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+# SHARPNESS-AWARE MINIMIZATION FOR EFFICIENTLY IMPROVING GENERALIZATION
+
+Pierre Foret ∗   
+Google Research   
+pierreforet@google.com   
+Ariel Kleiner   
+Google Research   
+akleiner@google.com
+
+Hossein Mobahi Google Research hmobahi@google.com
+
+Behnam Neyshabur Blueshift, Alphabet neyshabur@google.com
+
+# ABSTRACT
+
+In today’s heavily overparameterized models, the value of the training loss provides few guarantees on model generalization ability. Indeed, optimizing only the training loss value, as is commonly done, can easily lead to suboptimal model quality. Motivated by prior work connecting the geometry of the loss landscape and generalization, we introduce a novel, effective procedure for instead simultaneously minimizing loss value and loss sharpness. In particular, our procedure, Sharpness-Aware Minimization (SAM), seeks parameters that lie in neighborhoods having uniformly low loss; this formulation results in a minmax optimization problem on which gradient descent can be performed efficiently. We present empirical results showing that SAM improves model generalization across a variety of benchmark datasets (e.g., CIFAR- $\{ 1 0 , 1 0 0 \}$ , ImageNet, finetuning tasks) and models, yielding novel state-of-the-art performance for several. Additionally, we find that SAM natively provides robustness to label noise on par with that provided by state-of-the-art procedures that specifically target learning with noisy labels. We open source our code at https: //github.com/google-research/sam.
+
+# 1 INTRODUCTION
+
+Modern machine learning’s success in achieving ever better performance on a wide range of tasks has relied in significant part on ever heavier overparameterization, in conjunction with developing ever more effective training algorithms that are able to find parameters that generalize well. Indeed, many modern neural networks can easily memorize the training data and have the capacity to readily overfit (Zhang et al., 2016). Such heavy overparameterization is currently required to achieve stateof-the-art results in a variety of domains (Tan & Le, 2019; Kolesnikov et al., 2020; Huang et al., 2018). In turn, it is essential that such models be trained using procedures that ensure that the parameters actually selected do in fact generalize beyond the training set.
+
+Unfortunately, simply minimizing commonly used loss functions (e.g., cross-entropy) on the training set is typically not sufficient to achieve satisfactory generalization. The training loss landscapes of today’s models are commonly complex and non-convex, with a multiplicity of local and global minima, and with different global minima yielding models with different generalization abilities (Shirish Keskar et al., 2016). As a result, the choice of optimizer (and associated optimizer settings) from among the many available (e.g., stochastic gradient descent (Nesterov, 1983), Adam (Kingma & Ba, 2014), RMSProp (Hinton et al.), and others (Duchi et al., 2011; Dozat, 2016; Martens & Grosse, 2015)) has become an important design choice, though understanding of its relationship to model generalization remains nascent (Shirish Keskar et al., 2016; Wilson et al., 2017; Shirish Keskar & Socher, 2017; Agarwal et al., 2020; Jacot et al., 2018). Relatedly, a panoply of methods for modifying the training process have been proposed, including dropout (Srivastava et al., 2014), batch normalization (Ioffe & Szegedy, 2015), stochastic depth (Huang et al., 2016), data augmentation (Cubuk et al., 2018), and mixed sample augmentations (Zhang et al., 2017; Harris et al., 2020).
+
+![](images/44409d10327df0afb8908a10a8459bac79c05856067c1ec0db0ff073c0f0a127.jpg)  
+Figure 1: (left) Error rate reduction obtained by switching to SAM. Each point is a different dataset / model / data augmentation. (middle) A sharp minimum to which a ResNet trained with SGD converged. (right) A wide minimum to which the same ResNet trained with SAM converged.
+
+The connection between the geometry of the loss landscape—in particular, the flatness of minima— and generalization has been studied extensively from both theoretical and empirical perspectives (Shirish Keskar et al., 2016; Dziugaite & Roy, 2017; Jiang et al., 2019). While this connection has held the promise of enabling new approaches to model training that yield better generalization, practical efficient algorithms that specifically seek out flatter minima and furthermore effectively improve generalization on a range of state-of-the-art models have thus far been elusive (e.g., see (Chaudhari et al., 2016; Izmailov et al., 2018); we include a more detailed discussion of prior work in Section 5).
+
+We present here a new efficient, scalable, and effective approach to improving model generalization ability that directly leverages the geometry of the loss landscape and its connection to generalization, and is powerfully complementary to existing techniques. In particular, we make the following contributions:
+
+• We introduce Sharpness-Aware Minimization (SAM), a novel procedure that improves model generalization by simultaneously minimizing loss value and loss sharpness. SAM functions by seeking parameters that lie in neighborhoods having uniformly low loss value (rather than parameters that only themselves have low loss value, as illustrated in the middle and righthand images of Figure 1), and can be implemented efficiently and easily.   
+• We show via a rigorous empirical study that using SAM improves model generalization ability across a range of widely studied computer vision tasks (e.g., CIFAR- $\{ 1 0 , ~ 1 0 0 \}$ , ImageNet, finetuning tasks) and models, as summarized in the lefthand plot of Figure 1. For example, applying SAM yields novel state-of-the-art performance for a number of alreadyintensely-studied tasks, such as ImageNet, CIFAR- $\mathbf { \bar { \{ 1 0 , ~ 1 0 0 \} } }$ , SVHN, Fashion-MNIST, and the standard set of image classification finetuning tasks (e.g., Flowers, Stanford Cars, Oxford Pets, etc).   
+• We show that SAM furthermore provides robustness to label noise on par with that provided by state-of-the-art procedures that specifically target learning with noisy labels.   
+• Through the lens provided by SAM, we further elucidate the connection between loss sharpness and generalization by surfacing a promising new notion of sharpness, which we term m-sharpness.
+
+Section 2 below derives the SAM procedure and presents the resulting algorithm in full detail. Section 3 evaluates SAM empirically, and Section 4 further analyzes the connection between loss sharpness and generalization through the lens of SAM. Finally, we conclude with an overview of related work and a discussion of conclusions and future work in Sections 5 and 6, respectively.
+
+# 2 SHARPNESS-AWARE MINIMIZATION (SAM)
+
+Throughout the paper, we denote scalars as $a$ , vectors as $^ { a }$ , matrices as $\pmb { A }$ , sets as $\mathcal { A }$ , and equality by definition as $\triangleq$ . Given a training dataset ${ \mathcal { S } } \triangleq \cup _ { i = 1 } ^ { n } \{ ( \pmb { x } _ { i } , \pmb { y } _ { i } ) \}$ drawn i.i.d. from distribution $\mathcal { D }$ , we seek to learn a model that generalizes well. In particular, consider a family of models parameterized by $\pmb { w } \in \mathcal { W } \subseteq \mathbb { R } ^ { d }$ ; given a per-data-point loss function $l : \mathcal { W } \times \mathcal { X } \times \mathcal { Y }  \mathbb { R } _ { + }$ , we define the training set loss Having $\begin{array} { r } { L _ { S } ( \pmb { w } ) \triangleq \frac { 1 } { n } \sum _ { i = 1 } ^ { n } l ( \pmb { w } , \pmb { x } _ { i } , \pmb { y } _ { i } ) } \end{array}$ and the population loss odel training is to select $L _ { \mathcal { D } } ( \pmb { w } ) \triangleq \mathbb { E } _ { ( \pmb { x } , \pmb { y } ) \sim D } [ l ( \pmb { w } , \pmb { x } , \pmb { y } ) ]$ $s$ $\pmb { w }$ population loss $\scriptstyle L _ { \mathcal { D } } ( \pmb { w } )$ .
+
+Utilizing $L _ { S } ( w )$ as an estimate of $L _ { \mathcal { D } } ( \mathbf { \boldsymbol { w } } )$ motivates the standard approach of selecting parameters $\pmb { w }$ by solving $\mathrm { m i n } _ { w } L _ { S } ( w )$ (possibly in conjunction with a regularizer on $\pmb { w }$ ) using an optimization procedure such as SGD or Adam. Unfortunately, however, for modern overparameterized models such as deep neural networks, typical optimization approaches can easily result in suboptimal performance at test time. In particular, for modern models, $L _ { S } ( w )$ is typically non-convex in $\pmb { w }$ , with multiple local and even global minima that may yield similar values of $L _ { S } ( w )$ while having significantly different generalization performance (i.e., significantly different values of $L _ { \mathcal { D } } ( \mathbf { \boldsymbol { w } } )$ ).
+
+Motivated by the connection between sharpness of the loss landscape and generalization, we propose a different approach: rather than seeking out parameter values $\pmb { w }$ that simply have low training loss value $L _ { S } ( w )$ , we seek out parameter values whose entire neighborhoods have uniformly low training loss value (equivalently, neighborhoods having both low loss and low curvature). The following theorem illustrates the motivation for this approach by bounding generalization ability in terms of neighborhood-wise training loss (full theorem statement and proof in Appendix A):
+
+Theorem (stated informally) 1. For any $\rho > 0$ , with high probability over training set $s$ generated from distribution $\mathcal { D }$ ,
+
+$$
+L _ { \mathcal { D } } ( \pmb { w } ) \leq \operatorname* { m a x } _ { \| \pmb { \epsilon } \| _ { 2 } \leq \rho } L _ { S } ( \pmb { w } + \pmb { \epsilon } ) + h ( \| \pmb { w } \| _ { 2 } ^ { 2 } / \rho ^ { 2 } ) ,
+$$
+
+where $h : \mathbb { R } _ { + } \to \mathbb { R } _ { + }$ is a strictly increasing function (under some technical conditions on $L _ { \mathcal { D } } ( \boldsymbol { w } ) ,$ ).
+
+To make explicit our sharpness term, we can rewrite the right hand side of the inequality above as
+
+$$
+[ \operatorname* { m a x } _ { \| \epsilon \| _ { 2 } \le \rho } L _ { S } ( \pmb { w } + \epsilon ) - L _ { S } ( \pmb { w } ) ] + L _ { S } ( \pmb { w } ) + h ( \| \pmb { w } \| _ { 2 } ^ { 2 } / \rho ^ { 2 } ) .
+$$
+
+The term in square brackets captures the sharpness of $L _ { S }$ at $\pmb { w }$ by measuring how quickly the training loss can be increased by moving from $\pmb { w }$ to a nearby parameter value; this sharpness term is then summed with the training loss value itself and a regularizer on the magnitude of $\pmb { w }$ . Given that the specific function $h$ is heavily influenced by the details of the proof, we substitute the second term with $\lambda | | w | | _ { 2 } ^ { 2 }$ for a hyperparameter $\lambda$ , yielding a standard L2 regularization term. Thus, inspired by the terms from the bound, we propose to select parameter values by solving the following SharpnessAware Minimization (SAM) problem:
+
+$$
+\operatorname* { m i n } _ { \pmb { w } } L _ { S } ^ { S A M } ( \pmb { w } ) + \lambda | | \pmb { w } | | _ { 2 } ^ { 2 } \mathrm { w h e r e } L _ { S } ^ { S A M } ( \pmb { w } ) \triangleq \operatorname* { m a x } _ { | | \epsilon | | _ { p } \leq \rho } L _ { S } ( \pmb { w } + \epsilon ) ,
+$$
+
+where $\rho \geq 0$ is a hyperparameter and $p \in [ 1 , \infty ]$ (we have generalized slightly from an L2-norm to a $p$ -norm in the maximization over $\epsilon$ , though we show empirically in appendix C.5 that $p = 2$ is typically optimal). Figure 1 shows1 the loss landscape for a model that converged to minima found by minimizing either $L _ { S } ( w )$ or $L _ { S } ^ { S A M } ( w )$ , illustrating that the sharpness-aware loss prevents the model from converging to a sharp minimum.
+
+In order to minimize $L _ { S } ^ { S A M } ( w )$ , we derive an efficient and effective approximation to $\nabla _ { \pmb { w } } L _ { S } ^ { S A M } ( \pmb { w } )$ by differentiating through the inner maximization, which in turn enables us to apply stochastic gradient descent directly to the SAM objective. Proceeding down this path, we first approximate the inner maximization problem via a first-order Taylor expansion of $L _ { S } ( w + \epsilon )$ w.r.t. $\epsilon$ around 0, obtaining
+
+$$
+\epsilon ^ { * } ( w ) \overset { \triangleq } { \underset { \| \epsilon \| _ { p } \leq \rho } { \operatorname { \mathbb { I } } } } \operatorname* { m a x } L _ { S } ( w + \epsilon ) \approx \underset { \| \epsilon \| _ { p } \leq \rho } { \operatorname { \arg m a x } } L _ { S } ( w ) + \epsilon ^ { T } \nabla _ { w } L _ { S } ( w ) = \underset { \| \epsilon \| _ { p } \leq \rho } { \operatorname { \arg m a x } } \epsilon ^ { T } \nabla _ { w } L _ { S } ( w ) .
+$$
+
+In turn, the value $\hat { \epsilon } ( w )$ that solves this approximation is given by the solution to a classical dual norm problem $( | \cdot | ^ { q - 1 }$ denotes elementwise absolute value and power)2:
+
+$$
+\begin{array} { r } { \hat { \epsilon } ( \pmb { w } ) = \rho \operatorname { s i g n } \left( \nabla _ { \pmb { w } } L _ { S } ( \pmb { w } ) \right) | \nabla _ { \pmb { w } } L _ { S } ( \pmb { w } ) | ^ { q - 1 } / \bigg ( \| \nabla _ { \pmb { w } } L _ { S } ( \pmb { w } ) \| _ { q } ^ { q } \bigg ) ^ { 1 / p } } \end{array}
+$$
+
+where $1 / p + 1 / q = 1$ . Substituting back into equation (1) and differentiating, we then have
+
+$$
+\begin{array} { c } { \nabla _ { w } L _ { \mathcal { S } } ^ { S A M } ( { \pmb w } ) \approx \nabla _ { { \pmb w } } L _ { \mathcal { S } } ( { \pmb w } + \hat { \epsilon } ( { \pmb w } ) ) = \displaystyle \frac { d ( { \pmb w } + \hat { \epsilon } ( { \pmb w } ) ) } { d { \pmb w } } \nabla _ { { \pmb w } } L _ { \mathcal { S } } ( { \pmb w } ) | _ { { \pmb w } + \hat { \epsilon } ( { \pmb w } ) } } \\ { = \nabla _ { { \pmb w } } L _ { \mathcal { S } } ( { \pmb w } ) | _ { { \pmb w } + \hat { \epsilon } ( { \pmb w } ) } + \displaystyle \frac { d \hat { \epsilon } ( { \pmb w } ) } { d { \pmb w } } \nabla _ { { \pmb w } } L _ { \mathcal { S } } ( { \pmb w } ) | _ { { \pmb w } + \hat { \epsilon } ( { \pmb w } ) } . } \end{array}
+$$
+
+This approximation to $\nabla _ { \pmb { w } } L _ { S } ^ { S A M } ( \pmb { w } )$ can be straightforwardly computed via automatic differentiation, as implemented in common libraries such as JAX, TensorFlow, and PyTorch. Though this computation implicitly depends on the Hessian of $L _ { S } ( w )$ because $\hat { \epsilon } ( w )$ is itself a function of $\mathrm { \nabla } \mathrm { \nabla } \varpi { L s } ( w )$ , the Hessian enters only via Hessian-vector products, which can be computed tractably without materializing the Hessian matrix. Nonetheless, to further accelerate the computation, we drop the second-order terms. obtaining our final gradient approximation:
+
+$$
+\nabla _ { \pmb { w } } L _ { S } ^ { S A M } ( \pmb { w } ) \approx \nabla _ { \pmb { w } } L _ { S } ( \pmb { w } ) | _ { \pmb { w } + \hat { \epsilon } ( \pmb { w } ) } .
+$$
+
+As shown by the results in Section 3, this approximation (without the second-order terms) yields an effective algorithm. In Appendix C.4, we additionally investigate the effect of instead including the second-order terms; in that initial experiment, including them surprisingly degrades performance, and further investigating these terms’ effect should be a priority in future work.
+
+We obtain the final SAM algorithm by applying a standard numerical optimizer such as stochastic gradient descent (SGD) to the SAM objective $\check { L _ { S } ^ { S A M } } ( w )$ , using equation 3 to compute the requisite objective function gradients. Algorithm 1 gives pseudo-code for the full SAM algorithm, using SGD as the base optimizer, and Figure 2 schematically illustrates a single SAM parameter update.
+
+Input: Training set $s$ , $\cup _ { i = 1 } ^ { n } \{ ( { \pmb x } _ { i } , { \pmb y } _ { i } ) \}$ , Loss function $l : \mathcal { W } \times \mathcal { X } \times \mathcal { Y }  \mathbb { R } _ { + }$ , Batch size $b$ , Step size $\eta > 0$ , Neighborhood size $\rho > 0$ .
+
+Output: Model trained with SAM Initialize weights $\pmb { w } _ { 0 }$ , $t = 0$ ; while not converged do
+
+Sample batch $B = \{ ( { \pmb x } _ { 1 } , { \pmb y } _ { 1 } ) , . . . ( { \pmb x } _ { b } , { \pmb y } _ { b } ) \}$ ;   
+Compute gradient $\nabla _ { w } L _ { B } ( w )$ of the batch’s training loss;   
+Compute $\hat { \epsilon } ( w )$ per equation 2;   
+Compute gradient approximation for the SAM objective (equation 3): $\pmb { g } = \mathcal { \bar { \nabla } } _ { w } L _ { B } ( \pmb { w } ) | _ { \pmb { w } + \hat { \epsilon } ( \pmb { w } ) }$ ;   
+Update weights: $\mathbf { } \mathbf { } \mathbf { } w _ { t + 1 } = \mathbf { } w _ { t } - \eta \mathbf { } g$ ;   
+$t = t + 1$ ;
+
+end return ${ \pmb w } _ { t }$
+
+Algorithm 1: SAM algorithm
+
+![](images/52e169c6b48de6607068b4d204d8336ae3e10850598281e2806272424b53f639.jpg)  
+Figure 2: Schematic of the SAM parameter update.
+
+# 3 EMPIRICAL EVALUATION
+
+In order to assess SAM’s efficacy, we apply it to a range of different tasks, including image classification from scratch (including on CIFAR-10, CIFAR-100, and ImageNet), finetuning pretrained models, and learning with noisy labels. In all cases, we measure the benefit of using SAM by simply replacing the optimization procedure used to train existing models with SAM, and computing the resulting effect on model generalization. As seen below, SAM materially improves generalization performance in the vast majority of these cases.
+
+# 3.1 IMAGE CLASSIFICATION FROM SCRATCH
+
+We first evaluate SAM’s impact on generalization for today’s state-of-the-art models on CIFAR-10 and CIFAR-100 (without pretraining): WideResNets with ShakeShake regularization (Zagoruyko & Komodakis, 2016; Gastaldi, 2017) and PyramidNet with ShakeDrop regularization (Han et al., 2016; Yamada et al., 2018). Note that some of these models have already been heavily tuned in prior work and include carefully chosen regularization schemes to prevent overfitting; therefore, significantly improving their generalization is quite non-trivial. We have ensured that our implementations’ generalization performance in the absence of SAM matches or exceeds that reported in prior work (Cubuk et al., 2018; Lim et al., 2019)
+
+All results use basic data augmentations (horizontal flip, padding by four pixels, and random crop). We also evaluate in the setting of more advanced data augmentation methods such as cutout regularization (Devries & Taylor, 2017) and AutoAugment (Cubuk et al., 2018), which are utilized by prior work to achieve state-of-the-art results.
+
+SAM has a single hyperparameter $\rho$ (the neighborhood size), which we tune via a grid search over $\{ 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 , 0 . \bar { 2 } , 0 . 5 \}$ using $10 \%$ of the training set as a validation set3. Please see appendix C.1 for the values of all hyperparameters and additional training details. As each SAM weight update requires two backpropagation operations (one to compute $\hat { \epsilon } ( w )$ and another to compute the final gradient), we allow each non-SAM training run to execute twice as many epochs as each SAM training run, and we report the best score achieved by each non-SAM training run across either the standard epoch count or the doubled epoch count4. We run five independent replicas of each experimental condition for which we report results (each with independent weight initialization and data shuffling), reporting the resulting mean error (or accuracy) on the test set, and the associated $9 5 \%$ confidence interval. Our implementations utilize JAX (Bradbury et al., 2018), and we train all models on a single host having 8 Nvidia $\mathrm { V 1 0 0 \ G P U s }$ . To compute the SAM update when parallelizing across multiple accelerators, we divide each data batch evenly among the accelerators, independently compute the SAM gradient on each accelerator, and average the resulting sub-batch SAM gradients to obtain the final SAM update.
+
+As seen in Table 1, SAM improves generalization across all settings evaluated for CIFAR-10 and CIFAR-100. For example, SAM enables a simple WideResNet to attain $1 . 6 \%$ test error, versus $2 . 2 \%$ error without SAM. Such gains have previously been attainable only by using more complex model architectures (e.g., PyramidNet) and regularization schemes (e.g., Shake-Shake, ShakeDrop); SAM provides an easily-implemented, model-independent alternative. Furthermore, SAM delivers improvements even when applied atop complex architectures that already use sophisticated regularization: for instance, applying SAM to a PyramidNet with ShakeDrop regularization yields $1 0 . 3 \%$ error on CIFAR-100, which is, to our knowledge, a new state-of-the-art on this dataset without the use of additional data.
+
+Beyond CIFAR- $\{ 1 0 , 1 0 0 \}$ , we have also evaluated SAM on the SVHN (Netzer et al., 2011) and Fashion-MNIST datasets (Xiao et al., 2017). Once again, SAM enables a simple WideResNet to achieve accuracy at or above the state-of-the-art for these datasets: $0 . 9 9 \%$ error for SVHN, and $3 . 5 9 \%$ for Fashion-MNIST. Details are available in appendix B.1.
+
+To assess SAM’s performance at larger scale, we apply it to ResNets (He et al., 2015) of different depths (50, 101, 152) trained on ImageNet (Deng et al., 2009). In this setting, following prior work (He et al., 2015; Szegedy et al., 2015), we resize and crop images to 224-pixel resolution, normalize them, and use batch size 4096, initial learning rate 1.0, cosine learning rate schedule, SGD optimizer with momentum 0.9, label smoothing of 0.1, and weight decay 0.0001. When applying SAM, we use $\rho = 0 . 0 5$ (determined via a grid search on ResNet-50 trained for 100 epochs). We train all models on ImageNet for up to 400 epochs using a Google Cloud TPUv3 and report top-1 and top-5 test error rates for each experimental condition (mean and $9 5 \%$ confidence interval across 5 independent runs).
+
+<table><tr><td colspan="2"></td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td></tr><tr><td>Model</td><td>Augmentation</td><td>SAM</td><td>SGD</td><td>SAM</td><td>SGD</td></tr><tr><td>WRN-28-10 (200 epochs) WRN-28-10 (200 epochs)</td><td>Basic Cutout</td><td>2.7±0.1 2.3±0.1</td><td>3.5±0.1 2.6±0.1</td><td>16.5±0.2 14.9±0.2</td><td>18.8±0.2 16.9±0.1</td></tr><tr><td>WRN-28-10 (200 epochs) WRN-28-10 (1800 epochs)</td><td>AA Basic</td><td>2.1±&lt;0.1 2.4±0.1</td><td>2.3±0.1 3.5±0.1</td><td>13.6±0.2 16.3±0.2</td><td>15.8±0.2 19.1±0.1</td></tr><tr><td>WRN-28-10 (1800 epochs) WRN-28-10 (1800 epochs)</td><td>Cutout AA</td><td>2.1±0.1 1.6±0.1</td><td>2.7±0.1 2.2±&lt;0.1</td><td>14.0±0.1 12.8±0.2</td><td>17.4±0.1 16.1±0.2</td></tr><tr><td>Shake-Shake (26 2x96d) Shake-Shake (26 2x96d)</td><td>Basic Cutout</td><td>2.3±&lt;0.1 2.0±&lt;0.1</td><td>2.7±0.1 2.3±0.1</td><td>15.1±0.1 14.2±0.2</td><td>17.0±0.1 15.7±0.2</td></tr><tr><td>Shake-Shake (26 2x96d)</td><td>AA</td><td>1.6±&lt;0.1</td><td>1.9±0.1</td><td>12.8±0.1</td><td>14.1±0.2</td></tr><tr><td>PyramidNet</td><td>Basic</td><td>2.7±0.1</td><td>4.0±0.1</td><td>14.6±0.4</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>19.7±0.3</td></tr><tr><td>PyramidNet</td><td>Cutout</td><td>1.9±0.1</td><td>2.5±0.1</td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>12.6±0.2</td><td>16.4±0.1</td></tr><tr><td>PyramidNet</td><td>AA</td><td>1.6±0.1</td><td>1.9±0.1</td><td>11.6±0.1</td><td>14.6±0.1</td></tr><tr><td>PyramidNet+ShakeDrop</td><td>Basic</td><td>2.1±0.1</td><td>2.5±0.1</td><td>13.3±0.2</td><td></td></tr><tr><td>PyramidNet+ShakeDrop</td><td>Cutout</td><td></td><td></td><td></td><td>14.5±0.1</td></tr><tr><td></td><td></td><td>1.6±&lt;0.1</td><td>1.9±0.1</td><td>11.3±0.1</td><td>11.8±0.2</td></tr><tr><td>PyramidNet+ShakeDrop</td><td>AA</td><td>1.4±&lt;0.1</td><td>1.6±&lt;0.1</td><td>10.3±0.1</td><td>10.6±0.1</td></tr></table>
+
+Table 1: Results for SAM on state-of-the-art models on CIFAR- $\{ 1 0 , 1 0 0 \}$ (WRN $=$ WideResNet;   
+AA $=$ AutoAugment; SGD is the standard non-SAM procedure used to train these models).
+
+As seen in Table 2, SAM again consistently improves performance, for example improving the ImageNet top-1 error rate of ResNet-152 from $2 0 . 3 \%$ to $1 8 . 4 \%$ . Furthermore, note that SAM enables increasing the number of training epochs while continuing to improve accuracy without overfitting. In contrast, the standard training procedure (without SAM) generally significantly overfits as training extends from 200 to 400 epochs.
+
+Table 2: Test error rates for ResNets trained on ImageNet, with and without SAM.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Epoch</td><td colspan="2">SAM</td><td colspan="2">Standard Training (No SAM)</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top-1</td><td>Top-5</td></tr><tr><td rowspan="3">ResNet-50</td><td>100</td><td>22.5±0.1</td><td>6.28±0.08</td><td>22.9±0.1</td><td>6.62±0.11</td></tr><tr><td>200</td><td>21.4±0.1</td><td>5.82±0.03</td><td>22.3±0.1</td><td>6.37±0.04</td></tr><tr><td>400</td><td>20.9±0.1</td><td>5.51±0.03</td><td>22.3±0.1</td><td>6.40±0.06</td></tr><tr><td rowspan="3">ResNet-101</td><td>100</td><td>20.2±0.1</td><td>5.12±0.03</td><td>21.2±0.1</td><td>5.66±0.05</td></tr><tr><td>200</td><td>19.4±0.1</td><td>4.76±0.03</td><td>20.9±0.1</td><td>5.66±0.04</td></tr><tr><td>400</td><td>19.0±&lt;0.01</td><td>4.65±0.05</td><td>22.3±0.1</td><td>6.41±0.06</td></tr><tr><td rowspan="3">ResNet-152</td><td>100</td><td>19.2±&lt;0.01</td><td>4.69±0.04</td><td>20.4±&lt;0.0</td><td>5.39±0.06</td></tr><tr><td>200</td><td>18.5±0.1</td><td>4.37±0.03</td><td>20.3±0.2</td><td>5.39±0.07</td></tr><tr><td>400</td><td>18.4±&lt;0.01</td><td>4.35±0.04</td><td>20.9±&lt;0.0</td><td>5.84±0.07</td></tr></table>
+
+# 3.2 FINETUNING
+
+Transfer learning by pretraining a model on a large related dataset and then finetuning on a smaller target dataset of interest has emerged as a powerful and widely used technique for producing highquality models for a variety of different tasks. We show here that SAM once again offers considerable benefits in this setting, even when finetuning extremely large, state-of-the-art, already high-performing models.
+
+In particular, we apply SAM to finetuning EfficentNet-b7 (pretrained on ImageNet) and EfficientNet-L2 (pretrained on ImageNet plus unlabeled JFT; input resolution 475) (Tan & Le, 2019; Kornblith et al., 2018; Huang et al., 2018). We initialize these models to publicly available checkpoints6 trained with RandAugment $8 4 . 7 \%$ accuracy on ImageNet) and NoisyStudent $8 8 . 2 \%$ accuracy on ImageNet), respectively. We finetune these models on each of several target datasets by training each model starting from the aforementioned checkpoint; please see the appendix for details of the hyperparameters used. We report the mean and $9 5 \%$ confidence interval of top-1 test error over 5 independent runs for each dataset.
+
+As seen in Table 3, SAM uniformly improves performance relative to finetuning without SAM. Furthermore, in many cases, SAM yields novel state-of-the-art performance, including $0 . 3 0 \%$ error on CIFAR-10, $3 . 9 2 \%$ error on CIFAR-100, and $1 1 . 3 9 \%$ error on ImageNet.
+
+Table 3: Top-1 error rates for finetuning EfficientNet-b7 (left; ImageNet pretraining only) and EfficientNet-L2 (right; pretraining on ImageNet plus additional data, such as JFT) on various downstream tasks. Previous state-of-the-art (SOTA) includes EfficientNet (EffNet) (Tan & Le, 2019), Gpipe (Huang et al., 2018), DAT (Ngiam et al., 2018), BiT-M/L (Kolesnikov et al., 2020), KDforAA (Wei et al., 2020), TBMSL-Net (Zhang et al., 2020), and ViT (Dosovitskiy et al., 2020).   
+
+<table><tr><td>Dataset</td><td>EffNet-b7 + SAM</td><td>EffNet-b7</td><td>Prev. SOTA (ImageNet only)</td><td>EffNet-L2 + SAM</td><td>EffNet-L2</td><td>Prev. SOTA</td></tr><tr><td>FGVC_Aircraft</td><td>6.80±0.06</td><td>8.15±0.08</td><td>5.3(TBMSL-Net)</td><td>4.82±0.08</td><td>5.80±0.1</td><td>5.3 (TBMSL-Net)</td></tr><tr><td>Flowers</td><td>0.63±0.02</td><td>1.16±0.05</td><td>0.7 (BiT-M)</td><td>0.35±0.01</td><td>0.40±0.02</td><td>0.37 (EffNet)</td></tr><tr><td>Oxford_IIIT_Pets</td><td>3.97±0.04</td><td>4.24±0.09</td><td>4.1 (Gpipe)</td><td>2.90±0.04</td><td>3.08±0.04</td><td>4.1 (Gpipe)</td></tr><tr><td>Stanford_Cars</td><td>5.18±0.02</td><td>5.94±0.06</td><td>5.0 (TBMSL-Net)</td><td>4.04±0.03</td><td>4.93±0.04</td><td>3.8 (DAT)</td></tr><tr><td>CIFAR-10</td><td>0.88±0.02</td><td>0.95±0.03</td><td>1(Gpipe)</td><td>0.30±0.01</td><td>0.34±0.02</td><td>0.63 (BiT-L)</td></tr><tr><td>CIFAR-100</td><td>7.44±0.06</td><td>7.68±0.06</td><td>7.83 (BiT-M)</td><td>3.92±0.06</td><td>4.07±0.08</td><td>6.49 (BiT-L)</td></tr><tr><td>Birdsnap</td><td>13.64±0.15</td><td>14.30±0.18</td><td>15.7 (EffNet)</td><td>9.93±0.15</td><td>10.31±0.15</td><td>14.5 (DAT)</td></tr><tr><td>Food101</td><td>7.02±0.02</td><td>7.17±0.03</td><td>7.0 (Gpipe)</td><td>3.82±0.01</td><td>3.97±0.03</td><td>4.7 (DAT)</td></tr><tr><td>ImageNet</td><td>15.14±0.03</td><td>15.3</td><td>14.2 (KDforAA)</td><td>11.39±0.02</td><td>11.8</td><td>11.45 (ViT)</td></tr></table>
+
+The fact that SAM seeks out model parameters that are robust to perturbations suggests SAM’s potential to provide robustness to noise in the training set (which would perturb the training loss landscape). Thus, we assess here the degree of robustness that SAM provides to label noise.
+
+In particular, we measure the effect of applying SAM in the classical noisy-label setting for CIFAR-10, in which a fraction of the training set’s labels are randomly flipped; the test set remains unmodified (i.e., clean). To ensure valid comparison to prior work, which often utilizes architectures specialized to the noisy-label setting, we train a simple model of similar size (ResNet-32) for 200 epochs, following Jiang et al. (2019). We evaluate five variants of model training: standard SGD, SGD with Mixup (Zhang et al., 2017), SAM, and ”bootstrapped” variants of SGD with Mixup and SAM (wherein the model is first trained as usual and then retrained from scratch on the labels predicted by the initially trained model). When applying SAM, we use $\rho = 0 . 1$ for all noise levels except $80 \%$ , for which we use $\rho = 0 . 0 5$ for more stable convergence. For the Mixup baselines, we tried all values of $\alpha \in \{ 1 , 8 , 1 6 , 3 2 \}$ and conservatively report the best score for each noise level.
+
+Table 4: Test accuracy on the clean test set for models trained on CIFAR-10 with noisy labels. Lower block is our implementation, upper block gives scores from the literature, per Jiang et al. (2019).   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Noise rate (%)</td></tr><tr><td>20</td><td>40</td><td>60</td><td>80</td></tr><tr><td>Sanchez et al. (2019)</td><td>94.0</td><td>92.8</td><td>90.3</td><td>74.1</td></tr><tr><td>Zhang &amp; Sabuncu (2018)</td><td>89.7</td><td>87.6</td><td>82.7</td><td>67.9</td></tr><tr><td>Lee et al. (2019)</td><td>87.1</td><td>81.8</td><td>75.4</td><td>-</td></tr><tr><td>Chen et al. (2019)</td><td>89.7</td><td>-</td><td>-</td><td>52.3</td></tr><tr><td>Huang et al. (2019)</td><td>92.6</td><td>90.3</td><td>43.4</td><td>-</td></tr><tr><td>MentorNet (2017)</td><td>92.0</td><td>91.2</td><td>74.2</td><td>60.0</td></tr><tr><td>Mixup (2017)</td><td>94.0</td><td>91.5</td><td>86.8</td><td>76.9</td></tr><tr><td>MentorMix (2019)</td><td>95.6</td><td>94.2</td><td>91.3</td><td>81.0</td></tr><tr><td>SGD</td><td>84.8</td><td>68.8</td><td>48.2</td><td>26.2</td></tr><tr><td>Mixup</td><td>93.0</td><td>90.0</td><td>83.8</td><td>70.2</td></tr><tr><td>Bootstrap + Mixup</td><td>93.3</td><td>92.0</td><td>87.6</td><td>72.0</td></tr><tr><td>SAM</td><td>95.1</td><td>93.4</td><td>90.5</td><td>77.9</td></tr><tr><td>Bootstrap + SAM</td><td>95.4</td><td>94.2</td><td>91.8</td><td>79.9</td></tr></table>
+
+As seen in Table 4, SAM provides a high degree of robustness to label noise, on par with that provided by state-of-the art procedures that specifically target learning with noisy labels. Indeed, simply training a model with SAM outperforms all prior methods specifically targeting label noise robustness, with the exception of MentorMix (Jiang et al., 2019). However, simply bootstrapping SAM yields performance comparable to that of MentorMix (which is substantially more complex).
+
+![](images/9371d92a9e14163ccd6ef179e59711a9d141c6db9274be38df469de6219e43c6.jpg)  
+Figure 3: (left) Evolution of the spectrum of the Hessian during training of a model with standard SGD (lefthand column) or SAM (righthand column). (middle) Test error as a function of $\rho$ for different values of $m$ . (right) Predictive power of $m$ -sharpness for the generalization gap, for different values of $m$ (higher means the sharpness measure is more correlated with actual generalization gap).
+
+# 4 SHARPNESS AND GENERALIZATION THROUGH THE LENS OF SAM
+
+# 4.1 $m$ -SHARPNESS
+
+Though our derivation of SAM defines the SAM objective over the entire training set, when utilizing SAM in practice, we compute the SAM update per-batch (as described in Algorithm 1) or even by averaging SAM updates computed independently per-accelerator (where each accelerator receives a subset of size $m$ of a batch, as described in Section 3). This latter setting is equivalent to modifying the SAM objective (equation 1) to sum over a set of independent $\epsilon$ maximizations, each performed on a sum of per-data-point losses on a disjoint subset of $m$ data points, rather than performing the $\epsilon$ maximization over a global sum over the training set (which would be equivalent to setting $m$ to the total training set size). We term the associated measure of sharpness of the loss landscape $m$ -sharpness.
+
+To better understand the effect of $m$ on SAM, we train a small ResNet on CIFAR-10 using SAM with a range of values of $m$ . As seen in Figure 3 (middle), smaller values of $m$ tend to yield models having better generalization ability. This relationship fortuitously aligns with the need to parallelize across multiple accelerators in order to scale training for many of today’s models.
+
+Intriguingly, the $m$ -sharpness measure described above furthermore exhibits better correlation with models’ actual generalization gaps as $m$ decreases, as demonstrated by Figure 3 (right)7. In particular, this implies that $m$ -sharpness with $m < n$ yields a better predictor of generalization than the full-training-set measure suggested by Theorem 1 in Section 2 above, suggesting an interesting new avenue of future work for understanding generalization.
+
+# 4.2 HESSIAN SPECTRA
+
+Motivated by the connection between geometry of the loss landscape and generalization, we constructed SAM to seek out minima of the training loss landscape having both low loss value and low curvature (i.e., low sharpness). To further confirm that SAM does in fact find minima having low curvature, we compute the spectrum of the Hessian for a WideResNet40-10 trained on CIFAR-10 for 300 steps both with and without SAM (without batch norm, which tends to obscure interpretation of the Hessian), at different epochs during training. Due to the parameter space’s dimensionality, we approximate the Hessian spectrum using the Lanczos algorithm of Ghorbani et al. (2019).
+
+Figure 3 (left) reports the resulting Hessian spectra. As expected, the models trained with SAM converge to minima having lower curvature, as seen in the overall distribution of eigenvalues, the maximum eigenvalue $\left( \lambda _ { \operatorname* { m a x } } \right)$ at convergence (approximately 24 without SAM, 1.0 with SAM), and the bulk of the spectrum (the ratio $\lambda _ { \operatorname* { m a x } } / \lambda _ { 5 }$ , commonly used as a proxy for sharpness (Jastrzebski et al., 2020); up to 11.4 without SAM, and 2.6 with SAM).
+
+# 5 RELATED WORK
+
+The idea of searching for “flat” minima can be traced back to Hochreiter & Schmidhuber (1995), and its connection to generalization has seen significant study (Shirish Keskar et al., 2016; Dziugaite & Roy, 2017; Neyshabur et al., 2017; Dinh et al., 2017). In a recent large scale empirical study, Jiang et al. (2019) studied 40 complexity measures and showed that a sharpness-based measure has highest correlation with generalization, which motivates penalizing sharpness. Hochreiter & Schmidhuber (1997) was perhaps the first paper on penalizing the sharpness, regularizing a notion related to Minimum Description Length (MDL). Other ideas which also penalize sharp minima include operating on diffused loss landscape (Mobahi, 2016) and regularizing local entropy (Chaudhari et al., 2016). Another direction is to not penalize the sharpness explicitly, but rather average weights during training; Izmailov et al. (2018) showed that doing so can yield flatter minima that can also generalize better. However, the measures of sharpness proposed previously are difficult to compute and differentiate through. In contrast, SAM is highly scalable as it only needs two gradient computations per iteration. The concurrent work of Sun et al. (2020) focuses on resilience to random and adversarial corruption to expose a model’s vulnerabilities; this work is perhaps closest to ours. Our work has a different basis: we develop SAM motivated by a principled starting point in generalization, clearly demonstrate SAM’s efficacy via rigorous large-scale empirical evaluation, and surface important practical and theoretical facets of the procedure (e.g., $m$ -sharpness). The notion of all-layer margin introduced by Wei & Ma (2020) is closely related to this work; one is adversarial perturbation over the activations of a network and the other over its weights, and there is some coupling between these two quantities.
+
+# 6 DISCUSSION AND FUTURE WORK
+
+In this work, we have introduced SAM, a novel algorithm that improves generalization by simultaneously minimizing loss value and loss sharpness; we have demonstrated SAM’s efficacy through a rigorous large-scale empirical evaluation. We have surfaced a number of interesting avenues for future work. On the theoretical side, the notion of per-data-point sharpness yielded by $m$ -sharpness (in contrast to global sharpness computed over the entire training set, as has typically been studied in the past) suggests an interesting new lens through which to study generalization. Methodologically, our results suggest that SAM could potentially be used in place of Mixup in robust or semi-supervised methods that currently rely on Mixup (giving, for instance, MentorSAM). We leave to future work a more in-depth investigation of these possibilities.
+
+# 7 ACKNOWLEDGMENTS
+
+We thank our colleagues at Google — Atish Agarwala, Xavier Garcia, Dustin Tran, Yiding Jiang, Basil Mustafa, Samy Bengio — for their feedback and insightful discussions. We also thank the JAX and FLAX teams for going above and beyond to support our implementation. We are grateful to Sven Gowal for his help in replicating EfficientNet using JAX, and Justin Gilmer for his implementation of the Lanczos algorithm8 used to generate the Hessian spectra. We thank Niru Maheswaranathan for his matplotlib mastery. We also thank David Samuel for providing a PyTorch implementation of $\mathrm { \bf S A M ^ { 9 } }$ .
+
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+# A APPENDIX
+
+# A.1 PAC BAYESIAN GENERALIZATION BOUND
+
+Below, we state a generalization bound based on sharpness.
+
+Theorem 2. For any $\rho > 0$ and any distribution $\mathcal { D }$ , with probability $1 - \delta$ over the choice of the training set $s \sim \mathcal { D }$ ,
+
+$$
+L _ { \mathcal { D } } ( \boldsymbol { w } ) \leq \operatorname* { m a x } _ { \| \epsilon \| _ { 2 } \leq \rho } L _ { S } ( \boldsymbol { w } + \epsilon ) + \sqrt { \frac { k \log \left( 1 + \frac { \| \boldsymbol { w } \| _ { 2 } ^ { 2 } } { \rho ^ { 2 } } \left( 1 + \sqrt { \frac { \log ( n ) } { k } } \right) ^ { 2 } \right) + 4 \log \frac { n } { \delta } + \tilde { O } ( 1 ) } { n - 1 } }
+$$
+
+where $n = | S |$ , $k$ is the number of parameters and we assumed $L _ { \mathcal { D } } ( \boldsymbol { w } ) \leq \mathbb { E } _ { \epsilon _ { i } \sim \mathcal { N } ( 0 , \rho ) } [ L _ { \mathcal { D } } ( \boldsymbol { w } + \boldsymbol { \epsilon } ) ] ,$ .
+
+The condition $L _ { \mathcal { D } } ( \boldsymbol { w } ) \le \mathbb { E } _ { \epsilon _ { i } \sim \mathcal { N } ( 0 , \rho ) } [ L _ { \mathcal { D } } ( \boldsymbol { w } + \boldsymbol { \epsilon } ) ]$ means that adding Gaussian perturbation should not decrease the test error. This is expected to hold in practice for the final solution but does not necessarily hold for any $\textbf { \em w }$ .
+
+Proof. First, note that the right hand side of the bound in the theorem statement is lower bounded by $\sqrt { k \log ( 1 + \| \pmb { w } \| _ { 2 } ^ { 2 } / \rho ^ { 2 } ) / ( 4 n ) }$ which is greater than 1 when $\| \pmb { w } \| _ { 2 } ^ { 2 } > \rho ^ { 2 } ( \exp ( 4 n / k ) - 1 )$ . In that case, the right hand side becomes greater than 1 in which case the inequality holds trivially. Therefore, in the rest of the proof, we only consider the case when $\| \pmb { w } \| _ { 2 } ^ { 2 } \leq \rho ^ { \bar { 2 } } ( \mathrm { e x p } ( 4 n / k ) - 1 )$ .
+
+The proof technique we use here is inspired from Chatterji et al. (2020). Using PAC-Bayesian generalization bound McAllester (1999) and following Dziugaite & Roy (2017), the following generalization bound holds for any prior $\mathcal { P }$ over parameters with probability $1 - \delta$ over the choice of the training set $s$ , for any posterior $\mathcal { Q }$ over parameters:
+
+$$
+\mathbb { E } _ { \pmb { w } \sim \mathcal { A } } [ L _ { \mathcal { D } } ( \pmb { w } ) ] \le \mathbb { E } _ { \pmb { w } \sim \mathcal { Q } } [ L _ { S } ( \pmb { w } ) ] + \sqrt { \frac { K L ( \mathcal { Q } | | \mathcal { P } ) + \log \frac { n } { \delta } } { 2 ( n - 1 ) } }
+$$
+
+Moreover, if $\mathscr P = \mathcal N ( \mu _ { P } , \sigma _ { P } ^ { 2 } I )$ and $\mathcal { Q } = \mathcal { N } ( \pmb { \mu } _ { Q } , \sigma _ { Q } ^ { 2 } \pmb { I } )$ , then the KL divergence can be written as follows:
+
+$$
+K L ( \mathcal { P } | | \mathcal { Q } ) = \frac { 1 } { 2 } \bigg [ \frac { k \sigma _ { Q } ^ { 2 } + \| \pmb { \mu _ { P } } - \pmb { \mu _ { Q } } \| _ { 2 } ^ { 2 } } { \sigma _ { P } ^ { 2 } } - k + k \log \bigg ( \frac { \sigma _ { P } ^ { 2 } } { \sigma _ { Q } ^ { 2 } } \bigg ) \bigg ]
+$$
+
+Given a posterior standard deviation $\sigma _ { Q }$ , one could choose a prior standard deviation $\sigma _ { P }$ to minimize the above KL divergence and hence the generalization bound by taking the derivative10 of the above
+
+KL with respect to $\sigma _ { P }$ and setting it to zero. We would then have $\sigma _ { P } ^ { * } { } ^ { 2 } = \sigma _ { Q } ^ { 2 } + \| \pmb { \mu } _ { P } - \pmb { \mu } _ { Q } \| _ { 2 } ^ { 2 } / k$ However, since $\sigma _ { P }$ should be chosen before observing the training data $s$ and $\mu _ { Q } , \sigma _ { Q }$ could depend on $s$ , we are not allowed to optimize $\sigma _ { P }$ in this way. Instead, one can have a set of predefined values for $\sigma _ { P }$ and pick the best one in that set. See Langford & Caruana (2002) for the discussion around this technique. Given fixed $a , b > 0$ , let $T = \{ c \exp ( ( 1 - j ) / k ) | j \in \mathbb { N } \}$ be that predefined set of values for $\sigma _ { P } ^ { \dot { 2 } }$ . If for any $j \in \mathbb N$ , the above PAC-Bayesian bound holds for $\sigma _ { P } ^ { 2 } = \dot { c } \exp ( ( 1 -$ $j ) / k )$ with probability $1 - \delta _ { j }$ with $\begin{array} { r } { \delta _ { j } ~ = ~ \frac { 6 \delta } { \pi ^ { 2 } j ^ { 2 } } } \end{array}$ π2j2 , then by the union bound, all above bounds hold simultaneously with probability at least $\begin{array} { r } { 1 - \bar { \sum _ { j = 1 } ^ { \infty } } \frac { 6 \delta } { \pi ^ { 2 } j ^ { 2 } } = 1 - \delta } \end{array}$ .
+
+Let $\sigma _ { Q } = \rho$ , $\pmb { \mu } _ { Q } = \pmb { w }$ and $\pmb { \mu } _ { P } = \mathbf { 0 }$ . Therefore, we have:
+
+$$
+\sigma _ { Q } ^ { 2 } + \| \pmb { \mu } _ { P } - \pmb { \mu } _ { Q } \| _ { 2 } ^ { 2 } / k \leq \rho ^ { 2 } + \| \pmb { w } \| _ { 2 } ^ { 2 } / k \leq \rho ^ { 2 } ( 1 + \exp ( 4 n / k ) )
+$$
+
+We now consider the bound that corresponds to $j = \lfloor 1 - k \log ( ( \rho ^ { 2 } + \| \pmb { w } \| _ { 2 } ^ { 2 } / k ) / c ) \rfloor$ . We can ensure that $j \in \mathbb N$ using inequality equation 7 and by setting $c = \rho ^ { 2 } ( 1 + \exp ( 4 n / k ) )$ . Furthermore, for $\sigma _ { P } ^ { 2 } = c \exp ( ( 1 - j ) / k )$ , we have:
+
+$$
+\rho ^ { 2 } + \| \pmb { w } \| _ { 2 } ^ { 2 } / k \leq \sigma _ { P } ^ { 2 } \leq \exp ( 1 / k ) \left( \rho ^ { 2 } + \| \pmb { w } \| _ { 2 } ^ { 2 } / k \right)
+$$
+
+Therefore, using the above value for $\sigma _ { P }$ , $\mathrm { K L }$ divergence can be bounded as follows:
+
+$$
+\begin{array} { r l } & { K L ( \mathcal { P } | | \mathcal { X } ) = \frac { 1 } { 2 } \bigg [ \frac { k \sigma _ { Q } ^ { 2 } + \| \mu _ { P } - \mu _ { Q } \| _ { 2 } ^ { 2 } } { \sigma _ { P } ^ { 2 } } - k + k \log \bigg ( \frac { \sigma _ { P } ^ { 2 } } { \sigma _ { Q } ^ { 2 } } \bigg ) \bigg ] } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ &  \quad \quad \quad \ \end{array}
+$$
+
+Given the bound that corresponds to $j$ holds with probability $1 - \delta _ { j }$ for $\begin{array} { r } { \delta _ { j } = \frac { 6 \delta } { \pi ^ { 2 } j ^ { 2 } } } \end{array}$ 6δπ2j2 , the log term in the bound can be written as:
+
+$$
+\begin{array} { r l } & { \log \frac { n } { \delta _ { j } } = \log \frac { n } { \delta } + \log \frac { \pi ^ { 2 } j ^ { 2 } } { 6 } } \\ & { \qquad \le \log \frac { n } { \delta } + \log \frac { \pi ^ { 2 } k ^ { 2 } \log ^ { 2 } ( c / ( \rho ^ { 2 } + \| w \| _ { 2 } ^ { 2 } / k ) ) } { 6 } } \\ & { \qquad \le \log \frac { n } { \delta } + \log \frac { \pi ^ { 2 } k ^ { 2 } \log ^ { 2 } ( c / \rho ^ { 2 } ) } { 6 } } \\ & { \qquad \le \log \frac { n } { \delta } + \log \frac { \pi ^ { 2 } k ^ { 2 } \log ^ { 2 } ( 1 + \exp ( 4 n / k ) ) } { 6 } } \\ & { \qquad \le \log \frac { n } { \delta } + \log \frac { \pi ^ { 2 } k ^ { 2 } ( 2 + 4 n / k ) ^ { 2 } } { 6 } } \\ & { \qquad \le \log \frac { n } { \delta } + \log \frac { \pi ^ { 2 } k ^ { 2 } ( 2 + 4 n / k ) ^ { 2 } } { 6 } } \\ & { \qquad \le \log \frac { n } { \delta } + 2 \log ( 6 n + 3 k ) } \end{array}
+$$
+
+Therefore, the generalization bound can be written as follows:
+
+$$
+\begin{array} { r } { \tilde { \Sigma } _ { \epsilon _ { i } \sim N ( 0 , \sigma ) } [ L _ { \mathcal { D } } ( w + \epsilon ) ] \leq \mathbb { E } _ { \epsilon _ { i } \sim \mathcal { N } ( 0 , \sigma ) } [ L _ { S } ( w + \epsilon ) ] + \sqrt { \frac { \frac { 1 } { 4 } k \log \left( 1 + \frac { \| w \| _ { 2 } ^ { 2 } } { k \sigma ^ { 2 } } \right) + \frac { 1 } { 4 } + \log \frac { n } { \delta } + 2 \log \left( 6 n + \sigma \right) } { n - 1 } } } \end{array}
+$$
+
+In the above bound, we have $\epsilon _ { i } \sim \mathcal { N } ( 0 , \sigma )$ . Therefore, $\| \epsilon \| _ { 2 } ^ { 2 }$ has chi-square distribution and by Lemma 1 in Laurent & Massart (2000), we have that for any positive $t$ :
+
+$$
+P ( \| \epsilon \| _ { 2 } ^ { 2 } - k \sigma ^ { 2 } \ge 2 \sigma ^ { 2 } \sqrt { k t } + 2 t \sigma ^ { 2 } ) \le \exp ( - t )
+$$
+
+Therefore, with probability $1 - 1 / \sqrt { n }$ we have that:
+
+$$
+\| \epsilon \| _ { 2 } ^ { 2 } \leq \sigma ^ { 2 } ( 2 \ln ( { \sqrt { n } } ) + k + 2 { \sqrt { k \ln ( { \sqrt { n } } ) } } ) \leq \sigma ^ { 2 } k \left( 1 + { \sqrt { \frac { \ln ( n ) } { k } } } \right) ^ { 2 } \leq \rho ^ { 2 }
+$$
+
+Substituting the above value for $\sigma$ back to the inequality and using theorem’s assumption gives us following inequality:
+
+$$
+\begin{array} { r l } & { L _ { \mathcal { G } } ( w ) \leq ( 1 - 1 / \sqrt { n } ) \displaystyle \operatorname* { m a x } _ { \| c \| \geq \rho } L s ( w + \epsilon ) + 1 / \sqrt { n } } \\ & { \quad \quad + \sqrt { \frac { 1 } { 4 } k \log \left( 1 + \frac { \| w \| _ { 2 } ^ { 2 } } { \rho ^ { 2 } } \left( 1 + \sqrt { \frac { \log ( n ) } { k } } \right) ^ { 2 } \right) + \log \frac { n } { \delta } + 2 \log ( 6 n + 3 k ) } } \\ & { \quad \quad \leq \displaystyle \operatorname* { m a x } _ { \| \epsilon \| \geq \rho } L s ( w + \epsilon ) + } \\ & { \quad \quad + \sqrt { \displaystyle \operatorname* { k i m } _ { \rho } \left( 1 + \frac { \| w \| _ { 2 } ^ { 2 } } { \rho ^ { 2 } } \left( 1 + \sqrt { \frac { \log ( n ) } { k } } \right) ^ { 2 } \right) + 4 \log \frac { n } { \delta } + 8 \log ( 6 n + 3 k ) } } \end{array}
+$$
+
+# B ADDITIONAL EXPERIMENTAL RESULTS
+
+# B.1 SVHN AND FASHION-MNIST
+
+We report in table 5 results obtained on SVHN and Fashion-MNIST datasets. On these datasets, SAM allows a simple WideResnet to reach or push state-of-the-art accuracy $0 . 9 9 \%$ error rate for SVHN, $3 . 5 9 \%$ for Fashion-MNIST).
+
+For SVHN, we used all the available data (73257 digits for training $\sec + 5 3 1 1 3 1$ additional samples). For auto-augment, we use the best policy found on this dataset as described in (Cubuk et al., 2018) plus cutout (Devries & Taylor, 2017). For Fashion-MNIST, the auto-augmentation line correspond to cutout only.
+
+Table 5: Results on SVHN and Fashion-MNIST.   
+
+<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>SVHN</td><td rowspan=1 colspan=1>Fashion-MNIST</td></tr><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Augmentation</td><td rowspan=1 colspan=1>SAM     Baseline</td><td rowspan=1 colspan=1>SAM     Baseline</td></tr><tr><td rowspan=1 colspan=1>Wide-ResNet-28-10Wide-ResNet-28-10</td><td rowspan=1 colspan=1>BasicAuto augment</td><td rowspan=1 colspan=1>1.42±0.02  1.58±0.030.99±0.01  1.14±0.04</td><td rowspan=1 colspan=1>3.98±0.05  4.57±0.073.61±0.06  3.86±0.14</td></tr><tr><td rowspan=1 colspan=1>Shake-Shake (26 2x96d)Shake-Shake (26 2x96d)</td><td rowspan=1 colspan=1>BasicAuto augment</td><td rowspan=1 colspan=1>1.44±0.02  1.58±0.051.07±0.02  1.03±0.02</td><td rowspan=1 colspan=1>3.97±0.09  4.37±0.063.59±0.01  3.76±0.07</td></tr></table>
+
+# C EXPERIMENT DETAILS
+
+# C.1 HYPERPARAMETERS FOR EXPERIMENTS
+
+We report in table 6 the hyper-parameters selected by gridsearch for the CIFAR experiments, and the ones for SVHN and Fashion-MNIST in 7. For CIFAR10, CIFAR100, SVHN and Fashion-MNIST, we use a batch size of 256 and determine the learning rate and weight decay used to train each model via a joint grid search prior to applying SAM; all other model hyperparameter values are identical to those used in prior work.
+
+For the Imagenet results (Resnet models), the models are trained for 100, 200 or 400 epochs on Google Cloud TPUv3 32 cores with a batch size of 4096. The initial learning rate is set to 1.0 and decayed using a cosine schedule. Weight decay is set to 0.0001 with SGD optimizer and momentum $= 0 . 9$ .
+
+Table 6: Hyper-parameter used to produce the CIFAR- $\{ 1 0 , 1 0 0 \}$ results   
+
+<table><tr><td>CIFAR Dataset</td><td>LR</td><td>WD</td><td>p (CIFAR-10)</td><td>p (CIFAR-100)</td></tr><tr><td>WRN 28-10 (200 epochs)</td><td>0.1</td><td>0.0005</td><td>0.05</td><td>0.1</td></tr><tr><td>WRN 28-10 (1800 epochs)</td><td>0.05</td><td>0.001</td><td>0.05</td><td>0.1</td></tr><tr><td>WRN26-2x6ShakeShake</td><td>0.02</td><td>0.0010</td><td>0.02</td><td>0.05</td></tr><tr><td>Pyramid vanilla</td><td>0.05</td><td>0.0005</td><td>0.05</td><td>0.2</td></tr><tr><td>Pyramid ShakeDrop (CIFAR-10)</td><td>0.02</td><td>0.0005</td><td>0.05</td><td>1</td></tr><tr><td>Pyramid ShakeDrop (CIFAR-100)</td><td>0.05</td><td>0.0005</td><td>1</td><td>0.05</td></tr></table>
+
+Table 7: Hyper-parameter used to produce the SVHN and Fashion-MNIST results   
+
+<table><tr><td></td><td>LR</td><td>WD</td><td>p</td></tr><tr><td rowspan="2">SVHN</td><td>WRN 0.01</td><td>0.0005</td><td>0.01</td></tr><tr><td>ShakeShake 0.01</td><td>0.0005</td><td>0.01</td></tr><tr><td rowspan="2">Fashion</td><td>WRN 0.1</td><td>0.0005</td><td>0.05</td></tr><tr><td>ShakeShake 0.1</td><td>0.0005</td><td>0.02</td></tr></table>
+
+Finally, for the noisy label experiments, we also found $\rho$ by gridsearch, computing the accuracy on a (non-noisy) validation set composed of a random subset of $10 \%$ of the usual CIFAR training samples. We report the validation accuracy of the bootstrapped version of SAM for different levels of noise and different $\rho$ in table 8.
+
+<table><tr><td></td><td>20%</td><td>40%</td><td>60%</td><td>80%</td></tr><tr><td>0</td><td>15.0%</td><td>31.2%</td><td>52.3%</td><td>73.5%</td></tr><tr><td>0.01</td><td>13.7%</td><td>28.7%</td><td>50.1%</td><td>72.9%</td></tr><tr><td>0.02</td><td>12.8%</td><td>27.8%</td><td>48.9%</td><td>73.1%</td></tr><tr><td>0.05</td><td>11.6%</td><td>25.6%</td><td>47.1%</td><td>21.0%</td></tr><tr><td>0.1</td><td>4.6%</td><td>6.0%</td><td>8.7%</td><td>56.1%</td></tr><tr><td>0.2</td><td>5.3%</td><td>7.4%</td><td>23.3%</td><td>77.1%</td></tr><tr><td>0.5</td><td>17.6%</td><td>40.9%</td><td>80.1%</td><td>89.9%</td></tr></table>
+
+Table 8: Validation accuracy of the bootstrapped-SAM for different levels of noise and different $\rho$
+
+# C.2 FINETUNING DETAILS
+
+Weights are initialized to the values provided by the publicly available checkpoints, except the last dense layer, which change size to accomodate the new number of classes, that is randomly initialized. We train all models with weight decay $1 e ^ { - 5 }$ as suggested in (Tan & Le, 2019), but we reduce the learning rate to 0.016 as the models tend to diverge for higher values. We use a batch size of 1024 on Google Cloud TPUv3 64 cores and cosine learning rate decay. Because other works train with batch size of 256, we train for 5k steps instead of $2 0 \mathrm { k }$ . We freeze the batch norm statistics and use them for normalization, effectively using the batch norm as we would at test time 11 We train the models using SGD with momentum 0.9 and cosine learning rate decay. For Efficientnet-L2, we use this time a batch size 512 to save memory and adjusted the number of training steps accordingly. For CIFAR, we use the same autoaugment policy as in the previous experiments. We do not use data augmentation for the other datasets, applying the same preprocessing as for the Imagenet experiments. We also scale down the learning rate to 0.008 as the batch size is now twice as small. We used Google Cloud TPUv3 128 cores. All other parameters stay the same. For Imagenet, we trained both models from checkpoint for 10 epochs using a learning rate of 0.1 and $\rho = 0 . 0 5$ . We do not randomly initialize the last layer as we did for the other datasets, but instead use the weights included in the checkpoint.
+
+# C.3 EXPERIMENTAL RESULTS WITH $\rho = 0 . 0 5$
+
+A big sensitivity to the choice of hyper-parameters would make a method less easy to use. To demonstrate that SAM performs even when $\rho$ is not finely tuned, we compiled the table for the CIFAR and the finetuning experiments using $\rho = 0 . 0 5$ . Please note that we already used $\rho = 0 . 0 5$ for all Imagenet experiments. We report those scores in table 9 and 10.
+
+Table 9: Results for the Cifar10/Cifar100 experiments, using $\rho \quad = \quad 0 . 0 5$ for all models/datasets/augmentations   
+
+<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>Cifar10</td><td rowspan=1 colspan=3>Cifar100</td></tr><tr><td rowspan=1 colspan=2>Model                         Augmentation</td><td rowspan=1 colspan=1>p=0.05 SGD</td><td rowspan=1 colspan=3>rho=0.05  SGD</td></tr><tr><td rowspan=4 colspan=1>WRN-28-10 (200 epochs)WRN-28-10 (200 epochs)WRN-28-10 (200 epochs)</td><td rowspan=4 colspan=1>BasicCutoutAA</td><td rowspan=4 colspan=1>2.7      3.52.3      2.62.1      2.3</td><td rowspan=1 colspan=3>16.5     18.8</td></tr><tr><td rowspan=1 colspan=1></td><td></td><td rowspan=1 colspan=1>14.9     16.9</td></tr><tr><td rowspan=2 colspan=1></td><td></td><td></td></tr><tr><td rowspan=1 colspan=3>13.6     15.8</td></tr><tr><td rowspan=2 colspan=1>WRN-28-10 (1800 epochs)WRN-28-10 (1800 ep0chs)WRN-28-10 (1800 epochs)</td><td rowspan=2 colspan=1>BasicCutoutAA</td><td rowspan=2 colspan=1>2.4      3.52.1      2.71.6      2.2</td><td rowspan=1 colspan=2>16.3</td><td rowspan=1 colspan=1>16.3     19.1</td></tr><tr><td rowspan=1 colspan=3>14.0     17.412.8     16.1</td><td rowspan=1 colspan=2>14.0     17.4</td></tr><tr><td rowspan=2 colspan=1>WRN 26-2x6 ssWRN 26-2x6 ssWRN 26-2x6 ss</td><td rowspan=2 colspan=1>BasicCutoutAA</td><td rowspan=2 colspan=1>2.4      2.72.0      2.31.7      1.9</td><td rowspan=1 colspan=3>15.1     17.0</td></tr><tr><td rowspan=1 colspan=3>14.2     15.712.8     14.1</td></tr><tr><td rowspan=3 colspan=1>PyramidNetPyramidNetPyramidNet</td><td rowspan=3 colspan=1>BasicCutoutAA</td><td rowspan=3 colspan=1>2.1      4.01.6      2.51.4      1.9</td><td rowspan=1 colspan=3>15.4     19.7</td></tr><tr><td rowspan=1 colspan=3>13.1     16.4</td></tr><tr><td rowspan=1 colspan=3>12.1     14.6</td></tr><tr><td rowspan=3 colspan=1>PyramidNet+ShakeDropPyramidNet+ShakeDropPyramidNet+ShakeDrop</td><td rowspan=3 colspan=1>BasicCutoutAA</td><td rowspan=1 colspan=1>2.1      2.5</td><td rowspan=1 colspan=3>13.3     14.5</td></tr><tr><td rowspan=2 colspan=1>1.6      1.91.4      1.6</td><td rowspan=1 colspan=3>11.3     11.8</td></tr><tr><td rowspan=1 colspan=3>10.3     10.6</td></tr></table>
+
+Table 10: Results for the the finetuning experiments, using $\rho = 0 . 0 5$ for all datasets.   
+
+<table><tr><td>Dataset</td><td>Efficientnet-b7 + SAM (optimal)</td><td>Efficientnet-b7 + SAM (p= 0.05)</td><td>Efficientnet-b7</td></tr><tr><td>FGVC_Aircraft</td><td>6.80</td><td>7.06</td><td>8.15</td></tr><tr><td>Flowers</td><td>0.63</td><td>0.81</td><td>1.16</td></tr><tr><td>Oxford_IIIT_Pets</td><td>3.97</td><td>4.15</td><td>4.24</td></tr><tr><td>Stanford_Cars</td><td>5.18</td><td>5.57</td><td>5.94</td></tr><tr><td>cifar10</td><td>0.88</td><td>0.88</td><td>0.95</td></tr><tr><td>cifar100</td><td>7.44</td><td>7.56</td><td>7.68</td></tr><tr><td>Birdsnap</td><td>13.64</td><td>13.64</td><td>14.30</td></tr><tr><td>Food101</td><td>7.02</td><td>7.06</td><td>7.17</td></tr></table>
+
+# C.4 ABLATION OF THE SECOND ORDER TERMS
+
+As described in section 2, computing the gradient of the sharpness aware objective yield some second order terms that are more expensive to compute. To analyze this ablation more in depth, we trained a Wideresnet- $4 0 { \mathrm { x } } 2 $ on CIFAR-10 using SAM with and without discarding the second order terms during training. We report the cosine similarity of the two updates in figure 5, along the training trajectory of both experiments. We also report the training error rate (evaluated at $\pmb { w } + \hat { \epsilon } ( \pmb { w } ) )$ ) and the test error rate (evaluated at $\pmb { w }$ ).
+
+We observe that during the first half of the training, discarding the second order terms does not impact the general direction of the training, as the cosine similarity between the first and second order updates are very close to 1. However, when the model nears convergence, the similarity between both types of updates becomes weaker. Fortunately, the model trained without the second order terms reaches a lower test error, showing that the most efficient method is also the one providing the best generalization on this example. The reason for this is quite unclear and should be analyzed in follow up work.
+
+![](images/5684223aff1e4cedcd5f1739713e059827bc48fc2e509395a2a9cd59d0446107.jpg)  
+Figure 4: Training and test error for the first and second order version of the algorithm.
+
+![](images/a19aa5fbf5b3fd9913f2d80b7cd508b35fa606a8cb42f4891b575187ff24a869.jpg)  
+Figure 5: Cosine similarity between the first and second order updates.
+
+# C.5 CHOICE OF P-NORM
+
+Our theorem is derived for $p = 2$ , although generalizations can be considered for $p \in [ 1 , + \infty ]$ (the expression of the bound becoming way more involved). Empirically, we validate that the choice $p = 2$ is optimal by training a wide resnet on cifar10 with SAM for $p = \infty$ (in which case we have $\hat { \epsilon } ( \pmb { w } ) = \bar { \rho } \mathrm { s i g n } ( \nabla _ { \pmb { w } } L _ { S } ( \pmb { w } ) ) )$ and $p = 2$ (giving $\begin{array} { r } { \hat { \pmb { \epsilon } } ( \pmb { w } ) = \frac { \rho } { | | \nabla _ { \pmb { w } } L _ { S } ( \pmb { w } ) | | _ { 2 } ^ { 2 } } \big ( \nabla _ { \pmb { w } } L _ { S } ( \pmb { w } ) \big ) \big ) } \end{array}$ . We do not consider the case $p = 1$ which would give us a perturbation on a single weight. As an additional ablation study, we also use random weight perturbations of a fixed Euclidean norm: $\begin{array} { r } { \hat { \epsilon } ( w ) = \frac { \rho } { | | z | | _ { 2 } ^ { 2 } } z } \end{array}$ with $\boldsymbol { z } \sim \mathcal { N } ( \mathbf { 0 } , \boldsymbol { I } _ { d } )$ . We report the test accuracy of the model in figure 6.
+
+![](images/9650a9f81d0db360f5864fb08e006a63ba375a6e675bc2d0e79c17a91953970c.jpg)  
+Figure 6: Test accuracy for a wide resnet trained on CIFAR10 with SAM, for different perturbation norms.
+
+We observe that adversarial perturbations outperform random perturbations, and that using $p = 2$ yield superior accuracy on this example.
+
+# C.6 SEVERAL ITERATIONS IN THE INNER MAXIMIZATION
+
+To empirically verify that the linearization of the inner problem is sensible, we trained a WideResnet on the CIFAR datasets using a variant of SAM that performs several iterations of projected gradient ascent to estimate max $L ( w + \epsilon )$ . We report the evolution of max $L ( w + \epsilon ) - L ( w )$ during training (where $L$ stands for the training error rate computed on the current batch) in Figure 7, along with the test accuracy and the estimated sharpness $\begin{array} { r } { ( \operatorname* { m a x } _ { \epsilon } L ( w + \epsilon ) - L ( w ) ) } \end{array}$ at the end of training in Table 11; we report means and standard deviations across 20 runs.
+
+For most of the training, one projected gradient step (as used in standard SAM) is sufficient to obtain a good approximation of the $\epsilon$ found with multiple inner maximization steps. We however observe that this approximation becomes weaker near convergence, where doing several iterations of projected gradient ascent yields a better $\epsilon$ (for example, on CIFAR-10, the maximum loss found on each batch is about $3 \%$ more when doing 5 steps of inner maximization, compared to when doing a single step). That said, as seen in Table 11, the test accuracy is not strongly affected by the number of inner maximization iterations, though on CIFAR-100 it does seem that several steps outperform a single step in a statistically significant way.
+
+![](images/11b80d159ae0d784fbaa0290896af30214fe571e71e465d4198ce34f9e47c0bc.jpg)  
+Figure 7: Evolution of $\operatorname* { m a x } _ { \epsilon } L ( w + \epsilon ) - L ( w )$ vs. training step, for different numbers of inner projected gradient steps.
+
+<table><tr><td rowspan="2">Numberof projected gradient steps</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td></tr><tr><td>Test error</td><td>Estimated sharpness</td><td>Test error</td><td>Estimated sharpness</td></tr><tr><td>1</td><td>2.77±0.03</td><td>0.17±0.03</td><td>16.72±0.08</td><td>0.82±0.05</td></tr><tr><td>2</td><td>2.76±0.03</td><td>0.82±0.03</td><td>16.59±0.08</td><td>1.83±0.05</td></tr><tr><td>3</td><td>2.73±0.04</td><td>1.49±0.05</td><td>16.62±0.09</td><td>2.36±0.03</td></tr><tr><td>5</td><td>2.77±0.03</td><td>2.26±0.05</td><td>16.60±0.06</td><td>2.82±0.04</td></tr></table>
+
+Table 11: Test error rate and estimated sharpness $\begin{array} { r } { ( \operatorname* { m a x } _ { \epsilon } L ( w + \epsilon ) - L ( w ) ) } \end{array}$ at the end of the training.

parse/train/H1lK_lBtvS/H1lK_lBtvS.md

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+# CLASSIFICATION-BASED ANOMALY DETECTION FOR GENERAL DATA
+
+Liron Bergman Yedid Hoshen School of Computer Science and Engineering The Hebrew University of Jerusalem, Israel
+
+# ABSTRACT
+
+Anomaly detection, finding patterns that substantially deviate from those seen previously, is one of the fundamental problems of artificial intelligence. Recently, classification-based methods were shown to achieve superior results on this task. In this work, we present a unifying view and propose an open-set method, GOAD, to relax current generalization assumptions. Furthermore, we extend the applicability of transformation-based methods to non-image data using random affine transformations. Our method is shown to obtain state-of-the-art accuracy and is applicable to broad data types. The strong performance of our method is extensively validated on multiple datasets from different domains.
+
+# 1 INTRODUCTION
+
+Detecting anomalies in perceived data is a key ability for humans and for artificial intelligence. Humans often detect anomalies to give early indications of danger or to discover unique opportunities. Anomaly detection systems are being used by artificial intelligence to discover credit card fraud, for detecting cyber intrusion, alert predictive maintenance of industrial equipment and for discovering attractive stock market opportunities. The typical anomaly detection setting is a one class classification task, where the objective is to classify data as normal or anomalous. The importance of the task stems from being able to raise an alarm when detecting a different pattern from those seen in the past, therefore triggering further inspection. This is fundamentally different from supervised learning tasks, in which examples of all data classes are observed.
+
+There are different possible scenarios for anomaly detection methods. In supervised anomaly detection, we are given training examples of normal and anomalous patterns. This scenario can be quite well specified, however obtaining such supervision may not be possible. For example in cyber security settings, we will not have supervised examples of new, unknown computer viruses making supervised training difficult. On the other extreme, fully unsupervised anomaly detection, obtains a stream of data containing normal and anomalous patterns and attempts to detect the anomalous data. In this work we deal with the semi-supervised scenario. In this setting, we have a training set of normal examples (which contains no anomalies). After training the anomaly detector, we detect anomalies in the test data, containing both normal and anomalous examples. This supervision is easy to obtain in many practical settings and is less difficult than the fully-unsupervised case.
+
+Many anomaly detection methods have been proposed over the last few decades. They can be broadly classified into reconstruction and statistically based methods. Recently, deep learning methods based on classification have achieved superior results. Most semi-supervised classificationbased methods attempt to solve anomaly detection directly, despite only having normal training data. One example is: Deep-SVDD (Ruff et al., 2018) - one-class classification using a learned deep space. Another type of classification-based methods is self-supervised i.e. methods that solve one or more classification-based auxiliary tasks on the normal training data, and this is shown to be useful for solving anomaly detection, the task of interest e.g. (Golan & El-Yaniv, 2018). Self-supervised classification-based methods have been proposed with the object of image anomaly detection, but we show that by generalizing the class of transformations they can apply to all data types.
+
+In this paper, we introduce a novel technique, GOAD, for anomaly detection which unifies current state-of-the-art methods that use normal training data only and are based on classification. Our method first transforms the data into $M$ subspaces, and learns a feature space such that inter-class separation is larger than intra-class separation. For the learned features, the distance from the cluster center is correlated with the likelihood of anomaly. We use this criterion to determine if a new data point is normal or anomalous. We also generalize the class of transformation functions to include affine transformation which allows our method to generalize to non-image data. This is significant as tabular data is probably the most important for applications of anomaly detection. Our method is evaluated on anomaly detection on image and tabular datasets (cyber security and medical) and is shown to significantly improve over the state-of-the-art.
+
+# 1.1 PREVIOUS WORKS
+
+Anomaly detection methods can be generally divided into the following categories:
+
+Reconstruction Methods: Some of the most common anomaly detection methods are reconstructionbased. The general idea behind such methods is that every normal sample should be reconstructed accurately using a limited set of basis functions, whereas anomalous data should suffer from larger reconstruction costs. The choice of features, basis and loss functions differentiates between the different methods. Some of the earliest methods use: nearest neighbors (Eskin et al., 2002), low-rank PCA (Jolliffe, 2011; Candes et al., 2011) or K-means (Hartigan & Wong, 1979) as the reconstruction \` basis. Most recently, neural networks were used (Sakurada & Yairi, 2014; Xia et al., 2015) for learning deep basis functions for reconstruction. Another set of recent methods (Schlegl et al., 2017; Deecke et al., 2018) use GANs to learn a reconstruction basis function. GANs suffer from mode-collapse and are difficult to invert, which limits the performance of such methods.
+
+Distributional Methods: Another set of commonly used methods are distribution-based. The main theme in such methods is to model the distribution of normal data. The expectation is that anomalous test data will have low likelihood under the probabilistic model while normal data will have higher likelihoods. Methods differ in the features used to describe the data and the probabilistic model used to estimate the normal distribution. Some early methods used Gaussian or Gaussian mixture models. Such models will only work if the data under the selected feature space satisfies the probabilistic assumptions implicied by the model. Another set of methods used non-parametric density estimate methods such as kernel density estimate (Parzen, 1962). Recently, deep learning methods (autoencoders or variational autoencoders) were used to learn deep features which are sometimes easier to model than raw features (Yang et al., 2017). DAGMM introduced by Zong et al. (2018) learn the probabilistic model jointly with the deep features therefore shaping the features space to better conform with the probabilistic assumption.
+
+Classification-Based Methods: Another paradigm for anomaly detection is separation between space regions containing normal data from all other regions. An example of such approach is One-Class SVM (Scholkopf et al., 2000), which trains a classifier to perform this separation. Learning a good feature space for performing such separation is performed both by the classic kernel methods as well as by the recent deep learning approach (Ruff et al., 2018). One of the main challenges in unsupervised (or semi-supervised) learning is providing an objective for learning features that are relevant to the task of interest. One method for learning good representations in a self-supervised way is by training a neural network to solve an auxiliary task for which obtaining data is free or at least very inexpensive. Auxiliary tasks for learning high-quality image features include: video frame prediction (Mathieu et al., 2016), image colorization (Zhang et al., 2016; Larsson et al., 2016), puzzle solving (Noroozi & Favaro, 2016) - predicting the correct order of random permuted image patches. Recently, Gidaris et al. (2018) used a set of image processing transformations (rotation by 0, 90, 180, 270 degrees around the image axis, and predicted the true image orientation has been used to learn high-quality image features. Golan & El-Yaniv (2018), have used similar image-processing task prediction for detecting anomalies in images. This method has shown good performance on detecting images from anomalous classes. In this work, we overcome some of the limitations of previous classification-based methods and extend their applicability of self-supervised methods to general data types. We also show that our method is more robust to adversarial attacks.
+
+# 2 CLASSIFICATION-BASED ANOMALY DETECTION
+
+Classification-based methods have dominated supervised anomaly detection. In this section we will analyse semi-supervised classification-based methods:
+
+Let us assume all data lies in space $R ^ { L }$ (where $L$ is the data dimension). Normal data lie in subspace $X \subset R ^ { L }$ . We assume that all anomalies lie outside $X$ . To detect anomalies, we would therefore like to build a classifier $C$ , such that $C ( x ) = 1$ if $x \in X$ and $C ( x ) = 0$ if $x \in R ^ { L } \backslash X$ .
+
+One-class classification methods attempt to learn $C$ directly as $P ( x \in X )$ . Classical approaches have learned a classifier either in input space or in a kernel space. Recently, Deep-SVDD (Ruff et al., 2018) learned end-to-end to i) transform the data to an isotropic feature space $f ( x )$ ii) fit the minimal hypersphere of radius $R$ and center $c _ { 0 }$ around the features of the normal training data. Test data is classified as anomalous if the following normality score is positive: $\| f ( x ) - \bar { c } _ { 0 } \| ^ { 2 } - R ^ { 2 }$ . Learning an effective feature space is not a simple task, as the trivial solution of $f ( x ) = 0 \ \forall \ x$ results in the smallest hypersphere, various tricks are used to avoid this possibility.
+
+Geometric-transformation classification (GEOM), proposed by Golan & El-Yaniv (2018) first transforms the normal data subspace $X$ into $M$ subspaces $X _ { 1 } . . X _ { M }$ . This is done by transforming each image $x \in X$ using $M$ different geometric transformations (rotation, reflection, translation) into $T ( x , 1 ) . . T ( x , M )$ . Although these transformations are image specific, we will later extend the class of transformations to all affine transformations making this applicable to non-image data. They set an auxiliary task of learning a classifier able to predict the transformation label $m$ given transformed data point $T ( x , m )$ . As the training set consists of normal data only, each sample is $x \in X$ and the transformed sample is in $\cup _ { m } X _ { m }$ . The method attempts to estimate the following conditional probability:
+
+$$
+P ( m ^ { \prime } | T ( x , m ) ) = \frac { P ( T ( x , m ) \in X _ { m ^ { \prime } } ) P ( m ^ { \prime } ) } { \sum _ { \tilde { m } } P ( T ( x , m ) \in X _ { \tilde { m } } ) P ( \tilde { m } ) } = \frac { P ( T ( x , m ) \in X _ { m ^ { \prime } } ) } { \sum _ { \tilde { m } } P ( T ( x , m ) \in X _ { \tilde { m } } ) }
+$$
+
+Where the second equality follows by design of the training set, and where every training sample is transformed exactly once by each transformation leading to equal priors.
+
+For anomalous data $x \in R ^ { L } \backslash X$ , by construction of the subspace, if the transformations $T$ are oneto-one, it follows that the transformed sample does not fall in the appropriate subspace: $T ( x , m ) \in$ $R _ { \mathrm { ~ - ~ } } ^ { L } \backslash X _ { m }$ . GEOM uses $P ( m | T ( x , m ) )$ as a score for determining if $x$ is anomalous i.e. that $x \in$ $R ^ { L } \backslash X$ . GEOM gives samples with low probabilities $P ( m | T ( x , m ) )$ high anomaly scores.
+
+A significant issue with this methodology, is that the learned classifier $P ( m ^ { \prime } | T ( x , m ) )$ is only valid for samples $x \in X$ which were found in the training set. For $x \in R ^ { L } \backslash \mathrm { X }$ we should in fact have $P ( T ( x , m ) \in X _ { m ^ { \prime } } ) = 0$ for all $m = 1 . . M$ (as the transformed $x$ is not in any of the subsets). This makes the anomaly score $P ( m ^ { \prime } | T ( x , m ) )$ have very high variance for anomalies.
+
+One way to overcome this issue is by using examples of anomalies $x _ { a }$ and training $P ( m | T ( x , m ) ) =$ $\frac { 1 } { M }$ on anomalous data. This corresponds to the supervised scenario and was recently introduced as Outlier Exposure (Hendrycks et al., 2018). Although getting such supervision is possible for some image tasks (where large external datasets can be used) this is not possible in the general case e.g. for tabular data which exhibits much more variation between datasets.
+
+# 3 DISTANCE-BASED MULTIPLE TRANSFORMATION CLASSIFICATION
+
+We propose a novel method to overcome the generalization issues highlighted in the previous section by using ideas from open-set classification (Bendale & Boult, 2016). Our approach unifies one-class and transformation-based classification methods. Similarly to GEOM, we transform $X$ to $X _ { 1 } . . X _ { M }$ . We learn a feature extractor $f ( x )$ using a neural network, which maps the original input data into a feature representation. Similarly to deep OC methods, we model each subspace $X _ { m }$ mapped to the feature space $\{ f ( x ) | x \in X _ { m } \}$ as a sphere with center $c _ { m }$ . The probability of data point $x$ after transformation $m$ is parameterized by $\begin{array} { r } { P ( T ( x , m ) \in X _ { m } ^ { \prime } ) = \frac { 1 } { Z } e ^ { - ( f ( T ( x , m ) ) - c _ { m } ^ { \prime } ) ^ { 2 } } } \end{array}$ . The classifier predicting transformation $m$ given a transformed point is therefore:
+
+$$
+P ( m ^ { \prime } | T ( x , m ) ) = \frac { e ^ { - \| f ( T ( x , m ) ) - c _ { m ^ { \prime } } \| ^ { 2 } } } { \sum _ { \tilde { m } } e ^ { - \| f ( T ( x , m ) ) - c _ { \tilde { m } } \| ^ { 2 } } }
+$$
+
+The centers $c _ { m }$ are given by the average feature over the training set for every transformation i.e. $\begin{array} { r } { c _ { m } = \frac { 1 } { N } \sum _ { x \in X } f ( \hat { T ( x , m ) } ) } \end{array}$ . One option is to directly learn $f$ by optimizing cross-entropy between
+
+$P ( m ^ { \prime } | T ( x , m ) )$ and the correct label on the normal training set. In practice we obtained better results by training $f$ using the center triplet loss (He et al., 2018), which learns supervised clusters with low intra-class variation, and high-inter-class variation by optimizing the following loss function (where $s$ is a margin regularizing the distance between clusters):
+
+$$
+L = \sum _ { i } \operatorname* { m a x } ( \| f ( T ( x _ { i } , m ) ) - c _ { m } \| ^ { 2 } + s - m i n _ { m ^ { \prime } \neq m } \| f ( T ( x _ { i } , m ) ) - c _ { m ^ { \prime } } \| ^ { 2 } , 0 )
+$$
+
+Having learned a feature space in which the different transformation subspaces are well separated, we use the probability in Eq. 2 as a normality score. However, for data far away from the normal distributions, the distances from the means will be large. A small difference in distance will make the classifier unreasonably certain of a particular transformation. To add a general prior for uncertainty far from the training set, we add a small regularizing constant $\epsilon$ to the probability of each transformation. This ensures equal probabilities for uncertain regions:
+
+$$
+\tilde { P } ( m ^ { \prime } | T ( x , m ) ) = \frac { e ^ { - \| f ( T ( x , m ) ) - c _ { m ^ { \prime } } \| ^ { 2 } } + \epsilon } { \sum _ { \tilde { m } } e ^ { - \| f ( T ( x , m ) ) - c _ { \tilde { m } } \| ^ { 2 } } + M \cdot \epsilon }
+$$
+
+At test time we transform each sample by the $M$ transformations. By assuming independence between transformations, the probability that $x$ is normal (i.e. $x \in X$ ) is the product of the probabilities that all transformed samples are in their respective subspace. For log-probabilities the total score is given by:
+
+$$
+S c o r e ( x ) = - \log P ( x \in X ) = - \sum _ { m } \log \tilde { P } ( T ( x , m ) \in X _ { m } ) = - \sum _ { m } \log \tilde { P } ( m | T ( x , m ) )
+$$
+
+The score computes the degree of anomaly of each sample. Higher scores indicate a more anomalous sample.
+
+<table><tr><td>Algorithm 1 GOAD: Training Algorithm</td></tr><tr><td>Input: Normal training data x1,x2...X N</td></tr><tr><td>Transformations T(,1),T(,2)..T(, M)</td></tr><tr><td>Output: Feature extractor f, centers C1, C2...CM T(xi,1),T(xi,2)..T(xi,M) ← xi</td></tr><tr><td>// Transform each sample by all transformations 1 to M Find f,c1, C2...CM that optimize the triplet loss in Eq. 3</td></tr></table>
+
+# Algorithm 2 GOAD: Evaluation Algorithm
+
+Input: Test sample: $x$ , feature extractor: $f$ , centers: c1, c2...cM , transformations: $T ( \ u , \ d ^ { 1 } ) , T ( \ u , \ d ^ { 2 } ) . . . T ( \ u , \ d M )$   
+Output: Score(x)   
+$T ( x , 1 ) , T ( x , 2 ) . . . T ( x , M ) \gets x$   
+// Transform test sample by all transformations 1 to M   
+$P ( m | T ( x , m ) )  f ( T ( x , m ) ) , c _ { 1 } , c _ { 2 } . . . c _ { M }$ // Likelihood of predicting the correct transformation (Eq. 4)   
+$S c o r e ( x ) \gets P ( 1 | T ( x , 1 ) ) , P ( 2 | T ( x , 2 ) ) . . . P ( M | T ( x , M ) )$ // Aggregate probabilities to compute anomaly score (Eq. 5)
+
+# 4 PARAMETERIZING THE SET OF TRANSFORMATIONS
+
+Geometric transformations have been used previously for unsupervised feature learning by Gidaris et al. (2018) as well as by GEOM (Golan & El-Yaniv, 2018) for classification-based anomaly detection. This set of transformations is hand-crafted to work well with convolutional neural networks (CNNs) which greatly benefit from preserving neighborhood between pixels. This is however not a requirement for fully-connected networks.
+
+Anomaly detection often deals with non-image datasets e.g. tabular data. Tabular data is very commonly used on the internet e.g. for cyber security or online advertising. Such data consists of both discrete and continuous attributes with no particular neighborhoods or order. The data is onedimensional and rotations do not naturally generalize to it. To allow transformation-based methods to work on general data types, we therefore need to extend the class of transformations.
+
+We propose to generalize the set of transformations to the class of affine transformations (where we have a total of $M$ transformations):
+
+$$
+T ( x , m ) = W _ { m } x + b _ { m }
+$$
+
+It is easy to verify that all geometric transformations in Golan & El-Yaniv (2018) (rotation by a multiple of 90 degrees, flips and translations) are a special case of this class ( $x$ in this case is the set of image pixels written as a vector). The affine class is however much more general than mere permutations, and allows for dimensionality reduction, non-distance preservation and random transformation by sampling $W$ , $b$ from a random distribution.
+
+Apart from reduced variance across different dataset types where no apriori knowledge on the correct transformation classes exists, random transformations are important for avoiding adversarial examples. Assume an adversary wishes to change the label of a particular sample from anomalous to normal or vice versa. This is the same as requiring that $\tilde { P } ( m ^ { \prime } | T ( x , m ) )$ has low or high probability for $m ^ { \prime } = m$ . If $T$ is chosen deterministically, the adversary may create adversarial examples against the known class of transformations (even if the exact network parameters are unknown). Conversely, if $T$ is unknown, the adversary must create adversarial examples that generalize across different transformations, which reduces the effectiveness of the attack.
+
+To summarize, generalizing the set of transformations to the affine class allows us to: generalize to non-image data, use an unlimited number of transformations and choose transformations randomly which reduces variance and defends against adversarial examples.
+
+# 5 EXPERIMENTS
+
+We perform experiments to validate the effectiveness of our distance-based approach and the performance of the general class of transformations we introduced for non-image data.
+
+# 5.1 IMAGE EXPERIMENTS
+
+Cifar10: To evaluate the performance of our method, we perform experiments on the Cifar10 dataset. We use the same architecture and parameter choices of Golan & El-Yaniv (2018), with our distance-based approach. We use the standard protocol of training on all training images of a single digit and testing on all test images. Results are reported in terms of AUC. In our method, we used a margin of $s = 0 . 1$ (we also run GOAD with $s = 1$ , shown in the appendix). Similarly to He et al. (2018), to stabilize training, we added a softmax $^ +$ cross entropy loss, as well as $L _ { 2 }$ norm regularization for the extracted features $f ( x )$ . We compare our method with the deep oneclass method of Ruff et al. (2018) as well as Golan & El-Yaniv (2018) without and with Dirichlet weighting. We believe the correct comparison is without Dirichlet post-processing, as we also do not use it in our method. Our distance based approach outperforms the SOTA approach by Golan & El-Yaniv (2018), both with and without Dirichlet (which seems to improve performance on a few classes). This gives evidence for the importance of considering the generalization behavior outside the normal region used in training. Note that we used the same geometric transformations as Golan & El-Yaniv (2018). Random affine matrices did not perform competitively as they are not pixel order preserving, this information is effectively used by CNNs and removing this information hurts performance. This is a special property of CNN architectures and image/time series data. As a rule of thumb, fully-connected networks are not pixel order preserving and can fully utilize random affine matrices.
+
+Table 1: Anomaly Detection Accuracy on Cifar10 (ROC-AUC %)   
+
+<table><tr><td rowspan="2">Class</td><td colspan="4">Method</td></tr><tr><td>Deep-SVDD</td><td>GEOM (no Dirichlet)</td><td>GEOM (w. Dirichlet)</td><td>Ours</td></tr><tr><td>0</td><td>61.7 ± 1.3</td><td>76.0 ± 0.8</td><td>74.7 ± 0.4</td><td>77.2 ± 0.6</td></tr><tr><td>1</td><td>65.9 ± 0.7</td><td>83.0 ± 1.6</td><td>95.7 ± 0.0</td><td>96.7 ± 0.2</td></tr><tr><td>2</td><td>50.8 ± 0.3</td><td>79.5 ± 0.7</td><td>78.1 ± 0.4</td><td>83.3 ± 1.4</td></tr><tr><td>3</td><td>59.1 ± 0.4</td><td>71.4 ± 0.9</td><td>72.4 ± 0.5</td><td>77.7 ± 0.7</td></tr><tr><td>4</td><td>60.9 ± 0.3</td><td>83.5 ± 1.0</td><td>87.8 ± 0.2</td><td>87.8 ± 0.7</td></tr><tr><td>5</td><td>65.7 ± 0.8</td><td>84.0 ± 0.3</td><td>87.8 ± 0.1</td><td>87.8 ± 0.6</td></tr><tr><td>6</td><td>67.7 ± 0.8</td><td>78.4 ± 0.7</td><td>83.4 ± 0.5</td><td>90.0 ± 0.6</td></tr><tr><td>7</td><td>67.3 ± 0.3</td><td>89.3 ± 0.5</td><td>95.5 ± 0.1</td><td>96.1 ± 0.3</td></tr><tr><td>8</td><td>75.9 ± 0.4</td><td>88.6 ± 0.6</td><td>93.3 ± 0.0</td><td>93.8 ± 0.9</td></tr><tr><td>9</td><td>73.1 ± 0.4</td><td>82.4 ± 0.7</td><td>91.3 ± 0.1</td><td>92.0 ± 0.6</td></tr><tr><td>Average</td><td>64.8</td><td>81.6</td><td>86.0</td><td>88.2</td></tr></table>
+
+Table 2: Anomaly Detection Accuracy on FashionMNIST (ROC-AUC %)   
+
+<table><tr><td rowspan="2">Class</td><td colspan="4">Method</td></tr><tr><td>Deep-SVDD</td><td>GEOM (no Dirichlet)</td><td>GEOM (w. Dirichlet)</td><td>Ours</td></tr><tr><td>0</td><td>98.2</td><td>77.8 ± 5.9</td><td>99.4 ± 0.0</td><td>94.1 ± 0.9</td></tr><tr><td>1</td><td>90.3</td><td>79.1 ± 16.3</td><td>97.6 ± 0.1</td><td>98.5 ± 0.3</td></tr><tr><td>2</td><td>90.7</td><td>80.8 ± 6.9</td><td>91.1 ± 0.2</td><td>90.8 ± 0.4</td></tr><tr><td>3</td><td>94.2</td><td>79.2 ± 9.1</td><td>89.9 ± 0.4</td><td>91.6 ± 0.9</td></tr><tr><td>4</td><td>89.4</td><td>77.8 ± 3.3</td><td>92.1 ± 0.0</td><td>91.4 ± 0.3</td></tr><tr><td>5</td><td>91.8</td><td>58.0 ± 29.4</td><td>93.4 ± 0.9</td><td>94.8 ± 0.5</td></tr><tr><td>6</td><td>83.4</td><td>73.6 ± 8.7</td><td>83.3 ± 0.1</td><td>83.4 ± 0.4</td></tr><tr><td>7</td><td>98.8</td><td>87.4 ± 11.4</td><td>98.9 ± 0.1</td><td>97.9 ± 0.4</td></tr><tr><td>8</td><td>91.9</td><td>84.6 ± 5.6</td><td>90.8 ± 0.1</td><td>98.9 ± 0.1</td></tr><tr><td>9</td><td>99.0</td><td>99.5 ± 0.0</td><td>99.2 ± 0.0</td><td>99.2 ± 0.3</td></tr><tr><td>Average</td><td>92.8</td><td>79.8</td><td>93.5</td><td>94.1</td></tr></table>
+
+FasionMNIST: In Tab. 2, we present a comparison between our method (GOAD) and the strongest baseline methods (Deep SVDD and GEOM) on the FashionMNIST dataset. We used exactly the same setting as Golan & El-Yaniv (2018). GOAD was run with $s \ = \ 1$ . OCSVM and GEOM with Dirichlet were copied from their paper. We run their method without Dirichlet and presented it in the table (we verified the implementation by running their code with Dirichlet and replicated the numbers in the paper). It appears that GEOM is quite dependent on Dirichlet for this dataset, whereas we do not use it at all. GOAD outperforms all the baseline methods.
+
+Adversarial Robustness: Let us assume an attack model where the attacker knows the architecture and the normal training data and is trying to minimally modify anomalies to look normal. We examine the merits of two settings i) the adversary knows the transformations used (non-random) ii) the adversary uses another set of transformations. To measure the benefit of the randomized transformations, we train three networks A, B, C. Networks A and B use exactly the same transformations but random parameter initialization prior to training. Network C is trained using other randomly selected transformations. The adversary creates adversarial examples using PGD (Madry et al., 2017) based on network A (making anomalies appear like normal data). On Cifar10, we randomly selected 8 transformations from the full set of 72 for $A$ and $B$ , another randomly selected 8 transformations are used for $C$ . We measure the increase of false classification rate on the adversarial examples using the three networks. The average increase in performance of classifying transformation correctly on anomalies (causing lower anomaly scores) on the original network $A$ was $1 2 . 8 \%$ , the transfer performance for B causes an increase by $5 . 0 \%$ on network $B$ which shared the same set of transformation, and $3 \%$ on network $C$ that used other rotations. This shows the benefits of using random transformations.
+
+Table 3: Anomaly Detection Accuracy $( \% )$   
+
+<table><tr><td>Method</td><td colspan="7">Dataset</td></tr><tr><td></td><td colspan="2">Arrhythmia</td><td colspan="2">Thyroid</td><td colspan="2">KDD</td><td colspan="2">KDDRev</td></tr><tr><td></td><td>F1 Score</td><td>0</td><td>F1 Score</td><td>0</td><td>F1 Score</td><td>0</td><td>F1 Score</td><td>0</td></tr><tr><td>OC-SVM</td><td>45.8</td><td></td><td>38.9</td><td></td><td>79.5</td><td></td><td>83.2</td><td></td></tr><tr><td>E2E-AE</td><td>45.9</td><td></td><td>11.8</td><td></td><td>0.3</td><td></td><td>74.5</td><td></td></tr><tr><td>LOF</td><td>50.0</td><td>0.0</td><td>52.7</td><td>0.0</td><td>83.8</td><td>5.2</td><td>81.6</td><td>3.6</td></tr><tr><td>DAGMM</td><td>49.8</td><td></td><td>47.8</td><td></td><td>93.7</td><td></td><td>93.8</td><td></td></tr><tr><td>FB-AE</td><td>51.5</td><td>1.6</td><td>75.0</td><td>0.8</td><td>92.7</td><td>0.3</td><td>95.9</td><td>0.4</td></tr><tr><td>GOAD(Ours)</td><td>52.0</td><td>2.3</td><td>74.5</td><td>1.1</td><td>98.4</td><td>0.2</td><td>98.9</td><td>0.3</td></tr></table>
+
+# 5.2 TABULAR DATA EXPERIMENTS
+
+Datasets: We evaluate on small-scale medical datasets Arrhythmia, Thyroid as well as large-scale cyber intrusion detection datasets KDD and KDDRev. Our configuration follows that of Zong et al. (2018). Categorical attributes are encoded as one-hot vectors. For completeness the datasets are described in the appendix A.2. We train all compared methods on $5 0 \%$ of the normal data. The methods are evaluated on $5 0 \%$ of the normal data as well as all the anomalies.
+
+Baseline methods: The baseline methods evaluated are: One-Class SVM (OC-SVM, Scholkopf et al. (2000)), End-to-End Autoencoder (E2E-AE), Local Outlier Factor (LOF, Breunig et al. (2000)). We also evaluated deep distributional method DAGMM (Zong et al., 2018), choosing their strongest variant. To compare against ensemble methods e.g. Chen et al. (2017), we implemented the Feature Bagging Autoencoder (FB-AE) with autoencoders as the base classifier, feature bagging as the source of randomization, and average reconstruction error as the anomaly score. OC-SVM, E2E-AE and DAGMM results are directly taken from those reported by Zong et al. (2018). LOF and FB-AE were computed by us.
+
+Implementation of $G O A D$ : We randomly sampled transformation matrices using the normal distribution for each element. Each matrix has dimensionality $L \times r$ , where $L$ is the data dimension and $r$ is a reduced dimension. For Arryhthmia and Thyroid we used $r = 3 2$ , for KDD and KDDrev we used $r = 1 2 8$ and $r = 6 4$ respectively, the latter due to high memory requirements. We used 256 tasks for all datasets apart from KDD (64) due to high memory requirements. We set the bias term to 0. For $C$ we used fully-connected hidden layers and leaky-ReLU activations (8 hidden nodes for the small datasets, 128 and 32 for KDDRev and KDD). We optimized using ADAM with a learning rate of 0.001. Similarly to He et al. (2018), to stabilize the triplet center loss training, we added a softmax $^ +$ cross entropy loss. We repeated the large-scale experiments 5 times, and the small scale GOAD experiments 500 times (due to the high variance). We report the mean and standard deviation $( \sigma )$ . Following the protocol in Zong et al. (2018), the decision threshold value is chosen to result in the correct number of anomalies e.g. if the test set contains $N _ { a }$ anomalies, the threshold is selected so that the highest $N _ { a }$ scoring examples are classified as anomalies. True positives and negatives are evaluated in the usual way. Some experiments copied from other papers did not measure standard variation and we kept the relevant cell blank.
+
+# Results
+
+Arrhythmia: The Arrhythmia dataset was the smallest examined. A quantitative comparison on this dataset can be seen in Tab. 3. OC-SVM and DAGMM performed reasonably well. Our method is comparable to FB-AE. A linear classifier $C$ performed better than deeper networks (which suffered from overfitting). Early stopping after a single epoch generated the best results.
+
+Thyroid: Thyroid is a small dataset, with a low anomaly to normal ratio and low feature dimensionality. A quantitative comparison on this dataset can be seen in Tab. 3. Most baselines performed about equally well, probably due to the low dimensionality. On this dataset, we also found that early stopping after a single epoch gave the best results. The best results on this dataset, were obtained with a linear classifier. Our method is comparable to FB-AE and beat all other baselines by a wide margin.
+
+![](images/a525875e49259446f38693aa0662d0b42abc1102c3d160bf98a7b77cf54d2a65.jpg)  
+Figure 1: Left: Classification error for our method and DAGMM as a function of percentage of the anomalous examples in the training set (on the KDDCUP99 dataset). Our method consistently outperforms the baseline. Right: Classification error as a function of the number of transformations (on the KDDRev dataset). The error and instability decrease as a function of the number of transformations. For both, lower is better.
+
+KDDCUP99: The UCI KDD $1 0 \%$ dataset is the largest dataset examined. A quantitative comparison on this dataset can be seen in Tab. 3. The strongest baselines are FB-AE and DAGMM. Our method significantly outperformed all baselines. We found that large datasets have different dynamics from very small datasets. On this dataset, deep networks performed the best. We also, did not need early stopping. The results are reported after 25 epochs.
+
+KDD-Rev: The KDD-Rev dataset is a large dataset, but smaller than KDDCUP99 dataset. A quantitative comparison on this dataset can be seen in Tab. 3. Similarly to KDDCUP99, the best baselines are FB-AE and DAGMM, where FB-AE significantly outperforms DAGMM. Our method significantly outperformed all baselines. Due to the size of the dataset, we did not need early stopping. The results are reported after 25 epochs.
+
+Adversarial Robustness: Due to the large number of transformations and relatively small networks, adversarial examples are less of a problem for tabular data. PGD generally failed to obtain adversarial examples on these datasets. On KDD, transformation classification accuracy on anomalies was increased by $3 . 7 \%$ for the network the adversarial examples were trained on, $1 . 3 \%$ when transferring to the network with the same transformation and only $0 . 2 \%$ on the network with other randomly selected transformations. This again shows increased adversarial robustness due to random transformations.
+
+# Further Analysis
+
+Contaminated Data: This paper deals with the semi-supervised scenario i.e. when the training dataset contains only normal data. In some scenarios, such data might not be available but instead we might have a training dataset that contains a small percentage of anomalies. To evaluate the robustness of our method to this unsupervised scenario, we analysed the KDDCUP99 dataset, when $X \%$ of the training data is anomalous. To prepare the data, we used the same normal training data as before and added further anomalous examples. The test data consists of the same proportions as before. The results are shown in Fig. 1. Our method significantly outperforms DAGMM for all impurity values, and degrades more graceful than the baseline. This attests to the effectiveness of our approach. Results for the other datasets are presented in Fig. 3, showing similar robustness to contamination.
+
+Number of Tasks: One of the advantages of GOAD, is the ability to generate any number of tasks. We present the anomaly detection performance on the KDD-Rev dataset with different numbers of tasks in Fig. 1. We note that a small number of tasks (less than 16) leads to poor results. From 16 tasks, the accuracy remains stable. We found that on the smaller datasets (Thyroid, Arrhythmia) using a larger number of transformations continued to reduce $F _ { 1 }$ score variance between differently initialized runs (Fig. 2).
+
+# 6 DISCUSSION
+
+Openset vs. Softmax: The openset-based classification presented by GOAD resulted in performance improvement over the closed-set softmax approach on Cifar10 and FasionMNIST. In our experiments, it has also improved performance in KDDRev. Arrhythmia and Thyroid were comparable. As a negative result, performance of softmax was better on KDD $F _ { 1 } = 0 . 9 9$ ).
+
+Choosing the margin parameter $s$ : GOAD is not particularly sensitive to the choice of margin parameter $s$ , although choosing $s$ that is too small might cause some instability. We used a fixed value of $s = 1$ in our experiments, and recommend this value as a starting point.
+
+Other transformations: GOAD can also work with other types of transformations such as rotations or permutations for tabular data. In our experiments, we observed that these transformation types perform comparably but a little worse than affine transformations.
+
+Unsupervised training: Although most of our results are semi-supervised i.e. assume that no anomalies exist in the training set, we presented results showing that our method is more robust than strong baselines to a small percentage of anomalies in the training set. We further presented results in other datasets showing that our method degrades gracefully with a small amount of contamination. Our method might therefore be considered in the unsupervised settings.
+
+Deep vs. shallow classifiers: Our experiments show that for large datasets deep networks are beneficial (particularly for the full KDDCUP99), but are not needed for smaller datasets (indicating that deep learning has not benefited the smaller datasets). For performance critical operations, our approach may be used in a linear setting. This may also aid future theoretical analysis of our method.
+
+# 7 CONCLUSION
+
+In this paper, we presented a method for detecting anomalies for general data. This was achieved by training a classifier on a set of random auxiliary tasks. Our method does not require knowledge of the data domain, and we are able to generate an arbitrary number of random tasks. Our method significantly improve over the state-of-the-art.
+
+# REFERENCES
+
+Arthur Asuncion and David Newman. Uci machine learning repository, 2007.
+
+Abhijit Bendale and Terrance E Boult. Towards open set deep networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1563–1572, 2016.
+
+Markus M Breunig, Hans-Peter Kriegel, Raymond T $\mathrm { N g }$ , and Jorg Sander. Lof: identifying density- ¨ based local outliers. In ACM sigmod record, volume 29, pp. 93–104. ACM, 2000.
+
+Emmanuel J Candes, Xiaodong Li, Yi Ma, and John Wright. Robust principal component analysis? \` JACM, 2011.
+
+Jinghui Chen, Saket Sathe, Charu Aggarwal, and Deepak Turaga. Outlier detection with autoencoder ensembles. In ICDM, 2017.
+
+Lucas Deecke, Robert Vandermeulen, Lukas Ruff, Stephan Mandt, and Marius Kloft. Anomaly detection with generative adversarial networks. In ICLR, 2018.
+
+Eleazar Eskin, Andrew Arnold, Michael Prerau, Leonid Portnoy, and Sal Stolfo. A geometric framework for unsupervised anomaly detection. In Applications of data mining in computer security, pp. 77–101. Springer, 2002.
+
+Spyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. ICLR, 2018.
+
+Izhak Golan and Ran El-Yaniv. Deep anomaly detection using geometric transformations. In NeurIPS, 2018.
+
+John A Hartigan and Manchek A Wong. Algorithm as 136: A k-means clustering algorithm. Journal of the Royal Statistical Society. Series C (Applied Statistics), 1979.
+
+Xinwei He, Yang Zhou, Zhichao Zhou, Song Bai, and Xiang Bai. Triplet-center loss for multiview 3d object retrieval. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1945–1954, 2018.
+
+Dan Hendrycks, Mantas Mazeika, and Thomas G Dietterich. Deep anomaly detection with outlier exposure. arXiv preprint arXiv:1812.04606, 2018.
+
+Ian Jolliffe. Principal component analysis. Springer, 2011.
+
+Gustav Larsson, Michael Maire, and Gregory Shakhnarovich. Learning representations for automatic colorization. In ECCV, 2016.
+
+Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. arXiv preprint arXiv:1706.06083, 2017.
+
+Michael Mathieu, Camille Couprie, and Yann LeCun. Deep multi-scale video prediction beyond mean square error. ICLR, 2016.
+
+Mehdi Noroozi and Paolo Favaro. Unsupervised learning of visual representations by solving jigsaw puzzles. In ECCV, 2016.
+
+Emanuel Parzen. On estimation of a probability density function and mode. The annals of mathematical statistics, 1962.
+
+Shebuti Rayana. ODDS library http://odds.cs.stonybrook.edu, 2016.
+
+Lukas Ruff, Nico Gornitz, Lucas Deecke, Shoaib Ahmed Siddiqui, Robert Vandermeulen, Alexander Binder, Emmanuel Muller, and Marius Kloft. Deep one-class classification. In ¨ ICML, 2018.
+
+Mayu Sakurada and Takehisa Yairi. Anomaly detection using autoencoders with nonlinear dimensionality reduction. In MLSD. ACM, 2014.
+
+Thomas Schlegl, Philipp Seebock, Sebastian M Waldstein, Ursula Schmidt-Erfurth, and Georg ¨ Langs. Unsupervised anomaly detection with generative adversarial networks to guide marker discovery. In International Conference on Information Processing in Medical Imaging, 2017.
+
+Bernhard Scholkopf, Robert C Williamson, Alex J Smola, John Shawe-Taylor, and John C Platt. Support vector method for novelty detection. In NIPS, 2000.
+
+Yan Xia, Xudong Cao, Fang Wen, Gang Hua, and Jian Sun. Learning discriminative reconstructions for unsupervised outlier removal. In ECCV, 2015.
+
+Bo Yang, Xiao Fu, Nicholas D Sidiropoulos, and Mingyi Hong. Towards k-means-friendly spaces: Simultaneous deep learning and clustering. In ICML, 2017.
+
+Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In ECCV, 2016.
+
+Bo Zong, Qi Song, Martin Renqiang Min, Wei Cheng, Cristian Lumezanu, Daeki Cho, and Haifeng Chen. Deep autoencoding gaussian mixture model for unsupervised anomaly detection. ICLR, 2018.
+
+# A APPENDIX
+
+# A.1 IMAGE EXPERIMENTS
+
+Sensitive to margin s: We run Cifar10 experiments with $s = 0 . 1$ and $s = 1$ and presented the results in Fig. 4. The results were not affected much by the margin parameter. This is in-line with the rest of our empirical observations that GOAD is not very sensitive to the margin parameter.
+
+Table 4: Anomaly Detection Accuracy on Cifar10 $( \% )$   
+
+<table><tr><td rowspan="2">Class</td><td colspan="3">Method</td></tr><tr><td>GEOM (w. Dirichlet)</td><td>GOAD(s = 0.1)</td><td>GOAD(1.0)</td></tr><tr><td>0</td><td>74.7 ± 0.4</td><td>77.2 ± 0.6</td><td>77.9 ± 0.7</td></tr><tr><td>1</td><td>95.7 ± 0.0</td><td>96.7 ± 0.2</td><td>96.4 ± 0.9</td></tr><tr><td>2</td><td>78.1 ± 0.4</td><td>83.3 ± 1.4</td><td>81.8 ± 0.8</td></tr><tr><td>3</td><td>72.4 ± 0.5</td><td>77.7 ± 0.7</td><td>77.0 ± 0.7</td></tr><tr><td>4</td><td>87.8 ± 0.2</td><td>87.8 ± 0.7</td><td>87.7 ± 0.5</td></tr><tr><td>5</td><td>87.8 ± 0.1</td><td>87.8 ± 0.6</td><td>87.8 ± 0.7</td></tr><tr><td>6</td><td>83.4 ± 0.5</td><td>90.0 ± 0.6</td><td>90.9 ± 0.5</td></tr><tr><td>7</td><td>95.5 ± 0.1</td><td>96.1 ± 0.3</td><td>96.1 ± 0.2</td></tr><tr><td>8</td><td>93.3 ± 0.0</td><td>93.8 ± 0.9</td><td>93.3 ± 0.1</td></tr><tr><td>9</td><td>91.3 ± 0.1</td><td>92.0 ± 0.6</td><td>92.4 ± 0.3</td></tr><tr><td>Average</td><td>86.0</td><td>88.2</td><td>88.1</td></tr></table>
+
+# A.2 TABULAR DATASETS
+
+Following the evaluation protocol of Zong et al. (2018), 4 datasets are used in this comparison:
+
+Arrhythmia: A cardiology dataset from the UCI repository (Asuncion & Newman, 2007) containing attributes related to the diagnosis of cardiac arrhythmia in patients. The datasets consists of 16 classes: class 1 are normal patients, 2-15 contain different arrhythmia conditions, and class 16 contains undiagnosed cases. Following the protocol established by ODDS (Rayana, 2016), the smallest classes: 3, 4, 5, 7, 8, 9, 14, 15 are taken to be anomalous and the rest normal. Also following ODDS, the categorical attributes are dropped, the final attributes total 274.
+
+Thyroid: A medical dataset from the UCI repository (Asuncion & Newman, 2007), containing attributes related to whether a patient is hyperthyroid. Following ODDS (Rayana, 2016), from the 3 classes of the dataset, we designate hyperfunction as the anomalous class and the rest as normal. Also following ODDS only the 6 continuous attributes are used.
+
+KDD: The KDD Intrusion Detection dataset was created by an extensive simulation of a US Air Force LAN network. The dataset consists of the normal and 4 simulated attack types: denial of service, unauthorized access from a remote machine, unauthorized access from local superuser and probing. The dataset consists of around 5 million TCP connection records. Following the evaluation protocol in Zong et al. (2018), we use the UCI KDD $1 0 \%$ dataset, which is a subsampled version of the original dataset. The dataset contains 41 different attributes. 34 are continuous and 7 are categorical. Following Zong et al. (2018), we encode the categorical attributes using 1-hot encoding.
+
+Following Zong et al. (2018), we evaluate two different settings for the KDD dataset:
+
+KDDCUP99: In this configuration we use the entire UCI $1 0 \%$ dataset. As the non-attack class consists of only $2 0 \%$ of the dataset, it is treated as the anomaly in this case, while attacks are treated as normal.
+
+KDDCUP99-Rev: To better correspond to the actual use-case, in which the non-attack scenario is normal and attacks are anomalous, Zong et al. (2018) also evaluate on the reverse configuration, in which the attack data is sub-sampled to consist of $2 5 \%$ of the number of non-attack samples. The attack data is in this case designated as anomalous (the reverse of the KDDCUP99 dataset).
+
+In all the above datasets, the methods are trained on $5 0 \%$ of the normal data. The methods are evaluated on $5 0 \%$ of the normal data as well as all the anomalies.
+
+# A.3 NUMBER OF TASKS
+
+We provide plots of the number of auxiliary tasks vs. the anomaly detection accuracy (measured by $F _ { 1 , }$ ) for all datasets. The results are presented in Fig. 2. Performance increases rapidly up to a certain number of tasks (around 16). Afterwards more tasks reduce the variance of $F _ { 1 }$ scores between runs.
+
+![](images/55ef42d627c9ba88b9baf86d4f5c32923b4d5808311593e446f3e1f9d9b39111.jpg)  
+Figure 2: Plots of the number of auxiliary tasks vs. the anomaly detection accuracy (measured by $F _ { 1 }$ ) a) Arrhythmia b) Thyroid c) KDDRev d) KDDCup99 Accuracy often increases with the number of tasks, although the rate diminishes with the number of tasks.
+
+![](images/1714725f825a4c3d69f4543913fba5904bbb19a3a0b1c21569169b1ff2e913bb.jpg)  
+Figure 3: Plots of the degree of contamination vs. the anomaly detection accuracy (measured by $F _ { 1 }$ ) (left) KDDRev (center) KDDCup99 (right) Arrhythmia. GOAD is generally robust to the degree of contamination.
+
+# A.4 CONTAMINATION EXPERIMENTS
+
+We conduct contamination experiments for 3 datasets. Thyroid was omitted due to not having a sufficient number of anomalies. The protocol is different than that of KDDRev as we do not have unused anomalies for contamination. Instead, we split the anomalies into train and test. Train anomalies are used for contamination, test anomalies are used for evaluation. As DAGMM did not present results for the other datasets, we only present GOAD. GOAD was reasonably robust to contamination on KDD, KDDRev and Arrhythmia. The results are presented in Fig. 3

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+    {
+        "type": "text",
+        "text": "CLASSIFICATION-BASED ANOMALY DETECTION FOR GENERAL DATA ",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+    },
+    {
+        "type": "text",
+        "text": "Liron Bergman Yedid Hoshen School of Computer Science and Engineering The Hebrew University of Jerusalem, Israel ",
+        "bbox": [
+            183,
+            170,
+            485,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            452,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anomaly detection, finding patterns that substantially deviate from those seen previously, is one of the fundamental problems of artificial intelligence. Recently, classification-based methods were shown to achieve superior results on this task. In this work, we present a unifying view and propose an open-set method, GOAD, to relax current generalization assumptions. Furthermore, we extend the applicability of transformation-based methods to non-image data using random affine transformations. Our method is shown to obtain state-of-the-art accuracy and is applicable to broad data types. The strong performance of our method is extensively validated on multiple datasets from different domains. ",
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            426,
+            336,
+            443
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Detecting anomalies in perceived data is a key ability for humans and for artificial intelligence. Humans often detect anomalies to give early indications of danger or to discover unique opportunities. Anomaly detection systems are being used by artificial intelligence to discover credit card fraud, for detecting cyber intrusion, alert predictive maintenance of industrial equipment and for discovering attractive stock market opportunities. The typical anomaly detection setting is a one class classification task, where the objective is to classify data as normal or anomalous. The importance of the task stems from being able to raise an alarm when detecting a different pattern from those seen in the past, therefore triggering further inspection. This is fundamentally different from supervised learning tasks, in which examples of all data classes are observed. ",
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+        "text": "There are different possible scenarios for anomaly detection methods. In supervised anomaly detection, we are given training examples of normal and anomalous patterns. This scenario can be quite well specified, however obtaining such supervision may not be possible. For example in cyber security settings, we will not have supervised examples of new, unknown computer viruses making supervised training difficult. On the other extreme, fully unsupervised anomaly detection, obtains a stream of data containing normal and anomalous patterns and attempts to detect the anomalous data. In this work we deal with the semi-supervised scenario. In this setting, we have a training set of normal examples (which contains no anomalies). After training the anomaly detector, we detect anomalies in the test data, containing both normal and anomalous examples. This supervision is easy to obtain in many practical settings and is less difficult than the fully-unsupervised case. ",
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+    {
+        "type": "text",
+        "text": "Many anomaly detection methods have been proposed over the last few decades. They can be broadly classified into reconstruction and statistically based methods. Recently, deep learning methods based on classification have achieved superior results. Most semi-supervised classificationbased methods attempt to solve anomaly detection directly, despite only having normal training data. One example is: Deep-SVDD (Ruff et al., 2018) - one-class classification using a learned deep space. Another type of classification-based methods is self-supervised i.e. methods that solve one or more classification-based auxiliary tasks on the normal training data, and this is shown to be useful for solving anomaly detection, the task of interest e.g. (Golan & El-Yaniv, 2018). Self-supervised classification-based methods have been proposed with the object of image anomaly detection, but we show that by generalizing the class of transformations they can apply to all data types. ",
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+        "type": "text",
+        "text": "In this paper, we introduce a novel technique, GOAD, for anomaly detection which unifies current state-of-the-art methods that use normal training data only and are based on classification. Our method first transforms the data into $M$ subspaces, and learns a feature space such that inter-class separation is larger than intra-class separation. For the learned features, the distance from the cluster center is correlated with the likelihood of anomaly. We use this criterion to determine if a new data point is normal or anomalous. We also generalize the class of transformation functions to include affine transformation which allows our method to generalize to non-image data. This is significant as tabular data is probably the most important for applications of anomaly detection. Our method is evaluated on anomaly detection on image and tabular datasets (cyber security and medical) and is shown to significantly improve over the state-of-the-art. ",
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+    {
+        "type": "text",
+        "text": "1.1 PREVIOUS WORKS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Anomaly detection methods can be generally divided into the following categories: ",
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+        "type": "text",
+        "text": "Reconstruction Methods: Some of the most common anomaly detection methods are reconstructionbased. The general idea behind such methods is that every normal sample should be reconstructed accurately using a limited set of basis functions, whereas anomalous data should suffer from larger reconstruction costs. The choice of features, basis and loss functions differentiates between the different methods. Some of the earliest methods use: nearest neighbors (Eskin et al., 2002), low-rank PCA (Jolliffe, 2011; Candes et al., 2011) or K-means (Hartigan & Wong, 1979) as the reconstruction \\` basis. Most recently, neural networks were used (Sakurada & Yairi, 2014; Xia et al., 2015) for learning deep basis functions for reconstruction. Another set of recent methods (Schlegl et al., 2017; Deecke et al., 2018) use GANs to learn a reconstruction basis function. GANs suffer from mode-collapse and are difficult to invert, which limits the performance of such methods. ",
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+    {
+        "type": "text",
+        "text": "Distributional Methods: Another set of commonly used methods are distribution-based. The main theme in such methods is to model the distribution of normal data. The expectation is that anomalous test data will have low likelihood under the probabilistic model while normal data will have higher likelihoods. Methods differ in the features used to describe the data and the probabilistic model used to estimate the normal distribution. Some early methods used Gaussian or Gaussian mixture models. Such models will only work if the data under the selected feature space satisfies the probabilistic assumptions implicied by the model. Another set of methods used non-parametric density estimate methods such as kernel density estimate (Parzen, 1962). Recently, deep learning methods (autoencoders or variational autoencoders) were used to learn deep features which are sometimes easier to model than raw features (Yang et al., 2017). DAGMM introduced by Zong et al. (2018) learn the probabilistic model jointly with the deep features therefore shaping the features space to better conform with the probabilistic assumption. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Classification-Based Methods: Another paradigm for anomaly detection is separation between space regions containing normal data from all other regions. An example of such approach is One-Class SVM (Scholkopf et al., 2000), which trains a classifier to perform this separation. Learning a good feature space for performing such separation is performed both by the classic kernel methods as well as by the recent deep learning approach (Ruff et al., 2018). One of the main challenges in unsupervised (or semi-supervised) learning is providing an objective for learning features that are relevant to the task of interest. One method for learning good representations in a self-supervised way is by training a neural network to solve an auxiliary task for which obtaining data is free or at least very inexpensive. Auxiliary tasks for learning high-quality image features include: video frame prediction (Mathieu et al., 2016), image colorization (Zhang et al., 2016; Larsson et al., 2016), puzzle solving (Noroozi & Favaro, 2016) - predicting the correct order of random permuted image patches. Recently, Gidaris et al. (2018) used a set of image processing transformations (rotation by 0, 90, 180, 270 degrees around the image axis, and predicted the true image orientation has been used to learn high-quality image features. Golan & El-Yaniv (2018), have used similar image-processing task prediction for detecting anomalies in images. This method has shown good performance on detecting images from anomalous classes. In this work, we overcome some of the limitations of previous classification-based methods and extend their applicability of self-supervised methods to general data types. We also show that our method is more robust to adversarial attacks. ",
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+    {
+        "type": "text",
+        "text": "2 CLASSIFICATION-BASED ANOMALY DETECTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Classification-based methods have dominated supervised anomaly detection. In this section we will analyse semi-supervised classification-based methods: ",
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+        "type": "text",
+        "text": "Let us assume all data lies in space $R ^ { L }$ (where $L$ is the data dimension). Normal data lie in subspace $X \\subset R ^ { L }$ . We assume that all anomalies lie outside $X$ . To detect anomalies, we would therefore like to build a classifier $C$ , such that $C ( x ) = 1$ if $x \\in X$ and $C ( x ) = 0$ if $x \\in R ^ { L } \\backslash X$ . ",
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+    {
+        "type": "text",
+        "text": "One-class classification methods attempt to learn $C$ directly as $P ( x \\in X )$ . Classical approaches have learned a classifier either in input space or in a kernel space. Recently, Deep-SVDD (Ruff et al., 2018) learned end-to-end to i) transform the data to an isotropic feature space $f ( x )$ ii) fit the minimal hypersphere of radius $R$ and center $c _ { 0 }$ around the features of the normal training data. Test data is classified as anomalous if the following normality score is positive: $\\| f ( x ) - \\bar { c } _ { 0 } \\| ^ { 2 } - R ^ { 2 }$ . Learning an effective feature space is not a simple task, as the trivial solution of $f ( x ) = 0 \\ \\forall \\ x$ results in the smallest hypersphere, various tricks are used to avoid this possibility. ",
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+    {
+        "type": "text",
+        "text": "Geometric-transformation classification (GEOM), proposed by Golan & El-Yaniv (2018) first transforms the normal data subspace $X$ into $M$ subspaces $X _ { 1 } . . X _ { M }$ . This is done by transforming each image $x \\in X$ using $M$ different geometric transformations (rotation, reflection, translation) into $T ( x , 1 ) . . T ( x , M )$ . Although these transformations are image specific, we will later extend the class of transformations to all affine transformations making this applicable to non-image data. They set an auxiliary task of learning a classifier able to predict the transformation label $m$ given transformed data point $T ( x , m )$ . As the training set consists of normal data only, each sample is $x \\in X$ and the transformed sample is in $\\cup _ { m } X _ { m }$ . The method attempts to estimate the following conditional probability: ",
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+    {
+        "type": "equation",
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+        "text": "$$\nP ( m ^ { \\prime } | T ( x , m ) ) = \\frac { P ( T ( x , m ) \\in X _ { m ^ { \\prime } } ) P ( m ^ { \\prime } ) } { \\sum _ { \\tilde { m } } P ( T ( x , m ) \\in X _ { \\tilde { m } } ) P ( \\tilde { m } ) } = \\frac { P ( T ( x , m ) \\in X _ { m ^ { \\prime } } ) } { \\sum _ { \\tilde { m } } P ( T ( x , m ) \\in X _ { \\tilde { m } } ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Where the second equality follows by design of the training set, and where every training sample is transformed exactly once by each transformation leading to equal priors. ",
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+        "type": "text",
+        "text": "For anomalous data $x \\in R ^ { L } \\backslash X$ , by construction of the subspace, if the transformations $T$ are oneto-one, it follows that the transformed sample does not fall in the appropriate subspace: $T ( x , m ) \\in$ $R _ { \\mathrm { ~ - ~ } } ^ { L } \\backslash X _ { m }$ . GEOM uses $P ( m | T ( x , m ) )$ as a score for determining if $x$ is anomalous i.e. that $x \\in$ $R ^ { L } \\backslash X$ . GEOM gives samples with low probabilities $P ( m | T ( x , m ) )$ high anomaly scores. ",
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+        "type": "text",
+        "text": "A significant issue with this methodology, is that the learned classifier $P ( m ^ { \\prime } | T ( x , m ) )$ is only valid for samples $x \\in X$ which were found in the training set. For $x \\in R ^ { L } \\backslash \\mathrm { X }$ we should in fact have $P ( T ( x , m ) \\in X _ { m ^ { \\prime } } ) = 0$ for all $m = 1 . . M$ (as the transformed $x$ is not in any of the subsets). This makes the anomaly score $P ( m ^ { \\prime } | T ( x , m ) )$ have very high variance for anomalies. ",
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+        "type": "text",
+        "text": "One way to overcome this issue is by using examples of anomalies $x _ { a }$ and training $P ( m | T ( x , m ) ) =$ $\\frac { 1 } { M }$ on anomalous data. This corresponds to the supervised scenario and was recently introduced as Outlier Exposure (Hendrycks et al., 2018). Although getting such supervision is possible for some image tasks (where large external datasets can be used) this is not possible in the general case e.g. for tabular data which exhibits much more variation between datasets. ",
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+    {
+        "type": "text",
+        "text": "3 DISTANCE-BASED MULTIPLE TRANSFORMATION CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We propose a novel method to overcome the generalization issues highlighted in the previous section by using ideas from open-set classification (Bendale & Boult, 2016). Our approach unifies one-class and transformation-based classification methods. Similarly to GEOM, we transform $X$ to $X _ { 1 } . . X _ { M }$ . We learn a feature extractor $f ( x )$ using a neural network, which maps the original input data into a feature representation. Similarly to deep OC methods, we model each subspace $X _ { m }$ mapped to the feature space $\\{ f ( x ) | x \\in X _ { m } \\}$ as a sphere with center $c _ { m }$ . The probability of data point $x$ after transformation $m$ is parameterized by $\\begin{array} { r } { P ( T ( x , m ) \\in X _ { m } ^ { \\prime } ) = \\frac { 1 } { Z } e ^ { - ( f ( T ( x , m ) ) - c _ { m } ^ { \\prime } ) ^ { 2 } } } \\end{array}$ . The classifier predicting transformation $m$ given a transformed point is therefore: ",
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+        "type": "equation",
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+        "text": "$$\nP ( m ^ { \\prime } | T ( x , m ) ) = \\frac { e ^ { - \\| f ( T ( x , m ) ) - c _ { m ^ { \\prime } } \\| ^ { 2 } } } { \\sum _ { \\tilde { m } } e ^ { - \\| f ( T ( x , m ) ) - c _ { \\tilde { m } } \\| ^ { 2 } } }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The centers $c _ { m }$ are given by the average feature over the training set for every transformation i.e. $\\begin{array} { r } { c _ { m } = \\frac { 1 } { N } \\sum _ { x \\in X } f ( \\hat { T ( x , m ) } ) } \\end{array}$ . One option is to directly learn $f$ by optimizing cross-entropy between ",
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+        "type": "text",
+        "text": "$P ( m ^ { \\prime } | T ( x , m ) )$ and the correct label on the normal training set. In practice we obtained better results by training $f$ using the center triplet loss (He et al., 2018), which learns supervised clusters with low intra-class variation, and high-inter-class variation by optimizing the following loss function (where $s$ is a margin regularizing the distance between clusters): ",
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+        "img_path": "images/8c29c40ae8fa25dc4ce37b7e6ed5c6197721ba279e7825f04465c3208bf34820.jpg",
+        "text": "$$\nL = \\sum _ { i } \\operatorname* { m a x } ( \\| f ( T ( x _ { i } , m ) ) - c _ { m } \\| ^ { 2 } + s - m i n _ { m ^ { \\prime } \\neq m } \\| f ( T ( x _ { i } , m ) ) - c _ { m ^ { \\prime } } \\| ^ { 2 } , 0 )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Having learned a feature space in which the different transformation subspaces are well separated, we use the probability in Eq. 2 as a normality score. However, for data far away from the normal distributions, the distances from the means will be large. A small difference in distance will make the classifier unreasonably certain of a particular transformation. To add a general prior for uncertainty far from the training set, we add a small regularizing constant $\\epsilon$ to the probability of each transformation. This ensures equal probabilities for uncertain regions: ",
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+        "img_path": "images/d4007e81b04063f06342e2082f1542c16dde2b90c8d873947805faebad470bf8.jpg",
+        "text": "$$\n\\tilde { P } ( m ^ { \\prime } | T ( x , m ) ) = \\frac { e ^ { - \\| f ( T ( x , m ) ) - c _ { m ^ { \\prime } } \\| ^ { 2 } } + \\epsilon } { \\sum _ { \\tilde { m } } e ^ { - \\| f ( T ( x , m ) ) - c _ { \\tilde { m } } \\| ^ { 2 } } + M \\cdot \\epsilon }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "At test time we transform each sample by the $M$ transformations. By assuming independence between transformations, the probability that $x$ is normal (i.e. $x \\in X$ ) is the product of the probabilities that all transformed samples are in their respective subspace. For log-probabilities the total score is given by: ",
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+    {
+        "type": "equation",
+        "img_path": "images/ca10bd4b0eb0fecd44b7c77fae07b3ad4f54ab7ffbb70bea861d462f16b75aaf.jpg",
+        "text": "$$\nS c o r e ( x ) = - \\log P ( x \\in X ) = - \\sum _ { m } \\log \\tilde { P } ( T ( x , m ) \\in X _ { m } ) = - \\sum _ { m } \\log \\tilde { P } ( m | T ( x , m ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The score computes the degree of anomaly of each sample. Higher scores indicate a more anomalous sample. ",
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+    {
+        "type": "table",
+        "img_path": "images/82d9844f0da008c7dbb53d376c795ea7d6a3ddc8a491de2867b7fd0d3909e33d.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm 1 GOAD: Training Algorithm</td></tr><tr><td>Input: Normal training data x1,x2...X N</td></tr><tr><td>Transformations T(,1),T(,2)..T(, M)</td></tr><tr><td>Output: Feature extractor f, centers C1, C2...CM T(xi,1),T(xi,2)..T(xi,M) ← xi</td></tr><tr><td>// Transform each sample by all transformations 1 to M Find f,c1, C2...CM that optimize the triplet loss in Eq. 3</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Algorithm 2 GOAD: Evaluation Algorithm ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Input: Test sample: $x$ , feature extractor: $f$ , centers: c1, c2...cM , transformations: $T ( \\ u , \\ d ^ { 1 } ) , T ( \\ u , \\ d ^ { 2 } ) . . . T ( \\ u , \\ d M )$   \nOutput: Score(x)   \n$T ( x , 1 ) , T ( x , 2 ) . . . T ( x , M ) \\gets x$   \n// Transform test sample by all transformations 1 to M   \n$P ( m | T ( x , m ) )  f ( T ( x , m ) ) , c _ { 1 } , c _ { 2 } . . . c _ { M }$ // Likelihood of predicting the correct transformation (Eq. 4)   \n$S c o r e ( x ) \\gets P ( 1 | T ( x , 1 ) ) , P ( 2 | T ( x , 2 ) ) . . . P ( M | T ( x , M ) )$ // Aggregate probabilities to compute anomaly score (Eq. 5) ",
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+    {
+        "type": "text",
+        "text": "4 PARAMETERIZING THE SET OF TRANSFORMATIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Geometric transformations have been used previously for unsupervised feature learning by Gidaris et al. (2018) as well as by GEOM (Golan & El-Yaniv, 2018) for classification-based anomaly detection. This set of transformations is hand-crafted to work well with convolutional neural networks (CNNs) which greatly benefit from preserving neighborhood between pixels. This is however not a requirement for fully-connected networks. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Anomaly detection often deals with non-image datasets e.g. tabular data. Tabular data is very commonly used on the internet e.g. for cyber security or online advertising. Such data consists of both discrete and continuous attributes with no particular neighborhoods or order. The data is onedimensional and rotations do not naturally generalize to it. To allow transformation-based methods to work on general data types, we therefore need to extend the class of transformations. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "We propose to generalize the set of transformations to the class of affine transformations (where we have a total of $M$ transformations): ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/869b7e7febd53b2dabaffd12d60090edd7ce06db27c4999c09e1c942311d8dff.jpg",
+        "text": "$$\nT ( x , m ) = W _ { m } x + b _ { m }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "It is easy to verify that all geometric transformations in Golan & El-Yaniv (2018) (rotation by a multiple of 90 degrees, flips and translations) are a special case of this class ( $x$ in this case is the set of image pixels written as a vector). The affine class is however much more general than mere permutations, and allows for dimensionality reduction, non-distance preservation and random transformation by sampling $W$ , $b$ from a random distribution. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Apart from reduced variance across different dataset types where no apriori knowledge on the correct transformation classes exists, random transformations are important for avoiding adversarial examples. Assume an adversary wishes to change the label of a particular sample from anomalous to normal or vice versa. This is the same as requiring that $\\tilde { P } ( m ^ { \\prime } | T ( x , m ) )$ has low or high probability for $m ^ { \\prime } = m$ . If $T$ is chosen deterministically, the adversary may create adversarial examples against the known class of transformations (even if the exact network parameters are unknown). Conversely, if $T$ is unknown, the adversary must create adversarial examples that generalize across different transformations, which reduces the effectiveness of the attack. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "To summarize, generalizing the set of transformations to the affine class allows us to: generalize to non-image data, use an unlimited number of transformations and choose transformations randomly which reduces variance and defends against adversarial examples. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We perform experiments to validate the effectiveness of our distance-based approach and the performance of the general class of transformations we introduced for non-image data. ",
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+    {
+        "type": "text",
+        "text": "5.1 IMAGE EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Cifar10: To evaluate the performance of our method, we perform experiments on the Cifar10 dataset. We use the same architecture and parameter choices of Golan & El-Yaniv (2018), with our distance-based approach. We use the standard protocol of training on all training images of a single digit and testing on all test images. Results are reported in terms of AUC. In our method, we used a margin of $s = 0 . 1$ (we also run GOAD with $s = 1$ , shown in the appendix). Similarly to He et al. (2018), to stabilize training, we added a softmax $^ +$ cross entropy loss, as well as $L _ { 2 }$ norm regularization for the extracted features $f ( x )$ . We compare our method with the deep oneclass method of Ruff et al. (2018) as well as Golan & El-Yaniv (2018) without and with Dirichlet weighting. We believe the correct comparison is without Dirichlet post-processing, as we also do not use it in our method. Our distance based approach outperforms the SOTA approach by Golan & El-Yaniv (2018), both with and without Dirichlet (which seems to improve performance on a few classes). This gives evidence for the importance of considering the generalization behavior outside the normal region used in training. Note that we used the same geometric transformations as Golan & El-Yaniv (2018). Random affine matrices did not perform competitively as they are not pixel order preserving, this information is effectively used by CNNs and removing this information hurts performance. This is a special property of CNN architectures and image/time series data. As a rule of thumb, fully-connected networks are not pixel order preserving and can fully utilize random affine matrices. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/285a962f5deb61c1ad0aacae6b75a25c2be448196fc6415f90434277b74afb63.jpg",
+        "table_caption": [
+            "Table 1: Anomaly Detection Accuracy on Cifar10 (ROC-AUC %) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Class</td><td colspan=\"4\">Method</td></tr><tr><td>Deep-SVDD</td><td>GEOM (no Dirichlet)</td><td>GEOM (w. Dirichlet)</td><td>Ours</td></tr><tr><td>0</td><td>61.7 ± 1.3</td><td>76.0 ± 0.8</td><td>74.7 ± 0.4</td><td>77.2 ± 0.6</td></tr><tr><td>1</td><td>65.9 ± 0.7</td><td>83.0 ± 1.6</td><td>95.7 ± 0.0</td><td>96.7 ± 0.2</td></tr><tr><td>2</td><td>50.8 ± 0.3</td><td>79.5 ± 0.7</td><td>78.1 ± 0.4</td><td>83.3 ± 1.4</td></tr><tr><td>3</td><td>59.1 ± 0.4</td><td>71.4 ± 0.9</td><td>72.4 ± 0.5</td><td>77.7 ± 0.7</td></tr><tr><td>4</td><td>60.9 ± 0.3</td><td>83.5 ± 1.0</td><td>87.8 ± 0.2</td><td>87.8 ± 0.7</td></tr><tr><td>5</td><td>65.7 ± 0.8</td><td>84.0 ± 0.3</td><td>87.8 ± 0.1</td><td>87.8 ± 0.6</td></tr><tr><td>6</td><td>67.7 ± 0.8</td><td>78.4 ± 0.7</td><td>83.4 ± 0.5</td><td>90.0 ± 0.6</td></tr><tr><td>7</td><td>67.3 ± 0.3</td><td>89.3 ± 0.5</td><td>95.5 ± 0.1</td><td>96.1 ± 0.3</td></tr><tr><td>8</td><td>75.9 ± 0.4</td><td>88.6 ± 0.6</td><td>93.3 ± 0.0</td><td>93.8 ± 0.9</td></tr><tr><td>9</td><td>73.1 ± 0.4</td><td>82.4 ± 0.7</td><td>91.3 ± 0.1</td><td>92.0 ± 0.6</td></tr><tr><td>Average</td><td>64.8</td><td>81.6</td><td>86.0</td><td>88.2</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/a04fddc1d8f59b6be0eb80f0f5b0195bfa67ccdb36da46fd134fd39d173cd99f.jpg",
+        "table_caption": [
+            "Table 2: Anomaly Detection Accuracy on FashionMNIST (ROC-AUC %) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Class</td><td colspan=\"4\">Method</td></tr><tr><td>Deep-SVDD</td><td>GEOM (no Dirichlet)</td><td>GEOM (w. Dirichlet)</td><td>Ours</td></tr><tr><td>0</td><td>98.2</td><td>77.8 ± 5.9</td><td>99.4 ± 0.0</td><td>94.1 ± 0.9</td></tr><tr><td>1</td><td>90.3</td><td>79.1 ± 16.3</td><td>97.6 ± 0.1</td><td>98.5 ± 0.3</td></tr><tr><td>2</td><td>90.7</td><td>80.8 ± 6.9</td><td>91.1 ± 0.2</td><td>90.8 ± 0.4</td></tr><tr><td>3</td><td>94.2</td><td>79.2 ± 9.1</td><td>89.9 ± 0.4</td><td>91.6 ± 0.9</td></tr><tr><td>4</td><td>89.4</td><td>77.8 ± 3.3</td><td>92.1 ± 0.0</td><td>91.4 ± 0.3</td></tr><tr><td>5</td><td>91.8</td><td>58.0 ± 29.4</td><td>93.4 ± 0.9</td><td>94.8 ± 0.5</td></tr><tr><td>6</td><td>83.4</td><td>73.6 ± 8.7</td><td>83.3 ± 0.1</td><td>83.4 ± 0.4</td></tr><tr><td>7</td><td>98.8</td><td>87.4 ± 11.4</td><td>98.9 ± 0.1</td><td>97.9 ± 0.4</td></tr><tr><td>8</td><td>91.9</td><td>84.6 ± 5.6</td><td>90.8 ± 0.1</td><td>98.9 ± 0.1</td></tr><tr><td>9</td><td>99.0</td><td>99.5 ± 0.0</td><td>99.2 ± 0.0</td><td>99.2 ± 0.3</td></tr><tr><td>Average</td><td>92.8</td><td>79.8</td><td>93.5</td><td>94.1</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "FasionMNIST: In Tab. 2, we present a comparison between our method (GOAD) and the strongest baseline methods (Deep SVDD and GEOM) on the FashionMNIST dataset. We used exactly the same setting as Golan & El-Yaniv (2018). GOAD was run with $s \\ = \\ 1$ . OCSVM and GEOM with Dirichlet were copied from their paper. We run their method without Dirichlet and presented it in the table (we verified the implementation by running their code with Dirichlet and replicated the numbers in the paper). It appears that GEOM is quite dependent on Dirichlet for this dataset, whereas we do not use it at all. GOAD outperforms all the baseline methods. ",
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+    },
+    {
+        "type": "text",
+        "text": "Adversarial Robustness: Let us assume an attack model where the attacker knows the architecture and the normal training data and is trying to minimally modify anomalies to look normal. We examine the merits of two settings i) the adversary knows the transformations used (non-random) ii) the adversary uses another set of transformations. To measure the benefit of the randomized transformations, we train three networks A, B, C. Networks A and B use exactly the same transformations but random parameter initialization prior to training. Network C is trained using other randomly selected transformations. The adversary creates adversarial examples using PGD (Madry et al., 2017) based on network A (making anomalies appear like normal data). On Cifar10, we randomly selected 8 transformations from the full set of 72 for $A$ and $B$ , another randomly selected 8 transformations are used for $C$ . We measure the increase of false classification rate on the adversarial examples using the three networks. The average increase in performance of classifying transformation correctly on anomalies (causing lower anomaly scores) on the original network $A$ was $1 2 . 8 \\%$ , the transfer performance for B causes an increase by $5 . 0 \\%$ on network $B$ which shared the same set of transformation, and $3 \\%$ on network $C$ that used other rotations. This shows the benefits of using random transformations. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2a33ee3cf1a1e66de2b0a395f3586c029351c39701f4c13d271141c8dd2e4086.jpg",
+        "table_caption": [
+            "Table 3: Anomaly Detection Accuracy $( \\% )$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td colspan=\"7\">Dataset</td></tr><tr><td></td><td colspan=\"2\">Arrhythmia</td><td colspan=\"2\">Thyroid</td><td colspan=\"2\">KDD</td><td colspan=\"2\">KDDRev</td></tr><tr><td></td><td>F1 Score</td><td>0</td><td>F1 Score</td><td>0</td><td>F1 Score</td><td>0</td><td>F1 Score</td><td>0</td></tr><tr><td>OC-SVM</td><td>45.8</td><td></td><td>38.9</td><td></td><td>79.5</td><td></td><td>83.2</td><td></td></tr><tr><td>E2E-AE</td><td>45.9</td><td></td><td>11.8</td><td></td><td>0.3</td><td></td><td>74.5</td><td></td></tr><tr><td>LOF</td><td>50.0</td><td>0.0</td><td>52.7</td><td>0.0</td><td>83.8</td><td>5.2</td><td>81.6</td><td>3.6</td></tr><tr><td>DAGMM</td><td>49.8</td><td></td><td>47.8</td><td></td><td>93.7</td><td></td><td>93.8</td><td></td></tr><tr><td>FB-AE</td><td>51.5</td><td>1.6</td><td>75.0</td><td>0.8</td><td>92.7</td><td>0.3</td><td>95.9</td><td>0.4</td></tr><tr><td>GOAD(Ours)</td><td>52.0</td><td>2.3</td><td>74.5</td><td>1.1</td><td>98.4</td><td>0.2</td><td>98.9</td><td>0.3</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "5.2 TABULAR DATA EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Datasets: We evaluate on small-scale medical datasets Arrhythmia, Thyroid as well as large-scale cyber intrusion detection datasets KDD and KDDRev. Our configuration follows that of Zong et al. (2018). Categorical attributes are encoded as one-hot vectors. For completeness the datasets are described in the appendix A.2. We train all compared methods on $5 0 \\%$ of the normal data. The methods are evaluated on $5 0 \\%$ of the normal data as well as all the anomalies. ",
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+    },
+    {
+        "type": "text",
+        "text": "Baseline methods: The baseline methods evaluated are: One-Class SVM (OC-SVM, Scholkopf et al. (2000)), End-to-End Autoencoder (E2E-AE), Local Outlier Factor (LOF, Breunig et al. (2000)). We also evaluated deep distributional method DAGMM (Zong et al., 2018), choosing their strongest variant. To compare against ensemble methods e.g. Chen et al. (2017), we implemented the Feature Bagging Autoencoder (FB-AE) with autoencoders as the base classifier, feature bagging as the source of randomization, and average reconstruction error as the anomaly score. OC-SVM, E2E-AE and DAGMM results are directly taken from those reported by Zong et al. (2018). LOF and FB-AE were computed by us. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Implementation of $G O A D$ : We randomly sampled transformation matrices using the normal distribution for each element. Each matrix has dimensionality $L \\times r$ , where $L$ is the data dimension and $r$ is a reduced dimension. For Arryhthmia and Thyroid we used $r = 3 2$ , for KDD and KDDrev we used $r = 1 2 8$ and $r = 6 4$ respectively, the latter due to high memory requirements. We used 256 tasks for all datasets apart from KDD (64) due to high memory requirements. We set the bias term to 0. For $C$ we used fully-connected hidden layers and leaky-ReLU activations (8 hidden nodes for the small datasets, 128 and 32 for KDDRev and KDD). We optimized using ADAM with a learning rate of 0.001. Similarly to He et al. (2018), to stabilize the triplet center loss training, we added a softmax $^ +$ cross entropy loss. We repeated the large-scale experiments 5 times, and the small scale GOAD experiments 500 times (due to the high variance). We report the mean and standard deviation $( \\sigma )$ . Following the protocol in Zong et al. (2018), the decision threshold value is chosen to result in the correct number of anomalies e.g. if the test set contains $N _ { a }$ anomalies, the threshold is selected so that the highest $N _ { a }$ scoring examples are classified as anomalies. True positives and negatives are evaluated in the usual way. Some experiments copied from other papers did not measure standard variation and we kept the relevant cell blank. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Results ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Arrhythmia: The Arrhythmia dataset was the smallest examined. A quantitative comparison on this dataset can be seen in Tab. 3. OC-SVM and DAGMM performed reasonably well. Our method is comparable to FB-AE. A linear classifier $C$ performed better than deeper networks (which suffered from overfitting). Early stopping after a single epoch generated the best results. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Thyroid: Thyroid is a small dataset, with a low anomaly to normal ratio and low feature dimensionality. A quantitative comparison on this dataset can be seen in Tab. 3. Most baselines performed about equally well, probably due to the low dimensionality. On this dataset, we also found that early stopping after a single epoch gave the best results. The best results on this dataset, were obtained with a linear classifier. Our method is comparable to FB-AE and beat all other baselines by a wide margin. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/a525875e49259446f38693aa0662d0b42abc1102c3d160bf98a7b77cf54d2a65.jpg",
+        "image_caption": [
+            "Figure 1: Left: Classification error for our method and DAGMM as a function of percentage of the anomalous examples in the training set (on the KDDCUP99 dataset). Our method consistently outperforms the baseline. Right: Classification error as a function of the number of transformations (on the KDDRev dataset). The error and instability decrease as a function of the number of transformations. For both, lower is better. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "KDDCUP99: The UCI KDD $1 0 \\%$ dataset is the largest dataset examined. A quantitative comparison on this dataset can be seen in Tab. 3. The strongest baselines are FB-AE and DAGMM. Our method significantly outperformed all baselines. We found that large datasets have different dynamics from very small datasets. On this dataset, deep networks performed the best. We also, did not need early stopping. The results are reported after 25 epochs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "KDD-Rev: The KDD-Rev dataset is a large dataset, but smaller than KDDCUP99 dataset. A quantitative comparison on this dataset can be seen in Tab. 3. Similarly to KDDCUP99, the best baselines are FB-AE and DAGMM, where FB-AE significantly outperforms DAGMM. Our method significantly outperformed all baselines. Due to the size of the dataset, we did not need early stopping. The results are reported after 25 epochs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Adversarial Robustness: Due to the large number of transformations and relatively small networks, adversarial examples are less of a problem for tabular data. PGD generally failed to obtain adversarial examples on these datasets. On KDD, transformation classification accuracy on anomalies was increased by $3 . 7 \\%$ for the network the adversarial examples were trained on, $1 . 3 \\%$ when transferring to the network with the same transformation and only $0 . 2 \\%$ on the network with other randomly selected transformations. This again shows increased adversarial robustness due to random transformations. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Further Analysis ",
+        "text_level": 1,
+        "bbox": [
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+            672,
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+            688
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Contaminated Data: This paper deals with the semi-supervised scenario i.e. when the training dataset contains only normal data. In some scenarios, such data might not be available but instead we might have a training dataset that contains a small percentage of anomalies. To evaluate the robustness of our method to this unsupervised scenario, we analysed the KDDCUP99 dataset, when $X \\%$ of the training data is anomalous. To prepare the data, we used the same normal training data as before and added further anomalous examples. The test data consists of the same proportions as before. The results are shown in Fig. 1. Our method significantly outperforms DAGMM for all impurity values, and degrades more graceful than the baseline. This attests to the effectiveness of our approach. Results for the other datasets are presented in Fig. 3, showing similar robustness to contamination. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Number of Tasks: One of the advantages of GOAD, is the ability to generate any number of tasks. We present the anomaly detection performance on the KDD-Rev dataset with different numbers of tasks in Fig. 1. We note that a small number of tasks (less than 16) leads to poor results. From 16 tasks, the accuracy remains stable. We found that on the smaller datasets (Thyroid, Arrhythmia) using a larger number of transformations continued to reduce $F _ { 1 }$ score variance between differently initialized runs (Fig. 2). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Openset vs. Softmax: The openset-based classification presented by GOAD resulted in performance improvement over the closed-set softmax approach on Cifar10 and FasionMNIST. In our experiments, it has also improved performance in KDDRev. Arrhythmia and Thyroid were comparable. As a negative result, performance of softmax was better on KDD $F _ { 1 } = 0 . 9 9$ ). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Choosing the margin parameter $s$ : GOAD is not particularly sensitive to the choice of margin parameter $s$ , although choosing $s$ that is too small might cause some instability. We used a fixed value of $s = 1$ in our experiments, and recommend this value as a starting point. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Other transformations: GOAD can also work with other types of transformations such as rotations or permutations for tabular data. In our experiments, we observed that these transformation types perform comparably but a little worse than affine transformations. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised training: Although most of our results are semi-supervised i.e. assume that no anomalies exist in the training set, we presented results showing that our method is more robust than strong baselines to a small percentage of anomalies in the training set. We further presented results in other datasets showing that our method degrades gracefully with a small amount of contamination. Our method might therefore be considered in the unsupervised settings. ",
+        "bbox": [
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+            295,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Deep vs. shallow classifiers: Our experiments show that for large datasets deep networks are beneficial (particularly for the full KDDCUP99), but are not needed for smaller datasets (indicating that deep learning has not benefited the smaller datasets). For performance critical operations, our approach may be used in a linear setting. This may also aid future theoretical analysis of our method. ",
+        "bbox": [
+            174,
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+            428
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            448,
+            318,
+            464
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we presented a method for detecting anomalies for general data. This was achieved by training a classifier on a set of random auxiliary tasks. Our method does not require knowledge of the data domain, and we are able to generate an arbitrary number of random tasks. Our method significantly improve over the state-of-the-art. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            824,
+            297,
+            839
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A.1 IMAGE EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            854,
+            370,
+            869
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sensitive to margin s: We run Cifar10 experiments with $s = 0 . 1$ and $s = 1$ and presented the results in Fig. 4. The results were not affected much by the margin parameter. This is in-line with the rest of our empirical observations that GOAD is not very sensitive to the margin parameter. ",
+        "bbox": [
+            174,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/b7e393c23356639d081dcc799153c43f9f730554091ec460168f2990951191b9.jpg",
+        "table_caption": [
+            "Table 4: Anomaly Detection Accuracy on Cifar10 $( \\% )$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Class</td><td colspan=\"3\">Method</td></tr><tr><td>GEOM (w. Dirichlet)</td><td>GOAD(s = 0.1)</td><td>GOAD(1.0)</td></tr><tr><td>0</td><td>74.7 ± 0.4</td><td>77.2 ± 0.6</td><td>77.9 ± 0.7</td></tr><tr><td>1</td><td>95.7 ± 0.0</td><td>96.7 ± 0.2</td><td>96.4 ± 0.9</td></tr><tr><td>2</td><td>78.1 ± 0.4</td><td>83.3 ± 1.4</td><td>81.8 ± 0.8</td></tr><tr><td>3</td><td>72.4 ± 0.5</td><td>77.7 ± 0.7</td><td>77.0 ± 0.7</td></tr><tr><td>4</td><td>87.8 ± 0.2</td><td>87.8 ± 0.7</td><td>87.7 ± 0.5</td></tr><tr><td>5</td><td>87.8 ± 0.1</td><td>87.8 ± 0.6</td><td>87.8 ± 0.7</td></tr><tr><td>6</td><td>83.4 ± 0.5</td><td>90.0 ± 0.6</td><td>90.9 ± 0.5</td></tr><tr><td>7</td><td>95.5 ± 0.1</td><td>96.1 ± 0.3</td><td>96.1 ± 0.2</td></tr><tr><td>8</td><td>93.3 ± 0.0</td><td>93.8 ± 0.9</td><td>93.3 ± 0.1</td></tr><tr><td>9</td><td>91.3 ± 0.1</td><td>92.0 ± 0.6</td><td>92.4 ± 0.3</td></tr><tr><td>Average</td><td>86.0</td><td>88.2</td><td>88.1</td></tr></table>",
+        "bbox": [
+            267,
+            127,
+            730,
+            330
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 TABULAR DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            361,
+            362,
+            376
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Following the evaluation protocol of Zong et al. (2018), 4 datasets are used in this comparison: ",
+        "bbox": [
+            174,
+            388,
+            794,
+            404
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Arrhythmia: A cardiology dataset from the UCI repository (Asuncion & Newman, 2007) containing attributes related to the diagnosis of cardiac arrhythmia in patients. The datasets consists of 16 classes: class 1 are normal patients, 2-15 contain different arrhythmia conditions, and class 16 contains undiagnosed cases. Following the protocol established by ODDS (Rayana, 2016), the smallest classes: 3, 4, 5, 7, 8, 9, 14, 15 are taken to be anomalous and the rest normal. Also following ODDS, the categorical attributes are dropped, the final attributes total 274. ",
+        "bbox": [
+            174,
+            410,
+            825,
+            494
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Thyroid: A medical dataset from the UCI repository (Asuncion & Newman, 2007), containing attributes related to whether a patient is hyperthyroid. Following ODDS (Rayana, 2016), from the 3 classes of the dataset, we designate hyperfunction as the anomalous class and the rest as normal. Also following ODDS only the 6 continuous attributes are used. ",
+        "bbox": [
+            174,
+            501,
+            825,
+            556
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "KDD: The KDD Intrusion Detection dataset was created by an extensive simulation of a US Air Force LAN network. The dataset consists of the normal and 4 simulated attack types: denial of service, unauthorized access from a remote machine, unauthorized access from local superuser and probing. The dataset consists of around 5 million TCP connection records. Following the evaluation protocol in Zong et al. (2018), we use the UCI KDD $1 0 \\%$ dataset, which is a subsampled version of the original dataset. The dataset contains 41 different attributes. 34 are continuous and 7 are categorical. Following Zong et al. (2018), we encode the categorical attributes using 1-hot encoding. ",
+        "bbox": [
+            174,
+            564,
+            825,
+            661
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Following Zong et al. (2018), we evaluate two different settings for the KDD dataset: ",
+        "bbox": [
+            176,
+            669,
+            728,
+            683
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "KDDCUP99: In this configuration we use the entire UCI $1 0 \\%$ dataset. As the non-attack class consists of only $2 0 \\%$ of the dataset, it is treated as the anomaly in this case, while attacks are treated as normal. ",
+        "bbox": [
+            176,
+            689,
+            821,
+            731
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "KDDCUP99-Rev: To better correspond to the actual use-case, in which the non-attack scenario is normal and attacks are anomalous, Zong et al. (2018) also evaluate on the reverse configuration, in which the attack data is sub-sampled to consist of $2 5 \\%$ of the number of non-attack samples. The attack data is in this case designated as anomalous (the reverse of the KDDCUP99 dataset). ",
+        "bbox": [
+            176,
+            739,
+            825,
+            795
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In all the above datasets, the methods are trained on $5 0 \\%$ of the normal data. The methods are evaluated on $5 0 \\%$ of the normal data as well as all the anomalies. ",
+        "bbox": [
+            173,
+            801,
+            821,
+            829
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.3 NUMBER OF TASKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            853,
+            351,
+            867
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We provide plots of the number of auxiliary tasks vs. the anomaly detection accuracy (measured by $F _ { 1 , }$ ) for all datasets. The results are presented in Fig. 2. Performance increases rapidly up to a certain number of tasks (around 16). Afterwards more tasks reduce the variance of $F _ { 1 }$ scores between runs. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            922
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/55ef42d627c9ba88b9baf86d4f5c32923b4d5808311593e446f3e1f9d9b39111.jpg",
+        "image_caption": [
+            "Figure 2: Plots of the number of auxiliary tasks vs. the anomaly detection accuracy (measured by $F _ { 1 }$ ) a) Arrhythmia b) Thyroid c) KDDRev d) KDDCup99 Accuracy often increases with the number of tasks, although the rate diminishes with the number of tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            104,
+            790,
+            479
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/1714725f825a4c3d69f4543913fba5904bbb19a3a0b1c21569169b1ff2e913bb.jpg",
+        "image_caption": [
+            "Figure 3: Plots of the degree of contamination vs. the anomaly detection accuracy (measured by $F _ { 1 }$ ) (left) KDDRev (center) KDDCup99 (right) Arrhythmia. GOAD is generally robust to the degree of contamination. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            561,
+            779,
+            666
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.4 CONTAMINATION EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            752,
+            441,
+            766
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We conduct contamination experiments for 3 datasets. Thyroid was omitted due to not having a sufficient number of anomalies. The protocol is different than that of KDDRev as we do not have unused anomalies for contamination. Instead, we split the anomalies into train and test. Train anomalies are used for contamination, test anomalies are used for evaluation. As DAGMM did not present results for the other datasets, we only present GOAD. GOAD was reasonably robust to contamination on KDD, KDDRev and Arrhythmia. The results are presented in Fig. 3 ",
+        "bbox": [
+            173,
+            777,
+            825,
+            862
+        ],
+        "page_idx": 11
+    }
+]

parse/train/SkgbmyHFDS/SkgbmyHFDS.md

@@ -0,0 +1,332 @@
+# WHAT CAN LEARNED INTRINSIC REWARDS CAPTURE?
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Reinforcement learning agents can include different components, such as policies, value functions, state representations, and environment models. Any or all of these can be the loci of knowledge, i.e., structures where knowledge, whether given or learned, can be deposited and reused. Regardless of its composition, the objective of an agent is behave so as to maximise the sum of a suitable scalar function of state: the reward. As far as the learning algorithm is concerned, these rewards are typically given and immutable. In this paper we instead consider the proposition that the reward function itself may be a good locus of knowledge. This is consistent with a common use, in the literature, of hand-designed intrinsic rewards to improve the learning dynamics of an agent. We adopt the multi-lifetime setting of the Optimal Rewards Framework, and propose to meta-learn an intrinsic reward function from experience that allows agents to maximise their extrinsic rewards accumulated until the end of their lifetimes. Rewards as a locus of knowledge provide guidance on “what” the agent should strive to do rather than on knowledge of “how” the agent should behave that is more directly captured in policies or value functions for example. Thus, our focus here is on demonstrating the following: (1) that it is feasible to meta-learn good reward functions, (2) to show that the learned reward functions can capture interesting kinds of “what” knowledge, and (3) that because of the indirectness of this form of knowledge the learned reward functions can generalise to other kinds of agents and to changes in the dynamics of the environment.
+
+Reinforcement learning agents can store knowledge in their policies, value functions, state representations, and models of the environment dynamics. These components can be the loci of knowledge in the sense that they are structures in which knowledge, either learned from experience by the agent’s algorithm or given by the agent-designer, can be deposited and reused. The objective of the agent is defined by a reward function, and the goal is to learn to act so as to optimise cumulative rewards. In this paper we consider the proposition that the reward function itself is a good locus of knowledge. This is unusual in that most prior work treats the reward as given and immutable, at least as far as the learning algorithm is concerned. At the same time, especially in challenging reinforcement-learning problems, agent designers do find it convenient to modify the reward function given to the agent to facilitate learning. It is therefore useful to distinguish between two kinds of reward functions (Singh et al., 2010): extrinsic rewards define the task and capture the designer’s preferences over agent behaviour, whereas intrinsic rewards serve as helpful signals to improve the learning dynamics of the agent. Intrinsic rewards are typically hand-designed and then often added to the immutable extrinsic rewards to form the reward optimised by the agent.
+
+Most existing work on intrinsic rewards falls into two broad categories: task-dependent and taskindependent. Both are typically designed by hand. Hand-designing task-dependent rewards can be fraught with difficulty as even minor misalignment between the actual reward and the intended bias can lead to unintended and sometimes catastrophic consequences (Clark & Amodei, 2016). Task-independent intrinsic rewards are also typically hand-designed, often based on an intuitive understanding of animal/human behaviour or on heuristics on desired exploratory behaviour. It can, however, be hard to match such task-independent intrinsic rewards to the specific learning dynamics induced by the interaction between agent and environment. The motivation for this paper is our interest in the comparatively under-explored possibility of learned (not hand-designed) taskdependent intrinsic rewards (see Zheng et al., 2018, for previous work).
+
+We emphasise that it is not our objective to show that rewards are a better locus of learned knowledge than others; the best locus likely depends on the kind of knowledge that is most useful in a given task. In particular, knowledge captured in rewards provides guidance on “what” the agent should strive to do while knowledge captured in policies provides guidance on “how” an agent should behave. Knowledge about “what” captured in rewards is indirect and thus slower to make an impact on behaviour because it takes effect through learning, while knowledge about “how” can directly have an immediate impact on behaviour. At the same time, because of its indirectness the former can generalise better to changes in dynamics and learning architectures. Therefore, instead of comparing different loci of knowledge, the purpose of this paper is to show that it is feasible to capture useful learned knowledge in rewards and to study the kinds of knowledge that can be captured.
+
+How should we measure the usefulness of a learned reward function? Ideally, we would like to measure the effect the learned reward function has on the learning dynamics. Of course, learning happens over multiple episodes, indeed it happens over an entire lifetime. Therefore, we choose lifetime return, the cumulative extrinsic reward obtained by the agent over its entire lifetime, as the main objective. To this end, we adopt the multi-lifetime setting of the Optimal Rewards Framework (Singh et al., 2009) in which an agent is initialised randomly at the start of each lifetime and then faces a stationary or non-stationary task drawn from some distribution. In this setting, the only knowledge that is transferred across lifetimes is the reward instead of the policy. Specifically, the goal is to learn a single intrinsic reward function that, when used to adapt the agent’s policy using a standard episodic RL algorithm, ends up optimising the cumulative extrinsic reward over its lifetime.
+
+In previous work, good reward functions were found via exhaustive search, limiting the range of applicability of the framework. Here, we develop a more scalable gradient-based method ( $\mathrm { { X u } }$ et al., 2018c) for learning the intrinsic rewards by exploiting the fact the interaction between the policy update and the reward function is differentiable (Zheng et al., 2018). Since it is infeasible to backpropgate through the full computation graph that spans across the entire lifetime, we truncate the unrolled computation graph of learning updates up to some horizon. However, we handle the long-term credit assignment by using a lifetime value function that estimates the remaining lifetime return, which needs to take into account changing policies. Our main scientific contributions are a sequence of empirical studies on carefully designed environments that show how our learned intrinsic rewards can capture interesting regularities in the interaction between a learning agent and an environment sampled from a distribution, and how the learned intrinsic reward can generalise to changed dynamics and agent architectures. Collectively, our contributions present an effective approach to the discovery of intrinsic rewards that can help an agent optimise the extrinsic rewards collected in a lifetime.
+
+# 1 RELATED WORK
+
+Hand-designed Rewards There is a long history of work on designing rewards to accelerate learning in reinforcement learning (RL). Reward shaping aims to design task-specific rewards towards known optimal behaviours, typically requiring domain knowledge. Both the benefits (Randlov & Alstrm, 1998; Ng et al., 1999; Harutyunyan et al., 2015) and the difficulty (Clark & Amodei, 2016) of task-specific reward shaping have been studied. On the other hand, many intrinsic rewards have been proposed to encourage exploration, inspired by animal behaviours. Examples include prediction error (Schmidhuber, 1991b; Gordon & Ahissar, 2011; Mirolli & Baldassarre, 2013; Pathak et al., 2017; Schmidhuber, 1991a), surprise (Itti & Baldi, 2006), weight change (Linke et al., 2019), and state-visitation counts (Sutton, 1990; Poupart et al., 2006; Strehl & Littman, 2008; Bellemare et al., 2016; Ostrovski et al., 2017). Although these kinds of intrinsic rewards are not domain-specific, they are often not well-aligned with the task that the agent tries to solve, and ignores the effect on the agent’s learning dynamics. In contrast, our work aims to learn intrinsic rewards from data that take into account the agent’s learning dynamics without requiring prior knowledge from a human.
+
+Rewards Learned from Data There have been a few attempts to learn useful intrinsic rewards from data. The optimal reward framework (Singh et al., 2009) proposed to learn an optimal reward function that allows agents to solve a distribution of tasks quickly using random search. We revisit this problem in this paper and propose a more scalable gradient-based approach. Although there have been follow-up works (Sorg et al., 2010; Guo et al., 2016) that uses a gradient-based method, they consider a non-parameteric policy using Monte-Carlo Tree Search (MCTS). Our work is closely related to LIRPG (Zheng et al., 2018) which proposed a meta-gradient method to learn intrinsic rewards. However, LIRPG considers a single task in a single lifetime with a myopic episode return objective, which is limited in that it does not allow exploration across episodes or generalisation to different agents.
+
+![](images/3ef9c31f414d6956d1f1b75935eeddc3f4d63e06416cbadbda17f6bae752e2c6.jpg)  
+Figure 1: Illustration of the proposed intrinsic reward learning framework. The intrinsic reward $r _ { \eta }$ is used to update the agent’s parameter $\theta _ { i }$ throughout its lifetime which consists of many episodes. The goal is to find the optimal intrinsic reward parameters $\eta ^ { * }$ across many lifetimes that maximises the lifetime return $( G ^ { \mathrm { l i f e } } )$ given any randomly initialised agents and possibly non-stationary tasks drawn from some distribution $p ( \mathcal T )$ .
+
+Meta-learning for Exploration Meta-learning (Schmidhuber et al., 1996; Thrun & Pratt, 1998) has recently received considerable attention in RL. Recent advances include few-shot adaptation (Finn et al., 2017a), few-shot imitation (Finn et al., 2017b; Duan et al., 2017), model adaptation (Clavera et al., 2018), and inverse RL (Xu et al., 2018a). In particular, our work is closely related to the prior work on meta-learning good exploration strategies (Wang et al., 2016; Duan et al., 2016; Stadie et al., 2018; Xu et al., 2018b) in that both perform temporal credit assignment across episode boundaries by maximising rewards accumulated beyond an episode. Unlike the prior work that aims to learn an exploratory policy, our framework indirectly drives exploration via a reward function which can be reused by different learning agents as we show in this paper (Section 5.1).
+
+Meta-learning of Agent Update There have been a few studies that directly meta-learn how to update the agent’s parameters via meta-parameters including discount factor and returns (Xu et al., 2018c), auxiliary tasks (Schlegel et al., 2018; Veeriah et al., 2019), unsupervised learning rules (Metz et al., 2019), and RL objectives (Chebotar et al., 2019). Our work also belongs to this category in that our meta-parameters are the reward function used in the agent’s update. In particular, our multilifetime formulation is similar to $\mathrm { { M L ^ { 3 } } }$ (Chebotar et al., 2019). However, we consider the long-term lifetime return as objective to perform cross-episode temporal credit assignment as opposed to the myopic episodic objective in ML3.
+
+# 2 THE OPTIMAL REWARD PROBLEM
+
+We first introduce some terminology.
+
+• Agent: A learning system interacting with an environment. On each step $t$ the agent selects an action $a _ { t }$ and receives from the environment an observation $s _ { t + 1 }$ and an extrinsic reward $r _ { t + 1 }$ defined by a task $\tau$ . The agent chooses actions based on a policy $\pi _ { \boldsymbol { \theta } } \big ( a _ { t } | \boldsymbol { s } _ { t } \big )$ parameterised by $\theta$ . • Episode: A finite sequence of agent-environment interactions until the end of the episode defined by the task. An episode return is defined as: $\begin{array} { r } { G ^ { \mathrm { e p } } = \sum _ { t = 0 } ^ { T _ { \mathrm { e p } } - 1 } \gamma ^ { t } r _ { t + 1 } } \end{array}$ , where $\gamma$ is a discount factor, and the random variable $T _ { \mathrm { e p } }$ gives the finite number of steps until the end of the episode. • Lifetime: A finite sequence of agent-environment interactions until the end of training deent-des, where ner, which can include is a discount factor, and ltiple episodes. The lifetime return is is the number of steps in the lifetime. $G ^ { \mathrm { { \bar { i } f e } } } =$ $\scriptstyle \sum _ { t = 0 } ^ { T - 1 } \gamma ^ { t } r _ { t + 1 }$ $\gamma$ $T$ • Intrinsic reward: A reward function $r _ { \eta } ( \tau _ { t + 1 } )$ parameterised by $\eta$ , where $\begin{array} { r l } { \tau _ { t } } & { { } = } \end{array}$ $( s _ { 0 } , a _ { 0 } , r _ { 1 } , d _ { 1 } , s _ { 1 } , \ldots , r _ { t } , d _ { t } , s _ { t } )$ is a lifetime history with (binary) episode terminations $d _ { i }$ .
+
+The Optimal Reward Problem (Singh et al., 2010), illustrated in Figure 1, aims to learn the parameters of the intrinsic reward such that the resulting rewards achieve a learning dynamic for an RL agent that maximises the lifetime (extrinsic) return on tasks drawn from some distribution. Formally, the optimal reward function is defined as:
+
+$$
+\eta ^ { * } = \underset { \eta } { \arg \operatorname* { m a x } } J ( \eta ) = \underset { \eta } { \arg \operatorname* { m a x } } \mathbb { E } _ { \theta _ { 0 } \sim \Theta , \mathcal { T } \sim p ( \mathcal { T } ) } \left[ \mathbb { E } _ { \tau \sim p _ { \eta } ( \tau \mid \theta _ { 0 } ) } \left[ G ^ { \mathrm { l i f e } } \right] \right] ,
+$$
+
+where $\Theta$ and $p ( \mathcal T )$ are an initial policy distribution and a distribution over possibly non-stationary tasks respectively, and $\begin{array} { r } { G ^ { \mathrm { l i f e } } = \sum _ { t = 0 } ^ { { \bar { T } } - 1 } \dot { \gamma } ^ { t } r _ { t + 1 } } \end{array}$ is a lifetime return. The likelihood of a lifetime history
+
+Algorithm 1 Learning intrinsic rewards across multiple lifetimes via meta-gradient
+
+Input: $p ( \mathcal { T } )$ : Task distribution, $\Theta$ : Randomly-initialised policy distribution   
+Initialise intrinsic reward function $\eta$ and lifetime value function $\phi$   
+repeat Initialise task $\tau \sim p ( \mathcal { T } )$ and policy $\theta \sim \Theta$ while lifetime not ended do $\theta _ { 0 } \gets \theta$ for $k = 1 , 2 , \ldots , N$ do Generate a trajectory using πθk−1 Update policy $\theta _ { k } \gets \theta _ { k - 1 } + \alpha \nabla _ { \theta _ { k - 1 } } J _ { \eta } ( \theta _ { k - 1 } )$ using intrinsic rewards $r _ { \eta }$ (Eq. 2) end for Update intrinsic reward function $\eta$ using Eq. 3 Update lifetime value function $\phi$ using Eq. 4 $\boldsymbol { \theta } \dot { }  \boldsymbol { \theta } _ { N }$ end while   
+until $\eta$ converges
+
+$\tau$ is $\begin{array} { r } { p _ { \eta } ( \tau \vert \theta _ { 0 } ) = p ( s _ { 0 } ) \prod _ { t = 0 } ^ { T - 1 } \pi _ { \theta _ { t } } ( a _ { t } \vert s _ { t } ) p ( d _ { t + 1 } , r _ { t + 1 } , s _ { t + 1 } \vert s _ { t } , a _ { t } ) , } \end{array}$ , where $\theta _ { t } = f ( \theta _ { t - 1 } , \eta )$ is a policy parameter as updated with update function $f$ , which is policy gradient in this paper.1 Note that the optimisation of $\eta$ spans multiple lifetimes, each of which can span multiple episodes.
+
+Using the lifetime return $G ^ { \mathrm { l i f e } }$ as the objective instead of the conventional episodic return $G ^ { \mathrm { e p } }$ allows exploration across multiple episodes as long as the lifetime return is maximised in the long run. In particular, when the lifetime is defined as a fixed number of episodes, we find that the lifetime return objective is sometimes more beneficial than the episodic return objective even in terms of the episodic return performance measure. However, different objectives (e.g., final episode return) can be considered depending on the definition of what a good reward function is.
+
+# 3 META-LEARNING INTRINSIC REWARD
+
+We propose a meta-gradient approach $\mathrm { { X u } }$ et al., 2018c; Zheng et al., 2018) to solve the optimal reward problem. At a high-level, we sample a new task $\tau$ and a new random policy parameter $\theta$ at each lifetime iteration. We then simulate an agent’s lifetime by updating the parameter $\theta$ using an intrinsic reward function $r _ { \eta }$ (Section 3.1) with policy gradient (Section 3.2). In the meantime, we compute the meta-gradient by taking into account the effect of the intrinsic rewards on the policy parameters to update the intrinsic reward function with a lifetime value function (Section 3.3). Algorithm 1 gives an overview of our algorithm. The following sections describe the details.
+
+# 3.1 INTRINSIC REWARD AND LIFETIME VALUE FUNCTION ARCHITECTURES
+
+The intrinsic reward function is a recurrent neural network (RNN) parameterised by $\eta$ , which produces a scalar reward on arriving in state $s _ { t }$ by taking into account the history of an agent’s lifetime $\tau _ { t } = ( s _ { 0 } , a _ { 0 } , r _ { 1 } , d _ { 1 } , s _ { 1 } , . . . , r _ { t } , d _ { t } , s _ { t } )$ . We claim that giving the lifetime history across episodes as input is crucial for balancing exploration and exploitation, for instance by capturing how frequently a certain state is visited to determine an exploration bonus reward. The lifetime value function is a separate recurrent neural network parameterised by $\phi$ , which takes the same inputs as the intrinsic reward function and produces a scalar value estimation of the expected future return within the lifetime.
+
+# 3.2 POLICY UPDATE (θ)
+
+Each agent interacts with an environment and a task sampled from a distribution $\tau \sim p ( \mathcal { T } )$ . However, instead of directly maximising the extrinsic rewards defined by the task, the agent maximises
+
+the intrinsic rewards $( r _ { \eta } )$ by using policy gradient (Williams, 1992; Sutton et al., 2000):
+
+$$
+J _ { \eta } ( \theta ) = \mathbb { E } _ { \theta } \bigg [ \sum _ { t = 0 } ^ { T _ { \mathrm { e p } } - 1 } \bar { \gamma } ^ { t } r _ { \eta } ( \tau _ { t + 1 } ) \bigg ] \qquad \quad \nabla _ { \theta } J _ { \eta } ( \theta ) = \mathbb { E } _ { \theta } \bigg [ G _ { \eta , t } ^ { \mathrm { e p } } \nabla _ { \theta } \log \pi _ { \theta } ( a | s ) \bigg ] ,
+$$
+
+where $r _ { \eta } ( \tau _ { t + 1 } )$ is the intrinsic reward at time $t$ , and = PTep−1k=t γ¯k−trη(τk+1) is the return of the intrinsic rewards accumulated over an episode with discount factor $\bar { \gamma }$ .
+
+3.3 INTRINSIC REWARD $( \eta )$ AND LIFETIME VALUE FUNCTION $( \phi )$ UPDATE
+
+To update the intrinsic reward parameters $\eta$ , we directly take a meta-gradient ascent step using the overall objective (Equation 1). Specifically, the gradient is (see Appendix A for derivation):
+
+$$
+\nabla _ { \eta } J ( \eta ) = \mathbb { E } _ { \theta _ { 0 } \sim \Theta , \mathcal { T } \sim p ( T ) } \biggl [ \mathbb { E } _ { \tau _ { t } \sim p ( \tau _ { t } | \eta , \theta _ { 0 } ) } \biggl [ G _ { t } ^ { \mathrm { l i f e } } \nabla _ { \theta _ { t } } \log \pi _ { \theta _ { t } } ( a _ { t } | s _ { t } ) \nabla _ { \eta } \theta _ { t } \biggr ] \biggr ] ,
+$$
+
+where PT −1k=t γk−trk+1 is a lifetime return based on the extrinsic rewards of task T with discount factor $\gamma$ . The chain rule is used to get the meta-gradient $( \nabla _ { \boldsymbol { \eta } } \theta _ { t } )$ as in previous work (Zheng et al., 2018). The computation graph of this procedure is illustrated in Figure 1.
+
+Computing the true meta-gradient in Equation 3 requires backpropagation through the entire lifetime, which is infeasible as each lifetime can involve more than thousands of policy updates. To partially address this issue, we truncate the meta-gradient after $N$ policy updates but approximate the lifetime return $G _ { t } ^ { \mathrm { l i f e } , \phi } \approx G _ { t } ^ { \mathrm { l i f e } }$ using a lifetime value function $V _ { \phi } ( \tau )$ parameterised by $\phi$ , which is learned using a temporal difference learning from -step trajectory:
+
+$$
+G _ { t } ^ { \mathrm { l i f e } , \phi } = \sum _ { k = 0 } ^ { n - 1 } \gamma ^ { k } r _ { t + k + 1 } + \gamma ^ { n } V _ { \phi } ( \tau _ { t + n } ) \qquad \phi = \phi + \alpha ^ { \prime } ( G _ { t } ^ { \mathrm { l i f e } , \phi } - V _ { \phi } ( \tau _ { t } ) ) \nabla _ { \phi } V _ { \phi } ( \tau _ { t } ) .
+$$
+
+In our empirical work, we found that the lifetime value estimates were crucial to allow the intrinsic reward to perform long-term credit assignments across episodes.
+
+# 3.4 CONNECTION TO STANDARD RL FRAMEWORKS
+
+The policy learning problem specified in Section 3.2 deviates from the standard Markov Decision Process (MDP) framework because the intrinsic reward function is a function of the lifetime history rather than a function of states or state-action pairs. Consequently, from the memoryless policy’s perspective, the rewards are non-stationary. However, we can also view the combination of the intrinsic reward function and the policy as a joint lifetime-history-based policy parameterised by $\eta$ and $\theta$ (see derivation in Appendix A). From this perspective, the overall learning problem specified in Section 3.3 can be formulated as an MDP with history as state (recall, we use RNNs for the intrinsic reward function). As a result, standard temporal-difference learning methods are applicable to learning lifetime value functions.
+
+# 4 EMPIRICAL INVESTIGATIONS: FEASIBILITY AND USEFULNESS
+
+We present the results from our empirical investigations in two sections. For the results in this section, the experiments and domains are designed to answer the following research questions:
+
+• What kind of knowledge can be learned by the intrinsic reward? • How does the distribution of tasks drive the form of the intrinsic reward? • What is the benefit of the lifetime return objective over the episode return? • When is it important to provide the lifetime history as input to the intrinsic reward?
+
+We investigate these research questions in various grid-world domains illustrated in Figure 2. For each domain, we trained an intrinsic reward function across many lifetimes and evaluated it by training an agent using the learned reward. We implemented the following baselines.
+
+• Extrinsic-EP: A policy is trained with extrinsic rewards to maximise the episode return.
+
+![](images/0d553e8ac18d76f927e644f7009853d9b0c6f94c94b1695efe43449e074db081.jpg)  
+Figure 2: Illustration of domains. (a) The agent needs to find the goal location which gives a positive reward, but the goal is not visible to the agent. (b) Each object (A, B, and C) gives rewards. (c) The agent is required to first collect the key and visit one of the boxes (A, B, and C) to receive the corresponding reward. All objects are placed to random locations after every episode.
+
+![](images/3776bc779886d5dac24e55fb5b1c51d390913ad1ff73748f06236bc60e04bf9e.jpg)  
+Figure 3: Evaluation of different reward functions averaged over 30 seeds. The learning curves show agents trained with our intrinsic reward (blue), with the extrinsic reward with the episodic return objective (orange) and the lifetime return objective (brown), and with a count-based exploration reward (green). The dashed line corresponds to a hand-designed near-optimal exploration strategy.
+
+• Extrinsic-LIFE: A policy is trained with extrinsic rewards to maximise the lifetime return.   
+• Count-based (Strehl & Littman, 2008): A policy is trained with extrinsic rewards and countbased exploration bonus rewards.   
+• ICM (Pathak et al., 2017): A policy is trained with extrinsic rewards and curiosity rewards based on an inverse dynamics model.
+
+Note that these baselines, unlike the learned intrinsic rewards, do not transfer any knowledge across different lifetimes. Throughout Sections 4.1-4.4, we focus on analysing what kind of knowledge is learned by the intrinsic reward depending on the nature of environments. We discuss the benefit of using the lifetime return and considering the lifetime history when learning the intrinsic reward in Section 4.5. The details of implementation and hyperparameters are described in Appendix B.
+
+# 4.1 EXPLORING UNCERTAIN STATES
+
+We designed ‘Empty Rooms’ (Figure 2a) to see whether the intrinsic reward can learn to encourage exploration of uncertain states like novelty-based exploration methods. The goal is to visit an invisible goal location, which is fixed within each lifetime but varies across lifetimes. Episode terminates when the goal is reached. Each lifetime consists of 200 episodes. From the agent’s perspective, its policy should visit the locations suggested by the intrinsic reward. From the intrinsic reward’s perspective, it should encourage the agent to go to unvisited locations to locate the goal, and once the goal is located to exploit that knowledge for the rest of that lifetime.
+
+Figure 3 shows that our learned intrinsic reward was more efficient than extrinsic rewards and countbased exploration when training a new agent. We observed that the intrinsic reward learned two interesting strategies as visualised in Figure 4. While the goal is not found, it encourages exploration of unvisited locations, because it learned the prior that there exists a rewarding goal location somewhere. Once the goal is found the intrinsic reward encourages the agent to exploit it without further exploration, because it learned that there is only one goal. This result shows that curiosity about uncertain states can naturally emerge when various states can be rewarding in a domain, even when the rewarding states are fixed within an agent’s lifetime.
+
+![](images/8217eed6c0b57029eee8e8f9e77ef8b33e619f413a56532eef4020fad068d1f8.jpg)  
+Figure 4: Visualisation of the first 3000 steps of an agent trained with different reward functions in Empty Rooms. (a) The blue and yellow squares represent the agent and the hidden goal, respectively. (b) The learned reward encourages the agent to visit many locations if the goal is not found (top). However, when the goal is found early, the intrinsic reward makes the agent exploit it without further exploration (bottom). (c) An agent trained only with extrinsic rewards explores poorly. (d-e) Both the count-based and ICM rewards tend to encourage exploration (top) but hinders exploitation when the goal is found (bottom).
+
+![](images/89e3cb8c01df9a2f42ec95376a6857278e77bbe709b8b81b607cd1bed3a95658.jpg)  
+Figure 5: Visualisation of the learned intrinsic reward in Random ABC, where the extrinsic rewards for A, B, and C are 0.2, -0.5, and 0.1 respectively. Each figure shows the sum of intrinsic rewards for a trajectory towards each object (A, B, and C). In the first episode, the intrinsic reward encourages the agent to explore A. In the second episode, the intrinsic reward encourages exploring C if A is visited (top) or vice versa (bottom). In episode 3, after both A and C are explored, the intrinsic reward encourages to revisit A (both top and bottom).
+
+# 4.2 EXPLORING UNCERTAIN OBJECTS AND AVOIDING HARMFUL OBJECTS
+
+In the previous domain, we considered uncertainty of where the reward (or goal location) is. We now consider dealing with uncertainty about the value of different objects. In the ‘Random ABC’ environment (see Figure 2b), for each lifetime the rewards for objects A, B, and C are uniformly sampled from $[ - 1 , \bar { 1 ] }$ , $[ - 0 . 5 , 0 ]$ , and $[ 0 , 0 . 5 ]$ respectively but are held fixed within the lifetime. A good intrinsic reward should learn that: 1) B should be avoided, 2) A and C have uncertain rewards, hence require systematic exploration (first go to one and then the other), and 3) once it is determined which of the two A or C is better, exploit that knowledge by encouraging the agent to repeatedly go to that object for the rest of the lifetime.
+
+Figure 3 shows that the agent learned a near-optimal exploration-and-then-exploitation method with the learned intrinsic reward. Note that the agent cannot pass information about the reward for objects across episodes, as usual in reinforcement learning. The intrinsic reward can propagate such information across episodes and help the agent explore or exploit appropriately. We visualised the learned intrinsic reward for different actions sequences in Figure 5. The intrinsic rewards encourage the agent to explore towards A and C in the first few episodes. Once A and C are explored, the agent exploits the largest rewarding object. Throughout training, the agent is discouraged to visit B through negative intrinsic rewards. These results show that avoidance and curiosity about uncertain objects can potentially emerge if the environment has various or fixed rewarding objects.
+
+![](images/d39f89e801a85bc9490e7fc902efe3aa86ddb279c408622b85b12c7dccf4762f.jpg)
+
+![](images/d8f7c140ad7661ab691e5bb31131cb1a0d41b0fd3f4282a2a451b25894ec1263.jpg)  
+Figure 6: Visualisation of the agent’s intrinsic and extrinsic rewards (left) and the entropy of its policy (right) on Non-stationary ABC. The task changes at $5 0 0 0 \mathrm { { t h } }$ episode (dashed vertical line). The intrinsic reward gives a negative reward even before the task changes (green rectangle) and makes the policy less peaky (entropy increases). As a result, the agent quickly adapts to the change.   
+Figure 7: Evaluation of different intrinsic reward architectures and objectives. For ‘LSTM’ the reward network has an LSTM taking the lifetime history as input. For ‘FF’ a feed-forward reward network takes only the current time-step. ‘Lifetime’ and ‘Episode’ means the lifetime and episodic return as objective respectively.
+
+# 4.3 EXPLOITING INVARIANT CAUSAL RELATIONSHIP
+
+To see how the intrinsic reward deals with causal relationship between objects, we designed ‘KeyBox’, which is similar to Random ABC except that there is a key in the room (see Figure 2c). The agent needs to collect the key first to open one of the boxes (A, B, and C) and receive the corresponding reward. The rewards for the objects are sampled from the same distribution as Random ABC. The key itself gives a neutral reward of 0. Moreover, the locations of the agent, the key, and the boxes are randomly sampled for each episode. As a result, the state space contains more than 3 billion distinct states and thus is infeasible to fully enumerate. Figure 3 shows that learned intrinsic reward leads to a near-optimal exploration. The agent trained with extrinsic rewards did not learn to open any box. The intrinsic reward captures that the key is necessary to open any box, which is true across many lifetimes of training. This demonstrates that the intrinsic reward can capture causal relationships between objects when the domain has this kind of invariant dynamics.
+
+# 4.4 DEALING WITH NON-STATIONARITY
+
+We investigated how the intrinsic reward deals with non-stationarity of tasks within a lifetime in our ‘Non-stationary ABC’ environment. Rewards are as follows: for A is either 1 or $- 1$ , for $\mathbf { B }$ i s $- 0 . 5$ , for C is the negative value of the reward for A. The rewards of A and C are swapped every 250 episodes. Each lifetime lasts 1000 episodes. Figure 3 shows that the agent with the learned intrinsic reward quickly recovered its performance when the task changes, whereas the baselines take more time to recover. Figure 6 shows how the learned intrinsic reward encourages the learning agent to react to the changing rewards. Interestingly, the intrinsic reward has learned to prepare for the change by giving negative rewards to the exploitation policy of the agent a few episodes before the task changes. In other words, the intrinsic reward starts to discourage the agent to commit to the current best rewarding object, thereby increasing entropy in the current policy in anticipation of the change, eventually making it easier to adapt quickly. This shows that the intrinsic reward can capture the (regularly) repeated non-stationarity across many lifetimes and make the agent intrinsically motivated not to commit too firmly to a policy, in anticipation of changes in the environment.
+
+# 4.5 ABLATION STUDY
+
+To study relative benefits of the proposed technical ideas, we conducted an ablation study 1) by replacing the long-term lifetime return objective $( G ^ { \mathrm { l i f e } } )$ with the episodic return $( G ^ { \mathrm { e p } } )$ and 2) by restricting the input of the reward network to the current time-step instead of the entire lifetime history. Figure 7 shows that the lifetime history was crucial to achieve good performance. This is reasonable because all domains require some past information (e.g., current object rewards in Random ABC, visited locations in Empty Rooms) to provide useful exploration strategies. It is also shown that the lifetime return objective was beneficial on Random ABC, Non-stationary ABC, and Key-Box. These domains require exploration across multiple episodes in order to find the optimal policy. For example, collecting an uncertain object (e.g., object A in Random ABC) is necessary even if the episode terminates with a negative reward. The episodic value function would directly penalise such an under-performed exploratory episode when computing meta-gradient, which prevents the intrinsic reward from learning to encourage exploration across episodes. On the other hand, such behaviour can be encouraged by the lifetime value function as long as it provides useful information to maximise the lifetime return in the long term.
+
+![](images/8fcb4aa069f767786613a956c467e2a4f59d7fd2e129dfad1fa480d8a8dc3a06.jpg)  
+Figure 8: Comparison to policy transfer methods.
+
+# 5 EMPIRICAL INVESTIGATIONS: GENERALISATION VIA REWARDS
+
+As noted above, rewards capture knowledge about what an agent’s goals should be rather than how it should behave. At the same time, transferring the latter in the form of policies is also feasible in our domains presented above. Here we confirm that by implementing and presenting results for the following two meta-learning methods:
+
+• MAML (Finn et al., 2017a): A policy meta-learned from a distributions of tasks such that it can adapt quickly to the given task after a few parameter updates. • $\mathtt { R L } ^ { 2 }$ (Duan et al., 2016; Wang et al., 2016): An LSTM policy unrolled over the entire lifetime to maximise the lifetime return, which is pre-trained on a distributions of tasks.
+
+Although all the methods we implemented including ours are designed to learn useful knowledge from a distribution of tasks, they have different objectives. Specifically, the objective of our method is to learn knowledge that is useful for training “randomly-initialised policies” by capturing “what to do”, whereas the goal of policy transfer methods is to directly transfer a useful policy for fast task adaptation by transferring “how to do” knowledge. In fact, it can be more efficient to transfer and reuse pre-trained policies instead of restarting from a random policy and learning using the learned rewards given a new task. Figure 8 indeed shows that $\mathrm { { R L ^ { 2 } } }$ performs better than our intrinsic reward approach. It is also shown that MAML and $\mathrm { { R L ^ { 2 } } }$ achieve good performance from the beginning, as they have already learned how to navigate the grid worlds and how to achieve the goals of the tasks. In our method, on the other hand, the agent starts from a random policy and relies on the learned intrinsic reward which indirectly tells it what to do. Nevertheless, our method outperforms MAML and achieves a comparable asymptotic performance to $\mathrm { { R L ^ { 2 } } }$ .
+
+# 5.1 GENERALISATION TO DIFFERENT AGENT-ENVIRONMENT INTERFACES
+
+In fact, our method can be interpreted as an instance of $\mathtt { R L } ^ { 2 }$ with a particular decomposition of parameters $\boldsymbol { \theta }$ and $\eta$ ), which uses policy gradient as a recurrent update (see Figure 1). While this modular structure may not be more beneficial than $\mathrm { { R L ^ { 2 } } }$ when evaluated with the same agent-environment interface, such a decomposition provides clear semantics of each module: the policy $\mathbf { \eta } ^ { ( \theta ) }$ captures “how to do” while the intrinsic reward $( \eta )$ captures “what to do”, and this enables interesting kinds of generalisations as we show below. Specifically, we show that “what” knowledge captured by the intrinsic reward can be reused by many different learning agents as follows.
+
+Generalisation to unseen action spaces We first evaluated the learned intrinsic reward on new action spaces. Specifically, the intrinsic reward was used to train new agents with either 1) permuted actions, where the semantics of left/right and up/down are reversed, or 2) extended actions, with 4 additional actions that move diagonally. Figure 9a shows that the intrinsic reward provided useful rewards to new agents with different actions, even when these were not trained with those actions. This is possible because the intrinsic reward assigns rewards to the agent’s state changes rather than its actions. In other words, the intrinsic reward captures “what to do”, which makes it possible to generalise to new actions, as long as the goal remains the same. On the other hand, it is unclear how to generalise $\mathtt { R L } ^ { 2 }$ and MAML in this way.
+
+![](images/db6945122086cc6de484fa5658b93fcb3df4004e11ddf53f3e966a1900cf73fd.jpg)  
+Figure 9: Generalisation to new agent-environment interfaces in Random ABC. (a) ‘Permuted’ agents have different action semantics. ‘Extended’ agents have additional actions. (b) ‘AC-Intrinsic’ is the original actorcritic agent trained with the intrinsic reward. ‘Q-Intrinsic’ is a Q-learning agent with the intrinsic reward learned from actor-critic agents. ‘Q-Extrinsic’ is the Q-learning agent with the extrinsic reward. (c) shows the performances of the policy transfer baselines with permuted actions during evaluation.
+
+Generalisation to unseen learning algorithms We further investigated how general the knowledge captured by the intrinsic reward is by evaluating the learned intrinsic reward on agents with different learning algorithms. In particular, after training the intrinsic reward from actor-critic agents, we evaluated it by training new agents through Q-learning while using the learned intrinsic reward as denoted by ‘Q-Intrinsic’ in Figure 9b. Interestingly, it turns out that the learned intrinsic reward is general enough to be useful for Q-learning agents, even though it was trained for actor-critic agents. Again, it is unclear how to generalise $\mathrm { { R L } ^ { 2 } }$ and MAML in this way.
+
+Comparison to policy transfer While it wasn’t possible to apply the learned policy from $\mathtt { R L } ^ { 2 }$ and MAML when we extended the action space and when we changed the learning algorithm, we can do so when we keep the same number of actions and just permute them. As shown in Figure ${ 9 \mathrm { c } }$ , both $\mathrm { { R L ^ { 2 } } }$ and MAML generalise poorly when the action space is permuted for Random ABC, because the transferred policies are highly biased to the original action space. Again, this result highlights the difference between “what to do” knowledge captured by our approach and “how to do” knowledge captured by policies.
+
+# 6 CONCLUSION
+
+We revisited the optimal reward problem (Singh et al., 2009) and proposed a more scalable gradientbased method for learning intrinsic rewards. Through several proof-of-concept experiments, we showed that the learned non-stationary intrinsic reward can capture regularities within a distribution of environments or, over time, within a non-stationary environment. As a result, they were capable of encouraging both exploratory and exploitative behaviour across multiple episodes. In addition, some task-independent notions of intrinsic motivation such as curiosity emerged when they were effective for the distribution over tasks across lifetimes the agent was trained on. We also showed that the learned intrinsic rewards can generalise to different agent-environment interfaces such as different action spaces and different learning algorithms, whereas policy transfer methods fail to generalise. This highlights the difference between the “what” kind of knowledge captured by rewards and the “how” kind of knowledge captured by policies. The flexibility and range of knowledge captured by intrinsic rewards in our proof-of-concept experiments encourages further work towards combining different loci of knowledge to achieve greater practical benefits.
+
+# REFERENCES
+
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+
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+
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+
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+
+# A DERIVATION OF INTRINSIC REWARD UPDATE
+
+Following the conventional notation in RL, we define $v _ { T } ( \tau _ { t } | \eta , \theta _ { 0 } )$ as the state-value function that estimates the expected future lifetime return given the lifetime history $\tau _ { t }$ , the task $\tau$ , initial policy parameters $\theta _ { 0 }$ and the intrinsic reward parameters $\eta$ . Specially, $v _ { T } ( \tau _ { 0 } | \eta , \theta _ { 0 } )$ denotes the expected lifetime return at the starting state, i.e.,
+
+$$
+v _ { T } ( \tau _ { 0 } | \eta , \theta _ { 0 } ) = \mathbb { E } _ { \tau \sim p _ { \eta } ( \tau | \theta _ { 0 } ) } \left[ G ^ { \mathrm { l i f e } } \right] ,
+$$
+
+where $G ^ { \mathrm { l i f e } }$ denotes the lifetime return in task $\tau$ . We also define the action-value function $q _ { T } ( \tau _ { t } , a _ { t } | \eta , \theta _ { 0 } )$ accordingly as the expected future lifetime return given the lifetime history $\tau _ { t }$ and an action $a _ { t }$ .
+
+The objective function of the optimal reward problem is defined as:
+
+$$
+\begin{array} { r l } & { J ( \eta ) = \mathbb { E } _ { \theta _ { 0 } \sim \Theta , \mathcal { T } \sim p ( \mathcal { T } ) } \left[ \mathbb { E } _ { \tau \sim p _ { \eta } ( \tau \mid \theta _ { 0 } ) } \left[ G ^ { \mathrm { l i f e } } \right] \right] } \\ & { \qquad = \mathbb { E } _ { \theta _ { 0 } \sim \Theta , \mathcal { T } \sim p ( \mathcal { T } ) } \left[ v _ { \mathcal { T } } ( \tau _ { 0 } \vert \eta , \theta _ { 0 } ) \right] , } \end{array}
+$$
+
+where $\Theta$ and $p ( \mathcal { T } )$ are an initial policy distribution and a task distribution respectively.
+
+Assuming the task $\tau$ and the initial policy parameters $\theta _ { 0 }$ are given, we omit $\tau$ and $\theta _ { 0 }$ for the rest of equations for simplicity. Let $\pi _ { \eta } ( \cdot | \tau _ { t } ) = \dot { \pi } _ { \theta _ { t } } \bar { ( \cdot | s _ { t } ) }$ be the probability distribution over actions at time $t$ given the history $\tau _ { t }$ , where $\dot { \theta _ { t } } = f _ { \eta } ( \tau _ { t } , \theta _ { 0 } )$ is the policy parameters at time $t$ in the lifetime. We can derive the meta-gradient with respect to $\eta$ by the following:
+
+$$
+\begin{array} { r l } &  \begin{array} { r l } & { ( \gamma _ { 1 } , \gamma _ { 2 } ) } \\ & { = \nabla _ { x } \cdot \nabla _ { x } \cdot \rho ( x _ { 2 } ) ) \cdot } \\ & { = \nabla _ { x } \cdot \left[ \sum _ { \alpha } \cdot \nabla _ { x } \cdot ( \alpha ) \ln ^ { \beta } [ \gamma _ { \alpha } ] \cdot \rho _ { \alpha } ( x _ { \beta } , \alpha , \eta ) \right] } \\ & { = \nabla _ { x } \cdot \left[ \nabla _ { x } \cdot \sigma _ { \alpha } ( x _ { \beta } ) \mathrm { e r f o r } [ \gamma _ { \alpha } , \alpha , \eta ) \right] } \\ & { \quad - \frac { 1 } { \sqrt { \alpha } } \left[ \nabla _ { x } \cdot \sigma _ { \alpha } ( x _ { \beta } ) \mathrm { e r f o r } [ \gamma _ { \alpha } , \alpha , \eta ) ] - \sigma _ { \alpha } ( x _ { \beta } ) \cdot \nabla _ { x } \cdot \rho _ { \alpha } ( x _ { \beta } ) \cdot \sigma _ { \alpha } ( x _ { \beta } ) \right] } \\ & { = - \frac { 1 } { \sqrt { \alpha } } \left[ \nabla _ { x } \cdot \sigma _ { \alpha } ( x _ { \beta } ) \mathrm { e r f o r } [ \gamma _ { \alpha } , \gamma _ { \alpha } , \eta ) ] \cdot \Pi _ { \alpha \eta \gamma } \cdot \sigma _ { \alpha } ( x _ { \beta } ) \cdot \sigma _ { \alpha } ( x _ { \beta } ) \right] \cdot \nabla _ { x } \cdot \rho _ { \alpha } ( x _ { \beta } ) \cdot \sigma _ { \alpha } ( x _ { \eta } ) \cdot \sigma _ { \alpha } ( x _ { \eta } ) \cdot \sigma _ { \alpha } ( x _ { \beta } ) \cdot } \\ & { \quad - \frac { 1 } { \sqrt { \alpha } } \left[ \nabla _ { x } \cdot \sigma _ { \alpha } ( x _ { \beta } ) \mathrm { e r f o r } [ \gamma _ { \alpha } , \eta ] \cdot \sigma _ { \alpha } ( x _ { \eta } ) ) \right] \cdot \Pi _ { \alpha \eta \gamma } \cdot \rho _ { \alpha } ( x _ { \beta } ) \cdot \nabla _ { x } \cdot \rho _ { \alpha } ( x _ { \eta } ) \cdot \nabla _ { x } \cdot \rho _ { \alpha } ( x _ { \eta } ) \cdot \sigma _ { \alpha } ( x _ { \beta } ) \right] } \\ &  = \frac { 1 } { \sqrt { \alpha } } \left[ \nabla _ { x } \cdot \sigma _ { \alpha } ( x _ { \beta } ) \mathrm { e r f o r } [ \gamma _ { \alpha } , \eta  \end{array} \end{array}
+$$
+
+where $\begin{array} { r } { G _ { t } = \sum _ { k = t } ^ { T - 1 } r _ { k } } \end{array}$ is the lifetime return given the history $\tau _ { t }$ , and we assume the discount factor $\gamma = 1$ for brevity. Thus, the derivative of the overall objective is:
+
+$$
+\begin{array} { r } { \nabla _ { \eta } J ( \eta ) = \mathbb { E } _ { \theta _ { 0 } \sim \Theta , \mathcal { T } \sim p ( \mathcal { T } ) } \left[ \mathbb { E } _ { \tau _ { t } \sim p ( \tau _ { t } \mid \eta , \theta _ { 0 } ) } \left[ G _ { t } \nabla _ { \theta _ { t } } \log \pi _ { \theta _ { t } } ( a _ { t } | s _ { t } ) \nabla _ { \eta } \theta _ { t } \right] \right] . } \end{array}
+$$
+
+# B EXPERIMENTAL DETAILS
+
+# B.1 IMPLEMENTATION DETAILS
+
+We used mini-batch update to reduce the variance of meta-gradient estimation. Specifically, we ran 64 lifetimes in parallel, each with a randomly sample task and randomly initialised policy parameters. We took the average of the meta-gradients from each lifetime to compute the update to the intrinsic reward parameters $( \eta )$ . We ran $2 \times 1 0 ^ { 5 }$ updates to $\eta$ at training time. We used arctan activation on the output of the intrinsic reward. The hyperparameters used for each domain are described in Table 1.
+
+Table 1: Hyperparameters.   
+
+<table><tr><td>Hyperparameters</td><td>Empty Rooms</td><td>Random ABC</td><td>Key-Box</td><td>Non-stationary ABC</td></tr><tr><td>Time limit per episode</td><td>100</td><td>10</td><td>100</td><td>10</td></tr><tr><td>Numberof episodes per lifetime</td><td>200</td><td>50</td><td>5000</td><td>1000</td></tr><tr><td>Inner unroll length</td><td>8</td><td>4</td><td>16</td><td>4</td></tr><tr><td>Entropy regularisation</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.05</td></tr><tr><td>Policy architecture</td><td colspan="4">Conv(16)-FC(64)</td></tr><tr><td>Policy optimiser</td><td colspan="4">SGD SGD</td></tr><tr><td>Policy learning rate</td><td colspan="4">Adam 0.1 0.1 0.001</td></tr><tr><td>Reward architecture</td><td colspan="4">Conv(16)-FC(64)-LSTM(64) Adam</td></tr><tr><td>Reward optimiser Reward learning rate</td><td colspan="4">0.001</td></tr><tr><td>Outer unroll length</td><td colspan="4">5</td></tr><tr><td>Inner discount factor</td><td colspan="4">0.9</td></tr><tr><td>Outer discounter factor</td><td colspan="4"></td></tr><tr><td></td><td colspan="4">0.99</td></tr></table>
+
+# B.2 DOMAINS
+
+We will consider five task distributions, instantiated within one of the three main gridworld domains shown in Figure 2. In all cases the agent has four actions available, corresponding to moving up, down, left and right. However the topology of the gridworld and the reward structure may vary.
+
+# B.2.1 EMPTY ROOMS
+
+Figure 2a shows the layout of the Empty Rooms domain. There are four rooms in this domain. The agent always starts at the centre of the top-left room. One and only one cell is rewarding, which is called the goal. The goal is invisible. The goal location is sampled uniformly from all cells at the beginning of each lifetime. An episode terminates when the agent reaches the goal location or a time limit of 100 steps is reached. Each lifetime consists of 200 episodes. The agent needs to explore all rooms to find the goal and then goes to the goal afterwards.
+
+# B.2.2 ABC WORLD
+
+Figure 2b shows the layout of the ABC World domain. There is a single 5 by 5 room, with three objects (denoted by A, B, C). All object provides reward upon reaching them. An episode terminates when the agent reaches an object or a time limit of 10 steps is reached. We consider three different versions of this environment: Fixed ABC, Random $A B C$ and Non-stationary ABC. In the Fixed ABC environment, each lifetime has 200 episodes. The reward associated with each object is fixed across lifetimes. Specifically, the rewards for objects A, B, and C are 1, $- 0 . 5$ , and 0.5 respectively. The optimal policy is to always collect A. In the Random ABC environment, each lifetime has 50 episodes. The reward associated with each object is randomly sampled for each lifetime and is held fixed within a lifetime. Thus, the environment is stationary from an agent’s perspective but non-stationary from the reward function’s perspective. Specifically, the rewards for A, B, and C are uniformly sampled from $[ - 1 , 1 ]$ , $[ - 0 . 5 , 0 ]$ , and [0.0.5] respectively. The optimal behaviour is to explore A and C at the beginning of a lifetime to assess which is the better, and then commits to the better one for all subsequent episode. In the non-stationary ABC environment, each lifetime has
+
+![](images/3566109a904a4c2bf69ea4b8440f6405ed3491a41000a33b790269d9b3556327.jpg)  
+Figure 10: Evaluation of different rewards in the Fixed ABC domain. The $\mathbf { X }$ -axis shows the number of episodes within a single lifetime; the y-axis measures the episode return.
+
+1000 episodes. The rewards for A, B, and C are 1, $- 0 . 5$ , and $- 1$ respectively. The rewards for A and C swap every 250 episodes.
+
+# B.2.3 KEY BOX WORLD
+
+Figure 2c shows the Key Box World domain. In this domain, there is a key and three boxes, A, B, and C. In order to open any box, the agent must pick up the key first. The rewards for A, B, and C are uniformly sampled from $[ - 1 , 1 ]$ , $[ - 0 . 5 , 0 ]$ , and $[ 0 , 0 . 5 ]$ respectively for each lifetime. An episode terminates when the agent opens a box or a time limit of 100 steps is reached. Each lifetime consists of 5000 episodes.
+
+# B.3 HAND-DESIGNED NEAR-OPTIMAL EXPLORATION STRATEGY FOR RANDOM ABC
+
+We hand-designed a heuristic strategy for the Random ABC domain. We assume the agent has the prior knowledge that B is always bad and A and C have uncertain rewards. Therefore, the heuristic is to go to A in the first episode, go to C in the second episode, and then go to the better one in the remaining episodes in the lifetime. We view this heuristic as an upper-bound because it always finds the best object and can arbitrarily control the agent’s behaviour.
+
+# C ADDITIONAL EMPIRICAL RESULT
+
+# C.1 EXPLOITING OPTIMAL BEHAVIOUR ON A FIXED TASK
+
+To investigate what the intrinsic reward learns in a fixed task, we designed the ‘Fixed ABC’ environment (see Figure 2b). The reward for each object (A, B, and C) is fixed within and across lifetimes. When the agent collects an object, it receives the corresponding reward of 1, $- 0 . 5$ , or 0.5 for object A, B, or C respectively, and the episode terminates. Each lifetime contains 200 episodes. The optimal policy is to always collect A. The optimal reward should capture the regularity of the environment that object A has the highest reward and drive the agent towards object A.
+
+Figure 10 shows that agents trained with the learned intrinsic reward learn optimal policies within a few episodes. This indicates that the intrinsic reward memorises the fixed optimal behaviour during training and assigns rewards accordingly to aid learning during evaluation. The result on Fixed ABC is not particularly surprising. In a fixed task in a stationary environment, an optimal reward function does not need to encourage exploration, and helps the agent to directly learn the optimal behaviour as quickly as possible, similarly to reward shaping.

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+        "type": "text",
+        "text": "WHAT CAN LEARNED INTRINSIC REWARDS CAPTURE? ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+    },
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+        "text": "Reinforcement learning agents can include different components, such as policies, value functions, state representations, and environment models. Any or all of these can be the loci of knowledge, i.e., structures where knowledge, whether given or learned, can be deposited and reused. Regardless of its composition, the objective of an agent is behave so as to maximise the sum of a suitable scalar function of state: the reward. As far as the learning algorithm is concerned, these rewards are typically given and immutable. In this paper we instead consider the proposition that the reward function itself may be a good locus of knowledge. This is consistent with a common use, in the literature, of hand-designed intrinsic rewards to improve the learning dynamics of an agent. We adopt the multi-lifetime setting of the Optimal Rewards Framework, and propose to meta-learn an intrinsic reward function from experience that allows agents to maximise their extrinsic rewards accumulated until the end of their lifetimes. Rewards as a locus of knowledge provide guidance on “what” the agent should strive to do rather than on knowledge of “how” the agent should behave that is more directly captured in policies or value functions for example. Thus, our focus here is on demonstrating the following: (1) that it is feasible to meta-learn good reward functions, (2) to show that the learned reward functions can capture interesting kinds of “what” knowledge, and (3) that because of the indirectness of this form of knowledge the learned reward functions can generalise to other kinds of agents and to changes in the dynamics of the environment. ",
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+    {
+        "type": "text",
+        "text": "Reinforcement learning agents can store knowledge in their policies, value functions, state representations, and models of the environment dynamics. These components can be the loci of knowledge in the sense that they are structures in which knowledge, either learned from experience by the agent’s algorithm or given by the agent-designer, can be deposited and reused. The objective of the agent is defined by a reward function, and the goal is to learn to act so as to optimise cumulative rewards. In this paper we consider the proposition that the reward function itself is a good locus of knowledge. This is unusual in that most prior work treats the reward as given and immutable, at least as far as the learning algorithm is concerned. At the same time, especially in challenging reinforcement-learning problems, agent designers do find it convenient to modify the reward function given to the agent to facilitate learning. It is therefore useful to distinguish between two kinds of reward functions (Singh et al., 2010): extrinsic rewards define the task and capture the designer’s preferences over agent behaviour, whereas intrinsic rewards serve as helpful signals to improve the learning dynamics of the agent. Intrinsic rewards are typically hand-designed and then often added to the immutable extrinsic rewards to form the reward optimised by the agent. ",
+        "bbox": [
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+            547,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Most existing work on intrinsic rewards falls into two broad categories: task-dependent and taskindependent. Both are typically designed by hand. Hand-designing task-dependent rewards can be fraught with difficulty as even minor misalignment between the actual reward and the intended bias can lead to unintended and sometimes catastrophic consequences (Clark & Amodei, 2016). Task-independent intrinsic rewards are also typically hand-designed, often based on an intuitive understanding of animal/human behaviour or on heuristics on desired exploratory behaviour. It can, however, be hard to match such task-independent intrinsic rewards to the specific learning dynamics induced by the interaction between agent and environment. The motivation for this paper is our interest in the comparatively under-explored possibility of learned (not hand-designed) taskdependent intrinsic rewards (see Zheng et al., 2018, for previous work). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We emphasise that it is not our objective to show that rewards are a better locus of learned knowledge than others; the best locus likely depends on the kind of knowledge that is most useful in a given task. In particular, knowledge captured in rewards provides guidance on “what” the agent should strive to do while knowledge captured in policies provides guidance on “how” an agent should behave. Knowledge about “what” captured in rewards is indirect and thus slower to make an impact on behaviour because it takes effect through learning, while knowledge about “how” can directly have an immediate impact on behaviour. At the same time, because of its indirectness the former can generalise better to changes in dynamics and learning architectures. Therefore, instead of comparing different loci of knowledge, the purpose of this paper is to show that it is feasible to capture useful learned knowledge in rewards and to study the kinds of knowledge that can be captured. ",
+        "bbox": [
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+        "text": "How should we measure the usefulness of a learned reward function? Ideally, we would like to measure the effect the learned reward function has on the learning dynamics. Of course, learning happens over multiple episodes, indeed it happens over an entire lifetime. Therefore, we choose lifetime return, the cumulative extrinsic reward obtained by the agent over its entire lifetime, as the main objective. To this end, we adopt the multi-lifetime setting of the Optimal Rewards Framework (Singh et al., 2009) in which an agent is initialised randomly at the start of each lifetime and then faces a stationary or non-stationary task drawn from some distribution. In this setting, the only knowledge that is transferred across lifetimes is the reward instead of the policy. Specifically, the goal is to learn a single intrinsic reward function that, when used to adapt the agent’s policy using a standard episodic RL algorithm, ends up optimising the cumulative extrinsic reward over its lifetime. ",
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+    {
+        "type": "text",
+        "text": "In previous work, good reward functions were found via exhaustive search, limiting the range of applicability of the framework. Here, we develop a more scalable gradient-based method ( $\\mathrm { { X u } }$ et al., 2018c) for learning the intrinsic rewards by exploiting the fact the interaction between the policy update and the reward function is differentiable (Zheng et al., 2018). Since it is infeasible to backpropgate through the full computation graph that spans across the entire lifetime, we truncate the unrolled computation graph of learning updates up to some horizon. However, we handle the long-term credit assignment by using a lifetime value function that estimates the remaining lifetime return, which needs to take into account changing policies. Our main scientific contributions are a sequence of empirical studies on carefully designed environments that show how our learned intrinsic rewards can capture interesting regularities in the interaction between a learning agent and an environment sampled from a distribution, and how the learned intrinsic reward can generalise to changed dynamics and agent architectures. Collectively, our contributions present an effective approach to the discovery of intrinsic rewards that can help an agent optimise the extrinsic rewards collected in a lifetime. ",
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+    {
+        "type": "text",
+        "text": "1 RELATED WORK ",
+        "text_level": 1,
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+        "text": "Hand-designed Rewards There is a long history of work on designing rewards to accelerate learning in reinforcement learning (RL). Reward shaping aims to design task-specific rewards towards known optimal behaviours, typically requiring domain knowledge. Both the benefits (Randlov & Alstrm, 1998; Ng et al., 1999; Harutyunyan et al., 2015) and the difficulty (Clark & Amodei, 2016) of task-specific reward shaping have been studied. On the other hand, many intrinsic rewards have been proposed to encourage exploration, inspired by animal behaviours. Examples include prediction error (Schmidhuber, 1991b; Gordon & Ahissar, 2011; Mirolli & Baldassarre, 2013; Pathak et al., 2017; Schmidhuber, 1991a), surprise (Itti & Baldi, 2006), weight change (Linke et al., 2019), and state-visitation counts (Sutton, 1990; Poupart et al., 2006; Strehl & Littman, 2008; Bellemare et al., 2016; Ostrovski et al., 2017). Although these kinds of intrinsic rewards are not domain-specific, they are often not well-aligned with the task that the agent tries to solve, and ignores the effect on the agent’s learning dynamics. In contrast, our work aims to learn intrinsic rewards from data that take into account the agent’s learning dynamics without requiring prior knowledge from a human. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Rewards Learned from Data There have been a few attempts to learn useful intrinsic rewards from data. The optimal reward framework (Singh et al., 2009) proposed to learn an optimal reward function that allows agents to solve a distribution of tasks quickly using random search. We revisit this problem in this paper and propose a more scalable gradient-based approach. Although there have been follow-up works (Sorg et al., 2010; Guo et al., 2016) that uses a gradient-based method, they consider a non-parameteric policy using Monte-Carlo Tree Search (MCTS). Our work is closely related to LIRPG (Zheng et al., 2018) which proposed a meta-gradient method to learn intrinsic rewards. However, LIRPG considers a single task in a single lifetime with a myopic episode return objective, which is limited in that it does not allow exploration across episodes or generalisation to different agents. ",
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+        "img_path": "images/3ef9c31f414d6956d1f1b75935eeddc3f4d63e06416cbadbda17f6bae752e2c6.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of the proposed intrinsic reward learning framework. The intrinsic reward $r _ { \\eta }$ is used to update the agent’s parameter $\\theta _ { i }$ throughout its lifetime which consists of many episodes. The goal is to find the optimal intrinsic reward parameters $\\eta ^ { * }$ across many lifetimes that maximises the lifetime return $( G ^ { \\mathrm { l i f e } } )$ given any randomly initialised agents and possibly non-stationary tasks drawn from some distribution $p ( \\mathcal T )$ . "
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Meta-learning for Exploration Meta-learning (Schmidhuber et al., 1996; Thrun & Pratt, 1998) has recently received considerable attention in RL. Recent advances include few-shot adaptation (Finn et al., 2017a), few-shot imitation (Finn et al., 2017b; Duan et al., 2017), model adaptation (Clavera et al., 2018), and inverse RL (Xu et al., 2018a). In particular, our work is closely related to the prior work on meta-learning good exploration strategies (Wang et al., 2016; Duan et al., 2016; Stadie et al., 2018; Xu et al., 2018b) in that both perform temporal credit assignment across episode boundaries by maximising rewards accumulated beyond an episode. Unlike the prior work that aims to learn an exploratory policy, our framework indirectly drives exploration via a reward function which can be reused by different learning agents as we show in this paper (Section 5.1). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Meta-learning of Agent Update There have been a few studies that directly meta-learn how to update the agent’s parameters via meta-parameters including discount factor and returns (Xu et al., 2018c), auxiliary tasks (Schlegel et al., 2018; Veeriah et al., 2019), unsupervised learning rules (Metz et al., 2019), and RL objectives (Chebotar et al., 2019). Our work also belongs to this category in that our meta-parameters are the reward function used in the agent’s update. In particular, our multilifetime formulation is similar to $\\mathrm { { M L ^ { 3 } } }$ (Chebotar et al., 2019). However, we consider the long-term lifetime return as objective to perform cross-episode temporal credit assignment as opposed to the myopic episodic objective in ML3. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 THE OPTIMAL REWARD PROBLEM ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We first introduce some terminology. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Agent: A learning system interacting with an environment. On each step $t$ the agent selects an action $a _ { t }$ and receives from the environment an observation $s _ { t + 1 }$ and an extrinsic reward $r _ { t + 1 }$ defined by a task $\\tau$ . The agent chooses actions based on a policy $\\pi _ { \\boldsymbol { \\theta } } \\big ( a _ { t } | \\boldsymbol { s } _ { t } \\big )$ parameterised by $\\theta$ . • Episode: A finite sequence of agent-environment interactions until the end of the episode defined by the task. An episode return is defined as: $\\begin{array} { r } { G ^ { \\mathrm { e p } } = \\sum _ { t = 0 } ^ { T _ { \\mathrm { e p } } - 1 } \\gamma ^ { t } r _ { t + 1 } } \\end{array}$ , where $\\gamma$ is a discount factor, and the random variable $T _ { \\mathrm { e p } }$ gives the finite number of steps until the end of the episode. • Lifetime: A finite sequence of agent-environment interactions until the end of training deent-des, where ner, which can include is a discount factor, and ltiple episodes. The lifetime return is is the number of steps in the lifetime. $G ^ { \\mathrm { { \\bar { i } f e } } } =$ $\\scriptstyle \\sum _ { t = 0 } ^ { T - 1 } \\gamma ^ { t } r _ { t + 1 }$ $\\gamma$ $T$ • Intrinsic reward: A reward function $r _ { \\eta } ( \\tau _ { t + 1 } )$ parameterised by $\\eta$ , where $\\begin{array} { r l } { \\tau _ { t } } & { { } = } \\end{array}$ $( s _ { 0 } , a _ { 0 } , r _ { 1 } , d _ { 1 } , s _ { 1 } , \\ldots , r _ { t } , d _ { t } , s _ { t } )$ is a lifetime history with (binary) episode terminations $d _ { i }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The Optimal Reward Problem (Singh et al., 2010), illustrated in Figure 1, aims to learn the parameters of the intrinsic reward such that the resulting rewards achieve a learning dynamic for an RL agent that maximises the lifetime (extrinsic) return on tasks drawn from some distribution. Formally, the optimal reward function is defined as: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/5235d5297ba1fa6302040ae32b4b2e4edac40de65127169094aed5656f2068fa.jpg",
+        "text": "$$\n\\eta ^ { * } = \\underset { \\eta } { \\arg \\operatorname* { m a x } } J ( \\eta ) = \\underset { \\eta } { \\arg \\operatorname* { m a x } } \\mathbb { E } _ { \\theta _ { 0 } \\sim \\Theta , \\mathcal { T } \\sim p ( \\mathcal { T } ) } \\left[ \\mathbb { E } _ { \\tau \\sim p _ { \\eta } ( \\tau \\mid \\theta _ { 0 } ) } \\left[ G ^ { \\mathrm { l i f e } } \\right] \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "where $\\Theta$ and $p ( \\mathcal T )$ are an initial policy distribution and a distribution over possibly non-stationary tasks respectively, and $\\begin{array} { r } { G ^ { \\mathrm { l i f e } } = \\sum _ { t = 0 } ^ { { \\bar { T } } - 1 } \\dot { \\gamma } ^ { t } r _ { t + 1 } } \\end{array}$ is a lifetime return. The likelihood of a lifetime history ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Learning intrinsic rewards across multiple lifetimes via meta-gradient ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Input: $p ( \\mathcal { T } )$ : Task distribution, $\\Theta$ : Randomly-initialised policy distribution   \nInitialise intrinsic reward function $\\eta$ and lifetime value function $\\phi$   \nrepeat Initialise task $\\tau \\sim p ( \\mathcal { T } )$ and policy $\\theta \\sim \\Theta$ while lifetime not ended do $\\theta _ { 0 } \\gets \\theta$ for $k = 1 , 2 , \\ldots , N$ do Generate a trajectory using πθk−1 Update policy $\\theta _ { k } \\gets \\theta _ { k - 1 } + \\alpha \\nabla _ { \\theta _ { k - 1 } } J _ { \\eta } ( \\theta _ { k - 1 } )$ using intrinsic rewards $r _ { \\eta }$ (Eq. 2) end for Update intrinsic reward function $\\eta$ using Eq. 3 Update lifetime value function $\\phi$ using Eq. 4 $\\boldsymbol { \\theta } \\dot { }  \\boldsymbol { \\theta } _ { N }$ end while   \nuntil $\\eta$ converges ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\tau$ is $\\begin{array} { r } { p _ { \\eta } ( \\tau \\vert \\theta _ { 0 } ) = p ( s _ { 0 } ) \\prod _ { t = 0 } ^ { T - 1 } \\pi _ { \\theta _ { t } } ( a _ { t } \\vert s _ { t } ) p ( d _ { t + 1 } , r _ { t + 1 } , s _ { t + 1 } \\vert s _ { t } , a _ { t } ) , } \\end{array}$ , where $\\theta _ { t } = f ( \\theta _ { t - 1 } , \\eta )$ is a policy parameter as updated with update function $f$ , which is policy gradient in this paper.1 Note that the optimisation of $\\eta$ spans multiple lifetimes, each of which can span multiple episodes. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Using the lifetime return $G ^ { \\mathrm { l i f e } }$ as the objective instead of the conventional episodic return $G ^ { \\mathrm { e p } }$ allows exploration across multiple episodes as long as the lifetime return is maximised in the long run. In particular, when the lifetime is defined as a fixed number of episodes, we find that the lifetime return objective is sometimes more beneficial than the episodic return objective even in terms of the episodic return performance measure. However, different objectives (e.g., final episode return) can be considered depending on the definition of what a good reward function is. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "3 META-LEARNING INTRINSIC REWARD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We propose a meta-gradient approach $\\mathrm { { X u } }$ et al., 2018c; Zheng et al., 2018) to solve the optimal reward problem. At a high-level, we sample a new task $\\tau$ and a new random policy parameter $\\theta$ at each lifetime iteration. We then simulate an agent’s lifetime by updating the parameter $\\theta$ using an intrinsic reward function $r _ { \\eta }$ (Section 3.1) with policy gradient (Section 3.2). In the meantime, we compute the meta-gradient by taking into account the effect of the intrinsic rewards on the policy parameters to update the intrinsic reward function with a lifetime value function (Section 3.3). Algorithm 1 gives an overview of our algorithm. The following sections describe the details. ",
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+    {
+        "type": "text",
+        "text": "3.1 INTRINSIC REWARD AND LIFETIME VALUE FUNCTION ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The intrinsic reward function is a recurrent neural network (RNN) parameterised by $\\eta$ , which produces a scalar reward on arriving in state $s _ { t }$ by taking into account the history of an agent’s lifetime $\\tau _ { t } = ( s _ { 0 } , a _ { 0 } , r _ { 1 } , d _ { 1 } , s _ { 1 } , . . . , r _ { t } , d _ { t } , s _ { t } )$ . We claim that giving the lifetime history across episodes as input is crucial for balancing exploration and exploitation, for instance by capturing how frequently a certain state is visited to determine an exploration bonus reward. The lifetime value function is a separate recurrent neural network parameterised by $\\phi$ , which takes the same inputs as the intrinsic reward function and produces a scalar value estimation of the expected future return within the lifetime. ",
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+    {
+        "type": "text",
+        "text": "3.2 POLICY UPDATE (θ) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Each agent interacts with an environment and a task sampled from a distribution $\\tau \\sim p ( \\mathcal { T } )$ . However, instead of directly maximising the extrinsic rewards defined by the task, the agent maximises ",
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+    {
+        "type": "text",
+        "text": "the intrinsic rewards $( r _ { \\eta } )$ by using policy gradient (Williams, 1992; Sutton et al., 2000): ",
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+    {
+        "type": "equation",
+        "img_path": "images/d68e53b0511267349526b944e92339973e355c6a5ae3d530e7153743d511cd59.jpg",
+        "text": "$$\nJ _ { \\eta } ( \\theta ) = \\mathbb { E } _ { \\theta } \\bigg [ \\sum _ { t = 0 } ^ { T _ { \\mathrm { e p } } - 1 } \\bar { \\gamma } ^ { t } r _ { \\eta } ( \\tau _ { t + 1 } ) \\bigg ] \\qquad \\quad \\nabla _ { \\theta } J _ { \\eta } ( \\theta ) = \\mathbb { E } _ { \\theta } \\bigg [ G _ { \\eta , t } ^ { \\mathrm { e p } } \\nabla _ { \\theta } \\log \\pi _ { \\theta } ( a | s ) \\bigg ] ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $r _ { \\eta } ( \\tau _ { t + 1 } )$ is the intrinsic reward at time $t$ , and = PTep−1k=t γ¯k−trη(τk+1) is the return of the intrinsic rewards accumulated over an episode with discount factor $\\bar { \\gamma }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 INTRINSIC REWARD $( \\eta )$ AND LIFETIME VALUE FUNCTION $( \\phi )$ UPDATE ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To update the intrinsic reward parameters $\\eta$ , we directly take a meta-gradient ascent step using the overall objective (Equation 1). Specifically, the gradient is (see Appendix A for derivation): ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/230c0b2bccd3aa0766f55efa26c056456e7428af4d5f8d4aff4ffed153bccae4.jpg",
+        "text": "$$\n\\nabla _ { \\eta } J ( \\eta ) = \\mathbb { E } _ { \\theta _ { 0 } \\sim \\Theta , \\mathcal { T } \\sim p ( T ) } \\biggl [ \\mathbb { E } _ { \\tau _ { t } \\sim p ( \\tau _ { t } | \\eta , \\theta _ { 0 } ) } \\biggl [ G _ { t } ^ { \\mathrm { l i f e } } \\nabla _ { \\theta _ { t } } \\log \\pi _ { \\theta _ { t } } ( a _ { t } | s _ { t } ) \\nabla _ { \\eta } \\theta _ { t } \\biggr ] \\biggr ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where PT −1k=t γk−trk+1 is a lifetime return based on the extrinsic rewards of task T with discount factor $\\gamma$ . The chain rule is used to get the meta-gradient $( \\nabla _ { \\boldsymbol { \\eta } } \\theta _ { t } )$ as in previous work (Zheng et al., 2018). The computation graph of this procedure is illustrated in Figure 1. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Computing the true meta-gradient in Equation 3 requires backpropagation through the entire lifetime, which is infeasible as each lifetime can involve more than thousands of policy updates. To partially address this issue, we truncate the meta-gradient after $N$ policy updates but approximate the lifetime return $G _ { t } ^ { \\mathrm { l i f e } , \\phi } \\approx G _ { t } ^ { \\mathrm { l i f e } }$ using a lifetime value function $V _ { \\phi } ( \\tau )$ parameterised by $\\phi$ , which is learned using a temporal difference learning from -step trajectory: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/e40131f0b3282d925507372f6c514a7d27962595cc9952b6af08f6361a0dcf07.jpg",
+        "text": "$$\nG _ { t } ^ { \\mathrm { l i f e } , \\phi } = \\sum _ { k = 0 } ^ { n - 1 } \\gamma ^ { k } r _ { t + k + 1 } + \\gamma ^ { n } V _ { \\phi } ( \\tau _ { t + n } ) \\qquad \\phi = \\phi + \\alpha ^ { \\prime } ( G _ { t } ^ { \\mathrm { l i f e } , \\phi } - V _ { \\phi } ( \\tau _ { t } ) ) \\nabla _ { \\phi } V _ { \\phi } ( \\tau _ { t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In our empirical work, we found that the lifetime value estimates were crucial to allow the intrinsic reward to perform long-term credit assignments across episodes. ",
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+    {
+        "type": "text",
+        "text": "3.4 CONNECTION TO STANDARD RL FRAMEWORKS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The policy learning problem specified in Section 3.2 deviates from the standard Markov Decision Process (MDP) framework because the intrinsic reward function is a function of the lifetime history rather than a function of states or state-action pairs. Consequently, from the memoryless policy’s perspective, the rewards are non-stationary. However, we can also view the combination of the intrinsic reward function and the policy as a joint lifetime-history-based policy parameterised by $\\eta$ and $\\theta$ (see derivation in Appendix A). From this perspective, the overall learning problem specified in Section 3.3 can be formulated as an MDP with history as state (recall, we use RNNs for the intrinsic reward function). As a result, standard temporal-difference learning methods are applicable to learning lifetime value functions. ",
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+    {
+        "type": "text",
+        "text": "4 EMPIRICAL INVESTIGATIONS: FEASIBILITY AND USEFULNESS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We present the results from our empirical investigations in two sections. For the results in this section, the experiments and domains are designed to answer the following research questions: ",
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+        "type": "text",
+        "text": "• What kind of knowledge can be learned by the intrinsic reward? • How does the distribution of tasks drive the form of the intrinsic reward? • What is the benefit of the lifetime return objective over the episode return? • When is it important to provide the lifetime history as input to the intrinsic reward? ",
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+    {
+        "type": "text",
+        "text": "We investigate these research questions in various grid-world domains illustrated in Figure 2. For each domain, we trained an intrinsic reward function across many lifetimes and evaluated it by training an agent using the learned reward. We implemented the following baselines. ",
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+        "type": "text",
+        "text": "• Extrinsic-EP: A policy is trained with extrinsic rewards to maximise the episode return. ",
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+    {
+        "type": "image",
+        "img_path": "images/0d553e8ac18d76f927e644f7009853d9b0c6f94c94b1695efe43449e074db081.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of domains. (a) The agent needs to find the goal location which gives a positive reward, but the goal is not visible to the agent. (b) Each object (A, B, and C) gives rewards. (c) The agent is required to first collect the key and visit one of the boxes (A, B, and C) to receive the corresponding reward. All objects are placed to random locations after every episode. "
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+        "type": "image",
+        "img_path": "images/3776bc779886d5dac24e55fb5b1c51d390913ad1ff73748f06236bc60e04bf9e.jpg",
+        "image_caption": [
+            "Figure 3: Evaluation of different reward functions averaged over 30 seeds. The learning curves show agents trained with our intrinsic reward (blue), with the extrinsic reward with the episodic return objective (orange) and the lifetime return objective (brown), and with a count-based exploration reward (green). The dashed line corresponds to a hand-designed near-optimal exploration strategy. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• Extrinsic-LIFE: A policy is trained with extrinsic rewards to maximise the lifetime return.   \n• Count-based (Strehl & Littman, 2008): A policy is trained with extrinsic rewards and countbased exploration bonus rewards.   \n• ICM (Pathak et al., 2017): A policy is trained with extrinsic rewards and curiosity rewards based on an inverse dynamics model. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that these baselines, unlike the learned intrinsic rewards, do not transfer any knowledge across different lifetimes. Throughout Sections 4.1-4.4, we focus on analysing what kind of knowledge is learned by the intrinsic reward depending on the nature of environments. We discuss the benefit of using the lifetime return and considering the lifetime history when learning the intrinsic reward in Section 4.5. The details of implementation and hyperparameters are described in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "4.1 EXPLORING UNCERTAIN STATES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We designed ‘Empty Rooms’ (Figure 2a) to see whether the intrinsic reward can learn to encourage exploration of uncertain states like novelty-based exploration methods. The goal is to visit an invisible goal location, which is fixed within each lifetime but varies across lifetimes. Episode terminates when the goal is reached. Each lifetime consists of 200 episodes. From the agent’s perspective, its policy should visit the locations suggested by the intrinsic reward. From the intrinsic reward’s perspective, it should encourage the agent to go to unvisited locations to locate the goal, and once the goal is located to exploit that knowledge for the rest of that lifetime. ",
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+    {
+        "type": "text",
+        "text": "Figure 3 shows that our learned intrinsic reward was more efficient than extrinsic rewards and countbased exploration when training a new agent. We observed that the intrinsic reward learned two interesting strategies as visualised in Figure 4. While the goal is not found, it encourages exploration of unvisited locations, because it learned the prior that there exists a rewarding goal location somewhere. Once the goal is found the intrinsic reward encourages the agent to exploit it without further exploration, because it learned that there is only one goal. This result shows that curiosity about uncertain states can naturally emerge when various states can be rewarding in a domain, even when the rewarding states are fixed within an agent’s lifetime. ",
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+        "img_path": "images/8217eed6c0b57029eee8e8f9e77ef8b33e619f413a56532eef4020fad068d1f8.jpg",
+        "image_caption": [
+            "Figure 4: Visualisation of the first 3000 steps of an agent trained with different reward functions in Empty Rooms. (a) The blue and yellow squares represent the agent and the hidden goal, respectively. (b) The learned reward encourages the agent to visit many locations if the goal is not found (top). However, when the goal is found early, the intrinsic reward makes the agent exploit it without further exploration (bottom). (c) An agent trained only with extrinsic rewards explores poorly. (d-e) Both the count-based and ICM rewards tend to encourage exploration (top) but hinders exploitation when the goal is found (bottom). "
+        ],
+        "image_footnote": [],
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+        "img_path": "images/89e3cb8c01df9a2f42ec95376a6857278e77bbe709b8b81b607cd1bed3a95658.jpg",
+        "image_caption": [
+            "Figure 5: Visualisation of the learned intrinsic reward in Random ABC, where the extrinsic rewards for A, B, and C are 0.2, -0.5, and 0.1 respectively. Each figure shows the sum of intrinsic rewards for a trajectory towards each object (A, B, and C). In the first episode, the intrinsic reward encourages the agent to explore A. In the second episode, the intrinsic reward encourages exploring C if A is visited (top) or vice versa (bottom). In episode 3, after both A and C are explored, the intrinsic reward encourages to revisit A (both top and bottom). "
+        ],
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+    {
+        "type": "text",
+        "text": "4.2 EXPLORING UNCERTAIN OBJECTS AND AVOIDING HARMFUL OBJECTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the previous domain, we considered uncertainty of where the reward (or goal location) is. We now consider dealing with uncertainty about the value of different objects. In the ‘Random ABC’ environment (see Figure 2b), for each lifetime the rewards for objects A, B, and C are uniformly sampled from $[ - 1 , \\bar { 1 ] }$ , $[ - 0 . 5 , 0 ]$ , and $[ 0 , 0 . 5 ]$ respectively but are held fixed within the lifetime. A good intrinsic reward should learn that: 1) B should be avoided, 2) A and C have uncertain rewards, hence require systematic exploration (first go to one and then the other), and 3) once it is determined which of the two A or C is better, exploit that knowledge by encouraging the agent to repeatedly go to that object for the rest of the lifetime. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 3 shows that the agent learned a near-optimal exploration-and-then-exploitation method with the learned intrinsic reward. Note that the agent cannot pass information about the reward for objects across episodes, as usual in reinforcement learning. The intrinsic reward can propagate such information across episodes and help the agent explore or exploit appropriately. We visualised the learned intrinsic reward for different actions sequences in Figure 5. The intrinsic rewards encourage the agent to explore towards A and C in the first few episodes. Once A and C are explored, the agent exploits the largest rewarding object. Throughout training, the agent is discouraged to visit B through negative intrinsic rewards. These results show that avoidance and curiosity about uncertain objects can potentially emerge if the environment has various or fixed rewarding objects. ",
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+        "image_caption": [],
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+        "page_idx": 7
+    },
+    {
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+        "image_caption": [
+            "Figure 6: Visualisation of the agent’s intrinsic and extrinsic rewards (left) and the entropy of its policy (right) on Non-stationary ABC. The task changes at $5 0 0 0 \\mathrm { { t h } }$ episode (dashed vertical line). The intrinsic reward gives a negative reward even before the task changes (green rectangle) and makes the policy less peaky (entropy increases). As a result, the agent quickly adapts to the change. ",
+            "Figure 7: Evaluation of different intrinsic reward architectures and objectives. For ‘LSTM’ the reward network has an LSTM taking the lifetime history as input. For ‘FF’ a feed-forward reward network takes only the current time-step. ‘Lifetime’ and ‘Episode’ means the lifetime and episodic return as objective respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 EXPLOITING INVARIANT CAUSAL RELATIONSHIP ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To see how the intrinsic reward deals with causal relationship between objects, we designed ‘KeyBox’, which is similar to Random ABC except that there is a key in the room (see Figure 2c). The agent needs to collect the key first to open one of the boxes (A, B, and C) and receive the corresponding reward. The rewards for the objects are sampled from the same distribution as Random ABC. The key itself gives a neutral reward of 0. Moreover, the locations of the agent, the key, and the boxes are randomly sampled for each episode. As a result, the state space contains more than 3 billion distinct states and thus is infeasible to fully enumerate. Figure 3 shows that learned intrinsic reward leads to a near-optimal exploration. The agent trained with extrinsic rewards did not learn to open any box. The intrinsic reward captures that the key is necessary to open any box, which is true across many lifetimes of training. This demonstrates that the intrinsic reward can capture causal relationships between objects when the domain has this kind of invariant dynamics. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.4 DEALING WITH NON-STATIONARITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We investigated how the intrinsic reward deals with non-stationarity of tasks within a lifetime in our ‘Non-stationary ABC’ environment. Rewards are as follows: for A is either 1 or $- 1$ , for $\\mathbf { B }$ i s $- 0 . 5$ , for C is the negative value of the reward for A. The rewards of A and C are swapped every 250 episodes. Each lifetime lasts 1000 episodes. Figure 3 shows that the agent with the learned intrinsic reward quickly recovered its performance when the task changes, whereas the baselines take more time to recover. Figure 6 shows how the learned intrinsic reward encourages the learning agent to react to the changing rewards. Interestingly, the intrinsic reward has learned to prepare for the change by giving negative rewards to the exploitation policy of the agent a few episodes before the task changes. In other words, the intrinsic reward starts to discourage the agent to commit to the current best rewarding object, thereby increasing entropy in the current policy in anticipation of the change, eventually making it easier to adapt quickly. This shows that the intrinsic reward can capture the (regularly) repeated non-stationarity across many lifetimes and make the agent intrinsically motivated not to commit too firmly to a policy, in anticipation of changes in the environment. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.5 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To study relative benefits of the proposed technical ideas, we conducted an ablation study 1) by replacing the long-term lifetime return objective $( G ^ { \\mathrm { l i f e } } )$ with the episodic return $( G ^ { \\mathrm { e p } } )$ and 2) by restricting the input of the reward network to the current time-step instead of the entire lifetime history. Figure 7 shows that the lifetime history was crucial to achieve good performance. This is reasonable because all domains require some past information (e.g., current object rewards in Random ABC, visited locations in Empty Rooms) to provide useful exploration strategies. It is also shown that the lifetime return objective was beneficial on Random ABC, Non-stationary ABC, and Key-Box. These domains require exploration across multiple episodes in order to find the optimal policy. For example, collecting an uncertain object (e.g., object A in Random ABC) is necessary even if the episode terminates with a negative reward. The episodic value function would directly penalise such an under-performed exploratory episode when computing meta-gradient, which prevents the intrinsic reward from learning to encourage exploration across episodes. On the other hand, such behaviour can be encouraged by the lifetime value function as long as it provides useful information to maximise the lifetime return in the long term. ",
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+    {
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+        "img_path": "images/8fcb4aa069f767786613a956c467e2a4f59d7fd2e129dfad1fa480d8a8dc3a06.jpg",
+        "image_caption": [
+            "Figure 8: Comparison to policy transfer methods. "
+        ],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 EMPIRICAL INVESTIGATIONS: GENERALISATION VIA REWARDS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As noted above, rewards capture knowledge about what an agent’s goals should be rather than how it should behave. At the same time, transferring the latter in the form of policies is also feasible in our domains presented above. Here we confirm that by implementing and presenting results for the following two meta-learning methods: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• MAML (Finn et al., 2017a): A policy meta-learned from a distributions of tasks such that it can adapt quickly to the given task after a few parameter updates. • $\\mathtt { R L } ^ { 2 }$ (Duan et al., 2016; Wang et al., 2016): An LSTM policy unrolled over the entire lifetime to maximise the lifetime return, which is pre-trained on a distributions of tasks. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Although all the methods we implemented including ours are designed to learn useful knowledge from a distribution of tasks, they have different objectives. Specifically, the objective of our method is to learn knowledge that is useful for training “randomly-initialised policies” by capturing “what to do”, whereas the goal of policy transfer methods is to directly transfer a useful policy for fast task adaptation by transferring “how to do” knowledge. In fact, it can be more efficient to transfer and reuse pre-trained policies instead of restarting from a random policy and learning using the learned rewards given a new task. Figure 8 indeed shows that $\\mathrm { { R L ^ { 2 } } }$ performs better than our intrinsic reward approach. It is also shown that MAML and $\\mathrm { { R L ^ { 2 } } }$ achieve good performance from the beginning, as they have already learned how to navigate the grid worlds and how to achieve the goals of the tasks. In our method, on the other hand, the agent starts from a random policy and relies on the learned intrinsic reward which indirectly tells it what to do. Nevertheless, our method outperforms MAML and achieves a comparable asymptotic performance to $\\mathrm { { R L ^ { 2 } } }$ . ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "5.1 GENERALISATION TO DIFFERENT AGENT-ENVIRONMENT INTERFACES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In fact, our method can be interpreted as an instance of $\\mathtt { R L } ^ { 2 }$ with a particular decomposition of parameters $\\boldsymbol { \\theta }$ and $\\eta$ ), which uses policy gradient as a recurrent update (see Figure 1). While this modular structure may not be more beneficial than $\\mathrm { { R L ^ { 2 } } }$ when evaluated with the same agent-environment interface, such a decomposition provides clear semantics of each module: the policy $\\mathbf { \\eta } ^ { ( \\theta ) }$ captures “how to do” while the intrinsic reward $( \\eta )$ captures “what to do”, and this enables interesting kinds of generalisations as we show below. Specifically, we show that “what” knowledge captured by the intrinsic reward can be reused by many different learning agents as follows. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Generalisation to unseen action spaces We first evaluated the learned intrinsic reward on new action spaces. Specifically, the intrinsic reward was used to train new agents with either 1) permuted actions, where the semantics of left/right and up/down are reversed, or 2) extended actions, with 4 additional actions that move diagonally. Figure 9a shows that the intrinsic reward provided useful rewards to new agents with different actions, even when these were not trained with those actions. This is possible because the intrinsic reward assigns rewards to the agent’s state changes rather than its actions. In other words, the intrinsic reward captures “what to do”, which makes it possible to generalise to new actions, as long as the goal remains the same. On the other hand, it is unclear how to generalise $\\mathtt { R L } ^ { 2 }$ and MAML in this way. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/db6945122086cc6de484fa5658b93fcb3df4004e11ddf53f3e966a1900cf73fd.jpg",
+        "image_caption": [
+            "Figure 9: Generalisation to new agent-environment interfaces in Random ABC. (a) ‘Permuted’ agents have different action semantics. ‘Extended’ agents have additional actions. (b) ‘AC-Intrinsic’ is the original actorcritic agent trained with the intrinsic reward. ‘Q-Intrinsic’ is a Q-learning agent with the intrinsic reward learned from actor-critic agents. ‘Q-Extrinsic’ is the Q-learning agent with the extrinsic reward. (c) shows the performances of the policy transfer baselines with permuted actions during evaluation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            102,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Generalisation to unseen learning algorithms We further investigated how general the knowledge captured by the intrinsic reward is by evaluating the learned intrinsic reward on agents with different learning algorithms. In particular, after training the intrinsic reward from actor-critic agents, we evaluated it by training new agents through Q-learning while using the learned intrinsic reward as denoted by ‘Q-Intrinsic’ in Figure 9b. Interestingly, it turns out that the learned intrinsic reward is general enough to be useful for Q-learning agents, even though it was trained for actor-critic agents. Again, it is unclear how to generalise $\\mathrm { { R L } ^ { 2 } }$ and MAML in this way. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Comparison to policy transfer While it wasn’t possible to apply the learned policy from $\\mathtt { R L } ^ { 2 }$ and MAML when we extended the action space and when we changed the learning algorithm, we can do so when we keep the same number of actions and just permute them. As shown in Figure ${ 9 \\mathrm { c } }$ , both $\\mathrm { { R L ^ { 2 } } }$ and MAML generalise poorly when the action space is permuted for Random ABC, because the transferred policies are highly biased to the original action space. Again, this result highlights the difference between “what to do” knowledge captured by our approach and “how to do” knowledge captured by policies. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We revisited the optimal reward problem (Singh et al., 2009) and proposed a more scalable gradientbased method for learning intrinsic rewards. Through several proof-of-concept experiments, we showed that the learned non-stationary intrinsic reward can capture regularities within a distribution of environments or, over time, within a non-stationary environment. As a result, they were capable of encouraging both exploratory and exploitative behaviour across multiple episodes. In addition, some task-independent notions of intrinsic motivation such as curiosity emerged when they were effective for the distribution over tasks across lifetimes the agent was trained on. We also showed that the learned intrinsic rewards can generalise to different agent-environment interfaces such as different action spaces and different learning algorithms, whereas policy transfer methods fail to generalise. This highlights the difference between the “what” kind of knowledge captured by rewards and the “how” kind of knowledge captured by policies. The flexibility and range of knowledge captured by intrinsic rewards in our proof-of-concept experiments encourages further work towards combining different loci of knowledge to achieve greater practical benefits. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "text": "Ronald J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Mach. Learn., 8(3-4):229–256, May 1992. ISSN 0885-6125. doi: 10.1007/ BF00992696.   \nKelvin Xu, Ellis Ratner, Anca Dragan, Sergey Levine, and Chelsea Finn. Learning a prior over intent via meta-inverse reinforcement learning. arXiv preprint arXiv:1805.12573, 2018a.   \nTianbing Xu, Qiang Liu, Liang Zhao, and Jian Peng. Learning to explore with meta-policy gradient. arXiv preprint arXiv:1803.05044, 2018b.   \nZhongwen Xu, Hado P van Hasselt, and David Silver. Meta-gradient reinforcement learning. In Advances in Neural Information Processing Systems, pp. 2396–2407, 2018c.   \nZeyu Zheng, Junhyuk Oh, and Satinder Singh. On learning intrinsic rewards for policy gradient methods. In Advances in Neural Information Processing Systems, pp. 4644–4654, 2018. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A DERIVATION OF INTRINSIC REWARD UPDATE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            581,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Following the conventional notation in RL, we define $v _ { T } ( \\tau _ { t } | \\eta , \\theta _ { 0 } )$ as the state-value function that estimates the expected future lifetime return given the lifetime history $\\tau _ { t }$ , the task $\\tau$ , initial policy parameters $\\theta _ { 0 }$ and the intrinsic reward parameters $\\eta$ . Specially, $v _ { T } ( \\tau _ { 0 } | \\eta , \\theta _ { 0 } )$ denotes the expected lifetime return at the starting state, i.e., ",
+        "bbox": [
+            173,
+            132,
+            825,
+            189
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/40b8a93d5629a1e4d921518583d3d87d9e3b37f0c7f3e838e4f6f181a48cb29a.jpg",
+        "text": "$$\nv _ { T } ( \\tau _ { 0 } | \\eta , \\theta _ { 0 } ) = \\mathbb { E } _ { \\tau \\sim p _ { \\eta } ( \\tau | \\theta _ { 0 } ) } \\left[ G ^ { \\mathrm { l i f e } } \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            195,
+            616,
+            217
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $G ^ { \\mathrm { l i f e } }$ denotes the lifetime return in task $\\tau$ . We also define the action-value function $q _ { T } ( \\tau _ { t } , a _ { t } | \\eta , \\theta _ { 0 } )$ accordingly as the expected future lifetime return given the lifetime history $\\tau _ { t }$ and an action $a _ { t }$ . ",
+        "bbox": [
+            174,
+            224,
+            826,
+            267
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The objective function of the optimal reward problem is defined as: ",
+        "bbox": [
+            174,
+            273,
+            616,
+            289
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/48a49896f5fcde1f98b416d986d4e4695676898ad930ed94b7e75e72c48ff136.jpg",
+        "text": "$$\n\\begin{array} { r l } & { J ( \\eta ) = \\mathbb { E } _ { \\theta _ { 0 } \\sim \\Theta , \\mathcal { T } \\sim p ( \\mathcal { T } ) } \\left[ \\mathbb { E } _ { \\tau \\sim p _ { \\eta } ( \\tau \\mid \\theta _ { 0 } ) } \\left[ G ^ { \\mathrm { l i f e } } \\right] \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { \\theta _ { 0 } \\sim \\Theta , \\mathcal { T } \\sim p ( \\mathcal { T } ) } \\left[ v _ { \\mathcal { T } } ( \\tau _ { 0 } \\vert \\eta , \\theta _ { 0 } ) \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            294,
+            645,
+            335
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $\\Theta$ and $p ( \\mathcal { T } )$ are an initial policy distribution and a task distribution respectively. ",
+        "bbox": [
+            166,
+            340,
+            740,
+            356
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Assuming the task $\\tau$ and the initial policy parameters $\\theta _ { 0 }$ are given, we omit $\\tau$ and $\\theta _ { 0 }$ for the rest of equations for simplicity. Let $\\pi _ { \\eta } ( \\cdot | \\tau _ { t } ) = \\dot { \\pi } _ { \\theta _ { t } } \\bar { ( \\cdot | s _ { t } ) }$ be the probability distribution over actions at time $t$ given the history $\\tau _ { t }$ , where $\\dot { \\theta _ { t } } = f _ { \\eta } ( \\tau _ { t } , \\theta _ { 0 } )$ is the policy parameters at time $t$ in the lifetime. We can derive the meta-gradient with respect to $\\eta$ by the following: ",
+        "bbox": [
+            173,
+            362,
+            825,
+            419
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1f4f601db8561c43ec985b996e8517ec35db58a930f4c61c39b3b653a97dd63a.jpg",
+        "text": "$$\n\\begin{array} { r l } &  \\begin{array} { r l } & { ( \\gamma _ { 1 } , \\gamma _ { 2 } ) } \\\\ & { = \\nabla _ { x } \\cdot \\nabla _ { x } \\cdot \\rho ( x _ { 2 } ) ) \\cdot } \\\\ & { = \\nabla _ { x } \\cdot \\left[ \\sum _ { \\alpha } \\cdot \\nabla _ { x } \\cdot ( \\alpha ) \\ln ^ { \\beta } [ \\gamma _ { \\alpha } ] \\cdot \\rho _ { \\alpha } ( x _ { \\beta } , \\alpha , \\eta ) \\right] } \\\\ & { = \\nabla _ { x } \\cdot \\left[ \\nabla _ { x } \\cdot \\sigma _ { \\alpha } ( x _ { \\beta } ) \\mathrm { e r f o r } [ \\gamma _ { \\alpha } , \\alpha , \\eta ) \\right] } \\\\ & { \\quad - \\frac { 1 } { \\sqrt { \\alpha } } \\left[ \\nabla _ { x } \\cdot \\sigma _ { \\alpha } ( x _ { \\beta } ) \\mathrm { e r f o r } [ \\gamma _ { \\alpha } , \\alpha , \\eta ) ] - \\sigma _ { \\alpha } ( x _ { \\beta } ) \\cdot \\nabla _ { x } \\cdot \\rho _ { \\alpha } ( x _ { \\beta } ) \\cdot \\sigma _ { \\alpha } ( x _ { \\beta } ) \\right] } \\\\ & { = - \\frac { 1 } { \\sqrt { \\alpha } } \\left[ \\nabla _ { x } \\cdot \\sigma _ { \\alpha } ( x _ { \\beta } ) \\mathrm { e r f o r } [ \\gamma _ { \\alpha } , \\gamma _ { \\alpha } , \\eta ) ] \\cdot \\Pi _ { \\alpha \\eta \\gamma } \\cdot \\sigma _ { \\alpha } ( x _ { \\beta } ) \\cdot \\sigma _ { \\alpha } ( x _ { \\beta } ) \\right] \\cdot \\nabla _ { x } \\cdot \\rho _ { \\alpha } ( x _ { \\beta } ) \\cdot \\sigma _ { \\alpha } ( x _ { \\eta } ) \\cdot \\sigma _ { \\alpha } ( x _ { \\eta } ) \\cdot \\sigma _ { \\alpha } ( x _ { \\beta } ) \\cdot } \\\\ & { \\quad - \\frac { 1 } { \\sqrt { \\alpha } } \\left[ \\nabla _ { x } \\cdot \\sigma _ { \\alpha } ( x _ { \\beta } ) \\mathrm { e r f o r } [ \\gamma _ { \\alpha } , \\eta ] \\cdot \\sigma _ { \\alpha } ( x _ { \\eta } ) ) \\right] \\cdot \\Pi _ { \\alpha \\eta \\gamma } \\cdot \\rho _ { \\alpha } ( x _ { \\beta } ) \\cdot \\nabla _ { x } \\cdot \\rho _ { \\alpha } ( x _ { \\eta } ) \\cdot \\nabla _ { x } \\cdot \\rho _ { \\alpha } ( x _ { \\eta } ) \\cdot \\sigma _ { \\alpha } ( x _ { \\beta } ) \\right] } \\\\ &  = \\frac { 1 } { \\sqrt { \\alpha } } \\left[ \\nabla _ { x } \\cdot \\sigma _ { \\alpha } ( x _ { \\beta } ) \\mathrm { e r f o r } [ \\gamma _ { \\alpha } , \\eta  \\end{array} \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            207,
+            435,
+            787,
+            729
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { G _ { t } = \\sum _ { k = t } ^ { T - 1 } r _ { k } } \\end{array}$ is the lifetime return given the history $\\tau _ { t }$ , and we assume the discount factor $\\gamma = 1$ for brevity. Thus, the derivative of the overall objective is: ",
+        "bbox": [
+            173,
+            734,
+            823,
+            765
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/54a12cda25a0b54874cbe1df6ee3bef404f641f45efcc3eb522643229884f80d.jpg",
+        "text": "$$\n\\begin{array} { r } { \\nabla _ { \\eta } J ( \\eta ) = \\mathbb { E } _ { \\theta _ { 0 } \\sim \\Theta , \\mathcal { T } \\sim p ( \\mathcal { T } ) } \\left[ \\mathbb { E } _ { \\tau _ { t } \\sim p ( \\tau _ { t } \\mid \\eta , \\theta _ { 0 } ) } \\left[ G _ { t } \\nabla _ { \\theta _ { t } } \\log \\pi _ { \\theta _ { t } } ( a _ { t } | s _ { t } ) \\nabla _ { \\eta } \\theta _ { t } \\right] \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            770,
+            735,
+            791
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            419,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.1 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            135,
+            408,
+            150
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We used mini-batch update to reduce the variance of meta-gradient estimation. Specifically, we ran 64 lifetimes in parallel, each with a randomly sample task and randomly initialised policy parameters. We took the average of the meta-gradients from each lifetime to compute the update to the intrinsic reward parameters $( \\eta )$ . We ran $2 \\times 1 0 ^ { 5 }$ updates to $\\eta$ at training time. We used arctan activation on the output of the intrinsic reward. The hyperparameters used for each domain are described in Table 1. ",
+        "bbox": [
+            174,
+            161,
+            825,
+            246
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/501f2292c05ccdb02e24bf93cdaf2aa25cf2f3c865bcc3bf66bd24659b47795f.jpg",
+        "table_caption": [
+            "Table 1: Hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameters</td><td>Empty Rooms</td><td>Random ABC</td><td>Key-Box</td><td>Non-stationary ABC</td></tr><tr><td>Time limit per episode</td><td>100</td><td>10</td><td>100</td><td>10</td></tr><tr><td>Numberof episodes per lifetime</td><td>200</td><td>50</td><td>5000</td><td>1000</td></tr><tr><td>Inner unroll length</td><td>8</td><td>4</td><td>16</td><td>4</td></tr><tr><td>Entropy regularisation</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.05</td></tr><tr><td>Policy architecture</td><td colspan=\"4\">Conv(16)-FC(64)</td></tr><tr><td>Policy optimiser</td><td colspan=\"4\">SGD SGD</td></tr><tr><td>Policy learning rate</td><td colspan=\"4\">Adam 0.1 0.1 0.001</td></tr><tr><td>Reward architecture</td><td colspan=\"4\">Conv(16)-FC(64)-LSTM(64) Adam</td></tr><tr><td>Reward optimiser Reward learning rate</td><td colspan=\"4\">0.001</td></tr><tr><td>Outer unroll length</td><td colspan=\"4\">5</td></tr><tr><td>Inner discount factor</td><td colspan=\"4\">0.9</td></tr><tr><td>Outer discounter factor</td><td colspan=\"4\"></td></tr><tr><td></td><td colspan=\"4\">0.99</td></tr></table>",
+        "bbox": [
+            179,
+            285,
+            816,
+            477
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.2 DOMAINS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            503,
+            287,
+            518
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We will consider five task distributions, instantiated within one of the three main gridworld domains shown in Figure 2. In all cases the agent has four actions available, corresponding to moving up, down, left and right. However the topology of the gridworld and the reward structure may vary. ",
+        "bbox": [
+            174,
+            531,
+            825,
+            574
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.2.1 EMPTY ROOMS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            590,
+            338,
+            604
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 2a shows the layout of the Empty Rooms domain. There are four rooms in this domain. The agent always starts at the centre of the top-left room. One and only one cell is rewarding, which is called the goal. The goal is invisible. The goal location is sampled uniformly from all cells at the beginning of each lifetime. An episode terminates when the agent reaches the goal location or a time limit of 100 steps is reached. Each lifetime consists of 200 episodes. The agent needs to explore all rooms to find the goal and then goes to the goal afterwards. ",
+        "bbox": [
+            173,
+            616,
+            825,
+            700
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.2.2 ABC WORLD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            718,
+            326,
+            732
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 2b shows the layout of the ABC World domain. There is a single 5 by 5 room, with three objects (denoted by A, B, C). All object provides reward upon reaching them. An episode terminates when the agent reaches an object or a time limit of 10 steps is reached. We consider three different versions of this environment: Fixed ABC, Random $A B C$ and Non-stationary ABC. In the Fixed ABC environment, each lifetime has 200 episodes. The reward associated with each object is fixed across lifetimes. Specifically, the rewards for objects A, B, and C are 1, $- 0 . 5$ , and 0.5 respectively. The optimal policy is to always collect A. In the Random ABC environment, each lifetime has 50 episodes. The reward associated with each object is randomly sampled for each lifetime and is held fixed within a lifetime. Thus, the environment is stationary from an agent’s perspective but non-stationary from the reward function’s perspective. Specifically, the rewards for A, B, and C are uniformly sampled from $[ - 1 , 1 ]$ , $[ - 0 . 5 , 0 ]$ , and [0.0.5] respectively. The optimal behaviour is to explore A and C at the beginning of a lifetime to assess which is the better, and then commits to the better one for all subsequent episode. In the non-stationary ABC environment, each lifetime has ",
+        "bbox": [
+            173,
+            742,
+            825,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/3566109a904a4c2bf69ea4b8440f6405ed3491a41000a33b790269d9b3556327.jpg",
+        "image_caption": [
+            "Figure 10: Evaluation of different rewards in the Fixed ABC domain. The $\\mathbf { X }$ -axis shows the number of episodes within a single lifetime; the y-axis measures the episode return. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            339,
+            103,
+            656,
+            227
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "1000 episodes. The rewards for A, B, and C are 1, $- 0 . 5$ , and $- 1$ respectively. The rewards for A and C swap every 250 episodes. ",
+        "bbox": [
+            173,
+            294,
+            823,
+            321
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.2.3 KEY BOX WORLD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            337,
+            357,
+            352
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Figure 2c shows the Key Box World domain. In this domain, there is a key and three boxes, A, B, and C. In order to open any box, the agent must pick up the key first. The rewards for A, B, and C are uniformly sampled from $[ - 1 , 1 ]$ , $[ - 0 . 5 , 0 ]$ , and $[ 0 , 0 . 5 ]$ respectively for each lifetime. An episode terminates when the agent opens a box or a time limit of 100 steps is reached. Each lifetime consists of 5000 episodes. ",
+        "bbox": [
+            174,
+            361,
+            825,
+            431
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.3 HAND-DESIGNED NEAR-OPTIMAL EXPLORATION STRATEGY FOR RANDOM ABC ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            449,
+            771,
+            463
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We hand-designed a heuristic strategy for the Random ABC domain. We assume the agent has the prior knowledge that B is always bad and A and C have uncertain rewards. Therefore, the heuristic is to go to A in the first episode, go to C in the second episode, and then go to the better one in the remaining episodes in the lifetime. We view this heuristic as an upper-bound because it always finds the best object and can arbitrarily control the agent’s behaviour. ",
+        "bbox": [
+            174,
+            474,
+            825,
+            545
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL EMPIRICAL RESULT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            565,
+            483,
+            580
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "C.1 EXPLOITING OPTIMAL BEHAVIOUR ON A FIXED TASK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            597,
+            583,
+            611
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "To investigate what the intrinsic reward learns in a fixed task, we designed the ‘Fixed ABC’ environment (see Figure 2b). The reward for each object (A, B, and C) is fixed within and across lifetimes. When the agent collects an object, it receives the corresponding reward of 1, $- 0 . 5$ , or 0.5 for object A, B, or C respectively, and the episode terminates. Each lifetime contains 200 episodes. The optimal policy is to always collect A. The optimal reward should capture the regularity of the environment that object A has the highest reward and drive the agent towards object A. ",
+        "bbox": [
+            174,
+            622,
+            825,
+            705
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Figure 10 shows that agents trained with the learned intrinsic reward learn optimal policies within a few episodes. This indicates that the intrinsic reward memorises the fixed optimal behaviour during training and assigns rewards accordingly to aid learning during evaluation. The result on Fixed ABC is not particularly surprising. In a fixed task in a stationary environment, an optimal reward function does not need to encourage exploration, and helps the agent to directly learn the optimal behaviour as quickly as possible, similarly to reward shaping. ",
+        "bbox": [
+            173,
+            712,
+            825,
+            797
+        ],
+        "page_idx": 15
+    }
+]

parse/train/T3kmOP_cMFB/T3kmOP_cMFB.md

@@ -0,0 +1,314 @@
+# BOOSTING ONE-POINT DERIVATIVE-FREE ONLINE OPTIMIZATION VIA RESIDUAL FEEDBACK
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Zeroth-order optimization (ZO) typically relies on two-point feedback to estimate the unknown gradient of the objective function, which queries the objective function value twice at each time instant. However, if the objective function is time-varying, as in online optimization, two-point feedback can not be used. In this case, the gradient can be estimated using one-point feedback that queries a single function value at each time instant, although at the expense of producing gradient estimates with large variance. In this work, we propose a new one-point feedback method for online optimization that estimates the objective function gradient using the residual between two feedback points at consecutive time instants. We study the regret bound of ZO with residual feedback for both convex and nonconvex online optimization problems. Specifically, for both Lipschitz and smooth functions, we show that using residual feedback produces gradient estimates with much smaller variance compared to conventional one-point feedback methods, which improves the learning rate. Our regret bound for ZO with residual feedback is tighter than the existing regret bound for ZO with conventional one-point feedback and relies on weaker assumptions, which suggests that ZO with our proposed residual feedback can better track the optimizer of online optimization problems. We provide numerical experiments that demonstrate that ZO with residual feedback significantly outperforms existing one-point feedback methods in practice.
+
+# 1 INTRODUCTION
+
+Zeroth-order optimization (ZO) algorithms have been widely used to solve online optimization problems where first or second order information (i.e., gradient or Hessian information) is unavailable at each time instant. Such problems arise, e.g., in online learning and involve adversarial training Chen et al. (2017) and reinforcement learning Fazel et al. (2018); Malik et al. (2018) among others. The goal in online optimization is to minimize a sequence of time-varying objective functions $\{ f _ { t } ( \bar { x } ) \} _ { t = 1 : T }$ , where the value $f _ { t } ( x _ { t } )$ is revealed to the agent after an action $x _ { t }$ is selected and is used to adapt the agent’s future strategy. Since the future objective functions are not known a priori, the performance of the online decision process can be measured using notions of regret, generally defined as the difference between the total cost incurred by the decision selected by the agent online and the cost of the fixed or varying optimal decision that a clairvoyant agent could select.
+
+Perhaps the most popular zeroth-order gradient estimator is the two-point estimator that has been extensively studied in Agarwal et al. (2010); Ghadimi & Lan (2013); Duchi et al. (2015); Ghadimi et al. (2016); Bach & Perchet (2016); Nesterov & Spokoiny (2017); Gao et al. (2018); Roy et al. (2019). Specifically, the two-point estimator queries the function value $f _ { t } ( x )$ for twice, for two different realizations of the decision variables, and uses the difference in these function values to estimate the desired gradient, as illustrated by the equation
+
+$$
+\widetilde { g } _ { t } ^ { ( 2 ) } ( x ) = \frac { u } { \delta } \Big ( f _ { t } ( x + \delta u ) - f _ { t } ( x ) \Big ) ,
+$$
+
+where $\delta > 0$ is a parameter and $u \sim \mathcal { N } ( 0 , I )$ . However, the two-point gradient estimator can not be used for the solution of non-stationary online optimization problems that arise frequently, e.g., in online learning. The reason is that in these non-stationary online optimization problems, the objective
+
+function being queried is time-varying, and hence only a single function value can be sampled at a given time instant. In this case, the following one-point feedback can be used
+
+$$
+( \mathrm { O n e - p o i n t ~ f e e d b a c k } ) \colon \widetilde { g } _ { t } ^ { ( 1 ) } ( x ) = \frac { u } { \delta } f _ { t } ( x + \delta u ) ,
+$$
+
+which queries the objective function $f _ { t } ( x )$ only once at each time instant. One-point feedback was first proposed and analyzed in Flaxman et al. (2005) for the solution of online convex optimization problems. Saha & Tewari (2011); Hazan & Levy (2014); Dekel et al. (2015) showed that the regret of convex online optimization methods using one-point gradient estimation can be improved assuming smoothness or strong convexity of the objective functions and using self-concordant regularization. More recently, Gasnikov et al. (2017) developed such regret bounds for stochastic convex problems. On the other hand, Hazan et al. (2016) characterized the convergence of one-point zeroth-order methods for static stochastic non-convex optimization problems. However, as shown in these studies, a limitation of one-point feedback is that the resulting gradient estimator has large variance and, therefore, induces large regret. In addition, the regret analysis for ZO with one-point feedback usually requires the strong assumption that the function value is uniformly upper bounded over time, so this method can not be used for practical non-stationary optimization problems.
+
+Contributions: In this paper, we propose a novel one-point gradient estimator for zeroth-order online optimization and develop new regret bounds to study its performance. Specifically, our contributions are as follows. We propose a new one-point feedback scheme which requires a single function evaluation at each time instant. This feedback scheme estimates the gradient using the residual between two consecutive feedback points and we refer to it as residual feedback. We show that our residual feedback induces a smaller gradient estimation variance than the conventional one-point feedback scheme in Flaxman et al. (2005); Gasnikov et al. (2017). Furthermore, we provide regret bounds for online convex optimization with our proposed residual feedback estimator. Our analysis relies on a weaker assumption than the one needed in the case of the conventional one-point estimator, and our proposed regret bounds are tighter especially when the value of the objective function is large. In addition, we provide regret bounds for online non-convex optimization with residual feedback. Finally, we present numerical experiments that demonstrate that the proposed residual-feedback estimator significantly outperforms the conventional one-point method in its ability to track the time-varying optimizers of online learning problems. To the best of our knowledge, this is the first time a one-point zeroth-order method is theoretically studied for online non-convex optimization problems. It is also the first time that a one-point gradient estimator demonstrates comparable empirical performance to that of the two-point method. We note that two-point estimators can only be used to solve online non-stationary learning problems in simulations, where the system can be hard coded to be fixed during two queries of the objective function values at two different decision variables.
+
+Related work: Zeroth-order methods have been used to solve many different types of optimization problems. For example, Balasubramanian & Ghadimi (2018) apply ZO to solve a set-constrained optimization problem where the projection onto the constraint set is non-trivial. Gorbunov et al. (2018); Ji et al. (2019) apply a variance-reduced technique and acceleration schemes to achieve better convergence speed in ZO. Wang et al. (2018) improve the dependence of the iteration complexity on the dimension of the problem under an additional sparsity assumption on the gradient of the objective function. And Hajinezhad & Zavlanos (2018); Tang & Li (2019) apply zeroth-order oracles to distributed optimization problems when only bandit feedbacks are available at each local agents. Our proposed residual feedback oracle can be used to solve such online optimization problems as well. Also related is work by Zhang et al. (2015) that considers non-convex online bandit optimization problems with a single query at each time step. However, this method employs the exploration and exploitation bandit learning framework and the proposed analysis is restricted to a special class of non-convex objective functions. Finally, Agarwal et al. (2011); Hazan & Li (2016); Bubeck et al. (2017) study online bandit algorithms using ellipsoid methods. In particular, these methods induce heavy computation per step and achieve regret bounds that have bad dependence on the problem dimension. As a comparison, our one-point method is computation light and achieves regret bounds that have better dependence on the problem dimension.
+
+# 2 PRELIMINARIES AND RESIDUAL FEEDBACK
+
+We first introduce the classes of Lipschitz and smooth functions.
+
+Definition 2.1 (Lipschitz functions). The class of Lipschtiz-continuous functions $C ^ { 0 , 0 }$ satisfies: for any $f \in C ^ { 0 , 0 }$ , $| f ( x ) - f ( y ) | \leq L _ { 0 } \| x - y \|$ , 8x, $y \in \bar { \mathbb { R } } ^ { d }$ , where $L _ { 0 } > 0$ is the Lipschitz parameter. The class of smooth functions $C ^ { 1 , 1 }$ satisfies: for any ${ \bf { \bar { f } } } \in C ^ { 1 , 1 }$ , $\| \nabla f ( x ) - \nabla f ( y ) \| \leq L _ { 1 } \| x - y \|$ , $\forall x , y \in$ $\mathbb { R } ^ { d }$ , where $L _ { 1 } > 0$ is the smoothness parameter.
+
+In ZO, the objective is to estimate the first-order gradient of a function using zeroth-order oracles. Necessarily, we need to perturb the function around the current point along all the directions uniformly in order to estimate the gradient. This motivates us to consider the Gaussian-smoothed version of the function $f$ as introduced in Nesterov & Spokoiny (2017), $f _ { \delta } ( x ) : = \mathbb { E } _ { u \sim \mathcal { N } ( 0 , 1 ) } [ f ( x + \delta u ) ]$ , where the coordinates of the vector $u$ are i.i.d standard Gaussian random variables. The following bounds on the approximation error of the function $f _ { \delta } ( x )$ have been developed in Nesterov & Spokoiny (2017).
+
+Lemma 2.2. Consider a function $f$ and its smoothed version $f _ { \delta }$ . It holds that
+
+$$
+f _ { \delta } ( x ) - f ( x ) | \leq \{ \delta L _ { 0 } \sqrt { d } , \ i f f \in C ^ { 0 , 0 } ,  \quad a n d \| \nabla f _ { \delta } ( x ) - \nabla f ( x ) \| \leq \delta L _ { 1 } ( d + 3 ) ^ { 3 / 2 } , \ i f f \in C ^ { 1 , 1 } . 
+$$
+
+The smoothed function $f _ { \delta } ( x )$ satisfies the following amenable property Nesterov & Spokoiny (2017).
+
+Lemma 2.3. If $f \in C ^ { 0 , 0 }$ is $L _ { 0 }$ -Lipschitz, then $f _ { \delta } \in C ^ { 1 , 1 }$ with Lipschitz constant $L _ { 1 } = \sqrt { d } \delta ^ { - 1 } L _ { 0 }$ .
+
+Consider the following online bandit optimization problem.
+
+$$
+\operatorname* { m i n } _ { x \in \mathcal { X } } \sum _ { t = 0 } ^ { T - 1 } f _ { t } ( x ) ,
+$$
+
+where $\mathcal { X } \subset \mathbb { R } ^ { d }$ is a convex set and $\cdot$ is a random sequence of objective functions. In this setting, the objective functions $\{ f _ { t } \} _ { t }$ are unknown a priori and their derivatives are unavailable. At time $t$ , a new objective function $f _ { t }$ is randomly generated independent of an agent’s decisions, and then the agent queries the objective function value at certain perturbed points and use them to update the current policy parameters. The goal of the agent is to minimize a certain regret function.
+
+Such an online setting often occurs in non-stationary learning scenarios where either the system is time-varying on its own or a single query of the function $f _ { t }$ changes the system state (i.e., $f _ { t }$ changes to $f _ { t + 1 } )$ ). In this non-stationary setting, the conventional two-point feedback scheme is known to be impractical as it requires to evaluate $f _ { t }$ at two different points at the same time $t$ . Instead, it is natural to use the one-point feedback scheme (2) in Gasnikov et al. (2017). However, the gradient estimate based on the above one-point feedback induces a large variance that leads to a large regret. In this paper, we focus on such an one-point derivative-free setting and propose the following novel one-point residual feedback scheme for estimating the gradient with reduced variance.
+
+$$
+\mathrm {  ~ \ k b } \mathrm {  ~ \cdot ~ } \widetilde { g } _ { t } ( x _ { t } ) : = \frac { u _ { t } } { \delta } \big ( f _ { t } ( x _ { t } + \delta u _ { t } ) - f _ { t - 1 } ( x _ { t - 1 } + \delta u _ { t - 1 } ) \big ) ,
+$$
+
+where $u _ { t - 1 } , u _ { t } \sim { \mathcal { N } } ( 0 , I )$ are independent random vectors. To elaborate, the residual feedback in (3) queries $f _ { t }$ at a single perturbed point $x _ { t } + \delta u _ { t }$ , and then subtracts it by $f _ { t - 1 } ( x _ { t - 1 } + \delta u _ { t - 1 } )$ obtained from the previous iteration. We name such a scheme as one-point residual feedback. Next, we explore some basic properties of the residual feedback. We first show that this estimator is an unbiased gradient estimate of the smoothed function $f _ { \delta , t }$ .
+
+Lemma 2.4. The residual feedback satisfies $\mathbb { E } \left[ \widetilde { g } _ { t } ( x _ { t } ) \right] = \nabla f _ { \delta , t } ( x _ { t } )$ for all $x _ { t } \in \mathcal { X }$ and $t$ .
+
+Proof. By the fact that $u _ { t }$ has zero mean and is independent from $u _ { t - 1 }$ and $x _ { t - 1 }$
+
+We consider the following ZO algorithm with residual feedback
+
+$$
+x _ { t + 1 } = \Pi _ { \mathcal { X } } \big ( x _ { t } - \eta \tilde { g } _ { t } ( x _ { t } ) \big ) ,
+$$
+
+where $\eta$ is the learning rate and $\Pi _ { \mathcal { X } }$ is the projection operator onto the set $\mathcal { X }$ . The update (4) can be implemented assuming that the objective function can be queried at points outside the feasible set $\cdot$ , similar to the methods considered in Duchi et al. (2015); Bach & Perchet (2016); Gasnikov et al. (2017). Note that it is possible to modify the update (4) so that the iterates are guaranteed to be within the feasible set $\mathcal { X }$ . This modification and related analysis can be found in Section $_ \mathrm { H }$ in the supplementary material. The requirement that the objective function is evaluated at feasible points in derivative-free optimization algorithms has also been considered in Bubeck et al. (2017); Bilenne et al. (2020). Specifically, Bubeck et al. (2017) develop the so called ellipsoid method, which requires computation of an ellipsoid containing the optimizer at each time step. On the other hand, almost concurrently with this work, Bilenne et al. (2020) proposed a similar oracle as in (3) for a static convex optimization problem with specific objective and constraint functions. Next, we bound the second moment of the gradient estimate based on the residual feedback.
+
+Lemma 2.5 (Second moment). Assume that $f _ { t } \in C ^ { 0 , 0 }$ with Lipschitz constant $L _ { 0 }$ for all time $t$ Then, under the ZO update rule in (4), the second moment of the residual feedback satisfies: for all $t$
+
+$$
+\begin{array} { c } { \displaystyle \mathbb { E } [ \| \widetilde { g } _ { t } ( x _ { t } ) \| ^ { 2 } ] \leq \frac { 4 d L _ { 0 } ^ { 2 } \eta ^ { 2 } } { \delta ^ { 2 } } \mathbb { E } [ \| \widetilde { g } _ { t - 1 } ( x _ { t - 1 } ) \| ^ { 2 } ] + D _ { t } , } \\ { \displaystyle } \\ { w h e r e ~ D _ { t } : = 1 6 L _ { 0 } ^ { 2 } ( d + 4 ) ^ { 2 } + \frac { 2 d } { \delta ^ { 2 } } \mathbb { E } \big [ \big ( f _ { t } ( x _ { t - 1 } + \delta u _ { t - 1 } ) - f _ { t - 1 } ( x _ { t - 1 } + \delta u _ { t - 1 } ) \big ) ^ { 2 } \big ] . } \end{array}
+$$
+
+The above lemma shows that the second moment of the residual feedback can be bounded by a perturbed contraction, provided that we choose $\eta$ and $\delta$ such that the contracting rate $\dot { \alpha } = \dot { 4 } d L _ { 0 } ^ { 2 } \eta ^ { 2 } \delta ^ { - 2 } < 1$ . As we show later in the analysis, such a contraction property leads to a small variance of the residual feedback that helps reduce the regret of the online ZO algorithm.
+
+# 3 ZO WITH RESIDUAL FEEDBACK FOR ONLINE CONVEX OPTIMIZATION
+
+In this section, we consider the online bandit problem (P) where the sequence of functions $\{ f _ { t } \} _ { t = 0 : T - 1 }$ are all convex. In particular, we are interested in analyzing the following static regret of the algorithm.
+
+$$
+R _ { T } : = \mathbb { E } \Big [ \sum _ { t = 0 } ^ { T - 1 } f _ { t } ( x _ { t } ) - \operatorname* { m i n } _ { x \in \mathscr { X } } \sum _ { t = 0 } ^ { T - 1 } f _ { t } ( x ) \Big ] .
+$$
+
+We make the following assumption on the non-stationary of the online learning problem.
+
+Assumption 3.1 (Bounded variation). There exists $V _ { f } > 0$ such that for all $\cdot$ ,
+
+$$
+\begin{array} { r } { \mathbb { E } \big [ | f _ { t } ( x _ { t - 1 } + \delta u _ { t - 1 } ) - f _ { t - 1 } ( x _ { t - 1 } + \delta u _ { t - 1 } ) | ^ { 2 } \big ] \leq V _ { f } ^ { 2 } , } \end{array}
+$$
+
+here the expectation is taken over $x _ { t - 1 }$ , the random vector $\cdot$ and the random functions $\cdot$
+
+Intuitively, we assume the squared variation of the objective function between two consecutive time instants is uniformly bounded over time. We note that this assumption is much weaker than the uniformly bounded function value assumption, i.e., $\mathbb { E } \big [ f _ { t } ( x ) ^ { 2 } \big ] \ \leq \ \dot { B } ^ { 2 } , \forall t , x \in \mathcal { X }$ , which is used in the analysis of ZO with the conventional one-point feedback Gasnikov et al. (2017). In particular, under Assumption 3.1, the perturbation term in Lemma 2.5 can be bounded as $D _ { t } \ \leq$ $\bar { 1 } 6 L _ { 0 } ^ { 2 } ( d + 4 ) ^ { 2 } + 2 d V _ { . } ^ { 2 } \delta ^ { - 2 }$ . Then, by telescoping the contraction inequality, we obtain the following bound for the second moment of the residual-feedback gradient estimate,
+
+$$
+\mathbb { E } [ \| \tilde { g } _ { t } ( x _ { t } ) \| ^ { 2 } ] \le \operatorname* { m a x } \Big \{ \mathbb { E } [ \| \tilde { g } _ { 0 } ( x _ { 0 } ) \| ^ { 2 } ] , \frac { 1 } { 1 - \alpha } \Big ( 1 6 L _ { 0 } ^ { 2 } ( d + 4 ) ^ { 2 } + \frac { 2 d } { \delta ^ { 2 } } V _ { f } ^ { 2 } \Big ) \Big \} .
+$$
+
+In practice, $\delta$ is usually chosen to be sufficiently small, and the above bound is dominated by $\mathcal { O } ( \dot { d } \delta ^ { - 2 } V _ { f } ^ { 2 } )$ , which is much smaller than the second moment bound of the conventional one-point feedback $\mathcal { O } ( d \delta ^ { - 2 } B ^ { 2 } )$ ( $B ^ { 2 }$ is the uniform bound of the second moment of $f _ { t }$ over time). For example, consider the time-varying objective functions, $f _ { 0 } ( x ) = 1 / 2 x ^ { 2 }$ and $f _ { t } ( x ) = f _ { t - 1 } ( x ) + n _ { t }$ , where $\cdot$ is Gaussian noise with zero mean at time $t$ . Then, it can be verified that Assumption 3.1 holds with a finite $\cdot$ whereas the second moment of $f _ { t } ( x )$ is unbounded over time. This suggests that the variance of the residual feedback can be significantly smaller than that of the conventional one-point feedback.
+
+Next, we first consider the case where the objective function $f _ { t }$ is convex and Lipschitz. Based on the above characterization of the second moment of residual feedback, we obtain the following regret bound for ZO with residual feedback.
+
+Theorem 3.2 (Regret for Convex Lipschitz $f _ { t }$ ). Let Assumption 3.1 hold. Assume that $f _ { t } \in C ^ { 0 , 0 }$ is convex with Lipschitz constant $L _ { 0 }$ for all $t$ and $\| x _ { 0 } - x ^ { * } \| \leq R$ . Run $Z O$ with residual feedback for $T > R ^ { 2 }$ iterations with $\eta = R ^ { \frac { 3 } { 2 } } ( 2 \sqrt { 2 } L _ { 0 } \sqrt { d } T ^ { \frac { 3 } { 4 } } ) ^ { - 1 }$ and $\delta = \sqrt { R } T ^ { - \frac { 1 } { 4 } }$ . Then, we have that
+
+$$
+R _ { T } \leq \sqrt { 2 } L _ { 0 } \sqrt { d R } T ^ { \frac { 3 } { 4 } } + \frac { \mathbb { E } \left[ \| \tilde { g } _ { 0 } ( x _ { 0 } ) \| ^ { 2 } \right] R ^ { \frac { 3 } { 2 } } } { 2 \sqrt { 2 d } L _ { 0 } T ^ { \frac { 3 } { 4 } } } + 8 \sqrt { 2 } \frac { ( d + 4 ) ^ { 2 } } { \sqrt { d } } L _ { 0 } R ^ { \frac { 3 } { 2 } } T ^ { \frac { 1 } { 4 } }
+$$
+
+Asymptotically, we have $R _ { T } = \mathcal { O } ( ( L _ { 0 } + { L _ { 0 } } ^ { - 1 } V _ { f } ^ { 2 } ) \sqrt { d R } T ^ { \frac { 3 } { 4 } } )$ .
+
+To the best of our knowledge, the best known regret for ZO with the conventional one-point feedback is of the order $\mathcal { O } ( \sqrt { d L _ { 0 } R B } T ^ { \frac { 3 } { 4 } } )$ Gasnikov et al. (2017). Therefore, our regret bound is tighter if the function variation satisfies $V _ { f } ^ { 2 } \le \mathcal { O } ( B ^ { \frac { 1 } { 2 } } L _ { 0 } ^ { \frac { 3 } { 2 } } )$ . Essentially, using the proposed residual feedback gradient estimator, the regret of $\mathrm { \Delta } \breve { Z } { \mathrm O }$ no longer depends on the uniform bound of the function value, which can be huge in practice. Instead, our regret only relies on how fast the function varies over time.
+
+Remark 3.3. We note that the complexity bound in Theorem 3.2 generally depends on the values of the Lipschitz parameters $\cdot$ , $\cdot$ and the constant $V _ { f } ^ { 2 }$ . Specifically, choose $\eta = R ^ { \frac { 3 } { 2 } } ( 2 \sqrt { 2 } L _ { 0 } \sqrt { d } T ^ { \frac { 3 } { 4 } } ) ^ { - 1 }$ and $L _ { 0 } ^ { 2 q - 1 } V _ { f } ^ { 2 } ) \sqrt { d R } T ^ { \frac { 3 } { 4 } } )$ $\cdot$ ) when with $\cdot$ $q > 0$ as a tuning parameter, and we obtain that . If $L _ { 0 } ~ < ~ 1$ , we can choose $q = 1$ $R _ { T } = \mathcal { O } ( { ( L _ { 0 } + L _ { 0 } } ^ { 1 - q } +$ to achieve the bound $-$ . On the other hand, if $\cdot$ , we can choose $\cdot$ to achieve the bound $R _ { T } = { \mathcal O } ( ( L _ { 0 } + L _ { 0 } { } ^ { - 1 } V _ { f } ^ { 2 } ) \sqrt { d R } T ^ { \frac { 3 } { 4 } } )$ ). We note that the dependence of the bounds in Theorems 3.4, 4.2 and 4.3 on $L _ { 0 } , L _ { 1 }$ can also be optimized in a similar way by properly choosing $\delta$ .
+
+Next, we present the regret of ZO with residual feedback for convex smooth objective functions.
+
+Theorem 3.4 (Regret for Convex Smooth $f _ { t }$ ). Let Assumption 3.1 hold. Assume that $f _ { t } ( x ) \in$ $C ^ { 0 , 0 } \cap C ^ { 1 , 1 }$ is convex with Lipschitz constant $L _ { 0 }$ and smoothness constant $L _ { 1 }$ for all $t$ , and assume that $\| x _ { 0 } - x ^ { * } \| \leq R .$ Run $Z O$ with residual feedback for $T > R ^ { 2 }$ iterations with $\eta = R ^ { \frac { 4 } { 3 } } ( 2 \sqrt { 2 } L _ { 0 } d ^ { \frac { 2 } { 3 } } T ^ { \frac { 2 } { 3 } } ) ^ { - 1 }$ and $\delta = R ^ { \frac { 1 } { 3 } } d ^ { - { \frac { 1 } { 6 } } } T ^ { - { \frac { 1 } { 6 } } }$ . Then, we have that
+
+$$
+\begin{array} { c } { { R _ { T } \leq \sqrt { 2 } L _ { 0 } d ^ { \frac 2 3 } R ^ { \frac 2 3 } T ^ { \frac 2 3 } + \frac { \mathbb { E } \left[ \| \tilde { g } _ { 0 } ( x _ { 0 } ) \| ^ { 2 } \right] R ^ { \frac 4 3 } } { 2 \sqrt { 2 } L _ { 0 } d ^ { \frac 2 3 } T ^ { \frac 2 3 } } + 8 \sqrt { 2 } L _ { 0 } \frac { ( d + 4 ) ^ { 2 } } { d ^ { \frac 2 3 } } R ^ { \frac 4 3 } T ^ { \frac 1 3 } } }  \\ { { + 2 L _ { 1 } d ^ { \frac 2 3 } R ^ { \frac 2 3 } T ^ { \frac 2 3 } + \sqrt { 2 } L _ { 0 } ^ { - 1 } d ^ { \frac 2 3 } R ^ { \frac 2 3 } V _ { f } ^ { 2 } T ^ { \frac 2 3 } . } } \end{array}
+$$
+
+Asymptotically, the above regret bound is in the order of $\mathcal { O } ( ( L _ { 0 } + L _ { 1 } + L _ { 0 } { } ^ { - 1 } V _ { f } ^ { 2 } ) ( d R T ) ^ { \frac { 2 } { 3 } } )$ .
+
+To the best of our knowledge, the best known regret for ZO with the conventional one-point feedback in the convex smooth case is of the order $\mathcal { O } ( L _ { 1 } ^ { \frac { 1 } { 3 } } ( d R B T ) ^ { \frac { 2 } { 3 } } )$ Gasnikov et al. (2017). Therefore, our regret bound is tighter if the function variation satisfies $V _ { f } ^ { 2 } \leq \mathcal { O } ( B ^ { \frac { 2 } { 3 } } L _ { 1 } ^ { \frac { 1 } { 3 } } L _ { 0 } )$ . Our numerical experiments show that ZO with residual feedback always outperforms ZO with the conventional one-point feedback in practice.
+
+# 4 ZO WITH RESIDUAL FEEDBACK FOR ONLINE NONCONVEX OPTIMIZATION
+
+In this section, we analyze the regret of ZO with residual feedback in solving the unconstrained online bandit problem (P) with nonconvex functions. Throughout this section, we make the following assumption regarding the objective functions.
+
+Assumption 4.1. There exist $\cdot$ such that the following conditions hold for all $t$ .
+
+1. $\begin{array} { r } { \sum _ { t = 1 } ^ { T } \mathbb { E } [ f _ { \delta , t } ( x _ { t } ) - f _ { \delta , t - 1 } ( x _ { t } ) ] \leq W _ { T } } \end{array}$ , where the expectation is taken with respect to $\cdot$ and the random smoothed objective functions $\cdot$ , $f _ { \delta , t }$ . 2. $\begin{array} { r } { \sum _ { t = 1 } ^ { T } \mathbb { E } [ | f _ { t } ( x _ { t - 1 } + \delta u _ { t - 1 } ) - f _ { t - 1 } ( x _ { t - 1 } + \delta u _ { t - 1 } ) | ^ { 2 } ] \leq \tilde { W } _ { T } } \end{array}$ , where the expectation is taken with respect to $\cdot$ , the random vector $u _ { t - 1 }$ and the random objective functions $f _ { t - 1 }$ , $\cdot$ .
+
+The above two conditions measure the accumulated first-order and second-order function variations, as also adopted by Roy et al. (2019).
+
+Next, we consider the case where $\{ f _ { t } \} _ { t }$ are nonconvex and Lipschitz continuous functions. Since the objective function $f _ { t }$ is not necessarily differentiable, i.e., $\nabla f ( t )$ is not well defined, we define the regret as the accumulated gradient of the smoothed function, i.e., $\begin{array} { r } { R _ { g , \delta } ^ { T } : = \sum _ { t = 0 } ^ { T - 1 } \mathbb { E } [ \| \nabla f _ { \delta , t } ( x _ { t } ) \| ^ { 2 } ] } \end{array}$ . In addition, it is often required that the smoothed function $f _ { \delta , t }$ is close to the original function $f _ { t }$ such that $| f _ { \delta , t } ( x ) - f _ { t } ( x ) | \leq \epsilon _ { f }$ for all $t$ . To satisfy this condition, we need to choose $\delta \le ( \sqrt { d } L _ { 0 } ) ^ { - 1 } \epsilon _ { f }$ according to Lemma 2.2. We obtain the following regret bound for ZO with residual feedback.
+
+Theorem 4.2 (Nonconvex Lipschitz $f _ { t }$ ). Let Assumptions 4.1 hold. Assume that $f _ { t } \in C ^ { 0 , 0 }$ with Lipschitz constant $L _ { 0 }$ and that $f _ { t }$ is bounded below by $f _ { t } ^ { * }$ for all $t$ . Run $Z O$ with residual feedback for $T > ( d \epsilon _ { f } ) ^ { - 1 }$ iterations with $\eta = \epsilon _ { f } ^ { \frac { 3 } { 2 } } ( 2 \sqrt { 2 } L _ { 0 } ^ { 2 } d ^ { \frac { 3 } { 2 } } T ^ { \frac { 1 } { 2 } } ) ^ { - 1 }$ and $\delta = \epsilon _ { f } ( d ^ { \frac { 1 } { 2 } } L _ { 0 } ) ^ { - 1 }$ . Then, we have that
+
+$$
+\begin{array} { r l } & { R _ { g , \delta } ^ { T } \leq 2 \sqrt { 2 } L _ { 0 } ^ { 2 } \big ( \mathbb { E } [ f _ { \delta , 0 } ( x _ { 0 } ) ] - f _ { \delta , T } ^ { * } + W _ { T } \big ) d ^ { \frac { 3 } { 2 } } \epsilon _ { f } ^ { - \frac { 3 } { 2 } } T ^ { \frac { 1 } { 2 } } + \frac { \epsilon _ { f } ^ { \frac { 1 } { 2 } } \mathbb { E } \big [ \| \tilde { g } _ { 0 } ( x _ { 0 } ) \| ^ { 2 } \big ] } { 2 \sqrt { 2 d T } } } \\ & { \quad \quad \quad \quad + 4 \sqrt { 2 } L _ { 0 } \epsilon _ { f } ^ { \frac { 1 } { 2 } } \frac { \big ( d + 4 \big ) ^ { 2 } } { d ^ { \frac { 1 } { 2 } } } T ^ { \frac { 1 } { 2 } } + \frac { L _ { 0 } ^ { 2 } } { \sqrt { 2 } } \frac { d ^ { \frac { 3 } { 2 } } \widetilde { W } _ { T } } { \epsilon _ { f } ^ { \frac { 3 } { 2 } } T ^ { \frac { 1 } { 2 } } } . } \end{array}
+$$
+
+Asymptotically, we have $R _ { g , \delta } ^ { T } = \mathcal { O } ( d ^ { \frac { 3 } { 2 } } L _ { 0 } ^ { 2 } \epsilon _ { f } ^ { - \frac { 3 } { 2 } } ( W _ { T } + \widetilde { W } _ { T } T ^ { - 1 } ) T ^ { \frac { 1 } { 2 } } + d ^ { \frac { 3 } { 2 } } L _ { 0 } \epsilon _ { f } ^ { \frac { 1 } { 2 } } T ^ { \frac { 1 } { 2 } } ) .$
+
+Based on Theorem 4.2, we observe that the regret bound satisfies $R _ { g , \delta } ^ { T } / T  0$ whenever $W _ { T } =$ $o ( T ^ { \frac { 1 } { 2 } } \epsilon _ { f } ^ { \frac { 3 } { 2 } } )$ and $\widetilde { W } _ { T } = o ( T ^ { \frac { 3 } { 2 } } \epsilon _ { f } ^ { \frac { 3 } { 2 } } )$ . In particular, if the bounded variation Assumption 3.1 holds, then we have $\widetilde { W } _ { T } \leq \mathcal { O } ( T V _ { f } ^ { 2 } )$ , and it suffices to let $T ^ { - \frac { 1 } { 2 } } \epsilon _ { f } ^ { - \frac { 3 } { 2 } } = o ( 1 )$ .
+
+Next, we consider the nonconvex and smooth problem and study the regret $R _ { g } ^ { T } : =$ $\begin{array} { r l } {  { \sum _ { t = 0 } ^ { T - 1 } \mathbb { E } [ \| \nabla f _ { t } ( x _ { t } ) \| ^ { 2 } ] } \quad } & { { } } \end{array}$ . We obtain the following regret for ZO with residual-feedback.
+
+Theorem 4.3 (Nonconvex smooth $f _ { t }$ ). Let Assumptions 4.1 hold. Assume that $f _ { t } \in C ^ { 0 , 0 } \cap C ^ { 1 , 1 }$ with Lipschitz constant $L _ { 0 }$ and smoothness constant $L _ { 1 }$ and that $f _ { t }$ is bounded below by $f _ { t } ^ { * }$ for all $t$ . Run $Z O$ with residual feedback for $T$ iterations with $\eta = ( 2 \sqrt { 2 } L _ { 0 } d ^ { \frac { 4 } { 3 } } T ^ { \frac { 1 } { 2 } } ) ^ { - 1 }$ and $\delta = ( d ^ { \frac { 5 } { 6 } } T ^ { \frac { 1 } { 4 } } ) ^ { - 1 }$ . Then,
+
+$$
+\begin{array} { r } { R _ { g } ^ { T } \leq 4 \sqrt { 2 } L _ { 0 } \big ( \mathbb { E } [ f _ { \delta , 0 } ( x _ { 0 } ) ] - f _ { \delta , T } ^ { * } + W _ { T } \big ) d ^ { \frac { 4 } { 3 } } T ^ { \frac { 1 } { 2 } } + \frac { L _ { 1 } \mathbb { E } \big [ \| \tilde { g } _ { 0 } ( x _ { 0 } ) \| ^ { 2 } \big ] } { \sqrt { 2 } L _ { 0 } d ^ { \frac { 4 } { 3 } } T ^ { \frac { 1 } { 2 } } } } \\ { + 8 \sqrt { 2 } L _ { 1 } L _ { 0 } \frac { ( d + 4 ) ^ { 2 } } { d ^ { \frac { 4 } { 3 } } } T ^ { \frac { 1 } { 2 } } + \frac { \sqrt { 2 } L _ { 1 } } { L _ { 0 } } d ^ { \frac { 4 } { 3 } } \widetilde { W } _ { T } + 2 L _ { 1 } ^ { 2 } \frac { ( d + 3 ) ^ { 3 } } { d ^ { \frac { 5 } { 3 } } } T ^ { \frac { 1 } { 2 } } . } \end{array}
+$$
+
+Asymptotically, the above regret bound is in the order of $\mathcal { O } ( d ^ { \frac { 4 } { 3 } } L _ { 0 } W _ { T } T ^ { \frac { 1 } { 2 } } + d ^ { \frac { 4 } { 3 } } L _ { 1 } { L _ { 0 } } ^ { - 1 } \widetilde { W } _ { T } )$
+
+Based on Theorem 4.3, we observe that the regret bound satisfies $R _ { g } ^ { T } / T \to 0$ whenever $W _ { T } = o ( T ^ { \frac { 1 } { 2 } } )$ and $\widetilde { W } _ { T } = o ( T )$ . We note that these requirements of $W _ { T } , \widetilde { W } _ { T }$ are more relaxed than those in the nonsmooth case, as they do not rely on the small parameter $\epsilon _ { f }$ .
+
+# 5 ZO WITH RESIDUAL FEEDBACK FOR STOCHASTIC ONLINE OPTIMIZATION
+
+In this section, we generalize the residual feedback to solve stochastic online bandit problems. Since its regret analysis follows the same proof logic as that of ZO with residual feedback, we only introduce the key technical lemmas and comment on the proof difference. The stochastic online bandit problems are formulated as follows.
+
+$$
+\operatorname* { m i n } _ { x \in \mathcal { X } } \sum _ { t = 0 } ^ { T - 1 } \mathbb { E } [ F _ { t } ( x ; \xi _ { t } ) ] , \quad \mathrm { w h e r e } \ \mathbb { E } [ F _ { t } ( x ; \xi _ { t } ) ] = f _ { t } ( x ) , \forall t ,
+$$
+
+where $\xi _ { t }$ denotes a certain noise that is independent of $\cdot$ . Different from the previous deterministic online setting, the agent in the stochastic setting can only query noisy evaluations of the function.
+
+This covers the scenarios where the agent does not have access to the underlying data distribution. To solve the above stochastic online problem, we propose the following stochastic residual feedback
+
+$$
+\widetilde { g } _ { t } ( x _ { t } ) : = \frac { u _ { t } } { \delta } \big ( F _ { t } ( x _ { t } + \delta u _ { t } ; \xi _ { t } ) - F _ { t - 1 } ( x _ { t - 1 } + \delta u _ { t - 1 } ; \xi _ { t - 1 } ) \big ) ,
+$$
+
+where $\xi _ { t - 1 }$ and $\xi _ { t }$ are independent random samples that are sampled in the iterations $t - 1$ and $t$ , respectively. Since the noisy function value $F ( x ; \xi _ { t } )$ is an unbiased estimate of the objective function $f _ { t } ( x )$ , it is straightforward to show that (13) is an unbiased gradient estimate of the function $f _ { \delta , t } ( \boldsymbol { x } )$ .
+
+To analyze the regret of ZO with stochastic residual feedback, we first consider the convex setting and make the following assumption that bounds the variation of the stochastic functions.
+
+Assumption 5.1. (Bounded stochastic variation) There exists $V _ { f , \xi } > 0$ such that for all $\cdot$ ,
+
+$$
+-
+$$
+
+where the expectation is taken with respect to $x _ { t - 1 }$ , the random vector $\cdot$ and the random objective functions $\cdot$ , $\cdot$ .
+
+The above assumption generalizes Assumption 3.1 to the stochastic setting. The bound $V _ { f , \xi } ^ { 2 }$ controls
+
+The following lemma characterizes the second moment of the stochastic residual feedback.
+
+Lemma 5.2. Assume $F ( x , \xi ) \in C ^ { 0 , 0 }$ with Lipschitz constant $L _ { 0 }$ for all $\xi$ . Then, under the ZO update rule, we have that
+
+$$
+\begin{array} { r l r } {  { \operatorname { \mathbb { E } } [ \| \widetilde { g } _ { t } ( x _ { t } ) \| ^ { 2 } ] \le \frac { 4 d L _ { 0 } ^ { 2 } \eta ^ { 2 } } { \delta ^ { 2 } } \mathbb { E } [ \| \widetilde { g } _ { t } ( x _ { t - 1 } ) \| ^ { 2 } ] + D _ { t , \xi } , } }  \\ & { } & \\ & { } & { \ \cdot D _ { t , \xi } : = 1 6 L _ { 0 } ^ { 2 } ( d + 4 ) ^ { 2 } + \frac { 2 d } { \delta ^ { 2 } } \mathbb { E } [ \big ( F _ { t } ( x _ { t - 1 } + \delta u _ { t - 1 } , \xi _ { t } ) - F _ { t - 1 } ( x _ { t - 1 } + \delta u _ { t - 1 } , \xi _ { t - 1 } ) \big ) ^ { 2 } ] . } \end{array}
+$$
+
+Observe that the above second moment bound is very similar to that in Lemma 2.5, and the only difference is the perturbation term. In particular, the perturbation term $D _ { t , \xi }$ can be further bounded by leveraging Assumption 5.1, and the resulting second moment bound is almost the same as that in eq. (8) for the deterministic case (simply replace $V _ { f }$ in eq. (8) by $V _ { f , \xi }$ ). Therefore, the regret analysis of ZO with stochastic residual feedback is the same as that of ZO with residual feedback in the deterministic online setting. Consequently, ZO with stochastic residual feedback achieves almost the same regret bounds as those in Theorems 3.2 and 3.4, and one simply needs to replace $V _ { f }$ by $V _ { f , \xi }$ .
+
+For the nonconvex setting, we adopt the following assumption that generalizes Assumption 4.1.
+
+Assumption 5.3. There exists $W _ { T } , \tilde { W } _ { T , \xi } > 0$ such that the following two conditions hold for all $\cdot$ $\begin{array} { r } { \sum _ { t = 1 } ^ { T } \mathbb { E } [ f _ { \delta , t } ( x _ { t } ) - f _ { \delta , t - 1 } ( x _ { t } ) ] \leq W _ { T } } \end{array}$ , where the expectation is taken with respect to $\cdot$ and the random smoothed objective functions $\cdot$ , $f _ { \delta , t }$ .
+
+2. $\begin{array} { r } { \sum _ { t = 1 } ^ { T } \mathbb { E } [ | F _ { t } ( x _ { t - 1 } + \delta u _ { t - 1 } ; \xi _ { t } ) - F _ { t - 1 } ( x _ { t - 1 } + \delta u _ { t - 1 } ; \xi _ { t - 1 } ) | ^ { 2 } ] \leq \tilde { W } _ { T , \xi } } \end{array}$ , where the expectation is taken with respect to $x _ { t - 1 }$ , the random vector $\cdot$ and the random objective functions $F _ { t - 1 } ( \cdot , \xi _ { t - 1 } )$ , $F _ { t } ( \cdot , \xi _ { t } )$ .
+
+Then, following the same proof logic as that of Theorems 4.2 and 4.3, on can obtain similar regret bounds for ZO with stochastic residual feedback (simply replace $W _ { T } , \widetilde { W } _ { T }$ in Theorems 4.2 and 4.3 by $W _ { T , \xi } , \widetilde { W } _ { T , \xi }$ , respectively).
+
+# 6 NUMERICAL EXPERIMENTS
+
+In this section, we compare the performance of ZO with one-point, two-point and residual feedback in solving two non-stationary reinforcement learning problems, i.e., LQR control and resource allocation, in which either the reward or transition functions are varying over episodes.
+
+# 6.1 NONSTATINOARY LQR CONTROL
+
+We consider an LQR problem with noisy system dynamics. The static version of this problem is considered in Fazel et al. (2018); Malik et al. (2018). Specifically, consider a system whose state
+
+![](images/172f027f5c953de0a0f456e836adae097a4157c0967aa9cb40f9bca3e5050279.jpg)  
+Figure 1: The regrets of applying the proposed residual one-point feedback (3) (blue), the two-point oracle in Bach & Perchet (2016) (orange) and the conventional one-point oracle in Gasnikov et al. (2017) (green) to online policy optimization for the nonstationary LQR problem. In (a), the regrets $\textstyle \sum _ { t = 0 } ^ { T } | V ( K _ { t } ) - V ( K ^ { * } ) |$ of three methods are presented. In (b), the variance of the gradient estimates given by three methods are presented. The two point method (orange) is infeasible to use in practice and is presented here to serve as the simulating benchmark.
+
+$x _ { k } \in \mathbb { R } ^ { n _ { x } }$ at step $k$ is subject to a transition function $x _ { k + 1 } = A _ { t } x _ { k } + B _ { t } u _ { k } + w _ { k }$ , where $u _ { k } \in \mathbb { R } ^ { n _ { u } }$ is the action at step $k$ , and $A _ { t } \in \mathbb R ^ { n _ { x } \times n _ { x } }$ and $B _ { t } \in \mathbb { R } ^ { n _ { x } \times n _ { u } }$ are dynamical matrices in episode $t$ These matrices are unknown and changing over episodes. The vector $w _ { k }$ is the noise on the state transition. Specifically, the entries of the dynamical matrices $\cdot$ and $\cdot$ at episode 0 are randomly generated from a Gaussian distribution $\mathcal { N } ( 0 , 0 . 1 ^ { 2 } )$ . Then, we generate the time-varying dynamical matrices as $A _ { t + 1 } = A _ { t } + 0 . 0 1 M _ { t }$ and $-$ , where $M _ { t }$ and $N _ { t }$ are random matrices whose entries are uniformly sampled from [0,1]. Moreover, consider a state feedback policy $u _ { k } = K _ { t } x _ { k }$ , where $K _ { t } \in \mathbb { R } ^ { n _ { u } \times n _ { x } }$ is the policy parameter that is fixed within episode $t$ . Within $K _ { t } ^ { * }$ so thatepisode the discounted accuis minimized, where d cost fuis the d ionunt $\begin{array} { r } { V _ { t } ( K ) : = \mathbb { E } \big [ \sum _ { k = 0 } ^ { H - 1 } \gamma ^ { k } ( x _ { k } ^ { T } Q x _ { k } + u _ { k } ^ { T } R u _ { k } ) \big ] } \end{array}$ $t$ $\gamma \leq 1$   
+$H$ $K _ { t } ^ { * }$ that $V _ { t } ( K _ { t } ) - V _ { t } ( K _ { t } ^ { * } )$ is small in every episode.
+
+We apply the conventional one-point method in Gasnikov et al. (2017) and the proposed residualfeedback method (13) to solve the above non-stationary LQR problem. The performance of the twopoint method in Bach & Perchet (2016) is also presented as a benchmark, although it is impractical in non-stationary scenarios. This is because the two-point method in Bach & Perchet (2016) requires to evaluate value function $V _ { t }$ for two different policy functions at two consecutive episodes. However, evaluating the value function $V _ { t }$ for a given policy during episode $t$ requires to collect samples by executing this policy. Then, during the subsequent episode $t + 1$ , since the problem is non-stationary, the dynamic matrices change to $A _ { t + 1 } , B _ { t + 1 }$ and so does the value function $\cdot$ . Therefore, it is not possible to evaluate the same value function $\cdot$ at two different episodes and, as a result, the two-point method in Bach & Perchet (2016) is not applicable here. Each algorithm is run for 10 trials, and the stepsizes are optimized respectively. The accumulated regrets $\begin{array} { r } { \sum _ { t = 0 } ^ { T - 1 } | V ( K _ { t } ) - V ( K ^ { * } ) | } \end{array}$ of these algorithms are presented in Figure 1(a). We observe that the residual feedback method achieves a much lower regret than the conventional one-point method and has a comparable performance to that of the impractical two-point method. Moreover, we present in Figure 1(b) the estimated variance of the gradient estimates of these three methods at the policy iterates over episodes. It can be seen that the variance of our proposed residual-feedback is close to the impractical two-point feedback and is much smaller than that of the conventional one-point feedback. This observation justifies our theoretical characterization of the second moment of the residual feedback.
+
+# 6.2 NONSTATIONARY RESOURCE ALLOCATION
+
+We consider a multi-stage resource allocation problem with time-varying sensitivity to the lack of resource supply. Specifically, 16 agents are located on a $4 \times 4$ grid. During episode $t$ , at step $k$ , agent $i$ stores $m _ { i } ( k )$ amount of resources and has a demand for resources in the amount of $d _ { i } ( k )$ . Also, agent $i$ decides to send a fraction of resources $a _ { i j } ( k ) \in [ 0 , 1 ]$ to its neighbors
+
+![](images/9f9d6c9a052d224f533ef42168b212bda74f45eaed2a072c49dfb0e8c4d29dce.jpg)  
+Figure 2: The costs during each episode by applying the proposed residual one-point feedback (3) (blue), the two-point oracle in Bach & Perchet (2016) (orange) and the conventional one-point oracle in Gasnikov et al. (2017) (green) to solve the non-stationary resource allocation problem are presented. In (a), the varying cost $J _ { t } ( \theta _ { t } )$ of three methods are presented. In (b), the variance of the gradient estimates at agent 1 given by three methods are presented. The two point method (orange) is infeasible to use in practice and is presented here to serve as the simulating benchmark.
+
+$j \in \mathcal N _ { i }$ on the grid. The local amount of resources and demands of agent $i$ evolve as $m _ { i } ( k + 1 ) =$ $\begin{array} { r } { m _ { i } ( k ) - \sum _ { j \in \mathcal { N } _ { i } } a _ { i j } ( k ) m _ { i } ( k ) + \sum _ { j \in \mathcal { N } _ { i } } a _ { j i } ( k ) m _ { j } ( k ) - d _ { i } ( k ) } \end{array}$ and $d _ { i } ( \bar { k } ) = \psi _ { i } \sin ( \omega _ { i } k + \dot { \phi } _ { i } ) + \dot { w } _ { i , k }$ , where $w _ { i , k }$ is the noise in the demand. At each step $k$ , agent $i$ receives a local cost $r _ { i , t } ( k )$ , such that $r _ { i , t } ( k ) = 0$ when $m _ { i } ( k ) \geq 0$ and $r _ { i , t } ( k ) = \zeta _ { t } m _ { i } ( k ) ^ { 2 }$ when $m _ { i } ( k ) < 0$ , where $\zeta _ { t }$ represents the varying sensitivity of the agents to the lack of supply during episode $t$ . Let agent $i$ makes its decisions according to a parameterized policy function $\pi _ { i , t } \big ( o _ { i } ; \theta _ { i , t } \big ) : \mathcal { O } _ { i } \to [ 0 , 1 ] ^ { | \mathcal { N } _ { i } | }$ , where $\theta _ { i , t }$ is the parameter of the policy function $\pi _ { i , t }$ at episode $t$ , $o _ { i } \in { \mathcal { O } } _ { i }$ denotes agent $i$ ’s local observation. Specifically, we let so that the accumul $o _ { i } ( k ) = [ m _ { i } ( k ) , d _ { i } ( k ) ] ^ { T }$ arying optimal policyduring each episode is $\begin{array} { r } { \left. J _ { t } ( \theta _ { t } ) \right. = \sum _ { i = 1 } ^ { 1 6 } \sum _ { k = 0 } ^ { H } \gamma ^ { k } r _ { i , t } ( k ) } \end{array}$   
+$\theta _ { t } = [ \dots , \theta _ { i , t } , \dots ]$ $H$   
+at each episode, and $\gamma$ is the discount factor.
+
+In Figure 2(a), we present the costs achieved during each episode $J _ { t } ( \theta _ { t } )$ with 10 trials using ZO with the residual-feedback, one-point and the impractical two-point feedback. It can be seen that our proposed residual-feedback achieves a cost $J _ { t } ( \theta _ { t } )$ that is as low as the cost achieved by the impractical two-point feedback in such a non-stationary environment. In particular, both residual and two-point feedback perform much better than the conventional one-point feedback. Moreover, Figure 2(b) compares the estimated variances of these feedback schemes, and one can observe that the variance of the residual feedback is comparable to that of the two-point feedback and is much smaller than that of the conventional one-point feedback.
+
+# 7 CONCLUSION
+
+In this paper, we proposed a residual one-point feedback oracle for zeroth-order online learning problems, which estimates the gradient of the time-varying objective function using a single query of the function value at each time instant. We showed that the regret bound of the proposed residual feedback estimator can be much lower than that of the conventional one-point method in online convex optimization setting. In addition, we studied the gradient size regret bound of the residualfeedback estimator when it is applied to the online non-convex optimization problems. Numerical experiments on two non-stationary reinforcement learning problems were conducted and the proposed residual-feedback estimator was shown to significantly outperform the conventional one-point method in non-stationary online learning problems.
+
+# REFERENCES
+
+Alekh Agarwal, Ofer Dekel, and Lin Xiao. Optimal algorithms for online convex optimization with multi-point bandit feedback. In COLT, pp. 28–40. Citeseer, 2010.
+
+Alekh Agarwal, Dean P Foster, Daniel J Hsu, Sham M Kakade, and Alexander Rakhlin. Stochastic convex optimization with bandit feedback. In Advances in Neural Information Processing Systems, pp. 1035–1043, 2011.
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+Francis Bach and Vianney Perchet. Highly-smooth zero-th order online optimization. In Conference on Learning Theory, pp. 257–283, 2016.
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+Krishnakumar Balasubramanian and Saeed Ghadimi. Zeroth-order (non)-convex stochastic optimization via conditional gradient and gradient updates. In Advances in Neural Information Processing Systems, pp. 3455–3464, 2018.
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+Olivier Bilenne, Panayotis Mertikopoulos, and Elena-Veronica Belmega. Fast optimization with zeroth-order feedback in distributed, multi-user mimo systems. IEEE Transactions on Signal Processing, 2020.
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+Sébastien Bubeck, Nicolo Cesa-Bianchi, et al. Regret analysis of stochastic and nonstochastic multi-armed bandit problems. Foundations and Trends $\textsuperscript { \textregistered }$ in Machine Learning, 5(1):1–122, 2012.
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+Sébastien Bubeck, Yin Tat Lee, and Ronen Eldan. Kernel-based methods for bandit convex optimization. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 72–85, 2017.
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+Pin-Yu Chen, Huan Zhang, Yash Sharma, Jinfeng Yi, and Cho-Jui Hsieh. Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models. In Proceedings of the 10th ACM Workshop on Artificial Intelligence and Security, pp. 15–26, 2017.
+
+Ofer Dekel, Ronen Eldan, and Tomer Koren. Bandit smooth convex optimization: Improving the bias-variance tradeoff. In Advances in Neural Information Processing Systems, pp. 2926–2934, 2015.
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+John C Duchi, Michael I Jordan, Martin J Wainwright, and Andre Wibisono. Optimal rates for zero-order convex optimization: The power of two function evaluations. IEEE Transactions on Information Theory, 61(5):2788–2806, 2015.
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+Maryam Fazel, Rong Ge, Sham Kakade, and Mehran Mesbahi. Global convergence of policy gradient methods for the linear quadratic regulator. In Proceedings of the 35th International Conference on Machine Learning, volume 80, 2018.
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+Abraham D Flaxman, Adam Tauman Kalai, and H Brendan McMahan. Online convex optimization in the bandit setting: gradient descent without a gradient. In Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 385–394. Society for Industrial and Applied Mathematics, 2005.
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+Xiand Gao, Xiaobo Li, and Shuzhong Zhang. Online learning with non-convex losses and nonstationary regret. In International Conference on Artificial Intelligence and Statistics, pp. 235–243, 2018.
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+Alexander V Gasnikov, Ekaterina A Krymova, Anastasia A Lagunovskaya, Ilnura N Usmanova, and Fedor A Fedorenko. Stochastic online optimization. single-point and multi-point non-linear multi-armed bandits. convex and strongly-convex case. Automation and remote control, 78(2): 224–234, 2017.
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+Saeed Ghadimi and Guanghui Lan. Stochastic first-and zeroth-order methods for nonconvex stochastic programming. SIAM Journal on Optimization, 23(4):2341–2368, 2013.
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+Saeed Ghadimi, Guanghui Lan, and Hongchao Zhang. Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization. Mathematical Programming, 155(1-2):267–305, 2016.
+
+Eduard Gorbunov, Pavel Dvurechensky, and Alexander Gasnikov. An accelerated method for derivative-free smooth stochastic convex optimization. arXiv preprint arXiv:1802.09022, 2018.
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+Davood Hajinezhad and Michael M Zavlanos. Gradient-free multi-agent nonconvex nonsmooth optimization. In 2018 IEEE Conference on Decision and Control (CDC), pp. 4939–4944. IEEE, 2018.
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+Elad Hazan and Kfir Levy. Bandit convex optimization: Towards tight bounds. In Advances in Neural Information Processing Systems, pp. 784–792, 2014.
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+Elad Hazan and Yuanzhi Li. An optimal algorithm for bandit convex optimization. arXiv preprint arXiv:1603.04350, 2016.
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+Elad Hazan, Kfir Yehuda Levy, and Shai Shalev-Shwartz. On graduated optimization for stochastic non-convex problems. In International conference on machine learning, pp. 1833–1841, 2016.
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+Kaiyi Ji, Zhe Wang, Yi Zhou, and Yingbin Liang. Improved zeroth-order variance reduced algorithms and analysis for nonconvex optimization. arXiv preprint arXiv:1910.12166, 2019.
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+Dhruv Malik, Ashwin Pananjady, Kush Bhatia, Koulik Khamaru, Peter L Bartlett, and Martin J Wainwright. Derivative-free methods for policy optimization: Guarantees for linear quadratic systems. arXiv preprint arXiv:1812.08305, 2018.
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+Yurii Nesterov. Introductory lectures on convex optimization: A basic course, volume 87. Springer Science & Business Media, 2013.
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+Yurii Nesterov and Vladimir Spokoiny. Random gradient-free minimization of convex functions. Foundations of Computational Mathematics, 17(2):527–566, 2017.
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+Abhishek Roy, Krishnakumar Balasubramanian, Saeed Ghadimi, and Prasant Mohapatra. Multipoint bandit algorithms for nonstationary online nonconvex optimization. arXiv preprint arXiv:1907.13616, 2019.
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+Ankan Saha and Ambuj Tewari. Improved regret guarantees for online smooth convex optimization with bandit feedback. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, pp. 636–642, 2011.
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+Yujie Tang and Na Li. Distributed zero-order algorithms for nonconvex multi-agent optimization. In 2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 781–786. IEEE, 2019.
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+Yining Wang, Simon Du, Sivaraman Balakrishnan, and Aarti Singh. Stochastic zeroth-order optimization in high dimensions. In International Conference on Artificial Intelligence and Statistics, pp. 1356–1365, 2018.
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+Lijun Zhang, Tianbao Yang, Rong Jin, and Zhi-Hua Zhou. Online bandit learning for a special class of non-convex losses. In AAAI, pp. 3158–3164, 2015.

parse/train/T3kmOP_cMFB/T3kmOP_cMFB_content_list.json

@@ -0,0 +1,1587 @@
+[
+    {
+        "type": "text",
+        "text": "BOOSTING ONE-POINT DERIVATIVE-FREE ONLINE OPTIMIZATION VIA RESIDUAL FEEDBACK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            128,
+            823,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            202,
+            398,
+            228
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            265,
+            544,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zeroth-order optimization (ZO) typically relies on two-point feedback to estimate the unknown gradient of the objective function, which queries the objective function value twice at each time instant. However, if the objective function is time-varying, as in online optimization, two-point feedback can not be used. In this case, the gradient can be estimated using one-point feedback that queries a single function value at each time instant, although at the expense of producing gradient estimates with large variance. In this work, we propose a new one-point feedback method for online optimization that estimates the objective function gradient using the residual between two feedback points at consecutive time instants. We study the regret bound of ZO with residual feedback for both convex and nonconvex online optimization problems. Specifically, for both Lipschitz and smooth functions, we show that using residual feedback produces gradient estimates with much smaller variance compared to conventional one-point feedback methods, which improves the learning rate. Our regret bound for ZO with residual feedback is tighter than the existing regret bound for ZO with conventional one-point feedback and relies on weaker assumptions, which suggests that ZO with our proposed residual feedback can better track the optimizer of online optimization problems. We provide numerical experiments that demonstrate that ZO with residual feedback significantly outperforms existing one-point feedback methods in practice. ",
+        "bbox": [
+            232,
+            295,
+            766,
+            559
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            584,
+            336,
+            601
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zeroth-order optimization (ZO) algorithms have been widely used to solve online optimization problems where first or second order information (i.e., gradient or Hessian information) is unavailable at each time instant. Such problems arise, e.g., in online learning and involve adversarial training Chen et al. (2017) and reinforcement learning Fazel et al. (2018); Malik et al. (2018) among others. The goal in online optimization is to minimize a sequence of time-varying objective functions $\\{ f _ { t } ( \\bar { x } ) \\} _ { t = 1 : T }$ , where the value $f _ { t } ( x _ { t } )$ is revealed to the agent after an action $x _ { t }$ is selected and is used to adapt the agent’s future strategy. Since the future objective functions are not known a priori, the performance of the online decision process can be measured using notions of regret, generally defined as the difference between the total cost incurred by the decision selected by the agent online and the cost of the fixed or varying optimal decision that a clairvoyant agent could select. ",
+        "bbox": [
+            174,
+            616,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Perhaps the most popular zeroth-order gradient estimator is the two-point estimator that has been extensively studied in Agarwal et al. (2010); Ghadimi & Lan (2013); Duchi et al. (2015); Ghadimi et al. (2016); Bach & Perchet (2016); Nesterov & Spokoiny (2017); Gao et al. (2018); Roy et al. (2019). Specifically, the two-point estimator queries the function value $f _ { t } ( x )$ for twice, for two different realizations of the decision variables, and uses the difference in these function values to estimate the desired gradient, as illustrated by the equation ",
+        "bbox": [
+            174,
+            762,
+            825,
+            845
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/201a46f6e7c2b03ceccbe6535373106f0ec62ffefbca4c22e796ae517f16bd83.jpg",
+        "text": "$$\n\\widetilde { g } _ { t } ^ { ( 2 ) } ( x ) = \\frac { u } { \\delta } \\Big ( f _ { t } ( x + \\delta u ) - f _ { t } ( x ) \\Big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            455,
+            849,
+            694,
+            877
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "where $\\delta > 0$ is a parameter and $u \\sim \\mathcal { N } ( 0 , I )$ . However, the two-point gradient estimator can not be used for the solution of non-stationary online optimization problems that arise frequently, e.g., in online learning. The reason is that in these non-stationary online optimization problems, the objective ",
+        "bbox": [
+            176,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "function being queried is time-varying, and hence only a single function value can be sampled at a given time instant. In this case, the following one-point feedback can be used ",
+        "bbox": [
+            174,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c05c861967df2b230b15a5a23bb5e0bda4cf3cd1e146b208df385f4602ae0670.jpg",
+        "text": "$$\n( \\mathrm { O n e - p o i n t ~ f e e d b a c k } ) \\colon \\widetilde { g } _ { t } ^ { ( 1 ) } ( x ) = \\frac { u } { \\delta } f _ { t } ( x + \\delta u ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            135,
+            656,
+            161
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "which queries the objective function $f _ { t } ( x )$ only once at each time instant. One-point feedback was first proposed and analyzed in Flaxman et al. (2005) for the solution of online convex optimization problems. Saha & Tewari (2011); Hazan & Levy (2014); Dekel et al. (2015) showed that the regret of convex online optimization methods using one-point gradient estimation can be improved assuming smoothness or strong convexity of the objective functions and using self-concordant regularization. More recently, Gasnikov et al. (2017) developed such regret bounds for stochastic convex problems. On the other hand, Hazan et al. (2016) characterized the convergence of one-point zeroth-order methods for static stochastic non-convex optimization problems. However, as shown in these studies, a limitation of one-point feedback is that the resulting gradient estimator has large variance and, therefore, induces large regret. In addition, the regret analysis for ZO with one-point feedback usually requires the strong assumption that the function value is uniformly upper bounded over time, so this method can not be used for practical non-stationary optimization problems. ",
+        "bbox": [
+            174,
+            165,
+            825,
+            332
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contributions: In this paper, we propose a novel one-point gradient estimator for zeroth-order online optimization and develop new regret bounds to study its performance. Specifically, our contributions are as follows. We propose a new one-point feedback scheme which requires a single function evaluation at each time instant. This feedback scheme estimates the gradient using the residual between two consecutive feedback points and we refer to it as residual feedback. We show that our residual feedback induces a smaller gradient estimation variance than the conventional one-point feedback scheme in Flaxman et al. (2005); Gasnikov et al. (2017). Furthermore, we provide regret bounds for online convex optimization with our proposed residual feedback estimator. Our analysis relies on a weaker assumption than the one needed in the case of the conventional one-point estimator, and our proposed regret bounds are tighter especially when the value of the objective function is large. In addition, we provide regret bounds for online non-convex optimization with residual feedback. Finally, we present numerical experiments that demonstrate that the proposed residual-feedback estimator significantly outperforms the conventional one-point method in its ability to track the time-varying optimizers of online learning problems. To the best of our knowledge, this is the first time a one-point zeroth-order method is theoretically studied for online non-convex optimization problems. It is also the first time that a one-point gradient estimator demonstrates comparable empirical performance to that of the two-point method. We note that two-point estimators can only be used to solve online non-stationary learning problems in simulations, where the system can be hard coded to be fixed during two queries of the objective function values at two different decision variables. ",
+        "bbox": [
+            173,
+            338,
+            826,
+            616
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Related work: Zeroth-order methods have been used to solve many different types of optimization problems. For example, Balasubramanian & Ghadimi (2018) apply ZO to solve a set-constrained optimization problem where the projection onto the constraint set is non-trivial. Gorbunov et al. (2018); Ji et al. (2019) apply a variance-reduced technique and acceleration schemes to achieve better convergence speed in ZO. Wang et al. (2018) improve the dependence of the iteration complexity on the dimension of the problem under an additional sparsity assumption on the gradient of the objective function. And Hajinezhad & Zavlanos (2018); Tang & Li (2019) apply zeroth-order oracles to distributed optimization problems when only bandit feedbacks are available at each local agents. Our proposed residual feedback oracle can be used to solve such online optimization problems as well. Also related is work by Zhang et al. (2015) that considers non-convex online bandit optimization problems with a single query at each time step. However, this method employs the exploration and exploitation bandit learning framework and the proposed analysis is restricted to a special class of non-convex objective functions. Finally, Agarwal et al. (2011); Hazan & Li (2016); Bubeck et al. (2017) study online bandit algorithms using ellipsoid methods. In particular, these methods induce heavy computation per step and achieve regret bounds that have bad dependence on the problem dimension. As a comparison, our one-point method is computation light and achieves regret bounds that have better dependence on the problem dimension. ",
+        "bbox": [
+            173,
+            622,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES AND RESIDUAL FEEDBACK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            878,
+            568,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We first introduce the classes of Lipschitz and smooth functions. ",
+        "bbox": [
+            173,
+            909,
+            594,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.1 (Lipschitz functions). The class of Lipschtiz-continuous functions $C ^ { 0 , 0 }$ satisfies: for any $f \\in C ^ { 0 , 0 }$ , $| f ( x ) - f ( y ) | \\leq L _ { 0 } \\| x - y \\|$ , 8x, $y \\in \\bar { \\mathbb { R } } ^ { d }$ , where $L _ { 0 } > 0$ is the Lipschitz parameter. The class of smooth functions $C ^ { 1 , 1 }$ satisfies: for any ${ \\bf { \\bar { f } } } \\in C ^ { 1 , 1 }$ , $\\| \\nabla f ( x ) - \\nabla f ( y ) \\| \\leq L _ { 1 } \\| x - y \\|$ , $\\forall x , y \\in$ $\\mathbb { R } ^ { d }$ , where $L _ { 1 } > 0$ is the smoothness parameter. ",
+        "bbox": [
+            173,
+            102,
+            825,
+            160
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In ZO, the objective is to estimate the first-order gradient of a function using zeroth-order oracles. Necessarily, we need to perturb the function around the current point along all the directions uniformly in order to estimate the gradient. This motivates us to consider the Gaussian-smoothed version of the function $f$ as introduced in Nesterov & Spokoiny (2017), $f _ { \\delta } ( x ) : = \\mathbb { E } _ { u \\sim \\mathcal { N } ( 0 , 1 ) } [ f ( x + \\delta u ) ]$ , where the coordinates of the vector $u$ are i.i.d standard Gaussian random variables. The following bounds on the approximation error of the function $f _ { \\delta } ( x )$ have been developed in Nesterov & Spokoiny (2017). ",
+        "bbox": [
+            173,
+            170,
+            826,
+            256
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2.2. Consider a function $f$ and its smoothed version $f _ { \\delta }$ . It holds that ",
+        "bbox": [
+            174,
+            258,
+            684,
+            273
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/157108b4efc58a6c31a7d87ee539a2e6c1f228c61cd25698cde2abfdb8c8fa4f.jpg",
+        "text": "$$\nf _ { \\delta } ( x ) - f ( x ) | \\leq \\{ \\delta L _ { 0 } \\sqrt { d } , \\ i f f \\in C ^ { 0 , 0 } ,  \\quad a n d \\| \\nabla f _ { \\delta } ( x ) - \\nabla f ( x ) \\| \\leq \\delta L _ { 1 } ( d + 3 ) ^ { 3 / 2 } , \\ i f f \\in C ^ { 1 , 1 } . \n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            280,
+            833,
+            315
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The smoothed function $f _ { \\delta } ( x )$ satisfies the following amenable property Nesterov & Spokoiny (2017). ",
+        "bbox": [
+            173,
+            329,
+            821,
+            344
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2.3. If $f \\in C ^ { 0 , 0 }$ is $L _ { 0 }$ -Lipschitz, then $f _ { \\delta } \\in C ^ { 1 , 1 }$ with Lipschitz constant $L _ { 1 } = \\sqrt { d } \\delta ^ { - 1 } L _ { 0 }$ . ",
+        "bbox": [
+            173,
+            349,
+            818,
+            366
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider the following online bandit optimization problem. ",
+        "bbox": [
+            173,
+            376,
+            563,
+            392
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/11c43993656a0f9dac83913599c22aa8edf77aed463f136736d393d7e74e79be.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal { X } } \\sum _ { t = 0 } ^ { T - 1 } f _ { t } ( x ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            444,
+            397,
+            549,
+            441
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { X } \\subset \\mathbb { R } ^ { d }$ is a convex set and $\\cdot$ is a random sequence of objective functions. In this setting, the objective functions $\\{ f _ { t } \\} _ { t }$ are unknown a priori and their derivatives are unavailable. At time $t$ , a new objective function $f _ { t }$ is randomly generated independent of an agent’s decisions, and then the agent queries the objective function value at certain perturbed points and use them to update the current policy parameters. The goal of the agent is to minimize a certain regret function. ",
+        "bbox": [
+            173,
+            446,
+            825,
+            518
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Such an online setting often occurs in non-stationary learning scenarios where either the system is time-varying on its own or a single query of the function $f _ { t }$ changes the system state (i.e., $f _ { t }$ changes to $f _ { t + 1 } )$ ). In this non-stationary setting, the conventional two-point feedback scheme is known to be impractical as it requires to evaluate $f _ { t }$ at two different points at the same time $t$ . Instead, it is natural to use the one-point feedback scheme (2) in Gasnikov et al. (2017). However, the gradient estimate based on the above one-point feedback induces a large variance that leads to a large regret. In this paper, we focus on such an one-point derivative-free setting and propose the following novel one-point residual feedback scheme for estimating the gradient with reduced variance. ",
+        "bbox": [
+            173,
+            523,
+            826,
+            636
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aab92e094424557c9600908d7e074d448fb4bc15ac1779bea8d1ffa4922d6915.jpg",
+        "text": "$$\n\\mathrm {  ~ \\ k b } \\mathrm {  ~ \\cdot ~ } \\widetilde { g } _ { t } ( x _ { t } ) : = \\frac { u _ { t } } { \\delta } \\big ( f _ { t } ( x _ { t } + \\delta u _ { t } ) - f _ { t - 1 } ( x _ { t - 1 } + \\delta u _ { t - 1 } ) \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            641,
+            748,
+            669
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $u _ { t - 1 } , u _ { t } \\sim { \\mathcal { N } } ( 0 , I )$ are independent random vectors. To elaborate, the residual feedback in (3) queries $f _ { t }$ at a single perturbed point $x _ { t } + \\delta u _ { t }$ , and then subtracts it by $f _ { t - 1 } ( x _ { t - 1 } + \\delta u _ { t - 1 } )$ obtained from the previous iteration. We name such a scheme as one-point residual feedback. Next, we explore some basic properties of the residual feedback. We first show that this estimator is an unbiased gradient estimate of the smoothed function $f _ { \\delta , t }$ . ",
+        "bbox": [
+            173,
+            674,
+            826,
+            744
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2.4. The residual feedback satisfies $\\mathbb { E } \\left[ \\widetilde { g } _ { t } ( x _ { t } ) \\right] = \\nabla f _ { \\delta , t } ( x _ { t } )$ for all $x _ { t } \\in \\mathcal { X }$ and $t$ . ",
+        "bbox": [
+            169,
+            747,
+            761,
+            767
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proof. By the fact that $u _ { t }$ has zero mean and is independent from $u _ { t - 1 }$ and $x _ { t - 1 }$ ",
+        "bbox": [
+            173,
+            780,
+            705,
+            796
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We consider the following ZO algorithm with residual feedback ",
+        "bbox": [
+            173,
+            809,
+            593,
+            825
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b91148f67d653b812430cd329f5fef6a0ad76836a0bb29f089cc66d1c76f68b4.jpg",
+        "text": "$$\nx _ { t + 1 } = \\Pi _ { \\mathcal { X } } \\big ( x _ { t } - \\eta \\tilde { g } _ { t } ( x _ { t } ) \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            508,
+            829,
+            692,
+            849
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta$ is the learning rate and $\\Pi _ { \\mathcal { X } }$ is the projection operator onto the set $\\mathcal { X }$ . The update (4) can be implemented assuming that the objective function can be queried at points outside the feasible set $\\cdot$ , similar to the methods considered in Duchi et al. (2015); Bach & Perchet (2016); Gasnikov et al. (2017). Note that it is possible to modify the update (4) so that the iterates are guaranteed to be within the feasible set $\\mathcal { X }$ . This modification and related analysis can be found in Section $_ \\mathrm { H }$ in the supplementary material. The requirement that the objective function is evaluated at feasible points in derivative-free optimization algorithms has also been considered in Bubeck et al. (2017); Bilenne et al. (2020). Specifically, Bubeck et al. (2017) develop the so called ellipsoid method, which requires computation of an ellipsoid containing the optimizer at each time step. On the other hand, almost concurrently with this work, Bilenne et al. (2020) proposed a similar oracle as in (3) for a static convex optimization problem with specific objective and constraint functions. Next, we bound the second moment of the gradient estimate based on the residual feedback. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2.5 (Second moment). Assume that $f _ { t } \\in C ^ { 0 , 0 }$ with Lipschitz constant $L _ { 0 }$ for all time $t$ Then, under the ZO update rule in (4), the second moment of the residual feedback satisfies: for all $t$ ",
+        "bbox": [
+            173,
+            204,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2759904bfd352313a6c20689a11449fe744d6720edf40601b3ac80a4e7f048f1.jpg",
+        "text": "$$\n\\begin{array} { c } { \\displaystyle \\mathbb { E } [ \\| \\widetilde { g } _ { t } ( x _ { t } ) \\| ^ { 2 } ] \\leq \\frac { 4 d L _ { 0 } ^ { 2 } \\eta ^ { 2 } } { \\delta ^ { 2 } } \\mathbb { E } [ \\| \\widetilde { g } _ { t - 1 } ( x _ { t - 1 } ) \\| ^ { 2 } ] + D _ { t } , } \\\\ { \\displaystyle } \\\\ { w h e r e ~ D _ { t } : = 1 6 L _ { 0 } ^ { 2 } ( d + 4 ) ^ { 2 } + \\frac { 2 d } { \\delta ^ { 2 } } \\mathbb { E } \\big [ \\big ( f _ { t } ( x _ { t - 1 } + \\delta u _ { t - 1 } ) - f _ { t - 1 } ( x _ { t - 1 } + \\delta u _ { t - 1 } ) \\big ) ^ { 2 } \\big ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            241,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The above lemma shows that the second moment of the residual feedback can be bounded by a perturbed contraction, provided that we choose $\\eta$ and $\\delta$ such that the contracting rate $\\dot { \\alpha } = \\dot { 4 } d L _ { 0 } ^ { 2 } \\eta ^ { 2 } \\delta ^ { - 2 } < 1$ . As we show later in the analysis, such a contraction property leads to a small variance of the residual feedback that helps reduce the regret of the online ZO algorithm. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 ZO WITH RESIDUAL FEEDBACK FOR ONLINE CONVEX OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we consider the online bandit problem (P) where the sequence of functions $\\{ f _ { t } \\} _ { t = 0 : T - 1 }$ are all convex. In particular, we are interested in analyzing the following static regret of the algorithm. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a3722db08d45714ccea9e14d25af72e8e9e262103bf049b8ce703391467bc14.jpg",
+        "text": "$$\nR _ { T } : = \\mathbb { E } \\Big [ \\sum _ { t = 0 } ^ { T - 1 } f _ { t } ( x _ { t } ) - \\operatorname* { m i n } _ { x \\in \\mathscr { X } } \\sum _ { t = 0 } ^ { T - 1 } f _ { t } ( x ) \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            464,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We make the following assumption on the non-stationary of the online learning problem. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3.1 (Bounded variation). There exists $V _ { f } > 0$ such that for all $\\cdot$ , ",
+        "bbox": [
+            171,
+            541,
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+            556
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e7a1da09b2609f331b59b9782c3679a9c52a34d7299ce6d1c764ea6942c0621e.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\big [ | f _ { t } ( x _ { t - 1 } + \\delta u _ { t - 1 } ) - f _ { t - 1 } ( x _ { t - 1 } + \\delta u _ { t - 1 } ) | ^ { 2 } \\big ] \\leq V _ { f } ^ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            564,
+            679,
+            583
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "here the expectation is taken over $x _ { t - 1 }$ , the random vector $\\cdot$ and the random functions $\\cdot$ ",
+        "bbox": [
+            194,
+            590,
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+            606
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, we assume the squared variation of the objective function between two consecutive time instants is uniformly bounded over time. We note that this assumption is much weaker than the uniformly bounded function value assumption, i.e., $\\mathbb { E } \\big [ f _ { t } ( x ) ^ { 2 } \\big ] \\ \\leq \\ \\dot { B } ^ { 2 } , \\forall t , x \\in \\mathcal { X }$ , which is used in the analysis of ZO with the conventional one-point feedback Gasnikov et al. (2017). In particular, under Assumption 3.1, the perturbation term in Lemma 2.5 can be bounded as $D _ { t } \\ \\leq$ $\\bar { 1 } 6 L _ { 0 } ^ { 2 } ( d + 4 ) ^ { 2 } + 2 d V _ { . } ^ { 2 } \\delta ^ { - 2 }$ . Then, by telescoping the contraction inequality, we obtain the following bound for the second moment of the residual-feedback gradient estimate, ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9bdcfe254c4b3970221a6bca3c679a89c783e07517e1ab036a504db6ebc6dffd.jpg",
+        "text": "$$\n\\mathbb { E } [ \\| \\tilde { g } _ { t } ( x _ { t } ) \\| ^ { 2 } ] \\le \\operatorname* { m a x } \\Big \\{ \\mathbb { E } [ \\| \\tilde { g } _ { 0 } ( x _ { 0 } ) \\| ^ { 2 } ] , \\frac { 1 } { 1 - \\alpha } \\Big ( 1 6 L _ { 0 } ^ { 2 } ( d + 4 ) ^ { 2 } + \\frac { 2 d } { \\delta ^ { 2 } } V _ { f } ^ { 2 } \\Big ) \\Big \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            723,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In practice, $\\delta$ is usually chosen to be sufficiently small, and the above bound is dominated by $\\mathcal { O } ( \\dot { d } \\delta ^ { - 2 } V _ { f } ^ { 2 } )$ , which is much smaller than the second moment bound of the conventional one-point feedback $\\mathcal { O } ( d \\delta ^ { - 2 } B ^ { 2 } )$ ( $B ^ { 2 }$ is the uniform bound of the second moment of $f _ { t }$ over time). For example, consider the time-varying objective functions, $f _ { 0 } ( x ) = 1 / 2 x ^ { 2 }$ and $f _ { t } ( x ) = f _ { t - 1 } ( x ) + n _ { t }$ , where $\\cdot$ is Gaussian noise with zero mean at time $t$ . Then, it can be verified that Assumption 3.1 holds with a finite $\\cdot$ whereas the second moment of $f _ { t } ( x )$ is unbounded over time. This suggests that the variance of the residual feedback can be significantly smaller than that of the conventional one-point feedback. ",
+        "bbox": [
+            173,
+            760,
+            825,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Next, we first consider the case where the objective function $f _ { t }$ is convex and Lipschitz. Based on the above characterization of the second moment of residual feedback, we obtain the following regret bound for ZO with residual feedback. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2 (Regret for Convex Lipschitz $f _ { t }$ ). Let Assumption 3.1 hold. Assume that $f _ { t } \\in C ^ { 0 , 0 }$ is convex with Lipschitz constant $L _ { 0 }$ for all $t$ and $\\| x _ { 0 } - x ^ { * } \\| \\leq R$ . Run $Z O$ with residual feedback for $T > R ^ { 2 }$ iterations with $\\eta = R ^ { \\frac { 3 } { 2 } } ( 2 \\sqrt { 2 } L _ { 0 } \\sqrt { d } T ^ { \\frac { 3 } { 4 } } ) ^ { - 1 }$ and $\\delta = \\sqrt { R } T ^ { - \\frac { 1 } { 4 } }$ . Then, we have that ",
+        "bbox": [
+            174,
+            102,
+            825,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fcd95db8640f95a00979ad1a3223be730e4500361be5812e909ba769ab1ba194.jpg",
+        "text": "$$\nR _ { T } \\leq \\sqrt { 2 } L _ { 0 } \\sqrt { d R } T ^ { \\frac { 3 } { 4 } } + \\frac { \\mathbb { E } \\left[ \\| \\tilde { g } _ { 0 } ( x _ { 0 } ) \\| ^ { 2 } \\right] R ^ { \\frac { 3 } { 2 } } } { 2 \\sqrt { 2 d } L _ { 0 } T ^ { \\frac { 3 } { 4 } } } + 8 \\sqrt { 2 } \\frac { ( d + 4 ) ^ { 2 } } { \\sqrt { d } } L _ { 0 } R ^ { \\frac { 3 } { 2 } } T ^ { \\frac { 1 } { 4 } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            155,
+            725,
+            217
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Asymptotically, we have $R _ { T } = \\mathcal { O } ( ( L _ { 0 } + { L _ { 0 } } ^ { - 1 } V _ { f } ^ { 2 } ) \\sqrt { d R } T ^ { \\frac { 3 } { 4 } } )$ . ",
+        "bbox": [
+            173,
+            222,
+            575,
+            241
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge, the best known regret for ZO with the conventional one-point feedback is of the order $\\mathcal { O } ( \\sqrt { d L _ { 0 } R B } T ^ { \\frac { 3 } { 4 } } )$ Gasnikov et al. (2017). Therefore, our regret bound is tighter if the function variation satisfies $V _ { f } ^ { 2 } \\le \\mathcal { O } ( B ^ { \\frac { 1 } { 2 } } L _ { 0 } ^ { \\frac { 3 } { 2 } } )$ . Essentially, using the proposed residual feedback gradient estimator, the regret of $\\mathrm { \\Delta } \\breve { Z } { \\mathrm O }$ no longer depends on the uniform bound of the function value, which can be huge in practice. Instead, our regret only relies on how fast the function varies over time. ",
+        "bbox": [
+            173,
+            250,
+            826,
+            342
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark 3.3. We note that the complexity bound in Theorem 3.2 generally depends on the values of the Lipschitz parameters $\\cdot$ , $\\cdot$ and the constant $V _ { f } ^ { 2 }$ . Specifically, choose $\\eta = R ^ { \\frac { 3 } { 2 } } ( 2 \\sqrt { 2 } L _ { 0 } \\sqrt { d } T ^ { \\frac { 3 } { 4 } } ) ^ { - 1 }$ and $L _ { 0 } ^ { 2 q - 1 } V _ { f } ^ { 2 } ) \\sqrt { d R } T ^ { \\frac { 3 } { 4 } } )$ $\\cdot$ ) when with $\\cdot$ $q > 0$ as a tuning parameter, and we obtain that . If $L _ { 0 } ~ < ~ 1$ , we can choose $q = 1$ $R _ { T } = \\mathcal { O } ( { ( L _ { 0 } + L _ { 0 } } ^ { 1 - q } +$ to achieve the bound $-$ . On the other hand, if $\\cdot$ , we can choose $\\cdot$ to achieve the bound $R _ { T } = { \\mathcal O } ( ( L _ { 0 } + L _ { 0 } { } ^ { - 1 } V _ { f } ^ { 2 } ) \\sqrt { d R } T ^ { \\frac { 3 } { 4 } } )$ ). We note that the dependence of the bounds in Theorems 3.4, 4.2 and 4.3 on $L _ { 0 } , L _ { 1 }$ can also be optimized in a similar way by properly choosing $\\delta$ . ",
+        "bbox": [
+            173,
+            345,
+            826,
+            465
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Next, we present the regret of ZO with residual feedback for convex smooth objective functions. ",
+        "bbox": [
+            173,
+            474,
+            802,
+            491
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.4 (Regret for Convex Smooth $f _ { t }$ ). Let Assumption 3.1 hold. Assume that $f _ { t } ( x ) \\in$ $C ^ { 0 , 0 } \\cap C ^ { 1 , 1 }$ is convex with Lipschitz constant $L _ { 0 }$ and smoothness constant $L _ { 1 }$ for all $t$ , and assume that $\\| x _ { 0 } - x ^ { * } \\| \\leq R .$ Run $Z O$ with residual feedback for $T > R ^ { 2 }$ iterations with $\\eta = R ^ { \\frac { 4 } { 3 } } ( 2 \\sqrt { 2 } L _ { 0 } d ^ { \\frac { 2 } { 3 } } T ^ { \\frac { 2 } { 3 } } ) ^ { - 1 }$ and $\\delta = R ^ { \\frac { 1 } { 3 } } d ^ { - { \\frac { 1 } { 6 } } } T ^ { - { \\frac { 1 } { 6 } } }$ . Then, we have that ",
+        "bbox": [
+            174,
+            493,
+            825,
+            555
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4a1d2de24319dfcbd529ed474f4462c92da04db43e3ada63d6172b43e1195048.jpg",
+        "text": "$$\n\\begin{array} { c } { { R _ { T } \\leq \\sqrt { 2 } L _ { 0 } d ^ { \\frac 2 3 } R ^ { \\frac 2 3 } T ^ { \\frac 2 3 } + \\frac { \\mathbb { E } \\left[ \\| \\tilde { g } _ { 0 } ( x _ { 0 } ) \\| ^ { 2 } \\right] R ^ { \\frac 4 3 } } { 2 \\sqrt { 2 } L _ { 0 } d ^ { \\frac 2 3 } T ^ { \\frac 2 3 } } + 8 \\sqrt { 2 } L _ { 0 } \\frac { ( d + 4 ) ^ { 2 } } { d ^ { \\frac 2 3 } } R ^ { \\frac 4 3 } T ^ { \\frac 1 3 } } }  \\\\ { { + 2 L _ { 1 } d ^ { \\frac 2 3 } R ^ { \\frac 2 3 } T ^ { \\frac 2 3 } + \\sqrt { 2 } L _ { 0 } ^ { - 1 } d ^ { \\frac 2 3 } R ^ { \\frac 2 3 } V _ { f } ^ { 2 } T ^ { \\frac 2 3 } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            560,
+            730,
+            622
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Asymptotically, the above regret bound is in the order of $\\mathcal { O } ( ( L _ { 0 } + L _ { 1 } + L _ { 0 } { } ^ { - 1 } V _ { f } ^ { 2 } ) ( d R T ) ^ { \\frac { 2 } { 3 } } )$ . ",
+        "bbox": [
+            169,
+            628,
+            777,
+            647
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge, the best known regret for ZO with the conventional one-point feedback in the convex smooth case is of the order $\\mathcal { O } ( L _ { 1 } ^ { \\frac { 1 } { 3 } } ( d R B T ) ^ { \\frac { 2 } { 3 } } )$ Gasnikov et al. (2017). Therefore, our regret bound is tighter if the function variation satisfies $V _ { f } ^ { 2 } \\leq \\mathcal { O } ( B ^ { \\frac { 2 } { 3 } } L _ { 1 } ^ { \\frac { 1 } { 3 } } L _ { 0 } )$ . Our numerical experiments show that ZO with residual feedback always outperforms ZO with the conventional one-point feedback in practice. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 ZO WITH RESIDUAL FEEDBACK FOR ONLINE NONCONVEX OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            756,
+            821,
+            773
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we analyze the regret of ZO with residual feedback in solving the unconstrained online bandit problem (P) with nonconvex functions. Throughout this section, we make the following assumption regarding the objective functions. ",
+        "bbox": [
+            173,
+            787,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.1. There exist $\\cdot$ such that the following conditions hold for all $t$ . ",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1. $\\begin{array} { r } { \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ f _ { \\delta , t } ( x _ { t } ) - f _ { \\delta , t - 1 } ( x _ { t } ) ] \\leq W _ { T } } \\end{array}$ , where the expectation is taken with respect to $\\cdot$ and the random smoothed objective functions $\\cdot$ , $f _ { \\delta , t }$ . 2. $\\begin{array} { r } { \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ | f _ { t } ( x _ { t - 1 } + \\delta u _ { t - 1 } ) - f _ { t - 1 } ( x _ { t - 1 } + \\delta u _ { t - 1 } ) | ^ { 2 } ] \\leq \\tilde { W } _ { T } } \\end{array}$ , where the expectation is taken with respect to $\\cdot$ , the random vector $u _ { t - 1 }$ and the random objective functions $f _ { t - 1 }$ , $\\cdot$ . ",
+        "bbox": [
+            173,
+            859,
+            825,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The above two conditions measure the accumulated first-order and second-order function variations, as also adopted by Roy et al. (2019). ",
+        "bbox": [
+            171,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Next, we consider the case where $\\{ f _ { t } \\} _ { t }$ are nonconvex and Lipschitz continuous functions. Since the objective function $f _ { t }$ is not necessarily differentiable, i.e., $\\nabla f ( t )$ is not well defined, we define the regret as the accumulated gradient of the smoothed function, i.e., $\\begin{array} { r } { R _ { g , \\delta } ^ { T } : = \\sum _ { t = 0 } ^ { T - 1 } \\mathbb { E } [ \\| \\nabla f _ { \\delta , t } ( x _ { t } ) \\| ^ { 2 } ] } \\end{array}$ . In addition, it is often required that the smoothed function $f _ { \\delta , t }$ is close to the original function $f _ { t }$ such that $| f _ { \\delta , t } ( x ) - f _ { t } ( x ) | \\leq \\epsilon _ { f }$ for all $t$ . To satisfy this condition, we need to choose $\\delta \\le ( \\sqrt { d } L _ { 0 } ) ^ { - 1 } \\epsilon _ { f }$ according to Lemma 2.2. We obtain the following regret bound for ZO with residual feedback. ",
+        "bbox": [
+            173,
+            138,
+            826,
+            231
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.2 (Nonconvex Lipschitz $f _ { t }$ ). Let Assumptions 4.1 hold. Assume that $f _ { t } \\in C ^ { 0 , 0 }$ with Lipschitz constant $L _ { 0 }$ and that $f _ { t }$ is bounded below by $f _ { t } ^ { * }$ for all $t$ . Run $Z O$ with residual feedback for $T > ( d \\epsilon _ { f } ) ^ { - 1 }$ iterations with $\\eta = \\epsilon _ { f } ^ { \\frac { 3 } { 2 } } ( 2 \\sqrt { 2 } L _ { 0 } ^ { 2 } d ^ { \\frac { 3 } { 2 } } T ^ { \\frac { 1 } { 2 } } ) ^ { - 1 }$ and $\\delta = \\epsilon _ { f } ( d ^ { \\frac { 1 } { 2 } } L _ { 0 } ) ^ { - 1 }$ . Then, we have that ",
+        "bbox": [
+            174,
+            233,
+            823,
+            284
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/65701bcedbf9b14e08ba7d9750af3b370caaf5e69078ea7f12a295f191767c6a.jpg",
+        "text": "$$\n\\begin{array} { r l } & { R _ { g , \\delta } ^ { T } \\leq 2 \\sqrt { 2 } L _ { 0 } ^ { 2 } \\big ( \\mathbb { E } [ f _ { \\delta , 0 } ( x _ { 0 } ) ] - f _ { \\delta , T } ^ { * } + W _ { T } \\big ) d ^ { \\frac { 3 } { 2 } } \\epsilon _ { f } ^ { - \\frac { 3 } { 2 } } T ^ { \\frac { 1 } { 2 } } + \\frac { \\epsilon _ { f } ^ { \\frac { 1 } { 2 } } \\mathbb { E } \\big [ \\| \\tilde { g } _ { 0 } ( x _ { 0 } ) \\| ^ { 2 } \\big ] } { 2 \\sqrt { 2 d T } } } \\\\ & { \\quad \\quad \\quad \\quad + 4 \\sqrt { 2 } L _ { 0 } \\epsilon _ { f } ^ { \\frac { 1 } { 2 } } \\frac { \\big ( d + 4 \\big ) ^ { 2 } } { d ^ { \\frac { 1 } { 2 } } } T ^ { \\frac { 1 } { 2 } } + \\frac { L _ { 0 } ^ { 2 } } { \\sqrt { 2 } } \\frac { d ^ { \\frac { 3 } { 2 } } \\widetilde { W } _ { T } } { \\epsilon _ { f } ^ { \\frac { 3 } { 2 } } T ^ { \\frac { 1 } { 2 } } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            291,
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+            377
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Asymptotically, we have $R _ { g , \\delta } ^ { T } = \\mathcal { O } ( d ^ { \\frac { 3 } { 2 } } L _ { 0 } ^ { 2 } \\epsilon _ { f } ^ { - \\frac { 3 } { 2 } } ( W _ { T } + \\widetilde { W } _ { T } T ^ { - 1 } ) T ^ { \\frac { 1 } { 2 } } + d ^ { \\frac { 3 } { 2 } } L _ { 0 } \\epsilon _ { f } ^ { \\frac { 1 } { 2 } } T ^ { \\frac { 1 } { 2 } } ) .$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Based on Theorem 4.2, we observe that the regret bound satisfies $R _ { g , \\delta } ^ { T } / T  0$ whenever $W _ { T } =$ $o ( T ^ { \\frac { 1 } { 2 } } \\epsilon _ { f } ^ { \\frac { 3 } { 2 } } )$ and $\\widetilde { W } _ { T } = o ( T ^ { \\frac { 3 } { 2 } } \\epsilon _ { f } ^ { \\frac { 3 } { 2 } } )$ . In particular, if the bounded variation Assumption 3.1 holds, then we have $\\widetilde { W } _ { T } \\leq \\mathcal { O } ( T V _ { f } ^ { 2 } )$ , and it suffices to let $T ^ { - \\frac { 1 } { 2 } } \\epsilon _ { f } ^ { - \\frac { 3 } { 2 } } = o ( 1 )$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Next, we consider the nonconvex and smooth problem and study the regret $R _ { g } ^ { T } : =$ $\\begin{array} { r l } {  { \\sum _ { t = 0 } ^ { T - 1 } \\mathbb { E } [ \\| \\nabla f _ { t } ( x _ { t } ) \\| ^ { 2 } ] } \\quad } & { { } } \\end{array}$ . We obtain the following regret for ZO with residual-feedback. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.3 (Nonconvex smooth $f _ { t }$ ). Let Assumptions 4.1 hold. Assume that $f _ { t } \\in C ^ { 0 , 0 } \\cap C ^ { 1 , 1 }$ with Lipschitz constant $L _ { 0 }$ and smoothness constant $L _ { 1 }$ and that $f _ { t }$ is bounded below by $f _ { t } ^ { * }$ for all $t$ . Run $Z O$ with residual feedback for $T$ iterations with $\\eta = ( 2 \\sqrt { 2 } L _ { 0 } d ^ { \\frac { 4 } { 3 } } T ^ { \\frac { 1 } { 2 } } ) ^ { - 1 }$ and $\\delta = ( d ^ { \\frac { 5 } { 6 } } T ^ { \\frac { 1 } { 4 } } ) ^ { - 1 }$ . Then, ",
+        "bbox": [
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+            523,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee776d3c5ddac57bc2e30abb1ae8960b47db5f9c6e6591f9a6744e4a6577cfc3.jpg",
+        "text": "$$\n\\begin{array} { r } { R _ { g } ^ { T } \\leq 4 \\sqrt { 2 } L _ { 0 } \\big ( \\mathbb { E } [ f _ { \\delta , 0 } ( x _ { 0 } ) ] - f _ { \\delta , T } ^ { * } + W _ { T } \\big ) d ^ { \\frac { 4 } { 3 } } T ^ { \\frac { 1 } { 2 } } + \\frac { L _ { 1 } \\mathbb { E } \\big [ \\| \\tilde { g } _ { 0 } ( x _ { 0 } ) \\| ^ { 2 } \\big ] } { \\sqrt { 2 } L _ { 0 } d ^ { \\frac { 4 } { 3 } } T ^ { \\frac { 1 } { 2 } } } } \\\\ { + 8 \\sqrt { 2 } L _ { 1 } L _ { 0 } \\frac { ( d + 4 ) ^ { 2 } } { d ^ { \\frac { 4 } { 3 } } } T ^ { \\frac { 1 } { 2 } } + \\frac { \\sqrt { 2 } L _ { 1 } } { L _ { 0 } } d ^ { \\frac { 4 } { 3 } } \\widetilde { W } _ { T } + 2 L _ { 1 } ^ { 2 } \\frac { ( d + 3 ) ^ { 3 } } { d ^ { \\frac { 5 } { 3 } } } T ^ { \\frac { 1 } { 2 } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            577,
+            727,
+            651
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Asymptotically, the above regret bound is in the order of $\\mathcal { O } ( d ^ { \\frac { 4 } { 3 } } L _ { 0 } W _ { T } T ^ { \\frac { 1 } { 2 } } + d ^ { \\frac { 4 } { 3 } } L _ { 1 } { L _ { 0 } } ^ { - 1 } \\widetilde { W } _ { T } )$ ",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Based on Theorem 4.3, we observe that the regret bound satisfies $R _ { g } ^ { T } / T \\to 0$ whenever $W _ { T } = o ( T ^ { \\frac { 1 } { 2 } } )$ and $\\widetilde { W } _ { T } = o ( T )$ . We note that these requirements of $W _ { T } , \\widetilde { W } _ { T }$ are more relaxed than those in the nonsmooth case, as they do not rely on the small parameter $\\epsilon _ { f }$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 ZO WITH RESIDUAL FEEDBACK FOR STOCHASTIC ONLINE OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we generalize the residual feedback to solve stochastic online bandit problems. Since its regret analysis follows the same proof logic as that of ZO with residual feedback, we only introduce the key technical lemmas and comment on the proof difference. The stochastic online bandit problems are formulated as follows. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6f522417fc0b9e2fb7694b092178848a30947207e761b436649d64ffcbfbcd8c.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal { X } } \\sum _ { t = 0 } ^ { T - 1 } \\mathbb { E } [ F _ { t } ( x ; \\xi _ { t } ) ] , \\quad \\mathrm { w h e r e } \\ \\mathbb { E } [ F _ { t } ( x ; \\xi _ { t } ) ] = f _ { t } ( x ) , \\forall t ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            847,
+            684,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\xi _ { t }$ denotes a certain noise that is independent of $\\cdot$ . Different from the previous deterministic online setting, the agent in the stochastic setting can only query noisy evaluations of the function. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This covers the scenarios where the agent does not have access to the underlying data distribution. To solve the above stochastic online problem, we propose the following stochastic residual feedback ",
+        "bbox": [
+            171,
+            103,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bf25681c14fa1c4d0c9f31c6bc9d36384640e34ef357259658654a1804c1425d.jpg",
+        "text": "$$\n\\widetilde { g } _ { t } ( x _ { t } ) : = \\frac { u _ { t } } { \\delta } \\big ( F _ { t } ( x _ { t } + \\delta u _ { t } ; \\xi _ { t } ) - F _ { t - 1 } ( x _ { t - 1 } + \\delta u _ { t - 1 } ; \\xi _ { t - 1 } ) \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            137,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\xi _ { t - 1 }$ and $\\xi _ { t }$ are independent random samples that are sampled in the iterations $t - 1$ and $t$ , respectively. Since the noisy function value $F ( x ; \\xi _ { t } )$ is an unbiased estimate of the objective function $f _ { t } ( x )$ , it is straightforward to show that (13) is an unbiased gradient estimate of the function $f _ { \\delta , t } ( \\boldsymbol { x } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To analyze the regret of ZO with stochastic residual feedback, we first consider the convex setting and make the following assumption that bounds the variation of the stochastic functions. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assumption 5.1. (Bounded stochastic variation) There exists $V _ { f , \\xi } > 0$ such that for all $\\cdot$ , ",
+        "bbox": [
+            171,
+            251,
+            763,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad51ff703ce51665de94042123a498b61803aa9a7743f2cf7079cf2f9d01f9be.jpg",
+        "text": "$$\n-\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            268,
+            715,
+            290
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where the expectation is taken with respect to $x _ { t - 1 }$ , the random vector $\\cdot$ and the random objective functions $\\cdot$ , $\\cdot$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The above assumption generalizes Assumption 3.1 to the stochastic setting. The bound $V _ { f , \\xi } ^ { 2 }$ controls ",
+        "bbox": [
+            174,
+            333,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The following lemma characterizes the second moment of the stochastic residual feedback. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.2. Assume $F ( x , \\xi ) \\in C ^ { 0 , 0 }$ with Lipschitz constant $L _ { 0 }$ for all $\\xi$ . Then, under the ZO update rule, we have that ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cadc4c1f3d95a7267ac6c3e8ba1d3533c121773b88bb510dbd15d9f60d9f4dc8.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\operatorname { \\mathbb { E } } [ \\| \\widetilde { g } _ { t } ( x _ { t } ) \\| ^ { 2 } ] \\le \\frac { 4 d L _ { 0 } ^ { 2 } \\eta ^ { 2 } } { \\delta ^ { 2 } } \\mathbb { E } [ \\| \\widetilde { g } _ { t } ( x _ { t - 1 } ) \\| ^ { 2 } ] + D _ { t , \\xi } , } }  \\\\ & { } & \\\\ & { } & { \\ \\cdot D _ { t , \\xi } : = 1 6 L _ { 0 } ^ { 2 } ( d + 4 ) ^ { 2 } + \\frac { 2 d } { \\delta ^ { 2 } } \\mathbb { E } [ \\big ( F _ { t } ( x _ { t - 1 } + \\delta u _ { t - 1 } , \\xi _ { t } ) - F _ { t - 1 } ( x _ { t - 1 } + \\delta u _ { t - 1 } , \\xi _ { t - 1 } ) \\big ) ^ { 2 } ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            411,
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+            469
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Observe that the above second moment bound is very similar to that in Lemma 2.5, and the only difference is the perturbation term. In particular, the perturbation term $D _ { t , \\xi }$ can be further bounded by leveraging Assumption 5.1, and the resulting second moment bound is almost the same as that in eq. (8) for the deterministic case (simply replace $V _ { f }$ in eq. (8) by $V _ { f , \\xi }$ ). Therefore, the regret analysis of ZO with stochastic residual feedback is the same as that of ZO with residual feedback in the deterministic online setting. Consequently, ZO with stochastic residual feedback achieves almost the same regret bounds as those in Theorems 3.2 and 3.4, and one simply needs to replace $V _ { f }$ by $V _ { f , \\xi }$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For the nonconvex setting, we adopt the following assumption that generalizes Assumption 4.1. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assumption 5.3. There exists $W _ { T } , \\tilde { W } _ { T , \\xi } > 0$ such that the following two conditions hold for all $\\cdot$ $\\begin{array} { r } { \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ f _ { \\delta , t } ( x _ { t } ) - f _ { \\delta , t - 1 } ( x _ { t } ) ] \\leq W _ { T } } \\end{array}$ , where the expectation is taken with respect to $\\cdot$ and the random smoothed objective functions $\\cdot$ , $f _ { \\delta , t }$ . ",
+        "bbox": [
+            173,
+            599,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            625,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "2. $\\begin{array} { r } { \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ | F _ { t } ( x _ { t - 1 } + \\delta u _ { t - 1 } ; \\xi _ { t } ) - F _ { t - 1 } ( x _ { t - 1 } + \\delta u _ { t - 1 } ; \\xi _ { t - 1 } ) | ^ { 2 } ] \\leq \\tilde { W } _ { T , \\xi } } \\end{array}$ , where the expectation is taken with respect to $x _ { t - 1 }$ , the random vector $\\cdot$ and the random objective functions $F _ { t - 1 } ( \\cdot , \\xi _ { t - 1 } )$ , $F _ { t } ( \\cdot , \\xi _ { t } )$ . ",
+        "bbox": [
+            174,
+            657,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Then, following the same proof logic as that of Theorems 4.2 and 4.3, on can obtain similar regret bounds for ZO with stochastic residual feedback (simply replace $W _ { T } , \\widetilde { W } _ { T }$ in Theorems 4.2 and 4.3 by $W _ { T , \\xi } , \\widetilde { W } _ { T , \\xi }$ , respectively). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 NUMERICAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            779,
+            436,
+            795
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare the performance of ZO with one-point, two-point and residual feedback in solving two non-stationary reinforcement learning problems, i.e., LQR control and resource allocation, in which either the reward or transition functions are varying over episodes. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.1 NONSTATINOARY LQR CONTROL ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            446,
+            883
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We consider an LQR problem with noisy system dynamics. The static version of this problem is considered in Fazel et al. (2018); Malik et al. (2018). Specifically, consider a system whose state ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/172f027f5c953de0a0f456e836adae097a4157c0967aa9cb40f9bca3e5050279.jpg",
+        "image_caption": [
+            "Figure 1: The regrets of applying the proposed residual one-point feedback (3) (blue), the two-point oracle in Bach & Perchet (2016) (orange) and the conventional one-point oracle in Gasnikov et al. (2017) (green) to online policy optimization for the nonstationary LQR problem. In (a), the regrets $\\textstyle \\sum _ { t = 0 } ^ { T } | V ( K _ { t } ) - V ( K ^ { * } ) |$ of three methods are presented. In (b), the variance of the gradient estimates given by three methods are presented. The two point method (orange) is infeasible to use in practice and is presented here to serve as the simulating benchmark. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            104,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "$x _ { k } \\in \\mathbb { R } ^ { n _ { x } }$ at step $k$ is subject to a transition function $x _ { k + 1 } = A _ { t } x _ { k } + B _ { t } u _ { k } + w _ { k }$ , where $u _ { k } \\in \\mathbb { R } ^ { n _ { u } }$ is the action at step $k$ , and $A _ { t } \\in \\mathbb R ^ { n _ { x } \\times n _ { x } }$ and $B _ { t } \\in \\mathbb { R } ^ { n _ { x } \\times n _ { u } }$ are dynamical matrices in episode $t$ These matrices are unknown and changing over episodes. The vector $w _ { k }$ is the noise on the state transition. Specifically, the entries of the dynamical matrices $\\cdot$ and $\\cdot$ at episode 0 are randomly generated from a Gaussian distribution $\\mathcal { N } ( 0 , 0 . 1 ^ { 2 } )$ . Then, we generate the time-varying dynamical matrices as $A _ { t + 1 } = A _ { t } + 0 . 0 1 M _ { t }$ and $-$ , where $M _ { t }$ and $N _ { t }$ are random matrices whose entries are uniformly sampled from [0,1]. Moreover, consider a state feedback policy $u _ { k } = K _ { t } x _ { k }$ , where $K _ { t } \\in \\mathbb { R } ^ { n _ { u } \\times n _ { x } }$ is the policy parameter that is fixed within episode $t$ . Within $K _ { t } ^ { * }$ so thatepisode the discounted accuis minimized, where d cost fuis the d ionunt $\\begin{array} { r } { V _ { t } ( K ) : = \\mathbb { E } \\big [ \\sum _ { k = 0 } ^ { H - 1 } \\gamma ^ { k } ( x _ { k } ^ { T } Q x _ { k } + u _ { k } ^ { T } R u _ { k } ) \\big ] } \\end{array}$ $t$ $\\gamma \\leq 1$   \n$H$ $K _ { t } ^ { * }$ that $V _ { t } ( K _ { t } ) - V _ { t } ( K _ { t } ^ { * } )$ is small in every episode. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We apply the conventional one-point method in Gasnikov et al. (2017) and the proposed residualfeedback method (13) to solve the above non-stationary LQR problem. The performance of the twopoint method in Bach & Perchet (2016) is also presented as a benchmark, although it is impractical in non-stationary scenarios. This is because the two-point method in Bach & Perchet (2016) requires to evaluate value function $V _ { t }$ for two different policy functions at two consecutive episodes. However, evaluating the value function $V _ { t }$ for a given policy during episode $t$ requires to collect samples by executing this policy. Then, during the subsequent episode $t + 1$ , since the problem is non-stationary, the dynamic matrices change to $A _ { t + 1 } , B _ { t + 1 }$ and so does the value function $\\cdot$ . Therefore, it is not possible to evaluate the same value function $\\cdot$ at two different episodes and, as a result, the two-point method in Bach & Perchet (2016) is not applicable here. Each algorithm is run for 10 trials, and the stepsizes are optimized respectively. The accumulated regrets $\\begin{array} { r } { \\sum _ { t = 0 } ^ { T - 1 } | V ( K _ { t } ) - V ( K ^ { * } ) | } \\end{array}$ of these algorithms are presented in Figure 1(a). We observe that the residual feedback method achieves a much lower regret than the conventional one-point method and has a comparable performance to that of the impractical two-point method. Moreover, we present in Figure 1(b) the estimated variance of the gradient estimates of these three methods at the policy iterates over episodes. It can be seen that the variance of our proposed residual-feedback is close to the impractical two-point feedback and is much smaller than that of the conventional one-point feedback. This observation justifies our theoretical characterization of the second moment of the residual feedback. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.2 NONSTATIONARY RESOURCE ALLOCATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We consider a multi-stage resource allocation problem with time-varying sensitivity to the lack of resource supply. Specifically, 16 agents are located on a $4 \\times 4$ grid. During episode $t$ , at step $k$ , agent $i$ stores $m _ { i } ( k )$ amount of resources and has a demand for resources in the amount of $d _ { i } ( k )$ . Also, agent $i$ decides to send a fraction of resources $a _ { i j } ( k ) \\in [ 0 , 1 ]$ to its neighbors ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/9f9d6c9a052d224f533ef42168b212bda74f45eaed2a072c49dfb0e8c4d29dce.jpg",
+        "image_caption": [
+            "Figure 2: The costs during each episode by applying the proposed residual one-point feedback (3) (blue), the two-point oracle in Bach & Perchet (2016) (orange) and the conventional one-point oracle in Gasnikov et al. (2017) (green) to solve the non-stationary resource allocation problem are presented. In (a), the varying cost $J _ { t } ( \\theta _ { t } )$ of three methods are presented. In (b), the variance of the gradient estimates at agent 1 given by three methods are presented. The two point method (orange) is infeasible to use in practice and is presented here to serve as the simulating benchmark. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$j \\in \\mathcal N _ { i }$ on the grid. The local amount of resources and demands of agent $i$ evolve as $m _ { i } ( k + 1 ) =$ $\\begin{array} { r } { m _ { i } ( k ) - \\sum _ { j \\in \\mathcal { N } _ { i } } a _ { i j } ( k ) m _ { i } ( k ) + \\sum _ { j \\in \\mathcal { N } _ { i } } a _ { j i } ( k ) m _ { j } ( k ) - d _ { i } ( k ) } \\end{array}$ and $d _ { i } ( \\bar { k } ) = \\psi _ { i } \\sin ( \\omega _ { i } k + \\dot { \\phi } _ { i } ) + \\dot { w } _ { i , k }$ , where $w _ { i , k }$ is the noise in the demand. At each step $k$ , agent $i$ receives a local cost $r _ { i , t } ( k )$ , such that $r _ { i , t } ( k ) = 0$ when $m _ { i } ( k ) \\geq 0$ and $r _ { i , t } ( k ) = \\zeta _ { t } m _ { i } ( k ) ^ { 2 }$ when $m _ { i } ( k ) < 0$ , where $\\zeta _ { t }$ represents the varying sensitivity of the agents to the lack of supply during episode $t$ . Let agent $i$ makes its decisions according to a parameterized policy function $\\pi _ { i , t } \\big ( o _ { i } ; \\theta _ { i , t } \\big ) : \\mathcal { O } _ { i } \\to [ 0 , 1 ] ^ { | \\mathcal { N } _ { i } | }$ , where $\\theta _ { i , t }$ is the parameter of the policy function $\\pi _ { i , t }$ at episode $t$ , $o _ { i } \\in { \\mathcal { O } } _ { i }$ denotes agent $i$ ’s local observation. Specifically, we let so that the accumul $o _ { i } ( k ) = [ m _ { i } ( k ) , d _ { i } ( k ) ] ^ { T }$ arying optimal policyduring each episode is $\\begin{array} { r } { \\left. J _ { t } ( \\theta _ { t } ) \\right. = \\sum _ { i = 1 } ^ { 1 6 } \\sum _ { k = 0 } ^ { H } \\gamma ^ { k } r _ { i , t } ( k ) } \\end{array}$   \n$\\theta _ { t } = [ \\dots , \\theta _ { i , t } , \\dots ]$ $H$   \nat each episode, and $\\gamma$ is the discount factor. ",
+        "bbox": [
+            173,
+            393,
+            826,
+            556
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Figure 2(a), we present the costs achieved during each episode $J _ { t } ( \\theta _ { t } )$ with 10 trials using ZO with the residual-feedback, one-point and the impractical two-point feedback. It can be seen that our proposed residual-feedback achieves a cost $J _ { t } ( \\theta _ { t } )$ that is as low as the cost achieved by the impractical two-point feedback in such a non-stationary environment. In particular, both residual and two-point feedback perform much better than the conventional one-point feedback. Moreover, Figure 2(b) compares the estimated variances of these feedback schemes, and one can observe that the variance of the residual feedback is comparable to that of the two-point feedback and is much smaller than that of the conventional one-point feedback. ",
+        "bbox": [
+            174,
+            563,
+            825,
+            674
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
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+        "text": "In this paper, we proposed a residual one-point feedback oracle for zeroth-order online learning problems, which estimates the gradient of the time-varying objective function using a single query of the function value at each time instant. We showed that the regret bound of the proposed residual feedback estimator can be much lower than that of the conventional one-point method in online convex optimization setting. In addition, we studied the gradient size regret bound of the residualfeedback estimator when it is applied to the online non-convex optimization problems. Numerical experiments on two non-stationary reinforcement learning problems were conducted and the proposed residual-feedback estimator was shown to significantly outperform the conventional one-point method in non-stationary online learning problems. ",
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parse/train/rkl3m1BFDB/rkl3m1BFDB.md

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+# EXPLORATORY NOT EXPLANATORY: COUNTERFACTUAL ANALYSIS OF SALIENCY MAPS FOR DEEP REINFORCEMENT LEARNING
+
+Akanksha Atrey, Kaleigh Clary & David Jensen
+
+University of Massachusetts Amherst {aatrey,kclary,jensen}@cs.umass.edu
+
+# ABSTRACT
+
+Saliency maps are frequently used to support explanations of the behavior of deep reinforcement learning (RL) agents. However, a review of how saliency maps are used in practice indicates that the derived explanations are often unfalsifiable and can be highly subjective. We introduce an empirical approach grounded in counterfactual reasoning to test the hypotheses generated from saliency maps and assess the degree to which they correspond to the semantics of RL environments. We use Atari games, a common benchmark for deep RL, to evaluate three types of saliency maps. Our results show the extent to which existing claims about Atari games can be evaluated and suggest that saliency maps are best viewed as an exploratory tool rather than an explanatory tool.
+
+# 1 INTRODUCTION
+
+Saliency map methods are a popular visualization technique that produce heatmap-like output highlighting the importance of different regions of some visual input. They are frequently used to explain how deep networks classify images in computer vision applications (Simonyan et al., 2014; Zeiler & Fergus, 2014; Springenberg et al., 2015; Ribeiro et al., 2016; Dabkowski & Gal, 2017; Fong & Vedaldi, 2017; Selvaraju et al., 2017; Shrikumar et al., 2017; Smilkov et al., 2017; Zhang et al., 2018) and to explain how agents choose actions in reinforcement learning (RL) applications (Bogdanovic et al., 2015; Wang et al., 2016; Zahavy et al., 2016; Greydanus et al., 2017; Iyer et al., 2018; Sundar, 2018; Yang et al., 2018; Annasamy & Sycara, 2019).
+
+Saliency methods in computer vision and reinforcement learning use similar procedures to generate these maps. However, the temporal and interactive nature of RL systems presents a unique set of opportunities and challenges. Deep models in reinforcement learning select sequential actions whose effects can interact over long time periods. This contrasts strongly with visual classification tasks, in which deep models merely map from images to labels. For RL systems, saliency maps are often used to assess an agent’s internal representations and behavior over multiple frames in the environment, rather than to assess the importance of specific pixels in classifying images.
+
+Despite their common use to explain agent behavior, it is unclear whether saliency maps provide useful explanations of the behavior of deep RL agents. Some prior work has evaluated the applicability of saliency maps for explaining the behavior of image classifiers (Samek et al., 2017; Adebayo et al., 2018; Kindermans et al., 2019), but there is not a corresponding literature evaluating the applicability of saliency maps for explaining RL agent behavior.
+
+In this work, we develop a methodology grounded in counterfactual reasoning to empirically evaluate the explanations generated using saliency maps in deep RL. Specifically, we:
+
+C1 Survey the ways in which saliency maps have been used as evidence in explanations of deep RL agents.   
+C2 Describe a new interventional method to evaluate the inferences made from saliency maps.   
+C3 Experimentally evaluate how well the pixel-level inferences of saliency maps correspond to the semantic-level inferences of humans.
+
+![](images/d24986a4fd0af4067ac5f69695e78829d7b8a9d393c1347674cecaf674fe09a5.jpg)  
+Figure 1: Three examples of perturbation saliency maps generated from the same model on different inputs: (a) a frame taken from an episode of agent play in Breakout, (b) the same frame with the brick pattern reflected across the vertical axis, and (c) the original frame with the ball, paddle and brick pattern reflected across the vertical axis. The blue and red regions represent their importance in action selection and reward estimation from the current state, respectively. The pattern and intensity of saliency around the tunnel is not symmetric in either reflection intervention, indicating that a popular hypothesis (agents learn to aim at tunnels) does not hold for all possible tunnels.
+
+# 2 INTERPRETING SALIENCY MAPS IN DEEP RL
+
+Consider the saliency maps generated from a deep RL agent trained to play the Atari game Breakout. The goal of Breakout is to use the paddle to keep the ball in play so it hits bricks, eliminating them from the screen. Figure 1a shows a sample frame with its corresponding saliency. Note the high salience on the missing section of bricks (“tunnel”) in Figure 1a.
+
+Creating a tunnel to target bricks at the top layers is one of the most high-profile examples of agent behavior being explained according to semantic, human-understandable concepts (Mnih et al., 2015). Given the intensity of saliency on the tunnel in 1a, it may seem reasonable to infer that this saliency map provides evidence that the agent has learned to aim at tunnels. If this is the case, moving the horizontal position of the tunnel should lead to similar saliency patterns on the new tunnel. However, Figures 1b and 1c show that the salience pattern is not preserved. Neither the presence of the tunnel, nor the relative positioning of the ball, paddle, and tunnel, are responsible for the intensity of the saliency observed in Figure 1a.
+
+# 2.1 SALIENCY MAPS AS INTERVENTIONS
+
+Examining how some of the technical details of reinforcement learning interact with saliency maps can help explain both the potential utility and the potential pitfalls of interpreting saliency maps. RL methods enable agents to learn how to act effectively within an environment by repeated interaction with that environment. Certain states in the environment give the agent positive or negative reward. The agent learns a policy, a mapping between states and actions according to these reward signals. The goal is to learn a policy that maximizes the discounted sum of rewards received while acting in the environment (Sutton & Barto, 1998). Deep reinforcement learning uses deep neural networks to represent policies. These models enable interaction with environments requiring high-dimensional state inputs (e.g., Atari games).
+
+Consider the graphical model in Figure 2a representing the deep RL system for a vision-based game environment. Saliency maps are produced by performing some kind of intervention $M$ on this system and calculating the difference in logits produced by the original and modified images. The interventions used to calculate saliency for deep RL are performed at the pixel level (red node and arrow in Figure 2a). These interventions change the conditional probability distribution of “Pixels” by giving it another parent (Pearl, 2000).
+
+Functionally, this can be accomplished through a variety of means, including changing the color of the pixel (Simonyan et al., 2014), adding a gray mask (Zeiler & Fergus, 2014), blurring a small region (Greydanus et al., 2017), or masking objects with the background color (Iyer et al., 2018). The interventions $M$ are used to simulate the effect of the absence of the pixel(s) on the network’s output. Note however that these interventions change the image in a way that is inconsistent with the generative process $F$ . They are not “naturalistic” interventions. This type of intervention produces images for which the learned network function may not be well-defined.
+
+![](images/7f1d7a6d104a13366f9073758cb0cf2e07a2f1450f97cd0b218a0177d1f90b4b.jpg)  
+Figure 2: (a) Causal graphical model of the relationships between an RL agent (yellow plate) and an image-based environment (blue plate). The environment maintains some (usually latent) game state. Some function $F$ produces a high-dimensional pixel representation of game state (“Pixels”). The learned network takes this pixel image and produces logits used to select an action. Temporally extended sequences of this action selection procedure result in observed agent behavior.“M” represents interventions made by saliency maps. Such interventions are not naturalistic and are inconsistent with the generative process F; (b) conceptual diagram of how a human observer infers explanations. Hypotheses (“Claims”) about the semantic features identified by the learned policy are proposed by reasoning backwards about what representation, often latent, might jointly produce the observed saliency pattern and agent behavior.
+
+# 2.2 EXPLANATIONS FROM SALIENCY MAPS
+
+To form explanations of agent behavior, human observers combine information from saliency maps, agent behavior, and semantic concepts. Figure 2b shows a system diagram of how these components interact. We note that semantic concepts are often identified visually from the pixel output as the game state is typically latent.
+
+Counterfactual reasoning has been identified as a particularly effective way to present explanations of the decision boundaries of deep models (Mittelstadt et al., 2019). Humans use counterfactuals to reason about the enabling conditions of particular outcomes, as well as to identify situations where the outcome would have occurred even in the absence of some action or condition (de Graaf & Malle, 2017; Byrne, 2019). Saliency maps provide a kind of pixel-level counterfactual, but if the goal is to explain agent behavior according to semantic concepts, interventions at the pixel level seem unlikely to be sufficient.
+
+Since many semantic concepts may map to the same set of pixels, it may be difficult to identify the functional relationship between changes in pixels and changes in network output according to semantic concepts or game state (Chalupka et al., 2015). Researchers may be interpreting differences in network outputs as evidence of differences in semantic concepts. However, changes in pixels do not guarantee changes in semantic concepts or game state.
+
+In terms of changes to pixels, semantic concepts, and game state, we distinguish among three classes of interventions: distortion, semantics-preserving, and fat-hand (see Table 1). Semantics-preserving and fat-hand interventions are defined with respect to a specific set of semantic concepts. Fat-hand interventions change game state in such a way that the semantic concepts of interest are also altered.
+
+The pixel-level manipulations used to produce saliency maps primarily result in distortion interventions, though some saliency methods (e.g., object-based) may conceivably produce semanticspreserving or fat-hand interventions as well. Pixel-level interventions are not guaranteed to produce changes in semantic concepts, so counterfactual evaluations that apply semantics-preserving interventions may be a more appropriate approach for precisely testing hypotheses of behavior.
+
+<table><tr><td rowspan="2">Intervention</td><td colspan="3">Change in...</td><td rowspan="2">Examples</td></tr><tr><td>pixels</td><td>game state</td><td>semantic concepts</td></tr><tr><td>Distortion</td><td>vv√</td><td>X</td><td>×</td><td>Adversarial ML (Szegedy et al., 2014)</td></tr><tr><td>Semantics-preserving</td><td></td><td>√</td><td>×</td><td>Reflection across a line of symmetry</td></tr><tr><td>Fat-hand</td><td></td><td>√</td><td>√</td><td>Teleporting the agent to a new position</td></tr></table>
+
+Table 1: Categories of interventions on images. Distortion interventions change pixels without changing game state or semantic concepts. Pixel perturbations in adversarial ML add adversarial noise to images to change the output of the network without making human-perceptible changes in the image. Semantics-preserving interventions are manipulations of game state that result in an image that preserves some semantic concept of interest. Reflections across lines of symmetry typically alter aspects of game state, but do not meaningfully change any semantic information about the scene. “Fat-hand” interventions are manipulations intended to measure the effect of some specific treatment, but which unintentionally alter other relevant aspects of the system. The term is drawn from the literature on causal modeling (Scheines, 2005).
+
+# 3 SURVEY OF USAGE OF SALIENCY MAPS IN DEEP RL LITERATURE
+
+To assess how saliency maps are typically used to make inferences regarding agent behavior, we surveyed recent conference papers in deep RL. We focused our pool of papers on those that use saliency maps to generate explanations or make claims regarding agent behavior. Our search criteria consisted of examining papers that cited work that first described any of the following four types of saliency maps:
+
+Jacobian Saliency. Wang et al. (2016) extend gradient-based saliency maps to deep RL by computing the Jacobian of the output logits with respect to a stack of input images.
+
+Perturbation Saliency. Greydanus et al. (2017) generate saliency maps by perturbing the original input image using a Gaussian blur of the image and measure changes in policy from removing information from a region.
+
+Object Saliency. Iyer et al. (2018) use template matching, a common computer vision technique (Brunelli, 2009), to detect (template) objects within an input image and measure salience through changes in Q-values for masked and unmasked objects.
+
+Attention Saliency. Most recently, attention-based saliency mapping methods have been proposed to generate interpretable saliency maps (Mott et al., 2019; Nikulin et al., 2019).
+
+From a set of 90 papers, we found 46 claims drawn from 11 papers that cited and used saliency maps as evidence in their explanations of agent behavior. The full set of claims are given in Appendix C.
+
+# 3.1 SURVEY RESULTS
+
+We found three categories of saliency map usage, summarized in Table 2. First, all claims interpret salient areas as a proxy for agent focus. For example, a claim about a Breakout agent notes that the network is focusing on the paddle and little else (Greydanus et al., 2017).
+
+<table><tr><td></td><td>Discuss Focus</td><td>Generate Explanation</td><td>Evaluate Explanation</td></tr><tr><td> Jacobian</td><td>21</td><td>19</td><td>0</td></tr><tr><td>Perturbation</td><td>11</td><td>9</td><td>1</td></tr><tr><td>Object</td><td>5</td><td>4</td><td>2</td></tr><tr><td>Attention</td><td>9</td><td>8</td><td>0</td></tr><tr><td>Total Claims</td><td>46</td><td>40</td><td>3</td></tr></table>
+
+Second, $87 \%$ of the claims in our survey propose hypotheses about the features of the learned policy by reasoning backwards about what representation might jointly produce the observed saliency pattern and agent
+
+Table 2: Summary of the survey on usage of saliency maps in deep RL. Columns represent categories of saliency map usage, and rows represent categories of saliency map methods, with each cell denoting the number of claims in those categories. Individual claims may be counted in multiple columns.
+
+behavior. These types of claims either develop an a priori explanation of behavior and evaluate it using saliency, or they propose an ad hoc explanation after observing saliency to reason about how the agent is using salient areas. One a priori claim notes that the displayed score is the only differing factor between two states and evaluates that claim by noting that saliency focuses on these pixels (Zahavy et al., 2016). An ad hoc claim about a racing game notes that the agent is recognizing a time-of-day cue from the background color and acting to prepare for a new race (Yang et al., 2018).
+
+Finally, only $7 \%$ (3 out of 46) of the claims drawn from saliency maps are accompanied by additional or more direct experimental evidence. One of these attempts to corroborate the interpreted saliency behavior by obtaining additional saliency samples from multiple runs of the game. The other two attempt to manipulate semantics in the pixel input to assess the agent’s response by, for example, adding an additional object to verify a hypothesis about memorization (Annasamy & Sycara, 2019).
+
+# 3.2 COMMON PITFALLS IN CURRENT USAGE
+
+In the course of the survey, we also observed several more qualitative characteristics of how saliency maps are routinely used.
+
+Subjectivity. Recent critiques of machine learning have already noted a worrying tendency to conflate speculation and explanation (Lipton & Steinhardt, 2018). Saliency methods are not designed to formalize an abstract human-understandable concept such as “aiming” in Breakout, and they do not provide a means to quantitatively compare semantically meaningful consequences of agent behavior. This leads to subjectivity in the conclusions drawn from saliency maps.
+
+Unfalsiability. One hallmark of a scientific hypothesis or claim is falsifiability (Popper, 1959). If a claim is false, its falsehood should be identifiable from some conceivable experiment or observation. One of the most disconcerting practices identified in the survey is the presentation of unfalsifiable interpretations of saliency map patterns. An example: “A diver is noticed in the saliency map but misunderstood as an enemy and being shot at” (see Appendix C). It is unclear how we might falsify an abstract concept such as “misunderstanding”.
+
+Cognitive Biases. Current theory and evidence from cognitive science implies that humans learn complex processes, such as video games, by categorizing objects into abstract classes and by inferring causal relationships among instances of those classes (Tenenbaum & Niyogi, 2003; Dubey et al., 2018). Our survey suggests that researchers infer that: (1) salient regions map to learned representations of semantic concepts (e.g., ball, paddle), and (2) the relationships among the salient regions map to high-level behaviors (e.g., tunnel-building, aiming). Researchers’ expectations impose a strong bias on both the existence and nature of these mappings.
+
+# 4 METHODOLOGY
+
+Our survey indicates that many researchers use saliency maps as an explanatory tool to infer the representations and processes behind an agent’s behavior. However, the extent to which such inferences are valid has not been empirically evaluated under controlled conditions.
+
+In this section, we show how to generate falsifiable hypotheses from saliency maps and propose an intervention-based approach to verify the hypotheses generated from saliency maps. We intervene on game state to produce counterfactual semantic conditions. This provides a concrete methodology to assess the relationship between saliency and learned semantic representations.
+
+Building Falsifiable Hypotheses from Saliency Maps. Though saliency maps may not relate directly to semantic concepts, they may still be an effective tool for exploring hypotheses about agent behavior. As we show schematically in Figure 2b, claims or explanations informed by saliency maps have three components: semantic concepts, saliency, and behavior. Recall that our survey indicates that researchers often attempt to infer aspects of the network’s learned representations from saliency patterns. Let $X$ be a subset of the semantic concepts that can be inferred from the input image. Let $B$ represent behavior, or aggregate actions, over temporally extended sequences of frames, and let $R$ be a representation that is a function of some pixels that the agent learns during training.
+
+To create scientific claims from saliency maps, we recommend using a relatively standard pattern which facilitates objectivity and falsifiability:
+
+{concept set $X \}$ is salient $\Longrightarrow$ agent has learned {representation $R \}$ resulting in {behavior $B \}$
+
+Consider the Breakout brick reflection example presented in Section 2. The hypothesis introduced (“the agent has learned to aim at tunnels”) can be reformulated as: bricks are salient $\Longrightarrow$ agent has learned to identify a partially complete tunnel resulting in maneuvering the paddle to hit the ball toward that region. Stating hypotheses in this format implies falsifiable claims amenable to empirical analysis.
+
+Counterfactual Evaluation of Claims. As indicated in Figure 2, the learned representation and pixel input share a relationship with saliency maps generated over a sequence of frames. Given that the representation learned is static, the relationship between the learned representation and saliency should be invariant under different manipulations of pixel input. We use this property to assess saliency under counterfactual conditions.
+
+We generate counterfactual conditions by intervening on the RL environment. Prior work has focused on manipulating the pixel input. However, this does not modify the underlying latent game state. Instead, we intervene directly on game state. In the do-calculus formalism (Pearl, 2000), this shifts the intervention node in Figure 2a to game state, which leaves the generative process $F$ of the pixel image intact.
+
+We employ TOYBOX, a set of fully parameterized implementation of Atari games (Foley et al., 2018), to generate interventional data under counterfactual conditions. The interventions are dependent on the mapping between semantic concepts and learned representations in the hypotheses. Given a mapping between concept set $X$ and a learned representation $R$ , any intervention would require meaningfully manipulating the state in which $X$ resides to assess the saliency on $X$ under the semantic treatment applied. Saliency on $x \in X$ is defined as the average saliency over a bounding-box1 around $x$ .
+
+Since the learned policies should be semantically invariant under manipulations of the RL environment, by intervening on state, we can verify whether the counterfactual states produce expected patterns of saliency on the associated concept set $X$ . If the counterfactual saliency maps reflect similar saliency patterns, this provides stronger evidence that the observed saliency indicates the agent has learned representation R corresponding to semantic concept set $X$ .
+
+# 5 EVALUATION OF HYPOTHESES ON AGENT BEHAVIOR
+
+We conduct three case studies to evaluate hypotheses about the relationship between semantic concepts and semantic processes formed from saliency maps. Each case study uses observed saliency maps to identify hypotheses in the format described in Section 4. The hypotheses were generated by watching multiple episodes and noting atypical, interesting or popular behaviors from saliency maps. In each case study, we produce Jacobian, perturbation and object saliency maps from the same set of counterfactual states. We include examples of each map in Appendix A. Using TOYBOX allows us to produce counterfactual states and to generate saliency maps in these altered states. The case studies are conducted on two Atari games, Breakout and Amidar.2 The deterministic nature of both games allows some stability in the way we interpret the network’s action selection. Each map is produced from an agent trained with A2C (Mnih et al., 2016) using a CNN-based (Mnih et al., 2015) OpenAI Baselines implementation (Dhariwal et al., 2017) with default hyperparameters (see Appendix B for more details). Our choice of model is arbitrary. The emphasis of this work is on methods of explanation, not the explanations themselves.
+
+![](images/629a43cd2b78712258ef1a367e25a56117ae138a08807c67277a71bb37b07d66.jpg)  
+Figure 3: Interventions on brick configurations in Breakout. (a) saliency after shifting the brick positions by some pixels where shift ${ \boldsymbol { \mathbf { \mathit { \sigma } } } } = 0$ represents the original frame; (b) saliency after shifting the brick positions, ball, and paddle to the left. The pattern and intensity of saliency around the tunnel is not symmetric in the reflection interventions.
+
+Case Study 1: Breakout Brick Translation. Here we evaluate the behavior from Section 2:
+
+Hypothesis 1: $\{ { \mathrm { b r i c k s } } \}$ are salient $\Longrightarrow$ agent has learned to {identify a partially complete tunnel $\}$ resulting in {maneuvering the paddle to hit the ball toward that region}.
+
+To evaluate this hypothesis, we intervene on the state by translating the brick configurations horizontally. Because the semantic concepts relating to the tunnel are preserved under translation, we expect salience will be nearly invariant to the horizontal translation of the brick configuration. Figure 3a depicts saliency after intervention. Salience on the tunnel is less pronounced under left translation, and more pronounced under right translation. Since the paddle appears on the right, we additionally move the ball and paddle to the far left (Figure 3b).
+
+Conclusion. Temporal association (e.g. formation of a tunnel followed by higher saliency) does not generally imply causal dependence. In this case, tunnel formation and salience appear to be confounded by location or, at least, the dependence of these phenomena are highly dependent on location.
+
+Case Study 2: Amidar Score. Amidar is a Pac-Man-like game in which an agent attempts to completely traverse a series of passages while avoiding enemies. The yellow sprite that indicates the location of the agent is almost always salient in Amidar. Surprisingly, the displayed score is often as salient as the yellow sprite throughout the episode with varying levels of intensity. This can lead to multiple hypotheses about the agent’s learned representation: (1) the agent has learned to associate increasing score with higher reward; (2) due to the deterministic nature of Amidar, the agent has created a lookup table that associates its score and its actions. We can summarize these as follows:
+
+Hypothesis 2: $\{ { \mathrm { s c o r e } } \}$ is salient $\Longrightarrow$ agent has learned to {use score as a guide to traverse the board} resulting in {successfully following similar paths in games}.
+
+To evaluate hypothesis 2, we designed four interventions on score:
+
+• intermittent reset: modify the score to 0 every $x \in [ 5 , 2 0 ]$ timesteps.   
+• random varying: modify the score to a random number between [1,200] every $x \in [ 5 , 2 0 ]$ timesteps.   
+• fixed: select a score from [0,200] and fix it for the whole game.   
+• decremented: modify score to be 3000 initially and decrement score by $d \in [ 1 , 2 0 ]$ at every timestep.
+
+Figures 4a and 4b show the result of intervening on displayed score on reward and saliency intensity, measured as the average saliency over a $2 5 \mathrm { x } 1 5$ bounding box, respectively for the first 1000 timesteps of an episode. The mean is calculated over 50 samples. If an agent died before 1000 timesteps, the last reward was extended for the remainder of the timesteps and saliency was set to zero.
+
+![](images/41f10c8a507d15458379a76e60f802540aed7015c8c31ce044e0c0a6b020476b.jpg)  
+Figure 4: Interventions on displayed score in Amidar. The legend in (b) applies to all figures. (a) reward over time for different interventions on displayed score; (b) object saliency on displayed score over time; (c) correlation between the differences in reward and object saliency from the original trajectory. Interventions on displayed score result in differing levels of degraded performance but produce similar saliency maps, suggesting that agent behavior as measured by rewards is underdetermined by salience.
+
+Using reward as a summary of agent behavior, different interventions on score produce different agent behavior. Total accumulated reward differs over time for all interventions, typically due to early agent death. However, salience intensity patterns of all interventions follow the original trajectory very closely. Different interventions on displayed score cause differing degrees of degraded performance (Figure 4a) despite producing similar saliency maps (Figure 4b), indicating that agent behavior is underdetermined by salience. Specifically, the salience intensity patterns are similar for the control, fixed, and decremented scores, while the non-ordered score interventions result in degraded performance. Figure 4c indicates only very weak correlations between the difference-in-reward and difference-in-saliency-under-intervention as compared to the original trajectory. Correlation coefficients range from 0.041 to 0.274, yielding insignificant p-values for all but one intervention. See full results in Appendix E.1, Table 6.
+
+Similar trends are noted for Jacobian and perturbation saliency methods in Appendix E.1.
+
+Conclusion. The existence of a high correlation between two processes (e.g., incrementing score and persistence of saliency) does not imply causation. Interventions can be useful in identifying the common cause leading to the high correlation.
+
+Case Study 3: Amidar Enemy Distance. Enemies are salient in Amidar at varying times. From visual inspection, we observe that enemies close to the player tend to have higher saliency. Accordingly, we generate the following hypothesis:
+
+Hypothesis 3: {enemy} is salient $\Longrightarrow$ agent has learned to {identify enemies close to it} resulting in {successful avoidance of enemy collision $\}$ .
+
+Without directly intervening on the game state, we can first identify whether the player-enemy distance and enemy saliency is correlated using observational data. We collect 1000 frames of an episode of Amidar and record the Manhattan distance between the midpoints of the player and enemies, represented by $7 \mathbf { x } 7$ bounding boxes, along with the object salience of each enemy. Figure 5a shows the distance of each enemy to the player over time with saliency intensity represented by the shaded region. Figure 5b shows the correlation between the distance to each enemy and the corresponding saliency. Correlation coefficients and significance values are reported in Table 3. It is clear that there is no correlation between saliency and distance of each enemy to the player.
+
+Given that statistical dependence is almost always a necessary pre-condition for causation, we expect that there will not be any causal dependence. To further examine this, we intervene on enemy positions of salient enemies at each timestep by moving the enemy closer and farther away from the player. Figure 5c contains these results. Given Hypothesis 3, we would expect to see an increasing trend in saliency for enemies closer to the player. However, the size of the effect is close to 0 (see Table 3). In addition, we find no correlation in the enemy distance experiments for the Jacobian or perturbation saliency methods (included in Appendix E.2).
+
+![](images/ad8aeea642a97fc5cb88562219e46739d458b571c80f8a970c340eebf36e4bd3.jpg)  
+Figure 5: Interventions on enemy location in Amidar. The legend in (b) applies to all figures. (a) the distance-to-player of each enemy in Amidar, observed over time, where saliency intensity is represented by the shaded region around each line; (b) the distance-to-player and saliency, with linear regressions, observed for each enemy; (c) variation in enemy saliency when enemy position is varied by intervention. The plots suggest that there is no substantial correlation and no causal dependence between distance-to-player and object saliency.
+
+Table 3: Numeric results from regression analysis for the observational and interventional results in Figures 5b and c. The results indicate a very small strength of effect (slope) for both observational and interventional data and a small correlation coefficient $( r )$ , suggesting that there is, at best, only a very weak causal dependence of saliency on distance-to-player.   
+
+<table><tr><td></td><td colspan="3">Observational</td><td colspan="3">Interventional</td></tr><tr><td>Enemy</td><td>slope</td><td>p-value</td><td>r</td><td>slope</td><td>p-value</td><td>r</td></tr><tr><td>1</td><td>-0.001</td><td>0.26</td><td>-0.036</td><td>-0.001</td><td>0.97</td><td>-0.001</td></tr><tr><td>2</td><td>-0.001</td><td>0.35</td><td>-0.298</td><td>0.013</td><td>0.68</td><td>0.039</td></tr><tr><td>3</td><td>0.004</td><td>1.59e-4</td><td>0.119</td><td>-0.008</td><td>0.79</td><td>-0.022</td></tr><tr><td>4</td><td>-0.008</td><td>8.26e-19</td><td>-0.275</td><td>-0.011</td><td>0.75</td><td>-0.028</td></tr><tr><td>5</td><td>0.001</td><td>0.13</td><td>0.047</td><td>-0.033</td><td>0.47</td><td>-0.063</td></tr></table>
+
+Conclusion. Spurious correlations, or misinterpretations of existing correlation, can occur between two processes (e.g. correlation between player-enemy distance and saliency), and human observers are susceptible to identifying spurious correlations (Simon, 1954). Spurious correlations can sometimes be identified from observational analysis without requiring interventional analysis.
+
+# 6 DISCUSSION AND RELATED WORK
+
+Thinking counterfactually about the explanations generated from saliency maps facilitates empirical evaluation of those explanations. The experiments above show some of the difficulties in drawing conclusions from saliency maps. These include the tendency of human observers to incorrectly infer association between observed processes, the potential for experimental evidence to contradict seemingly obvious observational conclusions, and the challenges of potential confounding in temporal processes.
+
+One of the main conclusions from this evaluation is that saliency maps are an exploratory tool rather than an explanatory tool for evaluating agent behavior in deep RL. Saliency maps alone cannot be reliably used to infer explanations and instead require other supporting tools. This can include combining evidence from saliency maps with other explanation methods or employing a more experimental approach to evaluation of saliency maps such as the approach demonstrated in the case studies above.
+
+The framework for generating falsifiable hypotheses suggested in Section 4 can assist with designing more specific and falsifiable explanations. The distinction between the components of an explanation, particularly the semantic concept set $X$ , learned representation $R$ and observed behavior $B$ , can further assist in experimental evaluation. Note that the semantic space devised by an agent might be quite different from the semantic space given by the latent factors of the environment. It is crucial to note that this mismatch is one aspect of what plays out when researchers create hypotheses about agent behavior, and the methodology we provide in this work demonstrates how to evaluate hypotheses that reflect that mismatch.
+
+Generalization of Proposed Methodology. The methodology presented in this work can be easily extended to other vision-based domains in deep RL. Particularly, the framework of the graphical model introduced in Figure 2a applies to all domains where the input to the network is image data. An extended version of the model for Breakout can be found in Appendix 7.
+
+We propose intervention-based experimentation as a primary tool to evaluate the hypotheses generated from saliency maps. Yet, alternative methods can identify a false hypothesis even earlier. For instance, evaluating statistical dependence alone can provide strong evidence against causal dependence (e.g., Case Study 3). In this work, we employ a particularly capable simulation environment (TOYBOX). However, limited forms of evaluation may be possible in non-intervenable environments, though they may be more tedious to implement. For instance, each of the interventions conducted in Case Study 1 can be produced in an observation-only environment by manipulating the pixel input (Brunelli, 2009; Chalupka et al., 2015). Developing more experimental systems for evaluating explanations is an open area of research.
+
+This work analyzes explanations generated from feed-forward deep RL agents. However, the proposed methodology is not model dependent, and aspects of the approach will carry over to recurrent deep RL agents. The proposed methodology would not work well for repeated interventions on recurrent deep RL agents due to their capacity for memorization.
+
+Explanations in Deep RL. Prior work has introduced alternatives to the use of saliency maps to support explanation of deep RL agents. Some of these methods also use counterfactual reasoning to develop explanations. TOYBOX was developed to support experimental evaluation and behavioral tests of deep RL models (Tosch et al., 2019). Olson et al. (2019) use a generative deep learning architecture to produce counterfactual states resulting in the agent taking a different action. Others have proposed alternative methods for developing semantically meaningful interpretations of agent behavior. Juozapaitis et al. (2019) use reward decomposition to attribute policy behaviors according to semantically meaningful components of reward. Verma et al. (2018) use domain-specific languages for policy representation, allowing for human-readable policy descriptions.
+
+Evaluation and Critiques of Saliency Maps. Prior work in the deep network literature has evaluated and critiqued saliency maps. Kindermans et al. (2019) and Adebayo et al. (2018) demonstrate the utility of saliency maps by adding random variance in input. Seo et al. (2018) provide a theoretical justification of saliency and hypothesize that there exists a correlation between gradients-based saliency methods and model interpretation. Samek et al. (2017) and Hooker et al. (2019) present evaluations of existing saliency methods for image classification.
+
+# 7 CONCLUSIONS
+
+We conduct a survey of uses of saliency maps, propose a methodology to evaluate saliency maps, and examine the extent to which the agent’s learned representations can be inferred from saliency maps. We investigate how well the pixel-level inferences of saliency maps correspond to the semantic concept-level inferences of human-level interventions. Our results show saliency maps cannot be trusted to reflect causal relationships between semantic concepts and agent behavior. We recommend saliency maps to be used as an exploratory tool, not explanatory tool.
+
+# ACKNOWLEDGMENTS
+
+Thanks to Emma Tosch, Amanda Gentzel, Deep Chakraborty, Abhinav Bhatia, Blossom Metevier, Chris Nota, Karthikeyan Shanmugam and the anonymous ICLR reviewers for thoughtful comments and contributions. This material is based upon work supported by the United States Air Force under Contract No, FA8750-17-C-0120. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Air Force.
+
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+Ziyu Wang, Tom Schaul, Matteo Hessel, Hado Van Hasselt, Marc Lanctot, and Nando De Freitas. Dueling network architectures for deep reinforcement learning. ICML, 2016.   
+Zhao Yang, Song Bai, Li Zhang, and Philip HS Torr. Learn to interpret atari agents. arXiv preprint arXiv:1812.11276, 2018.   
+Tom Zahavy, Nir Ben-Zrihem, and Shie Mannor. Graying the black box: Understanding dqns. 2016.   
+Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In ECCV, 2014.   
+Jianming Zhang, Sarah Adel Bargal, Zhe Lin, Jonathan Brandt, Xiaohui Shen, and Stan Sclaroff. Top-down neural attention by excitation backprop. International Journal of Computer Vision, 2018.
+
+# APPENDICES
+
+# A SALIENCY METHODS
+
+Figure 6 shows example saliency maps of the three saliency methods evaluated in this work, namely perturbation, object and Jacobian, for Amidar.
+
+![](images/c651e505a1ae3c52943f1c7084b25351cbc2263e87a9a173b3d8ebcf044e7d5b.jpg)  
+Figure 6: Examples of (a) perturbation saliency method (Greydanus et al., 2017), (b) object saliency method (Iyer et al., 2018), and (c) Jacobian saliency method (Wang et al., 2016), for Amidar.
+
+# B MODEL
+
+We use the OpenAI Baselines’ implementation (Dhariwal et al., 2017) of an A2C model (Mnih et al., 2016) to train the RL agents on Breakout and Amidar. The model uses the CNN architecture proposed by Mnih et al. (2015). Each agent is trained for 40 million iterations using RMSProp with default hyperparameters (Table 4).
+
+Table 4: Hyperparameters used in training the A2C model.   
+
+<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Learning rate</td><td>7e-4</td></tr><tr><td>Learning rate schedule</td><td>Linear</td></tr><tr><td># Iterations</td><td>40,000,000</td></tr><tr><td>Value function coefficient</td><td>0.5</td></tr><tr><td>Policy function coefficient</td><td>0.01</td></tr><tr><td>RMSProp epsilon</td><td>1e-5</td></tr><tr><td>RMSProp decay</td><td>0.99</td></tr><tr><td>Reward discounting parameter</td><td>0.99</td></tr><tr><td>Max gradient (clip)</td><td>0.5</td></tr></table>
+
+# C SURVEY OF USAGE OF SALIENCY MAPS IN DEEP RL LITERATURE
+
+We conducted a survey of recent literature to assess how saliency maps are used to interpret agent behavior in deep RL. We began our search by focusing on work citing the following four types of saliency maps: Jacobian (Wang et al., 2016), perturbation (Greydanus et al., 2017), object (Iyer et al., 2018) and attention (Mott et al., 2019). Papers were selected if they employed saliency maps to create explanations regarding agent behavior. This resulted in selecting 46 claims from 11 papers. These 11 papers have appeared at ICML (3), NeurIPS (1), AAAI (2), ArXiv (3), OpenReview (1) and as a thesis (1). There are several model-specific saliency mapping methods that we excluded from our survey.
+
+Following is the full set of claims. All claims are for Atari games. The Reason column represents whether an explanation for agent behavior was provided (Y/N) and the Exp column represents whether an experiment was conducted to evaluate the explanation (Y/N).
+
+<table><tr><td>Claim</td><td>Game</td><td>Saliency Type</td><td>Reason</td><td>Exp</td></tr><tr><td colspan="3">Greydanus et al. (2017)</td><td></td><td></td></tr><tr><td>“The agent is positioning its own paddle,which allows it to return the ball at a specific angle."</td><td>Pong</td><td>Perturbation</td><td>Y</td><td>N</td></tr><tr><td>"Interestingly, from the saliency we see that the agent attends to very little besides its own paddle: not even the ball."</td><td>Pong</td><td>Perturbation</td><td>N</td><td>N</td></tr><tr><td>“After the agent has executed the kill shot, we see that saliency centers entirely around the ball. This makes sense since at this point neither paddle can alter the outcome and their positions are irrele-</td><td>Pong</td><td>Perturbation</td><td>Y</td><td>N</td></tr><tr><td>vant." “It appears that the deep RL agent is exploiting the deterministic nature of the Pong environment. It has learned that it can obtain a reward with high certainty upon executing a precise series of ac- tions.This insight...gives evidence that the agent</td><td>Pong</td><td>Perturbation</td><td>Y</td><td>N</td></tr><tr><td>is not robust and has overfit to the particular op- ponent." "[The agent] had learned a sophisticated aim- ing strategy during which first the actor and then the critic would ‘track’ a target. Aiming begins when the actor highlights a particular alien in blue..Aiming ends with the agent shooting at the</td><td>Space- Invaders</td><td>Perturbation</td><td>Y</td><td>N</td></tr><tr><td>new target." “The critic highlights the target in anticipation of an upcoming reward."</td><td>Space- Invaders</td><td>PerturbationY</td><td></td><td>N</td></tr><tr><td>“Notice that both actor and critic tend to monitor the area above the ship. This may be useful for de- termining whether the ship is protected from en-</td><td>Space- Invaders</td><td>Perturbation</td><td>Y</td><td>N</td></tr><tr><td>emy fire or has a clear shot at enemies."</td><td></td><td></td><td></td><td></td></tr><tr><td>“We found that the agent enters and exits a ‘tun- neling mode’ over the course of a single frame. Once the tunneling location becomes salient, it re- mains so until the tunnel is finished."</td><td></td><td>BreakoutPerturbationN</td><td></td><td>N</td></tr><tr><td colspan="5">ceive those pixels as input."</td></tr><tr><td colspan="5">Bogdanovic et al. (2015) “The line of cars in the upper right are far away</td></tr><tr><td>and the agent correctly ignores them in favour of focusing on the much more dangerous cars in the lower left."</td><td>Freeway</td><td>Jacobian</td><td>Y</td><td>N</td></tr><tr><td>“Firstly, the agent ignores irrelevant features from mountains， sky， the mileage board and empty grounds,and relies on information from the race track to make decisions. Specifically,the agent</td><td>Yang et al. (2018) Enduro</td><td>Binary Jacobian</td><td>Y</td><td>N</td></tr><tr><td>keeps separate two categories of objects on the race track,i.e., cars and the player." “On the one hand, the agent locates the player and a local area around it for avoiding immediate col- lisions with cars. On the other hand,the agent locates the next potential collision targets at dif-</td><td>Enduro</td><td>Binary Jacobian</td><td>Y</td><td>N</td></tr><tr><td>ferent locations, particularly the remote ones." “Near the completion of the current goal,the agent celebrates in advance. As shown from Fig. 4(d) to Fig. 4(f), the left gaze loses its focus on cars and diverts to the mileage board starting when only 13 cars remain before completion. The pre- vious car tracker now picks up on the important</td><td>Enduro</td><td>Binary Jacobian</td><td>Y</td><td>N</td></tr><tr><td>information that it is close to victory,and fixates on the countdown." “Upon reaching the target, the agent does not re- ceive reward signals until the next day starts.Dur- ing this period the agent learns to output no-op</td><td>Enduro</td><td>Binary Jacobian</td><td>Y</td><td>N</td></tr><tr><td>actions, corresponding to not playing the game." “When slacking happens, the agent considers the flag signs as important and the road not. The com- plete reverse in focus as compared to the normal case explains this shift in policy. The flags out-</td><td>Enduro</td><td>Binary Jacobian</td><td>Y</td><td>N</td></tr><tr><td>weigh the road in importance, since they are signs of absolute zero return." "(Prepping) As it turns out, the agent recognizes that the time is dawn (right before morning when race starts) from the unique colours of the light gray sky and orange mountains, therefore the</td><td>Enduro</td><td>Binary Jacobian</td><td>Y</td><td>N</td></tr><tr><td>agent gets ready early for a head start in the new race."</td><td></td><td></td><td></td><td></td></tr><tr><td>“When smog partially blocks the forward view, the left gaze loses its focus on cars. It strays off the road into some empty area."</td><td>Enduro</td><td>Binary Jacobian</td><td>Y</td><td>N</td></tr><tr><td>"In Fig. 5(a), the left gaze detects the two ghosts</td><td>Pacman</td><td>Binary Jacobian</td><td>Y</td><td>N</td></tr><tr><td>on the upper-right corner of the map. Therefore, ms pacman,as located by the right gaze,stays in the mid-left section to safely collect dense re- wards." "In Fig._5(b),the left gaze locks in on all three</td><td>Pacman</td><td>Binary</td><td>Y</td><td>N</td></tr><tr><td>vulnerable ghosts in the mid-right section,as ms pacman chases after them." “In Fig. 5(c)， the left gaze detects a newly- appeared cherry at the lower-left warp tunnel en-</td><td>Pacman</td><td>Jacobian Binary Jacobian</td><td>Y</td><td>N</td></tr><tr><td>trance.Ms pacman immediately enters the closest opposite tunnel entrance in the shortest path to the cherry." “In Fig. 5(d), the right gaze locates ms pacman</td><td>Pacman</td><td>Binary</td><td>Y</td><td>N</td></tr><tr><td>entering the upper-right tunnel. In this case, the left gaze no longer detects a moving object, but predicts the upper-left tunnel as the exiting point." “As shown in Fig. 5(g), the left gaze locates the last pellet when ms pacman is in the mid-section</td><td>Pacman</td><td>Jacobian Binary</td><td>Y</td><td>N</td></tr><tr><td>of the maze. Therefore ms pacman moves towards the pellet." “In Fig. 5(h), a red ghost appears in the left gaze</td><td>Pacman</td><td>Jacobian Binary</td><td>Y</td><td>N</td></tr><tr><td>close to the pellet, causing ms pacman to deviate to the right.” “After changing course, the ghosts approach ms pacman from all directions as shown in Fig. 5(i).</td><td>Pacman</td><td>Jacobian Binary Jacobian</td><td>Y</td><td>N</td></tr><tr><td>Even though the agent detects all the ghosts (they appear in the gazes), ms pacman has no route to escape.” “The left gaze often focuses on white ice blocks</td><td></td><td></td><td></td><td></td></tr><tr><td>that are the destinations of jumping.” “Fig. 6(b) shows the sub-task of the player enter-</td><td>Frostbite Frostbite</td><td>Binary Jacobian Binary</td><td>N Y</td><td>N N</td></tr><tr><td>ing the igloo,after jumping over white ice blocks for building it. The player must avoid the bear when running for the igloo.” “As it shows, the igloo is two jumps away from</td><td></td><td>Jacobian</td><td></td><td></td></tr><tr><td>completion,and the left gaze focuses on the igloo in advance for preparing to enter." Annasamy &amp; Sycara (2019)</td><td>Frostbite</td><td>Binary Jacobian</td><td>Y</td><td>N</td></tr><tr><td>“For example,in MsPacman, since visualizations suggest that the agent may be memorizing pac- mans positions (also maybe ghosts and other ob- jects),we simply add an extra pellet adjacent to a trajectory seen during training (Figure 7a). The agent does not clear the additional pellet and sim- ply continues to execute actions performed during training (Figure 7b).”</td><td>Pacman</td><td>Object</td><td>Y</td><td>Y</td></tr><tr><td>“Similarly, in case of SpaceInvaders, the agent has a strong bias towards shooting from the leftmost- end (seen in Figure 4). This helps in clearing the triangle like shape and moving to the next level (Figure 7d). However,when triangular positions of spaceships are inverted, the agent repeats the same strategy of trying to shoot from left and fails to clear ships (Figure 7c).”</td><td>Space- Invaders</td><td>Object</td><td>Y</td><td>Y</td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5">Goel et al. (2018) “In our Breakout results,the network learns toBreakout</td></tr><tr><td>split the paddle into a left and right side,and does not move the middle portion." “On Beam Rider, our network achieves low loss after learning to segment the games light beams,</td><td>Beam</td><td>Object Object</td><td>Y Y</td><td>N N</td></tr><tr><td>however these beams are purely visual effects and are unimportant for action selection." “Consequently， the game enemies，which are much smaller in size,are ignored by the network, and the resulting learned representation is not that</td><td>Rider Beam Rider</td><td> Object</td><td>N</td><td>N</td></tr><tr><td>useful for a reinforcement learning agent." “There is a very notable difference between the policy saliency between the two models,where</td><td>van der Wal et al. (2018)</td><td>CarRacing Perturbation</td><td>Y</td><td>N</td></tr><tr><td>the former one only pays limited attention to the road and almost no attention to the engine indica- tor,the opposite from fA3C-LSTM. Explicitly,it means masking any regions from the input does not cause much perturbation to the policy when trained with continuous space as targets,likely be- cause the real consequence from a small change in action, e.g. no braking (a3= O) versus braking (a3= O.3),can be very substantial but numerically too subtle for the network to capture during opti- mization on the continuous spectrum." Rupprecht et al. (2018)</td><td></td><td></td><td></td><td></td></tr><tr><td>“Analyzing the visualizations on Seaquest, weSeaQuestPerturbationY make an interesting observation. When maximiz- ing the Q-value for the actions,in many samples we see a low or very low oxygen meter. In these cases the submarine would need to ascend to the surface to avoid suffocation. Although the up ac- tion is the only sensible choice in this case,we also obtain visualized low oxygen states for all other actions.This implies that the agent has not understood the importance of resurfacing when the oxygen is low. We then run several roll outs</td><td></td><td></td><td></td><td>Y</td></tr><tr><td>mies.” “The most dominant pattern we observe is that the model learns to attend to task-relevant things in the scene. In most ATARI games that usually means that the player is one of the foci of atten- tion,as well as enemies,power-ups and the score itself (which is an important factor in the calculat- ing the value function)."</td><td>Mott et al. (2019) SeaQuest</td><td>Attention</td><td>N</td><td>N</td></tr><tr><td colspan="5">Published asa conference paperatICLR 2020</td></tr><tr><td>“Figure 4 shows a examples of this in Ms Pac- man and Alien in the both games the model scans through possible paths,making sure there are no enemies or ghosts ahead. We observe that when it does see a ghost,another path is produced or</td><td>Pacman</td><td>Attention</td><td>Y</td><td>N</td></tr><tr><td>executed in order to avoid it." “In many games we observe that the agent learns to place trip-wires at strategic points in space such that if a game object crosses them a specific ac- tion is taken.For example,in Space Invaders two such trip wires are following the player ship on both sides such that if a bullet crosses one of them the agent immediately evades them by moving to-</td><td>Space In-Attention vaders</td><td></td><td>Y</td><td>N</td></tr><tr><td>wards the opposite direction." “Another example is Breakout where we can see it working in two stages.First the attention is spread out around the general area of the ball, then fo- cuses into a localized line.Once the ball crosses that line the agent moves towards the ball."</td><td>Breakout</td><td>tAttention</td><td>Y</td><td>N</td></tr><tr><td>“As can be seen, the system uses the two modes to make its decisions, some of the heads are con- tent specific looking for opponent cars. Some are mixed, scanning the horizon for incoming cars and when found, tracking them,and some are lo-</td><td>Enduro</td><td>Attention</td><td>Y</td><td>N</td></tr><tr><td>cation based queries,scanning the area right in front of the player for anything the crosses its path (a trip-wire which moves with the player). “Comparing the attention agent to the baseline agent,we see that the attention agent is sensitive to more focused areas along the possible future trajectory. The baseline agent is more focused on the area immediately in front of the player (for the policy saliency) and on the score,while the atten- tion agent focuses more specifically on the path the agent will follow (for the policy） and on pos-</td><td>Pacman</td><td>Perturbation</td><td>Y</td><td>N</td></tr><tr><td colspan="5">sible future longer term paths (for the value)." Nikulin et al. (2019) “Figs.4(a)-(b) show Dense FLS digging a tunnel</td></tr><tr><td rowspan="2">through blocks in Breakout. The model focuses its attention on the end of the tunnel as soon as it is complete, suggesting that it sees shooting the ball through the tunnel as a good strategy." "Figs. 4(c)-(d) depict the same concept of tun- neling per-formed by theSparse FLSmodel. Note how it focuses attention on the upper part of the screen after destroying multiple bricks from the top. This attention does not go away after the ball moves elsewhere (not shown in the images).We speculate that this is how the agent models tun- neling:rather than having a high-level concept of digging a tunnel,it simply strikes wherever it has managed to strike already."</td><td>Breakout Breakout</td><td>tAttention tAttention</td><td>Y</td><td>N</td></tr><tr><td></td><td></td><td>Y</td><td>N</td></tr><tr><td>“Figs. 4(e)-(f) illustrate how the Dense FLSSeaquestAttention model playing Seaquest has learned to attend to in-game objects and, importantly, the oxygen bar at the bottom of the screen. As the oxygen bar is nearing depletion,attention focuses around it,and the submarine reacts by rising to refill its air sup- ply.”</td><td></td><td></td><td>Y</td><td>N</td></tr><tr><td>“Figs.4(g)-(h) are two consecutive frames where an agent detects a target appearing from the left side of the screen.The bottom part of the screen- shots shows how attention in the bottom left cor- ner lights up as soon as a tiny part of the target, only a few pixels wide,appears from the left edge of the screen. In the next frame, the agent will turn left and shoot the target (not shown here).How- ever, the agent completely ignores targets in the top part of the screen,and its attention does not move as they move (also not shown)."</td><td>BreakoutAttention</td><td></td><td>Y</td><td>N</td></tr><tr><td>“The value stream learns to pay attention to theEnduro road.The advantage stream learns to pay attention only when there are cars immediately in front, so as to avoid collisions."</td><td>Wang et al. (2016)</td><td>Jacobian</td><td>Y</td><td>N</td></tr></table>
+
+# D GENERALIZATION OF CAUSAL GRAPHICAL MODEL
+
+The causal graphical model in Figure 2a can be extended to different domains in RL where the input to the model is an image. This requires extending game state to include the underlying state variables. Figure 7 shows the extension for Breakout using an A2C model. Note, game state includes the underlying state variables for Breakout and logits were split into action logits and value as outputted by A2C.
+
+![](images/aba634d387e463bf19071cad79dae41ef1ef8d57cc66283ab483ea28a3925087.jpg)  
+Figure 7: Causal graphical model for Breakout.
+
+<table><tr><td></td><td colspan="2">Object</td><td colspan="2">Perturbation</td><td colspan="2">Jacobian</td></tr><tr><td>Intervention</td><td>r</td><td>p</td><td>r</td><td>p</td><td>r</td><td>p</td></tr><tr><td>intermittent_reset</td><td>0.274</td><td>1.09e-18</td><td>-0.10</td><td>1.52e-3</td><td>0.149</td><td>2.26e-6</td></tr><tr><td>random_varying</td><td>0.142</td><td>6.95e-6</td><td>-0.02</td><td>0.49</td><td>0.281</td><td>1.21e-19</td></tr><tr><td>fixed</td><td>0.041</td><td>0.20</td><td>-0.02</td><td>0.44</td><td>0.286</td><td>2.51e-20</td></tr><tr><td>decremented</td><td>0.119</td><td>0.15e-3</td><td>-0.09</td><td>6.73e-3</td><td>0.261</td><td>5.31e-17</td></tr></table>
+
+Table 6: Numeric results representing Pearson’s correlation coefficient and $p$ -value for the differences in reward and saliency (object, perturbation and Jacobian) from the original trajectory for each intervention. Results show small correlation coefficients $( r )$ suggesting that there is a weak relationship between the differences in reward and saliency for the interventions.
+
+![](images/ae97a6aad613cec873d366d70b765ce2ba41fc288b0c5017b0abda514483532a.jpg)  
+Figure 8: Interventions on displayed score in Amidar (legend in (b) applies to all figures). (a) reward over time for different interventions on displayed score; (b) perturbation saliency on displayed score over time; (c) correlation between the differences in reward and perturbation saliency from the original trajectory. Interventions on displayed score result in differing levels of degraded performance but produce similar saliency maps, suggesting that agent behavior as measured by rewards is underdetermined by salience.
+
+# E EVALUATION OF HYPOTHESES ON AGENT BEHAVIOR
+
+# E.1 CASE STUDY 2: AMIDAR SCORE
+
+We further evaluated the effects of interventions on displayed score (Section 5) in Amidar on perturbation and Jacobian saliency maps. These results are presented in Figures 8 and 9, respectively.
+
+We also evaluated Pearson’s correlation between the differences in reward and saliency with the original trajectory for all three methods (see Table 6). The results support the correlation plots in Figures 4c, 8c and ${ 9 \mathrm { c } }$ .
+
+# E.2 CASE STUDY 3: AMIDAR ENEMY DISTANCE
+
+We further evaluated the relationship between player-enemy distance and saliency (Section 5) in Amidar on perturbation and Jacobian saliency maps. These results are presented in Figures 10 and 11, respectively. Jacobian saliency performed the worst for the intervention-based experiment, suggesting that there is no impact of player-enemy distance on saliency.
+
+Regression analysis between distance and perturbation and Jacobian saliency can be found in Tables 7 and 8 respectively. The results support the lack of correlation between the observational and interventional distributions. Note, enemy 1 is more salient throughout the game compared to the other four enemies resulting in a larger interventional sample size.
+
+![](images/b3514d89d1faedc81eea3bd36c3516d4d6b005bea776bf64f2d109a8dda6f1fb.jpg)  
+Figure 9: Interventions on displayed score in Amidar for Jacobian saliency (legend in (b) applies to all figures). (a) reward over time for different interventions on displayed score; (b) saliency on displayed score over time; (c) correlation between the differences in reward and saliency from the original trajectory. Interventions on displayed score result in differing levels of degraded performance but produce similar saliency maps, suggesting that agent behavior as measured by rewards is underdetermined by salience.
+
+![](images/0486764197abcf9e3c235a8e333ecd6275594f3493d737ebeb36ee804d5b818c.jpg)  
+Figure 10: Interventions on enemy location in Amidar (legend in (b) applies to all figures). (a) the distance-to-player of each enemy in Amidar, observed over time, where saliency intensity is represented by the shaded region around each line; (b) the distance-to-player and saliency, with linear regressions, observed for each enemy; (c) variation in enemy saliency when enemy position is varied by intervention. The plots suggest that there is no substantial correlation and no causal dependence between distance-to-player and perturbation saliency.
+
+![](images/11a9f28eebef4863d9cc91f516bc90a31e41deb7da10b6639c3ec2fa1fb833ef.jpg)  
+Figure 11: Interventions on enemy location in Amidar (legend in (b) applies to all figures). (a) the distance-to-player of each enemy in Amidar, observed over time, where saliency intensity is represented by the shaded region around each line; (b) the distance-to-player and saliency, with linear regressions, observed for each enemy; (c) variation in enemy saliency when enemy position is varied by intervention. The plots suggest that there is no substantial correlation and no causal dependence between distance-to-player and Jacobian saliency.
+
+<table><tr><td></td><td colspan="3">Observational</td><td colspan="3">Interventional</td></tr><tr><td>Enemy</td><td>Slope</td><td>p-value</td><td>r</td><td>Slope</td><td>p-value</td><td>r</td></tr><tr><td>1</td><td>-0.052</td><td>7.04e-10</td><td>-0.204</td><td>-0.374</td><td>0.01</td><td>-0.126</td></tr><tr><td>2</td><td>-0.043</td><td>1.09e-9</td><td>-0.191</td><td>0.536</td><td>0.16</td><td>0.132</td></tr><tr><td>3</td><td>-0.065</td><td>6.04e-20</td><td>-0.283</td><td>-0.039</td><td>0.92</td><td>-0.008</td></tr><tr><td>4</td><td>-0.103</td><td>2.57e-25</td><td>-0.320</td><td>-0.005</td><td>0.98</td><td>0.003</td></tr><tr><td>5</td><td>-0.062</td><td>3.44e-12</td><td>-0.217</td><td>-0.083</td><td>0.71</td><td>0.032</td></tr></table>
+
+Table 7: Numeric results from regression analysis for the observational and interventional results in Figures 10b and c. The results indicate a very small strength of effect (slope) for both observational and interventional data and a small correlation coefficient $( r )$ , suggesting that there is, at best, only a very weak causal dependence of saliency on distance-to-player.
+
+<table><tr><td></td><td colspan="3">Observational</td><td colspan="3">Interventional</td></tr><tr><td>Enemy</td><td>Slope</td><td> p-value</td><td>r</td><td>Slope</td><td>p-value</td><td>r</td></tr><tr><td>1</td><td>-0.008</td><td>0.08</td><td>-0.055</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td>-0.029</td><td>4.17e-6</td><td>-0.145</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td>-0.031</td><td>3.57e-4</td><td>-0.113</td><td>0</td><td>0</td><td>0</td></tr><tr><td>２34</td><td>-0.023</td><td>5.97e-3</td><td>-0.087</td><td>0</td><td>0</td><td>0</td></tr><tr><td>5</td><td>-0.013</td><td>0.02</td><td>-0.076</td><td>0</td><td>0</td><td>0</td></tr></table>
+
+Table 8: Numeric results from regression analysis for the observational and interventional results in Figures 11b and c. The results indicate a very small strength of effect (slope) for both observational and interventional data and a small correlation coefficient $( r )$ , suggesting that there is, at best, only a very weak causal dependence of saliency on distance-to-player.