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+# A RISK-SENSITIVE POLICY GRADIENT METHOD
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Standard deep reinforcement learning (DRL) agents aim to maximize expected reward, considering collected experiences equally in formulating a policy. This differs from human decision-making, where gains and losses are valued differently and outlying outcomes are given increased consideration. It also wastes an opportunity for the agent to modulate behavior based on distributional context. Several approaches to distributional DRL have been investigated, with one popular strategy being to evaluate the projected distribution of returns for possible actions. We propose a more direct approach, whereby the distribution of full-episode outcomes is optimized to maximize a chosen function of its cumulative distribution function (CDF). This technique allows for outcomes to be weighed based on relative quality, does not require modification of the reward function to modulate agent behavior, and may be used for both continuous and discrete action spaces. We show how to achieve an asymptotically consistent estimate of the policy gradient for a broad class of CDF-based objectives via sampling, subsequently incorporating variance reduction measures to facilitate effective on-policy learning. We use the resulting algorithm to train agents with different “risk profiles” in penalty-based formulations of six OpenAI Safety Gym environments, observing that moderate emphasis on improvement in training scenarios where the agent performs poorly both increases the accumulation of positive rewards and decreases the frequency of incurred penalties. We found that, in all environments tested, the same risk profile can be used to produce both stronger overall performance than standard Proximal Policy Optimization (PPO) and higher levels of positive reward than PPO constrained by Lagrangians to maintain the same cost levels.
+
+# 1 INTRODUCTION
+
+While deep reinforcement learning (DRL) has been used to master an impressive array of simulated tasks in controlled settings, it has not yet been widely adopted for high-stakes, real-world applications. One reason for this gap is the lack of distributional perspective in standard artificial agents. Endowing agents with such perspective could make their decision-making more robust, potentially leading to increased safety, increased trust from humans, and more widespread real-world adoption.
+
+In reinforcement learning (RL), risk arises due to uncertainty around the possible outcomes of an agent’s future actions. It is a result of randomness in the operating environment, mismatch between training and test conditions, and the inherent randomness of a stochastic policy. Risk-sensitive policies, or those that consider more than a mean over the distribution of possible outcomes, offer the potential for added robustness under uncertain and dynamic conditions. There is an evolving landscape of algorithmic paradigms for handling risk in RL, from constraint-based approaches adapted from optimal control (Achiam et al., 2017; Chow et al., 2019; Ray et al., 2019; Zhong et al., 2020) to adversarial approaches emerging from AI Safety (Garc´ıa & Fernandez, 2015; Amodei et al., 2016). ´ Within this landscape, learning approaches that optimize distributional measures offer the ability to express design preferences over the full distribution of potential outcomes, through the specification of a risk-sensitivity criterion.
+
+Distributional RL has been studied for value-based methods, with a popular strategy being to use the distributional Bellman equation to estimate the distribution of Q-values for each member of a discrete set of potential actions (Bellemare et al., 2017; Dabney et al., 2018a;b). However, distributional RL has not been widely explored for policy gradient methods, which could permit direct optimization of risk-sensitive measures and naturally accommodate both discrete and continuous action spaces.
+
+In the following, we introduce a novel framework for risk-sensitive learning using policy gradients. Our approach allows agents to be trained with different risk profiles through design-time specification of both utility and weight functions, with the latter being defined over the estimated distribution of full-episode rewards. This framework enables agent-based learning that captures aspects of human decision-making, such as overemphasis of rare occurrences and diminishing marginal utility relative to a reference outcome (Kahneman & Tversky, 1979). It also allows implementation of another key strategy of human learning: emphasizing improvement on tasks where one is deficient. We demonstrate the ability of our algorithm to use this strategy to improve performance relative to both unconstrained and constrained methods in six OpenAI Safety Gym environments (Ray et al., 2019).
+
+# 2 RELATED WORK
+
+Constrained RL offers a set of approaches to safe exploration (Garc´ıa & Fernandez (2015)) that aim ´ to enforce explicit constraints throughout the learning process via methods including Lagrangian constraints (Ray et al. (2019)) and constraint coefficients (Achiam et al. (2017)). Differing from the safe exploration scenario, we here consider problems with distinct training and test phases where agent performance is to be evaluated. Our experiments indicate that risk-sensitive learning can offer performance improvements over constrained learning in scenarios where safety constraints need not be enforced during training.
+
+Distributional approaches to risk-sensitive RL have primarily been explored in the value-based setting. Therein, the value distribution has been explicitly modeled through categorical techniques (Bellemare et al., 2017) or quantile regression (Dabney et al., 2018a) and used to improve both value predictions and overall performance. Recent works utilize distributional modeling in the actor-critic setting to enable application to continuous action spaces, again demonstrating improved performance over baseline approaches (Ma et al., 2020; Zhang et al., 2021; Duan et al., 2021). In value-based approaches, risk-sensitivity criteria are applied at run time as a nonlinear warping of the estimated value distribution.
+
+Policy gradient approaches offer additional promise for risk-sensitive RL, but require direct optimization of a parameterized policy with respect to a distributional objective. Some existing methods are limited to a specific class of learning objective, such as the set of concave risk measures that allow a globally-optimal solution (Tamar et al., 2015; Zhong et al., 2020). Others allow a broader class of measures but are more restrictive in the class of policies that can be represented (Prashanth et al., 2016). We aim for a risk-sensitive policy gradient approach that both offers significant flexibility in the choice of learning objective and can learn policies parameterized by a deep neural network.
+
+Various measures have been considered in the context of risk-sensitive RL, including exponential utility (Pratt, 1964), percentile performance criteria (Wu & Lin, 1999), value-at-risk (Leavens, 1945), conditional value-at-risk (Rockafellar & Uryasev, 2000), and prospect theory (Kahneman & Tversky, 1979). In this work, we consider a class of risk-sensitivity measures motivated by Cumulative Prospect Theory (CPT) (Tversky & Kahneman, 1992). CPT uniquely models two key aspects of human decision-making: (1) a utility function $u$ , computed relative to a reference point that induces more risk-averse behavior in the presence of gains than losses and (2) a weight function $w$ that prioritizes outlying events. Specific forms of $u$ and $w$ are given in Tversky & Kahneman (1992) (and in Appendix A.4), but the general form of CPT admits a wide variety of risk-sensitive objectives.
+
+Here we show how to train agents to optimize this class of objectives through sampling-based estimation of their policy gradients and requisite variance reduction. The final algorithm resembles well-known on-policy approaches such as Proximal Policy Optimization (Schulman et al., 2017b) and is similarly widely applicable. Although we do not explore it here, the incorporation of an appropriate risk-sensitivity criterion could additionally enable risk-aware exploration and adversarial training for increased robustness (Pinto et al., 2017; Parisi et al., 2019; Zhang et al., 2020).
+
+# 3 RISK-SENSITIVE POLICY OPTIMIZATION
+
+In this section we formalize the class of distributional objectives to be considered, derive a samplingbased approximation of its policy gradient, enact variance reduction on this estimate, and use the result to produce a practical learning algorithm.
+
+# 3.1 PRELIMINARIES: PROBLEM AND NOTATION
+
+Standard deep reinforcement learning seeks to maximize the expected reward of an agent over encountered trajectories; that is, it maximizes the objective
+
+$$
+J ( \theta ) = E _ { \tau \sim p _ { \theta } ( \tau ) } \Big [ \sum _ { t } r ( \mathbf { s } _ { t } , \mathbf { a } _ { t } ) \Big ] .
+$$
+
+Here $p _ { \theta } ( \tau )$ is the distribution over trajectories $\tau \equiv \mathbf { s } _ { 1 } , \mathbf { a } _ { 1 } , \dots , \mathbf { s } _ { T } , \mathbf { a } _ { T }$ induced by a policy parameterized by $\theta$ ; $\mathbf { s } _ { t } , \mathbf { a } _ { t }$ , and $r ( \mathbf { s } _ { t } , \mathbf { a } _ { t } )$ denote the state, action, and reward at time $t$ , respectively. To enable the incorporation of distributional context, we instead consider the objective
+
+$$
+J ( \theta ) = \int _ { - \infty } ^ { + \infty } u ( r ( \tau ) ) \frac { d } { d r ( \tau ) } \biggl ( w ( P _ { \theta } ( r ( \tau ) ) \biggr ) d r ( \tau ) ,
+$$
+
+where $u ( r ( \tau ) )$ is the utility associated with full-trajectory reward $\begin{array} { r } { r ( \tau ) \equiv \sum _ { t } r ( \mathbf { s } _ { t } , \mathbf { a } _ { t } ) } \end{array}$ and $w$ is a piecewise differentiable weighting function of the CDF of trajectory reward $P _ { \theta } ( r ( \tau ) ) ~ =$ $\int _ { - \infty } ^ { r ( \tau ) } p _ { \theta } ( r ^ { \prime } ) d r ^ { \prime }$ .
+
+Equation 2 is inspired by CPT (Tversky & Kahneman, 1992), which includes a pair of integrals of this form. It was chosen for its generality; by using different utility functions $u$ and weight functions $w$ one may represent all of the risk measures mentioned in Section 2 and all of the risk measures evaluated by Dabney et al. (2018a). The form (2) reduces to (1) when $u$ and $w$ are both the identity mapping. While designed for the episodic setting, the objective (2) may be considered for infinite horizons through the choice of appropriately long windows.
+
+# 3.2 RISK-SENSITIVE POLICY GRADIENT
+
+To optimize the objective (2), we first derive an approximation to its gradient with respect to the policy parameters $\theta$ . Working toward a representation that can be sampled, we assert the independence of the reward on $\theta$ and use the chain rule to write
+
+$$
+\nabla _ { \theta } J ( \theta ) = \int _ { - \infty } ^ { \infty } u ( r ( \tau ) ) \frac { d } { d r ( \tau ) } \biggl ( w ^ { \prime } ( P _ { \theta } ( r ( \tau ) ) ) \nabla _ { \theta } P _ { \theta } ( r ( \tau ) ) \biggr ) d r ( \tau ) ,
+$$
+
+where $w ^ { \prime }$ is the derivative of $w$ with respect to $P _ { \theta } ( r ( \tau ) )$ . The gradient of the CDF may be written as follows:
+
+$$
+\begin{array} { l } { \displaystyle \nabla _ { \theta } P _ { \theta } ( r ( \tau ) ) = \nabla _ { \theta } \int _ { - \infty } ^ { r ( \tau ) } p _ { \theta } ( r ^ { \prime } ) d r ^ { \prime } = \nabla _ { \theta } \int _ { \tau ^ { \prime } } H ( r ( \tau ) - r ( \tau ^ { \prime } ) ) p _ { \theta } ( \tau ^ { \prime } ) d \tau ^ { \prime } } \\ { \displaystyle = \int _ { \tau ^ { \prime } } H ( r ( \tau ) - r ( \tau ^ { \prime } ) ) \nabla _ { \theta } p _ { \theta } ( \tau ^ { \prime } ) d \tau ^ { \prime } = \int _ { \tau ^ { \prime } } H ( r ( \tau ) - r ( \tau ^ { \prime } ) ) p _ { \theta } ( \tau ^ { \prime } ) \nabla _ { \theta } \log p _ { \theta } ( \tau ^ { \prime } ) d \tau ^ { \prime } . } \end{array}
+$$
+
+Here the first equality is the integral representation of $P _ { \theta } ( r ( \tau ) )$ , the second uses the Heaviside step function to select all trajectories with total reward $\leq r ( \tau )$ , the third follows from the independence of reward on $\theta$ , and the fourth follows from the expression for the derivative of the natural logarithm. In the following, we also use the complementary expression
+
+$$
+\nabla _ { \theta } P _ { \theta } ( r ( \tau ) ) = \nabla _ { \theta } \bigg ( 1 - \int _ { r ( \tau ) } ^ { \infty } p _ { \theta } ( r ^ { \prime } ) d r ^ { \prime } \bigg ) = - \int _ { \tau ^ { \prime } } H ( r ( \tau ^ { \prime } ) - r ( \tau ) ) p _ { \theta } ( \tau ^ { \prime } ) \nabla _ { \theta } \log p _ { \theta } ( \tau ^ { \prime } ) d \tau ^ { \prime } .
+$$
+
+Either form, or a combination of the two, may be substituted into (3) and the result sampled over $N$ trajectories by first ordering trajectories $i = 1 \ldots N$ by increasing reward $r ( \tau )$ . Then
+
+$$
+\nabla _ { \theta } J ( \theta ) \approx \sum _ { i = 1 } ^ { N } u ( r ( \tau _ { i } ) ) \bigg ( w ^ { \prime } \bigg ( \frac { i } { N } \bigg ) \nabla _ { \theta } P _ { \theta } ( r ( \tau _ { i } ) ) - w ^ { \prime } \bigg ( \frac { i - 1 } { N } \bigg ) \nabla _ { \theta } P _ { \theta } ( r ( \tau _ { i - 1 } ) ) \bigg ) ,
+$$
+
+where the term $w ^ { \prime } ( 0 ) \nabla _ { \theta } P _ { \theta } \big ( r ( \tau _ { 0 } ) \big ) \equiv 0$ . This ordering scheme produces an asymptotically consistent estimate, as shown in the context of CPT value estimation by Prashanth et al. (2016). $\nabla _ { \theta } P _ { \theta } ( r ( \tau _ { i } ) )$ may be sampled in one of two ways, based on either (4) or (5):
+
+$$
+\nabla _ { \theta } P _ { \theta } ( r ( \tau _ { i } ) ) \approx \frac { 1 } { N } \sum _ { j = 1 } ^ { i } \sum _ { t = 1 } ^ { T _ { j } } \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) \approx - \frac { 1 } { N } \sum _ { j = i + 1 } ^ { N } \sum _ { t = 1 } ^ { T _ { j } } \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) .
+$$
+
+The expression (6) may be used to train a policy that optimizes the distributional objective (2) in a manner similar to REINFORCE (Williams, 1992).
+
+# 3.3 VARIANCE REDUCTION
+
+Reducing the variance of sample-based gradient estimates enables faster learning. Here we take several steps to reduce the variance of (6), as has been done with the policy gradient estimate of REINFORCE (Williams, 1992).
+
+First, note that cross-trajectory terms of the form $f ( \tau _ { i } , \mathbf { a } _ { j , t } , \mathbf { s } _ { j , t } ) \stackrel { } { = } u ( r ( \tau _ { i } ) ) \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } )$ while nonzero, do not contribute to the gradient estimate in expectation when $i \neq j$ . A proof of this assertion is given in Appendix A.1. Using (4) for the first term of (6) and (5) for the second allows us to write
+
+$$
+\begin{array} { r l r } {  { \nabla _ { \theta } J ( \theta ) \approx \sum _ { i = 1 } ^ { N } u ( r ( \tau _ { i } ) ) \bigg ( w ^ { \prime } \bigg ( \frac { i } { N } \bigg ) \frac { 1 } { N } \sum _ { j = 1 } ^ { i } \sum _ { t = 1 } ^ { T _ { j } } \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) } } \\ & { } & { \qquad + w ^ { \prime } \bigg ( \frac { i - 1 } { N } \bigg ) \frac { 1 } { N } \sum _ { j = i } ^ { N } \sum _ { t = 1 } ^ { T _ { j } } \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) \bigg ) . } \end{array}
+$$
+
+Removing cross-trajectory terms gives
+
+$$
+\overline { { \nabla _ { \theta } J ( \theta ) } } \approx \frac { 1 } { N } \sum _ { i = 1 } ^ { N } u ( r ( \tau _ { i } ) ) \bigg ( \bigg ( w ^ { \prime } \bigg ( \frac { i } { N } \bigg ) + w ^ { \prime } \bigg ( \frac { i - 1 } { N } \bigg ) \bigg ) \sum _ { t = 1 } ^ { T _ { i } } \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } ) .
+$$
+
+Note that the weight coefficients $\begin{array} { r } { ( w ^ { \prime } ( \frac { i } { N } ) + w ^ { \prime } ( \frac { i - 1 } { N } ) ) } \end{array}$ should be normalized over each batch. The expression (9) is equivalent to (6) in expectation, but with reduced variance (see Appendix A.1 for justification). It has a clear intuition – trajectories are assigned utilities based on their rewards and their contributions to the gradient are scaled by the derivative of the weight function, just as they are in Cumulative Prospect Theory (Tversky & Kahneman, 1992).
+
+Standard variance reduction techniques may be applied to this simplified form. Without further assumption or introduction of additional bias, a static baseline $b$ can be employed:
+
+$$
+\nabla _ { \theta } J ( \theta ) \approx \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \left( u ( r ( \tau _ { i } ) ) - b \right) \left( w ^ { \prime } \left( \frac { i } { N } \right) + w ^ { \prime } \left( \frac { i - 1 } { N } \right) \right) \sum _ { t = 1 } ^ { T _ { i } } \nabla _ { \theta } \log \pi _ { \theta } \big ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } \big )
+$$
+
+Justification for this assertion is given in Appendix A.2. Learning may be further improved if we additionally assume that utility may be allocated on a per-step basis. In this case, per-step utilities are computed as the difference between what the full-episode utility would be if the episode were to end at a given time step and what it would have been had the episode ended at the previous time step. While not applicable in cases where episode utility is adjusted based on final outcome, this assumption has the significant benefit of modeling the temporal allocation of rewards and aligns with the standard formulation of RL. With it, the variance of (9) may be further reduced through the incorporation of utility-to-go and a state-dependent baseline $V _ { \phi } ( \mathbf { s } _ { i , t } )$ :
+
+$$
+\nabla _ { \theta } J ( \theta ) \approx \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \left[ w ^ { \prime } \left( \frac { i } { N } \right) + w ^ { \prime } \left( \frac { i - 1 } { N } \right) \right] \sum _ { t = 1 } ^ { T _ { i } } \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } ) \left[ \sum _ { t ^ { \prime } = t } ^ { T _ { i } } u ( \mathbf { s } _ { i , t ^ { \prime } } , \mathbf { a } _ { i , t ^ { \prime } } ) - V _ { \phi } ( \mathbf { s } _ { i , t } ) \right] .
+$$
+
+Here $u ( \mathbf { s } _ { i , t ^ { \prime } } , \mathbf { a } _ { i , t ^ { \prime } } )$ is the per-step utility. The value function $V _ { \phi } ( \mathbf { s } _ { i , t } )$ is parameterized by $\phi$ and trained via regression to minimize
+
+$$
+\mathcal { L } ( \phi ) = \sum _ { i , t } \bigg ( V _ { \phi } ( \mathbf { s } _ { i , t } ) - \sum _ { t ^ { \prime } = t } ^ { T _ { i } } u ( \mathbf { s } _ { i , t ^ { \prime } } , \mathbf { a } _ { i , t ^ { \prime } } ) \bigg ) ^ { 2 } .
+$$
+
+A standard argument, similar to the approach taken in (Achiam, 2018), can be used to show that the incorporation of utility-to-go does not change the expected value of (9). The addition of a state-dependent baseline also does not introduce additional bias, as justified in Appendix A.2.
+
+Finally, discount factors, bootstrapping, and trust regions may be used to provide additional variance reduction, just as they are in conventional on-policy learning (Appendix A.2). These measures may introduce additional bias to the policy gradient estimate, but typically lead to more sampleefficient learning. In our experiments, we evaluate the use of generalized advantage estimation (GAE; (Schulman et al., 2016)) based on the utility-to-go as well as the clipping-based trust regions of Proximal Policy Optimization (PPO; (Schulman et al., 2017b)). Incorporating these in the policy gradient yields
+
+$$
+\nabla _ { \theta } J ( \theta ) \approx \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \left( w ^ { \prime } \left( \frac { i } { N } \right) + w ^ { \prime } \left( \frac { i - 1 } { N } \right) \right) \sum _ { t = 1 } ^ { T _ { i } } \nabla _ { \theta } L _ { \mathrm { c l i p } } \bigg ( \log \pi _ { \theta } ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } ) , A _ { u } ^ { \pi } ( \mathbf { s } _ { i , t } , \mathbf { a } _ { i , t } ) \bigg ) ,
+$$
+
+where $A _ { u } ^ { \pi } ( \mathbf { s } _ { i , t } , \mathbf { a } _ { i , t } )$ is the standard GAE except with per-step utilities in place of rewards. Trust regions are implemented similarly to PPO, pessimistically clipping policy updates to be within a multiplicative factor of $1 \pm \epsilon$ of the existing policy:
+
+$$
+\begin{array} { r l } & { L _ { \mathrm { c l i p } } = \operatorname* { m i n } \Bigg ( \log \pi _ { \boldsymbol \theta } \big ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } \big ) A _ { u } ^ { \pi } \big ( \mathbf { s } _ { i , t } , \mathbf { a } _ { i , t } \big ) , } \\ & { \qquad \log \bigg ( \mathrm { c l i p } \bigg ( \frac { \pi _ { \boldsymbol \theta } \big ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } \big ) } { \pi _ { \theta _ { \mathrm { o l d } } } \big ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } \big ) } , 1 \pm \epsilon \bigg ) \pi _ { \theta _ { \mathrm { o l d } } } \big ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } \big ) \Bigg ) A _ { u } ^ { \pi } \big ( \mathbf { s } _ { i , t } , \mathbf { a } _ { i , t } \big ) \Bigg ) . } \end{array}
+$$
+
+They are used to perform multiple policy updates using the same batch of data, providing learning that is no longer strictly on-policy but that can be significantly more sample efficient. When following this route, we apply the same early stopping mechanism, based on the Kullback-Leibler divergence $( D _ { \mathrm { K L } } )$ between old and new policies, as was used by Ray et al. (2019).
+
+# 3.4 LEARNING ALGORITHM
+
+The above sample-based estimate of the policy gradient may be used to train agents to maximize distributional objectives of the form (2). The resulting method, Cumulative Prospect Proximal Policy Optimization (C3PO), is given in Algorithm 1 and mirrors standard on-policy learning.
+
+<table><tr><td>Algorithm 1 Cumulative Prospect Proximal Policy Optimization (C3PO)</td></tr><tr><td>Require: Policy: initial parameters 0o,learning rate αθ,updates per batch Mθ</td></tr><tr><td>Require: Value: initial parameters o,learning rate α𝜙,updates per batch MΦ Require:Early stopping threshold DKL,stop, discount factor y</td></tr><tr><td>for k = 1,2,... do</td></tr><tr><td>Collect set of episodes Dk = {Ti} by running policy π(0k) in the environment</td></tr><tr><td>Compute per-step utilities u(Si,t, ai,t)</td></tr><tr><td>Fit value function by regression:</td></tr><tr><td>for m=1,...M do</td></tr><tr><td>(v(i)-∑（)） ←Φ+aV∑T∑it(</td></tr><tr><td>end for</td></tr><tr><td></td></tr><tr><td>Update utility-based advantage estimates A&quot;(s,a), using new V(s)</td></tr><tr><td>Compute weight coefficients based on ordered episode outcomes and normalize</td></tr><tr><td>Update policy, using KL-based early stopping:</td></tr><tr><td>for m=1,...Mθ do if DKL(πθ/πold）&lt;DkL, stop then</td></tr><tr><td>N</td></tr><tr><td>θ←θ+aθ∑1(w′（）+w′（1)）∑=1 VLeip(logθ(ai,tlsi,t),A（(sit,ai,t)) else</td></tr></table>
+
+![](images/77aa2a8e4e8903a0e8c73d9178f7e58cb69c1b08ab6f8e6fde7356f5b778ee51.jpg)  
+Figure 1: Example weight functions and their resulting coefficients in the policy gradient estimate (9). In these plots, outcomes increase in quality from left to right. As in CPT (Tversky & Kahneman, 1992), weight coefficients are proportional to the derivative of the weight function.
+
+Beyond the utility and weight components, Algorithm 1 differs from conventional methods in the requirement to collect full episodes of data in each batch. This requirement can be removed if outcomes can be defined over partial rather than full episodes, an assumption that is often viable and matches human decision-making. For instance, while out of scope for this work, our approach could be applied to the Atari suite (Bellemare et al., 2013) by considering the outcomes of fixed-length windows and restructuring Algorithm 1 to mimic minibatch PPO (Schulman et al., 2017b).
+
+# 4 EXPERIMENTS
+
+To evaluate our approach, we sought to both establish that it can effectively optimize different distributional objectives and explore the impact of using different objectives on agent outcomes. We found the OpenAI Safety Gym (Ray et al., 2019) to be suitable for these purposes. Safety Gym is a configurable suite of continuous, multidimensional control tasks wherein different types of robots must navigate through obstacles with different dynamics to perform different tasks. By including both positive and negative events in each training scenario, it allowed us to evaluate how our various agents handled risk. Safety Gym is also highly stochastic: the locations of the goals and obstacles are randomized, leading to outcome variability and forcing the agent to learn a generalized navigation strategy.
+
+Safety Gym logs adverse events but does not incorporate them into the reward function. As our method relies solely on the training signal from the reward, we assigned each logged adverse event a fixed, negative reward contribution in experiments using it or other unconstrained agents. Our initial experiments were conducted with a reward contribution of $- 0 . 0 2 5$ , which was found to allow agents to prioritize reaching goals but deter them from collisions with obstacles. To further emphasize obstacle avoidance, we doubled this contribution to $- 0 . 0 5$ in our experiments using cautious weightings. These choices and the role they play are further discussed in Section 5.
+
+To highlight distributional differences, we focused on the publicly available, obstacle-rich level 2 environments.1 Avoiding the longer compute time of the “Doggo” robot, we evaluated the “Point” and “Car” robots on each task (“Goal”, “Button”, and “Push”). Further details on these environments and our rationale for choosing them are given in Appendix A.3.
+
+In all experiments, we evaluated five random seeds and matched the hyperparameters used in the baselines accompanying Safety Gym (Ray et al., 2019) as closely as possible. The neural networks used to model both policy and value were multilayer perceptrons (MLPs), with two hidden layers of 256 units each and tanh activations. As in Ray et al. (2019), the policy network outputs the mean values of a multivariate gaussian with diagonal covariance. The control variances are optimized but independent of state. The full complement of variance reduction measures were used throughout; see Appendix A.4 for experimental justification of this choice.
+
+# 4.1 DIFFERING OBJECTIVES
+
+Agent performance was explored under four different distributional objectives. In addition to expected reward and CPT (configured to match the original form of Tversky & Kahneman (1992) and as given in Appendix A.4), we optimized for cautious ( $\gamma = 0 . 7 5 )$ and aggressive $( \eta = - 0 . 7 5 )$ versions of the distortion risk measure proposed in Wang (2000). This measure is defined as $w ( p ) = \Phi ( \Phi ^ { - 1 } ( p ) + \eta )$ where $\Phi$ and $\Phi ^ { - 1 }$ are the standard normal cumulative distribution function and its inverse. While we found this form to be convenient, the “Pow” metric in Dabney et al. (2018a) or any other set of similarly shaped $w$ curves should achieve a similar effect. In experiments using the objective from Tversky & Kahneman (1992), the reference point was taken to be the mean episode reward of the current batch, matching the tendency of humans to change their standards over time. The four weight functions and their resulting coefficients in (9) are shown in Figure 1.
+
+![](images/e0e22e1a977d26c084e5a479a42b9426590fced223729c1eece0c6e3436890d2.jpg)  
+Figure 2: Impact of different distributional objectives in one environment (CarButton2). The shading in the first 2 plots (and subsequent learning curves) reflects the standard deviation associated with running over 5 random seeds. Left: Net reward (positive reward minus penalty) throughout learning. Middle: Average number of cost events per episode during training (lower is better). Right: Agent outcome distribution in testing (with sampling turned off). The cautious (Wang $( \eta = 0 . 7 5 )$ ) weighting shows higher reward and lower cost once trained.
+
+Plots of the total rewards (including penalties) in training, average cost events per episode in training, and outcome distributions in testing are shown for one environment in Figure 2 and for two additional environments in Appendix A.5. The trends were fairly consistent over the three environments evaluated in this manner. While no explicit effort was made to handle cost (agents were given only the sum of positive rewards and penalties), the cautious and aggressive weightings consistently accumulated relatively low and high costs, respectively. The cautious $( \mathrm { W a n g } ( \eta = 0 . 7 5 ) )$ ) agent typically also generated the highest positive and total rewards after 10 millions steps of training.
+
+To generate the histograms in Figures 2, 5, 8, 9, and 13 as well as the numbers in Table 1, the trained agents were deployed on a set of 5000 test episodes – 1000 for each of the 5 networks learned using different random seeds in training. The resulting distributions therefore include contributions from both aleatoric and epistemic uncertainty. Sampling was turned off, allowing the agents to choose their perceived optimal action at each time step. In this context the benefit of emphasizing the lower part of the outcome distribution (i.e., cautious weighting) became more pronounced, in part because the methods that emphasize poor outcomes tended to maintain higher policy entropy (Appendix A.5).
+
+# 4.2 CAUTIOUS WEIGHTINGS
+
+To further explore the apparent benefits of cautious weightings in Safety Gym, we trained a series of variably cautious agents by tuning $\eta$ in the risk-averse weight function proposed by Wang (2000). Histograms of their episode rewards in testing are given in Appendix A.5 and summarized in Table 1. In these environments, agent performance – both in terms of improving the lower end of the reward distribution and on average – was seen to generally improve with increasing $\eta$ until the range $\eta \in [ 0 . 7 5 , 1 . 2 5 ]$ , subsequently degrading. Additional comparisons were made with PPO (Schulman et al., 2017b), which unsurprisingly was found to closely track performance of the “Uniform” agent. We found that naively incorporating cautious weightings into PPO improved its performance (row $\mathrm { P P O } + \mathrm { W a n g } ( 0 . 7 5 )$ in Table 1), though not to the level of the full C3PO method with $\eta = 0 . 7 5$ .
+
+We then pursued a set of longer runs to compare C3PO with the cautious objective from Wang (2000) to both unconstrained and constrained benchmarks. Here we did not tune $\eta$ , keeping it fixed at 0.75 for all experiments. Comparisons with unconstrained methods for three environments are given in Figures 3, 4, and 5 and for the remaining three environments in Appendix A.6. In addition to PPO, we compared performance with Trust Region Policy Optimization (TRPO; Schulman et al. (2017a)) as configured in Ray et al. (2019). Since this TRPO configuration generally outperformed the PPO configuration in Ray et al. (2019) from which we derived the hyperparameters for C3PO, we would expect C3PO to be at a disadvantage compared to TRPO. However, we found C3PO had the highest
+
+<table><tr><td rowspan="2"></td><td colspan="4">PointButton2</td><td colspan="4">CarGoal2</td><td colspan="4">CarButton2</td></tr><tr><td>Mean</td><td>Std</td><td>Q=0.5</td><td>Q=.05</td><td>Mean</td><td>Std</td><td>Q=0.5</td><td>Q=.05</td><td>Mean</td><td>Std</td><td>Q=0.5</td><td>Q=.05</td></tr><tr><td>Uniform</td><td>22.5</td><td>7.1</td><td>22.8</td><td>10.7</td><td>18.1</td><td>6.6</td><td>18.0</td><td>7.4</td><td>12.4</td><td>9.1</td><td>14.0</td><td>-6.1</td></tr><tr><td> CPT Value</td><td>16.5</td><td>7.3</td><td>16.3</td><td>4.9</td><td>13.9</td><td>7.7</td><td>14.6</td><td>-0.5</td><td>9.5</td><td>10.7</td><td>11.2</td><td>-11.0</td></tr><tr><td>Wang (-0.75)</td><td>17.5</td><td>6.1</td><td>17.6</td><td>7.6</td><td>13.5</td><td>6.9</td><td>14.0</td><td>0.9</td><td>6.4</td><td>10.7</td><td>8.9</td><td>-15.5</td></tr><tr><td>Wang (0.5)</td><td>23.3</td><td>6.3</td><td>23.3</td><td>13.1</td><td>18.5</td><td>6.0</td><td>18.8</td><td>8.2</td><td>12.9</td><td>9.0</td><td>14.4</td><td>-5.6</td></tr><tr><td>Wang (0.75)</td><td>24.2</td><td>6.7</td><td>24.4</td><td>13.7</td><td>19.0</td><td>6.4</td><td>19.3</td><td>8.2</td><td>14.3</td><td>9.5</td><td>15.8</td><td>-4.4</td></tr><tr><td>Wang (1.0)</td><td>24.7</td><td>6.0</td><td>24.9</td><td>15.1</td><td>20.3</td><td>6.9</td><td>21.3</td><td>7.2</td><td>11.4</td><td>11.0</td><td>13.8</td><td>-12.2</td></tr><tr><td>Wang (1.25)</td><td>25.4</td><td>6.1</td><td>25.4</td><td>15.9</td><td>17.3</td><td>6.7</td><td>17.8</td><td>5.6</td><td>12.7</td><td>10.4</td><td>14.8</td><td>-8.3</td></tr><tr><td>Wang (1.50)</td><td>23.6</td><td>5.9</td><td>23.7</td><td>14.1</td><td>16.6</td><td>7.5</td><td>17.2</td><td>3.1</td><td>10.1</td><td>12.0</td><td>13.3</td><td>-17.9</td></tr><tr><td>Wang (1.75)</td><td>23.4</td><td>6.2</td><td>23.5</td><td>13.5</td><td>12.5</td><td>8.0</td><td>13.0</td><td>-0.6</td><td>7.3</td><td>13.0</td><td>11.1</td><td>-22.8</td></tr><tr><td>PPO°</td><td>19.0</td><td>6.4</td><td>18.9</td><td>9.1</td><td>15.8</td><td>6.0</td><td>15.8</td><td>5.8</td><td>8.3</td><td>9.5</td><td>9.5</td><td>-9.8</td></tr><tr><td>PPO + Wang(0.75)</td><td>20.4</td><td>8.4</td><td>21.7</td><td>2.6</td><td>17.7</td><td>6.7</td><td>18.0</td><td>6.3</td><td>12.1</td><td>9.5</td><td>13.6</td><td>-6.8</td></tr></table>
+
+Table 1: Testing statistics for episode rewards achieved by agents trained over 10 million steps with different distributional objectives. $Q = 0 . 5$ is the median and $Q = 0 . 0 5$ refers to the location of the 0.05 quantile. Blue bold-face represents the best performance for a given environment; in all cases these occur for moderately cautious weightings $( \eta \in [ 0 . 7 5 , 1 . 2 5 ] )$ .
+
+![](images/bb501da34dad9ffd280e1e0abf0d5aecacef9ec4aebd9042e3c91ece0f75c297.jpg)  
+Figure 3: Average episode reward (including penalty) over training for different learning approaches in three different environments. C3PO with Wang $\langle \eta = 0 . 7 5 \rangle$ weighting outperforms others.
+
+average reward (including penalty) in five of the six environments and lowest average cost in five of the six environments. In addition, agents that used the cautious weightings tended to have more stable and repeatable training, as evidenced by the tight distribution of their learning curves. This tightness was found to reflect a lack of negative outlier episodes and potentially lower epistemic uncertainty throughout training. Finally, note that the use of a nonzero penalty for cost events resulted in significantly lower incurred costs than were observed with unconstrained agents trained without a penalty (Ray et al., 2019). PPO and TRPO were seen to reach similar cost levels without a penalty; these levels are indicated by red dashed lines in the cost figures.
+
+Comparisons with versions of PPO and TRPO that use Lagrangian constraints (PPO-Lagrangian and TRPO-Lagrangian; Ray et al. (2019)) to match the cost level of C3PO are shown in Figure 6 and Appendix A.7 . We see that, given the same level of cost incurred per episode, agents trained using C3PO consistently achieve higher levels of reward than those trained with PPO-Lagrangian and TRPO-Lagrangian. As above, training is seen to be more stable and repeatable using our risksensitive method. Additional comparisons were generated with Constrained Policy Optimization (CPO; Achiam et al. (2017)), but are not shown in Figure 6 because they failed to maintain the cost levels of the other methods. For completeness, they are given in Appendix A.7.
+
+![](images/b480abf87ed601bfa9179249c5ec3dd7d87906197280fa6f22c1a694668b308f.jpg)  
+Figure 4: Average number of penalty events per episode (lower is better) over training for different learning approaches in three different environments. The horizontal lines reflect the cost levels reached by both PPO and TRPO training with zero penalty in Ray et al. (2019).
+
+![](images/1ea98699dfc873c29da3ef3b46e960c5a5e7468be941a26dfa56083a0f5b663d.jpg)  
+Figure 5: Testing reward distributions (including penalty; sampling turned off) for long training runs of three Safety Gym environments.
+
+![](images/590e3d8733c49cdf2b37370bfa39c5fd34856e12ec257bbfea7255b8db3a9693.jpg)  
+Figure 6: Comparison of positive contributions to episode reward during training for our approach (yellow) and Lagrangian methods configured to have the same cost level.
+
+# 5 DISCUSSION
+
+The analysis above allows for sample-based policy gradient estimates of a broad class of distributional objectives. Variance reduction measures were shown to enable efficient optimization based on these estimates (Appendix A.4). However, it was not seen to be the case that a given distributional objective could be most effectively optimized directly. Instead, the best results were generally obtained through moderate emphasis on improving negative training outcomes (Table 1).
+
+To understand this behavior, consider the interplay of optimization and exploration in the training of cautious and aggressive agents. Cautious weightings continually emphasize the lower part of the outcome distribution, pushing that part of the distribution upward and adjusting behavior the most where it is most necessary. Once a part of the state space where the agent is deficient is adequately addressed, a different part of the state space takes its place. Policy entropy remains high because of the emphasis on problematic situations, ensuring adequate exploration. This trend continues with increasing $\eta$ , until the point where the agent begins to ignore high quality training outcomes too much. Conversely, aggressive weightings continually emphasize the best outcomes in the distribution. When an already strong outcome is given increased attention, it is likely to stay at the top. Hence agents trained with aggressive weightings tend to become myopic, obsessing over a fraction of the state space while neglecting the rest of it. They tend to explore inadequately and ironically fail to attain better top-end performance than more cautious weightings.
+
+Given the consistent performance gains observed using our method, we propose that it represents a useful option for improving the performance and stability of on-policy learners. This should be particularly true in the presence of a meaningful trade-off between positive and negative reward terms and when there is significant stochasticity. While our approach does add an additional hyperparameter – the shaping constant $\eta$ – one value for that hyperparameter was seen to provide gains across all environments tested. While our approach does not provide for a direct choice of cost limit as constrained methods do, it is simpler to implement and was consistently seen to be more performant for the cost level it reached. It is also likely possible to use cautious weightings in conjunction with constrained RL, though this has not yet been investigated.
+
+# 6 CONCLUSIONS
+
+In this work, we proposed a risk-sensitive learning algorithm based on a policy gradient estimate for a broad class of distributional objectives. When configured to emphasize improvement in scenarios where the agent performs poorly, we found our method to compare favorably with existing unconstrained and constrained on-policy learners.
+
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+
+# A APPENDIX
+
+# A.1 EVALUATION OF CROSS-TRAJECTORY TERMS IN POLICY GRADIENT
+
+In this section, we first show that the cross-trajectory terms in our policy gradient estimate (6) have an expectation value of 0. We then argue that their removal leads to a policy gradient estimate with reduced variance.
+
+Lemma 1. Cross-trajectory terms of the form $f ( \tau _ { i } , \mathbf { a } _ { j , t } , \mathbf { s } _ { j , t } ) = u ( r ( \tau _ { i } ) ) \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } )$ , where $i \neq j$ , do not contribute to the gradient estimate (6) in expectation.
+
+Proof. First, note that
+
+$$
+\begin{array} { r l } & { E _ { \tau _ { i } \sim p _ { \theta } ( \tau ) , \tau _ { j } \sim p _ { \theta } ( \tau ) } f ( \tau _ { i } , \mathbf { a } _ { j , t } , \mathbf { s } _ { j , t } ) = E _ { \tau _ { i } \sim p _ { \theta } ( \tau ) , \tau _ { j } \sim p _ { \theta } ( \tau ) } u ( r ( \tau _ { i } ) ) \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) } \\ & { \qquad = E _ { \tau _ { i } \sim p _ { \theta } ( \tau ) } \Bigg [ u ( r ( \tau _ { i } ) ) E _ { \tau _ { j } \sim p _ { \theta } ( \tau ) } \Bigg [ \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) \Bigg | \tau _ { i } \Bigg ] \Bigg ] } \end{array}
+$$
+
+Then consider the innermost expectation:
+
+$$
+\begin{array} { r l } { \mathbb { E } _ { \tau _ { j } \sim \mathbb { P } _ { \theta } ( \tau ) } \Bigg [ \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) \Bigg | \tau _ { i } \Bigg ] = \displaystyle \int _ { \mathbf { s } _ { j , t } , \mathbf { a } _ { j , t } } p ( \mathbf { s } _ { j , t } , \mathbf { a } _ { j , t } | \tau _ { \theta } , \tau _ { j } ) \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) d \mathbf { a } _ { j , t } d \mathbf { s } _ { j , t } } \\ { = \displaystyle \int _ { \mathbf { s } _ { j , t } } p ( \mathbf { s } _ { j , t } | \pi _ { \theta } , \tau _ { i } ) \int _ { \mathbf { a } _ { j , t } } \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) d \mathbf { a } _ { j , t } d \mathbf { s } _ { j , t } } \\ { = \displaystyle \int _ { \mathbf { s } _ { j , t } } p ( \mathbf { s } _ { j , t } | \pi _ { \theta } , \tau _ { i } ) \int _ { \mathbf { a } _ { j , t } } \nabla _ { \theta } \pi _ { \theta } ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) d \mathbf { a } _ { j , t } d \mathbf { s } _ { j , t } } \\ { = \displaystyle \int _ { \mathbf { s } _ { j , t } } p ( \mathbf { s } _ { j , t } | \pi _ { \theta } , \tau _ { i } ) \nabla _ { \theta } \int _ { \mathbf { a } _ { j , t } } \pi ( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } ) d \mathbf { a } _ { j , t } d \mathbf { s } _ { j , t } } \\ { = \displaystyle \int _ { \mathbf { s } _ { j , t } } p ( \mathbf { s } _ { j , t } | \pi _ { \theta } , \tau _ { i } ) \nabla _ { \theta } \log \left( \mathbf { a } _ { j , t } | \mathbf { s } _ { j , t } \right) d \mathbf { a } _ { j , t } d \mathbf { s } _ { j , t } } \\  = \displaystyle \int _ { \mathbf { s } _ { j , t } , | \pi _ { \theta } , \tau _ { i } | }  \end{array}
+$$
+
+To see why removal of cross-trajectory terms leads to reduced variance, consider that the full expression (6) may be written as the sum of terms of the form $\begin{array} { r l } { f ( \tau _ { i } , \mathbf { a } _ { j , t } , \mathbf { s } _ { j , t } ) } & { { } = } \end{array}$ $u ( r ( \tau _ { i } ) \bar { ) } _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { j , t } , \bar { \mathbf { s } _ { j , t } } )$ . Its variance is the sum of the total variance from terms where $i = j$ , the total variance from terms where $i \neq j$ , and a term proportional to the covariance of these two totals. However, because each term in the covariance contains at least one trajectory that differs from the rest, the above reasoning may be applied to argue that the covariance is 0. Hence, the removal of the cross-trajectory terms lowers the variance of the policy gradient estimate by the variance of the cross-trajectory terms.
+
+A.2 INTRODUCTION OF STATIC AND STATE-DEPENDENT BASELINES
+
+Lemma 2. A static baseline of the utility may be added to the policy gradient estimate (9) without introduction of bias.
+
+Proof. The additional term is 0 in expectation as
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { \tau _ { \tau } \sim p _ { \theta } ( \tau ) } \left[ b \Bigg ( w ^ { \prime } \left( \frac { i } { n } \right) + w ^ { \prime } \left( \frac { i - 1 } { n } \right) \Bigg ) \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { i } , | \mathbf { s } _ { i } , t ) \right] } \\ & { \qquad = b \Bigg ( w ^ { \prime } \left( \frac { i } { n } \right) + w ^ { \prime } \left( \frac { i - 1 } { n } \right) \Bigg ) \int _ { \mathbf { s } _ { i , t } , \mathbf { a } _ { i } , t } p ( \mathbf { s } _ { i , t } , \mathbf { a } _ { i } , t | \pi _ { \theta } ) \Bigg [ \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { i } , | \mathbf { s } _ { i } , t ) \Bigg ] d \mathbf { a } _ { i , t } d \mathbf { s } _ { i , t } } \\ & { \qquad = b \Bigg ( w ^ { \prime } \left( \frac { i } { n } \right) + w ^ { \prime } \left( \frac { i - 1 } { n } \right) \Bigg ) \int _ { \mathbf { s } _ { i , t } } p ( \mathbf { s } _ { i , t } | \pi _ { \theta } ) \int _ { \mathbf { a } _ { i , t } } \pi _ { \theta } ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } ) \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i } , t ) d \mathbf { a } _ { i , t } d \mathbf { s } _ { i , t } } \\ & { \qquad = b \Bigg ( w ^ { \prime } \left( \frac { i } { n } \right) + w ^ { \prime } \left( \frac { i - 1 } { n } \right) \Bigg ) \int _ { \mathbf { s } _ { i , t } } p ( \mathbf { s } _ { i , t } | \pi _ { \theta } ) \nabla _ { \theta } \int _ { \mathbf { a } _ { i , t } } \pi _ { \theta } ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } ) d \mathbf { a } _ { i , t } d \mathbf { s } _ { i , t } } \\ &  \qquad = b \Bigg ( w ^ { \prime } \left( \frac { i } { n } \right) + w ^ { \prime } \left( \frac { i - 1 } { n } \right) \Bigg ) \int _ { \mathbf { s } _ { i , t } } p ( \mathbf { s } _ { i , t } | \pi _ { \theta } ) \nabla _ { \theta } \mathbf { 1 } _ { i , t } d \end{array}
+$$
+
+The contribution of the weight terms $\begin{array} { r } { ( w ^ { \prime } ( \frac { i } { n } ) + w ^ { \prime } ( \frac { i - 1 } { n } ) ) } \end{array}$ may be pulled out of the integral between the first and second line because of its independence on both state and action. This term is fixed for a given trajectory by the rank of its reward amongst the rewards accumulated on all trajectories in the current batch.
+
+In our variance reduction experiments (Appendix A.4), the “Base” agent uses $b$ equal to the mean of full-episode utility in the current batch.
+
+As described in Section 3.3, we may further adjust the policy gradient estimate through introduction of per-step utilities. In this case, we may justify the use of a state-dependent baseline through the following.
+
+Lemma 3. A state-dependent baseline $V _ { \phi } ( \mathbf { s } _ { i , t } )$ may be added to the policy gradient estimate (9) without introduction of bias, if per-step utilities are assumed.
+
+Proof. The additional term is 0 in expectation as
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { \tau _ { i } \sim p _ { 0 } ( \tau ) } \Big [ \Bigg ( w ^ { \prime } \left( \frac { i } { n } \right) + w ^ { \prime } \left( \frac { i - 1 } { n } \right) \Big ) \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } ) V _ { \phi } ( \mathbf { s } _ { i , t } ) \Big ] } \\ & { \qquad = \Bigg ( w ^ { \prime } \left( \frac { i } { n } \right) + w ^ { \prime } \left( \frac { i - 1 } { n } \right) \Bigg ) \int _ { \mathbf { s } _ { i , t } , \mathbf { s } _ { i , t } } p ( \mathbf { s } _ { i , t } , \mathbf { a } _ { i , t } | \pi _ { \theta } ) V _ { \phi } ( \mathbf { s } _ { i , t } ) \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } ) d \mathbf { a } _ { i , t } d \mathbf { s } _ { i , t } } \\ & { \qquad = \Bigg ( w ^ { \prime } \left( \frac { i } { n } \right) + w ^ { \prime } \left( \frac { i - 1 } { n } \right) \Bigg ) \int _ { \mathbf { s } _ { i , t } } p ( \mathbf { s } _ { i , t } | \pi _ { \theta } ) V _ { \phi } ( \mathbf { s } _ { i , t } ) \int _ { \mathbf { a } _ { i , t } } \pi _ { \theta } ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } ) \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } ) d \mathbf { a } _ { i , t } } \\ & { \qquad = \Bigg ( w ^ { \prime } \left( \frac { i } { n } \right) + w ^ { \prime } \left( \frac { i - 1 } { n } \right) \Bigg ) \int _ { \mathbf { s } _ { i , t } } p ( \mathbf { s } _ { i , t } | \pi _ { \theta } ) V _ { \phi } ( \mathbf { s } _ { i , t } ) \nabla _ { \theta } \left( \mathbf { a } _ { i , t } | \mathbf { s } _ { i , t } \right) d \mathbf { a } _ { i , t } d \mathbf { s } _ { i , t } } \\ &  \qquad = \Bigg ( w ^ { \prime } \left( \frac { i } { n } \right) + w ^ { \prime } \left( \frac { i - 1 } { n } \right) \Bigg ) \int _ { \mathbf { s } _ { i , t } } p ( \mathbf  s \end{array}
+$$
+
+The rationale for pulling the $w ^ { \prime }$ terms out of the integral is the same as in Lemma 2.
+
+Finally, we note that the ability to pull the contribution of the weight terms $\begin{array} { r } { ( w ^ { \prime } ( \frac { i } { n } ) + w ^ { \prime } ( \frac { i - 1 } { n } ) ) } \end{array}$ to the front of Equation 11 allows us to formulate advantage estimates based on per-step utility. Bootstrap estimates of the value function $V _ { \phi } ( \mathbf { s } _ { i , t } )$ and Generalized Advantage Estimation as in Schulman et al. (2016) can be conducted exactly as they are in standard on-policy learning, if rewards are replaced by per-step utilities.
+
+# A.3 ADDITIONAL INFORMATION ON SAFETY GYM
+
+As mentioned in Section 4, we chose to evaluate our approach using the OpenAI Safety Gym (Ray et al., 2019). The choice was governed by our desire to test in conditions with clear cost-benefit trade-offs, significant stochasticity, adequate complexity, and available benchmarks. While our methods are not limited to particular task types or observation/action spaces, we found Safety Gym to be suitable for exploring their potential.
+
+The six environments chosen were the most obstacle-rich of the publicly available environments that used the “Point” and “Car” robots. The Point robot is constrained to the 2D plane and has two control dimensions: one for moving forward/backward and one for turning. The Car robot also has two control dimensions, corresponding to independently actuated parallel wheels. It has a freely rotating wheel and, while it is not constrained to the 2D plane, typically remains in it. While we expect our results to extend to the remaining default robot, “Doggo”, we did not experiment with it because of the order of magnitude longer training times it exhibited in Ray et al. (2019).
+
+Several types of obstacles and tasks were present in the environments we evaluated. In all cases, the robot is given a fixed amount of time (1000 steps) to complete the prescribed task as many times as possible and is motivated by both sparse and dense reward contributions. In the “Goal” environments, the robot must navigate to a series of randomly-assigned goal positions, with a new target being assigned as soon as a goal is reached. In the “Button” environments, the robot must reach and press a sequence of goal buttons while avoiding other buttons. In the “Push” task, the robot must push a box to a series of goal positions. The set of obstacles are different for each task; among the three environments there are a total of five different constraint elements (hazards, vases, incorrect buttons, pillars, and gremlins), each with different dynamics. See Ray et al. (2019) for further details.
+
+# A.4 EMPIRICAL PERFORMANCE OF VARIANCE REDUCTION MEASURES
+
+To gauge the impact of the variance reduction techniques outlined in Section 3.3, we evaluated their performance in maximizing the value function of Cumulative Prospect Theory (Tversky & Kahneman, 1992). As mentioned in Section 3.1, this function has two integrals of the form (2):
+
+![](images/e4fc7555f5d13748a1481c4e76a2449b69e4d2ae8e481a159d249b1af8716792.jpg)  
+Figure 7: Impact of variance reduction measures on optimization of the CPT value function. Here “Base” refers to the risk-sensitive policy gradient estimate (10), “UTG” adds utility-to-go and a neural network baseline (11), “GAE” incorporates generalized advantage estimation, and “TR” implements trust regions via clipping. Shading represents the variation over five random seeds.
+
+$$
+\begin{array} { l } { \displaystyle { J ( \theta ) = - \int _ { - \infty } ^ { \infty } u ^ { - } ( r ( \tau ) ) \frac { d } { d r ( \tau ) } \bigg ( w ^ { - } ( P _ { \theta } ( r ( \tau ) ) ) \bigg ) d r ( \tau ) } } \\ { \displaystyle { + \int _ { - \infty } ^ { \infty } u ^ { + } ( r ( \tau ) ) \frac { d } { d r ( \tau ) } \bigg ( - w ^ { + } ( 1 - P _ { \theta } ( r ( \tau ) ) ) \bigg ) d r ( \tau ) } } \end{array}
+$$
+
+In Tversky & Kahneman (1992), the utility functions are computed relative to a reference point and reflect the tendency of humans to be more risk-averse in the presence of gains than in the presence of losses. The weight functions $\{ w ^ { + } , w ^ { - } \}$ model our inclination to emphasize the best and worst possible outcomes in our decision-making.
+
+More specifically, in these experiments we used the piecewise utility functions $u ^ { + } ( r ) = H ( r -$ $r _ { 0 } ) ( r - r _ { 0 } ) ^ { \sigma }$ and $u ^ { - } = \lambda H ( r _ { 0 } - r ) ( r _ { 0 } - r ) ^ { \sigma }$ with static reference $r _ { 0 } = 1 0$ , $\sigma = 0 . 8 8$ , and $\lambda = 2 . 2 5$ . The weight function $\begin{array} { r } { w ( p ) = \frac { p ^ { \eta } } { ( p ^ { \eta } + ( 1 - p ) ^ { \eta } ) ^ { \frac { 1 } { \eta } } } } \end{array}$ was used, where $\eta = 0 . 6 1$ for $r < r _ { 0 }$ and $\eta = 0 . 6 9$ for $r \geq r _ { 0 }$ . Four methods were evaluated, incorporating progressive amounts of variance reduction:
+
+• Base: Risk-sensitive policy gradient with a static baseline (10)
+
+• UTG: Base with utility-to-go and a neural network baseline (11)
+
+• GAE: UTG with generalized advantage estimation ((13) without clipping)
+
+• TR: GAE with trust regions ((13) with clipping (14))
+
+As shown in Figure 7, the incorporation of these techniques increased the sample efficiency of the CPT value optimization significantly. Consequently, we used the full complement (TR) in all other experiments.
+
+# A.5 DIFFERING OBJECTIVES: ADDITIONAL RESULTS
+
+Below we include results for all environments for the experiments described in Section 4.1.
+
+![](images/81ec3e345b1af717ace27eb04a70b8099728900fe08a0cfddaef39054e279403.jpg)  
+Figure 8: Impact of different distributional objectives in remaining two environments of initial trials. The shading reflects the standard deviation associated with running over 5 random seeds. Left: Net reward (positive reward minus penalty) throughout learning. Middle: Average number of cost events per episode during training (lower is better). Right: Agent outcome distribution in testing (i.e., with sampling turned off).
+
+![](images/59f9e15291612540513e9a339525bfe082627947977ab0f09ef0a60afb6db7e4.jpg)  
+Figure 9: Agent outcome distributions across trials run over increasingly cautious ( $\dot { \eta }$ increasing) objectives. Distributions correspond to results shown in Table 1.
+
+In addition, we note the trend of policy entropies with different distributional objectives. In general, more cautious weightings maintain higher entropy for longer than more aggressive weightings. Note that these plots represent an upper bound because they do not account for action clipping by the environment; see Ray et al. (2019) for details.
+
+![](images/00ab35e3a5313b6d13471e09bf379ea0297414422ff0001e0a1ee8e3d079506d.jpg)  
+Figure 10: Policy entropy progression during training for three environments. Shading reflects the observed variation over 5 random seeds.
+
+# A.6 ADDITIONAL COMPARISONS WITH UNCONSTRAINED METHODS
+
+Below are plots of average episode reward and average number of episode cost events throughout training for the remainder of the environments on which we conducted long runs (Section 4.2). Also included are histograms of testing performance for those runs.
+
+![](images/4e0bde6c535c84b3123dd3fcc102743d54ee7eb04ce8ee2c18002eb2e37c1a66.jpg)  
+Figure 11: Average episode reward (including penalty) over training for different unconstrained learning approaches in remaining three environments.
+
+![](images/d1c2fdfe7eb2dab990a76db14e17c5a4577c034dc9ac5e533dc1af2133f48d77.jpg)  
+Figure 12: Average number of cost events per episode (lower is better) over training for different unconstrained learning approaches in remaining three environments. As above, the “zero-penalty” line refers to the level reached by PPO and TRPO trained with no penalty in the reward (Ray et al., 2019).
+
+![](images/92c7a36dbcab49314f44bc5df77ef04e277e3258109d0f0552b1bdccdf5113e3.jpg)  
+Figure 13: Testing reward distributions (including penalty; sampling turned off) for long training runs in the remaining three Safety Gym environments. In five of the six environments, C3PO with $\eta = 0 . 7 5$ provides tangible benefit. A smaller $\eta$ is likely required to improve performance on CarPush2.
+
+# A.7 ADDITIONAL COMPARISONS WITH CONSTRAINED METHODS
+
+As mentioned in Section 4.2, we compared the performance of C3PO with constrained methods by setting the cost limit of the constrained methods to match the cost level attained by C3PO. Here we provide
+
+• the positive reward plots for the remaining three Safety Gym environments studied, • the cost plots for each of the six environments, and
+
+• all plots for Constrained Policy Optimization (CPO; Achiam et al. (2017)).
+
+The intent of the cost plots of Figure 15 is to show rough consistency between the cost levels of our approach and Lagrangian methods configured to have the same cost limit. This is verified, but other trends should be noted. First, while the Lagrangian-based methods typically follow the cost constraint well, they cannot satisfy it in each batch. Second, our approach tends to have comparable or lower cost rates throughout training. This safe exploration metric, defined in Ray et al. (2019), refers to the average cost per episode over all of training up to a given point.
+
+Results related to Constrained Policy Optimization (CPO; Achiam et al. (2017)) are included here but not in the main text because, consistent with (Ray et al., 2019), we were not able to configure CPO to respect the cost levels of the other constrained methods. Here we show the cost levels reached by CPO compared with C3PO (Figure 16) as well as a comparison of the average episode rewards of the two (Figure 17). For the latter, we employed the penalty scaling used by C3PO to enable a fair comparison.
+
+![](images/24de7d6235c5be2e5e1dc0b014bc5127dafbe58f7faa61aa080822bd5d15716a.jpg)  
+Figure 14: Comparison of positive contributions to episode reward during training for our approach (yellow) and Lagrangian methods configured to have the same cost level. The plots for the other three environments are shown in Figure 6.
+
+![](images/0a1c8a5e024f2f99938d54a3a75070e0d0ca8e8cf1be356e2020459e46f946b6.jpg)  
+Figure 15: Comparison of cost incurred (lower is better) during training for our approach and Lagrangian methods configured to have the same cost level. As intended, cost levels are consistently matched between the methods. As above, the “zero-penalty” line refers to the level reached by PPO and TRPO trained with no penalty in the reward (Ray et al., 2019).
+
+![](images/6423517191387cafba4cc4f1f5f46b465712c3f21dbd55084697f87212ff8ad3.jpg)  
+Figure 16: Comparison of cost incurred (lower is better) during training for our method and Constrained Policy Optimization (Achiam et al., 2017) configured to have a matching cost limit. Results are consistent with Ray et al. (2019). As above, the “zero-penalty” line refers to the level reached by unconstrained PPO and TRPO trained with no penalty in the reward (Ray et al., 2019).
+
+![](images/c52be513072a68cb4cd41cf8eaae3878b94349cfe8d6349a99d42264d32f3dfd.jpg)  
+Figure 17: Comparison of average episode reward (including penalty) for C3PO and CPO.
+
+# A.8 SUPPLEMENTARY MATERIALS
+
+The code used to produce these results is included in our Supplementary Materials.

parse/dev/iAWNOXfLz0/iAWNOXfLz0_content_list.json

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+    {
+        "type": "text",
+        "text": "AnoFormer: Time Series Anomaly Detection using Transformer-based GAN with Two-Step Masking ",
+        "text_level": 1,
+        "bbox": [
+            192,
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+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Time series anomaly detection is a task that determines whether an unseen signal is normal or abnormal, and it is a crucial function in various real-world applications. Typical approach is to learn normal data representation using generative models, like Generative Adversarial Network (GAN), to discriminate between normal and abnormal signals. Recently, a few studies actively adopt transformer to model time series data, but there is no transformer-based GAN framework for time series anomaly detection. As a pioneer work, we propose a new transformerbased GAN framework, called AnoFormer, and its effective training strategy for better representation learning. Specifically, we improve the detection ability of our model by introducing two-step masking strategies. The first step is Random masking: we design a random mask pool to hide parts of the signal randomly. This allows our model to learn the representation of normal data. The second step is Exclusive and Entropy-based Re-masking: we propose a novel refinement step to provide feedback to accurately model the exclusive and uncertain parts in the first step. We empirically demonstrate the effectiveness of re-masking step that our model generates more normal-like signals robustly. Extensive experiments on various datasets show that AnoFormer significantly outperforms the state-of-the-art methods in time series anomaly detection. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "19 1 Introduction ",
+        "text_level": 1,
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+        "text": "20 Time series anomaly detection is a crucial technology to prevent potential risks and financial losses   \n21 in a variety of areas, such as detecting anomalies on sensor data of large-scale plants [1], ECG   \n22 monitoring [2], and the network traffic analysis [3]. To deal with this task, from the classic methods   \n23 [4, 5, 6] to the recent deep learning-based methods [2, 7, 8, 9, 10, 11, 12, 13], many studies have   \n24 focused on unsupervised learning methods due to the lack of labeled anomalies and highly nonlinear   \n25 temporal dependencies.   \n26 One of major deep learning-based approaches is a reconstruction-based method. It typically uses an   \n27 autoencoder (AE) or Generative Adversarial Network (GAN) to learn the representation of normal   \n28 data and to reconstruct a normal-like signal from an input always. As a backbone network, the   \n29 existing studies widely utilize CNN (Convolutional Neural Networks) [2] or RNN (Recurrent Neural   \n30 Networks) [8, 9, 10]. More recently, there have been attempts to apply transformer [14] to time   \n31 series anomaly detection, and it shows remarkable performances [11]. In this work, we also adopt   \n32 transformer to embed time series representation, but design an adversarial framework for anomaly   \n33 detection.   \n34 If we devise GAN using a transformer encoder, we expect that the model learns normal time series   \n35 data and eventually generates real normal-like signals. However, there is a major issue. Unlike the   \n36 AE structure, a pure transformer encoder-based generator does not have a compressed latent space,   \n37 i.e., it makes the model find the trivial solution, just copying an input and pasting to the output for the   \n38 reconstruction. Therefore, we need a new training method for the generator to learn the distribution   \n39 of normal time series data. To address this issue, we introduce a novel two-step masking strategy.   \n40 From this approach, the next question is where to mask an input signal to detect anomalies effectively.   \n41 Understandably, in order to make the normal-like output, the best masking positions are abnormal   \n42 points in the input signal. It is a challenging to mask the abnormal areas selectively because we do   \n43 not know where the abnormal parts are in advance.   \n44 In this paper, we propose AnoFormer, which is a novel transformer-based GAN utilizing a pure   \n45 transformer encoder only. To learn data representation effectively, we adopt a masking strategy. We   \n46 first train transformer-based GAN with random masking (Step 1) for representation learning of the   \n47 normal time series data. While filling the randomly masked parts of the input at Step 1, the model   \n48 learns the distribution of normal data effectively. In Step 1 alone, all parts of the input signal cannot   \n49 be considered, and this randomness is a big problem in anomaly detection. Therefore, we solve this   \n50 problem by re-masking the exclusive parts of Step 1. Also, to find the best masking positions, we   \n51 calculate entropy from the attention maps of transformer blocks and re-mask the parts with high   \n52 entropy that is likely to be abnormal points with high uncertainty. This exclusive and entropy-based   \n53 re-masking (Step 2) provides feedback for better representation learning, eventually improving the   \n54 anomaly detection performance. We experimentally prove that the proposed two-step masking is   \n55 essential for AnoFormer to solve anomaly detection problem successfully. ",
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+    {
+        "type": "text",
+        "text": "56 Our contributions can be summarized as follows: ",
+        "bbox": [
+            150,
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+            381
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose a simple yet effective transformer-based GAN framework having a generator and a discriminator for unsupervised time series anomaly detection, called AnoFormer. Moreover, we present pre-processing and embedding methods for our framework to deal with time series data effectively.   \n• We introduce a new two-step masking method to encode the distribution of normal time series data. A newly proposed entropy-based re-masking helps our model to provide the feedback to the uncertain parts based on entropy. From the extensive ablations, we empirically verify that our two-step masking makes our model robust and successfully embed the representation of normal time series data.   \n• AnoFormer achieves new state-of-the-art results with significant improvements on various unsupervised time series anomaly detection datasets: NeurIPS-TS, MIT-BIH, 2D-gesture, and Power-demand. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "69 2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            618,
+            321,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "70 Generative models using transformer have been proposed and applied to diverse domains, e.g.,   \n71 computer vision [15, 16, 17, 18], natural language processing [19, 20], and sequence modeling   \n72 [21, 22]. In particular, these models are used to solve various tasks in the image domain, such   \n73 as scene generation [15, 16, 23], saliency prediction [18], semantic segmentation [24], and sketch   \n74 synthesis [25]. Moreover, transformer is presented to solve graph-to-sequence transduction task   \n75 using graph neural network [26], text generation task [27], and time series forecasting task with   \n76 the modified self-attention mechanism [28]. We also utilize transformer to construct a generative   \n77 framework, i.e., having both a generator and a discriminator. In this framework, we propose an   \n78 appropriate embedding method and loss form to effectively solve the anomaly detection problems.   \n79 Many studies have used masking to the transformer architecture for effective representation learning.   \n80 Including BERT [29], which proposes the Masked Language Model (MLM) technique to pretrain   \n81 the language representation, many studies also adopt the masking methods, like [30] for action   \n82 recognition, [30] for text classification task, [31] for text log anomaly detection, [32, 33] for visual   \n83 representation learning. In [34], the CNN-based model learns the semantic context features by using   \n84 a multi-scale mask across the whole image with different scales for anomaly detection in image   \n85 domain. We also use masking in our transformer-based GAN for time series anomaly detection, but   \n86 unlike the above studies, we propose the two-step masking strategy for training and test to provide   \n87 feedback that boosts the model to generate the uncertain parts successfully. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/7115bd0fec0ddd2e3db41bbe484d7135d01fb4409712ae48ff3ddde0e1a8c791.jpg",
+        "image_caption": [
+            "Figure 1: Overview of the proposed AnoFormer. For simplicity, this figure shows the univariate case. In Step 1 (random masking), a pre-processed input $\\tilde { \\boldsymbol { X } }$ is masked with a randomly selected mask from a predefined mask pool. After passing the masked input $\\tilde { \\pmb X } _ { m } ^ { 1 }$ to the generator, $\\hat { X } _ { 1 }$ is generated as the output by passing through embedding, transformer encoder, and inverse embedding layers. In Step 2 (exclusive and entropy-based re-masking), based on the entropy calculated from attention maps of all layers in Step 1, $\\tilde { \\boldsymbol { X } }$ is re-masked and ${ \\hat { \\mathbf { X } } } _ { 2 }$ is generated again from the generator. Final output $\\hat { \\pmb X }$ is constructed via the combination of the masked parts of Step 1 and Step 2. With an aid of a critic, the generator is able to generate more normal-like signals. Here, $\\mathcal { L } _ { a d v }$ is the adversarial loss, including $\\mathcal { L } _ { a d v } ^ { g }$ and $\\mathcal { L } _ { a d v } ^ { c }$ . Note that the critic is used only for the train time. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "88 3 AnoFormer ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "89 In this section, we propose AnoFormer for unsupervised time series anomaly detection. We first   \n90 define the target task including an algorithm procedure briefly in Section 3.1. We then describe how   \n91 to construct a transformer-based GAN framework based on a transformer encoder in Section 3.2.   \n92 Next, we introduce two different masking steps for our model to encode time series data effectively   \n93 in Section 3.3. Finally, we present the whole training scheme of AnoFormer in Section 3.4. Figure 1   \n94 shows the overall architecture of AnoFormer. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.1 Problem Definition ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Let $\\begin{array} { r c l } { { \\textbf {  { X } } } } & { { = } } & { { \\left\\{ { \\pmb x } _ { 1 } , { \\pmb x } _ { 2 } , \\cdot \\cdot \\cdot , { \\pmb x } _ { T } \\right\\} \\ \\in \\ \\mathbb { R } ^ { T \\times n } } } \\end{array}$ be an input signal of $T$ lengths, where $\\begin{array} { r l } { \\mathbf { \\boldsymbol { x } } _ { t } } & { { } = } \\end{array}$ $\\left\\{ \\pmb { x } _ { t } ^ { 1 } , \\pmb { x } _ { t } ^ { 2 } , \\cdot \\cdot \\cdot , \\pmb { x } _ { t } ^ { n } \\right\\} \\in \\mathbb { R } ^ { n }$ at time step $t$ is a vector of dimension $n$ . Since it is easier to get normal time series data compared to abnormal ones, we train a generator $\\mathbf { G }$ and a discriminator $\\mathrm { D }$ using only normal data without any label in an unsupervised manner. After training, for each unseen signal $\\boldsymbol { X }$ , which can be normal or abnormal, the generator $G$ generates a normal-like signal $\\hat { X }$ . From the generated signal, we can determine whether the observed signal $\\boldsymbol { X }$ is normal or not based on the reconstruction errors between the given signal $\\boldsymbol { X }$ and the generated signal $\\hat { X }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 Transformer-based GAN for Time Series Data ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Pre-Processing. To deal with an input signal for a transformer encoder, we need a pre-processing step that makes the input signal discrete tokens. To this end, we normalize each time series input $\\boldsymbol { X }$ between $^ { - 1 }$ and 1 by using the min-max scaling. Then, we quantize the normalized real value within a specific range $[ 0 , K )$ , where the integer $K$ is a hyperparameter controlling the quantization resolution, and use the corresponding integer value as a token. Let $\\tilde { \\pmb X } \\in \\mathbb { R } ^ { T \\times n }$ be the pre-processed signal. We set $K = 4 0 0$ for all the experiments, in which the pre-processed signal $\\tilde { X }$ looks almost like the input $\\boldsymbol { X }$ . In total of $K$ tokens (quantization levels), we add a [MASK] token to utilize it for both training and test. To sum up, the input signal $\\boldsymbol { X }$ is pre-processed to be $\\tilde { \\boldsymbol { X } }$ by applying scaling and quantization sequentially. ",
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+    },
+    {
+        "type": "text",
+        "text": "113 Embedding. An embedding step embeds discrete tokens into the embedding vectors. Here we use   \n114 a token embedding layer to map each token to the corresponding entry in an embedding weight   \n115 $W ^ { e } \\in \\mathbb { R } ^ { ( K + 1 ) \\times d }$ . We denote $z ^ { \\prime } \\in \\mathbb { R } ^ { T \\times d }$ as the output token embedding, where $d$ is an embedding ",
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+    },
+    {
+        "type": "text",
+        "text": "dimension. We add a sinusoidal positional embedding 116 $\\pmb { p } ^ { \\prime }$ to the token embedding $z ^ { \\prime }$ to allow the 117 model to attend relative positions as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5af77b482249728d1d3b1213b8c68e8fb98458f84094d50b6a344e4c958b26b8.jpg",
+        "text": "$$\nz = z ^ { \\prime } + p ^ { \\prime } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "118 Transformer Encoder. A transformer encoder uses the embedding $z \\in \\mathbb { R } ^ { T \\times d }$ as the input, and   \n119 outputs $\\hat { \\boldsymbol { z } } \\in \\mathbb { R } ^ { T \\times d }$ . Each block of the transformer encoder contains a multi-head self-attention layer   \n120 and a feed-forward network, followed by a residual connection and a layer normalization. Through   \n121 the self-attention mechanism, it is possible to attend the relevant information of each time step at   \n122 once, while multiple attention heads can consider different periodicities in time series data [35].   \n123 Inverse Embedding. We need to invert the output $\\hat { z }$ into the original form of time series, $\\hat { \\boldsymbol { X } } \\in \\mathbb { R } ^ { T \\times n }$ .   \n124 To this end, we introduce an inverse embedding layer to our model. We calculate the cosine similarity   \n125 between the output embedding $\\hat { z }$ and the embedding weight $\\pmb { W } ^ { e }$ taken from the token embedding   \n126 layer, and apply the softmax operation as follows: ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/a71b20f479bdb45b2a158c976d47f18fc5c33907e7e8c0dc71181902094cdccd.jpg",
+        "text": "$$\n\\hat { \\pmb { p } } = s o f t m a x \\left( \\frac { \\hat { \\boldsymbol { z } } \\cdot \\mathbf { W } ^ { e ^ { \\top } } } { \\lVert \\hat { \\boldsymbol { z } } \\rVert \\left. \\mathbf { W } ^ { e ^ { \\top } } \\right. } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "127 From the above equation, we obtain the probability distribution $\\hat { \\pmb { p } } \\in \\mathbb { R } ^ { T \\times K }$ , where $\\hat { p } _ { t , k }$ means the   \n128 probability that $k$ will be selected in the range of $[ 0 , K )$ except the [MASK] token at the position $t$ .   \n129 We then extract an index $\\hat { \\mathbf { x } } _ { t }$ of the maximum probability for each time step $t \\in [ 1 , 2 , \\cdots , T ]$ , using   \n130 the soft-argmax operation as follows: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/1f0624b289e50399c4da019a9c61d1a31f18cc34b9a45c33163c93d109cada7c.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\pmb { x } } _ { t } = s o f t - a r g m a x ( \\hat { \\pmb { p } } _ { t } ) = \\sum _ { i = 0 } ^ { K - 1 } \\frac { e ^ { \\beta \\hat { \\pmb { p } } _ { t , i } } } { \\sum _ { j = 0 } ^ { K - 1 } e ^ { \\beta \\hat { \\pmb { p } } _ { t , j } } } i , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "131 where $\\beta$ is a sufficiently large value, such as 1000. Then, the indices in all time steps are concatenated to reconstruct the quantized output 132 $\\hat { X }$ as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ab0c7cc7cebf2dd842944d7342c971da98806d9028673833a5d313d10a9f2d71.jpg",
+        "text": "$$\n\\hat { \\pmb X } = \\left\\{ \\hat { \\pmb x } _ { 1 } , \\hat { \\pmb x } _ { 2 } , \\hat { \\star } \\hat { \\star } \\hat { \\ b } _ { T } \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            415,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "133 Transformer-based GAN Framework. To enhance the generation quality of $\\hat { X }$ , we design an   \n134 adversarial framework using transformer encoders. Following the notation of WGAN-GP [36], from   \n135 now on we use the term critic $C$ instead of the discriminator $D$ . Same as the generator $G$ , we   \n136 construct the critic $C$ using the transformer encoder, but in the critic $C$ , a [CLS] token is added in   \n137 front of the input tokens for classification. After passing through the transformer encoder, the linear   \n138 classifier outputs the critic score using only the [CLS] token. While classifying the real input $\\tilde { \\boldsymbol X }$ and   \n139 the fake output $\\hat { X }$ , the critic $C$ guides the generator $G$ to reconstruct more normal-like signal $\\hat { X }$ . As   \n140 a result, our model can distinguish $\\tilde { \\boldsymbol X }$ whether it is normal or abnormal according to the difference   \n141 between the input signal $\\tilde { \\boldsymbol { X } }$ and the reconstructed signal $\\hat { X }$ from the generator $G$ at test time. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 Two-Step Masking for Time Series Encoding ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "143 In the previous section, we introduce the transformer-based GAN framework for time series data.   \n144 However, we empirically find that the representation learning of the proposed transformer-based   \n145 GAN is not possible because the generator $G$ just copies the input as the output always. Inspired by   \n146 recent studies [35, 32, 33] that effectively learn the representation through masking in transformer,   \n147 we propose two different masking steps during training and test time: 1) random masking and 2)   \n148 exclusive and entropy-based re-masking. We experimentally demonstrate that the proposed two-step   \n149 masking is essential for our framework to learn the distribution of normal time series data successfully.   \n150 In the following content, we describe how to mask the input effectively in each step with details. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Step 1: Random Masking. As the first step, we partially hide the input signal $\\tilde { \\boldsymbol { X } }$ using a randomly selected mask from a mask pool. To construct the mask pool, we design a single mask in which the mask and the non-mask sections alternately appear. We then generate multiple masks by applying sliding window to the single mask, and group them as the mask pool. The composition of the mask pool depends on a length $l _ { m }$ of a single mask section, a ratio $r _ { m }$ of all mask parts, and a stride $s _ { m }$ for the sliding window. The number of masks $n _ { m }$ in the predefined mask pool is determined as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1f94296985c1d7f521193683b0c1c827be1c81f84a5411bd838bb326105e9871.jpg",
+        "text": "$$\nn _ { m } = 2 \\times \\left\\lceil { \\frac { l _ { m } } { s _ { m } } } \\right\\rceil .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "157 Using the above equation, we generate the enough number of masks in the pool to cover all sections   \n158 of the signal. During the train and test time, the mask is randomly selected in the predefined mask   \n159 pool per each signal, and the generator $G$ reconstructs $\\hat { X } _ { 1 }$ from the masked input $\\bar { \\tilde { \\mathbf { X } } } _ { m } ^ { 1 }$ .   \n160 Step 2: Exclusive and Entropy-based Re-Masking. After $\\hat { \\pmb X } _ { 1 }$ is generated from Step 1, we again   \n161 mask the exclusive parts that are not covered in Step 1 for our model to consider all parts of the   \n162 input. To avoid the error accumulation, here we re-mask the input $\\tilde { \\boldsymbol X }$ , instead of the first output $\\hat { X } _ { 1 }$ .   \n163 In addition, we provide feedback to our model by re-masking the parts that the model considers   \n164 uncertain during Step 1. To this end, we get an attention map from each layer of the generator as   \n165 follows: ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/d1b2da5425c010147d359fa4bf881dd73b7d48888cdb790074b7946637573a44.jpg",
+        "text": "$$\n\\begin{array} { c } { { A ^ { l , h } = s o f t m a x \\left( \\displaystyle \\frac { Q ^ { h } K ^ { h ^ { T } } } { \\sqrt { d } } \\right) , } } \\\\ { { A ^ { l } = \\displaystyle \\frac { 1 } { H } \\sum _ { h = 1 } ^ { H } A ^ { l , h } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "166 where $l \\in [ 1 , 2 , \\cdots , L ]$ and $A ^ { l }$ is the attention map in the $l$ -th layer, calculated by the average of all   \n167 attention maps for individual heads, $A ^ { l , h }$ . This layer-wise attention map determines how much a   \n168 specific time step focuses on the other parts of the input per signal. In this context, the uniformly   \n169 distributed attention means that the model does not know which connections are valuable [37], i.e.,   \n170 the prediction is uncertain. To quantify the uncertainty, we calculate an entropy $H _ { \\hat { X } _ { 1 } }$ of the masked   \n171 input ${ \\hat { X } } _ { 1 }$ as follows: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7b7cef18302eefc34af94f806207219f75b629b4fbd97050526d0f3fca1cf4de.jpg",
+        "text": "$$\n\\begin{array} { r } { H ( t ) = - \\displaystyle \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } \\sum _ { j = 1 } ^ { T } A _ { t , j } ^ { l } \\log A _ { t , j } ^ { l } , } \\\\ { H _ { \\hat { \\pmb X } _ { 1 } } = \\{ H ( 1 ) , H ( 2 ) , \\cdots , H ( T ) \\} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "172 To provide feedback on the high entropy parts, we re-mask $50 \\%$ of the parts already masked in Step   \n173 1. Then the generator $G$ re-generates the second output ${ \\hat { \\mathbf { X } } } _ { 2 }$ from the masked signal ${ \\tilde { \\bf X } } _ { 2 }$ . Finally,   \n174 we combine the masked parts generated from Step 1 and the ones from Step 2 to construct the final   \n175 output $\\hat { X }$ . If there are overlapped parts between Step 1 and Step 2, the parts of Step 2 are used. From   \n176 this re-masking step, we experimentally prove that our model becomes robust to unexplored and   \n177 uncertain parts within a fixed model size. We also use the same random masking and re-masking   \n178 strategies at test time. ",
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+    {
+        "type": "text",
+        "text": "3.4 Training AnoFormer ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To train AnoFormer, we apply the cross-entropy loss to reconstruct the same input 80 $\\tilde { \\boldsymbol X }$ from the final output 81 $\\hat { X }$ as follows: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b4dee98012c340c8a46652695b059bebc84adf953184b1b4ff256f743f55ad7c.jpg",
+        "text": "$$\n\\mathcal { L } _ { r e c } = - \\sum _ { i = 1 } ^ { T } \\sum _ { j = 1 } ^ { K } \\tilde { X } _ { i , j } \\cdot \\log \\left( \\hat { p } _ { i , j } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where 182 $\\tilde { X } _ { i , j }$ denotes the one-hot label vector from the input and $\\hat { p } _ { i , j }$ denotes the probability distribu183 tion of the final output $\\hat { X }$ . Using $\\hat { X }$ from the generator $G$ during two-step masking, the critic $C$ tries 184 to minimize the following loss function: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb6197e84fbc8da51a2c79bfb2a11c09abb4d2438239c477be947a36a46e5cbb.jpg",
+        "text": "$$\n\\begin{array} { c } { { X ^ { \\prime } = \\epsilon \\tilde { X } + ( 1 - \\epsilon ) \\hat { X } , } } \\\\ { { { \\mathcal { L } } _ { C , a d v } = \\left( \\mathbb { E } \\left[ C \\left( \\hat { X } \\right) \\right] - \\mathbb { E } \\left[ C \\left( \\tilde { X } \\right) \\right] \\right) + \\lambda \\mathbb { E } _ { X ^ { \\prime } \\sim P _ { X ^ { \\prime } } } \\left[ \\left( \\left. \\nabla _ { X ^ { \\prime } } C \\left( X ^ { \\prime } \\right) \\right. _ { 2 } - 1 \\right) ^ { 2 } \\right] , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "185 where $\\epsilon$ is randomly chosen between zero and one. The first term measures the Wasserstein distance   \n186 and the second term is the gradient penalty, where $X ^ { \\prime }$ is a random sample from $P _ { X ^ { \\prime } }$ to enforce the   \n187 Lipschitz constraint. The coefficient is a harmonic parameter to balance the Wasserstein distance and   \n188 the gradient penalty, where we use the value of 10. The loss function of the generator $G$ is as follows:   \n189 ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/845ba6154e501fc6c41df3d8cd7a97fb64f6b65c0ddb77884bd9d824b46f13c5.jpg",
+        "text": "$$\n\\mathcal { L } _ { a d v } ^ { g } = - \\mathbb { E } \\left[ C \\left( \\hat { \\boldsymbol { X } } \\right) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/f5d35d824e0f51fe78877b9551cfdf57efc90b96896aa2acf52cca926903bd36.jpg",
+        "table_caption": [
+            "Table 1: Quantitative comparisons in four datasets. For all of the metrics, a higher value indicates a better performance. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Metric</td><td rowspan=\"2\">Base Architecture</td><td rowspan=\"2\">Method</td><td colspan=\"6\">NeurIPS-TS</td><td rowspan=\"2\">MIT-BIH</td><td rowspan=\"2\">2D-gesture</td><td rowspan=\"2\">Power-demand</td></tr><tr><td>Global</td><td>Contextual</td><td>Shapelet</td><td>Seasonal</td><td>Trend</td><td>Average</td></tr><tr><td rowspan=\"6\">AUROC</td><td>CNN</td><td>BeatGAN</td><td>0.9753</td><td>0.6128</td><td>0.7398</td><td>0.9742</td><td>1.0000</td><td>0.8372</td><td>0.9475</td><td>0.7256</td><td>0.5796</td></tr><tr><td>RNN</td><td>TadGAN</td><td>1.0000</td><td>0.4285</td><td>0.9834</td><td>0.9744</td><td>0.9327</td><td>0.9726</td><td>0.8256</td><td>0.5294</td><td>0.8438</td></tr><tr><td></td><td>RAE-ensemble</td><td>0.5226</td><td>0.9348</td><td>0.9244</td><td>0.9625</td><td>0.7246</td><td>0.8138</td><td>-</td><td>0.7808</td><td>0.6587</td></tr><tr><td></td><td>RAMED</td><td>0.5265</td><td>0.9325</td><td>0.9084</td><td>0.9628</td><td>0.7259</td><td>0.8112</td><td>-</td><td>0.7839</td><td>0.6787</td></tr><tr><td>Transformer</td><td>Anomaly Transformer</td><td>0.9931</td><td>0.6224</td><td>0.7407</td><td>0.9332</td><td>0.9976</td><td>0.8400</td><td>0.8108</td><td>0.7868</td><td>0.7739</td></tr><tr><td></td><td>AnoFormer (Ours)</td><td>1.0000</td><td>0.9758</td><td>0.9900</td><td>0.9985</td><td>0.9985</td><td>0.9911</td><td>0.9552</td><td>0.8407</td><td>0.8667</td></tr><tr><td rowspan=\"6\">AUPRC</td><td>CNN</td><td>BeatGAN</td><td>0.9855</td><td>0.7051</td><td>0.6817</td><td>0.9748</td><td>1.0000</td><td>0.9634</td><td>0.9143</td><td>0.4952</td><td>0.1228</td></tr><tr><td>RNN</td><td>TadGAN</td><td>1.0000</td><td>0.3603</td><td>0.9565</td><td>0.9754</td><td>0.8731</td><td>0.9806</td><td>0.4621</td><td>0.4367</td><td>0.3098</td></tr><tr><td></td><td>RAE-ensemble</td><td>0.0453</td><td>0.8297</td><td>0.8159</td><td>0.9191</td><td>0.1378</td><td>0.5496</td><td>-</td><td>0.5287</td><td>0.1400</td></tr><tr><td></td><td>RAMED</td><td>0.0443</td><td>0.8223</td><td>0.6873</td><td>0.9109</td><td>0.1291</td><td>0.5188</td><td>-</td><td>0.5331</td><td>0.1627</td></tr><tr><td>Transformer</td><td>Anomaly Transformer</td><td>0.9959</td><td>0.6957</td><td>0.6630</td><td>0.9364</td><td>0.9978</td><td>0.9639</td><td>0.5603</td><td>0.5607</td><td>0.4967</td></tr><tr><td></td><td>AnoFormer (Ours)</td><td>1.0000</td><td>0.9854</td><td>0.9901</td><td>0.9985</td><td>0.9987</td><td>0.9982</td><td>0.9187</td><td>0.6142</td><td>0.5584</td></tr><tr><td rowspan=\"6\">F1 score</td><td>CNN</td><td>BeatGAN</td><td>0.9345</td><td>0.7348</td><td>0.6136</td><td>0.9487</td><td>1.0000</td><td>0.9008</td><td>0.8015</td><td>0.4941</td><td>0.2266</td></tr><tr><td>RNN</td><td>TadGAN</td><td>1.0000</td><td>0.3590</td><td>0.9331</td><td>0.9844</td><td>0.8170</td><td>0.9380</td><td>0.5289</td><td>0.4138</td><td>0.5714</td></tr><tr><td></td><td>RAE-ensemble</td><td>0.0853</td><td>0.8343</td><td>0.7750</td><td>0.9181</td><td>0.3889</td><td>0.6003</td><td>-</td><td>0.5511</td><td>0.2678</td></tr><tr><td></td><td>RAMED</td><td>0.0838</td><td>0.8272</td><td>0.6203</td><td>0.8782</td><td>0.4040</td><td>0.5627</td><td>-</td><td>0.5633</td><td>0.2934</td></tr><tr><td>Transformer</td><td>Anomaly Transformer</td><td>0.9751</td><td>0.7358</td><td>0.6115</td><td>0.8730</td><td>0.9958</td><td>0.9014</td><td>0.5446</td><td>0.6486</td><td>0.6053</td></tr><tr><td></td><td>AnoFormer (Ours)</td><td>1.0000</td><td>0.9400</td><td>0.9696</td><td>0.9913</td><td>0.9974</td><td>0.9798</td><td>0.8410</td><td>0.6667</td><td>0.6226</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "which makes the critic 190 $C$ not be able to classify the generated $\\hat { X }$ . To sum up, the proposed AnoFormer 191 is trained via the following loss functions for the generator $G$ and the critic $C$ : ",
+        "bbox": [
+            143,
+            371,
+            826,
+            400
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/187b51cdbd7631c63c3feb414ee4b75917ff3893f7590288cb292855f9a1ea88.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\mathcal { L } _ { G } = \\lambda _ { r e c } \\mathcal { L } _ { r e c } + \\lambda _ { a d v } \\mathcal { L } _ { a d v } ^ { g } , } } \\\\ { { \\mathcal { L } _ { C } = \\mathcal { L } _ { a d v } ^ { c } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            405,
+            596,
+            443
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "192 where we set $\\lambda _ { r e c }$ and $\\lambda _ { a d v }$ as 1. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Datasets. We evaluated AnoFormer on four real-world benchmarks: 1) MIT-BIH 1 contains 48 ECG records of test subjects from Beth Israel Hospital, 2) 2D-gesture contains time series of X and Y coordinates of an actor’s right hand, 3) Power-demand is a dataset measuring the power comsumption for the Dutch research facility, and 4) NeurIPS-TS 2 [38] is a synthetic dataset including five different time series anomaly scenarios as point-global, point-contextual, pattern-shapelet, pattern-seasonal, and pattern-trend. More details on each dataset are summarized in Appendix A. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Baselines. We compared our model with various baselines, including CNN, RNN, and transformerbased reconstruction models. BeatGAN [2] and TadGAN [8] are CNN and LSTM-based GAN models, respectively. RAE-ensemble [9] is an ensemble of RNNs with sparse skip connections in autoencoder. RAMED [10] additionally uses the multiresolution decoding based on RAE-ensemble. Anomaly Transformer [11] develops the transformer architecture to utilize association information. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Implementation Details. For both the generator $G$ and the critic $C$ , we utilized the basic transformer encoders with 9 and 6 layers for MIT-BIH, and 4 and 2 layers for other datasets, respectively. The embedding dimension and the number of heads are 128 and 8, respectively. The mask length $l _ { m }$ is about $10 \\%$ of the sequence length $T$ , and the mask stride $s _ { m }$ is about half of the mask length $l _ { m }$ We used Adam optimizer with initial learning rate, momentum $\\beta _ { 1 }$ , and $\\beta _ { 2 }$ as 0.0001, 0.5, and 0.999, respectively. We implemented our model using PyTorch and trained on a NVIDIA RTX 3090 GPU. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Quantitative Results ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Table 1 shows the anomaly detection performances of each baseline on three different real-world datasets (i.e., MIT-BIH, 2D-gesture, and Power-demand), and a synthetic dataset (i.e., NeurIPS-TS [38]). Overall, RNN or transformer-based models showed high performances except MIT-BIH. In MIT-BIH, BeatGAN showed the second-best performances among all the benchmarks. In case of the proposed AnoFormer, this model outperformed all the baselines in four different datasets. In ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/47b408cd47c8108b2ae4f17f6296ca554263ab559cf561bb93d4146ced1092e0.jpg",
+        "image_caption": [
+            "Figure 2: Output visualization in Point-Contextual (NeurIPS-TS) and MIT-BIH datasets. Left: visualization of abnormal input and the normal-like output. Middle: reconstruction results of random masking (blue). Right: reconstruction results of exclusive (green) and entropy-based (red) re-masking. Vest viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "217 particular, our model performed well on NeurIPS-TS containing five types of outliers, and it means   \n218 AnoFormer is robust to the various types of outliers. AnoFormer achieved the state-of-the-art results   \n219 from small datasets (e.g., 2D-gesture and Power-demand) with about 1,000 training sets to large   \n220 datasets (e.g., NeurIPS-TS and MIT-BIH) with about tens of thousands of training sets, and from   \n221 univariate to multivariate cases. The experimental results demonstrate that the proposed transformer  \n222 based GAN framework with the two-step masking strategy is effective to reconstruct normal time   \n223 series data for anomaly detection. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 Qualitative Results ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Figure 2 shows the qualitative examples of AnoFormer. First column (Figure 2(a) and Figure 2(d)) shows the abnormal examples of point-contextual of NeurIPS-TS and MIT-BIH datasets, respectively. The other columns show that the proposed rnadom masking and re-masking strategies actually provide feedback to our framework. For example, the incorrectly copied parts in Step 1 was refined by the entropy-based re-masking (please see the black circles in the figure). As shown in the figure, when the abnormal inputs were received, the model generated the normal-like outputs. Therefore, AnoFormer can detect the abnormal points through the difference between the input and the output. ",
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+    {
+        "type": "text",
+        "text": "4.3 Ablation Study ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We conducted various ablation studies to analyze the effectiveness of the proposed transformerbased GAN framework and two-step masking. All of the ablation studies were performed on the Point-Contextual dataset of NeurIPS-TS, since it is the most difficult task to detect the anomalies out of the five types of outliers. Figure 2 shows an example of Point-Contextual dataset, which has the small glitches as the outliers. In Appendix B, we additionally examined the sensitivity of each hyperparameter newly adopted in our model. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3.1 Transformer-based GAN Framework ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            790,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We first investigated the effectiveness of the transformer-based adversarial framework in our model. In this experiment, we used BeatGAN as a CNN-based baseline. Table 2 shows the ablation results when the generator and the critic use different backbone networks, such as CNN, and transformer. As shown in the table, the transformer-based generator showed higher performances on all of metrics with large margins than the CNN-based generator. Interestingly, we empirically found that there was no synergy when using CNN-based critic with the transformer-based generator. It means, it is not helpful for the transformer-based generator to construct the critic with an inappropriate baseline. On ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/bca7feaa2fd9a1a9c3563a63edcdca89c832506b75f674128bf51e540bbb3be4.jpg",
+        "table_caption": [
+            "Table 2: Ablation study of the proposed transformer-based GAN. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Generator</td><td>Critic</td><td>AUROC</td><td>AUPRC</td><td>F1 score</td></tr><tr><td>CNN</td><td>CNN</td><td>0.6128</td><td>0.7051</td><td>0.7348</td></tr><tr><td>Transformer</td><td>-</td><td>0.9572</td><td>0.9735</td><td>0.9093</td></tr><tr><td>Transformer</td><td>CNN</td><td>0.9510</td><td>0.9675</td><td>0.9026</td></tr><tr><td>Transformer</td><td>Transformer</td><td>0.9758</td><td>0.9854</td><td>0.9400</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/a411a39e6ddc34da1423d6ffd0537b736f9ceac713076e3012f497d263d9d769.jpg",
+        "table_caption": [
+            "Table 3: Ablation study of the proposed two-step masking. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Step1</td><td>Step 2</td><td>AUROC</td><td>AUPRC</td><td>F1 score</td></tr><tr><td>-</td><td>-</td><td>0.5000</td><td>0.3602</td><td>0.2386</td></tr><tr><td>Random</td><td>■</td><td>0.8557</td><td>0.7959</td><td>0.7548</td></tr><tr><td>Mask pool</td><td>■</td><td>0.9109</td><td>0.8651</td><td>0.8200</td></tr><tr><td>Mask pool</td><td>Mask pool (50%)</td><td>0.9277</td><td>0.8590</td><td>0.7808</td></tr><tr><td>Mask pool</td><td>Exclusive (50%)</td><td>0.9489</td><td>0.9622</td><td>0.9057</td></tr><tr><td>Mask pool</td><td>Exclusive +Random (75%)</td><td>0.9709</td><td>0.9466</td><td>0.9004</td></tr><tr><td>Mask pool</td><td>Exclusive + Anomaly score (75%)</td><td>0.9747</td><td>0.9533</td><td>0.9119</td></tr><tr><td>Mask pool</td><td>Exclusive+ Entropy (75%)</td><td>0.9758</td><td>0.9854</td><td>0.9400</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "247 the other hand, the transformer-based critic showed better performances than the baseline without   \n248 the critic, which means it encourages the generated output to be close to the normal signal. From   \n249 this result, we demonstrate that our transformer-based GAN framework trained with the proposed   \n250 masking strategy is effective to reconstruct normal time series data for anomaly detection. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3.2 Two-Step Masking ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            481,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As shown in Table 3, we investigated the effect of two-step masking in our model. The first row means a naive form of the transformer-based GAN without any masking. The result was 0.5 of AUROC, which means the naive transformer-based GAN cannot distinguish between normal and abnormal signals at all. To overcome this critical issue, we adopted various masking strategies. First, we investigated the masking for Step 1. Here, Random means a fully random masking without any predefined mask pool. Mask Pool means our predefined mask pool defined in Section 3.3. The results showed that regardless of the masking strategy, masking itself during training and test enabled the transformer-based GAN to effectively learn the distribution of normal time series data. Moreover, we confirmed that Mask Pool is much better than Random masking, because each mask in the mask pool definitely covers the different parts from each other, providing a complementary effect. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "262 Next, we conducted in-depth experiments to evaluate and compare different re-masking strategies in   \n263 Step 2. Mask pool method in Step 1 can be also used for re-masking. Exclusive method re-masks the   \n264 exclusive parts of the random mask selected in Step 1. From the results, we found that re-masking   \n265 improved the detection ability of our model, and especially, exclusive masking strategy was really   \n266 effective. This is because the model can consider the characteristic of whole signal during two-step   \n267 masking. To provide more feedback to our model, we additionally re-masked the masked parts in   \n268 Step 1. We experimented the following three cases: 1) Random method re-masks the signal randomly,   \n269 2) Anomaly score method re-masks the parts with high anomaly scores, and 3) Entropy method   \n270 re-masks the parts with high entropy values. The results showed that masking the uncertain parts   \n271 provided the proper feedback to our model, resulting in the highest scores among all the baselines.   \n272 Therefore, we confirmed that the entropy-based re-masking is more effective than the other additional   \n273 masking methods. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 Discussion ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            837,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "275 We further conducted analysis to demonstrate the effectiveness of the proposed two-step masking   \n276 strategy. To confirm the importance of Step 2, we compared our method with the absence of Step 2.   \n277 Here, we also used Point-Contextual dataset in NeurIPS-TS for analysis.   \n278 The Effectiveness of Re-Masking. First, we investigated the validity of re-masking step. We reported   \n279 the average AUROC and std for all masks in the predefined mask pool in Figure 3. In Step 1, there   \n280 was a problem that the standard deviation (std) was too high because both training and test time   \n281 had randomness in selecting the mask parts. By re-masking through Step 2, the randomness of the   \n282 masked parts was eliminated and the std was significantly reduced. The performance of anomaly   \n283 detection also increased with a large margin by referring to the entire signal.   \n284 Analysis on Entropy-based Re-Masking. To understand the effectiveness of entropy-based re  \n285 masking intuitively, we visualized the relation between entropy and anomaly score in Figure 3(b) and   \n286 Figure 3(c) for both cases of normal and abnormal. Since the entropy-based method re-masked the   \n287 parts selected in Step 1, we measured the anomaly score only in the parts corresponding to Step 1. We   \n288 found two meaningful insights through the analysis. First, the higher the entropy, the more incorrect   \n289 signal the model generates. In training phase, the high anomaly score means that the model generates   \n290 output incorrectly, because only normal data is used. From the result of Figure 3(b), the entropy was   \n291 also high in the parts with high anomaly score, which means that the model did not generate signals   \n292 well in the parts with high entropy during Step 1. This is because in the high entropy the attention is   \n293 uniformly distributed and the meaningful connection is not learned. By re-masking these parts in Step   \n294 2, anomaly score was significantly reduces, which means that the model reconstructed the normal   \n295 data well in the training process. Second, the entropy-based masking improves the discriminative   \n296 ability between normal and abnormal in test time. As shown in Figure 3(c), likewise in the case of   \n297 normal, the higher the entropy, the higher the anomaly score in abnormal case. However, there was   \n298 also a part with a high entropy and a low anomaly score. These parts mean that the model copied   \n299 the abnormal input as it was without making it normal. It is possible to provide feedback in both   \n300 cases with entropy-based re-masking. By re-masking the parts with high entropy, anomaly score was   \n301 considerably increased, and it means the parts that were not well generated due to copying in Step 1   \n302 were well re-generated. This makes it possible to further discriminate between normal and abnormal   \n303 through anomaly score. In fact, a large anomaly score does not mean getting close to normal data. We   \n304 used a NeurIPS-TS dataset to see if the output gets closer to normal data through re-masking. Since   \n305 we synthesized abnormal datasets by injecting sporadic outliers in an additive manner, we could   \n306 easily get the original normal version of the abnormal data. From this, we confirmed that the output   \n307 was correctly getting closer to the original normal through Step 2. We further experimented about   \n308 which layer’s entropy information should be used? From the results in Appendix C, we calculated   \n309 entropy from all layers and averaged them. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/ed4625d277a62b4ec5b0601fff9fc4d34c1c943f2d384494de37abcbde8e24f6.jpg",
+        "image_caption": [
+            "Figure 3: (a) AUROC and std of all masks in the predefined mask pool. (b) Analysis on the entropy-based re-masking strategy (normal-case). (c) Analysis on the entropy-based re-masking strategy (abnormal-case). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "0 6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "311 In this paper, we introduce AnoFormer, a novel transformer-based GAN for time series anomaly   \n312 detection. To learn time series data directly with our model, we propose pre-processing and embedding   \n313 methods suitable for time series data. A new training scheme based on two-step masking enables   \n314 AnoFormer to embed the representation of normal signals. Especially, the exclusive and entropy  \n315 based re-masking method significantly improves the anomaly detection performances on several   \n316 benchmark datasets. From the extensive experiments, we empirically demonstrate that our model   \n317 is really effective to solve time series anomaly detection. As future work, we plan to study novel   \n318 techniques for shorter inference time, and deal with time series data longer than an hour or a day. ",
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parse/dev/kTcRljax0x9/kTcRljax0x9.md

@@ -0,0 +1,234 @@
+# ASSESSING REINFORCEMENT LEARNING POLICIES VIA NATURAL CORRUPTIONS AT THE EDGE OF IMPERCEPTIBILITY
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Deep reinforcement learning algorithms have recently achieved significant success in learning high-performing policies from purely visual observations. The ability to perform end-to-end learning from raw high dimensional input alone has led to deep reinforcement learning algorithms being deployed in a variety of fields. Thus, understanding and improving the ability of deep reinforcement learning policies to generalize to unseen data distributions is of critical importance. Much recent work has focused on assessing the generalization of deep reinforcement learning policies by introducing specifically crafted adversarial perturbations to their inputs. In this paper, we approach this problem from another perspective and propose a framework to assess the generalization skills of trained deep reinforcement learning policies. Rather than focusing on worst-case analysis of distribution shift, our approach is based on black-box perturbations that correspond to minimal semantically meaningful natural changes to the environment or the agent’s visual observation system ranging from brightness to compression artifacts. We demonstrate that the perceptual similarity distance of the minimal natural perturbations is orders of magnitude smaller than the perceptual similarity distance of the adversarial perturbations to the unperturbed observations (i.e. minimal natural perturbations are perceptually more similar to the unperturbed states than the adversarial perturbations), while causing larger degradation in the policy performance. Furthermore, we investigate state-of-the-art adversarial training methods and show that adversarially trained deep reinforcement learning policies are more sensitive to almost all of the natural perturbations compared to vanilla trained policies. Lastly, we highlight that our framework captures a diverse set of bands in the Fourier spectrum; thus providing a better overall understanding of the policy’s generalization capabilities. We believe our work can be crucial towards building resilient and generalizable deep reinforcement learning policies.
+
+# 1 INTRODUCTION
+
+Following the initial work of Mnih et al. (2015), the use of DNNs as function approximators in reinforcement learning has led to a dramatic increase in the capabilities of RL agents Schulman et al. (2017); Lillicrap et al. (2015). In particular, these developments allow for the direct learning of strong policies from raw, high-dimensional inputs (i.e. visual observations). With the successes of these new methods come new challenges regarding the robustness and generalization capabilities of deep reinforcement learning agents.
+
+Szegedy et al. (2014) showed that specifically crafted imperceptible perturbations can lead to misclassification in image classification. After this initial work a new research area emerged to investigate the abilities of deep neural networks against specifically crafted adversarial examples. While various works studied many different ways to compute these examples (Carlini & Wagner, 2017; Madry et al., 2018; Goodfellow et al., 2015; Kurakin et al., 2016), several works focused on studying ways to increase the robustness against such specifically crafted perturbations, based on training with the existence of such perturbations (Madry et al., 2018; Tramer et al., 2018; Goodfellow et al., \` 2015; Xie & Yuille, 2020).
+
+As image classification suffered from this vulnerability towards worst-case distributional shift in the input, a series of work conducted in deep reinforcement learning showed that deep neural policies are also susceptible to specifically crafted imperceptible perturbations (Huang et al., 2017; Kos & Song, 2017; Pattanaik et al., 2018; Lin et al., 2017; Sun et al., 2020; Korkmaz, 2021). While one line of work put effort on exploring these vulnerabilities in deep neural policies, another line in parallel focused making them robust and reliable via adversarial training (Pinto et al., 2017; Mandlekar et al., 2017; Huan et al., 2020).
+
+While adversarial perturbations and adversarial training provide a notion of robustness for trained deep neural policies, in this paper we approach the resilience problem of the deep neural policies from a wider perspective, and propose a framework to test a more generic sense of robustness towards minimal perceptually similar perturbations1. To be able to achieve this we go beyond $\ell _ { p }$ -norm bounded pixel perturbations and include semantically meaningful minimal realistic perturbations. By this approach we seek answers to the following questions: (i) How perceptually similar are minimal semantically meaningful perturbed states to the original unperturbed states, and how does this compare to $\ell _ { p }$ -norm bounded adversarially perturbed states? (ii) What are the differences between adversarial perturbations and minimal natural perturbations introduced to the policy observation in terms of performance degradation of the trained deep reinforcement learning policy? (iii) How does state-of-the-art adversarial training affect the performance degradation caused by perceptually similar minimal natural perturbations compared to vanilla training? To be able answer these questions, in this work we focus on the notion of robustness of trained deep reinforcement learning agents and make the following contributions:
+
+• We propose a framework consisting of a diverse set of minimalistic (i.e. perceptually similar) semantically meaningful natural perturbations.   
+• We run multiple experiments in the Arcade Learning Environment (ALE) in various games with high dimensional state representation and provide the relationship between the perceptual similarities to unperturbed states under our proposed natural perturbation framework and adversarial perturbations.   
+• We compare our proposed framework with the state-of-the-art adversarial method based on $\ell _ { p }$ -norm changes, and we show that our natural perturbation framework is competitive in degrading the performance of the deep reinforcement learning agent with lower perceptual similarity distance.   
+• We inspect state-of-the-art adversarial training under our proposed framework, and demonstrate that the adversarially trained models become more vulnerable to various natural perturbations compared to vanilla trained models.   
+• Finally, we investigate the frequency domain of our framework and state-of-the-art targeted attacks. We show that our framework captures different bands of the frequency spectrum, thus yielding a better estimate of the model robustness.
+
+# 2 BACKGROUND AND RELATED WORK
+
+# 2.1 PRELIMINARIES
+
+In this paper we consider Markov Decision Processes (MDPs) given by a tuple $( S , A , P , r , \gamma , s _ { 0 } )$ . The reinforcement learning agent interacts with the MDP by observing states $s \in S$ , and then taking actions $a \in A$ . Here $s _ { 0 }$ represents the initial state of the agent, and $\gamma$ represents the discount factor. The probability of transitioning to state $s ^ { \prime }$ when the agent takes action $a$ in state $s$ is determined by the Markovian transition kernel $P : S \times A \times S \to \mathbb { R }$ . The reward received by the agent when taking action $a$ in state $s$ is given by the reward function $r : S \times A \to \mathbb { R }$ . The goal of the expected cumulative discounted reward with the environment. This goal is achieve the agent is to learn a policy $\pi _ { \theta } : S \times A \to \mathbb { R }$ $\scriptstyle \sum _ { t = 0 } ^ { T - 1 } \gamma ^ { t } r ( s _ { t } , a _ { t } )$ which takes an action that the agent receives viastate-action value function $a$ in state that maximizes $Q ( s , a ) =$ $\mathbb { E } _ { \pi } [ \sum _ { t = 0 } ^ { T - 1 } \gamma ^ { t } r ( s _ { t } , a _ { t } ) | s = s _ { 0 } , a _ { 0 } = 0 ]$ assigning a value to each state-action pair. We use $\mathcal { F } ( s )$ to denote the 2D discrete Fourier transform of state $s$ in which each frequency is computed via $\begin{array} { r } { \mathcal { F } ( m , n ) = \sum _ { k = 0 } ^ { M - 1 } \sum _ { l = 0 } ^ { N - 1 } s ( k , l ) e ^ { - j 2 \pi ( m k / M + n l / N ) } } \end{array}$ where $k$ and $l$ are the coordinates of the state, and $M$ and $N$ correspond to ranges of the 2D state representation.
+
+# 2.2 CRAFTING ADVERSARIAL PERTURBATIONS
+
+Szegedy et al. (2014) proposed to minimize the distance between the original image and adversarially produced image to create adversarial perturbations. The authors used box-constrained L-BFGS to solve this optimization problem. Goodfellow et al. (2015) introduced the fast gradient method (FGM)
+
+$$
+x _ { \mathrm { a d v } } = x + \epsilon \cdot \frac { \nabla _ { x } J ( x , y ) } { | | \nabla _ { x } J ( x , y ) | | _ { p } } ,
+$$
+
+for crafting adversarial examples in image classification by taking the gradient of the cost function $J ( x , y )$ used to train the neural network in the direction of the input, where $x$ is the input, $y$ is the output label, and $J ( x , y )$ is the cost function for image classification. Carlini & Wagner (2017) introduced targeted attacks in the image classification domain based on distance minimization between the adversarial image and the original image while targeting a particular label. In the deep reinforcement learning domain the Carlini & Wagner (2017) formulation is
+
+$$
+\begin{array} { r l } & { \underset { s _ { \mathrm { a d v } } \in D _ { \epsilon , p } ( s ) } { \operatorname* { m i n } } \Vert s _ { \mathrm { a d v } } - s \Vert _ { p } } \\ & { \mathrm { s u b j e c t ~ t o ~ } \underset { a } { \operatorname { a r g m a x } } Q ( s , a ) \neq \underset { a } { \operatorname { a r g m a x } } Q ( s _ { \mathrm { a d v } } , a ) } \end{array}
+$$
+
+where $s$ is the unperturbed input, $S _ { \mathrm { a d v } }$ is the adversarially perturbed input, $a ^ { * } ( s )$ is the action taken in the unperturbed state, and $a ^ { * } ( s _ { \mathrm { a d v } } ) = \arg \operatorname* { m a x } _ { a } Q ( s _ { \mathrm { a d v } } , a )$ is the action taken in the adversarial state. This formulation attempts to minimize the distance to the original state, constrained to states leading to sub-optimal actions as determined by the $Q$ -network. In contrast to adversarial attacks, in our proposed threat model we will not need any information on the cost function used to train the network, the $Q$ -network of the trained agent, or access to the visited states themselves.
+
+# 2.3 ADVERSARIAL APPROACH IN DEEP REINFORCEMENT LEARNING
+
+The first adversarial attacks on deep reinforcement learning introduced by Huang et al. (2017) and Kos & Song (2017) adapted FGSM from image classification to the deep reinforcement learning setting. Subsequently, Mandlekar et al. (2017) used FGSM perturbations for adversarial training of deep reinforcement learning agents. Pinto et al. (2017); Gleave et al. (2020) focused on modeling the interaction between the adversary and the agent, while Lin et al. (2017); Sun et al. (2020) focused on strategically timing when (i.e. in which state) to attack an agent using perturbations computed with the Carlini & Wagner (2017) adversarial formulation. Quite recently, Huan et al. (2020) proposed to model this dynamic as a State-Adversarial Markov Decision Process (SA-MDP), and the authors claimed the SA-MDP model provides theoretically justified robust deep reinforcement learning agents.
+
+# 2.4 PERCEPTUAL SIMILARITY DISTANCE
+
+Zhang et al. (2018) found that internal activations of networks trained for high-level tasks correspond to human perceptual judgements across different network architectures Iandola et al. (2016), Krizhevsky et al. (2012), Simonyan & Zisserman (2015) without calibration. Furthermore, the authors propose a method to measure the perceptual distance between two images with the Learned Perceptual Image Patch Similarity (LPIPS) metric. We compare the distance between adversarial states ${ \boldsymbol { s } } _ { \mathrm { a d v } }$ and the original states $s$ with the LPIPS metric. We refer to the LPIPS metric as $\mathcal { P } _ { \mathrm { s i m i l a r i t y } }$ throughout the paper. $\mathcal { P } _ { \mathrm { s i m i l a r i t y } } ( s , s _ { \mathrm { a d v } } )$ returns the distance between $s$ and ${ \boldsymbol { s } } _ { \mathrm { a d v } }$ based on network activations. Zhang et al. (2018) show that $\mathcal { P } _ { \mathrm { s i m i l a r i t y } }$ results in a reliable approximation of human perception.
+
+In more detail, the LPIPS metric in Zhang et al. (2018) is given by measuring the $\ell _ { 2 }$ distance between a normalized version of the activations of the neural network at several internal convolutional layers. For each convolutional layer $l$ let $W _ { l }$ be the width, $H _ { l }$ the height, and $C _ { l }$ the number of channels. Further, let $y ^ { l } \in \mathbb { R } ^ { W _ { l } \times H _ { l } \times C _ { l } }$ denote the vector of activations in convolutional layer $l$ .
+
+To compute the perceptual similarity distance between two states $s$ and $s _ { 0 }$ , first calculate the internal activations $y ^ { \hat { l } } , y _ { 0 } ^ { l } \ { \in } \ \mathbb { R } ^ { W _ { l } \times H _ { l } \times C _ { l } ^ { \bullet } }$ (corresponding to $s$ and $s _ { 0 }$ respectively) for $L$ internal layers. Second, unit-normalize the activation vectors in the channel dimension, and denote the resulting normalized activations by $\hat { y } ^ { l }$ and $\hat { y } _ { 0 } ^ { l }$ . Next scale each channel in $\hat { y } ^ { l }$ and $\hat { y } _ { 0 } ^ { l }$ by the same, fixed weight vector $w _ { l } \in \mathbb { R } ^ { C _ { l } }$ . Here $w _ { l }$ can either be a learned vector of weights for layer $l$ or the vector of all ones if no scaling is desired. The last step is then to compute the perceptual similarity distance by first averaging the $\ell _ { 2 }$ distance between the scaled activations over the spatial dimensions, and then summing over the $L$ layers. Formally,
+
+$$
+{ \mathcal { P } } _ { \mathrm { s i m i l a r i t y } } \big ( s , s _ { 0 } \big ) = \sum _ { l } { \frac { 1 } { H _ { l } W _ { l } } \sum _ { h , w } \left\| w _ { l } \odot \big ( \hat { y } _ { h w } ^ { l } - \hat { y } _ { 0 h w } ^ { l } \big ) \right\| _ { 2 } ^ { 2 } }
+$$
+
+# 2.5 IMPACT
+
+To be able to compare between different algorithms and different games the performance degradation of the deep reinforcement learning policy is defined as the normalized impact of an adversary on the agent:
+
+$$
+\mathcal { T } = \frac { \mathrm { S c o r e } _ { \mathrm { c l e a n } } - \mathrm { S c o r e } _ { \mathrm { a d v } } } { \mathrm { S c o r e } _ { \mathrm { c l e a n } } - \mathrm { S c o r e } _ { \mathrm { m i n } } ^ { \mathrm { f u x e d } } } .
+$$
+
+Scorefixedmin is a fixed minimum score for a game, ${ \mathrm { S c o r e } } _ { \mathrm { a d v } }$ and $\mathbf { S c o r e } _ { \mathrm { c l e a n } }$ are the scores of the agent with and without any modification to the agent’s observations system respectively.
+
+# 3 A GENERALIZATION TESTING FRAMEWORK WITH NATURAL PERTURBATIONS
+
+In our paper we focus on robustness and generalization issues that deep reinforcement learning policies encounter with a contrasting view compared to prior work focusing on worst-case distributional shift within an imperceptibility bound (see Section 2.3). We propose a baseline to evaluate deep reinforcement learning policies with realistic and minimal corruptions to the environment with which they interact. We essentially juxtapose adversarial perturbations and natural corruptions with respect to their perceptual similarity distance (see Section 2.4) to the original states and their degree of impact on the policy performance. More importantly, we question the imperceptibility of $\ell _ { p }$ -norm bounded adversarial perturbations in terms of perceptual similarity distance, and compare this imperceptibility notion to natural perturbations. While we categorize adversarial perturbations also as a component in the framework majorly concentrated on the high frequencies, we embed several realistic perturbations that aim to cover diverse bands in the frequency spectrum. We highlight that prior work focused on the presence of a strong adversary model that requires prior access to training details of the agent’s neural network Huang et al. (2017); Korkmaz (2021), real time access to the agent’s perception system Pattanaik et al. (2018); Kos & Song (2017), and highly computationally demanding adversarial formulations for computing simultaneous perturbations Lin et al. (2017); Sun et al. (2020). From the security point of view we emphasize that natural corruptions at the edge of imperceptibility can be more dangerous than a strong adversary assumption 2 without carrying any of these requirements.
+
+In our model we examine several natural environmental changes such as: changes in the brightness of the environment, blurring of the observation, slight rotation of the observation, several geometric transformations and compression artifacts. These changes from our model can be easily linked to naturally occurring changes in the environment3. In Table 1 we compare our proposed framework with the state-of-the-art targeted adversarial attack proposed by Carlini & Wagner (2017) in terms of perceptual similarity distances, and the impacts on the policy performance. While in the remainder of this section we explain in detail each component of our proposed framework, we provide all the experimental details in Section 5, as well as results on policy gradients, performance degradation in the time domain, and complementary results for Section 5 in the appendix.
+
+![](images/645e2c179f1e81f16c3049bee457fd7714c16bc7569b48ad9acd77a7ef956d7f.jpg)  
+Figure 1: Original frame and environmental modifications. Columns: original frame, shifting, rotation, perspective transformation, blurring, compression artifacts. brightness and contrast. Rows: JamesBond, Pong and BankHeist.
+
+Note that the natural corruptions considered in our framework are as minimalistic as possible. Most of the perturbations from the proposed framework cannot be recognized by human perception (see Figure 1). More formally, the perceptual similarity distances for each corruption, and the resulting policy performance degradation, are given in Table 1.
+
+Brightness and Contrast: To inspect the effects of low frequency corruptions we included brightness and contrast level changes using linear brightness and contrast transformation,
+
+$$
+s _ { \mathrm { a d v } } ( i , j ) = s ( i , j ) \cdot \alpha + \beta ,
+$$
+
+where $s ( i , j )$ is the $i j ^ { \mathrm { t h } }$ pixel of state $s$ , and $\alpha$ and $\beta$ are the linear brightness parameters. In Table 1 we show the impacts and perceptual similarity distances with corresponding $\alpha , \beta$ values. In all of the games except BankHeist brightness and contrast change results in higher impact than the Carlini & Wagner (2017) formulation, while the perceptual similarity distance of brightness and contrast is lower in every game.
+
+Blurring: To observe the effects of high frequency corruptions we included blurring in our framework. In particular, median bluring4 is a nonlinear noise removal technique that replaces the original pixel value with the median pixel value of its neighbouring pixels. A kernel size $k$ means that the median is computed over a $k \times k$ neighborhood of the original pixel. Only in BankHeist and TimePilot we observe that the perceptual similarity distance required for blurring is higher compared to adversarial perturbations to be able to cause higher impact on policy performance (see Table 1). For the rest of the games impact is higher and perceptual similarity distance is lower for blurring.
+
+Rotation: Rotation is one of the most fundamental geometric changes in an environment which we incorporate in our framework. In Table 1 we show impact values and perceptual similarity distance with corresponding rotation angle in degrees. In all of the games except Pong rotation results in higher impact and orders of magnitude lower perceptual similarity distance than the Carlini & Wagner (2017) formulation. In Pong the impact is comparable and the perceptual similarity distance is lower by a factor of 6.
+
+Table 1: Impacts on the policy performance, perceptual similarity distances $\mathcal { P } _ { \mathrm { s i m i l a r i t y } }$ to the unperturbed states, and raw scores for Carlini & Wagner (2017) formulation and natural perturbation framework components. We report all of the results with the standard error of the mean.   
+
+<table><tr><td>Games</td><td>BankHeist</td><td>JamesBond</td><td>Pong</td><td>Riverraid</td><td>TimePilot</td></tr><tr><td>Carlini&amp;Wagner Impact</td><td>0.982±0.009</td><td>0.451±0.231</td><td>0.995±0.014</td><td>0.928±0.030</td><td>0.567 ±0.159</td></tr><tr><td>Brightness&amp;Contrast Impact</td><td>0.966± 0.030</td><td>0.913 ±0.047</td><td>1.0±0.009</td><td>0.951 ±0.016</td><td>0.663±0.239</td></tr><tr><td>Blurring Impact</td><td>0.979±0.009</td><td>0.635±0.200</td><td>1.0±0.000</td><td>0.946±0.015</td><td>0.589±0.150</td></tr><tr><td>Rotation Impact</td><td>0.997±0.004</td><td>0.635±0.189</td><td>0.99±0.015</td><td>0.942±0.042</td><td>0.581±0.158</td></tr><tr><td>Shifting Impact</td><td>0.985 ±0.005</td><td>0.865±0.140</td><td>1.0±0.00</td><td>0.935 ±0.023</td><td>0.623±0.199</td></tr><tr><td>Compression Artifacts Impact</td><td>0.980 ±0.013</td><td>0.884 ±0.128</td><td>0.962±0.032</td><td>0.803 ±0.051</td><td>0.578 ±0.271</td></tr><tr><td>Perspective Transform Impact</td><td>0.998±0.003</td><td>0.865±0.087</td><td>0.996±0.009</td><td>0.968±0.006</td><td>0.624±0.198</td></tr><tr><td>Carlini&amp;Wagner Psimilarity</td><td>0.0657±0.0073</td><td>0.2622±0.0312</td><td>0.6134±0.0271</td><td>0.2714±0.0285</td><td>0.1336± 0.0231</td></tr><tr><td>Brightness&amp;Contrast Psimilarity</td><td>0.0307±0.0039</td><td>0.011± 0.0003</td><td>0.2190± 0.0046</td><td>0.2147±0.0212</td><td>0.1045± 0.0031</td></tr><tr><td>Blurring Psimilarity</td><td>0.1672±0.0192</td><td>0.0707±0.0074</td><td>0.0351±0.0072</td><td>0.1442±0.0107</td><td>0.2014±0.0645</td></tr><tr><td>Rotation Psimilarity</td><td>0.0520±0.0070</td><td>0.0275±0.0016</td><td>0.1020±0.0115</td><td>0.0422± 0.0033</td><td>0.1020±0.0115</td></tr><tr><td>Shifting Psimilarity</td><td>0.0492±0.0046</td><td>0.0650±0.0092</td><td>0.2455±0.0432</td><td>0.0945±0.0032</td><td>0.1167±0.0121</td></tr><tr><td>Compression Artifacts Psimilarity</td><td>0.0240±0.0037</td><td>0.1325±0.0301</td><td>0.2506±0.0559</td><td>0.2250±0.0202</td><td>0.1592±0.0369</td></tr><tr><td>Perspective Transform Psimilarity</td><td>0.0398±0.0067</td><td>0.012±0.0007</td><td>0.0140±0.0018</td><td>0.0422±0.0016</td><td>0.0440±0.0050</td></tr><tr><td>Carlini&amp;WagnerRaw Scores</td><td>15.0±2.549</td><td>285.0±25.495</td><td>-20.8±0.189</td><td>1168.0± 140.696</td><td>4090.0±347.979</td></tr><tr><td>Brightness&amp;Contrast Raw Scores</td><td>17.0±1.651</td><td>45.0±6.846</td><td>-21.0±0.000</td><td>744.0±76.957</td><td>3180.0±711.027</td></tr><tr><td>Blurring Raw Scores</td><td>18.0±3.405</td><td>190.0±33.015</td><td>-21.0±0.000</td><td>820.0±72.013</td><td>3880.0±329.484</td></tr><tr><td>Rotation Raw Scores</td><td>2.0±1.264</td><td>190.0± 27.203</td><td>-20.6±0.209</td><td>873.0±201.866</td><td>3150.0±482.959</td></tr><tr><td>Shifting Raw Scores</td><td>13.0±1.449</td><td>70.0±20.248</td><td>-21.0±0.000</td><td>988.0± 89.057</td><td>3560.0± 437.538</td></tr><tr><td>Compression ArtifactsRaw Scores Perspective Transform Raw Scores</td><td>17.0±3.478</td><td>60.0±18.439</td><td>-19.4±0.428</td><td>2589.0±389.679</td><td>3980.0±593.936</td></tr><tr><td></td><td>1.0±0.948</td><td>75.0±12.649</td><td>-20.9±0.126</td><td>486.0±29.127</td><td>3550.0±435.028</td></tr><tr><td>Brightness&amp;Contrast [α,β]</td><td>[1.2,40]</td><td>[0.9,20]</td><td>[1.7,40]</td><td>[2.4,-275]</td><td>[2.4,-260]</td></tr><tr><td>Blurring Kernel Size</td><td>5</td><td>3</td><td>3</td><td>5</td><td>5</td></tr><tr><td>Rotation Degree</td><td>1.4</td><td>1.6</td><td>3</td><td>1.8</td><td>5</td></tr><tr><td>Shifting[ti,tj]</td><td>[1,1]</td><td>[0,1]</td><td>[2,1]</td><td>[1,2]</td><td>[2.2]</td></tr><tr><td>Perspective Transform Norm</td><td>1</td><td>1</td><td>3</td><td>2</td><td>3</td></tr></table>
+
+Shifting: We included several plausible geometric transformations in our natural perturbation framework, the first of which is shifting. In more detail, shifting an image moves the elements of the image matrix along any dimension by any number of elements. For this modification we shift the inputs in the $x$ or $y$ direction with as few pixels shifted as possible. We use $[ t _ { i } , t _ { j } ]$ to denote the distance shifted, where $t _ { i }$ is in the direction of $x$ and $t _ { j }$ is in the direction of $y$ . In Table 1 we show the impact values and perceptual similarity distances for both Carlini & Wagner (2017) and shifting with corresponding $[ t _ { i } , t _ { j } ]$ values. For all of the games shifting yields higher impact and lower perceptual similarity distance.
+
+Compression Artifacts: With this natural perturbation component we look at JPEG compression artifacts caused by the discrete cosine transform (DCT) resulting in the loss of high frequency components (ringing and blocking). In Table 1 we show the impact values and perceptual similarities of Carlini & Wagner (2017) and compression artifacts (CA). Only in Pong and Riverraid do we observe a lower impact than Carlini & Wagner (2017) while the perceptual similarity distance is significantly smaller for compression artifacts. While in TimePilot the perceptual similarity distance is higher, in the rest of the games compression artifacts result in higher impact and lower perceptual similarity distance compared to Carlini & Wagner (2017).
+
+Perspective Transformation: The final component of our proposed natural perturbation framework is perspective transformation. Given four points in the plane defining a convex quadrangle, there is a unique perspective transformation mapping the corners of the square to these four points5. We define the norm of a perspective transformation as the maximum distance that one of the corners of the square moves under this mapping. Note that for most of the games the perspective norm is small (see Table 1). Hence, the changes caused by the perspective transform are imperceptible (e.g. fourth column of Figure 1). Furthermore, for all the games we observe perspective transformation yields higher impact and lower perceptual similarity distance than the Carlini & Wagner (2017) formulation.
+
+![](images/c30a5ebe5a8f6ed96e6cf33e91031eb6818a31a770b28cb51b7d34c5823dd9c3.jpg)  
+Figure 2: Performance drop of adversarially trained deep reinforcement learning policy and vanilla trained deep reinforcement learning policy under the changes in rotation, compression artifacts, and contrast.
+
+# 4 ADVERSARIAL TRAINING UNDER NATURAL CORRUPTIONS
+
+In this section we investigate state-of-the-art adversarially trained deep reinforcement learning policies within our proposed natural perturbation framework. In particular, we test State Adversarial Double Deep Q-Network, a state-of-the-art algorithm (see Section 2.3). In this paper the authors propose using what they call a state-adversarial MDP to model adversarial attacks in deep reinforcement learning. Based on this model they develop methods to regularize Double Deep Q-Network policies to be more robust to adversarial attacks. In more detail, letting $B ( s )$ be the $\ell _ { p }$ -norm ball of radius $\epsilon$ , this regularization is achieved by adding,
+
+$$
+\mathcal { R } ( \theta ) = \operatorname* { m a x } \{ \operatorname* { m a x } _ { \hat { s } \in B ( s ) } \operatorname* { m a x } _ { a \neq a ^ { * } ( s ) } Q _ { \theta } ( \hat { s } , a ) - Q _ { \theta } ( \hat { s } , a ^ { * } ( s ) ) , - c \} .
+$$
+
+to the temporal difference loss used in standard DQN. In particular, for a sample of the form $( s , a , r , s ^ { \prime } )$ the loss is
+
+$$
+\mathcal { L } ( \theta ) = L _ { H } \left( r + \gamma \operatorname* { m a x } _ { a ^ { \prime } } Q ^ { \mathrm { t a r g e t } } ( s ^ { \prime } , a ^ { \prime } ) - Q _ { \theta } ( s , a ) \right) + \mathcal { R } ( \theta )
+$$
+
+where $L _ { H }$ is the Huber loss.
+
+Table 2 shows the impact values of the components of our proposed framework for the vanilla trained agent and the adversarially trained agent. We find that while the adversarially trained model gains robustness against blurring, no additional robustness is gained against any other component of the framework under adversarial training. Furthermore, in Figure 2 and Figure 3 we show the effect of varying the degrees for rotation, $\alpha$ for contrast, $\beta$ for brightness, and jpeg quality $\kappa$ for compression artifacts. We find that, as these parameters are varied, the vanilla trained agent is more robust than the adversarially trained one. For example, modifying brightness with $\beta$ in the range 3.1 to 20.0 causes impact close to 1.0 (i.e. total failure) for the adversarially trained policy, but has negligible impact on the vanilla trained policy. Thus, not only does our proposed framework capture semantically meaningful perturbations that are not captured by adversarial robustness, but additionally adversarial training actively harms robustness to some of the natural perturbations from our proposed framework.
+
+![](images/a6e117b8a55972f97737fb94c1a6f776016e91d7dfcbec374f9aeca50629b9b6.jpg)  
+Figure 3: Performance drop of adversarially trained model and vanilla trained model to the changes in brightness.
+
+The results in Figure 2 and Figure 3 demonstrate that, across a wide range of parameters, adversarially trained neural policies are less robust to natural perturbations than vanilla trained policies. This occurs despite the fact that the central purpose of adversarial training is to increase robustness to imperceptible perturbations, where imperceptibility is measured by $\ell _ { p }$ -norm. Our results indicate that an increase in robustness to $\ell _ { p }$ -norm bounded perturbations can come at the cost of a loss in robustness to other natural types of imperceptible corruptions. These results call into question the use of adversarial training for the creation of robust deep reinforcement learning policies, and in particular the use of $\ell _ { p }$ -norm bounds as a metric of imperceptibility.
+
+The fact that deep reinforcement learning policies are being widely deployed in many different domains: self-driving automobiles Dosovitsky et al. (2017); Wolf et al. (2017), drug design Pereira et al. (2021); Popova et al. (2018), autonomous aerial vehicles Zhang et al. (2020), medical diagnosis and treatment Thananjeyan et al. (2017); Yauney & Pratik (2018), natural language processing He et al. (2016); Jaques et al. (2017), and industrial control and security Wang et al. (2019); Duan et al. (2020), brings the term “robustness” into question. The decrease in resilience to overall distributional shift that “certified robust” adversarial training methods encounter demonstrates the need for further investigation into how robustness should be defined.
+
+Table 2: Impacts of adversarially and vanilla trained policies with natural perturbation framework: brightness &contrast, blurring, rotation, shifting, compression artifacts and perspective transform.   
+
+<table><tr><td>Environment Training Method</td><td>BankHeist Adversarially Trained</td><td>BankHeist Vanilla Trained</td><td>Pong Vanilla Trained</td><td>Pong Adversarially Trained</td></tr><tr><td>Brightness&amp;Contrast (D)</td><td>0.881±0.010</td><td>0.971±0.030</td><td>0.996±0.009</td><td>1.0±0.000</td></tr><tr><td>Compression Artifacts (I)</td><td>0.960±0.0014</td><td>0.984±0.013</td><td>0.962±0.032</td><td>1.0±0.000</td></tr><tr><td>Perspective Transform (Z)</td><td>1.0±0.000</td><td>1.0±0.003</td><td>0.996±0.009</td><td>0.992±0.0034</td></tr><tr><td>Blurring ()</td><td>0.003±0.002</td><td>0.983±0.009</td><td>1.0±0.000</td><td>0.805±0.123</td></tr><tr><td>Rotation (T)</td><td>1.0±0.000</td><td>1.0±0.004</td><td>0.99±0.015</td><td>1.0±0.000</td></tr><tr><td>Shifting (Z)</td><td>1.0±0.000</td><td>0.989±0.005</td><td>1.0±0.000</td><td>1.0±0.000</td></tr></table>
+
+# 5 PERTURBATIONS IN THE FOURIER DOMAIN
+
+In this section we provide frequency analysis of our proposed framework and state-of-the-art adversarial formulations. The purpose of this analysis is to provide quantitative evidence that natural perturbations cover a broader concept of robustness than adversarial perturbations alone. In particular, we demonstrate that each natural perturbation has distinctly different effects in the Fourier spectrum, both from other natural corruptions and from adversarial perturbations. Furthermore, we quantify these effects by measuring, for each type of perturbation, the change in total Fourier energy at each spatial frequency level. Aside from outlining our methodology, Section 5 serves the purpose of explaining results obtained in Section 4. In particular, training techniques (e.g. adversarial training) solely focusing on building robustness towards high spatial frequency corruptions become more vulnerable towards corruptions in different band of the spectrum.
+
+![](images/cae3eb783dca1b500f5ec6970823a62f059205060d247c5bbd7e2eb75525bb52.jpg)  
+Figure 4: Rows: $\mathcal { F } ( s )$ for BankHeist, $\mathcal { F } ( s )$ for Riverraid. Columns: unperturbed state, Carlini & Wagner, brightness and contrast, blurring, rotation, shifting, perspective transformation, compression artifacts.
+
+In Figure 4 we show the Fourier spectrum of the original state $s$ and the perturbed states ${ \boldsymbol { s } } _ { \mathrm { a d v } }$ from our proposed framework based on natural perturbations and Carlini & Wagner (2017) formulated perturbations. In these spectrums the magnitude of the spatial frequencies increases by moving outward from the center, and the center of the image represents the Fourier basis function where spatial frequencies are zero. We provide more detailed description of $\mathcal { F } ( s )$ in Section 2.1. To investigate which type of perturbations occupy which band in the Fourier domain we compute total energy $\mathcal { E } ( f )$ for all basis functions whose maximum spatial frequency is $f$ . Hence, Figure 5 shows the power spectral density of the original state compared to perturbed states computed via components from our proposed natural perturbation framework and Carlini $\&$ Wagner (2017). Figure 5 demonstrates that each component from our natural perturbation framework occupies different bands in the Fourier domain. In particular, in Figure 5 we observe that while the Carlini & Wagner (2017) formulation increases the magnitude of the higher frequencies, compression artifacts decrease the magnitude of the high frequency band. On the other hand, brightness and contrast decreases the magnitude of the low frequency band, and shifting increases the mid-band. Blurring decreases the mid-band and high frequencies together, and perspective transformation decreases the low frequencies and high frequencies while increasing the mid-band.
+
+Figure 5 shows that our proposed framework indeed captures a broader set of directions in the frequency domain. Thus, capturing the susceptibilities towards perturbations in different bands of the frequency domain represents a wider notion of robustness compared to only focusing on worstcase distributional shifts.
+
+![](images/0250fd9ab1722285efb853d4454959ce839bb298e8808c9a20b2dd37f13bc988.jpg)  
+Figure 5: Riverraid total energy $\mathcal { E } ( f )$ spectrum with various perturbations: Carlini & Wagner, compression artifacts, brightness and contrast, perspective transformation, shifting, rotation.
+
+Experimental Details: In our experiments the vanilla trained deep neural policies are trained with Double Deep Q- Network Wang et al. (2016) and the adversarially trained deep neural policy is trained via the theoretically justified State-Adversarial MDP modelled State-Adversarial Double Deep Q-Network (SA-DDQN) (see Section 2.3) in the OpenAI Gym Brockman et al. (2016) Arcade Learning Environment Bellemare et al. (2013). We evaluate several trained policies from Arcade Learning Environment with our proposed framework averaged over 10 episodes. In all of our tables and figures we include the means and the standard error of the mean values. See more details in the appendix.
+
+# 6 CONCLUSION
+
+In this paper we studied a realistic threat model based on basic environmental changes and proposed a framework to asses the generalization capabilities of deep reinforcement learning policies. We compared our natural perturbation framework with the state-of-the-art adversarial attacks in the Arcade Learning Environment (ALE). We questioned the imperceptibility notion of the $\ell _ { p }$ -norm bounded adversarial perturbations, and demonstrated that the states with minimal natural perturbations are more perceptually similar to the unperturbed states compared to adversarial ones. Moreover, we demonstrated that our framework achieves higher impact on policy performance with lower perceptual similarity distance without having access to the policy training details, real time access to the agent’s memory and perception system, and computationally demanding adversarial formulations to compute simultaneous perturbations. Furthermore, we showed that each component of our framework contains distinct bands in the frequency domain, resulting in a better estimate of the generalization capabilities of trained agents. Most importantly, we investigated state-of-the-art adversarial training methods and found that vanilla trained policies are more robust than adversarially trained policies to minimal natural perturbations. We think that the robustness of the trained deep neural policies should be investigated in a more diverse spectrum and we believe our framework can be instrumental towards generalization and robustification of deep reinforcement learning algorithms.
+
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+        "text": "Anonymous authors Paper under double-blind review ",
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+        "type": "text",
+        "text": "ABSTRACT ",
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+        "text": "Deep reinforcement learning algorithms have recently achieved significant success in learning high-performing policies from purely visual observations. The ability to perform end-to-end learning from raw high dimensional input alone has led to deep reinforcement learning algorithms being deployed in a variety of fields. Thus, understanding and improving the ability of deep reinforcement learning policies to generalize to unseen data distributions is of critical importance. Much recent work has focused on assessing the generalization of deep reinforcement learning policies by introducing specifically crafted adversarial perturbations to their inputs. In this paper, we approach this problem from another perspective and propose a framework to assess the generalization skills of trained deep reinforcement learning policies. Rather than focusing on worst-case analysis of distribution shift, our approach is based on black-box perturbations that correspond to minimal semantically meaningful natural changes to the environment or the agent’s visual observation system ranging from brightness to compression artifacts. We demonstrate that the perceptual similarity distance of the minimal natural perturbations is orders of magnitude smaller than the perceptual similarity distance of the adversarial perturbations to the unperturbed observations (i.e. minimal natural perturbations are perceptually more similar to the unperturbed states than the adversarial perturbations), while causing larger degradation in the policy performance. Furthermore, we investigate state-of-the-art adversarial training methods and show that adversarially trained deep reinforcement learning policies are more sensitive to almost all of the natural perturbations compared to vanilla trained policies. Lastly, we highlight that our framework captures a diverse set of bands in the Fourier spectrum; thus providing a better overall understanding of the policy’s generalization capabilities. We believe our work can be crucial towards building resilient and generalizable deep reinforcement learning policies. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Following the initial work of Mnih et al. (2015), the use of DNNs as function approximators in reinforcement learning has led to a dramatic increase in the capabilities of RL agents Schulman et al. (2017); Lillicrap et al. (2015). In particular, these developments allow for the direct learning of strong policies from raw, high-dimensional inputs (i.e. visual observations). With the successes of these new methods come new challenges regarding the robustness and generalization capabilities of deep reinforcement learning agents. ",
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+        "type": "text",
+        "text": "Szegedy et al. (2014) showed that specifically crafted imperceptible perturbations can lead to misclassification in image classification. After this initial work a new research area emerged to investigate the abilities of deep neural networks against specifically crafted adversarial examples. While various works studied many different ways to compute these examples (Carlini & Wagner, 2017; Madry et al., 2018; Goodfellow et al., 2015; Kurakin et al., 2016), several works focused on studying ways to increase the robustness against such specifically crafted perturbations, based on training with the existence of such perturbations (Madry et al., 2018; Tramer et al., 2018; Goodfellow et al., \\` 2015; Xie & Yuille, 2020). ",
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+        "type": "text",
+        "text": "As image classification suffered from this vulnerability towards worst-case distributional shift in the input, a series of work conducted in deep reinforcement learning showed that deep neural policies are also susceptible to specifically crafted imperceptible perturbations (Huang et al., 2017; Kos & Song, 2017; Pattanaik et al., 2018; Lin et al., 2017; Sun et al., 2020; Korkmaz, 2021). While one line of work put effort on exploring these vulnerabilities in deep neural policies, another line in parallel focused making them robust and reliable via adversarial training (Pinto et al., 2017; Mandlekar et al., 2017; Huan et al., 2020). ",
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+        "text": "While adversarial perturbations and adversarial training provide a notion of robustness for trained deep neural policies, in this paper we approach the resilience problem of the deep neural policies from a wider perspective, and propose a framework to test a more generic sense of robustness towards minimal perceptually similar perturbations1. To be able to achieve this we go beyond $\\ell _ { p }$ -norm bounded pixel perturbations and include semantically meaningful minimal realistic perturbations. By this approach we seek answers to the following questions: (i) How perceptually similar are minimal semantically meaningful perturbed states to the original unperturbed states, and how does this compare to $\\ell _ { p }$ -norm bounded adversarially perturbed states? (ii) What are the differences between adversarial perturbations and minimal natural perturbations introduced to the policy observation in terms of performance degradation of the trained deep reinforcement learning policy? (iii) How does state-of-the-art adversarial training affect the performance degradation caused by perceptually similar minimal natural perturbations compared to vanilla training? To be able answer these questions, in this work we focus on the notion of robustness of trained deep reinforcement learning agents and make the following contributions: ",
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+        "text": "• We propose a framework consisting of a diverse set of minimalistic (i.e. perceptually similar) semantically meaningful natural perturbations.   \n• We run multiple experiments in the Arcade Learning Environment (ALE) in various games with high dimensional state representation and provide the relationship between the perceptual similarities to unperturbed states under our proposed natural perturbation framework and adversarial perturbations.   \n• We compare our proposed framework with the state-of-the-art adversarial method based on $\\ell _ { p }$ -norm changes, and we show that our natural perturbation framework is competitive in degrading the performance of the deep reinforcement learning agent with lower perceptual similarity distance.   \n• We inspect state-of-the-art adversarial training under our proposed framework, and demonstrate that the adversarially trained models become more vulnerable to various natural perturbations compared to vanilla trained models.   \n• Finally, we investigate the frequency domain of our framework and state-of-the-art targeted attacks. We show that our framework captures different bands of the frequency spectrum, thus yielding a better estimate of the model robustness. ",
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND AND RELATED WORK ",
+        "text_level": 1,
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+        "text": "2.1 PRELIMINARIES ",
+        "text_level": 1,
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+        "text": "In this paper we consider Markov Decision Processes (MDPs) given by a tuple $( S , A , P , r , \\gamma , s _ { 0 } )$ . The reinforcement learning agent interacts with the MDP by observing states $s \\in S$ , and then taking actions $a \\in A$ . Here $s _ { 0 }$ represents the initial state of the agent, and $\\gamma$ represents the discount factor. The probability of transitioning to state $s ^ { \\prime }$ when the agent takes action $a$ in state $s$ is determined by the Markovian transition kernel $P : S \\times A \\times S \\to \\mathbb { R }$ . The reward received by the agent when taking action $a$ in state $s$ is given by the reward function $r : S \\times A \\to \\mathbb { R }$ . The goal of the expected cumulative discounted reward with the environment. This goal is achieve the agent is to learn a policy $\\pi _ { \\theta } : S \\times A \\to \\mathbb { R }$ $\\scriptstyle \\sum _ { t = 0 } ^ { T - 1 } \\gamma ^ { t } r ( s _ { t } , a _ { t } )$ which takes an action that the agent receives viastate-action value function $a$ in state that maximizes $Q ( s , a ) =$ $\\mathbb { E } _ { \\pi } [ \\sum _ { t = 0 } ^ { T - 1 } \\gamma ^ { t } r ( s _ { t } , a _ { t } ) | s = s _ { 0 } , a _ { 0 } = 0 ]$ assigning a value to each state-action pair. We use $\\mathcal { F } ( s )$ to denote the 2D discrete Fourier transform of state $s$ in which each frequency is computed via $\\begin{array} { r } { \\mathcal { F } ( m , n ) = \\sum _ { k = 0 } ^ { M - 1 } \\sum _ { l = 0 } ^ { N - 1 } s ( k , l ) e ^ { - j 2 \\pi ( m k / M + n l / N ) } } \\end{array}$ where $k$ and $l$ are the coordinates of the state, and $M$ and $N$ correspond to ranges of the 2D state representation. ",
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+        "text": "2.2 CRAFTING ADVERSARIAL PERTURBATIONS ",
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+        "text": "Szegedy et al. (2014) proposed to minimize the distance between the original image and adversarially produced image to create adversarial perturbations. The authors used box-constrained L-BFGS to solve this optimization problem. Goodfellow et al. (2015) introduced the fast gradient method (FGM) ",
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+        "img_path": "images/c1f9cd2ae1b835b5633d25bbe4b136bd42c0a128653a041f3978caa61327842c.jpg",
+        "text": "$$\nx _ { \\mathrm { a d v } } = x + \\epsilon \\cdot \\frac { \\nabla _ { x } J ( x , y ) } { | | \\nabla _ { x } J ( x , y ) | | _ { p } } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "for crafting adversarial examples in image classification by taking the gradient of the cost function $J ( x , y )$ used to train the neural network in the direction of the input, where $x$ is the input, $y$ is the output label, and $J ( x , y )$ is the cost function for image classification. Carlini & Wagner (2017) introduced targeted attacks in the image classification domain based on distance minimization between the adversarial image and the original image while targeting a particular label. In the deep reinforcement learning domain the Carlini & Wagner (2017) formulation is ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/777492679d894a0a39a89d9d5e8a32c3b5e63191eb9624d9534f943d2b44d2e2.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\underset { s _ { \\mathrm { a d v } } \\in D _ { \\epsilon , p } ( s ) } { \\operatorname* { m i n } } \\Vert s _ { \\mathrm { a d v } } - s \\Vert _ { p } } \\\\ & { \\mathrm { s u b j e c t ~ t o ~ } \\underset { a } { \\operatorname { a r g m a x } } Q ( s , a ) \\neq \\underset { a } { \\operatorname { a r g m a x } } Q ( s _ { \\mathrm { a d v } } , a ) } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $s$ is the unperturbed input, $S _ { \\mathrm { a d v } }$ is the adversarially perturbed input, $a ^ { * } ( s )$ is the action taken in the unperturbed state, and $a ^ { * } ( s _ { \\mathrm { a d v } } ) = \\arg \\operatorname* { m a x } _ { a } Q ( s _ { \\mathrm { a d v } } , a )$ is the action taken in the adversarial state. This formulation attempts to minimize the distance to the original state, constrained to states leading to sub-optimal actions as determined by the $Q$ -network. In contrast to adversarial attacks, in our proposed threat model we will not need any information on the cost function used to train the network, the $Q$ -network of the trained agent, or access to the visited states themselves. ",
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+    {
+        "type": "text",
+        "text": "2.3 ADVERSARIAL APPROACH IN DEEP REINFORCEMENT LEARNING",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The first adversarial attacks on deep reinforcement learning introduced by Huang et al. (2017) and Kos & Song (2017) adapted FGSM from image classification to the deep reinforcement learning setting. Subsequently, Mandlekar et al. (2017) used FGSM perturbations for adversarial training of deep reinforcement learning agents. Pinto et al. (2017); Gleave et al. (2020) focused on modeling the interaction between the adversary and the agent, while Lin et al. (2017); Sun et al. (2020) focused on strategically timing when (i.e. in which state) to attack an agent using perturbations computed with the Carlini & Wagner (2017) adversarial formulation. Quite recently, Huan et al. (2020) proposed to model this dynamic as a State-Adversarial Markov Decision Process (SA-MDP), and the authors claimed the SA-MDP model provides theoretically justified robust deep reinforcement learning agents. ",
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+    {
+        "type": "text",
+        "text": "2.4 PERCEPTUAL SIMILARITY DISTANCE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Zhang et al. (2018) found that internal activations of networks trained for high-level tasks correspond to human perceptual judgements across different network architectures Iandola et al. (2016), Krizhevsky et al. (2012), Simonyan & Zisserman (2015) without calibration. Furthermore, the authors propose a method to measure the perceptual distance between two images with the Learned Perceptual Image Patch Similarity (LPIPS) metric. We compare the distance between adversarial states ${ \\boldsymbol { s } } _ { \\mathrm { a d v } }$ and the original states $s$ with the LPIPS metric. We refer to the LPIPS metric as $\\mathcal { P } _ { \\mathrm { s i m i l a r i t y } }$ throughout the paper. $\\mathcal { P } _ { \\mathrm { s i m i l a r i t y } } ( s , s _ { \\mathrm { a d v } } )$ returns the distance between $s$ and ${ \\boldsymbol { s } } _ { \\mathrm { a d v } }$ based on network activations. Zhang et al. (2018) show that $\\mathcal { P } _ { \\mathrm { s i m i l a r i t y } }$ results in a reliable approximation of human perception. ",
+        "bbox": [
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+        "type": "text",
+        "text": "In more detail, the LPIPS metric in Zhang et al. (2018) is given by measuring the $\\ell _ { 2 }$ distance between a normalized version of the activations of the neural network at several internal convolutional layers. For each convolutional layer $l$ let $W _ { l }$ be the width, $H _ { l }$ the height, and $C _ { l }$ the number of channels. Further, let $y ^ { l } \\in \\mathbb { R } ^ { W _ { l } \\times H _ { l } \\times C _ { l } }$ denote the vector of activations in convolutional layer $l$ . ",
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+        "type": "text",
+        "text": "To compute the perceptual similarity distance between two states $s$ and $s _ { 0 }$ , first calculate the internal activations $y ^ { \\hat { l } } , y _ { 0 } ^ { l } \\ { \\in } \\ \\mathbb { R } ^ { W _ { l } \\times H _ { l } \\times C _ { l } ^ { \\bullet } }$ (corresponding to $s$ and $s _ { 0 }$ respectively) for $L$ internal layers. Second, unit-normalize the activation vectors in the channel dimension, and denote the resulting normalized activations by $\\hat { y } ^ { l }$ and $\\hat { y } _ { 0 } ^ { l }$ . Next scale each channel in $\\hat { y } ^ { l }$ and $\\hat { y } _ { 0 } ^ { l }$ by the same, fixed weight vector $w _ { l } \\in \\mathbb { R } ^ { C _ { l } }$ . Here $w _ { l }$ can either be a learned vector of weights for layer $l$ or the vector of all ones if no scaling is desired. The last step is then to compute the perceptual similarity distance by first averaging the $\\ell _ { 2 }$ distance between the scaled activations over the spatial dimensions, and then summing over the $L$ layers. Formally, ",
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+        "img_path": "images/1301541b70b019bcb8f2cf48ef469f5f633d20d449b7e5fef3a970c5e8ade218.jpg",
+        "text": "$$\n{ \\mathcal { P } } _ { \\mathrm { s i m i l a r i t y } } \\big ( s , s _ { 0 } \\big ) = \\sum _ { l } { \\frac { 1 } { H _ { l } W _ { l } } \\sum _ { h , w } \\left\\| w _ { l } \\odot \\big ( \\hat { y } _ { h w } ^ { l } - \\hat { y } _ { 0 h w } ^ { l } \\big ) \\right\\| _ { 2 } ^ { 2 } }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "2.5 IMPACT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To be able to compare between different algorithms and different games the performance degradation of the deep reinforcement learning policy is defined as the normalized impact of an adversary on the agent: ",
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+        "text": "$$\n\\mathcal { T } = \\frac { \\mathrm { S c o r e } _ { \\mathrm { c l e a n } } - \\mathrm { S c o r e } _ { \\mathrm { a d v } } } { \\mathrm { S c o r e } _ { \\mathrm { c l e a n } } - \\mathrm { S c o r e } _ { \\mathrm { m i n } } ^ { \\mathrm { f u x e d } } } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Scorefixedmin is a fixed minimum score for a game, ${ \\mathrm { S c o r e } } _ { \\mathrm { a d v } }$ and $\\mathbf { S c o r e } _ { \\mathrm { c l e a n } }$ are the scores of the agent with and without any modification to the agent’s observations system respectively. ",
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+    {
+        "type": "text",
+        "text": "3 A GENERALIZATION TESTING FRAMEWORK WITH NATURAL PERTURBATIONS ",
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+        "text": "In our paper we focus on robustness and generalization issues that deep reinforcement learning policies encounter with a contrasting view compared to prior work focusing on worst-case distributional shift within an imperceptibility bound (see Section 2.3). We propose a baseline to evaluate deep reinforcement learning policies with realistic and minimal corruptions to the environment with which they interact. We essentially juxtapose adversarial perturbations and natural corruptions with respect to their perceptual similarity distance (see Section 2.4) to the original states and their degree of impact on the policy performance. More importantly, we question the imperceptibility of $\\ell _ { p }$ -norm bounded adversarial perturbations in terms of perceptual similarity distance, and compare this imperceptibility notion to natural perturbations. While we categorize adversarial perturbations also as a component in the framework majorly concentrated on the high frequencies, we embed several realistic perturbations that aim to cover diverse bands in the frequency spectrum. We highlight that prior work focused on the presence of a strong adversary model that requires prior access to training details of the agent’s neural network Huang et al. (2017); Korkmaz (2021), real time access to the agent’s perception system Pattanaik et al. (2018); Kos & Song (2017), and highly computationally demanding adversarial formulations for computing simultaneous perturbations Lin et al. (2017); Sun et al. (2020). From the security point of view we emphasize that natural corruptions at the edge of imperceptibility can be more dangerous than a strong adversary assumption 2 without carrying any of these requirements. ",
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+        "text": "In our model we examine several natural environmental changes such as: changes in the brightness of the environment, blurring of the observation, slight rotation of the observation, several geometric transformations and compression artifacts. These changes from our model can be easily linked to naturally occurring changes in the environment3. In Table 1 we compare our proposed framework with the state-of-the-art targeted adversarial attack proposed by Carlini & Wagner (2017) in terms of perceptual similarity distances, and the impacts on the policy performance. While in the remainder of this section we explain in detail each component of our proposed framework, we provide all the experimental details in Section 5, as well as results on policy gradients, performance degradation in the time domain, and complementary results for Section 5 in the appendix. ",
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+        "image_caption": [
+            "Figure 1: Original frame and environmental modifications. Columns: original frame, shifting, rotation, perspective transformation, blurring, compression artifacts. brightness and contrast. Rows: JamesBond, Pong and BankHeist. "
+        ],
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+        "text": "Note that the natural corruptions considered in our framework are as minimalistic as possible. Most of the perturbations from the proposed framework cannot be recognized by human perception (see Figure 1). More formally, the perceptual similarity distances for each corruption, and the resulting policy performance degradation, are given in Table 1. ",
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+    {
+        "type": "text",
+        "text": "Brightness and Contrast: To inspect the effects of low frequency corruptions we included brightness and contrast level changes using linear brightness and contrast transformation, ",
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+    {
+        "type": "equation",
+        "img_path": "images/c6a0d5cf4689af9e2925e9efc2214a28e3250693eb0e64d736713df1f74e1060.jpg",
+        "text": "$$\ns _ { \\mathrm { a d v } } ( i , j ) = s ( i , j ) \\cdot \\alpha + \\beta ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $s ( i , j )$ is the $i j ^ { \\mathrm { t h } }$ pixel of state $s$ , and $\\alpha$ and $\\beta$ are the linear brightness parameters. In Table 1 we show the impacts and perceptual similarity distances with corresponding $\\alpha , \\beta$ values. In all of the games except BankHeist brightness and contrast change results in higher impact than the Carlini & Wagner (2017) formulation, while the perceptual similarity distance of brightness and contrast is lower in every game. ",
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+        "type": "text",
+        "text": "Blurring: To observe the effects of high frequency corruptions we included blurring in our framework. In particular, median bluring4 is a nonlinear noise removal technique that replaces the original pixel value with the median pixel value of its neighbouring pixels. A kernel size $k$ means that the median is computed over a $k \\times k$ neighborhood of the original pixel. Only in BankHeist and TimePilot we observe that the perceptual similarity distance required for blurring is higher compared to adversarial perturbations to be able to cause higher impact on policy performance (see Table 1). For the rest of the games impact is higher and perceptual similarity distance is lower for blurring. ",
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+        "type": "text",
+        "text": "Rotation: Rotation is one of the most fundamental geometric changes in an environment which we incorporate in our framework. In Table 1 we show impact values and perceptual similarity distance with corresponding rotation angle in degrees. In all of the games except Pong rotation results in higher impact and orders of magnitude lower perceptual similarity distance than the Carlini & Wagner (2017) formulation. In Pong the impact is comparable and the perceptual similarity distance is lower by a factor of 6. ",
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+        "table_caption": [
+            "Table 1: Impacts on the policy performance, perceptual similarity distances $\\mathcal { P } _ { \\mathrm { s i m i l a r i t y } }$ to the unperturbed states, and raw scores for Carlini & Wagner (2017) formulation and natural perturbation framework components. We report all of the results with the standard error of the mean. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Games</td><td>BankHeist</td><td>JamesBond</td><td>Pong</td><td>Riverraid</td><td>TimePilot</td></tr><tr><td>Carlini&amp;Wagner Impact</td><td>0.982±0.009</td><td>0.451±0.231</td><td>0.995±0.014</td><td>0.928±0.030</td><td>0.567 ±0.159</td></tr><tr><td>Brightness&amp;Contrast Impact</td><td>0.966± 0.030</td><td>0.913 ±0.047</td><td>1.0±0.009</td><td>0.951 ±0.016</td><td>0.663±0.239</td></tr><tr><td>Blurring Impact</td><td>0.979±0.009</td><td>0.635±0.200</td><td>1.0±0.000</td><td>0.946±0.015</td><td>0.589±0.150</td></tr><tr><td>Rotation Impact</td><td>0.997±0.004</td><td>0.635±0.189</td><td>0.99±0.015</td><td>0.942��0.042</td><td>0.581±0.158</td></tr><tr><td>Shifting Impact</td><td>0.985 ±0.005</td><td>0.865±0.140</td><td>1.0±0.00</td><td>0.935 ±0.023</td><td>0.623±0.199</td></tr><tr><td>Compression Artifacts Impact</td><td>0.980 ±0.013</td><td>0.884 ±0.128</td><td>0.962±0.032</td><td>0.803 ±0.051</td><td>0.578 ±0.271</td></tr><tr><td>Perspective Transform Impact</td><td>0.998±0.003</td><td>0.865±0.087</td><td>0.996±0.009</td><td>0.968±0.006</td><td>0.624±0.198</td></tr><tr><td>Carlini&amp;Wagner Psimilarity</td><td>0.0657±0.0073</td><td>0.2622±0.0312</td><td>0.6134±0.0271</td><td>0.2714±0.0285</td><td>0.1336± 0.0231</td></tr><tr><td>Brightness&amp;Contrast Psimilarity</td><td>0.0307±0.0039</td><td>0.011± 0.0003</td><td>0.2190± 0.0046</td><td>0.2147±0.0212</td><td>0.1045± 0.0031</td></tr><tr><td>Blurring Psimilarity</td><td>0.1672±0.0192</td><td>0.0707±0.0074</td><td>0.0351±0.0072</td><td>0.1442±0.0107</td><td>0.2014±0.0645</td></tr><tr><td>Rotation Psimilarity</td><td>0.0520±0.0070</td><td>0.0275±0.0016</td><td>0.1020±0.0115</td><td>0.0422± 0.0033</td><td>0.1020±0.0115</td></tr><tr><td>Shifting Psimilarity</td><td>0.0492±0.0046</td><td>0.0650±0.0092</td><td>0.2455±0.0432</td><td>0.0945±0.0032</td><td>0.1167±0.0121</td></tr><tr><td>Compression Artifacts Psimilarity</td><td>0.0240±0.0037</td><td>0.1325±0.0301</td><td>0.2506±0.0559</td><td>0.2250±0.0202</td><td>0.1592±0.0369</td></tr><tr><td>Perspective Transform Psimilarity</td><td>0.0398±0.0067</td><td>0.012±0.0007</td><td>0.0140±0.0018</td><td>0.0422±0.0016</td><td>0.0440±0.0050</td></tr><tr><td>Carlini&amp;WagnerRaw Scores</td><td>15.0±2.549</td><td>285.0±25.495</td><td>-20.8±0.189</td><td>1168.0± 140.696</td><td>4090.0±347.979</td></tr><tr><td>Brightness&amp;Contrast Raw Scores</td><td>17.0±1.651</td><td>45.0±6.846</td><td>-21.0±0.000</td><td>744.0±76.957</td><td>3180.0±711.027</td></tr><tr><td>Blurring Raw Scores</td><td>18.0±3.405</td><td>190.0±33.015</td><td>-21.0±0.000</td><td>820.0±72.013</td><td>3880.0±329.484</td></tr><tr><td>Rotation Raw Scores</td><td>2.0±1.264</td><td>190.0± 27.203</td><td>-20.6±0.209</td><td>873.0±201.866</td><td>3150.0±482.959</td></tr><tr><td>Shifting Raw Scores</td><td>13.0±1.449</td><td>70.0±20.248</td><td>-21.0±0.000</td><td>988.0± 89.057</td><td>3560.0± 437.538</td></tr><tr><td>Compression ArtifactsRaw Scores Perspective Transform Raw Scores</td><td>17.0±3.478</td><td>60.0±18.439</td><td>-19.4±0.428</td><td>2589.0±389.679</td><td>3980.0±593.936</td></tr><tr><td></td><td>1.0±0.948</td><td>75.0±12.649</td><td>-20.9±0.126</td><td>486.0±29.127</td><td>3550.0±435.028</td></tr><tr><td>Brightness&amp;Contrast [α,β]</td><td>[1.2,40]</td><td>[0.9,20]</td><td>[1.7,40]</td><td>[2.4,-275]</td><td>[2.4,-260]</td></tr><tr><td>Blurring Kernel Size</td><td>5</td><td>3</td><td>3</td><td>5</td><td>5</td></tr><tr><td>Rotation Degree</td><td>1.4</td><td>1.6</td><td>3</td><td>1.8</td><td>5</td></tr><tr><td>Shifting[ti,tj]</td><td>[1,1]</td><td>[0,1]</td><td>[2,1]</td><td>[1,2]</td><td>[2.2]</td></tr><tr><td>Perspective Transform Norm</td><td>1</td><td>1</td><td>3</td><td>2</td><td>3</td></tr></table>",
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+        "type": "text",
+        "text": "Shifting: We included several plausible geometric transformations in our natural perturbation framework, the first of which is shifting. In more detail, shifting an image moves the elements of the image matrix along any dimension by any number of elements. For this modification we shift the inputs in the $x$ or $y$ direction with as few pixels shifted as possible. We use $[ t _ { i } , t _ { j } ]$ to denote the distance shifted, where $t _ { i }$ is in the direction of $x$ and $t _ { j }$ is in the direction of $y$ . In Table 1 we show the impact values and perceptual similarity distances for both Carlini & Wagner (2017) and shifting with corresponding $[ t _ { i } , t _ { j } ]$ values. For all of the games shifting yields higher impact and lower perceptual similarity distance. ",
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+    {
+        "type": "text",
+        "text": "Compression Artifacts: With this natural perturbation component we look at JPEG compression artifacts caused by the discrete cosine transform (DCT) resulting in the loss of high frequency components (ringing and blocking). In Table 1 we show the impact values and perceptual similarities of Carlini & Wagner (2017) and compression artifacts (CA). Only in Pong and Riverraid do we observe a lower impact than Carlini & Wagner (2017) while the perceptual similarity distance is significantly smaller for compression artifacts. While in TimePilot the perceptual similarity distance is higher, in the rest of the games compression artifacts result in higher impact and lower perceptual similarity distance compared to Carlini & Wagner (2017). ",
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+    {
+        "type": "text",
+        "text": "Perspective Transformation: The final component of our proposed natural perturbation framework is perspective transformation. Given four points in the plane defining a convex quadrangle, there is a unique perspective transformation mapping the corners of the square to these four points5. We define the norm of a perspective transformation as the maximum distance that one of the corners of the square moves under this mapping. Note that for most of the games the perspective norm is small (see Table 1). Hence, the changes caused by the perspective transform are imperceptible (e.g. fourth column of Figure 1). Furthermore, for all the games we observe perspective transformation yields higher impact and lower perceptual similarity distance than the Carlini & Wagner (2017) formulation. ",
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+        "image_caption": [
+            "Figure 2: Performance drop of adversarially trained deep reinforcement learning policy and vanilla trained deep reinforcement learning policy under the changes in rotation, compression artifacts, and contrast. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4 ADVERSARIAL TRAINING UNDER NATURAL CORRUPTIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we investigate state-of-the-art adversarially trained deep reinforcement learning policies within our proposed natural perturbation framework. In particular, we test State Adversarial Double Deep Q-Network, a state-of-the-art algorithm (see Section 2.3). In this paper the authors propose using what they call a state-adversarial MDP to model adversarial attacks in deep reinforcement learning. Based on this model they develop methods to regularize Double Deep Q-Network policies to be more robust to adversarial attacks. In more detail, letting $B ( s )$ be the $\\ell _ { p }$ -norm ball of radius $\\epsilon$ , this regularization is achieved by adding, ",
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+    {
+        "type": "equation",
+        "img_path": "images/144f91a40843330cb86de1382a4131b58b7737153ef741977c7d70d54084a146.jpg",
+        "text": "$$\n\\mathcal { R } ( \\theta ) = \\operatorname* { m a x } \\{ \\operatorname* { m a x } _ { \\hat { s } \\in B ( s ) } \\operatorname* { m a x } _ { a \\neq a ^ { * } ( s ) } Q _ { \\theta } ( \\hat { s } , a ) - Q _ { \\theta } ( \\hat { s } , a ^ { * } ( s ) ) , - c \\} .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "to the temporal difference loss used in standard DQN. In particular, for a sample of the form $( s , a , r , s ^ { \\prime } )$ the loss is ",
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+        "img_path": "images/28d33da085eb96ef1ebccf770bc0377ef4d2646e1fb187a5e3d11ec6c290efd5.jpg",
+        "text": "$$\n\\mathcal { L } ( \\theta ) = L _ { H } \\left( r + \\gamma \\operatorname* { m a x } _ { a ^ { \\prime } } Q ^ { \\mathrm { t a r g e t } } ( s ^ { \\prime } , a ^ { \\prime } ) - Q _ { \\theta } ( s , a ) \\right) + \\mathcal { R } ( \\theta )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $L _ { H }$ is the Huber loss. ",
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+    {
+        "type": "text",
+        "text": "Table 2 shows the impact values of the components of our proposed framework for the vanilla trained agent and the adversarially trained agent. We find that while the adversarially trained model gains robustness against blurring, no additional robustness is gained against any other component of the framework under adversarial training. Furthermore, in Figure 2 and Figure 3 we show the effect of varying the degrees for rotation, $\\alpha$ for contrast, $\\beta$ for brightness, and jpeg quality $\\kappa$ for compression artifacts. We find that, as these parameters are varied, the vanilla trained agent is more robust than the adversarially trained one. For example, modifying brightness with $\\beta$ in the range 3.1 to 20.0 causes impact close to 1.0 (i.e. total failure) for the adversarially trained policy, but has negligible impact on the vanilla trained policy. Thus, not only does our proposed framework capture semantically meaningful perturbations that are not captured by adversarial robustness, but additionally adversarial training actively harms robustness to some of the natural perturbations from our proposed framework. ",
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+    {
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+        "img_path": "images/a6e117b8a55972f97737fb94c1a6f776016e91d7dfcbec374f9aeca50629b9b6.jpg",
+        "image_caption": [
+            "Figure 3: Performance drop of adversarially trained model and vanilla trained model to the changes in brightness. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "The results in Figure 2 and Figure 3 demonstrate that, across a wide range of parameters, adversarially trained neural policies are less robust to natural perturbations than vanilla trained policies. This occurs despite the fact that the central purpose of adversarial training is to increase robustness to imperceptible perturbations, where imperceptibility is measured by $\\ell _ { p }$ -norm. Our results indicate that an increase in robustness to $\\ell _ { p }$ -norm bounded perturbations can come at the cost of a loss in robustness to other natural types of imperceptible corruptions. These results call into question the use of adversarial training for the creation of robust deep reinforcement learning policies, and in particular the use of $\\ell _ { p }$ -norm bounds as a metric of imperceptibility. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The fact that deep reinforcement learning policies are being widely deployed in many different domains: self-driving automobiles Dosovitsky et al. (2017); Wolf et al. (2017), drug design Pereira et al. (2021); Popova et al. (2018), autonomous aerial vehicles Zhang et al. (2020), medical diagnosis and treatment Thananjeyan et al. (2017); Yauney & Pratik (2018), natural language processing He et al. (2016); Jaques et al. (2017), and industrial control and security Wang et al. (2019); Duan et al. (2020), brings the term “robustness” into question. The decrease in resilience to overall distributional shift that “certified robust” adversarial training methods encounter demonstrates the need for further investigation into how robustness should be defined. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 2: Impacts of adversarially and vanilla trained policies with natural perturbation framework: brightness &contrast, blurring, rotation, shifting, compression artifacts and perspective transform. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Environment Training Method</td><td>BankHeist Adversarially Trained</td><td>BankHeist Vanilla Trained</td><td>Pong Vanilla Trained</td><td>Pong Adversarially Trained</td></tr><tr><td>Brightness&amp;Contrast (D)</td><td>0.881±0.010</td><td>0.971±0.030</td><td>0.996±0.009</td><td>1.0±0.000</td></tr><tr><td>Compression Artifacts (I)</td><td>0.960±0.0014</td><td>0.984±0.013</td><td>0.962±0.032</td><td>1.0±0.000</td></tr><tr><td>Perspective Transform (Z)</td><td>1.0±0.000</td><td>1.0±0.003</td><td>0.996±0.009</td><td>0.992±0.0034</td></tr><tr><td>Blurring ()</td><td>0.003±0.002</td><td>0.983±0.009</td><td>1.0±0.000</td><td>0.805±0.123</td></tr><tr><td>Rotation (T)</td><td>1.0±0.000</td><td>1.0±0.004</td><td>0.99±0.015</td><td>1.0±0.000</td></tr><tr><td>Shifting (Z)</td><td>1.0±0.000</td><td>0.989±0.005</td><td>1.0±0.000</td><td>1.0±0.000</td></tr></table>",
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+        "type": "text",
+        "text": "5 PERTURBATIONS IN THE FOURIER DOMAIN ",
+        "text_level": 1,
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+        "text": "In this section we provide frequency analysis of our proposed framework and state-of-the-art adversarial formulations. The purpose of this analysis is to provide quantitative evidence that natural perturbations cover a broader concept of robustness than adversarial perturbations alone. In particular, we demonstrate that each natural perturbation has distinctly different effects in the Fourier spectrum, both from other natural corruptions and from adversarial perturbations. Furthermore, we quantify these effects by measuring, for each type of perturbation, the change in total Fourier energy at each spatial frequency level. Aside from outlining our methodology, Section 5 serves the purpose of explaining results obtained in Section 4. In particular, training techniques (e.g. adversarial training) solely focusing on building robustness towards high spatial frequency corruptions become more vulnerable towards corruptions in different band of the spectrum. ",
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+        "image_caption": [
+            "Figure 4: Rows: $\\mathcal { F } ( s )$ for BankHeist, $\\mathcal { F } ( s )$ for Riverraid. Columns: unperturbed state, Carlini & Wagner, brightness and contrast, blurring, rotation, shifting, perspective transformation, compression artifacts. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "In Figure 4 we show the Fourier spectrum of the original state $s$ and the perturbed states ${ \\boldsymbol { s } } _ { \\mathrm { a d v } }$ from our proposed framework based on natural perturbations and Carlini & Wagner (2017) formulated perturbations. In these spectrums the magnitude of the spatial frequencies increases by moving outward from the center, and the center of the image represents the Fourier basis function where spatial frequencies are zero. We provide more detailed description of $\\mathcal { F } ( s )$ in Section 2.1. To investigate which type of perturbations occupy which band in the Fourier domain we compute total energy $\\mathcal { E } ( f )$ for all basis functions whose maximum spatial frequency is $f$ . Hence, Figure 5 shows the power spectral density of the original state compared to perturbed states computed via components from our proposed natural perturbation framework and Carlini $\\&$ Wagner (2017). Figure 5 demonstrates that each component from our natural perturbation framework occupies different bands in the Fourier domain. In particular, in Figure 5 we observe that while the Carlini & Wagner (2017) formulation increases the magnitude of the higher frequencies, compression artifacts decrease the magnitude of the high frequency band. On the other hand, brightness and contrast decreases the magnitude of the low frequency band, and shifting increases the mid-band. Blurring decreases the mid-band and high frequencies together, and perspective transformation decreases the low frequencies and high frequencies while increasing the mid-band. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 shows that our proposed framework indeed captures a broader set of directions in the frequency domain. Thus, capturing the susceptibilities towards perturbations in different bands of the frequency domain represents a wider notion of robustness compared to only focusing on worstcase distributional shifts. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/0250fd9ab1722285efb853d4454959ce839bb298e8808c9a20b2dd37f13bc988.jpg",
+        "image_caption": [
+            "Figure 5: Riverraid total energy $\\mathcal { E } ( f )$ spectrum with various perturbations: Carlini & Wagner, compression artifacts, brightness and contrast, perspective transformation, shifting, rotation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Experimental Details: In our experiments the vanilla trained deep neural policies are trained with Double Deep Q- Network Wang et al. (2016) and the adversarially trained deep neural policy is trained via the theoretically justified State-Adversarial MDP modelled State-Adversarial Double Deep Q-Network (SA-DDQN) (see Section 2.3) in the OpenAI Gym Brockman et al. (2016) Arcade Learning Environment Bellemare et al. (2013). We evaluate several trained policies from Arcade Learning Environment with our proposed framework averaged over 10 episodes. In all of our tables and figures we include the means and the standard error of the mean values. See more details in the appendix. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper we studied a realistic threat model based on basic environmental changes and proposed a framework to asses the generalization capabilities of deep reinforcement learning policies. We compared our natural perturbation framework with the state-of-the-art adversarial attacks in the Arcade Learning Environment (ALE). We questioned the imperceptibility notion of the $\\ell _ { p }$ -norm bounded adversarial perturbations, and demonstrated that the states with minimal natural perturbations are more perceptually similar to the unperturbed states compared to adversarial ones. Moreover, we demonstrated that our framework achieves higher impact on policy performance with lower perceptual similarity distance without having access to the policy training details, real time access to the agent’s memory and perception system, and computationally demanding adversarial formulations to compute simultaneous perturbations. Furthermore, we showed that each component of our framework contains distinct bands in the frequency domain, resulting in a better estimate of the generalization capabilities of trained agents. Most importantly, we investigated state-of-the-art adversarial training methods and found that vanilla trained policies are more robust than adversarially trained policies to minimal natural perturbations. We think that the robustness of the trained deep neural policies should be investigated in a more diverse spectrum and we believe our framework can be instrumental towards generalization and robustification of deep reinforcement learning algorithms. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+            287,
+            117
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "text": "Multi-Objective Online Learning ",
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "1 This paper presents a systematic study of multi-objective online learning. We first   \n2 formulate the framework of Multi-Objective Online Convex Optimization, which   \n3 encompasses two novel multi-objective regret definitions. The regret definitions   \n4 build upon an equivalent transformation of the multi-objective dynamic regret   \n5 based on the commonly used Pareto suboptimality gap metric in zero-order multi  \n6 objective bandits, making it amenable to be optimized via first-order iterative   \n7 methods. To motivate the algorithm design, we give an explicit example in which   \n8 equipping OMD with the vanilla min-norm solver for gradient composition will   \n9 incur a linear regret, which shows that only regularizing the iterates, as in single  \n10 objective online learning, is not enough to guarantee sublinear regrets in the multi  \n11 objective setting. To resolve this issue, we propose a novel min-regularized-norm   \n12 solver that regularizes the composite weights. Combining min-regularized-norm   \n13 with OMD results in the Doubly Regularized Online Mirror Multiple Descent   \n14 algorithm. We further derive both the static and dynamic regret bounds for the   \n15 proposed algorithm, each of which matches the corresponding optimal bound in the   \n16 single-objective setting. Extensive experiments on both simulation and real-world   \n17 datasets verify the effectiveness of the proposed algorithm. ",
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+        "type": "text",
+        "text": "18 1 Introduction ",
+        "text_level": 1,
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+        "text": "19 Traditional optimization methods for machine learning are usually designed to optimize a single   \n20 objective. However, in many real-world applications, we are often required to optimize multiple   \n21 correlated objectives concurrently. For example, in autonomous driving [12, 20], the self-driving   \n22 vehicles need to solve multiple tasks such as self-localization and object identification at the same   \n23 time. In online advertising [21, 22], advertisers need to determine the exposure of items to different   \n24 users to maximize both the Click-Through Rate (CTR) and the Post-Click Conversion Rate (CVR).   \n25 In many multi-objective scenarios, the objectives may conflict with each other [15]. Hence, there may   \n26 not exist any single solution that optimizes all the objectives simultaneously. For example, in online   \n27 advertising, merely optimizing CTR or CVR will degrade the performance of the other [21, 22].   \n28 Multi-objective optimization (MOO) [23, 6] is concerned with optimizing multiple conflicting   \n29 objectives simultaneously. It seeks Pareto optimality, where no single objective can be improved   \n30 without hurting the performance of the others. Many different methods for MOO have been proposed,   \n31 including evolutionary methods [26, 39], scalarization methods [9], and gradient-based iterative   \n32 methods [7]. Recently, the Multiple Gradient Descent Algorithm (MGDA) and its variants have been   \n33 introduced to the training of multi-task deep neural networks and achieved great empirical success   \n34 [29], making them regain a significant amount of research interest [17, 33, 18]. These methods   \n35 compute a composite gradient based on the gradient information of all the individual objectives   \n36 and then apply the composite gradient to update the model parameters. The composite weights are   \n37 determined by a min-norm solver [7] which yields a common descent direction of all the objectives.   \n38 However, compared to the increasingly wide application prospect, the gradient-based iterative   \n39 algorithms are relatively understudied, especially for the online learning setting. Multi-objective   \n40 online learning is of essential importance due to reasons in two folds. First, due to the data explosion in   \n41 many real-world scenarios such as web applications, making in-time predictions requires performing   \n42 online learning. Second, the theoretical investigation of multi-objective online learning will lay a solid   \n43 foundation for the design of new optimizers for multi-task deep neural networks. This is analogous to   \n44 the single-objective setting, where nearly all the optimizers for training DNNs are initially analyzed   \n45 in the online setting, such as AdaGrad [8], Adam [16], and AMSGrad [28].   \n46 In this paper, we give a systematic study of multi-objective online learning. To begin with, we   \n47 formulate the framework of Multi-Objective Online Convex Optimization (MO-OCO). The first   \n48 major challenge is the lack of regret definitions in the multi-objective setting. To tackle this challenge,   \n49 we need appropriate discrepancy metrics that can be used in the regret definitions, which evaluate the   \n50 gap between any two vector losses by producing scalar values. Intuitively, the Pareto suboptimality   \n51 gap (PSG) metric, which is frequently used in zero-order multi-objective bandits [30, 19], is a very   \n52 promising candidate. It can yield scalarized distances from any vector loss to a given comparator set.   \n53 We can thus define the multi-objective regret by simply plugging in PSG as the discrepancy metric.   \n54 However, as a metric designed purely from the geometric view, PSG is intrinsically difficult to be   \n55 optimized directly via gradient-based iterative methods. To resolve this problem, for the PSG-based   \n56 multi-objective dynamic regret, we derive its equivalent unconstrained max-min form via a highly   \n57 non-trivial transformation. This form is intuitive to the design of first-order multi-objective online   \n58 algorithms, indicating that we should select a convex combination of the gradients at each round.   \n59 Unfortunately, for the PSG-based static variant, such an equivalence does not exist. To remedy this   \n60 issue, we make extensions of the dynamic variant by fixing the comparator set and the composite   \n61 weights, which yields an appropriate definition of the multi-objective static regret.   \n62 Based on the MO-OCO framework, we develop a novel multi-objective online algorithm termed   \n63 Doubly Regularized Online Mirror Multiple Descent. The key module of the algorithm is the gradient   \n64 composition scheme, which calculates a composite gradient in the form of a convex combination of   \n65 the gradients of all objectives. Intuitively, the most direct way to determine the composite weights is   \n66 to apply the min-norm solver [7] commonly used in offline multi-objective optimization. However,   \n67 directly applying min-norm is not workable in the online setting. Specifically, the composite weights   \n68 in min-norm are only determined by the gradients at the current round. In the online setting, since   \n69 the gradients can be adversarial, they may result in undesired composite weights, further producing   \n70 a composite gradient that reversely optimizes the loss. To rigorously verify this point, we give a   \n71 showcase in which equipping OMD with vanilla min-norm even incurs a linear regret, showing that   \n72 only regularizing the iterate, as in OMD, is not enough to guarantee sublinear regrets in the multi  \n73 objective setting. To fix this issue, we devise a novel min-regularized-norm solver with an explicit   \n74 regularization on composite weights. Equipping it with OMD results in our proposed algorithm.   \n75 We then conduct the theoretical analysis for our proposed algorithm. We derive a multi-objective static   \n76 regret bound $O ( \\sqrt { T } )$ and a multi-objective dynamic regret bound $O ( V _ { T } ^ { 1 / 3 } T ^ { 2 / 3 } )$ for DR-OMMD.   \n77 Both bounds match the optimal bounds in the single-objective setting [11, 34]. Our analysis also   \n78 shows that DR-OMMD attains a lower regret than linearization with fixed composite weights.   \n79 To evaluate the effectiveness of DR-OMMD, we conduct extensive experiments on both simulation   \n80 datasets and real-world datasets. We first elaborate simulation experiments, in which we find   \n81 that DR-OMMD attains lower regret than vanilla min-norm and linearization, which verifies the   \n82 superiority of the min-regularized-norm solver. We then realize adaptive regularization via multi  \n83 objective optimization on real-world datasets, and find that adaptive regularization with DR-OMMD   \n84 significantly outperforms fixed regularization with linearization.   \n85 In summary, in this paper, we give the first systematic study of multi-objective online learning, which   \n86 encompasses a novel framework, a new algorithm, and corresponding non-trivial theoretical analysis.   \n87 We believe that this work paves the way for future research on more advanced multiple-objective   \n88 optimization algorithms, which may inspire the design of new optimizers for multi-task deep learning. ",
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+        "text": "89 2 Preliminaries ",
+        "text_level": 1,
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+        "text": "90 In this section, we briefly review the necessary background knowledge of online convex optimization   \n91 and multi-objective optimization.   \n93 Online Convex Optimization (OCO) [38, 11] is the most commonly adopted framework for   \n94 designing online learning algorithms. It can be viewed as a structured repeated game between a   \n95 learner and an adversary. At each round $t \\in \\{ 1 , \\ldots , T \\}$ , the learner is required to generate a decision   \n96 $x _ { t }$ from a convex compact set $\\mathcal { X } \\subset \\mathbb { R } ^ { n }$ . Then the adversary replies the learner with a convex function   \n97 $f _ { t } : \\mathcal { X }  \\mathbb { R }$ and the learner suffers the loss $f _ { t } ( x _ { t } )$ . The goal of the learner is to minimize the regret   \n98 with respect to the best fixed decision in hindsight, i.e., ",
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+        "text": "$$\nR _ { S } ( T ) = \\sum _ { t = 1 } ^ { T } f _ { t } ( x _ { t } ) - \\operatorname* { m i n } _ { x ^ { * } \\in \\mathcal { X } } \\sum _ { t = 1 } ^ { T } f _ { t } ( x ^ { * } ) .\n$$",
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+        "text": "99 Note that the above regret is the static regret [10], which compares the learner’s cumulative loss   \n100 with that of a fixed decision. There is another version of regret, namely the dynamic regret [10, 34],   \n101 which compares the learner’s cumulative loss with that of a sequence of local optimal decisions, i.e., ",
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+        "text": "$$\nR _ { D } ( T ) = \\sum _ { t = 1 } ^ { T } f _ { t } ( x _ { t } ) - \\sum _ { t = 1 } ^ { T } \\operatorname* { m i n } _ { x _ { t } ^ { * } \\in \\mathcal { X } } f _ { t } ( x _ { t } ^ { * } ) .\n$$",
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+        "text": "102 Any meaningful regret is required to be sublinear in $T$ , i.e., $\\begin{array} { r } { \\operatorname* { l i m } _ { T \\to \\infty } R _ { S / D } ( T ) / T = 0 } \\end{array}$ , which implies   \n103 that when $T$ is large enough, the learner can perform as well as the best fixed decision in hindsight   \n104 (for static regret) or the local optimal decision at each round (for dynamic regret).   \n105 Online Mirror Descent (OMD) [11] is a classic first-order online learning algorithm. At each round   \n106 $t \\in \\{ 1 , \\ldots , T \\}$ , OMD yields its decision using the following formula ",
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+        "text": "$$\n\\begin{array} { r } { x _ { t + 1 } = \\underset { x \\in \\mathcal { X } } { \\arg \\operatorname* { m i n } } \\eta \\langle \\nabla f _ { t } ( x _ { t } ) , x \\rangle + B _ { R } ( x , x _ { t } ) , } \\end{array}\n$$",
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+        "text": "107 where $\\eta$ is the step size, $R : \\mathcal { X }  \\mathbb { R }$ is the regularization function, and $B _ { R } ( x , x ^ { \\prime } ) = R ( x ) - R ( x ^ { \\prime } ) -$   \n108 $\\langle \\nabla R ( x ^ { \\prime } ) , x - x ^ { \\prime } \\rangle$ is the Bregman divergence induced from $R$ . As a meta-algorithm, by instantiating   \n109 different regularization functions, OMD can induce two important algorithms, i.e., Online Gradient   \n110 Descent [38, 13] and Online Exponentiated Gradient [11]. ",
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+        "text": "11 2.2 Multi-Objective Optimization ",
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+        "text": "112 Multiple-objective optimization (MOO) is concerned with solving the problems of optimizing   \n113 multiple objectives simultaneously [39, 29]. In general, since different objectives may conflict with   \n114 each other, there is no single solution that can optimize all the objectives at the same time. Instead,   \n115 MOO seeks to find solutions that achieve Pareto optimality. Next, we exposit Pareto optimality and   \n116 related definitions more formally using a vector-valued loss $H = ( h ^ { 1 } , \\ldots , \\overline { { { h ^ { m } } } } ) ^ { \\top }$ as objectives, where   \n117 $m \\geq 2$ and $h ^ { i } : { \\mathcal { K } }  \\mathbb { R }$ , $i \\in \\{ 1 , \\ldots , m \\}$ , $\\kappa \\subset \\mathbb { R }$ , is the $i$ -th loss function. ",
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+        "text": "Definition 2.1 (Pareto optimality). (a) For any two solutions $x , x ^ { \\prime } \\in \\mathcal { K }$ , we say that $x$ dominates $x ^ { \\prime }$ , denoted as $\\boldsymbol { x } \\prec \\boldsymbol { x } ^ { \\prime }$ or $x ^ { \\prime } \\succ x$ , if $h ^ { i } ( x ) \\leq h ^ { i } ( x ^ { \\prime } )$ for all $i$ , and there exists one $i$ such that $h ^ { i } ( x ) < h ^ { i } ( x ^ { \\prime } )$ ; otherwise, we say that $x$ does not dominate $x ^ { \\prime }$ , denoted as $x \\not \\prec x ^ { \\prime }$ or $x ^ { \\prime } \\nsimeq x$ . ",
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+        "text": "(b) A solution $x ^ { * } \\in \\kappa$ is called Pareto optimal if it is not dominated by any other solution in $\\kappa$ . ",
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+        "text": "122 There may exist multiple Pareto optimal solutions. For example, it is easy to show that the optimizer   \n123 of any single objective, i.e., $x _ { i } ^ { * } \\in \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { K } } h ^ { i } ( x ) , i \\in \\{ \\bar { 1 } , \\ldots , m \\}$ , is Pareto optimal. Different   \n124 Pareto optimal solutions reflect different trade-offs among the objectives [17].   \n25 Definition 2.2 (Pareto front). (a) All Pareto optimal solutions form the Pareto set ${ \\mathcal { P } } _ { \\kappa } ( H )$ .   \n26 (b) The image of ${ \\mathcal { P } } _ { \\kappa } ( H )$ constitutes the Pareto front, denoted as $\\mathcal { P } ( H ) = \\{ H ( x ) \\mid x \\in \\mathcal { P } _ { K } ( H ) \\} .$   \n127 Now that we have established the notion of optimality in MOO, we proceed to introduce the metrics   \n128 that measure the discrepancy of an arbitrary solution $x \\in \\kappa$ from being optimal. Recall that, in the   \n129 single-objective setting with merely one loss function $h : \\mathcal { Q }  \\mathbb { R }$ , where $\\mathcal { Q } \\subset \\mathbb { R }$ , for any $z \\in \\mathcal { Q }$ ,   \n130 the loss gap $h ( z ) - \\mathrm { { m i n } } _ { z ^ { \\prime \\prime } \\in \\mathcal { Q } } h ( z ^ { \\prime \\prime } )$ is directly the discrepancy measure. However, in MOO with   \n131 more than one loss, for any $x \\in \\kappa$ , the loss gap $H ( x ) - \\overline { { H ( x ^ { \\prime \\prime } ) } }$ , where $x ^ { \\prime \\prime } \\in { \\mathcal { P } } _ { \\kappa } ( H )$ , is a vector.   \n132 Intuitionally, the desired discrepancy metric shall scalarize the vector-valued loss gap and yield   \n133 the value 0 for any Pareto optimal solution. In general, there are two commonly used discrepancy   \n134 metrics in MOO, i.e. Pareto suboptimality gap (PSG) [30] and Hypervolume (HV) [4]. As HV is a   \n135 volume-based metric, it is more difficult to optimize or analyze via iterative algorithms [36]. Hence   \n136 in this paper, we adopt PSG, which has been extensively used in multi-objective bandits [30, 19].   \n137 Definition 2.3 (Pareto suboptimality gap). For any $x \\in \\kappa$ , the Pareto suboptimality gap to a given   \n138 comparator set $\\kappa ^ { * } \\subset \\kappa$ , denoted as $\\Delta ( x ; K ^ { * } , H )$ , is defined as the minimal scalar $\\epsilon \\geq 0$ that needs   \n139 to be subtracted from all entries of $H ( x )$ , such that $H ( x ) - \\epsilon \\mathbf { 1 }$ is not dominated by any point in $\\kappa ^ { * }$ ,   \n140 where 1 denotes the all-one vector in $\\mathbb { R } ^ { m }$ , i.e.,1 ",
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+        "text": "$$\n\\Delta ( x ; K ^ { * } , H ) = \\operatorname* { i n f } _ { \\epsilon \\geq 0 } \\epsilon , \\quad \\mathrm { s . t . } \\forall x ^ { \\prime \\prime } \\in K ^ { * } , \\exists i \\in \\{ 1 , . . . , m \\} , h ^ { i } ( x ) - \\epsilon < h ^ { i } ( x ^ { \\prime \\prime } ) .\n$$",
+        "text_format": "latex",
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+        "text": "141 Clearly, PSG is a distance-based discrepancy metric that motivated from a purely geometric viewpoint.   \n142 In practice, the comparator set $\\kappa ^ { * }$ is often set to be the Pareto set ${ \\mathcal { P } } _ { \\kappa } ( H )$ [30]. Then for any $x \\in \\kappa$ ,   \n143 its PSG is always non-negative and equals to zero if and only if $x \\in { \\mathcal { P } } _ { \\kappa } ( H )$ .   \n144 Multiple Gradient Descent Algorithm (MGDA) is an offline first-order algorithm for MOO [9, 7].   \n145 146 At each iteration for each objectiv $l \\in \\{ 1 , \\ldots , L \\}$ $i \\in \\{ 1 , \\ldots , m \\}$ $L$ is the number of iterations), it first c then derive the composite gradient $\\begin{array} { r } { g _ { l } ^ { c o \\bar { m } p } = \\sum _ { i = 1 } ^ { \\bar { m } } \\lambda _ { l } ^ { i } \\nabla h ^ { i } ( x _ { l } ) } \\end{array}$ $\\nabla h ^ { i } ( x _ { l } )$   \n147 the convex combination of these multiple gradients; it applies ${ \\dot { \\boldsymbol g } _ { l } } ^ { c o m p }$ to execute the gradient descent   \n148 step to update the decision, i.e., $x _ { l + 1 } = x _ { l } - \\eta g _ { l } ^ { c o m p }$ gcompl , where η is the step size. The core part of   \n149 MGDA is the module that determines the composite weights $\\lambda _ { l } = ( \\lambda _ { l } ^ { 1 } , \\ldots , \\lambda _ { l } ^ { m } )$ , which is given as ",
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+        "text": "$$\n\\lambda _ { l } = \\arg \\operatorname* { m i n } _ { \\lambda _ { l } \\in \\mathcal { S } _ { m } } \\| \\sum _ { i = 1 } ^ { m } \\lambda _ { l } ^ { i } \\nabla h ^ { i } ( x _ { l } ) \\| _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
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+        "text": "150 where $\\begin{array} { r } { \\mathcal { S } _ { m } = \\{ \\lambda \\in \\mathbb { R } ^ { m } | \\sum _ { i = 1 } ^ { m } \\lambda ^ { i } = 1 , \\lambda ^ { i } \\geq 0 , i \\in \\{ 1 , \\dots , m \\} \\} } \\end{array}$ denotes the probabilistic simplex in   \n151 $\\mathbb { R } ^ { m }$ . This is a min-norm solver which finds the weights in the simplex that yields the minimum $L _ { 2 }$   \n152 norm of the composite gradient. Thus MGDA is also called the min-norm method. Existing works   \n153 [7, 29] have shown that MGDA is guaranteed to decrease all the objectives simultaneously until it   \n154 reaches a Pareto optimal decision (under the convex setting where all $h ^ { i }$ are convex functions). ",
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+        "text": "3 Multi-Objective Online Convex Optimization ",
+        "text_level": 1,
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+        "text": "In this section, we formally formulate the framework of multi-objective optimization in the online setting, termed Multi-Objective Online Convex Optimization (MO-OCO). ",
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+        "text": "Framework overview. We tailor the famous online convex optimization (OCO) framework to the multi-objective setting, which can be viewed as a repeated game between an online learner and the adversarial environment. At each round $t \\in \\{ 1 , \\ldots , T \\}$ , the learner generates a decision $x _ { t }$ from a given convex compact decision set $\\mathcal { X } \\subset \\mathbb { R } ^ { n }$ . Then the adversary replies the decision with a vector loss function $F _ { t } ( \\bar { x } ) : \\mathcal { X }  \\mathbb { R } ^ { m }$ , where its $i$ -th component $f _ { t } ^ { i } ( x ) \\ \\bar { : } \\ x \\ \\to \\ \\mathbb { R }$ belongs to the $i$ -th objective, and the learner suffers the loss $F _ { t } ( x _ { t } ) \\in \\mathbb { R } ^ { m }$ . The goal of the learner is to generate a sequence of decisions $\\{ x _ { t } \\} _ { t = 1 } ^ { T }$ so that the cumulative loss $\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } )$ can be optimized. ",
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+        "text": "165 Recall that, in the single-objective setting, the performance metric $\\begin{array} { r } { R ( T ) = \\sum _ { t = 1 } ^ { T } ( f _ { t } ( x _ { t } ) - f _ { t } ( z _ { t } ) ) } \\end{array}$   \n166 i.e., the regret, compares the actual decisions with some comparator $z _ { t } \\in \\mathcal { X }$ at each round $t$ . For   \n167 the static regret, all $z _ { t }$ are identically set as the fixed optimal decision $x ^ { * }$ w.r.t. all losses in hindsight,   \n168 i.e., $\\begin{array} { r } { z _ { t } \\equiv x ^ { * } \\in \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } } \\sum _ { t = 1 } ^ { T } f _ { t } ( x ) } \\end{array}$ . For the dynamic regret, each $z _ { t }$ is selected as the optimal   \n169 decision $\\boldsymbol { x } _ { t } ^ { * }$ w.r.t. the instantaneous loss $f _ { t }$ at that round, i.e., $z _ { t } = x _ { t } ^ { * } \\in \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } } f _ { t } ( x )$ .   \n170 In analogy, we can define the multi-objective regret as $\\begin{array} { r } { R ( T ) = \\sum _ { t = 1 } ^ { T } \\Delta _ { t } } \\end{array}$ , where each $\\Delta _ { t }$ compares   \n171 . However, in general, no single decision can   \n172 optimize all the objectives at the same time. Hence, it is natural to compare $x _ { t }$ with a group of Pareto   \n173 optimal decisions, which constitute a comparator set $\\mathcal { C } _ { t } \\subset \\mathcal { X }$ . To measure the discrepancy between $x _ { t }$   \n174 and $\\mathcal { C } _ { t }$ , we further introduce the Pareto suboptimality gap (PSG) [30] $\\Delta ( x _ { t } ; \\mathcal { C } _ { t } , F _ { t } )$ . Then the multi  \n175 objective regret can be defined as $\\begin{array} { r } { R ( T ) = \\sum _ { t = 1 } ^ { T } \\Delta ( x _ { t } ; \\mathcal { C } _ { t } , F _ { t } ) } \\end{array}$ . Now we can formulate the static or   \n176 the dynamic variant by specifying the comparator set $\\mathcal { C } _ { t }$ at each round. Specifically, by setting all $\\mathcal { C } _ { t }$ to   \n177 be the Pareto set $\\mathcal { X } ^ { \\ast }$ of the cumulative loss $\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t }$ , we formulate the multi-objective static regret   \n178 $\\begin{array} { r } { R _ { \\mathrm { M O S } } ( T ) = \\sum _ { t = 1 } ^ { T } \\Delta ( x _ { t } ; \\mathcal { X } ^ { \\ast } , F _ { t } ) } \\end{array}$ $F _ { t }$ . By setting each bjective dynamic $\\mathcal { C } _ { t }$ to bgret $\\begin{array} { r } { R _ { \\mathrm { M O D } } ( T ) = \\sum _ { t = 1 } ^ { T } \\Delta ( x _ { t } ; \\mathcal { X } _ { t } ^ { \\ast } , F _ { t } ) } \\end{array}$ $\\mathcal { X } _ { t } ^ { \\ast }$ neous.   \n180 Recall that PSG is a zero-order metric motivated in a purely geometric sense, namely, its calculation   \n181 needs to solve a constrained optimization problem with an unknown boundary $f _ { t } ^ { i } ( x ^ { \\prime \\prime } ) , \\forall x ^ { \\prime \\prime } \\in { \\mathcal { C } } _ { t }$   \n182 Hence, it is not straightforward to design a first-order algorithm to optimize PSG, not to mention   \n183 the regret analysis. To motivate algorithm design and analysis, we investigate the two variants in   \n184 more detail. We begin with the dynamic variant, since we find that it has an equivalent form, which is   \n185 intuitive and has a strong implication on the design of effective online multiple gradient algorithms. ",
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+        "text": "An equivalent form of the dynamic regret. Surprisingly, the multi-objective dynamic regret $R _ { \\mathrm { M O D } }$ can be transformed into an unconstrained max-min form. The derivation utilizes Pareto optimality of $\\mathcal { X } _ { t } ^ { \\ast }$ and is highly non-trivial, which is deferred to the appendix due to the space limit. ",
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+        "text": "189 Proposition 3.1. The multi-objective dynamic regret has an equivalent form, i.e., ",
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+        "text": "$$\nR _ { \\mathrm { M O D } } ( T ) = \\operatorname* { s u p } _ { \\stackrel { x _ { t } ^ { * } \\in \\mathcal { X } _ { t } ^ { * } , \\ } { 1 \\leq t \\leq T } } \\operatorname* { i n f } _ { \\stackrel { x \\in S _ { m } } { 1 \\leq t \\leq T } } \\sum _ { t = 1 } ^ { T } \\lambda _ { t } ^ { * } { ^ { \\top } ( F _ { t } ( x _ { t } ) - F _ { t } ( x _ { t } ^ { * } ) ) } .\n$$",
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+        "text": "190 Remark. (i) The above form can be understood as a variant of the standard dynamic regret regarding   \n191 $\\{ \\lambda _ { t } ^ { * } ^ { \\top } F _ { t } \\} _ { t = 1 } ^ { T }$ , whereas $\\lambda _ { t } ^ { * }$ are unknown to the learner. This provides an intuition that we can gen  \n192 erate weights $\\lambda _ { t } \\in \\boldsymbol { S } _ { m }$ at each round and optimize $\\{ \\lambda _ { t } F _ { t } \\} _ { t = 1 } ^ { T }$ via single-objective techniques. For   \n193 first-order algorithms, it is equivalent to selecting a convex combination of individual gradients and   \n194 then applying the composite gradient to model update. Undoubtedly, how to generate the weights $\\lambda _ { t }$   \n195 needs some careful designs, which will be explicated later in the algorithm section. ",
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+        "text": "(ii) When $m \\ = \\ 1$ , we have $S _ { m } ~ = ~ \\{ 1 \\}$ and $\\begin{array} { r } { \\mathcal { X } _ { t } ^ { * } ~ = ~ \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } } F _ { t } ( x ) } \\end{array}$ . Hence $R _ { \\mathrm { M O D } } ( T ) ~ =$ $\\begin{array} { r } { \\sum _ { t = 1 } ^ { T } ( F _ { t } ( x _ { t } ) - \\operatorname* { m i n } _ { x \\in \\mathcal { X } } F _ { t } ( x ) ) } \\end{array}$ , which is exactly the single-objective dynamic regret $R _ { D } ( T )$ . ",
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+        "type": "text",
+        "text": "198 An alternative form of the static regret. Unfortunately, for $R _ { \\mathrm { M O S } }$ , the above equivalence form   \n199 does not exist. Here is the reason. In $R _ { \\mathrm { M O S } }$ , the comparator set $\\mathcal { X } ^ { \\ast }$ is the Pareto set of the cumulative   \n200 loss $\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t }$ rather than the instantaneous loss $F _ { t }$ . Hence, at some specific round $t$ , the decision   \n201 $x _ { t }$ may Pareto dominate all points in w.r.t. the instantaneous $F _ { t }$ , and we would expect the   \n202 metric $\\Delta _ { t }$ to be negative. However, PSG (or other commonly used metrics such as Hypervolume)   \n203 204 yields no, we have $R _ { \\mathrm { M O S } }$ th , w $R _ { S }$ . For example, whenh can be much looser   \n$m = 1$ $\\begin{array} { r } { R _ { \\operatorname { M O S } } ( T ) = \\operatorname* { s u p } _ { x ^ { * } \\in \\mathcal { X } ^ { * } } \\sum _ { t = 1 } ^ { T } \\operatorname* { m a x } \\{ F _ { t } ( x _ { t } ) - F _ { t } ( x ^ { * } ) , 0 \\} } \\end{array}$   \n205 than the static regret $\\begin{array} { r } { R _ { S } ( T ) = \\operatorname* { s u p } _ { x ^ { * } \\in \\mathcal { X } ^ { * } } \\sum _ { t = 1 } ^ { T } ( F _ { t } ( x _ { t } ) - F _ { t } ( x ^ { * } ) ) } \\end{array}$ . Hence the analysis of $R _ { \\mathrm { M O S } }$ is   \n206 intrinsically complex if we use existing discrepancy metrics that always yield non-negative values.   \n207 Enlightened by Proposition 3.1, we can formulate the static regret in a different way, i.e., by modifying   \n208 the equivalent form of dynamic regret. Recall that in Proposition 3.1, at each round $t$ , the comparator   \n209 $\\boldsymbol { x } _ { t } ^ { * }$ is selected from the Pareto set $\\mathcal { X } _ { t } ^ { \\ast }$ of the instantaneous loss $F _ { t }$ , and the weights $\\lambda _ { t } ^ { * }$ are generated   \n210 from $S _ { m }$ . To formulate the static variant, we can use a fixed comparator $x ^ { * }$ from the Pareto set $\\mathcal { X } ^ { \\ast }$ of   \n211 the cumulative loss $\\sum _ { t } F _ { t }$ and fixed weights $\\lambda ^ { * } \\in S _ { m }$ at all rounds. Now the static variant takes ",
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+        "img_path": "images/759e8230542b99bdaacb7436137bb1f0c245c8706ac39c7fa62d608bdfe0a42e.jpg",
+        "text": "$$\nR _ { \\mathrm { M O S } } ( T ) : = \\operatorname* { s u p } _ { x ^ { * } \\in \\mathcal { X } ^ { * } } \\operatorname* { i n f } _ { \\lambda ^ { * } \\in \\mathcal { S } _ { m } } { \\lambda ^ { * } } ^ { \\top } ( \\sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } ) - \\sum _ { t = 1 } ^ { T } F _ { t } ( x ^ { * } ) ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "212 Remark. (i) $R _ { \\mathrm { M O S } } ( T )$ has a clear physical meaning that optimizing it will impose the cumulative loss 213 $\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } )$ to reach the Pareto front ${ \\mathcal { P } } ^ { * }$ . See more details in Appendix C. ",
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+        "type": "text",
+        "text": "(ii) When 214 $m = 1$ , $S _ { m } = \\{ 1 \\}$ and $\\mathcal { X } ^ { \\ast }$ reduces to $\\begin{array} { r } { \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } } \\sum _ { t = 1 } ^ { T } F _ { t } ( x ) } \\end{array}$ . Therein $R _ { \\mathrm { M O S } } ( T ) =$ 15 $\\begin{array} { r } { \\sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } ) - \\operatorname* { m i n } _ { x ^ { * } \\in \\mathcal { X } ^ { * } } \\sum _ { t = 1 } ^ { T } F _ { t } ( x ^ { * } ) } \\end{array}$ x∈X t=1 , which reduces to the single-objective static regret $R _ { S } ( T )$ ",
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+        "type": "text",
+        "text": "216 4 Online Mirror Multiple Descent ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we present the Online Mirror Multiple Descent (OMMD) algorithm, the protocol of which is given in Algorithm 1. At each round $t$ , the learner first computes the gradient of the loss regarding each objective, then determines the composite weights of all these gradients, and finally applies the composite gradient to the online mirror descent step. ",
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+    {
+        "type": "text",
+        "text": "4.1 Vanilla Min-Norm May Incur Linear Regrets ",
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+        "text": "222 The core module of OMMD is the composition of multiple gradients. For simplicity, we represent the gradients at round 223 $t$ in a matrix form $\\nabla F _ { t } ( x _ { t } ) = [ \\nabla \\mathsf { \\bar { f } } _ { t } ^ { 1 } ( \\bar { x } _ { t } ) , \\ldots , \\nabla f _ { t } ^ { m } ( x _ { t } ) ] \\in \\bar { \\mathbb { R } } ^ { \\mathsf { \\bar { n } } \\times m }$ . Then the ",
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+        "type": "text",
+        "text": "1: Input: Convex set $\\mathcal { X }$ , time horizon $T$ , regularization parameter $\\alpha _ { t }$ , learning rate $\\eta _ { t }$ , regulariza tion function $R$ , user preference $\\lambda _ { 0 }$ .   \n2: Initialize: $x _ { 1 } \\in \\mathcal { X }$ .   \n3: for $t = 1 , \\dots , T$ do   \n4: Predict $x _ { t }$ and receive a loss function $F _ { t } : \\mathcal { X }  \\mathbb { R } ^ { m }$ .   \n5: Compute the multiple gradients $\\nabla F _ { t } ( x _ { t } ) = [ \\nabla f _ { t } ^ { 1 } ( x _ { t } ) , \\ldots , \\nabla f _ { t } ^ { m } ( x _ { t } ) ] \\in \\mathbb { R } ^ { n \\times m } .$ .   \n6: Determine the weights for the gradient composition via min-regularized-norm $\\lambda _ { t } = \\operatorname * { \\bar { a r g m i n } } _ { \\lambda \\in { \\cal S } _ { m } } \\| \\nabla \\dot { F _ { t } } ( x _ { t } ) \\lambda \\| _ { 2 } ^ { 2 } + \\alpha \\| \\lambda - \\stackrel { \\smile } { \\lambda } _ { 0 } \\| _ { 1 } .$   \n7: Compute the composite gradient $g _ { t } = \\nabla F _ { t } ( x _ { t } ) \\lambda _ { t }$ .   \n8: Perform online mirror descent using $g _ { t }$ $x _ { t + 1 } = \\underset { x \\in \\mathcal { X } } { \\arg \\operatorname* { m i n } } \\eta \\langle g _ { t } , x \\rangle + B _ { R } ( x , x _ { t } ) .$ ",
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+        "type": "text",
+        "text": "9: end for ",
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+        "type": "text",
+        "text": "224 composite gradient is given as $g _ { t } = \\nabla F _ { t } ( x _ { t } ) \\lambda _ { t }$ , where $\\lambda _ { t }$ is the composite weights. As illustrated in   \n225 Preliminary, the min-norm method in MGDA [7, 29] is a classic method to determine the composite   \n226 weights in the offline setting, which results in a common descent direction that can descend all the   \n227 losses simultaneously. Thus, it is tempting to consider applying it to the online setting.   \n228 However, directly applying the min-norm method to the online setting is not workable, which may   \n229 even incur linear regrets of the resulting algorithms. The rationale is as follows. In the vanilla   \n230 min-norm method, the composite weights $\\lambda _ { t }$ are determined solely by the gradients $\\nabla F _ { t } ( x _ { t } )$ at the   \n231 current round $t$ , hence they are very sensitive to the instantaneous loss $F _ { t }$ . In the online setting,   \n232 the losses at each round can be adversarially chosen, and thus the corresponding gradients can be   \n233 adversarial. These adversarial gradients may result in undesired composite weights, which may   \n234 further produce a composite gradient that even deteriorates the next prediction. In the following,   \n235 we provide a problem instance in which min-norm incurs a linear regret. We extend OMD to the   \n236 multi-objective setting, where the composite weights are directly yielded by min-norm [11].   \n237 Problem instance. We consider a two-objective problem. The decision domain is $\\mathcal { X } = \\{ ( u , v ) ~ |$   \n238 $\\begin{array} { r } { u + v \\leq \\frac { 1 } { 2 } , v - u \\leq \\frac { 1 } { 2 } , v \\geq 0 \\} } \\end{array}$ and the loss function at each round is ",
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+        "type": "equation",
+        "img_path": "images/0d5a171f637adb9f2eb0528d59a5870302286e4146b4f9aa99c52bd39766479f.jpg",
+        "text": "$$\nF _ { t } ( x ) = \\left\\{ \\begin{array} { l l } { ( \\| x - a \\| ^ { 2 } , \\| x - b \\| ^ { 2 } ) , ~ t = 2 k - 1 , } & { ~ k = 1 , 2 , . . . ; } \\\\ { ( \\| x - b \\| ^ { 2 } , \\| x - c \\| ^ { 2 } ) , ~ t = 2 k , } & { ~ k = 1 , 2 , . . . , } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "239 where $a = ( - 2 , - 1 ) , b = ( 0 , 1 ) , c = ( 2 , - 1 )$ . For simplicity, we first analyze the case where the   \n240 total time horizon $T$ is an even number. Then we can compute the Pareto set of the cumulative   \n241 loss $\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t }$ , i.e., $\\begin{array} { r } { \\mathcal { X } ^ { * } = \\{ ( u , 0 ) \\mid - \\frac { 1 } { 2 } \\leq u \\leq \\frac { 1 } { 2 } \\} } \\end{array}$ , which locates at the $x$ -axis. For conciseness of   \n242 analysis, we instantiate OMD with L2-regularization, which results in the simple OGD algorithm   \n243 [24]. We start at an arbitrary point $x _ { 1 } = ( u _ { 1 } , v _ { 1 } ) \\in \\mathcal { X }$ satisfying $v _ { 1 } > 0$ . At each round $t$ , suppose   \n244 the decision $x _ { t } = ( u _ { t } , v _ { t } ) \\in \\mathcal { X }$ , then the gradients of each objective w.r.t. $x _ { t }$ can be calculated as ",
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+        "img_path": "images/14cb2aa94f549718bcec12a508e4b0249ddaad2970271250ebf8e7e7304f85c8.jpg",
+        "text": "$$\ng _ { t } ^ { 1 } = { \\left\\{ \\begin{array} { l l } { ( 2 u _ { t } + 4 , ~ 2 v _ { t } + 2 ) , } & { t = 2 k - 1 ; } \\\\ { ( 2 u _ { t } , } & { 2 v _ { t } - 2 ) , } & { t = 2 k . } \\end{array} \\right. } \\qquad g _ { t } ^ { 2 } = { \\left\\{ \\begin{array} { l l } { ( 2 u _ { t } , } & { 2 v _ { t } - 2 ) , } & { t = 2 k - 1 ; } \\\\ { ( 2 u _ { t } - 4 , } & { 2 v _ { t } + 2 ) , } & { t = 2 k . } \\end{array} \\right. }\n$$",
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+        "text": "245 Since $\\begin{array} { r } { 0 \\leq v _ { t } \\leq \\frac { 1 } { 2 } } \\end{array}$ , we observe that the second entry of either gradient alternates between positive   \n246 and negative. By using min-norm, the composite weights $\\lambda _ { t }$ can be computed as ",
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+        "text": "$$\n\\lambda _ { t } = \\left\\{ { \\begin{array} { l l } { ( ( 1 - u _ { t } - v _ { t } ) / 4 , } & { ( 3 + u _ { t } + v _ { t } ) / 4 ) , t = 2 k - 1 ; } \\\\ { ( ( 3 - u _ { t } + v _ { t } ) / 4 , } & { ( 1 + u _ { t } - v _ { t } ) / 4 ) , t = 2 k . } \\end{array} } \\right.\n$$",
+        "text_format": "latex",
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+        "text": "247 We observe that both entries of composite weights alternative between above $\\frac { 1 } { 2 }$ and below $\\frac { 1 } { 2 }$ , and   \n248 $\\| \\lambda _ { t + 1 } - \\lambda _ { t } \\| _ { 1 } \\geq 1$ . Recall that $\\| \\lambda _ { t } \\| _ { 1 } = 1$ , hence the composite weights at two consecutive rounds   \n249 change radically. The resulting composite gradient takes ",
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+        "text": "$$\ng _ { t } ^ { c o m p } = \\left\\{ \\begin{array} { l l } { { ( u _ { t } - v _ { t } + 1 , ~ } } & { { - u _ { t } + v _ { t } - 1 ) , t = 2 k - 1 ; } } \\\\ { { ( - u _ { t } - v _ { t } - 1 , } } & { { - u _ { t } - v _ { t } - 1 ) , t = 2 k . } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "250 The fluctuating composite weights mix with the positive and negative second entries of gradients,   \n251 making the second entry of $g _ { t } ^ { c \\bar { o } m p }$ always negative, i.e., $- u _ { t } + v _ { t } - 1 < 0$ and $- u _ { t } - v _ { t } - 1 < 0$   \n252 Hence ${ \\bf { \\bar { \\it g } } } _ { t } ^ { c o m p }$ actually drives $x _ { t }$ away from the Pareto set $\\mathcal { X } ^ { \\ast }$ that coincides with the $x$ -axis. This   \n253 essentially reversely optimizes the loss, hence increases the regret. In fact, we can prove that it even   \n254 incurs a linear regret2. Due to the lack of space, we leave the proof of linear regret when $T$ is an odd   \n255 number in the appendix. The above results of the problem instance are summarized as follows. ",
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+        "type": "text",
+        "text": "Proposition 4.1. For OMD equipped with vanilla min-norm, there exists a multi-objective online convex optimization problem, in which the resulting algorithm incurs a linear regret. ",
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+        "type": "text",
+        "text": "258 Remark. Stability is a basic requirement to guarantee meaningful regrets in online learning [25].   \n259 In the single-objective setting, directly regularizing the iterate $x _ { t }$ (e.g., OMD) is already enough.   \n260 However, as shown in the above analysis, only regularizing $x _ { t }$ is not enough to attain sublinear regrets   \n261 in the multi-objective setting, since there is another source of instability, i.e., the composite weights,   \n262 that affects the direction of the composite gradient. Therefore, in multi-objective online learning,   \n263 besides regularizing the iterates, we also need to explicitly regularize the composite weights. ",
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+        "type": "text",
+        "text": "4.2 Doubly Regularized Online Mirror Multiple Descent ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Enlightened by the design of regularization in FTRL [25], we consider the regularizer $r ( \\lambda , \\lambda _ { 0 } )$ , where $\\lambda _ { 0 }$ is the pre-defined composite weight that may reflect the user preference. This results in a new solver called min-regularized-norm, i.e., ",
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+        "text": "$$\n\\lambda _ { t } = \\underset { \\lambda \\in S _ { m } } { \\arg \\operatorname* { m i n } } \\| \\nabla F _ { t } ( x _ { t } ) \\lambda \\| _ { 2 } ^ { 2 } + \\alpha r ( \\lambda , \\lambda _ { 0 } ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "268 where $\\alpha$ is the strength of regularization. Equipping OMD with the new solver, we derive the   \n269 proposed online algorithm. Note that beyond the regularization on the iterate $x _ { t }$ that is intrinsic in   \n270 online learning, there is another regularization on the composite weights $\\lambda _ { t }$ in min-regularized norm.   \n271 Both regularizations are fundamental and they together ensure the stability in the multi-objective   \n272 online setting. Hence we call the algorithm Doubly Regularized OMMD (DR-OMMD).   \n273 In principle, $r$ can take various forms such as $L _ { 1 }$ -norm, $L _ { 2 }$ -norm and KL divergence etc. Here   \n274 we adopt $L _ { 1 }$ -norm since it aligns well with the simplex constraint of $\\lambda$ . Min-regularized-norm   \n275 can be computed very efficiently, since it has a closed-form solution when $m = 2$ . Specifically,   \n276 suppose the gradients at round $t$ are $g _ { t } ^ { 1 }$ and $g _ { t } ^ { 2 }$ . Set $\\gamma _ { L } = ( g _ { 2 } ^ { \\top } ( g _ { 2 } - g _ { 1 } ) - \\alpha ) / \\Vert g _ { 2 } - g _ { 1 } \\Vert ^ { 2 }$ and   \n277 $\\gamma _ { R } = ( g _ { 2 } ^ { \\top } ( g _ { 2 } - g _ { 1 } ) + \\alpha ) / \\Vert g _ { 2 } - g _ { 1 } \\Vert ^ { 2 }$ . Given any $\\lambda _ { 0 } = ( \\gamma _ { 0 } , 1 - \\gamma _ { 0 } ) \\in S _ { 2 }$ , we can compute the   \n278 composite weights $\\lambda _ { t }$ as $( \\gamma _ { t } , 1 - \\gamma _ { t } )$ where ",
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+        "text": "$$\n\\gamma _ { t } = \\operatorname* { m a x } \\{ \\operatorname* { m i n } \\{ \\gamma _ { t } ^ { \\prime \\prime } , 1 \\} , 0 \\} , \\quad \\mathrm { w h e r e } \\ \\gamma _ { t } ^ { \\prime \\prime } = \\operatorname* { m a x } \\{ \\operatorname* { m i n } \\{ \\gamma _ { 0 } , \\gamma _ { R } \\} , \\gamma _ { L } \\} .\n$$",
+        "text_format": "latex",
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+        "text": "79 In addition, when $m > 2$ , since the feasible region $S _ { m }$ is a simplex, we can introduce a Frank-Wolfe   \n80 solver [14] to compute the composite weights. See the protocol and more details in Appendix D. ",
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+        "text": "Compared to vanilla min-norm, the composite weights in min-regularized-norm are not fully determined by the adversarial gradients. The resulting relative stability of composite weights make the composite gradients more robust to the adversarial environment. In the following, we give a general analysis and prove that DR-OMMD indeed guarantees sublinear regrets. ",
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+        "text": "4.3 Analysis ",
+        "text_level": 1,
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+        "text": "We now analyze the static regret and the dynamic regret of DR-OMMD. Our analysis is based on the following commonly used assumptions [13, 11]. ",
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+        "type": "text",
+        "text": "Assumption 4.2 (Bregman divergence). The regularization function $R$ is 1-strongly convex. In addition, the Bregman divergence is $\\gamma$ -Lipschitz continuous, i.e., $B _ { R } ( x , z ) - B _ { R } ( \\bar { y } , z ) \\leq \\gamma \\| x -$ $y \\| , \\forall x , y , z \\in \\mathrm { d o m } R$ , where $\\mathrm { d o m } R$ is the domain of $R$ and satisfies $\\mathcal { X } \\subset \\mathrm { d o m } R \\subset \\mathbb { R } ^ { n }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.3 (Lipschitz continuity). For each $i \\in \\{ 1 , \\ldots , m \\}$ , there exists some positive and finite $G$ such that, the $i$ -th loss $f _ { t } ^ { i }$ at each round $t \\in \\{ 1 , \\ldots , T \\}$ is $G$ -Lipschitz continuous w.r.t. $\\| \\cdot \\|$ , i.e., $| f _ { t } ^ { i } ( x ) - f _ { t } ^ { i } ( x ^ { \\prime } ) | \\leq G \\| x - \\bar { x } ^ { \\prime } \\|$ . Note that in the convex setting, this assumption leads to bounded gradients, i.e., $\\| \\nabla f _ { t } ^ { i } ( x ) \\| _ { * } \\leq G$ for any $t \\in \\{ 1 , \\ldots , T \\} , i \\in \\{ 1 , \\ldots , m \\} , x \\in \\mathcal { X }$ . ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "295 We first provide the static regret bound. The proof is left to the appendix due to the lack of space. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.4. Suppose the diameter of 296 $\\mathcal { X }$ is bounded by $D$ . Assume $F _ { t }$ is bounded, i.e., $| f _ { t } ^ { i } ( x ) | \\leq$ 297 $F , \\forall x \\in \\mathcal { X } , t \\in \\{ \\bar { 1 } , \\dots , T \\} , i \\in \\{ 1 , \\dots , m \\}$ . For any $\\lambda _ { 0 } \\in  { S _ { m } }$ , DR-OMMD attains ",
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+            109,
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/244348e8a04debb30b6e01f798772fa63f1a766a31f073cc4754407ab786b25e.jpg",
+        "text": "$$\nR _ { \\mathrm { M O S } } ( T ) \\leq \\frac { 1 } { \\eta } B _ { R } ( x ^ { * } , x _ { 1 } ) + \\frac { \\eta } { 2 } \\sum _ { t = 1 } ^ { T } ( \\Vert \\nabla F _ { t } ( x _ { t } ) \\lambda _ { t } \\Vert _ { 2 } ^ { 2 } + \\frac { 4 F } { \\eta } \\Vert \\lambda _ { t } - \\lambda _ { 0 } \\Vert _ { 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Remark. (i) Linearization with weights $\\lambda _ { 0 } \\in \\mathcal { S } _ { m }$ can be viewed as single-objective optimization on scalar loss $\\lambda _ { 0 } ^ { \\top } F _ { t }$ , whose gradient is $g _ { t } = \\nabla F _ { t } ( x _ { t } ) \\lambda _ { 0 }$ . Hence we can directly borrow the tight bound of OMD (Theorem 6.8 in [27]) and derive a bound $\\begin{array} { r } { \\frac { 1 } { \\eta } B _ { R } ( x ^ { * } , x _ { 1 } ) + \\sum _ { t = 1 } ^ { T } \\frac { \\eta _ { t } } { 2 } \\| \\nabla F _ { t } ( x _ { t } ) \\lambda _ { 0 } \\| _ { 2 } ^ { 2 } } \\end{array}$ $\\lambda _ { t }$ r linearization. In co, the bound becomes $\\begin{array} { r } { \\frac { 1 } { \\eta } B _ { R } ( x ^ { * } , x _ { 1 } ) + \\frac { \\eta } { 2 } \\sum _ { t = 1 } ^ { T } \\operatorname* { m i n } _ { \\lambda \\in { \\cal S } _ { m } } \\{ \\| \\nabla F _ { t } ( x _ { t } ) \\lambda \\| ^ { 2 } + \\alpha \\| \\lambda - \\lambda _ { 0 } \\| _ { 1 } \\} . } \\end{array}$ $\\alpha = 4 F / \\eta$ ulation of, which is smaller than that of linearization. Note that the lower regret of DR-OMMD compared to linearization is also empirically verified in our experiments (see Figure 1). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "(ii) When $\\begin{array} { r } { \\eta = \\frac { \\hat { \\sqrt { 2 \\gamma D } } } { G \\sqrt { T } } , \\alpha = \\frac { 4 F } { \\eta } } \\end{array}$ , the bound is in the order of $O ( \\sqrt { T } )$ . It matches the optimal static single-objective regret bound w.r.t. $T$ [11] (see more details in Appendix E). ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Then we turn to the dynamic regret. Our analysis relies on an additional assumption [2, 32, 5]. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.5 (Temporal variability). For each $i \\in \\{ 1 , \\ldots , m \\}$ , there exists some positive and finite $V _ { T }$ such that $\\begin{array} { r } { \\sum _ { t = 1 } ^ { T - 1 } \\operatorname* { s u p } _ { x \\in \\mathcal { X } } | f _ { t } ^ { i } ( x ) - f _ { t + 1 } ^ { i } ( x ) | \\leq V _ { T } } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.6. Assume the step size satisfies 310 $\\begin{array} { r } { \\frac { 4 V _ { T } } { G ^ { 2 } T } \\leq \\eta \\leq \\frac { 4 V _ { T } } { G ^ { 2 } } } \\end{array}$ . Then under all the above assumptions, 311 for any preference $\\lambda _ { 0 } \\in  { S _ { m } }$ , OMMD with min-regularized-norm attains ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/95e535e5a9e21f4de7055d2b2a383d0d2930960b2c050fd2203447d57c6ea6c9.jpg",
+        "text": "$$\nR _ { \\mathrm { M O D } } ( T ) \\leq \\frac { \\eta G ^ { 2 } T } { 2 } + \\frac { 4 \\gamma D V _ { T } } { \\eta ^ { 2 } G ^ { 2 } } + \\frac { \\eta } { 2 } \\sum _ { t = 1 } ^ { T } ( \\Vert \\nabla F _ { t } ( x _ { t } ) \\lambda _ { t } \\Vert _ { 2 } ^ { 2 } + \\frac { 8 F G ^ { 2 } T } { V _ { T } } \\Vert \\lambda _ { t } - \\lambda _ { 0 } \\Vert _ { 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Remark. When 312 $\\begin{array} { r } { \\eta = \\frac { 2 } { G } ( \\frac { \\gamma D V _ { T } } { G T } ) ^ { 1 / 3 } , \\alpha = \\frac { 8 F G ^ { 2 } T } { V _ { T } } } \\end{array}$ , the bound is in the order of $O ( T ^ { 2 / 3 } V _ { T } ^ { 1 / 3 } )$ , matching 313 the best attainable single-objective dynamic regret bound [2, 35] (see more details in Appendix E). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we conduct extensive experiments to evaluate the effectiveness of DR-OMMD. We consider two baselines: (i) linearization performs single-objective online learning on the linearized loss $\\lambda _ { 0 } ^ { \\top } F _ { t }$ at each round $t$ , where the weights $\\lambda _ { 0 } \\in  { S _ { m } }$ are given beforehand; note that it is equivalent to computing composite gradients with fixed weights $\\lambda _ { t } \\equiv \\lambda _ { 0 }$ . (ii) min-norm equips OMD with vanilla min-norm [7] for gradient composition. ",
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+    {
+        "type": "text",
+        "text": "5.1 Simulation Experiments: Tracking the Pareto Front ",
+        "text_level": 1,
+        "bbox": [
+            165,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As summarized in Figure 1 (a), the goal is to track two points $\\xi _ { t } ^ { 1 } , \\xi _ { t } ^ { 2 }$ cycling along a circle ${ \\mathcal { C } } = \\{ \\xi \\in { }$ $\\mathbb { R } ^ { 2 } \\mid \\| \\xi \\| _ { 2 } = 1 \\}$ . For each $i \\in \\{ 1 , 2 \\}$ , $\\xi _ { t } ^ { i } = ( \\cos \\theta _ { t } ^ { i } , \\sin \\bar { \\theta } _ { t } ^ { i } )$ is determined by some angle $\\theta _ { t } ^ { i }$ . We set a positive integer $P ^ { i }$ as the rotating period of $\\xi _ { t } ^ { i }$ , which is unknown to the learner. The two points are initialized by $\\theta _ { 1 } ^ { 1 } = 0$ and $\\theta _ { 1 } ^ { 2 } = \\pi / 2$ and move as follows: at each round $t$ , for each $i \\in \\{ 1 , 2 \\}$ , the adversary independently samples an angle $\\delta _ { t } ^ { i }$ from a Gaussian distribution $\\mathcal { N } ( { 2 \\pi } / { P ^ { i } } , { 1 } / { \\sqrt { P ^ { i } } } )$ , then moves the $i$ -th point to $\\xi _ { t + 1 } ^ { i } \\bar { \\mathbf { \\xi } } = ( \\cos \\theta _ { t + 1 } ^ { i } , \\sin \\theta _ { t + 1 } ^ { i } )$ where $\\theta _ { t + 1 } ^ { i } = \\theta _ { t } ^ { i } - \\delta _ { t } ^ { i }$ . Note that $\\mathbb { E } \\theta _ { t + 1 } ^ { i } =$ $\\theta _ { 1 } ^ { i } + 2 \\pi t / P ^ { i }$ , hence in average $\\xi _ { t } ^ { i }$ rotates clockwise with a period of $P ^ { i }$ . At each round $t$ , the learner 1 generates a decision $x _ { t }$ from a $L 2$ -norm ball $\\mathcal { X } = \\{ x \\in \\mathbb { R } ^ { 2 ^ { \\cdot } } | \\ \\| x \\| _ { 2 } \\leq 2 \\}$ . Then it acquires $\\xi _ { t } ^ { 1 } , \\xi _ { t } ^ { 2 }$ and suffer the losses $f _ { t } ^ { i } ( x _ { t } ) = \\| x _ { t } - \\xi _ { t } ^ { i } \\| _ { 2 } ^ { 2 } / 2 , i \\in \\{ 1 , 2 \\}$ . In this problem, the Pareto set of $F _ { t } = ( f _ { t } ^ { \\mathrm { i } } , f _ { t } ^ { 2 } )$ is exactly the line segment between $\\xi _ { t } ^ { 1 }$ and $\\xi _ { t } ^ { 2 }$ , i.e., $\\mathcal { X } _ { t } ^ { * } = \\{ \\lambda \\xi _ { t } ^ { 1 } + ( 1 - \\lambda ) \\xi _ { t } ^ { 2 } \\ | \\ \\lambda \\in [ 0 , 1 ] \\}$ . A t each round $t$ , PSG measures the squared distance between $x _ { t }$ and $\\mathcal { X } _ { t } ^ { \\ast }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "332 We run $T = 1 0 , 0 0 0$ rounds. To simulate the pattern drift, we set $P ^ { 1 } = 1 0 , P ^ { 2 } = 2 0$ at the first   \n333 $T _ { 1 } = 3 , 0 0 0$ rounds, and $P ^ { 1 } = 2 0 , P ^ { 2 } = 1 0$ at the last $T _ { 2 } = 7 , 0 0 0$ rounds. For linearization,   \n334 the weights $\\lambda _ { 0 } = ( \\lambda _ { 0 } ^ { 1 } , 1 - \\lambda _ { 0 } ^ { 1 } )$ are decided via a grid search $\\lambda _ { 0 } ^ { 1 } \\in \\{ 0 , 0 . 1 , . . . , 1 \\}$ ; we consider   \n335 three variants: lin- $^ { 1 }$ uses the optimal $\\lambda _ { 0 }$ for the first $T _ { 1 }$ rounds, lin-2 uses the optimal $\\lambda _ { 0 }$ for the   \n336 last $T _ { 2 }$ rounds, and lin-opt uses the optimal $\\lambda _ { 0 }$ for all $T$ rounds. For DR-OMMD, for fairness of ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/a3b4326a03d8acf25d9ced0df3ebeaa1dfb1dd10d00da99c7cc03a87bd49e358.jpg",
+        "image_caption": [
+            "Figure 1: Simulation setup and results. (a) The targets $\\xi _ { t } ^ { 1 } , \\xi _ { t } ^ { 2 }$ cycle along the circle. The Pareto set at each round is the line segment $[ \\xi _ { t } ^ { 1 } , \\xi _ { t } ^ { 2 } ]$ PSG measures the distance from $x _ { t }$ to $[ \\xi _ { t } ^ { 1 } , \\xi _ { t } ^ { 2 } ]$ (b) Performance of DR-OMMD and baselines. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/30ee82d59d0ad7951da9536a53b3265f80f8ed0fa2dccfd60c48a4136590979b.jpg",
+        "image_caption": [
+            "Figure 2: Results to verify the effectiveness of adaptive regularization on protein. (a) Performance of DR-OMMD and linearization under varying $\\lambda _ { 0 } = ( \\lambda _ { 0 } ^ { 1 } , 1 - \\lambda _ { 0 } ^ { 1 } )$ . (b) Performance using the optimal weights $\\lambda _ { 0 } = ( 0 . 1 , 0 . 9 )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "comparison we use the same $\\lambda _ { 0 }$ of lin-opt. The learning rates $\\eta$ in all algorithms and the parameter $\\alpha$ in DR-OMMD follow the corresponding theories (e.g., Theorem 4.6). In this experiment, since the loss functions are manually designed, the value of $V _ { T }$ can be directly calculated. Note that in some scenarios where $V _ { T }$ is unknown, we can conduct a grid search and utilize a meta-algorithm to handle the unknown $V _ { T }$ [37, 1], similar to the single-objective setting. From the results in Figure 1 (b), we find that DR-OMMD achieves the lowest PSG, showing its ability to track the Pareto front; meanwhile, min-norm appears very unstable in the online setting, even worse than linearization. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2 Convex Experiments: Adaptive Regularization via Multi-Objective Optimization ",
+        "text_level": 1,
+        "bbox": [
+            161,
+            382,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In many real-world online scenarios, regularization is often adopted to avoid overfitting. A standard way is to add a term $r ( x )$ to the loss $f _ { t } ( x )$ at each round and optimize the regularized loss $f _ { t } ( x ) +$ $\\sigma r ( x )$ [24], where $\\sigma$ is treated as a hyperparameter that needs to be fixed beforehand. The formalism of multi-objective online learning provides a novel way to realize regularization. Since $r ( x )$ measures the complexity of $x$ , it can be regarded as the second objective alongside the primary goal $f _ { t } ( x )$ . We can construct a vector loss $F _ { t } ( \\bar { x ) = ( f _ { t } ( x ) , r ( x ) ) }$ at each round and thereby cast regularized online learning into a bi-objective online optimization problem. Compared to fixed regularization, the new approach effectively chooses the regularization strength $\\sigma _ { t } = \\bar { \\lambda } _ { t } ^ { 2 } / \\lambda _ { t } ^ { 1 }$ in an adaptive way. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "353 We use two large-scale online benchmark datasets. (i) protein is a bioinformatics dataset for protein   \n354 type classification [31], which has 17 thousand instances with 357 features. (ii) covtype is a biological   \n355 dataset collected from a non-stationary environment for forest cover type prediction [3], which has   \n356 50 thousand instances with 54 features. For both tasks, we set the logistic loss of classification as   \n357 the first objective, and the squared $L 2$ -norm of model parameters as the second objective. Since the   \n358 ultimate goal of regularization is to enhance predictive performance, we adopt the average loss as the   \n359 performance metric, namely $\\textstyle \\sum _ { t \\leq T } l _ { t } ( x _ { t } ) / { \\bar { T } }$ , where $l _ { t } ( x _ { t } )$ is the classification loss at round $t$ . ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We adopt a $L 2$ -norm ball centered at the origin with diameter $K = 1 0 0$ as the decision set. The learning rates are decided by a grid search over $\\{ 0 . 1 , 0 . 2 , \\ldots , 3 . 0 \\}$ . For DR-OMMD, the parameter $\\alpha$ is simply set as 0.1. For fixed regularization, the strength $\\sigma = ( 1 - \\lambda _ { 0 } ^ { 1 } ) / \\lambda _ { 0 } ^ { 1 }$ is determined by the some preference $\\lambda _ { 0 } ^ { 1 } \\in [ 0 , 1 ]$ , which is essentially linearization with weights $\\overset { \\vartriangle } { \\lambda _ { 0 } } = ( \\lambda _ { 0 } ^ { 1 } , 1 - \\lambda _ { 0 } ^ { \\bar { 1 } } )$ . We run both algorithms with varying initial weights $\\lambda _ { 0 } ^ { 1 } \\in \\{ 0 , 0 . 1 , . . . , 1 \\}$ . In Figure 2, we plot (a) their final performance w.r.t. the choice of $\\lambda _ { 0 }$ and (b) their learning curves with desirable $\\lambda _ { 0 }$ (e.g., (0.1, 0.9) on protein). Other results are deferred to the appendix due to the lack of space. The results show that DR-OMMD consistently outperforms fixed regularization. ",
+        "bbox": [
+            161,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            762,
+            305,
+            779
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we give a systematic study of multi-objective optimization in the online setting. We first formulate the framework of Multi-Objective Online Convex Optimization. Then we devise the Doubly Regularized Online Mirror Multiple Descent algorithm, which has a special design for gradient composition in online learning, namely min-regularized-norm. We provide non-trivial regret bounds for DR-OMMD and conduct extensive experiments to demonstrate its effectiveness. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Limitations. As the first step of studying multiple gradient algorithm in online learning, we conduct our analysis in the convex setting. Although it does not affect the usage in the non-convex setting (see empirical validation in Appendix F), we can give a formal non-convex analysis in the future. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Dheeraj Baby and Yu-Xiang Wang. Optimal dynamic regret in exp-concave online learning. In Conference on Learning Theory, pages 359–409. PMLR, 2021.   \n[2] Omar Besbes, Yonatan Gur, and Assaf Zeevi. Non-stationary stochastic optimization. Operations Research, 63(5):1227–1244, 2015.   \n[3] Jock A Blackard and Denis J Dean. Comparative accuracies of artificial neural networks and discriminant analysis in predicting forest cover types from cartographic variables. Computers and Electronics in Agriculture, 24(3):131–151, 1999.   \n[4] Lucas Bradstreet. The hypervolume indicator for multi-objective optimisation: calculation and use. University of Western Australia Perth, 2011.   \n[5] Nicolò Campolongo and Francesco Orabona. A closer look at temporal variability in dynamic online learning. arXiv preprint arXiv:2102.07666, 2021.   \n[6] Kalyanmoy Deb. Multi-objective optimization. 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Tracking slowly moving clairvoyant: Optimal dynamic regret of online learning with true and noisy gradient. In International Conference on Machine Learning, pages 449–457, 2016.   \n[33] Tianhe Yu, Saurabh Kumar, Abhishek Gupta, Sergey Levine, Karol Hausman, and Chelsea Finn. Gradient surgery for multi-task learning. In Advances in Neural Information Processing Systems, pages 5824–5836, 2020.   \n[34] Lijun Zhang, Shiyin Lu, and Zhi-Hua Zhou. Adaptive online learning in dynamic environments. In Advances in Neural Information Processing Systems, pages 1330–1340, 2018.   \n[35] Lijun Zhang, Tianbao Yang, Zhi-Hua Zhou, et al. Dynamic regret of strongly adaptive methods. In International Conference on Machine Learning, pages 5882–5891, 2018.   \n[36] Richard Zhang and Daniel Golovin. Random hypervolume scalarizations for provable multiobjective black box optimization. In International Conference on Machine Learning, pages 11096–11105, 2020.   \n[37] Peng Zhao, Yu-Jie Zhang, Lijun Zhang, and Zhi-Hua Zhou. Dynamic regret of convex and smooth functions. Advances in Neural Information Processing Systems, 33:12510–12520, 2020.   \n[38] Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In International Conference on Machine Learning, pages 928–936, 2003.   \n[39] Eckart Zitzler and Lothar Thiele. Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach. IEEE transactions on Evolutionary Computation, 3(4):257–271, 1999. ",
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parse/dev/pfNyExj7z2/pfNyExj7z2.md

@@ -0,0 +1,334 @@
+# VECTOR-QUANTIZED IMAGE MODELING WITH IMPROVED VQGAN
+
+Jiahui Yu Xin Li Jing Yu Koh Han Zhang Ruoming Pang James Qin Alexander Ku Yuanzhong Xu Jason Baldridge Yonghui Wu
+
+Google Research
+
+jiahuiyu@google.com
+
+# ABSTRACT
+
+Pretraining language models with next-token prediction on massive text corpora has delivered phenomenal zero-shot, few-shot, transfer learning and multi-tasking capabilities on both generative and discriminative language tasks. Motivated by this success, we explore a Vector-quantized Image Modeling (VIM) approach that involves pretraining a Transformer to predict rasterized image tokens autoregressively. The discrete image tokens are encoded from a learned Vision-Transformerbased VQGAN (ViT-VQGAN). We first propose multiple improvements over vanilla VQGAN from architecture to codebook learning, yielding better efficiency and reconstruction fidelity. The improved ViT-VQGAN further improves vectorquantized image modeling tasks, including unconditional, class-conditioned image generation and unsupervised representation learning. When trained on ImageNet at $2 5 6 \times 2 5 6$ resolution, we achieve Inception Score (IS) of 175.1 and Frechet Inception Distance (FID) of 4.17, a dramatic improvement over the vanilla ´ VQGAN, which obtains 70.6 and 17.04 for IS and FID, respectively. Based on ViT-VQGAN and unsupervised pretraining, we further evaluate the pretrained Transformer by averaging intermediate features, similar to Image GPT (iGPT). This ImageNet-pretrained VIM-L significantly beats iGPT-L on linear-probe accuracy from $6 0 . 3 \%$ to $7 3 . 2 \%$ for a similar model size. ViM-L also outperforms iGPT-XL which is trained with extra web image data and larger model size.
+
+# 1 INTRODUCTION
+
+Natural language processing (NLP) has recently experienced dramatic improvements from learning general-purpose representations by pretraining language models on unlabeled text corpora. This strategy has produced large performance gains for a wide range of natural language generation (NLG) and natural language understanding (NLU) tasks (Dai & Le, 2015; Radford et al., 2018; 2019; Brown et al., 2020). Conceptually, generative pretraining models the data density $P ( X )$ in a tractable way, with the hope of also helping discriminative tasks of $P ( { Y \vert } X )$ (Lasserre et al., 2006); importantly, there are no limitations on whether the signals are from the language domain or others, such as vision.
+
+In computer vision, in contrast, most recent unsupervised or self-supervised learning research focuses on applying different random augmentations to images, with the pretraining objective to distinguish image instances (Chen et al., 2020b; He et al., 2020; Chen et al., 2020d; Grill et al., 2020; Chen et al., $2 0 2 0 \mathrm { c }$ ; Caron et al., 2021). The quality of learned representation relies on manually chosen augmentations, such as random brightness, cropping, blurring, and others. Chen et al. (2020a) explored GPT-style (Radford et al., 2018) generative pretraining on images by autoregressively predicting pixels without incorporating knowledge of the 2D structure. Each pixel is represented as a 9-bit value created by clustering (R, G, B) pixel values, using k-means with $_ { \mathrm { k } = 5 1 2 }$ . Unfortunately, this color encoding does not scale to typical image resolutions as it entails very long sequences to represent the image (e.g., $2 2 4 \times 2 2 4$ resolution leads to 50,176 tokens per image), and this demands much more memory and computation for training, compared to language models. As a result, iGPT’s maximum resolution is $6 4 \times 6 4$ for image recognition at scale—which severely limits its representation capabilities.
+
+![](images/dcffd61f6310583296baec12b1619c518511d41255d7c41866c6e4b3aab1f278.jpg)  
+Figure 1: Overview of ViT-VQGAN (left) and Vector-quantized Image Modeling (right) for both image generation and image understanding.
+
+Remarkable image generation results have been achieved by pre-quantizing images into discrete latent variables and modeling them autoregressively, including VQVAE (Oord et al., 2017), DALLE (Ramesh et al., 2021) and VQGAN (Esser et al., 2021). In these approaches, a convolution neural network (CNN) is learned to auto-encode an image and a second stage CNN or Transformer is learned to model the density of encoded latent variables. These have been proved effective for image generation, but few studies have evaluated the learned representation in discriminative tasks (Ramesh et al., 2021; Esser et al., 2021).
+
+We explore an approach we refer to as Vector-quantized Image Modeling (VIM) and apply it to both image generation and image understanding tasks. VIM follows a two-stage approach:
+
+• Stage 1: Image Quantization. Given an image of resolution $2 5 6 \times 2 5 6$ , a VisionTransformer-based VQGAN encodes it into $3 2 \times 3 2$ discretized latent codes where the codebook size is 8192. We propose multiple improvements–from architecture to codebook learning–to VQGAN (Esser et al., 2021). The resulting ViT-VQGAN is more efficient and improves reconstruction fidelity in terms of pixel-wise reconstruction metrics, Inception Score (IS) and Frechet Inception Distance (FID). ViT-VQGAN is trained end-to-end on ´ image-only data with combined objective functions of logit-laplace loss, $\ell _ { 2 }$ loss, adversarial loss and perceptual loss (Johnson et al., 2016; Zhang et al., 2018). • Stage 2: Vector-quantized Image Modeling. We train a Transformer model to predict rasterized $3 2 \times 3 2 = 1 0 2 4$ image tokens autoregressively, where image tokens are encoded by a learned Stage 1 ViT-VQGAN. For unconditional image synthesis or unsupervised learning, we pretrain a decoder-only Transformer model to predict the next token. For class-conditioned image synthesis, a class-id token is prepended before the image tokens. To evaluate the quality of unsupervised learning, we average the intermediate Transformer features and learn a linear head to predict the logit of the classes (a.k.a., linear-probe).
+
+We show that one key component for improving both image generation and image understanding with VIM is to have a better image quantizer with respect to both computational efficiency and reconstruction quality. An efficient quantizer can speed up Stage 2 training, where random augmentations are applied first to an image, followed by the encoder of image quantizer to obtain the input tokens. Moreover, an image quantizer with better reconstruction quality can reduce information loss compared with the original image in pixel space, which is critical for image understanding tasks.
+
+The evaluations of our proposed ViT-VQGAN and VIM are studied with three aspects. (1) We evaluate the image quantizer based on reconstruction quality metrics including $\ell _ { 1 }$ distance, $\ell _ { 2 }$ distance, log-laplace distance, as well as Inception Score (IS) and Frechet Inception Distance (FID) ´ of reconstructed images. (2) We evaluate the capabilities of the learned quantizer for unconditional or class-conditioned image synthesis based on FID and IS, and compare with other methods. (3) We rely on linear-probe accuracy to evaluate representations with the common intuition that good features should linearly separate the classes of downstream tasks.
+
+# 2 RELATED WORK
+
+Image Synthesis. Image generation has received much attention with the progress of deep generative models, including Generative Adversarial Networks (GANs) (Goodfellow et al., 2014; Zhang et al., 2019b), Variational Autoencoders (VAEs) (Kingma & Welling, 2014; Vahdat & Kautz, 2020), Diffusion Models (Song & Ermon, 2019; Dhariwal & Nichol, 2021) and Autoregressive Models (van den Oord et al., 2016; Parmar et al., 2018). Unlike many autogressive methods which generate sequence directly in pixel space, VQVAE (van den Oord et al., 2017; Razavi et al., 2019) decomposes the image generation process into two stages: the first stage trains a vector quantized autoencoder with image reconstruction objective to convert an image into a shorter sequence of discrete tokens. Then the second stage learns an autoregressive model, e.g., PixelSNAIL (Chen et al., 2018), to model the underlying distribution of token sequences. Driven by the effectiveness of VQVAE and progress in sequence modeling (Vaswani et al., 2017; Devlin et al., 2019), many approaches follow the two-stage paradigm. DALL-E (Ramesh et al., 2021) improves token prediction in second stage by using Transformers (Vaswani et al., 2017), resulting in a strong text-to-image synthesis model. VQGAN (Esser et al., 2021) further uses adversarial loss and perceptual loss (Johnson et al., 2016; Zhang et al., 2018) to train a better autoencoder in the first stage to synthesize greater detail in images.
+
+Image Recognition with Generative Pretraining. Many image generation models (Goodfellow et al., 2014; Kingma & Welling, 2014; Radford et al., 2016; Donahue et al., 2017; Higgins et al., 2017) have been studied for their capabilities in representation learning. However, their performance is usually not superior to competing self-supervised approaches that solve auxiliary classification tasks (Noroozi & Favaro, 2016a; Gidaris et al., 2018a; van den Oord et al., 2018). BigBiGAN (Donahue & Simonyan, 2019a) first demonstrated that a generation-based model can match other self-supervised methods in representation learning on ImageNet. iGPT (Chen et al., 2020a) uses the autoregressive objective to learn a giant transformer that directly predicts pixel values, producing even more competitive results. Compared to iGPT, our method first tokenizes the original image into discrete image tokens and then trains a transformer to predict them. As a result, our approach obtains comparable results with smaller model and less data. Similar to our method in predicting image tokens, BEiT (Bao et al., 2021) follows pre-training scheme of BERT Devlin et al. (2019) by learning to recover randomly masked image tokens with a bidirectional transformer. Unlike BEiT, we explore vector-quantized image modeling for image generation in addition to image recognition.
+
+# 3 VECTOR-QUANTIZED IMAGES WITH VIT-VQGAN
+
+The Vector-quantized Variational AutoEncoder (VQVAE) (van den Oord et al., 2017) is a CNNbased auto-encoder whose latent space is a matrix of discrete learnable variables, trained end-to-end via straight-through estimation. Esser et al. (2021) introduce VQGAN, a model which improves upon VQVAE by introducing an adversarial loss produced by a discriminator. Below, we introduce further improvements to VQGAN that boost efficiency and enhance reconstruction quality.
+
+# 3.1 VQGAN WITH VISION TRANSFORMERS
+
+The core network architectures used by both VQVAE and VQGAN to encode and reconstruct images are CNNs. VQGAN introduces transformer-like elements in the form of non-local attention block (Zhang et al., 2019a), allowing it to capture distant interactions with fewer layers. We propose taking this approach one step further by replacing the CNN encoder and decoder with Vision Transformer (ViT) (Dosovitskiy et al., 2020), as shown in Figure 1. Given sufficient data (for which unlabeled image data is plentiful) we find that ViT-VQGAN is less constrained by the inductive priors imposed by convolutions. Furthermore, ViT-VQGAN yields better computational efficiency on accelerators, and produces higher quality reconstructions, as shown in Table 1.
+
+The encoder of ViT-VQGAN first maps $8 \times 8$ non-overlapping image patches into image tokens, followed by Transformer blocks, encoding a $2 5 6 \times 2 5 6$ resolution image into a $3 2 \times 3 2 { = } 1 0 2 4$ token sequence. The decoder performs the inverse operation, mapping each image token from latent variables back to $8 \times 8$ image patches and regrouping them into a $2 5 6 \times 2 5 6$ image (see Figure 1). At the output of transformer blocks, we apply a two-layer feed-forward network with a tanh activation layer in the middle. No activation is applied at the output of ViT-VQGAN encoder or decoder. We find that this simple approach yields high quality reconstructions without any noticeable grid artifacts.
+
+Table 1: ViT-VQGAN achieves better speed-quality trade-offs compared with CNN-VQGAN. This in turn further speeds up Stage 2 training. Throughputs are benchmarked with the same 128 CloudTPUv4 devices.   
+
+<table><tr><td>Architecture</td><td>Model Size (encoder-decoder)</td><td>Throughput 个 (imgs/sec)</td><td>l2loss (1e-2)</td><td>Logit-Laplace loss ↓</td><td>FID↓</td><td>IS↑</td></tr><tr><td>ViT-VQGAN</td><td>Small-Small</td><td>1520</td><td>3.34</td><td>-2.44</td><td>1.99</td><td>184.4</td></tr><tr><td>CNN-VQGAN</td><td>Channels ×1</td><td>946</td><td>3.81</td><td>-2.36</td><td>2.26</td><td>178.7</td></tr><tr><td>ViT-VQGAN</td><td>Base-Base</td><td>960</td><td>3.09</td><td>-2.54</td><td>1.55</td><td>190.2</td></tr><tr><td>CNN-VQGAN</td><td>Channels × 2</td><td>400</td><td>3.44</td><td>-2.46</td><td>1.91</td><td>183.4</td></tr><tr><td>ViT-VQGAN</td><td> Small-Large</td><td>384</td><td>2.88</td><td>-2.58</td><td>1.28</td><td>192.3</td></tr></table>
+
+# 3.2 CODEBOOK LEARNING
+
+Vanilla VQVAEs usually suffer from low codebook usage due to the poor initialization of the codebook. Therefore, during training a significant portion of codes are rarely used, or dead. The reduction in effective codebook size results in worse reconstructions in stage 1 quantizer training and poor diversity in stage 2 for image synthesis. As a result, VQGAN (Esser et al., 2021) relies on top$k$ and top- $p$ (nucleus) sampling heuristics (Holtzman et al., 2020) with a default codebook size of 1024 to obtain best results for image synthesis. We propose two improvements that can significantly encourage the codebook usage even with a larger codebook size of 8192. During image synthesis, we perform simple sampling with temperature of 1.0 without top- $k$ and top- $p$ heuristics.
+
+The training objective of vector-quantization is defined as follows:
+
+$$
+L _ { \mathrm { V Q } } = \| \mathbf { s g } [ z _ { e } ( x ) ] - e \| _ { 2 } ^ { 2 } + \beta \| z _ { e } ( x ) - \mathbf { s g } [ e ] \| _ { 2 } ^ { 2 } .
+$$
+
+Here, $\operatorname { s g } ( x ) \equiv x$ , $\begin{array} { r } { { \frac { \mathrm { d } } { \mathrm { d } x } } \mathrm { s g } ( x ) \equiv 0 } \end{array}$ is the stop-gradient operator, $\beta$ is a commitment loss hyperparameter set to 0.25 in all our experiments, and $e$ is the codebook vector. The quantized codebook index is determined by looking up the codebook vector closest to the input features $z _ { e } ( x )$ in terms of the Euclidean distance, yielding $i = \mathrm { a r g m i n } _ { j } \| z _ { e } ( x ) - e _ { j } \| _ { 2 } ^ { 2 }$ .
+
+Factorized codes. We introduce a linear projection from the output of the encoder to a lowdimensional latent variable space for code index lookup (e.g., reduced from a 768-d vector to a 32-d or 8-d vector per code) and find it has an immediate boost of codebook usage. The factorization can be viewed as decoupling code lookup and code embedding: we lookup the the closest variable encoded from input on a lower-dimensional lookup space and then project the matched latent code to the high-dimensional embedding space. Our experiments show reducing dimension of lookup space from 256-d to 32-d consistently improves reconstruction quality. A detailed illustration is provided in the supplementary materials.
+
+$\ell _ { 2 }$ -normalized codes. We also apply $\ell _ { 2 }$ normalization on the encoded latent variables $z _ { e } ( x )$ and codebook latent variables $e$ . The codebook variables are initialized from a normal distribution. By mapping all latent variables on a sphere, the Euclidean distance of $\ell _ { 2 }$ -normalized latent variables $| | \ell _ { 2 } ( \bar { z } _ { e } ( \bar { x } ) ) - \ell _ { 2 } ( e _ { j } ) | | _ { 2 } ^ { 2 }$ evolves to the cosine similarity of two vectors between $z _ { e } ( x )$ and $e$ , further improving training stability and reconstruction quality shown in our experiments.
+
+# 3.3 VIT-VQGAN TRAINING LOSSES
+
+We use a combination of logit-laplace loss, $\ell _ { 2 }$ loss, perceptual loss (Johnson et al., 2016; Zhang et al., 2018) based on VGG network (Simonyan & Zisserman, 2014) and GAN loss with architecture of StyleGAN discriminator (Karras et al., 2020). Loss balancing weights are configured with a hyper-parameter sweep to optimize image reconstruction quality, codebook usage, FID and Inception Score. After the sweep, we apply the same set of hyper-parameters of training losses to all datasets including CelebA-HQ, FFHQ, and ImageNet. Logit-Laplace loss can be viewed as normalized $\ell _ { 1 }$ loss which assumes the noise at the pixel level is laplace-distributed while $\ell _ { 2 }$ loss assumes the noise is of a Gaussian distribution. We find logit-laplace loss contributes to codebook usage while $\ell _ { 2 }$ loss and perceptual loss significantly contribute to FID. The final loss combination we used by default is $L = L _ { \mathrm { { V Q } } } + 0 . 1 L _ { \mathrm { { A d v } } } + 0 . 1 L _ { \mathrm { { P e r c e p t u a l } } } + 0 . 1 L _ { \mathrm { { L o g i t } \mathrm { { - } \mathrm { { l a p l a c e } } } } } + 1 . 0 L _ { \mathrm { { 2 } } } .$ .
+
+Table 2: Transformer architectures of Stage 1 ViT-VQGAN and Stage 2 VIM.   
+
+<table><tr><td>Model</td><td>Size</td><td>#Params</td><td>#Blocks</td><td>#Heads</td><td>Model Dim</td><td>Hidden Dim</td><td>Dropout</td><td>#Tokens</td></tr><tr><td>ViT-VQGAN</td><td>Small</td><td>32M</td><td>8</td><td>8</td><td>512</td><td>2048</td><td>0.0</td><td>1024</td></tr><tr><td>ViT-VQGAN</td><td>Base</td><td>91M</td><td>12</td><td>12</td><td>768</td><td>3072</td><td>0.0</td><td>1024</td></tr><tr><td>ViT-VQGAN</td><td>Large</td><td>599M</td><td>32</td><td>16</td><td>1280</td><td>5120</td><td>0.0</td><td>1024</td></tr><tr><td>VIM</td><td>Base</td><td>650M</td><td>24</td><td>16</td><td>1536</td><td>6144</td><td>0.1</td><td>1024</td></tr><tr><td>VIM</td><td>Large</td><td>1697M</td><td>36</td><td>32</td><td>2048</td><td>8192</td><td>0.1</td><td>1024</td></tr></table>
+
+One caveat on the VGG-based perceptual loss is that the VGG network is pretrained with supervised classification loss, so the supervision might leak into Stage 2 for linear-probe accuracy measurement. Thus, for all of our reported unsupervised learning results, we exclude the perceptual loss during ViT-VQGAN training. For all unconditional and class-conditioned image synthesis, we use ViTVQGAN quantizers trained with perceptual loss, as it leads to higher-fidelity reconstructions.
+
+# 4 VECTOR-QUANTIZED IMAGE MODELING
+
+With a learned ViT-VQGAN, images are encoded into discrete latent code ids flattened in the raster order, similar to Image GPT (Chen et al., 2020a). A decoder-only Transformer model is used to model the density of image data $P ( x )$ autoregressively as
+
+$$
+P ( x ) = \prod _ { i = 1 } ^ { n } P ( x _ { i } | x _ { 1 } , x _ { 2 } , . . . , x _ { i - 1 } ; \theta ) ,
+$$
+
+where $\theta$ is learnable weights. The training objective is to minimize the negative log-likelihood of the data $L = \mathbb { E } _ { x \in X } [ - l o g P ( x ) ]$ .
+
+Table 2 summarizes the architecture configurations for the Transformers. We first embed discrete image token ids into a learnable embedding space at each position, with an additive learnable 2D positional embedding. Both embedding dimensions are the same as model dimension. We apply a stack of Transformer blocks to the inputs with causal attention over the entire sequence. A dropout ratio of 0.1 is used in all residual, activation and attention outputs. At the final layer of all Transformer blocks, we apply an additional layer normalization.
+
+# 4.1 IMAGE SYNTHESIS
+
+With a pretrained generative Transformer model, unconditional image generation is achieved by simply sampling token-by-token from the output softmax distribution. All samples used for both qualitative and quantitative results are obtained without temperature reduction. The sampled tokens are then fed into the decoder of ViT-VQGAN to decode output images. Our default Stage 1 ViTVQGAN encodes input images of resolution $2 5 6 \times 2 5 6$ into $3 2 \times 3 2$ latent codes with a codebook size 8192, while Stage 2 Transformer takes the flattened image tokens with total a length of 1024.
+
+Class-conditioned ImageNet generation is also a widely used benchmark for measuring capabiltiy of models for image synthesis. We extend the unconditional generation to class-conditioned generation by prepending a class-id token before the image tokens. Separate embedding layers are learned from scratch for class-id token and image tokens, with the embedding dimension the same as the Transformer model dimension. During sampling, a class-id token is provided at the first position to decode the remaining image tokens autoregressively.
+
+# 4.2 UNSUPERVISED LEARNING
+
+For the image understanding task, we feed all image tokens of the input into a pretrained Transformer, and get a sequence of 1024 token features. Similar to Image GPT (Chen et al., 2020a), we take a layer output at a specific block $l$ over total blocks $L$ , average over the sequence of token features (frozen) and insert a softmax layer (learnable) projecting averaged feature to class logits. We only take one specific Transformer block output instead of concatenating different block outputs as in iGPT (Chen et al., 2020a). We find that most discriminating feature for the linear-probe is typically near the middle of all Transformer blocks.
+
+Table 3: Frechet Inception Distance (FID) between reconstructed validation split and original val- ´ idation split on ImageNet, CelebA-HQ and FFHQ. ∗ denotes models trained with Gumbel-Softmax reparameterization as in Ramesh et al. (2021). ∗∗ denotes models trained with multi-scale hierarchical codebook as in Razavi et al. (2019).   
+
+<table><tr><td>Model</td><td>Dataset</td><td>Latent Size</td><td>dimZ</td><td>FID on Validation</td></tr><tr><td>DALL-E dVAE</td><td>Web data</td><td>32 ×32</td><td>8192</td><td>32.00</td></tr><tr><td>VQGAN</td><td>ImageNet</td><td>16 ×16</td><td>1024</td><td>7.94</td></tr><tr><td>VQGAN</td><td>ImageNet</td><td>16 ×16</td><td>16384</td><td>4.98</td></tr><tr><td>VQGAN*</td><td>ImageNet</td><td>32 × 32</td><td>8192</td><td>1.49</td></tr><tr><td>VQGAN**</td><td>ImageNet</td><td>64×64&amp;32×32</td><td>512</td><td>1.45</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>ImageNet</td><td>32×32</td><td>8192</td><td>1.28</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>CelebA-HQ</td><td>32 ×32</td><td>8192</td><td>4.66</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>FFHQ</td><td>32 ×32</td><td>8192</td><td>3.13</td></tr></table>
+
+# 5 EXPERIMENTS
+
+# 5.1 IMAGE QUANTIZATION
+
+We train the proposed ViT-VQGAN on three datasets separately, CelebA-HQ (Karras et al., 2019), FFHQ (Karras et al., 2019), and ImageNet (Krizhevsky et al., 2012). For CelebA-HQ and FFHQ, we follow the default train and validation split as VQGAN (Esser et al., 2021).1 For Stage 1 image quantization, three different architecture sizes are experimented, Small, Base and Large for either encoder or decoder, as defined in Table 2. The smallest ViT-VQGAN-SS has a Small-size encoder and Small-size decoder, while ViT-VQGAN-BB has a Base-size encoder and Base-size decoder. The largest ViT-VQGAN-SL has an asymmetric Small-size encoder and Large-size decoder, with the motivation that Stage 2 training only requires forward propagation of the encoder of ViT-VQGAN (in inference/decoding for image synthesis, the decoder of ViT-VQGAN is still required to decode images from codes predicted during Stage 2).
+
+We train all ViT-VQGAN models with a training batch size of 256 distributed across 128 CloudTPUv4 for a total 500,000 training steps. For both ViT-VQGAN and StyleGAN discriminator, Adam optimizer (Kingma & Ba, 2014) is used with $\beta _ { 1 } = 0 . 9$ and $\beta _ { 2 } = 0 . 9 9$ with the learning rate linearly warming up to a peak value of $1 \times 1 0 ^ { - 4 }$ over 50,000 steps and then decaying to $5 \times \mathrm { \overline { { 1 } } 0 ^ { - 5 } }$ over the remaining 450,000 steps with a cosine schedule. We use a decoupled weight decay (Loshchilov & Hutter, 2017) of $1 \dot { \times } 1 0 ^ { - 4 }$ for both ViT-VQGAN and StyleGAN discriminator. All models are trained with an input image resolution $2 5 6 \times 2 5 6$ on CloudTPUv4.
+
+Table 3 shows FID between reconstructed images and original images in the validation split on ImageNet, CelebA-HQ and FFHQ datasets. Without multi-scale hierarchical codebook or gumbelsoftmax, ViT-VQGAN is able to achieve better FID with a large codebook size of 8192 compared with vanilla VQGAN.
+
+Table 4 provides extensive ablations on the proposed modifications, with empirical results on mean $\ell _ { 1 }$ distance, $\ell _ { 2 }$ distance, logit-laplace distance, Inception Score and FID on ImageNet. Among different model sizes, ViT-VQGAN-SS (small-encoder, small-decoder) performs worse than ViTVQGAN-BB (base-encoder, base-decoder) and ViT-VQGAN-SL (small-encoder, large-decoder), but achieves much better throughput. The CNN-based VQGAN architecture is worse in both quality and throughput compared with ViT-based VQGAN. The StyleGAN-based discriminator (Karras et al., 2019) is more stable and yields better reconstruction quality than PatchGAN (Isola et al., 2017)
+
+Table 4: Ablation study on ViT-VQGAN. The codebook usage is calculated as the percentage of used codes given a batch of 256 test images averaged over the entire test set.   
+
+<table><tr><td></td><td>groe ereerg rrr repeeeg</td><td>JAeeettect</td><td>Jirrriisi</td><td></td><td>pineg Lir aeaer</td><td>↑（2-0）</td><td>↑（-01）</td><td> 9der-18g</td><td></td><td>→ D</td><td>eeesn googopeg</td><td>↑ndysnou</td></tr><tr><td>Ablation on</td><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>32</td><td>√</td><td>3.06</td><td>3.09 -2.54</td><td>190.2</td><td></td><td>1.55 96%</td><td>960</td></tr><tr><td>Model Size</td><td>Small Small</td><td>Small</td><td>ViT ViT</td><td>StyleGAN</td><td>32</td><td>√ 3.22</td><td>3.34</td><td>-2.44</td><td>184.4</td><td>1.99</td><td>95%</td><td>1520</td></tr><tr><td rowspan="2"></td><td rowspan="2"></td><td rowspan="2">Large</td><td rowspan="2"></td><td rowspan="2">StyleGAN</td><td rowspan="2">32</td><td rowspan="2">√ 2.93</td><td rowspan="2">2.88</td><td rowspan="2">-2.58</td><td rowspan="2">192.3</td><td rowspan="2">1.28</td><td rowspan="2">95%</td><td rowspan="2">384</td></tr><tr><td>√ 3.45 2.82</td></tr><tr><td>Architecture</td><td>1 Base</td><td>1 Base</td><td>CNN ViT</td><td>StyleGAN PatchGAN</td><td>32 32</td><td>√</td><td>3.81 2.58</td><td>-2.36 -2.62</td><td>178.7 165.6</td><td>2.26 3.88</td><td>63% 89%</td><td>946 1227</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>4.28</td><td>-2.38</td><td>160.1</td><td>3.68</td><td>4%</td><td></td></tr><tr><td rowspan="8">Codebook Learning</td><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>256 128</td><td>1</td><td>3.60 3.41</td><td></td><td></td><td></td><td></td><td>954</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td></td><td></td><td>3.93</td><td>-2.44</td><td>173.9</td><td>2.77</td><td>14%</td><td>960</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>64</td><td></td><td>3.18 3.37</td><td>-2.49</td><td>179.5 191.2</td><td>2.50</td><td>37%</td><td>960</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>16</td><td></td><td>3.00</td><td>2.96 -2.54</td><td></td><td>1.50</td><td>95%</td><td>960</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>8</td><td></td><td>2.98</td><td>2.92 -2.55</td><td>189.5</td><td>1.52</td><td>96%</td><td>960</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>4</td><td></td><td>3.55</td><td>4.18 -2.37</td><td>143.8</td><td>3.68 5.44</td><td>96%</td><td>960</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>32</td><td>&gt;&gt;&gt;&gt;&gt;x</td><td>4.13</td><td>5.41</td><td>-2.20</td><td>123.6</td><td>2%</td><td>960</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 5: FID comparison with unconditional image synthesis on CelebA-HQ and FFHQ.   
+
+<table><tr><td colspan="2">CelebA-HQ 256 × 256</td><td colspan="2">FFHQ 256× 256</td></tr><tr><td>Method</td><td>FID↓</td><td>Method</td><td>FID↓</td></tr><tr><td>GLOW (Kingma &amp; Dhariwal, 2018)</td><td>69.0</td><td>VDVAE (t = 0.7) (Child,2021)</td><td>38.8</td></tr><tr><td>NVAE (Vahdat &amp; Kautz,2020)</td><td>40.3</td><td>VDVAE (t = 1.0)</td><td>33.5</td></tr><tr><td>PIONEER (Heljakka etal.,2018)</td><td>25.3</td><td>VDVAE (t = 0.8)</td><td>29.8</td></tr><tr><td>NCPVAE (Aneja et al.,2020)</td><td>24.8</td><td>VDVAE (t = 0.9)</td><td>28.5</td></tr><tr><td>VAEBM (Xiao et al., 2021)</td><td>20.4</td><td>VQGAN+P.SNAIL</td><td>21.9</td></tr><tr><td>Style ALAE (Pidhorskyi et al., 2020)</td><td>19.2</td><td>BigGAN</td><td>12.4</td></tr><tr><td>DC-VAE (Parmar et al., 2021)</td><td>15.8</td><td>U-Net GAN (Schonfeld et al., 2020)</td><td>10.9</td></tr><tr><td>PGGAN(Karras et al.,2018)</td><td>8.0</td><td>StyleGAN2 (Karras etal., 2020)</td><td>3.8</td></tr><tr><td>VQGAN (w/ top-k sampling)</td><td>10.2</td><td>VQGAN (w/ top-k sampling)</td><td>9.6</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>7.0</td><td>ViT-VQGAN (Ours)</td><td>5.3</td></tr></table>
+
+(which was used for VQGAN). For codebook learning, factorized codes with low-dimensional latent variables consistently achieve better reconstruction quality when the latent dimension is reduced from 256 to 16 or 8. Moreover, removing $\ell _ { 2 }$ -normalization leads to much worse results.
+
+# 5.2 IMAGE SYNTHESIS
+
+On top of the pre-learned ViT-VQGAN image quantizer, we train stage 2 transformer models for unconditional and class-conditioned image synthesis and compare with previous work. We use a default model size of ViT-VQGAN-SS (small-encoder, small-decoder) for stage 1 and VIM-Large for stage 2 (model architectures are listed in Table 2), as we find it beneficial to put more computation in stage 2 while keeping stage 1 transformers lightweight. We also present a model size ablation study and comparison with VQGAN in the Appendix. Models are trained with a global training batch size of 1024 for a total of 450,000 training steps. We use Adam optimizer (Kingma & Ba, 2014) with $\beta _ { 1 } = 0 . 9$ and $\beta _ { 2 } = 0 . 9 6$ with the learning rate linearly warming up to a peak constant value of $4 . 5 \times 1 0 ^ { - 4 }$ over the first 5000 steps and then exponentially decaying to $1 \times \mathrm { 1 0 ^ { - 5 } }$ starting from 80,000 steps. To save memory, we use a factorized version of Adam, Adafactor (Shazeer & Stern, 2018), with the first moment quantized into Int8 and factorized second moments. No other techniques like mixed-precision training, model sharding, or gradient compression is used. All models are trained with an input image resolution $2 5 6 \times 2 5 6$ on CloudTPUv4.
+
+<table><tr><td>Model</td><td>Acceptance Rate</td><td>FID</td><td>IS</td></tr><tr><td>Validation data</td><td>1.0</td><td>1.62</td><td>235.0</td></tr><tr><td>DCTransformer (Nash et al.,2021)</td><td>1.0</td><td>36.5</td><td>n/a</td></tr><tr><td>BigGAN (Brock et al.,2019)</td><td>1.0</td><td>7.53</td><td>168.6</td></tr><tr><td>BigGAN-deep</td><td>1.0</td><td>6.84</td><td>203.6</td></tr><tr><td>IDDPM (Nichol &amp; Dhariwal,2021)</td><td>1.0</td><td>12.3</td><td>n/a</td></tr><tr><td>ADM-G, no guid. (Dhariwal &amp; Nichol, 2021)</td><td>1.0</td><td>10.94</td><td>101.0</td></tr><tr><td>ADM-G, 1.0 guid.</td><td>1.0</td><td>4.59</td><td>186.7</td></tr><tr><td>VQVAE-2 (Razavi et al.,2019)</td><td>1.0</td><td>~31</td><td>~45</td></tr><tr><td>VQGAN (Esser et al.,2021)</td><td>1.0</td><td>17.04</td><td>70.6</td></tr><tr><td>VQGAN</td><td>0.5</td><td>10.26</td><td>125.5</td></tr><tr><td>VQGAN</td><td>0.25</td><td>7.35</td><td>188.6</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>1.0</td><td>4.17</td><td>175.1</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>0.5</td><td>3.04</td><td>227.4</td></tr></table>
+
+Table 6: FID comparison for class-conditional image synthesis on ImageNet with resolution $2 5 6 \times$ 256. Acceptance rate shows results based on ResNet-101 classifier-based rejection sampling.
+
+![](images/2d4bd8b401b7977b3c4ee742b5204d1709b675afa13b9a6ea9cac79c8b6d5b86.jpg)  
+Figure 2: Uncurated set of samples from class-conditioned image generation trained on ImageNet. Top row shows sampled class ids while bottom row shows fine-grained dog species from class id 184 to 189. More samples are shown in Appendix.
+
+Our main results on unconditional image synthesis on CelebA-HQ and FFHQ are summarized in Table 5. Without top- $k$ and top- $p$ (nucleus) sampling heuristics, we achieve FID of 7.0 on CelebAHQ and 5.3 on FFHQ, significantly better than VQGAN (Esser et al., 2021). Table 6 shows classconditioned image synthesis models on ImageNet, following Section 4.1. Based on ViT-VQGANSS, we achieve IS of 175.1 and FID of 4.17, improving over the IS of 70.6 and FID of 17.04 with vanilla VQGAN. When applied with classifier-based rejection sampling, the ViT-VQGAN based model further achieves best FID of 3.04 and best Inception Score of 321.7. Qualitative results are sampled and shown in Figure 2 (see the Appendix for more).
+
+# 5.3 UNSUPERVISED LEARNING
+
+After the generative pretraining to autoregressively model the density of ViT-VQGAN quantized image tokens, we evaluate the learned representation under the common linear protocol on ImageNet classification. We follow the same training hyper-parameters as the unconditional image synthesis models on ImageNet, and use ViT-VQGAN-SS image quantizer for better training throughput. As discussed in Section 3.3, the ViT-VQGAN-SS image quantizer is trained without perceptual loss for unsupervised learning (perceptual loss is based on a supervised VGG network trained on ImageNet, which may make comparison unfair). We apply an average pooling over the token features at a specific transformer block $l$ from totally $L$ blocks. Similar to findings reported in iGPT (Chen et al., 2020a), the representations from the middle transformer blocks has better linear-probe accuracy (more study can be found in Appendix). Specifically, we use the Transformer block of index 15 (36 blocks in total) for VIM-Large and index 10 (24 blocks in total) for VIM-Base (architecture configurations are listed in Table 2).
+
+Table 7: Linear-probe accuracy with different unsupervised learning methods on ImageNet. DALLE dVAE (Ramesh et al., 2021) image quantizer is trained with extra image data. VIM-Large is trained without dropout in transformers.   
+
+<table><tr><td>Method</td><td></td><td>#Tokens</td><td>Features</td><td>Params</td><td>Top-1 ↑</td></tr><tr><td rowspan="15">Brirrniir irnrriiiirig</td><td>Jigsaw (Noroozi &amp; Favaro,2016b)</td><td></td><td>4096</td><td>94M</td><td>44.6</td></tr><tr><td>RelativePosition (Doersch et al., 2015)</td><td></td><td>4096</td><td>94M</td><td>51.4</td></tr><tr><td>Rotation (Gidaris et al.,2018b)</td><td></td><td>8192</td><td>86M</td><td>55.4</td></tr><tr><td>AMDIM (Bachman et al.,2019)</td><td></td><td>8192</td><td>626M</td><td>68.1</td></tr><tr><td>CPC v2 (Henaff, 2020)</td><td></td><td>4096</td><td>303M</td><td>71.5</td></tr><tr><td>MoCo (He et al.,2020)</td><td></td><td>8192</td><td>375M</td><td>68.6</td></tr><tr><td>SimCLR (Chen et al., 2020b)</td><td></td><td>8192</td><td>375M</td><td>76.5</td></tr><tr><td>SwAV (Caron et al., 2020)</td><td></td><td>2048</td><td>93M</td><td>75.3</td></tr><tr><td>DINO (Caron et al., 2021)</td><td></td><td>2048</td><td>85M</td><td>75.3</td></tr><tr><td>BYOL (Grill et al., 2020)</td><td></td><td>8192</td><td>375M</td><td>78.6</td></tr><tr><td rowspan="4">Greeeed rierieeg</td><td>BiGAN (Donahue et al., 2016)</td><td></td><td>512</td><td>138M</td><td>31.0</td></tr><tr><td>BigBiGAN (Donahue &amp; Simonyan,2019b)</td><td></td><td>4096</td><td>86M</td><td>56.6</td></tr><tr><td>BigBiGAN</td><td>=</td><td>16384</td><td>344M</td><td>61.3</td></tr><tr><td>iGPT-L (Chen et al., 2020a)</td><td>32× 32</td><td>1536</td><td>1362M</td><td>60.3</td></tr><tr><td>iGPT-L</td><td></td><td>48×48</td><td>1536</td><td>1362M</td><td>65.2</td></tr><tr><td>iGPT-XL (extra data)</td><td></td><td>64×64</td><td>3072</td><td>6801M</td><td>68.7</td></tr><tr><td></td><td> iGPT-XL (extra data, feature ensemble)</td><td>64 × 64</td><td>5×3072</td><td>6801M</td><td>72.0</td></tr><tr><td></td><td>VIM-Base (Ours)</td><td>32 × 32</td><td>1024</td><td>650M</td><td>65.1</td></tr><tr><td></td><td>VIM-Large (Ours)</td><td>32 × 32</td><td>2048</td><td>1697M</td><td>73.2</td></tr><tr><td></td><td>VIM-Base + DALL-E dVAE quantizer</td><td>32 ×32</td><td>1024</td><td>650M</td><td>63.8 (-1.3)</td></tr><tr><td></td><td>VIM-Base + CNN-VQGAN quantizer</td><td>32×32</td><td>1024</td><td>650M</td><td>61.8 (-3.3)</td></tr></table>
+
+Table 7 shows the comparisons among different approaches divided into two categories: discriminative pretraining methods to distinguish among cropped or augmented image patches; and generative pretraining methods to generate image pixels or patches. The linear-probe accuracy of our proposed VIM with ViT-VQGAN are superior to other generative pretraining approaches like iGPT, and competitive with discriminative pretraining methods like BYOL (Grill et al., 2020) and DINO (Caron et al., 2021). Specifically, ImageNet-pretrained VIM-L significantly improves over iGPT-L, increasing linear-probe accuracy from $6 0 . 3 \%$ to $7 3 . 2 \%$ for a similar model size. VIM-L also outperforms iGPT-XL, which is larger and trained with extra web image data. Moreover, we also compare different stage 1 quantizers including CNN-based VQGAN and pretrained DALL-E dVAE (trained with extra web-scale image data)2 in Table 7; these results are all worse than ViT-VQGAN quantizer, suggesting the importance of the multiple changes defined in Section 3.
+
+# 6 ETHICS
+
+Tasks involving generation raise a number of issues that should be considered, such as possible biases in underlying models and data—especially with respect to capabilities for people with different demographic backgrounds. The three datasets used in this paper–ImageNet, CelebA-HQ, and FFHQ–are all widely used in the literature, but it is worthwhile highlighting their unique natures and recent scholarship around them.
+
+The FFHQ dataset3 contains 70,000 images collected from Flickr, all of which have licenses appropriate for sharing, and the data maintainers provide means for individuals to opt-out of inclusion in the dataset. FFHQ was specifically collected to cover a broad range of demographics with respect to faces of people. This is confirmed in our models’ generated examples, which cover a broad range of perceived ages, genders and ethnicities. Nevertheless, Balakrishnan et al. (2020) provide an extensive analysis of multiple forms of bias in datasets (including CelebA-HQ and FFHQ) and algorithms for face generation; not only do they find imbalances in skin tone in FFHQ, but also correlations between multiple attributes such as skin tone and hair length. Based on this and other factors such as privacy and copyright, they argue that synthetically-created face datasets, for which multiple attributes can be controlled, is an important direction of investment and general inquiry.
+
+The CelebA-HQ dataset covers celebrities, which brings a consequent bias toward images of attractive people who are mostly in age range of twenty to forty years old. Esser et al. (2020) discusses these biases in details, and they furthermore project images from the FFHQ dataset onto CelebAHQ: the main effect of which is to produce images of younger people with features conforming more to norms of celebrities popular in the United States of America. Our model’s generations appear to have a similar bias as derived from training on CelebA-HQ. Neverethless, they do show broad coverage of different perceived genders and ethnicities, but with age skewed to the 20-40 year old range.
+
+ImageNet is, of course, quite pervasive in computer vision. In this paper, we learn to generate images given ImageNet class labels; these labels mostly concern animals, plants and things. People are sometimes generated when conditioning on classes such as sunglasses since the training data images contain people wearing sunglasses, but the generated images contain few depictions of people overall. Nevertheless, it is important to recognize that ImageNet itself was created with biases in terms of image selection and label annotation as a result of its process of creation (Denton et al., 2021). Given this, results present on ImageNet cover a significant, but nonetheless biased, sample of the kinds of scenes and objects one might encounter across the entire world.
+
+There are also potential problematic aspects of image generation models, as demonstrated with biases found in the PULSE model (Menon et al., 2020) (see Section 6) and in model correlations with human biases found in social psychology (Steed & Caliskan, 2021), as well as with possible uses of such models to create fake media (Westerlund, 2019).
+
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+
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+# A LINEAR-PROBE ON IMAGENET
+
+We show linear-probe accuracy from different layers in a pretrained VIM-Base Transformer model in Figure 3. Similar to iGPT (Chen et al., 2020a), we also find the last few layers may not be the best layers for discriminative features, as the generative pretraining objective is to recover the original image tokens. The linearprobe accuracy increases quickly from the first transformer output, reaches its peak at middle layers, and finally decreases for the last few blocks. Interestingly, we find for both VIMBase and VIM-Large, the middle transformer block has the near-best result. This observation connects the transformer model to an encoderdecoder model where the encoder encodes image tokens into high-level semantic features and the decoder takes feature information to gener
+
+![](images/1d84e809d5636b23d7ceb5e8d59293c393c6e1ce9679abdcb3fefcdce313b433.jpg)  
+Figure 3: Linear-probe accuracy from different layers in a pretrained VIM-Base Transformer model.
+
+ate output image tokens. We leave for future study regrading the interpretability of pretrained VIM models.
+
+# B MODEL SIZES OF CLASS-CONDITIONED IMAGENET SYNTHESIS
+
+We also present results of different sizes of Stage 2 Transformers for class-conditioned image synthesis and compare with VQGAN (Esser et al., 2021)4 summarized in Table 8.   
+
+<table><tr><td>Model</td><td>Stage-2 Transformer Size</td><td>#Tokens</td><td>FID</td><td>IS</td></tr><tr><td>Validation data</td><td>/</td><td>=</td><td>1.62</td><td>235.0</td></tr><tr><td>VQGAN (Esser et al.,2021)</td><td>1.4B</td><td>16 ×16</td><td>17.04</td><td>70.6</td></tr><tr><td>ViT-VQGAN+VIM-Base</td><td>650M</td><td>16 ×16</td><td>11.20</td><td>97.2</td></tr><tr><td>ViT-VQGAN + VIM-Large</td><td>1.7B</td><td>16 ×16</td><td>5.3</td><td>149.9</td></tr><tr><td>ViT-VQGAN + VIM-Base</td><td>650M</td><td>32 × 32</td><td>8.81</td><td>110.8</td></tr><tr><td>ViT-VQGAN + VIM-Large</td><td>1.7B</td><td>32 ×32</td><td>4.17</td><td>175.1</td></tr></table>
+
+Table 8: FID comparison for class-conditional image synthesis on ImageNet with different Transformer sizes in Stage 2. Results are reported without rejection sampling.
+
+# C IMPLEMENTATION DETAILS OF FACTORIZED CODEBOOK
+
+As we introduced in Section 3.2, we use a linear projection to reduce the encoded embedding to a low-dimensional variable space for code lookup. A detailed illustration is shown in Figure 4.
+
+![](images/808468d18686ea4c5749fcf0e24a77e6508b7620f5bbe5bc1144fb5f4eb09778.jpg)  
+Figure 4: Illustration of factorized codes and codebook details.
+
+![](images/0cf2583848d23fed0a25567a05ff4f7a7593c2be749af32a27ee19d63132e4bd.jpg)  
+Figure 5: Uncurated set of samples from class-conditioned generation trained on ImageNet.
+
+Scale
+
+![](images/2260824a7c774887327e7ed39a63230563e7ec24f408abf7f020883a92447b17.jpg)  
+Figure 6: Uncurated set of samples from class-conditioned generation trained on ImageNet.

parse/dev/pfNyExj7z2/pfNyExj7z2_content_list.json

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+    {
+        "type": "text",
+        "text": "VECTOR-QUANTIZED IMAGE MODELING WITH IMPROVED VQGAN ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jiahui Yu Xin Li Jing Yu Koh Han Zhang Ruoming Pang James Qin Alexander Ku Yuanzhong Xu Jason Baldridge Yonghui Wu ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Google Research ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "jiahuiyu@google.com ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            246,
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+            261
+        ],
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+    },
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+        "text": "Pretraining language models with next-token prediction on massive text corpora has delivered phenomenal zero-shot, few-shot, transfer learning and multi-tasking capabilities on both generative and discriminative language tasks. Motivated by this success, we explore a Vector-quantized Image Modeling (VIM) approach that involves pretraining a Transformer to predict rasterized image tokens autoregressively. The discrete image tokens are encoded from a learned Vision-Transformerbased VQGAN (ViT-VQGAN). We first propose multiple improvements over vanilla VQGAN from architecture to codebook learning, yielding better efficiency and reconstruction fidelity. The improved ViT-VQGAN further improves vectorquantized image modeling tasks, including unconditional, class-conditioned image generation and unsupervised representation learning. When trained on ImageNet at $2 5 6 \\times 2 5 6$ resolution, we achieve Inception Score (IS) of 175.1 and Frechet Inception Distance (FID) of 4.17, a dramatic improvement over the vanilla ´ VQGAN, which obtains 70.6 and 17.04 for IS and FID, respectively. Based on ViT-VQGAN and unsupervised pretraining, we further evaluate the pretrained Transformer by averaging intermediate features, similar to Image GPT (iGPT). This ImageNet-pretrained VIM-L significantly beats iGPT-L on linear-probe accuracy from $6 0 . 3 \\%$ to $7 3 . 2 \\%$ for a similar model size. ViM-L also outperforms iGPT-XL which is trained with extra web image data and larger model size. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Natural language processing (NLP) has recently experienced dramatic improvements from learning general-purpose representations by pretraining language models on unlabeled text corpora. This strategy has produced large performance gains for a wide range of natural language generation (NLG) and natural language understanding (NLU) tasks (Dai & Le, 2015; Radford et al., 2018; 2019; Brown et al., 2020). Conceptually, generative pretraining models the data density $P ( X )$ in a tractable way, with the hope of also helping discriminative tasks of $P ( { Y \\vert } X )$ (Lasserre et al., 2006); importantly, there are no limitations on whether the signals are from the language domain or others, such as vision. ",
+        "bbox": [
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+        "text": "In computer vision, in contrast, most recent unsupervised or self-supervised learning research focuses on applying different random augmentations to images, with the pretraining objective to distinguish image instances (Chen et al., 2020b; He et al., 2020; Chen et al., 2020d; Grill et al., 2020; Chen et al., $2 0 2 0 \\mathrm { c }$ ; Caron et al., 2021). The quality of learned representation relies on manually chosen augmentations, such as random brightness, cropping, blurring, and others. Chen et al. (2020a) explored GPT-style (Radford et al., 2018) generative pretraining on images by autoregressively predicting pixels without incorporating knowledge of the 2D structure. Each pixel is represented as a 9-bit value created by clustering (R, G, B) pixel values, using k-means with $_ { \\mathrm { k } = 5 1 2 }$ . Unfortunately, this color encoding does not scale to typical image resolutions as it entails very long sequences to represent the image (e.g., $2 2 4 \\times 2 2 4$ resolution leads to 50,176 tokens per image), and this demands much more memory and computation for training, compared to language models. As a result, iGPT’s maximum resolution is $6 4 \\times 6 4$ for image recognition at scale—which severely limits its representation capabilities. ",
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+    {
+        "type": "image",
+        "img_path": "images/dcffd61f6310583296baec12b1619c518511d41255d7c41866c6e4b3aab1f278.jpg",
+        "image_caption": [
+            "Figure 1: Overview of ViT-VQGAN (left) and Vector-quantized Image Modeling (right) for both image generation and image understanding. "
+        ],
+        "image_footnote": [],
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+    {
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+        "text": "Remarkable image generation results have been achieved by pre-quantizing images into discrete latent variables and modeling them autoregressively, including VQVAE (Oord et al., 2017), DALLE (Ramesh et al., 2021) and VQGAN (Esser et al., 2021). In these approaches, a convolution neural network (CNN) is learned to auto-encode an image and a second stage CNN or Transformer is learned to model the density of encoded latent variables. These have been proved effective for image generation, but few studies have evaluated the learned representation in discriminative tasks (Ramesh et al., 2021; Esser et al., 2021). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We explore an approach we refer to as Vector-quantized Image Modeling (VIM) and apply it to both image generation and image understanding tasks. VIM follows a two-stage approach: ",
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+    {
+        "type": "text",
+        "text": "• Stage 1: Image Quantization. Given an image of resolution $2 5 6 \\times 2 5 6$ , a VisionTransformer-based VQGAN encodes it into $3 2 \\times 3 2$ discretized latent codes where the codebook size is 8192. We propose multiple improvements–from architecture to codebook learning–to VQGAN (Esser et al., 2021). The resulting ViT-VQGAN is more efficient and improves reconstruction fidelity in terms of pixel-wise reconstruction metrics, Inception Score (IS) and Frechet Inception Distance (FID). ViT-VQGAN is trained end-to-end on ´ image-only data with combined objective functions of logit-laplace loss, $\\ell _ { 2 }$ loss, adversarial loss and perceptual loss (Johnson et al., 2016; Zhang et al., 2018). • Stage 2: Vector-quantized Image Modeling. We train a Transformer model to predict rasterized $3 2 \\times 3 2 = 1 0 2 4$ image tokens autoregressively, where image tokens are encoded by a learned Stage 1 ViT-VQGAN. For unconditional image synthesis or unsupervised learning, we pretrain a decoder-only Transformer model to predict the next token. For class-conditioned image synthesis, a class-id token is prepended before the image tokens. To evaluate the quality of unsupervised learning, we average the intermediate Transformer features and learn a linear head to predict the logit of the classes (a.k.a., linear-probe). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We show that one key component for improving both image generation and image understanding with VIM is to have a better image quantizer with respect to both computational efficiency and reconstruction quality. An efficient quantizer can speed up Stage 2 training, where random augmentations are applied first to an image, followed by the encoder of image quantizer to obtain the input tokens. Moreover, an image quantizer with better reconstruction quality can reduce information loss compared with the original image in pixel space, which is critical for image understanding tasks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The evaluations of our proposed ViT-VQGAN and VIM are studied with three aspects. (1) We evaluate the image quantizer based on reconstruction quality metrics including $\\ell _ { 1 }$ distance, $\\ell _ { 2 }$ distance, log-laplace distance, as well as Inception Score (IS) and Frechet Inception Distance (FID) ´ of reconstructed images. (2) We evaluate the capabilities of the learned quantizer for unconditional or class-conditioned image synthesis based on FID and IS, and compare with other methods. (3) We rely on linear-probe accuracy to evaluate representations with the common intuition that good features should linearly separate the classes of downstream tasks. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Image Synthesis. Image generation has received much attention with the progress of deep generative models, including Generative Adversarial Networks (GANs) (Goodfellow et al., 2014; Zhang et al., 2019b), Variational Autoencoders (VAEs) (Kingma & Welling, 2014; Vahdat & Kautz, 2020), Diffusion Models (Song & Ermon, 2019; Dhariwal & Nichol, 2021) and Autoregressive Models (van den Oord et al., 2016; Parmar et al., 2018). Unlike many autogressive methods which generate sequence directly in pixel space, VQVAE (van den Oord et al., 2017; Razavi et al., 2019) decomposes the image generation process into two stages: the first stage trains a vector quantized autoencoder with image reconstruction objective to convert an image into a shorter sequence of discrete tokens. Then the second stage learns an autoregressive model, e.g., PixelSNAIL (Chen et al., 2018), to model the underlying distribution of token sequences. Driven by the effectiveness of VQVAE and progress in sequence modeling (Vaswani et al., 2017; Devlin et al., 2019), many approaches follow the two-stage paradigm. DALL-E (Ramesh et al., 2021) improves token prediction in second stage by using Transformers (Vaswani et al., 2017), resulting in a strong text-to-image synthesis model. VQGAN (Esser et al., 2021) further uses adversarial loss and perceptual loss (Johnson et al., 2016; Zhang et al., 2018) to train a better autoencoder in the first stage to synthesize greater detail in images. ",
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+    {
+        "type": "text",
+        "text": "Image Recognition with Generative Pretraining. Many image generation models (Goodfellow et al., 2014; Kingma & Welling, 2014; Radford et al., 2016; Donahue et al., 2017; Higgins et al., 2017) have been studied for their capabilities in representation learning. However, their performance is usually not superior to competing self-supervised approaches that solve auxiliary classification tasks (Noroozi & Favaro, 2016a; Gidaris et al., 2018a; van den Oord et al., 2018). BigBiGAN (Donahue & Simonyan, 2019a) first demonstrated that a generation-based model can match other self-supervised methods in representation learning on ImageNet. iGPT (Chen et al., 2020a) uses the autoregressive objective to learn a giant transformer that directly predicts pixel values, producing even more competitive results. Compared to iGPT, our method first tokenizes the original image into discrete image tokens and then trains a transformer to predict them. As a result, our approach obtains comparable results with smaller model and less data. Similar to our method in predicting image tokens, BEiT (Bao et al., 2021) follows pre-training scheme of BERT Devlin et al. (2019) by learning to recover randomly masked image tokens with a bidirectional transformer. Unlike BEiT, we explore vector-quantized image modeling for image generation in addition to image recognition. ",
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+    {
+        "type": "text",
+        "text": "3 VECTOR-QUANTIZED IMAGES WITH VIT-VQGAN ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The Vector-quantized Variational AutoEncoder (VQVAE) (van den Oord et al., 2017) is a CNNbased auto-encoder whose latent space is a matrix of discrete learnable variables, trained end-to-end via straight-through estimation. Esser et al. (2021) introduce VQGAN, a model which improves upon VQVAE by introducing an adversarial loss produced by a discriminator. Below, we introduce further improvements to VQGAN that boost efficiency and enhance reconstruction quality. ",
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+    {
+        "type": "text",
+        "text": "3.1 VQGAN WITH VISION TRANSFORMERS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The core network architectures used by both VQVAE and VQGAN to encode and reconstruct images are CNNs. VQGAN introduces transformer-like elements in the form of non-local attention block (Zhang et al., 2019a), allowing it to capture distant interactions with fewer layers. We propose taking this approach one step further by replacing the CNN encoder and decoder with Vision Transformer (ViT) (Dosovitskiy et al., 2020), as shown in Figure 1. Given sufficient data (for which unlabeled image data is plentiful) we find that ViT-VQGAN is less constrained by the inductive priors imposed by convolutions. Furthermore, ViT-VQGAN yields better computational efficiency on accelerators, and produces higher quality reconstructions, as shown in Table 1. ",
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+        "text": "The encoder of ViT-VQGAN first maps $8 \\times 8$ non-overlapping image patches into image tokens, followed by Transformer blocks, encoding a $2 5 6 \\times 2 5 6$ resolution image into a $3 2 \\times 3 2 { = } 1 0 2 4$ token sequence. The decoder performs the inverse operation, mapping each image token from latent variables back to $8 \\times 8$ image patches and regrouping them into a $2 5 6 \\times 2 5 6$ image (see Figure 1). At the output of transformer blocks, we apply a two-layer feed-forward network with a tanh activation layer in the middle. No activation is applied at the output of ViT-VQGAN encoder or decoder. We find that this simple approach yields high quality reconstructions without any noticeable grid artifacts. ",
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+    {
+        "type": "table",
+        "img_path": "images/3c67b3d968b68ec6900c6d41a2e7a1ba3316d5a8ef2798d024cdaece7a3c37ea.jpg",
+        "table_caption": [
+            "Table 1: ViT-VQGAN achieves better speed-quality trade-offs compared with CNN-VQGAN. This in turn further speeds up Stage 2 training. Throughputs are benchmarked with the same 128 CloudTPUv4 devices. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture</td><td>Model Size (encoder-decoder)</td><td>Throughput 个 (imgs/sec)</td><td>l2loss (1e-2)</td><td>Logit-Laplace loss ↓</td><td>FID↓</td><td>IS↑</td></tr><tr><td>ViT-VQGAN</td><td>Small-Small</td><td>1520</td><td>3.34</td><td>-2.44</td><td>1.99</td><td>184.4</td></tr><tr><td>CNN-VQGAN</td><td>Channels ×1</td><td>946</td><td>3.81</td><td>-2.36</td><td>2.26</td><td>178.7</td></tr><tr><td>ViT-VQGAN</td><td>Base-Base</td><td>960</td><td>3.09</td><td>-2.54</td><td>1.55</td><td>190.2</td></tr><tr><td>CNN-VQGAN</td><td>Channels × 2</td><td>400</td><td>3.44</td><td>-2.46</td><td>1.91</td><td>183.4</td></tr><tr><td>ViT-VQGAN</td><td> Small-Large</td><td>384</td><td>2.88</td><td>-2.58</td><td>1.28</td><td>192.3</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
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+        "text": "3.2 CODEBOOK LEARNING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Vanilla VQVAEs usually suffer from low codebook usage due to the poor initialization of the codebook. Therefore, during training a significant portion of codes are rarely used, or dead. The reduction in effective codebook size results in worse reconstructions in stage 1 quantizer training and poor diversity in stage 2 for image synthesis. As a result, VQGAN (Esser et al., 2021) relies on top$k$ and top- $p$ (nucleus) sampling heuristics (Holtzman et al., 2020) with a default codebook size of 1024 to obtain best results for image synthesis. We propose two improvements that can significantly encourage the codebook usage even with a larger codebook size of 8192. During image synthesis, we perform simple sampling with temperature of 1.0 without top- $k$ and top- $p$ heuristics. ",
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+    {
+        "type": "text",
+        "text": "The training objective of vector-quantization is defined as follows: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/4c8c30328e0595cac30cb68ffcad93728f8c8cb82aa8b52d677393c414bcca58.jpg",
+        "text": "$$\nL _ { \\mathrm { V Q } } = \\| \\mathbf { s g } [ z _ { e } ( x ) ] - e \\| _ { 2 } ^ { 2 } + \\beta \\| z _ { e } ( x ) - \\mathbf { s g } [ e ] \\| _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here, $\\operatorname { s g } ( x ) \\equiv x$ , $\\begin{array} { r } { { \\frac { \\mathrm { d } } { \\mathrm { d } x } } \\mathrm { s g } ( x ) \\equiv 0 } \\end{array}$ is the stop-gradient operator, $\\beta$ is a commitment loss hyperparameter set to 0.25 in all our experiments, and $e$ is the codebook vector. The quantized codebook index is determined by looking up the codebook vector closest to the input features $z _ { e } ( x )$ in terms of the Euclidean distance, yielding $i = \\mathrm { a r g m i n } _ { j } \\| z _ { e } ( x ) - e _ { j } \\| _ { 2 } ^ { 2 }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Factorized codes. We introduce a linear projection from the output of the encoder to a lowdimensional latent variable space for code index lookup (e.g., reduced from a 768-d vector to a 32-d or 8-d vector per code) and find it has an immediate boost of codebook usage. The factorization can be viewed as decoupling code lookup and code embedding: we lookup the the closest variable encoded from input on a lower-dimensional lookup space and then project the matched latent code to the high-dimensional embedding space. Our experiments show reducing dimension of lookup space from 256-d to 32-d consistently improves reconstruction quality. A detailed illustration is provided in the supplementary materials. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$\\ell _ { 2 }$ -normalized codes. We also apply $\\ell _ { 2 }$ normalization on the encoded latent variables $z _ { e } ( x )$ and codebook latent variables $e$ . The codebook variables are initialized from a normal distribution. By mapping all latent variables on a sphere, the Euclidean distance of $\\ell _ { 2 }$ -normalized latent variables $| | \\ell _ { 2 } ( \\bar { z } _ { e } ( \\bar { x } ) ) - \\ell _ { 2 } ( e _ { j } ) | | _ { 2 } ^ { 2 }$ evolves to the cosine similarity of two vectors between $z _ { e } ( x )$ and $e$ , further improving training stability and reconstruction quality shown in our experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 VIT-VQGAN TRAINING LOSSES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We use a combination of logit-laplace loss, $\\ell _ { 2 }$ loss, perceptual loss (Johnson et al., 2016; Zhang et al., 2018) based on VGG network (Simonyan & Zisserman, 2014) and GAN loss with architecture of StyleGAN discriminator (Karras et al., 2020). Loss balancing weights are configured with a hyper-parameter sweep to optimize image reconstruction quality, codebook usage, FID and Inception Score. After the sweep, we apply the same set of hyper-parameters of training losses to all datasets including CelebA-HQ, FFHQ, and ImageNet. Logit-Laplace loss can be viewed as normalized $\\ell _ { 1 }$ loss which assumes the noise at the pixel level is laplace-distributed while $\\ell _ { 2 }$ loss assumes the noise is of a Gaussian distribution. We find logit-laplace loss contributes to codebook usage while $\\ell _ { 2 }$ loss and perceptual loss significantly contribute to FID. The final loss combination we used by default is $L = L _ { \\mathrm { { V Q } } } + 0 . 1 L _ { \\mathrm { { A d v } } } + 0 . 1 L _ { \\mathrm { { P e r c e p t u a l } } } + 0 . 1 L _ { \\mathrm { { L o g i t } \\mathrm { { - } \\mathrm { { l a p l a c e } } } } } + 1 . 0 L _ { \\mathrm { { 2 } } } .$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/87ef8f604057b4abf1a2f09332fb5943fa00af5f6008c54f13d634537bfd59b3.jpg",
+        "table_caption": [
+            "Table 2: Transformer architectures of Stage 1 ViT-VQGAN and Stage 2 VIM. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Size</td><td>#Params</td><td>#Blocks</td><td>#Heads</td><td>Model Dim</td><td>Hidden Dim</td><td>Dropout</td><td>#Tokens</td></tr><tr><td>ViT-VQGAN</td><td>Small</td><td>32M</td><td>8</td><td>8</td><td>512</td><td>2048</td><td>0.0</td><td>1024</td></tr><tr><td>ViT-VQGAN</td><td>Base</td><td>91M</td><td>12</td><td>12</td><td>768</td><td>3072</td><td>0.0</td><td>1024</td></tr><tr><td>ViT-VQGAN</td><td>Large</td><td>599M</td><td>32</td><td>16</td><td>1280</td><td>5120</td><td>0.0</td><td>1024</td></tr><tr><td>VIM</td><td>Base</td><td>650M</td><td>24</td><td>16</td><td>1536</td><td>6144</td><td>0.1</td><td>1024</td></tr><tr><td>VIM</td><td>Large</td><td>1697M</td><td>36</td><td>32</td><td>2048</td><td>8192</td><td>0.1</td><td>1024</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "One caveat on the VGG-based perceptual loss is that the VGG network is pretrained with supervised classification loss, so the supervision might leak into Stage 2 for linear-probe accuracy measurement. Thus, for all of our reported unsupervised learning results, we exclude the perceptual loss during ViT-VQGAN training. For all unconditional and class-conditioned image synthesis, we use ViTVQGAN quantizers trained with perceptual loss, as it leads to higher-fidelity reconstructions. ",
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+    {
+        "type": "text",
+        "text": "4 VECTOR-QUANTIZED IMAGE MODELING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "With a learned ViT-VQGAN, images are encoded into discrete latent code ids flattened in the raster order, similar to Image GPT (Chen et al., 2020a). A decoder-only Transformer model is used to model the density of image data $P ( x )$ autoregressively as ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/af75e1b6e1a59d7efd98c666b5becb4609ce27c98b194de3631dfb2fc550295c.jpg",
+        "text": "$$\nP ( x ) = \\prod _ { i = 1 } ^ { n } P ( x _ { i } | x _ { 1 } , x _ { 2 } , . . . , x _ { i - 1 } ; \\theta ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\theta$ is learnable weights. The training objective is to minimize the negative log-likelihood of the data $L = \\mathbb { E } _ { x \\in X } [ - l o g P ( x ) ]$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 2 summarizes the architecture configurations for the Transformers. We first embed discrete image token ids into a learnable embedding space at each position, with an additive learnable 2D positional embedding. Both embedding dimensions are the same as model dimension. We apply a stack of Transformer blocks to the inputs with causal attention over the entire sequence. A dropout ratio of 0.1 is used in all residual, activation and attention outputs. At the final layer of all Transformer blocks, we apply an additional layer normalization. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 IMAGE SYNTHESIS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "With a pretrained generative Transformer model, unconditional image generation is achieved by simply sampling token-by-token from the output softmax distribution. All samples used for both qualitative and quantitative results are obtained without temperature reduction. The sampled tokens are then fed into the decoder of ViT-VQGAN to decode output images. Our default Stage 1 ViTVQGAN encodes input images of resolution $2 5 6 \\times 2 5 6$ into $3 2 \\times 3 2$ latent codes with a codebook size 8192, while Stage 2 Transformer takes the flattened image tokens with total a length of 1024. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Class-conditioned ImageNet generation is also a widely used benchmark for measuring capabiltiy of models for image synthesis. We extend the unconditional generation to class-conditioned generation by prepending a class-id token before the image tokens. Separate embedding layers are learned from scratch for class-id token and image tokens, with the embedding dimension the same as the Transformer model dimension. During sampling, a class-id token is provided at the first position to decode the remaining image tokens autoregressively. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 UNSUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For the image understanding task, we feed all image tokens of the input into a pretrained Transformer, and get a sequence of 1024 token features. Similar to Image GPT (Chen et al., 2020a), we take a layer output at a specific block $l$ over total blocks $L$ , average over the sequence of token features (frozen) and insert a softmax layer (learnable) projecting averaged feature to class logits. We only take one specific Transformer block output instead of concatenating different block outputs as in iGPT (Chen et al., 2020a). We find that most discriminating feature for the linear-probe is typically near the middle of all Transformer blocks. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/76bfbf1eb24a6e2680b1f4c7a1a4a9a9a77e50e2e9d66bc60cecd6deab9737a4.jpg",
+        "table_caption": [
+            "Table 3: Frechet Inception Distance (FID) between reconstructed validation split and original val- ´ idation split on ImageNet, CelebA-HQ and FFHQ. ∗ denotes models trained with Gumbel-Softmax reparameterization as in Ramesh et al. (2021). ∗∗ denotes models trained with multi-scale hierarchical codebook as in Razavi et al. (2019). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Dataset</td><td>Latent Size</td><td>dimZ</td><td>FID on Validation</td></tr><tr><td>DALL-E dVAE</td><td>Web data</td><td>32 ×32</td><td>8192</td><td>32.00</td></tr><tr><td>VQGAN</td><td>ImageNet</td><td>16 ×16</td><td>1024</td><td>7.94</td></tr><tr><td>VQGAN</td><td>ImageNet</td><td>16 ×16</td><td>16384</td><td>4.98</td></tr><tr><td>VQGAN*</td><td>ImageNet</td><td>32 × 32</td><td>8192</td><td>1.49</td></tr><tr><td>VQGAN**</td><td>ImageNet</td><td>64×64&amp;32×32</td><td>512</td><td>1.45</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>ImageNet</td><td>32×32</td><td>8192</td><td>1.28</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>CelebA-HQ</td><td>32 ×32</td><td>8192</td><td>4.66</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>FFHQ</td><td>32 ×32</td><td>8192</td><td>3.13</td></tr></table>",
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+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 IMAGE QUANTIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We train the proposed ViT-VQGAN on three datasets separately, CelebA-HQ (Karras et al., 2019), FFHQ (Karras et al., 2019), and ImageNet (Krizhevsky et al., 2012). For CelebA-HQ and FFHQ, we follow the default train and validation split as VQGAN (Esser et al., 2021).1 For Stage 1 image quantization, three different architecture sizes are experimented, Small, Base and Large for either encoder or decoder, as defined in Table 2. The smallest ViT-VQGAN-SS has a Small-size encoder and Small-size decoder, while ViT-VQGAN-BB has a Base-size encoder and Base-size decoder. The largest ViT-VQGAN-SL has an asymmetric Small-size encoder and Large-size decoder, with the motivation that Stage 2 training only requires forward propagation of the encoder of ViT-VQGAN (in inference/decoding for image synthesis, the decoder of ViT-VQGAN is still required to decode images from codes predicted during Stage 2). ",
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+    {
+        "type": "text",
+        "text": "We train all ViT-VQGAN models with a training batch size of 256 distributed across 128 CloudTPUv4 for a total 500,000 training steps. For both ViT-VQGAN and StyleGAN discriminator, Adam optimizer (Kingma & Ba, 2014) is used with $\\beta _ { 1 } = 0 . 9$ and $\\beta _ { 2 } = 0 . 9 9$ with the learning rate linearly warming up to a peak value of $1 \\times 1 0 ^ { - 4 }$ over 50,000 steps and then decaying to $5 \\times \\mathrm { \\overline { { 1 } } 0 ^ { - 5 } }$ over the remaining 450,000 steps with a cosine schedule. We use a decoupled weight decay (Loshchilov & Hutter, 2017) of $1 \\dot { \\times } 1 0 ^ { - 4 }$ for both ViT-VQGAN and StyleGAN discriminator. All models are trained with an input image resolution $2 5 6 \\times 2 5 6$ on CloudTPUv4. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 3 shows FID between reconstructed images and original images in the validation split on ImageNet, CelebA-HQ and FFHQ datasets. Without multi-scale hierarchical codebook or gumbelsoftmax, ViT-VQGAN is able to achieve better FID with a large codebook size of 8192 compared with vanilla VQGAN. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 4 provides extensive ablations on the proposed modifications, with empirical results on mean $\\ell _ { 1 }$ distance, $\\ell _ { 2 }$ distance, logit-laplace distance, Inception Score and FID on ImageNet. Among different model sizes, ViT-VQGAN-SS (small-encoder, small-decoder) performs worse than ViTVQGAN-BB (base-encoder, base-decoder) and ViT-VQGAN-SL (small-encoder, large-decoder), but achieves much better throughput. The CNN-based VQGAN architecture is worse in both quality and throughput compared with ViT-based VQGAN. The StyleGAN-based discriminator (Karras et al., 2019) is more stable and yields better reconstruction quality than PatchGAN (Isola et al., 2017) ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/51348d4ffd36ed642ee900149e5c4c1fc1b0af54302de4e5fa9ec117fa95942b.jpg",
+        "table_caption": [
+            "Table 4: Ablation study on ViT-VQGAN. The codebook usage is calculated as the percentage of used codes given a batch of 256 test images averaged over the entire test set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>groe ereerg rrr repeeeg</td><td>JAeeettect</td><td>Jirrriisi</td><td></td><td>pineg Lir aeaer</td><td>↑（2-0）</td><td>↑（-01）</td><td> 9der-18g</td><td></td><td>→ D</td><td>eeesn googopeg</td><td>↑ndysnou</td></tr><tr><td>Ablation on</td><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>32</td><td>√</td><td>3.06</td><td>3.09 -2.54</td><td>190.2</td><td></td><td>1.55 96%</td><td>960</td></tr><tr><td>Model Size</td><td>Small Small</td><td>Small</td><td>ViT ViT</td><td>StyleGAN</td><td>32</td><td>√ 3.22</td><td>3.34</td><td>-2.44</td><td>184.4</td><td>1.99</td><td>95%</td><td>1520</td></tr><tr><td rowspan=\"2\"></td><td rowspan=\"2\"></td><td rowspan=\"2\">Large</td><td rowspan=\"2\"></td><td rowspan=\"2\">StyleGAN</td><td rowspan=\"2\">32</td><td rowspan=\"2\">√ 2.93</td><td rowspan=\"2\">2.88</td><td rowspan=\"2\">-2.58</td><td rowspan=\"2\">192.3</td><td rowspan=\"2\">1.28</td><td rowspan=\"2\">95%</td><td rowspan=\"2\">384</td></tr><tr><td>√ 3.45 2.82</td></tr><tr><td>Architecture</td><td>1 Base</td><td>1 Base</td><td>CNN ViT</td><td>StyleGAN PatchGAN</td><td>32 32</td><td>√</td><td>3.81 2.58</td><td>-2.36 -2.62</td><td>178.7 165.6</td><td>2.26 3.88</td><td>63% 89%</td><td>946 1227</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>4.28</td><td>-2.38</td><td>160.1</td><td>3.68</td><td>4%</td><td></td></tr><tr><td rowspan=\"8\">Codebook Learning</td><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>256 128</td><td>1</td><td>3.60 3.41</td><td></td><td></td><td></td><td></td><td>954</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td></td><td></td><td>3.93</td><td>-2.44</td><td>173.9</td><td>2.77</td><td>14%</td><td>960</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>64</td><td></td><td>3.18 3.37</td><td>-2.49</td><td>179.5 191.2</td><td>2.50</td><td>37%</td><td>960</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>16</td><td></td><td>3.00</td><td>2.96 -2.54</td><td></td><td>1.50</td><td>95%</td><td>960</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>8</td><td></td><td>2.98</td><td>2.92 -2.55</td><td>189.5</td><td>1.52</td><td>96%</td><td>960</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>4</td><td></td><td>3.55</td><td>4.18 -2.37</td><td>143.8</td><td>3.68 5.44</td><td>96%</td><td>960</td></tr><tr><td>Base</td><td>Base</td><td>ViT</td><td>StyleGAN</td><td>32</td><td>&gt;&gt;&gt;&gt;&gt;x</td><td>4.13</td><td>5.41</td><td>-2.20</td><td>123.6</td><td>2%</td><td>960</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/422bb23bc60c859847af6ccfcd100585ca6e9f87402d463f3c4a5d88631aa644.jpg",
+        "table_caption": [
+            "Table 5: FID comparison with unconditional image synthesis on CelebA-HQ and FFHQ. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">CelebA-HQ 256 × 256</td><td colspan=\"2\">FFHQ 256× 256</td></tr><tr><td>Method</td><td>FID↓</td><td>Method</td><td>FID↓</td></tr><tr><td>GLOW (Kingma &amp; Dhariwal, 2018)</td><td>69.0</td><td>VDVAE (t = 0.7) (Child,2021)</td><td>38.8</td></tr><tr><td>NVAE (Vahdat &amp; Kautz,2020)</td><td>40.3</td><td>VDVAE (t = 1.0)</td><td>33.5</td></tr><tr><td>PIONEER (Heljakka etal.,2018)</td><td>25.3</td><td>VDVAE (t = 0.8)</td><td>29.8</td></tr><tr><td>NCPVAE (Aneja et al.,2020)</td><td>24.8</td><td>VDVAE (t = 0.9)</td><td>28.5</td></tr><tr><td>VAEBM (Xiao et al., 2021)</td><td>20.4</td><td>VQGAN+P.SNAIL</td><td>21.9</td></tr><tr><td>Style ALAE (Pidhorskyi et al., 2020)</td><td>19.2</td><td>BigGAN</td><td>12.4</td></tr><tr><td>DC-VAE (Parmar et al., 2021)</td><td>15.8</td><td>U-Net GAN (Schonfeld et al., 2020)</td><td>10.9</td></tr><tr><td>PGGAN(Karras et al.,2018)</td><td>8.0</td><td>StyleGAN2 (Karras etal., 2020)</td><td>3.8</td></tr><tr><td>VQGAN (w/ top-k sampling)</td><td>10.2</td><td>VQGAN (w/ top-k sampling)</td><td>9.6</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>7.0</td><td>ViT-VQGAN (Ours)</td><td>5.3</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "(which was used for VQGAN). For codebook learning, factorized codes with low-dimensional latent variables consistently achieve better reconstruction quality when the latent dimension is reduced from 256 to 16 or 8. Moreover, removing $\\ell _ { 2 }$ -normalization leads to much worse results. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 IMAGE SYNTHESIS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "On top of the pre-learned ViT-VQGAN image quantizer, we train stage 2 transformer models for unconditional and class-conditioned image synthesis and compare with previous work. We use a default model size of ViT-VQGAN-SS (small-encoder, small-decoder) for stage 1 and VIM-Large for stage 2 (model architectures are listed in Table 2), as we find it beneficial to put more computation in stage 2 while keeping stage 1 transformers lightweight. We also present a model size ablation study and comparison with VQGAN in the Appendix. Models are trained with a global training batch size of 1024 for a total of 450,000 training steps. We use Adam optimizer (Kingma & Ba, 2014) with $\\beta _ { 1 } = 0 . 9$ and $\\beta _ { 2 } = 0 . 9 6$ with the learning rate linearly warming up to a peak constant value of $4 . 5 \\times 1 0 ^ { - 4 }$ over the first 5000 steps and then exponentially decaying to $1 \\times \\mathrm { 1 0 ^ { - 5 } }$ starting from 80,000 steps. To save memory, we use a factorized version of Adam, Adafactor (Shazeer & Stern, 2018), with the first moment quantized into Int8 and factorized second moments. No other techniques like mixed-precision training, model sharding, or gradient compression is used. All models are trained with an input image resolution $2 5 6 \\times 2 5 6$ on CloudTPUv4. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/0f5fc7274130fa616b56310c7a07d6bb0de7bf2d946fd5b731b377840ca3fa5b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Acceptance Rate</td><td>FID</td><td>IS</td></tr><tr><td>Validation data</td><td>1.0</td><td>1.62</td><td>235.0</td></tr><tr><td>DCTransformer (Nash et al.,2021)</td><td>1.0</td><td>36.5</td><td>n/a</td></tr><tr><td>BigGAN (Brock et al.,2019)</td><td>1.0</td><td>7.53</td><td>168.6</td></tr><tr><td>BigGAN-deep</td><td>1.0</td><td>6.84</td><td>203.6</td></tr><tr><td>IDDPM (Nichol &amp; Dhariwal,2021)</td><td>1.0</td><td>12.3</td><td>n/a</td></tr><tr><td>ADM-G, no guid. (Dhariwal &amp; Nichol, 2021)</td><td>1.0</td><td>10.94</td><td>101.0</td></tr><tr><td>ADM-G, 1.0 guid.</td><td>1.0</td><td>4.59</td><td>186.7</td></tr><tr><td>VQVAE-2 (Razavi et al.,2019)</td><td>1.0</td><td>~31</td><td>~45</td></tr><tr><td>VQGAN (Esser et al.,2021)</td><td>1.0</td><td>17.04</td><td>70.6</td></tr><tr><td>VQGAN</td><td>0.5</td><td>10.26</td><td>125.5</td></tr><tr><td>VQGAN</td><td>0.25</td><td>7.35</td><td>188.6</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>1.0</td><td>4.17</td><td>175.1</td></tr><tr><td>ViT-VQGAN (Ours)</td><td>0.5</td><td>3.04</td><td>227.4</td></tr></table>",
+        "bbox": [
+            243,
+            101,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 6: FID comparison for class-conditional image synthesis on ImageNet with resolution $2 5 6 \\times$ 256. Acceptance rate shows results based on ResNet-101 classifier-based rejection sampling. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/2d4bd8b401b7977b3c4ee742b5204d1709b675afa13b9a6ea9cac79c8b6d5b86.jpg",
+        "image_caption": [
+            "Figure 2: Uncurated set of samples from class-conditioned image generation trained on ImageNet. Top row shows sampled class ids while bottom row shows fine-grained dog species from class id 184 to 189. More samples are shown in Appendix. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            363,
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+            522
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our main results on unconditional image synthesis on CelebA-HQ and FFHQ are summarized in Table 5. Without top- $k$ and top- $p$ (nucleus) sampling heuristics, we achieve FID of 7.0 on CelebAHQ and 5.3 on FFHQ, significantly better than VQGAN (Esser et al., 2021). Table 6 shows classconditioned image synthesis models on ImageNet, following Section 4.1. Based on ViT-VQGANSS, we achieve IS of 175.1 and FID of 4.17, improving over the IS of 70.6 and FID of 17.04 with vanilla VQGAN. When applied with classifier-based rejection sampling, the ViT-VQGAN based model further achieves best FID of 3.04 and best Inception Score of 321.7. Qualitative results are sampled and shown in Figure 2 (see the Appendix for more). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 UNSUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "After the generative pretraining to autoregressively model the density of ViT-VQGAN quantized image tokens, we evaluate the learned representation under the common linear protocol on ImageNet classification. We follow the same training hyper-parameters as the unconditional image synthesis models on ImageNet, and use ViT-VQGAN-SS image quantizer for better training throughput. As discussed in Section 3.3, the ViT-VQGAN-SS image quantizer is trained without perceptual loss for unsupervised learning (perceptual loss is based on a supervised VGG network trained on ImageNet, which may make comparison unfair). We apply an average pooling over the token features at a specific transformer block $l$ from totally $L$ blocks. Similar to findings reported in iGPT (Chen et al., 2020a), the representations from the middle transformer blocks has better linear-probe accuracy (more study can be found in Appendix). Specifically, we use the Transformer block of index 15 (36 blocks in total) for VIM-Large and index 10 (24 blocks in total) for VIM-Base (architecture configurations are listed in Table 2). ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6a1d3d8e08c316b9638eda1eb783af5e1411bed289b1b3453e8f51dd16529c24.jpg",
+        "table_caption": [
+            "Table 7: Linear-probe accuracy with different unsupervised learning methods on ImageNet. DALLE dVAE (Ramesh et al., 2021) image quantizer is trained with extra image data. VIM-Large is trained without dropout in transformers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td></td><td>#Tokens</td><td>Features</td><td>Params</td><td>Top-1 ↑</td></tr><tr><td rowspan=\"15\">Brirrniir irnrriiiirig</td><td>Jigsaw (Noroozi &amp; Favaro,2016b)</td><td></td><td>4096</td><td>94M</td><td>44.6</td></tr><tr><td>RelativePosition (Doersch et al., 2015)</td><td></td><td>4096</td><td>94M</td><td>51.4</td></tr><tr><td>Rotation (Gidaris et al.,2018b)</td><td></td><td>8192</td><td>86M</td><td>55.4</td></tr><tr><td>AMDIM (Bachman et al.,2019)</td><td></td><td>8192</td><td>626M</td><td>68.1</td></tr><tr><td>CPC v2 (Henaff, 2020)</td><td></td><td>4096</td><td>303M</td><td>71.5</td></tr><tr><td>MoCo (He et al.,2020)</td><td></td><td>8192</td><td>375M</td><td>68.6</td></tr><tr><td>SimCLR (Chen et al., 2020b)</td><td></td><td>8192</td><td>375M</td><td>76.5</td></tr><tr><td>SwAV (Caron et al., 2020)</td><td></td><td>2048</td><td>93M</td><td>75.3</td></tr><tr><td>DINO (Caron et al., 2021)</td><td></td><td>2048</td><td>85M</td><td>75.3</td></tr><tr><td>BYOL (Grill et al., 2020)</td><td></td><td>8192</td><td>375M</td><td>78.6</td></tr><tr><td rowspan=\"4\">Greeeed rierieeg</td><td>BiGAN (Donahue et al., 2016)</td><td></td><td>512</td><td>138M</td><td>31.0</td></tr><tr><td>BigBiGAN (Donahue &amp; Simonyan,2019b)</td><td></td><td>4096</td><td>86M</td><td>56.6</td></tr><tr><td>BigBiGAN</td><td>=</td><td>16384</td><td>344M</td><td>61.3</td></tr><tr><td>iGPT-L (Chen et al., 2020a)</td><td>32× 32</td><td>1536</td><td>1362M</td><td>60.3</td></tr><tr><td>iGPT-L</td><td></td><td>48×48</td><td>1536</td><td>1362M</td><td>65.2</td></tr><tr><td>iGPT-XL (extra data)</td><td></td><td>64×64</td><td>3072</td><td>6801M</td><td>68.7</td></tr><tr><td></td><td> iGPT-XL (extra data, feature ensemble)</td><td>64 × 64</td><td>5×3072</td><td>6801M</td><td>72.0</td></tr><tr><td></td><td>VIM-Base (Ours)</td><td>32 × 32</td><td>1024</td><td>650M</td><td>65.1</td></tr><tr><td></td><td>VIM-Large (Ours)</td><td>32 × 32</td><td>2048</td><td>1697M</td><td>73.2</td></tr><tr><td></td><td>VIM-Base + DALL-E dVAE quantizer</td><td>32 ×32</td><td>1024</td><td>650M</td><td>63.8 (-1.3)</td></tr><tr><td></td><td>VIM-Base + CNN-VQGAN quantizer</td><td>32×32</td><td>1024</td><td>650M</td><td>61.8 (-3.3)</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 7 shows the comparisons among different approaches divided into two categories: discriminative pretraining methods to distinguish among cropped or augmented image patches; and generative pretraining methods to generate image pixels or patches. The linear-probe accuracy of our proposed VIM with ViT-VQGAN are superior to other generative pretraining approaches like iGPT, and competitive with discriminative pretraining methods like BYOL (Grill et al., 2020) and DINO (Caron et al., 2021). Specifically, ImageNet-pretrained VIM-L significantly improves over iGPT-L, increasing linear-probe accuracy from $6 0 . 3 \\%$ to $7 3 . 2 \\%$ for a similar model size. VIM-L also outperforms iGPT-XL, which is larger and trained with extra web image data. Moreover, we also compare different stage 1 quantizers including CNN-based VQGAN and pretrained DALL-E dVAE (trained with extra web-scale image data)2 in Table 7; these results are all worse than ViT-VQGAN quantizer, suggesting the importance of the multiple changes defined in Section 3. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 ETHICS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Tasks involving generation raise a number of issues that should be considered, such as possible biases in underlying models and data—especially with respect to capabilities for people with different demographic backgrounds. The three datasets used in this paper–ImageNet, CelebA-HQ, and FFHQ–are all widely used in the literature, but it is worthwhile highlighting their unique natures and recent scholarship around them. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The FFHQ dataset3 contains 70,000 images collected from Flickr, all of which have licenses appropriate for sharing, and the data maintainers provide means for individuals to opt-out of inclusion in the dataset. FFHQ was specifically collected to cover a broad range of demographics with respect to faces of people. This is confirmed in our models’ generated examples, which cover a broad range of perceived ages, genders and ethnicities. Nevertheless, Balakrishnan et al. (2020) provide an extensive analysis of multiple forms of bias in datasets (including CelebA-HQ and FFHQ) and algorithms for face generation; not only do they find imbalances in skin tone in FFHQ, but also correlations between multiple attributes such as skin tone and hair length. Based on this and other factors such as privacy and copyright, they argue that synthetically-created face datasets, for which multiple attributes can be controlled, is an important direction of investment and general inquiry. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The CelebA-HQ dataset covers celebrities, which brings a consequent bias toward images of attractive people who are mostly in age range of twenty to forty years old. Esser et al. (2020) discusses these biases in details, and they furthermore project images from the FFHQ dataset onto CelebAHQ: the main effect of which is to produce images of younger people with features conforming more to norms of celebrities popular in the United States of America. Our model’s generations appear to have a similar bias as derived from training on CelebA-HQ. Neverethless, they do show broad coverage of different perceived genders and ethnicities, but with age skewed to the 20-40 year old range. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ImageNet is, of course, quite pervasive in computer vision. In this paper, we learn to generate images given ImageNet class labels; these labels mostly concern animals, plants and things. People are sometimes generated when conditioning on classes such as sunglasses since the training data images contain people wearing sunglasses, but the generated images contain few depictions of people overall. Nevertheless, it is important to recognize that ImageNet itself was created with biases in terms of image selection and label annotation as a result of its process of creation (Denton et al., 2021). Given this, results present on ImageNet cover a significant, but nonetheless biased, sample of the kinds of scenes and objects one might encounter across the entire world. ",
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A LINEAR-PROBE ON IMAGENET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            630,
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+            645
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We show linear-probe accuracy from different layers in a pretrained VIM-Base Transformer model in Figure 3. Similar to iGPT (Chen et al., 2020a), we also find the last few layers may not be the best layers for discriminative features, as the generative pretraining objective is to recover the original image tokens. The linearprobe accuracy increases quickly from the first transformer output, reaches its peak at middle layers, and finally decreases for the last few blocks. Interestingly, we find for both VIMBase and VIM-Large, the middle transformer block has the near-best result. This observation connects the transformer model to an encoderdecoder model where the encoder encodes image tokens into high-level semantic features and the decoder takes feature information to gener",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1d84e809d5636b23d7ceb5e8d59293c393c6e1ce9679abdcb3fefcdce313b433.jpg",
+        "image_caption": [
+            "Figure 3: Linear-probe accuracy from different layers in a pretrained VIM-Base Transformer model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            500,
+            655,
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+            824
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "ate output image tokens. We leave for future study regrading the interpretability of pretrained VIM models. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B MODEL SIZES OF CLASS-CONDITIONED IMAGENET SYNTHESIS",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/d24a65630b9f09a345b16a42be65ed57e5d1898fca149858925b11243c15790f.jpg",
+        "table_caption": [
+            "We also present results of different sizes of Stage 2 Transformers for class-conditioned image synthesis and compare with VQGAN (Esser et al., 2021)4 summarized in Table 8. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Stage-2 Transformer Size</td><td>#Tokens</td><td>FID</td><td>IS</td></tr><tr><td>Validation data</td><td>/</td><td>=</td><td>1.62</td><td>235.0</td></tr><tr><td>VQGAN (Esser et al.,2021)</td><td>1.4B</td><td>16 ×16</td><td>17.04</td><td>70.6</td></tr><tr><td>ViT-VQGAN+VIM-Base</td><td>650M</td><td>16 ×16</td><td>11.20</td><td>97.2</td></tr><tr><td>ViT-VQGAN + VIM-Large</td><td>1.7B</td><td>16 ×16</td><td>5.3</td><td>149.9</td></tr><tr><td>ViT-VQGAN + VIM-Base</td><td>650M</td><td>32 × 32</td><td>8.81</td><td>110.8</td></tr><tr><td>ViT-VQGAN + VIM-Large</td><td>1.7B</td><td>32 ×32</td><td>4.17</td><td>175.1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 8: FID comparison for class-conditional image synthesis on ImageNet with different Transformer sizes in Stage 2. Results are reported without rejection sampling. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C IMPLEMENTATION DETAILS OF FACTORIZED CODEBOOK ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "As we introduced in Section 3.2, we use a linear projection to reduce the encoded embedding to a low-dimensional variable space for code lookup. A detailed illustration is shown in Figure 4. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/808468d18686ea4c5749fcf0e24a77e6508b7620f5bbe5bc1144fb5f4eb09778.jpg",
+        "image_caption": [
+            "Figure 4: Illustration of factorized codes and codebook details. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/0cf2583848d23fed0a25567a05ff4f7a7593c2be749af32a27ee19d63132e4bd.jpg",
+        "image_caption": [
+            "Figure 5: Uncurated set of samples from class-conditioned generation trained on ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Scale ",
+        "bbox": [
+            184,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/2260824a7c774887327e7ed39a63230563e7ec24f408abf7f020883a92447b17.jpg",
+        "image_caption": [
+            "Figure 6: Uncurated set of samples from class-conditioned generation trained on ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            269,
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+        ],
+        "page_idx": 16
+    }
+]