1 0$ , $\rho = 2 8$ and $\beta = 8 / 3$ . The data consisted of 50 trajectories, each of length of 230 where the first 200 time points are used for training and the last 30 are used for testing. The observation model was a projection onto 10 dimensional space with Gaussian noise.We set the number of leaf nodes to be 4 and ran Gibbs for 1,000 samples; the last 50 samples were kept and we choose the sample that produced the highest log-likelihood to produce Fig. 3.
+
+The butterfly shape of the Lorenz attractor lends itself to being roughly approximated by two 2- dimensional ellipsoids; this is exactly what TrSLDS learns in the second level of the tree. As is evident from Fig. 5B, the two ellipsoids don’t capture the nuances of the dynamics. Thus, the model partitions each of the ellipsoids to obtain a finer description. We can see that embedding the system with a hierarchical tree-structured prior allows for the children to build off its parent’s approximations.
+
+# 5.3 NEURAL DATA
+
+To validate the model and inference procedure, we used the neural spike train data recorded from the primary visual cortex of an anesthetized macaque monkey collected by Graf et al. (2011). The dataset is composed of short trials where the monkey viewed periodic temporal pattern of motions of 72 orientations, each repeated 50 times. Dimensionality reduction of the dataset showed that for each orientation of the drifting grating stimulus, the neural response oscillates over time, but in a stimulus dependent geometry captured in 3-dimensions (Zhao & Park, 2017). We used 50 trials each from a subset of 4 stimulus orientations grouped in two (140 and 150 degrees vs. 230 and 240 degrees) where each trial contained 140 neurons. Out of the 140 neurons, we selected 63 well-tuned neurons. The spike trains were binarized with a $1 0 \mathrm { m s }$ window for Bernoulli observation model and we truncated the onset and offset neural responses, resulting in 111 time bins per trial.
+
+We fit TrSLDS with $K = 4$ leaf nodes and 3-dimensional continuous latent space; the sampler was run for 500 samples where the last sample was used to produce the results shown in Fig. 4. To obtain an initial estimate for $x _ { 0 : T }$ , we smoothed the spike trains using a Gaussian kernel and performed probabilistic PCA on the smoothed spike trains.
+
+From Fig. 4, it is evident that TrSLDS has learned a multi-scale view as expected. It is able to correctly distinguish between the two groups of orientations by assigning them to two different subtrees (green-yellow vs. red-orange). The leaf nodes of each subtree refines the periodic orbit further. From Fig. 4, we can see that TrSLDS also learns two limit cycles that are separated.
+
+![](images/7eb975bf6003ca2e20ee4e47c4dd18f8c8881b4ee1f18325322a7945c09ea31f.jpg)  
+Figure 4: Modeling primary visual cortex spike trains. (top) Example spike raster plots in response to a drifting grating of orientations 150 and 240 degrees. Our data consisted of 200 such trials. (bottom) The average inferred latent trajectories over time for orientations 140 and 150 degrees colored by the most likely discrete latent state. (right top) Same plotted in space. The model is able to separate the limit cycles for each orientation group (green-yellow vs. red-orange) and refine them further with the leaf nodes. (right bottom) Two model generated predictive trajectories showing two stable limit cycles that resemble the two periodic orbits.
+
+# 6 CONCLUSION
+
+In this paper, we propose tree-structured recurrent switching linear dynamical systems (TrSLDS) which is an extension of rSLDS (Linderman et al., 2017). The system relies on the use of treestructured stick-breaking to partition the space. The tree-structured stick-breaking paradigm naturally lends itself to imposing a hierarchical prior on the dynamics that respects the tree structure. This tree-structured prior allows for a multi-scale view of the system where one can query at different levels of the tree to see different scales of the resolution. We also developed a fully Bayesian sampler, which leverages the Pólya-Gamma augmentation, to learn the parameters of the model and infer latent states. The two synthetic experiments show that TrSLDS can recover a multi-scale view of the system, where the resolution of the system increase as we delve deeper into the tree. The analysis on the real neural data verifies that TrSLDS can find a multi-scale structure.
+
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+David Sussillo, Rafal Józefowicz, L. F Abbott, and Chethan Pandarinath. LFADS - Latent Factor Analysis via Dynamical Systems. CoRR, abs/1608.06315, 2016.
+
+Yuan Zhao and Il Memming Park. Interpretable nonlinear dynamic modeling of neural trajectories. In Advances in Neural Information Processing Systems (NIPS), 2016.
+
+Yuan Zhao and Il Memming Park. Variational Latent Gaussian Process for Recovering SingleTrial Dynamics from Population Spike Trains. Neural Computation, 29(5), May 2017. doi: 10.1162/NECO_a_00953.
+
+Yuan Zhao and Il Memming Park. Variational joint filtering. arXiv, abs/1707.09049, 2018.
+
+Onno Zoeter and Tom Heskes. Hierarchical visualization of time-series data using switching linear dynamical systems. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(10): 1202–1214, 2003.
+
+# A PROOF OF THEOREM 1
+
+Proof. Let $\tau$ be a balanced binary tree with $K$ leaf nodes. To show that the models are equal, it suffices to show the equivalence of the likelihood and the prior between models. For compactness, we drop the affine term and the $\mathrm { v e c } ( \cdot )$ operator. The likelihood of TrSLDS is
+
+$$
+p ( x _ { 1 : T } | \boldsymbol { z } _ { 1 : T } , \Theta ) = \prod _ { t = 1 } ^ { T } \mathcal { N } ( x _ { t } | x _ { t - 1 } + A _ { z _ { t } } x _ { t - 1 } , Q _ { z _ { t } } ) ,
+$$
+
+and the likelihood of the residual model is
+
+$$
+p ( x _ { 1 : T } | z _ { 1 : T } , \tilde { \Theta } ) = \prod _ { t = 1 } ^ { T } \mathcal { N } \left( x _ { t } | x _ { t - 1 } + \bar { A } _ { z _ { t } } x _ { t - 1 } , \tilde { Q } _ { z _ { t } } \right) .
+$$
+
+where $\bar { A } _ { z _ { t } }$ is defined in eq. (16). Substituting $\begin{array} { r } { A _ { z _ { t } } = \sum _ { j \in \mathrm { p a t h } ( z _ { t } ) } \tilde { A } _ { j } } \end{array}$ into eq. (27) equates the likelihoods. All that is left to do is to show the equality of the priors.
+
+We can express $\begin{array} { r } { A _ { n } = \sum _ { j \in \mathrm { p a t h } ( n ) } \tilde { A } _ { j } } \end{array}$ recursively
+
+$$
+A _ { n } = \tilde { A } _ { n } + A _ { \mathrm { p a r } ( n ) } .
+$$
+
+Plugging eq. (28) into $\ln p ( A _ { n } | A _ { \mathrm { p a r } ( n ) } )$
+
+$$
+\begin{array} { l } { \displaystyle \ln p \big ( A _ { n } \big | A _ { \mathrm { p a r } ( n ) } \big ) = - \frac { 1 } { 2 } \left( A _ { n } - A _ { \mathrm { p a r } ( n ) } \right) ^ { T } \Sigma _ { n } ^ { - 1 } \left( A _ { n } - A _ { \mathrm { p a r } ( n ) } \right) + \mathrm { C } } \\ { = - \frac { 1 } { 2 } \left( \tilde { A } _ { n } + A _ { \mathrm { p a r } ( n ) } - A _ { \mathrm { p a r } ( n ) } \right) ^ { T } \Sigma _ { n } ^ { - 1 } \left( \tilde { A } _ { n } + A _ { \mathrm { p a r } ( n ) } - A _ { \mathrm { p a r } ( n ) } \right) + \mathrm { C } } \\ { = - \frac { 1 } { 2 } \tilde { A } _ { n } ^ { T } \Sigma _ { n } ^ { - 1 } \tilde { A } _ { n } + \mathrm { C } } \\ { = - \frac { 1 } { 2 } \tilde { A } _ { n } ^ { T } \left( \lambda ^ { \mathrm { d e p t h } ( n ) } \Sigma _ { \epsilon } \right) ^ { - 1 } \tilde { A } _ { n } + \mathrm { C } } \end{array}
+$$
+
+where $\textrm { C }$ is a constant. Because $\Sigma _ { \epsilon } = \tilde { \Sigma } _ { \epsilon }$ and $\lambda = \widetilde { \lambda }$ , eq. (32) is equivalent to the kernel of $p ( { \tilde { A } } _ { n } )$ implying that the priors are equal. Since this is true $\forall n \in \mathcal { T }$ , the joint distributions of the two models are the same. □
+
+# B DETAILS ON BAYESIAN INFERENCE
+
+# B.1 HANDLING BERNOULLI OBSERVATIONS
+
+Suppose the observation of the system at time $t$ follows
+
+$$
+\begin{array} { c l c r } { { \displaystyle p ( \boldsymbol { y } _ { t } | \boldsymbol { x } _ { t } , \boldsymbol { \Psi } ) = \prod _ { n = 1 } ^ { N } \mathrm { B e r n } ( \boldsymbol { \sigma } ( \boldsymbol { v } _ { n , t } ) ) = \prod _ { n = 1 } ^ { N } \frac { ( e ^ { \boldsymbol { v } _ { n , t } } ) ^ { \boldsymbol { y } _ { n , t } } } { 1 + e ^ { \boldsymbol { v } _ { n , t } } } , } } \\ { { \boldsymbol { v } _ { n , t } = c _ { n } ^ { T } \boldsymbol { x } _ { t } + d _ { n } , } } \end{array}
+$$
+
+where $c _ { n } \in \mathbb { R } ^ { d _ { x } }$ , $d _ { n } \in \mathbb { R }$ . Equation 33 is of the same form as the left hand side of eq. (20), thus it is amenable to PG augmentation. We introduce PG axillary variables $\eta _ { n , t }$ . Conditioning on $\eta _ { 1 : N }$ eq. (33) becomes
+
+$$
+\begin{array} { l } { \displaystyle p ( y _ { t } | x _ { t } , \eta _ { 1 : N } ) = \prod _ { n = 1 } ^ { N } e ^ { - \frac { 1 } { 2 } ( \eta _ { n , t } v _ { n , t } - 2 \kappa _ { n , t } v _ { n , t } ) } } \\ { \displaystyle \propto \prod _ { n = 1 } ^ { N } \mathcal { N } ( c _ { n } ^ { T } x _ { t } + d _ { n } | \kappa _ { n , t } / \eta _ { n , t } , 1 / \eta _ { n , t } ) } \\ { \displaystyle = \mathcal { N } ( C x _ { t } + D | H _ { t } ^ { - 1 } \kappa _ { t } , H _ { t } ^ { - 1 } ) } \end{array}
+$$
+
+where $H _ { t } = \operatorname { d i a g } ( [ \eta _ { 1 , t } , \dotsc , \eta _ { N , t } ] )$ , $\kappa _ { t } = [ \kappa _ { 1 , t } , \ldots , \kappa _ { N , t } ]$ and $\begin{array} { r } { \kappa _ { n , t } = y _ { n , t } - \frac { 1 } { 2 } } \end{array}$
+
+The observation is now effectively Gaussian and can be incorporated into the message passing for $x _ { 1 : T }$ . The emission parameters are also conjugate with the augmented observation potential given a Matrix Normal prior. The conditional posterior on the axillary PG variables $\eta _ { n , t }$ also follows a PG distribution i.e. $\eta _ { n , t } \big | ( c _ { n } , d _ { n } ) , x _ { t } \sim \mathrm { P G } ( 1 , \overline { { \upsilon } } _ { n , t } )$ . Note that this augmentation scheme can also work for negative binomial, binomial, and multinomial observations (Polson et al., 2013; Linderman et al., 2015).
+
+# B.2 MESSAGE PASSING FOR $x _ { 1 : T }$
+
+Assuming that the observations, $y _ { 1 : T }$ , are linear and Gaussian, the posterior of the continuous latent states, $x _ { 0 : T }$ , conditioned on all the other variables is proportional to
+
+$$
+\prod _ { t = 1 } ^ { T } \psi ( x _ { t } , x _ { t - 1 } , z _ { t } ) \psi ( z _ { t } , x _ { t - 1 } ) \psi ( x _ { t } , y _ { t } )
+$$
+
+where $\psi ( x _ { t } , x _ { t - 1 } , z _ { t } )$ is the potential of the conditionally linear dynamics, $\psi ( x _ { t } , y _ { t } )$ is the potential of the observation and $\psi ( x _ { t - 1 } , z _ { t } )$ is the recurrence potential. $\psi ( x _ { t - 1 } , z _ { t } )$ is a product of all the internal nodes traversed at time $t$
+
+$$
+\psi ( x _ { t - 1 } , z _ { t } ) = \prod _ { n \in \mathrm { p a t h } ( z _ { t } ) \backslash \mathcal { Z } } \psi _ { n } ( x _ { t - 1 } , z _ { t } ) .
+$$
+
+If the potentials in eq. (38) were all linear and Gaussian, then we could efficiently sample from the posetrior of $x _ { 0 : T }$ by passing messsages forward through Kalman Filtering and then sampling backwards; the prescence of the recurrence potentials prevent this because they aren’t Gaussian. By augmenting the model with the PG r.v.’s, the recurrence potential at internal node $n$ becomes
+
+$$
+\psi _ { n } ( x _ { t - 1 } , z _ { t } , w _ { n , t - 1 } ) = \mathcal { N } ( R _ { n } ^ { T } x _ { t - 1 } + r _ { n } | \kappa _ { n , t - 1 } / \omega _ { n , t - 1 } , 1 / \omega _ { n , t - 1 } )
+$$
+
+which is effectively Gaussian , allowing for the use of the Kalman filter for message passing.
+
+# C INITIALIZATION
+
+We initialized the Gibbs sampler using the following initialization procedure: (i) probabilistic PCA was performed on the data, $y _ { 1 : T }$ to initialize the emission parameters, $\{ C , d \}$ and the continuous latent states, $x _ { 1 : T }$ . (ii) To initialize the dynamics of the nodes , $\Theta$ , and the hyperplanes, $\Gamma$ , we propose greedily fitting the proposed model using MSE as the loss function. We first optimize over the root node
+
+$$
+\underset { A _ { \epsilon } , b _ { \epsilon } } { \arg \operatorname* { m i n } } \frac { 1 } { T } \sum _ { t = 0 } ^ { T } \left\| x _ { t + 1 } - x _ { t } - A _ { \epsilon } x _ { t } - b _ { \epsilon } \right\| _ { 2 } ^ { 2 } ,
+$$
+
+and obtain $A _ { \epsilon } ^ { * } , b _ { \epsilon } ^ { * }$ (Note that $A _ { \epsilon } ^ { * } , b _ { \epsilon } ^ { * }$ can obtained in closed form by computing their corresponding OLS estimates). Fixing $A _ { \epsilon } ^ { * }$ and $b _ { \epsilon } ^ { * }$ , we then optimize over the second level in the tree
+
+$$
+\begin{array} { c } { \displaystyle \operatorname * { a r g m i n } _ { A _ { 1 } , b _ { 1 } , A _ { 2 } , b _ { 2 } , R _ { \epsilon } , r _ { \epsilon } } \frac { 1 } { T } \sum _ { t = 0 } ^ { T } \| x _ { t + 1 } - \sigma ( v _ { \epsilon } ) \hat { x } _ { 1 } - \sigma ( - v _ { \epsilon } ) \hat { x } _ { 2 } \| , } \\ { \hat { x } _ { i } = x _ { t } + \left( A _ { \epsilon } ^ { * } + A _ { i } \right) x _ { t } + ( b _ { \epsilon } ^ { * } + b _ { i } ) , } \\ { v _ { \epsilon } = R _ { \epsilon } ^ { T } x _ { t } + r _ { \epsilon } . } \end{array}
+$$
+
+This procedure would continue until we reach the leaf nodes of the tree. $\Theta ^ { * }$ and $\Gamma ^ { * }$ are then used to initialize the dynamics and the hyperplanes, respectively. In our simulations, we used stochastic gradient descent with momentum to perform the optimization. (iii) The discrete latent states, $z _ { 1 : T }$ , were initialized by performing hard classification using $\Gamma ^ { * }$ and the initial estimate of $x _ { 0 : T }$ .
+
+# D DEALING WITH ROTATIONAL INVARIANCE
+
+A well known problem with these types of model is it’s susceptibility to rotational and scaling transformation, thus we can only learn the dynamics up to an affine transformation Erosheva & Curtis (2017). During Gibbs sampling the parameters will continuously rotate and scale, which can slow down the mixing of the chains. One possible solution to the issue is if we constrained $C$ to have some special structure which would make the model identifiable; this would require sampling from manifolds which is usually inefficient. Similar to Geweke & Zhou (1996), we use the following procedure to prevent the samples from continuously rotating and scaling:
+
+• Once we obtain a sample from the conditional posterior of the emission parameters $\{ C , D \}$ , we normalize the columns of $C$ .   
+• RQ decomposition is performed on $C$ to obtain $U , O$ where $U \in \mathcal { R } ^ { d _ { y } \times d _ { x } }$ is an upper triangular matrix and $\mathcal { \dot { O } } \in \mathcal { R } ^ { d _ { x } \times d _ { x } }$ is an orthogonal matrix.   
+• We set $C = U$ and rotate all the parameters of the model using $O$ .
+
+![](images/db5128ea32c1ed553d4f2cad718af7d080559ae0e3e1cda443470d8a3e80d0ea.jpg)  
+E SCALABILITY AND COMPUTATIONAL COMPLEXITY OF THE INFERENCE   
+Figure 5: The logarithm of the joint density was computed for all the samples generated from the 3 TrSLDS and smoothed using a trailing moving average filter. The sampler seems to converge to a mode rather quickly for all the three instantiations of the TrSLDS.
+
+The rSLDS and the TrSLDS share the same linear time complexity for sampling the discrete and continuous states, and both models learn K-1 hyperplanes to weakly partition the space. Specifically, both models incur: an $\mathcal { O } ( T K )$ cost for sampling the discrete states, which increases to $\scriptstyle { \dot { \mathcal { O } } } ( T K ^ { 2 } )$ if we allow Markovian dependencies between discrete states; an $\mathcal { O } ( T D ^ { 3 } )$ cost ( $\mathrm { D }$ is the continuous state dimension) for sampling the continuous states, just like in a linear dynamical system; and $\Im { \mathcal { O } ( K D ^ { 3 } ) }$ cost for sampling the hyperplanes. The only additional cost of the TrSLDS stems from the hierarchical prior on state dynamics. Unlike the rSLDS, we impose a tree-structured prior on the dynamics to encourage similar dynamics between nearby nodes in the tree. Rather than sampling K dynamics parameters, we need to sample 2K-1. Since they are all related via a tree-structured Gaussian graphical model, the cost of an exact sample is $\mathcal { O } ( K \bar { D } ^ { 3 } )$ just as in the rSLDS, with the only difference being a constant factor of about 2. Thus, we obtain a multi-scale view of the underlying system with a negligible effect on the computational complexity.
+
+To see how the number of discrete latent states effects the convergence speed of the Gibbs sampler, we fit 3 TrSLDS, with $K = 2 , 4 , 8$ respectively, to a Lorenz Attractor described in Sec. but used 250 trajectories to train the model as opposed to 50. To assess convergence, we plotted the logarithm of the joint density as a function of Gibbs samples. The results are shown Fig. 5.
+
+# F SYNTHETIC NASCAR
+
+We ran the TrSLDS on the synthetic NASCAR
+
+![](images/cfeeba6d61bd2d40ab17765d02ae14fe7b28e6fa7e68c8565508e466a951e7f3.jpg)  
+Figure 6: TrSLDS applied to the synthetic NASCAR
+
+# G TREE SYNTHETIC NASCAR
+
+To check whether the sampler is mixing adequately, we test TrSLDS on a twist on the synthetic NASCAR
+
+![](images/4b226de744df071e054ce184313783f598a4f41d6427f6a788d7c3e8e822312c.jpg)  
+Figure 7: TrSLDS and rSLDS applied to the tree version of to the synthetic NASCAR

parse/train/S1eYKlrYvr/S1eYKlrYvr.md

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+# DIAGNOSING THE ENVIRONMENT BIAS IN VISION-AND-LANGUAGE NAVIGATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Vision-and-Language Navigation (VLN) requires an agent to follow naturallanguage instructions, explore the given environments, and reach the desired target locations. These step-by-step navigational instructions are extremely useful in navigating new environments that the agent does not know about previously. Most recent works that study VLN observe a significant performance drop when tested on unseen environments (i.e., environments not used in training), indicating that the neural agent models are highly biased towards training environments. Although this issue is considered as one of the major challenges in VLN research, it is still under-studied and needs a clearer explanation. In this work, we design novel diagnosis experiments via environment re-splitting and feature replacement, looking into possible reasons for this environment bias. We observe that neither the language nor the underlying navigational graph, but the low-level visual appearance conveyed by ResNet features directly affects the agent model and contributes to this environment bias in results. According to this observation, we explore several kinds of semantic representations which contain less low-level visual information, hence the agent learned with these features could be better generalized to unseen testing environments. Without modifying the baseline agent model and its training method, our explored semantic features significantly decrease the performance gap between seen and unseen on multiple datasets (i.e., $8 . 6 \%$ to $0 . 2 \%$ on R2R, $2 3 . 9 \%$ to $0 . 1 \%$ on R4R, and 3.74 to 0.17 on CVDN) and achieve competitive unseen results to previous state-of-the-art models.
+
+# 1 INTRODUCTION
+
+Vision-and-Language Navigation (VLN) tests an agent’s ability to follow complex natural language instructions as well as explore the given environments, so as to be able to reach the desired target locations. As shown in Fig. 1, the agent is put in an environment and given a detailed step-by-step navigational instruction. With these inputs, the agent needs to navigate the environment and find the correct path to the target location. In this work, we focus on the instruction-guided navigation (MacMahon et al., 2006; Anderson et al., 2018b; Misra et al., 2018; Blukis et al., 2018; Chen et al., 2019c) where detailed step-by-step navigational instructions are used (e.g., ‘Go outside the dining room and turn left ...’), in contrast to the target-oriented navigation (Gordon et al., 2018; Das et al., 2018; Mirowski et al., 2018; Yu et al., 2019) where only the target is referred (e.g., ‘Go to the kitchen’ or ‘Tell me the color of the bedroom’). Although these step-by-step instructions are overdetailed when navigating local areas (e.g., your home), they are actively used in unseen environments (e.g., your friend’s house, a new city) where the desired target is usually unknown to navigational agents. For this purpose, testing on unseen environments which are not used during agent-training is important and widely accepted by instruction-guided navigation datasets.
+
+Recent works propose different methods to improve generalizability of agents on these unseen testing environments; and most of the existing works (Anderson et al., 2018b; Wang et al., 2018b; Fried et al., 2018; Wang et al., 2019b; Ma et al., 2019a;b; Tan et al., 2019; Huang et al., 2019; Hu et al., 2019) observe a significant performance drop from seen environments (i.e., the environments used in training) to unseen environments (i.e., the environments not used in training), which indicates a strong bias in the model towards the training environments. While this performance gap is emphasized as one of the major challenges in current VLN research, the issue is still left unresolved and waits for an explicit explanation. Thus, in this paper, we aim to answer three questions to this environment bias: 1. Where (i.e., in which component) is the bias located? 2. Why does this bias exist? 3. How to eliminate this bias?
+
+![](images/0f93e130663742430e83365ba4e9f0ff9e6567b8ab35e7d864d7ce32ccfd97a3.jpg)  
+Figure 1: Vision-language-navigation: performance of the agent drops in unseen environments.
+
+To locate where the bias is, we start by showing that natural-language navigational instructions and underlying navigational graphs are not direct reasons for this performance gap. We then investigate the effect of environments on the agent’s performance. In order to conduct a detailed analysis, we resplit the environment and categorize the validation data into three sets based on their visibility to the training set: path-seen data intersecting with the training paths, path-unseen data using the training environments but away from the training paths, and env-unseen data using unseen environments (environments not used in training). By showing that the results gradually decrease from path-seen data to env-unseen data, we characterize the environment bias at three levels: path level, region level, and environment level.
+
+These three levels of environment biases indicate strong ‘spatial localities’ in the tasks of VLN, which are intuitively reasonable because environments and regions (e.g., houses and cities) usually have their own styles when built or decorated. We next want to analyze the detailed reason why this locality would further lead to a gap in seen versus unseen results. Our hypothesis is that the low-level information carried by the ResNet features (He et al., 2016) is the reason. To keep minimal low-level visual information and promote more high-level semantic information, we replace the ResNet features with the 1000 ImageNet classification probabilities. Although the semantic information encoded by these features is not accurate because of the shifted domain of images and labels, the same model with ImageNet-Labels features performs surprisingly well on various VLN datasets (i.e., Room-to-Room, R4R, and $\mathrm { C V D N ^ { 1 } }$ ). Most importantly, these noisy semantic features effectively eliminate the performance gap between seen and unseen environments, which suggests that the environment bias is attributed to the ResNet features as our hypothesis.
+
+Following the practice in using ImageNet labels as semantic features, we further provide a discussion on how the environment bias could be eliminated. For this, we employ advanced high-level semantic features which are more rational for the VLN domain. We explore three kinds of semantic features: (1) areas of detected object labels (Ren et al., 2015); (2) ground truth semantic views (Chang et al., 2017); and (3) learned semantic view features. We show that all of these semantic features significantly reduce the environment bias in multiple datasets and also achieve strong results in testing unseen environments. We hope this work encourages more investigation and research into improving the generalization of vision-language models to unseen real-world scenarios.
+
+# 2 RELATED WORK
+
+Vision-and-Language Navigation: Vision-and-language navigation is an emerging task in the vision-and-language area. A lot of datasets have been proposed in recent years, such as Roomto-Room (Anderson et al., 2018b), Room-for-Room (Jain et al., 2019), TouchDown (Chen et al., 2019c), CVDN (Thomason et al., 2019b), RERERE (Qi et al., 2019), House3D (Wu et al., 2018) and EQA (Das et al., 2018). Recent works (Thomason et al., 2019a; Wang et al., 2018b; Fried et al.,
+
+2018; Wang et al., 2019b; Ma et al., 2019a;b; Tan et al., 2019; Hu et al., 2019; Ke et al., 2019; Anderson et al., 2019) focusing on improving the performance of navigation models, especially in unseen testing environments, have helped to increase the navigational success rate.
+
+Domain Adaptation: The general setup of domain adaption contains two sets of data samples $\{ x _ { i } \} _ { x _ { i } \in X }$ and $\{ y _ { i } \} _ { y _ { i } \in Y }$ from two domains $X$ and $Y$ . Based on these samples, we could learn domain invariant feature with adversarial training (Goodfellow et al., 2014; Zhu et al., 2017; Long et al., 2018; Wang et al., 2019a; Hosseini-Asl et al., 2019; Zhang et al., 2019; Gong et al., 2019; Chen et al., 2019b) or learn a transfer function $f : X \to Y$ (Wang et al., 2018a; Chen et al., 2019a; Rozantsev et al., 2018). However, samples from the target domain may not be available (e.g., the testing environments in navigation should not be used in training) in applications. Thus, we try to give an interpretable explanation to why performance varies in different domains and design a robust feature for it without deliberately considering the target domain. Two methods in VLN, RCM (Wang et al., 2019b) and EnvDrop (Tan et al., 2019), explore the possibility of domain adaptation. Both works take the testing environments in training while RCM also uses testing instructions.
+
+Domain Generalization: In domain generalization (Blanchard et al., 2011), the goal is to predict the labels in the previous unseen domain. Similar to the test setting of VLN tasks, the testing data is unrevealed in training. Works have been proposed to learn the common features of the training domain (Muandet et al., 2013; Blanchard et al., 2017; Li et al., 2017; 2018; Carlucci et al., 2019; Deshmukh et al., 2019). In this paper, we focus on the domain generalization problem in VLN task, and try to find the reasons for the failures.
+
+# 3 VISION-AND-LANGUAGE NAVIGATION AND ITS ENVIRONMENT BIAS
+
+We first introduce the task of vision-and-language navigation (VLN) and briefly describe the neural agent models used in our work. We next survey previous works on multiple indoor navigation datasets to show that the environment bias is widely observed in current VLN research. Lastly, we claim that this bias also exists in the outdoor navigation tasks, if the agent is tested on unseen regions.
+
+# 3.1 VISION-AND-LANGUAGE NAVIGATION
+
+Tasks: As shown in Fig. 1, the goal of the VLN task is to train an agent to navigate a certain type of environments $\{ { \bf E } \}$ (e.g., indoor or outdoor environments) given the instruction I. Each environment $\mathbf { E }$ is an independent space, such as a room or a house, and consists of a set of viewpoints. Each viewpoint is represented as a panoramic image and can be decomposed into separate views $\{ o \}$ as inputs to the neural agent models. The viewpoints and their connectivity form the navigational graph. In practice, after being placed at a particular viewpoint and given the instruction in the beginning, at each time step, the agent can observe the panoramic image of the viewpoint where it is located, and choose to move along an edge of the graph to the next node (i.e., viewpoint) or stop. This navigational process produces a path (i.e., a list of viewpoints), and the performance of the agent is evaluated by whether it reaches the target location that the instruction indicates in the end.
+
+Neural Agent Models: Most instruction-guided navigational agents are built based on attentive encoder-decoder models (Bahdanau et al., 2015). The encoder reads the instructions while the decoder outputs actions based on the encoded instructions and perceived environments. Since the main purpose of this work is to understand the environment bias in vision-and-language navigation, we use a minimal representative neural agent model that achieves comparable results to previous works. Specifically, we adopt the panoramic-view neural agent model in Fried et al. (2018) (‘Follower’) with modifications from Tan et al. (2019) as our baseline model. We also exclude advanced training techniques (i.e., reinforcement learning and data augmentation) and only train the agent with imitation learning in all our experiments for the same purpose. More details in original papers.
+
+# 3.2 ENVIRONMENT BIAS IN INDOOR NAVIGATION
+
+In order to evaluate the generalizability of agent models, indoor vision-and-language navigation datasets (e.g., those collected from Matterport3D (Chang et al., 2017)) use disjoint sets of environments in training and testing. Most of the datasets provide two validation splits to verify the agent’s performance in both sets of environments: validation seen, which takes the data from training environments, and validation unseen, whose data is from new environments apart from the training environments.
+
+Table 1: Results show the performance gap between seen (‘Val Seen’) and unseen (‘Val Unseen’) environments in several VLN tasks. Room-to-Room and Room-for-Room are evaluated with ‘Success Rate’, CVDN is evaluated with ‘Goal Progress’, Touchdown is evaluated with ‘Task Completion’.   
+
+<table><tr><td rowspan="2">Task</td><td rowspan="2">Method</td><td colspan="3">Result</td></tr><tr><td>Val Seen</td><td>Val Unseen</td><td>Abs Gap |△|</td></tr><tr><td rowspan="10">Room-to-Room (Anderson et al.,2018b)</td><td>R2R (Anderson et al.,2018b)</td><td>38.6</td><td>21.8</td><td>16.8</td></tr><tr><td>RPA (Wang et al., 2018b)</td><td>42.9</td><td>24.6</td><td>18.3</td></tr><tr><td>S-Follower (Fried et al., 2018)</td><td>66.4</td><td>35.5</td><td>30.9</td></tr><tr><td>RCM(Wang et al.,2019b)</td><td>66.7</td><td>42.8</td><td>23.9</td></tr><tr><td>SMNA (Ma et al., 2019a)</td><td>67</td><td>45</td><td>22</td></tr><tr><td>Regretful (Ma et al.,2019b)</td><td>69</td><td>50</td><td>19</td></tr><tr><td>EnvDrop (Tan et al., 2019)</td><td>62.1</td><td>52.2</td><td>9.9</td></tr><tr><td>ALTR (Huang et al.,2019)</td><td>55.8</td><td>46.1</td><td>9.7</td></tr><tr><td>RN+Obj (Hu et al., 2019)</td><td>59.2</td><td>39.5</td><td>19.7</td></tr><tr><td>CG (Anderson et al., 2019) Our baseline</td><td>31</td><td>31</td><td>0</td></tr><tr><td>Our learned-semantic</td><td></td><td>56.1 53.1</td><td>47.5</td><td>8.6</td></tr><tr><td rowspan="4">Room-for-Room (Jain et al., 2019)</td><td></td><td></td><td>53.3</td><td>0.2</td></tr><tr><td>Speaker-Follower</td><td>51.9</td><td>23.8</td><td>28.1</td></tr><tr><td>RCM</td><td>55.5</td><td>28.6</td><td>26.9</td></tr><tr><td>Our baseline Our learned-semantic</td><td>54.6 36.2</td><td>30.7 36.1</td><td>23.9</td></tr><tr><td rowspan="3">CVDN (Thomason et al., 2019b)</td><td>NDH</td><td>5.92</td><td>2.10</td><td>0.1</td></tr><tr><td>Our baseline</td><td>5.97</td><td>2.23</td><td>3.82 3.74</td></tr><tr><td>Our learned-semantic</td><td>2.60</td><td>2.43</td><td>0.17</td></tr><tr><td rowspan="4">Touchdown (Chen et al., 2019c)</td><td></td><td>7.9 (dev)</td><td></td><td></td></tr><tr><td>GA (original split) RCONCAT (original split)</td><td>9.8 (dev)</td><td>5.5 (test) 10.7 (test)</td><td>1</td></tr><tr><td>Our baseline (original split)</td><td>15.0 (dev)</td><td>14.2 (test)</td><td>1</td></tr><tr><td>Our baseline (seen/unseen split)</td><td>17.5</td><td>5.3</td><td>1 12.2</td></tr></table>
+
+In the first part of Table 1, we list most of the previous works on the Room-to-Room dataset (Anderson et al., 2018b) and report the success rate under greedy decoding (i.e., without beam-search) on validation seen and validation unseen splits. The large absolute gaps (from $3 0 . 9 \%$ to $9 . 7 \%$ ) between the results of seen and unseen environments show that current neural agent models on R2R suffer from environment bias2. Besides Room-to-Room (R2R), we also analyze two newly-released indoor navigation datasets that were also collected from Matterport3D environments: Room-forRoom (R4R) (Jain et al., 2019) and Cooperative Vision-and-Dialog Navigation (CVDN) (Thomason et al., 2019b). As shown in the second and third parts of Table. 1, results drop significantly from seen to unseen environments (i.e., $2 6 . 9 \%$ on R4R and 3.74 on CVDN), indicating that agent models also suffer from the environment bias in these datasets. Lastly, we show the results (denoted as ‘ours’ in Table. 1) when the environment bias (reason analyzed in Sec. 5) is effectively eliminated by our learned semantic features (described in Sec. 6.3). As a result, the performance gaps are effectively decreased on all three datasets without changing the model and learning hyper-parameters, compared to our baselines (denoted as ‘Our baseline’) and previous works 3.
+
+# 3.3 ENVIRONMENT BIAS IN OUTDOOR NAVIGATION
+
+Since the three indoor navigational datasets in previous sections are collected from the Matterport3D environments (Chang et al., 2017), in order to show that the environment bias is a general phenomenon also existing in other kinds of environments, we investigate the outdoor navigation task from Touchdown dataset (Chen et al., 2019c), whose environments are taken from New York City. In the original data splits of Touchdown, the environment is not specifically divided into seen and unseen and only involved one city. Thus the trained agent is only tested on the training environments (similar to validation seen split). To reveal the environment bias in Touchdown dataset, we split the city environment according to latitude and create two sub-environments: ‘training’ and ‘unseen’. The data are then re-split into training, val-seen, and val-unseen, accordingly. We adapt our baseline R2R agent model with additional convolutional layers to fit this new task. As shown in the last part of Table. 1, when experimenting on the original data split, our baseline model achieves state-of-theart results on the original ‘dev’ set and ‘test’ set, proving the validity of our model in this dataset. However, the results on our re-split data (denoted as ‘Our baseline (seen/unseen split)’) still show a big drop from the ’training’ to the ’unseen’ sub-environment (from $1 7 . 5 \%$ to $5 . 3 \%$ ), indicating that environment bias is a broad issue.
+
+![](images/1182dd37143299aa3930ca49723be9c5056035f01d331ca93c462899b4d835d5.jpg)  
+Figure 2: The language ’distance’ distribution (defined by language scores) and its relationship to success rate.
+
+# 4 WHERE: THE EFFECT OF DIFFERENT TASK COMPONENTS
+
+In Sec. 3, we showed that current neural agent models are biased towards the training environments on multiple vision-and-language navigation (VLN) datasets. In this section, our goal is to locate the component of VLN tasks which this environment bias is attributed to. As one of the early-released and well-explored datasets of VLN, Room-to-Room (R2R) dataset (Anderson et al., 2018b) is used as the diagnosing dataset in the experiments. We start by showing that two possible candidates, the natural language instructions and the underlying navigational graph, do not directly contribute to the environment bias. Then the effect of visual environments is analyzed in detail.
+
+# 4.1 THE EFFECT OF NATURAL-LANGUAGE NAVIGATIONAL INSTRUCTIONS
+
+A common hypothesis is that the navigational instructions for unseen environments (e.g., val unseen) are much different from the training environments (i.e., training and val seen) due to the different objects and layouts in new environments; and this lingual difference thus leads to the performance gap. In this section, we analyze the distributions of success rate with regard to the relationship between validation data’s instructions and training instructions. In order to quantitatively evaluate this relationship, we define the ‘distances’ from a validating instruction to all training instructions as the phrase-matching metric. Suppose $x$ is a validating datum, $\mathbb { T }$ is the training set, and $\operatorname { i n s t } ( x )$ is the instruction of the datum $x$ , we use ROUGE-L (Lin, 2004) and BLEU-4 (Papineni et al., 2002) to
+
+![](images/2d21e90dbdc29d400646db803bfb2541045038f1ecb3bac90bbe6312a9ba0d9e.jpg)  
+Figure 3: Graph split: left is original data and right is re-splitting data. Black vertices are viewpoints visited during training; red paths are val seen $/$ val path-seen; blue paths are val path-unseen.
+
+calculate this ‘distance’:
+
+$$
+\operatorname { d i s } _ { \mathrm { R o U G E } } ( x , \mathbb { T } ) = \operatorname* { m i n } _ { t \in \mathbb { T } } \mathrm { R O U G E - L } \left( \operatorname { i n s t } ( x ) , \operatorname { i n s t } ( t ) \right)
+$$
+
+$$
+\mathrm { d i s } _ { \mathrm { B L E U } } ( x , \mathbb { T } ) = \mathrm { B L E U } \ – 4 \left( \mathrm { i n s t } ( x ) , \{ \mathrm { i n s t } ( t ) \} _ { t \in \mathbb { T } } \right)
+$$
+
+where we consider all the training instructions as references in calculating the BLEU-4 score.
+
+We show the distributions of success rates and distances in Fig. 2. As opposed to the hypothesis, we do not observe a significant difference between the distributions of ‘distances’ (as shown in Fig. 2 (a, b)) on seen validation and unseen validation. For the success rate distributions (in Fig. 2(c,d)), the performance is better on instructions with smaller ‘distances’ (i.e., higher BLEU-4/ROUGE-L scores w.r.t. the training instructions) on both validation splits. However, comparing two splits, with the same ‘distance’ to training instructions, seen validation still significantly outperforms the unseen validation set on success rate, which implies the existence of other reasons rather than language attributed to this performance gap.
+
+# 4.2 THE EFFECT OF UNDERLYING NAVIGATIONAL GRAPH
+
+As shown in Fig. 3, an environment could be considered as its underlying navigational graph with visual information (as in Fig. 1). In order to test whether the agent model could overfit to these navigational graphs (and thus be biased towards training environments), we follow the experiments in Hu et al. (2019) to train the agent without visual information. Specifically, we mask out the ResNet features with zero vectors thus the agent could only make the decision based on the instructions and the navigational graph. With our baseline model, the success rate is $3 8 . 5 \%$ on validation seen and $4 1 . 0 \%$ on validation unseen in this setting, which is consistent with the finding in Hu et al. (2019). Besides showing the relatively good performance of unseen split without visual contents (similar to Thomason et al. (2019a) and Hu et al. (2019)), we also want to emphasize the low performance gap between seen and unseen environments $2 . 5 \%$ compared to the $\bar { > } 1 0 \%$ gap in usual). Hence, we claim that the underlying graph is not a dominant reason for the environment bias.
+
+# 4.3 THE EFFECT OF VISUAL ENVIRONMENTS
+
+To show how the visual environments affect the agent’s performance, we analyze the results on unseen environments and in different spatial regions of the training environments. In order to give a detailed characterization of the effect of environments, we are going to reveal the spatial localities which are related to the agent’s performance at three different levels:
+
+• Path-level Locality: Agents are better at paths which intersect with the training paths. • Region-level Locality: Agents are better in regions which are closer to the training data. • Environment-level Locality: Agents perform better on training environments than on unseen environments.
+
+Table 2: Results on our re-splitting data showing the path-level and environment-level localities.   
+
+<table><tr><td colspan="2">Splitting Method</td><td rowspan="2">Train</td><td colspan="3">Validation</td></tr><tr><td></td><td></td><td>Path-seen</td><td>Path-unseen</td><td>Env-unseen</td></tr><tr><td rowspan="3">Environments</td><td>R2R</td><td>61</td><td>56</td><td>0</td><td>11</td></tr><tr><td>X-split</td><td>61</td><td>57</td><td>16</td><td>11</td></tr><tr><td>Z-split</td><td>61</td><td>56</td><td>29</td><td>11</td></tr><tr><td rowspan="3">Number of Data</td><td>R2R</td><td>14,025</td><td>1,020</td><td>0</td><td>2,349</td></tr><tr><td>X-split</td><td>11,631</td><td>1,230</td><td>1,098</td><td>2,349</td></tr><tr><td>Z-split</td><td>10,894</td><td>867</td><td>2,324</td><td>2.349</td></tr><tr><td rowspan="3">Success Rate</td><td>R2R</td><td>88.3</td><td>56.1</td><td>1</td><td>47.5</td></tr><tr><td>X-split</td><td>87.3</td><td>58.9</td><td>52.6</td><td>46.7</td></tr><tr><td>Z-split</td><td>94.7</td><td>62.5</td><td>47.8</td><td>42.4</td></tr></table>
+
+And the existence of these spatial locality inspires us to find the direct cause of the problem in Sec. 3.2. However, the original split of data is not fine-grained enough to separately reveal these spatial localities. To better illustrate this, we visualize the data from one environment of the Roomto-Room dataset in Fig. 3, where the vertices are viewpoints with visual information and edges are valid connections between viewpoints. The vertices highlighted with dark-black indicate the viewpoints which are used in training paths, and the red edges are the connections covered by original val-seen paths. As shown in Fig. 3, nearly all viewpoints in val-seen paths (vertices connected to red lines) are used as viewpoints in training data (vertices marked by dark-black). We thus cannot categorize the path-level and region-level localities. To bypass this, we propose a novel re-splitting method to create our diagnosis data splits.
+
+Structural Data Re-splitting We employ two kinds of structural data splitting methods based on the horizontal or vertical coordinates, denoted as $\mathbf { \epsilon } ^ { \bullet } \mathbf { X }$ -split’ and $^ { \bullet } \mathrm { Z }$ -split’, respectively. The $^ { 6 } \mathrm { Z }$ - split’ intuitively separates different floors in the houses and $\mathbf { \epsilon } ^ { \bullet } \mathbf { X }$ -split’ creates separate areas. When applying to the training environments in R2R dataset, we use one side of the splitting line (see the ‘X-splitting line’ Fig. 3) as the new training ‘environment’, and the other side as the path-unseen ‘environment’. In addition to this split of environments, we also re-split the original training data and val-seen data while keeping the val-unseen data the same. The data paths across the splitting line are dropped. As shown in the right part of Fig. 3, we create three new data splits: training split, val-path-seen split, and val-path-unseen split. The edges covered by the new val-path-unseen split are highlighted in blue, while the color style of training split and val-path-seen split (‘Black’ for viewpoints in training and ‘Red’ for edges in val path-seen) are the same. Since the amount of original val-seen data are inadequate to fill two new validation sets (val path-seen and val pathunseen), we bring some (original) training data into our new validation splits. The overall statistics of original splits and our new splits are shown in Table 2.4
+
+Existence of Path-level and Environment-level Localities For both splitting methods, we train our baseline model on the newly-split training set and evaluate on our three validation sets (denoted as $\mathbf { \epsilon } ^ { \bullet } \mathbf { X }$ -split’ or $^ { 6 } \mathrm { Z }$ -split’ rows in Table 2). The results of our baseline model on the original R2R (denoted as ‘R2R’ rows) splits are listed for comparison. As shown in Table. 2, the agent performs better on val path-seen than val path-unseen, which suggests that a path-level locality exists in current VLN agent models. Meanwhile, the results on val path-unseen are further higher than val env-unseen and it indicates the environment-level locality which is independent of the path-level locality.
+
+Existence of the Region-level Locality To further demonstrate region-level locality, we study how the success rate changes in different regions of the environment with respect to their distances to the training data, which is similar to the analysis of language ‘distance’ in Sec. 4.1. We first calculate the point-by-point shortest paths using the Dijkstra’s algorithm (Dijkstra, 1959), where the shortest distances between viewpoints $v$ and $v ^ { \prime }$ are denoted as the graph distance $\mathrm { d i s } _ { \mathrm { G R A P H } } ( v , v ^ { \prime } )$ . Based on this graph distance, we define the viewpoint distance disVIEWPOINT from a viewpoint $v$ to the training data $\mathbb { T }$ as $v$ ’s minimal graph distance to a viewpoint $v ^ { \prime }$ in training data. We then define the path distance $\mathrm { d i s } _ { \mathrm { P A T H } }$ from a validating data $x$ to the whole training data $\mathbb { T }$ as the maximal viewpoint distance in the path of $x$ :
+
+![](images/fb90a3896586324d3aca5aafa029005a6d8046801b735f91602e032818287107.jpg)  
+Figure 4: The success rate declines as the path moves further from training regions.
+
+$$
+\begin{array} { r l } { \left. { \mathrm { d i s } _ { \mathrm { P A T H } } ( x , \mathbb { T } ) = \operatorname* { m a x } _ { \boldsymbol { v } \in \mathrm { p a t h } ( x ) } \mathrm { d i s } _ { \mathrm { V I E W P O I N T } } ( \boldsymbol { v } , \mathbb { T } ) } } \\ & { = \operatorname* { m a x } _ { \boldsymbol { v } \in \mathrm { p a t h } ( x ) } \left\{ \begin{array} { l } { \qquad \mathrm { ~ m i n ~ } } \\ { \boldsymbol { v } ^ { \prime } \in \mathrm { p a t h } ( t ) } \end{array} \right. \mathrm { d i s } _ { \mathrm { G R A P H } } \left( \boldsymbol { v } , \boldsymbol { v } ^ { \prime } \right) \right\} } \end{array}
+$$
+
+We compute this path distance between paths in the env-seen validation set and training environments in our re-split data. As shown in Fig. 4, the success rate declines as the path moves further from the training environment on both re-splitting methods (i.e., $\mathbf { \hat { x } }$ -split’ and $^ { 6 } \mathrm { Z }$ -split’). As conclusion, the closer the path to the training data, the higher the agent performance is, which suggests the existence of region-level locality.
+
+# 5 WHY: WHAT INSIDE THE ENVIRONMENTS CONTRIBUTES TO THE BIAS?
+
+In Sec. 4, we locate the cause of performance gap in visual environments by excluding other potential reasons and categorizing the spatial localities. However, there are still multiple possible aspects inside the environment which could lead to these spatial localities, e.g., the object layout convention and the room connections. The agent model could be biased towards the training environments by over-fitting or memorizing these environment-specific characteristics. In this section, we want to identify which aspect directly contributes to the bias and draw the following conclusion: the environment bias is attributed to low-level visual information carried by the ResNet features.
+
+We first show an experiment that effectively decreases the gap between seen and unseen environments with minimal model modifications. We then clarify our conclusions based on the findings.
+
+5.1 AN INVESTIGATION EXPERIMENT: IMAGENET LABELS AS VISUAL FEATURES
+
+Suspecting that the over-fitting happens when the agent over-learns low-level features, we hope to find the replacement of ResNet 2048-features that contain minimal low-level information while preserving distinguishable visual contents. The most straightforward replacement is that instead of using mean-pooled features, we inject the frozen 1000-way classifying layer in ResNet pre-training, and use the probabilities of ImageNet labels as visual features. Shown as ‘ImageNet’ in Table. 3, the probability distribution almost closes the gap between seen and unseen. These results further constrain the reason of environment bias to the low-level ResNet features of image views. Combining with the findings of spatial localities, we suggest that environments (i.e., houses) and regions (i.e., rooms) usually have their own ‘style’. Thus the same semantic label (captured by ImageNet-1000 features) has different visual appearances (captured by ResNet features) in different environments or regions. As a result, ImageNet-1000 features, in spite of being noisy, are not distracted by low-level visual appearance and could generalize to unseen environments, while ResNet features could not.
+
+Although these ImageNet-1000 features decrease the performance gap, it has a disagreement with the VLN domain so that the validation unseen results of R4R and CVDN are slightly worse than baseline (and not much better for R2R). Hence it motivates us to find better semantic representations of environmental features that can both close the seen-unseen gap while also achieving state-of-theart on unseen results (which we discuss next).
+
+Table 3: Results showing that our semantic feature representations eliminate the performance gap in all three datasets.   
+
+<table><tr><td rowspan="2">Task</td><td colspan="3">Feature</td><td colspan="3">Result</td></tr><tr><td>Type</td><td>Name</td><td>Dim</td><td>Val Seen</td><td>Val Unseen</td><td>Abs Gap |△|</td></tr><tr><td rowspan="6">Room-to-Room</td><td>Baseline</td><td>ResNet NoDrop</td><td>2,048</td><td>54.5</td><td>38.2</td><td>16.3</td></tr><tr><td>Baseline</td><td>ResNet</td><td>2,048</td><td>56.1</td><td>47.5</td><td>8.6</td></tr><tr><td>Invesgation</td><td>ImageNet</td><td>1,000</td><td>47.1</td><td>48.2</td><td>1.1</td></tr><tr><td>Semantic</td><td>Detection</td><td>152</td><td>55.9</td><td>50.0</td><td>5.9</td></tr><tr><td>Semantic</td><td>Ground Truth</td><td>42</td><td>55.6</td><td>56.2</td><td>0.6</td></tr><tr><td>Semantic</td><td>Learned</td><td>42</td><td>53.1</td><td>53.3</td><td>0.2</td></tr><tr><td rowspan="6">R4R</td><td>Baseline</td><td>ResNet NoDrop</td><td>2,048</td><td>52.5</td><td>25.8</td><td>26.7</td></tr><tr><td>Baseline</td><td>ResNet</td><td>2,048</td><td>54.6</td><td>30.7</td><td>23.9</td></tr><tr><td>Investigation</td><td>ImageNet</td><td>1,000</td><td>28.7</td><td>28.9</td><td>0.2</td></tr><tr><td>Semantic</td><td>Detection</td><td>152</td><td>48.8</td><td>32.0</td><td>16.8</td></tr><tr><td>Semantic</td><td>Ground Truth</td><td>42</td><td>47.6</td><td>35.9</td><td>11.7</td></tr><tr><td>Semantic</td><td>Learned</td><td>42</td><td>36.2</td><td>36.1</td><td>0.1</td></tr><tr><td rowspan="6">CVDN</td><td>Baseline</td><td>ResNet NoDrop</td><td>2,048</td><td>5.88</td><td>2.14</td><td>3.74</td></tr><tr><td>Baseline</td><td>ResNet</td><td>2,048</td><td>5.97</td><td>2.23</td><td>3.74</td></tr><tr><td>Invesgation</td><td>ImageNet</td><td>1,000</td><td>3.22</td><td>2.08</td><td>1.14</td></tr><tr><td>Semantic</td><td>Detection</td><td>152</td><td>3.34</td><td>2.08</td><td>1.26</td></tr><tr><td>Semantic</td><td>Ground Truth</td><td>42</td><td>3.75</td><td>2.69</td><td>1.06</td></tr><tr><td>Semantic</td><td>Learned</td><td>42</td><td>2.60</td><td>2.43</td><td>0.17</td></tr></table>
+
+# 6 HOW: METHODOLOGY TO FIX THE ENVIRONMENT BIAS
+
+In the previous section (Sec. 5), we found that the environment bias is related to the low-level visual features (i.e., 2048-dim ResNet features). Following the findings we observed in Sec. 5.1, we build our agent on the features which are more correlated to the VLN environmental semantics than the ImageNet label features in Sec. 5.1. We first demonstrate our baseline results on three VLN datasets and then explore the advanced semantic feature replacements. As shown in Table 3, these advanced semantic features could effectively reduce the performance gap between seen and unseen environments and improve the unseen results compared to our strong baselines. The effectiveness of these semantic features supports our explanation of the environment bias in Sec. 5 and also suggests that future work in VLN tasks should think about such generalization issues.
+
+# 6.1 BASELINE
+
+In our baseline model, following the previous works we use the standard ResNet features as the representation of environments (Anderson et al., 2018b; Jain et al., 2019; Thomason et al., 2019b). These features come from the mean-pooled layer after the final convolutional layer of ResNet152 (He et al., 2016) pre-trained on ImageNet (Russakovsky et al., 2015). As shown in ‘Baseline’5 rows of Table. 3, val-seen results are significantly higher than val-unseen results in all three datasets. Note that our baseline method takes the ‘feature dropout’ technique demonstrated in Tan et al. (2019) (without back translation): the ResNet features are randomly masked by zero before used as inputs of the agent. Without this ‘feature dropout’ (denoted as ‘ResNet NoDrop’ in Table. 3), the gaps will increase in R2R and R4R, which suggests that this ‘feature dropout’ technique also helps to eliminate the low-level visual information over-fitting as we discussed in Sec. 5. However, the performance gap is still large, which leads us to the following discussions of semantic features.
+
+# 6.2 DETECTED OBJECTS AREAS
+
+During navigation, the objects in the environments are crucial since their matchings with the instruction often indicate the locations that can guide the agent, thus object detection results of the environments can provide relevant semantic information. In our work, we utilize the detection information generated by Faster R-CNN (Ren et al., 2015) to create the feature representations. Comparing to
+
+ImageNet-1000 features (Sec. 5.1), these detection features include more environmental information since the viewing images in VLN usually contain multiple objects. Instead of directly using classification probabilities of the labels from ResNet and different from the approach in $\mathrm { H u }$ et al. (2019) who utilized the embeddings of detected labels, we design our detection features f DETECT of each image view as the sum of the areas of detected objects weighted by detection confidence:
+
+$$
+\mathrm { f _ { \mathrm { { D E T E C T } } } } \mathrm { = } [ a _ { c _ { 1 } } , a _ { c _ { 2 } } , \dotsc , a _ { c _ { n } } ] ; \qquad a _ { c _ { i } } = \sum _ { \mathrm { o b j i s } ~ c _ { i } } \mathrm { { A r e a } ( o b j ) } \cdot \mathrm { { C o n f ( o b j ) } }
+$$
+
+where the $c _ { i }$ and $\boldsymbol { a } _ { c _ { i } }$ are the label and feature of each detected object, $\mathrm { A r e a } ( * )$ and $\operatorname { C o n f } ( * )$ are the area and confidence of each object. For implementation details, we use the Faster R-CNN (Ren et al., 2015) trained on Visual Genome (Krishna et al., 2017) provided in Bottom-Up Attention (Anderson et al., 2018a). To eliminate the labels irrelevant to VLN task, we calculate the total areas of each detection object among all environments and pick the labels that take up a relatively large proportion of the environments, creating features of dimension 152.6 Denoted as ‘Detection’ in Table 3, the performance gap is diminished with these detection features compared to baselines in all three datasets, indicating that changing the features to a higher semantic level has a positive effect on alleviating the environment bias. Meanwhile, the improvement of unseen validation results on R2R an R4R datasets suggests the better efficiency in the VLN task than the ImageNet labels.
+
+# 6.3 SEMANTIC SEGMENTATION
+
+Although the detection features can provide adequate semantic information for the agent to achieve comparable results as the baseline model, they do not fully utilize the visual information where the content left over from detection may contain useful knowledge for navigation. A better semantic representation is the semantic segmentation, which segments each view image on the pixel level and gives the label to each segment region, allowing us to utilize the semantics from the entire environment. Matterport3D (Chang et al., 2017) dataset provides the labeled semantic segmentation information of every scene and we take the rendered images from Tan et al. (2019)7. A comparison example of RGB images and semantic views is available in the Appendix. Since the semantic segmentation images are fine-grained and blurry in boundaries, we follow the design of detection features, using the areas of semantic classes in each image view as the semantic features (confidence is excluded since semantic segmentation does not provide this value). The areas are normalized to $[ 0 , 1 ]$ by dividing the area of the whole image region. We first assume that the semantic information is provided as additional environmental information and the results of the model using the ground truth semantic areas are shown in the ‘ground truth’ rows in Table. 3. We next study the situation where the semantic information is not available in testing environments thus the information needs to be learned from training environments. Thus we train a separate multi-layer perceptron to predict the areas of these semantic classes (details in Appendix), and the results of the model with these predicted semantics as features are shown in ‘learned’. As shown in Table. 3, both ‘ground truth’ and ‘learned’ semantic representations bring the performance of seen and unseen closer comparing to the baseline model, and the smallest performance gaps come from learned semantic segmentation features in all three datasets. The highest validation unseen success rates among all the proposed feature representations are also produced by semantic segmentation features, ‘learned’ semantic for R4R and ‘ground truth’ semantic for R2R and CVDN. Overall, among all the semantic representations we have explored, the semantic segmentation features are most effective in eliminating the environment bias.
+
+# 7 CONCLUSION
+
+In this paper, we focus on studying the performance gap between seen and unseen environments widely observed in vision-and-language navigation (VLN) tasks, trying to find where and why this environment bias exists and provide possible initial solutions. By designing the diagnosis experiments of environment re-splitting and feature replacement, we locate the environment bias to be in the low-level visual appearance; and we discuss semantic features that decrease the performance gap in three VLN datasets and achieve state-of-the-art results.
+
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+
+# A APPENDIX
+
+![](images/aeffcbb03294a1d8c8132f9852d689ea8d2bd272e7c64a054ba954b51b1c77a8.jpg)  
+Figure 5: Comparisons between RGB images and their semantic views.
+
+A.1 EXAMPLES OF RGB IMAGES AND SEMANTIC VIEWS
+
+In Fig. 5, we show a rendered semantic view from Tan et al. (2019) and its original RGB image. Different colors indicate different semantic segmentation areas and 40 semantic labels are considered in the Matterport3D dataset Chang et al. (2017).
+
+# A.2 DETAILS OF ‘LEARNED’ SEMANTIC TRAINING
+
+We use a multi-layer perceptron over the ResNet features to generate the ‘learned’ semantic features. The multi-layer perceptron includes three fully-connected layers with ReLU activation on the outputs of the first two layers. The input is the 2048-dim ResNet feature $f$ of each image view. The hidden sizes of the first two layers are 512 and 128. The final layer will output the 42-dim semantic feature $y$ that represents the areas of each semantic class. After the linear layers, we use the sigmoid function $\sigma$ to convert the output to the ratio of areas.
+
+$$
+\begin{array} { c } { { x _ { 1 } = \mathrm { R e L U } ( A _ { 1 } f + b _ { 1 } ) } } \\ { { x _ { 2 } = \mathrm { R e L U } ( A _ { 2 } x _ { 1 } + b _ { 2 } ) } } \\ { { y = \sigma ( A _ { 3 } x _ { 2 } + b _ { 3 } ) } } \end{array}
+$$
+
+The model is trained with ground truth semantic areas $y _ { \mathrm { A R E A } }$ (normalized to $[ 0 , 1 ] )$ and only the views in training environments are used in training. We minimize the binary cross-entropy loss between the ground truth areas $\{ y _ { i } ^ { * } \}$ and the predicted areas $\{ y _ { i } \}$ , where $i$ indicate the $i$ -th semantic class.
+
+$$
+\mathcal { L } = - \sum _ { i } \left( y _ { i } ^ { * } \log y _ { i } + \left( 1 - y _ { i } ^ { * } \right) \log \left( 1 - y _ { i } \right) \right)
+$$
+
+Dropout layers with a probability of 0.5 are added between fully-connected layers while training.   
+The sigmoid function $\sigma$ and the cross-entropy loss are combined to improve numerical stability.
+
+After the model is fitted, we freeze the weight and use it to predict the semantic features of all seen and unseen environments (i.e., environments for training, val-seen, and val-unseen data). The predicted features are then used as the input of our neural agent model for different datasets (i.e., R2R, R4R, and CVDN), and the neural agent models are the same except we change the input dimension from 2048 (the dimension of ResNet features) to 42 (the number of semantic classes).

parse/train/Skq89Scxx/Skq89Scxx.md

@@ -0,0 +1,263 @@
+# SGDR: STOCHASTIC GRADIENT DESCENT WITH WARM RESTARTS
+
+Ilya Loshchilov & Frank Hutter
+
+University of Freiburg   
+Freiburg, Germany,   
+{ilya,fh}@cs.uni-freiburg.de
+
+# ABSTRACT
+
+Restart techniques are common in gradient-free optimization to deal with multimodal functions. Partial warm restarts are also gaining popularity in gradientbased optimization to improve the rate of convergence in accelerated gradient schemes to deal with ill-conditioned functions. In this paper, we propose a simple warm restart technique for stochastic gradient descent to improve its anytime performance when training deep neural networks. We empirically study its performance on the CIFAR-10 and CIFAR-100 datasets, where we demonstrate new state-of-the-art results at $3 . 1 4 \%$ and $1 6 . 2 1 \%$ , respectively. We also demonstrate its advantages on a dataset of EEG recordings and on a downsampled version of the ImageNet dataset. Our source code is available at   
+https://github.com/loshchil/SGDR
+
+# 1 INTRODUCTION
+
+Deep neural networks (DNNs) are currently the best-performing method for many classification problems, such as object recognition from images (Krizhevsky et al., 2012a; Donahue et al., 2014) or speech recognition from audio data (Deng et al., 2013). Their training on large datasets (where DNNs perform particularly well) is the main computational bottleneck: it often requires several days, even on high-performance GPUs, and any speedups would be of substantial value.
+
+The training of a DNN with $n$ free parameters can be formulated as the problem of minimizing a function $f : \mathbb { R } ^ { n } \to \mathbb { R }$ . The commonly used procedure to optimize $f$ is to iteratively adjust $\pmb { x } _ { t } \in \mathbb { R } ^ { n }$ (the parameter vector at time step $t$ ) using gradient information $\nabla f _ { t } ( { \pmb x } _ { t } )$ obtained on a relatively small $t$ -th batch of $b$ datapoints. The Stochastic Gradient Descent (SGD) procedure then becomes an extension of the Gradient Descent (GD) to stochastic optimization of $f$ as follows:
+
+$$
+\begin{array} { r } { \pmb { x } _ { t + 1 } = \pmb { x } _ { t } - \eta _ { t } \nabla f _ { t } ( \pmb { x } _ { t } ) , } \end{array}
+$$
+
+where $\eta _ { t }$ is a learning rate. One would like to consider second-order information
+
+$$
+\begin{array} { r } { { \pmb x } _ { t + 1 } = { \pmb x } _ { t } - \eta _ { t } { \pmb H } _ { t } ^ { - 1 } \nabla f _ { t } ( { \pmb x } _ { t } ) , } \end{array}
+$$
+
+but this is often infeasible since the computation and storage of the inverse Hessian $\pmb { H } _ { t } ^ { - 1 }$ is intractable for large $n$ . The usual way to deal with this problem by using limited-memory quasiNewton methods such as L-BFGS (Liu & Nocedal, 1989) is not currently in favor in deep learning, not the least due to (i) the stochasticity of $\nabla f _ { t } ( { \pmb x } _ { t } )$ , (ii) ill-conditioning of $f$ and (iii) the presence of saddle points as a result of the hierarchical geometric structure of the parameter space (Fukumizu & Amari, 2000). Despite some recent progress in understanding and addressing the latter problems (Bordes et al., 2009; Dauphin et al., 2014; Choromanska et al., 2014; Dauphin et al., 2015), state-ofthe-art optimization techniques attempt to approximate the inverse Hessian in a reduced way, e.g., by considering only its diagonal to achieve adaptive learning rates. AdaDelta (Zeiler, 2012) and Adam (Kingma & Ba, 2014) are notable examples of such methods.
+
+![](images/2fdd7dc39a0f5c385884a9c0807862b1960fa602d0f7be2bde9952c4d44d5f22.jpg)  
+Figure 1: Alternative schedule schemes of learning rate $\eta _ { t }$ over batch index $t$ : default schemes with $\eta _ { 0 } = 0 . 1$ (blue line) and $\eta _ { 0 } = 0 . 0 5$ (red line) as used by Zagoruyko & Komodakis (2016); warm restarts simulated every $T _ { 0 } = 5 0$ (green line), $T _ { 0 } = 1 0 0$ (black line) and $T _ { 0 } = 2 0 0$ (grey line) epochs with $\eta _ { t }$ decaying during $i$ -th run from $\eta _ { m a x } ^ { i } = 0 . 0 5$ to $\eta _ { m i n } ^ { i } = 0$ according to eq. (5); warm restarts starting from epoch $T _ { 0 } = 1$ (dark green line) and $T _ { 0 } = 1 0$ (magenta line) with doubling $( T _ { m u l t } = 2$ ) periods $T _ { i }$ at every new warm restart.
+
+Intriguingly enough, the current state-of-the-art results on CIFAR-10, CIFAR-100, SVHN, ImageNet, PASCAL VOC and MS COCO datasets were obtained by Residual Neural Networks (He et al., 2015; Huang et al., 2016c; He et al., 2016; Zagoruyko & Komodakis, 2016) trained without the use of advanced methods such as AdaDelta and Adam. Instead, they simply use SGD with momentum 1:
+
+$$
+\begin{array} { r } { \pmb { \nu } _ { t + 1 } = \mu _ { t } \pmb { \nu } _ { t } - \eta _ { t } \nabla f _ { t } ( \pmb { x } _ { t } ) , } \\ { \pmb { x } _ { t + 1 } = \pmb { x } _ { t } + \pmb { \nu } _ { t + 1 } , } \end{array}
+$$
+
+where $\nu _ { t }$ is a velocity vector initially set to $\pmb { \theta }$ , $\eta _ { t }$ is a decreasing learning rate and $\mu _ { t }$ is a momentum rate which defines the trade-off between the current and past observations of $\nabla f _ { t } ( { \pmb x } _ { t } )$ . The main difficulty in training a DNN is then associated with the scheduling of the learning rate and the amount of L2 weight decay regularization employed. A common learning rate schedule is to use a constant learning rate and divide it by a fixed constant in (approximately) regular intervals. The blue line in Figure 1 shows an example of such a schedule, as used by Zagoruyko & Komodakis (2016) to obtain the state-of-the-art results on CIFAR-10, CIFAR-100 and SVHN datasets.
+
+In this paper, we propose to periodically simulate warm restarts of SGD, where in each restart the learning rate is initialized to some value and is scheduled to decrease. Four different instantiations of this new learning rate schedule are visualized in Figure 1. Our empirical results suggest that SGD with warm restarts requires $2 \times$ to $4 \times$ fewer epochs than the currently-used learning rate schedule schemes to achieve comparable or even better results. Furthermore, combining the networks obtained right before restarts in an ensemble following the approach proposed by Huang et al. (2016a) improves our results further to $3 . 1 4 \%$ for CIFAR-10 and $1 6 . 2 1 \%$ for CIFAR-100. We also demonstrate its advantages on a dataset of EEG recordings and on a downsampled version of the ImageNet dataset.
+
+# 2 RELATED WORK
+
+# 2.1 RESTARTS IN GRADIENT-FREE OPTIMIZATION
+
+When optimizing multimodal functions one may want to find all global and local optima. The tractability of this task depends on the landscape of the function at hand and the budget of function evaluations. Gradient-free optimization approaches based on niching methods (Preuss, 2015) usually can deal with this task by covering the search space with dynamically allocated niches of local optimizers. However, these methods usually work only for relatively small search spaces, e.g., $n < 1 0$ , and do not scale up due to the curse of dimensionality (Preuss, 2010). Instead, the current state-of-the-art gradient-free optimizers employ various restart mechanisms (Hansen, 2009; Loshchilov et al., 2012). One way to deal with multimodal functions is to iteratively sample a large number $\lambda$ of candidate solutions, make a step towards better solutions and slowly shape the sampling distribution to maximize the likelihood of successful steps to appear again (Hansen & Kern, 2004). The larger the $\lambda$ , the more global search is performed requiring more function evaluations. In order to achieve good anytime performance, it is common to start with a small $\lambda$ and increase it (e.g., by doubling) after each restart. This approach works best on multimodal functions with a global funnel structure and also improves the results on ill-conditioned problems where numerical issues might lead to premature convergence when $\lambda$ is small (Hansen, 2009).
+
+# 2.2 RESTARTS IN GRADIENT-BASED OPTIMIZATION
+
+Gradient-based optimization algorithms such as BFGS can also perform restarts to deal with multimodal functions (Ros, 2009). In large-scale settings when the usual number of variables $n$ is on the order of $1 0 ^ { 3 } - 1 0 ^ { 9 }$ , the availability of gradient information provides a speedup of a factor of $n$ w.r.t. gradient-free approaches. Warm restarts are usually employed to improve the convergence rate rather than to deal with multimodality: often it is sufficient to approach any local optimum to a given precision and in many cases the problem at hand is unimodal. Fletcher & Reeves (1964) proposed to flesh the history of conjugate gradient method every $n$ or $( n + 1 )$ iterations. Powell (1977) proposed to check whether enough orthogonality between $\nabla f ( { \pmb x } _ { t - 1 } )$ and $\nabla f ( \pmb { x } _ { t } )$ has been lost to warrant another warm restart. Recently, O’Donoghue & Candes (2012) noted that the iterates of accelerated gradient schemes proposed by Nesterov (1983; 2013) exhibit a periodic behavior if momentum is overused. The period of the oscillations is proportional to the square root of the local condition number of the (smooth convex) objective function. The authors showed that fixed warm restarts of the algorithm with a period proportional to the conditional number achieves the optimal linear convergence rate of the original accelerated gradient scheme. Since the condition number is an unknown parameter and its value may vary during the search, they proposed two adaptive warm restart techniques (O’Donoghue & Candes, 2012):
+
+• The function scheme restarts whenever the objective function increases.
+
+• The gradient scheme restarts whenever the angle between the momentum term and the negative gradient is obtuse, i.e, when the momentum seems to be taking us in a bad direction, as measured by the negative gradient at that point. This scheme resembles the one of Powell (1977) for the conjugate gradient method.
+
+O’Donoghue & Candes (2012) showed (and it was confirmed in a set of follow-up works) that these simple schemes provide an acceleration on smooth functions and can be adjusted to accelerate stateof-the-art methods such as FISTA on nonsmooth functions.
+
+Smith (2015; 2016) recently introduced cyclical learning rates for deep learning, his approach is closely-related to our approach in its spirit and formulation but does not focus on restarts.
+
+Yang & Lin (2015) showed that Stochastic subGradient Descent with restarts can achieve a linear convergence rate for a class of non-smooth and non-strongly convex optimization problems where the epigraph of the objective function is a polyhedron. In contrast to our work, they never increase the learning rate to perform restarts but decrease it geometrically at each epoch. To perform restarts, they periodically reset the current solution to the averaged solution from the previous epoch.
+
+# 3 STOCHASTIC GRADIENT DESCENT WITH WARM RESTARTS (SGDR)
+
+The existing restart techniques can also be used for stochastic gradient descent if the stochasticity is taken into account. Since gradients and loss values can vary widely from one batch of the data to another, one should denoise the incoming information: by considering averaged gradients and losses, e.g., once per epoch, the above-mentioned restart techniques can be used again.
+
+In this work, we consider one of the simplest warm restart approaches. We simulate a new warmstarted run / restart of SGD once $T _ { i }$ epochs are performed, where $i$ is the index of the run. Importantly, the restarts are not performed from scratch but emulated by increasing the learning rate $\eta _ { t }$ while the old value of $\mathbf { \boldsymbol { x } } _ { t }$ is used as an initial solution. The amount of this increase controls to which extent the previously acquired information (e.g., momentum) is used.
+
+Within the $i$ -th run, we decay the learning rate with a cosine annealing for each batch as follows:
+
+$$
+\eta _ { t } = \eta _ { m i n } ^ { i } + \frac { 1 } { 2 } ( \eta _ { m a x } ^ { i } - \eta _ { m i n } ^ { i } ) ( 1 + \cos ( \frac { T _ { c u r } } { T _ { i } } \pi ) ) ,
+$$
+
+where $\eta _ { m i n } ^ { i }$ and $\eta _ { m a x } ^ { i }$ are ranges for the learning rate, and $T _ { c u r }$ accounts for how many epochs have been performed since the last restart. Since $T _ { c u r }$ is updated at each batch iteration $t$ , it can take discredited values such as 0.1, 0.2, etc. Thus, $\eta _ { t } = \eta _ { m a x } ^ { i }$ when $t = 0$ and $T _ { c u r } = 0$ . Once $T _ { c u r } = T _ { i }$ , the cos function will output $- 1$ and thus $\eta _ { t } = \eta _ { m i n } ^ { i }$ . The decrease of the learning rate is shown in Figure 1 for fixed $T _ { i } = 5 0$ , $T _ { i } = 1 0 0$ and $T _ { i } ~ = ~ 2 0 0$ ; note that the logarithmic axis obfuscates the typical shape of the cosine function.
+
+In order to improve anytime performance, we suggest an option to start with an initially small $T _ { i }$ and increase it by a factor of $T _ { m u l t }$ at every restart (see, e.g., Figure 1 for $T _ { 0 } = 1 , T _ { m u l t } = 2$ and $T _ { 0 } = 1 0 , T _ { m u l t } = 2 )$ . It might be of great interest to decrease $\eta _ { m a x } ^ { i }$ and $\eta _ { m i n } ^ { i }$ at every new restart. However, for the sake of simplicity, here, we keep $\eta _ { m a x } ^ { i }$ and $\eta _ { m i n } ^ { i }$ the same for every $i$ to reduce the number of hyperparameters involved.
+
+Since our simulated warm restarts (the increase of the learning rate) often temporarily worsen performance, we do not always use the last $\mathbf { } _ { \pmb { x } _ { t } }$ as our recommendation for the best solution (also called the incumbent solution). While our recommendation during the first run (before the first restart) is indeed the last $\mathbf { } _ { \pmb { x } _ { t } }$ , our recommendation after this is a solution obtained at the end of the last performed run at ηt = ηimin. We emphasize that with the help of this strategy, our method does not require a separate validation data set to determine a recommendation.
+
+# 4 EXPERIMENTAL RESULTS
+
+# 4.1 EXPERIMENTAL SETTINGS
+
+We consider the problem of training Wide Residual Neural Networks (WRNs; see Zagoruyko & Komodakis (2016) for details) on the CIFAR-10 and CIFAR-100 datasets (Krizhevsky, 2009). We will use the abbreviation WRN-d- $k$ to denote a WRN with depth $d$ and width $k$ . Zagoruyko & Komodakis (2016) obtained the best results with a WRN-28-10 architecture, i.e., a Residual Neural Network with $d \ : = \ : 2 8$ layers and $k = 1 0$ times more filters per layer than used in the original Residual Neural Networks (He et al., 2015; 2016).
+
+The CIFAR-10 and CIFAR-100 datasets (Krizhevsky, 2009) consist of $3 2 \times 3 2$ color images drawn from 10 and 100 classes, respectively, split into 50,000 train and 10,000 test images. For image preprocessing Zagoruyko & Komodakis (2016) performed global contrast normalization and ZCA whitening. For data augmentation they performed horizontal flips and random crops from the image padded by 4 pixels on each side, filling missing pixels with reflections of the original image.
+
+For training, Zagoruyko & Komodakis (2016) used SGD with Nesterov’s momentum with initial learning rate set to $\eta _ { 0 } ~ = ~ 0 . 1$ , weight decay to 0.0005, dampening to 0, momentum to 0.9 and minibatch size to 128. The learning rate is dropped by a factor of 0.2 at 60, 120 and 160 epochs, with a total budget of 200 epochs. We reproduce the results of Zagoruyko & Komodakis (2016) with the same settings except that i) we subtract per-pixel mean only and do not use ZCA whitening; ii) we use SGD with momentum as described by eq. (3-4) and not Nesterov’s momentum.
+
+![](images/67fc79f25b3e3da44eb17b55361645c16f330c07cab651bdaa4489e1d819cf13.jpg)  
+Figure 2: Test errors on CIFAR-10 (left column) and CIFAR-100 (right column) datasets. Note that for SGDR we only plot the recommended solutions. The top and middle rows show the same results on WRN-28-10, with the middle row zooming into the good performance region of low test error. The bottom row shows performance with a wider network, WRN-28-20.
+
+The results of the default learning rate schedules of Zagoruyko & Komodakis (2016) with $\eta _ { 0 } = 0 . 1$ and $\eta _ { 0 } = 0 . 0 5$ are depicted by the blue and red lines, respectively. The schedules of $\eta _ { t }$ used in SGDR are shown with i) restarts every $T _ { 0 } = 5 0$ epochs (green line); ii) restarts every $T _ { 0 } = 1 0 0$ epochs (black line); iii) restarts every $T _ { 0 } = 2 0 0$ epochs (gray line); iv) restarts with doubling $T _ { m u l t } = 2 $ ) periods of restarts starting from the first epoch ( ${ { T } _ { 0 } } \ = \ 1$ , dark green line); and v) restarts with doubling $T _ { m u l t } = 2$ ) periods of restarts starting from the tenth epoch $T _ { 0 } = 1 0$ , magenta line).
+
+The schedule of $\eta _ { t }$ used by Zagoruyko & Komodakis (2016) is depicted by the blue line in Figure 1. The same schedule but with $\eta _ { 0 } = 0 . 0 5$ is depicted by the red line. The schedule of $\eta _ { t }$ used in SGDR is also shown in Figure 1, with two initial learning rates $T _ { 0 }$ and two restart doubling periods.
+
+Table 1: Test errors of different methods on CIFAR-10 and CIFAR-100 with moderate data augmentation (flip/translation). In the second column $k$ is a widening factor for WRNs. Note that the computational and memory resources used to train all WRN-28-10 are the same. In all other cases they are different, but WRNs are usually faster than original ResNets to achieve the same accuracy (e.g., up to a factor of 8 according to Zagoruyko & Komodakis (2016)). Bold text is used only to highlight better results and is not based on statistical tests (too few runs).   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>depth-k</td><td rowspan=1 colspan=1># params</td><td rowspan=1 colspan=1># runs</td><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>CIFAR-100</td></tr><tr><td rowspan=2 colspan=1>original-ResNet (He et al.,2015)</td><td rowspan=1 colspan=3>110</td><td rowspan=1 colspan=1>1.7M</td><td rowspan=1 colspan=1>mean of 5</td><td rowspan=1 colspan=1>6.43</td><td rowspan=1 colspan=1>25.16</td></tr><tr><td rowspan=1 colspan=3>1202</td><td rowspan=1 colspan=1>10.2M</td><td rowspan=1 colspan=1>mean of 5</td><td rowspan=1 colspan=1>7.93</td><td rowspan=1 colspan=1>27.82</td></tr><tr><td rowspan=2 colspan=1>stoc-depth (Huang et al., 2016c)</td><td rowspan=2 colspan=3>1101202</td><td rowspan=2 colspan=1>1.7M10.2M</td><td rowspan=2 colspan=1>1 run1 run</td><td rowspan=1 colspan=1>5.23</td><td rowspan=2 colspan=1>24.58n/a</td></tr><tr><td rowspan=1 colspan=2>02</td><td rowspan=1 colspan=1>4.91</td></tr><tr><td rowspan=3 colspan=1>pre-act-ResNet (He et al.,2016)</td><td rowspan=3 colspan=3>1101641001</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>1.7M</td><td rowspan=1 colspan=1>med. of5</td><td rowspan=1 colspan=1>6.37</td></tr><tr><td rowspan=2 colspan=1>1.7M10.2M</td><td rowspan=2 colspan=1>med. of 5med. of 5</td><td rowspan=1 colspan=1>5.46</td><td rowspan=2 colspan=1>24.3322.71</td></tr><tr><td rowspan=1 colspan=1>4.62</td></tr><tr><td rowspan=2 colspan=1>WRN (Zagoruyko &amp; Komodakis, 2016) with dropout</td><td rowspan=2 colspan=3>16-828-1028-10</td><td rowspan=2 colspan=1>11.0M36.5M36.5M</td><td rowspan=2 colspan=1>1 run1 run1 run</td><td rowspan=1 colspan=1>4.81</td><td rowspan=2 colspan=1>22.0720.5020.04</td></tr><tr><td rowspan=1 colspan=1>4.17n/a</td></tr><tr><td rowspan=2 colspan=1>WRN (ours)default with no = 0.1</td><td rowspan=1 colspan=3></td><td rowspan=2 colspan=1>36.5M</td><td rowspan=2 colspan=1>med. of 5</td><td rowspan=2 colspan=1>4.24</td><td rowspan=2 colspan=1>20.33</td></tr><tr><td rowspan=1 colspan=3>28-10</td></tr><tr><td rowspan=1 colspan=1>default with no = 0.05</td><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>4.13</td><td rowspan=1 colspan=1>20.21</td></tr><tr><td rowspan=1 colspan=1>T = 50,Tmult =1</td><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>4.17</td><td rowspan=1 colspan=1>19.99</td></tr><tr><td rowspan=1 colspan=1>To = 100,Tmult =1</td><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>4.07</td><td rowspan=1 colspan=1>19.87</td></tr><tr><td rowspan=1 colspan=1>To = 200,Tmult = 1</td><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>3.86</td><td rowspan=1 colspan=1>19.98</td></tr><tr><td rowspan=2 colspan=1>To =1,Tmult = 2To =10,Tmult = 2</td><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>4.09</td><td rowspan=1 colspan=1>19.74</td></tr><tr><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>4.03</td><td rowspan=1 colspan=1>19.58</td></tr><tr><td rowspan=1 colspan=1>default with no = 0.1</td><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>4.08</td><td rowspan=1 colspan=1>19.53</td></tr><tr><td rowspan=1 colspan=1>default with no = 0.05</td><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>3.96</td><td rowspan=1 colspan=1>19.67</td></tr><tr><td rowspan=1 colspan=1>To = 50,Tmult = 1</td><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>4.01</td><td rowspan=1 colspan=1>19.28</td></tr><tr><td rowspan=1 colspan=1>To = 100,Tmult = 1</td><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>3.77</td><td rowspan=1 colspan=1>19.24</td></tr><tr><td rowspan=3 colspan=1>To = 200,Tmult =1To =1,Tmult = 2To = 10,Tmult = 2</td><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>3.66</td><td rowspan=1 colspan=1>19.69</td></tr><tr><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>3.91</td><td rowspan=1 colspan=1>18.90</td></tr><tr><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>3.74</td><td rowspan=1 colspan=1>18.70</td></tr></table>
+
+# 4.2 SINGLE-MODEL RESULTS
+
+Table 1 shows that our experiments reproduce the results given by Zagoruyko & Komodakis (2016) for WRN-28-10 both on CIFAR-10 and CIFAR-100. These “default” experiments with $\eta _ { 0 } = 0 . 1$ and $\eta _ { 0 } = 0 . 0 5$ correspond to the blue and red lines in Figure 2. The results for $\eta _ { 0 } = 0 . 0 5$ show better performance, and therefore we use $\eta _ { 0 } = 0 . 0 5$ in our later experiments.
+
+SGDR with $T _ { 0 } = 5 0$ , $T _ { 0 } = 1 0 0$ and $T _ { 0 } = 2 0 0$ for $T _ { m u l t } = 1$ perform warm restarts every 50, 100 and 200 epochs, respectively. A single run of SGD with the schedule given by eq. (5) for $T _ { 0 } = 2 0 0$ shows the best results suggesting that the original schedule of WRNs might be suboptimal w.r.t. the test error in these settings. However, the same setting with $T _ { 0 } = 2 0 0$ leads to the worst anytime performance except for the very last epochs.
+
+SGDR with $T _ { 0 } = 1 \mathrm { , } T _ { m u l t } = 2$ and $T _ { 0 } = 1 0 , T _ { m u l t } = 2$ performs its first restart after 1 and 10 epochs, respectively. Then, it doubles the maximum number of epochs for every new restart. The main purpose of this doubling is to reach good test error as soon as possible, i.e., achieve good anytime performance. Figure 2 shows that this is achieved and test errors around $4 \%$ on CIFAR-10 and around $20 \%$ on CIFAR-100 can be obtained about 2-4 times faster than with the default schedule used by Zagoruyko & Komodakis (2016).
+
+![](images/9483ebc74be2a2955fa389608a8a576cae9c594a7486cafbf9d0969a764ed4dd.jpg)  
+Figure 3: Test errors of ensemble models built from $N$ runs of SGDR on WRN-28-10 with $M$ model snapshots per run made at epochs 150, 70 and 30 (right before warm restarts of SGDR as suggested by Huang et al. (2016a)). When $M { = } 1$ (respectively, $M { = } 2$ ), we aggregate probabilities of softmax layers of snapshot models at epoch index 150 (respectively, at epoch indexes 150 and 70).
+
+Table 2: Test errors of ensemble models on CIFAR-10 and CIFAR-100 datasets.   
+
+<table><tr><td></td><td>CIFAR-10</td><td>CIFAR-100</td></tr><tr><td>N = 1 run of WRN-28-10 with M = 1 snapshot (median of 16 runs)</td><td>4.03</td><td>19.57</td></tr><tr><td>N = 1 run of WRN-28-10 with M = 3 snapshots per run</td><td>3.51</td><td>17.75</td></tr><tr><td>N = 3 runs of WRN-28-10 with M = 3 snapshots per run</td><td>3.25</td><td>16.64</td></tr><tr><td>N = 16 runs of WRN-28-10 with M= 3 snapshots per run</td><td>3.14</td><td>16.21</td></tr></table>
+
+Since SGDR achieves good performance faster, it may allow us to train larger networks. We therefore investigated whether results on CIFAR-10 and CIFAR-100 can be further improved by making WRNs two times wider, i.e., by training WRN-28-20 instead of WRN-28-10. Table 1 shows that the results indeed improved, by about $0 . 2 5 \%$ on CIFAR-10 and by about $0 . 5 \substack { - 1 . 0 \% }$ on CIFAR-100. While network architecture WRN-28-20 requires roughly three-four times more computation than WRN-28-10, the aggressive learning rate reduction of SGDR nevertheless allowed us to achieve a better error rate in the same time on WRN-28-20 as we spent on 200 epochs of training on WRN28-10. Specifically, Figure 2 (right middle and right bottom) show that after only 50 epochs, SGDR (even without restarts, using $T _ { 0 } = 5 0 , T _ { m u l t } = 1 )$ achieved an error rate below $19 \%$ (whereas none of the other learning methods performed better than $1 9 . 5 \%$ on WRN-28-10). We therefore have hope that – by enabling researchers to test new architectures faster – SGDR’s good anytime performance may also lead to improvements of the state of the art.
+
+In a final experiment for SGDR by itself, Figure 7 in the appendix compares SGDR and the default schedule with respect to training and test performance. As the figure shows, SGDR optimizes training loss faster than the standard default schedule until about epoch 120. After this, the default schedule overfits, as can be seen by an increase of the test error both on CIFAR-10 and CIFAR-100 (see, e.g., the right middle plot of Figure 7). In contrast, we only witnessed very mild overfitting for SGDR.
+
+# 4.3 ENSEMBLE RESULTS
+
+Our initial arXiv report on SGDR (Loshchilov & Hutter, 2016) inspired a follow-up study by Huang et al. (2016a) in which the authors suggest to take $M$ snapshots of the models obtained by SGDR (in their paper referred to as cyclical learning rate schedule and cosine annealing cycles) right before $M$ last restarts and to use those to build an ensemble, thereby obtaining ensembles “for free” (in contrast to having to perform multiple independent runs). The authors demonstrated new state-ofthe-art results on CIFAR datasets by making ensembles of DenseNet models (Huang et al., 2016b). Here, we investigate whether their conclusions hold for WRNs used in our study. We used WRN28-10 trained by SGDR with $T _ { 0 } = 1 0 , T _ { m u l t } = 2$ as our baseline model.
+
+Figure 3 and Table 2 aggregate the results of our study. The original test error of $4 . 0 3 \%$ on CIFAR-10 and $1 9 . 5 7 \%$ on CIFAR-100 (median of 16 runs) can be improved to $3 . 5 1 \%$ on CIFAR-10 and $1 7 . 7 5 \%$ on CIFAR-100 when $M = 3$ snapshots are taken at epochs 30, 70 and 150: when the learning rate of SGDR with $T _ { 0 } = 1 0 , T _ { m u l t } = 2$ is scheduled to achieve 0 (see Figure 1) and the models are used with uniform weights to build an ensemble. To achieve the same result, one would have to aggregate $N = 3$ models obtained at epoch 150 of $N = 3$ independent runs (see $N = 3 , M = 1$ in Figure 3). Thus, the aggregation from snapshots provides a 3-fold speedup in these settings because additional ( $M > 1$ -th) snapshots from a single SGDR run are computationally free. Interestingly, aggregation of models from independent runs (when $N > 1$ and $M = 1$ ) does not scale up as well as from $M > 1$ snapshots of independent runs when the same number of models is considered: the case of $N = 3$ and $M = 3$ provides better performance than the cases of $M = 1$ with $N = 1 8$ and $N = 2 1$ . Not only the number of snapshots $M$ per run but also their origin is crucial. Thus, naively building ensembles from models obtained at last epochs only (i.e., $M = 3$ snapshots at epochs 148, 149, 150) did not improve the results (i.e., the baseline of $M = 1$ snapshot at 150) thereby confirming the conclusion of Huang et al. (2016a) that snapshots of SGDR provide a useful diversity of predictions for ensembles.
+
+Three runs $N = 3$ ) of SGDR with $M = 3$ snapshots per run are sufficient to greatly improve the results to $3 . 2 5 \%$ on CIFAR-10 and $1 6 . 6 4 \%$ on CIFAR-100 outperforming the results of Huang et al. (2016a). By increasing $N$ to 16 one can achieve $3 . 1 4 \%$ on CIFAR-10 and $1 6 . 2 1 \%$ on CIFAR-100. We believe that these results could be further improved by considering better baseline models than WRN-28-10 (e.g., WRN-28-20).
+
+# 4.4 EXPERIMENTS ON A DATASET OF EEG RECORDINGS
+
+To demonstrate the generality of SGDR, we also considered a very different domain: a dataset of electroencephalographic (EEG) recordings of brain activity for classification of actual right and left hand and foot movements of 14 subjects with roughly 1000 trials per subject. The best classification results obtained with the original pipeline based on convolutional neural networks [R. Schirrmeister et al. Convolutional neural networks for EEG analysis: Design choices, training strategies, and feature visualization., under review at Neuroimage] were used as our reference. First, we compared the baseline learning rate schedule with different settings of the total number of epochs and initial learning rates (see Figure 4). When 30 epochs were considered, we dropped the learning rate by a factor of 10 at epoch indexes 10, 15 and 20. As expected, with more epochs used and a similar (budget proportional) schedule better results can be achieved. Alternatively, one can consider SGDR and get a similar final performance while having a better anytime performance without defining the total budget of epochs beforehand.
+
+Similarly to our results on the CIFAR datasets, our experiments with the EEG data confirm that snapshots are useful and the median reference error (about $9 \%$ ) can be improved i) by $1 { - } 2 \%$ when model snapshots of a single run are considered, and ii) by $2 { - } 3 \%$ when model snapshots from both hyperparameter settings are considered. The latter would correspond to $N = 2$ in Section (4.3).
+
+# 4.5 PRELIMINARY EXPERIMENTS ON A DOWNSAMPLED IMAGENET DATASET
+
+In order to additionally validate our SGDR on a larger dataset, we constructed a downsampled version of the ImageNet dataset [P. Chrabaszcz, I. Loshchilov and F. Hutter. A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets., in preparation]. In contrast to earlier attempts (Pouransari & Ghili, 2015), our downsampled ImageNet contains exactly the same images from 1000 classes as the original ImageNet but resized with box downsampling to $3 2 \times 3 2$ pixels. Thus, this dataset is substantially harder than the original ImageNet dataset because the average number of pixels per image is now two orders of magnitude smaller. The new dataset is also more difficult than the CIFAR datasets because more classes are used and the relevant objects to be classified often cover only a tiny subspace of the image and not most of the image as in the CIFAR datasets.
+
+![](images/5dfdedd88ded2a58cd5402059f23bdeb3b206e82cfb5df1c9c28a04b45a45e3e.jpg)  
+Figure 4: (Top) Improvements obtained by the baseline learning rate schedule and SGDR w.r.t. the best known reference classification error on a dataset of electroencephalographic (EEG) recordings of brain activity for classification of actual right and left hand and foot movements of 14 subjects with roughly 1000 trials per subject. Both considered approaches were tested with the initial learning rate $l r = 0 . 0 2 5$ (Top-Left) and $l r = 0 . 0 5$ (Top-Right). Note that the baseline approach is considered with different settings of the total number of epochs: 30, 60, . . ., 480. (Bottom) SGDR with $l r = 0 . 0 2 5$ and $l r = 0 . 0 5$ without and with $M$ model snapshots taken at the last $M = n r / 2$ restarts, where $n r$ is the total number of restarts.
+
+We benchmarked SGD with momentum with the default learning rate schedule, SGDR with $T _ { 0 } =$ $1 , T _ { m u l t } = 2$ and SGDR with $T _ { 0 } = 1 0 , T _ { m u l t } = 2$ on WRN-28-10, all trained with 4 settings of the initial learning rate $\eta _ { m a x } ^ { i }$ : 0.050, 0.025, 0.01 and 0.005. We used the same data augmentation procedure as for the CIFAR datasets. Similarly to the results on the CIFAR datasets, Figure 5 shows that SGDR demonstrates better anytime performance. SGDR with $T _ { 0 } = 1 0 , T _ { m u l t } \stackrel { = } { = } 2 , \eta _ { m a x } ^ { i } =$ 0.01 achieves top-1 error of $3 9 . 2 4 \%$ and top-5 error of $1 7 . 1 7 \%$ matching the original results by AlexNets $4 0 . 7 \%$ and $1 8 . 2 \%$ , respectively) obtained on the original ImageNet with full-size images of ca. 50 times more pixels per image (Krizhevsky et al., 2012b). Interestingly, when the dataset is permuted only within 10 subgroups each formed from 100 classes, SGDR also demonstrates better results (see Figure 8 in the Supplementary Material). An interpretation of this might be that while the initial learning rate seems to be very important, SGDR reduces the problem of improper selection of the latter by scanning / annealing from the initial learning rate to 0.
+
+Clearly, longer runs (more than 40 epochs considered in this preliminary experiment) and hyperparameter tuning of learning rates, regularization and other hyperparameters shall further improve the results.
+
+![](images/edcfbdc72ca22c95d6c0b06c3715ba2b80be5349cf0ca5ae5f976abadf982d72.jpg)  
+Figure 5: Top-1 and Top-5 test errors obtained by SGD with momentum with the default learning rate schedule, SGDR with $T _ { 0 } = 1 , T _ { m u l t } = 2$ and SGDR with $T _ { 0 } = 1 0$ , $T _ { m u l t } = 2$ on WRN-28-10 trained on a version of ImageNet, with all images from all 1000 classes downsampled to $3 2 \times 3 2$ pixels. The same baseline data augmentation as for the CIFAR datasets is used. Four settings of the initial learning rate are considered: 0.050, 0.025, 0.01 and 0.005.
+
+# 5 DISCUSSION
+
+Our results suggest that even without any restarts the proposed aggressive learning rate schedule given by eq. (5) is competitive w.r.t. the default schedule when training WRNs on the CIFAR10 (e.g., for $T _ { 0 } = 2 0 0 , T _ { m u l t } = 1 )$ and CIFAR-100 datasets. In practice, the proposed schedule requires only two hyper-parameters to be defined: the initial learning rate and the total number of epochs.
+
+We found that the anytime performance of SGDR remain similar when shorter epochs are considered (see section 8.1 in the Supplemenary Material).
+
+One should not suppose that the parameter values used in this study and many other works with (Residual) Neural Networks are selected to demonstrate the fastest decrease of the training error. Instead, the best validation or $/$ and test errors are in focus. Notably, the validation error is rarely used when training Residual Neural Networks because the recommendation is defined by the final solution (in our approach, the final solution of each run). One could use the validation error to determine the optimal initial learning rate and then run on the whole dataset; this could further improve results.
+
+The main purpose of our proposed warm restart scheme for SGD is to improve its anytime performance. While we mentioned that restarts can be useful to deal with multi-modal functions, we do not claim that we observe any effect related to multi-modality.
+
+As we noted earlier, one could decrease $\eta _ { m a x } ^ { i }$ and $\eta _ { m i n } ^ { i }$ at every new warm restart to control the amount of divergence. If new restarts are worse than the old ones w.r.t. validation error, then one might also consider going back to the last best solution and perform a new restart with adjusted hyperparameters.
+
+Our results reproduce the finding by Huang et al. (2016a) that intermediate models generated by SGDR can be used to build efficient ensembles at no cost. This finding makes SGDR especially attractive for scenarios when ensemble building is considered.
+
+# 6 CONCLUSION
+
+In this paper, we investigated a simple warm restart mechanism for SGD to accelerate the training of DNNs. Our SGDR simulates warm restarts by scheduling the learning rate to achieve competitive results on CIFAR-10 and CIFAR-100 roughly two to four times faster. We also achieved new stateof-the-art results with SGDR, mainly by using even wider WRNs and ensembles of snapshots from
+
+SGDR’s trajectory. Future empirical studies should also consider the SVHN, ImageNet and MS COCO datasets, for which Residual Neural Networks showed the best results so far. Our preliminary results on a dataset of EEG recordings suggest that SGDR delivers better and better results as we carry out more restarts and use more model snapshots. The results on our downsampled ImageNet dataset suggest that SGDR might also reduce the problem of learning rate selection because the annealing and restarts of SGDR scan / consider a range of learning rate values. Future work should consider warm restarts for other popular training algorithms such as AdaDelta (Zeiler, 2012) and Adam (Kingma & Ba, 2014).
+
+Alternative network structures should be also considered; e.g., soon after our initial arXiv report (Loshchilov & Hutter, 2016), Zhang et al. (2016); Huang et al. (2016b); Han et al. (2016) reported that WRNs models can be replaced by more memory-efficient models. Thus, it should be tested whether our results for individual models and ensembles can be further improved by using their networks instead of WRNs. Deep compression methods (Han et al., 2015) can be used to reduce the time and memory costs of DNNs and their ensembles.
+
+# 7 ACKNOWLEDGMENTS
+
+This work was supported by the German Research Foundation (DFG), under the BrainLinksBrainTools Cluster of Excellence (grant number EXC 1086). We thank Gao Huang, Kilian Quirin Weinberger, Jost Tobias Springenberg, Mark Schmidt and three anonymous reviewers for their helpful comments and suggestions.
+
+# REFERENCES
+
+Antoine Bordes, Leon Bottou, and Patrick Gallinari. Sgd-qn: Careful quasi-newton stochastic gra-´ dient descent. The Journal of Machine Learning Research, 10:1737–1754, 2009.
+
+Anna Choromanska, Mikael Henaff, Michael Mathieu, Gerard Ben Arous, and Yann LeCun. The ´ loss surface of multilayer networks. arXiv preprint arXiv:1412.0233, 2014.
+
+Yann N Dauphin, Razvan Pascanu, Caglar Gulcehre, Kyunghyun Cho, Surya Ganguli, and Yoshua Bengio. Identifying and attacking the saddle point problem in high-dimensional non-convex optimization. In Advances in Neural Information Processing Systems, pp. 2933–2941, 2014.
+
+Yann N Dauphin, Harm de Vries, Junyoung Chung, and Yoshua Bengio. Rmsprop and equilibrated adaptive learning rates for non-convex optimization. arXiv preprint arXiv:1502.04390, 2015.
+
+L. Deng, G. Hinton, and B. Kingsbury. New types of deep neural network learning for speech recognition and related applications: An overview. In Proc. of ICASSP’13, 2013.
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+J. Donahue, Y. Jia, O. Vinyals, J. Hoffman, N. Zhang, E. Tzeng, and T. Darrell. Decaf: A deep convolutional activation feature for generic visual recognition. In Proc. of ICML’14, 2014.
+
+Reeves Fletcher and Colin M Reeves. Function minimization by conjugate gradients. The computer journal, 7(2):149–154, 1964.
+
+Kenji Fukumizu and Shun-ichi Amari. Local minima and plateaus in hierarchical structures of multilayer perceptrons. Neural Networks, 13(3):317–327, 2000.
+
+Dongyoon Han, Jiwhan Kim, and Junmo Kim. Deep pyramidal residual networks. arXiv preprint arXiv:1610.02915, 2016.
+
+Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149, 2015.
+
+Nikolaus Hansen. Benchmarking a BI-population CMA-ES on the BBOB-2009 function testbed. In Proceedings of the 11th Annual Conference Companion on Genetic and Evolutionary Computation Conference: Late Breaking Papers, pp. 2389–2396. ACM, 2009.
+
+Nikolaus Hansen and Stefan Kern. Evaluating the cma evolution strategy on multimodal test functions. In International Conference on Parallel Problem Solving from Nature, pp. 282–291. Springer, 2004.
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+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. arXiv preprint arXiv:1512.03385, 2015.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. arXiv preprint arXiv:1603.05027, 2016.
+
+Gao Huang, Yixuan Li, Geoff Pleiss, Zhuang Liu, John E. Hopcroft, and Kilian Q. Weinberger. Snapshot ensembles: Train 1, get m for free. ICLR 2017 submission, 2016a.
+
+Gao Huang, Zhuang Liu, and Kilian Q Weinberger. Densely connected convolutional networks. arXiv preprint arXiv:1608.06993, 2016b.
+
+Gao Huang, Yu Sun, Zhuang Liu, Daniel Sedra, and Kilian Weinberger. Deep networks with stochastic depth. arXiv preprint arXiv:1603.09382, 2016c.
+
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+A. Krizhevsky, I. Sutskever, and G. Hinton. Imagenet classification with deep convolutional neural networks. In Proc. of NIPS’12, pp. 1097–1105, 2012a.
+
+Alex Krizhevsky. Learning multiple layers of features from tiny images. 2009.
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+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012b.
+
+Dong C Liu and Jorge Nocedal. On the limited memory bfgs method for large scale optimization. Mathematical programming, 45(1-3):503–528, 1989.
+
+Ilya Loshchilov and Frank Hutter. SGDR: Stochastic Gradient Descent with Restarts. arXiv preprint arXiv:1608.03983, 2016.
+
+Ilya Loshchilov, Marc Schoenauer, and Michele Sebag. Alternative restart strategies for CMA-ES. In International Conference on Parallel Problem Solving from Nature, pp. 296–305. Springer, 2012.
+
+Yurii Nesterov. A method of solving a convex programming problem with convergence rate o (1/k2). In Soviet Mathematics Doklady, volume 27, pp. 372–376, 1983.
+
+Yurii Nesterov. Introductory lectures on convex optimization: A basic course, volume 87. Springer Science & Business Media, 2013.
+
+Brendan O’Donoghue and Emmanuel Candes. Adaptive restart for accelerated gradient schemes. arXiv preprint arXiv:1204.3982, 2012.
+
+Hadi Pouransari and Saman Ghili. Tiny imagenet visual recognition challenge. CS231 course at STANFORD, 2015.
+
+Michael James David Powell. Restart procedures for the conjugate gradient method. Mathematical programming, 12(1):241–254, 1977.
+
+Mike Preuss. Niching the CMA-ES via nearest-better clustering. In Proceedings of the 12th annual conference companion on Genetic and evolutionary computation, pp. 1711–1718. ACM, 2010.
+
+Mike Preuss. Niching methods and multimodal optimization performance. In Multimodal Optimization by Means of Evolutionary Algorithms, pp. 115–137. Springer, 2015.
+
+Raymond Ros. Benchmarking the bfgs algorithm on the bbob-2009 function testbed. In Proceedings of the 11th Annual Conference Companion on Genetic and Evolutionary Computation Conference: Late Breaking Papers, pp. 2409–2414. ACM, 2009.
+
+Leslie N Smith. No more pesky learning rate guessing games. arXiv preprint arXiv:1506.01186, 2015.   
+Leslie N Smith. Cyclical learning rates for training neural networks. arXiv preprint arXiv:1506.01186v3, 2016.   
+Tianbao Yang and Qihang Lin. Stochastic subgradient methods with linear convergence for polyhedral convex optimization. arXiv preprint arXiv:1510.01444, 2015.   
+Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016.   
+Matthew D Zeiler. Adadelta: An adaptive learning rate method. arXiv preprint arXiv:1212.5701, 2012.   
+K. Zhang, M. Sun, T. X. Han, X. Yuan, L. Guo, and T. Liu. Residual Networks of Residual Networks: Multilevel Residual Networks. ArXiv e-prints, August 2016.
+
+# 8 SUPPLEMENTARY MATERIAL
+
+![](images/2e914475e7ea34de85b07237a7c7c57ae81524b5e8a2ef32ab2cf99c2d03e7b8.jpg)  
+Figure 6: The median results of 5 runs for the best learning rate settings considered for WRN-28-1.
+
+8.1 50K VS 100K EXAMPLES PER EPOCH
+
+Our data augmentation procedure code is inherited from the Lasagne Recipe code for ResNets where flipped images are added to the training set. This doubles the number of training examples per epoch and thus might impact the results because hyperparameter values defined as a function of epoch index have a different meaning. While our experimental results given in Table 1 reproduced the results obtained by Zagoruyko & Komodakis (2016), here we test whether SGDR still makes sense for WRN-28-1 (i.e., ResNet with 28 layers) where one epoch corresponds to $5 0 \mathrm { k }$ training examples. We investigate different learning rate values for the default learning rate schedule (4 values out of [0.01, 0.025, 0.05, 0.1]) and SGDR (3 values out of [0.025, 0.05, 0.1]). In line with the results given in the main paper, Figure 6 suggests that SGDR is competitive in terms of anytime performance.
+
+![](images/2f7c89b5cef976568af3fa70f8131ab79330ccbf256f8dee4ff5b844ad58f173.jpg)  
+Figure 7: Training cross-entropy $^ +$ regularization loss (top row), test loss (middle row) and test error (bottom row) on CIFAR-10 (left column) and CIFAR-100 (right column).
+
+![](images/9d2d2cb02d1c568b6f7a9925194f61ec66a074860b937949ee3f6ce968211768.jpg)  
+Figure 8: Top-5 test errors obtained by SGD with momentum with the default learning rate schedule and SGDR with $T _ { 0 } = 1 \mathrm { , } T _ { m u l t } = 2$ on WRN-28-10 trained on a version of ImageNet, with all images from all 1000 classes downsampled to $3 2 \times 3 2$ pixels. The same baseline data augmentation as for the CIFAR datasets is used. Three settings of the initial learning rate are considered: 0.050, 0.015 and 0.005. In contrast to the experiments described in the main paper, here, the dataset is permuted only within 10 subgroups each formed from 100 classes which makes good generalization much harder to achieve for both algorithms. An interpretation of SGDR results given here might be that while the initial learning rate seems to be very important, SGDR reduces the problem of improper selection of the latter by scanning / annealing from the initial learning rate to 0.

parse/train/Skq89Scxx/Skq89Scxx_content_list.json

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+    {
+        "type": "text",
+        "text": "SGDR: STOCHASTIC GRADIENT DESCENT WITH WARM RESTARTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Ilya Loshchilov & Frank Hutter ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "University of Freiburg   \nFreiburg, Germany,   \n{ilya,fh}@cs.uni-freiburg.de ",
+        "bbox": [
+            184,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
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+            277
+        ],
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+    },
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+        "type": "text",
+        "text": "Restart techniques are common in gradient-free optimization to deal with multimodal functions. Partial warm restarts are also gaining popularity in gradientbased optimization to improve the rate of convergence in accelerated gradient schemes to deal with ill-conditioned functions. In this paper, we propose a simple warm restart technique for stochastic gradient descent to improve its anytime performance when training deep neural networks. We empirically study its performance on the CIFAR-10 and CIFAR-100 datasets, where we demonstrate new state-of-the-art results at $3 . 1 4 \\%$ and $1 6 . 2 1 \\%$ , respectively. We also demonstrate its advantages on a dataset of EEG recordings and on a downsampled version of the ImageNet dataset. Our source code is available at   \nhttps://github.com/loshchil/SGDR ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            336,
+            489
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks (DNNs) are currently the best-performing method for many classification problems, such as object recognition from images (Krizhevsky et al., 2012a; Donahue et al., 2014) or speech recognition from audio data (Deng et al., 2013). Their training on large datasets (where DNNs perform particularly well) is the main computational bottleneck: it often requires several days, even on high-performance GPUs, and any speedups would be of substantial value. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The training of a DNN with $n$ free parameters can be formulated as the problem of minimizing a function $f : \\mathbb { R } ^ { n } \\to \\mathbb { R }$ . The commonly used procedure to optimize $f$ is to iteratively adjust $\\pmb { x } _ { t } \\in \\mathbb { R } ^ { n }$ (the parameter vector at time step $t$ ) using gradient information $\\nabla f _ { t } ( { \\pmb x } _ { t } )$ obtained on a relatively small $t$ -th batch of $b$ datapoints. The Stochastic Gradient Descent (SGD) procedure then becomes an extension of the Gradient Descent (GD) to stochastic optimization of $f$ as follows: ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e71626ea7f98d77754c7d40c2b0eefd43425c57db99d01b1319f6cbb38b1717.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { x } _ { t + 1 } = \\pmb { x } _ { t } - \\eta _ { t } \\nabla f _ { t } ( \\pmb { x } _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta _ { t }$ is a learning rate. One would like to consider second-order information ",
+        "bbox": [
+            173,
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+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/12ba4179395fed59be54b08f28e911684595ca9dcbfdf21650be938d6434e56a.jpg",
+        "text": "$$\n\\begin{array} { r } { { \\pmb x } _ { t + 1 } = { \\pmb x } _ { t } - \\eta _ { t } { \\pmb H } _ { t } ^ { - 1 } \\nabla f _ { t } ( { \\pmb x } _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "but this is often infeasible since the computation and storage of the inverse Hessian $\\pmb { H } _ { t } ^ { - 1 }$ is intractable for large $n$ . The usual way to deal with this problem by using limited-memory quasiNewton methods such as L-BFGS (Liu & Nocedal, 1989) is not currently in favor in deep learning, not the least due to (i) the stochasticity of $\\nabla f _ { t } ( { \\pmb x } _ { t } )$ , (ii) ill-conditioning of $f$ and (iii) the presence of saddle points as a result of the hierarchical geometric structure of the parameter space (Fukumizu & Amari, 2000). Despite some recent progress in understanding and addressing the latter problems (Bordes et al., 2009; Dauphin et al., 2014; Choromanska et al., 2014; Dauphin et al., 2015), state-ofthe-art optimization techniques attempt to approximate the inverse Hessian in a reduced way, e.g., by considering only its diagonal to achieve adaptive learning rates. AdaDelta (Zeiler, 2012) and Adam (Kingma & Ba, 2014) are notable examples of such methods. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/2fdd7dc39a0f5c385884a9c0807862b1960fa602d0f7be2bde9952c4d44d5f22.jpg",
+        "image_caption": [
+            "Figure 1: Alternative schedule schemes of learning rate $\\eta _ { t }$ over batch index $t$ : default schemes with $\\eta _ { 0 } = 0 . 1$ (blue line) and $\\eta _ { 0 } = 0 . 0 5$ (red line) as used by Zagoruyko & Komodakis (2016); warm restarts simulated every $T _ { 0 } = 5 0$ (green line), $T _ { 0 } = 1 0 0$ (black line) and $T _ { 0 } = 2 0 0$ (grey line) epochs with $\\eta _ { t }$ decaying during $i$ -th run from $\\eta _ { m a x } ^ { i } = 0 . 0 5$ to $\\eta _ { m i n } ^ { i } = 0$ according to eq. (5); warm restarts starting from epoch $T _ { 0 } = 1$ (dark green line) and $T _ { 0 } = 1 0$ (magenta line) with doubling $( T _ { m u l t } = 2$ ) periods $T _ { i }$ at every new warm restart. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Intriguingly enough, the current state-of-the-art results on CIFAR-10, CIFAR-100, SVHN, ImageNet, PASCAL VOC and MS COCO datasets were obtained by Residual Neural Networks (He et al., 2015; Huang et al., 2016c; He et al., 2016; Zagoruyko & Komodakis, 2016) trained without the use of advanced methods such as AdaDelta and Adam. Instead, they simply use SGD with momentum 1: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c91be6b271b73067af0d6e2772d157391cf9785d53e93b04f637c954c79522a5.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { \\nu } _ { t + 1 } = \\mu _ { t } \\pmb { \\nu } _ { t } - \\eta _ { t } \\nabla f _ { t } ( \\pmb { x } _ { t } ) , } \\\\ { \\pmb { x } _ { t + 1 } = \\pmb { x } _ { t } + \\pmb { \\nu } _ { t + 1 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            579,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\nu _ { t }$ is a velocity vector initially set to $\\pmb { \\theta }$ , $\\eta _ { t }$ is a decreasing learning rate and $\\mu _ { t }$ is a momentum rate which defines the trade-off between the current and past observations of $\\nabla f _ { t } ( { \\pmb x } _ { t } )$ . The main difficulty in training a DNN is then associated with the scheduling of the learning rate and the amount of L2 weight decay regularization employed. A common learning rate schedule is to use a constant learning rate and divide it by a fixed constant in (approximately) regular intervals. The blue line in Figure 1 shows an example of such a schedule, as used by Zagoruyko & Komodakis (2016) to obtain the state-of-the-art results on CIFAR-10, CIFAR-100 and SVHN datasets. ",
+        "bbox": [
+            173,
+            643,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose to periodically simulate warm restarts of SGD, where in each restart the learning rate is initialized to some value and is scheduled to decrease. Four different instantiations of this new learning rate schedule are visualized in Figure 1. Our empirical results suggest that SGD with warm restarts requires $2 \\times$ to $4 \\times$ fewer epochs than the currently-used learning rate schedule schemes to achieve comparable or even better results. Furthermore, combining the networks obtained right before restarts in an ensemble following the approach proposed by Huang et al. (2016a) improves our results further to $3 . 1 4 \\%$ for CIFAR-10 and $1 6 . 2 1 \\%$ for CIFAR-100. We also demonstrate its advantages on a dataset of EEG recordings and on a downsampled version of the ImageNet dataset. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 RESTARTS IN GRADIENT-FREE OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            138,
+            529,
+            151
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "When optimizing multimodal functions one may want to find all global and local optima. The tractability of this task depends on the landscape of the function at hand and the budget of function evaluations. Gradient-free optimization approaches based on niching methods (Preuss, 2015) usually can deal with this task by covering the search space with dynamically allocated niches of local optimizers. However, these methods usually work only for relatively small search spaces, e.g., $n < 1 0$ , and do not scale up due to the curse of dimensionality (Preuss, 2010). Instead, the current state-of-the-art gradient-free optimizers employ various restart mechanisms (Hansen, 2009; Loshchilov et al., 2012). One way to deal with multimodal functions is to iteratively sample a large number $\\lambda$ of candidate solutions, make a step towards better solutions and slowly shape the sampling distribution to maximize the likelihood of successful steps to appear again (Hansen & Kern, 2004). The larger the $\\lambda$ , the more global search is performed requiring more function evaluations. In order to achieve good anytime performance, it is common to start with a small $\\lambda$ and increase it (e.g., by doubling) after each restart. This approach works best on multimodal functions with a global funnel structure and also improves the results on ill-conditioned problems where numerical issues might lead to premature convergence when $\\lambda$ is small (Hansen, 2009). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 RESTARTS IN GRADIENT-BASED OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            401,
+            540,
+            414
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Gradient-based optimization algorithms such as BFGS can also perform restarts to deal with multimodal functions (Ros, 2009). In large-scale settings when the usual number of variables $n$ is on the order of $1 0 ^ { 3 } - 1 0 ^ { 9 }$ , the availability of gradient information provides a speedup of a factor of $n$ w.r.t. gradient-free approaches. Warm restarts are usually employed to improve the convergence rate rather than to deal with multimodality: often it is sufficient to approach any local optimum to a given precision and in many cases the problem at hand is unimodal. Fletcher & Reeves (1964) proposed to flesh the history of conjugate gradient method every $n$ or $( n + 1 )$ iterations. Powell (1977) proposed to check whether enough orthogonality between $\\nabla f ( { \\pmb x } _ { t - 1 } )$ and $\\nabla f ( \\pmb { x } _ { t } )$ has been lost to warrant another warm restart. Recently, O’Donoghue & Candes (2012) noted that the iterates of accelerated gradient schemes proposed by Nesterov (1983; 2013) exhibit a periodic behavior if momentum is overused. The period of the oscillations is proportional to the square root of the local condition number of the (smooth convex) objective function. The authors showed that fixed warm restarts of the algorithm with a period proportional to the conditional number achieves the optimal linear convergence rate of the original accelerated gradient scheme. Since the condition number is an unknown parameter and its value may vary during the search, they proposed two adaptive warm restart techniques (O’Donoghue & Candes, 2012): ",
+        "bbox": [
+            174,
+            430,
+            825,
+            650
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• The function scheme restarts whenever the objective function increases. ",
+        "bbox": [
+            217,
+            670,
+            707,
+            684
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• The gradient scheme restarts whenever the angle between the momentum term and the negative gradient is obtuse, i.e, when the momentum seems to be taking us in a bad direction, as measured by the negative gradient at that point. This scheme resembles the one of Powell (1977) for the conjugate gradient method. ",
+        "bbox": [
+            217,
+            695,
+            823,
+            751
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "O’Donoghue & Candes (2012) showed (and it was confirmed in a set of follow-up works) that these simple schemes provide an acceleration on smooth functions and can be adjusted to accelerate stateof-the-art methods such as FISTA on nonsmooth functions. ",
+        "bbox": [
+            178,
+            770,
+            823,
+            811
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Smith (2015; 2016) recently introduced cyclical learning rates for deep learning, his approach is closely-related to our approach in its spirit and formulation but does not focus on restarts. ",
+        "bbox": [
+            176,
+            819,
+            823,
+            847
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Yang & Lin (2015) showed that Stochastic subGradient Descent with restarts can achieve a linear convergence rate for a class of non-smooth and non-strongly convex optimization problems where the epigraph of the objective function is a polyhedron. In contrast to our work, they never increase the learning rate to perform restarts but decrease it geometrically at each epoch. To perform restarts, they periodically reset the current solution to the averaged solution from the previous epoch. ",
+        "bbox": [
+            176,
+            854,
+            823,
+            922
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 STOCHASTIC GRADIENT DESCENT WITH WARM RESTARTS (SGDR) ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            766,
+            118
+        ],
+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "The existing restart techniques can also be used for stochastic gradient descent if the stochasticity is taken into account. Since gradients and loss values can vary widely from one batch of the data to another, one should denoise the incoming information: by considering averaged gradients and losses, e.g., once per epoch, the above-mentioned restart techniques can be used again. ",
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+    {
+        "type": "text",
+        "text": "In this work, we consider one of the simplest warm restart approaches. We simulate a new warmstarted run / restart of SGD once $T _ { i }$ epochs are performed, where $i$ is the index of the run. Importantly, the restarts are not performed from scratch but emulated by increasing the learning rate $\\eta _ { t }$ while the old value of $\\mathbf { \\boldsymbol { x } } _ { t }$ is used as an initial solution. The amount of this increase controls to which extent the previously acquired information (e.g., momentum) is used. ",
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+    {
+        "type": "text",
+        "text": "Within the $i$ -th run, we decay the learning rate with a cosine annealing for each batch as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c6dc5dabd64eec38ffcf2c6d0921853fb89bdccb49120b43a02b33d445126984.jpg",
+        "text": "$$\n\\eta _ { t } = \\eta _ { m i n } ^ { i } + \\frac { 1 } { 2 } ( \\eta _ { m a x } ^ { i } - \\eta _ { m i n } ^ { i } ) ( 1 + \\cos ( \\frac { T _ { c u r } } { T _ { i } } \\pi ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            309,
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+            340
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta _ { m i n } ^ { i }$ and $\\eta _ { m a x } ^ { i }$ are ranges for the learning rate, and $T _ { c u r }$ accounts for how many epochs have been performed since the last restart. Since $T _ { c u r }$ is updated at each batch iteration $t$ , it can take discredited values such as 0.1, 0.2, etc. Thus, $\\eta _ { t } = \\eta _ { m a x } ^ { i }$ when $t = 0$ and $T _ { c u r } = 0$ . Once $T _ { c u r } = T _ { i }$ , the cos function will output $- 1$ and thus $\\eta _ { t } = \\eta _ { m i n } ^ { i }$ . The decrease of the learning rate is shown in Figure 1 for fixed $T _ { i } = 5 0$ , $T _ { i } = 1 0 0$ and $T _ { i } ~ = ~ 2 0 0$ ; note that the logarithmic axis obfuscates the typical shape of the cosine function. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In order to improve anytime performance, we suggest an option to start with an initially small $T _ { i }$ and increase it by a factor of $T _ { m u l t }$ at every restart (see, e.g., Figure 1 for $T _ { 0 } = 1 , T _ { m u l t } = 2$ and $T _ { 0 } = 1 0 , T _ { m u l t } = 2 )$ . It might be of great interest to decrease $\\eta _ { m a x } ^ { i }$ and $\\eta _ { m i n } ^ { i }$ at every new restart. However, for the sake of simplicity, here, we keep $\\eta _ { m a x } ^ { i }$ and $\\eta _ { m i n } ^ { i }$ the same for every $i$ to reduce the number of hyperparameters involved. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since our simulated warm restarts (the increase of the learning rate) often temporarily worsen performance, we do not always use the last $\\mathbf { } _ { \\pmb { x } _ { t } }$ as our recommendation for the best solution (also called the incumbent solution). While our recommendation during the first run (before the first restart) is indeed the last $\\mathbf { } _ { \\pmb { x } _ { t } }$ , our recommendation after this is a solution obtained at the end of the last performed run at ηt = ηimin. We emphasize that with the help of this strategy, our method does not require a separate validation data set to determine a recommendation. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETTINGS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We consider the problem of training Wide Residual Neural Networks (WRNs; see Zagoruyko & Komodakis (2016) for details) on the CIFAR-10 and CIFAR-100 datasets (Krizhevsky, 2009). We will use the abbreviation WRN-d- $k$ to denote a WRN with depth $d$ and width $k$ . Zagoruyko & Komodakis (2016) obtained the best results with a WRN-28-10 architecture, i.e., a Residual Neural Network with $d \\ : = \\ : 2 8$ layers and $k = 1 0$ times more filters per layer than used in the original Residual Neural Networks (He et al., 2015; 2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The CIFAR-10 and CIFAR-100 datasets (Krizhevsky, 2009) consist of $3 2 \\times 3 2$ color images drawn from 10 and 100 classes, respectively, split into 50,000 train and 10,000 test images. For image preprocessing Zagoruyko & Komodakis (2016) performed global contrast normalization and ZCA whitening. For data augmentation they performed horizontal flips and random crops from the image padded by 4 pixels on each side, filling missing pixels with reflections of the original image. ",
+        "bbox": [
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+        "type": "text",
+        "text": "For training, Zagoruyko & Komodakis (2016) used SGD with Nesterov’s momentum with initial learning rate set to $\\eta _ { 0 } ~ = ~ 0 . 1$ , weight decay to 0.0005, dampening to 0, momentum to 0.9 and minibatch size to 128. The learning rate is dropped by a factor of 0.2 at 60, 120 and 160 epochs, with a total budget of 200 epochs. We reproduce the results of Zagoruyko & Komodakis (2016) with the same settings except that i) we subtract per-pixel mean only and do not use ZCA whitening; ii) we use SGD with momentum as described by eq. (3-4) and not Nesterov’s momentum. ",
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+    {
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+        "img_path": "images/67fc79f25b3e3da44eb17b55361645c16f330c07cab651bdaa4489e1d819cf13.jpg",
+        "image_caption": [
+            "Figure 2: Test errors on CIFAR-10 (left column) and CIFAR-100 (right column) datasets. Note that for SGDR we only plot the recommended solutions. The top and middle rows show the same results on WRN-28-10, with the middle row zooming into the good performance region of low test error. The bottom row shows performance with a wider network, WRN-28-20. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The results of the default learning rate schedules of Zagoruyko & Komodakis (2016) with $\\eta _ { 0 } = 0 . 1$ and $\\eta _ { 0 } = 0 . 0 5$ are depicted by the blue and red lines, respectively. The schedules of $\\eta _ { t }$ used in SGDR are shown with i) restarts every $T _ { 0 } = 5 0$ epochs (green line); ii) restarts every $T _ { 0 } = 1 0 0$ epochs (black line); iii) restarts every $T _ { 0 } = 2 0 0$ epochs (gray line); iv) restarts with doubling $T _ { m u l t } = 2 $ ) periods of restarts starting from the first epoch ( ${ { T } _ { 0 } } \\ = \\ 1$ , dark green line); and v) restarts with doubling $T _ { m u l t } = 2$ ) periods of restarts starting from the tenth epoch $T _ { 0 } = 1 0$ , magenta line). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The schedule of $\\eta _ { t }$ used by Zagoruyko & Komodakis (2016) is depicted by the blue line in Figure 1. The same schedule but with $\\eta _ { 0 } = 0 . 0 5$ is depicted by the red line. The schedule of $\\eta _ { t }$ used in SGDR is also shown in Figure 1, with two initial learning rates $T _ { 0 }$ and two restart doubling periods. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/99bd41e721edbaaca6dc0874a9e81855776daeb55ebc6ae871c16fc40d7a15b8.jpg",
+        "table_caption": [
+            "Table 1: Test errors of different methods on CIFAR-10 and CIFAR-100 with moderate data augmentation (flip/translation). In the second column $k$ is a widening factor for WRNs. Note that the computational and memory resources used to train all WRN-28-10 are the same. In all other cases they are different, but WRNs are usually faster than original ResNets to achieve the same accuracy (e.g., up to a factor of 8 according to Zagoruyko & Komodakis (2016)). Bold text is used only to highlight better results and is not based on statistical tests (too few runs). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>depth-k</td><td rowspan=1 colspan=1># params</td><td rowspan=1 colspan=1># runs</td><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>CIFAR-100</td></tr><tr><td rowspan=2 colspan=1>original-ResNet (He et al.,2015)</td><td rowspan=1 colspan=3>110</td><td rowspan=1 colspan=1>1.7M</td><td rowspan=1 colspan=1>mean of 5</td><td rowspan=1 colspan=1>6.43</td><td rowspan=1 colspan=1>25.16</td></tr><tr><td rowspan=1 colspan=3>1202</td><td rowspan=1 colspan=1>10.2M</td><td rowspan=1 colspan=1>mean of 5</td><td rowspan=1 colspan=1>7.93</td><td rowspan=1 colspan=1>27.82</td></tr><tr><td rowspan=2 colspan=1>stoc-depth (Huang et al., 2016c)</td><td rowspan=2 colspan=3>1101202</td><td rowspan=2 colspan=1>1.7M10.2M</td><td rowspan=2 colspan=1>1 run1 run</td><td rowspan=1 colspan=1>5.23</td><td rowspan=2 colspan=1>24.58n/a</td></tr><tr><td rowspan=1 colspan=2>02</td><td rowspan=1 colspan=1>4.91</td></tr><tr><td rowspan=3 colspan=1>pre-act-ResNet (He et al.,2016)</td><td rowspan=3 colspan=3>1101641001</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>1.7M</td><td rowspan=1 colspan=1>med. of5</td><td rowspan=1 colspan=1>6.37</td></tr><tr><td rowspan=2 colspan=1>1.7M10.2M</td><td rowspan=2 colspan=1>med. of 5med. of 5</td><td rowspan=1 colspan=1>5.46</td><td rowspan=2 colspan=1>24.3322.71</td></tr><tr><td rowspan=1 colspan=1>4.62</td></tr><tr><td rowspan=2 colspan=1>WRN (Zagoruyko &amp; Komodakis, 2016) with dropout</td><td rowspan=2 colspan=3>16-828-1028-10</td><td rowspan=2 colspan=1>11.0M36.5M36.5M</td><td rowspan=2 colspan=1>1 run1 run1 run</td><td rowspan=1 colspan=1>4.81</td><td rowspan=2 colspan=1>22.0720.5020.04</td></tr><tr><td rowspan=1 colspan=1>4.17n/a</td></tr><tr><td rowspan=2 colspan=1>WRN (ours)default with no = 0.1</td><td rowspan=1 colspan=3></td><td rowspan=2 colspan=1>36.5M</td><td rowspan=2 colspan=1>med. of 5</td><td rowspan=2 colspan=1>4.24</td><td rowspan=2 colspan=1>20.33</td></tr><tr><td rowspan=1 colspan=3>28-10</td></tr><tr><td rowspan=1 colspan=1>default with no = 0.05</td><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>4.13</td><td rowspan=1 colspan=1>20.21</td></tr><tr><td rowspan=1 colspan=1>T = 50,Tmult =1</td><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>4.17</td><td rowspan=1 colspan=1>19.99</td></tr><tr><td rowspan=1 colspan=1>To = 100,Tmult =1</td><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>4.07</td><td rowspan=1 colspan=1>19.87</td></tr><tr><td rowspan=1 colspan=1>To = 200,Tmult = 1</td><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>3.86</td><td rowspan=1 colspan=1>19.98</td></tr><tr><td rowspan=2 colspan=1>To =1,Tmult = 2To =10,Tmult = 2</td><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>4.09</td><td rowspan=1 colspan=1>19.74</td></tr><tr><td rowspan=1 colspan=3>28-10</td><td rowspan=1 colspan=1>36.5M</td><td rowspan=1 colspan=1>med. of 5</td><td rowspan=1 colspan=1>4.03</td><td rowspan=1 colspan=1>19.58</td></tr><tr><td rowspan=1 colspan=1>default with no = 0.1</td><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>4.08</td><td rowspan=1 colspan=1>19.53</td></tr><tr><td rowspan=1 colspan=1>default with no = 0.05</td><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>3.96</td><td rowspan=1 colspan=1>19.67</td></tr><tr><td rowspan=1 colspan=1>To = 50,Tmult = 1</td><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>4.01</td><td rowspan=1 colspan=1>19.28</td></tr><tr><td rowspan=1 colspan=1>To = 100,Tmult = 1</td><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>3.77</td><td rowspan=1 colspan=1>19.24</td></tr><tr><td rowspan=3 colspan=1>To = 200,Tmult =1To =1,Tmult = 2To = 10,Tmult = 2</td><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>3.66</td><td rowspan=1 colspan=1>19.69</td></tr><tr><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>3.91</td><td rowspan=1 colspan=1>18.90</td></tr><tr><td rowspan=1 colspan=3>28-20</td><td rowspan=1 colspan=1>145.8M</td><td rowspan=1 colspan=1>med. of 2</td><td rowspan=1 colspan=1>3.74</td><td rowspan=1 colspan=1>18.70</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 SINGLE-MODEL RESULTS",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 1 shows that our experiments reproduce the results given by Zagoruyko & Komodakis (2016) for WRN-28-10 both on CIFAR-10 and CIFAR-100. These “default” experiments with $\\eta _ { 0 } = 0 . 1$ and $\\eta _ { 0 } = 0 . 0 5$ correspond to the blue and red lines in Figure 2. The results for $\\eta _ { 0 } = 0 . 0 5$ show better performance, and therefore we use $\\eta _ { 0 } = 0 . 0 5$ in our later experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "SGDR with $T _ { 0 } = 5 0$ , $T _ { 0 } = 1 0 0$ and $T _ { 0 } = 2 0 0$ for $T _ { m u l t } = 1$ perform warm restarts every 50, 100 and 200 epochs, respectively. A single run of SGD with the schedule given by eq. (5) for $T _ { 0 } = 2 0 0$ shows the best results suggesting that the original schedule of WRNs might be suboptimal w.r.t. the test error in these settings. However, the same setting with $T _ { 0 } = 2 0 0$ leads to the worst anytime performance except for the very last epochs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "SGDR with $T _ { 0 } = 1 \\mathrm { , } T _ { m u l t } = 2$ and $T _ { 0 } = 1 0 , T _ { m u l t } = 2$ performs its first restart after 1 and 10 epochs, respectively. Then, it doubles the maximum number of epochs for every new restart. The main purpose of this doubling is to reach good test error as soon as possible, i.e., achieve good anytime performance. Figure 2 shows that this is achieved and test errors around $4 \\%$ on CIFAR-10 and around $20 \\%$ on CIFAR-100 can be obtained about 2-4 times faster than with the default schedule used by Zagoruyko & Komodakis (2016). ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/9483ebc74be2a2955fa389608a8a576cae9c594a7486cafbf9d0969a764ed4dd.jpg",
+        "image_caption": [
+            "Figure 3: Test errors of ensemble models built from $N$ runs of SGDR on WRN-28-10 with $M$ model snapshots per run made at epochs 150, 70 and 30 (right before warm restarts of SGDR as suggested by Huang et al. (2016a)). When $M { = } 1$ (respectively, $M { = } 2$ ), we aggregate probabilities of softmax layers of snapshot models at epoch index 150 (respectively, at epoch indexes 150 and 70). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/5c9a3578fca5f274aa703c2f856c646793f579affa7267b2932694eb45d9c4cf.jpg",
+        "table_caption": [
+            "Table 2: Test errors of ensemble models on CIFAR-10 and CIFAR-100 datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CIFAR-10</td><td>CIFAR-100</td></tr><tr><td>N = 1 run of WRN-28-10 with M = 1 snapshot (median of 16 runs)</td><td>4.03</td><td>19.57</td></tr><tr><td>N = 1 run of WRN-28-10 with M = 3 snapshots per run</td><td>3.51</td><td>17.75</td></tr><tr><td>N = 3 runs of WRN-28-10 with M = 3 snapshots per run</td><td>3.25</td><td>16.64</td></tr><tr><td>N = 16 runs of WRN-28-10 with M= 3 snapshots per run</td><td>3.14</td><td>16.21</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since SGDR achieves good performance faster, it may allow us to train larger networks. We therefore investigated whether results on CIFAR-10 and CIFAR-100 can be further improved by making WRNs two times wider, i.e., by training WRN-28-20 instead of WRN-28-10. Table 1 shows that the results indeed improved, by about $0 . 2 5 \\%$ on CIFAR-10 and by about $0 . 5 \\substack { - 1 . 0 \\% }$ on CIFAR-100. While network architecture WRN-28-20 requires roughly three-four times more computation than WRN-28-10, the aggressive learning rate reduction of SGDR nevertheless allowed us to achieve a better error rate in the same time on WRN-28-20 as we spent on 200 epochs of training on WRN28-10. Specifically, Figure 2 (right middle and right bottom) show that after only 50 epochs, SGDR (even without restarts, using $T _ { 0 } = 5 0 , T _ { m u l t } = 1 )$ achieved an error rate below $19 \\%$ (whereas none of the other learning methods performed better than $1 9 . 5 \\%$ on WRN-28-10). We therefore have hope that – by enabling researchers to test new architectures faster – SGDR’s good anytime performance may also lead to improvements of the state of the art. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In a final experiment for SGDR by itself, Figure 7 in the appendix compares SGDR and the default schedule with respect to training and test performance. As the figure shows, SGDR optimizes training loss faster than the standard default schedule until about epoch 120. After this, the default schedule overfits, as can be seen by an increase of the test error both on CIFAR-10 and CIFAR-100 (see, e.g., the right middle plot of Figure 7). In contrast, we only witnessed very mild overfitting for SGDR. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 ENSEMBLE RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our initial arXiv report on SGDR (Loshchilov & Hutter, 2016) inspired a follow-up study by Huang et al. (2016a) in which the authors suggest to take $M$ snapshots of the models obtained by SGDR (in their paper referred to as cyclical learning rate schedule and cosine annealing cycles) right before $M$ last restarts and to use those to build an ensemble, thereby obtaining ensembles “for free” (in contrast to having to perform multiple independent runs). The authors demonstrated new state-ofthe-art results on CIFAR datasets by making ensembles of DenseNet models (Huang et al., 2016b). Here, we investigate whether their conclusions hold for WRNs used in our study. We used WRN28-10 trained by SGDR with $T _ { 0 } = 1 0 , T _ { m u l t } = 2$ as our baseline model. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Figure 3 and Table 2 aggregate the results of our study. The original test error of $4 . 0 3 \\%$ on CIFAR-10 and $1 9 . 5 7 \\%$ on CIFAR-100 (median of 16 runs) can be improved to $3 . 5 1 \\%$ on CIFAR-10 and $1 7 . 7 5 \\%$ on CIFAR-100 when $M = 3$ snapshots are taken at epochs 30, 70 and 150: when the learning rate of SGDR with $T _ { 0 } = 1 0 , T _ { m u l t } = 2$ is scheduled to achieve 0 (see Figure 1) and the models are used with uniform weights to build an ensemble. To achieve the same result, one would have to aggregate $N = 3$ models obtained at epoch 150 of $N = 3$ independent runs (see $N = 3 , M = 1$ in Figure 3). Thus, the aggregation from snapshots provides a 3-fold speedup in these settings because additional ( $M > 1$ -th) snapshots from a single SGDR run are computationally free. Interestingly, aggregation of models from independent runs (when $N > 1$ and $M = 1$ ) does not scale up as well as from $M > 1$ snapshots of independent runs when the same number of models is considered: the case of $N = 3$ and $M = 3$ provides better performance than the cases of $M = 1$ with $N = 1 8$ and $N = 2 1$ . Not only the number of snapshots $M$ per run but also their origin is crucial. Thus, naively building ensembles from models obtained at last epochs only (i.e., $M = 3$ snapshots at epochs 148, 149, 150) did not improve the results (i.e., the baseline of $M = 1$ snapshot at 150) thereby confirming the conclusion of Huang et al. (2016a) that snapshots of SGDR provide a useful diversity of predictions for ensembles. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Three runs $N = 3$ ) of SGDR with $M = 3$ snapshots per run are sufficient to greatly improve the results to $3 . 2 5 \\%$ on CIFAR-10 and $1 6 . 6 4 \\%$ on CIFAR-100 outperforming the results of Huang et al. (2016a). By increasing $N$ to 16 one can achieve $3 . 1 4 \\%$ on CIFAR-10 and $1 6 . 2 1 \\%$ on CIFAR-100. We believe that these results could be further improved by considering better baseline models than WRN-28-10 (e.g., WRN-28-20). ",
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+    {
+        "type": "text",
+        "text": "4.4 EXPERIMENTS ON A DATASET OF EEG RECORDINGS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To demonstrate the generality of SGDR, we also considered a very different domain: a dataset of electroencephalographic (EEG) recordings of brain activity for classification of actual right and left hand and foot movements of 14 subjects with roughly 1000 trials per subject. The best classification results obtained with the original pipeline based on convolutional neural networks [R. Schirrmeister et al. Convolutional neural networks for EEG analysis: Design choices, training strategies, and feature visualization., under review at Neuroimage] were used as our reference. First, we compared the baseline learning rate schedule with different settings of the total number of epochs and initial learning rates (see Figure 4). When 30 epochs were considered, we dropped the learning rate by a factor of 10 at epoch indexes 10, 15 and 20. As expected, with more epochs used and a similar (budget proportional) schedule better results can be achieved. Alternatively, one can consider SGDR and get a similar final performance while having a better anytime performance without defining the total budget of epochs beforehand. ",
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+        "type": "text",
+        "text": "Similarly to our results on the CIFAR datasets, our experiments with the EEG data confirm that snapshots are useful and the median reference error (about $9 \\%$ ) can be improved i) by $1 { - } 2 \\%$ when model snapshots of a single run are considered, and ii) by $2 { - } 3 \\%$ when model snapshots from both hyperparameter settings are considered. The latter would correspond to $N = 2$ in Section (4.3). ",
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+    {
+        "type": "text",
+        "text": "4.5 PRELIMINARY EXPERIMENTS ON A DOWNSAMPLED IMAGENET DATASET",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In order to additionally validate our SGDR on a larger dataset, we constructed a downsampled version of the ImageNet dataset [P. Chrabaszcz, I. Loshchilov and F. Hutter. A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets., in preparation]. In contrast to earlier attempts (Pouransari & Ghili, 2015), our downsampled ImageNet contains exactly the same images from 1000 classes as the original ImageNet but resized with box downsampling to $3 2 \\times 3 2$ pixels. Thus, this dataset is substantially harder than the original ImageNet dataset because the average number of pixels per image is now two orders of magnitude smaller. The new dataset is also more difficult than the CIFAR datasets because more classes are used and the relevant objects to be classified often cover only a tiny subspace of the image and not most of the image as in the CIFAR datasets. ",
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+    {
+        "type": "image",
+        "img_path": "images/5dfdedd88ded2a58cd5402059f23bdeb3b206e82cfb5df1c9c28a04b45a45e3e.jpg",
+        "image_caption": [
+            "Figure 4: (Top) Improvements obtained by the baseline learning rate schedule and SGDR w.r.t. the best known reference classification error on a dataset of electroencephalographic (EEG) recordings of brain activity for classification of actual right and left hand and foot movements of 14 subjects with roughly 1000 trials per subject. Both considered approaches were tested with the initial learning rate $l r = 0 . 0 2 5$ (Top-Left) and $l r = 0 . 0 5$ (Top-Right). Note that the baseline approach is considered with different settings of the total number of epochs: 30, 60, . . ., 480. (Bottom) SGDR with $l r = 0 . 0 2 5$ and $l r = 0 . 0 5$ without and with $M$ model snapshots taken at the last $M = n r / 2$ restarts, where $n r$ is the total number of restarts. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We benchmarked SGD with momentum with the default learning rate schedule, SGDR with $T _ { 0 } =$ $1 , T _ { m u l t } = 2$ and SGDR with $T _ { 0 } = 1 0 , T _ { m u l t } = 2$ on WRN-28-10, all trained with 4 settings of the initial learning rate $\\eta _ { m a x } ^ { i }$ : 0.050, 0.025, 0.01 and 0.005. We used the same data augmentation procedure as for the CIFAR datasets. Similarly to the results on the CIFAR datasets, Figure 5 shows that SGDR demonstrates better anytime performance. SGDR with $T _ { 0 } = 1 0 , T _ { m u l t } \\stackrel { = } { = } 2 , \\eta _ { m a x } ^ { i } =$ 0.01 achieves top-1 error of $3 9 . 2 4 \\%$ and top-5 error of $1 7 . 1 7 \\%$ matching the original results by AlexNets $4 0 . 7 \\%$ and $1 8 . 2 \\%$ , respectively) obtained on the original ImageNet with full-size images of ca. 50 times more pixels per image (Krizhevsky et al., 2012b). Interestingly, when the dataset is permuted only within 10 subgroups each formed from 100 classes, SGDR also demonstrates better results (see Figure 8 in the Supplementary Material). An interpretation of this might be that while the initial learning rate seems to be very important, SGDR reduces the problem of improper selection of the latter by scanning / annealing from the initial learning rate to 0. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Clearly, longer runs (more than 40 epochs considered in this preliminary experiment) and hyperparameter tuning of learning rates, regularization and other hyperparameters shall further improve the results. ",
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+    {
+        "type": "image",
+        "img_path": "images/edcfbdc72ca22c95d6c0b06c3715ba2b80be5349cf0ca5ae5f976abadf982d72.jpg",
+        "image_caption": [
+            "Figure 5: Top-1 and Top-5 test errors obtained by SGD with momentum with the default learning rate schedule, SGDR with $T _ { 0 } = 1 , T _ { m u l t } = 2$ and SGDR with $T _ { 0 } = 1 0$ , $T _ { m u l t } = 2$ on WRN-28-10 trained on a version of ImageNet, with all images from all 1000 classes downsampled to $3 2 \\times 3 2$ pixels. The same baseline data augmentation as for the CIFAR datasets is used. Four settings of the initial learning rate are considered: 0.050, 0.025, 0.01 and 0.005. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our results suggest that even without any restarts the proposed aggressive learning rate schedule given by eq. (5) is competitive w.r.t. the default schedule when training WRNs on the CIFAR10 (e.g., for $T _ { 0 } = 2 0 0 , T _ { m u l t } = 1 )$ and CIFAR-100 datasets. In practice, the proposed schedule requires only two hyper-parameters to be defined: the initial learning rate and the total number of epochs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We found that the anytime performance of SGDR remain similar when shorter epochs are considered (see section 8.1 in the Supplemenary Material). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "One should not suppose that the parameter values used in this study and many other works with (Residual) Neural Networks are selected to demonstrate the fastest decrease of the training error. Instead, the best validation or $/$ and test errors are in focus. Notably, the validation error is rarely used when training Residual Neural Networks because the recommendation is defined by the final solution (in our approach, the final solution of each run). One could use the validation error to determine the optimal initial learning rate and then run on the whole dataset; this could further improve results. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The main purpose of our proposed warm restart scheme for SGD is to improve its anytime performance. While we mentioned that restarts can be useful to deal with multi-modal functions, we do not claim that we observe any effect related to multi-modality. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "As we noted earlier, one could decrease $\\eta _ { m a x } ^ { i }$ and $\\eta _ { m i n } ^ { i }$ at every new warm restart to control the amount of divergence. If new restarts are worse than the old ones w.r.t. validation error, then one might also consider going back to the last best solution and perform a new restart with adjusted hyperparameters. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our results reproduce the finding by Huang et al. (2016a) that intermediate models generated by SGDR can be used to build efficient ensembles at no cost. This finding makes SGDR especially attractive for scenarios when ensemble building is considered. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we investigated a simple warm restart mechanism for SGD to accelerate the training of DNNs. Our SGDR simulates warm restarts by scheduling the learning rate to achieve competitive results on CIFAR-10 and CIFAR-100 roughly two to four times faster. We also achieved new stateof-the-art results with SGDR, mainly by using even wider WRNs and ensembles of snapshots from ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "SGDR’s trajectory. Future empirical studies should also consider the SVHN, ImageNet and MS COCO datasets, for which Residual Neural Networks showed the best results so far. Our preliminary results on a dataset of EEG recordings suggest that SGDR delivers better and better results as we carry out more restarts and use more model snapshots. The results on our downsampled ImageNet dataset suggest that SGDR might also reduce the problem of learning rate selection because the annealing and restarts of SGDR scan / consider a range of learning rate values. Future work should consider warm restarts for other popular training algorithms such as AdaDelta (Zeiler, 2012) and Adam (Kingma & Ba, 2014). ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Alternative network structures should be also considered; e.g., soon after our initial arXiv report (Loshchilov & Hutter, 2016), Zhang et al. (2016); Huang et al. (2016b); Han et al. (2016) reported that WRNs models can be replaced by more memory-efficient models. Thus, it should be tested whether our results for individual models and ensembles can be further improved by using their networks instead of WRNs. Deep compression methods (Han et al., 2015) can be used to reduce the time and memory costs of DNNs and their ensembles. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "7 ACKNOWLEDGMENTS ",
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This work was supported by the German Research Foundation (DFG), under the BrainLinksBrainTools Cluster of Excellence (grant number EXC 1086). We thank Gao Huang, Kilian Quirin Weinberger, Jost Tobias Springenberg, Mark Schmidt and three anonymous reviewers for their helpful comments and suggestions. ",
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+        "text": "Leslie N Smith. No more pesky learning rate guessing games. arXiv preprint arXiv:1506.01186, 2015.   \nLeslie N Smith. Cyclical learning rates for training neural networks. arXiv preprint arXiv:1506.01186v3, 2016.   \nTianbao Yang and Qihang Lin. Stochastic subgradient methods with linear convergence for polyhedral convex optimization. arXiv preprint arXiv:1510.01444, 2015.   \nSergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016.   \nMatthew D Zeiler. Adadelta: An adaptive learning rate method. arXiv preprint arXiv:1212.5701, 2012.   \nK. Zhang, M. Sun, T. X. Han, X. Yuan, L. Guo, and T. Liu. Residual Networks of Residual Networks: Multilevel Residual Networks. ArXiv e-prints, August 2016. ",
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+    {
+        "type": "text",
+        "text": "8 SUPPLEMENTARY MATERIAL ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/2e914475e7ea34de85b07237a7c7c57ae81524b5e8a2ef32ab2cf99c2d03e7b8.jpg",
+        "image_caption": [
+            "Figure 6: The median results of 5 runs for the best learning rate settings considered for WRN-28-1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            349,
+            137,
+            637,
+            330
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "8.1 50K VS 100K EXAMPLES PER EPOCH ",
+        "bbox": [
+            176,
+            395,
+            467,
+            409
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Our data augmentation procedure code is inherited from the Lasagne Recipe code for ResNets where flipped images are added to the training set. This doubles the number of training examples per epoch and thus might impact the results because hyperparameter values defined as a function of epoch index have a different meaning. While our experimental results given in Table 1 reproduced the results obtained by Zagoruyko & Komodakis (2016), here we test whether SGDR still makes sense for WRN-28-1 (i.e., ResNet with 28 layers) where one epoch corresponds to $5 0 \\mathrm { k }$ training examples. We investigate different learning rate values for the default learning rate schedule (4 values out of [0.01, 0.025, 0.05, 0.1]) and SGDR (3 values out of [0.025, 0.05, 0.1]). In line with the results given in the main paper, Figure 6 suggests that SGDR is competitive in terms of anytime performance. ",
+        "bbox": [
+            173,
+            420,
+            825,
+            546
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/2f7c89b5cef976568af3fa70f8131ab79330ccbf256f8dee4ff5b844ad58f173.jpg",
+        "image_caption": [
+            "Figure 7: Training cross-entropy $^ +$ regularization loss (top row), test loss (middle row) and test error (bottom row) on CIFAR-10 (left column) and CIFAR-100 (right column). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            189,
+            823,
+            790
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/9d2d2cb02d1c568b6f7a9925194f61ec66a074860b937949ee3f6ce968211768.jpg",
+        "image_caption": [
+            "Figure 8: Top-5 test errors obtained by SGD with momentum with the default learning rate schedule and SGDR with $T _ { 0 } = 1 \\mathrm { , } T _ { m u l t } = 2$ on WRN-28-10 trained on a version of ImageNet, with all images from all 1000 classes downsampled to $3 2 \\times 3 2$ pixels. The same baseline data augmentation as for the CIFAR datasets is used. Three settings of the initial learning rate are considered: 0.050, 0.015 and 0.005. In contrast to the experiments described in the main paper, here, the dataset is permuted only within 10 subgroups each formed from 100 classes which makes good generalization much harder to achieve for both algorithms. An interpretation of SGDR results given here might be that while the initial learning rate seems to be very important, SGDR reduces the problem of improper selection of the latter by scanning / annealing from the initial learning rate to 0. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
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+        ],
+        "page_idx": 15
+    }
+]

parse/train/ZTFeSBIX9C/ZTFeSBIX9C.md

@@ -0,0 +1,352 @@
+# UNDERSTANDING AND IMPROVING LEXICAL CHOICE IN NON-AUTOREGRESSIVE TRANSLATION
+
+Liang Ding1∗, Longyue Wang2, Xuebo Liu3, Derek F. Wong3, Dacheng Tao1 & Zhaopeng $\mathbf { T } \mathbf { u } ^ { 2 }$
+
+1The University of Sydney 2Tencent AI Lab 3University of Macau {ldin3097,dacheng.tao}@sydney.edu.au, nlp2ct.xuebo@gmail.com, {vinnylywang,zptu}@tencent.com, derekfw@um.edu.com
+
+# ABSTRACT
+
+Knowledge distillation (KD) is essential for training non-autoregressive translation (NAT) models by reducing the complexity of the raw data with an autoregressive teacher model. In this study, we empirically show that as a side effect of this training, the lexical choice errors on low-frequency words are propagated to the NAT model from the teacher model. To alleviate this problem, we propose to expose the raw data to NAT models to restore the useful information of low-frequency words, which are missed in the distilled data. To this end, we introduce an extra Kullback-Leibler divergence term derived by comparing the lexical choice of NAT model and that embedded in the raw data. Experimental results across language pairs and model architectures demonstrate the effectiveness and universality of the proposed approach. Extensive analyses confirm our claim that our approach improves performance by reducing the lexical choice errors on low-frequency words. Encouragingly, our approach pushes the SOTA NAT performance on the WMT14 English-German and WMT16 Romanian-English datasets up to 27.8 and 33.8 BLEU points, respectively.
+
+# 1 INTRODUCTION
+
+When translating a word, translation models need to spend a substantial amount of its capacity in disambiguating its sense in the source language and choose a lexeme in the target language which adequately express its meaning (Choi et al., 2017; Tamchyna, 2017). However, neural machine translation (NMT) has a severe problem on lexical choice, since it usually has mistranslation errors on low-frequency words (Koehn & Knowles, 2017; Nguyen & Chiang, 2018; Gu et al., 2020).
+
+In recent years, there has been a growing interest in non-autoregressive translation (NAT, Gu et al., 2018), which improves decoding efficiency by predicting all tokens independently and simultaneously. Well-performed NAT models are generally trained on synthetic data distilled by autoregressive translation (AT) teachers instead of the raw training data (Figure 1(a)) (Stern et al., 2019; Lee et al., 2018; Ghazvininejad et al., 2019; Gu et al., 2019; Hao et al., 2021). Recent studies have revealed that knowledge distillation (KD) reduces the modes (i.e. multiple lexical choices for a source word) in the raw data by re-weighting the training examples (Furlanello et al., 2018; Tang et al.,
+
+<table><tr><td>SRC RAW-TGT KD-TGT</td><td>今天纽马基特的跑道湿软。 The going at Newmarket is soft... Today, Newmargot&#x27;s runway is soft ...</td></tr><tr><td>SRC RAW-TGT KD-TGT</td><td>纽马 基特赛马总是吸引. The Newmarket stakes is always ... The Newmarquette races always ...</td></tr><tr><td>SRC</td><td>在纽马基特3时45分那场中，我..</td></tr><tr><td>RAW-TGT KD-TGT</td><td>I&#x27;ve ...inthe3.45atNewmarket. I.. at 3:45 a.m. in Newmarquite.</td></tr></table>
+
+Table 1: All samples that contain the source word “纽 基 ” in raw and distilled training corpora, which 马 特are different in target sides (RAW-TGT vs. KD-TGT).
+
+2020), which lowers the intrinsic uncertainty (Ott et al., 2018) and learning difficulty for NAT (Zhou et al., 2020; Ren et al., 2020). However, the side effect of KD has not been fully studied. In this work, we investigate this problem from the perspective of lexical choice, which is at the core of machine translation.
+
+We argue that the lexical choice errors of AT teacher can be propagated to the NAT model via the distilled training data. To verify this hypothesis, we qualitatively compare raw and distilled training corpora. Table 1 lists all samples whose source sentences contain the place name “纽 基 ”. In the 马 特raw corpus (“RAW-TGT”), this low-frequency word totally occurs three times and corresponds to correct translation “Newmarket”. However, in the KD corpus (“KD-TGT”), the word is incorrectly translated into a person name “Newmargot” (Margot Robbie is an Australian actress) or organization name “Newmarquette” (Marquette is an university in Wisconsin) or even invalid one “Newmarquite”.
+
+Motivated by this finding, we explore NAT from the lexical choice perspective. We first validate our hypothesis by analyzing the lexical choice behaviors of NAT models (§3). Concretely, we propose a new metric AoLC (accuracy of lexical choice) to evaluate the lexical translation accuracy of a given NAT model. Experimental results across different language pairs show that NAT models trained on distilled data have higher accuracy of global lexical translation (AoLC↑), which results in better sequence generation. However, fine-grained analyses revealed that although KD improves the accuracy on high-frequency tokens, it meanwhile harms performance on low-frequency ones (Low freq. AoLC↓). And with the improvement of teacher models, this issue becomes more severe. We conclude that the lexical choice of the low-frequency tokens is a typical kind of lost information when using knowledge distillation from AT model.
+
+In order to rejuvenate this lost information in raw data, we propose to expose the raw data to the training of NAT models, which augments NAT models the ability to learn the lost knowledge by themselves. Specifically, we propose two bi-lingual lexical-level data-dependent priors (Word Alignment Distribution and Self-Distilled Distribution) extracted from raw data, which is integrated into NAT training via Kullback-Leibler divergence. Both approaches expose the lexical knowledge in the raw data to NAT, which makes it learn to restore the useful information of low-frequency words to accomplish the translation.
+
+We validated our approach on several datasets that widely used in previous studies (i.e. WMT14 En-De, WMT16 Ro-En, WMT17 Zh-En, and WAT17 Ja-En) and model architectures (i.e. MaskPredict (Ghazvininejad et al., 2019) and Levenshtein Transformer (Gu et al., 2019)). Experimental results show that the proposed method consistently improve translation performance over the standard NAT models across languages and advanced NAT architectures. The improvements come from the better lexical translation accuracy (low-frequency tokens in particular) of NAT models $\mathrm { ( A o L C \uparrow ) }$ ), which leads to less mis-translations and low-frequency words prediction errors. The main contributions of this work are:
+
+• Our study reveals the side effect of NAT models’ knowledge distillation on low-frequency lexicons, which makes the standard NAT training on the distilled data sub-optimal.   
+• We demonstrate the necessity of letting NAT models learn to distill lexical choices from the raw data by themselves.   
+• We propose an simple yet effective approach to accomplish this goal1, which are robustly applicable to several model architectures and language pairs.
+
+# 2 PRELIMINARIES
+
+# 2.1 NON-AUTOREGRESSIVE TRANSLATION
+
+The idea of NAT has been pioneered by Gu et al. (2018), which enables the inference process goes in parallel. Different from AT models that generate each target word conditioned on previously generated ones, NAT models break the autoregressive factorization and produce target words in parallel. Given a source sentence $\mathbf { x }$ , the probability of generating its target sentence y with length $T$ is calculated as:
+
+$$
+p ( \mathbf { y } | \mathbf { x } ) = p _ { L } ( T | \mathbf { x } ; \theta ) \prod _ { t = 1 } ^ { T } p ( \mathbf { y } _ { t } | \mathbf { x } ; \theta )
+$$
+
+where $p _ { L } ( \cdot )$ is a separate conditional distribution to predict the length of target sequence. During training, the negative loglikelihood loss function of NAT is accordingly $\mathcal { L } _ { \mathrm { N A T } } ( \boldsymbol { \theta } ) = - \log p ( \mathbf { y } | \mathbf { x } )$ . To bridge the performance gap between NAT and AT models, a variety approaches have been proposed, such as multi-turn refinement mechanism (Lee et al., 2018; Ghazvininejad et al., 2019; Gu et al., 2019; Kasai et al., 2020), rescoring with AT models (Wei et al., 2019; Ma et al., 2019; Sun et al., 2019), adding auxiliary signals to improve model capacity (Wang et al., 2019; Ran et al., 2019; Guo et al., 2019; Ding et al., 2020), and advanced training objective (Wei et al., 2019; Shao et al., 2019; Ma et al., 2020). Our work is complementary to theirs: while they focus on improving NAT models trained on the distilled data, we refine the NAT models by exploiting the knowledge in the raw data.
+
+Sentence-Level Knowledge Distillation NAT models suffer from the multimodality problem, in which the conditional independence assumption prevents a model from properly capturing the highly multimodal distribution of target translations. For example, one English source sentence “Thank you.” can be accurately translated into German as any one of “Danke.”, “Danke schon.” or “Vielen Dank.”, all of which occur in the training data. ¨
+
+To alleviate this problem, Gu et al. (2018) applied sequence-level KD (Kim & Rush, 2016) to construct a synthetic corpus, whose target sentences are generated by an AT model trained on the raw data, as shown in Figure 1(a). The NAT model is only trained on distilled data with lower modes, which makes it easily acquire more deterministic knowledge (e.g. one lexical choice for each source word). While separating KD and model training makes the pipeline simple and efficient, it has one potential threat: the re-weighted samples distilled with AT model may have lost some important information. Lee et al. (2020) show that distillation benefits the sequence generation but harms the density estimation. In this study, we exploit to bridge this gap by exposing the raw data to the training of NAT models, as shown in Figure 1(b).
+
+![](images/1b349c9d2c2f130186b4ff949bb980945ac1b95809cc11301fbd5fef10a0c345.jpg)  
+Figure 1: Comparison of existing two-step and our proposed NAT training scheme.
+
+# 2.2 EXPERIMENTAL SETUP
+
+Datasets Experiments were conducted on four widely-used translation datasets: WMT14 EnglishGerman (En-De, Vaswani et al. 2017), WMT16 Romanian-English (Ro-En, Gu et al. 2018), WMT17 Chinese-English (Zh-En, Hassan et al. 2018), and WAT17 Japanese-English (Ja-En, Morishita et al. 2017), which consist of 4.5M, 0.6M, 20M, and 2M sentence pairs, respectively. We use the same validation and test datasets with previous works for fair comparison. To avoid unknown words, we preprocessed data via BPE (Sennrich et al., 2016) with 32K merge operations. The $\mathrm { { G I Z A + + } }$ (Och & Ney, 2003) was employed to build word alignments for the training datasets. We evaluated the translation quality with BLEU (Papineni et al., 2002).
+
+NAT Models We validated our research hypotheses on two SOTA NAT models:
+
+• MaskPredict (MaskT, Ghazvininejad et al. 2019) that uses the conditional mask LM (Devlin et al., 2019) to iteratively generate the target sequence from the masked input. We followed its optimal settings to keep the iteration number be 10 and length beam be 5, respectively. • Levenshtein Transformer (LevT, Gu et al. 2019) that introduces three steps: deletion, placeholder prediction and token prediction. The decoding iterations in LevT adaptively depends on certain conditions.
+
+For regularization, we tune the dropout rate from [0.1, 0.2, 0.3] based on validation performance in each direction, and apply weight decay with 0.01 and label smoothing with $\epsilon = 0 . 1$ . We train batches of approximately 128K tokens using Adam (Kingma & Ba, 2015). The learning rate warms up to $5 \times \mathrm { 1 0 ^ { - 4 } }$ in the first 10K steps, and then decays with the inverse square-root schedule. We followed the common practices (Ghazvininejad et al., 2019; Kasai et al., 2020) to evaluate the translation performance on an ensemble of top 5 checkpoints to avoid stochasticity.
+
+Table 2: Results of different metrics on the MaskT model trained on different datasets. “KD (X)” denotes the distilled data produced by the AT model with X setting. “CoD” denotes the complexity of data metric proposed by Zhou et al. (2020), and “AoLC” is our proposed metric to evaluate the accuracy of lexical choice in NAT models.   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="3">En-De</td><td colspan="3">Zh-En</td><td colspan="3">Ja-En</td></tr><tr><td>CoD</td><td>AoLC</td><td>BLEU</td><td>CoD</td><td>AoLC</td><td>BLEU</td><td>CoD</td><td>AoLC</td><td>BLEU</td></tr><tr><td>Raw</td><td>3.53</td><td>74.3</td><td>24.6</td><td>5.11</td><td>68.5</td><td>22.6</td><td>3.92</td><td>73.1</td><td>27.8</td></tr><tr><td>KD (BASE)</td><td>1.85</td><td>75.5</td><td>26.5</td><td>3.23</td><td>71.8</td><td>23.6</td><td>2.80</td><td>74.7</td><td>28.4</td></tr><tr><td>KD (BIG)</td><td>1.77</td><td>76.3</td><td>27.0</td><td>3.01</td><td>72.7</td><td>24.2</td><td>2.47</td><td>75.3</td><td>28.9</td></tr></table>
+
+AT Teachers We closely followed previous works on NAT to apply sequence-level knowledge distillation (Kim & Rush, 2016) to reduce the modes of the training data. More precisely, to assess the effectiveness of our method under different of AT teachers, we trained three kinds of Transformer (Vaswani et al., 2017) models, including Transformer-BASE, Transformer-BIG and Transformer-STRONG. The main results employ LARGE for all directions except Ro-En, which is distilled by BASE. The architectures of Transformer-BIG and Transformer-STRONG are unchanged, but STRONG utilizes a large batch (458K tokens) training strategy.
+
+# 3 UNDERSTANDING LEXICAL CHOICE IN NAT MODELS
+
+# 3.1 EVALUATING LEXICAL CHOICE OF NAT MODELS
+
+Recently, Zhou et al. (2020) argue that knowledge distillation is necessary for the uncertain nature of the machine translation task. Accordingly, they propose a metric to estimate the complexity of the data $( C o D )$ , which is driven from an external word alignment model. They reveal that the distilled data is indeed less complex, which facilitates easier training for the NAT model. Inspired by this, we propose a metric to measure the lexical level accuracy of model predictions.
+
+Accuracy of Lexical Choice (AoLC) evaluates the accuracy of target lexicon chosen by a trained NAT model $M$ for each source word. Specifically, the model $M$ takes a source word $f$ as the input, and produce a hypothesis candidate list with their corresponding word confidence:
+
+$$
+\mathbf { P } _ { f } ^ { M } = \{ P ^ { M } ( e _ { 1 } | f ) , \dots , P ^ { M } ( e _ { | \mathbf { V } _ { t r g } | } | f ) \}
+$$
+
+where $\mathbf { V } _ { t r g }$ is the target side vocabularies over whole corpus. The AoLC score is calculated by averaging the probability of the gold target word $e _ { f }$ of each source word $f$ :
+
+$$
+A o L C = \frac { \sum _ { f \in \mathbf { V } _ { s r c } ^ { t e s t } } P ^ { M } ( e _ { f } | f ) } { | \mathbf { V } _ { s r c } ^ { t e s t } | }
+$$
+
+where $\mathbf { V } _ { s r c } ^ { t e s t }$ is the set of source side tokens in test set. Each gold word $e _ { f }$ is chosen with the help of the word alignment model $P _ { f } ^ { A }$ . The chosen procedure is as follows: Step 1) collecting the references of the source sentences that contains source word $f$ , and generating the target side word bag $\mathbb { B } _ { f }$ with these references. Step 2) Descending $P _ { f } ^ { A }$ in terms of alignment probabilities and looking up the word that first appears in $\mathbb { B } _ { f }$ as the gold word until the $\mathbb { B } _ { f }$ is traversed. Step 3) If the gold word is still not found, let the word with the highest alignment probability in $P _ { f } ^ { A }$ as the gold word. Generally, higher accuracy of lexical translation represents more confident of the predictions. We discuss the reliability of word alignment-based AoLC in Appendix A.1.
+
+# 3.2 GLOBAL EFFECT OF KNOWLEDGE DISTILLATION ON LEXICAL CHOICE
+
+In this section, we analyze the lexical choice behaviors of NAT models with our proposed AoLC. In particular, We evaluated three MaskT models, which are respectively trained on the raw data, AT-BASE and AT-BIG distilled data. We compared the AoLC with other two metrics (i.e. BLEU and CoD) on three different datasets (i.e. En-De, Zh-En and Ja-En). As shown in Table 2, KD is able to improve translation quality of NAT models (BLEU: KD(BIG) ${ \mathrm { > K D } }$ (BASE) ${ \mathrm { > R a w } }$ ) by increasing the lexical choice accuracy of data (AoLC: KD(BIG) ${ \tt > K D }$ (BASE) ${ \mathrm { > R a w } }$ ). As expected, NAT models trained on more deterministic data $( \mathrm { C o D } \downarrow )$ have lower lexical choice errors (AoLC↑) globally, resulting in better model generation performance (BLEU↑).
+
+3.3 DISCREPANCY BETWEEN HIGH- AND LOW-FREQUENCY WORDS ON LEXICAL CHOICE
+
+To better understand more detailed lexical change within data caused by distillation, we break down the lexicons to three categories in terms of frequency. And we revisit it from two angles: training data and translated data.
+
+We first visualize the changing of training data when adopting KD in terms of words frequency density.
+
+As shown in Figure 2, we find that the kurtosis of KD data distribution is higher than that of raw, which becomes more significant when adopting stronger teacher. The side effect is obvious, that is, the original high- / low-frequency words become more / fewer, making the distribution of training data more imbalance and skewed, which is problematic in data mining field (Chawla et al., 2004). This discrepancy may erode the translation performance of low-frequency words and generalization performance on other domains. Here we focus on lowfrequency words, and generalization performance degradation will be exploited in future work.
+
+![](images/dddd69e87f1d60170cfd9055d32bec4a6f19d5103ff92dddcd1b1cb1bdd80da0.jpg)  
+Figure 2: Comparison of the token frequency density (w.r.t the sampled tokens’ probability distribution) between Raw, $K D$ (Base) and $K D$ (Big) WMT14 En-De training data.quencycy
+
+In order to understand the detailed change during inference, we then analyze the lexical accuracy with different frequencies in the test set. We make the comprehensive comparison cross languages based on our proposed AoLC. As shownFrFrequeFrequency in Figure 3, as the teacher model becomes better, i.e. $\mathrm { K D ( b a s e ) { \to } K D ( b i g }$ ), the lexical choice of high-High High MHigh Med.
+
+frequency words becomes significantly more accurate (AoLC ↑) while that of low-frequency words808080 becomes worse (AoLC ↓). Through fine-grained analysis, we uncover this interesting discrepancy606060 between high- and low- frequency words. The same phenomena (lexical choice errors on lowfrequency words propagated from teacher model) also can be found in general cases, e.g. distillation404040 when training smaller AT models. Details can be found in Appendix A.2. To keep the accuracy of202020 high-frequency words and compensate for the imbalanced low-frequency words caused by KD, we present a simple yet effective approach below. En- En-De En-De Zh
+
+![](images/a49ead06b91e281953bc62e41853c855b922f9199f5d849e84ef0616fd90c617.jpg)  
+Figure 3: Accuracy of lexical choice (AoLC) for source words of different frequency.
+
+# 4 IMPROVING LEXICAL CHOICE IN NAT MODELS
+
+# 4.1 METHODOLOGY
+
+Our goal is to augment NAT models to learn needed lexical choices from the raw data to achieve better performance. To this end, we introduce an extra bilingual data-dependent prior objective to augment the current NAT models to distill the required lexical choices from the raw data. Specifically, we use Kullback-Leibler divergence to guide the probability distribution of model predictions $P ^ { \bar { M } } ( e | { \bf f } )$ to match the prior probability distributions $Q ( \cdot )$ :
+
+$$
+\mathcal { L } _ { p r i o r } = - \sum _ { e \in { \bf e } } \mathrm { K L } \big ( Q ( e | { \bf f } ) \big | \big | P ^ { M } ( e | { \bf f } ) \big )
+$$
+
+where f is the source sentence, and $\mathbf { e }$ is the target sentence. The bilingual prior distribution $Q ( \cdot )$ is derived from the raw data, which is independent of the model $M$ and will be described later. The final objective for training the NAT model becomes:
+
+$$
+\begin{array} { r } { \mathcal { L } = ( 1 - \lambda ) \mathcal { L } _ { N A T } + \lambda \mathcal { L } _ { p r i o r } } \end{array}
+$$
+
+in which the imitation rate $\lambda$ follows the logarithmic decay function:
+
+$$
+\lambda ( i ) = \left\{ \begin{array} { l l } { \frac { l o g ( \mathrm { I } / ( 2 ( i + 1 ) ) ) } { l o g ( \mathrm { I } / 2 ) } } & { i \leq \mathrm { I } / 2 } \\ { 0 } & { \mathrm { o t h e r s } } \end{array} \right.
+$$
+
+where $i$ is the current step, I is the total training step for distilled data. Accordingly, the NAT model is merely fed with the priori knowledge derived from the raw data at beginning. Along with training, the supervision signal of the prior information is getting weaker while that of the distilled data gradually prevails in the training objective. We run all models for 300K steps to ensure adequate training, thus the bilingual prior distributions will be exposed at the first 150K steps.
+
+Choices of Prior Distribution $Q ( \cdot )$ The goal of the prior objective is to guide the NAT models to learn to distill the lexical choices itself from the raw data. For each target word $e$ , we use the external word alignment to select the source word $f$ with the maximum alignment probability, and $Q ( \cdot )$ is rewritten as:
+
+$$
+Q ( e | \mathbf { f } ) = Q ( e | f )
+$$
+
+Specifically, we use two types of bilingual prior distributions:
+
+• Word Alignment Distribution $( W A D )$ is the distribution derived from the external word alignment $\mathbf { P } _ { f } ^ { D } = \{ { \breve { P } } ^ { D } ( e _ { 1 } | f ) , \dots , P ^ { D } ( e _ { N } | f ) \}$ where $\{ e _ { 1 } , \dots , e _ { N } \}$ are the set of target words aligned to the source word in the training data. We follow Hinton et al. (2015) to use the softmax temperature mechanism to map $\mathbf { P } _ { f } ^ { D }$ over the whole target vocabulary:
+
+$$
+Q ( e | f ) = \hat { \mathbf { P } } _ { f } ^ { D } = \frac { e x p ( \mathbf { P } _ { f } ^ { D } / \tau ) } { \sum _ { V _ { t g t } } e x p ( \mathbf { P } _ { f } ^ { D } / \tau ) }
+$$
+
+We tune the temperature from [0.5, 1, 2, 5] on WMT14 En-De dataset and use $\tau = 2$ as the default setting for incorporating word alignment distribution in all datasets.
+
+• Self-Distilled Distribution $( S D D )$ is the probability distribution for the source word $f$ , which is produced by a same NAT model pre-trained on raw data. Specifically, the model $M$ takes a source word $f$ as input and produces a probability distribution over whole words in target vocabulary:
+
+$$
+\mathbf { P } _ { f } ^ { M } = \{ P ^ { M } ( e _ { 1 } | f ) , \dots , P ^ { M } ( e _ { | \mathbf { V } _ { t r g } | } | f ) \}
+$$
+
+This prior distribution signal can be characterized as self-distilled lexicon level “born-again networks” (Furlanello et al., 2018) or self-knowledge distillation (Liu et al., 2020), where the teacher and student have the same neural architecture and model size, and yet surprisingly the student is able to surpass the teacher’s accuracy.
+
+Table 3: Ablation Study on raw data priors across different language pairs using the MaskT Model. “WAD” denotes word alignment distribution, and “SDD” denotes self-distilled distribution. “AoLC / LFT” denotes the lexical translation accuracies for all tokens / low-frequency tokens, respectively.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">En-De</td><td colspan="2">Zh-En</td><td colspan="2">Ja-En</td></tr><tr><td>AoLC /LFT</td><td>BLEU</td><td>AoLC /LFT</td><td>BLEU</td><td>AoLC /LFT</td><td>BLEU</td></tr><tr><td>AT-TEACHER</td><td>79.3 / 73.0</td><td>29.2</td><td>74.7 / 66.2</td><td>25.3</td><td>77.1/ 70.8</td><td>29.8</td></tr><tr><td>MaskT+KD</td><td>76.3 / 68.4</td><td>27.0</td><td>72.7 / 61.5</td><td>24.2</td><td>75.3 / 66.9</td><td>28.9</td></tr><tr><td>+WAD</td><td>77.5 / 71.9</td><td>27.4</td><td>73.4 / 64.5</td><td>24.8</td><td>76.3 / 69.0</td><td>29.4</td></tr><tr><td>+SDD</td><td>77.7 /72.2</td><td>27.5</td><td>73.5 / 64.7</td><td>24.9</td><td>76.1 / 68.6</td><td>29.3</td></tr><tr><td>+Both</td><td>78.1/ 72.4</td><td>27.8</td><td>74.0 / 65.0</td><td>25.2</td><td>76.6 / 69.1</td><td>29.6</td></tr></table>
+
+Table 4: Comparison with previous work on WMT14 En-De and WMT16 Ro-En datasets. “Iter.” column indicate the average number of refined iterations. “†” indicates statistically significant difference $( p < 0 . 0 5 )$ from baselines according to the statistical significance test (Collins et al., 2005).   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Iter. Speed</td><td rowspan="2"></td><td colspan="2">En-De</td><td colspan="2">Ro-En</td></tr><tr><td>AoLC</td><td>BLEU</td><td>AoLC</td><td>BLEU</td></tr><tr><td colspan="8">AT Models</td></tr><tr><td>Transformer-BASE (Ro-En Teacher)</td><td>n/a</td><td>1.0×</td><td></td><td>27.3</td><td></td><td>34.1</td></tr><tr><td>Transformer-BIG (En-De Teacher)</td><td>n/a</td><td>0.8×</td><td></td><td>29.2</td><td></td><td>n/a</td></tr><tr><td colspan="7">Existing NAT Models</td></tr><tr><td>NAT (Gu et al., 2018) Iterative NAT (Lee et al., 2018)</td><td>1.0</td><td>2.4×</td><td></td><td>19.2 21.6</td><td rowspan="4">n/a</td><td>31.4 30.2</td></tr><tr><td>DisCo (Kasai et al., 2020)</td><td>10.0 4.8</td><td>2.0× 3.2×</td><td></td><td>26.8</td><td>33.3</td></tr><tr><td>Mask-Predict (Ghazvininejad et al., 2019)</td><td>10.0</td><td>1.5×</td><td>n/a</td><td>27.0</td><td>33.3</td></tr><tr><td>Levenshtein (Gu et al., 2019)</td><td>2.5</td><td>3.5×</td><td></td><td>27.3</td><td>33.3</td></tr><tr><td colspan="8"></td></tr><tr><td>Mask-Predict</td><td>Our NAT Models</td><td></td><td>76.3</td><td>27.0</td><td>79.2</td><td>33.3</td></tr><tr><td>+Raw Data Prior</td><td>10.0</td><td>1.5×</td><td>78.1</td><td>27.8t</td><td>80.6</td><td>33.7</td></tr><tr><td>Levenshtein</td><td>2.5</td><td>3.5×</td><td>77.0</td><td>27.2</td><td>79.8</td><td>33.2</td></tr><tr><td>+RawData Prior</td><td></td><td></td><td>77.8</td><td>27.8t</td><td>80.9</td><td>33.8t</td></tr></table>
+
+# 4.2 EXPERIMENTAL RESULTS
+
+Ablation Study on Raw Data Prior Table 3 shows the results of our proposed two bilingual data dependent prior distributions across language pairs. The word alignment distribution (WAD) and self-distilled distribution (SDD) variants consistently improves performance over the vanilla two-step training scheme NAT model $\mathrm { ^ { * } N A T { + } K D ^ { \prime } }$ ) when used individually (averagely $+ 0 . 5$ BLEU point), and combining them $\hbar ^ { * } { + } \mathrm { B o t h } ^ { \prime \prime }$ ) by simply averaging the two distributions can achieve a further improvement (averagely $+ 0 . 9$ BLEU point). The improvements on translation performance are due to a increase of AoLC, especially for low-frequency tokens (averagely $+ 3 . 2$ ), which reconfirms our claim. Notably, averaging the two prior distributions could rectify each other, thus leading to a further increase. We explore the complementarity of two prior schemes in Section 4.3. In the following experiments, we use the combination of WAD and SDD as the default bilingual data dependent prior.
+
+Comparison with Previous Work Table 4 lists the results of previously competitive studies (Gu et al., 2018; Lee et al., 2018; Kasai et al., 2020; Ghazvininejad et al., 2019; Gu et al., 2019) on the widely-used WMT14 En-De and WMT16 Ro-En datasets. Clearly, our bilingual data-dependent prior significantly improves translation (BLEU↑) by substantially increasing the lexical choice accuracy (AoLC↑). It is worth noting that our approaches merely modify the training process, thus does not increase any latency (“Speed”), maintaining the intrinsic advantages of non-autoregressive generation.
+
+Table 7: Improvement of our approach over the MaskT $\mathrm { \Phi } _ { + \mathrm { K D } }$ model on AoLC.   
+
+<table><tr><td>Frequency</td><td>En-De</td><td>Zh-En</td><td>Ja-En</td></tr><tr><td>High</td><td>+1.3%</td><td>+0.3%</td><td>+1.3%</td></tr><tr><td>Medium</td><td>+0.2%</td><td>+0.1%</td><td>+0.9%</td></tr><tr><td>Low</td><td>+5.9%</td><td>+5.8%</td><td>+3.3%</td></tr><tr><td>All</td><td>+2.4%</td><td>+1.8%</td><td>+1.7%</td></tr></table>
+
+Table 8: Ratio of low-frequency target words in the MaskT model generated translations.   
+
+<table><tr><td>Model</td><td>En-De</td><td>Zh-En</td><td>Ja-En</td></tr><tr><td>NAT</td><td>10.3%</td><td>6.7%</td><td>9.4%</td></tr><tr><td>+KD</td><td>7.6%</td><td>4.2%</td><td>6.9%</td></tr><tr><td>+Ours</td><td>9.8%</td><td>6.1%</td><td>8.5%</td></tr></table>
+
+Comparison with Data Manipulation Strategies Instead of using the proposed priors, we also investigate two effective data manipulation strategies, i.e. Data Mixing and Curriculum Learning, to force the NAT model learns from both the raw and distilled data. For data mixing, we design two settings: a) Mix: simply combine the raw and distilled data, and then shuffle the mixed dataset. b) Tagged Mix: Inspired by successes of tagged back-translation (Caswell et al., 2019; Marie et al., 2020), we add tags to distinguish between KD and Raw sentences in the mixed dataset. For decay curriculum schedule, the NAT models learn more from raw data at the beginning and then learn more from KD as the training goes on. The details of curriculum can be found in Appendix A.3. As seen in Table 5, data mixing and decay curriculum schedule improve performance on both AoLC and BLEU, which confirm the necessity of exposing raw data to NAT models during training. Besides, our approach still outperforms those effective strategies, demonstrating the superiority of our learning scheme.
+
+Table 5: Performance of several data manipulation strategies on En-De dataset. Baseline is the $\mathbf { M a s k T + K D }$ model and Ours is our proposed approach.   
+
+<table><tr><td> Strategies</td><td>AoLC</td><td>BLEU</td></tr><tr><td>Baseline</td><td>76.3</td><td>27.0</td></tr><tr><td>Mix</td><td>76.6</td><td>27.2</td></tr><tr><td>Tagged Mix</td><td>77.1</td><td>27.4</td></tr><tr><td>Decay Curriculum</td><td>77.2</td><td>27.5</td></tr><tr><td>Ours</td><td>78.1</td><td>27.8</td></tr></table>
+
+# 4.3 EXPERIMENTAL ANALYSIS
+
+In this section, we conducted extensive analyses on the lexical choice to better understand our approach. Unless otherwise stated, results are reported on the MaskPredict models in Table 3.
+
+<table><tr><td>Model</td><td>BLEU</td><td>AoLC</td><td>Error</td></tr><tr><td>MaskT</td><td>22.6</td><td>68.5%</td><td>34.3%</td></tr><tr><td>+KD</td><td>24.2</td><td>72.7%</td><td>30.1%</td></tr><tr><td>+RDP</td><td>25.2</td><td>74.0%</td><td>28.2%</td></tr></table>
+
+Our approach improves translation performance by reducing mis-translation errors. The lexical choice ability of NAT models correlates to mistranslation errors, in which wrong lexicons are chosen to translate source words. To better understand whether our method alleviates the mis-translation problem, we
+
+Table 6: Subjective evaluation of mistranslation errors on the Zh-En dataset.
+
+assessed system output by human judgments. In particular, we randomly selected 50 sentences from the Zh-En testset, and manually labelled the words with lexical choice error. We defined the lexical choice error rate as $E / N$ , where $E$ is the number of lexical choice errors and $N$ is the number of content words in source sentences, since such errors mainly occur in translating content words. As seen in Table 6, our approache consistently improves BLEU scores by reducing the lexical choice errors, which confirm our claim. Additionally, AoLC metric correlates well with both the automatic BLEU score and the subjective evaluation, demonstrating its reasonableness.
+
+Our approach significantly improves the accuracy of lexical choice for low-frequency source words. As aforementioned discrepancy between high- & low-frquency words in Section 3.3, we focus on revealing the fine-grained lexical choice accuracy w.r.t our proposed AoLC. In Table 7, the majority of improvements is from the low-frequency words, confirming our hypothesis.
+
+Our approach generates translations that contain more low-frequency words. Besides improving the lexical choice of low-frequency words, our method results in more low-frequency words being recalled in the translation. In Table 8, although KD improves the translation, it biases the NAT model towards generating high-frequency tokens (Low freq.↓) while our method can not only correct this bias (averagely $+ 3 2 \%$ relative change), but also enhance translation (BLEU↑ in Table 3).
+
+Our proposed two priors complement each other by facilitating different tokens. As aforementioned in Table 3, combining two individual schemes can further increase the NAT performance. To explain how they complement each other, especially for low-frequency tokens, we classify low-frequency tokens into two categories according to their linguistic roles: content words (e.g. noun, verb, and adjective) and function words (e.g. preposition, determiner, and punctuation). The results are listed in Table 9. We show that WAD facilitates more on the understanding and generation of content tokens, while SDD brings more gains for function (i.e. content
+
+<table><tr><td rowspan="2">Prior</td><td colspan="2">AoLC on LFT</td><td colspan="2">Ratio of LFT</td></tr><tr><td>Content</td><td>Function</td><td>Content</td><td>Function</td></tr><tr><td>N/A</td><td>67.7%</td><td>70.1%</td><td>5.3%</td><td>2.4%</td></tr><tr><td>WAD</td><td>71.6%</td><td>72.9%</td><td>5.9%</td><td>2.5%</td></tr><tr><td>SDD</td><td>71.4%</td><td>74.3%</td><td>5.6%</td><td>3.4%</td></tr><tr><td>Both</td><td>71.6%</td><td>74.2%</td><td>6.2%</td><td>3.6%</td></tr></table>
+
+Table 9: AoLC and Ratio of different prior schemes on Low-Frequency Tokens (“LFT”). We list the performances on different linguistic roles, i.e. content words and function words. Note that Ratio of LFT means the ratio of low frequency tokens in generated translation. “N/A” means MaskT $\mathrm { + K D }$ baseline.
+
+free) tokens. We leave a more thorough exploration of this aspect for future work.
+
+Effect of Word Alignment Quality on Model Performance. Both the proposed AoLC and priors depend heavily on the quality of word alignment, we therefore design two weaker alignment scenarios to verify the robustness of our method.
+
+First, We adopt fast-align (Dyer et al., 2013), which is slightly weaker than $\mathrm { { G I Z A + + } }$ . Using fastalign, our methods can still achieve $+ 0 . 6$ and $+ 0 . 7$ improvements in terms of BLEU on En-De and Zh-En datasets, which are marginally lower than that using $\mathrm { { G I Z A + + } }$ (i.e. $+ 0 . 8$ and $+ 1 . 0$ BLEU). Encouragingly, we find that the improvements in translation accuracy on low-frequency words still hold $+ 5 . 5 \%$ and $+ 5 . 3 \%$ vs. $+ 5 . 9 \%$ and $+ 5 . 8 \%$ ), which demonstrates the robustness of our approach.
+
+In addition, we insert noises into the alignment distributions to deliberately reduce the alignment quality (Noise injection details can be found in Appendix A.4. The performances still significantly outperform the baseline, indicating that our method can tolerate alignment errors and maintain model performance to some extent.
+
+Effect of AT Teacher To further dissect the different effects when applying different AT teachers, we employ three teachers. Table 10 shows our method can enhance NAT models under variety of teacher-student scenarios, including base, big and strong teacher-guided models. Our approach obtains averagely $+ 0 . 7$ BLEU points, potentially complementary to the majority of existing work on improving knowledge distillation for NAT models.
+
+Table 10: Different teachers on the En-De dataset.   
+
+<table><tr><td colspan="2">AT Teacher</td><td colspan="2">NAT Model</td></tr><tr><td>Model</td><td>BLEU</td><td>Vanilla</td><td>+Prior △</td></tr><tr><td>Base</td><td>27.3</td><td>26.5</td><td>27.2 +0.7</td></tr><tr><td>Big</td><td>28.4</td><td>26.8</td><td>27.5 +0.7</td></tr><tr><td> Strong</td><td>29.2</td><td>27.0</td><td>27.8 +0.8</td></tr></table>
+
+# 5 RELATED WORK
+
+Understanding Knowledge Distillation for NAT Knowledge distillation is a crucial early step in the training of most NAT models. Ren et al. (2020) reveal that the difficulty of NAT heavily depends on the strongness of dependency among target tokens, and knowledge distillation reduces the token dependency in target sequence and thus improves the accuracy of NAT models. In the pioneering work of NAT, Gu et al. (2018) claim that NAT suffers from the multi-modality problem (i.e. multiple lexical translations for a source word), and knowledge distillation can simplify the dataset, which is empirically validated by Zhou et al. (2020). We confirm and extend these results, showing that the AT-distilled dataset indeed leads to more deterministic predictions but propagates the low-frequency lexical choices errors. To this end, we enhance the NAT lexical predictions by making them learn to distill knowledge from the raw data.
+
+Lexical Choice Problem in NMT Models Benefiting from continuous representations abstracted from the training data, NMT models have advanced the state of the art in the machine translation community. However, recent studies have revealed that NMT models suffer from inadequate translation (Tu et al., 2016), in which mis-translation error caused by the lexical choice problem is one main reason. For AT models, Arthur et al. (2016) alleviate this issue by integrating a count-based lexicon, and Nguyen & Chiang (2018) propose an additional lexical model, which is jointly trained with the AT model. The lexical choice problem is more serious for NAT models, since 1) the lexical choice errors (low-resource words in particular) of AT distillation will propagate to NAT models; and 2) NAT lacks target-side dependencies thus misses necessary target-side context. In this work, we alleviate this problem by solving the first challenge.
+
+# 6 CONCLUSION
+
+In this study, we investigated effects of KD on lexical choice in NAT. We proposed a new metric to evaluate lexical translation accuracy of NAT models, and found that 1) KD improves global lexical predictions; and 2) KD benefits the accuracy of high-frequency words but harms the low-frequency ones. There exists a discrepancy between high- and low-frequency words after adopting KD. To bridge this discrepancy, we exposed the useful information in raw data to the training of NAT models. Experiments show that our approach consistently and significantly improves translation performance across language pairs and model architectures. Extensive analyses reveal that our method reduces mistranslation errors, improves the accuracy of lexical choices for low-frequency source words, recalling more low-frequency words in the translations as well, which confirms our claim.
+
+# 7 ACKNOWLEDGMENTS
+
+This work was supported by Australian Research Council Projects under grants FL-170100117, DP-180103424, and IC-190100031. Xuebo and Derek were supported in part by the Science and Technology Development Fund, Macau SAR (Grant No. 0101/2019/A2), and the Multi-year Research Grant from the University of Macau (Grant No. MYRG2020-00054-FST). We also thank the anonymous reviewers for their insightful comments.
+
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+
+# A APPENDIX
+
+A.1 DISCUSSION ON THE RELIABILITY OF WORD ALIGNMENT-BASED AOLC
+
+We randomly select 20 sentence pairs from the Zh-En test set, which contains 576 source tokens. We use the trained word alignment model to produce alignments for the 20 sentence pairs, and then perform the gold word chosen procedure as described in Section 3.1. We manually evaluate these bilingual lexicons, and find that 551 out of 576 source words are aligned to reasonable equivalences (i.e. $96 \%$ accuracy). This demonstrates that it is reliable to calculate AoLC based on automatic word alignments.
+
+A.2 GENERAL CASES OF THE SIDE-EFFECT OF KNOWLEDGE DISTILLATION
+
+To verify the universality of our findings that lexical choice error will propagate from teacher model, we conduct the following experiments.
+
+In particular, we experiment AT-Base and AT-Small models on the En-De data, which are distilled by the AT-Strong model. Note that the AT-Small model consists of 256 model dimensions, 4 heads, 3 encoder and 3 decoder layers. As shown in Table 11, the same phenomena can be found in AT models when distillation is used. We leave a thorough exploration of this aspect for future work.
+
+<table><tr><td>Model</td><td>BLEU</td><td>AoLC on LFT</td><td>Ratio of LFT</td></tr><tr><td>AT-Base</td><td>27.3</td><td>72.5%</td><td>9.2%</td></tr><tr><td>+KD</td><td>27.8</td><td>68.4%</td><td>7.8%</td></tr><tr><td>AT-Small</td><td>21.6</td><td>61.8%</td><td>10.7%</td></tr><tr><td>+KD</td><td>23.5</td><td>59.3%</td><td>7.1%</td></tr></table>
+
+Table 11: Results of AT models on En-De when knowledge distillation is used. LFT denotes lowfrequency tokens and Ratio of LFT means the ratio of low-frequency tokens in generated translation.
+
+# A.3 DECAY CURRICULUM SETUP
+
+Specifically, the training process is divided into 5 phases, which differ at the constituent of training data. At Phase 1, all training examples are from the raw data; and at Phase 2, $7 5 \%$ of the training examples are from the raw data and the other $2 5 \%$ are from the distilled data (note that the two kinds of training examples should cover all source sentences). Similarly, the constituent ratios at the later phases are $( 5 0 \% , 5 0 \% )$ , $( 2 5 \% , 7 5 \% )$ , and $( 0 \% , 1 0 0 \% )$ .
+
+# A.4 NOISE INJECTION SETUP
+
+We swap the maximal probability tokens with other random tokens under the change ratio of $N \%$ . With $2 \%$ and $5 \%$ noises, our method respectively decreased by $- 0 . 1$ and -0.2 BLEU scores on En-De. The improvements in translation accuracy on low-frequency words are $+ 5 . 7 \%$ and $+ 5 . 3 \%$ , which is comparable to non-noisy one (i.e. $+ 5 . 9 \%$ ).

parse/train/ZTFeSBIX9C/ZTFeSBIX9C_content_list.json

@@ -0,0 +1,1892 @@
+[
+    {
+        "type": "text",
+        "text": "UNDERSTANDING AND IMPROVING LEXICAL CHOICE IN NON-AUTOREGRESSIVE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Liang Ding1∗, Longyue Wang2, Xuebo Liu3, Derek F. Wong3, Dacheng Tao1 & Zhaopeng $\\mathbf { T } \\mathbf { u } ^ { 2 }$ ",
+        "bbox": [
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+            169,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1The University of Sydney 2Tencent AI Lab 3University of Macau {ldin3097,dacheng.tao}@sydney.edu.au, nlp2ct.xuebo@gmail.com, {vinnylywang,zptu}@tencent.com, derekfw@um.edu.com ",
+        "bbox": [
+            186,
+            185,
+            774,
+            227
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            263,
+            544,
+            279
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Knowledge distillation (KD) is essential for training non-autoregressive translation (NAT) models by reducing the complexity of the raw data with an autoregressive teacher model. In this study, we empirically show that as a side effect of this training, the lexical choice errors on low-frequency words are propagated to the NAT model from the teacher model. To alleviate this problem, we propose to expose the raw data to NAT models to restore the useful information of low-frequency words, which are missed in the distilled data. To this end, we introduce an extra Kullback-Leibler divergence term derived by comparing the lexical choice of NAT model and that embedded in the raw data. Experimental results across language pairs and model architectures demonstrate the effectiveness and universality of the proposed approach. Extensive analyses confirm our claim that our approach improves performance by reducing the lexical choice errors on low-frequency words. Encouragingly, our approach pushes the SOTA NAT performance on the WMT14 English-German and WMT16 Romanian-English datasets up to 27.8 and 33.8 BLEU points, respectively. ",
+        "bbox": [
+            233,
+            295,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            530,
+            336,
+            545
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "When translating a word, translation models need to spend a substantial amount of its capacity in disambiguating its sense in the source language and choose a lexeme in the target language which adequately express its meaning (Choi et al., 2017; Tamchyna, 2017). However, neural machine translation (NMT) has a severe problem on lexical choice, since it usually has mistranslation errors on low-frequency words (Koehn & Knowles, 2017; Nguyen & Chiang, 2018; Gu et al., 2020). ",
+        "bbox": [
+            174,
+            560,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In recent years, there has been a growing interest in non-autoregressive translation (NAT, Gu et al., 2018), which improves decoding efficiency by predicting all tokens independently and simultaneously. Well-performed NAT models are generally trained on synthetic data distilled by autoregressive translation (AT) teachers instead of the raw training data (Figure 1(a)) (Stern et al., 2019; Lee et al., 2018; Ghazvininejad et al., 2019; Gu et al., 2019; Hao et al., 2021). Recent studies have revealed that knowledge distillation (KD) reduces the modes (i.e. multiple lexical choices for a source word) in the raw data by re-weighting the training examples (Furlanello et al., 2018; Tang et al., ",
+        "bbox": [
+            174,
+            638,
+            452,
+            875
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "table",
+        "img_path": "images/6d5aa67d0698a165ebbae3ec6c7a23bcb2545a59f0dced95e2e4d889970f76f6.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>SRC RAW-TGT KD-TGT</td><td>今天纽马基特的跑道湿软。 The going at Newmarket is soft... Today, Newmargot&#x27;s runway is soft ...</td></tr><tr><td>SRC RAW-TGT KD-TGT</td><td>纽马 基特赛马总是吸引. The Newmarket stakes is always ... The Newmarquette races always ...</td></tr><tr><td>SRC</td><td>在纽马基特3时45分那场中，我..</td></tr><tr><td>RAW-TGT KD-TGT</td><td>I&#x27;ve ...inthe3.45atNewmarket. I.. at 3:45 a.m. in Newmarquite.</td></tr></table>",
+        "bbox": [
+            464,
+            655,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Table 1: All samples that contain the source word “纽 基 ” in raw and distilled training corpora, which 马 特are different in target sides (RAW-TGT vs. KD-TGT). ",
+        "bbox": [
+            464,
+            814,
+            823,
+            856
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2020), which lowers the intrinsic uncertainty (Ott et al., 2018) and learning difficulty for NAT (Zhou et al., 2020; Ren et al., 2020). However, the side effect of KD has not been fully studied. In this work, we investigate this problem from the perspective of lexical choice, which is at the core of machine translation. ",
+        "bbox": [
+            174,
+            875,
+            823,
+            901
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We argue that the lexical choice errors of AT teacher can be propagated to the NAT model via the distilled training data. To verify this hypothesis, we qualitatively compare raw and distilled training corpora. Table 1 lists all samples whose source sentences contain the place name “纽 基 ”. In the 马 特raw corpus (“RAW-TGT”), this low-frequency word totally occurs three times and corresponds to correct translation “Newmarket”. However, in the KD corpus (“KD-TGT”), the word is incorrectly translated into a person name “Newmargot” (Margot Robbie is an Australian actress) or organization name “Newmarquette” (Marquette is an university in Wisconsin) or even invalid one “Newmarquite”. ",
+        "bbox": [
+            174,
+            138,
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+            236
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Motivated by this finding, we explore NAT from the lexical choice perspective. We first validate our hypothesis by analyzing the lexical choice behaviors of NAT models (§3). Concretely, we propose a new metric AoLC (accuracy of lexical choice) to evaluate the lexical translation accuracy of a given NAT model. Experimental results across different language pairs show that NAT models trained on distilled data have higher accuracy of global lexical translation (AoLC↑), which results in better sequence generation. However, fine-grained analyses revealed that although KD improves the accuracy on high-frequency tokens, it meanwhile harms performance on low-frequency ones (Low freq. AoLC↓). And with the improvement of teacher models, this issue becomes more severe. We conclude that the lexical choice of the low-frequency tokens is a typical kind of lost information when using knowledge distillation from AT model. ",
+        "bbox": [
+            173,
+            242,
+            825,
+            382
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In order to rejuvenate this lost information in raw data, we propose to expose the raw data to the training of NAT models, which augments NAT models the ability to learn the lost knowledge by themselves. Specifically, we propose two bi-lingual lexical-level data-dependent priors (Word Alignment Distribution and Self-Distilled Distribution) extracted from raw data, which is integrated into NAT training via Kullback-Leibler divergence. Both approaches expose the lexical knowledge in the raw data to NAT, which makes it learn to restore the useful information of low-frequency words to accomplish the translation. ",
+        "bbox": [
+            174,
+            388,
+            825,
+            486
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We validated our approach on several datasets that widely used in previous studies (i.e. WMT14 En-De, WMT16 Ro-En, WMT17 Zh-En, and WAT17 Ja-En) and model architectures (i.e. MaskPredict (Ghazvininejad et al., 2019) and Levenshtein Transformer (Gu et al., 2019)). Experimental results show that the proposed method consistently improve translation performance over the standard NAT models across languages and advanced NAT architectures. The improvements come from the better lexical translation accuracy (low-frequency tokens in particular) of NAT models $\\mathrm { ( A o L C \\uparrow ) }$ ), which leads to less mis-translations and low-frequency words prediction errors. The main contributions of this work are: ",
+        "bbox": [
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+            493,
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+            604
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Our study reveals the side effect of NAT models’ knowledge distillation on low-frequency lexicons, which makes the standard NAT training on the distilled data sub-optimal.   \n• We demonstrate the necessity of letting NAT models learn to distill lexical choices from the raw data by themselves.   \n• We propose an simple yet effective approach to accomplish this goal1, which are robustly applicable to several model architectures and language pairs. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
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+            733,
+            339,
+            750
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 NON-AUTOREGRESSIVE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            763,
+            486,
+            779
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The idea of NAT has been pioneered by Gu et al. (2018), which enables the inference process goes in parallel. Different from AT models that generate each target word conditioned on previously generated ones, NAT models break the autoregressive factorization and produce target words in parallel. Given a source sentence $\\mathbf { x }$ , the probability of generating its target sentence y with length $T$ is calculated as: ",
+        "bbox": [
+            174,
+            790,
+            825,
+            859
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c551767df106ac82f46d8ec6cdebdd292f24cc3acf5916fa6e317ee1e42a8a97.jpg",
+        "text": "$$\np ( \\mathbf { y } | \\mathbf { x } ) = p _ { L } ( T | \\mathbf { x } ; \\theta ) \\prod _ { t = 1 } ^ { T } p ( \\mathbf { y } _ { t } | \\mathbf { x } ; \\theta )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            857,
+            617,
+            900
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $p _ { L } ( \\cdot )$ is a separate conditional distribution to predict the length of target sequence. During training, the negative loglikelihood loss function of NAT is accordingly $\\mathcal { L } _ { \\mathrm { N A T } } ( \\boldsymbol { \\theta } ) = - \\log p ( \\mathbf { y } | \\mathbf { x } )$ . To bridge the performance gap between NAT and AT models, a variety approaches have been proposed, such as multi-turn refinement mechanism (Lee et al., 2018; Ghazvininejad et al., 2019; Gu et al., 2019; Kasai et al., 2020), rescoring with AT models (Wei et al., 2019; Ma et al., 2019; Sun et al., 2019), adding auxiliary signals to improve model capacity (Wang et al., 2019; Ran et al., 2019; Guo et al., 2019; Ding et al., 2020), and advanced training objective (Wei et al., 2019; Shao et al., 2019; Ma et al., 2020). Our work is complementary to theirs: while they focus on improving NAT models trained on the distilled data, we refine the NAT models by exploiting the knowledge in the raw data. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            229
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Sentence-Level Knowledge Distillation NAT models suffer from the multimodality problem, in which the conditional independence assumption prevents a model from properly capturing the highly multimodal distribution of target translations. For example, one English source sentence “Thank you.” can be accurately translated into German as any one of “Danke.”, “Danke schon.” or “Vielen Dank.”, all of which occur in the training data. ¨ ",
+        "bbox": [
+            173,
+            246,
+            826,
+            315
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To alleviate this problem, Gu et al. (2018) applied sequence-level KD (Kim & Rush, 2016) to construct a synthetic corpus, whose target sentences are generated by an AT model trained on the raw data, as shown in Figure 1(a). The NAT model is only trained on distilled data with lower modes, which makes it easily acquire more deterministic knowledge (e.g. one lexical choice for each source word). While separating KD and model training makes the pipeline simple and efficient, it has one potential threat: the re-weighted samples distilled with AT model may have lost some important information. Lee et al. (2020) show that distillation benefits the sequence generation but harms the density estimation. In this study, we exploit to bridge this gap by exposing the raw data to the training of NAT models, as shown in Figure 1(b). ",
+        "bbox": [
+            174,
+            315,
+            529,
+            536
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/1b349c9d2c2f130186b4ff949bb980945ac1b95809cc11301fbd5fef10a0c345.jpg",
+        "image_caption": [
+            "Figure 1: Comparison of existing two-step and our proposed NAT training scheme. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            547,
+            335,
+            820,
+            469
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            555,
+            375,
+            569
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Datasets Experiments were conducted on four widely-used translation datasets: WMT14 EnglishGerman (En-De, Vaswani et al. 2017), WMT16 Romanian-English (Ro-En, Gu et al. 2018), WMT17 Chinese-English (Zh-En, Hassan et al. 2018), and WAT17 Japanese-English (Ja-En, Morishita et al. 2017), which consist of 4.5M, 0.6M, 20M, and 2M sentence pairs, respectively. We use the same validation and test datasets with previous works for fair comparison. To avoid unknown words, we preprocessed data via BPE (Sennrich et al., 2016) with 32K merge operations. The $\\mathrm { { G I Z A + + } }$ (Och & Ney, 2003) was employed to build word alignments for the training datasets. We evaluated the translation quality with BLEU (Papineni et al., 2002). ",
+        "bbox": [
+            174,
+            582,
+            825,
+            694
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "NAT Models We validated our research hypotheses on two SOTA NAT models: ",
+        "bbox": [
+            173,
+            710,
+            707,
+            724
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• MaskPredict (MaskT, Ghazvininejad et al. 2019) that uses the conditional mask LM (Devlin et al., 2019) to iteratively generate the target sequence from the masked input. We followed its optimal settings to keep the iteration number be 10 and length beam be 5, respectively. • Levenshtein Transformer (LevT, Gu et al. 2019) that introduces three steps: deletion, placeholder prediction and token prediction. The decoding iterations in LevT adaptively depends on certain conditions. ",
+        "bbox": [
+            178,
+            737,
+            826,
+            827
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For regularization, we tune the dropout rate from [0.1, 0.2, 0.3] based on validation performance in each direction, and apply weight decay with 0.01 and label smoothing with $\\epsilon = 0 . 1$ . We train batches of approximately 128K tokens using Adam (Kingma & Ba, 2015). The learning rate warms up to $5 \\times \\mathrm { 1 0 ^ { - 4 } }$ in the first 10K steps, and then decays with the inverse square-root schedule. We followed the common practices (Ghazvininejad et al., 2019; Kasai et al., 2020) to evaluate the translation performance on an ensemble of top 5 checkpoints to avoid stochasticity. ",
+        "bbox": [
+            173,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/e5300f3331cffcbcc305392bf09bb43a28c6ace529404a2001869c4e62c8ecdc.jpg",
+        "table_caption": [
+            "Table 2: Results of different metrics on the MaskT model trained on different datasets. “KD (X)” denotes the distilled data produced by the AT model with X setting. “CoD” denotes the complexity of data metric proposed by Zhou et al. (2020), and “AoLC” is our proposed metric to evaluate the accuracy of lexical choice in NAT models. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"3\">En-De</td><td colspan=\"3\">Zh-En</td><td colspan=\"3\">Ja-En</td></tr><tr><td>CoD</td><td>AoLC</td><td>BLEU</td><td>CoD</td><td>AoLC</td><td>BLEU</td><td>CoD</td><td>AoLC</td><td>BLEU</td></tr><tr><td>Raw</td><td>3.53</td><td>74.3</td><td>24.6</td><td>5.11</td><td>68.5</td><td>22.6</td><td>3.92</td><td>73.1</td><td>27.8</td></tr><tr><td>KD (BASE)</td><td>1.85</td><td>75.5</td><td>26.5</td><td>3.23</td><td>71.8</td><td>23.6</td><td>2.80</td><td>74.7</td><td>28.4</td></tr><tr><td>KD (BIG)</td><td>1.77</td><td>76.3</td><td>27.0</td><td>3.01</td><td>72.7</td><td>24.2</td><td>2.47</td><td>75.3</td><td>28.9</td></tr></table>",
+        "bbox": [
+            183,
+            101,
+            818,
+            199
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "AT Teachers We closely followed previous works on NAT to apply sequence-level knowledge distillation (Kim & Rush, 2016) to reduce the modes of the training data. More precisely, to assess the effectiveness of our method under different of AT teachers, we trained three kinds of Transformer (Vaswani et al., 2017) models, including Transformer-BASE, Transformer-BIG and Transformer-STRONG. The main results employ LARGE for all directions except Ro-En, which is distilled by BASE. The architectures of Transformer-BIG and Transformer-STRONG are unchanged, but STRONG utilizes a large batch (458K tokens) training strategy. ",
+        "bbox": [
+            173,
+            290,
+            826,
+            388
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 UNDERSTANDING LEXICAL CHOICE IN NAT MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            407,
+            651,
+            424
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 EVALUATING LEXICAL CHOICE OF NAT MODELS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            439,
+            558,
+            454
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Recently, Zhou et al. (2020) argue that knowledge distillation is necessary for the uncertain nature of the machine translation task. Accordingly, they propose a metric to estimate the complexity of the data $( C o D )$ , which is driven from an external word alignment model. They reveal that the distilled data is indeed less complex, which facilitates easier training for the NAT model. Inspired by this, we propose a metric to measure the lexical level accuracy of model predictions. ",
+        "bbox": [
+            174,
+            465,
+            825,
+            536
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Accuracy of Lexical Choice (AoLC) evaluates the accuracy of target lexicon chosen by a trained NAT model $M$ for each source word. Specifically, the model $M$ takes a source word $f$ as the input, and produce a hypothesis candidate list with their corresponding word confidence: ",
+        "bbox": [
+            173,
+            550,
+            826,
+            593
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b6e8db68810bf55d2ac9fcd43988a30db76842273965b9707d834e7256c4369.jpg",
+        "text": "$$\n\\mathbf { P } _ { f } ^ { M } = \\{ P ^ { M } ( e _ { 1 } | f ) , \\dots , P ^ { M } ( e _ { | \\mathbf { V } _ { t r g } | } | f ) \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            599,
+            635,
+            619
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { V } _ { t r g }$ is the target side vocabularies over whole corpus. The AoLC score is calculated by averaging the probability of the gold target word $e _ { f }$ of each source word $f$ : ",
+        "bbox": [
+            173,
+            625,
+            826,
+            654
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/531872cf2295479f32f5c195d7642e38c8cdf19e7439d9707f58266d6e63758b.jpg",
+        "text": "$$\nA o L C = \\frac { \\sum _ { f \\in \\mathbf { V } _ { s r c } ^ { t e s t } } P ^ { M } ( e _ { f } | f ) } { | \\mathbf { V } _ { s r c } ^ { t e s t } | }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { V } _ { s r c } ^ { t e s t }$ is the set of source side tokens in test set. Each gold word $e _ { f }$ is chosen with the help of the word alignment model $P _ { f } ^ { A }$ . The chosen procedure is as follows: Step 1) collecting the references of the source sentences that contains source word $f$ , and generating the target side word bag $\\mathbb { B } _ { f }$ with these references. Step 2) Descending $P _ { f } ^ { A }$ in terms of alignment probabilities and looking up the word that first appears in $\\mathbb { B } _ { f }$ as the gold word until the $\\mathbb { B } _ { f }$ is traversed. Step 3) If the gold word is still not found, let the word with the highest alignment probability in $P _ { f } ^ { A }$ as the gold word. Generally, higher accuracy of lexical translation represents more confident of the predictions. We discuss the reliability of word alignment-based AoLC in Appendix A.1. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 GLOBAL EFFECT OF KNOWLEDGE DISTILLATION ON LEXICAL CHOICE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we analyze the lexical choice behaviors of NAT models with our proposed AoLC. In particular, We evaluated three MaskT models, which are respectively trained on the raw data, AT-BASE and AT-BIG distilled data. We compared the AoLC with other two metrics (i.e. BLEU and CoD) on three different datasets (i.e. En-De, Zh-En and Ja-En). As shown in Table 2, KD is able to improve translation quality of NAT models (BLEU: KD(BIG) ${ \\mathrm { > K D } }$ (BASE) ${ \\mathrm { > R a w } }$ ) by increasing the lexical choice accuracy of data (AoLC: KD(BIG) ${ \\tt > K D }$ (BASE) ${ \\mathrm { > R a w } }$ ). As expected, NAT models trained on more deterministic data $( \\mathrm { C o D } \\downarrow )$ have lower lexical choice errors (AoLC↑) globally, resulting in better model generation performance (BLEU↑). ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.3 DISCREPANCY BETWEEN HIGH- AND LOW-FREQUENCY WORDS ON LEXICAL CHOICE ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To better understand more detailed lexical change within data caused by distillation, we break down the lexicons to three categories in terms of frequency. And we revisit it from two angles: training data and translated data. ",
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+    {
+        "type": "text",
+        "text": "We first visualize the changing of training data when adopting KD in terms of words frequency density. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 2, we find that the kurtosis of KD data distribution is higher than that of raw, which becomes more significant when adopting stronger teacher. The side effect is obvious, that is, the original high- / low-frequency words become more / fewer, making the distribution of training data more imbalance and skewed, which is problematic in data mining field (Chawla et al., 2004). This discrepancy may erode the translation performance of low-frequency words and generalization performance on other domains. Here we focus on lowfrequency words, and generalization performance degradation will be exploited in future work. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/dddd69e87f1d60170cfd9055d32bec4a6f19d5103ff92dddcd1b1cb1bdd80da0.jpg",
+        "image_caption": [
+            "Figure 2: Comparison of the token frequency density (w.r.t the sampled tokens’ probability distribution) between Raw, $K D$ (Base) and $K D$ (Big) WMT14 En-De training data.quencycy "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In order to understand the detailed change during inference, we then analyze the lexical accuracy with different frequencies in the test set. We make the comprehensive comparison cross languages based on our proposed AoLC. As shownFrFrequeFrequency in Figure 3, as the teacher model becomes better, i.e. $\\mathrm { K D ( b a s e ) { \\to } K D ( b i g }$ ), the lexical choice of high-High High MHigh Med. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "frequency words becomes significantly more accurate (AoLC ↑) while that of low-frequency words808080 becomes worse (AoLC ↓). Through fine-grained analysis, we uncover this interesting discrepancy606060 between high- and low- frequency words. The same phenomena (lexical choice errors on lowfrequency words propagated from teacher model) also can be found in general cases, e.g. distillation404040 when training smaller AT models. Details can be found in Appendix A.2. To keep the accuracy of202020 high-frequency words and compensate for the imbalanced low-frequency words caused by KD, we present a simple yet effective approach below. En- En-De En-De Zh",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/a49ead06b91e281953bc62e41853c855b922f9199f5d849e84ef0616fd90c617.jpg",
+        "image_caption": [
+            "Figure 3: Accuracy of lexical choice (AoLC) for source words of different frequency. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4 IMPROVING LEXICAL CHOICE IN NAT MODELS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our goal is to augment NAT models to learn needed lexical choices from the raw data to achieve better performance. To this end, we introduce an extra bilingual data-dependent prior objective to augment the current NAT models to distill the required lexical choices from the raw data. Specifically, we use Kullback-Leibler divergence to guide the probability distribution of model predictions $P ^ { \\bar { M } } ( e | { \\bf f } )$ to match the prior probability distributions $Q ( \\cdot )$ : ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/ddb3e581197c795df1fdf09773d102bc57054e35813e072e167068c4f5c86b19.jpg",
+        "text": "$$\n\\mathcal { L } _ { p r i o r } = - \\sum _ { e \\in { \\bf e } } \\mathrm { K L } \\big ( Q ( e | { \\bf f } ) \\big | \\big | P ^ { M } ( e | { \\bf f } ) \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where f is the source sentence, and $\\mathbf { e }$ is the target sentence. The bilingual prior distribution $Q ( \\cdot )$ is derived from the raw data, which is independent of the model $M$ and will be described later. The final objective for training the NAT model becomes: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/7f7b88eb095a7c823f8b97c9f5f73bcd4ae38c50658fa845d3e72f5d0d42c5ae.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } = ( 1 - \\lambda ) \\mathcal { L } _ { N A T } + \\lambda \\mathcal { L } _ { p r i o r } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "in which the imitation rate $\\lambda$ follows the logarithmic decay function: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a47a4e44f8993159c113b4c389cde9b8662b1482e634a38d34e3ebfa00ca050d.jpg",
+        "text": "$$\n\\lambda ( i ) = \\left\\{ \\begin{array} { l l } { \\frac { l o g ( \\mathrm { I } / ( 2 ( i + 1 ) ) ) } { l o g ( \\mathrm { I } / 2 ) } } & { i \\leq \\mathrm { I } / 2 } \\\\ { 0 } & { \\mathrm { o t h e r s } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "where $i$ is the current step, I is the total training step for distilled data. Accordingly, the NAT model is merely fed with the priori knowledge derived from the raw data at beginning. Along with training, the supervision signal of the prior information is getting weaker while that of the distilled data gradually prevails in the training objective. We run all models for 300K steps to ensure adequate training, thus the bilingual prior distributions will be exposed at the first 150K steps. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Choices of Prior Distribution $Q ( \\cdot )$ The goal of the prior objective is to guide the NAT models to learn to distill the lexical choices itself from the raw data. For each target word $e$ , we use the external word alignment to select the source word $f$ with the maximum alignment probability, and $Q ( \\cdot )$ is rewritten as: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/7db3a664f26601f84441d38e81197f207fc5fb9551d75b6f3dbdac4afe8bcc4d.jpg",
+        "text": "$$\nQ ( e | \\mathbf { f } ) = Q ( e | f )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Specifically, we use two types of bilingual prior distributions: ",
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+    {
+        "type": "text",
+        "text": "• Word Alignment Distribution $( W A D )$ is the distribution derived from the external word alignment $\\mathbf { P } _ { f } ^ { D } = \\{ { \\breve { P } } ^ { D } ( e _ { 1 } | f ) , \\dots , P ^ { D } ( e _ { N } | f ) \\}$ where $\\{ e _ { 1 } , \\dots , e _ { N } \\}$ are the set of target words aligned to the source word in the training data. We follow Hinton et al. (2015) to use the softmax temperature mechanism to map $\\mathbf { P } _ { f } ^ { D }$ over the whole target vocabulary: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/7854e9f9dbe6c8ecf36b54e393fdf90b1ba22272da00b56af1e26c9349e4d837.jpg",
+        "text": "$$\nQ ( e | f ) = \\hat { \\mathbf { P } } _ { f } ^ { D } = \\frac { e x p ( \\mathbf { P } _ { f } ^ { D } / \\tau ) } { \\sum _ { V _ { t g t } } e x p ( \\mathbf { P } _ { f } ^ { D } / \\tau ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We tune the temperature from [0.5, 1, 2, 5] on WMT14 En-De dataset and use $\\tau = 2$ as the default setting for incorporating word alignment distribution in all datasets. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• Self-Distilled Distribution $( S D D )$ is the probability distribution for the source word $f$ , which is produced by a same NAT model pre-trained on raw data. Specifically, the model $M$ takes a source word $f$ as input and produces a probability distribution over whole words in target vocabulary: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/a7bcb4a22571a0b1ca7074fc344cf811072fe0aeee9f620ec4199984d9b46428.jpg",
+        "text": "$$\n\\mathbf { P } _ { f } ^ { M } = \\{ P ^ { M } ( e _ { 1 } | f ) , \\dots , P ^ { M } ( e _ { | \\mathbf { V } _ { t r g } | } | f ) \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This prior distribution signal can be characterized as self-distilled lexicon level “born-again networks” (Furlanello et al., 2018) or self-knowledge distillation (Liu et al., 2020), where the teacher and student have the same neural architecture and model size, and yet surprisingly the student is able to surpass the teacher’s accuracy. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8967daf33c8665b6cf04ea536d63f964a1c37dbcaa211523ce1fc497dcb4290e.jpg",
+        "table_caption": [
+            "Table 3: Ablation Study on raw data priors across different language pairs using the MaskT Model. “WAD” denotes word alignment distribution, and “SDD” denotes self-distilled distribution. “AoLC / LFT” denotes the lexical translation accuracies for all tokens / low-frequency tokens, respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">En-De</td><td colspan=\"2\">Zh-En</td><td colspan=\"2\">Ja-En</td></tr><tr><td>AoLC /LFT</td><td>BLEU</td><td>AoLC /LFT</td><td>BLEU</td><td>AoLC /LFT</td><td>BLEU</td></tr><tr><td>AT-TEACHER</td><td>79.3 / 73.0</td><td>29.2</td><td>74.7 / 66.2</td><td>25.3</td><td>77.1/ 70.8</td><td>29.8</td></tr><tr><td>MaskT+KD</td><td>76.3 / 68.4</td><td>27.0</td><td>72.7 / 61.5</td><td>24.2</td><td>75.3 / 66.9</td><td>28.9</td></tr><tr><td>+WAD</td><td>77.5 / 71.9</td><td>27.4</td><td>73.4 / 64.5</td><td>24.8</td><td>76.3 / 69.0</td><td>29.4</td></tr><tr><td>+SDD</td><td>77.7 /72.2</td><td>27.5</td><td>73.5 / 64.7</td><td>24.9</td><td>76.1 / 68.6</td><td>29.3</td></tr><tr><td>+Both</td><td>78.1/ 72.4</td><td>27.8</td><td>74.0 / 65.0</td><td>25.2</td><td>76.6 / 69.1</td><td>29.6</td></tr></table>",
+        "bbox": [
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+            101,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/4d4355687b4a9c1e664bcca572e4a959e9b3d840e31384345f11a02163ea8017.jpg",
+        "table_caption": [
+            "Table 4: Comparison with previous work on WMT14 En-De and WMT16 Ro-En datasets. “Iter.” column indicate the average number of refined iterations. “†” indicates statistically significant difference $( p < 0 . 0 5 )$ from baselines according to the statistical significance test (Collins et al., 2005). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Iter. Speed</td><td rowspan=\"2\"></td><td colspan=\"2\">En-De</td><td colspan=\"2\">Ro-En</td></tr><tr><td>AoLC</td><td>BLEU</td><td>AoLC</td><td>BLEU</td></tr><tr><td colspan=\"8\">AT Models</td></tr><tr><td>Transformer-BASE (Ro-En Teacher)</td><td>n/a</td><td>1.0×</td><td></td><td>27.3</td><td></td><td>34.1</td></tr><tr><td>Transformer-BIG (En-De Teacher)</td><td>n/a</td><td>0.8×</td><td></td><td>29.2</td><td></td><td>n/a</td></tr><tr><td colspan=\"7\">Existing NAT Models</td></tr><tr><td>NAT (Gu et al., 2018) Iterative NAT (Lee et al., 2018)</td><td>1.0</td><td>2.4×</td><td></td><td>19.2 21.6</td><td rowspan=\"4\">n/a</td><td>31.4 30.2</td></tr><tr><td>DisCo (Kasai et al., 2020)</td><td>10.0 4.8</td><td>2.0× 3.2×</td><td></td><td>26.8</td><td>33.3</td></tr><tr><td>Mask-Predict (Ghazvininejad et al., 2019)</td><td>10.0</td><td>1.5×</td><td>n/a</td><td>27.0</td><td>33.3</td></tr><tr><td>Levenshtein (Gu et al., 2019)</td><td>2.5</td><td>3.5×</td><td></td><td>27.3</td><td>33.3</td></tr><tr><td colspan=\"8\"></td></tr><tr><td>Mask-Predict</td><td>Our NAT Models</td><td></td><td>76.3</td><td>27.0</td><td>79.2</td><td>33.3</td></tr><tr><td>+Raw Data Prior</td><td>10.0</td><td>1.5×</td><td>78.1</td><td>27.8t</td><td>80.6</td><td>33.7</td></tr><tr><td>Levenshtein</td><td>2.5</td><td>3.5×</td><td>77.0</td><td>27.2</td><td>79.8</td><td>33.2</td></tr><tr><td>+RawData Prior</td><td></td><td></td><td>77.8</td><td>27.8t</td><td>80.9</td><td>33.8t</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Ablation Study on Raw Data Prior Table 3 shows the results of our proposed two bilingual data dependent prior distributions across language pairs. The word alignment distribution (WAD) and self-distilled distribution (SDD) variants consistently improves performance over the vanilla two-step training scheme NAT model $\\mathrm { ^ { * } N A T { + } K D ^ { \\prime } }$ ) when used individually (averagely $+ 0 . 5$ BLEU point), and combining them $\\hbar ^ { * } { + } \\mathrm { B o t h } ^ { \\prime \\prime }$ ) by simply averaging the two distributions can achieve a further improvement (averagely $+ 0 . 9$ BLEU point). The improvements on translation performance are due to a increase of AoLC, especially for low-frequency tokens (averagely $+ 3 . 2$ ), which reconfirms our claim. Notably, averaging the two prior distributions could rectify each other, thus leading to a further increase. We explore the complementarity of two prior schemes in Section 4.3. In the following experiments, we use the combination of WAD and SDD as the default bilingual data dependent prior. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Comparison with Previous Work Table 4 lists the results of previously competitive studies (Gu et al., 2018; Lee et al., 2018; Kasai et al., 2020; Ghazvininejad et al., 2019; Gu et al., 2019) on the widely-used WMT14 En-De and WMT16 Ro-En datasets. Clearly, our bilingual data-dependent prior significantly improves translation (BLEU↑) by substantially increasing the lexical choice accuracy (AoLC↑). It is worth noting that our approaches merely modify the training process, thus does not increase any latency (“Speed”), maintaining the intrinsic advantages of non-autoregressive generation. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b3ceae27e6b03a0605f1e361c056f24e3e0b8b9b3975ef27c65fb6b6f13209cd.jpg",
+        "table_caption": [
+            "Table 7: Improvement of our approach over the MaskT $\\mathrm { \\Phi } _ { + \\mathrm { K D } }$ model on AoLC. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Frequency</td><td>En-De</td><td>Zh-En</td><td>Ja-En</td></tr><tr><td>High</td><td>+1.3%</td><td>+0.3%</td><td>+1.3%</td></tr><tr><td>Medium</td><td>+0.2%</td><td>+0.1%</td><td>+0.9%</td></tr><tr><td>Low</td><td>+5.9%</td><td>+5.8%</td><td>+3.3%</td></tr><tr><td>All</td><td>+2.4%</td><td>+1.8%</td><td>+1.7%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/1add650ea315f2c038517926d0c3c49c923d76764c63b1f6e0533e554ff8665b.jpg",
+        "table_caption": [
+            "Table 8: Ratio of low-frequency target words in the MaskT model generated translations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>En-De</td><td>Zh-En</td><td>Ja-En</td></tr><tr><td>NAT</td><td>10.3%</td><td>6.7%</td><td>9.4%</td></tr><tr><td>+KD</td><td>7.6%</td><td>4.2%</td><td>6.9%</td></tr><tr><td>+Ours</td><td>9.8%</td><td>6.1%</td><td>8.5%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Comparison with Data Manipulation Strategies Instead of using the proposed priors, we also investigate two effective data manipulation strategies, i.e. Data Mixing and Curriculum Learning, to force the NAT model learns from both the raw and distilled data. For data mixing, we design two settings: a) Mix: simply combine the raw and distilled data, and then shuffle the mixed dataset. b) Tagged Mix: Inspired by successes of tagged back-translation (Caswell et al., 2019; Marie et al., 2020), we add tags to distinguish between KD and Raw sentences in the mixed dataset. For decay curriculum schedule, the NAT models learn more from raw data at the beginning and then learn more from KD as the training goes on. The details of curriculum can be found in Appendix A.3. As seen in Table 5, data mixing and decay curriculum schedule improve performance on both AoLC and BLEU, which confirm the necessity of exposing raw data to NAT models during training. Besides, our approach still outperforms those effective strategies, demonstrating the superiority of our learning scheme. ",
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+        "table_caption": [
+            "Table 5: Performance of several data manipulation strategies on En-De dataset. Baseline is the $\\mathbf { M a s k T + K D }$ model and Ours is our proposed approach. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td> Strategies</td><td>AoLC</td><td>BLEU</td></tr><tr><td>Baseline</td><td>76.3</td><td>27.0</td></tr><tr><td>Mix</td><td>76.6</td><td>27.2</td></tr><tr><td>Tagged Mix</td><td>77.1</td><td>27.4</td></tr><tr><td>Decay Curriculum</td><td>77.2</td><td>27.5</td></tr><tr><td>Ours</td><td>78.1</td><td>27.8</td></tr></table>",
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+    {
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+        "text": "4.3 EXPERIMENTAL ANALYSIS ",
+        "text_level": 1,
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+        "text": "In this section, we conducted extensive analyses on the lexical choice to better understand our approach. Unless otherwise stated, results are reported on the MaskPredict models in Table 3. ",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>BLEU</td><td>AoLC</td><td>Error</td></tr><tr><td>MaskT</td><td>22.6</td><td>68.5%</td><td>34.3%</td></tr><tr><td>+KD</td><td>24.2</td><td>72.7%</td><td>30.1%</td></tr><tr><td>+RDP</td><td>25.2</td><td>74.0%</td><td>28.2%</td></tr></table>",
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+        "type": "text",
+        "text": "Our approach improves translation performance by reducing mis-translation errors. The lexical choice ability of NAT models correlates to mistranslation errors, in which wrong lexicons are chosen to translate source words. To better understand whether our method alleviates the mis-translation problem, we ",
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+    {
+        "type": "text",
+        "text": "Table 6: Subjective evaluation of mistranslation errors on the Zh-En dataset. ",
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+    {
+        "type": "text",
+        "text": "assessed system output by human judgments. In particular, we randomly selected 50 sentences from the Zh-En testset, and manually labelled the words with lexical choice error. We defined the lexical choice error rate as $E / N$ , where $E$ is the number of lexical choice errors and $N$ is the number of content words in source sentences, since such errors mainly occur in translating content words. As seen in Table 6, our approache consistently improves BLEU scores by reducing the lexical choice errors, which confirm our claim. Additionally, AoLC metric correlates well with both the automatic BLEU score and the subjective evaluation, demonstrating its reasonableness. ",
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+    {
+        "type": "text",
+        "text": "Our approach significantly improves the accuracy of lexical choice for low-frequency source words. As aforementioned discrepancy between high- & low-frquency words in Section 3.3, we focus on revealing the fine-grained lexical choice accuracy w.r.t our proposed AoLC. In Table 7, the majority of improvements is from the low-frequency words, confirming our hypothesis. ",
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+    {
+        "type": "text",
+        "text": "Our approach generates translations that contain more low-frequency words. Besides improving the lexical choice of low-frequency words, our method results in more low-frequency words being recalled in the translation. In Table 8, although KD improves the translation, it biases the NAT model towards generating high-frequency tokens (Low freq.↓) while our method can not only correct this bias (averagely $+ 3 2 \\%$ relative change), but also enhance translation (BLEU↑ in Table 3). ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Our proposed two priors complement each other by facilitating different tokens. As aforementioned in Table 3, combining two individual schemes can further increase the NAT performance. To explain how they complement each other, especially for low-frequency tokens, we classify low-frequency tokens into two categories according to their linguistic roles: content words (e.g. noun, verb, and adjective) and function words (e.g. preposition, determiner, and punctuation). The results are listed in Table 9. We show that WAD facilitates more on the understanding and generation of content tokens, while SDD brings more gains for function (i.e. content",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Prior</td><td colspan=\"2\">AoLC on LFT</td><td colspan=\"2\">Ratio of LFT</td></tr><tr><td>Content</td><td>Function</td><td>Content</td><td>Function</td></tr><tr><td>N/A</td><td>67.7%</td><td>70.1%</td><td>5.3%</td><td>2.4%</td></tr><tr><td>WAD</td><td>71.6%</td><td>72.9%</td><td>5.9%</td><td>2.5%</td></tr><tr><td>SDD</td><td>71.4%</td><td>74.3%</td><td>5.6%</td><td>3.4%</td></tr><tr><td>Both</td><td>71.6%</td><td>74.2%</td><td>6.2%</td><td>3.6%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 9: AoLC and Ratio of different prior schemes on Low-Frequency Tokens (“LFT”). We list the performances on different linguistic roles, i.e. content words and function words. Note that Ratio of LFT means the ratio of low frequency tokens in generated translation. “N/A” means MaskT $\\mathrm { + K D }$ baseline. ",
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+    },
+    {
+        "type": "text",
+        "text": "free) tokens. We leave a more thorough exploration of this aspect for future work. ",
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+    },
+    {
+        "type": "text",
+        "text": "Effect of Word Alignment Quality on Model Performance. Both the proposed AoLC and priors depend heavily on the quality of word alignment, we therefore design two weaker alignment scenarios to verify the robustness of our method. ",
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+    {
+        "type": "text",
+        "text": "First, We adopt fast-align (Dyer et al., 2013), which is slightly weaker than $\\mathrm { { G I Z A + + } }$ . Using fastalign, our methods can still achieve $+ 0 . 6$ and $+ 0 . 7$ improvements in terms of BLEU on En-De and Zh-En datasets, which are marginally lower than that using $\\mathrm { { G I Z A + + } }$ (i.e. $+ 0 . 8$ and $+ 1 . 0$ BLEU). Encouragingly, we find that the improvements in translation accuracy on low-frequency words still hold $+ 5 . 5 \\%$ and $+ 5 . 3 \\%$ vs. $+ 5 . 9 \\%$ and $+ 5 . 8 \\%$ ), which demonstrates the robustness of our approach. ",
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+    {
+        "type": "text",
+        "text": "In addition, we insert noises into the alignment distributions to deliberately reduce the alignment quality (Noise injection details can be found in Appendix A.4. The performances still significantly outperform the baseline, indicating that our method can tolerate alignment errors and maintain model performance to some extent. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Effect of AT Teacher To further dissect the different effects when applying different AT teachers, we employ three teachers. Table 10 shows our method can enhance NAT models under variety of teacher-student scenarios, including base, big and strong teacher-guided models. Our approach obtains averagely $+ 0 . 7$ BLEU points, potentially complementary to the majority of existing work on improving knowledge distillation for NAT models. ",
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+    {
+        "type": "table",
+        "img_path": "images/5f004fec4f408ebb9a0fe4271df02ff698d3687cd8bf4634e24612b50f2fae4b.jpg",
+        "table_caption": [
+            "Table 10: Different teachers on the En-De dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">AT Teacher</td><td colspan=\"2\">NAT Model</td></tr><tr><td>Model</td><td>BLEU</td><td>Vanilla</td><td>+Prior △</td></tr><tr><td>Base</td><td>27.3</td><td>26.5</td><td>27.2 +0.7</td></tr><tr><td>Big</td><td>28.4</td><td>26.8</td><td>27.5 +0.7</td></tr><tr><td> Strong</td><td>29.2</td><td>27.0</td><td>27.8 +0.8</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Understanding Knowledge Distillation for NAT Knowledge distillation is a crucial early step in the training of most NAT models. Ren et al. (2020) reveal that the difficulty of NAT heavily depends on the strongness of dependency among target tokens, and knowledge distillation reduces the token dependency in target sequence and thus improves the accuracy of NAT models. In the pioneering work of NAT, Gu et al. (2018) claim that NAT suffers from the multi-modality problem (i.e. multiple lexical translations for a source word), and knowledge distillation can simplify the dataset, which is empirically validated by Zhou et al. (2020). We confirm and extend these results, showing that the AT-distilled dataset indeed leads to more deterministic predictions but propagates the low-frequency lexical choices errors. To this end, we enhance the NAT lexical predictions by making them learn to distill knowledge from the raw data. ",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Lexical Choice Problem in NMT Models Benefiting from continuous representations abstracted from the training data, NMT models have advanced the state of the art in the machine translation community. However, recent studies have revealed that NMT models suffer from inadequate translation (Tu et al., 2016), in which mis-translation error caused by the lexical choice problem is one main reason. For AT models, Arthur et al. (2016) alleviate this issue by integrating a count-based lexicon, and Nguyen & Chiang (2018) propose an additional lexical model, which is jointly trained with the AT model. The lexical choice problem is more serious for NAT models, since 1) the lexical choice errors (low-resource words in particular) of AT distillation will propagate to NAT models; and 2) NAT lacks target-side dependencies thus misses necessary target-side context. In this work, we alleviate this problem by solving the first challenge. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this study, we investigated effects of KD on lexical choice in NAT. We proposed a new metric to evaluate lexical translation accuracy of NAT models, and found that 1) KD improves global lexical predictions; and 2) KD benefits the accuracy of high-frequency words but harms the low-frequency ones. There exists a discrepancy between high- and low-frequency words after adopting KD. To bridge this discrepancy, we exposed the useful information in raw data to the training of NAT models. Experiments show that our approach consistently and significantly improves translation performance across language pairs and model architectures. Extensive analyses reveal that our method reduces mistranslation errors, improves the accuracy of lexical choices for low-frequency source words, recalling more low-frequency words in the translations as well, which confirms our claim. ",
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+    {
+        "type": "text",
+        "text": "7 ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This work was supported by Australian Research Council Projects under grants FL-170100117, DP-180103424, and IC-190100031. Xuebo and Derek were supported in part by the Science and Technology Development Fund, Macau SAR (Grant No. 0101/2019/A2), and the Multi-year Research Grant from the University of Macau (Grant No. MYRG2020-00054-FST). We also thank the anonymous reviewers for their insightful comments. ",
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+        "text": "Bingzhen Wei, Mingxuan Wang, Hao Zhou, Junyang Lin, and Xu Sun. Imitation learning for non-autoregressive neural machine translation. In ACL, 2019. ",
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+    {
+        "type": "text",
+        "text": "Chunting Zhou, Graham Neubig, and Jiatao Gu. Understanding knowledge distillation in nonautoregressive machine translation. In ICLR, 2020. ",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 DISCUSSION ON THE RELIABILITY OF WORD ALIGNMENT-BASED AOLC ",
+        "bbox": [
+            174,
+            133,
+            722,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We randomly select 20 sentence pairs from the Zh-En test set, which contains 576 source tokens. We use the trained word alignment model to produce alignments for the 20 sentence pairs, and then perform the gold word chosen procedure as described in Section 3.1. We manually evaluate these bilingual lexicons, and find that 551 out of 576 source words are aligned to reasonable equivalences (i.e. $96 \\%$ accuracy). This demonstrates that it is reliable to calculate AoLC based on automatic word alignments. ",
+        "bbox": [
+            174,
+            159,
+            826,
+            243
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 GENERAL CASES OF THE SIDE-EFFECT OF KNOWLEDGE DISTILLATION ",
+        "bbox": [
+            174,
+            258,
+            712,
+            273
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To verify the universality of our findings that lexical choice error will propagate from teacher model, we conduct the following experiments. ",
+        "bbox": [
+            174,
+            285,
+            825,
+            314
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In particular, we experiment AT-Base and AT-Small models on the En-De data, which are distilled by the AT-Strong model. Note that the AT-Small model consists of 256 model dimensions, 4 heads, 3 encoder and 3 decoder layers. As shown in Table 11, the same phenomena can be found in AT models when distillation is used. We leave a thorough exploration of this aspect for future work. ",
+        "bbox": [
+            176,
+            320,
+            825,
+            377
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/d43d17bdd9b7b4c338cba3cbbf491d153d3e48be0dcf24ac9bbb4610ebb6f197.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>BLEU</td><td>AoLC on LFT</td><td>Ratio of LFT</td></tr><tr><td>AT-Base</td><td>27.3</td><td>72.5%</td><td>9.2%</td></tr><tr><td>+KD</td><td>27.8</td><td>68.4%</td><td>7.8%</td></tr><tr><td>AT-Small</td><td>21.6</td><td>61.8%</td><td>10.7%</td></tr><tr><td>+KD</td><td>23.5</td><td>59.3%</td><td>7.1%</td></tr></table>",
+        "bbox": [
+            307,
+            390,
+            687,
+            477
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 11: Results of AT models on En-De when knowledge distillation is used. LFT denotes lowfrequency tokens and Ratio of LFT means the ratio of low-frequency tokens in generated translation. ",
+        "bbox": [
+            171,
+            486,
+            826,
+            515
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 DECAY CURRICULUM SETUP",
+        "text_level": 1,
+        "bbox": [
+            176,
+            539,
+            419,
+            554
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Specifically, the training process is divided into 5 phases, which differ at the constituent of training data. At Phase 1, all training examples are from the raw data; and at Phase 2, $7 5 \\%$ of the training examples are from the raw data and the other $2 5 \\%$ are from the distilled data (note that the two kinds of training examples should cover all source sentences). Similarly, the constituent ratios at the later phases are $( 5 0 \\% , 5 0 \\% )$ , $( 2 5 \\% , 7 5 \\% )$ , and $( 0 \\% , 1 0 0 \\% )$ . ",
+        "bbox": [
+            174,
+            564,
+            825,
+            635
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 NOISE INJECTION SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            652,
+            392,
+            666
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We swap the maximal probability tokens with other random tokens under the change ratio of $N \\%$ . With $2 \\%$ and $5 \\%$ noises, our method respectively decreased by $- 0 . 1$ and -0.2 BLEU scores on En-De. The improvements in translation accuracy on low-frequency words are $+ 5 . 7 \\%$ and $+ 5 . 3 \\%$ , which is comparable to non-noisy one (i.e. $+ 5 . 9 \\%$ ). ",
+        "bbox": [
+            174,
+            678,
+            826,
+            734
+        ],
+        "page_idx": 12
+    }
+]

parse/train/_WnwtieRHxM/_WnwtieRHxM.md

@@ -0,0 +1,715 @@
+# UNDERSTANDING THE ROLE OF IMPORTANCE WEIGHT-ING FOR DEEP LEARNING
+
+Da Xu   
+Walmart Labs   
+Sunnyvale, CA 94086, USA DaXu5180@gmail.com
+
+Yuting Ye Division of Biostatistics University of California, Berkeley Berkeley, CA 94720, USA yeyt@berkeley.edu
+
+Chuanwei Ruan ∗   
+Instacart   
+San Francisco, CA 94107, USA Ruanchuanwei@gmail.com
+
+# ABSTRACT
+
+The recent paper by Byrd & Lipton (2019), based on empirical observations, raises a major concern on the impact of importance weighting for the over-parameterized deep learning models. They observe that as long as the model can separate the training data, the impact of importance weighting diminishes as the training proceeds. Nevertheless, there lacks a rigorous characterization of this phenomenon. In this paper, we provide formal characterizations and theoretical justifications on the role of importance weighting with respect to the implicit bias of gradient descent and margin-based learning theory. We reveal both the optimization dynamics and generalization performance under deep learning models. Our work not only explains the various novel phenomenons observed for importance weighting in deep learning, but also extends to the studies where the weights are being optimized as part of the model, which applies to a number of topics under active research.
+
+# 1 INTRODUCTION
+
+Importance weighting is a standard tool for estimating a quantity under a target distribution while only the samples from some source distribution is accessible. It has been drawing extensive attention in the communities of statistics and machine learning. Causal inference for deep learning investigates heavily on the propensity score weighting method that applies the off-policy optimization with counterfactual estimator (Gilotte et al., 2018; Jiang & Li, 2016), modelling with observational feedback (Schnabel et al., 2016; Xu et al., 2020) and learning from controlled intervention (Swaminathan & Joachims, 2015). The importance weighting methods are also applied to characterize distribution shifts for deep learning models (Fang et al., 2020), with modern applications in such as the domain adaptation (Azizzadenesheli et al., 2019; Lipton et al., 2018) and learning from noisy labels (Song et al., 2020). Other usages include curriculum learning (Bengio et al., 2009) and knowledge distillation (Hinton et al., 2015), where the weights characterize the model confidence on each sample.
+
+To reduce the discrepancy between the source and target distribution for model training, a standard routine is to minimize a weighted risk (Rubinstein & Kroese, 2016). Many techniques have been developed to this end, and the common strategy is re-weighting the classes proportionally to the inverse of their frequencies (Huang et al., 2016; 2019; Wang et al., 2017). For example, Cui et al.
+
+(2019) proposes re-weighting by the inverse of effective number of samples. The focal loss (Lin et al., 2017) down-weights the well-classified examples, and the work by Li et al. (2019) suggests an improved technique which down-weights examples based on the magnitude of the gradients.
+
+Despite the empirical successes of various re-weighting methods, it is ultimately not clear how importance weighting lays influence from the theoretical standpoint. The recent study of Byrd & Lipton (2019) observes from experiments that there is little impact of importance weights on the converged deep neural network, if the data can be separated by the model using gradient descent. They connect this phenomenon to the implicit bias of gradient descent (Soudry et al., 2018) - a novel topic that studies why over-parameterized models trained on separable data is biased toward solutions that generalize well. Implicit bias of gradient descent has been observed and studied for linear model (Soudry et al., 2018; Ji & Telgarsky, 2018b), linear neural network (Ji & Telgarsky, 2018a; Gunasekar et al., 2018), two-layer neural network with homogeneous activation (Chizat & Bach, 2020) and smooth neural networks (Nacson et al., 2019; Lyu & Li, 2019). To summarize, those work reveals that the direction of the parameters (for linear predictor) and the normalized margin (for nonlinear predictor), regardless of the initialization, respectively converge to those of a max-margin solution. The pivotal role of margin for deep learning models has been explored actively after the long journey of understanding the generalization of over-parameterized neural networks (Bartlett et al., 2017; Golowich et al., 2018; Neyshabur et al., 2018). For instance, Wei et al. (2019) studies the margin of the neural networks for separable data under weak regularization. They show that the normalized margin also converges to the max-margin solution, and provide a generalization bound for a neural network that hinges on its margin.
+
+Although there are rich understandings for the implicit bias of gradient descent and the margin-based generalization, very few efforts are dedicated to studying how they adjust to the weighted empiricalrisk minimization (ERM) setting. The established results do not directly transfer since importance weighting can change both the optimization geometry and how the generalization is measured. In this paper, we fill in the gap by showing the impact of importance weighting on the implicit bias of gradient descent as well as the generalization performance. By studying the optimization dynamics of linear models, we first reveal the effect of importance weighting on the convergence speed under linearly separable data. When the data is not linearly separable, we characterize the unique role of importance weighting on defining the intercept term upon the implicit bias. We then investigate the non-linear neural network under a weak regularization as Wei et al. (2019). We provide a novel generalization bound that reflects how importance weighting leads to the interplay between the empirical risk and a compounding term that consists of the model complexity as well as the deviation between the source target distribution. Based on our theoretical results, we discuss several exploratory developments on importance weighting that are worthy of further investigations.
+
+• A good set of weights for learning can be inversely proportional to the hard-to-classify extent. For example, a sample that is close to (far from) the oracle decision boundary should have a large (small) weight.   
+• If the importance weights are jointly trained according to a weighting model, the impact of the weighting model eventually diminishes after showing strong correlation with the hard-to-classify extent such as margin.   
+• The usefulness of explicit regularization on weighted ERM can be studied, via their impact on the margin, on balancing the empirical loss and the distribution divergence.
+
+In summary, our contribution are three folds.
+
+• We characterize the impact of importance weighting on the implicit bias of gradient descent. • We find a generalization bound that hinges on the importance weights. For finite-step training, the role of importance weighting on the generalization bound is reflected in how the margin is affected, and how it balances the source and target distribution. • We propose several exploratory topics for importance weighting that worth further investigating from both the application and theoretical perspective.
+
+The rest of the paper is organized as follows. In Section 2, we introduce the background, preliminary results and the experimental setup. In Section 3 and 4, we demonstrate the influence of the importance weighting for linear and non-linear models in terms of the implicit bias of gradient descent and the generalization performance. We then discuss the extended investigations in Section 5.
+
+# 2 PRELIMINARIES
+
+We use bold-font letters for vectors and matrices, uppercase letters for random variables and distributions, and $\| \cdot \|$ to denote $\ell _ { 2 }$ norm when no confusion arises. We denote the training data by $\mathbf { \mathcal { D } } = \{ w _ { i } , \mathbf { x } _ { i } , y _ { i } \} _ { i = 1 } ^ { n }$ where $\mathbf { x } _ { i } \in \mathcal { X }$ denotes the features, $y _ { i }$ is binary or categorical, and the importance weight is bounded such that: $w _ { i } \in [ 1 / M , M ]$ for some $M > 1$ . We mention that the importance weights are often defined with respect to the source distribution $P _ { s }$ from which the training data is drawn, and the target distribution $P _ { t }$ . We do not make this assumption here because importance weighting is often applied for more general purposes. Therefore, $w _ { i }$ can be defined arbitrarily.
+
+We use $f ( { \pmb \theta } , { \bf x } )$ to denote the predictor and define $\mathcal { F } = \{ f ( \pmb \theta , \cdot ) | \theta \in \Theta \subset \mathbb { R } ^ { d } \}$ . For the sake of notation, we focus on the binary setting: $y _ { i } \in \{ - 1 , + 1 \}$ with $f ( { \pmb \theta } , { \mathbf x } ) \in \mathbb { R }$ . However, it will become clear later that our results can be easily extended to the multi-class setting. Consider the weighted empirical risk minimization (ERM) task with the risk given by $\begin{array} { r } { L ( \pmb { \theta } ; \mathbf { w } ) = 1 \bar { / } n \sum _ { i = 1 } ^ { n } w _ { i } \ell \big ( y _ { i } f ( \pmb { \theta } , \mathbf { \bar { x } } _ { i } ) \big ) } \end{array}$ for some non-negative loss function $\ell ( \cdot )$ . The weight-agnostic counterpart is denoted by: $L ( \pmb \theta ) =$ $\begin{array} { r } { 1 / n \sum _ { i = 1 } ^ { n } \ell \big ( y _ { i } f ( \pmb { \theta } , \mathbf { x } _ { i } ) \big ) . } \end{array}$ . We focus particularly on the exponential loss $\bar { \ell ( u ) } = \exp ( - u )$ and log loss $\ell ( u ) = \log ( 1 + \exp ( - u ) )$ . For the multi-class problem where $y _ { i } \in [ k ]$ , we extend our setup using the softmax function where the logits are now given by $\{ f _ { j } ( \pmb { \theta } , \mathbf { x } ) \} _ { j = 1 } ^ { k }$ . For optimization, we consider using gradient descent to minimize the total loss: $\pmb { \theta } ^ { ( t + 1 ) } ( \mathbf { w } ) = \mathcal { \bar { \pmb { \theta } } } ^ { ( t ) } ( \mathbf { w } ) - \eta _ { t } \nabla L ( \pmb { \theta } ; \mathbf { w } ) \big | _ { \pmb { \theta } = \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) } .$ where the learning rate $\eta _ { t }$ can be constant or step-dependent.
+
+# From parameter norm divergence to support vectors.
+
+Suppose $\mathcal { D }$ is separated by $f ( { \pmb \theta } ^ { ( t ) } , { \bf x } )$ after some point during training. The key factor that contributes to the implicit bias for both linear and non-linear predictor under a weak regularization 1 is that the norm of the parameters diverges after separation, i.e. $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \| \pmb { \theta } ^ { ( t ) } \| _ { 2 } = \infty } \end{array}$ , as a consequence of using gradient descent. Now we examine $\big \| \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) \big \| _ { 2 }$ . The heuristic is that if $\ell ( \cdot )$ is exponential-like, multiplying by $w _ { i }$ only changes its tail property up to a constant while the asymptotic behavior is not affected. In particular, the necessary conditions for norm divergence under gradient descent can be summarized by:
+
+• C1. The loss function $\ell ( \cdot )$ has a exponential tail behavior (that we formalize in Appendix A.1) such that $\begin{array} { r } { \operatorname* { l i m } _ { u  \infty } \ell ( - u ) = \operatorname* { l i m } _ { u  \infty } \nabla \ell ( - u ) = 0 } \end{array}$ ; • C2. The predictor $f ( { \pmb \theta } , { \bf x } )$ is $\alpha$ -homogeneous such that $f ( c \cdot \theta , { \bf x } ) = c ^ { \alpha } f ( \theta , { \bf x } ) , \forall c > 0 .$
+
+In addition, we need certain regularities from $f ( { \pmb \theta } , { \bf x } )$ to ensure the existence of critical points and the convergence of gradient descent:
+
+• C3. for any $\mathbf { x } \in \mathcal { X }$ , $f ( \cdot , \mathbf { x } )$ is $\beta$ -smooth and $l$ -Lipschitz on $\mathbb { R } ^ { d }$
+
+C1 can be satisfied by the exponential loss, log loss and cross entropy loss under the multi-class setting. For standard deep learning models such as multilayer perceptron (MLP), C2 implies that the activation functions are homogeneous such as ReLU and LeakyReLU, and bias terms are disallowed. C3 is a common technical assumptions whose practical implications are discussed in Appendix A.1. Among the three necessary conditions, importance weighting only affects C1 up to a constant, so its impact on the norm divergence diminishes in the asymptotic regime. The formal statement is provided as below.
+
+Claim 1. There exists a constant learning rate for gradient descent, such that for any w $\in$ $[ 1 / M , M ] ^ { n }$ , with a weak regularization, $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \| \dot { \pmb { \theta } } ^ { ( t ) } ( \mathbf { \check { w } } ) \| = \infty } \end{array}$ under C1-C3.
+
+Compared with the previous work, we extend the norm divergence result not only to weighted ERM but a more general setting where a weak regularization is considered. We defer the proof to Appendix A.1. A direct consequence of parameter norm divergence is that both the risk and the gradient are dominated by the terms with the smallest margin, i.e. $\mathrm { a r g m i n } _ { i } y _ { i } f ( \pmb \theta , \mathbf x _ { i } )$ , which are also referred to as the "support vectors". To make sense of this point, notice that both the risk and the gradient have the form of: $\begin{array} { r } { \sum _ { i } C _ { i } \exp \big ( - y _ { i } f ( \pmb { \theta } , \mathbf { x } _ { i } ) \big ) } \end{array}$ , where $C _ { i }$ are low-order terms. Since $f ( \pmb \theta , \mathbf x _ { i } ) = \| \pmb \theta \| _ { 2 } ^ { \alpha } f \big ( \pmb \theta / \| \pmb \theta \| _ { 2 } , \mathbf x _ { i } \big )$ due to the homogeneous assumption in C2, it holds that:
+
+![](images/5902d254c5b355031461e367805d97aaa06c75c49fd09b99bc7e4d377110729a.jpg)  
+Figure 1: (a). Linearly separable data; (b). Non-separable data; (c): Balanced moon-shaped nonlinear separable data; (d). Unbalance moon-shaped data after down-sampling both classes $20 \%$ for the blue class, and $80 \%$ for the orange class). We use solid line to denote the separating hyperplane of the trained linear model and shades to represent the decision boundary of trained nonlinear model.
+
+$\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \exp \big ( - y _ { i } f ( \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) , \mathbf { x } _ { i } ) \big ) \  \ 0 } \end{array}$ . Therefore, the decision boundaries may share certain characteristics with the support vector machine (SVM) since they rely on the same support vectors. As a matter of fact, the current understandings on the implicit bias of gradient descent are mostly established on the connection with hard-margin SVM:
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } \in \mathbb { R } ^ { d } } \| \pmb { \theta } \| _ { 2 } \quad \mathrm { s . t . } \quad y _ { i } f ( \pmb { \theta } , \mathbf { x } _ { i } ) \geq 1 \quad \forall i = 1 , 2 , \ldots , n ,
+$$
+
+whose optimization path coincides with the max-margin problem: $\begin{array} { r } { \operatorname* { m a x } _ { \| \pmb { \theta } \| _ { 2 } \leq 1 } \operatorname* { m i n } _ { i = 1 , \dots , n } y _ { i } f ( \pmb { \theta } , \mathbf { x } _ { i } ) } \end{array}$ as shown by Nacson et al. (2019). Define $\gamma ( \pmb \theta ) : = \operatorname* { m i n } _ { i } y _ { i } f ( \pmb \theta , \mathbf x _ { i } )$ . We use $\pmb { \theta } ^ { * }$ to denote the optimal solution and $\begin{array} { r } { \gamma ^ { * } = \gamma ( \pmb { \theta } ^ { * } ) : = \operatorname* { m i n } _ { i } y _ { i } f ( \pmb { \theta } ^ { * } , \mathbf { x } _ { i } ) } \end{array}$ to denote the corresponding margin.
+
+# Implicit bias of gradient descent.
+
+We start by considering the weight-agnostic setting. When $\mathcal { D }$ is linear separable, it is reasonable to conjecture that the separating hyperplane under a linear $f ( \pmb \theta , \cdot )$ overlaps with the solution of hard-margin SVM. Soudry et al. (2018) and Ji & Telgarsky (2018b) first show that $\| \pmb \theta ^ { ( t ) } \|$ converges in direction to $\pmb { \theta } ^ { * }$ , i.e. $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \pmb { \theta } ^ { ( t ) } / \| \pmb { \theta } ^ { ( t ) } \| _ { 2 } = \pmb { \theta } ^ { * } } \end{array}$ . For nonlinear predictors, however, the parameter direction is less meaningful. Instead, it has been pointed out that neural networks often achieve perfect separation of the training data (Zhang et al., 2016). Therefore, we are more interested in the margin whose pivoting role for the generalization of neural networks is studied extensively (Neyshabur et al., 2017; Bartlett et al., 2017; Golowich et al., 2018). Specifically, it has been show in Nacson et al. (2019) and Lyu & Li (2019) that the normalized margin, defined by $\tilde { \gamma } ( \pmb { \theta } ^ { ( t ) } ) : = \gamma \big ( \pmb { \theta } ^ { ( t ) } / \lVert \pmb { \theta } ^ { ( t ) } \rVert _ { 2 } \big )$ , converges to the maximum margin $\gamma ^ { * }$ without regularization.
+
+It becomes clear at this point that to understand the role of importance weighting for deep learning, we must characterize the impact of weights on the implicit bias since they reveal the optimization geometry and generalization performance. Formally, we address the following critical questions.
+
+• Q1. Does importance weighting modify the convergence results (convergence in direction for linear predictor and in normalized margin for nonlinear predictor)?   
+• If the convergence results remain unchanged, then: – Q2. in what way is importance weighting affecting the optimization process; – Q3. how does importance weighting influence the generalization from the source distribution to the target distribution?
+
+# Experiment setup.
+
+Throughout this paper, we use the regular regression model as linear predictor. The nonlinear predictor is a two-layer MLP with five hidden units and ReLU as the activation function. All the models are trained with gradient descent using 0.1 as learning rate. We use the exponential loss and the standard normal initialization. The generated datasets for our illustrative experiments are shown in Figure 1, which correspond to the different settings of our major topics.
+
+We begin with the linear predictors which allows more refined analysis on the gradient dynamics. Without loss of generality, we assume using the exponential loss. Also, we do not consider the weak regularization here since its practical impact on linear model is trivial when $\lambda  0$ (Rosset et al., $2 0 0 4 \mathrm { a } ; \mathrm { b } )$ , but it is not the case for nonlinear predictors. One sophistication with linear predictor is that the data may not be perfectly separated, as opposed to the nonlinear case where neural networks can in theory separate any non-degenerate data. With this kept in mind, we first assume $\mathcal { D }$ is linear separable and characterize the new convergence result in the following proposition.
+
+Proposition 1. With a constant learning rate $\eta _ { t } ~ \lesssim ~ \beta ^ { - 1 }$ , we consider normalizing the weights $\mathbf { w } \in [ \frac { 1 } { M } , M ] ^ { n }$ such that $\begin{array} { r } { \sum _ { i } \mathbf { w } _ { i } = 1 } \end{array}$ without loss of generality, it holds that:
+
+$$
+\Big | \frac { \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) } { \| \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) \| _ { 2 } } - \pmb { \theta } ^ { * } \Big | \lesssim \frac { \log n + D _ { K L } ( \pmb { p } ^ { * } \| \mathbf { w } ) + M } { \log t \cdot \gamma ^ { * } } ,
+$$
+
+where $\pmb { p } ^ { * } = [ p _ { 1 } ^ { * } , \ldots , p _ { n } ^ { * } ]$ characterizes the dual optimal for the hard-margin SVM such that $\pmb { \theta } ^ { * } =$ $\textstyle \sum _ { i = 1 } ^ { n } y _ { i } \mathbf { x } _ { i } \cdot p _ { i } ^ { * }$ and satisfies: $p _ { i } ^ { * } \geq 0$ and $\textstyle \sum _ { i = 1 } ^ { n } p _ { i } ^ { * } = 1$ . Here, $D _ { K L }$ is the Kullback-Leibler divergence.
+
+We leave the proof to Appendix A.2. We find that importance weighting does not change the convergence result as well as the $1 / \log t$ convergence rate. However, it does affect the convergence speed under the finite-step optimization. In particular, we show that the extra constant term induced by importance weighting is given by the KL-divergence between the (normalized) weights and the dual optimal of the hard-margin SVM, where samples with smaller margins usually have larger values. Therefore, importance weighting may accelerate gradient descent in finite-step optimization by matching weights with the inverse margin. As we show in Figure 2a and 2b, this type of "inversemargin weighted" design is able to accelerate the convergence and bring better performance under finite-step optimization.
+
+![](images/5b20d6dcbf9664345f652c4720f906f39c99796e6a64e484a379f22444a379a1.jpg)  
+Figure 2: (a): Epoch-wise training performances measured by the angle between the decision boundary (at that epoch) and the max-margin solution, using linear predictor on the linear separable data of Figure 1a; (b): Epoch-wise training performances measured by the average margin in the same setting as (a); (c). The generalization error on testing data (the remaining $80 \%$ of the orange class and $20 \%$ of the blue class that are not part of the down-sampling in Figure 1d) when the nonlinear model is trained under different class weights, as the training progresses; (d). The average margin for the nonlinear model on the non-linearly separable training data shown in Figure 1c, under different class weights, as the training progresses.
+
+When $\mathcal { D }$ is not linearly separable, the key insight is that we can always partition $\mathcal { D }$ into $\mathcal { D } _ { \mathrm { s e p } } \cup \mathcal { D } _ { \mathrm { n o n - s e p } }$ , where $\mathcal { D } _ { \mathrm { s e p } }$ is the maximal linear separable subset defined in Ji & Telgarsky (2018b). Let $\Pi _ { \mathrm { n o n - s e p } }$ be the (orthogonal) projection onto the subspace $S$ spanned by the $\mathbf { x } _ { i }$ ’s in $\mathcal { D } _ { \mathrm { n o n - s e p } }$ , and let $\Pi _ { \mathrm { s e p } }$ be the projection onto the orthogonal complement $S ^ { \perp }$ . The partition allows us to study the two projected parts independently since by the construction, we have $\begin{array} { r } { \mathbf { \tilde { \theta } } ^ { ( t ) } ( \mathbf { w } ) = \Pi _ { \mathrm { n o n - s e p } } \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) + \Pi _ { \mathrm { s e p } } \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) } \end{array}$ . It is intuitive that the optimization path of $\Pi _ { \mathrm { s e p } } \pmb { \theta } ^ { ( t ) } ( \mathbf { w } )$ behaves similarly to the linear separable case as in Proposition 1, so we can focus on the properties of $\Pi _ { \mathrm { n o n - s e p } } \pmb { \theta } ^ { ( t ) } ( \mathbf { w } )$ , which we summarize in the follow proposition.
+
+Proposition 2 (Informal). Let ${ \cal L } _ { n o n - s e p } ( \pmb \theta , \mathbf w )$ be the weighted risk defined on the non-separable subset, then with the constant learning rate:
+
+• $\begin{array} { r } { \tilde { \pmb \theta } ( \mathbf w ) = \arg \operatorname* { m i n } _ { \pmb \theta } L _ { n o n - s e p } ( \pmb \theta , \mathbf w ) } \end{array}$ is uniquely defined and $\left\| \tilde { \pmb { \theta } } ( \mathbf { w } ) \right\| _ { 2 } = \mathcal { O } ( 1 )$
+
+$$
+\begin{array} { r } { \bullet \left| \Pi _ { n o n - s e p } \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) - \widetilde { \pmb { \theta } } ( \mathbf { w } ) \right| \lesssim \frac { C \left( \left\| \widetilde { \pmb { \theta } } ( \mathbf { w } ) \right\| _ { 2 } \right) + \log ^ { 2 } t / \gamma _ { s e p } } { t } } \\ { o n \mathcal { D } _ { s e p } a n d C \left( \left\| \widetilde { \pmb { \theta } } ( \mathbf { w } ) \right) \right\| _ { 2 } ) = \mathcal { O } ( 1 ) . \qquad } \end{array}
+$$
+
+The formal statement, which involves how $\mathcal { D } _ { \mathrm { s e p } }$ is defined, is deferred to Appendix A.2 together with the proof. Proposition 2 informs that importance weighting uniquely defines the solution $\tilde { \pmb { \theta } } ( \mathbf { w } )$ on the non-separable subset of the data, to which $\Pi _ { \mathrm { n o n - s e p } } \pmb { \theta } ^ { ( t ) } ( \mathbf { w } )$ converges. Hence, we expect $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) = \tilde { \pmb { \theta } } ( \mathbf { w } ) + \pmb { \theta } _ { \mathrm { s e p } } ^ { * } } \end{array}$ , where $\theta _ { \mathrm { s e p } } ^ { * }$ is the solution on the separable subset $\mathcal { D } _ { \mathrm { s e p } }$ and thus its direction does not depend on w as implied by Proposition 1. We can therefore think of $\tilde { \pmb { \theta } } ( \mathbf { w } )$ as the intercept term where the weight controls how the intercept shifts on the subspace of the non-separable data. We also illustrate this finding in Figure 3. By far, we provide an in-depth understanding and our theoretical results fully explain the observations made in Byrd & Lipton (2019) on how importance weighting affects the implicit bias of gradient descent using linear predictors.
+
+![](images/37951384469b8c6c8124aa2043482d75fada6b3ad1858da117987f13a85e2c65.jpg)  
+Figure 3: The role of importance weighting on defining the intercept term in addition to the implicit bias for the linearly separable case, where the hyperplane shifts in the non-separable subspace depending on the class weights.
+
+# 4 IMPORTANCE WEIGHTING FOR NONLINEAR PREDICTOR
+
+Now we investigate the influence of importance weighting on non-linear predictors, e.g, the neural network. Here we are more interested in the regularized setting:
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } } L _ { \lambda } ( \pmb { \theta } ; \mathbf { w } ) : = L ( \pmb { \theta } , \mathbf { w } ) + \lambda \| \pmb { \theta } \| ^ { r } ,
+$$
+
+where $r \ > \ 0$ is fixed, $\lambda$ is the regularization coefficient. We use the notation: $\theta _ { \lambda } ( \mathbf { w } ) \ \in$ arg min $L _ { \lambda } ( \pmb { \theta } , \mathbf { w } )$ . Recall that $\begin{array} { r } { \gamma ^ { * } : = \operatorname* { m a x } _ { \| \pmb { \theta } \| \leq 1 } \operatorname* { m i n } _ { i } y _ { i } f ( \pmb { \theta } , \mathbf { x } _ { i } ) } \end{array}$ . Unlike the linear case, characterizing the gradient dynamics for nonlinear predictor is often insurmountable. Therefore, we mainly consider the asymptotic regime or the regime with sufficiently large $t$ . We omit the superscript in ${ \pmb \theta } ^ { ( i ) }$ when there is no confusion. The only assumption we need to make is that:
+
+A1. the data is separated by $f$ at some point during gradient descent, i.e. $\exists t \ > \ 0$ s.t. $y _ { i } f ( \pmb \theta ^ { ( t ) } , \mathbf x _ { i } ) > 0 , \forall i = 1 , \ldots , n$ . In addition, $y _ { i } f ( \pmb { \theta } ^ { * } , \mathbf x _ { i } ) \geq \gamma ^ { * } > 0$ for each $i$ .
+
+In Section 4.1, we show that by solving the equation 3 with an infinitesimal (weak) regularizer, gradient descent leads to the optimal margin $\gamma ^ { * }$ , regardless of the choice of the importance weights. In Section 4.2, we show that the the importance weighting affects the generalization bound via a multiplication factor as well as the margin in the finite-sample scenario.
+
+# 4.1 MARGIN IS INVARIANT TO IMPORTANCE WEIGHTING UNDER WEAK REGULARIZATION
+
+We show that for any bounded w, $\widetilde \gamma ( \pmb \theta _ { \lambda } ( \mathbf { w } ) ) : = \gamma ( \pmb \theta _ { \lambda } ( \mathbf { w } ) / \lVert \pmb \theta _ { \lambda } ( \mathbf { w } ) \rVert )$ converges to $\gamma ^ { * }$ as $\lambda$ decreases to zero. In practice, however, we might not obtain $\pmb { \theta } _ { \lambda } ( \mathbf { w } )$ in limited time. It is shown that as long as equation 3 is close enough to its optimum, the normalized margin of the associated $\pmb { \theta } ^ { \prime } ( \mathbf { w } )$ (under finite-step optimization) is lower bounded by $\gamma ^ { * }$ multiplied by a non-trivial factor. Formally,
+
+Proposition 3. Suppose C1-C3, A1 hold. For any $\pmb { w } \in [ 1 / M , M ] ^ { n }$ , it follows that • (Finite steps) There exists a $\begin{array} { r c l } { \lambda } & { : = } & { \lambda ( r , \alpha , \gamma ^ { * } , \mathbf { w } , c ) } \end{array}$ such that for $\pmb { \theta } ^ { \prime } ( \mathbf { w } )$ with $L _ { \lambda } ( \pmb \theta ^ { \prime } ( \mathbf { w } ) ; \mathbf { w } ) \leq \tau L _ { \lambda } ( \pmb \theta _ { \lambda } ( \mathbf { w } ) ; \mathbf { w } )$ and $\tau \leq 2$ , the associated normalized margin $\tilde { \gamma } ( \pmb { \theta } ^ { \prime } ( \mathbf { w } ) )$ satisfies $\begin{array} { r } { \tilde { \gamma } ( \pmb { \theta } ^ { \prime } ( \mathbf { w } ) ) \geq c \cdot \frac { \gamma ^ { * } } { \tau ^ { \alpha / r } } } \end{array}$ , where $\textstyle { \frac { 1 } { 1 0 } } \leq c < 1$ .
+
+This result is adapted from Wei et al. (2019), which relies on Claim 1. The proof is relegated to Appendix A.4.1. We see that importance weighting does not affect the asymptotic margin when $\lambda$ is sufficiently small. To get the intuition, note that when $\left| \left| \pmb { \theta } _ { \lambda } ( \mathbf { w } ) \right| \right|$ is large enough and $\lambda$ is small enough to be ignored, $L _ { \lambda } ( \pmb \theta _ { \lambda } ( \mathbf { w } ) , \mathbf { w } ) \approx \exp \big ( - \| \pmb \theta _ { \lambda } ( \mathbf { w } ) \| ^ { \alpha } \gamma _ { \lambda } \big )$ , which favors a large margin. In addition, even if $L _ { \lambda } ( \pmb { \theta } ^ { \prime } ( \mathbf { w } ) , \mathbf { w } )$ has not yet converged but close enough to its optimum, the corresponding normalized margin has a reasonable lower bound. We point out that this result does not rely on the choice of $\lambda$ . The assumption $L _ { \lambda } ( \pmb \theta ^ { \prime } ( \mathbf { w } ) ; \mathbf { w } ) \leq \tau L _ { \lambda } \big ( \pmb \theta _ { \lambda } ( \mathbf { w } ) ; \mathbf { w } \big )$ has already accounted for the major influence of importance weighting in terms of the optimization. That is, with a "good" set of importance weights, we can achieve this criteria (by approaching global optimum) faster. We leave detailed discussions to Section 5. Figure 2d also demonstrates that the choice of the importance weights has a significant influence on the convergence speed for the non-linear predictor.
+
+# 4.2 IMPORTANCE WEIGHTING AFFECTS THE GENERALIZATION BOUND
+
+Proposition 3 conjectures on the behavior of the margin corresponding to the optimum of $L _ { \lambda } ( \pmb { \theta } ; \mathbf { w } )$ , which does not rely on the sample size. To bridge the connection between importance weighting and the behavior of $f ( \pmb \theta , \cdot )$ in the finite-sample setting, we investigate the generalization bound of $f$ when the training sample distribution deviates from the testing sample distribution.
+
+Let $P _ { s }$ be the source distribution and $P _ { t }$ be the target distribution with the corresponding densities $p _ { s } ( \cdot )$ and $p _ { t } ( \cdot )$ . Assume that $P _ { s }$ and $P _ { t }$ have the same support. We consider the Pearson $\chi ^ { 2 }$ -divergence to measure the difference between $P _ { s }$ and $P _ { t }$ , i.e., $\begin{array} { r } { D _ { \chi ^ { 2 } } ( P _ { t } \| P _ { t } ) = \int \big [ ( d P _ { s } / d P _ { t } ) ^ { 2 } - 1 \big ] d P _ { s } } \end{array}$ . The training covariates $\mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { n }$ are generated from $P _ { s }$ , and the testing covariates are generated from $P _ { t }$ . Denote by $p _ { \mathrm { t r a i n } }$ and $p _ { \mathrm { t e s t } }$ the joint distribution of $\left( \mathbf { x } , y \right)$ for the training data and the testing data, respectively.
+
+We minimize equation 3 over the $H$ -layer feedforward neural network given by $f ^ { \mathrm { N N } } ( \pmb { \theta } , \mathbf { x } ) : =$ $W _ { H } \sigma ( W _ { H - 1 } \sigma ( \bar { \dots } \cdot \cdot \sigma ( W _ { 1 } \mathbf { x } ) \cdot \cdot \cdot ) )$ , where $\pmb \theta = [ W _ { 1 } , \cdots , W _ { H } ]$ are the parameter matrices and $\sigma ( \cdot )$ is the element-wise activation function such as ReLU. Denote by $\eta ( \mathbf { x } ) = p _ { t } ( \mathbf { x } ) / p _ { s } ( \mathbf { x } )$ . We show that the generalization performance is affected by importance weighting via the interplay between the empirical risk that hinges on $\eta$ , as well as a term that depends on the model complexity and the deviation of the target distribution from the source distribution.
+
+Theorem 1 (1). Assume $\sigma$ is 1-Lipschitz and 1-positive homogeneous. Then with probability at least $1 - \delta$ , we have
+
+$$
+\begin{array} { r l } & { \displaystyle _ { ( \mathbf { x } , y ) \sim p _ { \mathrm { r e r } } } ^ { \mathfrak { L } } \Big ( y f ^ { N N } ( \pmb { \theta } ( \mathbf { w } ) , \mathbf { x } ) \leq 0 \Big ) \leq } \\ & { \quad \underbrace { \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \eta ( \mathbf { x } _ { i } ) \mathbf { I } \Big ( y _ { i } f ^ { N N } ( \pmb { \theta } ( \mathbf { w } ) / \| \pmb { \theta } ( \mathbf { w } ) \| , \mathbf { x } _ { i } ) < \gamma \Big ) } _ { ( I ) } + \underbrace { C \cdot \sqrt { D _ { X ^ { 2 } } ( P _ { t } | | P _ { s } ) + 1 } } _ { ( I I ) } + \epsilon ( \gamma , n , \delta ) , } \end{array}
+$$
+
+where $( I )$ is the empirical risk, $( I I )$ reflects the compounding effect of the model complexity of the class of $H$ -layer neural networks and the deviation between target distribution and source distribution , $\begin{array} { r } { \epsilon ( \gamma , n , \delta ) = \sqrt { \frac { \log \log _ { 2 } \frac { 4 C } { \gamma } } { n } } + \sqrt { \frac { \log ( 1 / \delta ) } { n } } } \end{array}$ is a small quantity compared to $( I )$ and $( I I )$ . Here, $C : = \operatorname* { s u p } _ { \mathbf { x } } \| \mathbf { x } \|$ and $\gamma$ can take any positive value.
+
+The proof is deferred to Appendix A.4.2. Compared to Wei et al. (2019), the empirical risk (I) hinges on $\eta$ and there is an additional multiplier factor $\sqrt { D _ { \chi ^ { 2 } } ( P _ { t } | | P _ { s } ) + 1 }$ on (II). In the two discussions below, we argue that the role of importance weighting on the generalization bound in Theorem 1 is not only reflected in how the margin is affected, but also how it balances source and target distribution:
+
+1. Suppose $\pmb \theta ( \mathbf w )$ enables $f ^ { \mathrm { N N } }$ to separate the data. Let $\begin{array} { r } { \gamma _ { \pmb { \theta } ( \mathbf { w } ) } : = \operatorname* { m i n } _ { i } y _ { i } f ^ { \mathrm { N N } } \big ( \pmb { \theta } ( \mathbf { w } ) / \| \pmb { \theta } ( \mathbf { w } ) \| , \mathbf { x } _ { i } \big ) } \end{array}$ . In the generalization bound of Theorem 1, if we let $\gamma = \gamma _ { \pmb \theta ( \mathbf { w } ) }$ , then (I) vanishes and only (II) remains. In this case, the importance weights affects the generalization bound via $\gamma _ { \pmb \theta ( \mathbf w ) }$ in finite steps as discussed in Section 4.1. That is, within finite training steps, a good set of weights w can approach closer to $\gamma _ { \pmb \theta ( \mathbf w ) }$ than a bad set, and thus giving a better generalization performance. Also note that Theorem 1 holds for the non-separable cases as well.
+
+2. We point out that (II) is a strictly decreasing function, while (I) is a non-decreasing step function with respect to $\gamma$ . Therefore, there must exists a trade-off $\gamma$ that minimizes the sum of (I) and (II), which is usually attained at some $\gamma > \gamma _ { \pmb \theta ( \mathbf { w } ) }$ . When $\gamma$ grows, certain samples will activate $\mathbf { I } ( y _ { i } f ^ { \mathrm { N N } } ( \pmb \theta ( \mathbf { w } ) / \lVert \pmb \theta ( \mathbf { w } ) \rVert , \mathbf { x } _ { i } ) < \gamma )$ and inflate (I). The hope is that an initially activated sample (indicator term) in (I) corresponds to a small $\eta ( \mathbf { x } _ { i } )$ , while one with a large $\eta ( \mathbf { x } _ { i ^ { \prime } } )$ has a large value of $y _ { i ^ { \prime } } f ^ { \mathrm { N N } } ( \pmb { \theta } ( \mathbf { w } ) / \| \pmb { \theta } ( \mathbf { w } ) \| , \mathbf { x } _ { i ^ { \prime } } )$ and thus will be activated later. This can be achieved by aligning w with $\eta$ because a large weight on sample $i$ forces the decision boundary to drift away from this data point and gives a larger value of $y _ { i } f ^ { \mathrm { N N } } ( \pmb { \theta } ( \mathbf { w } ) / \Vert \pmb { \theta } ( \mathbf { w } ) \Vert , \mathbf { x } _ { i } )$ . Therefore, the generalization bound with w aligning with $\eta$ can be smaller than that with w deviating from $\eta$ .
+
+The empirical results in Figure 2c provides the numerical evidence that reflects the strong effects of importance weighting on the generalization behavior.
+
+# 5 EXTENSION
+
+# What makes a good set of weights for learning?
+
+We show in both Section 3 and 4 that importance weighting can affect how fast the classifier separates the data and converges to the max-margin solution. We also justify how the small-margin support vectors, who can think of as the hard-to-classify data points, are of significant importance. Imagine that we have access to an oracle that outputs the distance of each sample to the max-margin decision boundary. It is intuitive that by putting more weights on the small-margin samples, we "inform" gradient descent of their importance from the beginning and therefore accelerates the optimization. We also provide a rigorous result for linear predictor in Proposition 1. Our high-level intuition justifies a number of methodologies where people use various methods to measure the hardness of classifying a sample and use that as the weight, explicitly or implicitly. Examples include the curriculum learning (Bengio et al., 2009), mentor net (Jiang et al., 2018), co-teaching (Han et al., 2018) and knowledge distillation (Li et al., 2017; Hinton et al., 2015), where auxiliary models are employed (replacing the oracle) to represent the hardness of each data point.
+
+# The effect of jointly optimizing a weighting model
+
+It is not unusual that the importance weights, when depending on another model, is jointly trained with the classifier to achieve an better overall performance, such as the counterfactual modelling (Schnabel et al., 2016; Xu et al., 2020) and learning from noisy labels (Song et al., 2020). For the illustration purpose, we consider the following setup:
+
+$$
+\operatorname* { m i n i m i z e } _ { \psi , \theta } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } g ( \psi , \mathbf { x } _ { i } ) \cdot \ell \big ( y _ { i } f ( \pmb { \theta } , \mathbf { x } _ { i } ) \big ) , \quad \mathrm { s . t . } \quad \frac { 1 } { M } < g ( \psi , \mathbf { x } _ { i } ) < M ,
+$$
+
+where $g ( \psi , \mathbf { x } _ { i } )$ is the weighting model. By our main results, it is not difficult to conjecture that if the data is separable by $f$ , the convergence of $f$ to the max-margin solution will still hold and the weighting model $g ( \psi , \mathbf { x } _ { i } )$ will concentrate to a constant for all $i = 1 , \ldots , n$ . This is because the general convergence results are agnostic to the weights, so the weighting model will eventually be nullified. Also, during the beginning phase of training, the learned weights may correlate negatively to the margin (as it helps to speed up the convergence), and the correlation will diminish eventually as the weights converge to the same constant. The above conjectures are supported by the empirical evidence that we discuss in Figure 4. Therefore, jointly optimizing the weighting model may not change the convergence result but the speed of convergence is affected.
+
+# Interaction with explicit regularizations
+
+Deep learning models are often trained with explicit regularization. To see how they interact with importance weighting, we first check weather they alter the norm divergence in Claim 1. It is obvious that both the early stopping and strong regularization on $\lVert \pmb \theta \rVert$ prohibits the norm divergence, so $f ( \pmb \theta , \cdot )$ will not achieve the max-margin solution or even separate the training data. In such cases, as it has been observed by Byrd & Lipton (2019), the impact of importance weighting on $\theta _ { \lambda } ( \mathbf { w } )$ and $\tilde { \gamma } ( \pmb \theta _ { \lambda } ( \mathbf { w } ) )$ will be significant. However, this may not help generalization according to our arguments in Section 4.2, since the margins will be altered as well. Indeed, Zhang et al. (2016) shows that explicit regularizations may not lead to better generalization for neural networks. For the weighted ERM, Theorem 1 provides a powerful tool to characterize the trade-off induced by explicit regularizations via the margin size. Dropout, as an counter example, does not prohibit norm divergence and may not interfere with our main conclusions.
+
+![](images/83333996b965605835a5b3b3bb7718409d90b51f0337d3b44b18053651774091.jpg)  
+Figure 4: The left-five figures show that the distribution of the learned weights concentrates to a constant as the training progresses. The rightmost figure indicates the correlation pattern between margin and the learned weights: the correlation increases rapidly in the beginning, and then slowly decreases to zero (the process is much slower for nonlinear predictor so we only show the first part). Here, $g ( \mathbf { x } _ { i } ) = \sigma ( \psi ^ { \mathsf { T } } \mathbf { x } _ { i } + b ) + 1$ , where $\sigma ( \cdot )$ is the sigmoid function, the constant one is added to avoid numerical issues.
+
+# 6 DISCUSSION
+
+In this paper, we study the impact of importance weighting on the implicit bias of gradient descent as well as the generalization performance. Based on our theoretical findings, we propose the following future directions that are worth investigating from both the application and theoretical perspective: 1) Is there an optimal way to construct importance weights using such as the oracle margin? 2) How to correctly understand and utilize the role of a jointly-trained weighting model? 3) What is the combined effect of importance weighting and explicit regularizations for deep learning models?
+
+# REFERENCES
+
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+
+Peter L Bartlett, Dylan J Foster, and Matus J Telgarsky. Spectrally-normalized margin bounds for neural networks. In Advances in Neural Information Processing Systems, pp. 6240–6249, 2017.
+
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+# A APPENDIX
+
+We provide the omitted discussions, proofs, and extra numerical results in the appendix.
+
+# A.1 SUPPLEMENTARY MATERIAL FOR SECTION 2
+
+We discuss the exponential-tail behavior for loss functions, the piratical implication of condition C3 and the proof of Claim 1.
+
+# A.1.1 LOSS FUNCTION WITH EXPONENTIAL-TAIL BEHAVIOR
+
+Having a exponential decay on the tail of the loss function is essential for realizing the implicit bias of gradient descent, since we need $\ell ( u )$ behave like $\exp ( - u )$ as $u \to \infty$ . Soudry et al. (2018) first propose the notion of tight exponential tail, where the negative loss derivative $- \ell ^ { \prime } ( u )$ behave like:
+
+$$
+- \ell ^ { \prime } ( u ) \lesssim \big ( 1 + \exp ( - c _ { 1 } u ) \big ) e ^ { - u } \mathrm { ~ a n d ~ } - \ell ^ { \prime } ( u ) \gtrsim \big ( 1 - \exp ( - c _ { 2 } u ) \big ) e ^ { - u } ,
+$$
+
+for sufficiently large $u$ , where $c _ { 1 }$ and $c _ { 2 }$ are positive constants. There is also a smoothness assumption on $\ell ( \cdot )$ . It is obvious that under this definition, the tail behavior of the loss function is constraint from both sides by exponential-type functions.
+
+There is a more general (and perhaps more direct) definition of exponential-tail loss function Lyu & Li (2019), where $\ell ( u ) = \exp ( - f ( u ) )$ , such that:
+
+• $f$ is smooth and $f ^ { \prime } ( u ) \geq 0 , \forall u$ ;   
+• there exists $c > 0$ such that $f ^ { \prime } ( u ) u$ is non-decreasing for $u > c$ and $f ^ { \prime } ( u ) u  \infty$ as $u \to \infty$ .
+
+It is easy to verify that the exponential loss, log loss and cross-entropy loss satisfy both definitions. Since our focus is not to study the implicit bias of gradient descent, it suffice to work with the above loss functions.
+
+# A.1.2 PRACTICAL IMPLICATIONS OF CONDITION C3
+
+C3 asserts the Lipschitz and smoothness properties. The Lipschitz condition is rather mild assumption for neural networks, and several recent paper are dedicated to obtaining the Lipschitz constant of certain deep learning models (Fazlyab et al., 2019; Virmaux & Scaman, 2018).
+
+The $\beta$ -smooth condition, on the other hand, is more technical-driven such that we can analyze the gradient descent. In practice, neural networks with ReLU activation do not satisfy the smoothness condition. However, there are smooth homogeneous activation functions, such as the quadratic activation $\sigma ( x ) = x ^ { 2 }$ and higher-order ReLU activation $\sigma ( x ) = \mathrm { R e L U } ( x ) ^ { c }$ for $c > 2$ . Still, in our experiments, we use ReLU as the activation function for its convenience.
+
+# A.1.3 PROOF FOR CLAIM 1
+
+Soudry et al. (2018) and Ji & Telgarsky (2018b) show norm divergence for linear predictors, and the follow-up work by Ji & Telgarsky (2018a); Gunasekar et al. (2018) extend the result to linear neural networks. For nonlinear predictors such as multi-layer neural network with homogeneous activation, Nacson et al. (2019) and Lyu & Li (2019) prove the norm divergence for gradient descent in the absence of explicit regularization. Rosset et al. (2004a) and Wei et al. (2019) considers the weak regularization for linear and nonlinear predictors, however, they only study the property of the critical points instead of the gradient descent sequence.
+
+Proof. We first state a technical lemma that characterizes the dynamics of gradient descent.
+
+Lemma A.1 (Theorem E.10 of Lyu & Li (2019)). Under the conditions that:
+
+• $\ell ( \cdot )$ is given by the exponential loss, and $\ell \circ f ( \cdot , \mathbf { x } )$ is a smooth function on $\mathbb { R } ^ { d }$ for all $\mathbf { x } \in \mathcal { X }$ ;
+
+• $f ( { \pmb \theta } , { \bf x } )$ is $\alpha$ -homogeneous as in $^ { c 2 }$ ; • the data is separated by $f$ during gradient descent at some point $t _ { 0 }$ ; • the learning rate satisfy $\eta _ { t } : = \eta _ { 0 } \lesssim \Big ( L \big ( \pmb { \theta } ^ { ( t ) } ; \mathbf { w } \big ) \log \big ( 1 / L \big ( \pmb { \theta } ^ { ( t ) } ; \mathbf { w } \big ) \big ) ^ { 3 - 2 / \alpha } \Big ) ^ { - 1 } f ($ r all t,
+
+then under exponential loss we have:
+
+$$
+\frac { 1 } { L ( \pmb { \theta } ^ { ( t ) } ; \mathbf { w } ) ^ { 2 } \big ( \log \frac { 1 } { L ( \pmb { \theta } ^ { ( t ) } ; \mathbf { w } ) } \big ) ^ { 2 - 2 / \alpha } } \geq \frac { 1 } { 2 } \alpha ^ { 2 } \widetilde { \gamma } \big ( \pmb { \theta } ^ { ( t _ { 0 } ) } ( \mathbf { w } ) \big ) ^ { 2 / \alpha } \sum _ { i = t _ { 0 } } ^ { ( t ) } \eta _ { i } .
+$$
+
+To use the results of Lemma A.1, we simply need to show two things for weak regularization:
+
+• the total risk is still smooth and we still can achieve zero risk; • there exists a critical (stationary) point such that $\begin{array} { r } { \operatorname* { l i m } _ { \lambda \to 0 } L _ { \lambda } ( \pmb { \theta } ^ { * } ; \mathbf { w } ) = 0 . } \end{array}$
+
+Notice that the risk without regularization is a smooth function in terms of $\pmb \theta$ for all $\mathbf { x }$ , since the composition of smooth functions is still smooth. It is easy to see that adding a weak regularization, e.g. $\mathbf { \bar { \boldsymbol { \lambda } } } \mathbf { \| } \pmb { \theta } \| _ { 2 } ^ { r }$ for $r > 1$ , does not alter the smoothness condition as $\lambda  0$ . However, the weak $\ell _ { 1 }$ regularization will make the total risk non-smooth, and therefore we have excluded it from our discussion.
+
+For the second point, it is obvious that $\| \pmb \theta \| _ { 2 }  \infty$ is a critical point under exponential loss when $\lambda  0$ . Recall that:
+
+$$
+L _ { \lambda } ( \boldsymbol { \theta } ; \mathbf { w } ) = \frac { 1 } { n } \sum _ { i } w _ { i } \exp \big ( - y _ { i } f \big ( \boldsymbol { \theta } / \| \boldsymbol { \theta } \| _ { 2 } , \mathbf { x } _ { i } \big ) \cdot \| \boldsymbol { \theta } \| _ { 2 } \big ) \big ) + \lambda \| \boldsymbol { \theta } \| _ { 2 } ^ { r } ,
+$$
+
+and
+
+$$
+\nabla L _ { \lambda } ( \theta ; \mathbf { w } ) = \frac { 1 } { n } \sum _ { i } - w _ { i } \exp \Big ( - y _ { i } f \big ( \theta / \| \theta \| _ { 2 } , \mathbf { x } _ { i } \big ) \cdot \| \theta \| _ { 2 } \Big ) \cdot y _ { i } \nabla f \big ( \theta , \mathbf { x } _ { i } \big ) + \lambda \nabla \| \theta \| _ { 2 } ^ { r } .
+$$
+
+Therefore, for both the loss function and gradient, the main term decreases exponentially fast as $\lVert \pmb \theta \rVert _ { 2 }$ increases, while the remainder terms are only polynomial in $\lVert \pmb { \theta } \rVert _ { 2 }$ , so we can always find a small enough $\lambda$ that satisfy: $\begin{array} { r } { \operatorname* { l i m } _ { \lambda \to 0 } \operatorname* { l i m } _ { \| \pmb { \theta } \|  \infty } L _ { \lambda } ( \pmb { \theta } ; \mathbf { w } ) = 0 } \end{array}$ and $\begin{array} { r } { \operatorname* { l i m } _ { \lambda \to 0 } \operatorname* { l i m } _ { \parallel \theta \parallel \to \infty } \nabla L _ { \lambda } ( \theta ; \mathbf { w } ) = 0 } \end{array}$ , in the same fashion as we show in the (A.1) below.
+
+From a standard result of gradient descent on smooth function, which we summarize in Lemma A.2, gradient descent will always converge to a critical (stationary) point for the weighted ERM problem.
+
+Lemma A.2 (Lemma 10 of Soudry et al. (2018)). Let $L _ { \lambda } ( \pmb { \theta } ; \mathbf { w } )$ be a $B ( \mathbf { w } )$ -smooth non-negative objective. With a constant learning rate $\eta _ { 0 } \lesssim B ( \mathbf { w } ) ^ { - 1 }$ , the gradient descent sequence satisfies:
+
+$$
+\begin{array} { r l } & { \bullet \operatorname* { l i m } _ { t  \infty } \sum _ { i = 1 } ^ { t } \| \nabla L _ { \lambda } ( \pmb { \theta } ^ { ( t ) } ; \mathbf { w } ) \| < \infty ; } \\ & { \bullet \operatorname* { l i m } _ { t  \infty } \nabla L _ { \lambda } ( \pmb { \theta } ^ { ( t ) } ; \mathbf { w } ) = 0 . } \end{array}
+$$
+
+Now we need to show that under appropriate learning rate, which is specified in Lemma A.1, gradient descent converges to the stationary point that corresponds to the zero risk under weak regularization. Using the result from Lemma A.1, notice that if $L _ { \lambda } ( \pmb \theta ^ { ( t ) } ; \mathbf { w } )$ does not decrease to 0, then the denominator Lλ(θ(t); w)2 log 1L (θ(t);w) is bounded from below.
+
+However, there exists a constant learning rate such that $\textstyle \sum _ { i = t _ { 0 } } ^ { t } \eta _ { i } \to \infty$ as $t \to \infty$ , which leads to contradiction. Therefore, for weighted ERM with weak regularization, gradient descent converges to the stationary point where $L _ { \lambda } ( \pmb \theta ^ { ( \bar { t } ) } ; \mathbf { w } ) = 0$ .
+
+Finally, we show to make $L _ { \lambda } ( \pmb \theta ^ { ( t ) } ; \mathbf { w } )  0$ , we must have $\| \pmb \theta ^ { ( t ) } ( \mathbf { w } ) \|  \infty$ . We show by contradiction. Suppose $\left\| \pmb { \theta } ^ { ( t ) } ; \mathbf { w } ) \right\|$ is bounded from above by some constant $C > 0$ , for all $\lambda < \tilde { \lambda }$ that we choose later. So the loss function for each sample $i$ is bounded below by a positive value that depends on $\mathcal { O } \colon w _ { i } \exp \bigl ( - y _ { i } f \bigl ( \pmb \theta ^ { ( t ) } , \mathbf x \bigr ) \bigr ) \geq l ( C ) > 0 .$ . Hence, let $K : = \tilde { \lambda } ^ { - 1 / ( r + 1 ) }$ , then
+
+$$
+\begin{array} { l } { l ( C ) \leq L _ { \lambda } ( \pmb \theta _ { \lambda } ( \mathbf w ) ; \mathbf w ) \leq L _ { \lambda } ( K \pmb \theta ^ { * } ; \mathbf w ) } \\ { \leq M \exp \big ( - \tilde { \lambda } ^ { - \alpha / ( r + 1 ) } \cdot \gamma ^ { * } \big ) + \tilde { \lambda } ^ { 1 / ( 1 + r ) } ; } \end{array}
+$$
+
+and it easy obvious that $\mathrm { R H S } \to 0$ for a sufficiently small $\tilde { \lambda }$ , which contradicts $l ( C ) > 0$ . Hence, we have $\| \pmb \theta ^ { ( t ) } ( \mathbf { w } ) \|  \infty$ for all all $\lambda < \widetilde { \lambda }$ , which completes the proof.
+
+# A.2 SUPPLEMENTARY MATERIAL FOR SECTION 3
+
+We provide the proofs for Proposition 1 and 2 in this part of the appendix.
+
+# A.2.1 PROOF FOR PROPOSITION 1
+
+Proof. We first characterize the $1 / \log t$ rate using asymptotic arguments similar to that of Soudry et al. (2018). The key purpose here is to rigorously show that importance weighting plays a negligible role in the asymptotic regime. Let $\delta ( t )$ be the residual term at step $t$ :
+
+$$
+\pmb { \delta } ( t , \mathbf { w } ) : = \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) - \pmb { \theta } ^ { * } \log t .
+$$
+
+To show the $1 / \log t$ rate, we simply need to prove that $\| \delta ( t , \mathbf { w } ) \|$ is bounded for any $\mathbf { w } \in [ 1 / M , M ] ^ { n }$ . Notice that
+
+$$
+\| \delta ( t + 1 , \mathbf { w } ) \| ^ { 2 } = \left\| \delta ( t + 1 , \mathbf { w } ) - \delta ( t , \mathbf { w } ) \right\| ^ { 2 } + 2 \big ( \delta ( t + 1 , \mathbf { w } ) - \delta ( t , \mathbf { w } ) \big ) ^ { \mathsf { T } } \delta ( t , \mathbf { w } ) + \left\| \delta ( t , \mathbf { w } ) \right\| ^ { 2 } .
+$$
+
+For the first term, we have:
+
+$$
+\begin{array} { r l } & { \left\| \delta ( t + 1 , \mathbf { w } ) - \delta ( t , \mathbf { w } ) \right\| ^ { 2 } } \\ & { = \left\| \mathbf { \nabla } - \eta \nabla L \big ( \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) ; \mathbf { w } \big ) - \theta ^ { * } \big ( \log ( t + 1 ) - \log ( t ) \big ) \right\| ^ { 2 } } \\ & { = \eta ^ { 2 } \| - \eta \nabla L \big ( \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) ; \mathbf { w } \big ) \| + \| \theta ^ { * } \| ^ { 2 } \log ^ { 2 } ( 1 + 1 / t ) + 2 \eta \big ( \pmb { \theta } ^ { * } ) ^ { \top } \nabla L \big ( \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) ; \mathbf { w } \big ) \log ( 1 + 1 / t ) } \\ & { \leq \eta ^ { 2 } \big \| \nabla L \big ( \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) ; \mathbf { w } \big ) \big \| + \| \theta ^ { * } \| ^ { 2 } t ^ { - 2 } ; } \end{array}
+$$
+
+where in the last line we use:
+
+$$
+\begin{array} { r l } & { \bullet ( \pmb { \theta } ^ { * } ) ^ { \top } \nabla L \big ( \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) ; \mathbf { w } \big ) \ = \ \sum _ { i } - w _ { i } \exp ( - y _ { i } \pmb { \theta } ^ { * } \mathbf { x } _ { i } ) y _ { i } \pmb { \theta } ^ { * } \mathbf { x } _ { i } \ \leq \ 0 } \end{array}
+$$
+
+Also, from the first conclusion of Lemma A.2, we see that $\left\| \nabla L \big ( \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) ; \mathbf { w } \big ) \right\| \ = \ o ( 1 / t )$ , so $\left\| \delta ( t + 1 , \mathbf { w } ) - \delta ( t , \mathbf { w } ) \right\| ^ { 2 } = o ( 1 / t )$ and the running sum converges to some finite number:
+
+$$
+\sum _ { t = 1 } ^ { \infty } \left\| \delta ( t + 1 , \mathbf { w } ) - \delta ( t , \mathbf { w } ) \right\| ^ { 2 } = C _ { 0 } < \infty .
+$$
+
+We see that the role of the weights is totally negligible because $\pmb { \theta } ^ { * }$ separates the data (the second bullet point above). The same argument applies to the second term $2 \big ( \delta ( t + 1 , \mathbf { w } ) - \delta ( t , \mathbf { w } ) \big ) ^ { \sf T } \delta ( t , \mathbf { w } )$ , where w plays no part as long as $\pmb { \theta } ^ { * }$ separates the data. The detailed proof is technical, and we refer to Lemma 6 of Soudry et al. (2018), which states that:
+
+$$
+\big ( \delta ( t + 1 , \mathbf { w } ) - \delta ( t , \mathbf { w } ) \big ) ^ { \top } \delta ( t , \mathbf { w } ) = o ( 1 / t ) .
+$$
+
+Therefore, by applying tensorization, it holds that:
+
+$$
+\left\| \delta ( t , \mathbf { w } ) \right\| ^ { 2 } - \left\| \delta ( t = 0 , \mathbf { w } ) \right\| ^ { 2 } \leq C _ { 0 } + \sum _ { i = 1 } ^ { t } \left( \delta ( t + 1 , \mathbf { w } ) - \delta ( t , \mathbf { w } ) \right) ^ { \top } \delta ( t , \mathbf { w } ) < \infty ,
+$$
+
+hence $\left\| \delta ( t , \mathbf { w } ) \right\|$ is bounded and
+
+$$
+\| \delta ( t , \mathbf { w } ) \| / \log t = \mathcal { O } ( 1 / \log t ) , \quad \Big | \frac { \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) } { \| \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) \| _ { 2 } } - \pmb { \theta } ^ { * } \Big | = \mathcal { O } ( \frac { 1 } { \log t } ) .
+$$
+
+It is now obvious that under the asymptotic characterization of (A.2), the weights only play a negligible role since $\pmb { \theta } ^ { * }$ separate the data. However, the definition of $\delta$ under (A.2) also prohibits us from studying the finite-step behavior since it absorbs all the constant factors.
+
+Now we use the Fenchel-Young inequality to give a more precise characterization of the convergence speed. First of all, recall the max-margin problem for linear predictor has a dual representation for separable data according to the KKT condition for separable problem:
+
+$$
+\pmb { \theta } ^ { * } = y _ { i } \mathbf { X } _ { i } \cdot p _ { i } ^ { * } / \gamma ^ { * } ,
+$$
+
+where $p _ { i } ^ { * }$ is the dual optimal such that
+
+$$
+\gamma ^ { * } = - \operatorname* { m i n } \Big \{ \operatorname* { m a x } _ { i } - y _ { i } \mathbf { x } _ { i } ^ { \top } \pmb \theta \mathrm { \ s . t . } \ \lVert \pmb \theta \rVert = 1 \Big \} \equiv \operatorname* { m i n } \Big \{ \lVert y _ { i } \mathbf { X } _ { i } \cdot p _ { i } \rVert \mathrm { \ s . t . } \ p _ { i } \ge 0 , \sum _ { i } p _ { i } = 1 \Big \} .
+$$
+
+Now, we directly work with θ ( t ) ( w )( t ) − θ ∗  :
+
+$$
+\left| \frac { \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) } { \| \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) \| _ { 2 } } - \pmb { \theta } ^ { \ast } \right| ^ { 2 } = 2 - \frac { 2 \big \langle \pmb { \theta } ^ { \ast } , \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) \big \rangle } { \| \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) \| _ { 2 } } ,
+$$
+
+and from (A.4) and Fenchel-Young inequality we have:
+
+$$
+- \frac { \left. \theta ^ { * } , \theta ^ { ( t ) } ( \mathbf { w } ) \right. } { \| \theta ^ { ( t ) } ( \mathbf { w } ) \| _ { 2 } } = \frac { \left. p ^ { * } , - y _ { i } \mathbf { x } _ { i } ^ { ( \top ) } \theta ^ { ( t ) } ( \mathbf { w } ) \right. } { \gamma ^ { * } \| \theta ^ { ( t ) } ( \mathbf { w } ) \| _ { 2 } } \leq \frac { g ^ { * } \big ( p ^ { * } \big ) + g \big ( - y _ { i } \mathbf { x } _ { i } ^ { ( \top ) } \theta ^ { ( t ) } ( \mathbf { w } ) \big ) } { \gamma ^ { * } \| \theta ^ { ( t ) } ( \mathbf { w } ) \| _ { 2 } } ,
+$$
+
+where $g$ is a convex function with it conjugate function given by $g ^ { * }$ . To build the connections with the loss function and risk, we choose $g$ such that $\begin{array} { r } { g ( \pmb { u } ) \stackrel { = } { = } \log \frac { 1 } { n } \sum _ { i } w _ { i } \exp ( u _ { i } ) } \end{array}$ . As a consequence, by letting $u _ { i } = - y _ { i } \mathbf { x } _ { i } ^ { ( \top ) } \pmb \theta ^ { ( t ) }$ and $\pmb { u } = [ u _ { 1 } , \dots , u _ { n } ]$ , we have $g ( \pmb { u } ) = L ( \pmb \theta ^ { ( t ) } ; \mathbf w )$ .
+
+With simple algebraic computations, the conjugate function $g ^ { * } ( \pmb { p } )$ is given by:
+
+$$
+g ^ { \ast } ( \pmb { p } ) = \log n + \sum _ { i } p _ { i } \log \frac { p _ { i } } { w _ { i } } = D _ { K L } ( \pmb { p } | | \mathbf { w } ) + \log n .
+$$
+
+Plugging the above results to (A.5):
+
+$$
+\frac { 1 } { 2 } \Big | \frac { \theta ^ { ( t ) } ( \mathbf { w } ) } { \big \| \theta ^ { ( t ) } ( \mathbf { w } ) \big \| _ { 2 } } - \theta ^ { * } \Big | ^ { 2 } \leq 1 + \frac { \log L ( \theta ^ { ( t ) } ( \mathbf { w } ) ; \mathbf { w } ) } { \big \| \theta ^ { ( t ) } ( \mathbf { w } ) \big \| _ { 2 } \gamma ^ { * } } + \frac { \log n + D _ { K L } ( p \| \mathbf { w } ) } { \big \| \theta ^ { ( t ) } ( \mathbf { w } ) \big \| _ { 2 } \gamma ^ { * } }
+$$
+
+According the convergence analysis of Adaboost, we have the following technical lemma.
+
+Lemma A.3 (Schapire & Freund (2013)). Suppose $\ell$ is convex, $\ell ^ { \prime } \leq \ell _ { \mathrm { { : } } }$ , and $\ell ^ { \prime \prime } \leq \ell ,$ , with a linear predictor and a sufficiently small learning rate such that $\eta _ { t } L ( \pmb \theta ^ { ( t ) } ) \leq 1$ , then:
+
+$$
+L ( \pmb { \theta } ^ { ( t + 1 ) } ) \leq L ( \pmb { \theta } ^ { ( t ) } ) \Big ( 1 - \eta _ { t } L ( \pmb { \theta } ^ { ( t ) } ) \big ( 1 - \eta _ { t } L ( \pmb { \theta } ^ { ( t ) } ) / 2 \big ) \Big ( \frac { \| \nabla L ( \pmb { \theta } ^ { ( t ) } ) \| _ { 2 } } { L \big ( \pmb { \theta } ^ { ( t ) } \big ) } \Big ) ^ { 2 } \Big ) ,
+$$
+
+and thus
+
+$$
+\begin{array} { r l } & { L ( \pmb \theta ^ { ( t + 1 ) } ) \leq L ( \pmb \theta ^ { ( 0 ) } ) \exp \Big ( - \displaystyle \sum _ { j < t } \eta _ { t } L ( \pmb \theta ^ { ( j ) } ) \big ( 1 - \eta _ { j } L ( \pmb \theta ^ { ( j ) } ) / 2 \big ) \Big ( \frac { \| \nabla L ( \pmb \theta ^ { ( j ) } ) \| _ { 2 } } { L ( \pmb \theta ^ { ( j ) } ) } \Big ) ^ { 2 } \Big ) . } \\ & { } \\ & { \| \pmb \theta ^ { ( t + 1 ) } \| \leq \sum _ { j < t } \eta _ { t } L ( \pmb \theta ^ { ( j ) } ) \frac { \| \nabla L ( \pmb \theta ^ { ( j ) } ) \| _ { 2 } } { L ( \pmb \theta ^ { ( j ) } ) } . } \end{array}
+$$
+
+To use the results in Lemma A.3, we define the following shorthand notations. Let $a _ { t } ( \mathbf { w } ) : =$ ηtL(θ(t); w) and bt(w) := $b _ { t } ( \mathbf { w } ) : = \frac { \| \nabla L ( \pmb \theta ^ { ( t ) } ( \mathbf { w } ) ; \mathbf { w } ) \| _ { 2 } } { L ( \pmb \theta ^ { ( t ) } ( \mathbf { w } ) ; \mathbf { w } ) }$ k∇L(θ(t)(w); w)k2 . Now, (A.6) can be further given by:
+
+$$
+\begin{array} { r l } { \displaystyle \frac { 1 } { 2 } \Big | \frac { \theta ^ { ( t ) } ( \mathbf { w } ) } { \big \| \theta ^ { ( t ) } ( \mathbf { w } ) \big \| _ { 2 } } - \theta ^ { * } \Big | ^ { 2 } \leq 1 + \frac { \log L ( \theta ^ { ( 0 ) } ; \mathbf { w } ) } { \| \theta ^ { ( t ) } \| \gamma ^ { * } } - } & { } \\ { \displaystyle \qquad } & { \leq \frac { \sum _ { i = 0 } ^ { t - 1 } a _ { i } ( \mathbf { w } ) \big ( 1 - a _ { i } ( \mathbf { w } ) / 2 \big ) b _ { i } ( \mathbf { w } ) ^ { 2 } } { \big \| \theta ^ { ( i ) } \big \| \gamma ^ { * } } + \frac { \log n + D _ { K L } ( p \| \mathbf { w } ) } { \big \| \theta ^ { ( t ) } ( \mathbf { w } ) \big \| _ { 2 } \gamma ^ { * } } } \\ { \leq 1 - \frac { \sum _ { i = 1 } ^ { t - 1 } a _ { i } ( \mathbf { w } ) b _ { i } ^ { 2 } ( \mathbf { w } ) } { \| \theta ^ { ( i ) } \| \gamma ^ { * } } + \frac { 2 \sum _ { i = 1 } ^ { t - 1 } a _ { i } ^ { 2 } ( \mathbf { w } ) b _ { i } ^ { 2 } ( \mathbf { w } ) } { \| \theta ^ { ( i ) } \| \gamma ^ { * } } + \frac { \log n + D _ { K L } ( p \| \mathbf { w } ) } { \big \| \theta ^ { ( t ) } ( \mathbf { w } ) \big \| _ { 2 } \gamma ^ { * } } . } \end{array}
+$$
+
+Notice that Lemma A.3 also imply:
+
+$$
+\sum _ { i = 1 } ^ { t - 1 } a _ { i } ^ { 2 } ( \mathbf w ) b _ { i } ^ { 2 } ( \mathbf w ) = \sum _ { i = 1 } ^ { t - 1 } \eta _ { i } \| \nabla L ( \pmb \theta ^ { ( i ) } ( \mathbf w ) ; \mathbf w ) \| \leq 2 \sum _ { i = 1 } ^ { t - 1 } \Big ( L ( \pmb \theta ^ { ( i ) } ( \mathbf w ) ; \mathbf w ) - L ( \pmb \theta ^ { ( i + 1 ) } ( \mathbf w ) ; \mathbf w ) \Big ) ,
+$$
+
+which is bounded from above by $2 M$ . Finally, it is easy to verify that $b _ { t } ( \mathbf { w } ) \geq \gamma ^ { * }$ , and Lemma A.3 also implies that $\begin{array} { r } { \| \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) \| \leq \sum _ { i < t } a _ { i } ( \mathbf { w } ) b _ { i } ( \mathbf { w } ) } \end{array}$ . Finally, we simplify (A.9) to:
+
+$$
+\Big | \frac { \theta ^ { ( t ) } ( \mathbf { w } ) } { \big \| \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) \big \| _ { 2 } } - \pmb { \theta } ^ { * } \Big | ^ { 2 } \leq 2 \cdot \frac { \log n + D _ { K L } ( \pmb { p } \| \mathbf { w } ) + M } { \big \| \pmb { \theta } ^ { ( t ) } ( \mathbf { w } ) \big \| _ { 2 } \gamma ^ { * } } ,
+$$
+
+and obtain the desired result.
+
+# A.3 PROOF FOR PROPOSITION 2
+
+We first present a greedy approach for the construction of the maximal separable subset $\mathcal { D } _ { \mathrm { s e p } }$ , which is proposed by Ji & Telgarsky (2018b).
+
+For each sample $\left( \mathbf { x } _ { i } , y _ { i } \right)$ , if there exists a $\theta _ { i }$ such that $y _ { i } \pmb { \theta } _ { i } ^ { \top } \mathbf { x } _ { i } > 0$ and $\begin{array} { r } { \operatorname* { m i n } _ { j = 1 , \dots , n } y _ { j } \pmb { \theta } _ { i } ^ { \intercal } \mathbf { x } _ { j } \ge 0 } \end{array}$ , we add it to $\mathcal { D } _ { \mathrm { s e p } }$ . Otherwise, we add it to ${ \mathcal { D } } _ { \mathrm { n o n - s e p } }$ . To see why this approach work, first notice that by choosing $\begin{array} { r } { \pmb { \theta } _ { s e p } ^ { * ^ { \star } } = \sum _ { i \in \mathcal { D } } \pmb { \theta } _ { i } , \pmb { \theta } _ { s e p } ^ { * } } \end{array}$ separates the data in $\mathcal { D } _ { \mathrm { s e p } }$ . Then we check it is indeed maximal: for any $\pmb \theta$ that is correct on any $\left( \mathbf { x } _ { i } , y _ { i } \right)$ in $\mathcal { D } _ { \mathrm { n o n - s e p } }$ , there must also exist another $\left( \mathbf { x } _ { j } , y _ { j } \right)$ in $\mathcal { D } _ { \mathrm { n o n - s e p } }$ so $y _ { i } \pmb { \theta } _ { i } ^ { \top } \mathbf { x } _ { i } < 0$ , or otherwise $\left( \mathbf { x } _ { i } , y _ { i } \right)$ would have been in $\mathcal { D } _ { \mathrm { s e p } }$ .
+
+It has been shown in Ji & Telgarsky (2018b) that the risk is strongly convex on $\mathcal { D } _ { \mathrm { n o n - s e p } }$ under conditions that are satisfied by our setting.
+
+Lemma A.4 (Theorem 2.1 of Ji & Telgarsky (2018b)). If $\ell$ is twice differentiable, $\ell ^ { \prime \prime } > 0$ , $l \geq 0$ and $\begin{array} { r } { \operatorname* { l i m } _ { u  \infty } \ell ( u ) = 0 } \end{array}$ , then $\begin{array} { r } { L ( \pmb { \theta } ) = \sum _ { i } \frac { 1 } { n } \ell \big ( y _ { i } \mathbf { \dot { \theta } } ^ { \top } \mathbf { x } _ { i } \big ) } \end{array}$ is strongly convex on $\mathcal { D } _ { n o n - s e p }$ .
+
+Now we provide the proof for Proposition 2.
+
+Proof. The first part is a direct consequence of Lemma A.4, that $\begin{array} { r } { L ( \pmb \theta ; \mathbf w ) = \frac { 1 } { n } \sum _ { i } w _ { i } \exp ( - y _ { i } \pmb \theta ^ { \top } \mathbf x _ { i } ) } \end{array}$ is strongly convex on ${ \mathcal { D } } _ { \mathrm { n o n - s e p } }$ . Therefore, the optimum $\tilde { \pmb { \theta } } ( \mathbf { w } )$ is uniquely defined and $\lVert \tilde { \pmb { \theta } } ( \mathbf { w } ) \rVert = \mathcal { O } ( 1 )$ . To show the second part, we leverage a standard argument for gradient descent with smoothness condition.
+
+Lemma A.5 (Bubeck (2014)). Suppose $L ( \theta )$ is convex and $\beta$ -smooth. Then with learning rate $\eta _ { t } \leq \beta / 2$ , the sequence of gradient descent satisfies:
+
+$$
+L ( \pmb { \theta } ^ { ( t + 1 ) } ) \leq L ( \pmb { \theta } ^ { ( t ) } ) - \eta _ { t } \big ( 1 - \eta _ { t } \beta / 2 \big ) \lVert \pmb { \theta } ^ { ( t ) } ) \rVert ^ { 2 } .
+$$
+
+Then for any $\mathbf { z } \in \mathbb { R } ^ { d }$ :
+
+$$
+2 \sum _ { i = 0 } ^ { t - 1 } \eta _ { i } \left( L ( \pmb { \theta } ^ { ( i ) } ) - L ( \mathbf { z } ) \right) \leq \| \pmb { \theta } ^ { ( 0 ) } - \mathbf { z } \| ^ { 2 } - \| \pmb { \theta } ^ { ( t ) } - \mathbf { z } \| ^ { 2 } + \sum _ { i = 0 } ^ { t - 1 } \frac { \eta _ { i } } { 1 - \beta \eta _ { i } / 2 } \big ( L ( \pmb { \theta } ^ { ( i ) } ) - L ( \mathbf { z } ) \big ) .
+$$
+
+It is immediately clear that we may choose the $\mathbf { z }$ in Lemma A.5 such that it combines the optimal from $\mathcal { D } _ { \mathrm { s e p } }$ and $\mathcal { D } _ { \mathrm { n o n - s e p } }$ . In particular, we have shown that the optimal on ${ \mathcal { D } } _ { \mathrm { n o n - s e p } }$ is uniquely given by $\tilde { \pmb { \theta } } ( \mathbf { w } )$ . For $\mathcal { D } _ { \mathrm { s e p } }$ we assume the max-margin linear predictor is given by $\theta _ { s e p } ^ { * }$ (so $\| \pmb { \theta } _ { s e p } ^ { * } \| = 1 ,$ ). Therefore, according to Proposition 1, the optimum is given by $\log t \cdot \theta _ { s e p } ^ { * }$ .
+
+Now define
+
+$$
+\begin{array} { r } { \mathbf { z } : = \tilde { \pmb { \theta } } ( \mathbf { w } ) + \pmb { \theta } _ { s e p } ^ { * } \cdot \log t / \gamma _ { s e p } , } \end{array}
+$$
+
+where we add the extra constant $\gamma _ { s e p }$ , which is the maximum margin on the separable subset of the data, to simplify the following bound. Without loss of generality, we assume the features are bounded in $\| \cdot \| _ { 2 }$ norm such that $\| \mathbf { x } _ { i } \| _ { 2 } \leq 1$ . As a consequence:
+
+$$
+L ( \pmb \theta ; \mathbf w ) = L _ { \mathrm { n o n } \lnot \mathrm { e p } } ( \tilde { \pmb \theta } ( \mathbf w ) ; \mathbf w ) + L _ { \mathrm { s e p } } ( \mathbf z ) \le \operatorname* { i n f } _ { \pmb \theta } L ( \pmb \theta ; \mathbf w ) + n \exp ( \| \tilde { \pmb \theta } ( \mathbf w ) \| ) / t ,
+$$
+
+where we use $L _ { \mathrm { { n o n - s e p } } }$ and $L _ { \mathrm { s e p } }$ to denote the risk associated with $\mathcal { D } _ { \mathrm { n o n - s e p } }$ and $\mathcal { D } _ { \mathrm { s e p } }$ . To invoke Lemma A.5, first note that the required smoothness condition is guaranteed by Lemma A.3, i.e. in each step, the risk is $\eta _ { t } L ( \pmb \theta ^ { ( t ) } )$ -smooth. Without loss of generality, we assume $\eta _ { t } L ( \pmb \theta ^ { ( t ) } ) \leq \eta _ { t }$ . Therefore, according to Lemma A.5, we have:
+
+$$
+\begin{array} { r l } & { 2 \big ( \displaystyle \sum _ { i < t } \eta _ { j } \big ) \big ( L ( \pmb \theta ^ { ( i ) } ; \mathbf { w } ) - L ( \mathbf z ; \mathbf { w } ) \big ) } \\ & { \leq 2 \displaystyle \sum _ { i < t } \eta _ { j } \big ( L ( \pmb \theta ^ { ( i ) } ; \mathbf { w } ) - L ( \mathbf z ; \mathbf { w } ) \big ) + 2 \big ( L ( \pmb \theta ^ { ( i + 1 ) } ; \mathbf { w } ) - L ( \pmb \theta ^ { ( i ) } ; \mathbf { w } ) \big ) } \\ & { \leq 2 \displaystyle \sum _ { i < t } \eta _ { j } \big ( L ( \pmb \theta ^ { ( i ) } ; \mathbf { w } ) - L ( \mathbf z ; \mathbf { w } ) \big ) - \displaystyle \sum _ { i < t } \frac { \eta _ { i } } { 1 - \eta _ { i } / 2 } \big ( L ( \pmb \theta ^ { ( i ) } ; \mathbf { w } ) - L ( \pmb \theta ^ { ( i + 1 ) } ; \mathbf { w } ) \big ) } \\ & { \leq \| \pmb \theta ^ { ( 0 ) } - \mathbf z \| ^ { 2 } - \| \pmb \theta ^ { ( t ) } - \mathbf z \| ^ { 2 } \leq \| \mathbf { z } \| ^ { 2 } . } \end{array}
+$$
+
+Therefore, by our choice of $\mathbf { z }$ as well as the result in (A.10), we obtain the bound in terms of the risk:
+
+$$
+L ( \pmb \theta ^ { ( t ) } ; \mathbf w ) \le \operatorname* { i n f } _ { \pmb \theta } L ( \pmb \theta ; \mathbf w ) + \frac { \exp ( \tilde { \pmb \theta } ( \mathbf w ) ) } { t } + \frac { \| \tilde { \pmb \theta } ( \mathbf w ) \| ^ { 2 } + \log ^ { 2 } t / \gamma _ { \mathrm { s e p } } ^ { 2 } } { 2 \sum _ { i < t } \eta _ { i } } .
+$$
+
+Since we assume a constant learning rate, when $\textstyle \sum _ { i < t } \eta _ { i } = { \mathcal { O } } ( t )$ we can simplify the above result to:
+
+$$
+L ( \pmb \theta ^ { ( t ) } ; \mathbf w ) \le \operatorname* { i n f } _ { \pmb \theta } L ( \pmb \theta ; \mathbf w ) + \frac { C \big ( \| \tilde { \pmb \theta } ( \mathbf w ) \| \big ) + \log ^ { 2 } t / \gamma _ { \mathrm { s e p } } ^ { 2 } } { t } .
+$$
+
+Finally, from Lemma A.4 we known $L ( \pmb \theta ; \mathbf { w } )$ is strongly convex (which we assume to be $\omega$ -stronglyconvex). So the convergence in terms of the risk can be transformed to parameters:
+
+$$
+\begin{array} { l } { \displaystyle \big | \Pi _ { \mathrm { n o n - s e p } } \theta ^ { ( t ) } ( \mathbf { w } ) - \tilde { \theta } ( \mathbf { w } ) \big | \leq \frac { 2 } { \omega } \Big ( L _ { \mathrm { n o n - s e p } } ( \theta ^ { ( t ) } ( \mathbf { w } ) ; \mathbf { w } ) - L _ { \mathrm { n o n - s e p } } ( \tilde { \theta } ( \mathbf { w } ) ; \mathbf { w } ) \Big ) } \\ { \displaystyle \qquad \leq \frac { 2 } { \omega } \Big ( L ( \theta ^ { ( t ) } ( \mathbf { w } ) ; \mathbf { w } ) - \operatorname* { i n f } _ { \theta } L ( \theta ; \mathbf { w } ) \Big ) , } \end{array}
+$$
+
+which leads to our desired results.
+
+# A.4 SUPPLEMENTARY MATERIAL FOR SECTION 4
+
+In this section, we establish the detailed proofs of Proposition 3 and Theorem 1. Recall that the loss function we are interested in is:
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } } L _ { \lambda } ( \pmb { \theta } ; \mathbf { w } ) : = L ( \pmb { \theta } , \mathbf { w } ) + \lambda \| \pmb { \theta } \| ^ { r } ,
+$$
+
+Denote ${ \pmb \theta } _ { \lambda } ( { \bf w } ) \ \in \ \mathrm { a r g } \operatorname * { m i n } L _ { \lambda } ( { \pmb \theta } , { \bf w } ) , \ { \pmb \theta }$ $\theta ^ { * } \ = \ \arg \operatorname* { m a x } _ { \theta : \| \theta \| \leq 1 } \operatorname* { m a x } _ { i } y _ { i } f ( \theta , \mathbf { x } _ { i } ) )$ . Let $\gamma _ { \lambda } ( \mathbf { w } ) \ =$ $\mathrm { m a x } _ { i } y _ { i } f ( \pmb { \theta } _ { \lambda } ( \mathbf { w } ) / \Vert \pmb { \theta } _ { \lambda } ( \mathbf { w } ) \Vert , \mathbf { x } _ { i } )$ , $\gamma ^ { * } = \operatorname* { m a x } _ { i } y _ { i } f ( \pmb { \theta } ^ { * } , \mathbf { x } _ { i } )$ .
+
+A.4.1 PROOF OF PROPOSITION 3.
+
+We first restate the proposition.
+
+Proposition A.1. Suppose C1, C2, A1 hold. For any $\boldsymbol { \mathsf { \Sigma } } ^ { \prime } \in [ 1 / M , M ] ^ { n }$ , it follows that
+
+• (Asymptotic) $\mathrm { l i m } _ { \lambda \to 0 } \gamma _ { \lambda } ( \mathbf { w } )  \gamma ^ { * }$ .
+
+• (Finite steps) There exists $\begin{array} { r c l } { { a } } & { { \lambda } } & { { : = } } & { { \lambda ( r , \alpha , \gamma ^ { * } , \mathbf { w } , c ) } } \end{array}$ such that for $\pmb { \theta } ^ { \prime } ( \mathbf { w } )$ with $L _ { \lambda } ( \pmb \theta ^ { \prime } ( \mathbf w ) ; \mathbf w ) \leq \tau L _ { \lambda } ( \pmb \theta _ { \lambda } ( \mathbf { w } ) ; \mathbf { w } )$ and $\tau \leq 2$ , the associated margin $\tilde { \gamma } ( \pmb { \theta } ^ { \prime } ( \mathbf { w } ) )$ satisfies $\begin{array} { r } { \tilde { \gamma } ( \pmb { \theta } ^ { \prime } ( \mathbf { w } ) ) \geq c \cdot \frac { \gamma ^ { * } } { \tau ^ { \alpha / r } } } \end{array}$ , where $\textstyle { \frac { 1 } { 1 0 } } \leq c < 1$
+
+# Proof of the Asymptotic part:
+
+Proof. We first take consider the exponential loss $\ell ( u ) = \exp ( - u )$ . The log loss $\ell ( u ) = \log ( 1 +$ $\exp ( - u ) )$ can be shown in a similar fashion. Suppose the weights $\mathbf { w } = ( w _ { 1 } , \dots w _ { n } )$ are normalized so that $\textstyle \sum _ { i = 1 } ^ { n } w _ { i } = 1$ and $w _ { i } \geq 0$ . Consider
+
+$$
+\begin{array} { r c l } { L _ { \lambda } ( A \pmb \theta ; \mathbf { w } ) } & { = } & { \displaystyle \sum _ { i = 1 } ^ { n } w _ { i } \exp ( - A ^ { \alpha } \cdot y _ { i } f ( \pmb \theta ; \mathbf { x } _ { i } ) ) + \lambda A ^ { r } \| \pmb \theta \| ^ { r } } \\ & { \leq } & { \displaystyle \exp ( - A ^ { \alpha } \cdot \operatorname* { m a x } _ { i } ( y _ { i } f ( \pmb \theta ; \mathbf { x } _ { i } ) ) ) + \lambda A ^ { r } \| \pmb \theta \| ^ { r } , } \end{array}
+$$
+
+where $A > 0$ , and we disregard the $1 / n$ term in $L _ { \lambda }$ for the sake of notation. In addition, we have the lower bound
+
+$$
+\begin{array} { r l r } { L _ { \lambda } ( A \pmb \theta ; \mathbf { w } ) } & { \geq } & { w _ { i ^ { \prime } } \cdot \mathrm { e x p } ( - A ^ { \alpha } \cdot \underset { i } { \mathrm { m a x } } ( y _ { i } f ( \pmb \theta ; \mathbf { x } _ { i } ) ) ) + \lambda A ^ { r } \| \pmb \theta \| ^ { r } } \\ & { \geq } & { w _ { [ n ] } \cdot \mathrm { e x p } ( - A ^ { \alpha } \cdot \underset { i } { \mathrm { m a x } } ( y _ { i } f ( \pmb \theta ; \mathbf { x } _ { i } ) ) ) + \lambda A ^ { r } \| \pmb \theta \| ^ { r } , } \end{array}
+$$
+
+where $i ^ { \prime } = \arg \operatorname* { m i n } _ { i } y _ { i } f ( \pmb { \theta } ; \mathbf { x } _ { i } ) )$ , $w _ { [ n ] } = \operatorname* { m i n } _ { i } w _ { i }$ . By taking $A = \| \pmb { \theta } _ { \lambda } ( \mathbf { w } ) \|$ , $\pmb { \theta } = \pmb { \theta } ^ { * }$ in the upper bound and $A = 1$ , $\pmb \theta = \pmb \theta _ { \lambda } ( \mathbf { w } )$ in the lower bound , it follows that
+
+$$
+\begin{array} { r l } & { w _ { [ n ] } \cdot \exp ( - \| \pmb { \theta } _ { \lambda } ( \mathbf { w } ) \| ^ { \alpha } \gamma _ { \lambda } ( \mathbf { w } ) ) + \lambda \| \pmb { \theta } _ { \lambda } ( \mathbf { w } ) \| ^ { r } } \\ { \leq } & { L _ { \lambda } ( \mathbf { w } ) ( \pmb { \theta } _ { \lambda } ( \mathbf { w } ) ) } \\ { \leq } & { L _ { \lambda } ( \mathbf { w } ) ( \| \pmb { \theta } _ { \lambda } ( \mathbf { w } ) \| \pmb { \theta } ^ { * } ) } \\ { \leq } & { \exp ( - \| \pmb { \theta } _ { \lambda } ( \mathbf { w } ) \| ^ { \alpha } \cdot \gamma ^ { * } ) + \lambda \| \pmb { \theta } _ { \lambda } ( \mathbf { w } ) \| ^ { r } . } \end{array}
+$$
+
+It implies that
+
+$$
+w _ { [ n ] } \cdot \exp ( - \| \pmb \theta _ { \lambda } ( \mathbf w ) \| ^ { \alpha } \gamma _ { \lambda } ( \mathbf w ) ) \leq \exp ( - \| \pmb \theta _ { \lambda } ( \mathbf w ) \| ^ { \alpha } \cdot \gamma ^ { * } ) ,
+$$
+
+or
+
+$$
+w _ { [ n ] } \cdot \exp ( - \| \pmb { \theta } _ { \lambda } ( \mathbf { w } ) \| ^ { \alpha } ( \gamma ^ { * } - \gamma _ { \lambda } ( \mathbf { w } ) ) ) \leq 1 .
+$$
+
+By Claim 1 that $\| \pmb \theta _ { \lambda } ( \mathbf { w } ) \|  \infty$ as $\lambda  0$ (or Lemma C.4 in Wei et al. (2019)), the above inequality implies that $\gamma _ { \lambda } ( \mathbf { w } ) \to \gamma ^ { * }$ as $\lambda  0$ . □
+
+# Proof of the Finite steps part
+
+Proof. Consider $A = [ \textstyle { \frac { 1 } { \gamma ^ { * } } } \log ( ( \gamma ^ { * } ) ^ { r / \alpha } / \lambda ) ] ^ { 1 / \alpha }$ , it follows that
+
+$$
+\begin{array} { r c l } { { { \cal L } _ { \lambda } ( \pmb { \theta } ^ { \prime } ( { \bf w } ) , { \bf w } ) } } & { { \leq } } & { { \tau { \cal L } _ { \lambda } ( A \pmb { \theta } ^ { * } ) } } \\ { { } } & { { \leq } } & { { \tau \exp ( - A ^ { \alpha } \cdot \gamma ^ { * } ) + \tau \lambda A ^ { r } \qquad [ \mathrm { U p p e r ~ B o u n d ~ A . 4 . 1 } ] } } \\ { { } } & { { = } } & { { \displaystyle \frac { \lambda \tau } { ( \gamma ^ { * } ) ^ { r / \alpha } } \left( 1 + ( \log ( ( \gamma ^ { * } ) ^ { r / \alpha } / \lambda ) ) ^ { r / \alpha } \right) } } \end{array}
+$$
+
+Then by the lower bound A.4.1, it follows that
+
+$$
+w _ { [ n ] } \cdot \exp ( - \| \pmb \theta ^ { \prime } ( \mathbf w ) \| ^ { \alpha } \gamma ^ { \prime } ( \mathbf w ) ) \leq L _ { \lambda } ( \pmb \theta ^ { \prime } ( \mathbf w ) , \mathbf w ) \leq A . 1 5 ,
+$$
+
+where $\begin{array} { r } { \gamma ^ { \prime } ( \mathbf { w } ) = \operatorname* { m a x } _ { i } y _ { i } f ( \mathbf { w } ^ { \prime } / \Vert \mathbf { w } ^ { \prime } \Vert , \mathbf { x } _ { i } ) } \end{array}$ . Note $\lambda \| \pmb \theta ^ { \prime } ( \mathbf { w } ) \| ^ { r } \leq A . 1 5$ . It implies that
+
+$$
+\begin{array} { r l r } { \gamma ^ { \prime } ( \mathbf { w } ) } & { \geq } & { \frac { - \log ( A . 1 5 / w _ { [ n ] } ) } { \| \pmb { \theta } ^ { \prime } ( \mathbf { w } ) \| ^ { \alpha } } } \\ & { \geq } & { \frac { - \log ( \frac { \lambda \tau } { w _ { [ n ] } ( \gamma ^ { * } ) ^ { r / \alpha } } ( 1 + ( \log ( ( \gamma ^ { * } ) ^ { r / \alpha } / \lambda ) ) ^ { r / \alpha } ) ) } { \frac { \tau ^ { \alpha / r } } { \gamma ^ { * } } ( 1 + ( \log ( ( \gamma ^ { * } ) ^ { r / \alpha } / \lambda ) ) ^ { r / \alpha } ) ^ { \alpha / r } } } \end{array}
+$$
+
+Note that the numerator is at the scale $\log ( { \frac { 1 } { \lambda } } / \log { \frac { 1 } { \lambda } } )$ and the denominator is at the scale $\log { \frac { 1 } { \lambda } }$ . So for sufficiently small $\lambda = \lambda ( r , \alpha , \gamma ^ { * } , \mathbf { w } , c )$ , we have $\begin{array} { r } { \gamma ^ { \prime } ( \mathbf { w } ) \geq c \cdot \frac { \gamma ^ { * } } { \tau ^ { \alpha / r } } } \end{array}$ , where $\textstyle { \frac { 1 } { 1 0 } } \leq c < 1$ . We leave the details of finding out the dependency of $\lambda ( r , \alpha , \gamma ^ { * } , \mathbf { w } , c )$ on c to the readers, which is simply the basic analysis. □
+
+# A.4.2 PROOF OF THEOREM 1
+
+When the training distribution $p _ { \mathrm { t r a i n } }$ deviates from the testing distribution $p _ { \mathrm { t e s t } }$ , we develop the geof ralization bound that characterizes this deviation.from the training data and the testing data. Let $p _ { s }$ $p _ { t }$ ities and $\mathbf { x }$ $\begin{array} { r } { D ( P _ { t } \| P _ { s } ) = \int \big ( ( \frac { p _ { t } ( x ) } { p _ { s } ( x ) } ) ^ { 2 } - 1 \big ) p _ { s } ( x ) d x } \end{array}$ $\begin{array} { r } { \eta ( \mathbf { x } _ { i } ) = \frac { p _ { t } ( \mathbf { x } _ { i } ) } { p _ { s } ( \mathbf { x } _ { i } ) } } \end{array}$ . We first restate Theorem 1:
+
+Theorem A.1. Assume $\sigma$ is 1-Lipschitz and 1-positive homogeneous. Then with probability at least $1 - \delta$ , we have
+
+$$
+\begin{array} { r l } { \mathbf { \widetilde { \mathbf { \Gamma } } } _ { ( \mathbf { x } , y ) \sim p _ { t e r t } } ^ { \mathfrak { p } } \left( y f ^ { N N } ( \pmb { \theta } ( \mathbf { w } ) , \mathbf { x } ) \leq 0 \right) \leq } & { } \\ { \underbrace { \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \eta ( \mathbf { x } _ { i } ) \mathbf { I } \big ( y _ { i } f ^ { N N } ( \pmb { \theta } ( \mathbf { w } ) / \| \pmb { \theta } ( \mathbf { w } ) \| , \mathbf { x } _ { i } ) < \gamma \big ) } _ { ( I ) } + \underbrace { \frac { C \cdot \sqrt { D ( P _ { t } | | P _ { s } ) + 1 } } { \gamma \cdot H ^ { ( H - 1 ) / 2 } \sqrt { n } } } _ { ( I I ) } + \epsilon ( \gamma , n , \delta ) , } \end{array}
+$$
+
+where $( I )$ is the empirical risk, $( I I )$ reflects the compounding effect of the model complexity of the class of $H$ -layer neural networks and the deviation of the target distribution from the source distribution , $\begin{array} { r } { \epsilon ( \gamma , n , \delta ) = \sqrt { \frac { \log \log _ { 2 } \frac { 4 C } { \gamma } } { n } } + \sqrt { \frac { \log ( 1 / \delta ) } { n } } } \end{array}$ is a small quantity compared to $( I )$ and $( I I )$ . Here $C : = \operatorname* { s u p } _ { \mathbf { x } } \left\| \mathbf { x } \right\|$ ; $\gamma$ is any positive value.
+
+To prove Theorem A.1, we first establish a few lemmas.
+
+Lemma A.6. Consider an arbitrary function class $\mathcal { F }$ such that $\forall f \in { \mathcal { F } }$ we have $\begin{array} { r } { \sum _ { \mathbf { x } \in \mathcal { X } } | f ( \mathbf { x } ) | \le C } \end{array}$ Then, with probability at least $1 - \delta$ over the sample, for all margins $\gamma > 0$ and all $\bar { f } \in \mathcal { F }$ we have,
+
+$$
+\begin{array} { r l } & { \displaystyle \mathbb { P } _ { p _ { ( \mathbf { x } , y ) \sim p _ { t e t } } } \Big ( y f ( \mathbf { x } ) \le 0 \Big ) } \\ & { \le \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \eta ( \mathbf { x } _ { i } ) \mathbf { I } \big ( y _ { i } f ( \mathbf { x } _ { i } ) < \gamma \big ) + 4 \frac { \mathcal { R } _ { n , \eta } ( \mathcal { F } ) } { \gamma } + \sqrt { \frac { \log ( \log _ { 2 } \frac { 4 C } { \gamma } ) } { n } } + \sqrt { \frac { \log ( 1 / \delta ) } { 2 n } } , } \end{array}
+$$
+
+where $\begin{array} { r } { \mathcal { R } _ { n , \eta } ( \mathcal { F } ) = \mathbb { E } \Big [ \operatorname* { s u p } _ { f \in \mathcal { F } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \eta ( \mathbf { x } _ { i } ) f ( \mathbf { x } _ { i } ) \epsilon _ { i } \Big ] } \end{array}$ is the weighted Rademacher complexity $( \epsilon _ { i } ^ { \phantom { } } s$ are i.i.d Rademacher variables).
+
+Proof. This lemma is adapted from Theorem 1 of Koltchinskii et al. (2002) by considering the deviation of the testing distribution from the training distribution. Then it is obtained following Theorem 5 of Kakade et al. (2009). □
+
+Lemma A.7. Let $\mathcal { F } _ { H }$ be the class of real-valued networks of depth $H$ over the domain $\mathcal { X }$ , where each parameter matrix $W _ { h }$ has Frobenius norm at most $M _ { F } ( h )$ , and with an activation that is 1-Lipschitz, positive-homogeneous. Then,
+
+$$
+\mathcal { R } _ { n , \eta } ( \mathcal { F } _ { H } ) \leq \frac { C \cdot \sqrt { D ( P _ { t } | | P _ { s } ) + 1 + o ( \frac { 1 } { \sqrt { n } } ) } \cdot ( \sqrt { 2 \log { 2 H } } + 1 ) } { \sqrt { n } } \prod _ { h = 1 } ^ { H } M _ { F } ( h ) ,
+$$
+
+where $C : = \operatorname* { s u p } _ { x \in { \mathcal { X } } } \| \mathbf { x } \|$
+
+Proof. From Theorem 1 of Golowich et al. (2018), we arrive at
+
+$$
+n \mathcal { R } ( n , \eta ) ( \mathcal { F } _ { H } ) \leq \frac { 1 } { \lambda } \log \Big ( 2 ^ { H } \cdot \mathbb { E } _ { \epsilon } \Big ( M \lambda \| \sum _ { i = 1 } ^ { n } \epsilon _ { i } \eta ( \mathbf { x } _ { i } ) \mathbf { x } _ { i } \| \Big ) \Big ) ,
+$$
+
+where $\begin{array} { r } { M = \prod _ { h = 1 } ^ { H } M _ { F } ( h ) } \end{array}$ . Consider $\begin{array} { r } { Z : = M \cdot \| \sum _ { i = 1 } ^ { n } \epsilon _ { i } \eta ( \mathbf { x } _ { i } ) \mathbf { x } _ { i } \| } \end{array}$ that is a random function of the $n$ Rademacher variables. Then
+
+$$
+{ \frac { 1 } { \lambda } } \log \left\{ 2 ^ { H } \mathbb { E } \exp ( \lambda Z ) \right\} = { \frac { H \log ( 2 ) } { \lambda } } + { \frac { 1 } { \lambda } } \log \left\{ \mathbb { E } \exp \lambda ( Z - \mathbb { E } Z ) \right\} + \mathbb { E } Z .
+$$
+
+By Jensen’s inequality, we have
+
+$$
+\mathbb { E } [ Z ] \leq M \sqrt { \mathbb { E } _ { \epsilon } \| \sum _ { i = 1 } ^ { n } \epsilon _ { i } \eta ( \mathbf { x } _ { i } ) \mathbf { x } _ { i } \| ^ { 2 } } = M \sqrt { \sum _ { i = 1 } ^ { n } \eta ( \mathbf { x } _ { i } ) ^ { 2 } \| \mathbf { x } _ { i } \| ^ { 2 } } .
+$$
+
+In addition, we note that
+
+$$
+Z ( \epsilon _ { 1 } , \dots , \epsilon _ { i } , \dots , \epsilon _ { n } ) - Z ( \epsilon _ { 1 } , \dots , - \epsilon _ { i } , \dots , \epsilon _ { n } ) \leq 2 M \eta ( \mathbf { x } _ { i } ) \| \mathbf { x } _ { i } \| .
+$$
+
+By the bounded-difference condition (Boucheron et al., 2013), $Z$ is a sub-Gaussian with variance factor $\begin{array} { r } { v = \frac { 1 } { 4 } \sum _ { i = 1 } ^ { n } ( 2 M \eta ( \mathbf { x } _ { i } ) \| \mathbf { x } _ { i } \| ) ^ { 2 } = M ^ { 2 } \sum _ { i = 1 } ^ { n } \eta ( x _ { i } ) ^ { 2 } \| \mathbf { x } _ { i } \| ^ { 2 } } \end{array}$ . So
+
+$$
+\frac { 1 } { \lambda } \{ \mathbb { E } \exp \lambda ( Z - \mathbb { E } Z ) \} \leq \frac { \lambda M ^ { 2 } \sum _ { i = 1 } ^ { n } \eta ( \mathbf { x } _ { i } ) ^ { 2 } \| \mathbf { x } _ { i } \| ^ { 2 } } { 2 } .
+$$
+
+Taking $\begin{array} { r } { \lambda = \frac { \sqrt { 2 \log ( 2 ) H } } { M \sqrt { \sum _ { i = 1 } ^ { n } \eta ( \mathbf { x } _ { i } ) ^ { 2 } \| \mathbf { x } _ { i } \| ^ { 2 } } } } \end{array}$ , it follows that
+
+$$
+\begin{array} { r l } & { \frac { 1 } { \lambda } \{ 2 ^ { H } \mathbb { E } \exp \lambda Z \} } \\ & { \leq M ( \sqrt { 2 \log ( 2 ) H } + 1 ) \sqrt { \displaystyle \sum _ { i = 1 } ^ { n } \eta ( { \mathbf x } _ { i } ) ^ { 2 } \| { \mathbf x } _ { i } \| ^ { 2 } } \leq \sqrt { n } C M ( \sqrt { 2 \log ( 2 ) H } + 1 ) \sqrt { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \eta ( { \mathbf x } _ { i } ) ^ { 2 } } . } \end{array}
+$$
+
+By law of large number, $\begin{array} { r } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \eta ( \mathbf { x } _ { i } ) ^ { 2 } = D ( P _ { t } \| P _ { s } ) + 1 + o \big ( \frac { 1 } { \sqrt { n } } \big ) } \end{array}$ . The desired result follows.
+
+Lemma A.8. Suppose $f ^ { N N } ( \pmb { \theta } , \cdot )$ is a $H$ -layer neural network and $C = \operatorname* { s u p } _ { x \in { \mathcal { X } } } \| x \| _ { 2 }$ . Then, There exists another parameter $\tilde { \pmb { \theta } }$ s.t. $f ^ { N N } ( \pmb { \theta } / \| \pmb { \theta } \| , \mathbf { x } ) = f ^ { N N } ( \tilde { \pmb { \theta } } , \mathbf { x } )$ , for any $x \in \mathcal { X }$ and that
+
+• the parameter matrix of each layer of $f ^ { N N } ( \tilde { \pmb { \theta } } , \cdot )$ has a Frobenius norm no larger than $1 / { \sqrt { H } }$ . $\bullet \ \operatorname* { s u p } _ { x \in { \mathcal { X } } } f ^ { N N } ( \widetilde { \pmb { \theta } } , \cdot ) \leq C .$
+
+Proof. This lemma are obtained by reorganizing the proof of Lemma D3 and the proof of Proposition D.1 of Wei et al. (2019). □
+
+# Proof of Theorem A.1
+
+Proof. Theorem A.1 follows by Lemma A.6, A.7 and A.8.

parse/train/dgtpE6gKjHn/dgtpE6gKjHn.md

@@ -0,0 +1,523 @@
+# FEDBE: MAKING BAYESIAN MODEL ENSEMBLE APPLICABLE TO FEDERATED LEARNING
+
+Hong-You Chen The Ohio State University, USA chen.9301@osu.edu
+
+Wei-Lun Chao The Ohio State University, USA chao.209@osu.edu
+
+# ABSTRACT
+
+Federated learning aims to collaboratively train a strong global model by accessing users’ locally trained models but not their own data. A crucial step is therefore to aggregate local models into a global model, which has been shown challenging when users have non-i.i.d. data. In this paper, we propose a novel aggregation algorithm named FEDBE, which takes a Bayesian inference perspective by sampling higher-quality global models and combining them via Bayesian model Ensemble, leading to much robust aggregation. We show that an effective model distribution can be constructed by simply fitting a Gaussian or Dirichlet distribution to the local models. Our empirical studies validate FEDBE’s superior performance, especially when users’ data are not i.i.d. and when the neural networks go deeper. Moreover, FEDBE is compatible with recent efforts in regularizing users’ model training, making it an easily applicable module: you only need to replace the aggregation method but leave other parts of your federated learning algorithm intact.
+
+# 1 INTRODUCTION
+
+Modern machine learning algorithms are data and computation hungry. It is therefore desired to collect as many data and computational resources as possible, for example, from individual users (e.g., users’ smartphones and pictures taken on them), without raising concerns in data security and privacy. Federated learning has thus emerged as a promising learning paradigm, which leverages individuals’ computational powers and data securely — by only sharing their locally trained models with the server — to jointly optimize a global model (Konecnˇ y et al., 2016; Yang et al., 2019). \`
+
+Federated learning (FL) generally involves multiple rounds of communication between the server and clients (i.e., individual sites). Within each round, the clients first train their own models using their own data, usually with limited sizes. The server then aggregates these models into a single, global model. The clients then begin the next round of training, using the global model as the initialization.
+
+We focus on model aggregation, one of the most critical steps in FL. The standard method is FEDAVG (McMahan et al., 2017), which performs element-wise average over clients’ model weights. Assuming that each client’s data are sampled i.i.d. from their aggregated data, FEDAVG has been shown convergent to the ideal model trained in a centralized way using the aggregated data (Zinkevich et al., 2010; McMahan et al., 2017; Zhou & Cong, 2017). Its performance, however, can degrade drastically if such an assumption does not hold in practice (Karimireddy et al., 2020; Li et al., 2020b; Zhao et al., 2018): FEDAVG simply drifts away from the ideal model. Moreover, by only taking weight average, FEDAVG does not fully utilize the information among clients (e.g., variances), and may have negative effects on over-parameterized models like neural networks due to their permutation-invariant property in the weight space (Wang et al., 2020; Yurochkin et al., 2019).
+
+To address these issues, we propose a novel aggregation approach using Bayesian inference, inspired by (Maddox et al., 2019). Treating each client’s model as a possible global model, we construct a distribution of global models, from which weight average (i.e., FEDAVG) is one particular sample and many other global models can be sampled. This distribution enables Bayesian model ensemble — aggregating the outputs of a wide spectrum of global models for a more robust prediction. We show that Bayesian model ensemble can make more accurate predictions than weight average at a single round of communication, especially under the non i.i.d. client condition. Nevertheless, lacking a single global model that represents Bayesian model ensemble and can be sent back to clients, Bayesian model ensemble cannot directly benefit federated learning in a multi-round setting.
+
+We therefore present FEDBE, a learning algorithm that effectively incorporates Bayesian model Ensemble into federated learning. Following (Guha et al., 2019), we assume that the server has access to a set of unlabeled data, on which we can make predictions by model ensemble. This assumption can easily be satisfied: the server usually collects its own data for model validation, and collecting unlabeled data is simpler than labeled ones. (See section 6 for more discussion, including the privacy concern.) Treating the ensemble predictions as the “pseudo-labels” of the unlabeled data, we can then summarize model ensemble into a single global model by knowledge distillation (Hinton et al., 2015) — using the predicted labels (or probabilities or logits) as the teacher to train a student global model. The student global model can then be sent back to the clients to begin their next round of training1.
+
+We identify one key detail of knowledge distillation in FEDBE. In contrast to its common practice where the teacher is highly accurate and labeled data are accessible, the ensemble predictions in federated learning can be relatively noisy2. To prevent the student from over-fitting the noise, we apply stochastic weight average (SWA) (Izmailov et al., 2018) in distillation. SWA runs stochastic gradient descent (SGD) with a cyclical learning rate and averages the weights of the traversed models, allowing the traversed models to jump out of noisy local minimums, leading to a more robust student.
+
+We validate FEDBE on CIFAR-10/100 (Krizhevsky et al., 2009) and Tiny-ImageNet (Le & Yang, 2015) under different client conditions (i.e., i.i.d. and non-i.i.d. ones), using ConvNet (TensorFlow team, 2016), ResNet (He et al., 2016), and MobileNetV2 (Howard et al., 2017; Sandler et al., 2018). FEDBE consistently outperforms FEDAVG, especially when the neural network architecture goes deeper. Moreover, FEDBE can be compatible with existing FL algorithms that regularize clients’ learning or leverage server momentum (Li et al., 2020a; Sahu et al., 2018; Karimireddy et al., 2020; Hsu et al., 2019) and further improves upon them. Interestingly, even if the unlabeled server data have a different distribution or domain from the test data (e.g., taken from a different dataset), FEDBE can still maintain its accuracy, making it highly applicable in practice.
+
+# 2 RELATED WORK (MORE IN APPENDIX A)
+
+Federated learning (FL). In the multi-round setting, FEDAVG (McMahan et al., 2017) is the standard approach. Many works have studied its effectiveness and limitation regarding convergence, robustness, and communication cost, especially in the situations of non-i.i.d. clients. Please see Appendix A for a list of works. Many works proposed to improve FEDAVG. FEDPROX (Li et al., 2020a; Sahu et al., 2018), FEDDANE (Li et al., 2019), Yao et al. (2019), and SCAFFOLD (Karimireddy et al., 2020) designed better local training strategies to prevent clients’ model drifts. Zhao et al. (2018) studied the use of shared data between the server and clients to reduce model drifts. Reddi et al. (2020) and Hsu et al. (2019) designed better update rules for the global model by server momentum and adaptive optimization. Our FEDBE is complementary to and can be compatible with these efforts.
+
+In terms of model aggregation. Yurochkin et al. (2019) developed a Bayesian non-parametric approach to match clients’ weights before average, and FEDMA (Wang et al., 2020) improved upon it by iterative layer-wise matching. One drawback of FEDMA is its linear dependence of computation and communication on the network’s depth, not suitable for deeper models. Also, both methods are not yet applicable to networks with residual links and batch normalization (Ioffe & Szegedy, 2015). We improve aggregation via Bayesian ensemble and knowledge distillation, bypassing weight matching.
+
+Ensemble learning and knowledge distillation. Model ensemble is known to be more robust and accurate than individual base models (Zhou, 2012; Dietterich, 2000; Breiman, 1996). Several recent works (Anil et al., 2018; Guo et al., 2020; Chen et al., 2020) investigated the use of model ensemble and knowledge distillation (Hinton et al., 2015) in an online fashion to jointly learn multiple models, where the base models and distillation have access to the centralized labeled data or decentralized data of the same distribution. In contrast, client models in FL are learned with isolated and likely non-i.i.d. and limited data; our distillation is performed without labeled data. We thus propose to sample base models of higher quality for Bayesian ensemble and employ SWA for robust distillation.
+
+Knowledge distillation in FL. Guha et al. (2019) considered one-round FL and applied distillation to obtain a global model from the direct ensemble of clients’ models. A similar idea was used in (Papernot et al., 2017) in a different context. Our method can be viewed as an extension of (Guha et al., 2019) to multi-round FL, with higher-quality base models being sampled from a global distribution for more robust ensemble. Knowledge distillation was also used in (Li & Wang, 2019) and (Jeong et al., 2018) but for different purposes. Li & Wang (2019) performed ensemble distillation for each client, aiming to learn strong personalized models but not the global model. Jeong et al. (2018) aimed to speed up communication by sending averaged logits of clients’ data, not models, between clients and the server. The clients then use the aggregated logits to regularize local training via distillation. The accuracy, however, drops drastically compared to FEDAVG in exchange for faster communication. In contrast, we distill on the server using unlabeled data collected at the server, aiming to build a stronger global model. The most similar work to ours is a concurrent work by Lin et al. $( 2 0 2 0 ) ^ { 3 }$ , which also employs ensemble distillation on the server in a multi-round setting. Our work is notably different from all the above methods by taking the Bayesian perspective to sample better base models and investigating SWA for distillation, significantly improving the performance on multi-round FL.
+
+# 3 BAYESIAN MODEL ENSEMBLE FOR FEDERATED LEARNING
+
+# 3.1 BACKGROUND: FEDAVG
+
+Federated learning (FL) usually involves a server coordinating with many clients to jointly learn a global model without data sharing, in which FEDAVG (McMahan et al., 2017) in a standard approach. Denote by $s$ the set of clients, $\mathcal { D } _ { i } = \{ ( \boldsymbol { x } _ { n } , y _ { n } ) \} _ { n = 1 } ^ { N _ { i } }$ the labeled data of client $i$ , and $\bar { \mathbf { \Gamma } } _ { \bar { \mathbf { \Gamma } } } \bar { \mathbf { \Gamma } } _ { \bar { \mathbf { \Gamma } } }$ the weights of the current global model, FEDAVG starts with client training of all the clients in parallel, initializing each clients’ model ${ \pmb w } _ { i }$ with $\bar { \pmb w }$ and performing SGD for $K$ steps with a step size $\eta _ { l }$
+
+# Client training:
+
+$$
+\pmb { w } _ { i }  \pmb { w } _ { i } - \eta _ { l } \nabla \ell ( B _ { k } , \pmb { w } _ { i } ) , \mathrm { f o r } k = 1 , 2 , \cdots , K ,
+$$
+
+where $\ell$ is a loss function and $B _ { k }$ is the mini-batch sampled from $\mathcal { D } _ { i }$ at the $k$ th step. After receiving all the clients’ models $\{ w _ { i } ; i \in { \mathcal { S } } \}$ , given $\begin{array} { r } { \left. \mathcal { D } \right. = \sum _ { i } \mathsf { \bar { \vert } } \mathcal { D } _ { i } \vert } \end{array}$ , FEDAVG performs weight average to update the global model $\bar { \pmb w }$
+
+$$
+\mathbf { M o d e l \ a g g r e g a t i o n \ ( b y \ w e i g h t { a v e r a g e } ) } : \qquad \bar { w } \gets \sum _ { i } \frac { | { \mathcal { D } } _ { i } | } { | { \mathcal { D } } | } w _ { i } .
+$$
+
+With the updated global model $\bar { \mathbf { \Gamma } } _ { \bar { \mathbf { \Gamma } } } \bar { \mathbf { \Gamma } } _ { \bar { \mathbf { \Gamma } } }$ , FEDAVG then starts the next round of client training. The whole procedure of FEDAVG therefore iterates between Equation 1 and Equation 2, for $R$ rounds.
+
+In the case that $\mathcal { D } _ { i }$ is i.i.d. sampled from the aggregated data $\textstyle { \mathcal { D } } = \bigcup _ { i \in { \mathcal { S } } } { \mathcal { D } } _ { i }$ , FEDAVG has been shown convergent to the ideal model $\scriptstyle w ^ { \star }$ learned directly from $\mathcal { D }$ in a centralized manner (Stich, 2019; Haddadpour & Mahdavi, 2019; Khaled et al., 2020). In reality, however, the server has little control and knowledge about the clients. Each client may have different data distributions in the input (e.g., image distribution) or output (e.g., label distribution). Some clients may disconnect at certain rounds. All of these factors suggest the non-i.i.d. nature of federated learning in practice, under which the effectiveness of FEDAVG can largely degrade (Zhao et al., 2018; Li et al., $2 0 2 0 \mathrm { b }$ ; Hsu et al., 2019). For example, Karimireddy et al. (2020) show that $\bar { \pmb { w } }$ in Equation 2 can drift away from $\boldsymbol { w } ^ { \star }$ .
+
+# 3.2 A BAYESIAN PERSPECTIVE
+
+We propose to view the problem of model drift from a Bayesian perspective. In Bayesian learning, it is the posterior distribution $p ( \pmb { w } | \mathcal { D } )$ of the global model being learned, from which $\bar { \pmb w }$ and $\scriptstyle { { \pmb w } ^ { \star } }$ can be regarded as two particular samples (i.e., point estimates). Denote by $p ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { w } )$ the output probability of a global model $\pmb { w }$ , one approach to mitigate model drift is to perform Bayesian inference (Neal, 2012; Barber, 2012) for prediction, integrating the outputs of all possible models w.r.t. the posterior
+
+$$
+p ( y | \mathbf { x } ; \mathcal { D } ) = \int p ( y | \mathbf { x } ; \pmb { w } ) p ( \pmb { w } | \mathcal { D } ) d \pmb { w }
+$$
+
+rather than relying on a single point estimate. While Equation 3 is intractable in general, we can approximate it by the Monte Carlo method, sampling $M$ models for model ensemble
+
+$$
+p ( \boldsymbol { y } | \boldsymbol { x } ; \mathcal { D } ) \approx \frac { 1 } { M } \sum _ { m = 1 } ^ { M } p ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { w } ^ { ( m ) } ) , \mathrm { ~ w h e r e ~ } \boldsymbol { w } ^ { ( m ) } \sim p ( \boldsymbol { w } | \mathcal { D } ) .
+$$
+
+The question is: how to estimate $p ( \pmb { w } | \mathcal { D } )$ in federated learning, given merely client models $\{ w _ { i } \} ?$
+
+# 3.3 BAYESIAN MODEL ENSEMBLE WITH APPROXIMATED POSTERIORS
+
+We resort to a recently proposed idea, named stochastic weight average-Gaussian (SWAG) (Maddox et al., 2019), for estimating the posterior. SWAG employed a cyclical or constant learning rate in SGD, following SWA (Izmailov et al., 2018). SWAG then constructs a Gaussian distribution $\mathbf { \widetilde { \rho } } _ { p ( w | \mathcal { D } ) }$ by fitting the parameters to the model weights it traverses in SGD.
+
+In federated learning, by rewriting ${ \pmb w } _ { i }$ as $\bar { \mathbf { \pmb { w } } } - \mathbf { \nabla } _ { \mathbf { \lambda } }$ , where $\mathbf { } g _ { i } = - ( \boldsymbol { w } _ { i } - \bar { \boldsymbol { w } } )$ denotes the $K$ -step stochastic gradient on a mini-batch $\mathcal { D } _ { i } \subset \mathcal { D }$ (McMahan et al., 2017), we can indeed view each client’s model ${ \pmb w } _ { i }$ as taking $K$ -step SGD to traverse the weight space of global models.
+
+Gaussian. To this end, we propose to fit a diagonal Gaussian distribution $\begin{array} { r } { { \mathcal { N } } ( { \boldsymbol { \mu } } , { \boldsymbol { \Sigma } } _ { \mathrm { d i a g } } ) } \end{array}$ to the clients’ models $\{ w _ { i } \}$ following (Maddox et al., 2019),
+
+$$
+\mu = \sum _ { i } \frac { | { \mathcal D } _ { i } | } { | { \mathcal D } | } w _ { i } , \qquad \Sigma _ { \mathrm { d i a g } } = \mathrm { d i a g } \left( \sum _ { i } \frac { | { \mathcal D } _ { i } | } { | { \mathcal D } | } ( w _ { i } - \mu ) ^ { 2 } \right) ,
+$$
+
+from which we can sample $\{ \pmb { w } ^ { ( m ) } \sim \mathcal { N } ( \pmb { \mu } , \Sigma _ { \mathrm { d i a g } } ) \} _ { m = 1 } ^ { M }$ for model ensemble (cf. Equation 4). Here $( \cdot ) ^ { 2 }$ means taking element-wise square. We note that, both the clients’ models $\{ w _ { i } \}$ and FEDAVG $\bar { \pmb { w } }$ are possible samples from $\begin{array} { r } { { \mathcal { N } } ( { \boldsymbol { \mu } } , { \boldsymbol { \Sigma } } _ { \mathrm { d i a g } } ) } \end{array}$ .
+
+Dirichlet. We investigate another way to construct $p ( \pmb { w } | \mathcal { D } )$ , inspired by the fact that an averaged stochastic gradient is in general closer to the true gradient than individual stochastic gradients (Haddadpour & Mahdavi, 2019; Izmailov et al., 2018; Liang et al., 2019; Stich, 2019; Zhou & Cong, 2017). By viewing each client’s model as ${ \pmb w } _ { i } = \bar { \pmb w } - { \pmb g } _ { i }$ , such a fact suggests that a convex combination (i.e., weighted average) of clients’ models can lead to a better model than each client alone:
+
+$$
+w = \sum _ { i } \frac { \gamma _ { i } \lvert \mathcal { D } _ { i } \rvert } { \sum _ { i ^ { \prime } } \gamma _ { i ^ { \prime } } \lvert \mathcal { D } _ { i ^ { \prime } } \rvert } w _ { i } = \bar { w } - \sum _ { i } \frac { \gamma _ { i } \lvert \mathcal { D } _ { i } \rvert } { \sum _ { i ^ { \prime } } \gamma _ { i ^ { \prime } } \lvert \mathcal { D } _ { i ^ { \prime } } \rvert } g _ { i } ,
+$$
+
+where $\gamma = [ \gamma _ { 1 } , \cdot \cdot \cdot , \gamma _ { | S | } ] ^ { \top } \in \Delta ^ { | S | - 1 }$ is a vector on the $( | S | - 1 )$ -simplex. To this end, we use a Dirichlet distribution $\operatorname { D i r } ( \alpha )$ to model the distribution of $\gamma$ , from which we can then sample $\mathbf { \Delta } _ { \pmb { w } } ^ { ( m ) }$ by
+
+$$
+{ \pmb w } ^ { ( m ) } = \sum _ { i } \frac { \gamma _ { i } ^ { ( m ) } | { \mathcal D } _ { i } | } { \sum _ { i ^ { \prime } } \gamma _ { i ^ { \prime } } ^ { ( m ) } | { \mathcal D } _ { i ^ { \prime } } | } { \pmb w } _ { i } , \qquad \gamma ^ { ( m ) } \sim p ( \gamma ) = p ( \gamma _ { 1 } , \cdots , \gamma _ { | S | } ) = \frac { 1 } { \mathrm { B } ( { \alpha } ) } \prod _ { i } \gamma _ { i } ^ { \alpha _ { i } - 1 } ,
+$$
+
+where ${ \pmb { \alpha } } = [ \alpha _ { 1 } , \cdot \cdot \cdot , \alpha _ { | S | } ] ^ { \top } \succ { \bf 0 }$ is the parameter of a Dirichlet distribution, and $\mathbf { B } ( \alpha )$ is the multivariate beta function for normalization. We study different $_ { \pmb { \alpha } }$ in subsection C.1.
+
+To sum up, Bayesian model ensemble in federated learning takes the following two steps:
+
+• Construct $p ( \pmb { w } | \mathcal { D } )$ from the clients’ models $\{ w _ { i } \}$ (cf. Equation 5 or Equation 7) • Sample $\{ \pmb { w } ^ { ( m ) } \sim p ( \pmb { w } | \mathcal { D } ) \} _ { m = 1 } ^ { M }$ and perform ensemble (cf. Equation 4)
+
+Analysis. We validate Bayesian model ensemble with a three-class classification problem on the Swiss roll data in Figure 1 (a). We consider three clients with the same amount of training data: each has $8 0 \%$ data from one class and $2 0 \%$ from the other two classes, essentially a non-i.i.d. case. We apply FEDAVG to train a two-layer MLP for 10 rounds (each round with 2 epochs). We then show the test accuracy of models sampled from Equation 7 (with $\pmb { \alpha } = 0 . 5 \times \mathbf { 1 } _ { , }$ ) — the corners of the triangle (i.e., $\Delta ^ { 2 \cdot }$ ) in Figure 1 (b) correspond to the clients; the position inside the triangle corresponds to the $\gamma$ coefficients. We see that, the sampled models within the triangle usually have higher accuracy than the clients’ models. Surprisingly, the best performing model that can be sampled from a Dirichlet distribution is not FEDAVG (the center of the triangle), but the one drifting to the bottom right. This suggests that Bayesian model ensemble can lead to higher accuracy (by averaging over sampled models) than FEDAVG alone. Indeed, by combining 10 randomly sampled models via Equation 4, Bayesian model ensemble attains a $6 9 \%$ test accuracy, higher than $6 4 \%$ by FEDAVG. Figure 1 (c) further shows that the sampled models have a better alignment between the prediction confidence and accuracy than the clients’ (mean results of 3 clients or samples). See subsection C.1 for details.
+
+![](images/8b5e474b48abbbf749844dae8fa8f49b5d46c87bd69ff5f0e480c6d8e700aa98.jpg)  
+Figure 1: An illustration of models that can be sampled from a Dirichlet distribution (Equation 7). (a) A three-class toy data with three clients, each has non-i.i.d. imbalanced data. (b) We show the sampled model’s corresponding $\gamma$ (position in the triangle) and its test accuracy (color). FEDAVG is at the center; clients’ models are at corners. The best performing model (star) is not at the center, drifting away from FEDAVG. (c) Histograms of (in)correctly predicted examples at different confidences ( $\mathbf { \Delta x }$ -axis) by sampled models and clients.
+
+To further compare FEDAVG and ensemble, we linearize $p ( \boldsymbol { y } | \boldsymbol { x } ; \cdot )$ at $\bar { \pmb w }$ (Izmailov et al., 2018),
+
+$$
+p ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { w } ^ { ( m ) } ) = p ( \boldsymbol { y } | \boldsymbol { x } ; \bar { \boldsymbol { w } } ) + \langle \nabla p ( \boldsymbol { y } | \boldsymbol { x } ; \bar { \boldsymbol { w } } ) , \Omega ^ { ( m ) } \rangle + O ( \| \Omega ^ { ( m ) } \| ^ { 2 } ) ,
+$$
+
+where $\Omega ^ { ( m ) } = \pmb { w } ^ { ( m ) } - \bar { \pmb { w } }$ and $\langle \cdot , \cdot \rangle$ is the dot product. By averaging the sampled models, we arrive at
+
+$$
+\frac { 1 } { M } \sum _ { m } p ( y | x ; w ^ { ( m ) } ) - p ( y | x ; \bar { w } ) = \langle \nabla p ( y | x ; \bar { w } ) , \frac { 1 } { M } \sum _ { m } \Omega ^ { ( m ) } \rangle + O ( \Omega ^ { 2 } ) = O ( \Omega ^ { 2 } ) ,
+$$
+
+where $\Omega = \operatorname* { m a x } _ { m } \left. \Omega ^ { ( m ) } \right.$ . In federated learning, especially in the non-i.i.d. cases, $\Omega$ can be quite large. Bayesian ensemble thus can have a notable difference (improvement) compared to FEDAVG.
+
+# 4 FEDBE
+
+Bayesian model ensemble, however, cannot directly benefit multi-round federated learning, in which a single global model must be sent back to the clients to continue client training. We must translate the prediction rule of Bayesian model ensemble into a single global model.
+
+Server input :initial global model $\pmb { w }$ , SWA scheduler ηSWA, unlabeled data $\mathcal { U } = \{ \pmb { x } _ { j } \} _ { j = 1 } ^ { J }$ ;   
+Client i’s input :local step size $\eta _ { l }$ , local labeled data $\mathcal { D } _ { i }$ ;   
+for $r \gets 1$ to $R$ do Sample clients ${ \mathcal { S } } \subseteq \{ 1 , \cdots , N \}$ ; Communicate $\textbf { \em w }$ to all clients $i \in S$ ; for each client $i \in S$ in parallel do Initialize local model $\mathbf { \Delta } \mathbf { \psi } \mathbf { \Sigma } \mathbf { w } _ { i } \gets \mathbf { \Delta } \mathbf { w }$ ; ${ \boldsymbol { w } } _ { i } \gets$ Client trainin $\mathbf { \Omega } _ { \mathfrak { s } } ( \mathfrak { w } _ { i } , \mathcal { D } _ { i } , \eta _ { l } )$ ; [Equation 1] Communicate ${ \pmb w } _ { i }$ to the server; end Construct $\begin{array} { r } { \pmb { \bar { w } } = \sum _ { i \in \pmb { S } } \frac { | \mathcal { D } _ { i } | } { \sum _ { i ^ { \prime } \in \pmb { S } } | \mathcal { D } _ { i } ^ { \prime } | } \pmb { w } _ { i } } \end{array}$ ; Construct global model distribution $p ( \pmb { w } | \mathcal { D } )$ from $\{ w _ { i } ; i \in \mathcal { S } \}$ ; [Equation 5 or Equation 7] Sample $M$ global models $\{ \pmb { w } ^ { ( m ) } \sim p ( \pmb { w } | \mathcal { D } ) \} _ { m = 1 } ^ { M }$ ; Construct {w(m0)}M0m0= {w¯} ∪ {wi; i ∈ S} ∪ {w(m)}Mm= ; Construct $\mathcal { T } = \{ \pmb { x } _ { j } , \pmb { \hat { p } } _ { j } \} _ { j = 1 } ^ { J }$ , where pˆj = 10 Pm0 p(y|xj ; w(m0)); [Equation 4] Knowledge distillation: ${ \pmb w } \gets \mathrm { S W A } ( { \bar { \pmb w } } , { \mathcal T } , \eta _ { \mathrm { S W A } } )$ ;   
+end   
+Server output : $\textbf { \em w }$ .
+
+To this end, we make an assumption that we can access a set of unlabeled data $\mathcal { U } = \{ \pmb { x } _ { j } \} _ { j = 1 } ^ { J }$ at the server. This can easily be satisfied since collecting unlabeled data is simpler than labeled ones. We use $\mathcal { U }$ for two purposes. On one hand, we use $\mathcal { U }$ to memorize the prediction rule of Bayesian model ensemble, turning $\mathcal { U }$ into a pseudo-labeled set $\mathcal { T } = \{ ( \pmb { x } _ { j } , \bar { \hat { p } _ { j } } ) \} _ { j = 1 } ^ { J } ,$ where a prob $\begin{array} { r } { \hat { \pmb { p } } _ { j } = \frac { 1 } { M } \sum _ { m = 1 } ^ { M } p ( y | \pmb { x } _ { j } ; \pmb { w } ^ { ( m ) } ) } \end{array}$ isnd, we use $\tau$ as supervision to train a global model $\textbf { \em w }$ , aiming to mimic the prediction rule of Bayesian model ensemble on $\mathcal { U }$
+
+This process is reminiscent of knowledge distillation (Hinton et al., 2015) to transfer knowledge from a teacher model (in our case, the Bayesian model ensemble) to a student model (a single global model). Here we apply a cross entropy loss to learn $\begin{array} { r } { \pmb { w } : - \frac { 1 } { J } \sum _ { j } \hat { \pmb { p } } _ { j } ^ { \top } \log ( p ( y | \pmb { x } _ { j } ; \pmb { w } ) ) } \end{array}$ .
+
+SWA for knowledge distillation. Optimizing $\textbf { \em w }$ using standard SGD, however, may arrive at a suboptimal solution: the resulting $\pmb { w }$ can have much worse test accuracy than ensemble. We identify one major reason: the ensemble prediction $\hat { p } _ { j }$ can be noisy (e.g., arg $\operatorname* { m a x } _ { c } \hat { p } _ { j } [ c ]$ is not the true label of $\boldsymbol { \mathscr { x } } _ { j }$ ), especially in the early rounds of FL. The student model $\pmb { w }$ thus may over-fit the noise. We note that, this finding does not contradict our observations in subsection 3.3: Bayesian model ensemble has higher test accuracy than FEDAVG but is still far from being perfect (i.e., $1 0 0 \%$ accuracy). To address this issue, we apply SWA (Izmailov et al., 2018) to train $\pmb { w }$ . SWA employs a cyclical learning rate schedule in SGD by periodically imposing a sharp increase in step sizes and averages the weights of models it traverses, enabling $\pmb { w }$ to jump out of noisy local minimums. As will be shown in section 5, SWA consistently outperforms SGD in distilling the ensemble predictions into the global model.
+
+We name our algorithm FEDBE (Federated Bayesian Ensemble) and summarize it in algorithm 1. While knowledge distillation needs extra computation at the server, it is hardly a concern as the server is likely computationally rich. (See subsection D.1 for details.) We also empirically show that a small number of sampled models (e.g., $M = 1 0 \sim 2 0$ ) are already sufficient for FEDBE to be effective.
+
+# 5 EXPERIMENT
+
+# 5.1 SETUP (MORE DETAILS IN APPENDIX B)
+
+Datasets, models, and settings. We use CIFAR-10/100 (Krizhevsky et al., 2009), both contain 50K training and 10K test images, from 10 and 100 classes. We also use Tiny-ImageNet (Le & Yang, 2015), which has 500 training and 50 test images per class for 200 classes. We follow (McMahan et al., 2017) to use a ConvNet (LeCun et al., 1998) with 3 convolutional and 2 fully-connected layers. We also use ResNet-{20, 32, 44, 56} (He et al., 2016) and MobileNetV2 (Sandler et al., 2018). We split part of the training data to the server as the unlabeled data, distribute the rest to the clients, and evaluate on the test set. We report mean $\pm$ standard deviation (std) over five times of experiments.
+
+Implementation details. As mentioned in (McMahan et al., 2017; Wang et al., 2020; Li et al., 2020b), FEDAVG is sensitive to the local training epochs $E$ per round $\begin{array} { r } { \langle E = \lceil \frac { K \mid B _ { K } \mid } { \mid \mathcal { D } _ { i } \mid } \rceil } \end{array}$ in Equation 1). Thus, in each experiment, we first tune $E$ from $[ 1 , 5 , 1 0 , 2 0 , 3 0 , 4 0 ]$ for FEDAVG and adopt the same $E$ to FEDBE. Li et al. (2020b); Reddi et al. (2020) suggested that the local step size $\eta _ { l }$ (see Equation 1) must decay along the communication rounds in non-i.i.d. settings for convergence. We set the initial $\eta _ { l }$ as 0.01 and decay it by 0.1 at $30 \%$ and $60 \%$ of total rounds, respectively. Within each round of local training, we use SGD optimizer with weight decay and a 0.9 momentum and impose no further decay on local step sizes. Weight decay is crucial in local training (cf. subsection B.3). For ResNet and MobileNetV2, we use batch normalization (BN). See subsection C.5 for a discussion on using group normalization (GN) (Wu & He, 2018; Hsieh et al., 2020), which converges much slower.
+
+Baselines. Besides FEDAVG, we compare to one-round training with 200 local epochs followed by model ensemble at the end (1-Ensemble). We also compare to vanilla knowledge distillation $\mathbf { \widetilde { v } }$ -Distillation), which performs ensemble directly over clients’ models and uses a SGD momentum optimizer (with a batch size of 128 for 20 epochs) for distillation in each round. For fast convergence, we initialize distillation with the weight average of clients’ models and sharpen the pseudo label as $\begin{array} { r } { \hat { p } _ { j } [ c ]  \hat { p } _ { j } [ c ] ^ { 2 } / \sum _ { c ^ { \prime } } \hat { p } _ { j } [ c ^ { \prime } ] ^ { 2 } } \end{array}$ , similar to (Berthelot et al., 2019). We note that, v-Distillation is highly similar to (Lin et al., 2020) except for different hyper-parameters. We also compare to FEDPROX (Li et al., 2020a) and FEDAVGM (Hsu et al., 2019) on better local training and using server momentum.
+
+FEDBE. We focus on Gaussian (cf. Equation 5). Results with Dirichlet distributions are in subsection C.1. We sample $M { = } 1 0$ models and combine them with the weight average of clients and individual clients for ensemble. For distillation, we apply SWA (Izmailov et al., 2018), which uses a cyclical schedule with the step size $\eta _ { \mathrm { S W A } }$ decaying from 1e−3 to 4e−4, and collect models at the end of every cycle (every 25 steps) after the $2 5 0 \mathrm { t h }$ step. We follow other settings of v-Distillation (e.g., distill for 20 epochs per round). We average the collected models to obtain the global model.
+
+5.2 MAIN STUDIES: CIFAR-10 WITH NON-I.I.D. CLIENTS USING DEEP NEURAL NETWORKS
+
+Setup. We focus on CIFAR-10. We randomly split 10K training images to be the unlabeled data at the server. We distribute the remaining images to 10 clients with two non-i.i.d. cases. Step: Each client has 8 minor classes with 10 images per class, and 2 major classes with 1,960 images per class, inspired by (Cao et al., 2019). Dirichlet: We follow (Hsu et al., 2019) to simulate a heterogeneous partition for $N$ clients on $C$ classes. For class $c$ , we draw a $N$ -dim vector $\pmb { q } _ { c }$ from $\operatorname { D i r } ( 0 . 1 )$ and assign data to client $n$ proportionally to ${ \mathbf { } q } _ { c } [ n ]$ . The clients have different numbers of total images.
+
+Table 1: Mean±std of test accuracy $( \% )$ on non-i.i.d. CIFAR-10. $\star :$ : trained with 50K images without splitting.   
+
+<table><tr><td>Non-i.i.d. Type]</td><td>Method</td><td>ConvNet</td><td>ResNet20</td><td>ResNet32</td><td>ResNet44</td><td>ResNet56</td></tr><tr><td rowspan="5">Step</td><td>1-Ensemble</td><td>60.5±0.28</td><td>49.9±0.46</td><td>35.5±0.50</td><td>32.8±0.38</td><td>23.3±0.52</td></tr><tr><td>FEDAVG</td><td>72.0±0.25</td><td>70.2±0.17</td><td>66.5±0.36</td><td>60.5±0.26</td><td>51.4±0.15</td></tr><tr><td>v-Distillation</td><td>69.2±0.18</td><td>72.6±0.62</td><td>68.4±0.33</td><td>60.4±0.53</td><td>56.4±1.10</td></tr><tr><td>FEDBE (w/o SWA)</td><td>72.1±1.21</td><td>74.9±1.41</td><td>71.1±0.75</td><td>61.0±0.75</td><td>56.6±0.85</td></tr><tr><td>FEDBE</td><td>74.5±0.51 77.5±0.42</td><td></td><td>72.7±0.27</td><td>65.5±0.32</td><td>60.7±0.45</td></tr><tr><td rowspan="5">Dirichlet</td><td>1-Ensemble</td><td>63.3±0.56 45.2±1.06</td><td></td><td>39.5±0.78</td><td>31.5±0.77 27.2±0.65</td><td></td></tr><tr><td>FEDAVG</td><td>72.3±0.12</td><td>74.4±0.36</td><td>73.4±0.23</td><td></td><td>67.1±0.54 62.2±0.45</td></tr><tr><td>v-Distillation</td><td>67.7±0.98 73.1±0.78 70.8±0.64 66.9±0.85 62.8±0.66</td><td></td><td></td><td></td><td></td></tr><tr><td>FEDBE (w/o SWA)</td><td>70.1±0.42</td><td>75.9±0.56</td><td>73.9±0.55</td><td>68.2±0.72</td><td>63.2±0.71</td></tr><tr><td>FEDBE</td><td>73.9±0.45 78.2±0.36 77.7±0.45 71.5±0.38 67.0±0.30</td><td></td><td></td><td></td><td></td></tr><tr><td>Centralized*</td><td>SGD</td><td>84.5</td><td>91.7</td><td>92.6</td><td>93.1</td><td>93.4</td></tr></table>
+
+Table 2: Compatibility of FEDBE with FEDAVGM and FEDPROX on non-i.i.d. CIFAR-10.   
+
+<table><tr><td>Non-i.i.d. Type Method</td><td></td><td>ConvNet1</td><td>ResNet20</td><td>ResNet32</td><td>ResNet44</td><td>ResNet56</td></tr><tr><td rowspan="4">Step</td><td>FEDPROX</td><td></td><td></td><td></td><td></td><td>72.5±0.71 71.1±0.52 67.7±0.26 60.4±0.71 54.9±0.66</td></tr><tr><td> FEDBE +FEDPROX</td><td>74.9±0.38 77.7±0.45 72.9±0.44 64.5±0.37 60.1±0.62</td><td></td><td></td><td></td><td></td></tr><tr><td>FEDAVGM</td><td>72.3±0.55 73.2±0.57 70.0±0.62 59.9±0.65 52.7±0.49</td><td></td><td></td><td></td><td></td></tr><tr><td>FEDBE +FEDAVGM</td><td>74.5±0.47 78.0±0.46 73.6±0.50 65.5±0.40 59.7±0.51</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="4">Dirichlet</td><td>FEDPROX</td><td></td><td></td><td></td><td></td><td>72.6±0.38 76.1±0.49 73.4±0.51 68.1±0.79 60.9±0.46</td></tr><tr><td>FEDBE +FEDPROX</td><td>74.6±0.35 78.7±0.497</td><td></td><td>77.3±0.60 71.7±0.43 66.5±0.41</td><td></td><td></td></tr><tr><td>FEDAVGM</td><td>73.0±0.4376.5±0.4475.5±0.79 67.7±0.46 58.9±0.72</td><td></td><td></td><td></td><td></td></tr><tr><td> FEDBE +FEDAVGM</td><td>74.4±0.49 78.5±0.66 78.5±0.26 72.0±0.51 67.0±0.55</td><td></td><td></td><td></td><td></td></tr></table>
+
+Results. We implement all methods with 40 rounds4, except for the one-round Ensemble. We assume that all clients are connected at every round. We set the local batch size as 40. Table 1 summarizes the results. FEDBE outperforms the baselines by a notable margin. Compared to FEDAVG, FEDBE consistently leads to a $2 \sim 9 \%$ gain, which becomes larger as the network goes deeper. By comparing FEDBE to FEDBE (w/o SWA) and v-Distillation, we see the consistent improvement by SWA for distillation and Bayesian ensemble with sampled models. We note that, FEDAVG outperforms 1-Ensemble and is on a par with v-Distillation5, justifying (a) the importance of multi-round training; (b) the challenge of ensemble distillation. Please see subsection C.2 for an insightful analysis.
+
+Compatibility with existing efforts. Our improvement in model aggregation is compatible with recent efforts in better local training (Li et al., 2020a; Karimireddy et al., 2020) and using server momentum (Reddi et al., 2020; Hsu et al., 2019). Specifically, Reddi et al. (2020); Hsu et al. (2019) applied the server momentum to FEDAVG by treating FEDAVG in each round as a step of adaptive optimization. FEDBE can incorporate this idea by initializing distillation with their FEDAVG. Table 2 shows the results of FEDPROX (Li et al., 2020a) and FEDAVGM (Hsu et al., 2019), w/ or w/o FEDBE. FEDBE can largely improve them. The combination even outperforms FEDBE alone in many cases.
+
+Effects of Bayesian Ensemble. We focus on the Step setting. We compare different combinations of client models C: $\{ w _ { i } \}$ , client weight average A: $\bar { \pmb w }$ , and $M$ samples from Gaussian S: $\{ \pmb { w } ^ { ( m ) } \} _ { m = 1 } ^ { M }$ to construct the distillation targets $\tau$ for FEDBE in algorithm 1. As shown in Table 3, sampling global models for Bayesian ensemble improves the accuracy. Sampling $M = 1 0 \sim 2 0$ samples (plus weight average and clients to form ensemble) is sufficient to make FEDBE effective (see Figure 2).
+
+Table 3: FEDBE distillation targets. A: client average; C: clients; S: samples.   
+
+<table><tr><td>Distillation Targets丨ConvNet</td><td>tResNet20</td></tr><tr><td>s</td><td>72.6±0.28 73.4±0.46 73.1±0.46 75.2±0.61</td></tr><tr><td>S+A A+C</td><td>[73.9±0.33 76.1±0.47 73.0±0.3675.4±0.38</td></tr><tr><td>s+c S+A+C</td><td>74.0±0.66 77.9±0.56 77.5±0.42</td></tr><tr><td>Ground-truth labels|76.6±0.21</td><td>|74.5±0.51 80.2±0.23</td></tr></table>
+
+![](images/b5f32516f507e955c865a4efba4f0b2876b8af801f70cf43439f7aea5caaf110.jpg)  
+Figure 2: # of sampled models in FEDBE.
+
+![](images/820658658c106bfb911d5e5a9158770c019866551e8ab83365373652cc0cbd4f.jpg)  
+Figure 3: FEDAVG while monitoring the Bayesian ensemble.
+
+![](images/ef1e9939094a8ce4960071426ac567a42ab57e549476de74f58059bc2342d176.jpg)  
+Figure 4: # of layers (ConvNet). GT: with ground-truth targets.
+
+Table 4: FEDBE on non-i.i.d CIFAR-10 with different unlabeled data $\mathcal { U }$   
+
+<table><tr><td>Non-i.i.d. Type</td><td>u</td><td>|u</td><td>ConvNet</td><td>ResNet20</td><td>ResNet32</td><td>ResNet44</td><td>ResNet56</td></tr><tr><td rowspan="3">Step</td><td>CIFAR-10</td><td>10K</td><td>74.5±0.51</td><td>77.5±0.42</td><td>72.7±0.27</td><td>65.5±0.32</td><td>60.7±0.45</td></tr><tr><td>CIFAR-100</td><td>50K</td><td>74.4±0.45</td><td>78.2±0.58</td><td>72.2±0.35</td><td>65.1±0.37</td><td>61.0±0.49</td></tr><tr><td>Tiny-ImageNet</td><td>100K</td><td>74.5±0.64</td><td>77.1±0.51</td><td>72.3±0.43</td><td>64.5±0.51</td><td>60.9±0.32</td></tr><tr><td rowspan="3">Dirichlet</td><td>CIFAR-10</td><td>10K</td><td>73.9±0.45</td><td>78.2±0.36</td><td></td><td>77.7±0.45 71.5±0.38</td><td>67.0±0.30</td></tr><tr><td>CIFAR-100</td><td>50K</td><td>73.5±0.41</td><td>78.6±0.6376.5±0.61 72.0±0.71 66.9±0.57</td><td></td><td></td><td></td></tr><tr><td>Tiny-ImageNet</td><td>100K</td><td></td><td></td><td></td><td></td><td>74.0±0.35 78.2±0.72 76.7±0.52 71.6±0.66 67.3±0.32</td></tr></table>
+
+![](images/adfcf6e37c0e0a0345d70a245511f5bd209d74ccd1379009fc2acb379274002c.jpg)  
+Figure 5: Effects on varying the size and domains of the server dataset on CIFAR-10 experiments.
+
+Bayesian model ensemble vs. weight average for prediction. In Figure 3, we perform FEDAVG and show the test accuracy at every round, together with the accuracy by Bayesian model ensemble, on the Step-non-i.i.d CIFAR-10 experiment using ResNet20. That is, we take the clients’ models learned with FEDAVG to construct the distribution, sample models from it, and perform ensemble for the predictions. Bayesian model ensemble outperforms weight average at nearly all the rounds, even though it is noisy (i.e., not with $1 0 0 \%$ accuracy).
+
+Effects of unlabeled data. FEDBE utilizes unlabeled data $\mathcal { U }$ to enable knowledge distillation. Figure 5a studies the effect of $| \mathcal { U } |$ : we redo the same Step experiments but keep 25K training images away from clients and vary $| \mathcal { U } |$ in the server. FEDBE outperforms FEDAVG even with 1K unlabeled dataset (merely $4 \%$ of the total client data). We note that, FEDAVG (ResNet20) trained with the full 50K images only reaches $7 2 . 5 \%$ , worse than FEDBE, justifying that the gain by FEDBE is not simply from seeing more data. Adding more unlabeled data consistently but slightly improve FEDBE.
+
+We further investigate the situation that the unlabeled data come from a different domain or task. This is to simulate the cases that (a) the server has little knowledge about clients’ data and (b) the server cannot collect unlabeled data that accurately reflect the test data. In Table 4, we replace the unlabeled data to CIFAR-100 and Tiny-ImageNet. The accuracy matches or even outperforms using CIFAR-10, suggesting that out-of-domain unlabeled data are sufficient for FEDBE. The results also verify that FEDBE uses unlabeled data mainly as a medium for distillation, not a peep at future test data.
+
+In Figure 5b and Figure 5c, we investigate different sizes of CIFAR-100 or Tiny-ImageNet as the unlabeled data (cf. Table 4). We found that even with merely 2K unlabeled data, which is $5 \%$ of the total 40K CIFAR-10 labeled data and $2 \sim 4 \%$ of the original 50K-100K unlabeled data, FEDBE can already outperform FedAvg by a margin. This finding is aligned with what we have included in Figure 5a, where we showed that a small amount of unlabeled data is sufficient for FEDBE to be effective. Adding more unlabeled data can improve the accuracy but the gain is diminishing.
+
+Network depth. Unlike in centralized training that deeper models usually lead to higher accuracy (bottom row in Table 1, trained with 200 epochs), we observe an opposite trend in FL: all methods suffer accuracy drop when ResNets go deeper. This can be attributed to (a) local training over-fitting to small and non-i.i.d. data or (b) local models drifting away from each other. FEDBE suffers the least among all methods, suggesting it as a promising direction to resolve the problem. To understand the current limit, we conduct a study in Figure 4 by injecting more convolutional layers into ConvNet (Step setting). FEDAVG again degrades rapidly, while FEDBE is more robust. If we replace Bayesian ensemble by the CIFAR-10 ground truth labels as the distillation target, FEDBE improves with more layers added, suggesting that how to distill with noisy labeled targets is the key to improve FEDBE.
+
+Table 5: Partial participation (Tiny-ImageNet)   
+
+<table><tr><td></td><td>Method ResNet20</td><td>MobileNetV2</td></tr><tr><td rowspan="2">i.i.d</td><td>FEDAVG</td><td>32.4±0.68 26.1±0.98</td></tr><tr><td>FEDBE 35.4±0.58</td><td>28.9±1.15</td></tr><tr><td>non-i.i.d</td><td>FEDAVG FEDBE</td><td>127.5±0.78 25.5±1.23 32.4±0.81 27.8±0.99</td></tr></table>
+
+Table 6: Systems heterogeneity (non-i.i.d. CIFAR-10)   
+
+<table><tr><td>Method</td><td>ConvNet ResNet20</td></tr><tr><td>FEDAVG</td><td>70.6±0.46 69.9±0.59 64.0±0.50</td></tr><tr><td>FEDPROX</td><td>71.2±0.55 69.4±0.48 65.9±0.63</td></tr><tr><td>FEDBE</td><td>73.3±0.56 77.1±0.61 70.2±0.39</td></tr><tr><td>+FEDPROX</td><td>73.7±0.24 77.5±0.51 71.6±0.37</td></tr></table>
+
+# 5.3 PRACTICAL FEDERATED SYSTEMS
+
+Partial participation. We examine FEDBE in a more practical environment: (a) more clients are involved, (b) each client has fewer data, and (c) not all clients participate in every round. We consider a setting with 100 clients, in which 10 clients are randomly sampled at each round and iterates for 100 rounds, similar to (McMahan et al., 2017). We study both i.i.d. and non-i.i.d (Step) cases on Tiny-ImageNet, split 10K training images to the server, and distribute the rest to the clients. For the non-i.i.d case, each client has 2 major classes (351 images each) and 198 minor classes (1 image each). FEDBE outperforms FEDAVG (see Table 5). See subsection C.7 for results on CIFAR-100.
+
+Systems heterogeneity. In real-world FL, clients may have different computation resources, leading to systems heterogeneity (Li et al., 2020a). Specifically, some clients may not complete local training upon the time of scheduled aggregation, which might hurt the overall aggregation performance. We follow (Li et al., 2020a) to simulate the situation by assigning each client a local training epoch $E _ { i }$ , sampled uniformly from $( 0 , 2 0 ]$ , and aggregate their partially-trained models.Table 6 summarizes the results on non-i.i.d (Step) CIFAR-10. FEDBE outperforms FEDAVG and FEDPROX (Li et al., 2020a).
+
+# 6 DISCUSSION
+
+Privacy. Federated learning offers data privacy since the server has no access to clients’ data. It is worth noting that having unlabeled data not collected from the clients does not weaken the privacy if the server is benign, which is a general premise in the federated setting (McMahan et al., 2017). For instance, if the collected data are de-identified and the server does not intend to match them to clients, clients’ privacy is preserved. In contrast, if the server is adversarial and tries to infer clients’ information, federated learning can be vulnerable even without the unlabeled data: e.g., federated learning may not satisfy the requirement of differential privacy or robustness to membership attacks.
+
+Unlabeled data. Our assumption that the server has data is valid in many cases: e.g., a self-driving car company may collect its own data but also collaborate with customers to improve the system. Bassily et al. (2020a;b) also showed real cases where public data is available in differential privacy.
+
+# 7 CONCLUSION
+
+Weight average in model aggregation is one major barrier that limits the applicability of federated learning to i.i.d. conditions and simple neural network architectures. We address the issue by using Bayesian model ensemble for model aggregation, enjoying a much robust prediction at a very low cost of collecting unlabeled data. With the proposed FEDBE, we demonstrate the applicability of federated learning to deeper networks (i.e., ResNet20) and many challenging conditions.
+
+# ACKNOWLEDGMENTS
+
+We are thankful for the generous support of computational resources by Ohio Supercomputer Center and AWS Cloud Credits for Research.
+
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+# SUPPLEMENTARY MATERIAL
+
+We provide details omitted in the main paper.
+
+• Appendix A: additional related work (cf. section 2 of the main paper).   
+• Appendix B: details of experimental setups (cf. subsection 5.1 of the main paper).   
+• Appendix C: additional experimental results and analysis (cf. subsection 5.2 of the main paper).   
+• Appendix D: additional discussions (cf. section 4 and subsection 5.2 of the main paper).   
+• Appendix E: additional analysis to address reviewers’ comments.
+
+# A ADDITIONAL RELATED WORK
+
+Federated leatning (FL). In the multi-round federated setting, FEDAVG (McMahan et al., 2017) is the standard approach. Many works have studied its effectiveness and limitation regarding convergence (Khaled et al., 2020; Li et al., 2020b; Karimireddy et al., 2020; Li et al., 2020b; Liang et al., 2019; Stich, 2019; Zhao et al., 2018; Zhou & Cong, 2017; Haddadpour & Mahdavi, 2019), system robustness (Li et al., 2020a; Smith et al., 2017; Bonawitz et al., 2019), and communication cost (Konecnˇ y et al., 2016; Reisizadeh et al., 2019), especially for the situations that clients are not \` i.i.d. (Li et al., 2020b; Zhao et al., 2018; Li et al., 2020a; Sahu et al., 2018) and have different data distributions, stability, etc.
+
+Ensemble learning and stochastic weight average. Recent works (Huang et al., 2017; Draxler et al., 2018; Garipov et al., 2018) have developed efficient ways to obtain the base models for ensemble; e.g., by employing a dedicated learning rate schedule to sample models along a single pass of SGD training (Hsu et al., 2019). SWA (Maddox et al., 2019) applied the same learning rate schedule but simply took weight average over the base models to obtain a single strong model. We apply SWA, but for the purpose of learning with noisy labels in knowledge distillation.
+
+Bayesian deep learning. Bayesian approaches (Neal, 2012; Barber, 2012; Brochu et al., 2010) incorporate uncertainty in decision making by placing a distribution over model weights and marginalizing these models to form a whole predictive distribution. Our work is inspired by (Maddox et al., 2019), which constructs the distribution by fitting the parameters to traversed models along SGD training.
+
+Others. Our work is also related to semi-supervised learning (SSL) and unsupervised domain adaptation (UDA). SSL leverages unlabeled data to train a better model when limited labeled data are provided (Grandvalet & Bengio, 2005; Kingma et al., 2014; Tarvainen & Valpola, 2017; Berthelot et al., 2019; Zhu, 2005); UDA leverages unlabeled data to adapt a model trained from a source domain to a different but related target domain (Gong et al., 2012; 2014; Ganin et al., 2016; Saito et al., 2018; Ben-David et al., 2010). We also leverage unlabeled data, but for model aggregation. We note that, UDA and SLL generally assume the access to labeled (source) data, which is not the case in federated learning: the server cannot access clients’ labeled data.
+
+# B EXPERIMENTAL SETUPS
+
+# B.1 IMPLEMENTATION DETAILS
+
+As mentioned in the main paper (subsection 5.1), we select the number of local epochs, used in training a client model within one round of communication, according to the performance of FEDAVG. Other algorithms, like FEDBE and v-Distillation, then follow the same numbers. For ConvNet and ResNet experiments, we fixed $E = 2 0$ . We use $E = 1 0$ for MobileNet experiments. For all 1-Ensemble baselines, we tuned $E$ from $[ 1 0 , 2 0 , . . . , 2 0 0 ]$ .
+
+We observed that applying weight decay in local client training improves all FL methods, but the suitable hyper-parameter can be different for different methods on different network architectures. Tuning it specifically for each method is thus essential for fair comparisons. We search the weight decay hyper-parameter for each network and each method in [1e−3, 1e−4] with a validation set.
+
+For methods with distillation (FEDBE and v-Distillation), We tune the epochs for v-Distillation from [1, 5, 10, 20, 30, 40] and find 20 to be stable across different setups. We apply 20 to FEDBE as well.
+
+In constructing the pseudo-labeled data $\tau$ , we perform inference on the unlabeled data $\mathcal { U }$ in the server without data augmentation. We perform data augmentation in both local training on $\mathcal { D } _ { i }$ and knowledge distillation on $\tau$ . The $3 2 \times 3 2$ CIFAR images are padded 2 pixels each side, randomly flipped horizontally, and then randomly cropped back to $3 2 \times 3 2$ . The $6 4 \times 6 4$ Tiny-ImageNet images are padded 4 pixels each side, randomly flipped horizontally, and then randomly cropped back to $6 4 \times 6 4$ . In Table 4 of the main paper, we resize images of Tiny-ImageNet to $3 2 \times 3 2$ .
+
+For neural networks that contain batch normalization layers (Ioffe & Szegedy, 2015), we apply the same way as in section 3 to construct the global distribution for the layers and we observe no issues in our experiments. SWAG (Maddox et al., 2019) also reported that it performs stably even on very deep networks.
+
+# B.2 TRAINING FEDAVG
+
+For the local learning rate $\eta _ { l }$ , we observed that an appropriate value of $\eta _ { l }$ is important when training on the non-i.i.d local data. The local models cannot converge if the learning rate is set to a too large value (also shown in (Reddi et al., 2020)), and the models cannot reach satisfying performance within the local epochs $E$ with a too-small value as shown in Figure 6. Also, unlike the common practice of training a neural network with learning rate decay in the centralized setting, we observed that applying it within each round of local training (decay by 0.99 every step) results in much worse client models. The local models would need a large enough learning rate to converge with a fixed $E$ and we use 0.01 as the base learning rate for local training in all our experiments.
+
+Although decaying $\eta _ { l }$ within each round of local training is not helpful, decaying $\eta _ { l }$ along the rounds of communication could improve the performance. Wang et al. (2020) and Reddi et al. (2020) provided both theoretical and empirical studies suggesting that the local learning rate must decay along the communication rounds to alleviate client shifts in non-i.i.d setting. In our experiments, at the $r$ th round of communication, the local client training starts with a learning rate $\eta _ { l }$ , which is 0.01 if $r < 0 . 3 R$ , 0.001 if $0 . 3 R \le r < 0 . 6 R$ , and 0.0001 otherwise, where $R$ is the total rounds of communication. In Figure 7, we examined this schedule with different degrees of $\alpha$ in the Dirichlet-non-i.i.d setting. We observed consistent improvements and applied it to all our experiments in section 5.
+
+![](images/df7fe25bf28c31a93028abdade16061c6d3fd5697a3b25a3f2bb41a653b94c7a.jpg)  
+Figure 6: FEDAVG with ConvNet on Step-noni.i.d CIFAR-10 with or without learning rate decay within each round of local training.
+
+![](images/42865e7d930b8b983c97e1070af833fcac54d409e24ce10a306f3f41119bdbc9.jpg)  
+Figure 7: FEDAVG with ConvNet on Dirichletnon-i.i.d CIFAR-10 with or without learning rate decay at latter rounds of communication. We experimented with different values of $\alpha$ in $\operatorname { D i r } ( \alpha )$ .
+
+# B.3 EFFECTS OF WEIGHT DECAY IN LOCAL CLIENT TRAINING
+
+Federated learning on non-i.i.d data of clients is prone to model drift due to the deviation of local data distributions and is sensitive to the number of local epochs $E$ (Wang et al., 2020; Li et al., 2020b; McMahan et al., 2017). To prevent local training from over-fitting the local distribution, we apply $\ell _ { 2 }$ regularization as weight decay. In a different context of distributed deep learning, Sagawa et al. (2020) also showed that $\ell _ { 2 }$ regularization can improve the generalization by preventing the local model from perfectly fitting the non-i.i.d training data.
+
+Table 7: FEDBE with models sampled from a Dirichlet distribution on Step-non-i.i.d. CIFAR-10. We compare different $\alpha \times 1$ in setting the parameter of a Dirichlet distribution.   
+
+<table><tr><td>a</td><td>ConvNet</td><td>ResNet20</td></tr><tr><td>0.1</td><td>72.5±0.44</td><td>75.9±0.66</td></tr><tr><td>0.5</td><td>73.6±0.73</td><td>77.3±0.86</td></tr><tr><td>1</td><td>74.2±0.51</td><td>77.1±0.71</td></tr><tr><td>2</td><td>73.2±0.83</td><td>76.8±0.55</td></tr></table>
+
+As shown in Figure 8 where we compared FEDAVG and FEDBE with or without weight decay in local training, we found that weight decay not only leads to a higher test accuracy but also makes both algorithms more robust to the choice of local epochs $E$ .
+
+![](images/167809cd5e70eb8dcb7be26297b2324390e57ec44fc56fe98ad973cdaede931e.jpg)  
+Figure 8: FEDAVG and FEDBE with ConvNet on CIFAR-10 (Step-non-i.i.d) with or without weight decay in local training, for different numbers of local epochs $E$ .
+
+C EXPERIMENTAL RESULTS AND ANALYSIS
+
+C.1 GLOBAL MODEL SAMPLING IN FEDBE
+
+In the main paper, we mainly report accuracy by FEDBE with models sampled from a Gaussian distribution (cf. Equation 5 of the main paper). Here we report results using a Dirichlet distribution for model sampling (cf. Equation 7 of the main paper) in Table 7. We compare different ${ \pmb { \alpha } } = { \pmb { \alpha } } \times { \bf 1 }$ in setting the parameter of a Dirichlet distribution. We see that the accuracy is not sensitive to the change of $\alpha$ . Compared to Table 1 of the main paper, FEDBE with Dirichlet is slightly worse than FEDBE with Gaussian (by $\leq 0 . 5 \%$ ) but much better than FEDAVG.
+
+To further study the models sampled from the global model distribution constructed in FEDBE, we compare the prediction accuracy and confidences (the maximum values of the predicted probabilities) of clients’ models and sampled models. We show the histogram of correctly and incorrectly predicted test examples at different prediction confidences. As shown in Figure 9, we observe that clients’ models tend to be over-confident by assigning high confidences to wrong predictions. We hypothesize that it is because clients’ local training data are scarce and class-imbalanced. On the other hand, sampled models have much better alignment between confidences and accuracy (i.e., higher confidences, higher accuracy).
+
+C.2 ANALYSIS ON WEIGHT AVERAGE, (BAYESIAN) MODEL ENSEMBLE, AND DISTILLATION
+
+To investigate the difference of weight average, (Bayesian) model ensemble, and distillation in making predictions, we focus on one-round federated learning, in which the client models are trained in the same way regardless of what aggregation approach to be used. We experiment with Step-non-i.i.d. CIFAR-10 using ConvNet, and train the 10 local client models for 200 epochs. We then compare (a) weight average to combine the models, (b) model ensemble, and (c) Bayesian model ensemble with $M = 1 0$ extra samples beyond weight average and individual clients. For (b) and (c), we further apply knowledge distillation using the unlabeled server data to summarize the ensemble predictions into a single global model using SGD or SWA. We note that, method (a) is equivalent to one-round FEDAVG; method (b) without distillation is the same as 1-Ensemble; method (b) with SGD distillation is equivalent to one-round v-Distillation; method (c) with SWA is equivalent to one-round FEDBE. Table 8 shows the results with several interesting findings. First, without distillation, model ensemble clearly outperforms weight average and Bayesian model ensemble further adds a $2 \%$ gain, supporting our claims in section 3. Second, summarizing model ensemble into one global model largely degrades the accuracy, showing the challenges of applying ensemble distillation in federated learning. The $6 0 . 5 \%$ and $6 2 . 5 \%$ accuracy by ensemble, although relatively higher than others, may still be far from perfect to be used as distillation targets. Third, distillation with SWA outperforms SGD for both model ensemble and Bayesian model ensemble, justifying our proposed usage of SWA. Fourth, with or without distillation, Bayesian model ensemble always outperforms model ensemble, with very little cost of estimating the global model distribution and performing sampling. Fifth, even with the degraded accuracy after distillation, model ensemble and Bayesian model ensemble, after distilled into a single model, still outperforms weight average notably. Finally, although in the one-round setting we hardly see the advantage of distilling the model ensemble into a single model,
+
+![](images/b21dc80f376e6222c7eeec98f3c4e80923069de42652e385a1f23f4f9b1f58b1.jpg)  
+Figure 9: Histograms of correctly and incorrectly predicted examples (vertical axes) along the confidence values (the maximum values of the predicted probabilities). Upper row: Swiss roll dataset used in Figure 1 of main paper (averaged over 3 clients or sampled models); lower row: Step-non-i.i.d. CIFAR-10 (averaged over 10 clients or sampled models).
+
+Table 8: One-round federated learning on Step-non-i.i.d. CIFAR-10 with ConvNet. We compare different strategies to combine the clients’ local models, including weight average, (Bayesian) model ensemble, and ensemble distillation (with SGD or SWA).   
+
+<table><tr><td>Method</td><td colspan="3">Distillation</td></tr><tr><td></td><td>None</td><td>SGD</td><td>SWA</td></tr><tr><td>(a) Weight average</td><td>24.7±0.85</td><td></td><td>1</td></tr><tr><td>(b) Model ensemble</td><td>60.5±0.28</td><td>32.0±0.74</td><td>33.1±1.02</td></tr><tr><td>(c) Bayesian model ensemble</td><td>62.5±0.35</td><td>35.1±0.76</td><td>35.7±0.86</td></tr></table>
+
+# FEDAVG: inference on CIFAR-100
+
+![](images/9b4317a82452ab7bba083d6c66ed7dd9bc5b30a1adaeab11af3e9314e519e365.jpg)  
+FEDAVG: inference on CIFAR-10
+
+![](images/d52c42cdfa255223bb1b6cd310b9635d053d94a302d7693d9469baa9c33d6627.jpg)  
+FEDBE: inference on CIFAR-100
+
+![](images/e53b30366ecd6c926d4b48a1dbf8df791680199aab850c96244f19dda50d2461.jpg)  
+FEDBE: inference on CIFAR-10
+
+![](images/1f8cd8b6b99eb9b969d388fd2c1336d70916c3d3969886a7e4785506d3a47a29.jpg)  
+Figure 10: Feature visualization of FEDAVG and FEDBE models. All models are trained on Step-non-i.i.d. CIFAR-10, then inference on CIFAR-10/100 test sets. The features are colored with the ground-truth labels (CIFAR-10) or the predictions (CIFAR-100).
+
+with multiple rounds of communication as in the main paper (cf. subsection 5.2), its advantage becomes much clear—it allows the next-round local training to start from a better initialization (in comparison to weight average) and eventually leads to much higher accuracy than 1-Ensemble.
+
+# C.3 FEDBE VS. FEDAVG
+
+We provide further comparisons between FEDBE and FEDAVG. First, we perform Bayesian model ensemble at the end of FEDAVG training (Step-non-i.i.d. CIFAR-10). We achieve $7 2 . 5 \%$ with ResNet20, better than FEDAVG $( 7 0 . 2 \%$ in Table 1 of the main paper) but still worse than FEDBE $( 7 7 . 5 \% )$ , demonstrating the importance of incorporating Bayesian model ensemble into multi-round federated learning.
+
+To further analyze why FEDBE improves over FEDAVG, we train models on Step-non-i.i.d CIFAR-10, inference on the test sets, and visualize the features. We trained a FEDBE ResNet20 model for only 15 rounds such that the test accuracy is similar to a FEDAVG ResNet20 model trained for 40 rounds. In Figure 10, we plot their features using t-SNE (Maaten & Hinton, 2008) on the CIFAR-10/100 testing sets (consider 3 semantically different classes: automobile, cat, and frog) and color the features with the ground-truth labels of CIFAR-10 test set or the predictions of the models on CIFAR-100 test set. Interestingly, we observe that the features of FEDBE are more discriminative (separated) than the features of FEDAVG, especially on CIFAR-100, even if FEDBE and FEDAVG have similar test accuracy on CIFAR-10.
+
+We further discuss Table 3 of the main paper. We perform data augmentation on $\tau$ in knowledge distillation. This explains why we obtain improvement over FEDAVG when using the FEDAVG predictions as the target labels $( 7 2 . 6 \% / 7 3 . 4 \%$ vs. $7 2 . 0 \% / 7 0 . 2 \%$ in Table 1 of the main text, using ConvNet/ResNet20). We note that without data augmentation, using FEDAVG predictions as the target leads to zero gradients in knowledge distillation since we initialize the student model with FEDAVG. The results suggest the slight benefit of collecting unlabeled data for knowledge distillation in model aggregation.
+
+![](images/5a2f1c7d41971d3cb7995cb60077fe9ed62d603fb3b47b49db6b0f9468eda07a.jpg)  
+Figure 11: ResNet20 test accuracy on Step-non-i.i.d. CIFAR-10, with different numbers of epochs for distillation using SGD and SWA for FEDBE.
+
+# C.4 FEDBE WITH SWA AND SGD
+
+We found that distillation with SGD is more sensitive to noisy labels and the number of epochs. For ResNet20 (Table 1), FEDBE with FEDAVG $+ \textrm { C } + \textrm { S }$ w/o SWA (i.e., using SGD) achieves $7 4 . 9 \%$ with 20 epochs but $7 4 . 0 \%$ with 40 epoch. In contrast, FEDBE with SWA is much stable. As shown in Figure 11, the accuracy stays stable with more than 10 epochs being used, achieving $7 7 . 5 \%$ with 20 epochs and $7 7 . 3 \%$ with 40 epochs.
+
+# C.5 BATCH NORMALIZATION VS. GROUP NORMALIZATION
+
+Hsieh et al. (2020) showed that FEDAVG in non-i.i.d cases can be improved by replacing the batch normalization (BN) layers with group normalization (GN) layers (Wu & He, 2018). However, we observe that ResNets with GN converge much slower, which is consistent with the observations in (Zhang et al., 2020). In our CIFAR-10 (Step) experiments, FEDAVG using ResNet20 with GN can outperform that with BN slightly if both are trained with 200 rounds $( 7 6 . 4 \%$ vs. $7 4 . 6 \%$ ). FEDBE can further improve the performance: FEDBE with GN/BN achieves $7 9 . 6 \% / 8 0 . 2 \%$ .
+
+# C.6 COMPATIBILITY WITH SCAFFOLD
+
+SCAFFOLD (Karimireddy et al., 2020) is a recently proposed FL method to regularize local training. We experiment with SCAFFOLD on non-i.i.d. (Step) CIFAR-10. We find that SCAFFOLD cannot directly perform well with deeper networks (ResNet20: $5 9 . 4 \%$ ; ResNet32: $5 5 . 3 \%$ ). Nevertheless, FEDBE $+ \cal S { \bf C }$ AFFOLD can improve upon it, achieving $7 6 . 4 \%$ and $7 2 . 7 \%$ , respectively.
+
+To further analyze why FEDBE improves SCAFFOLD, we plot the test accuracy of SCAFFOLD vs. FEDBE $+ \cal S { \bf C }$ AFFOLD at every communication round. We see that both methods perform similarly in the early rounds. SCAFFOLD with weight average could not improve the accuracy after roughly 10 rounds. FEDBE $+ \mathrm { { S C } }$ AFFOLD, in contrast, performs Bayesian ensemble and distillation to obtain the global model, bypassing weight average and gradually improving the test accuracy. We therefore argue that, as long as FEDBE can improve SCAFFOLD slightly at every later round, the ultimate gain can be large. We also attribute the gain brought by FEDBE to the robustness of Bayesian ensemble for model aggregation.
+
+# C.7 PARTIAL PARTICIPATION ON CIFAR-100 (CF. SUBSECTION 5.3)
+
+We also conduct the experiments on CIFAR-100 with the non-i.i.d. Step setting. We consider a setting with 100 clients, in which 10 clients are randomly sampled at each round and iterates for 100 rounds, similar to (McMahan et al., 2017). We split 10K images from the 50K training images to the server, and distribute the remaining ones to the clients. Each client has 5 major classes (61 images each) and 95 minor classes (1 image each). Table 9 shows the results: FEDBE consistently outperforms FEDAVG.
+
+![](images/9ebcae6d76b61960f64c773ac3c83f7a81f8181506874595168378eb496b8531.jpg)  
+Figure 12: Step-non-i.i.d CIFAR-10 experiments accuracy curves of SCAFFOLD on ResNet20.
+
+Table 9: Non-i.i.d CIFAR-100  
+
+<table><tr><td>Method</td><td>ConvNet</td><td>ResNet20</td><td>ResNet32</td></tr><tr><td>FEDAVG</td><td>32.5±0.78</td><td>37.5±0.65</td><td>33.3±0.55</td></tr><tr><td>FEDBE</td><td>36.6±0.52</td><td>43.5±0.89</td><td>37.7±0.69</td></tr></table>
+
+# D DISCUSSION
+
+# D.1 EXTRA COMPUTATION COST
+
+FEDBE involves more computation compared to FEDAVG. The extra cost is on the server and no extra burden is on the clients. In practice, the server is assumed to be computationally rich so the extra training time is negligible w.r.t. communication time. Using a 2080 Ti GPU on CIFAR10 (ConvNet), building distributions and sampling takes 0.2s, inference of a model takes 2.4s, and distillation takes 10.4s. Constructing the ensemble predictions $\boldsymbol { \mathcal { T } } = \{ ( \boldsymbol { { \bf { x } } } _ { j } , \hat { p } _ { j } ) \} _ { j = 1 } ^ { J }$ , where $\begin{array} { r } { \hat { \pmb { p } } _ { j } \ = \ \frac { 1 } { M } \sum _ { m = 1 } ^ { M } p ( y | \pmb { x } _ { j } ; \pmb { w } ^ { ( m ) } ) } \end{array}$ , requires each $\pmb { w } ^ { ( m ) }$ to be evaluated on $\mathcal { U }$ , which can be easily parallelized in modern GPU machines. The convergence speed of the Monte Carlo approximation√ in Equation 4 is $1 / \sqrt { M }$ , yet we observe that $M = 1 0 \sim 2 0$ is sufficient for Bayesian model ensemble to be effective.
+
+# D.2 FEDAVG ON DEEPER NETWORKS
+
+Deeper models are known to be poorly calibrated (Guo et al., 2017), especially when trained on limited and imbalanced data. The loss surfaces can be non-convex (Garipov et al., 2018; Draxler et al., 2018). FEDAVG thus may not fuse clients well and may need significantly more rounds of communication with local training of small step sizes to prevent client’s model drifting.
+
+# E FURTHER ANALYSIS
+
+# E.1 TEST ACCURACY AT DIFFERENT ROUNDS
+
+We follow the experimental setup in section 5 and further show the test accuracy of the compared methods at different communication rounds (in total 40 rounds) in Figure 13. Specifically, we experiment with ResNet20 and ResNet32 for both the Step-non-i.i.d. and Dirichlet-non-i.i.d. settings on CIFAR-10 (the results at 40 rounds are the same as those listed in Table 1). FEDBE obtains the highest accuracy after roughly 10 rounds, and could gradually improve as more rounds are involved. Interestingly, v-Distillation, which performs ensemble directly over clients’ models without other sampled models, normally obtains the highest accuracy in the first 10 rounds, but is surpassed by FEDBE after that. We hypothesize that in the first 10 rounds, as the clients models are still not well trained, the constructed distributions may not be stable. We also note that except 1-Ensemble, the other methods with ensemble mostly outperform FEDAVG in the first 10 rounds, showing their robustness in aggregation.
+
+![](images/5e577bb15f682ee413bc5442a0e714aa725f501f04efc211962d1fd8652db1e0.jpg)  
+Figure 13: CIFAR-10 curves of test accuracy at different communication rounds. We study both non-i.i.d settings (Step and Dirichlet) using ResNet20 and ResNet32 (cf. subsection E.1).

parse/train/fgrc9OTuB_g/fgrc9OTuB_g.md

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+# An Analysis of Abstracted Model-Based Reinforcement Learning
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Many methods for Model-based Reinforcement Learning (MBRL) provide guaran  
+2 tees for the accuracy of the Markov decision process (MDP) model they can deliver.   
+3 At the same time, state abstraction techniques allow for a reduction of the size of   
+4 an MDP while maintaining a bounded loss with respect to the original problem.   
+5 It may come as a surprise, therefore, that no such guarantees are available when   
+6 combining both techniques, i.e., where MBRL merely observes abstract states. Our   
+7 theoretical analysis shows that abstraction can introduce a dependence between   
+8 samples collected online (i.e., in the real world), which invalidates most results   
+9 for MBRLs in this setting. Collecting samples using a simulator can avoid this   
+10 problem. We conclude that we should be careful when applying MBRL methods   
+11 to abstracted real-world data.
+
+# 12 1 Introduction
+
+13 When trying to find good solutions to MDPs using Reinforcement Learning (RL) a fundamental   
+14 problem is the exploration-exploitation dilemma: when to take actions to obtain more information,   
+15 and when to take actions that maximize reward based on the current knowledge. Tabular MBRL   
+16 methods have found good ways to deal with this dilemma [7, 28, 14].   
+17 However, MDPs can be very large, which can be problematic for these methods. One way to deal   
+18 with this is to reduce the size of the MDP. State abstractions are one way to do this [17, 1]. We   
+19 are interested in approximate state abstractions since they allow for greater reductions of the MDP,   
+20 though there is a trade-off with solution quality [1]. Specifically, we assume we have an approximate   
+21 model similarity abstraction function $\phi$ [1] that maps states to abstract states. The environment   
+22 returns states $s$ , but the agent receives $\phi ( s )$ , see Figure 1. This setting, which was considered before   
+23 [22, 2], is what we call Abstracted RL, and is the topic of this paper.   
+24 Abstracted RL corresponds to RL in a Partially   
+25 Observable MDP (POMDP), as previously de  
+26 scribed [5]. It is well known that policies for   
+27 POMDPs that only base their action on the last   
+28 observation $\phi ( s )$ could be arbitrarily bad [26].   
+29 However, when we assume that $\phi$ is an approx  
+30 imate model similarity abstraction [1] this worst   
+31 case may not apply: Based on the observed ab  
+32 stract states the agent learns an (empirical) abstract model. If we could show that this learned model   
+33 is close to an ‘abstract MDP’ (details in Section 2.2), we could give finite-sample guarantees on the   
+34 performance in the original MDP by combining results from MBRL and abstraction.   
+35 However, in MBRL, to guarantee (with high probability) that the learned model is close to the actual   
+36 environment model, it is typical (e.g., [28, 14]) to use concentration inequalities such as Theorem   
+37 2 from Weissman et al. [30]. But this theorem relies on independent and identically distributed   
+38 (i.i.d.) samples for each state-action pair. In this paper, we analyze online collection of such samples   
+39 in Abstracted RL and show that they are not independent1, which means that most guarantees for   
+40 existing MBRL methods do not hold in the online Abstracted RL setting.2   
+41 On the positive side, when we have access to a simulator, we show how this can be used to collect the   
+42 data such that the typical MBRL guarantees hold and we can learn an accurate model. We discuss   
+43 that emulating this in the real world is possible, but extremely sample inefficient, thus highlighting   
+44 the difficulty of assuming that we would have access to an i.i.d. dataset, as in some earlier works.
+
+![](images/558915b358a6bfb9d141a08438b36473b3a9748fe909488d6952bfbe3fae7ccf.jpg)  
+Figure 1: Abstracted RL, the agent observes $\bar { s } =$ $\phi ( s )$ instead of $s$ . Image based on Abel et al. [2]
+
+# 45 2 Preliminaries
+
+46 We assume the environment the agent is acting in can be represented by an infinite horizon MDP   
+47 $M : = \langle S , A , T , R , \gamma \rangle$ . Where $S$ is a finite set of states $s \in S$ , $A$ a finite set of actions $a \in A$ , $T$   
+48 a transition function $T ( s ^ { \prime } | s , a ) = \operatorname* { P r } ( s ^ { \prime } | s , a ) .$ $R$ a reward function $R ( s , a )$ which gives the reward   
+49 received when the agent executes action $a$ in state $s$ , and $\gamma$ the discount factor $( 0 \leq \gamma < 1 $ ).   
+50 In RL the goal of the agent is to find an optimal policy $\pi ^ { * } : S  A$ which maximizes the expectation   
+51 of the discounted cumulative reward. $V ^ { \pi } ( s )$ denotes the expected value of the discounted cumulative   
+52 reward under policy $\pi$ starting from state $s$ . Similarly, $Q ^ { \bar { \pi } } ( s , a )$ denotes the expected value of the   
+53 discounted cumulative reward when first taking action $a$ from state $s$ and then following policy $\pi$ .
+
+# 54 2.1 Model-Based RL
+
+55 MBRL methods learn a model from the experiences, these are obtained by the agent acting in the   
+56 MDP. The learned model is usually the empirical model, directly based on the experience the agent   
+57 obtains [7, 28, 14]. Per state-action pair the agent stores the next-states reached after taking action $a$   
+58 from state $s$ in sequence $Y _ { s , a } \colon Y _ { s , a } : \{ s ^ { \prime ( 1 ) } , s ^ { \bar { \prime } ( 2 ) } , \cdot \cdot \cdot , s ^ { \prime ( m ) } \}$ . We use $Y$ to refer to the collection of   
+59 all $Y _ { s , a }$ . From this the empirical, or learned, model $T _ { Y }$ is constructed, that just counts how often we   
+60 have seen the transition to a next-state, and normalizes this:
+
+$$
+\forall _ { s ^ { \prime } \in S } T _ { Y } ( s ^ { \prime } | s , a ) \triangleq \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \mathbb { 1 } \{ Y _ { s , a } ^ { ( i ) } = s ^ { \prime } \} ,
+$$
+
+where 61 $\mathbb { 1 } \{ \cdot \}$ denotes the indicator function of the specified event, i.e., it is 1 if $Y _ { s , a } ^ { ( i ) } = s ^ { \prime }$ and 0 62 otherwise.
+
+63 To give finite-sample guarantees on the accuracy of the estimate $T _ { Y }$ , 3 concentration bounds such as   
+64 Theorem 2.1 from Weissman et al. [30] are often used, e.g. in Strehl and Littman [28], Jaksch et al.   
+65 [14]. However, these typically make use of the fact that samples are i.i.d. In most MBRL settings this   
+66 is not a problem under some assumptions, e.g. when the MDP is communicating [25]. In this case   
+67 due to the Markov property the obtained samples are i.i.d.   
+68 In general, of course the hope is that with enough samples the learned model $T _ { Y }$ becomes accurate.   
+69 With accurate we mean that the distance between $T _ { Y } ( \cdot | s , a )$ and $T ( \cdot | s , a )$ will be small, where the   
+70 distance is measured using the $L _ { 1 }$ norm, defined as:
+
+$$
+| | T _ { Y } ( \cdot | s , a ) - T ( \cdot | s , a ) | | _ { 1 } \triangleq \sum _ { s ^ { \prime } \in S } | T _ { Y } ( s ^ { \prime } | s , a ) - T ( s ^ { \prime } | s , a ) | .
+$$
+
+71 Part of theorem 2.1 from Weissman et al. [30], slightly reworded, then gives a guarantee of accuracy:
+
+Lemma 1 (72 $L _ { 1 }$ inequality [30]). Let $\pmb { Y } _ { s , a } = Y ^ { ( 1 ) } , Y ^ { ( 2 ) } , \cdot \cdot \cdot , Y ^ { ( m ) }$ be i.i.d. random variables 73 distributed according to $T ( \cdot | s , a )$ . Then, for all $\epsilon > 0$ ,
+
+$$
+\operatorname* { P r } ( | | T _ { Y } ( \cdot | s , a ) - T ( \cdot | s , a ) | | _ { 1 } \geq \epsilon ) \leq ( 2 ^ { | S | } - 2 ) e ^ { - \frac { 1 } { 2 } m \epsilon ^ { 2 } } .
+$$
+
+74 In this way, MBRL can upper bound the probability that the learned model, based on $m$ samples, for   
+75 a state-action pair $( s , a ) \bar { T _ { Y } } ( \cdot | s , a )$ will be far away $( \geq \epsilon )$ from the true model $T ( \cdot | s , a )$ .   
+76 The situation is more subtle if the MDP is not communicating, i.e., if there exists $s _ { 1 } , s _ { 2 } \in S$ for which   
+77 there is no deterministic policy that eventually leads from $s _ { 1 }$ to $s _ { 2 }$ . This can create a dependence   
+78 between the samples [28]. Intuitively this happens because, if we look at one particular state-action   
+79 pair $( s , a )$ , there might be a transition to state $s ^ { \prime }$ such that the probability to return to $s$ is 0. Thus if   
+80 we would have $n$ outcomes of $( s , a )$ we would immediately know that at least $n - 1$ outcomes were   
+81 not state $s ^ { \prime }$ . Since as soon as we observe $s ^ { \prime }$ , we know the agent would not be able to return to state $s$ .   
+82 Strehl and Littman [28] show that in this setting it is still possible to use Lemma 3 as an upper bound.
+
+# 2.2 State abstraction for given models
+
+84 In the planning setting, where the model is known a priori, a state abstraction can be formulated as   
+85 a grouping or mapping from ground states to abstract states [18]. This is done with an abstraction   
+86 function $\phi$ , a surjective function that maps from ground states, $s \in S$ , to abstract states $\bar { s } \in \bar { S }$ :   
+87 $\phi ( s ) : S  { \bar { S } }$ . Here $\bar { S }$ is defined as $\bar { S } = \bar { \{ { \phi ( s ) \vert s \in S \} } }$ .We use the ¯ notation to refer to the abstract   
+88 space. We slightly overload notation and let $\bar { s }$ both denote an abstract state as well as the set of   
+89 ground states that map to the abstract state $\bar { s }$ , i.e., $\bar { s } = \{ g \in S \mid \phi ( g ) = \bar { s } \}$ , if $\bar { s } \in \bar { S }$ . The use should   
+90 be clear from the context. We define the probability to transition to an abstract state $\operatorname* { P r } ( \bar { s } ^ { \prime } | s , a )$ as   
+91 follows:
+
+$$
+\operatorname* { P r } ( { \bar { s } } ^ { \prime } | s , a ) \triangleq \sum _ { s ^ { \prime } \in { \bar { s } } ^ { \prime } } T ( s ^ { \prime } | s , a ) .
+$$
+
+92 This is a very general form of state abstraction, that clusters together states with different dynamics   
+93 into abstract states. Note that we do assume that the given state abstraction deterministically maps   
+94 states to an abstract state. This in contrast to some related work on problems with block structure   
+95 [10], where a Markov state can lead to multiple observations (abstract states in our terminology) that   
+96 need to be aggregated appropriately to result in a small MDP [4, 10].   
+97 Approximate model similarity abstraction Many different abstraction criteria exist [17], here we   
+98 focus on approximate model similarity abstraction [1]. In this abstraction two states can map to the   
+99 same abstract state if their behavior is similar, i.e., when the reward function and the transitions to   
+100 abstract states are close. Approximate model-similarity is defined as follows:
+
+101 Definition 1. An approximate model-similarity abstraction, $\phi _ { m o d e l , \eta } ,$ for fixed $\eta ,$ satisfies:
+
+$$
+\begin{array} { r l r } & { \phi _ { m o d e l , \eta } ( s _ { 1 } ) = \phi _ { m o d e l , \eta } ( s _ { 2 } ) \ } & { \Longrightarrow \ \forall _ { a } \left| R ( s _ { 1 } , a ) - R ( s _ { 2 } , a ) \right| \le \eta , } \\ & { } & { \forall _ { \bar { s } ^ { \prime } \in \bar { S } , a } \left| \mathrm { P r } ( \bar { s } ^ { \prime } | s _ { 1 } , a ) - \mathrm { P r } ( \bar { s } ^ { \prime } | s _ { 2 } , a ) \right| \le \eta . } \end{array}
+$$
+
+102 From now on we will just refer to $\phi _ { \mathrm { m o d e l } , \eta }$ as $\phi$ .
+
+103 We note that the abstraction we consider, approximate model-similarity abstraction, is still quite   
+104 generic. It can cluster together states that have different transition and reward functions. However, in   
+105 the online Abstracted RL setting, the differences in dynamics can cause a dependence between the   
+106 samples, as we will show in detail in section 3. E.g. looking at $( { \bar { s } } , a )$ , the probability that we reach a   
+107 state $s ^ { \prime }$ depends both on the probability that we reach a particular state $s \in { \bar { s } }$ and then state $s ^ { \prime }$ from $s$   
+108 Returning to abstraction of a given model, it is possible to construct an abstract MDP $\bar { M } _ { \omega }$ from   
+109 the model of an MDP $M$ and an abstraction function $\phi$ , where $\omega$ is an action-specific 4 weighting   
+110 function, defined as follows:   
+111 Definition 2. We refer to the weight associated with a ground state, $s \in S$ , and action, $a \in A$ , by   
+112 $\omega ( s , a )$ . We have: $\bar { \prime } _ { s \in S , \ a \in A } 0 \leq \bar { \omega } ( s , a ) \leq 1$ and $\begin{array} { r } { \sum _ { s ^ { \prime } \in \phi ( s ) } \omega ( s ^ { \prime } , a ) = 1 } \end{array}$ .   
+113 The weighting function can be used to create abstract transition and reward functions, which are a   
+114 weighted average of the ground function. In this way, from $M , \phi$ and any $\omega$ we can construct an   
+115 abstract MDP $\bar { M } _ { \omega }$ :
+
+Definition 3. Given an MDP 116 $M$ , $\phi _ { ; }$ , and $\omega$ , $\bar { M } _ { \omega } = \langle \bar { S } , A , \bar { T } _ { \omega } , \bar { R } _ { \omega } , \gamma \rangle$ is constructed as:
+
+$$
+\begin{array} { c } { { \bar { S } = \{ \phi ( s ) \mid s \in S \} , A = A , \gamma = \gamma , } } \\ { { \forall _ { \bar { s } \in \bar { S } , \ a \in A } \bar { R } _ { \omega } ( \bar { s } , a ) = \displaystyle \sum _ { s \in \bar { s } } \omega ( s , a ) R ( s , a ) , } } \\ { { \forall _ { \bar { s } , \bar { s } ^ { \prime } \in \bar { S } , \ a \in A } \bar { T } _ { \omega } ( \bar { s } ^ { \prime } | \bar { s } , a ) = \displaystyle \sum _ { s \in \bar { s } } \sum _ { s ^ { \prime } \in \bar { s } ^ { \prime } } \omega ( s , a ) T ( s ^ { \prime } | s , a ) . } } \end{array}
+$$
+
+An abstract MDP 117 $\bar { M } _ { \omega }$ is just an $M D P$ . This means we can use any planning method we like to find an optimal policy 118 $\bar { \pi } ^ { * }$ for $\bar { M } _ { \omega }$ .
+
+119 What we are interested in is the performance of a policy on the abstract space, when applied on the   
+120 original problem $M$ . Any policy on the abstract space $\bar { \pi }$ can be used in $M$ as follows $\bar { \pi } ( s ) \bar { : } = \bar { \pi } ( \phi ( s ) )$ ,   
+121 leading to $V ^ { \bar { \pi } ^ { * } }$ . It has been shown that we can upper bound the loss in performance due to using an   
+122 optimal policy for $\bar { M } _ { \omega }$ , $\bar { \pi } ^ { * }$ in $M$ instead of using the optimal solution for $M$ [8, 1, 29]:
+
+Lemma 2 (Lemma 4 [29]). An approximate model similarity abstraction (Definition $^ { l }$ ), has suboptimality bounded in $\begin{array} { r } { \eta \colon \forall _ { s \in S } V ^ { * } \bar { ( s ) } - V ^ { \bar { \pi } ^ { * } } ( s ) \le \frac { 2 \eta + 2 \gamma ( | \bar { S } | - 1 ) \eta } { ( 1 - \gamma ) ^ { 2 } } } \end{array}$ .
+
+# 125 3 Abstracted MBRL and the problem of online data collection
+
+126 As explained, we are interested in Abstracted RL, where we have an approximate model similarity   
+127 abstraction function $\phi$ and an MDP $M$ . The agent acts in $M$ but only observes $\phi ( s )$ using abstraction   
+128 function $\phi$ , as in Figure 1. This setting can also be seen as a POMDP, where the states are hidden and   
+129 there is a deterministic observation function, $o = \phi ( s )$ . However, in contrast to the usual POMDP   
+130 settings, we look for a myopic (memoryless) policy. While we know that in general this can lead to   
+131 arbitrarily bad results [26], in this case the value loss would be bounded in the planning setting by   
+132 Lemma 2. However, now we assume we are in the Abstracted RL setting, and the result for planning   
+133 may not hold for the learned model.   
+134 We assume we know $S , A , R , \gamma$ , and $\phi$ (and thus $\bar { S }$ ), but do not know the transition function.5 Since   
+135 we do not know the transition function we can neither simply do planning on $M$ nor can we construct   
+136 an abstract MDP, using Definition 3, and solve that. Instead, we let the agent interact with $M$ but   
+137 use $\phi$ to let the agent observe $\phi ( s )$ , instead of $s$ , and build a learned (abstract) model from the   
+138 observations it obtains. We show the general Abstracted MBRL procedure in Algorithm 1.
+
+The agent collects data for every abstract state-action pair 139 $( { \bar { s } } , a )$ , which is stored as sequences $\bar { Y } _ { \bar { s } , a }$
+
+$$
+\bar { Y } _ { \bar { s } , a } : \{ { \bar { s } } ^ { \prime ( 1 ) } , { \bar { s } } ^ { \prime ( 2 ) } , \cdot \cdot \cdot , { \bar { s } } ^ { \prime ( m ) } \} .
+$$
+
+Similar to before in (1), we construct a learned model 140 $\bar { T } _ { Y }$ , now looking at the abstract next-states that 141 were reached:
+
+$$
+\bar { T } _ { Y } ( \bar { s } ^ { \prime } | \bar { s } , a ) \triangleq \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \mathbb { 1 } \{ \bar { Y } _ { \bar { s } , a } ^ { ( i ) } = \bar { s } ^ { \prime } \} .
+$$
+
+142 If this model would be equal, or close, to the transition function $\bar { T } _ { \omega }$ of an abstract MDP $\bar { M } _ { \omega }$ , for   
+143 some valid $\omega$ , we could upper bound the loss in performance due to applying learned policy $\bar { \pi } ^ { * }$ to $M$   
+144 instead of the optimal policy $\pi ^ { * }$ [1, 29].   
+146 Our main question is: do the finite-sample model learning guarantees of MBRL algorithms still hold   
+147 in the Abstracted RL setting?
+
+<table><tr><td>Algorithm1 Procedure:Abstracted MBRL</td><td>Algorithm2 COLLECTSAMPLES Online</td></tr><tr><td>Input: M,Φ,δ,∈,π Y = COLLECTSAMPLES(M,𝜙,δ,∈π) The sampling results in sequences Ys,a, one for</td><td>Input: M,,δ,∈,π s = initial state // The number of samples m is based on the</td></tr><tr><td>every pair (s, a): Ys,=Φ(s(1）),.，,(s&#x27;(m))</td><td>simulator_analysis,Theorem 1. κ = δ/(|S||Al)</td></tr><tr><td>=(1),...,s(m) for all (s,a,s&#x27;) ∈ S × A × S do</td><td>m=[(2-ln() 2</td></tr><tr><td>m =} end for</td><td>for all s ∈ S do Ys,a=[] end for</td></tr><tr><td>My := (S,A,Ty,R,) π*=Value Iteration(My)</td><td>while min(s,a) |Ys,a| &lt; m do</td></tr><tr><td>Apply to M</td><td>s=(s) a=π(s) s&#x27; = Step(s,a)</td></tr></table>
+
+# 3.1 Online data collection
+
+149 In this section we follow the MBRL method from Algorithm 1, collecting samples online using   
+150 Algorithm 2.6 Starting from an initial state the agent follows a policy $\pi$ . Instead of observing the   
+151 states $s$ , the agent observes abstract states $\bar { s } = \phi ( s )$ , see Figure 1.   
+152 We make two important assumptions in order to make analysis possible. We assume that the MDP   
+153 is ergodic [25] 7 and that the policy assigns a positive probability to every action in every abstract   
+154 state. Together this can guarantee that Algorithm 2 can obtain any finite number of samples for every   
+155 abstract state-action pair within finite time.
+
+56 Our question is, can we still use Lemma 1 to guarantee that we learn an accurate model?
+
+157 Since we learn an abstract transition model $\bar { T } _ { Y }$ , we want to be able to guarantee that this learned   
+158 model will be close to the transition model of some abstract MDP. To define this transition model,   
+159 we first look at how the data is collected.   
+160 In the online data collection, a sample in $\bar { Y } _ { \bar { s } , a }$ is drawn when the agent takes action $a$ when it is   
+161 in a ground state $s \in \bar { s }$ . Specifically the $i$ -th abstract $\bar { Y } _ { \bar { s } , a } ^ { ( i ) } \ = \ \bar { s } ^ { \prime }$ is drawn from (ground) state   
+162 $X _ { \bar { s } , a } ^ { ( i ) } = s \in \bar { s }$ :
+
+$$
+\bar { Y } _ { \bar { s } , a } ^ { ( i ) } \sim \mathrm { P r } ( \cdot | X _ { \bar { s } , a } ^ { ( i ) } = s , a ) .
+$$
+
+Let 163 $X _ { \bar { s } , a } = ( X _ { \bar { s } , a } ^ { ( i ) } ) _ { i = 1 } ^ { m }$ denote the sequence of ground states $s \in { \bar { s } }$ from which the agent took action 164 . Each ground state gets a weight according to how often it was sampled from, which we formalize with the weighting function165 $\begin{array} { r } { \omega _ { X } \colon \forall _ { ( \bar { s } , a ) , s \in \bar { s } } \omega _ { X } \bigl ( s , a \bigr ) \triangleq \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \mathbb { 1 } \{ X _ { \bar { s } , a } ^ { ( i ) } = s \} } \end{array}$ . We use $\omega _ { X }$ to define 166 $\hat { T } _ { \omega _ { X } }$ analogous to (8):
+
+$$
+\forall _ { ( \bar { s } , a ) , \bar { s } ^ { \prime } } \bar { T } _ { \omega _ { X } } \left( \bar { s } ^ { \prime } | \bar { s } , a \right) = \sum _ { s \in \bar { s } } \omega _ { X } ( s , a ) \sum _ { s ^ { \prime } \in \bar { s } ^ { \prime } } T ( s ^ { \prime } | s , a ) .
+$$
+
+167 We want to have a concentration inequality to provide bounds on the deviation of the learned model   
+168 $\bar { T } _ { Y }$ from $\hat { T } _ { \omega _ { X } }$ , we refer to this inequality as the abstract L1 inequality, similar in form to (3):
+
+$$
+P ( | \bar { T } _ { Y } ( \cdot | \bar { s } , a ) - \bar { T } _ { \omega _ { X } } ( \cdot | \bar { s } , a ) | _ { 1 } \geq \epsilon ) \leq \delta ,
+$$
+
+where 169 $\bar { T } _ { Y } ( \cdot | \bar { s } , a )$ is defined according to (10) and $\hat { T } _ { \omega _ { X } }$ according to (12).
+
+170 If we could directly obtain i.i.d. samples from $\hat { T } _ { \omega _ { X } }$ and base our learned model $\bar { T } _ { Y }$ on the obtained   
+171 samples, then we would be able to show that the abstract L1 inequality holds by applying Lemma 1.   
+172 Since in this case, we would have $m$ i.i.d. samples per abstract state-action pair, distributed according   
+173 to $\bar { T } _ { \omega _ { X } } ( \cdot | \bar { s } , a )$ .
+
+the samples. Since every sample 175 in (11). These can have different176 $\bar { Y } ^ { ( i ) }$ was obtainributions if g action . $a$ from state ${ \cal X } _ { \bar { s } , a } ^ { ( i ) } = s \in \bar { s }$ $X _ { \bar { s } , a } ^ { ( i ) } \neq X _ { \bar { s } , a } ^ { ( j ) }$
+
+77 Non Identically Distributed While Lemma 1 assumes i.i.d. random variables, we show that it also   
+78 holds when the random variables are independent but not (necessarily) identically distributed.   
+179 Lemma 3. Let $\begin{array} { r l r } { X _ { \bar { s } , a } } & { { } = } & { s _ { 1 } , \cdot \cdot \cdot , s _ { m } } \end{array}$ be a sequence of states $ { \mathcal { S } } _ { s } \in  { \mathcal { S } } _ { } 0$ and let   
+180 $\bar { { \bf Y } } _ { \bar { s } , a } = \bar { { \cal Y } } ^ { ( 1 ) } , \bar { { \cal Y } } ^ { ( 2 ) } , \cdot \cdot \cdot , \bar { { \cal Y } } ^ { ( m ) }$ be independent random variables distributed according to   
+181 $\mathrm { P r } ( \cdot | s _ { 1 } , a ) , \cdot \cdot \cdot , \mathrm { P r } ( \cdot | s _ { m } , a )$ (Eqn. 4). Then, for all $\epsilon > 0$ ,
+
+$$
+\begin{array} { r } { \operatorname* { P r } ( | | \bar { T } _ { Y } ( \cdot | \bar { s } , a ) - \bar { T } _ { \omega _ { X } } ( \cdot | \bar { s } , a ) | | _ { 1 } \geq \epsilon ) \leq ( 2 ^ { | \bar { S } | } - 2 ) e ^ { - \frac { 1 } { 2 } m \epsilon ^ { 2 } } . } \end{array}
+$$
+
+182 The proof can be found in Appendix B. It mostly follows the proof by Weissman et al. [30], which uses   
+183 Hoeffding’s inequality [12] and the union bound [6].8 Lemma 3 shows that the fact that Hoeffding’s   
+184 inequality does not need identically distributed data can be carried over to the setting from Lemma 1.
+
+185 Independence We may be tempted to assume the samples are independent, i.e.,
+
+$$
+\forall _ { \bar { s } _ { 1 } , \cdots , \bar { s } _ { m } \in ( \bar { S } ) ^ { m } } \operatorname* { P r } ( \bar { Y } _ { \bar { s } , a } ^ { ( 1 ) } = \bar { s } _ { 1 } , \cdots , \bar { Y } _ { \bar { s } , a } ^ { ( m ) } = \bar { s } _ { m } ) = \operatorname* { P r } ( \bar { Y } _ { \bar { s } , a } ^ { ( 1 ) } = \bar { s } _ { 1 } ) \cdot \cdot \cdot P ( \bar { Y } _ { \bar { s } , a } ^ { ( m ) } = \bar { s } _ { m } )
+$$
+
+86 however, this may not be the case:
+
+Observation 1. When collecting samples online, i.e., based on Algorithm 2, the samples cannot be assumed to be independent.
+
+The following counterexample illustrates this.
+
+190 Counterexample To show that the samples may not be indepen  
+191 dent, we will give a counterexample. We use the example MDP and   
+192 abstraction in Figure 2, where we have 4 (ground) states, 3 abstract   
+193 states and only 1 action. We look at the transition probability from   
+194 abstract state $A$ $, { \bar { T } } _ { Y } ( \cdot | A )$ .   
+195 We will consider two samples and show that for at least one com  
+196 bination of $\bar { s } _ { 1 }$ and $\bar { s } _ { 2 }$ the samples are not independent. Consider   
+197 $\bar { s } _ { 1 } = \bar { s } _ { 2 } = B$ . That is, the first two times that we experience a   
+198 transition from the abstract state $A$ , we end up in $B$ .
+
+199 the starting state. Then we have and $\mathrm { P r } ( \bar { Y } _ { A } ^ { ( 1 ) } = B ) =$ $\operatorname* { P r } ( B | 1 ) = 0 . 6$
+
+$$
+\operatorname* { P r } ( \bar { Y } _ { A } ^ { ( 2 ) } = B ) = \sum _ { \bar { s } \in \bar { \cal S } } \operatorname* { P r } ( \bar { Y } _ { A } ^ { ( 2 ) } = B | \bar { Y } _ { A } ^ { ( 1 ) } = \bar { s } ) \operatorname* { P r } ( \bar { Y } _ { A } ^ { ( 1 ) } = \bar { s } )
+$$
+
+![](images/596348f5df8a7d9a095f28522a8ea9d1b1aecc37093cd955880be0e569bee57b.jpg)  
+Figure 2: Simple MDP, with only 1 action, and abstraction. The small circles are ground states (1,2,3,4). A, B and C are the abstract states. The numbers along the arrows show the transition probabilities, e.g. $P ( 3 | 1 ) = 0 . 6 $ .
+
+$$
+= 0 + 0 . 6 \cdot 0 . 6 + 0 . 4 \cdot 0 . 4 = 0 . 5 2 .
+$$
+
+So then we end up with: 201 $\operatorname* { P r } ( \bar { Y } _ { A } ^ { ( 1 ) } = B ) \operatorname* { P r } ( \bar { Y } _ { A } ^ { ( 2 ) } = B ) = 0 . 6 \cdot 0 . 5 2 = 0 . 3 2 1$ . And for the joint probability: 202 $\mathrm { P r } ( \bar { Y } _ { A } ^ { ( 1 ) } = B , \bar { Y } _ { A } ^ { ( 2 ) } = B ) = \mathrm { P r } ( \bar { Y } _ { A } ^ { ( 1 ) } = B ) \mathrm { P r } ( \bar { Y } _ { A } ^ { ( 2 ) } = B | \bar { Y } _ { A } ^ { ( 1 ) } = B ) = 0 . 6 \cdot 0 . 6 = 1$ 203 0.36.
+
+Thus we have that 204 independent. Lead $\operatorname* { P r } ( \bar { Y } _ { A } ^ { ( 1 ) } = B , \bar { Y } _ { A } ^ { ( 2 ) } = B ) \not = \operatorname* { P r } ( \bar { Y } _ { A } ^ { ( 1 ) } = B ) \operatorname* { P r } ( \bar { Y } _ { A } ^ { ( 2 ) } = B )$ , the samples are not
+
+6 Observation 2. As independence cannot be guaranteed, Lemmas 1 and 3 cannot be readily applied to show that the abstract $L l$ inequality holds.
+
+# 3.2 Simulator data collection
+
+Here we also want to give a guarantee in the form of the abstract L1 inequality from (13). While in the previous section we found this was not possible because the samples were dependent, here we assume that we have access to a simulator. To some extent this is not surprising, but to the best of our knowledge, this is the first work that explicitly shows how to combine MBRL and abstraction, using a simulator. We assume that this allows us to select (or move to) any state and draw a sample from its transition function. This we call the independent samples assumption:
+
+Assumption 1 (Independent samples). We assume we can obtain independent samples, e.g. for any state-action pair $( s , a )$ we can draw samples directly from its transition function $T ( \cdot | s , a )$ .
+
+217 In case a simulator of the MDP is available this is a reasonable assumption. For every $( \bar { s } , a )$ the   
+218 simulator sampling procedure (Algorithm 3 in Appendix B) selects a prototype $x _ { \bar { s } , a } \in \bar { s }$ to sample   
+219 from. We define a weighting function $\omega _ { x } ( s , a )$ that has weight 1 if $s$ is the prototype $x _ { \bar { s } , a }$ and 0   
+220 otherwise:
+
+$$
+\forall _ { ( \bar { s } , a ) , s \in \bar { s } } \omega _ { x } ( s , a ) \triangleq \mathbb { 1 } \{ s = x _ { \bar { s } , a } \} .
+$$
+
+221 Then we use this $\omega _ { x }$ to define the abstract transition function $\hat { T } _ { \omega _ { X } }$ according to (8). $\bar { T } _ { \omega _ { x } } ( \bar { s } ^ { \prime } | \bar { s } , a ) =$   
+222 $\begin{array} { r } { \sum _ { s ^ { \prime } \in \bar { s } ^ { \prime } } T \bigl ( s ^ { \prime } | s = x _ { \bar { s } , a } , a \bigr ) } \end{array}$ . This way the samples that we collect for one pair $( { \bar { s } } , a )$ are i.i.d., they are   
+223 independent because of our assumption of independent samples and identically distributed because   
+224 we sample from the prototype. This means we can use Lemma 1. We show that with the simulator   
+225 we can combine MBRL and abstraction, and still learn an accurate model, that is, we can guarantee   
+226 that $\bar { T } _ { Y }$ will be close to $\hat { T } _ { \omega _ { x } }$ , with high probability:
+
+Theorem 1. Under assumption $^ { l }$ , and following the procedure in Algorithm $^ { l }$ , with the data collection from Algorithm $^ 3$ (Appendix $B$ ), with inputs $| \bar { S } | , A , \epsilon$ and $\delta$ . For $\bar { T } _ { Y }$ constructed by the algorithm we have that with probability $1 - \delta$ , the following holds:
+
+$$
+\forall _ { ( \bar { s } , a ) } \vert \vert \bar { T } _ { Y } \big ( \cdot \vert \bar { s } , a \big ) - \bar { T } _ { \omega _ { x } } \big ( \cdot \vert \bar { s } , a \big ) \vert \vert _ { 1 } \leq \epsilon .
+$$
+
+230 By Assumption 1 we can obtain any number of independent samples for each abstract state action   
+231 pair $( \bar { s } , a )$ . Using Lemma 1 we can then derive the number of samples $m$ that is required for each   
+232 pair $( \bar { s } , a )$ such that, after applying a union bound, we obtain the bounds in (19). The full proof can   
+233 be found in Appendix B.
+
+# 234 4 Related work
+
+235 There is a lot of work that considers the combination of abstraction with either planning or (online)   
+236 RL. In a lot of these works the dependence of samples that arises in Abstracted RL is not an issue   
+237 due to various assumptions, similarly to how in MBRL dependence of samples is often not an issue   
+238 because of the Markov property and the assumption that the MDP is communicating [25]. Often   
+239 this is either due the assumption that data has been obtained i.i.d., the specific type of abstraction, or   
+240 because access to an MDP model is assumed.   
+241 One paper that does give a result for the Abstracted RL setting is the work by Abel et al. [2].   
+242 They show that in this setting R-MAX [7] no longer maintains its guarantees when paired with any   
+243 type of state-abstraction function, though their example is specifically for approximate Q-function   
+244 abstractions. They also show that the expected trajectory of a learning agent in a constructed   
+245 abstract MDP (Definition 3) is not the same as in Abstracted RL. Their work makes clear there is a   
+246 complication when combining MBRL and abstraction, here we further investigated the cause of this   
+247 complication, the dependence between samples.   
+248 For planning in constructed abstract MDPs, some main results for exact state-abstractions come   
+249 from Li et al. [18] and for approximate state-abstractions from Abel et al. [1]. The results from Abel   
+250 et al. [1] allow for quantifying an upper bound on performance for policies found in a constructed
+
+abstract MDP, as in section 2.2. Taïga et al. [29] build on this by giving a result for performing RL on top of the constructed abstract MDP. They provide upper bounds for this setting when using MBIE with exploratory bonus (MBIE-EB) [28]. In addition, they give an example to show that in this combination you cannot guarantee optimal performance in the original MDP. Still, they show that an upper bound on the loss in value can be given.
+
+Both Paduraru et al. [24] and Jiang et al. [15] deal with the issue of dependence by making the explicit assumption that samples are obtained i.i.d. Paduraru et al. [24] consider the setting where we are given a dataset for a continuous domain and then use discretization to aggregate states into abstract states. They then give PAC-style guarantees on the learned abstract model and the value that a policy based on this model can achieve in the real MDP. Instead of using the L1 deviation bound from Weissman et al. [30], Paduraru et al. [24] use a similar bound for i.i.d. samples by Devroye and Gyorfi [9], which requires a minimum amount of samples. Another difference is that their results calculate the probability that the model will be $\epsilon$ -accurate given a fixed dataset. They assume that the data has been gathered i.i.d., but our Lemma 3 shows that merely independent data would be enough. At the same time, our results show that when we collect data online in the Abstracted RL setting, their guarantees will not hold.
+
+Jiang et al. [15] operate in the abstraction selection setting, where the agent is provided with a set of abstraction functions (state representations). They do not assume that any of the abstraction functions results in a Markov model, but they do assume a given dataset, with data that was collected i.i.d. They give a bound directly on how accurate the Q-values based on the (implicitly) learned model will be, rather than on the accuracy of the model itself. As we showed, the assumption that the data is i.i.d. is not a trivial assumption, since it means the data cannot just have been collected online. They do mention that samples will not be strictly independent if a fixed exploration policy is used to collect data but do not mention what the implications are.
+
+There are quite a few other papers in the abstraction selection setting, several of these assume that the given set of state representations contains a Markov model [11, 19, 23]. Hallak et al. [11] give asymptotic guarantees for selecting the correct model and on building an exact MDP model. The assumption that there is an MDP model in the given set of representations is crucial in their analysis since for this ‘true model’ the samples are i.i.d. Similarly, both Maillard et al. [19] and Ortner et al. [23] also assume that the given set of state representations contains a Markov model. They create an algorithm for which they obtain regret bounds, their analysis also makes use of the Markov representation.
+
+Other work in the abstraction selection setting does not assume that the set of abstraction functions contains a Markov model [16, 22]. However, Ortner et al. [22] use Theorem 2.1 from Weissman et al. [30] that requires i.i.d. samples, which we have shown here cannot be guaranteed in this setting. Lattimore et al. [16] operate in a setting more general than MDPs, where the dynamics of the true environment depend arbitrarily on a history of actions, rewards, and observations. The agent gets as input a finite set of environments, one of which is the true environment. Since the input includes the full model of each environment, the agent does not have to learn a transition model. Instead, to obtain regret bounds, they directly compare the rewards the agent obtains to the expected rewards of the given environments and eliminate environments that are implausible given the observed rewards.
+
+Another way to deal with the issue of dependence is by looking at convergence in the limit [27, 13, 20]. Singh et al. [27] give an asymptotic result for the convergence of Q-learning and TD(0) in MDPs with soft state aggregation. Soft state aggregation means that a state $s$ belongs to a cluster $x$ with some probability $P ( x | s )$ , this means a state $s$ can belong to several clusters. The state-abstraction functions we consider are a special case of this, where each state is part of exactly one abstract state (or cluster). Their result relies on having a stationary policy that assigns a non-zero probability to every action in every state and the assumption that the MDP is ergodic. Together these imply there is a limiting state distribution, and using this they show convergence asymptotically. Our main interest is in finite-samples guarantees with policies that change due to exploration, whereas this work gives convergence guarantees in the limit using a fixed policy.
+
+Hutter [13] gives a variety of results focusing on both approximate and exact abstractions in environments without MDP assumptions. Several of these are in the planning setting, similar to those of Abel et al. [1]. Most relevant for us is their Theorem 12, which for online RL shows convergence in the limit of the empirical transition function under weak conditions, e.g. if the abstract process itself
+
+306 is an MDP. Under this condition however the problem reduces to RL in an (abstract) MDP, rather   
+307 than Abstracted RL.   
+308 Majeed and Hutter [20] build on the work by Hutter [13] and focus on the combination of model-free   
+309 RL and exact abstraction. They show that, under the condition of state uniformity, q-learning can be   
+310 shown to converge in the limit to the optimal solution. State uniformity means that histories that are   
+311 grouped together have the same optimal q-values. In contrast to our setting, they look at an exact   
+312 abstraction, extending it to approximate aggregation was left as an open question.   
+313 Other related work is in the area of MDPs with rich observations or block structure [4, 10]. However,   
+314 in that setting each observation can be generated only from a single hidden state, which means that   
+315 the issue of non-i.i.d. data due to abstraction does not arise. In contrast, in our setting multiple   
+316 (hidden) states generate the same observation. Azizzadenesheli et al. [4] state their setting can be   
+317 seen as an aggregation problem, where the observations can be aggregated to form a small (latent)   
+318 MDP. But in our case, we do not try to learn the MDP (as it is not small). Du et al. [10] describe   
+319 that their setting is similar to exact model similarity (or bisimulation), but we focus on approximate   
+320 model similarity which is what introduces the problems as described here.
+
+# 321 5 Discussion
+
+322 When collecting samples online in Abstracted RL, there is a potential dependence between samples,   
+323 meaning we cannot use the typically used concentration results that assume i.i.d. samples, e.g.   
+324 Theorem 2.1 from Weissman et al. [30], the empirical Bernstein inequality [3, 21] or the Chernoff   
+325 bound. In case the samples are only weakly dependent, it may be that concentration inequalities   
+326 for (weakly) dependent variables are a viable alternative through which we can come to guarantees   
+327 on the learned model. Alternatively, it may be possible to change the sampling process to ensure   
+328 independent samples. One way to ensure independent samples is to, as in the simulator setting, select   
+329 a prototype state and only use the samples collected from this state. Though in this case, we will be   
+330 discarding information when we reach a state $s \in { \bar { s } }$ that is not the prototype.   
+331 Our assumption on the simulator that we can go/reset to any state to draw samples from it can be   
+332 relaxed, though it may mean that the procedure takes considerably more time. Consider the case   
+333 where we cannot just reset the simulator to the state $s$ from which we want to sample, and instead, it   
+334 would behave like the MDP. In this case, we would have to take the right actions to arrive at the state   
+335 $s$ from which we would like to sample. Since we assume we do not know $T$ , this may take a long   
+336 time. This also shows the difficulty of assuming that in the MBRL setting somehow have access to an   
+337 i.i.d. dataset, as has been assumed in some earlier work [24, 15].
+
+# 338 6 Conclusion
+
+339 We analyzed Abstracted RL: the combination of MBRL and state abstraction when the model of   
+340 the MDP is not available. We have shown that in Abstracted RL samples obtained online cannot   
+341 be assumed to be independent. Since many current guarantees from MBRL methods rely on this   
+342 assumption, their guarantees do not hold in this setting. And in fact, no current methods exist that   
+343 give (correct) finite-sample quality guarantees for the models learned in this setting. This also means   
+344 that current results that rely on an i.i.d. assumption cannot be readily transferred to the Abstracted   
+345 RL setting.
+
+In addition, we show that with a simulator, since we can draw independent samples, it is still possible to give guarantees on the accuracy of the model. However, having access to a simulator may often not be possible. An important step is to see if the MBRL guarantees can be adapted to Abstracted RL for online sample collection.
+
+# References
+
+[1] David Abel, David Hershkowitz, and Michael Littman. Near optimal behavior via approximate state abstraction. In International Conference on Machine Learning, pages 2915–2923, 2016.
+
+[2] David Abel, Dilip Arumugam, Lucas Lehnert, and Michael Littman. State abstractions for lifelong reinforcement learning. In International Conference on Machine Learning, pages 10–19, 2018.   
+[3] Jean-Yves Audibert, Rémi Munos, and Csaba Szepesvári. Tuning bandit algorithms in stochastic environments. In International conference on algorithmic learning theory, pages 150–165. Springer, 2007.   
+[4] Kamyar Azizzadenesheli, Alessandro Lazaric, and Animashree Anandkumar. Reinforcement learning in rich-observation mdps using spectral methods. arXiv preprint arXiv:1611.03907, 2016.   
+[5] Aijun Bai, Siddharth Srivastava, and Stuart J Russell. Markovian state and action abstractions for mdps via hierarchical mcts. In IJCAI, pages 3029–3039, 2016.   
+[6] George Boole. An investigation of the laws of thought: on which are founded the mathematical theories of logic and probabilities. Dover Publications, 1854.   
+[7] Ronen I Brafman and Moshe Tennenholtz. R-max-a general polynomial time algorithm for near-optimal reinforcement learning. Journal of Machine Learning Research, 3(Oct):213–231, 2002.   
+[8] Richard Dearden and Craig Boutilier. Abstraction and approximate decision-theoretic planning. Artificial Intelligence, 89(1-2):219–283, 1997.   
+[9] L. Devroye and L. Gyorfi. Nonparametric Density Estimation: The L1 View. Wiley Interscience Series in Discrete Mathematics. Wiley, 1985.   
+[10] Simon Du, Akshay Krishnamurthy, Nan Jiang, Alekh Agarwal, Miroslav Dudik, and John Langford. Provably efficient rl with rich observations via latent state decoding. In International Conference on Machine Learning, pages 1665–1674. PMLR, 2019.   
+[11] Assaf Hallak, Dotan Di-Castro, and Shie Mannor. Model selection in markovian processes. In Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 374–382, 2013.   
+[12] Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301):13–30, 1963.   
+[13] Marcus Hutter. Extreme state aggregation beyond markov decision processes. Theoretical Computer Science, 650:73–91, 2016.   
+[14] Thomas Jaksch, Ronald Ortner, and Peter Auer. Near-optimal regret bounds for reinforcement learning. Journal of Machine Learning Research, 11(Apr):1563–1600, 2010.   
+[15] Nan Jiang, Alex Kulesza, and Satinder Singh. Abstraction selection in model-based reinforcement learning. In International Conference on Machine Learning, pages 179–188, 2015.   
+[16] Tor Lattimore, Marcus Hutter, Peter Sunehag, et al. The sample-complexity of general reinforcement learning. In Proceedings of the 30th International Conference on Machine Learning. Journal of Machine Learning Research, 2013.   
+[17] Lihong Li. A unifying framework for computational reinforcement learning theory. PhD thesis, Rutgers University-Graduate School-New Brunswick, 2009.   
+[18] Lihong Li, Thomas J Walsh, and Michael L Littman. Towards a unified theory of state abstraction for mdps. In ISAIM, 2006.   
+[19] Odalric-Ambrym Maillard, Phuong Nguyen, Ronald Ortner, and Daniil Ryabko. Optimal regret bounds for selecting the state representation in reinforcement learning. In International Conference on Machine Learning, pages 543–551. PMLR, 2013.   
+[20] Sultan Javed Majeed and Marcus Hutter. On q-learning convergence for non-markov decision processes. In IJCAI, pages 2546–2552, 2018.   
+[21] Andreas Maurer and Massimiliano Pontil. Empirical bernstein bounds and sample variance penalization. arXiv preprint arXiv:0907.3740, 2009.   
+[22] Ronald Ortner, Odalric-Ambrym Maillard, and Daniil Ryabko. Selecting near-optimal approximate state representations in reinforcement learning. In International Conference on Algorithmic Learning Theory, pages 140–154. Springer, 2014.   
+[23] Ronald Ortner, Matteo Pirotta, Alessandro Lazaric, Ronan Fruit, and Odalric-Ambrym Maillard. Regret bounds for learning state representations in reinforcement learning. In Advances in Neural Information Processing Systems, pages 12738–12748, 2019.   
+[24] Cosmin Paduraru, Robert Kaplow, Doina Precup, and Joelle Pineau. Model-based reinforcement learning with state aggregation. In 8th European Workshop on Reinforcement Learning, 2008.   
+[25] Martin L Puterman. Markov decision processes: discrete stochastic dynamic programming. John Wiley & Sons, 2014.   
+[26] Satinder P Singh, Tommi Jaakkola, and Michael I Jordan. Learning without state-estimation in partially observable markovian decision processes. In Machine Learning Proceedings 1994, pages 284–292. Elsevier, 1994.   
+[27] Satinder P Singh, Tommi Jaakkola, and Michael I Jordan. Reinforcement learning with soft state aggregation. In Advances in neural information processing systems, pages 361–368, 1995.   
+[28] Alexander L Strehl and Michael L Littman. An analysis of model-based interval estimation for markov decision processes. Journal of Computer and System Sciences, 74(8):1309–1331, 2008.   
+[29] Adrien Ali Taïga, Aaron Courville, and Marc G Bellemare. Approximate exploration through state abstraction. arXiv preprint arXiv:1808.09819, 2018.   
+[30] Tsachy Weissman, Erik Ordentlich, Gadiel Seroussi, Sergio Verdu, and Marcelo J Weinberger. Inequalities for the l1 deviation of the empirical distribution. Hewlett-Packard Labs, Tech. Rep, 2003.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] In Section 6 we describe one the limitations of our work.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] We state our general assumptions on the environment in Section 2 and more specific assumptions in Section 3, Section 3.1 and Section 3.2.   
+(b) Did you include complete proofs of all theoretical results? [Yes] In the Appendix.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [N/A]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [N/A]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [N/A]   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+    {
+        "type": "text",
+        "text": "An Analysis of Abstracted Model-Based Reinforcement Learning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            318,
+            535,
+            334
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Many methods for Model-based Reinforcement Learning (MBRL) provide guaran  \n2 tees for the accuracy of the Markov decision process (MDP) model they can deliver.   \n3 At the same time, state abstraction techniques allow for a reduction of the size of   \n4 an MDP while maintaining a bounded loss with respect to the original problem.   \n5 It may come as a surprise, therefore, that no such guarantees are available when   \n6 combining both techniques, i.e., where MBRL merely observes abstract states. Our   \n7 theoretical analysis shows that abstraction can introduce a dependence between   \n8 samples collected online (i.e., in the real world), which invalidates most results   \n9 for MBRLs in this setting. Collecting samples using a simulator can avoid this   \n10 problem. We conclude that we should be careful when applying MBRL methods   \n11 to abstracted real-world data. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "12 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            542,
+            312,
+            559
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "13 When trying to find good solutions to MDPs using Reinforcement Learning (RL) a fundamental   \n14 problem is the exploration-exploitation dilemma: when to take actions to obtain more information,   \n15 and when to take actions that maximize reward based on the current knowledge. Tabular MBRL   \n16 methods have found good ways to deal with this dilemma [7, 28, 14].   \n17 However, MDPs can be very large, which can be problematic for these methods. One way to deal   \n18 with this is to reduce the size of the MDP. State abstractions are one way to do this [17, 1]. We   \n19 are interested in approximate state abstractions since they allow for greater reductions of the MDP,   \n20 though there is a trade-off with solution quality [1]. Specifically, we assume we have an approximate   \n21 model similarity abstraction function $\\phi$ [1] that maps states to abstract states. The environment   \n22 returns states $s$ , but the agent receives $\\phi ( s )$ , see Figure 1. This setting, which was considered before   \n23 [22, 2], is what we call Abstracted RL, and is the topic of this paper.   \n24 Abstracted RL corresponds to RL in a Partially   \n25 Observable MDP (POMDP), as previously de  \n26 scribed [5]. It is well known that policies for   \n27 POMDPs that only base their action on the last   \n28 observation $\\phi ( s )$ could be arbitrarily bad [26].   \n29 However, when we assume that $\\phi$ is an approx  \n30 imate model similarity abstraction [1] this worst   \n31 case may not apply: Based on the observed ab  \n32 stract states the agent learns an (empirical) abstract model. If we could show that this learned model   \n33 is close to an ‘abstract MDP’ (details in Section 2.2), we could give finite-sample guarantees on the   \n34 performance in the original MDP by combining results from MBRL and abstraction.   \n35 However, in MBRL, to guarantee (with high probability) that the learned model is close to the actual   \n36 environment model, it is typical (e.g., [28, 14]) to use concentration inequalities such as Theorem   \n37 2 from Weissman et al. [30]. But this theorem relies on independent and identically distributed   \n38 (i.i.d.) samples for each state-action pair. In this paper, we analyze online collection of such samples   \n39 in Abstracted RL and show that they are not independent1, which means that most guarantees for   \n40 existing MBRL methods do not hold in the online Abstracted RL setting.2   \n41 On the positive side, when we have access to a simulator, we show how this can be used to collect the   \n42 data such that the typical MBRL guarantees hold and we can learn an accurate model. We discuss   \n43 that emulating this in the real world is possible, but extremely sample inefficient, thus highlighting   \n44 the difficulty of assuming that we would have access to an i.i.d. dataset, as in some earlier works. ",
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+        "img_path": "images/558915b358a6bfb9d141a08438b36473b3a9748fe909488d6952bfbe3fae7ccf.jpg",
+        "image_caption": [
+            "Figure 1: Abstracted RL, the agent observes $\\bar { s } =$ $\\phi ( s )$ instead of $s$ . Image based on Abel et al. [2] "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "45 2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            255,
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+            272
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "46 We assume the environment the agent is acting in can be represented by an infinite horizon MDP   \n47 $M : = \\langle S , A , T , R , \\gamma \\rangle$ . Where $S$ is a finite set of states $s \\in S$ , $A$ a finite set of actions $a \\in A$ , $T$   \n48 a transition function $T ( s ^ { \\prime } | s , a ) = \\operatorname* { P r } ( s ^ { \\prime } | s , a ) .$ $R$ a reward function $R ( s , a )$ which gives the reward   \n49 received when the agent executes action $a$ in state $s$ , and $\\gamma$ the discount factor $( 0 \\leq \\gamma < 1 $ ).   \n50 In RL the goal of the agent is to find an optimal policy $\\pi ^ { * } : S  A$ which maximizes the expectation   \n51 of the discounted cumulative reward. $V ^ { \\pi } ( s )$ denotes the expected value of the discounted cumulative   \n52 reward under policy $\\pi$ starting from state $s$ . Similarly, $Q ^ { \\bar { \\pi } } ( s , a )$ denotes the expected value of the   \n53 discounted cumulative reward when first taking action $a$ from state $s$ and then following policy $\\pi$ . ",
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+    {
+        "type": "text",
+        "text": "54 2.1 Model-Based RL ",
+        "text_level": 1,
+        "bbox": [
+            155,
+            419,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "55 MBRL methods learn a model from the experiences, these are obtained by the agent acting in the   \n56 MDP. The learned model is usually the empirical model, directly based on the experience the agent   \n57 obtains [7, 28, 14]. Per state-action pair the agent stores the next-states reached after taking action $a$   \n58 from state $s$ in sequence $Y _ { s , a } \\colon Y _ { s , a } : \\{ s ^ { \\prime ( 1 ) } , s ^ { \\bar { \\prime } ( 2 ) } , \\cdot \\cdot \\cdot , s ^ { \\prime ( m ) } \\}$ . We use $Y$ to refer to the collection of   \n59 all $Y _ { s , a }$ . From this the empirical, or learned, model $T _ { Y }$ is constructed, that just counts how often we   \n60 have seen the transition to a next-state, and normalizes this: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/95f1ba090291fa38736f0c9aceb27f7acf71544d0d86655d78b35d127871f89f.jpg",
+        "text": "$$\n\\forall _ { s ^ { \\prime } \\in S } T _ { Y } ( s ^ { \\prime } | s , a ) \\triangleq \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\mathbb { 1 } \\{ Y _ { s , a } ^ { ( i ) } = s ^ { \\prime } \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where 61 $\\mathbb { 1 } \\{ \\cdot \\}$ denotes the indicator function of the specified event, i.e., it is 1 if $Y _ { s , a } ^ { ( i ) } = s ^ { \\prime }$ and 0 62 otherwise. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "63 To give finite-sample guarantees on the accuracy of the estimate $T _ { Y }$ , 3 concentration bounds such as   \n64 Theorem 2.1 from Weissman et al. [30] are often used, e.g. in Strehl and Littman [28], Jaksch et al.   \n65 [14]. However, these typically make use of the fact that samples are i.i.d. In most MBRL settings this   \n66 is not a problem under some assumptions, e.g. when the MDP is communicating [25]. In this case   \n67 due to the Markov property the obtained samples are i.i.d.   \n68 In general, of course the hope is that with enough samples the learned model $T _ { Y }$ becomes accurate.   \n69 With accurate we mean that the distance between $T _ { Y } ( \\cdot | s , a )$ and $T ( \\cdot | s , a )$ will be small, where the   \n70 distance is measured using the $L _ { 1 }$ norm, defined as: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/84cf12444964b252df5da85163ee206dc2309fe139093926d68e330f4f3d3222.jpg",
+        "text": "$$\n| | T _ { Y } ( \\cdot | s , a ) - T ( \\cdot | s , a ) | | _ { 1 } \\triangleq \\sum _ { s ^ { \\prime } \\in S } | T _ { Y } ( s ^ { \\prime } | s , a ) - T ( s ^ { \\prime } | s , a ) | .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "71 Part of theorem 2.1 from Weissman et al. [30], slightly reworded, then gives a guarantee of accuracy: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1 (72 $L _ { 1 }$ inequality [30]). Let $\\pmb { Y } _ { s , a } = Y ^ { ( 1 ) } , Y ^ { ( 2 ) } , \\cdot \\cdot \\cdot , Y ^ { ( m ) }$ be i.i.d. random variables 73 distributed according to $T ( \\cdot | s , a )$ . Then, for all $\\epsilon > 0$ , ",
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/768fdf5c33a7eebcefcb45e121e72cec77795a80a3a527d54c9043424543c75a.jpg",
+        "text": "$$\n\\operatorname* { P r } ( | | T _ { Y } ( \\cdot | s , a ) - T ( \\cdot | s , a ) | | _ { 1 } \\geq \\epsilon ) \\leq ( 2 ^ { | S | } - 2 ) e ^ { - \\frac { 1 } { 2 } m \\epsilon ^ { 2 } } .\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "74 In this way, MBRL can upper bound the probability that the learned model, based on $m$ samples, for   \n75 a state-action pair $( s , a ) \\bar { T _ { Y } } ( \\cdot | s , a )$ will be far away $( \\geq \\epsilon )$ from the true model $T ( \\cdot | s , a )$ .   \n76 The situation is more subtle if the MDP is not communicating, i.e., if there exists $s _ { 1 } , s _ { 2 } \\in S$ for which   \n77 there is no deterministic policy that eventually leads from $s _ { 1 }$ to $s _ { 2 }$ . This can create a dependence   \n78 between the samples [28]. Intuitively this happens because, if we look at one particular state-action   \n79 pair $( s , a )$ , there might be a transition to state $s ^ { \\prime }$ such that the probability to return to $s$ is 0. Thus if   \n80 we would have $n$ outcomes of $( s , a )$ we would immediately know that at least $n - 1$ outcomes were   \n81 not state $s ^ { \\prime }$ . Since as soon as we observe $s ^ { \\prime }$ , we know the agent would not be able to return to state $s$ .   \n82 Strehl and Littman [28] show that in this setting it is still possible to use Lemma 3 as an upper bound. ",
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+    },
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+    },
+    {
+        "type": "text",
+        "text": "2.2 State abstraction for given models ",
+        "text_level": 1,
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+        "page_idx": 2
+    },
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+        "text": "84 In the planning setting, where the model is known a priori, a state abstraction can be formulated as   \n85 a grouping or mapping from ground states to abstract states [18]. This is done with an abstraction   \n86 function $\\phi$ , a surjective function that maps from ground states, $s \\in S$ , to abstract states $\\bar { s } \\in \\bar { S }$ :   \n87 $\\phi ( s ) : S  { \\bar { S } }$ . Here $\\bar { S }$ is defined as $\\bar { S } = \\bar { \\{ { \\phi ( s ) \\vert s \\in S \\} } }$ .We use the ¯ notation to refer to the abstract   \n88 space. We slightly overload notation and let $\\bar { s }$ both denote an abstract state as well as the set of   \n89 ground states that map to the abstract state $\\bar { s }$ , i.e., $\\bar { s } = \\{ g \\in S \\mid \\phi ( g ) = \\bar { s } \\}$ , if $\\bar { s } \\in \\bar { S }$ . The use should   \n90 be clear from the context. We define the probability to transition to an abstract state $\\operatorname* { P r } ( \\bar { s } ^ { \\prime } | s , a )$ as   \n91 follows: ",
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+        "text": "$$\n\\operatorname* { P r } ( { \\bar { s } } ^ { \\prime } | s , a ) \\triangleq \\sum _ { s ^ { \\prime } \\in { \\bar { s } } ^ { \\prime } } T ( s ^ { \\prime } | s , a ) .\n$$",
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+        "text": "92 This is a very general form of state abstraction, that clusters together states with different dynamics   \n93 into abstract states. Note that we do assume that the given state abstraction deterministically maps   \n94 states to an abstract state. This in contrast to some related work on problems with block structure   \n95 [10], where a Markov state can lead to multiple observations (abstract states in our terminology) that   \n96 need to be aggregated appropriately to result in a small MDP [4, 10].   \n97 Approximate model similarity abstraction Many different abstraction criteria exist [17], here we   \n98 focus on approximate model similarity abstraction [1]. In this abstraction two states can map to the   \n99 same abstract state if their behavior is similar, i.e., when the reward function and the transitions to   \n100 abstract states are close. Approximate model-similarity is defined as follows: ",
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+        "text": "101 Definition 1. An approximate model-similarity abstraction, $\\phi _ { m o d e l , \\eta } ,$ for fixed $\\eta ,$ satisfies: ",
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+        "text": "$$\n\\begin{array} { r l r } & { \\phi _ { m o d e l , \\eta } ( s _ { 1 } ) = \\phi _ { m o d e l , \\eta } ( s _ { 2 } ) \\ } & { \\Longrightarrow \\ \\forall _ { a } \\left| R ( s _ { 1 } , a ) - R ( s _ { 2 } , a ) \\right| \\le \\eta , } \\\\ & { } & { \\forall _ { \\bar { s } ^ { \\prime } \\in \\bar { S } , a } \\left| \\mathrm { P r } ( \\bar { s } ^ { \\prime } | s _ { 1 } , a ) - \\mathrm { P r } ( \\bar { s } ^ { \\prime } | s _ { 2 } , a ) \\right| \\le \\eta . } \\end{array}\n$$",
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+        "type": "text",
+        "text": "102 From now on we will just refer to $\\phi _ { \\mathrm { m o d e l } , \\eta }$ as $\\phi$ . ",
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+        "text": "103 We note that the abstraction we consider, approximate model-similarity abstraction, is still quite   \n104 generic. It can cluster together states that have different transition and reward functions. However, in   \n105 the online Abstracted RL setting, the differences in dynamics can cause a dependence between the   \n106 samples, as we will show in detail in section 3. E.g. looking at $( { \\bar { s } } , a )$ , the probability that we reach a   \n107 state $s ^ { \\prime }$ depends both on the probability that we reach a particular state $s \\in { \\bar { s } }$ and then state $s ^ { \\prime }$ from $s$   \n108 Returning to abstraction of a given model, it is possible to construct an abstract MDP $\\bar { M } _ { \\omega }$ from   \n109 the model of an MDP $M$ and an abstraction function $\\phi$ , where $\\omega$ is an action-specific 4 weighting   \n110 function, defined as follows:   \n111 Definition 2. We refer to the weight associated with a ground state, $s \\in S$ , and action, $a \\in A$ , by   \n112 $\\omega ( s , a )$ . We have: $\\bar { \\prime } _ { s \\in S , \\ a \\in A } 0 \\leq \\bar { \\omega } ( s , a ) \\leq 1$ and $\\begin{array} { r } { \\sum _ { s ^ { \\prime } \\in \\phi ( s ) } \\omega ( s ^ { \\prime } , a ) = 1 } \\end{array}$ .   \n113 The weighting function can be used to create abstract transition and reward functions, which are a   \n114 weighted average of the ground function. In this way, from $M , \\phi$ and any $\\omega$ we can construct an   \n115 abstract MDP $\\bar { M } _ { \\omega }$ : ",
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+        "text": "Definition 3. Given an MDP 116 $M$ , $\\phi _ { ; }$ , and $\\omega$ , $\\bar { M } _ { \\omega } = \\langle \\bar { S } , A , \\bar { T } _ { \\omega } , \\bar { R } _ { \\omega } , \\gamma \\rangle$ is constructed as: ",
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+        "text": "$$\n\\begin{array} { c } { { \\bar { S } = \\{ \\phi ( s ) \\mid s \\in S \\} , A = A , \\gamma = \\gamma , } } \\\\ { { \\forall _ { \\bar { s } \\in \\bar { S } , \\ a \\in A } \\bar { R } _ { \\omega } ( \\bar { s } , a ) = \\displaystyle \\sum _ { s \\in \\bar { s } } \\omega ( s , a ) R ( s , a ) , } } \\\\ { { \\forall _ { \\bar { s } , \\bar { s } ^ { \\prime } \\in \\bar { S } , \\ a \\in A } \\bar { T } _ { \\omega } ( \\bar { s } ^ { \\prime } | \\bar { s } , a ) = \\displaystyle \\sum _ { s \\in \\bar { s } } \\sum _ { s ^ { \\prime } \\in \\bar { s } ^ { \\prime } } \\omega ( s , a ) T ( s ^ { \\prime } | s , a ) . } } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "An abstract MDP 117 $\\bar { M } _ { \\omega }$ is just an $M D P$ . This means we can use any planning method we like to find an optimal policy 118 $\\bar { \\pi } ^ { * }$ for $\\bar { M } _ { \\omega }$ . ",
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+        "text": "119 What we are interested in is the performance of a policy on the abstract space, when applied on the   \n120 original problem $M$ . Any policy on the abstract space $\\bar { \\pi }$ can be used in $M$ as follows $\\bar { \\pi } ( s ) \\bar { : } = \\bar { \\pi } ( \\phi ( s ) )$ ,   \n121 leading to $V ^ { \\bar { \\pi } ^ { * } }$ . It has been shown that we can upper bound the loss in performance due to using an   \n122 optimal policy for $\\bar { M } _ { \\omega }$ , $\\bar { \\pi } ^ { * }$ in $M$ instead of using the optimal solution for $M$ [8, 1, 29]: ",
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+        "text": "Lemma 2 (Lemma 4 [29]). An approximate model similarity abstraction (Definition $^ { l }$ ), has suboptimality bounded in $\\begin{array} { r } { \\eta \\colon \\forall _ { s \\in S } V ^ { * } \\bar { ( s ) } - V ^ { \\bar { \\pi } ^ { * } } ( s ) \\le \\frac { 2 \\eta + 2 \\gamma ( | \\bar { S } | - 1 ) \\eta } { ( 1 - \\gamma ) ^ { 2 } } } \\end{array}$ . ",
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+        "type": "text",
+        "text": "125 3 Abstracted MBRL and the problem of online data collection ",
+        "text_level": 1,
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+        "text": "126 As explained, we are interested in Abstracted RL, where we have an approximate model similarity   \n127 abstraction function $\\phi$ and an MDP $M$ . The agent acts in $M$ but only observes $\\phi ( s )$ using abstraction   \n128 function $\\phi$ , as in Figure 1. This setting can also be seen as a POMDP, where the states are hidden and   \n129 there is a deterministic observation function, $o = \\phi ( s )$ . However, in contrast to the usual POMDP   \n130 settings, we look for a myopic (memoryless) policy. While we know that in general this can lead to   \n131 arbitrarily bad results [26], in this case the value loss would be bounded in the planning setting by   \n132 Lemma 2. However, now we assume we are in the Abstracted RL setting, and the result for planning   \n133 may not hold for the learned model.   \n134 We assume we know $S , A , R , \\gamma$ , and $\\phi$ (and thus $\\bar { S }$ ), but do not know the transition function.5 Since   \n135 we do not know the transition function we can neither simply do planning on $M$ nor can we construct   \n136 an abstract MDP, using Definition 3, and solve that. Instead, we let the agent interact with $M$ but   \n137 use $\\phi$ to let the agent observe $\\phi ( s )$ , instead of $s$ , and build a learned (abstract) model from the   \n138 observations it obtains. We show the general Abstracted MBRL procedure in Algorithm 1. ",
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+        "text": "The agent collects data for every abstract state-action pair 139 $( { \\bar { s } } , a )$ , which is stored as sequences $\\bar { Y } _ { \\bar { s } , a }$ ",
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+        "img_path": "images/6628f3fecd6dc3ab2e10069e004270ceba8faf0b88beca0068cc71ce91c337b0.jpg",
+        "text": "$$\n\\bar { Y } _ { \\bar { s } , a } : \\{ { \\bar { s } } ^ { \\prime ( 1 ) } , { \\bar { s } } ^ { \\prime ( 2 ) } , \\cdot \\cdot \\cdot , { \\bar { s } } ^ { \\prime ( m ) } \\} .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Similar to before in (1), we construct a learned model 140 $\\bar { T } _ { Y }$ , now looking at the abstract next-states that 141 were reached: ",
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+        "text": "$$\n\\bar { T } _ { Y } ( \\bar { s } ^ { \\prime } | \\bar { s } , a ) \\triangleq \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\mathbb { 1 } \\{ \\bar { Y } _ { \\bar { s } , a } ^ { ( i ) } = \\bar { s } ^ { \\prime } \\} .\n$$",
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+        "type": "text",
+        "text": "142 If this model would be equal, or close, to the transition function $\\bar { T } _ { \\omega }$ of an abstract MDP $\\bar { M } _ { \\omega }$ , for   \n143 some valid $\\omega$ , we could upper bound the loss in performance due to applying learned policy $\\bar { \\pi } ^ { * }$ to $M$   \n144 instead of the optimal policy $\\pi ^ { * }$ [1, 29].   \n146 Our main question is: do the finite-sample model learning guarantees of MBRL algorithms still hold   \n147 in the Abstracted RL setting? ",
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+        "table_body": "<table><tr><td>Algorithm1 Procedure:Abstracted MBRL</td><td>Algorithm2 COLLECTSAMPLES Online</td></tr><tr><td>Input: M,Φ,δ,∈,π Y = COLLECTSAMPLES(M,𝜙,δ,∈π) The sampling results in sequences Ys,a, one for</td><td>Input: M,,δ,∈,π s = initial state // The number of samples m is based on the</td></tr><tr><td>every pair (s, a): Ys,=Φ(s(1）),.，,(s&#x27;(m))</td><td>simulator_analysis,Theorem 1. κ = δ/(|S||Al)</td></tr><tr><td>=(1),...,s(m) for all (s,a,s&#x27;) ∈ S × A × S do</td><td>m=[(2-ln() 2</td></tr><tr><td>m =} end for</td><td>for all s ∈ S do Ys,a=[] end for</td></tr><tr><td>My := (S,A,Ty,R,) π*=Value Iteration(My)</td><td>while min(s,a) |Ys,a| &lt; m do</td></tr><tr><td>Apply to M</td><td>s=(s) a=π(s) s&#x27; = Step(s,a)</td></tr></table>",
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+        "type": "text",
+        "text": "3.1 Online data collection ",
+        "text_level": 1,
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+        "text": "149 In this section we follow the MBRL method from Algorithm 1, collecting samples online using   \n150 Algorithm 2.6 Starting from an initial state the agent follows a policy $\\pi$ . Instead of observing the   \n151 states $s$ , the agent observes abstract states $\\bar { s } = \\phi ( s )$ , see Figure 1.   \n152 We make two important assumptions in order to make analysis possible. We assume that the MDP   \n153 is ergodic [25] 7 and that the policy assigns a positive probability to every action in every abstract   \n154 state. Together this can guarantee that Algorithm 2 can obtain any finite number of samples for every   \n155 abstract state-action pair within finite time. ",
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+        "type": "text",
+        "text": "56 Our question is, can we still use Lemma 1 to guarantee that we learn an accurate model? ",
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+        "type": "text",
+        "text": "157 Since we learn an abstract transition model $\\bar { T } _ { Y }$ , we want to be able to guarantee that this learned   \n158 model will be close to the transition model of some abstract MDP. To define this transition model,   \n159 we first look at how the data is collected.   \n160 In the online data collection, a sample in $\\bar { Y } _ { \\bar { s } , a }$ is drawn when the agent takes action $a$ when it is   \n161 in a ground state $s \\in \\bar { s }$ . Specifically the $i$ -th abstract $\\bar { Y } _ { \\bar { s } , a } ^ { ( i ) } \\ = \\ \\bar { s } ^ { \\prime }$ is drawn from (ground) state   \n162 $X _ { \\bar { s } , a } ^ { ( i ) } = s \\in \\bar { s }$ : ",
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+        "text": "$$\n\\bar { Y } _ { \\bar { s } , a } ^ { ( i ) } \\sim \\mathrm { P r } ( \\cdot | X _ { \\bar { s } , a } ^ { ( i ) } = s , a ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Let 163 $X _ { \\bar { s } , a } = ( X _ { \\bar { s } , a } ^ { ( i ) } ) _ { i = 1 } ^ { m }$ denote the sequence of ground states $s \\in { \\bar { s } }$ from which the agent took action 164 . Each ground state gets a weight according to how often it was sampled from, which we formalize with the weighting function165 $\\begin{array} { r } { \\omega _ { X } \\colon \\forall _ { ( \\bar { s } , a ) , s \\in \\bar { s } } \\omega _ { X } \\bigl ( s , a \\bigr ) \\triangleq \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\mathbb { 1 } \\{ X _ { \\bar { s } , a } ^ { ( i ) } = s \\} } \\end{array}$ . We use $\\omega _ { X }$ to define 166 $\\hat { T } _ { \\omega _ { X } }$ analogous to (8): ",
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+        "text": "$$\n\\forall _ { ( \\bar { s } , a ) , \\bar { s } ^ { \\prime } } \\bar { T } _ { \\omega _ { X } } \\left( \\bar { s } ^ { \\prime } | \\bar { s } , a \\right) = \\sum _ { s \\in \\bar { s } } \\omega _ { X } ( s , a ) \\sum _ { s ^ { \\prime } \\in \\bar { s } ^ { \\prime } } T ( s ^ { \\prime } | s , a ) .\n$$",
+        "text_format": "latex",
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+        "text": "167 We want to have a concentration inequality to provide bounds on the deviation of the learned model   \n168 $\\bar { T } _ { Y }$ from $\\hat { T } _ { \\omega _ { X } }$ , we refer to this inequality as the abstract L1 inequality, similar in form to (3): ",
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+    {
+        "type": "equation",
+        "img_path": "images/0a218e49f0c3a309d8a6334cbee0d0b8d776ddbce3c0e85b3bf1e357a398d099.jpg",
+        "text": "$$\nP ( | \\bar { T } _ { Y } ( \\cdot | \\bar { s } , a ) - \\bar { T } _ { \\omega _ { X } } ( \\cdot | \\bar { s } , a ) | _ { 1 } \\geq \\epsilon ) \\leq \\delta ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where 169 $\\bar { T } _ { Y } ( \\cdot | \\bar { s } , a )$ is defined according to (10) and $\\hat { T } _ { \\omega _ { X } }$ according to (12). ",
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+        "type": "text",
+        "text": "170 If we could directly obtain i.i.d. samples from $\\hat { T } _ { \\omega _ { X } }$ and base our learned model $\\bar { T } _ { Y }$ on the obtained   \n171 samples, then we would be able to show that the abstract L1 inequality holds by applying Lemma 1.   \n172 Since in this case, we would have $m$ i.i.d. samples per abstract state-action pair, distributed according   \n173 to $\\bar { T } _ { \\omega _ { X } } ( \\cdot | \\bar { s } , a )$ . ",
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+    {
+        "type": "text",
+        "text": "the samples. Since every sample 175 in (11). These can have different176 $\\bar { Y } ^ { ( i ) }$ was obtainributions if g action . $a$ from state ${ \\cal X } _ { \\bar { s } , a } ^ { ( i ) } = s \\in \\bar { s }$ $X _ { \\bar { s } , a } ^ { ( i ) } \\neq X _ { \\bar { s } , a } ^ { ( j ) }$ ",
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+        "text": "77 Non Identically Distributed While Lemma 1 assumes i.i.d. random variables, we show that it also   \n78 holds when the random variables are independent but not (necessarily) identically distributed.   \n179 Lemma 3. Let $\\begin{array} { r l r } { X _ { \\bar { s } , a } } & { { } = } & { s _ { 1 } , \\cdot \\cdot \\cdot , s _ { m } } \\end{array}$ be a sequence of states $ { \\mathcal { S } } _ { s } \\in  { \\mathcal { S } } _ { } 0$ and let   \n180 $\\bar { { \\bf Y } } _ { \\bar { s } , a } = \\bar { { \\cal Y } } ^ { ( 1 ) } , \\bar { { \\cal Y } } ^ { ( 2 ) } , \\cdot \\cdot \\cdot , \\bar { { \\cal Y } } ^ { ( m ) }$ be independent random variables distributed according to   \n181 $\\mathrm { P r } ( \\cdot | s _ { 1 } , a ) , \\cdot \\cdot \\cdot , \\mathrm { P r } ( \\cdot | s _ { m } , a )$ (Eqn. 4). Then, for all $\\epsilon > 0$ , ",
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+        "img_path": "images/4577cfc46f9744b4b7bc030004cb3d2202bd64a049c92b0335ca750acab4998a.jpg",
+        "text": "$$\n\\begin{array} { r } { \\operatorname* { P r } ( | | \\bar { T } _ { Y } ( \\cdot | \\bar { s } , a ) - \\bar { T } _ { \\omega _ { X } } ( \\cdot | \\bar { s } , a ) | | _ { 1 } \\geq \\epsilon ) \\leq ( 2 ^ { | \\bar { S } | } - 2 ) e ^ { - \\frac { 1 } { 2 } m \\epsilon ^ { 2 } } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "182 The proof can be found in Appendix B. It mostly follows the proof by Weissman et al. [30], which uses   \n183 Hoeffding’s inequality [12] and the union bound [6].8 Lemma 3 shows that the fact that Hoeffding’s   \n184 inequality does not need identically distributed data can be carried over to the setting from Lemma 1. ",
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+        "type": "text",
+        "text": "185 Independence We may be tempted to assume the samples are independent, i.e., ",
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+        "type": "equation",
+        "img_path": "images/5418247315d88145e7104307a7aa737eb14484cf97de62d262176d68b832fd14.jpg",
+        "text": "$$\n\\forall _ { \\bar { s } _ { 1 } , \\cdots , \\bar { s } _ { m } \\in ( \\bar { S } ) ^ { m } } \\operatorname* { P r } ( \\bar { Y } _ { \\bar { s } , a } ^ { ( 1 ) } = \\bar { s } _ { 1 } , \\cdots , \\bar { Y } _ { \\bar { s } , a } ^ { ( m ) } = \\bar { s } _ { m } ) = \\operatorname* { P r } ( \\bar { Y } _ { \\bar { s } , a } ^ { ( 1 ) } = \\bar { s } _ { 1 } ) \\cdot \\cdot \\cdot P ( \\bar { Y } _ { \\bar { s } , a } ^ { ( m ) } = \\bar { s } _ { m } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "86 however, this may not be the case: ",
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+    {
+        "type": "text",
+        "text": "Observation 1. When collecting samples online, i.e., based on Algorithm 2, the samples cannot be assumed to be independent. ",
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+    {
+        "type": "text",
+        "text": "The following counterexample illustrates this. ",
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+        "text": "190 Counterexample To show that the samples may not be indepen  \n191 dent, we will give a counterexample. We use the example MDP and   \n192 abstraction in Figure 2, where we have 4 (ground) states, 3 abstract   \n193 states and only 1 action. We look at the transition probability from   \n194 abstract state $A$ $, { \\bar { T } } _ { Y } ( \\cdot | A )$ .   \n195 We will consider two samples and show that for at least one com  \n196 bination of $\\bar { s } _ { 1 }$ and $\\bar { s } _ { 2 }$ the samples are not independent. Consider   \n197 $\\bar { s } _ { 1 } = \\bar { s } _ { 2 } = B$ . That is, the first two times that we experience a   \n198 transition from the abstract state $A$ , we end up in $B$ . ",
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+    {
+        "type": "text",
+        "text": "199 the starting state. Then we have and $\\mathrm { P r } ( \\bar { Y } _ { A } ^ { ( 1 ) } = B ) =$ $\\operatorname* { P r } ( B | 1 ) = 0 . 6$ ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/07ac1775b5caa755d2bf12237cba58eaeeb7b568dc02e023a859d5bc1af069a2.jpg",
+        "text": "$$\n\\operatorname* { P r } ( \\bar { Y } _ { A } ^ { ( 2 ) } = B ) = \\sum _ { \\bar { s } \\in \\bar { \\cal S } } \\operatorname* { P r } ( \\bar { Y } _ { A } ^ { ( 2 ) } = B | \\bar { Y } _ { A } ^ { ( 1 ) } = \\bar { s } ) \\operatorname* { P r } ( \\bar { Y } _ { A } ^ { ( 1 ) } = \\bar { s } )\n$$",
+        "text_format": "latex",
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+        "type": "image",
+        "img_path": "images/596348f5df8a7d9a095f28522a8ea9d1b1aecc37093cd955880be0e569bee57b.jpg",
+        "image_caption": [
+            "Figure 2: Simple MDP, with only 1 action, and abstraction. The small circles are ground states (1,2,3,4). A, B and C are the abstract states. The numbers along the arrows show the transition probabilities, e.g. $P ( 3 | 1 ) = 0 . 6 $ . "
+        ],
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+        "type": "equation",
+        "img_path": "images/cc0866f4d0d2c49c9e73c48061eaf9a06b7abc4482efedbc35fe7927f8b0745c.jpg",
+        "text": "$$\n= 0 + 0 . 6 \\cdot 0 . 6 + 0 . 4 \\cdot 0 . 4 = 0 . 5 2 .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "So then we end up with: 201 $\\operatorname* { P r } ( \\bar { Y } _ { A } ^ { ( 1 ) } = B ) \\operatorname* { P r } ( \\bar { Y } _ { A } ^ { ( 2 ) } = B ) = 0 . 6 \\cdot 0 . 5 2 = 0 . 3 2 1$ . And for the joint probability: 202 $\\mathrm { P r } ( \\bar { Y } _ { A } ^ { ( 1 ) } = B , \\bar { Y } _ { A } ^ { ( 2 ) } = B ) = \\mathrm { P r } ( \\bar { Y } _ { A } ^ { ( 1 ) } = B ) \\mathrm { P r } ( \\bar { Y } _ { A } ^ { ( 2 ) } = B | \\bar { Y } _ { A } ^ { ( 1 ) } = B ) = 0 . 6 \\cdot 0 . 6 = 1$ 203 0.36. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Thus we have that 204 independent. Lead $\\operatorname* { P r } ( \\bar { Y } _ { A } ^ { ( 1 ) } = B , \\bar { Y } _ { A } ^ { ( 2 ) } = B ) \\not = \\operatorname* { P r } ( \\bar { Y } _ { A } ^ { ( 1 ) } = B ) \\operatorname* { P r } ( \\bar { Y } _ { A } ^ { ( 2 ) } = B )$ , the samples are not ",
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+    {
+        "type": "text",
+        "text": "6 Observation 2. As independence cannot be guaranteed, Lemmas 1 and 3 cannot be readily applied to show that the abstract $L l$ inequality holds. ",
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+    {
+        "type": "text",
+        "text": "3.2 Simulator data collection ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Here we also want to give a guarantee in the form of the abstract L1 inequality from (13). While in the previous section we found this was not possible because the samples were dependent, here we assume that we have access to a simulator. To some extent this is not surprising, but to the best of our knowledge, this is the first work that explicitly shows how to combine MBRL and abstraction, using a simulator. We assume that this allows us to select (or move to) any state and draw a sample from its transition function. This we call the independent samples assumption: ",
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+    {
+        "type": "text",
+        "text": "Assumption 1 (Independent samples). We assume we can obtain independent samples, e.g. for any state-action pair $( s , a )$ we can draw samples directly from its transition function $T ( \\cdot | s , a )$ . ",
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+        "type": "text",
+        "text": "217 In case a simulator of the MDP is available this is a reasonable assumption. For every $( \\bar { s } , a )$ the   \n218 simulator sampling procedure (Algorithm 3 in Appendix B) selects a prototype $x _ { \\bar { s } , a } \\in \\bar { s }$ to sample   \n219 from. We define a weighting function $\\omega _ { x } ( s , a )$ that has weight 1 if $s$ is the prototype $x _ { \\bar { s } , a }$ and 0   \n220 otherwise: ",
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+        "type": "equation",
+        "img_path": "images/186e27a55ec0c3d9b9b8bedbbbe881d04c1caa1b0df61ee8aa957460eb148dc5.jpg",
+        "text": "$$\n\\forall _ { ( \\bar { s } , a ) , s \\in \\bar { s } } \\omega _ { x } ( s , a ) \\triangleq \\mathbb { 1 } \\{ s = x _ { \\bar { s } , a } \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "221 Then we use this $\\omega _ { x }$ to define the abstract transition function $\\hat { T } _ { \\omega _ { X } }$ according to (8). $\\bar { T } _ { \\omega _ { x } } ( \\bar { s } ^ { \\prime } | \\bar { s } , a ) =$   \n222 $\\begin{array} { r } { \\sum _ { s ^ { \\prime } \\in \\bar { s } ^ { \\prime } } T \\bigl ( s ^ { \\prime } | s = x _ { \\bar { s } , a } , a \\bigr ) } \\end{array}$ . This way the samples that we collect for one pair $( { \\bar { s } } , a )$ are i.i.d., they are   \n223 independent because of our assumption of independent samples and identically distributed because   \n224 we sample from the prototype. This means we can use Lemma 1. We show that with the simulator   \n225 we can combine MBRL and abstraction, and still learn an accurate model, that is, we can guarantee   \n226 that $\\bar { T } _ { Y }$ will be close to $\\hat { T } _ { \\omega _ { x } }$ , with high probability: ",
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+    {
+        "type": "text",
+        "text": "Theorem 1. Under assumption $^ { l }$ , and following the procedure in Algorithm $^ { l }$ , with the data collection from Algorithm $^ 3$ (Appendix $B$ ), with inputs $| \\bar { S } | , A , \\epsilon$ and $\\delta$ . For $\\bar { T } _ { Y }$ constructed by the algorithm we have that with probability $1 - \\delta$ , the following holds: ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "equation",
+        "img_path": "images/dc17c2bd609e77a442df5ddfefb1c2beecc2f259987708231a605951aa2970f2.jpg",
+        "text": "$$\n\\forall _ { ( \\bar { s } , a ) } \\vert \\vert \\bar { T } _ { Y } \\big ( \\cdot \\vert \\bar { s } , a \\big ) - \\bar { T } _ { \\omega _ { x } } \\big ( \\cdot \\vert \\bar { s } , a \\big ) \\vert \\vert _ { 1 } \\leq \\epsilon .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "230 By Assumption 1 we can obtain any number of independent samples for each abstract state action   \n231 pair $( \\bar { s } , a )$ . Using Lemma 1 we can then derive the number of samples $m$ that is required for each   \n232 pair $( \\bar { s } , a )$ such that, after applying a union bound, we obtain the bounds in (19). The full proof can   \n233 be found in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "234 4 Related work ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "235 There is a lot of work that considers the combination of abstraction with either planning or (online)   \n236 RL. In a lot of these works the dependence of samples that arises in Abstracted RL is not an issue   \n237 due to various assumptions, similarly to how in MBRL dependence of samples is often not an issue   \n238 because of the Markov property and the assumption that the MDP is communicating [25]. Often   \n239 this is either due the assumption that data has been obtained i.i.d., the specific type of abstraction, or   \n240 because access to an MDP model is assumed.   \n241 One paper that does give a result for the Abstracted RL setting is the work by Abel et al. [2].   \n242 They show that in this setting R-MAX [7] no longer maintains its guarantees when paired with any   \n243 type of state-abstraction function, though their example is specifically for approximate Q-function   \n244 abstractions. They also show that the expected trajectory of a learning agent in a constructed   \n245 abstract MDP (Definition 3) is not the same as in Abstracted RL. Their work makes clear there is a   \n246 complication when combining MBRL and abstraction, here we further investigated the cause of this   \n247 complication, the dependence between samples.   \n248 For planning in constructed abstract MDPs, some main results for exact state-abstractions come   \n249 from Li et al. [18] and for approximate state-abstractions from Abel et al. [1]. The results from Abel   \n250 et al. [1] allow for quantifying an upper bound on performance for policies found in a constructed ",
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+        "type": "text",
+        "text": "abstract MDP, as in section 2.2. Taïga et al. [29] build on this by giving a result for performing RL on top of the constructed abstract MDP. They provide upper bounds for this setting when using MBIE with exploratory bonus (MBIE-EB) [28]. In addition, they give an example to show that in this combination you cannot guarantee optimal performance in the original MDP. Still, they show that an upper bound on the loss in value can be given. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Both Paduraru et al. [24] and Jiang et al. [15] deal with the issue of dependence by making the explicit assumption that samples are obtained i.i.d. Paduraru et al. [24] consider the setting where we are given a dataset for a continuous domain and then use discretization to aggregate states into abstract states. They then give PAC-style guarantees on the learned abstract model and the value that a policy based on this model can achieve in the real MDP. Instead of using the L1 deviation bound from Weissman et al. [30], Paduraru et al. [24] use a similar bound for i.i.d. samples by Devroye and Gyorfi [9], which requires a minimum amount of samples. Another difference is that their results calculate the probability that the model will be $\\epsilon$ -accurate given a fixed dataset. They assume that the data has been gathered i.i.d., but our Lemma 3 shows that merely independent data would be enough. At the same time, our results show that when we collect data online in the Abstracted RL setting, their guarantees will not hold. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Jiang et al. [15] operate in the abstraction selection setting, where the agent is provided with a set of abstraction functions (state representations). They do not assume that any of the abstraction functions results in a Markov model, but they do assume a given dataset, with data that was collected i.i.d. They give a bound directly on how accurate the Q-values based on the (implicitly) learned model will be, rather than on the accuracy of the model itself. As we showed, the assumption that the data is i.i.d. is not a trivial assumption, since it means the data cannot just have been collected online. They do mention that samples will not be strictly independent if a fixed exploration policy is used to collect data but do not mention what the implications are. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "There are quite a few other papers in the abstraction selection setting, several of these assume that the given set of state representations contains a Markov model [11, 19, 23]. Hallak et al. [11] give asymptotic guarantees for selecting the correct model and on building an exact MDP model. The assumption that there is an MDP model in the given set of representations is crucial in their analysis since for this ‘true model’ the samples are i.i.d. Similarly, both Maillard et al. [19] and Ortner et al. [23] also assume that the given set of state representations contains a Markov model. They create an algorithm for which they obtain regret bounds, their analysis also makes use of the Markov representation. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Other work in the abstraction selection setting does not assume that the set of abstraction functions contains a Markov model [16, 22]. However, Ortner et al. [22] use Theorem 2.1 from Weissman et al. [30] that requires i.i.d. samples, which we have shown here cannot be guaranteed in this setting. Lattimore et al. [16] operate in a setting more general than MDPs, where the dynamics of the true environment depend arbitrarily on a history of actions, rewards, and observations. The agent gets as input a finite set of environments, one of which is the true environment. Since the input includes the full model of each environment, the agent does not have to learn a transition model. Instead, to obtain regret bounds, they directly compare the rewards the agent obtains to the expected rewards of the given environments and eliminate environments that are implausible given the observed rewards. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Another way to deal with the issue of dependence is by looking at convergence in the limit [27, 13, 20]. Singh et al. [27] give an asymptotic result for the convergence of Q-learning and TD(0) in MDPs with soft state aggregation. Soft state aggregation means that a state $s$ belongs to a cluster $x$ with some probability $P ( x | s )$ , this means a state $s$ can belong to several clusters. The state-abstraction functions we consider are a special case of this, where each state is part of exactly one abstract state (or cluster). Their result relies on having a stationary policy that assigns a non-zero probability to every action in every state and the assumption that the MDP is ergodic. Together these imply there is a limiting state distribution, and using this they show convergence asymptotically. Our main interest is in finite-samples guarantees with policies that change due to exploration, whereas this work gives convergence guarantees in the limit using a fixed policy. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Hutter [13] gives a variety of results focusing on both approximate and exact abstractions in environments without MDP assumptions. Several of these are in the planning setting, similar to those of Abel et al. [1]. Most relevant for us is their Theorem 12, which for online RL shows convergence in the limit of the empirical transition function under weak conditions, e.g. if the abstract process itself ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "306 is an MDP. Under this condition however the problem reduces to RL in an (abstract) MDP, rather   \n307 than Abstracted RL.   \n308 Majeed and Hutter [20] build on the work by Hutter [13] and focus on the combination of model-free   \n309 RL and exact abstraction. They show that, under the condition of state uniformity, q-learning can be   \n310 shown to converge in the limit to the optimal solution. State uniformity means that histories that are   \n311 grouped together have the same optimal q-values. In contrast to our setting, they look at an exact   \n312 abstraction, extending it to approximate aggregation was left as an open question.   \n313 Other related work is in the area of MDPs with rich observations or block structure [4, 10]. However,   \n314 in that setting each observation can be generated only from a single hidden state, which means that   \n315 the issue of non-i.i.d. data due to abstraction does not arise. In contrast, in our setting multiple   \n316 (hidden) states generate the same observation. Azizzadenesheli et al. [4] state their setting can be   \n317 seen as an aggregation problem, where the observations can be aggregated to form a small (latent)   \n318 MDP. But in our case, we do not try to learn the MDP (as it is not small). Du et al. [10] describe   \n319 that their setting is similar to exact model similarity (or bisimulation), but we focus on approximate   \n320 model similarity which is what introduces the problems as described here. ",
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+    {
+        "type": "text",
+        "text": "321 5 Discussion ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "322 When collecting samples online in Abstracted RL, there is a potential dependence between samples,   \n323 meaning we cannot use the typically used concentration results that assume i.i.d. samples, e.g.   \n324 Theorem 2.1 from Weissman et al. [30], the empirical Bernstein inequality [3, 21] or the Chernoff   \n325 bound. In case the samples are only weakly dependent, it may be that concentration inequalities   \n326 for (weakly) dependent variables are a viable alternative through which we can come to guarantees   \n327 on the learned model. Alternatively, it may be possible to change the sampling process to ensure   \n328 independent samples. One way to ensure independent samples is to, as in the simulator setting, select   \n329 a prototype state and only use the samples collected from this state. Though in this case, we will be   \n330 discarding information when we reach a state $s \\in { \\bar { s } }$ that is not the prototype.   \n331 Our assumption on the simulator that we can go/reset to any state to draw samples from it can be   \n332 relaxed, though it may mean that the procedure takes considerably more time. Consider the case   \n333 where we cannot just reset the simulator to the state $s$ from which we want to sample, and instead, it   \n334 would behave like the MDP. In this case, we would have to take the right actions to arrive at the state   \n335 $s$ from which we would like to sample. Since we assume we do not know $T$ , this may take a long   \n336 time. This also shows the difficulty of assuming that in the MBRL setting somehow have access to an   \n337 i.i.d. dataset, as has been assumed in some earlier work [24, 15]. ",
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+        "type": "text",
+        "text": "338 6 Conclusion ",
+        "text_level": 1,
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+        "text": "339 We analyzed Abstracted RL: the combination of MBRL and state abstraction when the model of   \n340 the MDP is not available. We have shown that in Abstracted RL samples obtained online cannot   \n341 be assumed to be independent. Since many current guarantees from MBRL methods rely on this   \n342 assumption, their guarantees do not hold in this setting. And in fact, no current methods exist that   \n343 give (correct) finite-sample quality guarantees for the models learned in this setting. This also means   \n344 that current results that rely on an i.i.d. assumption cannot be readily transferred to the Abstracted   \n345 RL setting. ",
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+    {
+        "type": "text",
+        "text": "In addition, we show that with a simulator, since we can draw independent samples, it is still possible to give guarantees on the accuracy of the model. However, having access to a simulator may often not be possible. An important step is to see if the MBRL guarantees can be adapted to Abstracted RL for online sample collection. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+            171,
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+            266,
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+        ],
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+    },
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+        "type": "text",
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+        "text": "[2] David Abel, Dilip Arumugam, Lucas Lehnert, and Michael Littman. State abstractions for lifelong reinforcement learning. In International Conference on Machine Learning, pages 10–19, 2018.   \n[3] Jean-Yves Audibert, Rémi Munos, and Csaba Szepesvári. Tuning bandit algorithms in stochastic environments. In International conference on algorithmic learning theory, pages 150–165. Springer, 2007.   \n[4] Kamyar Azizzadenesheli, Alessandro Lazaric, and Animashree Anandkumar. Reinforcement learning in rich-observation mdps using spectral methods. arXiv preprint arXiv:1611.03907, 2016.   \n[5] Aijun Bai, Siddharth Srivastava, and Stuart J Russell. Markovian state and action abstractions for mdps via hierarchical mcts. In IJCAI, pages 3029–3039, 2016.   \n[6] George Boole. An investigation of the laws of thought: on which are founded the mathematical theories of logic and probabilities. Dover Publications, 1854.   \n[7] Ronen I Brafman and Moshe Tennenholtz. R-max-a general polynomial time algorithm for near-optimal reinforcement learning. Journal of Machine Learning Research, 3(Oct):213–231, 2002.   \n[8] Richard Dearden and Craig Boutilier. Abstraction and approximate decision-theoretic planning. Artificial Intelligence, 89(1-2):219–283, 1997.   \n[9] L. Devroye and L. Gyorfi. Nonparametric Density Estimation: The L1 View. Wiley Interscience Series in Discrete Mathematics. Wiley, 1985.   \n[10] Simon Du, Akshay Krishnamurthy, Nan Jiang, Alekh Agarwal, Miroslav Dudik, and John Langford. Provably efficient rl with rich observations via latent state decoding. In International Conference on Machine Learning, pages 1665–1674. PMLR, 2019.   \n[11] Assaf Hallak, Dotan Di-Castro, and Shie Mannor. Model selection in markovian processes. In Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 374–382, 2013.   \n[12] Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301):13–30, 1963.   \n[13] Marcus Hutter. Extreme state aggregation beyond markov decision processes. Theoretical Computer Science, 650:73–91, 2016.   \n[14] Thomas Jaksch, Ronald Ortner, and Peter Auer. Near-optimal regret bounds for reinforcement learning. Journal of Machine Learning Research, 11(Apr):1563–1600, 2010.   \n[15] Nan Jiang, Alex Kulesza, and Satinder Singh. Abstraction selection in model-based reinforcement learning. In International Conference on Machine Learning, pages 179–188, 2015.   \n[16] Tor Lattimore, Marcus Hutter, Peter Sunehag, et al. The sample-complexity of general reinforcement learning. In Proceedings of the 30th International Conference on Machine Learning. Journal of Machine Learning Research, 2013.   \n[17] Lihong Li. A unifying framework for computational reinforcement learning theory. PhD thesis, Rutgers University-Graduate School-New Brunswick, 2009.   \n[18] Lihong Li, Thomas J Walsh, and Michael L Littman. Towards a unified theory of state abstraction for mdps. In ISAIM, 2006.   \n[19] Odalric-Ambrym Maillard, Phuong Nguyen, Ronald Ortner, and Daniil Ryabko. Optimal regret bounds for selecting the state representation in reinforcement learning. In International Conference on Machine Learning, pages 543–551. PMLR, 2013.   \n[20] Sultan Javed Majeed and Marcus Hutter. On q-learning convergence for non-markov decision processes. In IJCAI, pages 2546–2552, 2018.   \n[21] Andreas Maurer and Massimiliano Pontil. Empirical bernstein bounds and sample variance penalization. arXiv preprint arXiv:0907.3740, 2009.   \n[22] Ronald Ortner, Odalric-Ambrym Maillard, and Daniil Ryabko. Selecting near-optimal approximate state representations in reinforcement learning. In International Conference on Algorithmic Learning Theory, pages 140–154. Springer, 2014.   \n[23] Ronald Ortner, Matteo Pirotta, Alessandro Lazaric, Ronan Fruit, and Odalric-Ambrym Maillard. Regret bounds for learning state representations in reinforcement learning. In Advances in Neural Information Processing Systems, pages 12738–12748, 2019.   \n[24] Cosmin Paduraru, Robert Kaplow, Doina Precup, and Joelle Pineau. Model-based reinforcement learning with state aggregation. In 8th European Workshop on Reinforcement Learning, 2008.   \n[25] Martin L Puterman. Markov decision processes: discrete stochastic dynamic programming. John Wiley & Sons, 2014.   \n[26] Satinder P Singh, Tommi Jaakkola, and Michael I Jordan. Learning without state-estimation in partially observable markovian decision processes. In Machine Learning Proceedings 1994, pages 284–292. Elsevier, 1994.   \n[27] Satinder P Singh, Tommi Jaakkola, and Michael I Jordan. Reinforcement learning with soft state aggregation. In Advances in neural information processing systems, pages 361–368, 1995.   \n[28] Alexander L Strehl and Michael L Littman. An analysis of model-based interval estimation for markov decision processes. Journal of Computer and System Sciences, 74(8):1309–1331, 2008.   \n[29] Adrien Ali Taïga, Aaron Courville, and Marc G Bellemare. Approximate exploration through state abstraction. arXiv preprint arXiv:1808.09819, 2018.   \n[30] Tsachy Weissman, Erik Ordentlich, Gadiel Seroussi, Sergio Verdu, and Marcelo J Weinberger. Inequalities for the l1 deviation of the empirical distribution. Hewlett-Packard Labs, Tech. Rep, 2003. ",
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parse/train/hMY6nm9lld/hMY6nm9lld.md

@@ -0,0 +1,290 @@
+# Predicting Molecular Conformation via Dynamic Graph Score Matching
+
+Shitong Luo\*1, Chence $\mathbf { S h i ^ { * 2 , 3 } }$ , Minkai $\mathbf { X } \mathbf { u } ^ { 2 , 3 }$ , Jian Tang2,4,5 1Peking University 2Mila - Québec AI Institute 3Université de Montréal $^ { 4 } \mathrm { H E C }$ Montréal 5CIFAR AI Research Chair luost@pku.edu.cn , chence.shi@umontreal.ca minkai.xu@umontreal.ca , jian.tang@hec.ca
+
+# Abstract
+
+Predicting stable 3D conformations from 2D molecular graphs has been a longstanding challenge in computational chemistry. Recently, machine learning approaches have demonstrated very promising results compared to traditional experimental and physics-based simulation methods. These approaches mainly focus on modeling the local interactions between neighboring atoms on the molecular graphs and overlook the long-range interactions between non-bonded atoms. However, these non-bonded atoms may be proximal to each other in 3D space, and modeling their interactions is of crucial importance to accurately determine molecular conformations, especially for large molecules and multi-molecular complexes. In this paper, we propose a new approach called Dynamic Graph Score Matching (DGSM) for molecular conformation prediction, which models both the local and long-range interactions by dynamically constructing graph structures between atoms according to their spatial proximity during both training and inference. Specifically, the DGSM directly estimates the gradient fields of the logarithm density of atomic coordinates according to the dynamically constructed graphs using score matching methods. The whole framework can be efficiently trained in an end-to-end fashion. Experiments across multiple tasks show that the DGSM outperforms state-of-theart baselines by a large margin, and it is capable of generating conformations for a broader range of systems such as proteins and multi-molecular complexes.
+
+# 1 Introduction
+
+Graph-based representations of molecules has become prevalent in a variety of tasks such as property prediction [13, 31] and molecule generation [17, 33, 45]. However, a more natural and intrinsic representation of a molecule is its 3D geometry or conformation, which represents a molecule as a set of 3D coordinates. The 3D representation of molecules is central to many tasks, such as molecular properties prediction and virtual screening. Nevertheless, determining the conformation of a molecule remains a challenging task — both computational approaches, e.g., molecular dynamics (MD) [9], and experimental approaches, e.g., crystallography, are expensive and time-consuming.
+
+Recently, machine learning approaches have demonstrated promising performance for molecular conformation generation. Pioneering methods such as GRAPHDG [36] and CGCF [43] first predict interatomic distances between bonded atoms and then solve 3D coordinates from the predicted distances via a post-processing algorithm. Very recently, Shi et al. proposed the CONFGF [34], which employs the score matching technique [38] to learn pseudo-forces between bonded atoms and iteratively applies the forces to a randomly initialized 3D structure until convergence. CONFGF gets rid of the two-stage fashion in prior works and significantly improves the performance. Nevertheless, these approaches have a common major limitation — they mainly focus on modeling the local interactions between bonded atoms defined by the input molecular graphs but fail to capture long-range interactions between non-bonded atoms1, as they only model distances (or gradients) between bonded atoms. While in molecular mechanics, the potential energy of a molecule that alters conformations can be modeled as a sum of four parts [22]:
+
+![](images/40aa5ba4168cefb1ae91527e2aef598f8197774fa9204092719b254b4b1a32df.jpg)  
+Figure 1: Three molecular systems where long-range interactions are crucial for their conformations.
+
+$$
+E = E _ { \mathrm { b o n d } } + E _ { \mathrm { a n g l e } } + E _ { \mathrm { t o r s i o n } } + E _ { \mathrm { n o n - b o n d e d } } ,
+$$
+
+where $E _ { \mathrm { b o n d } }$ , $E _ { \mathrm { a n g l e } }$ , and $E _ { \mathrm { t o r s i o n } }$ model local interactions between bonded atoms, which are modeled in previous methods [34, 36, 43]. Long-range interactions between non-bonded atoms, denoted as $E _ { \mathrm { n o n - b o n d e d } }$ , are also non-trivial, which shape the molecular geometry via non-negligible electrostatic forces or van der Waals forces, etc. For multi-molecular complexes, non-bonded interactions dominate complexes’ geometry. An ideal solution to conformation generation should therefore capture both the local and long-range interactions. In Figure 1, we present three typical molecular systems where long-range interactions play a key role in determining their conformations.
+
+To tackle the aforementioned challenge of modeling long-range interactions, in this paper, we propose the Dynamic Graph Score Matching (DGSM) for molecular conformation generation, following the principle of CONFGF [34] that learns the gradients of the logarithm density of atomic coordinates. Instead of relying on the static input molecular graph as existing work, the basic idea is to dynamically construct graph structures between atoms based on their spatial proximity during both training and inference. This allows the model to (1) dynamically learn molecular graph representations with evolved graph structures that take long-range interactions into consideration, and (2) dynamically determine a set of interatomic distances that contribute to gradients of the current atomic coordinates. Specifically, the edges in the dynamic graph consist of two parts. The first part of edges are determined by covalent bonds, which capture local interactions between atoms $\mathrm { E _ { b o n d } }$ , $E _ { \mathrm { a n g l e } }$ and $E _ { \mathrm { t o r s i o n } } \mathrm { , }$ ). The second part of edges are determined dynamically by spatial proximity between atoms at each training or sampling step, i.e., two atoms are connected as long as they are proximal, no matter whether they are bonded. Such a strategy is able to effectively capture non-local interactions $( E _ { \mathrm { n o n - b o n d e d } } )$ since the magnitude of long-range interactions is inversely correlated with distances between atoms [30]. It remains meanwhile scalable as we avoid connecting all the atom-pairs, which has quadratic complexity. In addition, modeling non-bonded interactions enable the model to sample conformations for multi-molecular complexes, which represent a broader range of problems.
+
+We conduct extensive experiments and compare DGSM against previous state-of-the-art methods on both standard conformation generation and property prediction tasks. Numerical results show that DGSM outperforms previous methods by a clear margin, confirming the benefit of modeling long-range interactions. Besides, to further demonstrate the advantage of DGSM, we bring attention to two more challenging tasks — protein sidechain conformation prediction and multi-molecular complex structure prediction. These two new tasks represent two classes of practical challenges: predicting structures for macro-molecules and multi-molecular complexes.
+
+# 2 Related Work
+
+Prior works on conformation generation mainly rely on molecular dynamics (MD) [9], where new conformations are sequentially generated based on an initial conformation and a physical model for interatomic potentials [25, 27]. Although capable of accurately sampling equilibrium conformations, these methods are computationally intensive, especially for large molecular systems [2, 35], e.g., proteins. Another category of approaches leverage distance geometry [8] and fix distances between atoms to idealized values heuristically [4], which are much faster but less accurate.
+
+Recently, a variety of deep generative models have been proposed for molecular conformation generation, which strike a good balance between computational efficiency and accuracy. Among these methods, Mansimov et al. [24] first propose a variational autoencoder to directly generate 3D atomic coordinates. Albeit simple, this method fails to model the roto-translation equivariance of molecular conformations, leading to unsatisfactory performance. To preserve roto-translation equivariance, Simm and Hernandez-Lobato [36] and Xu et al. [43] first model the molecular distance geometry and then reconstruct atomic coordinates from generated distances by solving an optimization problem. The state-of-the-art method CONFGF [34] estimates pseudo-forces acting on atoms and generates conformations via Langevin MCMC [42], which bypasses the two-stage fashion in previous works and enhances the performance significantly. Two concurrent works [12, 44] exist which generate conformations in end-to-end fashion via geometry elements assembly and bilevel programming respectively. Recently there has also been attempt to use reinforcement learning for conformation search [14]. Such a method is incapable of modeling bond lengths explicitly, and is fundamentally different from other approaches. To summarize, all of the previous methods focus mainly on modeling the local interactions based on the static input molecular graphs (or augmented graphs by adding auxiliary edges between atoms that are two- and three-hops away) and overlook the long-range non-bonded interactions between atoms. In contrast, our DGSM explicitly models both the local and long-range interactions via dynamic graph score matching and effectively addresses the above issue.
+
+# 3 Preliminaries
+
+# 3.1 Notations and Problem Formulation
+
+Notations. Let $\mathcal { G } = \langle \nu , \mathcal { E } \rangle$ be a molecular graph, where $\mathcal { V } = \{ v _ { 1 } , v _ { 2 } , \cdot \cdot \cdot , v _ { | \mathcal { V } | } \}$ is the set of nodes representing atoms, and $\mathcal { E } = \{ e _ { i j } \ | \ ( i , j ) \subseteq \mathcal { V } \times \mathcal { V } \}$ is the set of edges representing inter-atomic bonds in the molecule. Each node $v _ { i } \in \mathcal V$ is labeled with atomic attributes, e.g., the element type $Z _ { i }$ and the atomic coordinate $\boldsymbol { r } _ { i } \in \mathbb { R } ^ { 3 }$ . Each edge $e _ { i j } \in \mathcal { E }$ is labeled with a bond type. The conformation of the molecular graph $\mathcal { G }$ can be represented as a matrix $\pmb { R } \in \mathbb { R } ^ { | \nu | \times 3 }$ . The distances between all pairs of atoms can be represented as a matrix $D \in \mathbb { R } ^ { | \mathcal { V } | \times | \mathcal { V } | }$ , where $D _ { i j } : = d _ { i j } = \| \pmb { r } _ { i } - \pmb { r } _ { j } \| _ { 2 }$ denotes the Euclidean distance between the positions of $v _ { i }$ and $v _ { j }$ . Following previous work [34, 36, 43], we expand the original molecular graph by adding auxiliary edges between atoms that are second and third neighbors in $\mathcal { G }$ to reduce the degrees of freedom in 3D coordinates.
+
+Problem Formulation. Given a molecular graph $\mathcal { G } = \langle \nu , \mathcal { E } \rangle$ , the task of molecular conformation generation is the conditional generation of conformations $\pmb { R } = [ \pmb { r } _ { 1 } ; \pmb { r } _ { 2 } , \pmb { \cdot } \cdot \pmb { \cdot } ; \pmb { r } _ { | \pmb { \nu } | } ] \in \mathbb { R } ^ { | \mathcal { V } | \times 3 }$ based on $\mathcal { G }$ , while being able to capture long-range interactions between non-bonded atoms. Note that $\mathcal { G }$ may has multiple connected components, e.g., protein-ligand complexes and multi-molecular complexes.
+
+# 3.2 Score-Based Generative Modeling
+
+Score-based generative modeling [15, 38–40] is a class of generative models that has recently proven effective in a variety of tasks, ranging from image generation [15, 40], audio synthesis [7, 21] to shape generation [6, 23]. For any continuously differentiable probability density $p ( { \pmb x } )$ , we define its score function $\pmb { s } ( \pmb { x } )$ as $\nabla _ { \pmb { x } } \log p ( \pmb { x } )$ , i.e., the direction where the logarithm data density grows most rapidly. Score-based generative modeling perturbs the data with different levels of Gaussian noise and jointly estimates the score function of $\bar { p } ( { \pmb x } )$ using neural networks. Samples are then generated by sampling from a sequence of decreasing noise levels with Langevin dynamics [42].
+
+Formally, given a data distribution $p _ { \mathrm { d a t a } } ( \pmb { x } )$ , let $\{ \sigma _ { i } \} _ { i = 1 } ^ { L }$ be a sequence of noise levels that satisfies $\sigma _ { 1 } > \sigma _ { 2 } > \cdots > \sigma _ { L }$ and $\sigma _ { i } / \sigma _ { i - 1 } = \gamma$ . Consider a series of noise distributions $p _ { \sigma _ { i } } ( \tilde { \pmb { x } } \mid \pmb { x } ) : =$ $\mathcal { N } ( \tilde { \pmb { x } } ; \pmb { x } , \sigma _ { i } ^ { 2 } \pmb { I } )$ , and denote the corresponding perturbed data distribution as $\begin{array} { r } { p _ { \sigma _ { i } } ( \tilde { \pmb x } ) : = \int p _ { \sigma _ { i } } ( \tilde { \pmb x } \ | } \end{array}$ ${ \pmb x } ) p _ { \mathrm { d a t a } } ( { \pmb x } ) d { \pmb x }$ . Song and Ermon [38] propose to jointly approximate the score function of each noise level, denoted by ${ \bf s } _ { \pmb { \theta } } ( { \pmb x } , \sigma _ { i } )$ , with the following objective:
+
+$$
+\theta ^ { * } = \mathrm { a r g m i n } _ { \theta } \frac { 1 } { 2 L } \sum _ { i = 1 } ^ { L } \sigma _ { i } ^ { 2 } \mathbb { E } _ { p _ { \mathrm { d u a } } ( { \boldsymbol { x } } ) } \mathbb { E } _ { p _ { \sigma _ { i } } ( \tilde { { \boldsymbol { x } } } \mid { \boldsymbol { x } } ) } \left[ \left\| s _ { \theta } ( \tilde { { \boldsymbol { x } } } , \sigma _ { i } ) - \nabla _ { \tilde { { \boldsymbol { x } } } } \log p _ { \sigma _ { i } } ( \tilde { { \boldsymbol { x } } } \mid { \boldsymbol { x } } ) \right\| _ { 2 } ^ { 2 } \right] .
+$$
+
+![](images/a2c868ea8f3d1ee4db3f6d348db861fe287b6d0734ba6d8acc40b64a6e26774e.jpg)  
+Figure 2: The training procedure of the proposed DGSM. To model long-range interactions, the graph structures are dynamically determined by adding non-bonded edges based on added perturbations at each step. The bonded edges are marked by black border.
+
+Assuming sufficient data and model capacity, the optimal noise conditional score network ${ \pmb s } _ { \pmb \theta ^ { * } } \left( { \pmb x } , \sigma _ { i } \right)$ matches $\bar { \nabla } _ { \pmb { x } } \log p _ { \sigma _ { i } } ( \pmb { x } )$ almost everywhere [38]. After training score networks, Song and Ermon [38] run annealed Langevin dynamics for each $p _ { \sigma _ { i } } ( { \pmb x } )$ sequentially, where samples from each noise level serve as initializations for Langevin dynamics of the next noise level. Given $\sigma _ { L }$ small enough, the final samples from $p _ { \sigma _ { L } } ( \pmb { x } )$ approximate to samples from $p _ { \mathrm { d a t a } }$ under minor conditions.
+
+# 4 Model
+
+Our approach treats the conformation generation as sequentially moving atoms towards high-density regions guided by pseudo-forces, i.e., gradients of atoms. Following Shi et al. [34], we leverage the denoising score matching [38, 41] to approximate the gradients of the logarithm density of atomic coordinates, denoted as $\nabla _ { R } \log p ( R \mid \mathcal { G } )$ . To model atomic gradients that are sensitive to both the local and long-range interactions (Eq. 1) and inspired by the fact that long-range interactions decrease rapidly as distances increase, we propose to dynamically construct graph structures with non-bonded edges between atom pairs within a distance based on the current spatial proximity. In this way, we enable the model to effectively capture long-range non-bonded interactions while avoid connecting all atoms, which is computationally expensive. To ensure that the distribution of graph structures during training matches with that during generation, we devise a dynamic graph score matching algorithm, where graph structures are also dynamically determined during training depending on added perturbations. The whole framework is illustrated in Figure 2 and Figure 3. Below we describe the framework of score estimation for Cartesian coordinates in Section 4.1, the dynamic graph score matching algorithm in Section 4.2, and the generation procedure in Section 4.3.
+
+# 4.1 Score Estimation for Cartesian Coordinates
+
+Our goal is to learn the gradients of the logarithm density (score) of atomic coordinates, i.e., $\nabla _ { R } \log p ( R \mid \mathcal { G } )$ . Directly parameterizing score networks on absolute Cartesian coordinates with Graph Neural Networks (GNNs) [11, 13, 19, 29] relies on the arbitrary choice of rotation and translation [36, 43], which are non-essential degrees of freedom for effecting conformational changes in molecular systems. Therefore, we explicitly exclude them from the model, by first estimating scores for a set of dynamically determined interatomic distances, and then backpropagating gradients from distances to Cartesian coordinates via differentiation.
+
+Given a molecular graph $\mathcal { G } = \langle \nu , \mathcal { E } \rangle$ , the probability $p ( R \mid { \mathcal { G } } )$ of a conformation $\pmb { R }$ is subject to the Boltzmann distribution and is proportional to $\exp \left( - E ( R ) / k _ { B } T \right)$ , where $E ( R )$ is the conformational energy, $k _ { B }$ is the Boltzmann constant, and $T$ is the temperature. We assume the logarithm density of a conformation, i.e., the negative conformational energy up to a constant, can be parameterized as a function of interatomic distances, conditional on molecular graph $\mathcal { G }$ :
+
+$$
+\log p _ { \theta } ( R \mid \mathcal { G } ) : = f _ { \mathcal { G } } ( e _ { 1 } ( R ) , e _ { 2 } ( R ) , \cdot \cdot \cdot , e _ { K } ( R ) ) ,
+$$
+
+where $\{ e _ { k } : \mathbb { R } ^ { | \mathcal { V } | \times 3 }  \mathbb { R } \} _ { k = 1 } ^ { K }$ is a set of functions that calculate $K$ interatomic distances, which are invariant under the rotation and translation of $\pmb { R }$ . And $f _ { \mathcal { G } } : \mathbb { R } ^ { K }  \mathbb { R }$ is a graph neural network that predicts the negative conformational energy based on distances and the 2D graph representation $\mathcal { G }$ . Using interatomic distances for energy prediction is favored in existing literature [16, 20, 31, 34], as it preserves the 3D rotation and translation symmetries of molecular systems. Since $\{ e _ { k } ( { \pmb R } ) \} _ { k = 1 } ^ { K }$ are continuously differentiable with respect to the Cartesian coordinates $\pmb { R }$ , the gradients of logarithm density of interest, i.e., $\nabla _ { R } \log p _ { \theta } ( R | \mathcal { G } )$ , is interrelated with the logarithm density of each interatomic distance via chain rule:
+
+$$
+\begin{array} { l } { \displaystyle \forall i , s _ { \theta } ( { \pmb R } ) _ { i } : = \frac { \partial f _ { \mathcal { G } } \left( e _ { 1 } ( { \pmb R } ) , e _ { 2 } ( { \pmb R } ) , \cdots , e _ { K } ( { \pmb R } ) \right) } { \partial r _ { i } } } \\ { \displaystyle = \sum _ { k = 1 } ^ { K } \frac { \partial f _ { \mathcal { G } } \left( e _ { 1 } ( { \pmb R } ) , e _ { 2 } ( { \pmb R } ) , \cdots , e _ { K } ( { \pmb R } ) \right) } { \partial e _ { k } ( { \pmb R } ) } \cdot \frac { \partial e _ { k } ( { \pmb R } ) } { \partial r _ { i } } } \\ { \displaystyle = \sum _ { k = 1 } ^ { K } s _ { \theta } ( e _ { k } ( { \pmb R } ) ) \cdot \frac { \partial e _ { k } ( { \pmb R } ) } { \partial r _ { i } } , } \end{array}
+$$
+
+where ${ \displaystyle s _ { \theta } ( \mathbf { R } ) _ { i } }$ denotes $\nabla _ { r _ { i } } \log p _ { \theta } ( R \mid \mathcal { G } )$ , $\scriptstyle s _ { \theta } ( e _ { k } ( R ) )$ denotes $\nabla _ { e _ { k } ( R ) } \log p _ { \pmb \theta } ( \pmb R \mid \mathcal { G } )$ , and $\frac { \partial e _ { k } ( \pmb { R } ) } { \partial \pmb { r } _ { i } }$ can be calculated efficiently in closed form (see supplementary material for the full derivation).
+
+Motivated by the above equation, we first train a noise conditional score network to jointly predict the score of interatomic distances, i.e., $\{ s _ { \theta } ( e _ { k } ( { \pmb R } ) , \sigma ) \} _ { k = 1 } ^ { K }$ . After training the noise conditional score network, the gradients of the logarithm density of atomic coordinates, i.e., $s _ { \theta } ( R , \sigma )$ , can be estimated via Eq. 4. We have the following proposition (proof in supplementary material):
+
+Proposition 1 (Roto-Translation Equivariance). With the assumption that we can parameterize $\log p _ { \pmb \theta } ( \pmb R | \mathcal G )$ as a function of interatomic distances, conditional on molecular graph $\mathcal { G }$ (Eq. 3), the score function $s _ { \theta } ( R )$ defined in Eq. 4 is roto-translation equivariant.
+
+Remarkably, the choice of $\{ e _ { k } ( { \pmb R } ) \} _ { k = 1 } ^ { K }$ is flexible under this framework and can be carefully designed for specific goals. An ideal set of interatomic distances should capture both the local and long-range interactions between atoms (Eq. 1).
+
+Proposition 2 (Connection with CONFGF). The recent proposed CONFGF [34] is a special case of our approach, where they only model local distances between the first-order, the second-order, and the third-order neighbors, i.e., $\{ e _ { k } \} _ { k = 1 } ^ { K }$ map the conformation to a set of distances between bonded atoms. Therefore, CONFGF fails to capture long-range interactions between non-bonded atoms.
+
+# 4.2 Dynamic Graph Score Matching with Noise Conditional Score Networks
+
+In this section, we describe the proposed dynamic graph score matching for interatomic distances, with the goal of modeling both the local and long-range interactions. To ensure that the learned score functions cover all regions with different graph structures, we dynamically construct graph structures with non-bonded edges between atoms during training, based on added perturbations. Following Song and Ermon [38], we train a noise conditional score network to jointly estimate scores for perturbed distributions of a set of dynamically-determined interatomic distances, i.e., $\{ s _ { \pmb \theta } ( e _ { k } ( \pmb R ) , \sigma ) \} _ { k = 1 } ^ { K }$ , and parameterize the score network with the message passing neural network (MPNN) [13].
+
+Dynamic Score Matching. To capture long-range interactions between non-bonded atoms in a molecular system, a naive way is to treat the molecular graph as a fully-connected graph and model the gradients of logarithm density of distances between all pairs of atoms. However, such a practice is computationally expensive especially for large systems, e.g., proteins, and is sometimes unnecessary, e.g., van der Waals interactions decay rapidly as distances increase. As a remedy, we set a cutoff distance and assume each atom only interacts with all atoms within the cutoff distance, ignoring all interactions out of the considered sphere. This is a very popular strategy in computational chemistry that strikes a good balance between efficiency and accuracy [26, 31].
+
+Formally, consider a molecular graph $\mathcal { G } = \langle \nu , \mathcal { E } \rangle$ with distances between all pairs of atoms $\pmb { D } \in \mathbb { \Sigma }$ $\mathbb { R } ^ { | \nu | \times | \nu | }$ computed from its conformation $\pmb { R } \in \mathbb { R } ^ { | \nu | \times 3 }$ . For a given noise level $\sigma$ , we perturb the distances $_ { D }$ with Gaussian noise at each training step on the fly, and then augment the original graph structure with non-bonded edges between atom pairs within a certain threshold distance:
+
+$$
+\forall ( i , j ) , \tilde { D } _ { i j } \sim \mathcal { N } ( D _ { i j } , \sigma ^ { 2 } ) , \quad \mathcal { E } _ { \sigma } = \mathcal { E } \cup \{ e _ { i j } \mid \tilde { D } _ { i j } < \delta \} ,
+$$
+
+$$
+\begin{array} { r } { l _ { \sigma } = \{ D _ { i j } \mid e _ { i j } \in \mathcal { E } _ { \sigma } \} , \quad \tilde { d } _ { \sigma } = \{ \tilde { D } _ { i j } \mid e _ { i j } \in \mathcal { E } _ { \sigma } \} , } \end{array}
+$$
+
+where $\mathcal { E } _ { \sigma }$ is the constructed graph structure for noise level $\sigma$ , and $\delta$ is a hyper-parameter that controls the radius of long-range interactions. We empirically verify that a $1 0 \mathring \mathrm { A }$ cutoff is sufficient for systems we are studying, which is consistent with some experimental results in molecular dynamics [26]. $\scriptstyle d _ { \sigma }$ and $\tilde { d } _ { \sigma }$ denote the original and perturbed interatomic distances in augmented graph structure respectively. Hereafter, we omit the subscript for simplicity and use ${ \mathcal { E } } , d .$ , and $\tilde { d }$ instead, assuming all graphs are dynamically constructed during training and sampling.
+
+![](images/8f7d52f8c18d23459383561fc7802e024313762b231abd98f24b733b159cb4f9.jpg)  
+Figure 3: The generation procedure of the proposed DGSM via Langevin dynamics. The graph structure is dynamically constructed at each step of stochastic update based on the current conformation.
+
+With the above strategy, the graph structure of a specific molecular graph $\mathcal { G }$ is variadic depending on added perturbation, and all graph structures are possible as long as we sample sufficient enough noise. This will result in (1) a dynamically-determined graph structure for message passing and representation learning, which takes long-range interactions into consideration, and (2) a dynamicallydetermined set of interatomic distances, i.e., $\mathbf { \bar { \{ } }  e _ { k } ( { \pmb R } ) \} _ { k = 1 } ^ { K }$ , for score estimation, which contributes to gradients of atomic coordinates according to Eq. 4. Note that the vanilla implementation of Eq. 5 requires computing all distances between atom pairs. In practice, to avoid quadratic complexity, we pre-filter distant neighbors before adding perturbations for each atom by constructing radius graph with $2 \delta$ threshold, and empirically verify that it performs efficiently and effectively.
+
+Parameterizing with MPNNs. Let $\{ \sigma _ { i } \} _ { i = 1 } ^ { L }$ be a sequence of noise levels. Our goal is to learn a noise conditional score network to jointly estimate the scores of all perturbed distance distributions, i.e., $\forall \sigma \in \{ \sigma _ { i } \} _ { i = 1 } ^ { L } : s _ { \theta } ( \tilde { d } , \sigma ) \approx \mathring { \nabla _ { \tilde { d } } } \overrightarrow { \log p _ { \sigma } } ( \tilde { d } \mid \mathcal { G } ) .$ , where $\begin{array} { r } { p _ { \sigma } ( \tilde { d } \mid \tilde { \mathcal { G } } ) = \int p ( d \mid G ) \mathcal { N } ( \tilde { d } \mid d , \sigma ^ { 2 } I ) } \end{array}$ Following suggestions of Song and Ermon [38], we parameterize the score network with a MPNN as follows (see supplementary material for the full architecture):
+
+$$
+\begin{array} { r l } & { \quad h _ { i } ^ { \mathrm { n o d e } } = \mathrm { M P N N } ( \mathcal { G } , \mathcal { E } , \tilde { D } ) _ { i } , \quad \forall v _ { i } \in \mathcal { V } } \\ & { \quad h _ { i j } ^ { \mathrm { e d g e } } = \mathrm { C o n c a t } ( h _ { i } ^ { \mathrm { n o d e } } , h _ { j } ^ { \mathrm { n o d e } } ) , \quad \forall e _ { i j } \in \mathcal { E } _ { \sigma } } \\ & { s _ { \theta } ( \tilde { d } , \sigma ) _ { i j } = s _ { \theta } ( \tilde { d } ) _ { i j } / \sigma = \mathrm { M L P } ( h _ { i j } ^ { \mathrm { e d g e } } ) / \sigma , \quad \forall e _ { i j } \in \mathcal { E } _ { \sigma } } \end{array}
+$$
+
+molec where {hnodei }|V|i=1 are node embeddings compute and the perturbed distances, $\{ h _ { i j } ^ { \mathrm { e d g e } } \}$ MPNN based on the dynamically consare embeddings for each edge in $\mathcal { E }$ ucted, and $s _ { \theta } ( \tilde { d } , \sigma ) _ { i j }$ is the predicted score for interatomic distance $\tilde { D } _ { i j }$ $( e _ { i j } \in \mathcal { E }$ ). The noise conditional network can be jointly optimized with the following objective [38]:
+
+$$
+\theta ^ { * } = \operatorname * { a r g m i n } _ { \theta } \frac { 1 } { 2 L } \sum _ { i = 1 } ^ { L } \sigma _ { i } ^ { 2 } \mathbb { E } _ { p ( d | \mathcal { G } ) } \mathbb { E } _ { p _ { \sigma _ { i } } ( \tilde { d } | d , \mathcal { G } ) } \Big [ \Big \lVert \frac { s _ { \theta } ( \tilde { d } ) } { \sigma _ { i } } + \frac { \tilde { d } - d } { \sigma _ { i } ^ { 2 } } \Big \rVert _ { 2 } ^ { 2 } \Big ] ,
+$$
+
+where all expectations can be efficiently estimated using empirical averages.
+
+# 4.3 Generation
+
+After training the noise conditional score network, molecular conformations can be generated via annealed Langevin dynamics [38], guided by the gradients of atomic coordinates. The gradients can be computed via Eq. 4 based on dynamically constructed graph structures at each step of stochastic update, which allows model to effectively capture both the local and long-range interactions that contribute to the atomic gradients. Formally, given a molecular graph $\mathcal { G }$ , we first sample an initial conformation $\scriptstyle { R _ { 0 } }$ from a fixed prior distribution. We here take the prior distribution as a standard
+
+Gaussian $\mathcal { N } ( R _ { 0 } \mid \mathbf { 0 } , I )$ . Then, we update the conformation by running $T$ steps of Langevin dynamic to get a sample from each noise conditional score network ${ \bf \delta } _ { s _ { \theta } ( { \cal R } , \sigma _ { i } ) }$ sequentially with a special step size schedule $\alpha _ { i } = \varepsilon \cdot \sigma _ { i } ^ { 2 } / \sigma _ { L } ^ { 2 }$ . Samples from each noise level are used to initialize Langevin dynamics for the next noise level. At each sampling step $t$ , we first construct graph structures with non-bonded edges within a given distance $\delta$ based on the current pairwise distances $D _ { t - 1 }$ computed from $\mathbf { \delta } _ { R _ { t - 1 } }$ , and then get a set of interatomic distances $d _ { t - 1 }$ for score estimation:
+
+$$
+\mathcal { E } _ { t - 1 } = \mathcal { E } \cup \big \{ e _ { i j } ~ \big | ~ D _ { t - 1 , i j } < \delta \big \} , \quad d _ { t - 1 } = \big \{ D _ { t - 1 , i j } ~ \big | ~ e _ { i j } \in \mathcal { E } _ { t - 1 } \big \} .
+$$
+
+The conformation is then updated using the gradient information from the score network (Eq. 4). We provide the pseudo-code in Algorithm 1.
+
+# 5 Experiments
+
+Following previous works [34, 36, 43] on conformation generation, we evaluate the proposed DGSM using the following two standard tasks: Conformation Generation (Section 5.1), and Property Prediction (Section 5.2). To further demonstrate DGSM’s capability of modeling longrange interactions, we evaluate it on two more challenging benchmark tasks: Protein Sidechain Conformation Generation and Multi-molecular Complex Conformation Generation (Section 5.3). We describe experimental setups in task-specific sections.
+
+# Algorithm 1 Annealed Langevin dynamics [38]
+
+Require: $\mathcal { G } = \langle \nu , \mathcal { E } \rangle$ , $\{ \sigma _ { i } \} _ { i = 1 } ^ { L } , \delta , \varepsilon , T$ .
+
+1: Initialize conformation $\scriptstyle { R _ { 0 } }$   
+2: for $i \gets 1$ to $L$ do   
+3: $\alpha _ { i }  \varepsilon \cdot \sigma _ { i } ^ { 2 } / \sigma _ { L } ^ { 2 }$ $\triangleright \alpha _ { i }$ is the step size.   
+4: for $t \gets 1$ to $T$ do   
+5: $\mathcal { E } _ { t - 1 } , d _ { t - 1 } \gets \mathrm { a u g } ( \mathcal { E } , R _ { t - 1 } , \delta )$ . Eq. 8   
+6: $s _ { \theta } ( R _ { t - 1 } , \sigma _ { i } ) \gets \mathrm { g e t } ( s _ { \theta } ( d _ { t - 1 } , \sigma _ { i } ) ) \triangleright \mathrm { E q . } 4$   
+7: Draw $\boldsymbol { z } _ { t } \sim \mathcal { N } ( \mathbf { 0 } , I )$   
+8: $R _ { t } \gets R _ { t - 1 } + \alpha _ { i } s _ { \theta } ( R _ { t - 1 } , \sigma _ { i } ) + \sqrt { 2 \alpha _ { i } } z _ { t }$   
+9: end for   
+10: $R _ { 0 } \gets R _ { T }$   
+11: end for   
+Return: Generated conformation $R _ { T }$ .
+
+# 5.1 Conformation Generation
+
+Setup. This task evaluates the model’s capability to generate stable molecular conformations by measuring both accuracy and diversity of generated conformations. Following previous works [34, 43], we use the GEOM-QM9 and GEOM-Drugs [1] datasets for this task. We use the train-test split provided by [34]. The train splits of GEOM-QM9 and GEOM-Drugs both contain 40,000 molecules, each with 5 conformations for training, or 200,000 conformations in total. The test split of GEOM-QM9 contains 200 molecules with 22,408 conformations, and the test split of GEOM-Drugs contains 200 molecules with 14,324 conformations.
+
+We compare DGSM against 5 state-of-the-art baselines: RDKIT [28], CVGAE [24], GRAPHDG [36], CGCF [43] and CONFGF [34]. For each molecule in the test set, we sample twice as many conformations as its reference conformations. We use the matching score (MAT) to measure the accuracy of generated conformations, and coverage score (COV) to measure the diversity following [34, 43]. Both metrics are based on Root Mean Squared Deviations (RMSD) between molecules, taking symmetries into account (see supplementary material for the details of metrics).
+
+Results. We report the mean and median COV and MAT scores over all the molecules in the test split on GEOM-QM9 and GEOM-Drugs datasets. As shown in Table 1, DGSM consistently outperforms all the baselines. Notably, both DGSM and CONFGF are score-based models, but DGSM achieves better performance. The difference between them is that DGSM successfully takes long-range interactions into consideration via dynamic graph score matching. This confirms the significant benefit of modeling long-range interactions. We present several conformations generated by different approaches in Figure 4, which shows that DGSM successfully captures the long-range interactions in highlighted areas while the other baselines fail, resulting in distorted structures in those areas.
+
+Table 1: COV and MAT scores on GEOM-QM9 and GEOM-Drugs datasets. The threshold $\delta$ of COV score is $0 . 5 \mathring \mathrm { A }$ for GEOM-QM9 and $1 . 2 5 \mathring \mathrm { A }$ for GEOM-Drugs following $\mathrm { X u }$ et al. [43]. (↑): the higher the better. (↓): the lower the better.   
+
+<table><tr><td></td><td colspan="4">GEOM-QM9</td><td colspan="4">GEOM-Drugs</td></tr><tr><td></td><td colspan="4">COV (%, ↑)</td><td colspan="4">COV (%, ↑)</td></tr><tr><td>Method</td><td>Mean</td><td>Median</td><td>Mean</td><td>MAT (A,↓) Median</td><td>Mean</td><td>Median</td><td>Mean</td><td>MAT (A,↓) Median</td></tr><tr><td>RDKIT[28]</td><td>83.26</td><td>90.78</td><td>0.3447</td><td>0.2935</td><td>60.91</td><td>65.70</td><td>1.2026</td><td>1.1252</td></tr><tr><td>CVGAE [24]</td><td>0.09</td><td>0.00</td><td>1.6713</td><td>1.6088</td><td>0.00</td><td>0.00</td><td>3.0702</td><td>2.9937</td></tr><tr><td>GRAPHDG [36]</td><td>73.33</td><td>84.21</td><td>0.4245</td><td>0.3973</td><td>8.27</td><td>0.00</td><td>1.9722</td><td>1.9845</td></tr><tr><td>CGCF [43]</td><td>77.52</td><td>80.40</td><td>0.4206</td><td>0.3903</td><td>54.19</td><td>56.35</td><td>1.2575</td><td>1.2356</td></tr><tr><td>CONFGF[34]</td><td>88.49</td><td>94.13</td><td>0.2673</td><td>0.2685</td><td>62.15</td><td>70.93</td><td>1.1629</td><td>1.1596</td></tr><tr><td>DGSM</td><td>91.49</td><td>95.92</td><td>0.2139</td><td>0.2137</td><td>78.73</td><td>94.39</td><td>1.0154</td><td>0.9980</td></tr></table>
+
+![](images/d0a2cabb7c5639256dccf21a7dd886b8e4a87a4f1374a2c00f9f481477b87383.jpg)  
+Figure 4: Examples of conformations generated by different models based on four random molecular graphs from the test set of GEOM-Drugs. We present three reference conformations for each molecule, and visualize the best-aligned conformations generated by each method. Areas where long-range interactions should be modeled are highlighted in green.
+
+# 5.2 Property Prediction
+
+Setup. This task demonstrates how generative models for molecular conformations can be applied to property prediction as a downstream task. It also provides an assessment on the quality of generated conformations in a different light. We estimate the ensemble properties [1] of a molecular graph by aggregating its conformational properties following [34]. In specific, we first use the models
+
+Table 2: Mean absolute errors (MAE) of predicted ensemble properties in eV.   
+
+<table><tr><td>Method</td><td>E</td><td>Emin</td><td>A</td><td>△emin</td><td>△emax</td></tr><tr><td>RDKIT</td><td>0.9233</td><td>0.6585</td><td>0.3698</td><td>0.8021</td><td>0.2359</td></tr><tr><td>GRAPHDG</td><td>9.1027</td><td>0.8882</td><td>1.7973</td><td>4.1743</td><td>0.4776</td></tr><tr><td>CGCF</td><td>28.9661</td><td>2.8410</td><td>2.8356</td><td>10.6361</td><td>0.5954</td></tr><tr><td>CONFGF</td><td>2.7886</td><td>0.1765</td><td>0.4688</td><td>2.1843</td><td>0.1433</td></tr><tr><td>DGSM</td><td>1.0313</td><td>0.0761</td><td>0.1963</td><td>1.1811</td><td>0.1271</td></tr></table>
+
+to generate 50 conformations for each molecular graph in a subset of GEOM-QM9 [34], and use PSI4 [37], a quantum chemical toolkit, to calculate each conformation’s energy and HOMO-LUMO gap. Then, we calculate average energy $\overline { E }$ , lowest energy $E _ { \mathrm { m i n } }$ , average gap $\overline { { \Delta \epsilon } }$ , minimum gap $\Delta \epsilon _ { \mathrm { m i n } }$ and maximum gap $\Delta \epsilon _ { \mathrm { m a x } }$ from the conformational energy and gap. We evaluate the accuracy of estimated ensemble property by measuring their mean absolute errors (MAE) to the ground truth values. CVGAE is excluded in this task as its performance is poor, which is also reported in [36, 34].
+
+Results. Table 2 shows that DGSM outperforms other machine learning-based methods by a clear margin. DGSM’s estimation of average energy $\overline { E }$ and minimum gap $\Delta \epsilon _ { \mathrm { m i n } }$ is close to RDKIT but still outperforms the most competitive ML-based method CONFGF. The calculation of conformational energy is highly sensitive to changes in geometry — even a subtle deviation in bond lengths leads to significant energy change [36]. Therefore, the superior performance of DGSM indicates that it generates much more accurate conformations than other methods, leading to more accurate property estimation. This validates again the effectiveness of modeling long-range interactions.
+
+# 5.3 Large Molecule and Multi-molecular Modeling
+
+Protein Sidechain Conformation This task is to predict protein sidechain conformations based on its backbone structures. Compared to conventional molecular conformations generation in previous sections, the main challenge of this task is two-fold: (1) large number of atoms, which prohibits constructing complete graphs that grow quadratically to model long-range interactions.
+
+Table 3: RMSD of different approaches on sidechain conformation generation.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">RMSD</td></tr><tr><td>Mean (A)</td><td>Min (A)</td></tr><tr><td>CONFGF</td><td>3.38</td><td>3.11</td></tr><tr><td>DGSM</td><td>2.85 (↓15.7%)</td><td>2.61 (↓ 16.1%)</td></tr></table>
+
+(2) covalent bonds are sparse, which limits the power of the edge augmentation techniques in previous work. DGSM tackles these two challenges via dynamic graph score matching as introduced.
+
+![](images/ddb62042f2bd28694c95bb5a17d95b0004d17ab5ff7ec35601389ceff3b8c4fc.jpg)  
+Figure 5: (a) An example of the generated protein sidechain conformation with atomic-level coordinates. The ground-truth sidechain (blue) and the generated sidechain (red) are highlighted. (b) Conformations of two multi-molecular complexes generated by DGSM.
+
+We use the SidechainNet [18] dataset for this task and follow the official train-test splits. We compare DGSM with the state-of-the-art conformation generation model CONFGF. Despite that there are some machine learning-based methods specialized in protein sidechain structure prediction [10], they are built upon rotamer libraries [5], which incorporates a lot of domain knowledge. Thus, our method is not comparable with them. The main purpose of this task is to justify the effectiveness of DGSM for large molecules. For each protein, we generate 5 sidechain conformations with different initialization, and calculate the mean and min RMSD between the ground-truth conformation and the generated conformations. We report the overall mean and min RMSD scores by averaging scores of each protein in the test set in in Table 3, which shows that DGSM achieves better performance than previous state-of-the-art model. We also present an example in Figure 5(a), and we can see that the predicted conformation is consistent with the ground truth in major parts.
+
+Multi-molecular Complex Conformation This task is to predict conformations for multi-molecular complexes. A multi-molecular complex is made up of multiple molecules and there is no covalent bonds between them. Long-range interactions dominate the structure of multi-molecular complexes. The purpose of this task is to demonstrate DGSM’s potential application to a broader range of problems and provide a novel benchmark for conformation generation. We use the quantum chemical software xtb [3] to construct a dataset consisting of 24 water-organic complexes each with several hundreds of conformations, and leave out 4 complexes for testing (see supplementary material for details). We do not report RMSD-based metrics such as COV and MAT because the structures of multi-molecular complexes are highly flexible. Two set of generated examples are presented in Figure 5(b). We observe that water molecules are placed regularly around the solute organic molecule. Notably, hydrogen bonds (between water and the solute, and between water and water) are formed correctly. This can also be evidenced in the histogram of Hydrogen-Oxygen distances (Figure 6), where there is a peak between $1 . 5 \mathring \mathrm { A }$ and $2 . 5 \mathring \mathrm { A }$ , i.e., the range of hydrogen bond length between Hydrogen and Oxygen.
+
+![](images/7ac0bfb5e265be1599bc039c6c527547e5a21c558d824a8643a79268b2d55df7.jpg)  
+Figure 6: The distribution of Hydrogen-Oxygen distances. The first peak from the left is covalent bonds and the second peak is hydrogen bonds.
+
+# 6 Conclusion and Future Work
+
+We propose DGSM, a novel score-based approach for generating equilibrium molecular conformations. DGSM is capable of modeling both the local and long-range interactions in molecular systems, by dynamically constructing graph structures based on spatial proximity between atoms during both training and inference. We also devise a dynamic graph score matching algorithm to effectively estimate atomic gradients, where graph structures are dynamically determined depending on added perturbations. Extensive experiments over two standard tasks and two original tasks show that DGSM outperforms the state-of-the-art method by a large margin, confirming the significant benefit of modeling long-range interactions. In the future, we plan to apply our approach to the more challenging problem of protein structure prediction.
+
+# Acknowledgments and Disclosure of Funding
+
+We would like to thank all the reviewers for the insightful comments. This project is supported by the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant, the Canada CIFAR AI Chair Program, collaboration grants between Microsoft Research and Mila, Samsung Electronics Co., Ldt., Amazon Faculty Research Award, Tencent AI Lab Rhino-Bird Gift Fund and a NRC Collaborative R&D Project (AI4D-CORE-06). This project was also partially funded by IVADO Fundamental Research Project grant PRF-2019-3583139727.
+
+# References
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+        "type": "text",
+        "text": "Predicting Molecular Conformation via Dynamic Graph Score Matching ",
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+        "text": "Shitong Luo\\*1, Chence $\\mathbf { S h i ^ { * 2 , 3 } }$ , Minkai $\\mathbf { X } \\mathbf { u } ^ { 2 , 3 }$ , Jian Tang2,4,5 1Peking University 2Mila - Québec AI Institute 3Université de Montréal $^ { 4 } \\mathrm { H E C }$ Montréal 5CIFAR AI Research Chair luost@pku.edu.cn , chence.shi@umontreal.ca minkai.xu@umontreal.ca , jian.tang@hec.ca ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Predicting stable 3D conformations from 2D molecular graphs has been a longstanding challenge in computational chemistry. Recently, machine learning approaches have demonstrated very promising results compared to traditional experimental and physics-based simulation methods. These approaches mainly focus on modeling the local interactions between neighboring atoms on the molecular graphs and overlook the long-range interactions between non-bonded atoms. However, these non-bonded atoms may be proximal to each other in 3D space, and modeling their interactions is of crucial importance to accurately determine molecular conformations, especially for large molecules and multi-molecular complexes. In this paper, we propose a new approach called Dynamic Graph Score Matching (DGSM) for molecular conformation prediction, which models both the local and long-range interactions by dynamically constructing graph structures between atoms according to their spatial proximity during both training and inference. Specifically, the DGSM directly estimates the gradient fields of the logarithm density of atomic coordinates according to the dynamically constructed graphs using score matching methods. The whole framework can be efficiently trained in an end-to-end fashion. Experiments across multiple tasks show that the DGSM outperforms state-of-theart baselines by a large margin, and it is capable of generating conformations for a broader range of systems such as proteins and multi-molecular complexes. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "page_idx": 0
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+        "type": "text",
+        "text": "Graph-based representations of molecules has become prevalent in a variety of tasks such as property prediction [13, 31] and molecule generation [17, 33, 45]. However, a more natural and intrinsic representation of a molecule is its 3D geometry or conformation, which represents a molecule as a set of 3D coordinates. The 3D representation of molecules is central to many tasks, such as molecular properties prediction and virtual screening. Nevertheless, determining the conformation of a molecule remains a challenging task — both computational approaches, e.g., molecular dynamics (MD) [9], and experimental approaches, e.g., crystallography, are expensive and time-consuming. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Recently, machine learning approaches have demonstrated promising performance for molecular conformation generation. Pioneering methods such as GRAPHDG [36] and CGCF [43] first predict interatomic distances between bonded atoms and then solve 3D coordinates from the predicted distances via a post-processing algorithm. Very recently, Shi et al. proposed the CONFGF [34], which employs the score matching technique [38] to learn pseudo-forces between bonded atoms and iteratively applies the forces to a randomly initialized 3D structure until convergence. CONFGF gets rid of the two-stage fashion in prior works and significantly improves the performance. Nevertheless, these approaches have a common major limitation — they mainly focus on modeling the local interactions between bonded atoms defined by the input molecular graphs but fail to capture long-range interactions between non-bonded atoms1, as they only model distances (or gradients) between bonded atoms. While in molecular mechanics, the potential energy of a molecule that alters conformations can be modeled as a sum of four parts [22]: ",
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+        "img_path": "images/40aa5ba4168cefb1ae91527e2aef598f8197774fa9204092719b254b4b1a32df.jpg",
+        "image_caption": [
+            "Figure 1: Three molecular systems where long-range interactions are crucial for their conformations. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "equation",
+        "img_path": "images/f0e8d0dd07577f778170da6256bb983f06354a8f64768dd737ee187feb15d010.jpg",
+        "text": "$$\nE = E _ { \\mathrm { b o n d } } + E _ { \\mathrm { a n g l e } } + E _ { \\mathrm { t o r s i o n } } + E _ { \\mathrm { n o n - b o n d e d } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "where $E _ { \\mathrm { b o n d } }$ , $E _ { \\mathrm { a n g l e } }$ , and $E _ { \\mathrm { t o r s i o n } }$ model local interactions between bonded atoms, which are modeled in previous methods [34, 36, 43]. Long-range interactions between non-bonded atoms, denoted as $E _ { \\mathrm { n o n - b o n d e d } }$ , are also non-trivial, which shape the molecular geometry via non-negligible electrostatic forces or van der Waals forces, etc. For multi-molecular complexes, non-bonded interactions dominate complexes’ geometry. An ideal solution to conformation generation should therefore capture both the local and long-range interactions. In Figure 1, we present three typical molecular systems where long-range interactions play a key role in determining their conformations. ",
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+    {
+        "type": "text",
+        "text": "To tackle the aforementioned challenge of modeling long-range interactions, in this paper, we propose the Dynamic Graph Score Matching (DGSM) for molecular conformation generation, following the principle of CONFGF [34] that learns the gradients of the logarithm density of atomic coordinates. Instead of relying on the static input molecular graph as existing work, the basic idea is to dynamically construct graph structures between atoms based on their spatial proximity during both training and inference. This allows the model to (1) dynamically learn molecular graph representations with evolved graph structures that take long-range interactions into consideration, and (2) dynamically determine a set of interatomic distances that contribute to gradients of the current atomic coordinates. Specifically, the edges in the dynamic graph consist of two parts. The first part of edges are determined by covalent bonds, which capture local interactions between atoms $\\mathrm { E _ { b o n d } }$ , $E _ { \\mathrm { a n g l e } }$ and $E _ { \\mathrm { t o r s i o n } } \\mathrm { , }$ ). The second part of edges are determined dynamically by spatial proximity between atoms at each training or sampling step, i.e., two atoms are connected as long as they are proximal, no matter whether they are bonded. Such a strategy is able to effectively capture non-local interactions $( E _ { \\mathrm { n o n - b o n d e d } } )$ since the magnitude of long-range interactions is inversely correlated with distances between atoms [30]. It remains meanwhile scalable as we avoid connecting all the atom-pairs, which has quadratic complexity. In addition, modeling non-bonded interactions enable the model to sample conformations for multi-molecular complexes, which represent a broader range of problems. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We conduct extensive experiments and compare DGSM against previous state-of-the-art methods on both standard conformation generation and property prediction tasks. Numerical results show that DGSM outperforms previous methods by a clear margin, confirming the benefit of modeling long-range interactions. Besides, to further demonstrate the advantage of DGSM, we bring attention to two more challenging tasks — protein sidechain conformation prediction and multi-molecular complex structure prediction. These two new tasks represent two classes of practical challenges: predicting structures for macro-molecules and multi-molecular complexes. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Prior works on conformation generation mainly rely on molecular dynamics (MD) [9], where new conformations are sequentially generated based on an initial conformation and a physical model for interatomic potentials [25, 27]. Although capable of accurately sampling equilibrium conformations, these methods are computationally intensive, especially for large molecular systems [2, 35], e.g., proteins. Another category of approaches leverage distance geometry [8] and fix distances between atoms to idealized values heuristically [4], which are much faster but less accurate. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Recently, a variety of deep generative models have been proposed for molecular conformation generation, which strike a good balance between computational efficiency and accuracy. Among these methods, Mansimov et al. [24] first propose a variational autoencoder to directly generate 3D atomic coordinates. Albeit simple, this method fails to model the roto-translation equivariance of molecular conformations, leading to unsatisfactory performance. To preserve roto-translation equivariance, Simm and Hernandez-Lobato [36] and Xu et al. [43] first model the molecular distance geometry and then reconstruct atomic coordinates from generated distances by solving an optimization problem. The state-of-the-art method CONFGF [34] estimates pseudo-forces acting on atoms and generates conformations via Langevin MCMC [42], which bypasses the two-stage fashion in previous works and enhances the performance significantly. Two concurrent works [12, 44] exist which generate conformations in end-to-end fashion via geometry elements assembly and bilevel programming respectively. Recently there has also been attempt to use reinforcement learning for conformation search [14]. Such a method is incapable of modeling bond lengths explicitly, and is fundamentally different from other approaches. To summarize, all of the previous methods focus mainly on modeling the local interactions based on the static input molecular graphs (or augmented graphs by adding auxiliary edges between atoms that are two- and three-hops away) and overlook the long-range non-bonded interactions between atoms. In contrast, our DGSM explicitly models both the local and long-range interactions via dynamic graph score matching and effectively addresses the above issue. ",
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+    {
+        "type": "text",
+        "text": "3 Preliminaries ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 Notations and Problem Formulation ",
+        "text_level": 1,
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+        "text": "Notations. Let $\\mathcal { G } = \\langle \\nu , \\mathcal { E } \\rangle$ be a molecular graph, where $\\mathcal { V } = \\{ v _ { 1 } , v _ { 2 } , \\cdot \\cdot \\cdot , v _ { | \\mathcal { V } | } \\}$ is the set of nodes representing atoms, and $\\mathcal { E } = \\{ e _ { i j } \\ | \\ ( i , j ) \\subseteq \\mathcal { V } \\times \\mathcal { V } \\}$ is the set of edges representing inter-atomic bonds in the molecule. Each node $v _ { i } \\in \\mathcal V$ is labeled with atomic attributes, e.g., the element type $Z _ { i }$ and the atomic coordinate $\\boldsymbol { r } _ { i } \\in \\mathbb { R } ^ { 3 }$ . Each edge $e _ { i j } \\in \\mathcal { E }$ is labeled with a bond type. The conformation of the molecular graph $\\mathcal { G }$ can be represented as a matrix $\\pmb { R } \\in \\mathbb { R } ^ { | \\nu | \\times 3 }$ . The distances between all pairs of atoms can be represented as a matrix $D \\in \\mathbb { R } ^ { | \\mathcal { V } | \\times | \\mathcal { V } | }$ , where $D _ { i j } : = d _ { i j } = \\| \\pmb { r } _ { i } - \\pmb { r } _ { j } \\| _ { 2 }$ denotes the Euclidean distance between the positions of $v _ { i }$ and $v _ { j }$ . Following previous work [34, 36, 43], we expand the original molecular graph by adding auxiliary edges between atoms that are second and third neighbors in $\\mathcal { G }$ to reduce the degrees of freedom in 3D coordinates. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Problem Formulation. Given a molecular graph $\\mathcal { G } = \\langle \\nu , \\mathcal { E } \\rangle$ , the task of molecular conformation generation is the conditional generation of conformations $\\pmb { R } = [ \\pmb { r } _ { 1 } ; \\pmb { r } _ { 2 } , \\pmb { \\cdot } \\cdot \\pmb { \\cdot } ; \\pmb { r } _ { | \\pmb { \\nu } | } ] \\in \\mathbb { R } ^ { | \\mathcal { V } | \\times 3 }$ based on $\\mathcal { G }$ , while being able to capture long-range interactions between non-bonded atoms. Note that $\\mathcal { G }$ may has multiple connected components, e.g., protein-ligand complexes and multi-molecular complexes. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 Score-Based Generative Modeling ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Score-based generative modeling [15, 38–40] is a class of generative models that has recently proven effective in a variety of tasks, ranging from image generation [15, 40], audio synthesis [7, 21] to shape generation [6, 23]. For any continuously differentiable probability density $p ( { \\pmb x } )$ , we define its score function $\\pmb { s } ( \\pmb { x } )$ as $\\nabla _ { \\pmb { x } } \\log p ( \\pmb { x } )$ , i.e., the direction where the logarithm data density grows most rapidly. Score-based generative modeling perturbs the data with different levels of Gaussian noise and jointly estimates the score function of $\\bar { p } ( { \\pmb x } )$ using neural networks. Samples are then generated by sampling from a sequence of decreasing noise levels with Langevin dynamics [42]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Formally, given a data distribution $p _ { \\mathrm { d a t a } } ( \\pmb { x } )$ , let $\\{ \\sigma _ { i } \\} _ { i = 1 } ^ { L }$ be a sequence of noise levels that satisfies $\\sigma _ { 1 } > \\sigma _ { 2 } > \\cdots > \\sigma _ { L }$ and $\\sigma _ { i } / \\sigma _ { i - 1 } = \\gamma$ . Consider a series of noise distributions $p _ { \\sigma _ { i } } ( \\tilde { \\pmb { x } } \\mid \\pmb { x } ) : =$ $\\mathcal { N } ( \\tilde { \\pmb { x } } ; \\pmb { x } , \\sigma _ { i } ^ { 2 } \\pmb { I } )$ , and denote the corresponding perturbed data distribution as $\\begin{array} { r } { p _ { \\sigma _ { i } } ( \\tilde { \\pmb x } ) : = \\int p _ { \\sigma _ { i } } ( \\tilde { \\pmb x } \\ | } \\end{array}$ ${ \\pmb x } ) p _ { \\mathrm { d a t a } } ( { \\pmb x } ) d { \\pmb x }$ . Song and Ermon [38] propose to jointly approximate the score function of each noise level, denoted by ${ \\bf s } _ { \\pmb { \\theta } } ( { \\pmb x } , \\sigma _ { i } )$ , with the following objective: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ed534baee901dee1e55e16c98ba5692c221a1a51f7963f7fff9394371411cf8a.jpg",
+        "text": "$$\n\\theta ^ { * } = \\mathrm { a r g m i n } _ { \\theta } \\frac { 1 } { 2 L } \\sum _ { i = 1 } ^ { L } \\sigma _ { i } ^ { 2 } \\mathbb { E } _ { p _ { \\mathrm { d u a } } ( { \\boldsymbol { x } } ) } \\mathbb { E } _ { p _ { \\sigma _ { i } } ( \\tilde { { \\boldsymbol { x } } } \\mid { \\boldsymbol { x } } ) } \\left[ \\left\\| s _ { \\theta } ( \\tilde { { \\boldsymbol { x } } } , \\sigma _ { i } ) - \\nabla _ { \\tilde { { \\boldsymbol { x } } } } \\log p _ { \\sigma _ { i } } ( \\tilde { { \\boldsymbol { x } } } \\mid { \\boldsymbol { x } } ) \\right\\| _ { 2 } ^ { 2 } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/a2c868ea8f3d1ee4db3f6d348db861fe287b6d0734ba6d8acc40b64a6e26774e.jpg",
+        "image_caption": [
+            "Figure 2: The training procedure of the proposed DGSM. To model long-range interactions, the graph structures are dynamically determined by adding non-bonded edges based on added perturbations at each step. The bonded edges are marked by black border. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assuming sufficient data and model capacity, the optimal noise conditional score network ${ \\pmb s } _ { \\pmb \\theta ^ { * } } \\left( { \\pmb x } , \\sigma _ { i } \\right)$ matches $\\bar { \\nabla } _ { \\pmb { x } } \\log p _ { \\sigma _ { i } } ( \\pmb { x } )$ almost everywhere [38]. After training score networks, Song and Ermon [38] run annealed Langevin dynamics for each $p _ { \\sigma _ { i } } ( { \\pmb x } )$ sequentially, where samples from each noise level serve as initializations for Langevin dynamics of the next noise level. Given $\\sigma _ { L }$ small enough, the final samples from $p _ { \\sigma _ { L } } ( \\pmb { x } )$ approximate to samples from $p _ { \\mathrm { d a t a } }$ under minor conditions. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Model ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our approach treats the conformation generation as sequentially moving atoms towards high-density regions guided by pseudo-forces, i.e., gradients of atoms. Following Shi et al. [34], we leverage the denoising score matching [38, 41] to approximate the gradients of the logarithm density of atomic coordinates, denoted as $\\nabla _ { R } \\log p ( R \\mid \\mathcal { G } )$ . To model atomic gradients that are sensitive to both the local and long-range interactions (Eq. 1) and inspired by the fact that long-range interactions decrease rapidly as distances increase, we propose to dynamically construct graph structures with non-bonded edges between atom pairs within a distance based on the current spatial proximity. In this way, we enable the model to effectively capture long-range non-bonded interactions while avoid connecting all atoms, which is computationally expensive. To ensure that the distribution of graph structures during training matches with that during generation, we devise a dynamic graph score matching algorithm, where graph structures are also dynamically determined during training depending on added perturbations. The whole framework is illustrated in Figure 2 and Figure 3. Below we describe the framework of score estimation for Cartesian coordinates in Section 4.1, the dynamic graph score matching algorithm in Section 4.2, and the generation procedure in Section 4.3. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 Score Estimation for Cartesian Coordinates ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            609,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to learn the gradients of the logarithm density (score) of atomic coordinates, i.e., $\\nabla _ { R } \\log p ( R \\mid \\mathcal { G } )$ . Directly parameterizing score networks on absolute Cartesian coordinates with Graph Neural Networks (GNNs) [11, 13, 19, 29] relies on the arbitrary choice of rotation and translation [36, 43], which are non-essential degrees of freedom for effecting conformational changes in molecular systems. Therefore, we explicitly exclude them from the model, by first estimating scores for a set of dynamically determined interatomic distances, and then backpropagating gradients from distances to Cartesian coordinates via differentiation. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given a molecular graph $\\mathcal { G } = \\langle \\nu , \\mathcal { E } \\rangle$ , the probability $p ( R \\mid { \\mathcal { G } } )$ of a conformation $\\pmb { R }$ is subject to the Boltzmann distribution and is proportional to $\\exp \\left( - E ( R ) / k _ { B } T \\right)$ , where $E ( R )$ is the conformational energy, $k _ { B }$ is the Boltzmann constant, and $T$ is the temperature. We assume the logarithm density of a conformation, i.e., the negative conformational energy up to a constant, can be parameterized as a function of interatomic distances, conditional on molecular graph $\\mathcal { G }$ : ",
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c3c3a4567d7eac8e3ea3ac8364dd49b27d5b5c8fbf45a23abec040be83cd2db2.jpg",
+        "text": "$$\n\\log p _ { \\theta } ( R \\mid \\mathcal { G } ) : = f _ { \\mathcal { G } } ( e _ { 1 } ( R ) , e _ { 2 } ( R ) , \\cdot \\cdot \\cdot , e _ { K } ( R ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            814,
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+            832
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\{ e _ { k } : \\mathbb { R } ^ { | \\mathcal { V } | \\times 3 }  \\mathbb { R } \\} _ { k = 1 } ^ { K }$ is a set of functions that calculate $K$ interatomic distances, which are invariant under the rotation and translation of $\\pmb { R }$ . And $f _ { \\mathcal { G } } : \\mathbb { R } ^ { K }  \\mathbb { R }$ is a graph neural network that predicts the negative conformational energy based on distances and the 2D graph representation $\\mathcal { G }$ . Using interatomic distances for energy prediction is favored in existing literature [16, 20, 31, 34], as it preserves the 3D rotation and translation symmetries of molecular systems. Since $\\{ e _ { k } ( { \\pmb R } ) \\} _ { k = 1 } ^ { K }$ are continuously differentiable with respect to the Cartesian coordinates $\\pmb { R }$ , the gradients of logarithm density of interest, i.e., $\\nabla _ { R } \\log p _ { \\theta } ( R | \\mathcal { G } )$ , is interrelated with the logarithm density of each interatomic distance via chain rule: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3aaf98fd87223f93562ee7c2b5bb99256c8d6ec959dc3f260d9d6e1156dad484.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\forall i , s _ { \\theta } ( { \\pmb R } ) _ { i } : = \\frac { \\partial f _ { \\mathcal { G } } \\left( e _ { 1 } ( { \\pmb R } ) , e _ { 2 } ( { \\pmb R } ) , \\cdots , e _ { K } ( { \\pmb R } ) \\right) } { \\partial r _ { i } } } \\\\ { \\displaystyle = \\sum _ { k = 1 } ^ { K } \\frac { \\partial f _ { \\mathcal { G } } \\left( e _ { 1 } ( { \\pmb R } ) , e _ { 2 } ( { \\pmb R } ) , \\cdots , e _ { K } ( { \\pmb R } ) \\right) } { \\partial e _ { k } ( { \\pmb R } ) } \\cdot \\frac { \\partial e _ { k } ( { \\pmb R } ) } { \\partial r _ { i } } } \\\\ { \\displaystyle = \\sum _ { k = 1 } ^ { K } s _ { \\theta } ( e _ { k } ( { \\pmb R } ) ) \\cdot \\frac { \\partial e _ { k } ( { \\pmb R } ) } { \\partial r _ { i } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            133,
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+            255
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\displaystyle s _ { \\theta } ( \\mathbf { R } ) _ { i } }$ denotes $\\nabla _ { r _ { i } } \\log p _ { \\theta } ( R \\mid \\mathcal { G } )$ , $\\scriptstyle s _ { \\theta } ( e _ { k } ( R ) )$ denotes $\\nabla _ { e _ { k } ( R ) } \\log p _ { \\pmb \\theta } ( \\pmb R \\mid \\mathcal { G } )$ , and $\\frac { \\partial e _ { k } ( \\pmb { R } ) } { \\partial \\pmb { r } _ { i } }$ can be calculated efficiently in closed form (see supplementary material for the full derivation). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Motivated by the above equation, we first train a noise conditional score network to jointly predict the score of interatomic distances, i.e., $\\{ s _ { \\theta } ( e _ { k } ( { \\pmb R } ) , \\sigma ) \\} _ { k = 1 } ^ { K }$ . After training the noise conditional score network, the gradients of the logarithm density of atomic coordinates, i.e., $s _ { \\theta } ( R , \\sigma )$ , can be estimated via Eq. 4. We have the following proposition (proof in supplementary material): ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1 (Roto-Translation Equivariance). With the assumption that we can parameterize $\\log p _ { \\pmb \\theta } ( \\pmb R | \\mathcal G )$ as a function of interatomic distances, conditional on molecular graph $\\mathcal { G }$ (Eq. 3), the score function $s _ { \\theta } ( R )$ defined in Eq. 4 is roto-translation equivariant. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remarkably, the choice of $\\{ e _ { k } ( { \\pmb R } ) \\} _ { k = 1 } ^ { K }$ is flexible under this framework and can be carefully designed for specific goals. An ideal set of interatomic distances should capture both the local and long-range interactions between atoms (Eq. 1). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2 (Connection with CONFGF). The recent proposed CONFGF [34] is a special case of our approach, where they only model local distances between the first-order, the second-order, and the third-order neighbors, i.e., $\\{ e _ { k } \\} _ { k = 1 } ^ { K }$ map the conformation to a set of distances between bonded atoms. Therefore, CONFGF fails to capture long-range interactions between non-bonded atoms. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Dynamic Graph Score Matching with Noise Conditional Score Networks ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            521,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe the proposed dynamic graph score matching for interatomic distances, with the goal of modeling both the local and long-range interactions. To ensure that the learned score functions cover all regions with different graph structures, we dynamically construct graph structures with non-bonded edges between atoms during training, based on added perturbations. Following Song and Ermon [38], we train a noise conditional score network to jointly estimate scores for perturbed distributions of a set of dynamically-determined interatomic distances, i.e., $\\{ s _ { \\pmb \\theta } ( e _ { k } ( \\pmb R ) , \\sigma ) \\} _ { k = 1 } ^ { K }$ , and parameterize the score network with the message passing neural network (MPNN) [13]. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Dynamic Score Matching. To capture long-range interactions between non-bonded atoms in a molecular system, a naive way is to treat the molecular graph as a fully-connected graph and model the gradients of logarithm density of distances between all pairs of atoms. However, such a practice is computationally expensive especially for large systems, e.g., proteins, and is sometimes unnecessary, e.g., van der Waals interactions decay rapidly as distances increase. As a remedy, we set a cutoff distance and assume each atom only interacts with all atoms within the cutoff distance, ignoring all interactions out of the considered sphere. This is a very popular strategy in computational chemistry that strikes a good balance between efficiency and accuracy [26, 31]. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Formally, consider a molecular graph $\\mathcal { G } = \\langle \\nu , \\mathcal { E } \\rangle$ with distances between all pairs of atoms $\\pmb { D } \\in \\mathbb { \\Sigma }$ $\\mathbb { R } ^ { | \\nu | \\times | \\nu | }$ computed from its conformation $\\pmb { R } \\in \\mathbb { R } ^ { | \\nu | \\times 3 }$ . For a given noise level $\\sigma$ , we perturb the distances $_ { D }$ with Gaussian noise at each training step on the fly, and then augment the original graph structure with non-bonded edges between atom pairs within a certain threshold distance: ",
+        "bbox": [
+            174,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/55e9fef8e2b279c46460fe6c9c3f8c018492a3dc4e6dc42838f23be4756643c1.jpg",
+        "text": "$$\n\\forall ( i , j ) , \\tilde { D } _ { i j } \\sim \\mathcal { N } ( D _ { i j } , \\sigma ^ { 2 } ) , \\quad \\mathcal { E } _ { \\sigma } = \\mathcal { E } \\cup \\{ e _ { i j } \\mid \\tilde { D } _ { i j } < \\delta \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e9846ec7ac63039c1d526aaf92bb3ee30bf32811c0b9c38f9ab9112d74b354d9.jpg",
+        "text": "$$\n\\begin{array} { r } { l _ { \\sigma } = \\{ D _ { i j } \\mid e _ { i j } \\in \\mathcal { E } _ { \\sigma } \\} , \\quad \\tilde { d } _ { \\sigma } = \\{ \\tilde { D } _ { i j } \\mid e _ { i j } \\in \\mathcal { E } _ { \\sigma } \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            863,
+            666,
+            881
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { E } _ { \\sigma }$ is the constructed graph structure for noise level $\\sigma$ , and $\\delta$ is a hyper-parameter that controls the radius of long-range interactions. We empirically verify that a $1 0 \\mathring \\mathrm { A }$ cutoff is sufficient for systems we are studying, which is consistent with some experimental results in molecular dynamics [26]. $\\scriptstyle d _ { \\sigma }$ and $\\tilde { d } _ { \\sigma }$ denote the original and perturbed interatomic distances in augmented graph structure respectively. Hereafter, we omit the subscript for simplicity and use ${ \\mathcal { E } } , d .$ , and $\\tilde { d }$ instead, assuming all graphs are dynamically constructed during training and sampling. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/8f7d52f8c18d23459383561fc7802e024313762b231abd98f24b733b159cb4f9.jpg",
+        "image_caption": [
+            "Figure 3: The generation procedure of the proposed DGSM via Langevin dynamics. The graph structure is dynamically constructed at each step of stochastic update based on the current conformation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "With the above strategy, the graph structure of a specific molecular graph $\\mathcal { G }$ is variadic depending on added perturbation, and all graph structures are possible as long as we sample sufficient enough noise. This will result in (1) a dynamically-determined graph structure for message passing and representation learning, which takes long-range interactions into consideration, and (2) a dynamicallydetermined set of interatomic distances, i.e., $\\mathbf { \\bar { \\{ } }  e _ { k } ( { \\pmb R } ) \\} _ { k = 1 } ^ { K }$ , for score estimation, which contributes to gradients of atomic coordinates according to Eq. 4. Note that the vanilla implementation of Eq. 5 requires computing all distances between atom pairs. In practice, to avoid quadratic complexity, we pre-filter distant neighbors before adding perturbations for each atom by constructing radius graph with $2 \\delta$ threshold, and empirically verify that it performs efficiently and effectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Parameterizing with MPNNs. Let $\\{ \\sigma _ { i } \\} _ { i = 1 } ^ { L }$ be a sequence of noise levels. Our goal is to learn a noise conditional score network to jointly estimate the scores of all perturbed distance distributions, i.e., $\\forall \\sigma \\in \\{ \\sigma _ { i } \\} _ { i = 1 } ^ { L } : s _ { \\theta } ( \\tilde { d } , \\sigma ) \\approx \\mathring { \\nabla _ { \\tilde { d } } } \\overrightarrow { \\log p _ { \\sigma } } ( \\tilde { d } \\mid \\mathcal { G } ) .$ , where $\\begin{array} { r } { p _ { \\sigma } ( \\tilde { d } \\mid \\tilde { \\mathcal { G } } ) = \\int p ( d \\mid G ) \\mathcal { N } ( \\tilde { d } \\mid d , \\sigma ^ { 2 } I ) } \\end{array}$ Following suggestions of Song and Ermon [38], we parameterize the score network with a MPNN as follows (see supplementary material for the full architecture): ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b93b24051297495b2c10daa74bf846388ced7b4c4d648ef637aa1ee9c2c5c337.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\quad h _ { i } ^ { \\mathrm { n o d e } } = \\mathrm { M P N N } ( \\mathcal { G } , \\mathcal { E } , \\tilde { D } ) _ { i } , \\quad \\forall v _ { i } \\in \\mathcal { V } } \\\\ & { \\quad h _ { i j } ^ { \\mathrm { e d g e } } = \\mathrm { C o n c a t } ( h _ { i } ^ { \\mathrm { n o d e } } , h _ { j } ^ { \\mathrm { n o d e } } ) , \\quad \\forall e _ { i j } \\in \\mathcal { E } _ { \\sigma } } \\\\ & { s _ { \\theta } ( \\tilde { d } , \\sigma ) _ { i j } = s _ { \\theta } ( \\tilde { d } ) _ { i j } / \\sigma = \\mathrm { M L P } ( h _ { i j } ^ { \\mathrm { e d g e } } ) / \\sigma , \\quad \\forall e _ { i j } \\in \\mathcal { E } _ { \\sigma } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "molec where {hnodei }|V|i=1 are node embeddings compute and the perturbed distances, $\\{ h _ { i j } ^ { \\mathrm { e d g e } } \\}$ MPNN based on the dynamically consare embeddings for each edge in $\\mathcal { E }$ ucted, and $s _ { \\theta } ( \\tilde { d } , \\sigma ) _ { i j }$ is the predicted score for interatomic distance $\\tilde { D } _ { i j }$ $( e _ { i j } \\in \\mathcal { E }$ ). The noise conditional network can be jointly optimized with the following objective [38]: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d05f73ff4fbad165420099650f6e5fdb926ae1ba54e479f822b687efb8937b93.jpg",
+        "text": "$$\n\\theta ^ { * } = \\operatorname * { a r g m i n } _ { \\theta } \\frac { 1 } { 2 L } \\sum _ { i = 1 } ^ { L } \\sigma _ { i } ^ { 2 } \\mathbb { E } _ { p ( d | \\mathcal { G } ) } \\mathbb { E } _ { p _ { \\sigma _ { i } } ( \\tilde { d } | d , \\mathcal { G } ) } \\Big [ \\Big \\lVert \\frac { s _ { \\theta } ( \\tilde { d } ) } { \\sigma _ { i } } + \\frac { \\tilde { d } - d } { \\sigma _ { i } ^ { 2 } } \\Big \\rVert _ { 2 } ^ { 2 } \\Big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where all expectations can be efficiently estimated using empirical averages. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 Generation ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "After training the noise conditional score network, molecular conformations can be generated via annealed Langevin dynamics [38], guided by the gradients of atomic coordinates. The gradients can be computed via Eq. 4 based on dynamically constructed graph structures at each step of stochastic update, which allows model to effectively capture both the local and long-range interactions that contribute to the atomic gradients. Formally, given a molecular graph $\\mathcal { G }$ , we first sample an initial conformation $\\scriptstyle { R _ { 0 } }$ from a fixed prior distribution. We here take the prior distribution as a standard ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Gaussian $\\mathcal { N } ( R _ { 0 } \\mid \\mathbf { 0 } , I )$ . Then, we update the conformation by running $T$ steps of Langevin dynamic to get a sample from each noise conditional score network ${ \\bf \\delta } _ { s _ { \\theta } ( { \\cal R } , \\sigma _ { i } ) }$ sequentially with a special step size schedule $\\alpha _ { i } = \\varepsilon \\cdot \\sigma _ { i } ^ { 2 } / \\sigma _ { L } ^ { 2 }$ . Samples from each noise level are used to initialize Langevin dynamics for the next noise level. At each sampling step $t$ , we first construct graph structures with non-bonded edges within a given distance $\\delta$ based on the current pairwise distances $D _ { t - 1 }$ computed from $\\mathbf { \\delta } _ { R _ { t - 1 } }$ , and then get a set of interatomic distances $d _ { t - 1 }$ for score estimation: ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3c32243cbf342d28938d20f296be8bdcceb4e04326cadb0d0e036ea5b4cf3eca.jpg",
+        "text": "$$\n\\mathcal { E } _ { t - 1 } = \\mathcal { E } \\cup \\big \\{ e _ { i j } ~ \\big | ~ D _ { t - 1 , i j } < \\delta \\big \\} , \\quad d _ { t - 1 } = \\big \\{ D _ { t - 1 , i j } ~ \\big | ~ e _ { i j } \\in \\mathcal { E } _ { t - 1 } \\big \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The conformation is then updated using the gradient information from the score network (Eq. 4). We provide the pseudo-code in Algorithm 1. ",
+        "bbox": [
+            176,
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+        "type": "text",
+        "text": "5 Experiments ",
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+        "text": "Following previous works [34, 36, 43] on conformation generation, we evaluate the proposed DGSM using the following two standard tasks: Conformation Generation (Section 5.1), and Property Prediction (Section 5.2). To further demonstrate DGSM’s capability of modeling longrange interactions, we evaluate it on two more challenging benchmark tasks: Protein Sidechain Conformation Generation and Multi-molecular Complex Conformation Generation (Section 5.3). We describe experimental setups in task-specific sections. ",
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+        "text": "Algorithm 1 Annealed Langevin dynamics [38] ",
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+        "type": "text",
+        "text": "Require: $\\mathcal { G } = \\langle \\nu , \\mathcal { E } \\rangle$ , $\\{ \\sigma _ { i } \\} _ { i = 1 } ^ { L } , \\delta , \\varepsilon , T$ . ",
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+        "text": "1: Initialize conformation $\\scriptstyle { R _ { 0 } }$   \n2: for $i \\gets 1$ to $L$ do   \n3: $\\alpha _ { i }  \\varepsilon \\cdot \\sigma _ { i } ^ { 2 } / \\sigma _ { L } ^ { 2 }$ $\\triangleright \\alpha _ { i }$ is the step size.   \n4: for $t \\gets 1$ to $T$ do   \n5: $\\mathcal { E } _ { t - 1 } , d _ { t - 1 } \\gets \\mathrm { a u g } ( \\mathcal { E } , R _ { t - 1 } , \\delta )$ . Eq. 8   \n6: $s _ { \\theta } ( R _ { t - 1 } , \\sigma _ { i } ) \\gets \\mathrm { g e t } ( s _ { \\theta } ( d _ { t - 1 } , \\sigma _ { i } ) ) \\triangleright \\mathrm { E q . } 4$   \n7: Draw $\\boldsymbol { z } _ { t } \\sim \\mathcal { N } ( \\mathbf { 0 } , I )$   \n8: $R _ { t } \\gets R _ { t - 1 } + \\alpha _ { i } s _ { \\theta } ( R _ { t - 1 } , \\sigma _ { i } ) + \\sqrt { 2 \\alpha _ { i } } z _ { t }$   \n9: end for   \n10: $R _ { 0 } \\gets R _ { T }$   \n11: end for   \nReturn: Generated conformation $R _ { T }$ . ",
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+        "type": "text",
+        "text": "5.1 Conformation Generation ",
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+        "text": "Setup. This task evaluates the model’s capability to generate stable molecular conformations by measuring both accuracy and diversity of generated conformations. Following previous works [34, 43], we use the GEOM-QM9 and GEOM-Drugs [1] datasets for this task. We use the train-test split provided by [34]. The train splits of GEOM-QM9 and GEOM-Drugs both contain 40,000 molecules, each with 5 conformations for training, or 200,000 conformations in total. The test split of GEOM-QM9 contains 200 molecules with 22,408 conformations, and the test split of GEOM-Drugs contains 200 molecules with 14,324 conformations. ",
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+        "text": "We compare DGSM against 5 state-of-the-art baselines: RDKIT [28], CVGAE [24], GRAPHDG [36], CGCF [43] and CONFGF [34]. For each molecule in the test set, we sample twice as many conformations as its reference conformations. We use the matching score (MAT) to measure the accuracy of generated conformations, and coverage score (COV) to measure the diversity following [34, 43]. Both metrics are based on Root Mean Squared Deviations (RMSD) between molecules, taking symmetries into account (see supplementary material for the details of metrics). ",
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+        "text": "Results. We report the mean and median COV and MAT scores over all the molecules in the test split on GEOM-QM9 and GEOM-Drugs datasets. As shown in Table 1, DGSM consistently outperforms all the baselines. Notably, both DGSM and CONFGF are score-based models, but DGSM achieves better performance. The difference between them is that DGSM successfully takes long-range interactions into consideration via dynamic graph score matching. This confirms the significant benefit of modeling long-range interactions. We present several conformations generated by different approaches in Figure 4, which shows that DGSM successfully captures the long-range interactions in highlighted areas while the other baselines fail, resulting in distorted structures in those areas. ",
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+        "table_caption": [
+            "Table 1: COV and MAT scores on GEOM-QM9 and GEOM-Drugs datasets. The threshold $\\delta$ of COV score is $0 . 5 \\mathring \\mathrm { A }$ for GEOM-QM9 and $1 . 2 5 \\mathring \\mathrm { A }$ for GEOM-Drugs following $\\mathrm { X u }$ et al. [43]. (↑): the higher the better. (↓): the lower the better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">GEOM-QM9</td><td colspan=\"4\">GEOM-Drugs</td></tr><tr><td></td><td colspan=\"4\">COV (%, ↑)</td><td colspan=\"4\">COV (%, ↑)</td></tr><tr><td>Method</td><td>Mean</td><td>Median</td><td>Mean</td><td>MAT (A,↓) Median</td><td>Mean</td><td>Median</td><td>Mean</td><td>MAT (A,↓) Median</td></tr><tr><td>RDKIT[28]</td><td>83.26</td><td>90.78</td><td>0.3447</td><td>0.2935</td><td>60.91</td><td>65.70</td><td>1.2026</td><td>1.1252</td></tr><tr><td>CVGAE [24]</td><td>0.09</td><td>0.00</td><td>1.6713</td><td>1.6088</td><td>0.00</td><td>0.00</td><td>3.0702</td><td>2.9937</td></tr><tr><td>GRAPHDG [36]</td><td>73.33</td><td>84.21</td><td>0.4245</td><td>0.3973</td><td>8.27</td><td>0.00</td><td>1.9722</td><td>1.9845</td></tr><tr><td>CGCF [43]</td><td>77.52</td><td>80.40</td><td>0.4206</td><td>0.3903</td><td>54.19</td><td>56.35</td><td>1.2575</td><td>1.2356</td></tr><tr><td>CONFGF[34]</td><td>88.49</td><td>94.13</td><td>0.2673</td><td>0.2685</td><td>62.15</td><td>70.93</td><td>1.1629</td><td>1.1596</td></tr><tr><td>DGSM</td><td>91.49</td><td>95.92</td><td>0.2139</td><td>0.2137</td><td>78.73</td><td>94.39</td><td>1.0154</td><td>0.9980</td></tr></table>",
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+        "image_caption": [
+            "Figure 4: Examples of conformations generated by different models based on four random molecular graphs from the test set of GEOM-Drugs. We present three reference conformations for each molecule, and visualize the best-aligned conformations generated by each method. Areas where long-range interactions should be modeled are highlighted in green. "
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+    {
+        "type": "text",
+        "text": "5.2 Property Prediction ",
+        "text_level": 1,
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+        "text": "Setup. This task demonstrates how generative models for molecular conformations can be applied to property prediction as a downstream task. It also provides an assessment on the quality of generated conformations in a different light. We estimate the ensemble properties [1] of a molecular graph by aggregating its conformational properties following [34]. In specific, we first use the models ",
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+        "table_caption": [
+            "Table 2: Mean absolute errors (MAE) of predicted ensemble properties in eV. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>E</td><td>Emin</td><td>A</td><td>△emin</td><td>△emax</td></tr><tr><td>RDKIT</td><td>0.9233</td><td>0.6585</td><td>0.3698</td><td>0.8021</td><td>0.2359</td></tr><tr><td>GRAPHDG</td><td>9.1027</td><td>0.8882</td><td>1.7973</td><td>4.1743</td><td>0.4776</td></tr><tr><td>CGCF</td><td>28.9661</td><td>2.8410</td><td>2.8356</td><td>10.6361</td><td>0.5954</td></tr><tr><td>CONFGF</td><td>2.7886</td><td>0.1765</td><td>0.4688</td><td>2.1843</td><td>0.1433</td></tr><tr><td>DGSM</td><td>1.0313</td><td>0.0761</td><td>0.1963</td><td>1.1811</td><td>0.1271</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "to generate 50 conformations for each molecular graph in a subset of GEOM-QM9 [34], and use PSI4 [37], a quantum chemical toolkit, to calculate each conformation’s energy and HOMO-LUMO gap. Then, we calculate average energy $\\overline { E }$ , lowest energy $E _ { \\mathrm { m i n } }$ , average gap $\\overline { { \\Delta \\epsilon } }$ , minimum gap $\\Delta \\epsilon _ { \\mathrm { m i n } }$ and maximum gap $\\Delta \\epsilon _ { \\mathrm { m a x } }$ from the conformational energy and gap. We evaluate the accuracy of estimated ensemble property by measuring their mean absolute errors (MAE) to the ground truth values. CVGAE is excluded in this task as its performance is poor, which is also reported in [36, 34]. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Results. Table 2 shows that DGSM outperforms other machine learning-based methods by a clear margin. DGSM’s estimation of average energy $\\overline { E }$ and minimum gap $\\Delta \\epsilon _ { \\mathrm { m i n } }$ is close to RDKIT but still outperforms the most competitive ML-based method CONFGF. The calculation of conformational energy is highly sensitive to changes in geometry — even a subtle deviation in bond lengths leads to significant energy change [36]. Therefore, the superior performance of DGSM indicates that it generates much more accurate conformations than other methods, leading to more accurate property estimation. This validates again the effectiveness of modeling long-range interactions. ",
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+    {
+        "type": "text",
+        "text": "5.3 Large Molecule and Multi-molecular Modeling ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Protein Sidechain Conformation This task is to predict protein sidechain conformations based on its backbone structures. Compared to conventional molecular conformations generation in previous sections, the main challenge of this task is two-fold: (1) large number of atoms, which prohibits constructing complete graphs that grow quadratically to model long-range interactions. ",
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+    {
+        "type": "table",
+        "img_path": "images/17d1b8ffa92260c5a8e9efde34dbbcc253a2cc9c4fd9eb66dc3800eb8bc4840e.jpg",
+        "table_caption": [
+            "Table 3: RMSD of different approaches on sidechain conformation generation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">RMSD</td></tr><tr><td>Mean (A)</td><td>Min (A)</td></tr><tr><td>CONFGF</td><td>3.38</td><td>3.11</td></tr><tr><td>DGSM</td><td>2.85 (↓15.7%)</td><td>2.61 (↓ 16.1%)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "(2) covalent bonds are sparse, which limits the power of the edge augmentation techniques in previous work. DGSM tackles these two challenges via dynamic graph score matching as introduced. ",
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+        "img_path": "images/ddb62042f2bd28694c95bb5a17d95b0004d17ab5ff7ec35601389ceff3b8c4fc.jpg",
+        "image_caption": [
+            "Figure 5: (a) An example of the generated protein sidechain conformation with atomic-level coordinates. The ground-truth sidechain (blue) and the generated sidechain (red) are highlighted. (b) Conformations of two multi-molecular complexes generated by DGSM. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use the SidechainNet [18] dataset for this task and follow the official train-test splits. We compare DGSM with the state-of-the-art conformation generation model CONFGF. Despite that there are some machine learning-based methods specialized in protein sidechain structure prediction [10], they are built upon rotamer libraries [5], which incorporates a lot of domain knowledge. Thus, our method is not comparable with them. The main purpose of this task is to justify the effectiveness of DGSM for large molecules. For each protein, we generate 5 sidechain conformations with different initialization, and calculate the mean and min RMSD between the ground-truth conformation and the generated conformations. We report the overall mean and min RMSD scores by averaging scores of each protein in the test set in in Table 3, which shows that DGSM achieves better performance than previous state-of-the-art model. We also present an example in Figure 5(a), and we can see that the predicted conformation is consistent with the ground truth in major parts. ",
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+    {
+        "type": "text",
+        "text": "Multi-molecular Complex Conformation This task is to predict conformations for multi-molecular complexes. A multi-molecular complex is made up of multiple molecules and there is no covalent bonds between them. Long-range interactions dominate the structure of multi-molecular complexes. The purpose of this task is to demonstrate DGSM’s potential application to a broader range of problems and provide a novel benchmark for conformation generation. We use the quantum chemical software xtb [3] to construct a dataset consisting of 24 water-organic complexes each with several hundreds of conformations, and leave out 4 complexes for testing (see supplementary material for details). We do not report RMSD-based metrics such as COV and MAT because the structures of multi-molecular complexes are highly flexible. Two set of generated examples are presented in Figure 5(b). We observe that water molecules are placed regularly around the solute organic molecule. Notably, hydrogen bonds (between water and the solute, and between water and water) are formed correctly. This can also be evidenced in the histogram of Hydrogen-Oxygen distances (Figure 6), where there is a peak between $1 . 5 \\mathring \\mathrm { A }$ and $2 . 5 \\mathring \\mathrm { A }$ , i.e., the range of hydrogen bond length between Hydrogen and Oxygen. ",
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+        "image_caption": [
+            "Figure 6: The distribution of Hydrogen-Oxygen distances. The first peak from the left is covalent bonds and the second peak is hydrogen bonds. "
+        ],
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+    {
+        "type": "text",
+        "text": "6 Conclusion and Future Work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We propose DGSM, a novel score-based approach for generating equilibrium molecular conformations. DGSM is capable of modeling both the local and long-range interactions in molecular systems, by dynamically constructing graph structures based on spatial proximity between atoms during both training and inference. We also devise a dynamic graph score matching algorithm to effectively estimate atomic gradients, where graph structures are dynamically determined depending on added perturbations. Extensive experiments over two standard tasks and two original tasks show that DGSM outperforms the state-of-the-art method by a large margin, confirming the significant benefit of modeling long-range interactions. In the future, we plan to apply our approach to the more challenging problem of protein structure prediction. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We would like to thank all the reviewers for the insightful comments. This project is supported by the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant, the Canada CIFAR AI Chair Program, collaboration grants between Microsoft Research and Mila, Samsung Electronics Co., Ldt., Amazon Faculty Research Award, Tencent AI Lab Rhino-Bird Gift Fund and a NRC Collaborative R&D Project (AI4D-CORE-06). This project was also partially funded by IVADO Fundamental Research Project grant PRF-2019-3583139727. ",
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+    {
+        "type": "text",
+        "text": "References ",
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+    },
+    {
+        "type": "text",
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+# You Only Look at One Sequence: Rethinking Transformer in Vision through Object Detection
+
+Yuxin Fang 1∗ Bencheng Liao 1∗ Xinggang Wang 1† Jiemin Fang 2,1 Jiyang Qi 1 Rui Wu 3 Jianwei Niu 3 Wenyu Liu 1
+
+1 School of EIC, Huazhong University of Science & Technology   
+2 Institute of AI, Huazhong University of Science & Technology 3 Horizon Robotics {yxf, bcliao, xgwang}@hust.edu.cn
+
+# Abstract
+
+Can Transformer perform 2D object- and region-level recognition from a pure sequence-to-sequence perspective with minimal knowledge about the 2D spatial structure? To answer this question, we present You Only Look at One Sequence (YOLOS), a series of object detection models based on the vanilla Vision Transformer with the fewest possible modifications, region priors, as well as inductive biases of the target task. We find that YOLOS pre-trained on the mid-sized ImageNet- $1 k$ dataset only can already achieve quite competitive performance on the challenging COCO object detection benchmark, e.g., YOLOS-Base directly adopted from BERT-Base architecture can obtain 42.0 box AP on COCO val. We also discuss the impacts as well as limitations of current pre-train schemes and model scaling strategies for Transformer in vision through YOLOS. Code and pre-trained models are available at https://github.com/hustvl/YOLOS.
+
+# 1 Introduction
+
+Transformer [58] is born to transfer. In natural language processing (NLP), the dominant approach is to first pre-train Transformer on large, generic corpora for general language representation learning, and then fine-tune or adapt the model on specific target tasks [18]. Recently, Vision Transformer (ViT) 1 [21] demonstrates that canonical Transformer encoder architecture directly inherited from NLP can perform surprisingly well on image recognition at scale using modern vision transfer learning recipe [33]. Taking sequences of image patch embeddings as inputs, ViT can successfully transfer pre-trained general visual representations from sufficient scale to more specific image classification tasks with fewer data points from a pure sequence-to-sequence perspective.
+
+Since a pre-trained Transformer can be successfully fine-tuned on sentence-level tasks [7, 19] in NLP, as well as token-level tasks [48, 52], where models are required to produce fine-grained output at the token-level [18]. A natural question is: Can ViT transfer to more challenging object- and region-level target tasks in computer vision such as object detection other than image-level recognition?
+
+ViT-FRCNN [6] is the first to use a pre-trained ViT as the backbone for a Faster R-CNN [50] object detector. However, this design cannot get rid of the reliance on convolutional neural networks (CNNs)
+
+and strong 2D inductive biases, as ViT-FRCNN re-interprets the output sequences of ViT to 2D spatial feature maps and depends on region-wise pooling operations (i.e., RoIPool [23, 25] or RoIAlign [27]) as well as region-based CNN architectures [50] to decode ViT features for object- and region-level perception. Inspired by modern CNN design, some recent works [39, 59, 62, 65] introduce the pyramidal feature hierarchy, spatial locality, equivariant as well as invariant representations [24] to canonical Vision Transformer design, which largely boost the performance in dense prediction tasks including object detection. However, these architectures are performance-oriented and cannot reflect the properties of the canonical or vanilla Vision Transformer [21] directly inherited from Vaswani et al. [58]. Another series of work, the DEtection TRansformer (DETR) families [10, 72], use a random initialized Transformer to encode & decode CNN features for object detection, which does not reveal the transferability of a pre-trained Transformer.
+
+Intuitively, ViT is designed to model long-range dependencies and global contextual information instead of local and region-level relations. Moreover, ViT lacks hierarchical architecture as modern CNNs [26, 35, 53] to handle the large variations in the scale of visual entities [1, 37]. Based on the available evidence, it is still unclear whether a pure ViT can transfer pre-trained general visual representations from image-level recognition to the much more complicated 2D object detection task.
+
+To answer this question, we present You Only Look at One Sequence (YOLOS), a series of object detection models based on the canonical ViT architecture with the fewest possible modifications, region priors, as well as inductive biases of the target task injected. Essentially, the change from a pre-trained ViT to a YOLOS detector is embarrassingly simple: (1) YOLOS replaces one [CLS] token for image classification in ViT with one hundred [DET] tokens for object detection. (2) YOLOS replaces the image classification loss in ViT with the bipartite matching loss to perform object detection in a set prediction manner following Carion et al. [10], which can avoid re-interpreting the output sequences of ViT to 2D feature maps as well as prevent manually injecting heuristics and prior knowledge of object 2D spatial structure during label assignment [71]. Moreover, the prediction head of YOLOS can get rid of complex and diverse designs, which is as compact as a classification layer.
+
+Directly inherited from ViT [21], YOLOS is not designed to be yet another high-performance object detector, but to unveil the versatility and transferability of pre-trained canonical Transformer from image recognition to the more challenging object detection task. Concretely, our main contributions are summarized as follows:
+
+• We use the mid-sized ImageNet- $. 1 k$ [51] as the sole pre-training dataset, and show that a vanilla ViT [21] can be successfully transferred to perform the complex object detection task and produce competitive results on COCO [36] benchmark with the fewest possible modifications, i.e., by only looking at one sequence (YOLOS).   
+• For the first time, we demonstrate that 2D object detection can be accomplished in a pure sequence-to-sequence manner by taking a sequence of fixed-sized non-overlapping image patches as input. Among existing object detectors, YOLOS utilizes the minimal 2D inductive biases.   
+• For the vanilla ViT, we find the object detection results are quite sensitive to the pre-train scheme and the detection performance is far from saturating. Therefore the proposed YOLOS can be also used as a challenging benchmark task to evaluate different (label-supervised and self-supervised) pre-training strategies for ViT.
+
+# 2 You Only Look at One Sequence
+
+As for the model design, YOLOS closely follows the original ViT architecture [21], and is optimized for object detection in the same vein as Carion et al. [10]. YOLOS can be easily adapted to various canonical Transformer architectures available in NLP as well as in computer vision. This intentionally simple setup is not designed for better detection performance, but to exactly reveal characteristics of the Transformer family in object detection as unbiased as possible.
+
+# 2.1 Architecture
+
+An overview of the model is depicted in Fig. 1. Essentially, the change from a ViT to a YOLOS detector is simple: (1) YOLOS drops the [CLS] token for image classification and appends one hundred randomly initialized learnable detection tokens ([DET] tokens) to the input patch embeddings ([PATCH] tokens) for object detection. (2) During training, YOLOS replaces the image classification loss in ViT with the bipartite matching loss to perform object detection in a set prediction manner following Carion et al. [10].
+
+![](images/b9fa86f2f98eb89a87e5b972b72a23e150a85dd475478f66b08c585dd073e747.jpg)  
+Figure 1: YOLOS architecture overview. “Pat-Tok” refers to [PATCH] token, which is the embedding of a flattened image patch. “Det-Tok” refers to [DET] token, which is a learnable embedding for object binding. “PE” refers to positional embedding. During training, YOLOS produces an optimal bipartite matching between predictions from one hundred [DET] tokens and ground truth objects. During inference, YOLOS directly outputs the final set of predictions in parallel. The figure style is inspired by Dosovitskiy et al. [21].
+
+Stem. The canonical ViT [21] receives an 1D sequence of embedded tokens as the input. To handle 2D image inputs, we reshape the image $\mathbf { x } \in \mathbb { R } ^ { H \times W \times C }$ into a sequence of flattened 2D image patches $\mathbf { x } _ { \mathtt { P A T C H } } \in \mathbb { R } ^ { N \times \left( P ^ { 2 } \cdot C \right) }$ . Here, $( H , W )$ is the resolution of the input image, $C$ is the number of input channels, $( P , P )$ is the resolution of each image patch, and $\begin{array} { r } { \dot { N } = \frac { H \dot { W } } { P ^ { 2 } } } \end{array}$ is the resulting number of patches. Then we map $\mathbf { x } _ { \mathrm { P A T C H } }$ to $D$ dimensions with a trainable linear projection $\mathbf { E } \in \mathbb { R } ^ { \left( P ^ { 2 } \cdot C \right) \times D }$ . We refer to the output of this projection $\mathbf { x } _ { \mathrm { P A T C H } } \mathbf { E }$ as [PATCH] tokens. Meanwhile, one hundred randomly initialized learnable [DET] tokens $\mathbf { x } _ { \mathrm { D E T } } \in \mathbb { R } ^ { 1 0 0 \times D }$ are appended to the [PATCH] tokens. Position embeddings $\mathbf { P } \in \mathbb { R } ^ { ( \bar { N } + 1 0 \bar { 0 } ) \times D }$ are added to all the input tokens to retain positional information. We use the standard learnable 1D position embeddings following Dosovitskiy et al. [21]. The resulting sequence $\mathbf { z } _ { 0 }$ serves as the input of YOLOS Transformer encoder. Formally:
+
+$$
+\begin{array} { r } { \mathbf { z } _ { 0 } = \left[ \mathbf { x } _ { \mathtt { P A T C H } } ^ { \mathrm { 1 } } \mathbf { E } ; \cdot \cdot \cdot \ ; \mathbf { x } _ { \mathtt { P A T C H } } ^ { N } \mathbf { E } ; \mathbf { x } _ { \mathtt { D E T } } ^ { \mathrm { 1 } } ; \cdot \cdot \cdot ; \mathbf { x } _ { \mathtt { D E T } } ^ { \mathrm { 1 0 0 } } \right] + \mathbf { P } . } \end{array}
+$$
+
+Body. The body of YOLOS is basically the same as ViT, which consists of a stack of Transformer encoder layers only [58]. [PATCH] tokens and [DET] tokens are treated equally and they perform global interactions inside Transformer encoder layers.
+
+Each Transformer encoder layer consists of one multi-head self-attention (MSA) block and one MLP block. LayerNorm (LN) [2] is applied before every block, and residual connections [26] are applied after every block [3, 61]. The MLP contains one hidden layer with an intermediate GELU [29] non-linearity activation function. Formally, for the $\ell$ -th YOLOS Transformer encoder layer:
+
+$$
+\begin{array} { r l } & { \mathbf { z } _ { \ell } ^ { \prime } = \mathrm { M S A } \left( \mathrm { L N } \left( \mathbf { z } _ { \ell - 1 } \right) \right) + \mathbf { z } _ { \ell - 1 } , } \\ & { \mathbf { z } _ { \ell } = \mathrm { M L P } \left( \mathrm { L N } \left( \mathbf { z } _ { \ell } ^ { \prime } \right) \right) + \mathbf { z } _ { \ell } ^ { \prime } . } \end{array}
+$$
+
+Detector Heads. The detector head of YOLOS gets rid of complex and heavy designs, and is as neat as the image classification layer of ViT. Both the classification and the bounding box regression heads are implemented by one MLP with separate parameters containing two hidden layers with intermediate ReLU [41] non-linearity activation functions.
+
+Detection Token. We purposefully choose randomly initialized [DET] tokens as proxies for object representations to avoid inductive biases of 2D structure and prior knowledge about the task injected during label assignment. When fine-tuning on COCO, for each forward pass, an optimal bipartite matching between predictions generated by [DET] tokens and ground truth objects is established. This procedure plays the same role as label assignment [10, 71], but is unaware of the input 2D structure, i.e., YOLOS does not need to re-interpret the output sequence of ViT to an 2D feature maps for label assignment. Theoretically, it is feasible for YOLOS to perform any dimensional object detection without knowing the exact spatial structure and geometry, as long as the input is always flattened to a sequence in the same way for each pass.
+
+Fine-tuning at Higher Resolution. When fine-tuning on COCO, all the parameters are initialized from ImageNet- $. 1 k$ pre-trained weights except for the MLP heads for classification & bounding box regression as well as one hundred [DET] tokens, which are randomly initialized. During fine-tuning, the image has a much higher resolution than pre-training. We keep the patch size $P$ unchanged, i.e., $P \times P = 1 6 \times 1 6$ , which results in a larger effective sequence length. While ViT can handle arbitrary input sequence lengths, the positional embeddings need to adapt to the longer input sequences with various lengths. We perform 2D interpolation of the pre-trained position embeddings on the fly2.
+
+Inductive Bias. We carefully design the YOLOS architecture for the minimal additional inductive biases injection. The inductive biases inherent from ViT come from the patch extraction at the network stem part as well as the resolution adjustment for position embeddings [21]. Apart from that, YOLOS adds no non-degenerated (e.g., $3 \times 3$ or other non $1 \times 1$ ) convolutions upon ViT 3. From the representation learning perspective, we choose to use [DET] tokens to bind objects for final predictions to avoid additional 2D inductive biases as well as task-specific heuristics. The performance-oriented design inspired by modern CNN architectures such as pyramidal feature hierarchy, 2D local spatial attention as well as the region-wise pooling operation is not applied. All these efforts are meant to exactly unveil the versatility and transferability of pre-trained Transformers from image recognition to object detection in a pure sequence-to-sequence manner, with minimal knowledge about the input spatial structure and geometry.
+
+Comparisons with DETR. The design of YOLOS is deeply inspired by DETR [10]: YOLOS uses [DET] tokens following DETR as proxies for object representations to avoid inductive biases about 2D structures and prior knowledge about the task injected during label assignment, and YOLOS is optimized similarly as DETR.
+
+Meanwhile, there are some key differences between the two models: (1) DETR adopts a Transformer encoder-decoder architecture, while YOLOS chooses an encoder-only Transformer architecture. (2) DETR only employs pre-training on its CNN backbone but leaves the Transformer encoder & decoder being trained from random initialization, while YOLOS naturally inherits representations from any pre-trained canonical ViT. (3) DETR applies cross-attention between encoded image features and object queries with auxiliary decoding losses deeply supervised at each decoder layer, while YOLOS always looks at only one sequence for each encoder layer, without distinguishing [PATCH] tokens and [DET] tokens in terms of operations. Quantitative comparisons between the two are in Sec. 3.4.
+
+# 3 Experiments
+
+# 3.1 Setup
+
+Pre-training. We pre-train all YOLOS / ViT models on ImageNet- $1 k$ [51] dataset using the dataefficient training strategy suggested by Touvron et al. [57]. The parameters are initialized with a truncated normal distribution and optimized using AdamW [40]. The learning rate and batch size are $1 \times 1 0 ^ { - 3 }$ and 1024, respectively. The learning rate decay is cosine and the weight decay is 0.05. Rand-Augment [14] and random erasing [69] implemented by timm library [64] are used for data augmentation. Stochastic depth [32], Mixup [68] and Cutmix [66] are used for regularization.
+
+Fine-tuning. We fine-tune all YOLOS models on COCO object detection benchmark [36] in a similar way as Carion et al. [10]. All the parameters are initialized from ImageNet- $. 1 k$ pre-trained weights except for the MLP heads for classification $\&$ bounding box regression as well as one hundred [DET] tokens, which are randomly initialized. We train YOLOS on a single node with $8 \times 1 2 6$ GPUs. The learning rate and batch sizes are $2 . 5 \times 1 0 ^ { - 5 }$ and 8 respectively. The learning rate decay is cosine and the weight decay is $1 \times 1 0 ^ { - 4 }$ .
+
+As for data augmentation, we use multi-scale augmentation, resizing the input images such that the shortest side is at least 256 and at most 608 pixels while the longest at most 864 for tiny models. For small and base models, we resize the input images such that the shortest side is at least 480 and at most 800 pixels while the longest at most 1333. We also apply random crop augmentations during training following Carion et al. [10]. The number of [DET] tokens are 100 and we keep the loss function as well as loss weights the same as DETR, while we don’t apply dropout [54] or stochastic depth during fine-tuning since we find these regularization methods hurt performance.
+
+Model Variants. With available computational resources, we study several YOLOS variants. Detailed configurations are summarized in Tab. 1. The input patch size for all models is $1 6 \times 1 6$ . YOLOS-Ti (Tiny), -S (Small), and -B (Base) directly correspond to DeiT-Ti, -S, and -B [57]. From the model scaling perspective [20, 56, 60], the small and base models of YOLOS / DeiT can be seen as performing width scaling $( w )$ [30, 67] on the corresponding tiny model.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>DeiT [57]Model</td><td rowspan=1 colspan=1>Layers(Depth)</td><td rowspan=1 colspan=1>Embed. Dim.(Width)</td><td rowspan=1 colspan=1>Pre-train Resolution</td><td rowspan=1 colspan=1> Heads</td><td rowspan=1 colspan=1> Params.</td><td rowspan=1 colspan=1>FLOPs</td><td rowspan=1 colspan=1>f(Lin.)f(Att.)</td></tr><tr><td rowspan=1 colspan=1>YOLOS-TiYOLOS-SYOLOS-B</td><td rowspan=1 colspan=1>DeiT-TiDeiT-SDeiT-B</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>192384768</td><td rowspan=1 colspan=1>224</td><td rowspan=1 colspan=1>3612</td><td rowspan=1 colspan=1>5.7M22.1M86.4 M</td><td rowspan=1 colspan=1>1.2 G4.5G17.6G</td><td rowspan=1 colspan=1>5.911.823.5</td></tr><tr><td rowspan=1 colspan=1>YOLOS-S (dwr)YOLOS-S (dwr)</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>1914</td><td rowspan=1 colspan=1>240330</td><td rowspan=1 colspan=1>272240</td><td rowspan=1 colspan=1>66</td><td rowspan=1 colspan=1>13.7M19.0M</td><td rowspan=1 colspan=1>4.6G4.6G</td><td rowspan=1 colspan=1>5.08.8</td></tr></table>
+
+Table 1: Variants of YOLOS. “dwr” and “dwr” refer to uniform compound model scaling and fast model scaling, respectively. The “dwr” and “dwr” notations are inspired by Dollár et al. [20]. Note that all the numbers listed are for pre-training, which could change during fine-tuning, e.g., the resolution and FLOPs.
+
+Besides, we investigate two other model scaling strategies which proved to be effective in CNNs. The first one is uniform compound scaling (dwr) [20, 56]. In this case, the scaling is uniform w.r.t. FLOPs along all model dimensions (i.e., width $( w )$ , depth $( d )$ and resolution $( r ) _ { , }$ ). The second one is fast scaling $( d \mathbf { w } r )$ [20] that encourages primarily scaling model width (w), while scaling depth $( d )$ and resolution $( r )$ to a lesser extent w.r.t. FLOPs. During the ImageNet- $. 1 k$ pre-training phase, we apply dwr and dwr scaling to DeiT-Ti $\mathrm { ~ \sim ~ } 1 . 2 6$ FLOPs) and scale the model to $\sim 4 . 5 6$ FLOPs to align with the computations of DeiT-S. Larger models are left for future work.
+
+For canonical CNN architectures, the model complexity or FLOPs $( f )$ are proportional to $d w ^ { 2 } r ^ { 2 }$ [20]. Formally, $f ( \mathbf { C N N } ) \propto d w ^ { 2 } r ^ { 2 }$ . Different from CNN, there are two kinds of operations that contribute to the FLOPs of ViT. The first one is the linear projection (Lin.) or point-wise convolution, which fuses the information across different channels point-wisely via learnable parameters. The complexity is $f ( \mathtt { L i n . } ) \propto d w ^ { 2 } r ^ { 2 }$ , which is the same as $f ( \mathbf { C } \mathbb { N } \mathbb { N } )$ . The second one is the spatial attention (Att.), which aggregates the spatial information depth-wisely via computed attention weights. The complexity is $f ( \mathbf { A } \mathbf { t } \mathbf { \bar { t } } . ) \propto d w r ^ { \tilde { 4 } }$ , which grows quadratically with the input sequence length or number of pixels.
+
+Note that the available scaling strategies are designed for architectures with complexity $f \propto d w ^ { 2 } r ^ { 2 }$ , so theoretically the dwr as well as dwr model scaling are not directly applicable to ViT. However, during pre-training phase the resolution is relatively low, therefore $f ( \mathtt { L i n . } )$ dominates the FLOPs $\begin{array} { r } { ( \frac { f ( \mathrm { L i n . } ) } { f ( \mathrm { A t t . } ) } > 5 ) } \end{array}$ . Our experiments indicate that some model scaling properties of ViT are consistent with CNNs when $\textstyle { \frac { f ( \mathrm { L i n . } ) } { f ( \mathrm { A t t . } ) } }$ is large.
+
+# 3.2 The Effects of Pre-training
+
+We study the effects of different pre-training strategies (both label-supervised and self-supervised) when transferring ViT (DeiT-Ti and DeiT-S) from ImageNet- $. 1 k$ to the COCO object detection benchmark via YOLOS. For object detection, the input shorter size is 512 for tiny models and is 800 for small models during inference. The results are shown in Tab. 2 and Tab. 3.
+
+Table 2: The effects of label-supervised pre-training. “pFLOPs” refers to petaFLOPs $( \times 1 0 ^ { 1 5 } )$ ). “ImNet” refers to ImageNet- $1 k$ . “C” refers to the distillation method from Touvron et al. [57].   
+
+<table><tr><td>Model</td><td> Pre-train Method</td><td>Pre-train Epochs</td><td>Fine-tune Epochs</td><td>Pre-train pFLOPs</td><td>Fine-tune pFLOPs</td><td>Total pFLOPs</td><td>ImNet Top-1</td><td>AP</td></tr><tr><td rowspan="4">YOLOS-Ti</td><td>Rand. Init.</td><td>0</td><td>600</td><td>0</td><td>14.2 ×10²</td><td>14.2× 10²</td><td>1</td><td>19.7</td></tr><tr><td>Label Sup. [57]</td><td>200</td><td rowspan="2">300</td><td>3.1×102</td><td rowspan="2">7.1 × 10²</td><td>10.2×10²</td><td>71.2</td><td>26.9</td></tr><tr><td>Label Sup. [57]</td><td>300</td><td>4.7×10²</td><td>11.8×10²</td><td>72.2</td><td>28.7</td></tr><tr><td>Label Sup. () [57]</td><td>300</td><td></td><td>4.7×10²</td><td></td><td>11.8×10²</td><td>74.5</td><td>29.7</td></tr><tr><td rowspan="5">YOLOS-S</td><td>Rand. Init.</td><td>0</td><td>250</td><td>0</td><td>5.9×103</td><td>5.9×103</td><td>1</td><td>20.9</td></tr><tr><td>Label Sup. [57]</td><td>100</td><td rowspan="4">150</td><td>0.6×103 1.2×103</td><td></td><td>4.1 ×103</td><td>74.5</td><td>32.0</td></tr><tr><td>Label Sup. [57]</td><td>200</td><td></td><td></td><td>4.7 ×103</td><td>78.5</td><td>36.1</td></tr><tr><td>Label Sup. [57]</td><td>300</td><td>1.8×10</td><td>3.5 × 103</td><td>5.3×103</td><td>79.9</td><td>36.1</td></tr><tr><td>Label Sup. () [57]</td><td>300</td><td>1.8×103</td><td></td><td>5.3×10</td><td>81.2</td><td>37.2</td></tr></table>
+
+Table 3: Study of self-supervised pre-training on YOLOS-S.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Self Sup. Pre-train Method</td><td rowspan=1 colspan=1>Pre-train Epochs</td><td rowspan=1 colspan=1>Fine-tune Epochs</td><td rowspan=1 colspan=1>Linear Acc.</td><td rowspan=1 colspan=1>AP</td></tr><tr><td rowspan=1 colspan=1>YOLOS-S</td><td rowspan=1 colspan=1>MoCo-v3[13]DINO[11]</td><td rowspan=1 colspan=1>300800</td><td rowspan=1 colspan=1>150150</td><td rowspan=1 colspan=1>73.277.0</td><td rowspan=1 colspan=1>33.636.2</td></tr></table>
+
+Necessity of Pre-training. At least under prevalent transfer learning paradigms [10, 57], the pretraining is necessary in terms of computational efficiency. For both tiny and small models, we find that pre-training on ImageNet- $. 1 k$ saves the total theoretical forward pass computations (total pre-training FLOPs & total fine-tuning FLOPs) compared with training on COCO from random initialization (training from scratch [28]). Models trained from scratch with hundreds of epochs still lag far behind the pre-trained ViT even if given more total FLOPs budgets. This seems quite different from canonical modern CNN-based detectors, which can catch up with pre-trained counterparts quickly [28].
+
+Label-supervised Pre-training. For supervised pre-training with ImageNet- $. 1 k$ ground truth labels, we find that different-sized models prefer different pre-training schedules: 200 epochs pre-training for YOLOS-Ti still cannot catch up with 300 epochs pre-training even with a 300 epochs fine-tuning schedule, while for the small model 200 epochs pre-training provides feature representations as good as 300 epochs pre-training for transferring to the COCO object detection benchmark.
+
+With additional transformer-specific distillation $( ^ { 6 6 } \pmb { \tilde { m } } ^ { 3 } )$ introduced by Touvron et al. [57], the detection performance is further improved by $\sim 1$ AP for both tiny and small models, in part because exploiting a CNN teacher [47] during pre-training helps ViT adapt to COCO better. It is also promising to directly leverage [DET] tokens to help smaller YOLOS learn from larger YOLOS on COCO during fine-tuning in a similar way as Touvron et al. [57], we leave it for future work.
+
+Self-supervised Pre-training. The success of Transformer in NLP greatly benefits from large-scale self-supervised pre-training [18, 44, 45]. In vision, pioneering works [12, 21] train self-supervised Transformers following the masked auto-encoding paradigm in NLP. Recent works [11, 13] based on siamese networks show intriguing properties as well as excellent transferability to downstream tasks. Here we perform a preliminary transfer learning experiment on YOLOS-S using MoCo-v3 [13] and DINO [11] self-supervised pre-trained ViT weights in Tab. 3.
+
+The transfer learning performance of 800 epochs DINO self-supervised model on COCO object detection is on a par with 300 epochs DeiT label-supervised pre-training, suggesting great potentials of self-supervised pre-training for ViT on challenging object-level recognition tasks. Meanwhile, the transfer learning performance of MoCo-v3 is less satisfactory, in part for the MoCo-v3 weight is heavily under pre-trained. Note that the pre-training epochs of MoCo-v3 are the same as DeiT (300 epochs), which means that there is still a gap between the current state-of-the-art self-supervised pre-training approach and the prevalent label-supervised pre-training approach for YOLOS.
+
+YOLOS as a Transfer Learning Benchmark for ViT. From the above analysis, we conclude that the ImageNet- $. 1 k$ pre-training results cannot precisely reflect the transfer learning performance on COCO object detection. Compared with widely used image recognition transfer learning benchmarks such as CIFAR-10/100 [34], Oxford-IIIT Pets [43] and Oxford Flowers-102 [42], the performance of
+
+YOLOS on COCO is more sensitive to the pre-train scheme and the performance is far from saturating. Therefore it is reasonable to consider YOLOS as a challenging transfer learning benchmark to evaluate different (label-supervised or self-supervised) pre-training strategies for ViT.
+
+# 3.3 Pre-training and Transfer Learning Performance of Different Scaled Models
+
+We study the pre-training and the transfer learning performance of different model scaling strategies, i.e., width scaling $( w )$ , uniform compound scaling $( d w r )$ and fast scaling $( d \mathbf { w } r )$ . The models are scaled from $\sim 1 . 2 6$ to $\sim 4 . 5 6$ FLOPs regime for pre-training. Detailed model configurations and descriptions are given in Sec. 3.1 and Tab. 1.
+
+We pre-train all the models for 300 epochs on ImageNet- $. 1 k$ with input resolution determined by the corresponding scaling strategies, and then fine-tune these models on COCO for 150 epochs. Few literatures are available for resolution scaling in object detection, where the inputs are usually oblong in shape and the multi-scale augmentation [10, 27] is used as a common practice. Therefore for each model during inference, we select the smallest resolution (i.e., the shorter size) ranging in [480, 800] producing the highest box AP, which is 784 for dwr scaling and 800 for all the others. The results are summarized in Tab. 4.
+
+<table><tr><td rowspan="2">Scale</td><td colspan="4">Image Classification @ ImageNet-1k</td><td colspan="4">Object Detection @ COCO val</td></tr><tr><td>FLOPs</td><td>f（Lin.） f（Att.)</td><td>FPS</td><td>Top-1</td><td>FLOPs</td><td>（Lin.） f（Att.</td><td>FPS</td><td>AP</td></tr><tr><td>1</td><td>1.2 G</td><td>5.9</td><td>1315</td><td>72.2</td><td>81G</td><td>0.28</td><td>12.0</td><td>29.6</td></tr><tr><td>w</td><td>4.5 G</td><td>11.8</td><td>615</td><td>79.9</td><td>194 G</td><td>0.55</td><td>5.7</td><td>36.1</td></tr><tr><td>dwr</td><td>4.6G</td><td>5.0</td><td>386</td><td>80.5</td><td>163 G</td><td>0.35</td><td>4.5</td><td>36.2</td></tr><tr><td>dwr</td><td>4.6G</td><td>8.8</td><td>511</td><td>80.4</td><td>172G</td><td>0.49</td><td>5.7</td><td>37.6</td></tr></table>
+
+Table 4: Pre-training and transfer learning performance of different scaled models. FLOPs and FPS data of object detection are measured over the first 100 images of COCO val split during inference following Carion et al. [10]. FPS is measured with batch size 1 on a single 1080Ti GPU.
+
+Pre-training. Both dwr and dwr scaling can improve the accuracy compared with simple $w$ scaling, i.e., the DeiT-S baseline. Other properties of each scaling strategy are also consistent with CNNs [20, 56], e.g., $w$ scaling is the most speed friendly. dwr scaling achieves the strongest accuracy. dwr is nearly as fast as $w$ scaling and is on a par with dwr scaling in accuracy. Perhaps the reason why these CNN model scaling strategies are still appliable to ViT is that during pre-training the linear projection ( $1 \times 1$ convolution) dominates the model computations.
+
+Transfer Learning. The picture changes when transferred to COCO. The input resolution $r$ is much higher so the spatial attention takes over and linear projection part is no longer dominant in terms of FLOPs $\begin{array} { r } { \frac { f ( \mathrm { L i n . } ) } { f ( \mathrm { A t t . } ) } \propto \frac { w } { r ^ { 2 } } ) } \end{array}$ . Canonical CNN model scaling recipes do not take spatial attention computations into account. Therefore there is some inconsistency between pre-training and transfer learning performance: Despite being strong on ImageNet- $1 k$ , the dwr scaling achieves similar box AP as simple $w$ scaling. Meanwhile, the performance gain from dwr scaling on COCO cannot be clearly explained by the corresponding CNN scaling methodology that does not take $f ( \mathbf { A t t . } ) \propto d w r ^ { 4 }$ into account. The performance inconsistency between pre-training and transfer learning calls for novel model scaling strategies for ViT considering spatial attention complexity.
+
+# 3.4 Comparisons with CNN-based Object Detectors
+
+In previous sections, we treat YOLOS as a touchstone for the transferability of ViT. In this section, we consider YOLOS as an object detector and we compare YOLOS with some modern CNN detectors.
+
+Comparisons with Tiny-sized CNN Detectors. As shown in Tab. 5, the tiny-sized YOLOS model achieves impressive performance compared with well-established and highly-optimized CNN object detectors. YOLOS-Ti is strong in AP and competitive in FLOPs & FPS even though Transformer is not intentionally designed to optimize these factors. From the model scaling perspective [20, 56, 60], YOLOS-Ti can serve as a promising model scaling start point.
+
+Comparisons with DETR. The relations and differences in model design between YOLOS and DETR are given in Sec. 2.1, here we make quantitative comparisons between the two.
+
+Table 5: Comparisons with some tiny-sized modern CNN detectors. All models are trained to be fully converged. “Size” refers to input resolution for inference. FLOPs and FPS data are measured over the first 100 images of COCO val split during inference following Carion et al. [10]. FPS is measured with batch size 1 on a single 1080Ti GPU.   
+
+<table><tr><td>Method</td><td>Backbone</td><td>Size</td><td>AP</td><td>Params. (M)</td><td>FLOPs (G)</td><td>FPS</td></tr><tr><td>Y0LOv3-Tiny 49]</td><td>DarkNet [49]</td><td>416×416</td><td>16.6</td><td>8.9</td><td>5.6</td><td>330</td></tr><tr><td>YOLOv4-Tiny [60]</td><td>COSA [60]</td><td>416 × 416</td><td>21.7</td><td>6.1</td><td>7.0</td><td>371</td></tr><tr><td>YOLOS-Ti</td><td>DeiT-Ti( [57]</td><td>256×*</td><td>23.1</td><td>6.5</td><td>3.4</td><td>114</td></tr><tr><td>CenterNet [70]</td><td>ResNet-18 [26]</td><td>512 × 512</td><td>28.1</td><td>1</td><td>1</td><td>129</td></tr><tr><td>YOLOv4-Tiny (3l) [60]</td><td>COSA [60]</td><td>320 × 320</td><td>28.7</td><td>1</td><td>1</td><td>252</td></tr><tr><td>Def.DETR [72]</td><td>FBNet-V3[15]</td><td>800×*</td><td>27.9</td><td>12.2</td><td>12.3</td><td>35</td></tr><tr><td>YOLOS-Ti</td><td>DeiT-Ti () [57]</td><td>432×*</td><td>28.6</td><td>6.5</td><td>11.7</td><td>84</td></tr></table>
+
+<table><tr><td>Method</td><td>Backbone</td><td>Epochs</td><td>Size</td><td>AP</td><td>Params. (M)</td><td>FLOPs (G)</td><td>FPS</td></tr><tr><td>Def. DETR [72]</td><td>FBNet-V3[15]</td><td>150</td><td>800×*</td><td>27.5</td><td>12.2</td><td>12.3</td><td>35</td></tr><tr><td>YOLOS-Ti</td><td>DeiT-Ti [57]</td><td>300</td><td>512×*</td><td>28.7</td><td>6.5</td><td>18.8</td><td>60</td></tr><tr><td>YOLOS-Ti</td><td>DeiT-Ti() [57]</td><td>300</td><td>432×*</td><td>28.6</td><td>6.5</td><td>11.7</td><td>84</td></tr><tr><td>YOLOS-Ti</td><td>DeiT-Ti () [57]</td><td>300</td><td>528×*</td><td>30.0</td><td>6.5</td><td>20.7</td><td>51</td></tr><tr><td>DETR[10]</td><td>ResNet-18-DC5[26]</td><td rowspan="5">150</td><td>800×*</td><td>36.9</td><td>29</td><td>129</td><td>7.4</td></tr><tr><td>YOLOS-S</td><td>DeiT-S[57]</td><td>800×*</td><td>36.1</td><td>31</td><td>194</td><td>5.7</td></tr><tr><td>YOLOS-S</td><td>DeiT-S () [57]</td><td>800×*</td><td>37.2</td><td>31</td><td>194</td><td>5.7</td></tr><tr><td>YOLOS-S (dwr)</td><td>DeiT-S [57](dwr Scale [20])</td><td>704×*</td><td>37.2</td><td>28</td><td>123</td><td>7.7</td></tr><tr><td>YOLOS-S (dwr)</td><td>DeiT-S [57](dwr Scale [20])</td><td>784×*</td><td>37.6</td><td>28</td><td>172</td><td>5.7</td></tr><tr><td>DETR[10]</td><td>ResNet-101-DC5 [26]</td><td rowspan="2">150</td><td rowspan="2">800×*</td><td>42.5</td><td>60</td><td>253</td><td>5.3</td></tr><tr><td>YOLOS-B</td><td>DeiT-B() [57]</td><td>42.0</td><td>127</td><td>538</td><td>2.7</td></tr></table>
+
+Table 6: Comparisons with different DETR models. Tiny-sized models are trained to be fully converged. “Size” refers to input resolution for inference. FLOPs and FPS data are measured over the first 100 images of COCO val split during inference following Carion et al. [10]. FPS is measured with batch size 1 on a single 1080Ti GPU. The “ResNet-18-DC5” implantation is from timm library [64].
+
+As shown in Tab. 6, YOLOS-Ti still performs better than the DETR counterpart, while larger YOLOS models with width scaling become less competitive: YOLOS-S with more computations is $0 . 8 \mathrm { A P }$ lower compared with a similar-sized DETR model. Even worse, YOLOS-B cannot beat DETR with over $2 \times$ parameters and FLOPs. Even though YOLOS-S with dwr scaling is able to perform better than the DETR counterpart, the performance gain cannot be clearly explained as discussed in Sec. 3.3.
+
+Interpreting the Results. Although the performance is seemingly discouraging, the numbers are meaningful, as YOLOS is not purposefully designed for better performance, but designed to precisely reveal the transferability of ViT in object detection. E.g., YOLOS-B is directly adopted from the BERT-Base architecture [18] in NLP. This 12 layers, 768 channels Transformer along with its variants have shown impressive performance on a wide range of NLP tasks. We demonstrate that with minimal modifications, this kind of architecture can also be successfully transferred (i.e., $\mathbf { A P } = 4 2 . 0$ ) to the challenging COCO object detection benchmark in computer vision from a pure sequence-to-sequence perspective. The minimal modifications from YOLOS exactly reveal the versatility and generality of Transformer.
+
+# 3.5 Inspecting Detection Tokens
+
+![](images/63da8f8e3fb4e771aa06a9a128ef570a2e7b3949b1e5dd19e260d990e27d6718.jpg)  
+Figure 2: Visualization of all box predictions on all images from COCO val split for the first ten [DET] tokens. Each box prediction is represented as a point with the coordinates of its center normalized by each thumbnail image size. The points are color-coded so that blue points corresponds to small objects, green to medium objects and red to large objects. We observe that each [DET] token learns to specialize on certain regions and sizes. The visualization style is inspired by Carion et al. [10].
+
+![](images/84f5b527d1d90d74aefb7641fe880fca979703f452976ee62d7c9716969db305.jpg)  
+Figure 3: The statistics of all ground truth object categories (the red curve) and the statistics of all object category predictions from all [DET] tokens (the blue curve) on all images from COCO val split. The error bar of the blue curve represents the variability of the preference of different tokens for a given category, which is small. This suggests that different [DET] tokens are category insensitive.
+
+Qualitative Analysis on Detection Tokens. As an object detector, YOLOS uses [DET] tokens to represent detected objects. In general, we find that [DET] tokens are sensitive to object locations and sizes, while insensitive to object categories, as shown in Fig. 2 and Fig. 3.
+
+Quantitative Analysis on Detection Tokens. We give a quantitative analysis on the relation between $X =$ the cosine similarity of [DET] token pairs, and $Y =$ the corresponding predicted bounding box centers $\ell _ { 2 }$ distances. We use the Pearson correlation coefficient $\begin{array} { r } { \bar { \rho } _ { X , Y } = \frac { \mathbb { E } [ ( X - \mu _ { X } ) ( Y - \mu _ { Y } ) ] } { \sigma _ { X } \sigma _ { Y } } } \end{array}$ as a measure of linear correlation between variable $X$ and $Y$ , and we conduct this study on all predicted object pairs within each image in COCO val set averaged by all 5000 images. The result is $\rho _ { X , Y } = - 0 . 8 0$ . This means that [DET] tokens that are close to each other (i.e., with high cosine similarity) also lead to mostly nearby predictions (i.e., with short $\ell _ { 2 }$ distances, given $\rho _ { X , Y } < 0 $ ).
+
+We also conduct a quantitative study on the relation between $X =$ the cosine similarity of [DET] token pairs, and $Y =$ the corresponding cosine similarity of the output features of the classifier. The result is $\rho _ { X , Y } = - 0 . 0 7 $ , which is very close to 0. This means that there is no strong linear correlation between these two variables.
+
+Detaching Detection Tokens. To further understand the role [DET] tokens plays, we study impacts caused by detaching the [DET] tokens of YOLOS during training, i.e., we don’t optimize the parameters of the one hundred randomly initialized [DET] tokens. As shown in Tab. 7, detaching the [DET] tokens has a minor impact to AP. These results imply that [DET] tokens mainly serve as the information carrier for the [PATCH] tokens. Similar phenomena are also observed in Fang et al. [22].
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>[DET]Tokens Config</td><td rowspan=1 colspan=1>AP</td></tr><tr><td rowspan=1 colspan=1>YOLOS-Ti</td><td rowspan=1 colspan=1>Rand. Init.&amp; LearnableRand. Init.&amp; Detached</td><td rowspan=1 colspan=1>28.728.3</td></tr><tr><td rowspan=1 colspan=1>YOLOS-S</td><td rowspan=1 colspan=1>Rand. Init.&amp;LearnableRand.Init.&amp; Detached</td><td rowspan=1 colspan=1>36.136.4</td></tr></table>
+
+Table 7: Impacts of detaching the [DET] tokens of YOLOS during training.
+
+# 4 Related Work
+
+Vision Transformer for Object Detection. There has been a lot of interest in combining CNNs with forms of self-attention mechanisms [4] to improve object detection performance [9, 31, 63], while recent works trend towards augmenting Transformer with CNNs (or CNN design). Beal et al. [6] propose to use a pre-trained ViT as the feature extractor for a Faster R-CNN [50] object detector. Despite being effective, they fail to ablate the CNN architectures, region-wise pooling operations [23, 25, 27] as well as hand-crafted components such as dense anchors [50] and NMS. Inspired by modern CNN architecture, some works [39, 59, 62, 65] introduce the pyramidal feature hierarchy and locality to Vision Transformer design, which largely boost the performance in dense prediction tasks including object detection. However, these architectures are performance-oriented and cannot reflect the properties of the canonical or vanilla Vision Transformer [21] that directly inherited from Vaswani et al. [58]. Another series of work, the DEtection TRansformer (DETR) families [10, 72], use a random initialized Transformer to encode & decode CNN features for object detection, which does not reveal the transferability of a pre-trained Transformer.
+
+UP-DETR [16] is probably the first to study the effects of unsupervised pre-training in the DETR framework, which proposes an “object detection oriented” unsupervised pre-training task tailored for Transformer encoder & decoder in DETR. In this paper, we argue for the characteristics of a pre-trained vanilla ViT in object detection, which is rare in the existing literature.
+
+Pre-training and Fine-tuning of Transformer. The textbook-style usage of Transformer [58] follows a “pre-training & fine-tuning” paradigm. In NLP, Transformer-based models are often pretrained on large corpora and then fine-tuned for different tasks at hand [18, 44]. In computer vision, Dosovitskiy et al. [21] apply Transformer to image recognition at scale using modern vision transfer learning recipe [33]. They show that a standard Transformer encoder architecture is able to attain excellent results on mid-sized or small image recognition benchmarks (e.g, ImageNet- $. 1 k$ [51], CIFAR10/100 [34], etc.) when pre-trained at sufficient scale (e.g, JFT-300M [55], ImageNet- $2 1 k$ [17]). Touvron et al. [57] achieves competitive Top-1 accuracy by training Transformer on ImageNet- $. 1 k$ only, and is also capable of transferring to smaller datasets [34, 42, 43]. However, existing transfer learning literature of Transformer arrest in image-level recognition and does not touch more complex tasks in vision such as object detection, which is also widely used to benchmark CNNs transferability.
+
+Our work aims to bridge this gap. We study the performance and properties of ViT on the challenging COCO object detection benchmark [36] when pre-trained on the mid-sized ImageNet- $. 1 k$ dataset [51] using different strategies.
+
+# 5 Discussion
+
+Over recent years, the landscape of computer vision has been drastically transformed by Transformer, especially for recognition tasks [10, 21, 39, 57, 59]. Inspired by modern CNN design, some recent works [39, 59, 62, 65] introduce the pyramidal feature hierarchy as well as locality to vanilla ViT [21], which largely boost the performance in dense recognition tasks including object detection.
+
+We believe there is nothing wrong to make performance-oriented architectural designs for Transformer in vision, as choosing the right inductive biases and priors for target tasks is crucial for model design. However, we are more interested in designing and applying Transformer in vision following the spirit of NLP, i.e., pre-train the task-agnostic vanilla Vision Transformer for general visual representation learning first, and then fine-tune or adapt the model on specific target downstream tasks efficiently. Current state-of-the-art language models pre-trained on massive amounts of corpora are able to perform few-shot or even zero-shot learning, adapting to new scenarios with few or no labeled data [8, 38, 45, 46]. Meanwhile, prevalent pre-trained computer vision models, including various Vision Transformer variants, still need a lot of supervision to transfer to downstream tasks.
+
+We hope the introduction of Transformer can not only unify NLP and CV in terms of the architecture, but also in terms of the methodology. The proposed YOLOS is able to turn a pre-trained ViT into an object detector with the fewest possible modifications, but our ultimate goal is to adapt a pre-trained model to downstream vision tasks with the fewest possible costs. YOLOS still needs 150 epochs transfer learning to adapt a pre-trained ViT to perform object detection, and the detection results are far from saturating, indicating the pre-trained representation still has large room for improvement. We encourage the vision community to focus more on the general visual representation learning for the task-agnostic vanilla Transformer instead of the task-oriented architectural design of ViT. We hope one day, in computer vision, a universal pre-trained visual representation can be easily adapted to various understanding as well as generation tasks with the fewest possible costs.
+
+# 6 Conclusion
+
+In this paper, we have explored the transferability of the vanilla ViT pre-trained on mid-sized ImageNet- $1 k$ dataset to the more challenging COCO object detection benchmark. We demonstrate that 2D object detection can be accomplished in a pure sequence-to-sequence manner with minimal additional inductive biases. The performance on COCO is promising, and these preliminary results are meaningful, suggesting the versatility and generality of Transformer to various downstream tasks.
+
+# Acknowledgment
+
+This work is in part supported by NSFC (No. 61876212, No. 61733007, and No. 61773176) and the Zhejiang Laboratory under Grant 2019NB0AB02. We thank Zhuowen Tu for valuable suggestions.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Sec. 3.2, Sec. 3.3 and Sec. 3.4.   
+(b) Did you describe the limitations of your work? [Yes] See Sec. 3.2, Sec. 3.3 and Sec. 3.4.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Sec. 3.2 and Tab. 2 for the total theoretical computations analysis.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] .
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We include them in the supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Sec. 3.1.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Appendix.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Sec. 3.1 and Sec. 3.2.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes] In the supplementary material.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] In the supplementary material.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/nVofoXjTmA_/nVofoXjTmA__content_list.json

@@ -0,0 +1,1423 @@
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+    {
+        "type": "text",
+        "text": "You Only Look at One Sequence: Rethinking Transformer in Vision through Object Detection ",
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Yuxin Fang 1∗ Bencheng Liao 1∗ Xinggang Wang 1† Jiemin Fang 2,1 Jiyang Qi 1 Rui Wu 3 Jianwei Niu 3 Wenyu Liu 1 ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 School of EIC, Huazhong University of Science & Technology   \n2 Institute of AI, Huazhong University of Science & Technology 3 Horizon Robotics {yxf, bcliao, xgwang}@hust.edu.cn ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Can Transformer perform 2D object- and region-level recognition from a pure sequence-to-sequence perspective with minimal knowledge about the 2D spatial structure? To answer this question, we present You Only Look at One Sequence (YOLOS), a series of object detection models based on the vanilla Vision Transformer with the fewest possible modifications, region priors, as well as inductive biases of the target task. We find that YOLOS pre-trained on the mid-sized ImageNet- $1 k$ dataset only can already achieve quite competitive performance on the challenging COCO object detection benchmark, e.g., YOLOS-Base directly adopted from BERT-Base architecture can obtain 42.0 box AP on COCO val. We also discuss the impacts as well as limitations of current pre-train schemes and model scaling strategies for Transformer in vision through YOLOS. Code and pre-trained models are available at https://github.com/hustvl/YOLOS. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
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+        "type": "text",
+        "text": "Transformer [58] is born to transfer. In natural language processing (NLP), the dominant approach is to first pre-train Transformer on large, generic corpora for general language representation learning, and then fine-tune or adapt the model on specific target tasks [18]. Recently, Vision Transformer (ViT) 1 [21] demonstrates that canonical Transformer encoder architecture directly inherited from NLP can perform surprisingly well on image recognition at scale using modern vision transfer learning recipe [33]. Taking sequences of image patch embeddings as inputs, ViT can successfully transfer pre-trained general visual representations from sufficient scale to more specific image classification tasks with fewer data points from a pure sequence-to-sequence perspective. ",
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+    {
+        "type": "text",
+        "text": "Since a pre-trained Transformer can be successfully fine-tuned on sentence-level tasks [7, 19] in NLP, as well as token-level tasks [48, 52], where models are required to produce fine-grained output at the token-level [18]. A natural question is: Can ViT transfer to more challenging object- and region-level target tasks in computer vision such as object detection other than image-level recognition? ",
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+    {
+        "type": "text",
+        "text": "ViT-FRCNN [6] is the first to use a pre-trained ViT as the backbone for a Faster R-CNN [50] object detector. However, this design cannot get rid of the reliance on convolutional neural networks (CNNs) ",
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+    {
+        "type": "text",
+        "text": "and strong 2D inductive biases, as ViT-FRCNN re-interprets the output sequences of ViT to 2D spatial feature maps and depends on region-wise pooling operations (i.e., RoIPool [23, 25] or RoIAlign [27]) as well as region-based CNN architectures [50] to decode ViT features for object- and region-level perception. Inspired by modern CNN design, some recent works [39, 59, 62, 65] introduce the pyramidal feature hierarchy, spatial locality, equivariant as well as invariant representations [24] to canonical Vision Transformer design, which largely boost the performance in dense prediction tasks including object detection. However, these architectures are performance-oriented and cannot reflect the properties of the canonical or vanilla Vision Transformer [21] directly inherited from Vaswani et al. [58]. Another series of work, the DEtection TRansformer (DETR) families [10, 72], use a random initialized Transformer to encode & decode CNN features for object detection, which does not reveal the transferability of a pre-trained Transformer. ",
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Intuitively, ViT is designed to model long-range dependencies and global contextual information instead of local and region-level relations. Moreover, ViT lacks hierarchical architecture as modern CNNs [26, 35, 53] to handle the large variations in the scale of visual entities [1, 37]. Based on the available evidence, it is still unclear whether a pure ViT can transfer pre-trained general visual representations from image-level recognition to the much more complicated 2D object detection task. ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To answer this question, we present You Only Look at One Sequence (YOLOS), a series of object detection models based on the canonical ViT architecture with the fewest possible modifications, region priors, as well as inductive biases of the target task injected. Essentially, the change from a pre-trained ViT to a YOLOS detector is embarrassingly simple: (1) YOLOS replaces one [CLS] token for image classification in ViT with one hundred [DET] tokens for object detection. (2) YOLOS replaces the image classification loss in ViT with the bipartite matching loss to perform object detection in a set prediction manner following Carion et al. [10], which can avoid re-interpreting the output sequences of ViT to 2D feature maps as well as prevent manually injecting heuristics and prior knowledge of object 2D spatial structure during label assignment [71]. Moreover, the prediction head of YOLOS can get rid of complex and diverse designs, which is as compact as a classification layer. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Directly inherited from ViT [21], YOLOS is not designed to be yet another high-performance object detector, but to unveil the versatility and transferability of pre-trained canonical Transformer from image recognition to the more challenging object detection task. Concretely, our main contributions are summarized as follows: ",
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "• We use the mid-sized ImageNet- $. 1 k$ [51] as the sole pre-training dataset, and show that a vanilla ViT [21] can be successfully transferred to perform the complex object detection task and produce competitive results on COCO [36] benchmark with the fewest possible modifications, i.e., by only looking at one sequence (YOLOS).   \n• For the first time, we demonstrate that 2D object detection can be accomplished in a pure sequence-to-sequence manner by taking a sequence of fixed-sized non-overlapping image patches as input. Among existing object detectors, YOLOS utilizes the minimal 2D inductive biases.   \n• For the vanilla ViT, we find the object detection results are quite sensitive to the pre-train scheme and the detection performance is far from saturating. Therefore the proposed YOLOS can be also used as a challenging benchmark task to evaluate different (label-supervised and self-supervised) pre-training strategies for ViT. ",
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+        "type": "text",
+        "text": "2 You Only Look at One Sequence ",
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+        "text": "As for the model design, YOLOS closely follows the original ViT architecture [21], and is optimized for object detection in the same vein as Carion et al. [10]. YOLOS can be easily adapted to various canonical Transformer architectures available in NLP as well as in computer vision. This intentionally simple setup is not designed for better detection performance, but to exactly reveal characteristics of the Transformer family in object detection as unbiased as possible. ",
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+        "text": "2.1 Architecture ",
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+        "text": "An overview of the model is depicted in Fig. 1. Essentially, the change from a ViT to a YOLOS detector is simple: (1) YOLOS drops the [CLS] token for image classification and appends one hundred randomly initialized learnable detection tokens ([DET] tokens) to the input patch embeddings ([PATCH] tokens) for object detection. (2) During training, YOLOS replaces the image classification loss in ViT with the bipartite matching loss to perform object detection in a set prediction manner following Carion et al. [10]. ",
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+        "img_path": "images/b9fa86f2f98eb89a87e5b972b72a23e150a85dd475478f66b08c585dd073e747.jpg",
+        "image_caption": [
+            "Figure 1: YOLOS architecture overview. “Pat-Tok” refers to [PATCH] token, which is the embedding of a flattened image patch. “Det-Tok” refers to [DET] token, which is a learnable embedding for object binding. “PE” refers to positional embedding. During training, YOLOS produces an optimal bipartite matching between predictions from one hundred [DET] tokens and ground truth objects. During inference, YOLOS directly outputs the final set of predictions in parallel. The figure style is inspired by Dosovitskiy et al. [21]. "
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+        "type": "text",
+        "text": "Stem. The canonical ViT [21] receives an 1D sequence of embedded tokens as the input. To handle 2D image inputs, we reshape the image $\\mathbf { x } \\in \\mathbb { R } ^ { H \\times W \\times C }$ into a sequence of flattened 2D image patches $\\mathbf { x } _ { \\mathtt { P A T C H } } \\in \\mathbb { R } ^ { N \\times \\left( P ^ { 2 } \\cdot C \\right) }$ . Here, $( H , W )$ is the resolution of the input image, $C$ is the number of input channels, $( P , P )$ is the resolution of each image patch, and $\\begin{array} { r } { \\dot { N } = \\frac { H \\dot { W } } { P ^ { 2 } } } \\end{array}$ is the resulting number of patches. Then we map $\\mathbf { x } _ { \\mathrm { P A T C H } }$ to $D$ dimensions with a trainable linear projection $\\mathbf { E } \\in \\mathbb { R } ^ { \\left( P ^ { 2 } \\cdot C \\right) \\times D }$ . We refer to the output of this projection $\\mathbf { x } _ { \\mathrm { P A T C H } } \\mathbf { E }$ as [PATCH] tokens. Meanwhile, one hundred randomly initialized learnable [DET] tokens $\\mathbf { x } _ { \\mathrm { D E T } } \\in \\mathbb { R } ^ { 1 0 0 \\times D }$ are appended to the [PATCH] tokens. Position embeddings $\\mathbf { P } \\in \\mathbb { R } ^ { ( \\bar { N } + 1 0 \\bar { 0 } ) \\times D }$ are added to all the input tokens to retain positional information. We use the standard learnable 1D position embeddings following Dosovitskiy et al. [21]. The resulting sequence $\\mathbf { z } _ { 0 }$ serves as the input of YOLOS Transformer encoder. Formally: ",
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+    {
+        "type": "equation",
+        "img_path": "images/67c001c981a363e2a618fb8c573110b22bf77a7b64f3c86f2951806adc673a48.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { z } _ { 0 } = \\left[ \\mathbf { x } _ { \\mathtt { P A T C H } } ^ { \\mathrm { 1 } } \\mathbf { E } ; \\cdot \\cdot \\cdot \\ ; \\mathbf { x } _ { \\mathtt { P A T C H } } ^ { N } \\mathbf { E } ; \\mathbf { x } _ { \\mathtt { D E T } } ^ { \\mathrm { 1 } } ; \\cdot \\cdot \\cdot ; \\mathbf { x } _ { \\mathtt { D E T } } ^ { \\mathrm { 1 0 0 } } \\right] + \\mathbf { P } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Body. The body of YOLOS is basically the same as ViT, which consists of a stack of Transformer encoder layers only [58]. [PATCH] tokens and [DET] tokens are treated equally and they perform global interactions inside Transformer encoder layers. ",
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+        "type": "text",
+        "text": "Each Transformer encoder layer consists of one multi-head self-attention (MSA) block and one MLP block. LayerNorm (LN) [2] is applied before every block, and residual connections [26] are applied after every block [3, 61]. The MLP contains one hidden layer with an intermediate GELU [29] non-linearity activation function. Formally, for the $\\ell$ -th YOLOS Transformer encoder layer: ",
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+        "type": "equation",
+        "img_path": "images/b50b0532ae042351f6a4a30e3a677d8424f31b9e99674da979ada461063674f5.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbf { z } _ { \\ell } ^ { \\prime } = \\mathrm { M S A } \\left( \\mathrm { L N } \\left( \\mathbf { z } _ { \\ell - 1 } \\right) \\right) + \\mathbf { z } _ { \\ell - 1 } , } \\\\ & { \\mathbf { z } _ { \\ell } = \\mathrm { M L P } \\left( \\mathrm { L N } \\left( \\mathbf { z } _ { \\ell } ^ { \\prime } \\right) \\right) + \\mathbf { z } _ { \\ell } ^ { \\prime } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Detector Heads. The detector head of YOLOS gets rid of complex and heavy designs, and is as neat as the image classification layer of ViT. Both the classification and the bounding box regression heads are implemented by one MLP with separate parameters containing two hidden layers with intermediate ReLU [41] non-linearity activation functions. ",
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+        "type": "text",
+        "text": "Detection Token. We purposefully choose randomly initialized [DET] tokens as proxies for object representations to avoid inductive biases of 2D structure and prior knowledge about the task injected during label assignment. When fine-tuning on COCO, for each forward pass, an optimal bipartite matching between predictions generated by [DET] tokens and ground truth objects is established. This procedure plays the same role as label assignment [10, 71], but is unaware of the input 2D structure, i.e., YOLOS does not need to re-interpret the output sequence of ViT to an 2D feature maps for label assignment. Theoretically, it is feasible for YOLOS to perform any dimensional object detection without knowing the exact spatial structure and geometry, as long as the input is always flattened to a sequence in the same way for each pass. ",
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+        "type": "text",
+        "text": "Fine-tuning at Higher Resolution. When fine-tuning on COCO, all the parameters are initialized from ImageNet- $. 1 k$ pre-trained weights except for the MLP heads for classification & bounding box regression as well as one hundred [DET] tokens, which are randomly initialized. During fine-tuning, the image has a much higher resolution than pre-training. We keep the patch size $P$ unchanged, i.e., $P \\times P = 1 6 \\times 1 6$ , which results in a larger effective sequence length. While ViT can handle arbitrary input sequence lengths, the positional embeddings need to adapt to the longer input sequences with various lengths. We perform 2D interpolation of the pre-trained position embeddings on the fly2. ",
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+        "type": "text",
+        "text": "Inductive Bias. We carefully design the YOLOS architecture for the minimal additional inductive biases injection. The inductive biases inherent from ViT come from the patch extraction at the network stem part as well as the resolution adjustment for position embeddings [21]. Apart from that, YOLOS adds no non-degenerated (e.g., $3 \\times 3$ or other non $1 \\times 1$ ) convolutions upon ViT 3. From the representation learning perspective, we choose to use [DET] tokens to bind objects for final predictions to avoid additional 2D inductive biases as well as task-specific heuristics. The performance-oriented design inspired by modern CNN architectures such as pyramidal feature hierarchy, 2D local spatial attention as well as the region-wise pooling operation is not applied. All these efforts are meant to exactly unveil the versatility and transferability of pre-trained Transformers from image recognition to object detection in a pure sequence-to-sequence manner, with minimal knowledge about the input spatial structure and geometry. ",
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+        "type": "text",
+        "text": "Comparisons with DETR. The design of YOLOS is deeply inspired by DETR [10]: YOLOS uses [DET] tokens following DETR as proxies for object representations to avoid inductive biases about 2D structures and prior knowledge about the task injected during label assignment, and YOLOS is optimized similarly as DETR. ",
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+        "type": "text",
+        "text": "Meanwhile, there are some key differences between the two models: (1) DETR adopts a Transformer encoder-decoder architecture, while YOLOS chooses an encoder-only Transformer architecture. (2) DETR only employs pre-training on its CNN backbone but leaves the Transformer encoder & decoder being trained from random initialization, while YOLOS naturally inherits representations from any pre-trained canonical ViT. (3) DETR applies cross-attention between encoded image features and object queries with auxiliary decoding losses deeply supervised at each decoder layer, while YOLOS always looks at only one sequence for each encoder layer, without distinguishing [PATCH] tokens and [DET] tokens in terms of operations. Quantitative comparisons between the two are in Sec. 3.4. ",
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+    {
+        "type": "text",
+        "text": "3 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 Setup",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Pre-training. We pre-train all YOLOS / ViT models on ImageNet- $1 k$ [51] dataset using the dataefficient training strategy suggested by Touvron et al. [57]. The parameters are initialized with a truncated normal distribution and optimized using AdamW [40]. The learning rate and batch size are $1 \\times 1 0 ^ { - 3 }$ and 1024, respectively. The learning rate decay is cosine and the weight decay is 0.05. Rand-Augment [14] and random erasing [69] implemented by timm library [64] are used for data augmentation. Stochastic depth [32], Mixup [68] and Cutmix [66] are used for regularization. ",
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+        "type": "text",
+        "text": "Fine-tuning. We fine-tune all YOLOS models on COCO object detection benchmark [36] in a similar way as Carion et al. [10]. All the parameters are initialized from ImageNet- $. 1 k$ pre-trained weights except for the MLP heads for classification $\\&$ bounding box regression as well as one hundred [DET] tokens, which are randomly initialized. We train YOLOS on a single node with $8 \\times 1 2 6$ GPUs. The learning rate and batch sizes are $2 . 5 \\times 1 0 ^ { - 5 }$ and 8 respectively. The learning rate decay is cosine and the weight decay is $1 \\times 1 0 ^ { - 4 }$ . ",
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+    {
+        "type": "text",
+        "text": "As for data augmentation, we use multi-scale augmentation, resizing the input images such that the shortest side is at least 256 and at most 608 pixels while the longest at most 864 for tiny models. For small and base models, we resize the input images such that the shortest side is at least 480 and at most 800 pixels while the longest at most 1333. We also apply random crop augmentations during training following Carion et al. [10]. The number of [DET] tokens are 100 and we keep the loss function as well as loss weights the same as DETR, while we don’t apply dropout [54] or stochastic depth during fine-tuning since we find these regularization methods hurt performance. ",
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+    {
+        "type": "text",
+        "text": "Model Variants. With available computational resources, we study several YOLOS variants. Detailed configurations are summarized in Tab. 1. The input patch size for all models is $1 6 \\times 1 6$ . YOLOS-Ti (Tiny), -S (Small), and -B (Base) directly correspond to DeiT-Ti, -S, and -B [57]. From the model scaling perspective [20, 56, 60], the small and base models of YOLOS / DeiT can be seen as performing width scaling $( w )$ [30, 67] on the corresponding tiny model. ",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Variants of YOLOS. “dwr” and “dwr” refer to uniform compound model scaling and fast model scaling, respectively. The “dwr” and “dwr” notations are inspired by Dollár et al. [20]. Note that all the numbers listed are for pre-training, which could change during fine-tuning, e.g., the resolution and FLOPs. "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>DeiT [57]Model</td><td rowspan=1 colspan=1>Layers(Depth)</td><td rowspan=1 colspan=1>Embed. Dim.(Width)</td><td rowspan=1 colspan=1>Pre-train Resolution</td><td rowspan=1 colspan=1> Heads</td><td rowspan=1 colspan=1> Params.</td><td rowspan=1 colspan=1>FLOPs</td><td rowspan=1 colspan=1>f(Lin.)f(Att.)</td></tr><tr><td rowspan=1 colspan=1>YOLOS-TiYOLOS-SYOLOS-B</td><td rowspan=1 colspan=1>DeiT-TiDeiT-SDeiT-B</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>192384768</td><td rowspan=1 colspan=1>224</td><td rowspan=1 colspan=1>3612</td><td rowspan=1 colspan=1>5.7M22.1M86.4 M</td><td rowspan=1 colspan=1>1.2 G4.5G17.6G</td><td rowspan=1 colspan=1>5.911.823.5</td></tr><tr><td rowspan=1 colspan=1>YOLOS-S (dwr)YOLOS-S (dwr)</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>1914</td><td rowspan=1 colspan=1>240330</td><td rowspan=1 colspan=1>272240</td><td rowspan=1 colspan=1>66</td><td rowspan=1 colspan=1>13.7M19.0M</td><td rowspan=1 colspan=1>4.6G4.6G</td><td rowspan=1 colspan=1>5.08.8</td></tr></table>",
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+        "type": "text",
+        "text": "Besides, we investigate two other model scaling strategies which proved to be effective in CNNs. The first one is uniform compound scaling (dwr) [20, 56]. In this case, the scaling is uniform w.r.t. FLOPs along all model dimensions (i.e., width $( w )$ , depth $( d )$ and resolution $( r ) _ { , }$ ). The second one is fast scaling $( d \\mathbf { w } r )$ [20] that encourages primarily scaling model width (w), while scaling depth $( d )$ and resolution $( r )$ to a lesser extent w.r.t. FLOPs. During the ImageNet- $. 1 k$ pre-training phase, we apply dwr and dwr scaling to DeiT-Ti $\\mathrm { ~ \\sim ~ } 1 . 2 6$ FLOPs) and scale the model to $\\sim 4 . 5 6$ FLOPs to align with the computations of DeiT-S. Larger models are left for future work. ",
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+        "type": "text",
+        "text": "For canonical CNN architectures, the model complexity or FLOPs $( f )$ are proportional to $d w ^ { 2 } r ^ { 2 }$ [20]. Formally, $f ( \\mathbf { C N N } ) \\propto d w ^ { 2 } r ^ { 2 }$ . Different from CNN, there are two kinds of operations that contribute to the FLOPs of ViT. The first one is the linear projection (Lin.) or point-wise convolution, which fuses the information across different channels point-wisely via learnable parameters. The complexity is $f ( \\mathtt { L i n . } ) \\propto d w ^ { 2 } r ^ { 2 }$ , which is the same as $f ( \\mathbf { C } \\mathbb { N } \\mathbb { N } )$ . The second one is the spatial attention (Att.), which aggregates the spatial information depth-wisely via computed attention weights. The complexity is $f ( \\mathbf { A } \\mathbf { t } \\mathbf { \\bar { t } } . ) \\propto d w r ^ { \\tilde { 4 } }$ , which grows quadratically with the input sequence length or number of pixels. ",
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+        "type": "text",
+        "text": "Note that the available scaling strategies are designed for architectures with complexity $f \\propto d w ^ { 2 } r ^ { 2 }$ , so theoretically the dwr as well as dwr model scaling are not directly applicable to ViT. However, during pre-training phase the resolution is relatively low, therefore $f ( \\mathtt { L i n . } )$ dominates the FLOPs $\\begin{array} { r } { ( \\frac { f ( \\mathrm { L i n . } ) } { f ( \\mathrm { A t t . } ) } > 5 ) } \\end{array}$ . Our experiments indicate that some model scaling properties of ViT are consistent with CNNs when $\\textstyle { \\frac { f ( \\mathrm { L i n . } ) } { f ( \\mathrm { A t t . } ) } }$ is large. ",
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+    {
+        "type": "text",
+        "text": "3.2 The Effects of Pre-training ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We study the effects of different pre-training strategies (both label-supervised and self-supervised) when transferring ViT (DeiT-Ti and DeiT-S) from ImageNet- $. 1 k$ to the COCO object detection benchmark via YOLOS. For object detection, the input shorter size is 512 for tiny models and is 800 for small models during inference. The results are shown in Tab. 2 and Tab. 3. ",
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+        "type": "table",
+        "img_path": "images/2c5d725f459157bfba368093b266b1355b150d5917184f51ffc7681fb7678429.jpg",
+        "table_caption": [
+            "Table 2: The effects of label-supervised pre-training. “pFLOPs” refers to petaFLOPs $( \\times 1 0 ^ { 1 5 } )$ ). “ImNet” refers to ImageNet- $1 k$ . “C” refers to the distillation method from Touvron et al. [57]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td> Pre-train Method</td><td>Pre-train Epochs</td><td>Fine-tune Epochs</td><td>Pre-train pFLOPs</td><td>Fine-tune pFLOPs</td><td>Total pFLOPs</td><td>ImNet Top-1</td><td>AP</td></tr><tr><td rowspan=\"4\">YOLOS-Ti</td><td>Rand. Init.</td><td>0</td><td>600</td><td>0</td><td>14.2 ×10²</td><td>14.2× 10²</td><td>1</td><td>19.7</td></tr><tr><td>Label Sup. [57]</td><td>200</td><td rowspan=\"2\">300</td><td>3.1×102</td><td rowspan=\"2\">7.1 × 10²</td><td>10.2×10²</td><td>71.2</td><td>26.9</td></tr><tr><td>Label Sup. [57]</td><td>300</td><td>4.7×10²</td><td>11.8×10²</td><td>72.2</td><td>28.7</td></tr><tr><td>Label Sup. () [57]</td><td>300</td><td></td><td>4.7×10²</td><td></td><td>11.8×10²</td><td>74.5</td><td>29.7</td></tr><tr><td rowspan=\"5\">YOLOS-S</td><td>Rand. Init.</td><td>0</td><td>250</td><td>0</td><td>5.9×103</td><td>5.9×103</td><td>1</td><td>20.9</td></tr><tr><td>Label Sup. [57]</td><td>100</td><td rowspan=\"4\">150</td><td>0.6×103 1.2×103</td><td></td><td>4.1 ×103</td><td>74.5</td><td>32.0</td></tr><tr><td>Label Sup. [57]</td><td>200</td><td></td><td></td><td>4.7 ×103</td><td>78.5</td><td>36.1</td></tr><tr><td>Label Sup. [57]</td><td>300</td><td>1.8×10</td><td>3.5 × 103</td><td>5.3×103</td><td>79.9</td><td>36.1</td></tr><tr><td>Label Sup. () [57]</td><td>300</td><td>1.8×103</td><td></td><td>5.3×10</td><td>81.2</td><td>37.2</td></tr></table>",
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+        "type": "table",
+        "img_path": "images/625688be78761f4af696866f13b8d12ec5d2ad4be64376385c06b28e5beb0a4d.jpg",
+        "table_caption": [
+            "Table 3: Study of self-supervised pre-training on YOLOS-S. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Self Sup. Pre-train Method</td><td rowspan=1 colspan=1>Pre-train Epochs</td><td rowspan=1 colspan=1>Fine-tune Epochs</td><td rowspan=1 colspan=1>Linear Acc.</td><td rowspan=1 colspan=1>AP</td></tr><tr><td rowspan=1 colspan=1>YOLOS-S</td><td rowspan=1 colspan=1>MoCo-v3[13]DINO[11]</td><td rowspan=1 colspan=1>300800</td><td rowspan=1 colspan=1>150150</td><td rowspan=1 colspan=1>73.277.0</td><td rowspan=1 colspan=1>33.636.2</td></tr></table>",
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Necessity of Pre-training. At least under prevalent transfer learning paradigms [10, 57], the pretraining is necessary in terms of computational efficiency. For both tiny and small models, we find that pre-training on ImageNet- $. 1 k$ saves the total theoretical forward pass computations (total pre-training FLOPs & total fine-tuning FLOPs) compared with training on COCO from random initialization (training from scratch [28]). Models trained from scratch with hundreds of epochs still lag far behind the pre-trained ViT even if given more total FLOPs budgets. This seems quite different from canonical modern CNN-based detectors, which can catch up with pre-trained counterparts quickly [28]. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Label-supervised Pre-training. For supervised pre-training with ImageNet- $. 1 k$ ground truth labels, we find that different-sized models prefer different pre-training schedules: 200 epochs pre-training for YOLOS-Ti still cannot catch up with 300 epochs pre-training even with a 300 epochs fine-tuning schedule, while for the small model 200 epochs pre-training provides feature representations as good as 300 epochs pre-training for transferring to the COCO object detection benchmark. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "With additional transformer-specific distillation $( ^ { 6 6 } \\pmb { \\tilde { m } } ^ { 3 } )$ introduced by Touvron et al. [57], the detection performance is further improved by $\\sim 1$ AP for both tiny and small models, in part because exploiting a CNN teacher [47] during pre-training helps ViT adapt to COCO better. It is also promising to directly leverage [DET] tokens to help smaller YOLOS learn from larger YOLOS on COCO during fine-tuning in a similar way as Touvron et al. [57], we leave it for future work. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Self-supervised Pre-training. The success of Transformer in NLP greatly benefits from large-scale self-supervised pre-training [18, 44, 45]. In vision, pioneering works [12, 21] train self-supervised Transformers following the masked auto-encoding paradigm in NLP. Recent works [11, 13] based on siamese networks show intriguing properties as well as excellent transferability to downstream tasks. Here we perform a preliminary transfer learning experiment on YOLOS-S using MoCo-v3 [13] and DINO [11] self-supervised pre-trained ViT weights in Tab. 3. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The transfer learning performance of 800 epochs DINO self-supervised model on COCO object detection is on a par with 300 epochs DeiT label-supervised pre-training, suggesting great potentials of self-supervised pre-training for ViT on challenging object-level recognition tasks. Meanwhile, the transfer learning performance of MoCo-v3 is less satisfactory, in part for the MoCo-v3 weight is heavily under pre-trained. Note that the pre-training epochs of MoCo-v3 are the same as DeiT (300 epochs), which means that there is still a gap between the current state-of-the-art self-supervised pre-training approach and the prevalent label-supervised pre-training approach for YOLOS. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "YOLOS as a Transfer Learning Benchmark for ViT. From the above analysis, we conclude that the ImageNet- $. 1 k$ pre-training results cannot precisely reflect the transfer learning performance on COCO object detection. Compared with widely used image recognition transfer learning benchmarks such as CIFAR-10/100 [34], Oxford-IIIT Pets [43] and Oxford Flowers-102 [42], the performance of ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "YOLOS on COCO is more sensitive to the pre-train scheme and the performance is far from saturating. Therefore it is reasonable to consider YOLOS as a challenging transfer learning benchmark to evaluate different (label-supervised or self-supervised) pre-training strategies for ViT. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.3 Pre-training and Transfer Learning Performance of Different Scaled Models ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We study the pre-training and the transfer learning performance of different model scaling strategies, i.e., width scaling $( w )$ , uniform compound scaling $( d w r )$ and fast scaling $( d \\mathbf { w } r )$ . The models are scaled from $\\sim 1 . 2 6$ to $\\sim 4 . 5 6$ FLOPs regime for pre-training. Detailed model configurations and descriptions are given in Sec. 3.1 and Tab. 1. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We pre-train all the models for 300 epochs on ImageNet- $. 1 k$ with input resolution determined by the corresponding scaling strategies, and then fine-tune these models on COCO for 150 epochs. Few literatures are available for resolution scaling in object detection, where the inputs are usually oblong in shape and the multi-scale augmentation [10, 27] is used as a common practice. Therefore for each model during inference, we select the smallest resolution (i.e., the shorter size) ranging in [480, 800] producing the highest box AP, which is 784 for dwr scaling and 800 for all the others. The results are summarized in Tab. 4. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/f0d5921dff188ffdc15f57d2ce0ebdb3993db4b1988cbec5ed1ee0f7a335a236.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 4: Pre-training and transfer learning performance of different scaled models. FLOPs and FPS data of object detection are measured over the first 100 images of COCO val split during inference following Carion et al. [10]. FPS is measured with batch size 1 on a single 1080Ti GPU. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Scale</td><td colspan=\"4\">Image Classification @ ImageNet-1k</td><td colspan=\"4\">Object Detection @ COCO val</td></tr><tr><td>FLOPs</td><td>f（Lin.） f（Att.)</td><td>FPS</td><td>Top-1</td><td>FLOPs</td><td>（Lin.） f（Att.</td><td>FPS</td><td>AP</td></tr><tr><td>1</td><td>1.2 G</td><td>5.9</td><td>1315</td><td>72.2</td><td>81G</td><td>0.28</td><td>12.0</td><td>29.6</td></tr><tr><td>w</td><td>4.5 G</td><td>11.8</td><td>615</td><td>79.9</td><td>194 G</td><td>0.55</td><td>5.7</td><td>36.1</td></tr><tr><td>dwr</td><td>4.6G</td><td>5.0</td><td>386</td><td>80.5</td><td>163 G</td><td>0.35</td><td>4.5</td><td>36.2</td></tr><tr><td>dwr</td><td>4.6G</td><td>8.8</td><td>511</td><td>80.4</td><td>172G</td><td>0.49</td><td>5.7</td><td>37.6</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Pre-training. Both dwr and dwr scaling can improve the accuracy compared with simple $w$ scaling, i.e., the DeiT-S baseline. Other properties of each scaling strategy are also consistent with CNNs [20, 56], e.g., $w$ scaling is the most speed friendly. dwr scaling achieves the strongest accuracy. dwr is nearly as fast as $w$ scaling and is on a par with dwr scaling in accuracy. Perhaps the reason why these CNN model scaling strategies are still appliable to ViT is that during pre-training the linear projection ( $1 \\times 1$ convolution) dominates the model computations. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Transfer Learning. The picture changes when transferred to COCO. The input resolution $r$ is much higher so the spatial attention takes over and linear projection part is no longer dominant in terms of FLOPs $\\begin{array} { r } { \\frac { f ( \\mathrm { L i n . } ) } { f ( \\mathrm { A t t . } ) } \\propto \\frac { w } { r ^ { 2 } } ) } \\end{array}$ . Canonical CNN model scaling recipes do not take spatial attention computations into account. Therefore there is some inconsistency between pre-training and transfer learning performance: Despite being strong on ImageNet- $1 k$ , the dwr scaling achieves similar box AP as simple $w$ scaling. Meanwhile, the performance gain from dwr scaling on COCO cannot be clearly explained by the corresponding CNN scaling methodology that does not take $f ( \\mathbf { A t t . } ) \\propto d w r ^ { 4 }$ into account. The performance inconsistency between pre-training and transfer learning calls for novel model scaling strategies for ViT considering spatial attention complexity. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.4 Comparisons with CNN-based Object Detectors ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In previous sections, we treat YOLOS as a touchstone for the transferability of ViT. In this section, we consider YOLOS as an object detector and we compare YOLOS with some modern CNN detectors. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Comparisons with Tiny-sized CNN Detectors. As shown in Tab. 5, the tiny-sized YOLOS model achieves impressive performance compared with well-established and highly-optimized CNN object detectors. YOLOS-Ti is strong in AP and competitive in FLOPs & FPS even though Transformer is not intentionally designed to optimize these factors. From the model scaling perspective [20, 56, 60], YOLOS-Ti can serve as a promising model scaling start point. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Comparisons with DETR. The relations and differences in model design between YOLOS and DETR are given in Sec. 2.1, here we make quantitative comparisons between the two. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b83a7dd3e6718b016c28361127d54631344b20a73e3ae40ed565938360608597.jpg",
+        "table_caption": [
+            "Table 5: Comparisons with some tiny-sized modern CNN detectors. All models are trained to be fully converged. “Size” refers to input resolution for inference. FLOPs and FPS data are measured over the first 100 images of COCO val split during inference following Carion et al. [10]. FPS is measured with batch size 1 on a single 1080Ti GPU. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Backbone</td><td>Size</td><td>AP</td><td>Params. (M)</td><td>FLOPs (G)</td><td>FPS</td></tr><tr><td>Y0LOv3-Tiny 49]</td><td>DarkNet [49]</td><td>416×416</td><td>16.6</td><td>8.9</td><td>5.6</td><td>330</td></tr><tr><td>YOLOv4-Tiny [60]</td><td>COSA [60]</td><td>416 × 416</td><td>21.7</td><td>6.1</td><td>7.0</td><td>371</td></tr><tr><td>YOLOS-Ti</td><td>DeiT-Ti( [57]</td><td>256×*</td><td>23.1</td><td>6.5</td><td>3.4</td><td>114</td></tr><tr><td>CenterNet [70]</td><td>ResNet-18 [26]</td><td>512 × 512</td><td>28.1</td><td>1</td><td>1</td><td>129</td></tr><tr><td>YOLOv4-Tiny (3l) [60]</td><td>COSA [60]</td><td>320 × 320</td><td>28.7</td><td>1</td><td>1</td><td>252</td></tr><tr><td>Def.DETR [72]</td><td>FBNet-V3[15]</td><td>800×*</td><td>27.9</td><td>12.2</td><td>12.3</td><td>35</td></tr><tr><td>YOLOS-Ti</td><td>DeiT-Ti () [57]</td><td>432×*</td><td>28.6</td><td>6.5</td><td>11.7</td><td>84</td></tr></table>",
+        "bbox": [
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+            90,
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/c36156389b476b0a75dc64999832177fa821bc3d30a9dc26810861d0ed2914cf.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 6: Comparisons with different DETR models. Tiny-sized models are trained to be fully converged. “Size” refers to input resolution for inference. FLOPs and FPS data are measured over the first 100 images of COCO val split during inference following Carion et al. [10]. FPS is measured with batch size 1 on a single 1080Ti GPU. The “ResNet-18-DC5” implantation is from timm library [64]. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>Backbone</td><td>Epochs</td><td>Size</td><td>AP</td><td>Params. (M)</td><td>FLOPs (G)</td><td>FPS</td></tr><tr><td>Def. DETR [72]</td><td>FBNet-V3[15]</td><td>150</td><td>800×*</td><td>27.5</td><td>12.2</td><td>12.3</td><td>35</td></tr><tr><td>YOLOS-Ti</td><td>DeiT-Ti [57]</td><td>300</td><td>512×*</td><td>28.7</td><td>6.5</td><td>18.8</td><td>60</td></tr><tr><td>YOLOS-Ti</td><td>DeiT-Ti() [57]</td><td>300</td><td>432×*</td><td>28.6</td><td>6.5</td><td>11.7</td><td>84</td></tr><tr><td>YOLOS-Ti</td><td>DeiT-Ti () [57]</td><td>300</td><td>528×*</td><td>30.0</td><td>6.5</td><td>20.7</td><td>51</td></tr><tr><td>DETR[10]</td><td>ResNet-18-DC5[26]</td><td rowspan=\"5\">150</td><td>800×*</td><td>36.9</td><td>29</td><td>129</td><td>7.4</td></tr><tr><td>YOLOS-S</td><td>DeiT-S[57]</td><td>800×*</td><td>36.1</td><td>31</td><td>194</td><td>5.7</td></tr><tr><td>YOLOS-S</td><td>DeiT-S () [57]</td><td>800×*</td><td>37.2</td><td>31</td><td>194</td><td>5.7</td></tr><tr><td>YOLOS-S (dwr)</td><td>DeiT-S [57](dwr Scale [20])</td><td>704×*</td><td>37.2</td><td>28</td><td>123</td><td>7.7</td></tr><tr><td>YOLOS-S (dwr)</td><td>DeiT-S [57](dwr Scale [20])</td><td>784×*</td><td>37.6</td><td>28</td><td>172</td><td>5.7</td></tr><tr><td>DETR[10]</td><td>ResNet-101-DC5 [26]</td><td rowspan=\"2\">150</td><td rowspan=\"2\">800×*</td><td>42.5</td><td>60</td><td>253</td><td>5.3</td></tr><tr><td>YOLOS-B</td><td>DeiT-B() [57]</td><td>42.0</td><td>127</td><td>538</td><td>2.7</td></tr></table>",
+        "bbox": [
+            174,
+            267,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As shown in Tab. 6, YOLOS-Ti still performs better than the DETR counterpart, while larger YOLOS models with width scaling become less competitive: YOLOS-S with more computations is $0 . 8 \\mathrm { A P }$ lower compared with a similar-sized DETR model. Even worse, YOLOS-B cannot beat DETR with over $2 \\times$ parameters and FLOPs. Even though YOLOS-S with dwr scaling is able to perform better than the DETR counterpart, the performance gain cannot be clearly explained as discussed in Sec. 3.3. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Interpreting the Results. Although the performance is seemingly discouraging, the numbers are meaningful, as YOLOS is not purposefully designed for better performance, but designed to precisely reveal the transferability of ViT in object detection. E.g., YOLOS-B is directly adopted from the BERT-Base architecture [18] in NLP. This 12 layers, 768 channels Transformer along with its variants have shown impressive performance on a wide range of NLP tasks. We demonstrate that with minimal modifications, this kind of architecture can also be successfully transferred (i.e., $\\mathbf { A P } = 4 2 . 0$ ) to the challenging COCO object detection benchmark in computer vision from a pure sequence-to-sequence perspective. The minimal modifications from YOLOS exactly reveal the versatility and generality of Transformer. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.5 Inspecting Detection Tokens ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            746,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/63da8f8e3fb4e771aa06a9a128ef570a2e7b3949b1e5dd19e260d990e27d6718.jpg",
+        "image_caption": [
+            "Figure 2: Visualization of all box predictions on all images from COCO val split for the first ten [DET] tokens. Each box prediction is represented as a point with the coordinates of its center normalized by each thumbnail image size. The points are color-coded so that blue points corresponds to small objects, green to medium objects and red to large objects. We observe that each [DET] token learns to specialize on certain regions and sizes. The visualization style is inspired by Carion et al. [10]. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            790,
+            818,
+            830
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/84f5b527d1d90d74aefb7641fe880fca979703f452976ee62d7c9716969db305.jpg",
+        "image_caption": [
+            "Figure 3: The statistics of all ground truth object categories (the red curve) and the statistics of all object category predictions from all [DET] tokens (the blue curve) on all images from COCO val split. The error bar of the blue curve represents the variability of the preference of different tokens for a given category, which is small. This suggests that different [DET] tokens are category insensitive. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            92,
+            820,
+            223
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Qualitative Analysis on Detection Tokens. As an object detector, YOLOS uses [DET] tokens to represent detected objects. In general, we find that [DET] tokens are sensitive to object locations and sizes, while insensitive to object categories, as shown in Fig. 2 and Fig. 3. ",
+        "bbox": [
+            173,
+            301,
+            826,
+            344
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Quantitative Analysis on Detection Tokens. We give a quantitative analysis on the relation between $X =$ the cosine similarity of [DET] token pairs, and $Y =$ the corresponding predicted bounding box centers $\\ell _ { 2 }$ distances. We use the Pearson correlation coefficient $\\begin{array} { r } { \\bar { \\rho } _ { X , Y } = \\frac { \\mathbb { E } [ ( X - \\mu _ { X } ) ( Y - \\mu _ { Y } ) ] } { \\sigma _ { X } \\sigma _ { Y } } } \\end{array}$ as a measure of linear correlation between variable $X$ and $Y$ , and we conduct this study on all predicted object pairs within each image in COCO val set averaged by all 5000 images. The result is $\\rho _ { X , Y } = - 0 . 8 0$ . This means that [DET] tokens that are close to each other (i.e., with high cosine similarity) also lead to mostly nearby predictions (i.e., with short $\\ell _ { 2 }$ distances, given $\\rho _ { X , Y } < 0 $ ). ",
+        "bbox": [
+            174,
+            359,
+            825,
+            463
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We also conduct a quantitative study on the relation between $X =$ the cosine similarity of [DET] token pairs, and $Y =$ the corresponding cosine similarity of the output features of the classifier. The result is $\\rho _ { X , Y } = - 0 . 0 7 $ , which is very close to 0. This means that there is no strong linear correlation between these two variables. ",
+        "bbox": [
+            173,
+            468,
+            825,
+            523
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Detaching Detection Tokens. To further understand the role [DET] tokens plays, we study impacts caused by detaching the [DET] tokens of YOLOS during training, i.e., we don’t optimize the parameters of the one hundred randomly initialized [DET] tokens. As shown in Tab. 7, detaching the [DET] tokens has a minor impact to AP. These results imply that [DET] tokens mainly serve as the information carrier for the [PATCH] tokens. Similar phenomena are also observed in Fang et al. [22]. ",
+        "bbox": [
+            174,
+            540,
+            517,
+            679
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/a311873628d5d6daa34d8f8cff7c890e75d6bb3c6a023968645adee8ea1c2980.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 7: Impacts of detaching the [DET] tokens of YOLOS during training. "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>[DET]Tokens Config</td><td rowspan=1 colspan=1>AP</td></tr><tr><td rowspan=1 colspan=1>YOLOS-Ti</td><td rowspan=1 colspan=1>Rand. Init.&amp; LearnableRand. Init.&amp; Detached</td><td rowspan=1 colspan=1>28.728.3</td></tr><tr><td rowspan=1 colspan=1>YOLOS-S</td><td rowspan=1 colspan=1>Rand. Init.&amp;LearnableRand.Init.&amp; Detached</td><td rowspan=1 colspan=1>36.136.4</td></tr></table>",
+        "bbox": [
+            534,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            699,
+            321,
+            715
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Vision Transformer for Object Detection. There has been a lot of interest in combining CNNs with forms of self-attention mechanisms [4] to improve object detection performance [9, 31, 63], while recent works trend towards augmenting Transformer with CNNs (or CNN design). Beal et al. [6] propose to use a pre-trained ViT as the feature extractor for a Faster R-CNN [50] object detector. Despite being effective, they fail to ablate the CNN architectures, region-wise pooling operations [23, 25, 27] as well as hand-crafted components such as dense anchors [50] and NMS. Inspired by modern CNN architecture, some works [39, 59, 62, 65] introduce the pyramidal feature hierarchy and locality to Vision Transformer design, which largely boost the performance in dense prediction tasks including object detection. However, these architectures are performance-oriented and cannot reflect the properties of the canonical or vanilla Vision Transformer [21] that directly inherited from Vaswani et al. [58]. Another series of work, the DEtection TRansformer (DETR) families [10, 72], use a random initialized Transformer to encode & decode CNN features for object detection, which does not reveal the transferability of a pre-trained Transformer. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "UP-DETR [16] is probably the first to study the effects of unsupervised pre-training in the DETR framework, which proposes an “object detection oriented” unsupervised pre-training task tailored for Transformer encoder & decoder in DETR. In this paper, we argue for the characteristics of a pre-trained vanilla ViT in object detection, which is rare in the existing literature. ",
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+        "text": "Pre-training and Fine-tuning of Transformer. The textbook-style usage of Transformer [58] follows a “pre-training & fine-tuning” paradigm. In NLP, Transformer-based models are often pretrained on large corpora and then fine-tuned for different tasks at hand [18, 44]. In computer vision, Dosovitskiy et al. [21] apply Transformer to image recognition at scale using modern vision transfer learning recipe [33]. They show that a standard Transformer encoder architecture is able to attain excellent results on mid-sized or small image recognition benchmarks (e.g, ImageNet- $. 1 k$ [51], CIFAR10/100 [34], etc.) when pre-trained at sufficient scale (e.g, JFT-300M [55], ImageNet- $2 1 k$ [17]). Touvron et al. [57] achieves competitive Top-1 accuracy by training Transformer on ImageNet- $. 1 k$ only, and is also capable of transferring to smaller datasets [34, 42, 43]. However, existing transfer learning literature of Transformer arrest in image-level recognition and does not touch more complex tasks in vision such as object detection, which is also widely used to benchmark CNNs transferability. ",
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+        "type": "text",
+        "text": "Our work aims to bridge this gap. We study the performance and properties of ViT on the challenging COCO object detection benchmark [36] when pre-trained on the mid-sized ImageNet- $. 1 k$ dataset [51] using different strategies. ",
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+        "type": "text",
+        "text": "5 Discussion ",
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+        "type": "text",
+        "text": "Over recent years, the landscape of computer vision has been drastically transformed by Transformer, especially for recognition tasks [10, 21, 39, 57, 59]. Inspired by modern CNN design, some recent works [39, 59, 62, 65] introduce the pyramidal feature hierarchy as well as locality to vanilla ViT [21], which largely boost the performance in dense recognition tasks including object detection. ",
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+    {
+        "type": "text",
+        "text": "We believe there is nothing wrong to make performance-oriented architectural designs for Transformer in vision, as choosing the right inductive biases and priors for target tasks is crucial for model design. However, we are more interested in designing and applying Transformer in vision following the spirit of NLP, i.e., pre-train the task-agnostic vanilla Vision Transformer for general visual representation learning first, and then fine-tune or adapt the model on specific target downstream tasks efficiently. Current state-of-the-art language models pre-trained on massive amounts of corpora are able to perform few-shot or even zero-shot learning, adapting to new scenarios with few or no labeled data [8, 38, 45, 46]. Meanwhile, prevalent pre-trained computer vision models, including various Vision Transformer variants, still need a lot of supervision to transfer to downstream tasks. ",
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+        "type": "text",
+        "text": "We hope the introduction of Transformer can not only unify NLP and CV in terms of the architecture, but also in terms of the methodology. The proposed YOLOS is able to turn a pre-trained ViT into an object detector with the fewest possible modifications, but our ultimate goal is to adapt a pre-trained model to downstream vision tasks with the fewest possible costs. YOLOS still needs 150 epochs transfer learning to adapt a pre-trained ViT to perform object detection, and the detection results are far from saturating, indicating the pre-trained representation still has large room for improvement. We encourage the vision community to focus more on the general visual representation learning for the task-agnostic vanilla Transformer instead of the task-oriented architectural design of ViT. We hope one day, in computer vision, a universal pre-trained visual representation can be easily adapted to various understanding as well as generation tasks with the fewest possible costs. ",
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+        "text": "6 Conclusion ",
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+        "text": "In this paper, we have explored the transferability of the vanilla ViT pre-trained on mid-sized ImageNet- $1 k$ dataset to the more challenging COCO object detection benchmark. We demonstrate that 2D object detection can be accomplished in a pure sequence-to-sequence manner with minimal additional inductive biases. The performance on COCO is promising, and these preliminary results are meaningful, suggesting the versatility and generality of Transformer to various downstream tasks. ",
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+        "text": "Acknowledgment ",
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+        "text": "This work is in part supported by NSFC (No. 61876212, No. 61733007, and No. 61773176) and the Zhejiang Laboratory under Grant 2019NB0AB02. We thank Zhuowen Tu for valuable suggestions. ",
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+        "type": "text",
+        "text": "References ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Sec. 3.2, Sec. 3.3 and Sec. 3.4.   \n(b) Did you describe the limitations of your work? [Yes] See Sec. 3.2, Sec. 3.3 and Sec. 3.4.   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Sec. 3.2 and Tab. 2 for the total theoretical computations analysis.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] . ",
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parse/train/p5rMPjrcCZq/p5rMPjrcCZq.md

@@ -0,0 +1,341 @@
+# Only Train Once: A One-Shot Neural Network Training And Pruning Framework
+
+Tianyi Chen∗Microsofttiachen@microsoft.com
+
+Bo Ji National University of Singapore jibo@comp.nus.edu.sg
+
+Tianyu Ding Johns Hopkins University tding1@jhu.edu
+
+Biyi Fang Microsoft bif@microsoft.com
+
+Guanyi Wang Georgia Institute of Technology gwang93@gatech.edu
+
+Zhihui Zhu University of Denver zhihui.zhu@du.edu
+
+Luming Liang Microsoft lulian@microsoft.com
+
+Yixin Shi Microsoft yixshi@microsoft.com
+
+Sheng Yi Microsoft shengyi@microsoft.com
+
+Xiao Tu Microsoft xiaotu@microsoft.com
+
+# Abstract
+
+Structured pruning is a commonly used technique in deploying deep neural networks (DNNs) onto resource-constrained devices. However, the existing pruning methods are usually heuristic, task-specified, and require an extra fine-tuning procedure. To overcome these limitations, we propose a framework that compresses DNNs into slimmer architectures with competitive performances and significant FLOPs reductions by Only-Train-Once (OTO). OTO contains two keys: $( i )$ we partition the parameters of DNNs into zero-invariant groups, enabling us to prune zero groups without affecting the output; and $( i i )$ to promote zero groups, we then formulate a structured-sparsity optimization problem and propose a novel optimization algorithm, Half-Space Stochastic Projected Gradient (HSPG), to solve it, which outperforms the standard proximal methods on group sparsity exploration and maintains comparable convergence. To demonstrate the effectiveness of OTO, we train and compress full models simultaneously from scratch without finetuning for inference speedup and parameter reduction, and achieve state-of-the-art results on VGG16 for CIFAR10, ResNet50 for CIFAR10 and Bert for SQuAD and competitive result on ResNet50 for ImageNet. The source code is available at https://github.com/tianyic/only_train_once.
+
+# 1 Introduction
+
+Deep neural networks (DNNs) have been shown to be effective in various real applications (51; 28). It is widely acknowledged that large-scale DNN models not only learn faster but also outperform their slimmer counterparts. However, such heavy models pose a great challenge to the deployment stage due to their resource-consuming nature. In addressing this issue, many model compression techniques (5; 11) are proposed in the past decade that aim at compressing those large and complex models into slimmer and simpler ones while suffering negligible loss in performance.
+
+![](images/5122d4a224e0d94e46fc150d904c85791df8f37ce1b8e1861ab9a123843fa33e.jpg)  
+Figure 1: Overview of OTO. Without loss of generality, we illustrate OTO on a model with only vanilla convolutional layers, and for simplicity we only show Layeri with $m$ 3D filters and their biases. The key to its success is twofold: $( i )$ identify and partition the parameters of the model into zero-invariant groups (ZIGs); and $( i i )$ solve the structured-sparsity regularization problem using HSPG. Finally, we obtain the compressed model by directly pruning the zero groups, i.e., $\mathrm { Z I G } _ { m }$ .
+
+Pruning methods as one of the main categories of model compression, focus on identifying and pruning redundant structures via various mechanisms to achieve a slimmer architecture, and thus improve the interpretability of a DNN model (26; 11; 65). For example, (32; 33) adopt fine-grained pruning via $\ell _ { 1 }$ or $\ell _ { 2 }$ regularization, which prune the small-weight connections based on some hard threshold. (36; 57; 60) measure the importance of filters to accelerate the networks by removing insignificant feature maps. (37; 7) utilize reinforcement learning agent to predict compression action.
+
+Nevertheless, many of the existing pruning methods $( i )$ often rely on criteria based on heuristics or empirical cues, e.g., magnitude of a connection weight and importance score of a filter, to identify redundant parameters, which may cause divergence during optimization; $( i i )$ thus require complex multi-stage pipelines that involve either a retraining or fine-tuning procedure to regain the accuracy during constructing a slimmer model, which is time-consuming; and $( i i i )$ are specific to certain architectures or applications, and are consequently less applicable to various downstream scenarios. Recently, there have been a few efforts (14; 58; 8) to directly train the network with sparsity inducing regularizers, which provide generality and convergence guarantee. However, these approaches focus on either merely the individual sparsity of the parameters or the group sparsity of the filters, and thus cannot directly remove those zero components (still require subsequent fine-tuning) since the zeros are entangled with other commonly used components, e.g., bias, batch normalization or skip connection. Furthermore, the optimization algorithms used in (14; 58) lack sufficient capability to explore (group) sparsity in DNNs effectively and require a post-processing step to yield exact zeros.
+
+In this paper, we overcome the above limitations of existing pruning methods by proposing a one-shot neural network pruning framework, with which we are able to train a full heavy model from scratch only once, and obtain a slim architecture without fine-tuning while maintain high performance. As shown in Figure 1, the key to its success is twofold: $( i )$ we identify and partition the parameters of DNNs into zero-invariant groups (ZIGs), enabling us to prune redundant structures according to zero groups without affecting the output of the network; and $( i i )$ to promote zero groups, we formulate the pruning task as a structured-sparsity optimization problem and propose a novel optimization method, Half-Space Stochastic Projected Gradient (HSPG), to solve it, which outperforms the standard proximal methods on sparsity exploration and maintains comparable convergence. We highlight that both zero-invariant group partition and the novel optimization algorithm in promoting zero group lead to achieve one-shot neural network training and pruning regardless of its architecture.
+
+Our main contributions are summarized as follows.
+
+• One-Shot Training and Pruning. We propose OTO, a training and pruning framework that compresses a full neural network with competitive performance by Only-Train-Once, thereby one-shot. OTO dramatically simplifies the complex multi-stage training pipelines of the existing pruning approaches, fits various architectures and applications, and hence is generic and efficient.
+
+• Zero-Invariant Group. We define zero-invariant groups for neural networks. If a network is partitioned into ZIGs, it allows us to prune the zero groups without affecting the output, which results in one-shot pruning. Such property is applicable to various popular structures from plain fully connected layers to sophisticated ones such as residual blocks and multi-head attention.
+
+• Novel Structured-Sparsity Optimization Algorithm. We propose Half-Space Stochastic Projected Gradient (HSPG), a method that solves structured-sparsity inducing regularization problem. We show and analyze the superiority of HSPG in promoting zero groups of networks than the standard proximal methods and the competitive objective convergence in practice. The fact that ZIG and HSPG are designed agnostic to networks makes OTO generic to various applications.
+
+• Experimental Results. We train and compress full models simultaneously from scratch without fine-tuning for inference speedup and parameter reduction, and achieve state-of-the-art results on compression benchmark VGG for CIFAR10, ResNet50 for CIFAR10/ImageNet, Bert for SQuAD.
+
+# 2 Related Work
+
+Structured pruning focuses on identifying and pruning the redundant structures in a full model to achieve slimmer architectures for efficient model inference and storage (26; 32), where there have been numerous efforts dedicated. For CNN compression, the general procedure can be largely summarized as: (i) train a full model; (ii) identify and prune the redundant structures to build a slimmer model based on various criteria, including (structured) sparsity (58; 85; 14; 56; 102; 27; 102; 62; 91), Bayesian pruning (101; 65; 59; 81), ranking importance (54; 60; 41; 36; 57; 100), reinforcement learning (37; 7), adversarial robustness (76), scientific control (79), lottery ticket (23; 24; 72), joint quantization learning (80; 90), etc.; (iii) retrain or iteratively fine-tune the slimmer model to regain the accuracy regression during pruning. These methods cannot avoid the extra and usually timeconsuming fine-tuning step because the identified redundant structures, even parametrized with zeros, actually contribute to the model output, thereby additional fine-tuning step is an absolute necessity.
+
+For pruning Bert (82), knowledge distillation (40) and LayerDropout (21) shorten Bert by reducing the number of layers directly. Other methods (29; 75; 30) build slimmer Berts in the manner of individual sparsity, but require specially designed data structure for storage and computing library to take advantage of sparse data (31; 10), and typically cannot achieve inference speedup against the highly optimized library (16) for dense model due to the discontiguous memory allocation (9).
+
+The structured sparsity for weight pruning is the most relevant to the algorithm described in this paper. The existing structure learning works (58; 85; 14; 56; 102) have the respective disadvantages: (i) multiple trainings during the whole procedure since their group partition cannot isolate the impact of pruned structures to the model output; and (ii) heuristic post-processing to generate zero groups as the standard proximal methods (19; 87; 88; 12) and ADMM (100; 58; 4) defective on the sparsity exploration for deep learning (8), which may deteriorate the performance of the model significantly.
+
+Avoiding fine-tuning step during the whole pruning procedure is receiving more and more attentions because of its efficiency. In particular, SNIP (52) and GraSP (83) identify redundancy via salience scores at the initialization stage to construct pruned structures, then train the pruned models by the standard optimizers. SCP (48) isolates the impact of batch normalization, while lacks the consideration of more general DNN architectures.
+
+# 3 OTO
+
+In essence, OTO frames the network training and pruning as a structure learning problem. Given a full model $\mathcal { M }$ , OTO trains and compresses it simultaneously from scratch without fine-tuning, and achieves significant reduction in both FLOPs and parameters. Particularly, as stated in Algorithm 1, the trainable parameters of $\mathcal { M }$ are firstly partitioned into a ZIG set $\mathcal { G }$ (Section 3.1). We then construct and solve a structured-sparsity inducing optimization problem (Section 3.2) by proposed stochastic optimizer (HSPG) to seek a highly group-sparse solution $\pmb { x } _ { \mathrm { H S P G } } ^ { * }$ (Section 3.3). Lastly, we obtain a compressed model $\mathcal { M } ^ { \ast }$ by directly pruning these zero groups (Section 3.4).
+
+# Algorithm 1 Outline of OTO.
+
+1: Input: Full model $\mathcal { M }$ (no need to be pretrained).   
+2: Construct $\mathfrak { s }$ : Partition the trainable parameters of $\mathcal { M }$ into a ZIG set $\mathcal { G }$ .   
+3: Train: Train the model $\mathcal { M }$ using HSPG (Algorithm. 2) to obtain a group-sparse solution $\pmb { x } _ { \mathrm { H S P G } } ^ { * }$   
+4: Prune: Construct a slimmer model architecture M∗ by directly pruning zero groups of x∗HSPG.   
+5: Output: Compressed model $\mathcal { M } ^ { * }$ .
+
+# 3.1 Zero-Invariant Group
+
+The root cause of the existing methods having multi-stage training pipeline is that despite the pruned structure (e.g., 3D filter) being zeros, its associated structures (e.g., non-zero bias) still contribute to its corresponding output to the next layer (e.g., feature map). As a result, the model accuracy regresses, hence an extra step of fine-tuning is necessary. OTO avoids the necessity by partitioning the parameters of DNNs into a set of so-called zero-invariant groups (ZIGs) $\mathcal { G }$ defined as follows.
+
+Definition 1 (Zero-Invariant Groups (ZIGs)). For a layer-wise DNN, we partition its entire trainable parameters into disjoint groups $\mathcal { G } = \{ g \}$ . Then we call $\mathcal { G }$ zero-invariant groups (ZIGs) if each group $g \in { \mathcal { G } }$ is zero-invariant in the sense that all of the parameters in g being zeros results in its corresponding output to the next layer to be zeros as well.
+
+In effect, if and only if a DNN model is partitioned into a ZIG set $\mathcal { G }$ and one or more of its element $g$ are parameterized by zeros, the entire corresponding structures contribute none to the model outputs and hence can be pruned directly. Such partition is applicable to various structures of DNN models. Without loss of generality, we define and describe ZIG partition for three most popular structures: $( i )$ Conv-BN, (ii) Residual Block, and (iii) Fully Connected and Multi-Head Attention Layer.
+
+![](images/48438b786fc8e5517c936cd74cdfcff453877ce179a009c1570ac4ad34def872.jpg)
+
+(a) Conv-BN. $m$ denotes the number of channels in $\mathcal { O } ^ { l }$ (b) Residual block. $m$ denotes the number of output channels of the residual block.
+
+![](images/1b19210349c474bcb83508a0ecf6aedba3777bbbd3f3c4350f9684d6129d0442.jpg)
+
+![](images/a7c596082ec3463774fc5aac6315fb6292578b380ad98913bea369bca6c1dff5.jpg)  
+Figure 2: Zero-invariant group partition for three popular structures.
+
+(c) Fully connected layer (Left). Multi-head attention layer (Right). $m$ denotes the length of output vector.
+
+ZIG of Conv-BN. Convolutional layer (Conv) followed by batch-normalization layer (BN) is extensively used in DNN models. Figure 2a shows the ZIG partition for Conv-BN. The 4D filter tensor $\kappa ^ { l }$ is flattened into a filter matrix $\hat { \kappa } ^ { l }$ . During the forward pass, the input tensor $\boldsymbol { \mathcal { Z } ^ { l } }$ is transformed into the output tensor $\mathcal { O } ^ { l }$ of Conv and then into the input tensor of the $( l ^ { \bullet } + 1 ) ^ { t h }$ layer $\pmb { \mathcal { T } } ^ { l + 1 }$ by
+
+$$
+\mathcal { O } ^ { l }  \mathcal { T } ^ { l } \otimes \hat { \mathcal { K } } ^ { l } + b ^ { l } , \ \mathcal { Z } ^ { l + 1 }  \frac { a ( \mathcal { O } ^ { l } ) - \mu ^ { l } } { \sigma ^ { l } } \odot \gamma ^ { l } + \beta ^ { l } ,
+$$
+
+where denoted by $\otimes$ the convolutional operation, $\odot$ the element-wise multiplication and $a ( \cdot )$ the activation function. BN is parameterized by mean $\mu ^ { l }$ , standard deviation $\sigma ^ { l }$ , weight $\gamma ^ { l }$ and bias $\beta ^ { l }$ respectively. The activation function needs to be zero-invariant, i.e., $a ( \mathbf { 0 } ) = \mathbf { 0 }$ , where most instances satisfy, e.g., ReLU (25), PReLU (34), GELU (39) and LeakyReLU (89). Hence, each row of the flattened filter matrix $\hat { \kappa } ^ { l }$ and its bias $b ^ { l }$ belong to one ZIG because they being zeros results in their corresponding channel of $\mathcal { O } ^ { l }$ (i.e., feature map) to be zeros as well. Subsequently, $\gamma ^ { l }$ and $\beta ^ { l }$ of this corresponding channel in BN are also included into this ZIG to avoid the value shift (zero to non-zero) during normalization. Note that grouping these four sets of parameters channel-wisely makes Conv-BN zero-invariant regardless of the value of $\mu ^ { l }$ and $\sigma ^ { l }$ , and hence they are excluded from the ZIG. For illustration, each ZIG is highlighted in the same color (e.g., $g _ { 1 } ^ { l }$ in blue).
+
+ZIG of Residual Block. The residual block adds another layer of challenge because its output tensor is the summation of the outputs of two Conv-BNs. Figure 2b shows the ZIG partition for the residual block. As illustrated, before propagated to Conv3, the outputs of Conv1-BN1 and Conv2-BN2 are summarized and hence share the same dimension. As such, to make residual block zero-invariant, we group the four sets of parameters channel-wisely of both Conv1-BN1 and Conv2-BN2 into ZIGs, i.e., each row of ${ \hat { \kappa } } ^ { 1 }$ , $b ^ { \hat { 1 } }$ , $\gamma ^ { 1 }$ , $\beta ^ { 1 }$ of Conv1-BN1 and each row of ${ \hat { \kappa } } ^ { 2 }$ , $b ^ { 2 }$ , $\gamma ^ { 2 }$ , $\beta ^ { 2 }$ of Conv2-BN2. In Appendix A.1, we describe the zero-invariant group partition of ResNet50 in greater detail.
+
+ZIG of Fully Connected and Multi-Head Attention Layer. Figure 2c shows the ZIG partition for fully connected and multi-head attention layer. Particularly, we partition each row of weight matrix and its associated bias into a ZIG, and therefore any input element is turned to zero if that ZIG is parameterized with zeros, making the fully connected layer zero-invariant. Multi-head attention layer is the key building block of the transformer architectures (82). Its trainable parameters contain a weight matrix and bias vector, consisting of the sub-matrix and sub-vector of each head (we use two heads as an example). We form ZIG by grouping each row of every sub-matrix and sub-vector, i.e., each row of $w _ { h _ { 1 } }$ , $\boldsymbol { b } _ { h _ { 1 } }$ , $w _ { h _ { 2 } }$ and $b _ { h _ { 2 } }$ of $h _ { 1 }$ and $h _ { 2 }$ , respectively.
+
+Automatic ZIG Partition. Based on the above illustrating examples, we provide prescribed ZIG partition for the tested DNNs in Section 4. Furthermore, given an arbitrary DNN architecture, the procedure of partitioning variables into ZIGs could be automatically proceeded, wherein the key would be identifying the connections among various layers, then performing corresponding group partition. We will leave the automatic ZIG partition for arbitrary DNNs as future work.
+
+# 3.2 Structured-Sparsity Regularization
+
+We now formulate a structured-sparsity regularization problem over the ZIG set $\mathcal { G }$ for the trainable parameters of the full model $\mathcal { M }$ as follows
+
+$$
+\operatorname* { m i n i m i z e } _ { \pmb { x } \in \mathbb { R } ^ { n } } \psi ( \pmb { x } ) : = f ( \pmb { x } ) + \lambda r ( \pmb { x } ) , \ r ( \pmb { x } ) : = \sum _ { g \in \mathcal { G } } \| [ \pmb { x } ] _ { g } \| ,
+$$
+
+where $\lambda > 0$ is a weighting coefficient, $f ( { \pmb x } )$ is a task-specific loss function, and $r ( { \pmb x } )$ is an augmented structured-sparsity inducing regularization term encoding the topological structure of $\mathcal { M }$ over $\mathcal { G }$ . A larger $\lambda$ typically results in a higher group sparsity while sacrifices more on the bias of model estimation. We aim at computing a local optimum to achieve both low loss and high group sparsity.
+
+To induce group sparsity onto the solution of (2), there exist several candidates for $r ( { \pmb x } )$ , including mixed $\ell _ { 1 } / \ell _ { p }$ norm $( p > 1 )$ (1; 20) and group Minmax Concave Penalty (MCP) (96). Among these candidates, the mixed $\ell _ { 1 } / \ell _ { 2 }$ norm as defined in (2) is arguably the most popular choice in classical machine learning applications (1; 92), where $\lVert \cdot \rVert$ is the $\ell _ { 2 }$ -norm, and each component $g \in { \mathcal { G } }$ indexes a group of variables. In this paper, we will demonstrate the effectiveness of OTO by selecting $r ( { \pmb x } )$ as the mixed $\ell _ { 1 } / \ell _ { 2 }$ norm. We highlight OTO is applicable for other group sparsity regularizers as well.
+
+# 3.3 Half-Space Stochastic Projected Gradient (HSPG)
+
+To solve the non-smooth regularization problem as (2) in deep learning applications, the standard proximal method and the ADMM lack capability to effectively identify group sparsity; see the discussions later in this Section. Therefore, we propose a novel stochastic optimization algorithm so-called Half-Space Stochastic Projected Gradient (HSPG) to enhance the group sparsity exploration more effectively than the classical methods while maintain a similar convergence property.
+
+Outline. We state the outline of HSPG in Algorithm 2. It contains two stages: Initialization Stage and Group-Sparsity Stage. The first Initialization Stage employs Stochastic Gradient Descent (SGD) step to search for a good but usually non-sparse solution estimate. Then the second stage proceeds Half-Space step started with the non-sparse iterate to effectively exploit the group sparsity within a sequence of reduced spaces and converges to the group-sparse solutions. Half-Space step performs SGD update on free non-zero variables along with a novel projection operator so-called Half-Space Projection, which significantly outperforms the standard proximal operators on sparsity exploration.
+
+Initialization Stage. The Initialization Stage performs the vanilla SGD to find a good initial point for the subsequent Group-Sparsity Stage. At $k ^ { \mathit { \hat { t } h } }$ iteration, a stochastic gradient of $f$ , e.g., based on a mini-batch, is generated denoted as $\boldsymbol { \nabla } \tilde { f }$ . Since the group sparsity inducing regularizer $r ( { \pmb x } )$ in the form as (2) is non-smooth, we select a subgradient $\zeta ( \pmb { x } _ { k } )$ from its subdifferential $\partial r ( \pmb { x } _ { k } )$ to form a stochastic subgradient of $\psi ( \pmb { x } _ { k } )$ as $\nu ( { \pmb x } _ { k } ) : = \nabla \tilde { f } ( { \pmb x } _ { k } ) + \lambda \zeta ( { \pmb x } _ { k } )$ . We then compute the next iterate as $\pmb { x } _ { k + 1 } : = \pmb { x } _ { k } - \alpha _ { k } \nu ( \pmb { x } _ { k } )$ by subgradient descent update.
+
+![](images/67e7cbd3ef90c0abd50772be2db874a17a3bcb8ba70ef573ad987a889fefe45a.jpg)  
+Figure 3: Illustration of Half-Space Step with projection in (6), where $\mathcal { G } = \{ \{ 1 , 2 \} \}$ .
+
+Group-Sparsity Stage. The Group-Sparsity Stage is designed to effectively determine the groups of zero variables and capitalize convergence characteristic, which is in sharp contrast to other heuristic aggressive weight pruning methods that typically lack theoretical guarantees (55; 60). The intuition of Half-Space Step is to project $[ \boldsymbol { x } _ { k } ] _ { g }$ to zero only if $- [ { \pmb x } _ { k } ] _ { g }$ serves as a descent step to $\psi ( \pmb { x } _ { k } )$ , i.e., $- [ { \pmb x } _ { k } ] _ { g } ^ { \top } [ \nabla \psi ( { \pmb x } _ { k } ) ) ] _ { g } < 0$ , hence updating $[ { \pmb x } _ { k + 1 } ] _ { g }  [ { \pmb x } _ { k } ] _ { g } - [ { \pmb x } _ { k } ] _ { g } = 0$ still results in some progress to the optimality. In particular, we first define the following index sets for any $\pmb { x } \in \mathbb { R } ^ { n }$ :
+
+$$
+\begin{array} { r } { \mathcal { Z } ^ { 0 } ( \pmb { x } ) : = \{ g : g \in \mathcal { G } , [ \pmb { x } ] _ { g } = 0 \} \mathrm { a n d } \mathcal { Z } ^ { \neq 0 } ( \pmb { x } ) : = \{ g : g \in \mathcal { G } , [ \pmb { x } ] _ { g } \neq 0 \} , } \end{array}
+$$
+
+where $\mathcal { T } ^ { 0 } ( { \pmb x } )$ represents the indices of groups of zero variables at $_ { \textbf { \em x } }$ , and $\scriptstyle { \mathcal { T } } ^ { \neq 0 } ( { \pmb x } )$ indexes the groups of nonzero variables at $_ { \textbf { \em x } }$ . To proceed, we further define an artificial set that $_ { \textbf { \em x } }$ lies in:
+
+$$
+\begin{array} { r } { S ( { \boldsymbol x } ) : = \{ \mathbf 0 \} \bigcup \big \{ z \in \mathbb R ^ { n } : [ z ] _ { g } = \mathbf 0 \mathrm { ~ i f ~ } g \in \mathcal { X } ^ { 0 } ( { \boldsymbol x } ) , \mathrm { a n d ~ } [ z ] _ { g } ^ { \top } [ { \boldsymbol x } ] _ { g } \geq \epsilon \| [ { \boldsymbol x } ] _ { g } \| ^ { 2 } \mathrm { ~ i f ~ } g \in \mathcal { X } ^ { \neq 0 } ( { \boldsymbol x } ) \big \} , } \end{array}
+$$
+
+which consists of half-spaces and the origin. Here the parameter $\epsilon \geq 0$ controls how aggressively we promote group sparsity, and is typically fixed as zero in practice. Hence, $\pmb { x } \in S _ { k } : = \pmb { S } ( \pmb { x } _ { k } )$ only if: (i) $[ { \pmb x } ] _ { g }$ lies in the upper half-space for all $g \in { \mathcal { T } } ^ { \neq 0 } ( { \pmb x } _ { k } )$ for some prescribed $\epsilon \in [ 0 , 1 )$ as shown in Figure 3a; and (ii) $[ { \pmb x } ] _ { g }$ equals to zero for all $g \in \mathcal { T } ^ { 0 } ( { \pmb x } _ { k } )$ . Intuitively, $\scriptstyle { S _ { k } }$ establishes the region where important structures inhabit, thereby redundant structures vanish if falling outside.
+
+Ideally, the Initialization Stage has produced reasonably well but typically non-sparse iterate $\scriptstyle { \mathbf { { \mathit { x } } } } _ { k }$ nearby a group-sparse solution $\pmb { x } ^ { * }$ of problem (2), , i.e., the optimal distance $\| \pmb { x } _ { k } - \pmb { x } ^ { * } \|$ is sufficiently small. As seen in Appendix B, it further indicates that the group-sparse optimal solution $\pmb { x } ^ { * }$ inhabits $S _ { k }$ , and $S _ { k }$ has already covered the group-support of $\pmb { x } ^ { * }$ , i.e., $\mathcal { T } ^ { \neq 0 } ( { \pmb x } ^ { * } ) \subseteq \mathbb { Z } ^ { \neq 0 } ( { \pmb x } _ { k } ^ { * } )$ . Our goal now becomes minimizing $\psi ( { \pmb x } )$ over $\boldsymbol { S } _ { k }$ to identify the remaining zero groups, i.e., $\mathcal { T } ^ { 0 } ( \dot { \pmb { x } ^ { * } } ) / \mathcal { Z } ^ { 0 } ( \pmb { x } _ { k } )$ , which is formulated as the following problem:
+
+# Algorithm 2 Outline of HSPG for solving (2).
+
+1: Input: $\pmb { x } _ { 0 } \in \mathbb { R } ^ { n }$ , $\alpha _ { 0 } > 0 , \epsilon \in [ 0 , 1 )$ , and $N \in \mathbb { Z } ^ { + }$ .   
+2: Output: a group-sparse solution $\pmb { x } _ { \mathrm { H S P G } } ^ { * }$ from $\{ \boldsymbol { x } _ { k } \}$ .   
+3: for $k = 0 , 1 , 2 , \ldots$ . do   
+4: Compute a stochastic subgradient $\nu ( \pmb { x } _ { k } )$ of $\psi ( \pmb { x } _ { k } )$ .   
+5: if $k < N$ then   
+6: Subgradient Descent Update:   
+7: Set ${ \pmb x } _ { k + 1 }  { \pmb x } _ { k } - \alpha _ { k } { \pmb \nu } ( { \pmb x } _ { k } )$ .   
+8: else   
+9: Half-Space Update:   
+10: Set a trial iterate $\tilde { \pmb { x } } _ { k + 1 }$ as   
+$\begin{array} { r l } & { [ \tilde { \pmb { x } } _ { k + 1 } ] _ { \mathcal { T } ^ { \neq 0 } ( { \pmb x } _ { k } ) }  [ { \pmb x } _ { k } - \alpha _ { k } \nu ( { \pmb x } _ { k } ) ] _ { \mathcal { T } ^ { \neq 0 } ( { \pmb x } _ { k } ) } } \\ & { [ \tilde { \pmb { x } } _ { k + 1 } ] _ { \mathcal { T } ^ { 0 } ( { \pmb x } _ { k } ) }  \mathbf { 0 } . } \end{array}$   
+11: for each group $g$ in $\mathcal { G }$ do   
+12: $[ \pmb { x } _ { k + 1 } ] _ { g }  [ \mathrm { P r o j } _ { S _ { k } } ^ { H S } ( \tilde { \pmb { x } } _ { k + 1 } ) ] _ { g } .$ .   
+13: Update $\alpha _ { k + 1 }$ .
+
+$$
+{ \underset { \pmb { x } \in S _ { k } } { \mathrm { m i n i m i z e } } } \psi ( \pmb { x } ) = f ( \pmb { x } ) + \lambda r ( \pmb { x } ) .
+$$
+
+The next iterate $\scriptstyle { \pmb { x } } _ { k + 1 }$ is computed as an solution estimate of problem (5).
+
+Particularly, in Algorithm 2, $[ { \pmb x } _ { k + 1 } ] _ { { \mathcal { T } } ^ { 0 } ( { \pmb x } _ { k } ) } \equiv { \bf 0 }$ will not be updated, and only the entries in $\scriptstyle { \mathcal { T } } ^ { \neq 0 } ( { \pmb x } _ { k } )$ are free to move. Hence $\psi ( { \pmb x } )$ is smooth on $\scriptstyle { S _ { k } }$ , and (5) is a reduced space optimization problem. A standard way to solve problem (5) would be the stochastic gradient descent equipped with Euclidean projection (68). However, such a projected method rarely produces zero (group) variables, as the dense Euclidean projected point $\hat { \pmb x } _ { E } \neq { \bf 0 }$ illustrated in Figure 3a. To address, we introduce a novel half-space projection operator to effectively project an entire group of variables to zeros.
+
+As line 4 and 9-12 in Algorithm 2, we first approximate the (sub)gradient of $\psi$ on the free variables by $[ \nu ( \pmb { x } _ { k } ) ] _ { \pmb { \mathbb { T } } ^ { \neq 0 } ( \pmb { x } _ { k } ) }$ , then employ gradient descent over $\scriptstyle { \mathcal { T } } ^ { \neq 0 } ( { \pmb x } _ { k } )$ to compute a trial point $\widetilde { \pmb { x } } _ { k + 1 }$ which is passed into a fresh half-space projection operator $\mathrm { P r o j } _ { S _ { k } } ^ { H S } ( \cdot )$ defined as
+
+$$
+\begin{array} { r } { \left[ \mathrm { P r o j } _ { { \mathcal S } _ { k } } ^ { H S } ( z ) \right] _ { g } : = \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f } [ z ] _ { g } ^ { \top } [ \pmb { x } _ { k } ] _ { g } < \epsilon \left\| [ \pmb { x } _ { k } ] _ { g } \right\| ^ { 2 } , } \\ { [ z ] _ { g } } & { \mathrm { o t h e r w i s e } . } \end{array} \right. } \end{array}
+$$
+
+The above projector of form (6) is not the standard one in Euclidean sense2, and it has two advantages: $( i )$ the actual search direction $d _ { k } : = ( \mathrm { P r o j } _ { S _ { k } } ^ { H S } ( \tilde { \pmb { x } } _ { k + 1 } ) - \pmb { x } _ { k } ) / \alpha _ { k }$ performs as a descent direction to $\psi ( \pmb { x } _ { k } )$ , i.e., $[ \mathbf { \mathop { d } } _ { k } ] _ { g } ^ { \top } [ \nu ( \mathbf { \mathop { x } } _ { k } ) ) ] _ { g } < 0$ as $\theta < 9 0 ^ { \circ }$ in Figure 3a, hence the progress to the optimum is made via the sufficient decrease property drawn as Lemma 1 in Appendix B; then $( i i )$ it effectively projects entire groups of variables to zero if the inner product of corresponding entries is sufficiently small. In contrast, the Euclidean projection operator is far away effective to promote group sparsity.
+
+Superiority of HSPG on Group Sparsity Identification. We now intuitively illustrate the strength of HSPG on group sparsity exploration. In fact, the half-space projection (6) is a more effective sparsity promotion mechanism compared to the standard proximal methods. Particularly, it benefits from a much larger projection region to map a reference point $\hat { \pmb { x } } _ { k + 1 } : = \pmb { x } _ { k } - \alpha _ { k } \nabla \tilde { f } ( \pmb { x } _ { k } )$ or its variants to zero. As the 2D case described in Figure 3b, the projection regions of the state-of-the-art Prox-SG (19), Prox-SVRG (88), Prox-Spider (97) and SAGA (12) for (2) are $\ell _ { 2 }$ -balls with radius as $\alpha _ { k } \lambda$ . In deep learning applications, the step size $\alpha _ { k }$ is usually selected around $1 0 ^ { - 3 }$ to $1 0 ^ { - 4 }$ or even smaller for convergence. Together with the common setting of $\lambda \ll 1$ , their projection regions would vanish rapidly, resulting in the difficulties to produce group sparsity. As a sharp contrast, even though $\alpha _ { k } \lambda$ is near zero, the projection region of HSPG $\{ \pmb { x } : \pmb { x } _ { k } ^ { \top } \pmb { x } < ( \dot { \alpha _ { k } } \lambda + \epsilon \| \pmb { x } _ { k } \| ) \| \pmb { x } _ { k } \| \}$ (seen in Appendix B) is still an open half-space which contains those $\ell _ { 2 }$ balls as well as RDA (87)’s if $\epsilon$ is large enough. Conversely, vanilla ADMM alone lacks the mechanism to project a group of variables to zero, unless equips with extra post-processing step (100; 58). In Appendix B, we further reveal that HSPG still maintains the convergence to the optimality as drawn in Theorem 1. Moreover, we numerically demonstrate the superiority of HSPG in the sense of optimization in Appendix C.
+
+# 3.4 Pruning Without Fine-Tuning
+
+The group-sparse solution $\pmb { x } _ { \mathrm { H S P G } } ^ { * }$ over ZIGs to the full model $\mathcal { M }$ is leveraged to construct the slimmer model $\mathcal { M } ^ { * }$ . Particularly, we prune the redundant structures identified as zero groups $\mathcal { T } ^ { 0 }$ and retain non-zero groups $\mathcal { T } ^ { \neq 0 }$ in $\pmb { x } _ { \mathrm { H S P G } } ^ { * }$ . Because the parameters of full model are partitioned into ZIGs, the pruned structures contribute none to the model output. Therefore, given the same input, the slimmer model $\mathcal { M } ^ { * }$ computes the identical output as the full model $\mathcal { M }$ parameterized with $\pmb { x } _ { \mathrm { H S P G } } ^ { * }$ .
+
+# 4 Experiment
+
+In this section, we numerically demonstrate the effectiveness of OTO by one-shot training and pruning without fine-tuning on several benchmark compression tasks for CNNs, i.e., VGG16 (77) for CIFAR10 (49) and ResNet50 (35) for CIFAR10 (49) and ImagetNet (ILSVRC2012) (15). We also verify the scalibility of OTO onto Bert (82) evaluated on SQuAD (69). All datasets are free to academic usage and do not contain personally identifiable information or offensive content. CIFAR10 is under the MIT license, consisting of 50,000 training and 10,000 test images from 10 classes. ImagetNet is a large-scale dataset without license and contains about 1.2 million and 50,000 images in training and validation sets from 1,000 classes. SQuAD is under the CC BY-SA 4.0 license with about 100,000 question/answer pairs splitted into train/dev/test sets as $( 8 0 / 1 0 / 1 0 \%$ ). We conduct all experiments on a Nvidia RTX8000 GPU and provide implementation details in Appendix A.
+
+Table 1: VGG16 and VGG16-BN for CIFAR10. Convolutional layers are in bold.   
+
+<table><tr><td>Method</td><td>BN</td><td>Architecture</td><td>FLOPs</td><td>#of Params</td><td>Top-1 Acc.</td></tr><tr><td>Baseline</td><td>X</td><td>64-64-128-128-256-256-256-512-512-512-512-512-512-512-512</td><td>100%</td><td>100%</td><td>91.6%</td></tr><tr><td>SBP (65)</td><td>X</td><td>47-50-91-115-227-160-50-72-51-12-34-39-20-20-272</td><td>31.1%</td><td>5.9%</td><td>91.0%</td></tr><tr><td>BC (59)</td><td></td><td>51-62-125-128-228-129-38-13-9-6-5-6-6-6-20</td><td>38.5%</td><td>5.4%</td><td>91.0%</td></tr><tr><td>RBC (101)</td><td></td><td>43-62-120-120-182-113-40-12-20-11-6-9-10-10-22</td><td>32.3%</td><td>3.9%</td><td>90.5%</td></tr><tr><td>RBP (101)</td><td></td><td>50-63-123-108-104-57-23-14-9-8-6-7-11-11-12</td><td>28.6%</td><td>2.6%</td><td>91.0%</td></tr><tr><td>OTO</td><td>xxxx</td><td>21-45-82-110-109-68-37-13-9-7-3-5-8-170-344</td><td>16.3%</td><td>2.5%</td><td>91.0%</td></tr><tr><td>Baseline</td><td></td><td>64-64-128-128-256-256-256-512-512-512-512-512-512-512-512</td><td>100%</td><td>100%</td><td>93.2%</td></tr><tr><td>EC (55)</td><td></td><td>32-64-128-128-256-256-256-256-256-256-256-256-256-512-512</td><td>65.8%</td><td>37.0%</td><td>93.1%</td></tr><tr><td>Hinge (56)</td><td></td><td></td><td>60.9%</td><td>20.0%</td><td>93.6%</td></tr><tr><td>SCP(48)</td><td></td><td></td><td>33.8%</td><td>7.0%</td><td>93.8%</td></tr><tr><td>OTO</td><td></td><td>22-56-93-123-182-125-95-45-27-21-10-13-19-244-392</td><td>26.8%</td><td>5.5%</td><td>93.3%</td></tr></table>
+
+# 4.1 Deep Convolutional Neural Network
+
+The results on CNN experiments are summarized in Table 1, 2 and 4. In particular, we compare OTO to its state-of-the-art counterparts by Top-1/5 accuracy, remaining FLOPs and parameters against the corresponding baseline (full model). We report the numbers of other methods based on the corresponding literature and leave as ‘-’ if not reported. The best pruning results are marked as bold.
+
+VGG16 for CIFAR10. We consider the standard VGG16 and the version with batch normalization layer after each convolutional layer, referred to as VGG16-BN. OTO partitions the parameters into ZIGs following Section 3.1, then trains and prunes the model via HSPG, and finally constructs the slimmer model without fine-tuning. For VGG16, as shown in Table 1, the pruned architecture of OTO indicates that OTO identifies similar redundancy of the intermediate and late convolutional layers compared to other methods, but significantly more of the early convolutional layers. As a result, OTO achieves $8 3 . 7 \%$ $( 1 - 1 6 . 3 \% )$ FLOPs reduction and $9 7 . 5 \%$ $( \dot { 1 } - 2 . 5 \% )$ parameter reduction with the best Top-1 accuracy, which outperforms other state-of-the-arts significantly. For VGG16-BN, among all, OTO reduces FLOPs and parameters to the lowest $2 6 . 8 \%$ and $5 . 5 \%$ , respectively. EC (55) and Hinge (56) achieve the same level of Top-1 accuracy as OTO, but are substantially outperformed when it comes to FLOPs and parameter reduction. We further present the FLOPs reductions per layer of OTO in Table 7 of Appendix A.4.
+
+ResNet50 for CIFAR10. Since OTO is able to automatically learn a slimmer model of high performance, we compare it with two state-of-the-art automatic neural network compression frameworks, i.e., AMC (37) and ANNC (90). AMC trains a reinforcement learning agent to predict compression action for each layer environment. ANNC jointly proceeds pruning and quantization within energy
+
+Table 2: ResNet50 for CIFAR10.   
+
+<table><tr><td>Method</td><td>FLOPs</td><td>#of Params</td><td>Top-1 Acc.</td></tr><tr><td>Baseline</td><td>100%</td><td>100%</td><td>93.5%</td></tr><tr><td>AMC (37)</td><td>1</td><td>60.0%</td><td>93.6%</td></tr><tr><td>ANNC (90)</td><td>=</td><td>50.0%</td><td>95.0%</td></tr><tr><td>PruneTrain (61)</td><td>30.0%</td><td>1</td><td>93.1%</td></tr><tr><td>N2NSkip (78)</td><td>1</td><td>10.0%</td><td>94.4%</td></tr><tr><td>OTO</td><td>12.8%</td><td>8.8%</td><td>94.4%</td></tr></table>
+
+constraint. We conduct OTO on their shared experiment, i.e., ResNet50 on CIFAR10. ResNet50 includes both the standard convolutional layers and the layers with residual connections, which are partitioned into ZIGs following Section 3.1. We report the results in Table 2 along with other competitors from (61; 78). Based on the results, all methods achieve competitive validation accuracies, where most of them are even higher than the baseline reported in (37). OTO outperforms AMC, ANNC without quantization, PruneTrain and N2NSkip by using only $1 2 . 8 \%$ FLOPs and $8 . 8 \%$ parameters. Note that no FLOPs reduction is reported in (37) and (90). Finally, we highlight that OTO is flexible to incorporate quantization as the two techniques are complementary and will leave to future work.
+
+Ablation Study on Switching Parameter $N$ . We provide ablation study regarding the impact the switch (parameterized as $N$ ) between the initialization stage and the groupsparsity stage in Algorithm 1. In theory, as shown in Theorem 1 of Ap
+
+Table 3: OTO Under Different Switchings $( N = T , 2 T , 3 T )$ for VGG16, VGG16-BN and ResNet50 on CIFAR10   
+
+<table><tr><td>Backend</td><td>FLOPs</td><td>#of Params</td><td>Top-1 Acc.</td></tr><tr><td>VGG16</td><td>17.0% ± 1.4%</td><td>2.6% ± 0.4%</td><td>90.9%± 0.3%</td></tr><tr><td>VGG16-BN</td><td>25.4%±1.1%</td><td>5.0%± 0.5%</td><td>93.3% ±0.2%</td></tr><tr><td>ResNet50</td><td>12.9% ± 1.5%</td><td>8.5% ± 1.0%</td><td>94.2% ± 0.2%</td></tr></table>
+
+pendix B.4, the projection stage should start when the iterate falls nearby a group sparse local minimizer. In practice, we relax it to start the group sparsity stage once the iterate falling into some stationary status regarding the validation accuracy. As described in Appendix A.2, throughout all experiments, we periodically decay the learning rate per fixed number of epochs parameterized as $T$ .
+
+At the end of each $T$ epochs, we then proceed a statistical test similar to (98) but on the validation accuracy and find that the validation accuracy falls into stationarity near the late epochs of each period. Therefore, in our pruning experiments, we switch to the group-sparsity stage right after the first $T$ epochs. Table 3 describes the performance of OTO under varying switching parameters, from which we observe that OTO is not largely sensitive to the switching parameter if the group-sparsity stage starts after some stationary condition has been numerically satisfied.
+
+ResNet50 for ImageNet. We now evaluate OTO on ResNet50 for ImageNet. As shown in Table 4, OTO prunes $6 4 . 5 \% ( 1 \textrm { -- }$ $3 5 . 5 \%$ ) parameters to achieve $6 5 . 5 \% ( 1 - 3 4 . 5 \% )$ FLOPs reduction with only $1 . { \dot { 4 } } \% / 0 . 8 \%$ Top1/5 accuracy regression compared to the baseline. OTO consistently outperforms the majority of counterparts especially on the FLOPs reduction and the parameter reduction. We note that Hinge (56) prunes CNNs via structured-sparsity optimization by employing standard stochastic proximal gradient method. It
+
+Table 4: ResNet50 for ImageNet.   
+
+<table><tr><td>Method</td><td>FLOPs</td><td>#ofParams</td><td>Top-1 Acc.</td><td>Top-5 Acc.</td></tr><tr><td>Baseline</td><td>100%</td><td>100%</td><td>76.1%</td><td>92.9%</td></tr><tr><td>DDS-26 (43)</td><td>57.0%</td><td>61.2%</td><td>71.8%</td><td>91.9%</td></tr><tr><td>CP (38)</td><td>66.7%</td><td>1</td><td>72.3%</td><td>90.8%</td></tr><tr><td>ThiNet-50 (45)</td><td>44.2%</td><td>48.3%</td><td>71.0%</td><td>90.0%</td></tr><tr><td>RBP (101)</td><td>43.5%</td><td>48.0%</td><td>71.1%</td><td>90.0%</td></tr><tr><td>RRBP (101)</td><td>45.4%</td><td>1</td><td>73.0%</td><td>91.0%</td></tr><tr><td>SFP (36)</td><td>41.8%</td><td>=</td><td>74.6%</td><td>92.1%</td></tr><tr><td>Hinge (56)</td><td>46.6%</td><td>一</td><td>74.7%</td><td></td></tr><tr><td>GBN-50 (94)</td><td>44.9%</td><td>46.6%</td><td>75.2%</td><td>92.4%</td></tr><tr><td>GBN-60 (94)</td><td>59.5%</td><td>68.2%</td><td>76.2%</td><td>92.8%</td></tr><tr><td>Group-HS (2e-5) (91)</td><td>32.4%</td><td>=</td><td>75.2%</td><td>92.5%</td></tr><tr><td>Group-HS (1e-5) (91)</td><td>52.9%</td><td></td><td>76.4%</td><td>93.1%</td></tr><tr><td>ResRep (18)</td><td>45.5%</td><td></td><td>76.2%</td><td>92.9%</td></tr><tr><td>SCP (48)</td><td>45.7%</td><td></td><td>74.2%</td><td>92.0%</td></tr><tr><td>OTO</td><td>34.5%</td><td>35.5%</td><td>74.7%</td><td>92.1%</td></tr><tr><td>OTO*</td><td>34.5%</td><td>35.5%</td><td>75.1%</td><td>92.5%</td></tr></table>
+
+requires several trainings including fine-tuning the pruned model, because it partitions the parameters into non-ZIGs and relies on an empirical truncation mechanism to generate zero groups due to the weakness of proximal operator in deep learning applications (8). In contrast, OTO only trains and prunes the full model from scratch once and obtains better pruning results. The comparison between OTO and Hinge stand as evidence of the superiority of OTO due to ZIGs and HSPG. Furthermore, if with more training efforts, OTO reaches higher Top-1/5 accuracy marked as ∗ in Table 4 and becomes more competitive to stronger competitors, such as GBN (94), Group-HS (91) and ResRep (48).
+
+Representation of Deep Features of ImageNet. It is widely acknowledged that deep neural architectures could be treated as non-linear feature representation extractors. Therefore, we further study the feature representation extracted by OTO to demonstrate its generalizability to other visual applications besides image classification. Figure 4 shows the clustering results of ImageNet validation images using the deep feature extracted by both the baseline ResNet50 and the pruned ResNet50 by OTO. Specifically, we extract the deep features over the validation samples in ImageNet, i.e., the tensors fed into the fully connected layer, and project them onto a 2-dimensional space via PCA (47). For illustration, following the hierarchy of ImageNet (3), two sets of five classes are randomly selected3. We observe that the deep features of the pruned ResNet50 by OTO remain structured in the sense that distinct classes are well separated from each other. Over all 1000-class ImageNet validation images, OTO achieves $4 8 . 2 \%$ clustering accuracy compared to $4 2 . 5 \%$ of the baseline ResNet50 using $\mathbf { k }$ -means. Both observations indicate that with only $3 5 . 5 \%$ parameters and $3 4 . 5 \%$ FLOPs, the pruned ResNet50 is still able to extract highly discriminative deep features. We argue that during model compression, OTO not only achieves parameter and FLOPs reduction, but also preserves the ability of capturing perceptual properties (99). This is especially important in training and compressing models for many vision tasks, e.g., object detection (70; 71), frame interpolation (2; 17; 67) and video synthesis (84; 50). We leave the application of OTO to broader tasks to future work.
+
+# 4.2 Large-Scale Transformer
+
+We show the scalability of OTO by pruning the large-scale transformer Bert (82), evaluated on SQuAD, a question-answering benchmark (69). Bert mainly includes embedding layers, fully connected layers and multi-head attention layers. The fully connected layers and the multi-head attention layers are partitioned into ZIGs following Section 3.1. For fair comparisons, we follow the prior Bert compression works (14; 75) and do not prune the embedding layers.
+
+![](images/149863a9ad5e7f572f51efc22421bbc22dc4300af2734f2794e3c3e99cb4ad35.jpg)  
+Figure 4: Clustering results of ImageNet validation images using deep features extracted by full ResNet50 (left of a and b) and pruned ResetNet50 by OTO (right of a and b). The points are visualized by projecting deep features onto a two-dimensional space via PCA.
+
+To the best of our knowledge, OTO is the first work that compresses Bert by exploring group sparsity on individual layers and achieves significant parameter reduction and inference speedup4. In contrast, the existing works (29; 75; 30) prune individual parameters instead, i.e., the generated sparsity is not structured. Hence, the computed models typically do not have inference speedup (75), unless are executed by specialized hardware and sparse computing library (31; 10). As
+
+Table 5: Pruning Bert on SQuAD   
+
+<table><tr><td>Method</td><td>#ofParams</td><td>Exact</td><td>F1-score</td><td>SpeedUp</td></tr><tr><td>Baseline</td><td>100%</td><td>81.0%</td><td>88.3%</td><td>1×</td></tr><tr><td>MaP (75)</td><td>10.0%</td><td>67.7%</td><td>78.5%</td><td>1×*</td></tr><tr><td>MvP (75)</td><td>10.0%</td><td>71.9%</td><td>81.7%</td><td>1×*</td></tr><tr><td>ProxSSI (14)</td><td>83.4%t</td><td>72.3%</td><td>82.0%</td><td>1×</td></tr><tr><td>OTO</td><td>91.0%</td><td>75.0%</td><td>84.1%</td><td>1.1×</td></tr><tr><td>OTO</td><td>76.2%</td><td>72.3%</td><td>82.1%</td><td>1.2×</td></tr><tr><td>OTO</td><td>66.7%</td><td>71.9%</td><td>82.0%</td><td>1.3×</td></tr><tr><td>OTO</td><td>53.3%</td><td>71.4%</td><td>81.5%</td><td>1.5×</td></tr><tr><td>OTO</td><td>40.0%</td><td>70.9%</td><td>81.1%</td><td>1.8×</td></tr></table>
+
+\* Based on the statement in the official git repository of (75). † Approximate value based on the group sparsity reported in (14).
+
+shown in Table 5, under different group sparsity upper bound constraints, OTO reduces $9 \%$ t o $60 \%$ parameters and achieves up to $1 . 8 \times$ inference speedup based on the average model execution time 5. In comparison, despite that the pruned model contains $1 0 \%$ parameters, MaP and MvP (75) do not have any inference speedup. On the other hand, the structured sparsity on Bert is studied in (14) (referred to as ProxSSI), where an adaptive proximal method is proposed to yield group-sparse solution. Nonetheless, ProxSSI optimizes over non-ZIGs and relies on proximal operator to identify group sparsity. Therefore, the groups even parameterized with zeros have to be retained in the model rather than pruned. As a consequence, ProxSSI is not competitive to OTO on parameter reduction, and there is no reported inference speedup. Note that all the pruning methods achieve comparable exact match rate and F1-score.
+
+# 5 Conclusion And Future Work
+
+We propose OTO, a one-shot deep neural networks (DNNs) training and pruning framework, that compresses full DNNs into slimmer architectures with competitive performances and significant FLOPs and parameter reduction without fine-tuning. OTO contains two fundamentals: (i) partitions the trainable parameters of DNNs into zero-invariant groups (ZIGs), thereby pruning zero groups does not affect the model output, and (ii) trains by a novel optimizer, Half-Space Stochastic Projected Gradient (HSPG), which outperforms proximal methods on group sparsity exploration and maintains comparable convergence. We numerically demonstrate OTO on benchmark experiments, i.e., VGG16 for CIFAR10, ResNet50 for CIFAR10/ImageNet and Bert for SQuAD, and achieve state-of-theart pruning results. We leave automatically generating ZIGs for arbitrary DNNs, incorporating quantization and applying OTO to other tasks to future work.
+
+# References
+
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+        "text": "Only Train Once: A One-Shot Neural Network Training And Pruning Framework ",
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+        "text": "Structured pruning is a commonly used technique in deploying deep neural networks (DNNs) onto resource-constrained devices. However, the existing pruning methods are usually heuristic, task-specified, and require an extra fine-tuning procedure. To overcome these limitations, we propose a framework that compresses DNNs into slimmer architectures with competitive performances and significant FLOPs reductions by Only-Train-Once (OTO). OTO contains two keys: $( i )$ we partition the parameters of DNNs into zero-invariant groups, enabling us to prune zero groups without affecting the output; and $( i i )$ to promote zero groups, we then formulate a structured-sparsity optimization problem and propose a novel optimization algorithm, Half-Space Stochastic Projected Gradient (HSPG), to solve it, which outperforms the standard proximal methods on group sparsity exploration and maintains comparable convergence. To demonstrate the effectiveness of OTO, we train and compress full models simultaneously from scratch without finetuning for inference speedup and parameter reduction, and achieve state-of-the-art results on VGG16 for CIFAR10, ResNet50 for CIFAR10 and Bert for SQuAD and competitive result on ResNet50 for ImageNet. The source code is available at https://github.com/tianyic/only_train_once. ",
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+        "text": "1 Introduction ",
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+        "text": "Deep neural networks (DNNs) have been shown to be effective in various real applications (51; 28). It is widely acknowledged that large-scale DNN models not only learn faster but also outperform their slimmer counterparts. However, such heavy models pose a great challenge to the deployment stage due to their resource-consuming nature. In addressing this issue, many model compression techniques (5; 11) are proposed in the past decade that aim at compressing those large and complex models into slimmer and simpler ones while suffering negligible loss in performance. ",
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+            "Figure 1: Overview of OTO. Without loss of generality, we illustrate OTO on a model with only vanilla convolutional layers, and for simplicity we only show Layeri with $m$ 3D filters and their biases. The key to its success is twofold: $( i )$ identify and partition the parameters of the model into zero-invariant groups (ZIGs); and $( i i )$ solve the structured-sparsity regularization problem using HSPG. Finally, we obtain the compressed model by directly pruning the zero groups, i.e., $\\mathrm { Z I G } _ { m }$ . "
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+        "type": "text",
+        "text": "Pruning methods as one of the main categories of model compression, focus on identifying and pruning redundant structures via various mechanisms to achieve a slimmer architecture, and thus improve the interpretability of a DNN model (26; 11; 65). For example, (32; 33) adopt fine-grained pruning via $\\ell _ { 1 }$ or $\\ell _ { 2 }$ regularization, which prune the small-weight connections based on some hard threshold. (36; 57; 60) measure the importance of filters to accelerate the networks by removing insignificant feature maps. (37; 7) utilize reinforcement learning agent to predict compression action. ",
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+    {
+        "type": "text",
+        "text": "Nevertheless, many of the existing pruning methods $( i )$ often rely on criteria based on heuristics or empirical cues, e.g., magnitude of a connection weight and importance score of a filter, to identify redundant parameters, which may cause divergence during optimization; $( i i )$ thus require complex multi-stage pipelines that involve either a retraining or fine-tuning procedure to regain the accuracy during constructing a slimmer model, which is time-consuming; and $( i i i )$ are specific to certain architectures or applications, and are consequently less applicable to various downstream scenarios. Recently, there have been a few efforts (14; 58; 8) to directly train the network with sparsity inducing regularizers, which provide generality and convergence guarantee. However, these approaches focus on either merely the individual sparsity of the parameters or the group sparsity of the filters, and thus cannot directly remove those zero components (still require subsequent fine-tuning) since the zeros are entangled with other commonly used components, e.g., bias, batch normalization or skip connection. Furthermore, the optimization algorithms used in (14; 58) lack sufficient capability to explore (group) sparsity in DNNs effectively and require a post-processing step to yield exact zeros. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we overcome the above limitations of existing pruning methods by proposing a one-shot neural network pruning framework, with which we are able to train a full heavy model from scratch only once, and obtain a slim architecture without fine-tuning while maintain high performance. As shown in Figure 1, the key to its success is twofold: $( i )$ we identify and partition the parameters of DNNs into zero-invariant groups (ZIGs), enabling us to prune redundant structures according to zero groups without affecting the output of the network; and $( i i )$ to promote zero groups, we formulate the pruning task as a structured-sparsity optimization problem and propose a novel optimization method, Half-Space Stochastic Projected Gradient (HSPG), to solve it, which outperforms the standard proximal methods on sparsity exploration and maintains comparable convergence. We highlight that both zero-invariant group partition and the novel optimization algorithm in promoting zero group lead to achieve one-shot neural network training and pruning regardless of its architecture. ",
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+    {
+        "type": "text",
+        "text": "Our main contributions are summarized as follows. ",
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+    {
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+        "text": "• One-Shot Training and Pruning. We propose OTO, a training and pruning framework that compresses a full neural network with competitive performance by Only-Train-Once, thereby one-shot. OTO dramatically simplifies the complex multi-stage training pipelines of the existing pruning approaches, fits various architectures and applications, and hence is generic and efficient. ",
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+    {
+        "type": "text",
+        "text": "• Zero-Invariant Group. We define zero-invariant groups for neural networks. If a network is partitioned into ZIGs, it allows us to prune the zero groups without affecting the output, which results in one-shot pruning. Such property is applicable to various popular structures from plain fully connected layers to sophisticated ones such as residual blocks and multi-head attention. ",
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+    {
+        "type": "text",
+        "text": "• Novel Structured-Sparsity Optimization Algorithm. We propose Half-Space Stochastic Projected Gradient (HSPG), a method that solves structured-sparsity inducing regularization problem. We show and analyze the superiority of HSPG in promoting zero groups of networks than the standard proximal methods and the competitive objective convergence in practice. The fact that ZIG and HSPG are designed agnostic to networks makes OTO generic to various applications. ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "• Experimental Results. We train and compress full models simultaneously from scratch without fine-tuning for inference speedup and parameter reduction, and achieve state-of-the-art results on compression benchmark VGG for CIFAR10, ResNet50 for CIFAR10/ImageNet, Bert for SQuAD. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Structured pruning focuses on identifying and pruning the redundant structures in a full model to achieve slimmer architectures for efficient model inference and storage (26; 32), where there have been numerous efforts dedicated. For CNN compression, the general procedure can be largely summarized as: (i) train a full model; (ii) identify and prune the redundant structures to build a slimmer model based on various criteria, including (structured) sparsity (58; 85; 14; 56; 102; 27; 102; 62; 91), Bayesian pruning (101; 65; 59; 81), ranking importance (54; 60; 41; 36; 57; 100), reinforcement learning (37; 7), adversarial robustness (76), scientific control (79), lottery ticket (23; 24; 72), joint quantization learning (80; 90), etc.; (iii) retrain or iteratively fine-tune the slimmer model to regain the accuracy regression during pruning. These methods cannot avoid the extra and usually timeconsuming fine-tuning step because the identified redundant structures, even parametrized with zeros, actually contribute to the model output, thereby additional fine-tuning step is an absolute necessity. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For pruning Bert (82), knowledge distillation (40) and LayerDropout (21) shorten Bert by reducing the number of layers directly. Other methods (29; 75; 30) build slimmer Berts in the manner of individual sparsity, but require specially designed data structure for storage and computing library to take advantage of sparse data (31; 10), and typically cannot achieve inference speedup against the highly optimized library (16) for dense model due to the discontiguous memory allocation (9). ",
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+    {
+        "type": "text",
+        "text": "The structured sparsity for weight pruning is the most relevant to the algorithm described in this paper. The existing structure learning works (58; 85; 14; 56; 102) have the respective disadvantages: (i) multiple trainings during the whole procedure since their group partition cannot isolate the impact of pruned structures to the model output; and (ii) heuristic post-processing to generate zero groups as the standard proximal methods (19; 87; 88; 12) and ADMM (100; 58; 4) defective on the sparsity exploration for deep learning (8), which may deteriorate the performance of the model significantly. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Avoiding fine-tuning step during the whole pruning procedure is receiving more and more attentions because of its efficiency. In particular, SNIP (52) and GraSP (83) identify redundancy via salience scores at the initialization stage to construct pruned structures, then train the pruned models by the standard optimizers. SCP (48) isolates the impact of batch normalization, while lacks the consideration of more general DNN architectures. ",
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+    {
+        "type": "text",
+        "text": "3 OTO ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In essence, OTO frames the network training and pruning as a structure learning problem. Given a full model $\\mathcal { M }$ , OTO trains and compresses it simultaneously from scratch without fine-tuning, and achieves significant reduction in both FLOPs and parameters. Particularly, as stated in Algorithm 1, the trainable parameters of $\\mathcal { M }$ are firstly partitioned into a ZIG set $\\mathcal { G }$ (Section 3.1). We then construct and solve a structured-sparsity inducing optimization problem (Section 3.2) by proposed stochastic optimizer (HSPG) to seek a highly group-sparse solution $\\pmb { x } _ { \\mathrm { H S P G } } ^ { * }$ (Section 3.3). Lastly, we obtain a compressed model $\\mathcal { M } ^ { \\ast }$ by directly pruning these zero groups (Section 3.4). ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Outline of OTO. ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "1: Input: Full model $\\mathcal { M }$ (no need to be pretrained).   \n2: Construct $\\mathfrak { s }$ : Partition the trainable parameters of $\\mathcal { M }$ into a ZIG set $\\mathcal { G }$ .   \n3: Train: Train the model $\\mathcal { M }$ using HSPG (Algorithm. 2) to obtain a group-sparse solution $\\pmb { x } _ { \\mathrm { H S P G } } ^ { * }$   \n4: Prune: Construct a slimmer model architecture M∗ by directly pruning zero groups of x∗HSPG.   \n5: Output: Compressed model $\\mathcal { M } ^ { * }$ . ",
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+        "type": "text",
+        "text": "3.1 Zero-Invariant Group ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The root cause of the existing methods having multi-stage training pipeline is that despite the pruned structure (e.g., 3D filter) being zeros, its associated structures (e.g., non-zero bias) still contribute to its corresponding output to the next layer (e.g., feature map). As a result, the model accuracy regresses, hence an extra step of fine-tuning is necessary. OTO avoids the necessity by partitioning the parameters of DNNs into a set of so-called zero-invariant groups (ZIGs) $\\mathcal { G }$ defined as follows. ",
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+        "text": "Definition 1 (Zero-Invariant Groups (ZIGs)). For a layer-wise DNN, we partition its entire trainable parameters into disjoint groups $\\mathcal { G } = \\{ g \\}$ . Then we call $\\mathcal { G }$ zero-invariant groups (ZIGs) if each group $g \\in { \\mathcal { G } }$ is zero-invariant in the sense that all of the parameters in g being zeros results in its corresponding output to the next layer to be zeros as well. ",
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+        "type": "text",
+        "text": "In effect, if and only if a DNN model is partitioned into a ZIG set $\\mathcal { G }$ and one or more of its element $g$ are parameterized by zeros, the entire corresponding structures contribute none to the model outputs and hence can be pruned directly. Such partition is applicable to various structures of DNN models. Without loss of generality, we define and describe ZIG partition for three most popular structures: $( i )$ Conv-BN, (ii) Residual Block, and (iii) Fully Connected and Multi-Head Attention Layer. ",
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+        "img_path": "images/48438b786fc8e5517c936cd74cdfcff453877ce179a009c1570ac4ad34def872.jpg",
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+    {
+        "type": "text",
+        "text": "(a) Conv-BN. $m$ denotes the number of channels in $\\mathcal { O } ^ { l }$ (b) Residual block. $m$ denotes the number of output channels of the residual block. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "image",
+        "img_path": "images/a7c596082ec3463774fc5aac6315fb6292578b380ad98913bea369bca6c1dff5.jpg",
+        "image_caption": [
+            "Figure 2: Zero-invariant group partition for three popular structures. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(c) Fully connected layer (Left). Multi-head attention layer (Right). $m$ denotes the length of output vector. ",
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+    {
+        "type": "text",
+        "text": "ZIG of Conv-BN. Convolutional layer (Conv) followed by batch-normalization layer (BN) is extensively used in DNN models. Figure 2a shows the ZIG partition for Conv-BN. The 4D filter tensor $\\kappa ^ { l }$ is flattened into a filter matrix $\\hat { \\kappa } ^ { l }$ . During the forward pass, the input tensor $\\boldsymbol { \\mathcal { Z } ^ { l } }$ is transformed into the output tensor $\\mathcal { O } ^ { l }$ of Conv and then into the input tensor of the $( l ^ { \\bullet } + 1 ) ^ { t h }$ layer $\\pmb { \\mathcal { T } } ^ { l + 1 }$ by ",
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+    {
+        "type": "equation",
+        "img_path": "images/70ecdb70c7d0215b64202305a5d863c50a1169d4100e6cc619f5951d29caac8b.jpg",
+        "text": "$$\n\\mathcal { O } ^ { l }  \\mathcal { T } ^ { l } \\otimes \\hat { \\mathcal { K } } ^ { l } + b ^ { l } , \\ \\mathcal { Z } ^ { l + 1 }  \\frac { a ( \\mathcal { O } ^ { l } ) - \\mu ^ { l } } { \\sigma ^ { l } } \\odot \\gamma ^ { l } + \\beta ^ { l } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where denoted by $\\otimes$ the convolutional operation, $\\odot$ the element-wise multiplication and $a ( \\cdot )$ the activation function. BN is parameterized by mean $\\mu ^ { l }$ , standard deviation $\\sigma ^ { l }$ , weight $\\gamma ^ { l }$ and bias $\\beta ^ { l }$ respectively. The activation function needs to be zero-invariant, i.e., $a ( \\mathbf { 0 } ) = \\mathbf { 0 }$ , where most instances satisfy, e.g., ReLU (25), PReLU (34), GELU (39) and LeakyReLU (89). Hence, each row of the flattened filter matrix $\\hat { \\kappa } ^ { l }$ and its bias $b ^ { l }$ belong to one ZIG because they being zeros results in their corresponding channel of $\\mathcal { O } ^ { l }$ (i.e., feature map) to be zeros as well. Subsequently, $\\gamma ^ { l }$ and $\\beta ^ { l }$ of this corresponding channel in BN are also included into this ZIG to avoid the value shift (zero to non-zero) during normalization. Note that grouping these four sets of parameters channel-wisely makes Conv-BN zero-invariant regardless of the value of $\\mu ^ { l }$ and $\\sigma ^ { l }$ , and hence they are excluded from the ZIG. For illustration, each ZIG is highlighted in the same color (e.g., $g _ { 1 } ^ { l }$ in blue). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ZIG of Residual Block. The residual block adds another layer of challenge because its output tensor is the summation of the outputs of two Conv-BNs. Figure 2b shows the ZIG partition for the residual block. As illustrated, before propagated to Conv3, the outputs of Conv1-BN1 and Conv2-BN2 are summarized and hence share the same dimension. As such, to make residual block zero-invariant, we group the four sets of parameters channel-wisely of both Conv1-BN1 and Conv2-BN2 into ZIGs, i.e., each row of ${ \\hat { \\kappa } } ^ { 1 }$ , $b ^ { \\hat { 1 } }$ , $\\gamma ^ { 1 }$ , $\\beta ^ { 1 }$ of Conv1-BN1 and each row of ${ \\hat { \\kappa } } ^ { 2 }$ , $b ^ { 2 }$ , $\\gamma ^ { 2 }$ , $\\beta ^ { 2 }$ of Conv2-BN2. In Appendix A.1, we describe the zero-invariant group partition of ResNet50 in greater detail. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "ZIG of Fully Connected and Multi-Head Attention Layer. Figure 2c shows the ZIG partition for fully connected and multi-head attention layer. Particularly, we partition each row of weight matrix and its associated bias into a ZIG, and therefore any input element is turned to zero if that ZIG is parameterized with zeros, making the fully connected layer zero-invariant. Multi-head attention layer is the key building block of the transformer architectures (82). Its trainable parameters contain a weight matrix and bias vector, consisting of the sub-matrix and sub-vector of each head (we use two heads as an example). We form ZIG by grouping each row of every sub-matrix and sub-vector, i.e., each row of $w _ { h _ { 1 } }$ , $\\boldsymbol { b } _ { h _ { 1 } }$ , $w _ { h _ { 2 } }$ and $b _ { h _ { 2 } }$ of $h _ { 1 }$ and $h _ { 2 }$ , respectively. ",
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+    {
+        "type": "text",
+        "text": "Automatic ZIG Partition. Based on the above illustrating examples, we provide prescribed ZIG partition for the tested DNNs in Section 4. Furthermore, given an arbitrary DNN architecture, the procedure of partitioning variables into ZIGs could be automatically proceeded, wherein the key would be identifying the connections among various layers, then performing corresponding group partition. We will leave the automatic ZIG partition for arbitrary DNNs as future work. ",
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+    {
+        "type": "text",
+        "text": "3.2 Structured-Sparsity Regularization ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We now formulate a structured-sparsity regularization problem over the ZIG set $\\mathcal { G }$ for the trainable parameters of the full model $\\mathcal { M }$ as follows ",
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+    {
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+        "img_path": "images/52da9c495ca9fc9c0fd7b5b08119d7325aaf0bd0efe65b4de9588963507a0c09.jpg",
+        "text": "$$\n\\operatorname* { m i n i m i z e } _ { \\pmb { x } \\in \\mathbb { R } ^ { n } } \\psi ( \\pmb { x } ) : = f ( \\pmb { x } ) + \\lambda r ( \\pmb { x } ) , \\ r ( \\pmb { x } ) : = \\sum _ { g \\in \\mathcal { G } } \\| [ \\pmb { x } ] _ { g } \\| ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\lambda > 0$ is a weighting coefficient, $f ( { \\pmb x } )$ is a task-specific loss function, and $r ( { \\pmb x } )$ is an augmented structured-sparsity inducing regularization term encoding the topological structure of $\\mathcal { M }$ over $\\mathcal { G }$ . A larger $\\lambda$ typically results in a higher group sparsity while sacrifices more on the bias of model estimation. We aim at computing a local optimum to achieve both low loss and high group sparsity. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To induce group sparsity onto the solution of (2), there exist several candidates for $r ( { \\pmb x } )$ , including mixed $\\ell _ { 1 } / \\ell _ { p }$ norm $( p > 1 )$ (1; 20) and group Minmax Concave Penalty (MCP) (96). Among these candidates, the mixed $\\ell _ { 1 } / \\ell _ { 2 }$ norm as defined in (2) is arguably the most popular choice in classical machine learning applications (1; 92), where $\\lVert \\cdot \\rVert$ is the $\\ell _ { 2 }$ -norm, and each component $g \\in { \\mathcal { G } }$ indexes a group of variables. In this paper, we will demonstrate the effectiveness of OTO by selecting $r ( { \\pmb x } )$ as the mixed $\\ell _ { 1 } / \\ell _ { 2 }$ norm. We highlight OTO is applicable for other group sparsity regularizers as well. ",
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+    {
+        "type": "text",
+        "text": "3.3 Half-Space Stochastic Projected Gradient (HSPG) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To solve the non-smooth regularization problem as (2) in deep learning applications, the standard proximal method and the ADMM lack capability to effectively identify group sparsity; see the discussions later in this Section. Therefore, we propose a novel stochastic optimization algorithm so-called Half-Space Stochastic Projected Gradient (HSPG) to enhance the group sparsity exploration more effectively than the classical methods while maintain a similar convergence property. ",
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+    {
+        "type": "text",
+        "text": "Outline. We state the outline of HSPG in Algorithm 2. It contains two stages: Initialization Stage and Group-Sparsity Stage. The first Initialization Stage employs Stochastic Gradient Descent (SGD) step to search for a good but usually non-sparse solution estimate. Then the second stage proceeds Half-Space step started with the non-sparse iterate to effectively exploit the group sparsity within a sequence of reduced spaces and converges to the group-sparse solutions. Half-Space step performs SGD update on free non-zero variables along with a novel projection operator so-called Half-Space Projection, which significantly outperforms the standard proximal operators on sparsity exploration. ",
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+    {
+        "type": "text",
+        "text": "Initialization Stage. The Initialization Stage performs the vanilla SGD to find a good initial point for the subsequent Group-Sparsity Stage. At $k ^ { \\mathit { \\hat { t } h } }$ iteration, a stochastic gradient of $f$ , e.g., based on a mini-batch, is generated denoted as $\\boldsymbol { \\nabla } \\tilde { f }$ . Since the group sparsity inducing regularizer $r ( { \\pmb x } )$ in the form as (2) is non-smooth, we select a subgradient $\\zeta ( \\pmb { x } _ { k } )$ from its subdifferential $\\partial r ( \\pmb { x } _ { k } )$ to form a stochastic subgradient of $\\psi ( \\pmb { x } _ { k } )$ as $\\nu ( { \\pmb x } _ { k } ) : = \\nabla \\tilde { f } ( { \\pmb x } _ { k } ) + \\lambda \\zeta ( { \\pmb x } _ { k } )$ . We then compute the next iterate as $\\pmb { x } _ { k + 1 } : = \\pmb { x } _ { k } - \\alpha _ { k } \\nu ( \\pmb { x } _ { k } )$ by subgradient descent update. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/67e7cbd3ef90c0abd50772be2db874a17a3bcb8ba70ef573ad987a889fefe45a.jpg",
+        "image_caption": [
+            "Figure 3: Illustration of Half-Space Step with projection in (6), where $\\mathcal { G } = \\{ \\{ 1 , 2 \\} \\}$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Group-Sparsity Stage. The Group-Sparsity Stage is designed to effectively determine the groups of zero variables and capitalize convergence characteristic, which is in sharp contrast to other heuristic aggressive weight pruning methods that typically lack theoretical guarantees (55; 60). The intuition of Half-Space Step is to project $[ \\boldsymbol { x } _ { k } ] _ { g }$ to zero only if $- [ { \\pmb x } _ { k } ] _ { g }$ serves as a descent step to $\\psi ( \\pmb { x } _ { k } )$ , i.e., $- [ { \\pmb x } _ { k } ] _ { g } ^ { \\top } [ \\nabla \\psi ( { \\pmb x } _ { k } ) ) ] _ { g } < 0$ , hence updating $[ { \\pmb x } _ { k + 1 } ] _ { g }  [ { \\pmb x } _ { k } ] _ { g } - [ { \\pmb x } _ { k } ] _ { g } = 0$ still results in some progress to the optimality. In particular, we first define the following index sets for any $\\pmb { x } \\in \\mathbb { R } ^ { n }$ : ",
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+    {
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+        "img_path": "images/a16fe352c83966143499e83e35c4b11951ea0c3846fca57a73434a7f35ceef68.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { Z } ^ { 0 } ( \\pmb { x } ) : = \\{ g : g \\in \\mathcal { G } , [ \\pmb { x } ] _ { g } = 0 \\} \\mathrm { a n d } \\mathcal { Z } ^ { \\neq 0 } ( \\pmb { x } ) : = \\{ g : g \\in \\mathcal { G } , [ \\pmb { x } ] _ { g } \\neq 0 \\} , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { T } ^ { 0 } ( { \\pmb x } )$ represents the indices of groups of zero variables at $_ { \\textbf { \\em x } }$ , and $\\scriptstyle { \\mathcal { T } } ^ { \\neq 0 } ( { \\pmb x } )$ indexes the groups of nonzero variables at $_ { \\textbf { \\em x } }$ . To proceed, we further define an artificial set that $_ { \\textbf { \\em x } }$ lies in: ",
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+        "type": "equation",
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+        "text": "$$\n\\begin{array} { r } { S ( { \\boldsymbol x } ) : = \\{ \\mathbf 0 \\} \\bigcup \\big \\{ z \\in \\mathbb R ^ { n } : [ z ] _ { g } = \\mathbf 0 \\mathrm { ~ i f ~ } g \\in \\mathcal { X } ^ { 0 } ( { \\boldsymbol x } ) , \\mathrm { a n d ~ } [ z ] _ { g } ^ { \\top } [ { \\boldsymbol x } ] _ { g } \\geq \\epsilon \\| [ { \\boldsymbol x } ] _ { g } \\| ^ { 2 } \\mathrm { ~ i f ~ } g \\in \\mathcal { X } ^ { \\neq 0 } ( { \\boldsymbol x } ) \\big \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            534,
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+            556
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "which consists of half-spaces and the origin. Here the parameter $\\epsilon \\geq 0$ controls how aggressively we promote group sparsity, and is typically fixed as zero in practice. Hence, $\\pmb { x } \\in S _ { k } : = \\pmb { S } ( \\pmb { x } _ { k } )$ only if: (i) $[ { \\pmb x } ] _ { g }$ lies in the upper half-space for all $g \\in { \\mathcal { T } } ^ { \\neq 0 } ( { \\pmb x } _ { k } )$ for some prescribed $\\epsilon \\in [ 0 , 1 )$ as shown in Figure 3a; and (ii) $[ { \\pmb x } ] _ { g }$ equals to zero for all $g \\in \\mathcal { T } ^ { 0 } ( { \\pmb x } _ { k } )$ . Intuitively, $\\scriptstyle { S _ { k } }$ establishes the region where important structures inhabit, thereby redundant structures vanish if falling outside. ",
+        "bbox": [
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+            563,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Ideally, the Initialization Stage has produced reasonably well but typically non-sparse iterate $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }$ nearby a group-sparse solution $\\pmb { x } ^ { * }$ of problem (2), , i.e., the optimal distance $\\| \\pmb { x } _ { k } - \\pmb { x } ^ { * } \\|$ is sufficiently small. As seen in Appendix B, it further indicates that the group-sparse optimal solution $\\pmb { x } ^ { * }$ inhabits $S _ { k }$ , and $S _ { k }$ has already covered the group-support of $\\pmb { x } ^ { * }$ , i.e., $\\mathcal { T } ^ { \\neq 0 } ( { \\pmb x } ^ { * } ) \\subseteq \\mathbb { Z } ^ { \\neq 0 } ( { \\pmb x } _ { k } ^ { * } )$ . Our goal now becomes minimizing $\\psi ( { \\pmb x } )$ over $\\boldsymbol { S } _ { k }$ to identify the remaining zero groups, i.e., $\\mathcal { T } ^ { 0 } ( \\dot { \\pmb { x } ^ { * } } ) / \\mathcal { Z } ^ { 0 } ( \\pmb { x } _ { k } )$ , which is formulated as the following problem: ",
+        "bbox": [
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+            642,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 Outline of HSPG for solving (2). ",
+        "text_level": 1,
+        "bbox": [
+            436,
+            651,
+            735,
+            666
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: Input: $\\pmb { x } _ { 0 } \\in \\mathbb { R } ^ { n }$ , $\\alpha _ { 0 } > 0 , \\epsilon \\in [ 0 , 1 )$ , and $N \\in \\mathbb { Z } ^ { + }$ .   \n2: Output: a group-sparse solution $\\pmb { x } _ { \\mathrm { H S P G } } ^ { * }$ from $\\{ \\boldsymbol { x } _ { k } \\}$ .   \n3: for $k = 0 , 1 , 2 , \\ldots$ . do   \n4: Compute a stochastic subgradient $\\nu ( \\pmb { x } _ { k } )$ of $\\psi ( \\pmb { x } _ { k } )$ .   \n5: if $k < N$ then   \n6: Subgradient Descent Update:   \n7: Set ${ \\pmb x } _ { k + 1 }  { \\pmb x } _ { k } - \\alpha _ { k } { \\pmb \\nu } ( { \\pmb x } _ { k } )$ .   \n8: else   \n9: Half-Space Update:   \n10: Set a trial iterate $\\tilde { \\pmb { x } } _ { k + 1 }$ as   \n$\\begin{array} { r l } & { [ \\tilde { \\pmb { x } } _ { k + 1 } ] _ { \\mathcal { T } ^ { \\neq 0 } ( { \\pmb x } _ { k } ) }  [ { \\pmb x } _ { k } - \\alpha _ { k } \\nu ( { \\pmb x } _ { k } ) ] _ { \\mathcal { T } ^ { \\neq 0 } ( { \\pmb x } _ { k } ) } } \\\\ & { [ \\tilde { \\pmb { x } } _ { k + 1 } ] _ { \\mathcal { T } ^ { 0 } ( { \\pmb x } _ { k } ) }  \\mathbf { 0 } . } \\end{array}$   \n11: for each group $g$ in $\\mathcal { G }$ do   \n12: $[ \\pmb { x } _ { k + 1 } ] _ { g }  [ \\mathrm { P r o j } _ { S _ { k } } ^ { H S } ( \\tilde { \\pmb { x } } _ { k + 1 } ) ] _ { g } .$ .   \n13: Update $\\alpha _ { k + 1 }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cf2dcd166bd0484fdc57bf220c1eaec350d51b84b4e77946d8ede9afb2379c30.jpg",
+        "text": "$$\n{ \\underset { \\pmb { x } \\in S _ { k } } { \\mathrm { m i n i m i z e } } } \\psi ( \\pmb { x } ) = f ( \\pmb { x } ) + \\lambda r ( \\pmb { x } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            183,
+            854,
+            390,
+            877
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The next iterate $\\scriptstyle { \\pmb { x } } _ { k + 1 }$ is computed as an solution estimate of problem (5). ",
+        "bbox": [
+            174,
+            883,
+            421,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Particularly, in Algorithm 2, $[ { \\pmb x } _ { k + 1 } ] _ { { \\mathcal { T } } ^ { 0 } ( { \\pmb x } _ { k } ) } \\equiv { \\bf 0 }$ will not be updated, and only the entries in $\\scriptstyle { \\mathcal { T } } ^ { \\neq 0 } ( { \\pmb x } _ { k } )$ are free to move. Hence $\\psi ( { \\pmb x } )$ is smooth on $\\scriptstyle { S _ { k } }$ , and (5) is a reduced space optimization problem. A standard way to solve problem (5) would be the stochastic gradient descent equipped with Euclidean projection (68). However, such a projected method rarely produces zero (group) variables, as the dense Euclidean projected point $\\hat { \\pmb x } _ { E } \\neq { \\bf 0 }$ illustrated in Figure 3a. To address, we introduce a novel half-space projection operator to effectively project an entire group of variables to zeros. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As line 4 and 9-12 in Algorithm 2, we first approximate the (sub)gradient of $\\psi$ on the free variables by $[ \\nu ( \\pmb { x } _ { k } ) ] _ { \\pmb { \\mathbb { T } } ^ { \\neq 0 } ( \\pmb { x } _ { k } ) }$ , then employ gradient descent over $\\scriptstyle { \\mathcal { T } } ^ { \\neq 0 } ( { \\pmb x } _ { k } )$ to compute a trial point $\\widetilde { \\pmb { x } } _ { k + 1 }$ which is passed into a fresh half-space projection operator $\\mathrm { P r o j } _ { S _ { k } } ^ { H S } ( \\cdot )$ defined as ",
+        "bbox": [
+            174,
+            181,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7d6dbb387b267f57fa0d2e4d2f883f11ab182961f9ceafa83c8a50d769a78cfb.jpg",
+        "text": "$$\n\\begin{array} { r } { \\left[ \\mathrm { P r o j } _ { { \\mathcal S } _ { k } } ^ { H S } ( z ) \\right] _ { g } : = \\left\\{ \\begin{array} { l l } { 0 } & { \\mathrm { i f } [ z ] _ { g } ^ { \\top } [ \\pmb { x } _ { k } ] _ { g } < \\epsilon \\left\\| [ \\pmb { x } _ { k } ] _ { g } \\right\\| ^ { 2 } , } \\\\ { [ z ] _ { g } } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            234,
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+            270
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The above projector of form (6) is not the standard one in Euclidean sense2, and it has two advantages: $( i )$ the actual search direction $d _ { k } : = ( \\mathrm { P r o j } _ { S _ { k } } ^ { H S } ( \\tilde { \\pmb { x } } _ { k + 1 } ) - \\pmb { x } _ { k } ) / \\alpha _ { k }$ performs as a descent direction to $\\psi ( \\pmb { x } _ { k } )$ , i.e., $[ \\mathbf { \\mathop { d } } _ { k } ] _ { g } ^ { \\top } [ \\nu ( \\mathbf { \\mathop { x } } _ { k } ) ) ] _ { g } < 0$ as $\\theta < 9 0 ^ { \\circ }$ in Figure 3a, hence the progress to the optimum is made via the sufficient decrease property drawn as Lemma 1 in Appendix B; then $( i i )$ it effectively projects entire groups of variables to zero if the inner product of corresponding entries is sufficiently small. In contrast, the Euclidean projection operator is far away effective to promote group sparsity. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Superiority of HSPG on Group Sparsity Identification. We now intuitively illustrate the strength of HSPG on group sparsity exploration. In fact, the half-space projection (6) is a more effective sparsity promotion mechanism compared to the standard proximal methods. Particularly, it benefits from a much larger projection region to map a reference point $\\hat { \\pmb { x } } _ { k + 1 } : = \\pmb { x } _ { k } - \\alpha _ { k } \\nabla \\tilde { f } ( \\pmb { x } _ { k } )$ or its variants to zero. As the 2D case described in Figure 3b, the projection regions of the state-of-the-art Prox-SG (19), Prox-SVRG (88), Prox-Spider (97) and SAGA (12) for (2) are $\\ell _ { 2 }$ -balls with radius as $\\alpha _ { k } \\lambda$ . In deep learning applications, the step size $\\alpha _ { k }$ is usually selected around $1 0 ^ { - 3 }$ to $1 0 ^ { - 4 }$ or even smaller for convergence. Together with the common setting of $\\lambda \\ll 1$ , their projection regions would vanish rapidly, resulting in the difficulties to produce group sparsity. As a sharp contrast, even though $\\alpha _ { k } \\lambda$ is near zero, the projection region of HSPG $\\{ \\pmb { x } : \\pmb { x } _ { k } ^ { \\top } \\pmb { x } < ( \\dot { \\alpha _ { k } } \\lambda + \\epsilon \\| \\pmb { x } _ { k } \\| ) \\| \\pmb { x } _ { k } \\| \\}$ (seen in Appendix B) is still an open half-space which contains those $\\ell _ { 2 }$ balls as well as RDA (87)’s if $\\epsilon$ is large enough. Conversely, vanilla ADMM alone lacks the mechanism to project a group of variables to zero, unless equips with extra post-processing step (100; 58). In Appendix B, we further reveal that HSPG still maintains the convergence to the optimality as drawn in Theorem 1. Moreover, we numerically demonstrate the superiority of HSPG in the sense of optimization in Appendix C. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.4 Pruning Without Fine-Tuning ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The group-sparse solution $\\pmb { x } _ { \\mathrm { H S P G } } ^ { * }$ over ZIGs to the full model $\\mathcal { M }$ is leveraged to construct the slimmer model $\\mathcal { M } ^ { * }$ . Particularly, we prune the redundant structures identified as zero groups $\\mathcal { T } ^ { 0 }$ and retain non-zero groups $\\mathcal { T } ^ { \\neq 0 }$ in $\\pmb { x } _ { \\mathrm { H S P G } } ^ { * }$ . Because the parameters of full model are partitioned into ZIGs, the pruned structures contribute none to the model output. Therefore, given the same input, the slimmer model $\\mathcal { M } ^ { * }$ computes the identical output as the full model $\\mathcal { M }$ parameterized with $\\pmb { x } _ { \\mathrm { H S P G } } ^ { * }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 Experiment ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we numerically demonstrate the effectiveness of OTO by one-shot training and pruning without fine-tuning on several benchmark compression tasks for CNNs, i.e., VGG16 (77) for CIFAR10 (49) and ResNet50 (35) for CIFAR10 (49) and ImagetNet (ILSVRC2012) (15). We also verify the scalibility of OTO onto Bert (82) evaluated on SQuAD (69). All datasets are free to academic usage and do not contain personally identifiable information or offensive content. CIFAR10 is under the MIT license, consisting of 50,000 training and 10,000 test images from 10 classes. ImagetNet is a large-scale dataset without license and contains about 1.2 million and 50,000 images in training and validation sets from 1,000 classes. SQuAD is under the CC BY-SA 4.0 license with about 100,000 question/answer pairs splitted into train/dev/test sets as $( 8 0 / 1 0 / 1 0 \\%$ ). We conduct all experiments on a Nvidia RTX8000 GPU and provide implementation details in Appendix A. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/76f27be9b6d75452aca25091eb2d3e3468678c1803783f90b4ec6668d4a70f9d.jpg",
+        "table_caption": [
+            "Table 1: VGG16 and VGG16-BN for CIFAR10. Convolutional layers are in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>BN</td><td>Architecture</td><td>FLOPs</td><td>#of Params</td><td>Top-1 Acc.</td></tr><tr><td>Baseline</td><td>X</td><td>64-64-128-128-256-256-256-512-512-512-512-512-512-512-512</td><td>100%</td><td>100%</td><td>91.6%</td></tr><tr><td>SBP (65)</td><td>X</td><td>47-50-91-115-227-160-50-72-51-12-34-39-20-20-272</td><td>31.1%</td><td>5.9%</td><td>91.0%</td></tr><tr><td>BC (59)</td><td></td><td>51-62-125-128-228-129-38-13-9-6-5-6-6-6-20</td><td>38.5%</td><td>5.4%</td><td>91.0%</td></tr><tr><td>RBC (101)</td><td></td><td>43-62-120-120-182-113-40-12-20-11-6-9-10-10-22</td><td>32.3%</td><td>3.9%</td><td>90.5%</td></tr><tr><td>RBP (101)</td><td></td><td>50-63-123-108-104-57-23-14-9-8-6-7-11-11-12</td><td>28.6%</td><td>2.6%</td><td>91.0%</td></tr><tr><td>OTO</td><td>xxxx</td><td>21-45-82-110-109-68-37-13-9-7-3-5-8-170-344</td><td>16.3%</td><td>2.5%</td><td>91.0%</td></tr><tr><td>Baseline</td><td></td><td>64-64-128-128-256-256-256-512-512-512-512-512-512-512-512</td><td>100%</td><td>100%</td><td>93.2%</td></tr><tr><td>EC (55)</td><td></td><td>32-64-128-128-256-256-256-256-256-256-256-256-256-512-512</td><td>65.8%</td><td>37.0%</td><td>93.1%</td></tr><tr><td>Hinge (56)</td><td></td><td></td><td>60.9%</td><td>20.0%</td><td>93.6%</td></tr><tr><td>SCP(48)</td><td></td><td></td><td>33.8%</td><td>7.0%</td><td>93.8%</td></tr><tr><td>OTO</td><td></td><td>22-56-93-123-182-125-95-45-27-21-10-13-19-244-392</td><td>26.8%</td><td>5.5%</td><td>93.3%</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Deep Convolutional Neural Network ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The results on CNN experiments are summarized in Table 1, 2 and 4. In particular, we compare OTO to its state-of-the-art counterparts by Top-1/5 accuracy, remaining FLOPs and parameters against the corresponding baseline (full model). We report the numbers of other methods based on the corresponding literature and leave as ‘-’ if not reported. The best pruning results are marked as bold. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "VGG16 for CIFAR10. We consider the standard VGG16 and the version with batch normalization layer after each convolutional layer, referred to as VGG16-BN. OTO partitions the parameters into ZIGs following Section 3.1, then trains and prunes the model via HSPG, and finally constructs the slimmer model without fine-tuning. For VGG16, as shown in Table 1, the pruned architecture of OTO indicates that OTO identifies similar redundancy of the intermediate and late convolutional layers compared to other methods, but significantly more of the early convolutional layers. As a result, OTO achieves $8 3 . 7 \\%$ $( 1 - 1 6 . 3 \\% )$ FLOPs reduction and $9 7 . 5 \\%$ $( \\dot { 1 } - 2 . 5 \\% )$ parameter reduction with the best Top-1 accuracy, which outperforms other state-of-the-arts significantly. For VGG16-BN, among all, OTO reduces FLOPs and parameters to the lowest $2 6 . 8 \\%$ and $5 . 5 \\%$ , respectively. EC (55) and Hinge (56) achieve the same level of Top-1 accuracy as OTO, but are substantially outperformed when it comes to FLOPs and parameter reduction. We further present the FLOPs reductions per layer of OTO in Table 7 of Appendix A.4. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ResNet50 for CIFAR10. Since OTO is able to automatically learn a slimmer model of high performance, we compare it with two state-of-the-art automatic neural network compression frameworks, i.e., AMC (37) and ANNC (90). AMC trains a reinforcement learning agent to predict compression action for each layer environment. ANNC jointly proceeds pruning and quantization within energy ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/9d93d89387779aeb4a502dd0d0ba3cf1ddd45f37487d3ec66555f04ce3070326.jpg",
+        "table_caption": [
+            "Table 2: ResNet50 for CIFAR10. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>FLOPs</td><td>#of Params</td><td>Top-1 Acc.</td></tr><tr><td>Baseline</td><td>100%</td><td>100%</td><td>93.5%</td></tr><tr><td>AMC (37)</td><td>1</td><td>60.0%</td><td>93.6%</td></tr><tr><td>ANNC (90)</td><td>=</td><td>50.0%</td><td>95.0%</td></tr><tr><td>PruneTrain (61)</td><td>30.0%</td><td>1</td><td>93.1%</td></tr><tr><td>N2NSkip (78)</td><td>1</td><td>10.0%</td><td>94.4%</td></tr><tr><td>OTO</td><td>12.8%</td><td>8.8%</td><td>94.4%</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "constraint. We conduct OTO on their shared experiment, i.e., ResNet50 on CIFAR10. ResNet50 includes both the standard convolutional layers and the layers with residual connections, which are partitioned into ZIGs following Section 3.1. We report the results in Table 2 along with other competitors from (61; 78). Based on the results, all methods achieve competitive validation accuracies, where most of them are even higher than the baseline reported in (37). OTO outperforms AMC, ANNC without quantization, PruneTrain and N2NSkip by using only $1 2 . 8 \\%$ FLOPs and $8 . 8 \\%$ parameters. Note that no FLOPs reduction is reported in (37) and (90). Finally, we highlight that OTO is flexible to incorporate quantization as the two techniques are complementary and will leave to future work. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Ablation Study on Switching Parameter $N$ . We provide ablation study regarding the impact the switch (parameterized as $N$ ) between the initialization stage and the groupsparsity stage in Algorithm 1. In theory, as shown in Theorem 1 of Ap",
+        "bbox": [
+            174,
+            760,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/8d5c4a9b5fdfe2d45aa9f2a48e9df2c7a85965c547194bf46f49840bcd9f4059.jpg",
+        "table_caption": [
+            "Table 3: OTO Under Different Switchings $( N = T , 2 T , 3 T )$ for VGG16, VGG16-BN and ResNet50 on CIFAR10 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Backend</td><td>FLOPs</td><td>#of Params</td><td>Top-1 Acc.</td></tr><tr><td>VGG16</td><td>17.0% ± 1.4%</td><td>2.6% ± 0.4%</td><td>90.9%± 0.3%</td></tr><tr><td>VGG16-BN</td><td>25.4%±1.1%</td><td>5.0%± 0.5%</td><td>93.3% ±0.2%</td></tr><tr><td>ResNet50</td><td>12.9% ± 1.5%</td><td>8.5% ± 1.0%</td><td>94.2% ± 0.2%</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "pendix B.4, the projection stage should start when the iterate falls nearby a group sparse local minimizer. In practice, we relax it to start the group sparsity stage once the iterate falling into some stationary status regarding the validation accuracy. As described in Appendix A.2, throughout all experiments, we periodically decay the learning rate per fixed number of epochs parameterized as $T$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "At the end of each $T$ epochs, we then proceed a statistical test similar to (98) but on the validation accuracy and find that the validation accuracy falls into stationarity near the late epochs of each period. Therefore, in our pruning experiments, we switch to the group-sparsity stage right after the first $T$ epochs. Table 3 describes the performance of OTO under varying switching parameters, from which we observe that OTO is not largely sensitive to the switching parameter if the group-sparsity stage starts after some stationary condition has been numerically satisfied. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "ResNet50 for ImageNet. We now evaluate OTO on ResNet50 for ImageNet. As shown in Table 4, OTO prunes $6 4 . 5 \\% ( 1 \\textrm { -- }$ $3 5 . 5 \\%$ ) parameters to achieve $6 5 . 5 \\% ( 1 - 3 4 . 5 \\% )$ FLOPs reduction with only $1 . { \\dot { 4 } } \\% / 0 . 8 \\%$ Top1/5 accuracy regression compared to the baseline. OTO consistently outperforms the majority of counterparts especially on the FLOPs reduction and the parameter reduction. We note that Hinge (56) prunes CNNs via structured-sparsity optimization by employing standard stochastic proximal gradient method. It ",
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+        "img_path": "images/342a8328fc03a05e2a30461579dc92b4780677f7c59c72572e000bba3ea009cc.jpg",
+        "table_caption": [
+            "Table 4: ResNet50 for ImageNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>FLOPs</td><td>#ofParams</td><td>Top-1 Acc.</td><td>Top-5 Acc.</td></tr><tr><td>Baseline</td><td>100%</td><td>100%</td><td>76.1%</td><td>92.9%</td></tr><tr><td>DDS-26 (43)</td><td>57.0%</td><td>61.2%</td><td>71.8%</td><td>91.9%</td></tr><tr><td>CP (38)</td><td>66.7%</td><td>1</td><td>72.3%</td><td>90.8%</td></tr><tr><td>ThiNet-50 (45)</td><td>44.2%</td><td>48.3%</td><td>71.0%</td><td>90.0%</td></tr><tr><td>RBP (101)</td><td>43.5%</td><td>48.0%</td><td>71.1%</td><td>90.0%</td></tr><tr><td>RRBP (101)</td><td>45.4%</td><td>1</td><td>73.0%</td><td>91.0%</td></tr><tr><td>SFP (36)</td><td>41.8%</td><td>=</td><td>74.6%</td><td>92.1%</td></tr><tr><td>Hinge (56)</td><td>46.6%</td><td>一</td><td>74.7%</td><td></td></tr><tr><td>GBN-50 (94)</td><td>44.9%</td><td>46.6%</td><td>75.2%</td><td>92.4%</td></tr><tr><td>GBN-60 (94)</td><td>59.5%</td><td>68.2%</td><td>76.2%</td><td>92.8%</td></tr><tr><td>Group-HS (2e-5) (91)</td><td>32.4%</td><td>=</td><td>75.2%</td><td>92.5%</td></tr><tr><td>Group-HS (1e-5) (91)</td><td>52.9%</td><td></td><td>76.4%</td><td>93.1%</td></tr><tr><td>ResRep (18)</td><td>45.5%</td><td></td><td>76.2%</td><td>92.9%</td></tr><tr><td>SCP (48)</td><td>45.7%</td><td></td><td>74.2%</td><td>92.0%</td></tr><tr><td>OTO</td><td>34.5%</td><td>35.5%</td><td>74.7%</td><td>92.1%</td></tr><tr><td>OTO*</td><td>34.5%</td><td>35.5%</td><td>75.1%</td><td>92.5%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "requires several trainings including fine-tuning the pruned model, because it partitions the parameters into non-ZIGs and relies on an empirical truncation mechanism to generate zero groups due to the weakness of proximal operator in deep learning applications (8). In contrast, OTO only trains and prunes the full model from scratch once and obtains better pruning results. The comparison between OTO and Hinge stand as evidence of the superiority of OTO due to ZIGs and HSPG. Furthermore, if with more training efforts, OTO reaches higher Top-1/5 accuracy marked as ∗ in Table 4 and becomes more competitive to stronger competitors, such as GBN (94), Group-HS (91) and ResRep (48). ",
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+        "type": "text",
+        "text": "Representation of Deep Features of ImageNet. It is widely acknowledged that deep neural architectures could be treated as non-linear feature representation extractors. Therefore, we further study the feature representation extracted by OTO to demonstrate its generalizability to other visual applications besides image classification. Figure 4 shows the clustering results of ImageNet validation images using the deep feature extracted by both the baseline ResNet50 and the pruned ResNet50 by OTO. Specifically, we extract the deep features over the validation samples in ImageNet, i.e., the tensors fed into the fully connected layer, and project them onto a 2-dimensional space via PCA (47). For illustration, following the hierarchy of ImageNet (3), two sets of five classes are randomly selected3. We observe that the deep features of the pruned ResNet50 by OTO remain structured in the sense that distinct classes are well separated from each other. Over all 1000-class ImageNet validation images, OTO achieves $4 8 . 2 \\%$ clustering accuracy compared to $4 2 . 5 \\%$ of the baseline ResNet50 using $\\mathbf { k }$ -means. Both observations indicate that with only $3 5 . 5 \\%$ parameters and $3 4 . 5 \\%$ FLOPs, the pruned ResNet50 is still able to extract highly discriminative deep features. We argue that during model compression, OTO not only achieves parameter and FLOPs reduction, but also preserves the ability of capturing perceptual properties (99). This is especially important in training and compressing models for many vision tasks, e.g., object detection (70; 71), frame interpolation (2; 17; 67) and video synthesis (84; 50). We leave the application of OTO to broader tasks to future work. ",
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+    {
+        "type": "text",
+        "text": "4.2 Large-Scale Transformer ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We show the scalability of OTO by pruning the large-scale transformer Bert (82), evaluated on SQuAD, a question-answering benchmark (69). Bert mainly includes embedding layers, fully connected layers and multi-head attention layers. The fully connected layers and the multi-head attention layers are partitioned into ZIGs following Section 3.1. For fair comparisons, we follow the prior Bert compression works (14; 75) and do not prune the embedding layers. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/149863a9ad5e7f572f51efc22421bbc22dc4300af2734f2794e3c3e99cb4ad35.jpg",
+        "image_caption": [
+            "Figure 4: Clustering results of ImageNet validation images using deep features extracted by full ResNet50 (left of a and b) and pruned ResetNet50 by OTO (right of a and b). The points are visualized by projecting deep features onto a two-dimensional space via PCA. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge, OTO is the first work that compresses Bert by exploring group sparsity on individual layers and achieves significant parameter reduction and inference speedup4. In contrast, the existing works (29; 75; 30) prune individual parameters instead, i.e., the generated sparsity is not structured. Hence, the computed models typically do not have inference speedup (75), unless are executed by specialized hardware and sparse computing library (31; 10). As ",
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+    {
+        "type": "table",
+        "img_path": "images/dbefbea0bde8bd1b9228dab69ac94976c31babb393cd8c83f764e631118e867e.jpg",
+        "table_caption": [
+            "Table 5: Pruning Bert on SQuAD "
+        ],
+        "table_footnote": [
+            "\\* Based on the statement in the official git repository of (75). † Approximate value based on the group sparsity reported in (14). "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>#ofParams</td><td>Exact</td><td>F1-score</td><td>SpeedUp</td></tr><tr><td>Baseline</td><td>100%</td><td>81.0%</td><td>88.3%</td><td>1×</td></tr><tr><td>MaP (75)</td><td>10.0%</td><td>67.7%</td><td>78.5%</td><td>1×*</td></tr><tr><td>MvP (75)</td><td>10.0%</td><td>71.9%</td><td>81.7%</td><td>1×*</td></tr><tr><td>ProxSSI (14)</td><td>83.4%t</td><td>72.3%</td><td>82.0%</td><td>1×</td></tr><tr><td>OTO</td><td>91.0%</td><td>75.0%</td><td>84.1%</td><td>1.1×</td></tr><tr><td>OTO</td><td>76.2%</td><td>72.3%</td><td>82.1%</td><td>1.2×</td></tr><tr><td>OTO</td><td>66.7%</td><td>71.9%</td><td>82.0%</td><td>1.3×</td></tr><tr><td>OTO</td><td>53.3%</td><td>71.4%</td><td>81.5%</td><td>1.5×</td></tr><tr><td>OTO</td><td>40.0%</td><td>70.9%</td><td>81.1%</td><td>1.8×</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "shown in Table 5, under different group sparsity upper bound constraints, OTO reduces $9 \\%$ t o $60 \\%$ parameters and achieves up to $1 . 8 \\times$ inference speedup based on the average model execution time 5. In comparison, despite that the pruned model contains $1 0 \\%$ parameters, MaP and MvP (75) do not have any inference speedup. On the other hand, the structured sparsity on Bert is studied in (14) (referred to as ProxSSI), where an adaptive proximal method is proposed to yield group-sparse solution. Nonetheless, ProxSSI optimizes over non-ZIGs and relies on proximal operator to identify group sparsity. Therefore, the groups even parameterized with zeros have to be retained in the model rather than pruned. As a consequence, ProxSSI is not competitive to OTO on parameter reduction, and there is no reported inference speedup. Note that all the pruning methods achieve comparable exact match rate and F1-score. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 Conclusion And Future Work ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    },
+    {
+        "type": "text",
+        "text": "We propose OTO, a one-shot deep neural networks (DNNs) training and pruning framework, that compresses full DNNs into slimmer architectures with competitive performances and significant FLOPs and parameter reduction without fine-tuning. OTO contains two fundamentals: (i) partitions the trainable parameters of DNNs into zero-invariant groups (ZIGs), thereby pruning zero groups does not affect the model output, and (ii) trains by a novel optimizer, Half-Space Stochastic Projected Gradient (HSPG), which outperforms proximal methods on group sparsity exploration and maintains comparable convergence. We numerically demonstrate OTO on benchmark experiments, i.e., VGG16 for CIFAR10, ResNet50 for CIFAR10/ImageNet and Bert for SQuAD, and achieve state-of-theart pruning results. We leave automatically generating ZIGs for arbitrary DNNs, incorporating quantization and applying OTO to other tasks to future work. ",
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+    },
+    {
+        "type": "text",
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+        "bbox": [
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+    },
+    {
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parse/train/sCZbhBvqQaU/sCZbhBvqQaU.md

@@ -0,0 +1,367 @@
+# ROBUST REINFORCEMENT LEARNING ON STATE OBSERVATIONS WITH LEARNED OPTIMAL ADVERSARY
+
+Huan Zhang\*,1 Hongge Chen\*,2 Duane Boning2 Cho-Jui Hsieh1 1Department of Computer Science, UCLA 2Department of EECS, MIT huan@huan-zhang.com, chenhg@mit.edu, boning@mtl.mit.edu chohsieh@cs.ucla.edu \*Huan Zhang and Hongge Chen contributed equally.
+
+# ABSTRACT
+
+We study the robustness of reinforcement learning (RL) with adversarially perturbed state observations, which aligns with the setting of many adversarial attacks to deep reinforcement learning (DRL) and is also important for rolling out real-world RL agent under unpredictable sensing noise. With a fixed agent policy, we demonstrate that an optimal adversary to perturb state observations can be found, which is guaranteed to obtain the worst case agent reward. For DRL settings, this leads to a novel empirical adversarial attack to RL agents via a learned adversary that is much stronger than previous ones. To enhance the robustness of an agent, we propose a framework of alternating training with learned adversaries (ATLA), which trains an adversary online together with the agent using policy gradient following the optimal adversarial attack framework. Additionally, inspired by the analysis of state-adversarial Markov decision process (SA-MDP), we show that past states and actions (history) can be useful for learning a robust agent, and we empirically find a LSTM based policy can be more robust under adversaries. Empirical evaluations on a few continuous control environments show that ATLA achieves state-of-the-art performance under strong adversaries. Our code is available at https://github.com/huanzhang12/ATLA_robust_RL.
+
+# 1 INTRODUCTION
+
+Modern deep reinforcement learning agents (Mnih et al., 2015; Levine et al., 2015; Lillicrap et al., 2015; Silver et al., 2016; Fujimoto et al., 2018) typically use neuron networks as function approximators. Since the discovery of adversarial examples in image classification tasks (Szegedy et al., 2013), the vulnerabilities in DRL agents were first demonstrated in (Huang et al., 2017; Lin et al., 2017; Kos & Song, 2017) and further developed under more environments and different attack scenarios (Behzadan & Munir, 2017a; Pattanaik et al., 2018; Xiao et al., 2019). These attacks commonly add imperceptible noises into the observations of states, e.g., the observed environment slightly differs from true environment. This raises concerns for using RL in safety-crucial applications such as autonomous driving (Sallab et al., 2017; Voyage, 2019); additionally, the discrepancy between ground-truth states and agent observations also contributes to the “reality gap” - an agent working well in simulated environments may fail in real environments due to noises in observations (Jakobi et al., 1995; Muratore et al., 2019), as real-world sensing contains unavoidable noise (Brooks, 1992).
+
+We classify the weakness of a DRL agent on the perturbations of state observations into two classes: the vulnerability in function approximators, which typically originates from the highly non-linear and blackbox nature of neural networks; and intrinsic weakness of policy: even perfect features for states are extracted, an agent can still make mistakes due to an intrinsic weakness in its policy.
+
+For example, in the deep Q networks (DQNs) for Atari games, a large convolutional neural network (CNN) is used for extracting features from input frames. To act correctly, the network must extract crucial features: e.g., for the game of Pong, the position and velocity of the ball, which can observed by visualizing convolutional layers (Hausknecht & Stone, 2015; Guo et al., 2014). Many attacks to the DQN setting add imperceptible noises (Huang et al., 2017; Lin et al., 2017; Kos & Song, 2017; Behzadan & Munir, 2017a) that exploit the vulnerability of deep neural networks so that they extract wrong features, as we have seen in adversarial examples of image classification tasks. On the other
+
+<table><tr><td rowspan=1 colspan=1>0.59</td><td rowspan=1 colspan=1>0.66→-</td><td rowspan=1 colspan=1>0.73→</td><td rowspan=1 colspan=1>0.81*i↓</td><td rowspan=1 colspan=1>0.73↓</td><td rowspan=1 colspan=1>0.66↓</td></tr><tr><td rowspan=1 colspan=1>↑0.53</td><td rowspan=1 colspan=1>↑0.59</td><td rowspan=1 colspan=1>0.81R:-1.0↓</td><td rowspan=1 colspan=1>0.91i↓</td><td rowspan=1 colspan=1>0.81↓</td><td rowspan=1 colspan=1>0.73↓</td></tr><tr><td rowspan=1 colspan=1>0.59↓</td><td rowspan=1 colspan=1>0.81R:-1.0+</td><td rowspan=1 colspan=1>0.9→</td><td rowspan=1 colspan=1>111.011↓</td><td rowspan=1 colspan=1>0.9↓</td><td rowspan=1 colspan=1>0.81↓</td></tr><tr><td rowspan=1 colspan=1>0.66↓</td><td rowspan=1 colspan=1>0.73↓</td><td rowspan=1 colspan=1>1.0R:-1.0+</td><td rowspan=1 colspan=1>0.0R:1.0</td><td rowspan=1 colspan=1>.1.0</td><td rowspan=1 colspan=1>←0.9</td></tr><tr><td rowspan=1 colspan=1>0.73→</td><td rowspan=1 colspan=1>0.81→</td><td rowspan=1 colspan=1>0.9→</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>89</td><td rowspan=1 colspan=1>081</td></tr><tr><td rowspan=1 colspan=1>↑0.66</td><td rowspan=1 colspan=1>↑0.73</td><td rowspan=1 colspan=1>↑0.81</td><td rowspan=1 colspan=1>09</td><td rowspan=1 colspan=1>↑0.81</td><td rowspan=1 colspan=1>073</td></tr></table>
+
+![](images/fe64bcdbb7cbac0f93b218fff30826f9063b3fbc47b558fc86fa62de4758e121.jpg)
+
+(a) Path in unperturbed environment (found by policy iteration). Agent’s reward $= + 1$ . Black arrows and numbers show actions and value function of the agent.
+
+![](images/8c4a1bb85207df26b4d7dcaf5f94b69b3171d8692c58bae857f7a23dc59b275b.jpg)  
+(c) A robust POMDP policy solved by SARSOP (Kurniawati et al., 2008) under the same adversary. This policy is history dependent (Section 3.2).
+
+(b) Path under the optimal adversary. Agent’s reward $\quad = \quad - \infty$ . Red arrows and numbers show actions and value function of the optimal adversary (Section 3.1).
+
+Figure 1: We show an agent in gridworld environment trained with no function approximators, and its optimal policy is intrinsically not robust to perturbations of state observations. The red square and blue circle are the starting point and target (reward $+ 1$ ) of the agent, respectively. The green triangles are traps, with reward $^ { - 1 }$ once encountered. The adversary is allowed to perturb the observation to adjacent states along four directions: up, down, left, and right. Adversary earns $+ 1$ at traps and -1 at the target. We set $\gamma = 0 . 9$ for both agent and adversary. This example shows that the vulnerability of a RL agent does not only come from the errors in function approximators such as DNNs.
+
+hand, the fragile function approximation is not the only source of the weakness of a RL agent - in a finite-state Markov decision process (MDP), we can use tabular policy and value functions so there is no function approximation error. The agent can still be vulnerable to small perturbations on observations, e.g., perturbing the observation of a state to one of its four neighbors in a gridworldlike environment can prevent an agent from reaching its goal (Figure 1). To improve the robustness of RL, we need to take measures from both aspects — a more robust function approximator, and a policy aware of perturbations in observations.
+
+Techniques developed in enhancing the robustness of neural network (NN) classifiers can be applied to address the vulnerability in function approximators. Especially, for environments like Atari games with images as input and discrete actions as outputs, the policy network $\pi _ { \theta }$ behaves similarly to a classifier in test time. Thus, Fischer et al. (2019); Mirman et al. (2018a) utilized existing certified adversarial defense (Mirman et al., 2018b; Wong & Kolter, 2018; Gowal et al., 2018; Zhang et al., 2020a) approaches in supervised learning to enhance the robustness of DQN agents. Another successful approach (Zhang et al., 2020b) for both Atari and high-dimensional continuous control environment regularizes the smoothness of the learned policy such that $\begin{array} { r } { \operatorname* { m a x } _ { \hat { s } \in \mathcal { B } ( s ) } D ( \pi _ { \theta } ( s ) , \pi _ { \theta } ( \hat { s } ) ) } \end{array}$ is small for some divergence $D$ and $B ( s )$ is a neighborhood around $s$ . This maximization can be solved using a gradient based method or convex relaxations of NNs (Salman et al., 2019; Zhang et al., 2018; Xu et al., 2020), and then minimized by optimizing $\theta$ . Such an adversarial minimax regularization is in the same spirit as the ones used in some adversarial training approaches for (semi)supervised learning, e.g., TRADES (Zhang et al., 2019) and VAT (Miyato et al., 2015). However, regularizing the function approximators does not explicitly improve the intrinsic policy robustness.
+
+In this paper, we propose an orthogonal approach, alternating training with learned adversaries (ATLA), to enhance the robustness of DRL agents. We focus on dealing with the intrinsic weakness of the policy by learning an adversary online with the agent during training time, rather than directly regularizing function approximators. Our main contributions can be summarized as:
+
+• We follow the framework of state-adversarial Markov decision process (SA-MDP) and show how to learn an optimal adversary for perturbing observations. We demonstrate practical attacks under this formulation and obtain learned adversaries that are significantly stronger than previous ones. • We propose the alternating training with learned adversaries (ATLA) framework to improve the robustness of DRL agents. The difference between our approach and previous adversarial training approaches is that we use a stronger adversary, which is learned online together with the agent. • Our analysis on SA-MDP also shows that history can be important for learning a robust agent. We thus propose to use a LSTM based policy in the ATLA framework and find that it is more robust than policies parameterized as regular feedforward NNs.
+
+• We evaluate our approach empirically on four continuous control environments. We outperform explicit regularization based methods in a few environments, and our approach can also be directly combined with explicit regularizations on function approximators to achieve state-of-the-art results.
+
+# 2 RELATED WORK
+
+State-adversarial Markov decision process (SA-MDP) (Zhang et al., 2020b) characterizes the decision making problem under adversarial attacks on state observations. Most importantly, the true state in the environment is not perturbed by the adversary under this setting; for example, perturbing pixels in an Atari environment (Huang et al., 2017; Kos & Song, 2017; Lin et al., 2017; Behzadan & Munir, 2017a; Inkawhich et al., 2019) does not change the true location of an object in the game simulator. SA-MDP can characterize agent performance under natural or adversarial noise from sensor measurements. For example, GPS sensor readings on a car are naturally noisy, but the ground truth location of the car is not affected by the noise. Importantly, this setting is different from robust Markov decision process (RMDP) (Nilim & El Ghaoui, 2004; Iyengar, 2005), where the worst case transition probabilities of the environment are considered. “Robust reinforcement learning” in some works (Mankowitz et al., 2018; 2019) refer to this different definition of robustness in RMDP, and should not be confused with our setting of robustness against perturbations on state observations.
+
+Several works proposed methods to learn an adversary online together with an agent. RARL (Pinto et al., 2017) proposed to train an agent and an adversary under the two-player Markov game (Littman, 1994) setting. The adversary can change the environment states through actions directly applied to environment. The goal of RARL is to improve the robustness against environment parameter changes, such as mass, length or friction. Gleave et al. (2019) discussed the learning of an adversary using reinforcement learning to attack a victim agent, by taking adversarial actions that changes the environment and consequentially change the observation of the victim agent. Both Pinto et al. (2017); Gleave et al. (2019) conduct their attack under on the two-player Markov game framework, rather than considering perturbations on state observations. Besides, Li et al. (2019) consider a similar Markov game setting in multi-agent RL environments. The difference between these works and ours can be clearly seen in the setting where the adversary is fixed - under the framework of (Pinto et al., 2017; Gleave et al., 2019), the learning of agent is still a MDP, but in our setting, it becomes a harder POMDP problem (Section 3.2).
+
+Training DRL agents with perturbed state observations from adversaries have been investigated in a few works, sometimes referred to as adversarial training. Kos & Song (2017); Behzadan & Munir (2017b) used gradient based adversarial attacks to DQN agents and put adversarial frames into replay buffer. This approach is not very successful because for Atari environments the main source of weakness is likely to come from the function approximator, so an adversarial regularization framework such as (Zhang et al., 2020b; Qu et al., 2020) which directly controls the smoothness of the $Q$ function is more effective. For lower dimensional continuous control tasks such as the MuJoCo environments, Mandlekar et al. (2017); Pattanaik et al. (2018) conducted FGSM and multistep gradient based attacks during training time; however, their main focus was on the robustness against environment parameter changes and only limited evaluation on the adversarial attack setting was conducted with relatively weak adversaries. Zhang et al. (2020b) systematically tested this approach under newly proposed strong attacks, and found that it cannot reliably improve robustness. These early adversarial training approaches typically use gradients from a critic function. They are usually relatively weak, and not sufficient to lead to a robust policy under stronger attacks.
+
+The robustness of RL has also been investigated from other perspectives. For example, Tessler et al. (2019) study MDPs under action perturbations; Tan et al. (2020) use adversarial training on action space to enhance agent robustness under action perturbations. Besides, policy teaching (Zhang & Parkes, 2008; Zhang et al., 2009; Ma et al., 2019) and policy poisoning (Rakhsha et al., 2020; Huang & Zhu, 2019) manipulate the reward or cost signal during agent training time to induce a desired agent policy. Essentially, policy teaching is a training time “attack” with perturbed rewards from the environments (which can be analogous to data poisoning attacks in supervised learning settings), while our goal is to obtain a robust agent against test time adversarial attacks. All these settings differ from the setting of perturbing state observations discussed in our paper.
+
+# 3 METHODOLOGY
+
+In this section, we first discuss the case where the agent policy is fixed, and then the case where the adversary is fixed in SA-MDPs. This allows us to propose an alternating training framework to improve robustness of RL agents under perturbations on state observations.
+
+Notations and Background We use $s$ and $\mathcal { A }$ to represent the state space and the action space, respectively; ${ \mathcal { P } } ( S )$ defines the set of all possible probability measures on $s$ . We define a Markov decision process (MDP) as $( S , { \mathcal { A } } , R , p , \gamma )$ , where $R : \mathcal { S } \times \mathcal { A } \times \mathcal { S }  \mathbb { R }$ and $p : { \mathcal { S } } \times { \mathcal { A } }  { \mathcal { P } } ( { \mathcal { S } } )$ are two mappings represent the reward and transition probability. The transition probability at time step $t$ can be written as $p ( s ^ { \prime } | s , a ) \ = \ \mathrm { P r } \big ( s _ { t + 1 } \ = \ s ^ { \prime } | s _ { t } \ = \ s , a _ { t } \ = \ a \big )$ . Reward function is defined as the expected reward $R ( s , a , s ^ { \prime } ) : = \mathbb { E } [ r _ { t } | s _ { t } = s , a _ { t } = a , s _ { t + 1 } = s ^ { \prime } ]$ ]. $\gamma \in \ [ 0 , 1 ]$ is the discounting factor. We denote a stationary policy as $\pi : \mathcal { S }  \mathcal { P } ( \mathcal { A } )$ which is independent of history. We denote history $h _ { t }$ at time $t$ as $\left\{ s _ { 0 } , a _ { 0 } , \cdot \cdot \cdot , s _ { t - 1 } , a _ { t - 1 } , s _ { t } \right\}$ and $\mathcal { H }$ as the set of all histories. A history-dependent policy is defined as $\pi : \mathcal { H }  \mathcal { P } ( \mathcal { A } )$ . A partially observable Markov decision process (Astrom, 1965) (POMDP) can be defined as a 7-tuple $( \mathcal { S } , \dot { \mathcal { A } } , \mathcal { O } , \Omega , R , p , \gamma )$ where $\mathcal { O }$ is a set of observations and $\Omega$ is a set of conditional observation probabilities $p ( o | s )$ . Unlike MDPs, POMDPs typically require history-dependent optimal policies.
+
+To study the decision problem under adversaries on state observations, we use state-adversarial Markov decision process (SA-MDP) framework (Zhang et al., 2020b). In SA-MDP, an adversary $\nu : \mathcal { S } \to \mathcal { P } ( \mathcal { S } )$ is introduced to perturb the input state of an agent; however, the true environment state $s$ is unchanged (Figure 2). Formally, an SA-MDP is a 6-tuple $( S , A , B , R , p , \gamma )$ where $\boldsymbol { B }$ is a mapping from a state $s \in { \mathcal { S } }$ to a set of states $B ( s ) \in S$ . The agent sees the perturbed state $\hat { s } \sim \nu ( \cdot | s )$ and takes the action $\pi ( \boldsymbol a | \boldsymbol { \hat { s } } )$ accordingly. $\boldsymbol { B }$ limits the power of adversary: supp $( \nu ( \cdot | s ) ) \in B ( s )$ . The goal of SA-MDP is to solve an optimal policy $\pi ^ { * }$ under its optimal adversary $\nu ^ { * } ( \pi ^ { * } )$ ; an optimal adversary is defined as $\nu ^ { * } ( \pi )$ such that $\pi$ achieves the lowest possible expected discounted return (or value) on all states. Zhang et al. (2020b) did not give an explicit algorithm to solve SA-MDP and found that a stationary optimal policy need not exist.
+
+![](images/9062781f5b69002f4b718d4e852f4ca6668756a236c36f6895bc0e2135f42b70.jpg)  
+Figure 2: SA-MDP introduces an adversary on state observations in a MDP.
+
+# 3.1 FINDING THE OPTIMAL ADVERSARY UNDER A FIXED POLICY
+
+In this section, we discuss how to find an optimal adversary $\nu$ for a given policy $\pi$ . An optimal adversary leads to the worst case performance under bounded perturbation set $\boldsymbol { B }$ , and is an absolute lower bound of the expected cumulative reward an agent can receive. It is similar to the concept of “minimal adversarial example” in supervised learning tasks. We first show how to solve the optimal adversary in MDP setting and then apply it to the DRL settings.
+
+A technical lemma (Lemma 1) from Zhang et al. (2020b) shows that, from the adversary’s point of view, a fixed and stationary agent policy $\pi$ and the environment dynamics can be essentially merged into an MDP with redefined dynamics and reward functions:
+
+Lemma 1 (Zhang et al. (2020b)) Given an SA-MDP $M = ( S , \mathcal { A } , \mathcal { R } , \mathcal { B } , p , \gamma )$ and a fixed and stationary policy $\pi ( \cdot | \cdot )$ , there exists an MDP $\hat { M } = ( S , \hat { \mathcal { A } } , \hat { R } , \hat { p } , \gamma )$ such that the optimal policy of $\hat { M }$ is the optimal adversary $\nu$ for SA-MDP given the fixed $\pi$ , where $\hat { \mathcal { A } } = \mathcal { S }$ , and
+
+$$
+\hat { R } ( s , \hat { a } , s ^ { \prime } ) : = \mathbb { E } [ \hat { r } | s , \hat { a } , s ^ { \prime } ] = \left\{ \begin{array} { l l } { - \frac { \sum _ { a \in A } \pi ( a | \hat { a } ) p ( s ^ { \prime } | s , a ) R ( s , a , s ^ { \prime } ) } { \sum _ { a \in A } \pi ( a | \hat { a } ) p ( s ^ { \prime } | s , a ) } } & { f o r s , s ^ { \prime } \in S a n d \hat { a } \in B ( s ) \subset \hat { A } , } \\ { C } & { f o r s , s ^ { \prime } \in S a n d \hat { a } \notin B ( s ) . } \end{array} \right.
+$$
+
+where $C$ is a large negative constant, and
+
+$$
+\hat { p } ( s ^ { \prime } | s , \hat { a } ) = \sum _ { a \in \mathcal { A } } \pi ( a | \hat { a } ) p ( s ^ { \prime } | s , a ) \quad f o r s , s ^ { \prime } \in \mathcal { S } a n d \hat { a } \in \hat { \mathcal { A } } .
+$$
+
+The intuition behind Lemma 1 is that the adversary’s goal is to reduce the reward earned by the agent. Thus, when a reward $r _ { t }$ is received by the agent at time step $t$ , the adversary receives a negative reward of $\hat { r } _ { t } = - r _ { t }$ . To prevent the agent from taking actions outside of set $B ( s )$ , a large negative reward $C$ is assigned to these actions such that the optimal adversary cannot take them. For actions within $B ( s )$ , we calculate $\hat { R } ( s , \hat { a } , s ^ { \prime } )$ by its definition, $\hat { R } ( s , \hat { a } , s ^ { \prime } ) : = \mathbb { E } [ \hat { r } | s , \hat { a } , s ^ { \prime } ]$ which yields the term in Lemma 1. The proof can be found in Appendix B of Zhang et al. (2020b).
+
+After constructing the MDP $\hat { M }$ , it is possible to solve an optimal agent $\nu$ of $\hat { M }$ , which will be the optimal adversary on SA-MDP $M$ given policy $\pi$ . For MDPs, under mild regularity assumptions an optimal policy always exists (Puterman, 2014). In our case, the optimal policy on $\hat { M }$ corresponds to an optimal adversary in SA-MDP, which is the worst case perturbation for policy $\pi$ . As an illustration, in Figure 1, we show a GridWorld environment. The red square is the starting point. The blue circle and green triangles are the target and traps, respectively. When the agent hits the target, it earns reward $+ 1$ and the game stops and it earns reward $^ { - 1 }$ when it encounters a trap. We set $\gamma = 0 . 9$ for both agent and adversary. The adversary is allowed to perturb the observation to adjacent cells along four directions: up, down, left, and right. When there is no adversary, after running policy iteration, the agent can easily reach target and earn a reward of $+ 1$ , as in Figure 1a. However, if we train the adversary based on Lemma 1 and apply it to the agent, we are able to make the agent repeatedly encounter a trap. This leads to $- \infty$ reward for the agent and $+ \infty$ reward for the adversary as shown in Figure 1b.
+
+![](images/d2317beff8c847a01657a5489fa7ef0675325f4243304334d9434c6f729407a9.jpg)  
+Figure 3: Our “Optimal” Attack and Robust Sarsa attack (a previous strong attack proposed in Zhang et al. (2020b)) on Ant and HalfCheetah environments. Previous strong attacks make the agent fail and receive a small positive reward (less than 1/10 of the reward without attack). Our attack is strong enough to trick the agent into moving to the opposite direction, receiving a large negative reward.
+
+We now extend this Lemma 1 to the DRL setting. Since the learning of adversary is equivalent to solving an MDP, we parameterize the adversary as a neural network function and use any popular DRL algorithm to learn an “optimal” adversary. Here we quote the word “optimal” as we use function approximator to learn the agent so it’s no longer optimal, but we emphasize that it follows the SA-MDP framework of solving an optimal adversary. No existing adversarial attacks follow such a theoretically guided framework. We show our algorithm in Algorithm 1. Instead of learning to produce ${ \hat { s } } \in B ( s )$ directly, since $B ( s )$ is usually a small set nearby $s$ (e.g., $\mathcal { B } = \{ s ^ { \prime } | \| s - s ^ { \prime } \| _ { p } \leq \epsilon \}$ , our adversary learns a perturbation vector $\Delta$ , and we project $s + \Delta$ to $B ( s )$ .
+
+The first advantage of attacking a policy in this way is that it is strong - as we allow to optimize the adversary in an online loop of interactions with the agent policy and environment, and keep improving the adversary with a goal of receiving as less reward as possible. It is strong because it follows the theoretical framework of finding an optimal adversary, rather than using any heuristic to generate a perturbation. Empirically, in the cases demonstrated in Figure 3, previous strong attacks (e.g., Robust Sarsa attack) can successfully fail an agent and make it stop moving and receive a small positive reward; our learned attack can trick the agent into moving toward the opposite direction of the goal and receive a large negative reward. We also find that this attack can further reduce the reward of robustly trained agents, like SA-PPO (Zhang et al., 2020b).
+
+The second advantage of this attack is that it requires no gradient access to the policy itself; in fact, it treats the agent as part of the environment and only needs to run it in a blackbox. Previous attacks (e.g., Lin et al. (2017); Pattanaik et al. (2018); Xiao et al. (2019)) are mostly gradient based approach and need to access the values or gradients to a policy or value function. Even without access to gradients, the overall learning process is still just a MDP and we can apply any popular modern DRL methods to learn the adversary.
+
+# .2 FINDING THE OPTIMAL POLICY UNDER A FIXED ADVERSARY
+
+We now investigate SA-MDP when we fix the adversary $\nu$ and find an optimal policy. In Lemma 2, we show that this case SA-MDP becomes a POMDP:
+
+Lemma 2 (Optimal policy under fixed adversary) Given an SA-MDP $M \ : = \ : ( S , A , B , R , p , \gamma )$ and a fixed and stationary adversary $\nu ( \hat { s } | s )$ , there exists a POMDP $\bar { M } = ( S , \mathcal { A } , \Omega , O , R , p , \gamma )$
+
+Algorithm 1 Learning an “optimal” adversary for perturbations on state observations
+
+Input: Policy $\pi ( \cdot | s )$ under attack, number of iterations $N _ { i t e r }$ , batch size $B$ , perturbation set $B ( s )$   
+1: initialize adversary $\nu _ { \phi } ( \cdot | s )$ parameterized by a neural network with parameters $\phi$ ,   
+2: for $i = 1$ to $N _ { i t e r }$ do   
+3: $\mathcal { D }  A d \nu \_ T r a j ( \nu _ { \phi } , \pi , B )$ # collection of samples (for simplicity we ignore episodes here)   
+4: $\phi  \mathrm { P o l i c y O p t i m i z e r } ( \mathcal { D } , \phi )$   
+5: end for
+
+Function $A d \nu \_ T r a j ( \nu _ { \phi } , \pi , B )$ :
+
+6: $s  s _ { 0 }$ # Initial state   
+7: $\mathcal { D }  \emptyset$   
+8: for $b = 1$ to $B$ do   
+9: $\Delta \gets$ sample from $\nu _ { \phi } ( \cdot | s )$   
+10: $\hat { s } \gets \mathrm { P r o j } _ { \mathcal { B } } ( s ) ( s + \Delta )$ # projection will be a clipping for $\ell _ { \infty }$ norm set $B ( s )$   
+11: $a \gets$ sample from $\pi ( \cdot | \hat { s } )$   
+12: obtain current step reward $r _ { t }$ , next state $s ^ { \prime }$ from environment given action $a$   
+13: $\mathbf { \Lambda } _ { 3  S ^ { \prime } } ^ { \mathcal { D }  \mathcal { D } \cup ( s , \Delta , \mathbf { \bar { \Lambda } } ^ {  } r _ { t } , s ^ { \prime } ) }$ # state, action, reward and next state for the adversary   
+14:   
+15: end for   
+16: return $\mathcal { D }$
+
+such that the optimal policy of $\bar { M }$ is the optimal policy $\pi$ for SA-MDP given the fixed $\nu$ , where
+
+$$
+\Omega = \bigcup _ { s \in S } \{ s ^ { \prime } | s ^ { \prime } \in s u p p ( \nu ( \cdot | s ) ) \} , \qquad O ( o | s ) = \nu ( \hat { s } | s )
+$$
+
+where $\Omega$ is the set of observations, and $O$ defines the conditional observational probabilities (in our case it is conditioned only on $s$ and does not depend on actions). To prove Lemma 2, we construct the a POMDP with the observations defined on the support of all $\nu ( \cdot | s ) , s \in { \mathcal { S } }$ and the observation process is exactly the process of generating an adversarially perturbed state $\hat { s }$ . This POMDP is functionally identical to the original SA-MDP when $\nu$ is fixed. This lemma unveils the connection between POMDP and SA-MDP: SA-MDP can be seen as a version of “robust” POMDP where the policy needs to be robust under a set of observational processes (adversaries). SA-MDP is different from robust POMDP (RPOMDP) (Osogami, 2015; Rasouli & Saghafian, 2018), which optimizes for the worst case environment transitions.
+
+As a proof of concept, we use a modern POMDP solver, SARSOP (Kurniawati et al., 2008) to solve the GridWorld environment in Figure 1 to find a policy that can defeat the adversary. The POMDP solver produces a finite state controller (FSC) with 8 states (FSC is an efficient representation of history dependent policies). This FSC policy can almost eliminate the impact of the adversary and receive close to perfect reward, as shown in Figure 1c.
+
+Unfortunately, unlike MDPs, it is challenging to solve an optimal policy for POMDPs; state-of-theart solvers (Bai et al., 2014; Sunberg & Kochenderfer, 2017) can only work on relatively simple environments which are much smaller than those used in modern DRL. Thus, we do not aim to solve the optimal policy. We follow (Wierstra et al., 2007) to use recurrent policy gradient theorem on POMDPs and use LSTM as function approximators for the value and policy networks. We denote $h _ { t } = \left\{ \hat { s } _ { 0 } , a _ { 0 } , \hat { s } _ { 1 } , a _ { 1 } \cdot \cdot \cdot , \hat { s } _ { t } \right\}$ containing all history of states (perturbed states $\hat { s }$ in our setting) and actions. The policy $\pi$ parameterized by $\theta$ takes an action $a _ { t }$ given all observed history $h _ { t }$ , and $h _ { t }$ is typically encoded by a recurrent neural network (e.g., LSTM). The recurrent policy gradient theorem (Wierstra et al., 2007) shows that
+
+$$
+\nabla _ { \theta } J \approx \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \sum _ { t = 0 } ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { n } | h _ { t } ^ { n } ) r _ { t } ^ { n }
+$$
+
+where $N$ is the number of sampled episodes, $T$ is episode length (for notation similarity, we assume each episode has the same length), and $h _ { t } ^ { n }$ is the history of states for episode $n$ up to time $t$ , and $r _ { t } ^ { n }$ is the reward received for episode $n$ at time $t$ . We can then extend Eq. 2 to modern DRL algorithms such as proximal policy optimization (PPO), similarly as done in (Azizzadenesheli et al., 2018), by
+
+using the following loss function:
+
+$$
+J ( \theta ) \approx \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \sum _ { t = 0 } ^ { T } \left[ \operatorname* { m i n } \left( \frac { \pi _ { \theta } ( a _ { t } ^ { n } | h _ { t } ^ { n } ) } { \pi _ { \theta _ { \mathrm { o d d } } } ( a _ { t } ^ { n } | h _ { t } ^ { n } ) } A _ { h _ { t } ^ { n } } , \operatorname { c l i p } ( \frac { \pi _ { \theta } ( a _ { t } ^ { n } | h _ { t } ^ { n } ) } { \pi _ { \theta _ { \mathrm { o d d } } } ( a _ { t } ^ { n } | h _ { t } ^ { n } ) } , 1 - \epsilon , 1 + \epsilon ) A _ { h _ { t } ^ { n } } \right) \right] .
+$$
+
+where $A _ { h _ { t } ^ { n } }$ is a baseline advantage function for episode $n$ time step $t$ , which is based on a LSTM value function. $\epsilon$ is the clipping threshold in PPO. The loss can be optimized via a gradient based optimizer and $\theta _ { \mathrm { o l d } }$ is the old policy parameter before optimization iterations start. Although a LSTM or recurrent policy network has been used in the DRL setting in a few other works (Hausknecht & Stone, 2015; Azizzadenesheli et al., 2018), our focus is to improve agent robustness rather than learning a policy purely for POMDPs. In our empirical evaluation, we will compare feedforward and LSTM policies under our ATLA framework.
+
+# 3.3 ALTERNATING TRAINING WITH LEARNED ADVERSARIES (ATLA)
+
+As we have discussed in Section 3.1, we can solve an optimal adversary given any fixed policy. In our ATLA framework, we train such an adversary online with the agent: we first keep the agent and optimize the adversary; the adversary is also parameterized as a neural network. Then we keep the adversary and optimize the agent. Both adversary and agent can be updated using a policy gradient algorithm such as PPO. We show our full algorithm in Algorithm 2.
+
+Input: Environment $\mathcal { E }$ , number of iterations $N _ { i t e r }$ , and batch size $B$ .   
+1: Initialize the agent’s actor network $\pi ( a | \boldsymbol { \hat { s } } )$ with parameters $\theta$ .   
+2: Initialize the adversary’s actor network $\nu ( \hat { s } | s )$ with parameters $\phi$ .   
+3: for $i = 1$ to $N _ { i t e r }$ do   
+4: 5: for Run $j = 1$ $\pi _ { \theta }$ to with fixed $N _ { \pi }$ do $\nu _ { \phi }$ to collect a set of trajectories $\mathcal { D } _ { \pi } : = \left\{ \left( \hat { s } _ { t } ^ { k , j } , a _ { t } ^ { k , j } , r _ { t } ^ { k , j } , \hat { s } _ { t + 1 } ^ { k , j } \right) \right\} \big | _ { k = 1 } ^ { B } ,$ .   
+6: $\theta \gets$ PolicyOptimizer $( { \mathcal { D } } _ { \pi } , \theta )$   
+7: end for   
+8: for $j = 1$ to $N _ { \nu }$ do   
+9: $\mathcal { D } _ { \nu }  A d \nu \_ T r a j ( \nu _ { \phi } , \pi _ { \theta } , B )$ # Adv Traj defined in Algorithm 1   
+10: $\phi $ PolicyOptimizer $( \mathcal { D } _ { \nu } , \phi )$   
+11: end for   
+12: end for
+
+Our algorithm is designed to use a strong and learned adversary that tries to find intrinsic weakness of the policy, and to obtain a good reward the policy must learn to defeat such an adversary. In other words, it attempts to solve the SA-MDP problem directly rather than relying on explicit regularization on the function approximator like the approach in (Zhang et al., 2020b). In our empirical evaluation, we show that such regularization can be unhelpful in some environments and harmful for performance when evaluating the agent without attacks.
+
+The difference between our approach and previous adversarial training approaches such as (Pattanaik et al., 2018) is that we use a stronger adversary, learned online with the agent. Our empirical evaluation finds that using such a learned “optimal” adversary in training time allows the agent to learn a robust policy generalized to different types of strong adversarial attacks during test time. Additionally, it is important to distinguish between the original state $s$ and the perturbed state sˆ. We find that using $s$ instead of $\hat { s }$ to train the advantage function and policy of the agent leads to worse performance, as it does not follow the theoretical framework of SA-MDP.
+
+# 4 EXPERIMENTS
+
+“Optimal” attack on DRL agents1 In section 3.1 we show that it is possible to cast the optimal adversary finding problem as an MDP problem. In practice, the environment dynamics are unknown but model-free RL methods can be used to approximately find this optimal adversary. In this section, we use PPO to train an adversary on four OpenAI Gym MuJoCo continuous control environments.
+
+Table 1: Average episode rewards $\pm$ standard deviation over 50 episodes on PPO and SA-PPO agents. We report natural rewards (no attacks) and rewards under six adversarial attacks, including a simple random noise attack, the critic based attack in Pattanaik et al. (2018),MAD and RS attacks in Zhang et al. (2020b), Snooping attack proposed in Inkawhich et al. (2019), and the optimal attack proposed in this paper. In each row we bold the best (lowest) attack reward over all five attacks. “Optimal” attack is better than other attacks in all environments, sometimes by a large margin.   
+
+<table><tr><td rowspan="2">Env.</td><td rowspan="2">lnorm perturb- ation budget e</td><td rowspan="2">Method</td><td rowspan="2">Natural Reward</td><td colspan="6">Attack Reward</td></tr><tr><td>Critic</td><td>Random</td><td>MAD</td><td>Snooping</td><td>RS</td><td>“Optimal”</td></tr><tr><td rowspan="2">Hopper</td><td rowspan="2">0.075</td><td>PPO</td><td>3167±521</td><td>1464 ±523</td><td>2101±793</td><td>1410± 655</td><td>2234±1103</td><td>794±238</td><td>636±9</td></tr><tr><td>SA-PPO</td><td>3705±2</td><td>3789±15</td><td>2710± 801</td><td>2652±835</td><td>2509±838</td><td>1130±42</td><td>1076± 791</td></tr><tr><td rowspan="2">Walker2d</td><td rowspan="2">0.05</td><td>PPO</td><td>4472 ±635</td><td>3424 ±1295</td><td>3007 ±1200</td><td>2869 ±1271</td><td>2786±962</td><td>1336± 654</td><td>1086±516</td></tr><tr><td>SA-PPO</td><td>4487±61</td><td>4875±30</td><td>4867±39</td><td>3668± 1789</td><td>3928±1661</td><td>3808±138</td><td>2908±1136</td></tr><tr><td rowspan="2">Ant</td><td>0.15</td><td>PPO</td><td>5687 ±758</td><td>4934±1022</td><td>5261± 1005</td><td>1759±828</td><td>3668±547</td><td>268 ±227</td><td>-872± 436</td></tr><tr><td></td><td>SA-PPO</td><td>4292± 384</td><td>4805±128</td><td>4986±452</td><td>4662 ±522</td><td>4079±768</td><td>3412 ±1755</td><td>2511 ± 1117</td></tr><tr><td rowspan="2">HalfCheetah</td><td>0.15</td><td>PPO</td><td>7117±98</td><td>5761±119</td><td>5486±1378</td><td>1836±866</td><td>1637±843</td><td>489±758</td><td>-660± 219</td></tr><tr><td></td><td>SA-PPO</td><td>3632±20</td><td>3589± 21</td><td>3619±18</td><td>3624±23</td><td>3616±21</td><td>3283±20</td><td>3028±23</td></tr></table>
+
+Table 1 presents results on attacking vanilla PPO and robustly trained SA-PPO (Zhang et al., 2020b) agents. As a comparison, we also report the attack reward of five other baseline attacks: critic attack is based on (Pattanaik et al., 2018); random attack adds uniform random noise to state observations; MAD (maximal action difference) attack (Zhang et al., 2020b) maximizes the differences in action under perturbed states; RS (robust sarsa) attack is based on training robust action-value functions and is the strongest attack proposed in (Zhang et al., 2020b). Additionally, we include the blackbox Snooping attack (Inkawhich et al., 2019). For all attacks we consider $B ( s )$ as a $\ell _ { \infty }$ norm ball around $s$ with radius $\epsilon$ , set similarly as in (Zhang et al., 2020b). During testing, we run the agents without attacks as well as under attacks for 50 episodes and report the mean and standard deviation of episode rewards. In Table 1 our “optimal” attack achieves noticeably lower rewards than all the other five attacks. We illustrate a few examples of attacks in Figure 3. For RS and “optimal” attacks, we report the best (lowest) attack reward obtained from different hyper-parameters.
+
+Evaluation of ATLA In this experiment, we study the effectiveness of our proposed ATLA method. Specifically, we use PPO as our policy optimizer. For policy networks, we have two different structures: the original fully connected (MLP) structure, and an LSTM structure which takes historical observations. The LSTMs are trained using backpropagation through time for up to 100 steps. In Table 2 we include the following methods for comparisons:
+
+• PPO (vanilla) and PPO (LSTM): PPO with a feedforward NN or LSTM as the policy network. • SA-PPO (Zhang et al., 2020b): the state-of-the-art approach for improving the robustness of DRL in continuous control environments, using a smooth policy regularization on feedforward NNs solved by convex relaxations.   
+• Adversarial training using critic attack (Pattanaik et al., 2018): a previous work using critic based attack to generate adversarial observations in training time, and train a feedforward NN based agent with this relatively weak adversary.   
+• ATLA-PPO (MLP) and ATLA-PPO (LSTM): Our proposed method trained with a feedforward NN (MLP) or LSTM as the policy network. The agent and adversary are trained using PPO with independent value and policy networks. For simplicity, we set $N _ { \pi } = N _ { \nu } = 1$ in all settings. • ATLA-PPO (LSTM) $+ \mathrm { S A }$ reg: Based on ATLA-PPO (LSTM), but with an extra adversarial smoothness constraint similar to those in SA-PPO. We use a 2-step stochastic gradient Langevin dynamics (SGLD) to solve the minimax loss, as convex relaxations of LSTMs are expensive.
+
+For each agent, we report its “natural reward” (episode reward without attacks) and best attack reward in Table 2. To comprehensively evaluate the robustness of agents, the best attack reward is the lowest episode reward achieved by all six types attacks in Table 1, including our new “optimal” attack (these attacks include hundreds of independent adversaries for attacking a single agent, see Appendix A.1 for more details). For reproducibility, for each setup we train 21 agents, attack all of them and report the one with median robustness. We include detailed hyperparameters in A.5.
+
+In Table 2 we can see that vanilla PPO with MLP or LSTM are not robust. For feedforward (MLP) agent policies, critic based adversarial training (Pattanaik et al., 2018) is not very effective under our suite of strong adversaries and is sometimes only slightly better than vanilla PPO. ATLA-PPO (MLP) outperforms SA-PPO on Hopper and Walker2d and is also competitive on HalfCheetah; for high dimensional environments like Ant, the robust function approximator regularization in SA-PPO is more effective. For LSTM agent policies, compared to vanilla PPO (LSTM) agents,
+
+Table 2: Average episode rewards $\pm$ standard deviation over 50 episodes on ATLA agents and baselines. We report natural rewards (no attacks) and the best (lowest) attack rewards among six types of adversarial attacks, including a simple random noise attack, the critic based attack in (Pattanaik et al., 2018), MAD and RS attacks in Zhang et al. (2020b), Snooping attack proposed in Inkawhich et al. (2019), and the optimal attack proposed in this paper. For each environment, we bold the most robust agent. Since both RS attack and our “optimal” attack are parameterized attacks, the “best attack” column represents the worst case agent performance under hundreds of adversaries. See Appendix A.1 for more details.
+
+<table><tr><td rowspan=1 colspan=1>Env.</td><td rowspan=1 colspan=1>StateDimension</td><td rowspan=1 colspan=1>lnorm perturb-ation budget e</td><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>NaturalReward</td><td rowspan=1 colspan=1>BestAttack</td></tr><tr><td rowspan=1 colspan=1>Hopper</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>0.075</td><td rowspan=1 colspan=1>PPO (vanilla)SA-PPO (Zhang et al.,2020b)Pattanaik et al. (2018)ATLA-PPO (MLP)PPO (LSTM)ATLA-PPO (LSTM)ATLA-PPO (LSTM) +SA Reg</td><td rowspan=1 colspan=1>3167±5423705±22755±5822559 ±9583060± 639.33487±4523291± 600</td><td rowspan=1 colspan=1>636±91076± 791291±7976± 40784±481224± 1911772± 802</td></tr><tr><td rowspan=1 colspan=1>Walker2d</td><td rowspan=1 colspan=1>17</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>PPO (vanilla)SA-PPO (Zhang et al.,2020b)Pattanaik et al.(2018)ATLA-PPO (MLP)PPO (LSTM)ATLA-PPO (LSTM)ATLA-PPO (LSTM) +SA Reg</td><td rowspan=1 colspan=1>4472 ± 6354487± 614058± 14103138 ± 10612785± 11213920±1293842±475</td><td rowspan=1 colspan=1>1086±5162908±1136733±10122213± 9151259± 9373219 ±11323239±894</td></tr><tr><td rowspan=1 colspan=1>Ant</td><td rowspan=1 colspan=1>111</td><td rowspan=1 colspan=1>0.15</td><td rowspan=1 colspan=1>PPO (vanilla)SA-PPO (Zhang et al.,2020b)Pattanaik et al.(2018)ATLA-PPO (MLP)PPO (LSTM)ATLA-PPO (LSTM)ATLA-PPO (LSTM) +SA Reg</td><td rowspan=1 colspan=1>5687±7584292±3843469± 11394894±1235696±1655612±1305359±153</td><td rowspan=1 colspan=1>-872±4362511 ± 1117-672±10033±327-513 ±104716±2563765±101</td></tr><tr><td rowspan=1 colspan=1>HalfCheetah</td><td rowspan=1 colspan=1>17</td><td rowspan=1 colspan=1>0.15</td><td rowspan=1 colspan=1>PPO (vanilla)SA-PPO (Zhang et al.,2020b)Pattanaik et al. (2018)ATLA-PPO (MLP)PPO (LSTM)ATLA-PPO (LSTM)ATLA-PPO (LSTM) +SA Reg</td><td rowspan=1 colspan=1>7117±983632±205241± 11625417±495609±985766±1096157±852</td><td rowspan=1 colspan=1>-660± 2183028±23447±1922170± 2097-886±302485±14884806± 603</td></tr></table>
+
+ATLA-PPO (LSTM) can significantly improve agent robustness; a LSTM agent trained without a robust training procedure like ATLA cannot improve robustness. We find that LSTM agents tend to be more robust than their MLP counterparts, validating our findings in Section 3.2. ATLA-PPO (LSTM) is better than SA-PPO on Hopper and Walker2d. In all settings, especially for high dimensional environments like Ant, our ATLA approach that also includes State-Adversarial regularization (ATLA-PPO $+ \mathrm { S A }$ Reg) outperforms all other baselines, as this combination improves both the intrinsic robustness of policy and the robustness of function approximator.
+
+A robust function approximator can be insufficient For some environments, SA-PPO method has its limitations - even using an increasingly larger regularization parameter $\kappa$ (which controls how robust the function approximator needs to be), we still cannot reach the same performance as our ATLA agent (Figure 4). Additionally, when a large regularization is used, agent performance becomes much worse. In Figure 4, under the largest $\kappa = 1 . 0$ , the natural reward $( 1 4 3 6 \pm 9 6 )$ is much lower than other agents reported in Table 2.
+
+![](images/6455dd4e118fc93f94f546e20a3a04e897c80a992b1bfc3a75452989516bcd0b.jpg)  
+Figure 4: The performance under the strongest attack for SA-PPO Hopper with different regularization $\kappa$ . Even we increase regularization, it cannot outperform our ATLA agents.
+
+# 5 CONCLUSION
+
+In this paper, we first propose the optimal adversarial attack on state observations of RL agents, which is significantly stronger than many existing adversarial attacks. We then show the alternating training with learned adversaries (ATLA) framework to train an agent together with a learned optimal adversary to effectively improve agent robustness under attacks. We also show that a history dependent policy parameterized by a LSTM can be helpful for robustness. Our approach is orthogonal to existing regularization based techniques, and can be combined with state-adversarial regularization to achieve state-of-the-art robustness under strong adversarial attacks.
+
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+# A APPENDIX
+
+# A.1 FULL RESULTS OF ALL ENVIRONMENTS UNDER DIFFERENT TYPES OF ATTACKS
+
+In Table 2, we only include the best attack rewards (lowest rewards over all attacks). In Table 3 we list the rewards under each specific attack. Note that, Robust Sarsa (RS) attack and our “optimal” policy attack both have hyperparameters. For RS attack we use the same set of 30 different settings of hyperparameters as in (Zhang et al., 2020b) to train a robust value function to attack the network. The reported RS attack result for each agent is the strongest one over the 30 trained value functions. For Snooping based attack, we use the “imitator” attack proxy as it was the strongest reported in (Inkawhich et al., 2019), and we attack every step of the agent. The imitator is a MLP or LSTM network according to agent policy network. We use the same loss KL divergence function as in the MAD attacks for this Snooping attack. We first collect state-action pairs for 100 episodes to train the “imitators”, whose network structures are the same as the corresponding agents. In the test time, we first run MAD attack on it the “imitator” and then input the generated perturbed observation to the agent in a transfer attack fashion. For our “optimal” policy attack, the hyperparameters are PPO training parameters for the adversary (including the learning rate of the adversary policy network, learning rate of the adversary value network, the entropy regularization parameter and the ratio clip $\epsilon$ for PPO). We use a grid search of these hyperparameters to train an adversary that is as strong as possible, resulting in 100 to 200 adversaries produced for each agent. The reported optimal attack rewards is the lowest reward among all trained adversaries. Under this comprehensive adversarial evaluation, each agent is tested using hundreds of adversaries and the strongest adversary determines the true robustness of an agent.
+
+Table 3: Average episode rewards $\pm$ standard deviation over 50 episodes on five baselines and SAPPO (Zhang et al., 2020b). We report natural episode rewards (no attacks) and episode rewards under six adversarial attacks, including a simple random noise attack, the critic based attack in (Pattanaik et al., 2018), MAD and RS attacks in Zhang et al. (2020b), Snooping attack proposed in Inkawhich et al. (2019), and the optimal attack proposed in this paper. In each row we bold the best (lowest) attack reward over all five attacks. The row for the most robust method is highlighted.
+
+<table><tr><td>Env.</td><td>l norm perturb-</td><td>Method</td><td>Natural</td><td></td><td></td><td>Attack Reward</td><td></td><td></td><td></td><td>Best</td></tr><tr><td rowspan="3"></td><td rowspan="3">ation budget ε</td><td></td><td>Reward</td><td>Critic</td><td>Random</td><td>MAD</td><td>Snooping</td><td>RS</td><td>“Optimal” 636±9</td><td>Attack</td></tr><tr><td>PPO (vanilla)</td><td>3167±542</td><td>1464 ±523</td><td>2101±793</td><td>1410± 655</td><td>2234±1103</td><td>794±238</td><td></td><td>636</td></tr><tr><td>SA-PPO (Zhang et al., 2020b)</td><td>3705±2</td><td>3789± 15</td><td>2710±801 2265±502</td><td>2652±835</td><td>2509±838</td><td>1130 ±42 1219± 174</td><td>1076± 791</td><td>1076</td></tr><tr><td rowspan="7">Hopper</td><td rowspan="7">0.075</td><td>Pattanaik et al. (2018) ATLA-PPO (MLP)</td><td>2755±582 2559±958</td><td>2681± 555 3497±556</td><td>2153±882</td><td>1395±337 1679±676</td><td>1349±436 1769±562</td><td>2329±870</td><td>291±7 976±40</td><td>291 976</td></tr><tr><td>PPO (LSTM)</td><td>3060± 639.3</td><td>2705±986</td><td>2410± 786</td><td>2397±905</td><td>2234±1103</td><td>811± 74</td><td>784±48</td><td>784</td></tr><tr><td>ATLA-PPO (LSTM)</td><td>3487± 452</td><td></td><td></td><td></td><td>3130±692</td><td>1567± 347</td><td>1224± 191</td><td></td></tr><tr><td></td><td></td><td>3524±550</td><td>3474± 401</td><td>3081±754</td><td></td><td></td><td></td><td>1224</td></tr><tr><td>ATLA-PPO (LSTM)+ SAReg</td><td>3291± 600</td><td>2073± 824</td><td>3165±576</td><td>2814± 725</td><td>2857±724</td><td>2244± 618</td><td>1772±802</td><td>1772</td></tr><tr><td>PPO (vanilla)</td><td>4472 ± 635</td><td>3424 ± 1295</td><td>3007 ± 1200</td><td>2869 ± 1271</td><td>2786±962</td><td>1336 ± 654</td><td>1086±516</td><td>1086</td></tr><tr><td>SA-PPO (Zhang et al.,2020b)</td><td>4487± 61 4058± 1410</td><td>4875± 30 4058± 1410</td><td>4867± 39</td><td>3668±1789</td><td>3928±1661 2568±2044</td><td>3808±138 1713 ±1807</td><td>2908± 1136 733±1012</td><td>2908 733</td></tr><tr><td rowspan="6">Walker2d 0.05</td><td>Pattanaik et al. (2018) ATLA-PPO (MLP)</td><td>3138 ± 1061</td><td></td><td>2840±2018</td><td>2927±1954 2596±1005</td><td></td><td></td><td></td><td></td></tr><tr><td>PPO (LSTM)</td><td>2785±1121</td><td>3243± 1004</td><td>3384 ± 1056</td><td></td><td>2571±1084 2286±1156</td><td>3367± 1020 1259±937</td><td>2213±915</td><td>2213</td></tr><tr><td></td><td>3920±129</td><td>2730± 1082</td><td>2578 ± 1007</td><td>2471±1109</td><td></td><td></td><td>1523± 869</td><td>1259</td></tr><tr><td>ATLA-PPO (LSTM)</td><td></td><td>3915± 274</td><td>3779 ± 541</td><td>3963 ± 36</td><td>3716±666</td><td>3219 ± 1132</td><td>3463± 1016</td><td>3219</td></tr><tr><td>ATLA-PPO (LSTM) +SA Reg PPO (vanilla)</td><td>3842± 475</td><td>3884± 132 4934± 1022</td><td>3927± 368</td><td>3836± 492</td><td>3742±629</td><td>3239±894</td><td>3663± 707</td><td>3239</td></tr><tr><td>SA-PPO (Zhang et al., 2020b)</td><td>5687 ± 758 4292±384</td><td>4805± 128</td><td>5261± 1005 4986±452</td><td>1759± 828 4662 ±522</td><td>3668±547</td><td>268 ±227 3412 ±1755</td><td>-872 ± 436</td><td>-872</td></tr><tr><td rowspan="7">Ant</td><td rowspan="7">0.15</td><td>Pattanaik et al. (2018)</td><td>3469± 1139</td><td></td><td></td><td>1427± 625</td><td>4079±768 1336±644</td><td>1289± 777</td><td>2511 ± 1117 -672± 100</td><td>2511 -672</td></tr><tr><td>ATLA-PPO (MLP)</td><td></td><td>3469± 1139</td><td>2346± 459</td><td></td><td></td><td>842± 143</td><td></td><td></td></tr><tr><td>PPO (LSTM)</td><td>4894± 123</td><td>4427± 104</td><td>4541 ± 691</td><td>1891± 885</td><td>2862±1137</td><td></td><td>33±327</td><td>33</td></tr><tr><td></td><td>5696± 165</td><td>5519±114</td><td>5475 ± 691</td><td>3800±363</td><td>3723±1168</td><td>1069 ± 382</td><td>-513 ±104</td><td>-513</td></tr><tr><td>ATLA-PPO (LSTM)</td><td>5612±130</td><td>5196±134</td><td>5390± 704</td><td>3903± 217</td><td>4455±677</td><td>1096± 329</td><td>716±256</td><td>716</td></tr><tr><td>ATLA-PPO (LSTM) +SA Reg</td><td>5359 ±153</td><td>5295± 165</td><td>5366± 104</td><td>5240±170</td><td>5135±413</td><td>4136± 149</td><td>3765± 101</td><td>3765</td></tr><tr><td>PPO (vanilla)</td><td>7117± 98</td><td>5761±119</td><td>5486± 1378</td><td>1836±866</td><td>1637±843</td><td>489± 758</td><td>-660±218</td><td>-660</td></tr><tr><td rowspan="6">HalfCheetah</td><td rowspan="6">0.15</td><td>SA-PPO (Zhang et al.,2020b) Pattanaik et al. (2018)</td><td>3632±20</td><td>3589±21</td><td>3619± 18</td><td>3624±23 1773± 1248</td><td>3616±21</td><td>3283±20</td><td>3028±23</td><td>3028</td></tr><tr><td>ATLA-PPO (MLP)</td><td>5241± 1162 5417±49</td><td>5440± 676</td><td>2910± 1694</td><td>4623 ±1146</td><td>1465±726</td><td>1602± 1157 2170± 2097</td><td>447± 192</td><td>447</td></tr><tr><td></td><td>5609±98</td><td>5134± 38 4294± 112</td><td>5388±34 5395± 158</td><td>4768±106</td><td>4167±1507 4088±748</td><td>2899 ± 2006</td><td>2709±80 -886±30</td><td>2170</td></tr><tr><td>PPO (LSTM) ATLA-PPO (LSTM)</td><td>5766±109</td><td>4008±1031</td><td>5685 ± 107</td><td>4807± 154</td><td>4906±182</td><td>3458± 1338</td><td>2485±1488</td><td>-886 2485</td></tr><tr><td>ATLA-PPO (LSTM) +SA Reg</td><td>6157±852</td><td>5991± 209</td><td>6164±603</td><td>5790±174</td><td>5785±671</td><td>4806±603</td><td>5058±718</td><td>4806</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# A.2 AGENT PERFORMANCE DURING TRAINING
+
+In Table 3 we only report the agent performance at the end of training. In this subsection, we evaluate our agent performance during $20 \%$ , $40 \%$ , $60 \%$ and $80 \%$ of total training epochs using Robust Sarsa (RS) attacks. The results are presented in Table 4. The overall trend is that agents are getting stronger over time (“RS attack reward” is increasing), achieving better robustness in later checkpoints.
+
+# A.3 NETWORK STRUCTURE
+
+For fully connected networks, we use the same network as in (Zhang et al., 2020b), which contains 2 hidden layers with 64 hidden neurons each layer, for both policy and value networks, for both the agent and the adversary. For LSTM agents, we use a single layer LSTM with 64 hidden neurons, along with an input embedding layer projecting state dimension to 64 and an output layer projecting 64 to output dimension. For LSTM agents, when conducting the “optimal” attack, we also use a LSTM network for the adversary to ensure the adversary is powerful enough.
+
+Table 4: Natural and RS attack rewards of ATLA-PPO $( { \mathrm { L S T M } } ) +$ SA Reg checkpoints during training. We report Average rewards $\pm$ standard deviation over 50 episodes.   
+
+<table><tr><td rowspan=1 colspan=1>Environment</td><td rowspan=1 colspan=1>Reward</td><td rowspan=1 colspan=1>20%</td><td rowspan=1 colspan=2>40%</td><td rowspan=1 colspan=1>60%</td><td rowspan=1 colspan=1>80%</td><td rowspan=1 colspan=1>100%</td></tr><tr><td rowspan=1 colspan=1>Hopper</td><td rowspan=1 colspan=1>NaturalRewardRS Attack Reward</td><td rowspan=1 colspan=1>3440±11716±82</td><td rowspan=1 colspan=2>1161 ±485631±51</td><td rowspan=1 colspan=1>3013±5841089±501</td><td rowspan=1 colspan=1>3569±1613181±634</td><td rowspan=1 colspan=1>3291±6002244±618</td></tr><tr><td rowspan=1 colspan=1>Walker2d</td><td rowspan=1 colspan=1>NaturalRewardRS Attack Reward</td><td rowspan=1 colspan=1>989±254882±269</td><td rowspan=1 colspan=2>3506±1741744±347</td><td rowspan=1 colspan=1>2203±988739±531</td><td rowspan=1 colspan=1>3803±7262550±1020</td><td rowspan=1 colspan=1>3842±4753239±894</td></tr><tr><td rowspan=2 colspan=1>Ant</td><td rowspan=2 colspan=1>NaturalRewardRS Attack Reward</td><td rowspan=2 colspan=1>2634±1222216±171</td><td rowspan=2 colspan=2>4532±1061903±93</td><td rowspan=2 colspan=1>5007±1433040±241</td><td rowspan=2 colspan=1>5127±5423040±241</td><td rowspan=1 colspan=1>5393±139</td></tr><tr><td rowspan=1 colspan=1>3040±241</td><td rowspan=1 colspan=1>4136±149</td></tr><tr><td rowspan=2 colspan=1>HalfCheetah</td><td rowspan=2 colspan=1>NaturalRewardRS Attack Reward</td><td rowspan=2 colspan=1>4525±1403986±564</td><td rowspan=2 colspan=2>5567±1383986±564</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>5955±177</td><td rowspan=1 colspan=1>5956±181</td></tr><tr><td rowspan=1 colspan=1>4911±923</td><td rowspan=1 colspan=1>4571±1314</td><td></td></tr></table>
+
+# A.4 HYPERPARAMETER FOR THE LEARNING-BASED “OPTIMAL” ATTACK
+
+Our “optimal” attacks uses policy gradient methods to learn the optimal adversary during agent testing, and each learning process involves the selection of hyperparameters. Specifically, the hyperparameters include the learning rates of the adversary’s policy and value networks, the entropy coefficient, and the annealing of the learning rate. To reduce search space, for ATLA agents, the learning rates of the testing phase adversary’s policy and value networks are chosen ranging from $0 . 3 \mathrm { X }$ to 3X of the learning rates of adversary’s policy and value networks used in training. For other agents trained without an adversary, the learning rates of the testing phase adversary’s policy and value networks are chosen ranging from 0.3X to 3X of the learning rates of the agent’s policy and value networks. We tested both linearly annealed learning rate and non-annealing learning rate. The adversary’s entropy coefficient is chosen form 0 and 0.003. The final results reported in all tables are the best (lowest) reward achieved by the “optimal” attacks among all hyperparameter configurations. Typically this includes around 100 to 200 different adversaries trained with different hyperparameters. This guarantees the strength of this attack and allows a comprehensive evaluation of the robustness of all agents.
+
+# A.5 HYPERPARAMETERS FOR ATLA PERFORMANCE EVALUATION
+
+Hyperparameters for PPO (vanilla) For the Walker2d and Hopper environment, we use the same set of hyperparameters as in (Zhang et al., 2020b); the hyperparameters were originally from (Engstrom et al., 2020) and found using a grid search experiment. We found that this set of hyperparameters work well. For HalfCheetah and Ant environment, we use a grid search of hyperparameters, including the learning rate of the policy network, learning rate of the value network and the entropy bonus coefficient. For Hopper, Walker2d and HalfCheetah environments, we train for 2 million steps (2 million environment interactions). For Ant, we train for 10 million steps. Training for longer may slightly improve agent performance under no attacks, but has no impact for performance under strong adversarial attacks.
+
+Hyperparameters for PPO (LSTM) For PPO (LSTM), we conduct a smaller scale hyperparameter search. We search hyperparameter values that are close to the optimal ones found for the PPO vanilla agent. We train these LSTM agents for the same steps as those in vanilla PPO.
+
+Hyperparameters for SA-PPO We use the same value for all hyperparameters as in vanilla PPO except SA-PPO’s extra $\kappa$ for the strength of SA-PPO regularization. For $\kappa$ , we choose from $1 \times 1 0 ^ { - 6 }$ to 1. We train agents with each $\kappa \ : 2 1$ times and choose the $\kappa$ value whose median agent has the highest worst-case reward under all attacks.
+
+Hyperparameters for ATLA-PPO For ATLA-PPO, we have hyperparameters for both agent and adversary. We keep all agent hyperparameters the same as those in vanilla MLP/LSTM agents, except for the entropy bonus coefficient. We find that sometimes we need a larger entropy bonus coefficient in ATLA to allow sufficient exploration of the agent, as learning with an adversary is harder than learning in attack-free environments. For the adversary, we run a small-scale hyperparameter search on the learning rate of adversary policy and value networks, and the entropy bonus coefficient for the adversary. To reduce the number of hyperparameters for searching, we use values close to those of the agent. We set $N _ { \nu } = N _ { \pi } = 1$ in all experiments and did not tune this hyperparameter. For ATLA, we train 5 million steps for Hopper, Walker and HalfCheetah and 10 million steps for Ant. We find that similar to the observations in (Madry et al., 2018), training with an adversary typically requires more steps to converge, however in all our environments the training does reliably converge.
+
+Table 5: Hyperparameters for all environments and settings. For vanilla environments, we use the hyperparameters from Zhang et al. (2020b) and Engstrom et al. (2020) if they are available for that environment (Hopper and Walker2d). Other environments’ hyperparameter for the vanilla PPO model is found by a grid search. For SA-PPO and ATLA-PPO (MLP), the same set of hyperparameters as in the vanilla models are used, except that for SA-PPO we tune the parameter $\kappa$ and for ATLA-PPO (MLP) we tune the entropy bonus coefficients as well as learning rates for the adversary. For LSTM models, we first tune the vanilla LSTM PPO models and find the best learning rates, keep using them in all LSTM based models.   
+
+<table><tr><td>Env.</td><td>model</td><td>policy lr</td><td>val lr</td><td>entropy coeff.</td><td>K</td><td>adv. policy lr</td><td>adv.val lr</td><td>adv. entropy coeff.</td></tr><tr><td rowspan="6">Hopper</td><td>PPO(vanilla)</td><td>3e-4</td><td>2.5e-4</td><td>0</td><td>1</td><td>1</td><td>=</td><td>1</td></tr><tr><td>SA-PPO</td><td>3e-4</td><td>2.5e-4</td><td>0</td><td>0.03</td><td></td><td></td><td></td></tr><tr><td>ATLA-PPO (MLP)</td><td>3e-4</td><td>2.5e-4</td><td>0.01</td><td>1</td><td>0.001</td><td>0.0001</td><td>0.001</td></tr><tr><td>PPO (LSTM)</td><td>1e-3</td><td>3e-4</td><td>0.0</td><td>1</td><td>一</td><td></td><td></td></tr><tr><td>ATLA-PPO (LSTM)</td><td>1e-3</td><td>3e-4</td><td>0.01</td><td>1</td><td>0.01</td><td>0.01</td><td>0.001</td></tr><tr><td>ATLA-PPO (LSTM)+ SA Reg</td><td>1e-3</td><td>3e-4</td><td>0.01</td><td>0.3</td><td>0.003</td><td>0.01</td><td>0.003</td></tr><tr><td rowspan="6">Walker2d</td><td>PPO(vanilla)</td><td>4e-4</td><td>3e-4</td><td>0</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>SA-PPO</td><td>4e-4</td><td>3e-4</td><td>0</td><td>1</td><td>一</td><td>一</td><td>一</td></tr><tr><td>ATLA-PPO (MLP)</td><td>4e-4</td><td>3e-4</td><td>0.0003</td><td>1</td><td>0.0001</td><td>0.0001</td><td>0.002</td></tr><tr><td>PPO (LSTM)</td><td>1e-3</td><td>3e-2</td><td>0</td><td>1</td><td></td><td></td><td></td></tr><tr><td>ATLA-PPO (LSTM)</td><td>1e-3</td><td>3e-2</td><td>0.001</td><td>一</td><td>0.0003</td><td>0.03</td><td>0</td></tr><tr><td>ATLA-PPO (LSTM)+ SA Reg</td><td>1e-3</td><td>3e-2</td><td>0.001</td><td>0.3</td><td>0.003</td><td>0.03</td><td></td></tr><tr><td rowspan="6">Ant</td><td>PPO(vanilla)</td><td>5e-5</td><td>1e-5</td><td>0</td><td></td><td></td><td></td><td>0.001</td></tr><tr><td>SA-PPO</td><td>5e-5</td><td>1e-5</td><td>0</td><td>1 3e-3</td><td>1</td><td>1</td><td>1</td></tr><tr><td>ATLA-PPO (MLP)</td><td>5e-5</td><td>1e-5</td><td>3e-4</td><td>1</td><td>1e-05</td><td>3e-06</td><td>二 0</td></tr><tr><td>PPO (LSTM)</td><td>3e-4</td><td>3e-4</td><td>0</td><td>1</td><td></td><td>一</td><td>1</td></tr><tr><td>ATLA-PPO (LSTM)</td><td>3e-4</td><td>3e-4</td><td>0.0003</td><td>1</td><td>0.0003</td><td>0.0001</td><td>0.0003</td></tr><tr><td>ATLA-PPO (LSTM)+ SA Reg</td><td>3e-4</td><td>3e-4</td><td>0.003</td><td>0.1</td><td>0.0003</td><td>3e-05</td><td>3e-05</td></tr><tr><td rowspan="6">HalfCheetah</td><td>PPO(vanilla)</td><td>3e-4</td><td>1e-4</td><td>0</td><td>1</td><td>二</td><td>一</td><td></td></tr><tr><td>SA-PPO</td><td>3e-4</td><td>1e-4</td><td>0.1</td><td>1</td><td>一</td><td></td><td>1 1</td></tr><tr><td>ATLA-PPO (MLP)</td><td>3e-4</td><td>1e-4</td><td>0.0003</td><td>1</td><td>0.001</td><td>0.0003</td><td>0.003</td></tr><tr><td>PPO (LSTM)</td><td>1e-3</td><td>3e-4</td><td>0</td><td>1</td><td></td><td></td><td>二</td></tr><tr><td>ATLA-PPO (LSTM)</td><td>1e-3</td><td>3e-4</td><td>0.0003</td><td>一</td><td>0.003</td><td>0.001</td><td>0</td></tr><tr><td>ATLA-PPO (LSTM)+ SA Reg</td><td>1e-3</td><td>3e-4</td><td>0</td><td>0.03</td><td>0.003</td><td>0.003</td><td>0.0003</td></tr></table>
+
+Agent selection For each setup, we repeat the experiments using the same set of hyperparameters for 21 times due to the high performance variance in RL. We then attack all the agents using random, critic, MAD and RS attacks. We use the lowest reward among all attacks as a metric to rank those agents. Then, we select the agent with median robustness as our final agent. This final agemt is then attacked using the “optimal” attack to further reduce its reward. The numbers we report in Table 2 are not from the best runs, but the runs with median robustness. This is done to improve reproducibility as RL training process can have high variance.