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+# UNDERSTANDING SHORT-HORIZON BIAS IN STOCHASTIC META-OPTIMIZATION
+
+Yuhuai $\mathbf { W u } ^ { * }$ , Mengye $\mathbf { R e n } ^ { * }$ , Renjie Liao & Roger B. Grosse University of Toronto and Vector Institute {ywu, mren, rjliao, rgrosse}@cs.toronto.edu
+
+# ABSTRACT
+
+Careful tuning of the learning rate, or even schedules thereof, can be crucial to effective neural net training. There has been much recent interest in gradient-based meta-optimization, where one tunes hyperparameters, or even learns an optimizer, in order to minimize the expected loss when the training procedure is unrolled. But because the training procedure must be unrolled thousands of times, the metaobjective must be defined with an orders-of-magnitude shorter time horizon than is typical for neural net training. We show that such short-horizon meta-objectives cause a serious bias towards small step sizes, an effect we term short-horizon bias. We introduce a toy problem, a noisy quadratic cost function, on which we analyze short-horizon bias by deriving and comparing the optimal schedules for short and long time horizons. We then run meta-optimization experiments (both offline and online) on standard benchmark datasets, showing that meta-optimization chooses too small a learning rate by multiple orders of magnitude, even when run with a moderately long time horizon (100 steps) typical of work in the area. We believe short-horizon bias is a fundamental problem that needs to be addressed if metaoptimization is to scale to practical neural net training regimes.
+
+# 1 INTRODUCTION
+
+The learning rate is one of the most important and frustrating hyperparameters to tune in deep learning. Too small a value causes slow progress, while too large a value causes fluctuations or even divergence. While a fixed learning rate often works well for simpler problems, good performance on the ImageNet (Russakovsky et al., 2015) benchmark requires a carefully tuned schedule. A variety of decay schedules have been proposed for different architectures, including polynomial, exponential, staircase, etc. Learning rate decay is also required to achieve convergence guarantee for stochastic gradient methods under certain conditions (Bottou, 1998). Clever learning rate heuristics have resulted in large improvements in training efficiency (Goyal et al., 2017; Smith, 2017). A related hyperparameter is momentum; typically fixed to a reasonable value such as 0.9, careful tuning can also give significant performance gains (Sutskever et al., 2013). While optimizers such as Adam (Kingma & Ba, 2015) are often described as adapting coordinate-specific learning rates, in fact they also have global learning rate and momentum hyperparameters analogously to SGD, and tuning at least the learning rate can be important to good performance.
+
+In light of this, it is not surprising that there have been many attempts to adapt learning rates, either online during optimization (Schraudolph, 1999; Schaul et al., 2013), or offline by fitting a learning rate schedule (Maclaurin et al., 2015). More ambitiously, others have attempted to learn an optimizer (Andrychowicz et al., 2016; Li & Malik, 2017; Finn et al., 2017; Lv et al., 2017; Wichrowska et al., 2017; Metz et al., 2017). All of these approaches are forms of meta-optimization, where one defines a meta-objective (typically the expected loss after some number of optimization steps) and tunes the hyperparameters to minimize this meta-objective. But because gradient-based meta-optimization can require thousands of updates, each of which unrolls the entire base-level optimization procedure, the meta-optimization is thousands of times more expensive than the baselevel optimization. Therefore, the meta-objective must be defined with a much smaller time horizon (e.g. hundreds of updates) than we are ordinarily interested in for large-scale optimization. The hope is that the learned hyperparameters or optimizer will generalize well to much longer time horizons. Unfortunately, we show that this is not achieved in this paper. This is because of a strong tradeoff between short-term and long-term performance, which we refer to as short-horizon bias.
+
+In this work, we investigate the short-horizon bias both mathematically and empirically. First, we analyze a quadratic cost function with noisy gradients based on Schaul et al. (2013). We consider this a good proxy for neural net training because secondorder optimization algorithms have been shown to train neural networks in orders-of-magnitude fewer iterations (Martens, 2010), suggesting that much of the difficulty of SGD training can be explained by quadratic approximations to the cost. In our noisy quadratic problem, the dynamics of SGD with momentum can be analyzed exactly, allowing us to derive the greedy-optimal (i.e. 1-step horizon) learning rate and momentum in closed form, as well as to (locally) minimize the long-horizon loss using gradient descent. We analyze the differences between the short-horizon and long-horizon schedules.
+
+$$
+\mathcal { L } ( x , y ) = \textstyle { \frac { 1 } { 2 } } ( x ^ { 2 } + 1 0 0 y ^ { 2 } ) + \sigma ^ { 2 }
+$$
+
+![](images/ef23600a52c16eb77319be8389bad68d05bab5245212bf0c0bd832fba43c80f4.jpg)  
+Figure 1: Aggressive learning rate (red) followed by a decay schedule (yellow) wins over conservative learning rate (blue) by making more progress along the low curvature direction $x$ direction).
+
+Interestingly, when the noisy quadratic problem is either deterministic or spherical, greedy schedules are optimal. However, when the problem is both stochastic and badly conditioned (as is most neural net training), the greedy schedules decay the learning rate far too quickly, leading to slow convergence towards the optimum. This is because reducing the learning rate dampens the fluctuations along high curvature directions, giving it a large immediate reduction in loss. But this comes at the expense of long-run performance, because the optimizer fails to make progress along low curvature directions. This phenomenon is illustrated in Figure 1, a noisy quadratic problem in 2 dimensions, in which two learning rate schedule are compared: a small fixed learning rate (blue), versus a larger fixed learning rate (red) followed by exponential decay (yellow). The latter schedule initially has higher loss, but it makes more progress towards the optimum, such that it achieves an even smaller loss once the learning rate is decayed.
+
+Figure 2 shows this effect quantitatively for a noisy quadratic problem in 1000 dimensions (defined in Section 2.3). The solid lines show the loss after various numbers of steps of lookahead with a fixed learning rate; if this is used as the meta-objective, it favors small learning rates. The dashed curves show the loss if the same trajectories are followed by 50 steps with an exponentially decayed learning rate; these curves favor higher learning rates, and bear little obvious relationship to the solid ones. This illustrates the difficulty of selecting learning rates based on short-horizon information.
+
+The second part of our paper empirically investigates gradientbased meta-optimization for neural net training. We consider two idealized meta-optimization algorithms: an offline algorithm which fits a learning rate decay schedule by running optimization many times from scratch, and an online algorithm which adapts the learning rate during training. Since our interest is in studying the effect of the meta-objective itself rather than failures of meta-optimization, we give the metaoptimizers sufficient time to optimize their meta-objectives well. We show that short-horizon meta-optimizers, both online and offline, dramatically underperform a hand-tuned fixed learning rate, and sometimes cause the base-level optimization progress to slow to a crawl, even with moderately long time horizons (e.g. 100 or 1000 steps) similar to those used in prior work on gradient-based meta-optimization.
+
+In short, we expect that any meta-objective which does not correct for short-horizon bias will probably fail when run for a much longer time horizon than it was trained on. There are applications where short-horizon meta-optimization is directly useful, such as few-shot learning (Santoro et al., 2016; Ravi & Larochelle, 2017). In those settings, short-horizon bias is by definition not an issue. But much of the appeal of meta-optimization comes from the possibility of using it to speed up or simplify the training of large neural networks. In such settings, short-horizon bias is a fundamental obstacle that must be addressed for meta-optimization to be practically useful.
+
+![](images/356eca80c73a272154068fcac031704a35b7c72926af13d7a9ccc38e892b74ad.jpg)  
+LR α Figure 2: Short-horizon metaobjectives for the noisy quadratic problem. Solid: loss after $k$ updates with fixed learning rate. Dashed: loss after $k$ updates with fixed learning rate, followed by exponential decay.
+
+# 2 NOISY QUADRATIC PROBLEM
+
+In this section, we consider a toy problem which demonstrates the short-horizon bias and can be analyzed analytically. In particular, we borrow the noisy quadratic model of Schaul et al. (2013); the true function being optimized is a quadratic, but in each iteration we observe a noisy version with the correct curvature but a perturbed minimum. This can be equivalently viewed as noisy observations of the gradient, which are intended to capture the stochasticity of a mini-batch-based optimizer. We analyze the dynamics of SGD with momentum on this example, and compare the long-horizon-optimized and greedy-optimal learning rate schedules.
+
+# 2.1 BACKGROUND
+
+Approximating the cost surface of a neural network with a quadratic function has led to powerful insights and algorithms. Second-order optimization methods such as Newton-Raphson and natural gradient (Amari, 1998) iteratively minimize a quadratic approximation to the cost function. Hessianfree (H-F) optimization (Martens, 2010) is an approximate natural gradient method which tries to minimize a quadratic approximation using conjugate gradient. It can often fit deep neural networks in orders-of-magnitude fewer updates than SGD, suggesting that much of the difficulty of neural net optimization is captured by quadratic models. In the setting of Bayesian neural networks, quadratic approximations to the log-likelihood motivated the Laplace approximation (MacKay, 1992) and variational inference (Graves, 2011; Zhang et al., 2017). Koh & Liang (2017) used quadratic approximations to analyze the sensitivity of a neural network’s predictions to particular training labels, thereby yielding insight into adversarial examples.
+
+Such quadratic approximations to the cost function have also provided insights into learning rate and momentum adaptation. In a deterministic setting, under certain conditions, second-order optimization algorithms can be run with a learning rate of 1; for this reason, H-F was able to eliminate the need to tune learning rate or momentum hyperparameters. Martens & Grosse (2015) observed that for a deterministic quadratic cost function, greedily choosing the learning rate and momentum to minimize the error on the next step is equivalent to conjugate gradient (CG). Since CG achieves the minimum possible loss of any gradient-based optimizer on each iteration, the greedily chosen learning rates and momenta are optimal, in the sense that the greedy sequence achieves the minimum possible loss value of any sequence of learning rates and momenta. This property fails to hold in the stochastic setting, however, and as we show in this section, the greedy choice of learning rate and momentum can do considerably worse than optimal.
+
+Our primary interest in this work is to adapt scalar learning rate and momentum hyperparameters shared across all dimensions. Some optimizers based on diagonal curvature approximations (Kingma & Ba, 2015) have been motivated in terms of adapting dimension-specific learning rates, but in practice, one still needs to tune scalar learning rate and momentum hyperparameters. Even K-FAC (Martens & Grosse, 2015), which is based on more powerful curvature approximations, has scalar learning rate and momentum hyperparameters. Our analysis applies to all of these methods since they can be viewed as performing SGD in a preconditioned space.
+
+# 2.2 ANALYSIS
+
+# 2.2.1 NOTATIONS
+
+We will primarily focus on the SGD with momentum algorithm in this paper. The update is written as follows:
+
+$$
+\begin{array} { r l } & { \mathbf { v } ^ { ( t + 1 ) } = \boldsymbol { \mu } ^ { ( t ) } \mathbf { v } ^ { ( t ) } - \alpha ^ { ( t ) } \nabla _ { \pmb { \theta } ^ { ( t ) } } \mathcal { L } , } \\ & { \pmb { \theta } ^ { ( t + 1 ) } = \pmb { \theta } ^ { ( t ) } + \mathbf { v } ^ { ( t + 1 ) } , } \end{array}
+$$
+
+where $\mathcal { L }$ is the loss function, $t$ is the training step, and $\alpha ^ { ( t ) }$ is the learning rate. We call the gradient trace $\mathbf { v } ^ { ( t ) }$ “velocity”, and its decay constant $\mu ^ { ( t ) }$ “momentum”. We denote the ith coordinate of a vector $\mathbf { v }$ as $v _ { i }$ . When we focus on a single dimension, we sometimes drop the dimension subscripts. We also denote $A ( \cdot ) = \mathbb { E } [ \cdot ] ^ { 2 } + \mathbb { V } [ \cdot ]$ , where $\mathbb { E }$ and $\mathbb { V }$ denote expectation and variance respectively.
+
+# 2.2.2 PROBLEM FORMULATION
+
+We now define the noisy quadratic model, where in each iteration, the optimizer is given the gradient for a noisy version of a quadratic cost function, where the curvature is correct but the minimum is sampled stochastically from a Gaussian distribution. We assume WLOG that the Hessian is diagonal because SGD is a rotation invariant algorithm, and therefore the dynamics can be analyzed in a coordinate system corresponding to the eigenvectors of the Hessian. We make the further (nontrivial) assumption that the noise covariance is also diagonal.1 Mathematically, the stochastic cost function is written as:
+
+$$
+\hat { \mathcal { L } } ( \pmb { \theta } ) = \frac { 1 } { 2 } \sum _ { i } h _ { i } ( \theta _ { i } - c _ { i } ) ^ { 2 } ,
+$$
+
+where c is the stochastic minimum, and each $c _ { i }$ follows a Gaussian distribution with mean $\theta _ { i } ^ { * }$ and variance $\sigma _ { i } ^ { 2 }$ . The expected loss is given by:
+
+$$
+\mathcal { L } ( \pmb { \theta } ) = \mathbb { E } \left[ \hat { \mathcal { L } } ( \pmb { \theta } ) \right] = \frac { 1 } { 2 } \sum _ { i } h _ { i } \left( ( \theta _ { i } - \theta _ { i } ^ { * } ) ^ { 2 } + \sigma _ { i } ^ { 2 } \right) .
+$$
+
+The optimum of $\mathcal { L }$ is given by $\theta ^ { * } = \mathbb { E } [ \mathbf { c } ]$ ; we assume WLOG that $\pmb \theta ^ { * } = \mathbf 0$ . The stochastic gradient is given by $\begin{array} { r } { \frac { \partial \hat { \mathcal { L } } } { \partial \theta _ { i } } = h _ { i } ( \theta _ { i } - c _ { i } ) } \end{array}$ . Since the deterministic gradient is given by $\frac { \partial \mathcal { L } } { \partial \theta _ { i } } = h _ { i } \theta _ { i }$ , the stochastic gradient can be viewed as a noisy Gaussian observation of the deterministic gradient with variance $\bar { h } _ { i } ^ { 2 } \sigma _ { i } ^ { 2 }$ . This interpretation motivates the use of this noisy quadratic problem as a model of SGD dynamics.
+
+We treat the iterate ${ \pmb \theta } ^ { ( t ) }$ as a random variable (where the randomness comes from the sampled c’s); the expected loss in each iteration is given by
+
+$$
+\begin{array} { r l } & { \mathbb { E } \left[ \mathcal { L } ( \pmb { \theta } ^ { ( t ) } ) \right] = \mathbb { E } \left[ \frac { 1 } { 2 } \sum _ { i } h _ { i } \left( ( \pmb { \theta } _ { i } ^ { ( t ) } ) ^ { 2 } + \sigma _ { i } ^ { 2 } \right) \right] } \\ & { \quad \quad \quad = \displaystyle \frac { 1 } { 2 } \sum _ { i } h _ { i } \left( \mathbb { E } \left[ \theta _ { i } ^ { ( t ) } \right] ^ { 2 } + \mathbb { V } \left[ \theta _ { i } ^ { ( t ) } \right] + \sigma _ { i } ^ { 2 } \right) . } \end{array}
+$$
+
+# 2.2.3 OPTIMIZED AND GREEDY-OPTIMAL SCHEDULES
+
+We are interested in adapting a global learning rate $\alpha ^ { ( t ) }$ and a global momentum decay parameter $\mu ^ { ( t ) }$ for each time step $t$ . We first derive a recursive formula for the mean and variance of the iterates at each step, and then analyze the greedy-optimal schedule for $\alpha ^ { ( t ) }$ and $\mu ^ { ( t ) }$ .
+
+Several observations allow us to compactly model the dynamics of SGD with momentum on the noisy quadratic model. First, $\mathbb { E } [ \mathcal { L } ( \pmb { \theta } ^ { ( t ) } ) ]$ can be expressed in terms of $\mathbb { E } [ \theta _ { i } ]$ and $\mathbb { V } [ \theta _ { i } ]$ using Eqn. 5. Second, due to the diagonality of the Hessian and the noise covariance matrix, each coordinate evolves independently of the others. Third, the means and variances of the parameters $\theta _ { i } ^ { ( t ) }$ and the velocity $v _ { i } ^ { ( t ) }$ are functions of those statistics at the previous step.
+
+Because each dimension evolves independently, we now drop the dimension subscripts. Combining these observations, we model the dynamics of SGD with momentum as a deterministic recurrence relation with sufficient statistics $\mathbb { E } [ \theta ^ { ( t ) } ] , \mathbb { E } [ v ^ { ( t ) } ] , \mathbb { V } [ \theta ^ { ( t ) } ] , \mathbb { V } [ v ^ { ( t ) } ] ,$ , and $\Sigma _ { \theta , v } ^ { ( t ) } = \mathrm { C o v } ( \theta ^ { ( t ) } , v ^ { ( t ) } )$ . The dynamics are as follows:
+
+Theorem 1 (Mean and variance dynamics). The expectations of the parameter $\theta$ and the velocity $v$ are updated as,
+
+$$
+\begin{array} { r l } & { \mathbb { E } \left[ v ^ { ( t + 1 ) } \right] = \mu ^ { ( t ) } \mathbb { E } \left[ v ^ { ( t ) } \right] - ( \alpha ^ { ( t ) } h ) \mathbb { E } \left[ \theta ^ { ( t ) } \right] , } \\ & { \mathbb { E } \left[ \theta ^ { ( t + 1 ) } \right] = \mathbb { E } \left[ \theta ^ { ( t ) } \right] + \mathbb { E } \left[ v ^ { ( t + 1 ) } \right] . } \end{array}
+$$
+
+The variances of the parameter $\theta$ and the velocity v are updated as
+
+$$
+\begin{array} { r l } & { \mathbb { V } \left[ v ^ { ( t + 1 ) } \right] = \left( \mu ^ { ( t ) } \right) ^ { 2 } \mathbb { V } \left[ v ^ { ( t ) } \right] + \left( \alpha ^ { ( t ) } h \right) ^ { 2 } \mathbb { V } \left[ \theta ^ { ( t ) } \right] - 2 \mu ^ { ( t ) } \alpha ^ { ( t ) } h \Sigma _ { \theta , v } ^ { ( t ) } + \left( \alpha ^ { ( t ) } h \sigma \right) ^ { 2 } , } \\ & { \mathbb { V } \left[ \theta ^ { ( t + 1 ) } \right] = \left( 1 - 2 \alpha ^ { ( t ) } h \right) \mathbb { V } \left[ \theta ^ { ( t ) } \right] + \mathbb { V } \left[ v ^ { ( t + 1 ) } \right] + 2 \mu ^ { ( t ) } \Sigma _ { \theta , v } ^ { ( t ) } , } \\ & { \qquad \Sigma _ { \theta , v } ^ { ( t + 1 ) } = \mu ^ { ( t ) } \Sigma _ { \theta , v } ^ { ( t ) } - \alpha ^ { ( t ) } h \mathbb { V } \left[ \theta ^ { ( t ) } \right] + \mathbb { V } \left[ v ^ { ( t + 1 ) } \right] . } \end{array}
+$$
+
+By applying Theorem 1 recursively, we can obtain $\mathbb { E } [ { \pmb \theta } ^ { ( t ) } ]$ and $\mathbb { V } [ \pmb \theta ^ { ( t ) } ]$ , and hence $\mathbb { E } [ \mathcal { L } ( \pmb { \theta } ^ { ( t ) } ) ]$ , for every $t$ . Therefore, using gradient-based optimization, we can fit a locally optimal learning rate and momentum schedule, i.e. a sequence of values $\{ ( \alpha ^ { ( t ) } , \mu ^ { ( t ) } ) \} _ { t = 1 } ^ { T }$ which locally minimizes $\mathbb { E } [ \mathcal { L } ( \pmb { \theta } ^ { ( t ) } ) ]$ at some particular time $T$ . We refer to this as the optimized schedule.
+
+Furthermore, there is a closed-form solution for one-step lookahead, i.e., we can solve for the optimal learning rate $\alpha ^ { ( t ) * }$ and momentum $\mu ^ { ( t ) * }$ that minimizes $\mathbb { E } [ \mathcal { L } ( \pmb { \theta } ^ { ( t + 1 ) } ) ]$ given the statistics at time $t$ . We call this as the greedy-optimal schedule.
+
+Theorem 2 (Greedy-optimal learning rate and momentum). The greedy-optimal learning rate and momentum schedule is given by
+
+$$
+\begin{array} { r l } & { \boldsymbol { \alpha } ^ { ( t ) \ast } = \frac { \sum _ { i } h _ { i } ^ { 2 } A \left( \boldsymbol { \theta } _ { i } ^ { ( t ) } \right) \left[ \sum _ { j } h _ { j } A \left( \boldsymbol { v } _ { j } ^ { ( t ) } \right) \right] - \left( \sum _ { j } h _ { j } \mathbb { E } \left[ \boldsymbol { \theta } _ { j } ^ { ( t ) } \boldsymbol { v } _ { j } ^ { ( t ) } \right] \right) h _ { i } ^ { 2 } \mathbb { E } \left[ \boldsymbol { \theta } _ { i } ^ { ( t ) } \boldsymbol { v } _ { i } ^ { ( t ) } \right] } { \sum _ { i } h _ { i } ^ { 3 } \left[ A \left( \boldsymbol { \theta } _ { i } ^ { ( t ) } \right) + \sigma _ { i } ^ { 2 } \right] \left[ \sum _ { j } h _ { j } A \left( \boldsymbol { v } _ { j } ^ { ( t ) } \right) \right] - \left( \sum _ { j } h _ { j } ^ { 2 } \mathbb { E } \left[ \boldsymbol { \theta } _ { j } ^ { ( t ) } \boldsymbol { v } _ { j } ^ { ( t ) } \right] \right) h _ { i } ^ { 2 } \mathbb { E } \left[ \boldsymbol { \theta } _ { i } ^ { ( t ) } \boldsymbol { v } _ { i } ^ { ( t ) } \right] } , } \\ & { \boldsymbol { \mu } ^ { ( t ) \ast } = - \frac { \sum _ { i } h _ { i } \left( 1 - \boldsymbol { \alpha } ^ { ( t ) \ast } h _ { i } \right) \mathbb { E } \left[ \boldsymbol { \theta } _ { i } ^ { ( t ) } \boldsymbol { v } _ { i } ^ { ( t ) } \right] } { \sum _ { i } h _ { i } A \left( \boldsymbol { v } _ { i } ^ { ( t ) } \right) } . } \end{array}
+$$
+
+Note that Schaul et al. (2013) derived the greedy optimal learning rate for SGD, and Theorem 2 extends it to the greedy optimal learning rate and momentum for SGD with momentum.
+
+# 2.2.4 UNIVARIATE AND SPHERICAL CASES
+
+As noted in Section 2.1, Martens & Grosse (2015) found the greedy choice of $\alpha$ and $\mu$ to be optimal for gradient descent on deterministic quadratic objectives. We now show that the greedy schedule is also optimal for SGD without momentum in the case of univariate noisy quadratics, and hence also for multivariate ones with spherical Hessians and gradient covariances. In particular, the following holds for SGD without momentum on a univariate noisy quadratic:
+
+Theorem 3 (Optimal learning rate, univariate). For all $T \in \mathbb { N } ,$ , the sequence of learning rates $\{ \alpha ^ { ( t ) * } \} _ { t = 1 } ^ { T - 1 }$ that minimizes ${ \mathcal { L } } ( \theta ^ { ( T ) } )$ is given by
+
+$$
+\alpha ^ { ( t ) * } = \frac { A \left( \theta ^ { ( t ) } \right) } { h \left( A \left( \theta ^ { ( t ) } \right) + \sigma ^ { 2 } \right) } .
+$$
+
+Moreover, this agrees with the greedy-optimal learning rate schedule as derived by Schaul et al.   
+(2013).
+
+If the Hessian and the gradient covariance are both spherical, then each dimension evolves identically and independently according to the univariate dynamics. Of course, one is unlikely to encounter an optimization problem where both are exactly spherical. But some approximate secondorder optimizers, such as K-FAC, can be viewed as preconditioned SGD, i.e. SGD in a transformed space where the Hessian and the gradient covariance are better conditioned (Martens & Grosse, 2015). In principle, with a good enough preconditioner, the Hessian and the gradient covariance would be close enough to spherical that a greedy choice of $\alpha$ and $\mu$ would perform well. It will be interesting to investigate whether any practical optimization algorithms demonstrate this behavior.
+
+![](images/3dfd16cd7c142d2454bf5c210aaa94ab6f7a07045c6b178fade575e56d1ccb59.jpg)  
+Figure 3: Comparisons of the optimized learning rates and momenta trained by gradient descent (red), greedy learning rates and momenta (blue), and the optimized fixed learning rate and momentum (green) in both noisy (a) and deterministic (b) quadratic settings. In the deterministic case, our optimized schedule matched the greedy one, just as the theory predicts.
+
+# 2.3 EXPERIMENTS
+
+In this section, we compare the optimized and greedy-optimal schedules on a noisy quadratic problem. We chose a 1000 dimensional quadratic cost function with the curvature distribution from Li (2005), on which CG achieves its worst-case convergence rate. We assume that $\begin{array} { r } { h _ { i } = \mathbb { V } [ \frac { \partial \mathcal { L } } { \partial \theta _ { i } } ] } \end{array}$ , and hence $\begin{array} { r } { \sigma _ { i } ^ { 2 } = \frac { 1 } { h _ { i } } } \end{array}$ i; this choice is motivated by the observations that under certain assumptions, the Fisher information matrix is a good approximation to the Hessian matrix, but also reflects the covariance structure of the gradient noise (Martens, 2014). We computed the greedy-optimal schedules using Theorem 3. For the optimized schedules, we minimized the expected loss at time $T \ : = \ : 2 5 0$ using Adam using Adam (Kingma & Ba, 2015), with a learning rate 0.003 and 500 steps. We set an upper bound for the learning rate which prevented the loss component for any dimension from becoming larger than its initial value; this was needed because otherwise the optimized schedule allowed the loss to temporarily grow very large, a pathological solution which would be unstable on realistic problems. We also considered fixed learning rate and momentum, with the two hyperparameters fit using Adam. The training curves and the corresponding learning rates and momenta are shown in Figure 3(a). The optimized schedule achieved a much lower final expected loss value (4.25) than was obtained by the greedy-optimal schedule (63.86) or fixed schedule (42.19).
+
+We also show the sums of the losses along the 50 highest curvature directions and 50 lowest curvature directions. We find that under the optimized schedule, the losses along the high curvature directions hardly decrease initially. However, because it maintains a high learning rate, the losses along the low curvature directions decrease significantly. After 50 iterations, it begins decaying the learning rate, at which point it achieves a large drop in both the high-curvature and total losses. On the other hand, under the greedy-optimal schedule, the learning rates and momenta become small very early on, which immediately reduces the losses on the high curvature directions, and hence also the total loss. However, in the long term, since the learning rates are too small to make substantial progress along the low curvature directions, the total loss converged to a much higher value in the end. This gives valuable insight into the nature of the short-horizon bias in meta-optimization: shorthorizon objectives will often encourage the learning rate and momentum to decay quickly, so as to achieve the largest gain in the short term, but at the expense of long-run performance.
+
+It is interesting to compare this behavior with the deterministic case. We repeated the above experiment for a deterministic quadratic cost function (i.e. $\sigma _ { i } ^ { 2 } ~ = ~ 0 )$ with the same Hessian; results are shown in Figure 3(b). The greedy schedule matches the optimized one, as predicted by the analysis of Martens & Grosse (2015). This result illustrates that stochasticity is necessary for short-horizon bias to manifest. Interestingly, the learning rate and momentum schedules in the deterministic case are nearly flat, while the optimized schedules for the stochastic case are much more complex, suggesting that stochastic optimization raises a different set of issues for hyperparameter adaptation.
+
+# 3 GRADIENT-BASED META-OPTIMIZATION
+
+We now turn our attention to gradient-based hyperparameter optimization. A variety of approaches have been proposed which tune hyperparameters by doing gradient descent on a meta-objective (Schraudolph, 1999; Maclaurin et al., 2015; Andrychowicz et al., 2016). We empirically analyze an idealized version of a gradient-based meta-optimization algorithm called stochastic meta-descent (SMD) (Schraudolph, 1999). Our version of SMD is idealized in two ways: first, we drop the algorithmic tricks used in prior work, and instead allow the meta-optimizer more memory and computation than would be economical in practice. Second, we limit the representational power of our meta-model: whereas Andrychowicz et al. (2016) aimed to learn a full optimization algorithm, we focus on the much simpler problem of adapting learning rate and momentum hyperparameters, or schedules thereof. The aim of these two simplifications is that we would like to do a good enough job of optimizing the meta-objective that any base-level optimization failures can be attributed to deficiencies in the meta-objective itself (such as short-horizon bias) rather than incomplete metaoptimization.
+
+Despite these simplifications, we believe our experiments are relevant to practical meta-optimization algorithms which optimize the meta-objective less thoroughly. Since the goal of the metaoptimizer is to adapt two hyperparameters, it’s possible that poor meta-optimization could cause the hyperparameters to get stuck in regions that happen to perform well; indeed, we observed this phenomenon in some of our early explorations. But it would be dangerous to rely on poor meta-optimization, since improved meta-optimization methods would then lead to worse base-level performance, and tuning the meta-optimizer could become a roundabout way of tuning learning rates and momenta.
+
+We also believe our experiments are relevant to meta-optimization methods which aim to learn entire algorithms. Even if the learned algorithms don’t have explicit learning rate parameters, it’s possible for a learning rate schedule to be encoded into an algorithm itself; for instance, Adagrad (Duchi et al., 2011) implicitly uses a polynomial decay schedule because it sums rather than averages the squared derivatives in the denominator. Hence, one would need to worry about whether the metaoptimizer is implicitly fitting a learning rate schedule that’s optimized for short-term performance.
+
+# 3.1 BACKGROUND: STOCHASTIC META-DESCENT
+
+The high-level idea of stochastic meta-descent (SMD) (Schraudolph, 1999) is to perform gradient descent on the learning rate, or any other differentiable hyperparameters. This is feasible since any gradient based optimization algorithm can be unrolled as a computation graph (see Figure 4), and automatic differentiation is readily available in most deep learning libraries.
+
+There are two basic types of automatic differentiation (autodiff) methods: forward mode and reverse mode. In forward mode autodiff, directional derivatives are computed alongside the forward computation. In contrast, reverse mode autodiff (a.k.a. backpropagation) computes the gradients moving backwards through the computation graph. Meta-optimization using reverse mode can be computationally demanding due to memory constraints, since the parameters need to be stored at every step. Maclaurin et al. (2015) got around this by cleverly exploiting approximate reversibility to minimize the memory cost of activations. Since we are optimizing only two hyperparameters, however, forward mode autodiff can be done cheaply. Here, we provide the forward differentiation equations for obtaining the gradient of vanilla SGD learning rate. Let $\textstyle { \frac { d \pmb { \theta } _ { t } } { d \alpha } }$ be $\mathbf { \pmb { u } } _ { t }$ , and $\textstyle { \frac { d { \mathcal { L } } _ { t } } { d \alpha } }$ be $\alpha ^ { \prime }$ , an d the Hessian at step $t$ to be $H _ { t }$ . By chain rule, we get,
+
+![](images/00416bffcf2dc1cd4e28d89cfc98bc8506229745fceeb8c4d5bfe18b510ad4dc.jpg)  
+Figure 4: Regular SGD in the form of a computation graph. The learning rate parameter $\alpha$ is part of the differentiable computations.
+
+$$
+\begin{array} { r l } & { \alpha ^ { \prime } = \pmb { g } _ { t } \cdot \pmb { u } _ { t - 1 } , } \\ & { \pmb { u } _ { t } = \pmb { u } _ { t - 1 } - \pmb { g } _ { t } - \alpha H _ { t } \pmb { u } _ { t - 1 } . } \end{array}
+$$
+
+While the Hessian is infeasible to construct explicitly, the Hessian-vector product in Equation 9 can be computed efficiently using reverse-on-reverse (Werbos, 1988) or forward-on-reverse automatic differentiation (Pearlmutter, 1994), in time linear in the cost of the forward pass. See Schraudolph (2002) for more details.
+
+Using the gradients with respect to hyperparameters, as given in Eq. 9, we can apply gradient based meta-optimization, just like optimizing regular parameters. It is worth noting that, although SMD was originally proposed for optimizing vanilla SGD, in practice it can be applied to other optimization algorithms such as SGD with momentum or Adam (Kingma & Ba, 2015). Moreover, gradient-based optimizers other than SGD can be used for the meta-optimization as well.
+
+# Algorithm 1: Stochastic Meta-Descent
+
+The basic SMD algorithm is given as Algorithm 1. Here, $\alpha$ is a set of hyperparameters (e.g. learning rate), and $\alpha _ { 0 }$ are inital hyperparameter values; $\pmb { \theta }$ is a set of optimization intermediate variables, such as weights and velocities; $\eta$ is a set of metaoptimizer hyperparameters (e.g. meta learning rate). $\mathtt { B G r a d } ( y , x , d y )$ is the backward gradient function that computes the gradients
+
+# return $\alpha$
+
+of the loss function wrt. $\pmb \theta$ , and $\mathtt { F G r a d } ( y , x , d x )$ is the forward gradient function that accumulates the gradients of $\pmb \theta$ with respect to $\alpha$ . Step and MetaStep optimize regular parameters and hyperparameters, respectively, for one step using gradient-based methods. Additionally, $T$ is the lookahead window size, and $M$ is the number of meta updates.
+
+Simplifications from the original SMD algorithm. The original SMD algorithm (Schraudolph, 1999) fit coordinate-wise adaptive learning rates with intermediate gradients $( { \pmb u } _ { t } )$ accumulated throughout the process of training. Since computing separate directional derivatives for each coordinate using forward mode autodiff is computationally prohibitive, the algorithm used approximate updates. Both features introduced bias into the meta-gradients. We make several changes to the original algorithm. First, we tune only a global learning rate parameter. Second, we use exact forward mode accumulation because this is feasible for a single learning rate. Third, rather than accumulate directional derivatives during training, we compute the meta-updates on separate SGD trajectories simulated using fixed network parameters. Finally, we compute multiple meta-updates in order to ensure that the meta-objective is optimized sufficiently well. Together, these changes ensure unbiased meta-gradients, as well as careful optimization of the meta-objective, at the cost of high computational overhead. We do not recommend this approach as a practical SMD implementation, but rather as a way of understanding the biases in the meta-objective itself.
+
+![](images/290ddbc88701dfb2af3565c1e58b80be435c03c40b0541f619d7daeb5a6c2158.jpg)  
+Figure 5: Meta-objective surfaces and SMD trajectories (red) optimizing initial effective learning rate and decay exponent with horizons of {100, 1k, 5k, 20k} steps2. $2 . 5 \mathrm { k }$ random samples with Gaussian interpolation are used to illustrate the meta-objective surface.
+
+# 3.2 OFFLINE META-OPTIMIZATION
+
+To understand the sensitivity of the optimized hyperparameters to the horizon, we first carried out an offline experiment on a multi-layered perceptron (MLP) on MNIST (LeCun et al., 1998). Specifically, we fit learning rate decay schedules offline by repeatedly training the network, and a single meta-gradient was obtained from each training run.
+
+Learnable decay schedule. We used a parametric learning rate decay schedule known as inverse time decay (Welling & Teh, 2011): $\begin{array} { r } { \alpha _ { t } = \frac { \alpha _ { 0 } } { ( 1 + \frac { t } { K } ) ^ { \beta } } } \end{array}$ , where $\alpha _ { 0 }$ is the initial learning rate, $t$ is the number of training steps, $\beta$ is the learning rate decay exponent, and $K$ is the time constant. We jointly optimized $\alpha _ { 0 }$ and $\beta$ . We fixed $\mu = 0 . 9$ , $K = 5 0 0 0$ for simplicity.
+
+Experimental details. The network had two layers of 100 hidden units, with ReLU activations. Weights were initialized with a zero-mean Gaussian with standard deviation 0.1. We used a warm start from a network trained for 50 SGD with momentum steps, using $\alpha = 0 . 1 , \mu = 0 . 9$ . (We used a warm start because the dynamics are generally different at the very start of training.) For SMD optimization, we trained all hyperparameters in log space using Adam optimizer, with $5 \mathrm { k }$ meta steps.
+
+Figure 5 shows SMD optimization trajectories on the meta-objective surfaces, initialized with multiple random hyperparameter settings. The SMD trajectories appear to have converged to the global optimum.
+
+Importantly, the meta-objectives with longer horizons favored a much smaller learning rate decay exponent $\beta$ , leading to a more gradual decay schedule. The meta-objective surfaces were very different depending on the time horizon, and the final $\beta$ value differed by over two orders of magnitude between 100 and $2 0 \mathrm { k }$ step horizons.
+
+![](images/23429c32c7ad1ed0d1efe804496906bc94414cb75492aeaf508ed6a1432818c4.jpg)  
+Figure 6: Training curves with best learning rate schedules from meta-objective surfaces with {100, 1k, 5k, 20k} step horizons.
+
+We picked the best learning rate schedules from meta-objective surfaces (in Figure 5), and obtained the training curves of a network shown in Figure 6. The resulting training loss at 20k steps with the 100 step horizon was over three orders of magnitude larger than with the $2 0 \mathrm { k }$ step horizon. In general, short horizons gave better performance initially, but were surpassed by longer horizons. The differences in error were less drastic, but we see that the 100 step network was severely undertrained, and the 1k step network achieved noticeably worse test error than the longer-horizon ones.
+
+![](images/acc78549c8c59186928758fff7d00b9f7dda41370890589c0129d8434760d926.jpg)  
+Figure 7: Training curves and learning rates from online SMD with lookahead of 5 steps (blue), and hand-tuned fixed learning rate (red). Each blue curve corresponds to a different initial learning rate.
+
+# 3.3 ONLINE META-OPTIMIZATION
+
+In this section, we study whether online adaptation also suffers from short-horizon bias. Specifically, we used Algorithm 1) to adapt the learning rate and momentum hyperparameters online while a network is trained. We experimented with an MLP on MNIST and a CNN on CIFAR10 (Krizhevsky, 2009).
+
+Experimental details. For the MNIST experiments, we used an MLP network with two hidden layers of 100 units, with ReLU activations. Weights were initialized with a zero-mean Gaussian with standard deviation 0.1. For CIFAR-10 experiments, we used a CNN network adapted from Caffe (Jia et al., 2014), with 3 convolutional layers of filter size $3 \times 3$ and depth [32, 32, 64], and $2 \times 2$ max pooling with stride 2 after every convolution layer, and follwed by a fully connected hidden layer of 100 units. Meta-optimization was done with 100 steps of Adam for every 10 steps of regular training. We adapted the learning rate $\alpha$ and momentum $\mu$ . After $2 5 \mathrm { k }$ steps, adaptation was stopped, and we trained for another $2 5 \mathrm { k }$ steps with an exponentially decaying learning rate such that it reached 1e-4 on the last time step. We re-parameterized the learning rate with the effective learning rate $\textstyle \alpha _ { \mathrm { e f f } } = { \frac { \alpha } { 1 - \mu } }$ , and the momentum with $1 - \mu$ , so that they can be optimized more smoothly in the log space.
+
+Figure 7 shows training curves both with online SMD and with hand-tuned fixed learning rate and momentum hyperparameters. We show several SMD runs initialized from widely varying hyperparameters; all the SMD runs behaved similarly, suggesting it optimized the meta-objective efficiently enough. Under SMD, learning rates were quickly decreased to very small values, leading to slow progress in the long term, consistent with the noisy quadratic and offline adaptation experiments.
+
+As online SMD can be too conservative in the choice of learning rate, it is natural to ask whether removing the stochasticity in the lookahead sequence can fix the problem. We therefore considered online SMD where the entire lookahead trajectory used a single mini-batch, hence removing the stochasticity. As shown in Figure 8, this deterministic lookahead scheme led to the opposite problem: the adapted learning rates were very large, leading to instability. We conclude that the stochasticity of mini-batch training cannot be simply ignored in meta-optimization.
+
+# 4 CONCLUSION
+
+In this paper, we analyzed the problem of short-horizon bias in meta-optimization. We presented a noisy quadratic toy problem which we analyzed mathematically, and observed that the optimal learning rate schedule differs greatly from a greedy schedule that minimizes training loss one step ahead. While the greedy schedule tends to decay the learning rate drastically to reduce the loss on high curvature directions, the optimal schedule keeps a high learning rate in order to make steady progress on low curvature directions, and eventually achieves far lower loss. We showed that this bias stems from the combination of stochasticity and ill-conditioning: when the problem is either deterministic or spherical, the greedy learning rate schedule is globally optimal; however, when the problem is both stochastic and ill-conditioned (as is most neural net training), the greedy schedule performs poorly. We empirially verified the short-horizon bias in the context of neural net training by applying gradient based meta-optimization, both offline and online. We found the same pathological behaviors as in the noisy quadratic problem — a fast learning rate decay and poor long-run performance.
+
+![](images/969c6f30f61c66047da9a70bda0a2b9909cea11890c939b836f1c41d2bd44c3f.jpg)  
+Figure 8: Online SMD with deterministic lookahead of 5 steps (blue), compared with a manually tuned fixed learning rate (red). Other settings are the same as Figure 7.
+
+While our results suggest that meta-optimization should not be applied blindly, our noisy quadratic analysis also provides grounds for optimism: by removing ill-conditioning (by using a good preconditioner) and/or stochasticity (with large batch sizes or variance reduction techniques), it may be possible to enter the regime where short-horizon meta-optimization works well. It remains to be seen whether this is achievable with existing optimization algorithms.
+
+Acknowledgement YW is supported by a Google PhD Fellowship. RL is supported by Connaught International Scholarships.
+
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+
+# A PROOFS OF THEOREMS
+
+The proofs are organized as follows; we provide a proof to Theorem 1 in A.1, a proof to Theorem 2 in A.2 and a proof to Theorem 3 in A.3.
+
+# A.1 MODEL DYNAMICS
+
+Recall the stochastic gradient descent with momentum is defined as follows,
+
+$$
+\begin{array} { r l r } & { v ^ { ( t + 1 ) } = \mu ^ { ( t ) } v ^ { ( t ) } - \alpha ^ { ( t ) } ( h \theta ^ { ( t ) } + h \sigma \xi ) , } & { \xi \sim \mathcal { N } ( 0 , 1 ) } \\ & { \theta ^ { ( t + 1 ) } = \theta ^ { ( t ) } + v ^ { ( t + 1 ) } = \theta ^ { ( t ) } + \mu ^ { ( t ) } v ^ { ( t ) } - \alpha ^ { ( t ) } ( h \theta ^ { ( t ) } + h \sigma \xi ) } \\ & { } & { = ( 1 - \alpha ^ { ( t ) } h ) \theta ^ { ( t ) } + \mu ^ { ( t ) } v ^ { ( t ) } + h \sigma \xi . } \end{array}
+$$
+
+# A.1.1 DYNAMICS OF THE EXPECTATION
+
+We calculate the mean of the velocity $v ^ { ( t + 1 ) }$ ,
+
+$$
+\begin{array} { r l } & { \mathbb { E } \left[ v ^ { ( t + 1 ) } \right] = \mathbb { E } \left[ \mu ^ { ( t ) } v ^ { ( t ) } - \alpha ^ { ( t ) } h \theta ^ { ( t ) } \right] } \\ & { \qquad = \mu ^ { ( t ) } \mathbb { E } \left[ v ^ { ( t ) } \right] - \alpha ^ { ( t ) } h \mathbb { E } \left[ \theta ^ { ( t ) } \right] . } \end{array}
+$$
+
+We calculate the mean of the parameter $\theta ^ { ( t + 1 ) }$ ,
+
+$$
+\mathbb { E } \left[ \boldsymbol { \theta } ^ { ( t + 1 ) } \right] = \mathbb { E } \left[ \boldsymbol { \theta } ^ { ( t ) } \right] + \mathbb { E } \left[ \boldsymbol { v } ^ { ( t + 1 ) } \right] .
+$$
+
+Let’s assume the following initial conditions:
+
+$$
+\begin{array} { l } { { \mathbb E } \left[ v ^ { ( 0 ) } \right] = 0 } \\ { { \mathbb E } \left[ \theta ^ { ( 0 ) } \right] = E _ { 0 } . } \end{array}
+$$
+
+Then Eq.(10) and Eq.(11) describes how $\mathbb { E } \left[ \boldsymbol { \theta } ^ { ( t ) } \right] , \mathbb { E } \left[ \boldsymbol { v } ^ { ( t ) } \right]$ changes over time $t$
+
+# A.1.2 DYNAMICS OF THE VARIANCE
+
+We calculate the variance of the velocity $v ^ { ( t + 1 ) }$ ,
+
+$$
+\begin{array} { r l } & { \mathbb { V } \left[ v ^ { ( t + 1 ) } \right] = \mathbb { V } \left[ \mu ^ { ( t ) } v ^ { ( t ) } - \alpha ^ { ( t ) } h \theta ^ { ( t ) } \right] + ( \alpha ^ { ( t ) } h \sigma ) ^ { 2 } } \\ & { \qquad = ( \mu ^ { ( t ) } ) ^ { 2 } \mathbb { V } \left[ v ^ { ( t ) } \right] + ( \alpha ^ { ( t ) } h ) ^ { 2 } \mathbb { V } \left[ \theta ^ { ( t ) } \right] - 2 \mu ^ { ( t ) } \alpha ^ { ( t ) } h \cdot \mathrm { C o v } \left( \theta ^ { ( t ) } , v ^ { ( t ) } \right) + ( \alpha ^ { ( t ) } h \sigma ) ^ { 2 } . } \end{array}
+$$
+
+The variance of the parameter $\theta ^ { ( t + 1 ) }$ is given by,
+
+$$
+\mathbb { V } \left[ \boldsymbol { \theta } ^ { ( t + 1 ) } \right] = \mathbb { V } \left[ \boldsymbol { \theta } ^ { ( t ) } \right] + \mathbb { V } \left[ \boldsymbol { v } ^ { ( t + 1 ) } \right] + 2 \left( \mu ^ { ( t ) } \mathrm { C o v } \left( \boldsymbol { \theta } ^ { ( t ) } , \boldsymbol { v } ^ { ( t ) } \right) - \alpha ^ { ( t ) } h \mathbb { V } \left[ \boldsymbol { \theta } ^ { ( t ) } \right] \right) .
+$$
+
+We also need to derive how the covariance of $\theta$ and $v$ changes over time:
+
+$$
+\begin{array} { r l } & { \mathrm { C o v } \left( \theta ^ { ( t + 1 ) } , v ^ { ( t + 1 ) } \right) = \mathrm { C o v } \left( ( \theta ^ { ( t ) } + v ^ { ( t + 1 ) } ) , v ^ { ( t + 1 ) } \right) } \\ & { \qquad = \mathrm { C o v } \left( \theta ^ { ( t ) } , v ^ { ( t + 1 ) } \right) + \mathbb { V } \left[ v ^ { ( t + 1 ) } \right] } \\ & { \qquad = \mu ^ { ( t ) } \mathrm { C o v } \left( \theta ^ { ( t ) } , v ^ { ( t ) } \right) - \alpha ^ { ( t ) } h \mathbb { V } \left[ \theta ^ { ( t ) } \right] + \mathbb { V } \left[ v ^ { ( t + 1 ) } \right] . } \end{array}
+$$
+
+Let’s assume the following initial conditions:
+
+$$
+\begin{array} { c } { { \mathbb { V } \left[ v ^ { ( 0 ) } \right] = 0 } } \\ { { \mathbb { V } \left[ \theta ^ { ( 0 ) } \right] = V _ { 0 } } } \\ { { \mathrm { C o v } \left( \theta ^ { ( 0 ) } , v ^ { ( 0 ) } \right) = 0 . } } \end{array}
+$$
+
+Combining Eq.(12-14), we obtain the following dynamics (from $t = 0 , \ldots , T - 1 )$ :
+
+$$
+\begin{array} { c } { { \mathbb { V } \left[ v ^ { ( t + 1 ) } \right] = ( \mu ^ { ( t ) } ) ^ { 2 } \mathbb { V } \left[ v ^ { ( t ) } \right] + ( \alpha ^ { ( t ) } h ) ^ { 2 } \mathbb { V } \left[ \theta ^ { ( t ) } \right] - 2 \mu ^ { ( t ) } \alpha ^ { ( t ) } h \cdot \mathrm { C o v } \left( \theta ^ { ( t ) } , v ^ { ( t ) } \right) + ( \alpha ^ { ( t ) } h \alpha ^ { ( t ) } ) ^ { 2 } } } \\ { { \mathbb { V } \left[ \theta ^ { ( t + 1 ) } \right] = \mathbb { V } \left[ \theta ^ { ( t ) } \right] + \mathbb { V } \left[ v ^ { ( t + 1 ) } \right] + 2 \left( \mu ^ { ( t ) } \mathrm { C o v } \left( \theta ^ { ( t ) } , v ^ { ( t ) } \right) - \alpha ^ { ( t ) } h \mathbb { V } \left[ \theta ^ { ( t ) } \right] \right) } } \\ { { \mathrm { z o v } \left( \theta ^ { ( t + 1 ) } , v ^ { ( t + 1 ) } \right) = \mu ^ { ( t ) } \mathrm { C o v } \left( \theta ^ { ( t ) } , v ^ { ( t ) } \right) - \alpha ^ { ( t ) } h \mathbb { V } \left[ \theta ^ { ( t ) } \right] + \mathbb { V } \left[ v ^ { ( t + 1 ) } \right] . } } \end{array}
+$$
+
+# A.2 GREEDY OPTIMALITY
+
+# A.2.1 UNIVARIATE CASE
+
+The loss at time step $t$ is,
+
+$$
+\begin{array} { r l } { \mathfrak { L } ^ { ( \epsilon + 1 ) } = \frac { 1 } { 2 } h \left( \mathbb { E } \left[ \theta ^ { ( \mathfrak { t } + 1 ) } \right] ^ { 2 } + \mathbb { V } \left[ \theta ^ { ( \mathfrak { t } + 1 ) } \right] \right) } & { } \\ & { = \frac { 1 } { 2 } h \left[ \left( \mathbb { E } \left[ \theta ^ { ( \mathfrak { t } ) } \right] + \mu ^ { ( \mathfrak { t } ) } \mathbb { E } \left[ \varphi ^ { ( \mathfrak { t } ) } \right] - ( \alpha ^ { ( \mathfrak { t } ) } h ) \mathbb { E } \left[ \theta ^ { ( \mathfrak { t } ) } \right] \right) ^ { 2 } + \mathbb { V } \left[ \theta ^ { ( \mathfrak { t } ) } \right] + ( \mu ^ { ( \mathfrak { t } ) } ) ^ { 2 } \mathbb { V } \left[ \varphi ^ { ( \mathfrak { t } ) } \right] + ( \alpha ^ { ( \mathfrak { t } ) } h ) ^ { 2 } \right. } \\ & { \left. - 2 \mu ^ { ( \mathfrak { t } ) } \alpha ^ { ( \mathfrak { t } ) } h \cdot \mathbb { C } \cos \left( \theta ^ { ( \mathfrak { t } ) } , \varphi ^ { ( \mathfrak { t } ) } \right) + ( \alpha ^ { ( \mathfrak { t } ) } h \sigma ) ^ { 2 } + 2 \left( \mu ^ { ( \mathfrak { t } ) } \cos \left( \theta ^ { ( \mathfrak { t } ) } , \upsilon ^ { ( \mathfrak { t } ) } \right) - \alpha ^ { ( \mathfrak { t } ) } h \mathbb { V } \left[ \theta ^ { ( \mathfrak { t } ) } \right] \right) \right] } \\ & { = \frac { 1 } { 2 } h \bigg [ \left( ( 1 - \alpha ^ { ( \mathfrak { t } ) } h ) \mathbb { E } \left[ \theta ^ { ( \mathfrak { t } ) } \right] + \mu ^ { ( \mathfrak { t } ) } \mathbb { E } \left[ \varphi ^ { ( \mathfrak { t } ) } \right] \right) ^ { 2 } + ( 1 - \alpha ^ { ( \mathfrak { t } ) } h ) ^ { 2 } \mathbb { V } \left[ \theta ^ { ( \mathfrak { t } ) } \right] + ( \mu ^ { ( \mathfrak { t } ) } ) ^ { 2 } \mathbb { V } \left[ \upsilon ^ { ( \mathfrak { t } ) } \right] } \\ & { + 2 \mu ^ { ( \mathfrak { t } ) } ( 1 - \alpha ^ { ( \mathfrak { t } ) } h ) \mathrm { C o n } \left( \theta ^ { ( \mathfrak { t } ) } , \upsilon ^ { ( \mathfrak { t } ) } \right) + ( \alpha ^ { ( \mathfrak { t } ) } h \sigma ) ^ { 2 } \bigg ] } \\ &  = \frac \end{array}
+$$
+
+For simplicity, we denote $A ( \cdot ) = \mathbb { E } \left[ \cdot \right] ^ { 2 } + \mathbb { V } \left[ \cdot \right]$ , and notice that $\mathbb { E } \left[ \boldsymbol { \theta } ^ { ( t ) } \boldsymbol { v } ^ { ( t ) } \right] = \mathbb { E } \left[ \boldsymbol { \theta } ^ { ( t ) } \right] \mathbb { E } \left[ \boldsymbol { v } ^ { ( t ) } \right] +$ $\operatorname { C o v } \left( \theta ^ { ( t ) } , v ^ { ( t ) } \right)$ , hence,
+
+$$
+\ L _ { \lambda } ^ { ( t + 1 ) } = \frac 1 2 h \Big [ ( 1 - \alpha ^ { ( t ) } h ) ^ { 2 } A ( \theta ^ { ( t ) } ) + ( \mu ^ { ( t ) } ) ^ { 2 } A ( v ^ { ( t ) } ) + 2 \mu ^ { ( t ) } ( 1 - \alpha ^ { ( t ) } h ) \mathbb { E } \left[ \theta ^ { ( t ) } v ^ { ( t ) } \right] + ( \alpha ^ { ( t ) } h \sigma ) ^ { 2 } \Big ] .
+$$
+
+In order to find the optimal learning rate and momentum for minimizing $\mathcal { L } ^ { ( t + 1 ) }$ , we take the derivative with respect to $\alpha ^ { ( t ) }$ and $\mu ^ { ( t ) }$ , and set it to 0:
+
+$$
+\begin{array} { r l } & { \qquad \nabla _ { \alpha ^ { ( t ) } } \mathcal { L } ^ { ( t + 1 ) } = ( 1 - \alpha ^ { ( t ) } h ) A ( \theta ^ { ( t ) } ) \cdot ( - h ) - \mu ^ { ( t ) } h \mathbb { E } \left[ \theta ^ { ( t ) } v ^ { ( t ) } \right] + \alpha ^ { ( t ) } ( h \sigma ) ^ { 2 } = 0 } \\ & { \qquad \alpha ^ { ( t ) } h ( A ( \theta ^ { ( t ) } ) + \sigma ^ { 2 } ) - A ( \theta ^ { ( t ) } ) + \mu ^ { ( t ) } \mathbb { E } \left[ \theta ^ { ( t ) } v ^ { ( t ) } \right] } \\ & { \qquad \nabla _ { \mu ^ { ( t ) } } \mathcal { L } ^ { ( t + 1 ) } = \mu ^ { ( t ) } A ( v ^ { ( t ) } ) + ( 1 - \alpha ^ { ( t ) } h ) \mathbb { E } \left[ \theta ^ { ( t ) } v ^ { ( t ) } \right] = 0 } \\ & { \qquad \mu ^ { ( t ) } = - \frac { ( 1 - \alpha ^ { ( t ) } h ) \mathbb { E } \left[ \theta ^ { ( t ) } v ^ { ( t ) } \right] } { A ( \sigma ^ { ( t ) } ) } } \\ & { \qquad \alpha ^ { ( t ) } h ( A ( \theta ^ { ( t ) } ) + \sigma ^ { 2 } ) = A ( \theta ^ { ( t ) } ) - \frac { ( 1 - \alpha ^ { ( t ) } h ) \mathbb { E } \left[ \theta ^ { ( t ) } v ^ { ( t ) } \right] } { A ( \sigma ^ { ( t ) } ) } \mathbb { E } \left[ \theta ^ { ( t ) } v ^ { ( t ) } \right] } \\ & { \qquad \alpha ^ { ( t ) } \ast = \frac { A ( \theta ^ { ( t ) } ) A ( v ^ { ( t ) } ) } { h \left( A ( \theta ^ { ( t ) } ) \right) \left( A ( \theta ^ { ( t ) } ) + \sigma ^ { 2 } \right) - \mathbb { E } \left[ \theta ^ { ( t ) } v ^ { ( t ) } \right] ^ { 2 } } . } \end{array}
+$$
+
+# A.2.2 HIGH DIMENSION CASE
+
+The loss is the sum of losses along all directions:
+
+$$
+z ^ { ( t + 1 ) } = \sum _ { i } \frac { 1 } { 2 } h _ { i } \left[ ( 1 - \alpha ^ { ( t ) } h _ { i } ) ^ { 2 } A ( \theta _ { i } ^ { ( t ) } ) + ( \mu ^ { ( t ) } ) ^ { 2 } A ( v _ { i } ^ { ( t ) } ) + 2 \mu ^ { ( t ) } ( 1 - \alpha ^ { ( t ) } h _ { i } ) \mathbb { E } \left[ \theta _ { i } ^ { ( t ) } v _ { i } ^ { ( t ) } \right] + ( \alpha ^ { ( t ) } h _ { i } \sigma _ { i } ) ^ { 2 } \right]
+$$
+
+Now we obtain optimal learning rate and momentum by setting the derivative to 0,
+
+$$
+\begin{array} { r l } & { V _ { \alpha \beta \gamma } ( E ^ { ( 1 + 1 ) } - \sum _ { i = 1 } ^ { N } | ( 1 - \alpha ^ { ( 4 ) \theta } , \gamma ) | ^ { 2 } ) = ( \sum _ { j = 1 } ^ { N } ( \alpha ^ { ( 4 ) \theta } , \gamma ) - ( \gamma ^ { ( 4 ) } , \gamma ) - ( \gamma ^ { ( 4 ) } , \gamma ) - ( \gamma ^ { ( 4 ) } , \gamma ^ { ( 4 ) } , \gamma ) - ( - \alpha ^ { ( 4 ) \theta } , ( E ^ { ( 1 + 1 ) } - \gamma ) ) } \\ & { \quad + \gamma ^ { ( 4 ) \theta } \sum _ { i = 1 } ^ { N } ( ( \alpha ^ { ( 4 ) \theta } , \gamma ^ { ( 4 ) } - \gamma ) ^ { ( 4 ) } ) = ( \sum _ { j = 1 } ^ { N } ( ( \alpha ^ { ( 4 ) \theta } , \gamma ^ { ( 4 ) } - \gamma ^ { ( 4 ) } ) - \gamma ^ { ( 4 ) \theta } ) ( \sum _ { i = 1 } ^ { N } ( \alpha ^ { ( 4 ) \theta } , \gamma ^ { ( 4 ) } - \gamma ^ { ( 4 ) \theta } ) ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { V _ { \alpha \beta \gamma } ( E ^ { ( 1 + 1 ) } - \sum _ { j = 1 } ^ { N } \alpha ^ { ( 4 ) \theta } , \gamma \alpha ^ { ( 4 ) } ) + \sin ( - ( \alpha ^ { ( 4 ) \theta } , \gamma ) \mathbb { E } ^ { ( 1 + 1 ) } / ( \beta ^ { 2 } , \gamma ^ { ( 4 ) \theta } ) - ( \alpha ^ { ( 4 ) \theta } , \gamma ^ { ( 4 ) \theta } ) - ( \alpha ^ { ( 4 ) \theta } , \gamma ^ { ( 4 ) \theta } ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ &  \quad \quad \quad \quad \ \end{array}
+$$
+
+# A.3 UNIVARIATE OPTIMALITY IN SGD
+
+We now consider a dynamic programming approach to solve the problem. We formalize the optimization problem of $\left\{ \alpha _ { i } \right\}$ as follows. We first denote ${ \mathcal { L } } _ { \mathrm { m i n } }$ as the minimum expected loss at the last time step $T$ (i.e., under the optimal learning rate).
+
+$$
+\mathcal { L } _ { \operatorname* { m i n } } = \operatorname* { m i n } _ { \substack { \alpha ^ { ( t ) } , \alpha ^ { ( t + 1 ) } , \ldots , \alpha ^ { ( T - 1 ) } } } \mathbb { E } _ { \xi ^ { ( t ) } , \xi ^ { ( t + 1 ) } , \ldots , \xi ^ { ( T - 1 ) } } \left[ \mathcal { L } \big ( \theta ^ { ( T ) } \big ) \right] .
+$$
+
+Recall that the loss can be expressed in terms of the expectation and variance of $\theta$ . Denote $A ^ { ( t ) } =$ $( \mathbb { E } \left[ \theta ^ { ( t ) } \right] ) ^ { 2 } + \mathbb { V } \left[ \theta ^ { ( t ) } \right]$ . The final loss can be expressed in terms of $A _ { \operatorname* { m i n } } ^ { ( T ) }$ , obtained by using optimal learning rate schedule:
+
+$$
+\mathcal { L } _ { \mathrm { m i n } } = \frac { 1 } { 2 } h A _ { \mathrm { m i n } } ^ { ( T ) } + \sigma ^ { 2 } .
+$$
+
+As in Theorem 1, we derive the dynamics for SGD without momentum:
+
+$$
+\begin{array} { c } { \theta ^ { ( t ) } = ( 1 - \alpha ^ { ( t - 1 ) } h ) \theta ^ { ( t - 1 ) } + \alpha ^ { ( t - 1 ) } h \sigma \xi ^ { ( t - 1 ) } } \\ { \Rightarrow \left( \mathbb { E } \left[ \theta ^ { ( t ) } \right] , \mathbb { V } \left[ \theta ^ { ( t ) } \right] \right) = \left( ( 1 - \alpha ^ { ( t - 1 ) } h ) \mathbb { E } \left[ \theta ^ { ( t - 1 ) } \right] , ( 1 - \alpha ^ { ( t - 1 ) } h ) ^ { 2 } \mathbb { V } \left[ \theta ^ { ( t - 1 ) } \right] + ( \alpha ^ { ( t - 1 ) } h \sigma ) ^ { 2 } \right) . } \end{array}
+$$
+
+Thus, we can find a recurrence relation of the sequence $A ^ { ( t ) }$ :
+
+$$
+\begin{array} { r } { 4 ^ { ( t ) } = ( 1 - \alpha ^ { ( t - 1 ) } h ) ^ { 2 } \left( ( \mathbb { E } \left[ \theta ^ { ( t - 1 ) } \right] ) ^ { 2 } + \mathbb { V } \left[ \theta ^ { ( t - 1 ) } \right] ^ { 2 } \right) + ( \alpha ^ { ( t - 1 ) } h \sigma ) ^ { 2 } = ( 1 - \alpha ^ { ( t - 1 ) } h ) ^ { 2 } A _ { \operatorname* { m i n } } ^ { ( T - 1 ) } + ( \alpha ^ { ( t - 1 ) } h ) ^ { 2 } A _ { \operatorname* { m i n } } ^ { ( T - 1 ) } + \alpha ^ { ( t - 1 ) } h \sigma h . } \end{array}
+$$
+
+Since $A _ { \operatorname* { m i n } } ^ { ( T ) }$ is a function of $\alpha ^ { ( T - 1 ) * }$ . We can obtain optimal learning rate $\alpha ^ { ( T - 1 ) * }$ by taking the minderivative of ${ \mathcal { L } } _ { \mathrm { m i n } }$ w.r.t. $\alpha ^ { ( T - 1 ) }$ and setting it to zero:
+
+$$
+\begin{array} { r l } { \displaystyle \frac { d \mathcal { L } _ { \operatorname* { m i n } } } { d \alpha ^ { ( T - 1 ) } } = \frac { 1 } { 2 } h \frac { d A _ { \operatorname* { m i n } } ^ { ( T - 1 ) } } { d \alpha ^ { ( T - 2 ) } } = 0 } \\ { \displaystyle \Rightarrow } & { \frac { d A _ { \operatorname* { m i n } } ^ { ( T ) } } { d \alpha ^ { ( T - 1 ) } } = 0 } \\ { \displaystyle \Rightarrow } & { \alpha ^ { ( T - 1 ) * } = \frac { A _ { \operatorname* { m i n } } ^ { ( T - 1 ) } } { h ( A _ { \operatorname* { m i n } } ^ { ( T - 1 ) } + \sigma ^ { 2 } ) } . } \end{array}
+$$
+
+Thus we can write A(T )min i n terms of $A _ { \operatorname* { m i n } } ^ { ( T - 1 ) }$ and the optimal $\alpha ^ { ( T - 1 ) * }$ :
+
+$$
+\begin{array} { l } { { A _ { \mathrm { m i n } } ^ { ( T ) } = \left( 1 - { \frac { A _ { \mathrm { m i n } } ^ { ( T - 1 ) } } { A _ { \mathrm { m i n } } ^ { ( T - 1 ) } + \sigma ^ { 2 } } } \right) ^ { 2 } A _ { \mathrm { m i n } } ^ { ( T - 1 ) } + \left( { \frac { A _ { \mathrm { m i n } } ^ { ( T - 1 ) } } { A _ { \mathrm { m i n } } ^ { ( T - 1 ) } + \sigma ^ { 2 } } } \right) ^ { 2 } } } \\ { { \ \qquad = \left( { \frac { \sigma ^ { 2 } } { A _ { \mathrm { m i n } } ^ { ( T - 1 ) } + \sigma ^ { 2 } } } \right) ^ { 2 } A _ { \mathrm { m i n } } ^ { ( T - 1 ) } + \left( { \frac { A _ { \mathrm { m i n } } ^ { ( T - 1 ) } } { A _ { \mathrm { m i n } } ^ { ( T - 1 ) } + \sigma ^ { 2 } } } \right) ^ { 2 } } } \\ { { \ \qquad = { \frac { A _ { \mathrm { m i n } } ^ { ( T - 1 ) } \sigma ^ { 2 } } { A _ { \mathrm { m i n } } ^ { ( T - 1 ) } + \sigma ^ { 2 } } } . } } \end{array}
+$$
+
+Therefore,
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { L } _ { \mathrm { m i n } } = \frac { 1 } { 2 } h A _ { \mathrm { m i n } } ^ { ( T ) } + \sigma ^ { 2 } } \\ { \displaystyle \quad = \frac { 1 } { 2 } h \left( \frac { A _ { \mathrm { m i n } } ^ { ( T - 1 ) } \sigma ^ { 2 } } { A _ { \mathrm { m i n } } ^ { ( T - 1 ) } + \sigma ^ { 2 } } \right) + \sigma ^ { 2 } . } \end{array}
+$$
+
+We now generalize the above derivation. First rewrite ${ \mathcal { L } } _ { \mathrm { m i n } }$ in terms of $A _ { \operatorname* { m i n } } ^ { T - k }$ and calculate the optimal learning rate at time step $T - k$ .
+
+Theorem 4. For all $T \in \mathbb { N } ,$ , and $k \in \mathbb N$ , $1 \leq k \leq T$ , we have,
+
+$$
+\mathcal { L } _ { \mathrm { m i n } } = \frac { 1 } { 2 } h \left( \frac { A _ { \mathrm { m i n } } ^ { ( T - k ) } \sigma ^ { 2 } } { k A _ { \mathrm { m i n } } ^ { ( T - k ) } + \sigma ^ { 2 } } \right) + \sigma ^ { 2 } .
+$$
+
+Therefore, the optimal learning $\alpha ^ { ( t ) }$ at timestep $t$ is given as,
+
+$$
+\alpha ^ { ( t ) * } = \frac { A ^ { ( t ) } } { h ( A ^ { ( t ) } + \sigma ^ { 2 } ) } .
+$$
+
+Proof. The form of ${ \mathcal { L } } _ { \mathrm { m i n } }$ can be easily proven by induction on $k$ , and use the identity that,
+
+$$
+{ \frac { ( { \frac { a b } { a + b } } ) b } { k ( { \frac { a b } { a + b } } ) + b } } = { \frac { a b } { ( k + 1 ) a + b } } .
+$$
+
+The optimal learning rate then follows immediately by taking the derivative of ${ \mathcal { L } } _ { \mathrm { m i n } }$ w.r.t. $\alpha ^ { ( T - k - 1 ) }$ and setting it to zero. Note that the subscript min is omitted from $A ^ { ( t ) }$ in Eq.(17) as we assume all $A ^ { ( t ) }$ are obtained using optimal $\alpha ^ { * }$ , and hence minimum. □

parse/train/HkeeITEYDr/HkeeITEYDr.md

@@ -0,0 +1,376 @@
+# ROBUST REINFORCEMENT LEARNING WITH WASSERSTEIN CONSTRAINT
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Robust Reinforcement Learning aims to find the optimal policy with some extent of robustness to environmental dynamics. Existing learning algorithms usually enable the robustness though disturbing the current state or simulated environmental parameters in a heuristic way, which lack quantified robustness to the system dynamics (i.e. transition probability). To overcome this issue, we leverage Wasserstein distance to measure the disturbance to the reference transition kernel. With Wasserstein distance, we are able to connect transition kernel disturbance to the state disturbance, i.e. reduce an infinite-dimensional optimization problem to a finite-dimensional risk-aware problem. Through the derived risk-aware optimal Bellman equation, we show the existence of optimal robust policies, provide a sensitivity analysis for the perturbations, and then design a novel robust learning algorithm—Wasserstein Robust Advantage Actor-Critic algorithm (WRAAC). The effectiveness of the proposed algorithm is verified in the Cart-Pole environment.
+
+# 1 INTRODUCTION
+
+Robustness to environmental dynamics is an important topic in safe Reinforcement Learning. Take autonomous vehicle as an example. Autonomous vehicles have to adapt the complex real-world situations, but usually it is unlikely to cover all scenarios during training in real-world environments. To handle this issue, typically, a simulated environment are employed to help build a driving agent, however, the gap between the training and target environments makes the strategies trained with simulated environments sub-optimal to the real-world scenarios (Mannor et al., 2004; 2007). Learning robust policies from simulated environments is a challenging problem for safe Reinforcement Learning.
+
+For robust Reinforcement Learning algorithms, existing methods lie on two branches: One type of methods, borrowed from game theory, introduces an extra agent to disturb the simulated environmental parameters during training (Atkeson & Morimoto, 2003; Morimoto & Doya, 2005; Pinto et al., 2017; Rajeswaran et al., 2016). This method has to rely on the environmental characterization. The other type of methods disturbs the current state through Adversarial Examples (Huang et al., 2017; Kos & Song, 2017; Lin et al., 2017; Mandlekar et al., 2017; Pattanaik et al., 2018), which is more heuristic. Unfortunately, both methods are lack of theoretical guarantee to the robustness extent of transition dynamics.
+
+To address these issues, we design a Wasserstern constraint, which restricts the admissible transition probabilities within a Wasserstein ball centered at some reference transition dynamics. By applying the strong duality of Wasserstein distance (Santambrogio, 2015; Blanchet & Murthy, 2019), we are able to connect the disturbance on transition dynamics with the disturbance on the current state. As a result, the original infinite-dimensional robust optimal problem is reduced to some finitedimensional ordinary risk-aware RL problem. Through the moderated optimal Bellman equation, we prove the existence of robust optimal policies, provide the theoretical analyse on the performance of optimal policies, and design a corresponding —Wasserstein Robust Advantage Actor-Critic algorithm (WRAAC), which does not depend on the environmental characterization. In the experiments, we verified the robustness and effectiveness of the proposed algorithms in the Cart-Pole environment.
+
+The remainder of this paper is organized as follows. In Section 2, we briefly introduce some related work in Markov Decision Processes. In Section 3, we mainly describe the framework of Wasserstein robust Reinforcement Learning. In Section 4, we propose robust Advantage Actor-Critic algorithms according to the moderated robust Bellman equation. In Section 5, we perform experiments on the Cart-Pole environment to verify the effectiveness of our method. Finally, Section 6 concludes our study and provide possible future works.
+
+# 2 RELATED WORK
+
+In this section, we introduce some related work in the fields of MDPs. In robust MDP, the set of all possible transition kernels is called uncertainty set, which can be defined in various ways: one choice could be likelihood regions or entropy bounds of the environment parameters (White III & Eldeib, 1994; Nilim & El Ghaoui, 2005; Iyengar, 2005; Wiesemann et al., 2013); another choice is to constrain the deviation from a reference environment through some statistical distance. For example, Osogami (2012) discussed such robust problem where the uncertainty set are defined via KullbackLeibler divergence, and also uncover the relations between robust MDPs using $f$ -divergence constraint and risk-aware MDPs.
+
+Indeed, it was observed that since the robust MDP framework ignores probabilistic information of the uncertainty set, it can provide conservative solutions (Delage & Mannor, 2010; Xu & Mannor, 2007). Some papers consider bringing prior knowledge of dynamics to robust MDPs, and name such problem distributionally robust MDPs. Xu & Mannor (2010) discuss robust MDPs with prior information to estimate the confidence region of parameters abound, which is a moment-based constraint, and they also show that such distributionally robust problems can be reduced to standard robust MDP problems. Yang (2017; 2018) use Wasserstein distance to evaluate the difference among the prior distributions of transition probabilities. However, Yang’s algorithms are not appropriate for complex situations, because they need to estimate enough transition kernels to approximate prior distribution at each step.
+
+# 3 WASSERSTEIN ROBUST REINFORCEMENT LEARNING
+
+In this section, we specify the problem of interest, which is actually a minimax problem constrained by some Wassserstein-based uncertainty set. We start with introducing a general theoretical framework, i.e., robust Markov Decision Process, and then briefly recall the definition of Wasserstein distance between probability measures. Inspired by the strong duality brought by Wasserstein-based uncertainty set, the robust MDP is reformulated to some risk-aware MDP, making connections clear between robustness to dynamics and robustness to states.
+
+# 3.1 ROBUST MARKOV DECISION PROCESS
+
+Unlike ordinary Markov Decision Processes (MDPs), in robust MDP, environmental dynamics, including transition probabilities and rewards, might change over time (Nilim & El Ghaoui, 2004; 2005). Theoretically, such dynamics can be treated as stochastic changes within an uncertainty set. The objective of robust MDP is to find the optimal policy under the worst dynamics.
+
+Given discrete-time robust MDPs with continuous state and action spaces, without loss of generalization, we only consider the robustness to transition probabilities. Basic elements of robust MDPs include $( \mathcal { X } , \mathcal { A } , \mathcal { Q } , c )$ , where
+
+• $\mathcal { X }$ : state space, which is a Borel measurable metric space.   
+• $\mathcal { A }$ : action space, which is a Borel measurable space. Let $A ( x ) \in { \mathcal { A } }$ represent all the admissible actions at state $x \in X$ , and $\mathbb { K } _ { A }$ denote all the possible state-action pairs, i.e., $\mathbb { K } _ { A } = \{ ( x , a ) : x \in \mathcal { X } , a \in A ( x ) \}$ .   
+• $\mathcal { Q }$ : the uncertainty set that contains all possible transition kernels.   
+• c: $\mathbb { K } _ { A } \ \to \ \mathbb { R }$ , the immediate cost function. Generally we assume it is continuous and $c \in [ 0 , \bar { c } ]$ for some non-negative constant $\bar { c }$ .
+
+The robust system evolves in the following way. Let $ { n _ { \mathrm { ~ \scriptsize ~ \in ~ \mathbb ~ \mathbb \ N ~ } } }$ denote the current time and $x _ { n } ~ \in ~ \mathcal { X }$ the current state. Agent chooses an action $a _ { n } ~ \in ~ A ( x _ { n } )$ and environment selects a transition kernel $q _ { n }$ from the uncertainty set $\mathcal { Q }$ , respectively. Then at the next time $n + 1$ , an agent observes an immediate cost $c ( x _ { n } , a _ { n } )$ and a new state $x _ { n + 1 } \in { \mathcal { X } }$ which follows the distribution $q _ { n } ( \cdot | x _ { n } , a _ { n } )$ . The process repeats at each stage and produces trajectories in a form of $\omega = ( x _ { 0 } , a _ { 0 } , q _ { 0 } , c _ { 0 } , x _ { 1 } , a _ { 1 } , q _ { 1 } , c _ { 1 } , . . . )$ . Let $\Omega = ( \mathcal { X } \times \bar { \mathcal { A } } \times \mathcal { Q } \times [ 0 , \bar { c } ] ) ^ { \infty }$ denote all the trajectories. Let $\Omega _ { n } = \{ \omega _ { n } = ( x _ { 0 } , a _ { 0 } , q _ { 0 } , c _ { 0 } , x _ { 1 } , a _ { 1 } , q _ { 1 } , c _ { 1 } , . . . , x _ { n } ) \}$ denote all trajectories up to time $n$ and $\Omega _ { n } = \{ \tilde { \omega } _ { n } = ( x _ { 0 } , a _ { 0 } , q _ { 0 } , c _ { 0 } , x _ { 1 } , a _ { 1 } , q _ { 1 } , c _ { 1 } , . . . , x _ { n } , a _ { n } ) \}$ denote all trajectories up to time $n$ with action $a _ { n }$ .
+
+Correspondingly, a randomized policy is a series of stochastic kernels: ${ \boldsymbol \pi } = ( \pi _ { 0 } , \pi _ { 1 } , \pi _ { 2 } , . . . )$ where $\pi _ { n } ( \cdot | \omega _ { n } )$ is a probability measure over $A ( x _ { n } )$ . We name $\pi$ primal policy and use $\Pi$ to represent all such randomized policies. If $\pi _ { n } ( \cdot | \omega _ { n } ) = \pi _ { n } ( \cdot | x _ { n } )$ for $n \geq 0$ , we say the policy is Markov. If $\pi _ { n } \equiv \pi _ { 0 }$ for any $n \geq 0$ , this policy is stationary. If there exists measurable functions $f _ { n } : \Omega _ { n } \to { \mathcal { A } }$ such that $\pi _ { n } ( f _ { n } ( \omega _ { n } ) | \omega _ { n } ) \equiv 1$ , $n \geq 0$ , this policy is called deterministic. We denote the set of all such deterministic, stationary, Markov policies by $\mathbb { F }$ .
+
+The selection of transition kernels can be treated as a deterministic policy deployed by a secondary adversarial agent. Let $g = ( g _ { 0 } , g _ { 1 } , g _ { 2 } , . . . )$ with $g _ { n } : { \tilde { \Omega } } _ { n } \to \mathcal { Q }$ denote the adversarial policy. We use $\mathbb { G }$ to represent all such deterministic policies. Similarly, if $g _ { n } ( \cdot | \tilde { \omega } _ { n } ) = g _ { n } ( \cdot | x _ { n } , a _ { n } ) $ for all $n \geq 0$ , the policy is Markov, and if $g _ { n } \equiv g _ { 0 }$ for any $n \geq 0$ , the policy is stationary.
+
+Given the initial state $X _ { 0 } = x \in \mathcal { X }$ , primal policy $\pi \in \Pi$ and adversarial policy $g \in \mathbb { G }$ , applying the Ionescu-Tulcea theorem (Hernandez-Lerma´ $\&$ Lasserre, 2012a; Bertsekas & Shreve, 2004), there exist a probability measure $\mathbb { P } _ { x } ^ { \pi , g }$ on trajectory space. Let $\mathbb { E } _ { x } ^ { \pi , g }$ denote the corresponding expectation operation.
+
+As for the performance criterion, we consider the infinite-horizon discounted cost. Let $\gamma \in ( 0 , 1 )$ be the discounting factor. The discounted cost contributed by trajectory $\omega ~ \in ~ \Omega$ is $C _ { \gamma } ( \omega ) =$ $\Sigma _ { n = 0 } ^ { \infty } \gamma ^ { n } c ( x _ { n } , a _ { n } )$ . Given the initial state $x _ { 0 } = x$ , policies $\pi$ and $g$ , the expected infinite-horizon discounted cost is
+
+$$
+C _ { \gamma } ^ { \pi , g } ( x ) : = \mathbb { E } _ { x } ^ { \pi , g } [ \Sigma _ { n = 0 } ^ { \infty } \gamma ^ { n } c ( x _ { n } , a _ { n } ) ] .
+$$
+
+Robust MDPs aim to find the optimal policy $\pi ^ { * }$ for the agent under the worst realization of $g \in \mathbb { G }$ , which means that $\pi ^ { * }$ reaches
+
+$$
+\operatorname* { i n f } _ { \pi } \operatorname* { s u p } _ { g } C _ { \gamma } ^ { \pi , g } ( x ) .
+$$
+
+This minimax problem can be seen as a zero-sum game of two agents.
+
+# 3.2 WASSERSTEIN DISTANCE
+
+The popular Wasserstein distance is a special case of optimal transport costs, which measures the discrepancy between two probabilities in terms of minimum total costs associated with some transport function. For any two probability measures $Q$ and $P$ over the measurable space $( \mathcal { X } , B ( \mathcal { X } ) )$ , let $\Xi ( Q , P )$ denote the set of all joint distributions on $\mathcal { X } \times \mathcal { X }$ with $Q$ and $P$ are respective marginals. Each element in $\Xi ( Q , P )$ is called a coupling between $Q$ and $P$ . Let $\kappa : \mathcal { X } \times \mathcal { X }  [ 0 , \infty )$ be the transport cost function between two positions, which is non-negative, lower semi-continuous and satisfy $\kappa ( z , y ) = 0$ if and only if $z = y$ . Intuitively, the quantity $\kappa ( z , y )$ specifies the cost of transporting unit mass from $z$ in $\mathcal { X }$ to another element $y$ of $\mathcal { X }$ . Then the optimal transport total cost associated with $\kappa$ is defined as follows:
+
+$$
+D _ { \kappa } ( Q , P ) : = \operatorname* { i n f } _ { \substack { \xi \in \Xi ( Q , P ) } } \left\{ \int _ { \mathcal { X } \times \mathcal { X } } \kappa ( z , y ) d \xi ( z , y ) \right\} .
+$$
+
+Therefore, the optimal transport cost $D _ { \kappa } ( Q , P )$ corresponds to the lowest transport cost that can be obtained among all couplings between $Q$ and $P$ . Let the transport cost function $\kappa$ be some distance metric $d$ on $\mathcal { X }$ , and then it is actually the Wasserstein distance of first order. Wasserstein distance of order $p$ is defined as:
+
+$$
+W _ { p } ( Q , P ) : = \operatorname* { i n f } _ { \xi \in \Xi ( Q , P ) } \left\{ \int _ { \mathcal { X } \times \mathcal { X } } d ( z , y ) ^ { p } d \xi ( z , y ) \right\} ^ { \frac { 1 } { p } } , \ p \geq 1 .
+$$
+
+Unlike Kullback-Liebler divergence or other likelihood-based divergence measures, Wasserstein distance is a proper metric on the space of probabilities. More importantly, Wasserstein distance does not restrict probabilities to share the same support (Villani, 2008; Santambrogio, 2015). Let $d ( z , y ) = \parallel { z - y ^ { \ast } } \parallel _ { 2 } , \kappa ( z , y ) = \textstyle { \frac { 1 } { p } } \parallel { z - y } \parallel _ { 2 } ^ { p }$ and $\delta \stackrel { \star } { = } \frac { 1 } { p } \epsilon ^ { p }$ , the $\epsilon$ -Wasserstein ball of order $p$ and the $\delta$ -optimal-transport ball are identical:
+
+$$
+\{ Q : W _ { p } ( Q , P ) \leq \epsilon \} = \{ Q : D _ { \kappa } ( Q , P ) \leq \delta \} .
+$$
+
+Due to its superior statistical properties, Wasserstein-based uncertainty set has recently received a great deal of attention in DRSO problem (Gao & Kleywegt, 2016; Esfahani & Kuhn, 2018; Blanchet & Murthy, 2019), adversarial example (Sinha et al., 2017), and so on. We will apply it to robust RL.
+
+# 3.3 MAIN RESULT
+
+Let the uncertainty set $\mathcal { Q }$ be a $\epsilon$ -Wasserstein ball of order $p$ centered at some reference transition kernel $P$ :
+
+$$
+\begin{array} { r } { \mathcal { Q } = \{ Q : W _ { p } ( Q ( \cdot | x , a ) , P ( \cdot | x , a ) ) \leq \epsilon , \forall ( x , a ) \in \mathbb { K } _ { A } \} } \\ { = \{ Q : D _ { \kappa } ( Q ( \cdot | x , a ) , P ( \cdot | x , a ) ) \leq \delta , \forall ( x , a ) \in \mathbb { K } _ { A } \} , } \end{array}
+$$
+
+The radius $\epsilon$ or $\delta$ reflects the extent of adversarial perturbation to the reference transition kernel $P$ . The difference between our theoretical framework and Yang (2017; 2018) is that our framework is trying to find the optimal solution for the worst transition kernel within the Wasserstein ball, while theirs is trying to find the optimal solution for the worst distribution over transition kernels.
+
+Recall the state value function (1) at state $x _ { 0 }$ given primal policy $\pi$ and adversarial policy $g$ , we can rewrite the state value function as follows,
+
+$$
+\begin{array} { r l } & { C _ { \gamma } ^ { \pi , g } ( x _ { 0 } ) = \mathbb { E } _ { x _ { 0 } } ^ { \pi , g } [ \Sigma _ { n = 0 } ^ { \infty } \gamma ^ { n } c ( x _ { n } , a _ { n } ) ] } \\ & { \quad \quad \quad \quad = \mathbb { E } _ { x _ { 0 } } ^ { a _ { 0 } \sim \pi , q _ { 0 } } [ c ( x _ { 0 } , a _ { 0 } ) + \mathbb { E } _ { x _ { 1 } \sim q _ { 0 } ( \cdot \vert x _ { 0 } , a _ { 0 } ) } ^ { ( 1 ) } [ \Sigma _ { n = 1 } ^ { \infty } \gamma ^ { n } c ( x _ { n } , a _ { n } ) ] ] } \\ & { \quad \quad \quad \quad = \mathbb { E } _ { x _ { 0 } } ^ { a _ { 0 } \sim \pi , q _ { 0 } } [ c ( x _ { 0 } , a _ { 0 } ) + \gamma \int _ { x _ { 1 } \in \mathcal { X } } q _ { 0 } ( d x _ { 1 } \vert x _ { 0 } , a _ { 0 } ) C _ { \gamma } ^ { ( 1 ) \pi , ( ^ { 1 } ) g } ( x _ { 1 } ) ] , } \end{array}
+$$
+
+where ${ } ^ { ( 1 ) } \pi = ( \pi _ { 1 } , \pi _ { 2 } , . . . )$ and ${ } ^ { ( 1 ) } g = ( g _ { 1 } , g _ { 2 } , . . . )$ are the shift policies. Since $c$ is continuous and bounded, the value function is actually continuous in $\mathcal { X }$ and belongs to $[ 0 , \frac { \bar { c } } { 1 - \gamma } ]$ .
+
+Let $u : \mathcal { X }  \mathbb { R }$ be a measurable, upper semi-continuous function with $u \in [ 0 , \frac { \bar { c } } { 1 - \gamma } ]$ , and let $\mathbb { U }$ denote the set of all such functions. For state $x \in \mathcal { X }$ and action $a \in A ( x )$ . Consider the following operator $H ^ { a }$ defined on $\mathbb { U }$ :
+
+$$
+( H ^ { a } u ) ( x ) : = c ( x , a ) + \operatorname* { s u p } _ { Q \in { \mathcal Q } } \gamma \int _ { y \in { \mathcal X } } Q ( d y | x , a ) u ( y ) .
+$$
+
+Applying Lagrangian method and the strong duality property brought by Wasserstein distance (Blanchet $\&$ Murthy, 2019), we reformulate (5) to the following form:
+
+$$
+( H ^ { a } u ) ( x ) = \operatorname* { i n f } _ { \lambda \geq 0 } c ( x , a ) + \gamma \lambda \delta + \gamma \int _ { y \in \mathcal { X } } P ( d y | x , a ) [ \operatorname* { s u p } _ { z \in \mathcal { X } } ( u ( z ) - \lambda \kappa ( z , y ) ) ] .
+$$
+
+The significance of this strong dual representation lies on the fact that the operator $\mathrm { s u p } _ { \mathfrak { Q } }$ in eq. (5) is replaced by $\operatorname { s u p } _ { z \in { \mathcal { X } } }$ in eq. (6), which leads a much easier optimization algorithm. The righthand side of eq. (6) is a normal iterated-risk function. That is, it reduces the infinite-dimensional probability-searching problem (5) into an ordinary finite-dimensional optimization problem (6).
+
+It is easy to verify that $H ^ { a }$ maps $\mathbb { U }$ to $\mathbb { U }$ . Thus, given a state $x \in \mathcal { X }$ and agent policy $\pi$ , we have the following expected Bellman-form operator:
+
+$$
+\begin{array} { l } { { ( H ^ { \pi } u ) ( x ) : = \displaystyle \int _ { a \in A ( x ) } \pi ( d a | x ) H ^ { a } u ( x ) } } \\ { { \mathrm { ~ } = \displaystyle \operatorname* { i n f } _ { \lambda \geq 0 } \gamma \lambda \delta + \displaystyle \int _ { a \in A ( x ) } \pi ( d a | x ) [ c ( x , a ) + \gamma \displaystyle \int _ { \mathcal X } P ( d y | x , a ) [ \underset { z \in \mathcal X } { \operatorname* { s u p } } ( u ( z ) - \lambda \kappa ( z , y ) ) ] ] . } } \end{array}
+$$
+
+Similarly, $H ^ { \pi }$ maps $\mathbb { U }$ to $\mathbb { U }$ as well. Under the following Assumption 1, we are able to define the optimal iteration operator and show its contraction property.
+
+Assumption 1. $\mathcal { X }$ is a compact metric space. For any $x \in \mathcal { X }$ , $A ( x )$ is compact and $H ^ { a }$ is lower semi-continuous on $a \in A ( x )$ .
+
+Then, given an initial state $x \in \mathcal { X }$ , the following optimal operator over $\mathbb { U }$ is well-defined.
+
+$$
+\begin{array} { l } { \displaystyle ( H u ) ( x ) : = \operatorname* { i n f } _ { a \in A ( x ) } H ^ { a } u ( x ) \ } \\ { \displaystyle \qquad = \operatorname* { i n f } _ { a \in A ( x ) , \lambda \geq 0 } c ( x , a ) + \gamma \lambda \delta + \gamma \int _ { \mathcal X } P ( d y | x , a ) [ \underset { z \in \mathcal X } { \operatorname* { s u p } } ( u ( z ) - \lambda \kappa ( z , y ) ) ] . } \end{array}
+$$
+
+It is simple to verify that $H$ maps $\mathbb { U }$ to $\mathbb { U }$ . The contraction property of $H$ is shown in Lemma 1. We put the proof in the appendix.
+
+Lemma 1. $H$ is a contraction operator in $\mathbb { U }$ under $L _ { \infty }$ norm. There exists an unique element in $\mathbb { U }$ , denoted as $u ^ { * }$ , satisfying $H u ^ { * } = u ^ { * }$ .
+
+For any $u _ { 0 } \in \mathbb { U }$ , $u _ { n } : = H u _ { n - 1 } = H ^ { n } u _ { 0 } .$ . Due to the contraction, we have
+
+$$
+\operatorname* { l i m } _ { n \to \infty } u _ { n } = \operatorname* { l i m } _ { n \to \infty } H ^ { n } u _ { 0 } = u ^ { * } ,
+$$
+
+which indicates an iterative procedure of finding the optimal value function. Based on this optimal value function, we can demonstrate the existence of optimal policies, and single out an optimal policy who is deterministic, Markov and stationary, as shown in Theorem 1. We put the proof in the appendix.
+
+Theorem 1. There exists a deterministic Markov stationary policy $f \in \mathbb { F }$ that satisfies
+
+$$
+H ^ { f } u ^ { * } = H u ^ { * } = u ^ { * } .
+$$
+
+We now obtain the existence of an unique robust optimal value function, as well as a robust optimal policies, which is deterministic, Markov and stationary. Through an iterative procedure as (9), we can design corresponding algorithms for robust Reinforcement Learning.
+
+# 3.4 SENSITIVITY ANALYSIS
+
+Before going to the algorithm design, we present a sensitivity analysis for the optimal value function w.r.t. the radius $\delta$ and the Wasserstein order $p$ . Let $\lambda ^ { * }$ and $z ^ { * } ( y , \lambda ^ { * } ) = \mathrm { \bar { \ a r g m a x } } _ { z \in \mathcal { X } } ( u ( z ) -$ $\lambda ^ { * } \kappa ( z , y ) )$ be a solution of equation (8), and $\lambda ^ { * }$ is non-negative.
+
+If $\lambda ^ { * } = 0$ , which means the worst transition kernel is within our fixed $\epsilon$ -Wasserstein ball, equation (8) can be reduced to an ordinary problem:
+
+$$
+( H u ) ( x ) = \operatorname* { i n f } _ { a \in A ( x ) } c ( x , a ) + \gamma \operatorname* { s u p } _ { z \in \mathcal { X } } u ( z ) .
+$$
+
+Thus $u ^ { * }$ has nothing to do with $\delta$ or $p$ .
+
+If $\lambda ^ { * } > 0$ , via the envelop theorem, the gradient of optimal value function w.r.t. $\delta$ can be calculated as follows.
+
+$$
+\frac { \partial { u ^ { * } } ( x ) } { \partial { \delta } } = \gamma \lambda ^ { * } > 0 .
+$$
+
+This gradient remains positive. That is, the optimal value function increases as the volume of Wasserstein ball increases (remember that $\begin{array} { r } { \delta = \frac { 1 } { p } { \epsilon } ^ { p } } \end{array}$ and the value function represents the discounted cost). Similarly, via the envelop theorem, the gradient w.r.t. $p$ can be calculated as follows.
+
+$$
+\frac { \partial u ^ { * } ( x ) } { \partial p } = - \gamma \lambda ^ { * } \int _ { \mathcal { X } } P ( d y | x , a ) ( \log \parallel z ^ { * } ( y , \lambda ^ { * } ) - y \parallel _ { 2 } - \frac { 1 } { p } ) \frac { \parallel z ^ { * } ( y , \lambda ^ { * } ) - y \parallel _ { 2 } ^ { p } } { p } .
+$$
+
+Since $\lambda ^ { * } > 0$ , the worst transition kernel $Q ^ { * }$ satisfies $W _ { p } ( Q ^ { * } , P ) = \epsilon$ , i.e. $D _ { \kappa } ( Q ^ { * } , P ) = \delta .$ .1 Notice that calculating $z ^ { \ast } ( y , \lambda ^ { \ast } )$ for $y$ is actually trying to find an optimal transport map $\dot { T } _ { p } : \mathcal { X }  \mathcal { X }$ , which substantially perturbs $P$ to $Q ^ { * }$ . Recall that $u ^ { * }$ is upper semi-continuous and its domain is compact, and then we can actually regard $u ^ { * }$ as the Kantorovich potential (Villani, 2008) for a transport cost function $\lambda ^ { * } \kappa$ in the transport from $P$ to $Q ^ { * }$ . For $p > 1$ , $\lambda ^ { * } \kappa$ is strictly convex. Through theorem 1.17 in Santambrogio (2015), we can write the optimal transport map in an explicit way, as well as the gradient over $p$ .
+
+$$
+\begin{array} { c } { { z ^ { * } ( y , \lambda ^ { * } ) = T _ { p } ( y ) = y - ( \lambda ^ { * } ) ^ { - \frac { 1 } { p - 1 } } \parallel \nabla _ { y } u ^ { * } ( y ) \parallel ^ { - \frac { p - 2 } { p - 1 } } \nabla _ { y } u ^ { * } ( y ) , p > 1 . } } \\ { { \displaystyle \frac { u ^ { * } ( x ) } { \partial p } = - \frac { \gamma \lambda ^ { * } } { p ( p - 1 ) } \int _ { \mathcal X } P ( d y | x , a ) ( \log \parallel \frac { \nabla _ { y } u ^ { * } ( y ) } { \lambda ^ { * } } \parallel _ { 2 } - \frac { p - 1 } { p } ) \cdot \parallel \frac { \nabla _ { y } u ^ { * } ( y ) } { \lambda ^ { * } } \parallel _ { 2 } ^ { \frac { p } { p - 1 } } , p > 1 . } } \end{array}
+$$
+
+Thus when $\begin{array} { r } { \frac { 1 } { p } \leq 1 - \log \parallel \frac { \nabla _ { y } u ^ { * } ( y ) } { \lambda ^ { * } } \parallel _ { 2 } } \end{array}$ ∇yu∗(y)λ∗ k2 for all y ∈ X , the gradient over p is non-negative. Larger $\lambda ^ { * }$ makes non-negativity more likely to happen. Remember that $\lambda ^ { * }$ actually reflect the extent of robustness, i.e., larger $\lambda ^ { * }$ coincides with smaller radius $\epsilon$ . Intuitively, when the volume of Wasserstein ball is very small, the extent of perturbation at each point is small with high probability, making the gradient (11) positive. Thus in such situation, smaller $p$ is preferred.
+
+# 4 WASSERSTEIN ROBUST ADVANTANGE ACTOR-CRITIC ALGORITHMS
+
+In reinforcement learning, the agent does not know the precise environment dynamics, i.e., the transition kernel and immediate cost function are unknown. Some researchers leverage an adversarial agent to inject perturbations into environmental parameters during training procedures (Pinto et al., 2017). However, such methods have to work with pre-defined environmental parameters, and are lack of quantified robustness toward transition kernels. Other researchers borrow the idea of adversarial examples and disturb observed states in a heuristic way (Nguyen et al., 2015). They also lose the explanation of robustness towards system dynamics.
+
+Following the analysis in Section 3, we develop a robust Advantage Actor-Critic algorithm: a critic neural network with parameters $w$ , denoted by $u _ { w }$ , is employed to estimate value function; and an actor neural network with parameters $\theta$ , denoted by $\pi _ { \theta }$ , is designed as the primal policy. Rewrite equation (8):
+
+$$
+' H u _ { w } ) ( x ) = \operatorname* { i n f } _ { \theta , \lambda \geq 0 } \int _ { a \in A ( x ) } \pi _ { \theta } ( d a | x ) [ c ( x , a ) + \gamma \lambda \delta + \gamma \int _ { \mathcal { X } } P ( d y | x , a ) [ \underset { z \in \mathcal { X } } { \operatorname* { s u p } } ( u _ { w } ( z ) - \lambda \kappa ( z , y ) ) ] ] .
+$$
+
+Update for $z$ and $\lambda { : }$ Let $f _ { w } ( z ; y , \lambda ) : = u _ { w } ( z ) - \lambda \kappa ( z , y )$ where $\begin{array} { r } { \kappa ( z , y ) = \frac { 1 } { p } \parallel z - y \parallel ^ { p } , p \ge 1 } \end{array}$ Given $y \in \mathcal X$ and $\lambda \in [ 0 , \infty )$ , denote
+
+$$
+z _ { y , \lambda } : = \arg \operatorname* { m a x } _ { z \in \mathcal { X } } f _ { w } ( z ; y , \lambda ) .
+$$
+
+Initially, $z _ { y , \lambda }$ can be treated as the maximum perturbation to state $y \in \mathcal { X }$ , given the penalty $\lambda$ . The gradient of $f _ { w }$ over $z$ is: $\nabla _ { z } f _ { w } = \nabla _ { z } u _ { w } ( z ) - \lambda | | z - y | | ^ { p - 2 } ( z - y )$ . Let $G _ { w } ( \lambda ; x , a ) : =$ $\begin{array} { r } { \lambda \delta + \int _ { \mathcal { X } } P ( d y | x , a ) [ \mathrm { s u p } _ { z \in \mathcal { X } } ( u _ { w } ( z ) - \lambda \kappa ( z , y ) ) ] . } \end{array}$ and we get
+
+$$
+\lambda _ { x , a } : = \arg \operatorname* { m i n } _ { \lambda } G _ { w } ( \lambda ; x , a ) .
+$$
+
+Combining the envelope theorem, we can obtain the gradient of $G _ { w }$ w.r.t. $\lambda \colon \nabla _ { \lambda } G _ { w } \ = \ \delta -$ $\begin{array} { r } { \int _ { \mathcal { X } } P ( d y | x , a ) \kappa ( z _ { y , \lambda } , y ) } \end{array}$ . The expectation in the gradient can be approximated by Monte Carlo: take action $a$ at state $x$ for $n$ times; under the reference transition kernel $P$ , observe the next $y ^ { j }$ . $( x , a , c , y ^ { j } )$ , $j = 1 , 2 , \cdots , n$ ; and then we can approximate $\nabla _ { \lambda } G _ { w } \approx$ $\begin{array} { r } { \delta - \frac { 1 } { n } \sum _ { j = 1 } ^ { n } \kappa \bar { ( z _ { y ^ { j } , \lambda } , \dot { y } ^ { j } ) } } \end{array}$
+
+Critic Update Rule: Given state $x \in \mathcal { X }$ and policy $\pi _ { \theta }$ , let
+
+$$
+J ( \theta , w , x ) : = \int _ { a \in A ( x ) } \pi _ { \theta } ( d a | x ) [ c ( x , a ) + \gamma G _ { w } ( \lambda _ { x , a } ; x , a ) ] .
+$$
+
+To calculate $J$ , similarly, we leverage Monte Carlo, take actions $a _ { i } \sim \pi _ { \theta } ( \cdot | x )$ , $i = 1 , 2 , \cdots , m$ at the same state $x$ for $m$ times, observe $m$ “state-action” pairs $( x , a _ { i } )$ , $i = 1 , 2 , \cdots , m$ , and then approximate $\begin{array} { r } { J ( \theta , w , x ) \approx \frac { 1 } { m } \sum _ { i = 1 } ^ { m } [ c ( x , a _ { i } ) + \gamma G _ { w } ( \lambda _ { x , a } ; \overset { . } { x } , a _ { i } ) ] } \end{array}$ .
+
+Let $e ( x , a _ { i } ) : = c ( x , a _ { i } ) + \gamma G _ { w } ( \lambda _ { x , a } ; x , a _ { i } ) - u _ { w } ( x )$ , and $e ( x )$ denote the difference between the observed cost and the critic network:
+
+$$
+e ( x ) : = J ( \theta , w , x ) - u _ { w } ( x ) \approx \frac { 1 } { m } \sum _ { i = 1 } ^ { m } e ( x , a _ { i } ) = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } [ c ( x , a _ { i } ) + \gamma G _ { w } ( \lambda _ { x , a } ; x , a _ { i } ) - u _ { w } ( x ) ] .
+$$
+
+Through the envelope theorem, we can obtain the following gradient of $e ( x )$ w.r.t. $w$ :
+
+$$
+\begin{array} { l } { \displaystyle \nabla _ { w } e ( x ) = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \gamma \int _ { \mathcal { X } } P ( d y | x , a _ { i } ) \nabla _ { w } u _ { w } ( z _ { y , \lambda _ { x , a } } ) - \nabla _ { w } u _ { w } ( x ) } \\ { \displaystyle \approx \frac { \gamma } { m n } \sum _ { i = 1 } ^ { m } \sum _ { j = 1 } ^ { n } \nabla _ { w } u _ { w } ( z _ { y _ { i } ^ { j } , \lambda _ { x , a _ { i } } } ) - \nabla _ { w } u _ { w } ( x ) . } \end{array}
+$$
+
+Notice that we should actually update the critic network via minimizing ${ \textstyle \frac { 1 } { 2 } } e ( x ) ^ { 2 }$ , and the gradient is
+
+$$
+\nabla _ { w } \frac { 1 } { 2 } e ( x ) ^ { 2 } = e ( x ) \cdot \nabla _ { w } e ( x ) .
+$$
+
+In practice, we usually can let $m = n = 1$ to obtain faster convergence.
+
+Actor Update Rule: In classical AC algorithms, directly minimizing “state-action” value function $J ( \theta , w , x )$ may cause large variance and slow convergence, and optimizing the advantage function is a better choice instead. The advantage function is
+
+$$
+A ( x , a ) : = c ( x , a ) + \gamma G _ { w } ( \lambda _ { x , a } ; x , a ) - u _ { w } ( x ) = e ( x , a ) .
+$$
+
+Thus we can find the optimal $\theta$ via minimizing the expected advantage function $A ( x , \theta ) \ =$ $\begin{array} { r } { \int _ { a \in A ( x ) } \pi _ { \theta } ( d a | x ) e ( x , a ) } \end{array}$ . Similarly, we can approximate the gradient of $A$ w.r.t. $\theta$ as follows:
+
+$$
+\nabla _ { \theta } A ( x , \theta ) = \int _ { a \in A ( x ) } \pi _ { \theta } ( d a | x ) \nabla _ { \theta } \log \pi _ { \theta } ( d a | x ) e ( x , a ) \approx { \frac { 1 } { m } } \sum _ { i = 1 } ^ { m } \nabla _ { \theta } \log \pi _ { \theta } ( x , a _ { i } ) e ( x , a _ { i } ) .
+$$
+
+Finally, we obtain a corresponding Robust Advantage Actor-Critic algorithms. We name it Wasserstein Robust Advantage Actor-Critic algorithm with order $p$ , described in Algorithm 1 and Algorithm 2. Algorithm 1 is actually an inner loop that certifies the extent of perturbations, while Algorithm 2 finds the optimal policy in a normal way. Let the learning rates satisfy the RobbinsMonro condition (Robbins & Monro, 1951), and $\beta _ { 1 } \doteq o ( \beta _ { 2 } ) , \beta _ { 2 } = o ( \tilde { \beta _ { 3 } } ) , \beta _ { 3 } = o ( \tilde { \beta } _ { 4 } )$ , and via the multi-time-scales theory (Borkar, 2008), the convergence to a local minimum can be guaranteed.
+
+# 5 EXPERIMENTS
+
+In this section, we will verify WRAAC algorithm in Cart-Pole environment 2. State space has four dimensions, including cart position, cart velocity, pole angle and pole velocity at tip. There are only two admissible actions: left or right. The target is to prevent the pole from falling over.
+
+Our baseline includes the ordinary Advantage Actor-Critic algorithm. Policies are learnt under the default environment for WRAAC and the baseline. Then, we test the performances of these two policies under different environmental dynamics. We change the simulated environmental parameters such as gravity or pole-length to emulate different test dynamics. Note that the unit change on gravity and pole-length will result in different extents of the dynamic’s robustness.
+
+We apply WRAAC algorithm of order 2, and fix the degree of dynamical robustness at $\delta = 1 0$ . For each quadruple $( x , a , r , y )$ , if $y$ is not the last state of the trajectory, we set initial $\lambda$ be 0 and initial $z$ be $\begin{array} { r } { y + \delta \times ( 0 , \frac { 1 } { \sqrt { 2 6 } } , 0 , \frac { 5 } { \sqrt { 2 6 } } ) } \end{array}$ (designed according to the simulated dynamics of Cart-Pole). If $y$ is the last state, we set $\lambda \equiv 0$ and $z \equiv y$ . The baseline policy and WRAAC are tested in environments with different gravity or different pole-length, shown in Figure 1 and Figure 2.
+
+Remember that different parameters in the Cart-Pole environment have different effects to the dynamic’s robustness. We can see that our robust algorithm changes smoothly as parameter changes,
+
+# Algorithm 1 Calculating Perturbations.
+
+Input: $x \in \mathcal { X }$ , $w$ , $a \in A ( x )$ , $\delta \geq 0$ , $\lambda \geq 0$ , $e = 0$ , $g _ { e } = 0$ , discount factor $\alpha$ , order $p \geq 1$ , $\kappa = 0$ ,   
+learning rates $\beta _ { 1 }$ , $\beta _ { 2 }$ .   
+for $j = 1 , 2 , \cdots , n$ do collect roll-out $( x , a , c ^ { j } , y ^ { j } )$ . $z ^ { j } \gets y ^ { j }$ . $z$ update: $\begin{array} { r l } & { g _ { z }  \nabla _ { z } u _ { w } ( z ) - \lambda ( | | z ^ { j } - y ^ { j } | | ^ { p - 2 } ) ( z ^ { j } - y ^ { j } ) , } \\ & { z ^ { j }  z ^ { j } + \beta _ { 1 } \cdot g _ { z } , } \\ & { e  e + c ^ { j } + \alpha [ \lambda \delta + [ u _ { w } ( z ^ { j } ) - \lambda \frac { 1 } { p } | | z ^ { j } - y ^ { j } | | ^ { p } ] ] - u _ { w } ( x ) } \\ & { g _ { e }  g _ { e } + \alpha \nabla _ { w } u _ { w } ( z ) - \nabla _ { w } u _ { w } ( x ) } \\ & { \kappa  \kappa + \frac { 1 } { p } | | z - y ^ { j } | | ^ { p } , } \end{array}$   
+end for $\lambda$ update:   
+gλ ← δ − 1n κ,   
+λ ← λ + β2 · gλ ,   
+e = 1n ege = 1n ge   
+Input: $\overline { { x \in \mathcal { X } , \theta , w , \delta \geq 0 } }$ , discount factor $\gamma$ , order $p \geq 1$ , learning rates $\beta _ { 3 }$ , $\beta _ { 4 }$   
+for each step do $E = 0$ , $g _ { E } = 0$ . for $i = 1 , 2 , \cdots , m$ do sample $a _ { i } \sim \pi _ { \theta } ( \cdot | x )$ ; use Algorithm 1 and obtain $e , g _ { e }$ . $\begin{array} { l } { { e _ { i }  e } } \\ { { E  E + e } } \\ { { g _ { E }  g _ { E } + g _ { e } } } \end{array}$ end for $w$ update: $\begin{array} { r l } & { w \gets w - \beta _ { 3 } \cdot ( \frac { 1 } { m } E ) \cdot ( \frac { 1 } { m } g _ { E } ) } \\ & { \theta \mathop { \bf u p d a t e : } } \\ & { g _ { \theta } = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \nabla _ { \theta } \log \pi _ { \theta } ( x , a _ { i } ) e _ { i } } \\ & { \theta \gets \theta - \beta _ { 4 } \cdot g _ { \theta } } \end{array}$ state update: choose $\bar { a } \sim \pi _ { \theta } ( \cdot | x )$ , and collect roll-out $( x , a , c , y )$ . $x \gets y$   
+end for   
+Output: $\theta , w$ .
+
+while the baseline plunges. When the perturbation of parameter reaches some level (related with the fixed $\delta = 1 0$ ), our robust policy keeps the pole from falling over for a longer time, which indicates that our algorithm does learn some level of robustness, compared with baseline. If the perturbation of parameter is small, the baseline performs better, due to the fact that the perturbed environment is close to the default environment.
+
+# 6 CONCLUSIONS
+
+In this paper, we investigate the robust Reinforcement Learning with Wasserstein constraint. The derived theoretical framework can be reformulated into a tractable iterated-risk aware problem and the theoretical guarantee is then obtained by building connection between robustness to transition probabilities and robustness to states. Subsequently, we demonstrate the existence of optimal policies, provide a sensitivity analysis to reveal the effects of uncertainty set, and design a proper two-stage learning algorithm WRAAC. The experimental results on the Cart-Pole environment verified the effectiveness and robustness of our proposed approaches.
+
+![](images/a598949459870ee8e6009da7b178e94976fd301af0448a983b43b37b839d7a67.jpg)  
+Figure 1: Robustness to gravity.
+
+![](images/c5afc9d31e8c354fc4e577dc764b40a109c0fed76c22f56146f29a123ed93b56.jpg)  
+Figure 2: Robustness to length.
+
+Future works may favor a complete study for the effects of the radius of Wasserstein ball in our WRAAC algorithm. We are also interested in studying robust policy improvement in a data-driven situation where we only have access to the set of collected trajectories.
+
+# REFERENCES
+
+Christopher G Atkeson and Jun Morimoto. Nonparametric representation of policies and value functions: A trajectory-based approach. In Advances in neural information processing systems, pp. 1643–1650, 2003.
+
+Dimitir P Bertsekas and Steven Shreve. Stochastic optimal control: the discrete-time case. 2004.
+
+Jose Blanchet and Karthyek Murthy. Quantifying distributional model risk via optimal transport. Mathematics of Operations Research, 2019.
+
+Vivek S Borkar. Stochastic approximation: a dynamical systems viewpoint. Baptism’s 91 Witnesses, 2008.
+
+Erick Delage and Shie Mannor. Percentile optimization for markov decision processes with parameter uncertainty. Operations research, 58(1):203–213, 2010.
+
+Peyman Mohajerin Esfahani and Daniel Kuhn. Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations. Mathematical Programming, 171(1-2):115–166, 2018.
+
+Rui Gao and Anton J Kleywegt. Distributionally robust stochastic optimization with wasserstein distance. arXiv preprint arXiv:1604.02199, 2016.
+
+Onesimo Hern ´ andez-Lerma and Jean B Lasserre. ´ Discrete-time Markov control processes: basic optimality criteria, volume 30. Springer Science & Business Media, 2012a.
+
+Onesimo Hern ´ andez-Lerma and Jean B Lasserre. ´ Further topics on discrete-time Markov control processes, volume 42. Springer Science & Business Media, 2012b.
+
+Sandy Huang, Nicolas Papernot, Ian Goodfellow, Yan Duan, and Pieter Abbeel. Adversarial attacks on neural network policies. arXiv preprint arXiv:1702.02284, 2017.
+
+Garud N Iyengar. Robust dynamic programming. Mathematics of Operations Research, 30(2): 257–280, 2005.
+
+Jernej Kos and Dawn Song. Delving into adversarial attacks on deep policies. arXiv preprint arXiv:1705.06452, 2017.
+
+Yen-Chen Lin, Zhang-Wei Hong, Yuan-Hong Liao, Meng-Li Shih, Ming-Yu Liu, and Min Sun. Tactics of adversarial attack on deep reinforcement learning agents. arXiv preprint arXiv:1703.06748, 2017.
+
+Ajay Mandlekar, Yuke Zhu, Animesh Garg, Li Fei-Fei, and Silvio Savarese. Adversarially robust policy learning: Active construction of physically-plausible perturbations. In 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3932–3939. IEEE, 2017.
+
+Shie Mannor, Duncan Simester, Peng Sun, and John N Tsitsiklis. Bias and variance in value function estimation. In Proceedings of the twenty-first international conference on Machine learning, pp. 72. ACM, 2004.
+
+Shie Mannor, Duncan Simester, Peng Sun, and John N Tsitsiklis. Bias and variance approximation in value function estimates. Management Science, 53(2):308–322, 2007.
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+Jun Morimoto and Kenji Doya. Robust reinforcement learning. Neural computation, 17(2):335–359, 2005.
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+Anh Nguyen, Jason Yosinski, and Jeff Clune. Deep neural networks are easily fooled: High confidence predictions for unrecognizable images. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 427–436, 2015.
+
+Arnab Nilim and Laurent El Ghaoui. Robustness in markov decision problems with uncertain transition matrices. In Advances in Neural Information Processing Systems, pp. 839–846, 2004.
+
+Arnab Nilim and Laurent El Ghaoui. Robust control of markov decision processes with uncertain transition matrices. Operations Research, 53(5):780–798, 2005.
+
+Takayuki Osogami. Robustness and risk-sensitivity in markov decision processes. In Advances in Neural Information Processing Systems, pp. 233–241, 2012.
+
+Anay Pattanaik, Zhenyi Tang, Shuijing Liu, Gautham Bommannan, and Girish Chowdhary. Robust deep reinforcement learning with adversarial attacks. In Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, pp. 2040–2042. International Foundation for Autonomous Agents and Multiagent Systems, 2018.
+
+Lerrel Pinto, James Davidson, Rahul Sukthankar, and Abhinav Gupta. Robust adversarial reinforcement learning. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 2817–2826. JMLR. org, 2017.
+
+Aravind Rajeswaran, Sarvjeet Ghotra, Balaraman Ravindran, and Sergey Levine. Epopt: Learning robust neural network policies using model ensembles. arXiv preprint arXiv:1610.01283, 2016.
+
+Herbert Robbins and Sutton Monro. A stochastic approximation method. The annals of mathematical statistics, pp. 400–407, 1951.
+
+Filippo Santambrogio. Optimal transport for applied mathematicians. Birkauser, NY ¨ , pp. 99–102, 2015.
+
+Aman Sinha, Hongseok Namkoong, and John Duchi. Certifying some distributional robustness with principled adversarial training. arXiv preprint arXiv:1710.10571, 2017.
+
+Cedric Villani. ´ Optimal transport: old and new, volume 338. Springer Science & Business Media, 2008.
+
+Chelsea C White III and Hany K Eldeib. Markov decision processes with imprecise transition probabilities. Operations Research, 42(4):739–749, 1994.
+
+Wolfram Wiesemann, Daniel Kuhn, and Berc¸ Rustem. Robust markov decision processes. Mathematics of Operations Research, 38(1):153–183, 2013.
+
+Huan Xu and Shie Mannor. The robustness-performance tradeoff in markov decision processes. In Advances in Neural Information Processing Systems, pp. 1537–1544, 2007.
+
+Huan Xu and Shie Mannor. Distributionally robust markov decision processes. In Advances in Neural Information Processing Systems, pp. 2505–2513, 2010.
+
+Insoon Yang. A convex optimization approach to distributionally robust markov decision processes with wasserstein distance. IEEE control systems letters, 1(1):164–169, 2017.
+
+Insoon Yang. Wasserstein distributionally robust stochastic control: A data-driven approach. arXiv preprint arXiv:1812.09808, 2018.
+
+# A APPENDIX
+
+The trajectory space $( \Omega , { \mathcal { F } } )$ , where $\mathcal { F }$ is the $\sigma$ -algebra of $\Omega$ , satisfies
+
+$$
+\begin{array} { r l } { \bullet } & { \mathbb { P } _ { x } ^ { \pi , g } ( X _ { 0 } = x ) = 1 , } \\ { \bullet } & { \mathbb { P } _ { x } ^ { \pi , g } ( d a | \omega _ { n } ) = \pi _ { n } ( d a | \omega _ { n } ) , } \\ { \bullet } & { \mathbb { P } _ { x } ^ { \pi , g } ( d q | \tilde { \omega } _ { n } ) = \mathbb { 1 } ( g _ { n } ( \tilde { \omega } _ { n } ) \in d q ) , } \\ { \bullet } & { \mathbb { P } _ { x } ^ { \pi , g } ( X _ { n + 1 } \in d x | \omega _ { n } , a _ { n } , q _ { n } ) = q _ { n } ( X _ { n + 1 } \in d x | \omega _ { n } , a _ { n } ) . } \end{array}
+$$
+
+Proof of Lemma 1:
+
+Proof. (1) First, for $\{ u _ { 1 } , u _ { 2 } \} \subset \mathbb { U }$ , if $u _ { 1 } \geq u _ { 2 }$ , it’s easy to have $H u _ { 1 } \geq H u _ { 2 }$ , i.e., the operator $H$ is monotone about $u$ .   
+(2) For any real constant $C$ and $u \in \mathbb { U }$ , we can verify that $H ( u + C ) = H u + \gamma C$ .   
+(3) For any $u _ { 1 } \in \mathbb { U }$ , $u _ { 2 } \in \mathbb { U }$ , there is $u _ { 1 } \leq u _ { 2 } + | | u _ { 1 } - u _ { 2 } | | _ { \infty }$ . Combining (1) and (2), we have $H u _ { 1 } \leq H u _ { 2 } + \gamma | | u _ { 1 } - u _ { 2 } | | _ { \infty }$ , i.e., $H u _ { 1 } - H u _ { 2 } \leq \gamma | | u _ { 1 } - u _ { 2 } | | _ { \infty }$ . Thus $| | H u _ { 1 } - H u _ { 2 } | | _ { \infty } \leq$ $\gamma | | u _ { 1 } - u _ { 2 } | | _ { \infty }$ . Furthermore, since $\gamma \in ( 0 , 1 )$ , the operator $H$ has the contract property in $\mathbb { U }$ under $L _ { \infty }$ norm.
+
+(4) Via Banach fixed-point theorem, there exist an unique $u ^ { * } \in \mathbb { U }$ satisfying $H u ^ { * } = u ^ { * }$
+
+Proof of Theorem 1:
+
+Proof. Due to Assumption 1, for any $u \in \mathbb { U }$ , it is a measurable function on $\mathbb { K } _ { A }$ , an $( H ^ { a } u ) ( x )$ is lower semi-continuous w.r.t. $a$ . Based on the measurable selection theorem (see Lemma 8.3.8 in (Hernandez-Lerma & Lasserre, 2012b)), there is a deterministic Markov stationary policy ´ $f \in \mathbb { F }$ , satisfying $H ^ { f } u ^ { * } = H u ^ { * } = u ^ { * }$ . □

parse/train/HkeeITEYDr/HkeeITEYDr_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "ROBUST REINFORCEMENT LEARNING WITH WASSERSTEIN CONSTRAINT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            821,
+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Robust Reinforcement Learning aims to find the optimal policy with some extent of robustness to environmental dynamics. Existing learning algorithms usually enable the robustness though disturbing the current state or simulated environmental parameters in a heuristic way, which lack quantified robustness to the system dynamics (i.e. transition probability). To overcome this issue, we leverage Wasserstein distance to measure the disturbance to the reference transition kernel. With Wasserstein distance, we are able to connect transition kernel disturbance to the state disturbance, i.e. reduce an infinite-dimensional optimization problem to a finite-dimensional risk-aware problem. Through the derived risk-aware optimal Bellman equation, we show the existence of optimal robust policies, provide a sensitivity analysis for the perturbations, and then design a novel robust learning algorithm—Wasserstein Robust Advantage Actor-Critic algorithm (WRAAC). The effectiveness of the proposed algorithm is verified in the Cart-Pole environment. ",
+        "bbox": [
+            233,
+            270,
+            764,
+            463
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            500,
+            336,
+            516
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Robustness to environmental dynamics is an important topic in safe Reinforcement Learning. Take autonomous vehicle as an example. Autonomous vehicles have to adapt the complex real-world situations, but usually it is unlikely to cover all scenarios during training in real-world environments. To handle this issue, typically, a simulated environment are employed to help build a driving agent, however, the gap between the training and target environments makes the strategies trained with simulated environments sub-optimal to the real-world scenarios (Mannor et al., 2004; 2007). Learning robust policies from simulated environments is a challenging problem for safe Reinforcement Learning. ",
+        "bbox": [
+            174,
+            535,
+            825,
+            646
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For robust Reinforcement Learning algorithms, existing methods lie on two branches: One type of methods, borrowed from game theory, introduces an extra agent to disturb the simulated environmental parameters during training (Atkeson & Morimoto, 2003; Morimoto & Doya, 2005; Pinto et al., 2017; Rajeswaran et al., 2016). This method has to rely on the environmental characterization. The other type of methods disturbs the current state through Adversarial Examples (Huang et al., 2017; Kos & Song, 2017; Lin et al., 2017; Mandlekar et al., 2017; Pattanaik et al., 2018), which is more heuristic. Unfortunately, both methods are lack of theoretical guarantee to the robustness extent of transition dynamics. ",
+        "bbox": [
+            174,
+            652,
+            825,
+            763
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To address these issues, we design a Wasserstern constraint, which restricts the admissible transition probabilities within a Wasserstein ball centered at some reference transition dynamics. By applying the strong duality of Wasserstein distance (Santambrogio, 2015; Blanchet & Murthy, 2019), we are able to connect the disturbance on transition dynamics with the disturbance on the current state. As a result, the original infinite-dimensional robust optimal problem is reduced to some finitedimensional ordinary risk-aware RL problem. Through the moderated optimal Bellman equation, we prove the existence of robust optimal policies, provide the theoretical analyse on the performance of optimal policies, and design a corresponding —Wasserstein Robust Advantage Actor-Critic algorithm (WRAAC), which does not depend on the environmental characterization. In the experiments, we verified the robustness and effectiveness of the proposed algorithms in the Cart-Pole environment. ",
+        "bbox": [
+            174,
+            771,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The remainder of this paper is organized as follows. In Section 2, we briefly introduce some related work in Markov Decision Processes. In Section 3, we mainly describe the framework of Wasserstein robust Reinforcement Learning. In Section 4, we propose robust Advantage Actor-Critic algorithms according to the moderated robust Bellman equation. In Section 5, we perform experiments on the Cart-Pole environment to verify the effectiveness of our method. Finally, Section 6 concludes our study and provide possible future works. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            186
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            209,
+            344,
+            226
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce some related work in the fields of MDPs. In robust MDP, the set of all possible transition kernels is called uncertainty set, which can be defined in various ways: one choice could be likelihood regions or entropy bounds of the environment parameters (White III & Eldeib, 1994; Nilim & El Ghaoui, 2005; Iyengar, 2005; Wiesemann et al., 2013); another choice is to constrain the deviation from a reference environment through some statistical distance. For example, Osogami (2012) discussed such robust problem where the uncertainty set are defined via KullbackLeibler divergence, and also uncover the relations between robust MDPs using $f$ -divergence constraint and risk-aware MDPs. ",
+        "bbox": [
+            174,
+            242,
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+            353
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Indeed, it was observed that since the robust MDP framework ignores probabilistic information of the uncertainty set, it can provide conservative solutions (Delage & Mannor, 2010; Xu & Mannor, 2007). Some papers consider bringing prior knowledge of dynamics to robust MDPs, and name such problem distributionally robust MDPs. Xu & Mannor (2010) discuss robust MDPs with prior information to estimate the confidence region of parameters abound, which is a moment-based constraint, and they also show that such distributionally robust problems can be reduced to standard robust MDP problems. Yang (2017; 2018) use Wasserstein distance to evaluate the difference among the prior distributions of transition probabilities. However, Yang’s algorithms are not appropriate for complex situations, because they need to estimate enough transition kernels to approximate prior distribution at each step. ",
+        "bbox": [
+            174,
+            361,
+            825,
+            501
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 WASSERSTEIN ROBUST REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            523,
+            638,
+            539
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we specify the problem of interest, which is actually a minimax problem constrained by some Wassserstein-based uncertainty set. We start with introducing a general theoretical framework, i.e., robust Markov Decision Process, and then briefly recall the definition of Wasserstein distance between probability measures. Inspired by the strong duality brought by Wasserstein-based uncertainty set, the robust MDP is reformulated to some risk-aware MDP, making connections clear between robustness to dynamics and robustness to states. ",
+        "bbox": [
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+            555,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3.1 ROBUST MARKOV DECISION PROCESS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            659,
+            482,
+            672
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unlike ordinary Markov Decision Processes (MDPs), in robust MDP, environmental dynamics, including transition probabilities and rewards, might change over time (Nilim & El Ghaoui, 2004; 2005). Theoretically, such dynamics can be treated as stochastic changes within an uncertainty set. The objective of robust MDP is to find the optimal policy under the worst dynamics. ",
+        "bbox": [
+            174,
+            685,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Given discrete-time robust MDPs with continuous state and action spaces, without loss of generalization, we only consider the robustness to transition probabilities. Basic elements of robust MDPs include $( \\mathcal { X } , \\mathcal { A } , \\mathcal { Q } , c )$ , where ",
+        "bbox": [
+            176,
+            747,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• $\\mathcal { X }$ : state space, which is a Borel measurable metric space.   \n• $\\mathcal { A }$ : action space, which is a Borel measurable space. Let $A ( x ) \\in { \\mathcal { A } }$ represent all the admissible actions at state $x \\in X$ , and $\\mathbb { K } _ { A }$ denote all the possible state-action pairs, i.e., $\\mathbb { K } _ { A } = \\{ ( x , a ) : x \\in \\mathcal { X } , a \\in A ( x ) \\}$ .   \n• $\\mathcal { Q }$ : the uncertainty set that contains all possible transition kernels.   \n• c: $\\mathbb { K } _ { A } \\ \\to \\ \\mathbb { R }$ , the immediate cost function. Generally we assume it is continuous and $c \\in [ 0 , \\bar { c } ]$ for some non-negative constant $\\bar { c }$ . ",
+        "bbox": [
+            215,
+            804,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The robust system evolves in the following way. Let $ { n _ { \\mathrm { ~ \\scriptsize ~ \\in ~ \\mathbb ~ \\mathbb \\ N ~ } } }$ denote the current time and $x _ { n } ~ \\in ~ \\mathcal { X }$ the current state. Agent chooses an action $a _ { n } ~ \\in ~ A ( x _ { n } )$ and environment selects a transition kernel $q _ { n }$ from the uncertainty set $\\mathcal { Q }$ , respectively. Then at the next time $n + 1$ , an agent observes an immediate cost $c ( x _ { n } , a _ { n } )$ and a new state $x _ { n + 1 } \\in { \\mathcal { X } }$ which follows the distribution $q _ { n } ( \\cdot | x _ { n } , a _ { n } )$ . The process repeats at each stage and produces trajectories in a form of $\\omega = ( x _ { 0 } , a _ { 0 } , q _ { 0 } , c _ { 0 } , x _ { 1 } , a _ { 1 } , q _ { 1 } , c _ { 1 } , . . . )$ . Let $\\Omega = ( \\mathcal { X } \\times \\bar { \\mathcal { A } } \\times \\mathcal { Q } \\times [ 0 , \\bar { c } ] ) ^ { \\infty }$ denote all the trajectories. Let $\\Omega _ { n } = \\{ \\omega _ { n } = ( x _ { 0 } , a _ { 0 } , q _ { 0 } , c _ { 0 } , x _ { 1 } , a _ { 1 } , q _ { 1 } , c _ { 1 } , . . . , x _ { n } ) \\}$ denote all trajectories up to time $n$ and $\\Omega _ { n } = \\{ \\tilde { \\omega } _ { n } = ( x _ { 0 } , a _ { 0 } , q _ { 0 } , c _ { 0 } , x _ { 1 } , a _ { 1 } , q _ { 1 } , c _ { 1 } , . . . , x _ { n } , a _ { n } ) \\}$ denote all trajectories up to time $n$ with action $a _ { n }$ . ",
+        "bbox": [
+            173,
+            103,
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+            232
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Correspondingly, a randomized policy is a series of stochastic kernels: ${ \\boldsymbol \\pi } = ( \\pi _ { 0 } , \\pi _ { 1 } , \\pi _ { 2 } , . . . )$ where $\\pi _ { n } ( \\cdot | \\omega _ { n } )$ is a probability measure over $A ( x _ { n } )$ . We name $\\pi$ primal policy and use $\\Pi$ to represent all such randomized policies. If $\\pi _ { n } ( \\cdot | \\omega _ { n } ) = \\pi _ { n } ( \\cdot | x _ { n } )$ for $n \\geq 0$ , we say the policy is Markov. If $\\pi _ { n } \\equiv \\pi _ { 0 }$ for any $n \\geq 0$ , this policy is stationary. If there exists measurable functions $f _ { n } : \\Omega _ { n } \\to { \\mathcal { A } }$ such that $\\pi _ { n } ( f _ { n } ( \\omega _ { n } ) | \\omega _ { n } ) \\equiv 1$ , $n \\geq 0$ , this policy is called deterministic. We denote the set of all such deterministic, stationary, Markov policies by $\\mathbb { F }$ . ",
+        "bbox": [
+            173,
+            237,
+            825,
+            321
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The selection of transition kernels can be treated as a deterministic policy deployed by a secondary adversarial agent. Let $g = ( g _ { 0 } , g _ { 1 } , g _ { 2 } , . . . )$ with $g _ { n } : { \\tilde { \\Omega } } _ { n } \\to \\mathcal { Q }$ denote the adversarial policy. We use $\\mathbb { G }$ to represent all such deterministic policies. Similarly, if $g _ { n } ( \\cdot | \\tilde { \\omega } _ { n } ) = g _ { n } ( \\cdot | x _ { n } , a _ { n } ) $ for all $n \\geq 0$ , the policy is Markov, and if $g _ { n } \\equiv g _ { 0 }$ for any $n \\geq 0$ , the policy is stationary. ",
+        "bbox": [
+            174,
+            328,
+            825,
+            387
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given the initial state $X _ { 0 } = x \\in \\mathcal { X }$ , primal policy $\\pi \\in \\Pi$ and adversarial policy $g \\in \\mathbb { G }$ , applying the Ionescu-Tulcea theorem (Hernandez-Lerma´ $\\&$ Lasserre, 2012a; Bertsekas & Shreve, 2004), there exist a probability measure $\\mathbb { P } _ { x } ^ { \\pi , g }$ on trajectory space. Let $\\mathbb { E } _ { x } ^ { \\pi , g }$ denote the corresponding expectation operation. ",
+        "bbox": [
+            174,
+            392,
+            825,
+            449
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As for the performance criterion, we consider the infinite-horizon discounted cost. Let $\\gamma \\in ( 0 , 1 )$ be the discounting factor. The discounted cost contributed by trajectory $\\omega ~ \\in ~ \\Omega$ is $C _ { \\gamma } ( \\omega ) =$ $\\Sigma _ { n = 0 } ^ { \\infty } \\gamma ^ { n } c ( x _ { n } , a _ { n } )$ . Given the initial state $x _ { 0 } = x$ , policies $\\pi$ and $g$ , the expected infinite-horizon discounted cost is ",
+        "bbox": [
+            174,
+            455,
+            825,
+            511
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f38605f529f32401a715a26fd1f99d985e3e42c2d01d75607b1fc0496f037a06.jpg",
+        "text": "$$\nC _ { \\gamma } ^ { \\pi , g } ( x ) : = \\mathbb { E } _ { x } ^ { \\pi , g } [ \\Sigma _ { n = 0 } ^ { \\infty } \\gamma ^ { n } c ( x _ { n } , a _ { n } ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            511,
+            622,
+            530
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Robust MDPs aim to find the optimal policy $\\pi ^ { * }$ for the agent under the worst realization of $g \\in \\mathbb { G }$ , which means that $\\pi ^ { * }$ reaches ",
+        "bbox": [
+            171,
+            539,
+            823,
+            566
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5b2b193f6839d10a5c04598a82dd8e722ed823ea9b8ec933ff8c4f00cea1340.jpg",
+        "text": "$$\n\\operatorname* { i n f } _ { \\pi } \\operatorname* { s u p } _ { g } C _ { \\gamma } ^ { \\pi , g } ( x ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            442,
+            565,
+            557,
+            592
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This minimax problem can be seen as a zero-sum game of two agents. ",
+        "bbox": [
+            174,
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+            611
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 WASSERSTEIN DISTANCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            627,
+            392,
+            642
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The popular Wasserstein distance is a special case of optimal transport costs, which measures the discrepancy between two probabilities in terms of minimum total costs associated with some transport function. For any two probability measures $Q$ and $P$ over the measurable space $( \\mathcal { X } , B ( \\mathcal { X } ) )$ , let $\\Xi ( Q , P )$ denote the set of all joint distributions on $\\mathcal { X } \\times \\mathcal { X }$ with $Q$ and $P$ are respective marginals. Each element in $\\Xi ( Q , P )$ is called a coupling between $Q$ and $P$ . Let $\\kappa : \\mathcal { X } \\times \\mathcal { X }  [ 0 , \\infty )$ be the transport cost function between two positions, which is non-negative, lower semi-continuous and satisfy $\\kappa ( z , y ) = 0$ if and only if $z = y$ . Intuitively, the quantity $\\kappa ( z , y )$ specifies the cost of transporting unit mass from $z$ in $\\mathcal { X }$ to another element $y$ of $\\mathcal { X }$ . Then the optimal transport total cost associated with $\\kappa$ is defined as follows: ",
+        "bbox": [
+            173,
+            654,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a07eb9dae35e99ce5fea4180cabde2e3398abf783643999b3a446eb8af2ee4b0.jpg",
+        "text": "$$\nD _ { \\kappa } ( Q , P ) : = \\operatorname* { i n f } _ { \\substack { \\xi \\in \\Xi ( Q , P ) } } \\left\\{ \\int _ { \\mathcal { X } \\times \\mathcal { X } } \\kappa ( z , y ) d \\xi ( z , y ) \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            784,
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+            819
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Therefore, the optimal transport cost $D _ { \\kappa } ( Q , P )$ corresponds to the lowest transport cost that can be obtained among all couplings between $Q$ and $P$ . Let the transport cost function $\\kappa$ be some distance metric $d$ on $\\mathcal { X }$ , and then it is actually the Wasserstein distance of first order. Wasserstein distance of order $p$ is defined as: ",
+        "bbox": [
+            174,
+            825,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/67b97441099acbfa29a6a0e709db0080b9923cbc73995e01c2dcd1c0ca7e4135.jpg",
+        "text": "$$\nW _ { p } ( Q , P ) : = \\operatorname* { i n f } _ { \\xi \\in \\Xi ( Q , P ) } \\left\\{ \\int _ { \\mathcal { X } \\times \\mathcal { X } } d ( z , y ) ^ { p } d \\xi ( z , y ) \\right\\} ^ { \\frac { 1 } { p } } , \\ p \\geq 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            888,
+            699,
+            929
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unlike Kullback-Liebler divergence or other likelihood-based divergence measures, Wasserstein distance is a proper metric on the space of probabilities. More importantly, Wasserstein distance does not restrict probabilities to share the same support (Villani, 2008; Santambrogio, 2015). Let $d ( z , y ) = \\parallel { z - y ^ { \\ast } } \\parallel _ { 2 } , \\kappa ( z , y ) = \\textstyle { \\frac { 1 } { p } } \\parallel { z - y } \\parallel _ { 2 } ^ { p }$ and $\\delta \\stackrel { \\star } { = } \\frac { 1 } { p } \\epsilon ^ { p }$ , the $\\epsilon$ -Wasserstein ball of order $p$ and the $\\delta$ -optimal-transport ball are identical: ",
+        "bbox": [
+            173,
+            102,
+            825,
+            175
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9eb13fcbd7deae09b9686a9b01d6f25ec2a7c7b1b8536c705d55917ebdec7306.jpg",
+        "text": "$$\n\\{ Q : W _ { p } ( Q , P ) \\leq \\epsilon \\} = \\{ Q : D _ { \\kappa } ( Q , P ) \\leq \\delta \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            178,
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+            195
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Due to its superior statistical properties, Wasserstein-based uncertainty set has recently received a great deal of attention in DRSO problem (Gao & Kleywegt, 2016; Esfahani & Kuhn, 2018; Blanchet & Murthy, 2019), adversarial example (Sinha et al., 2017), and so on. We will apply it to robust RL. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 MAIN RESULT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            262,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let the uncertainty set $\\mathcal { Q }$ be a $\\epsilon$ -Wasserstein ball of order $p$ centered at some reference transition kernel $P$ : ",
+        "bbox": [
+            173,
+            289,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/699da1e35b0fcac829454d92e0f59937b789cc409652fef72676fbe8cea4414c.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { Q } = \\{ Q : W _ { p } ( Q ( \\cdot | x , a ) , P ( \\cdot | x , a ) ) \\leq \\epsilon , \\forall ( x , a ) \\in \\mathbb { K } _ { A } \\} } \\\\ { = \\{ Q : D _ { \\kappa } ( Q ( \\cdot | x , a ) , P ( \\cdot | x , a ) ) \\leq \\delta , \\forall ( x , a ) \\in \\mathbb { K } _ { A } \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            318,
+            679,
+            358
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The radius $\\epsilon$ or $\\delta$ reflects the extent of adversarial perturbation to the reference transition kernel $P$ . The difference between our theoretical framework and Yang (2017; 2018) is that our framework is trying to find the optimal solution for the worst transition kernel within the Wasserstein ball, while theirs is trying to find the optimal solution for the worst distribution over transition kernels. ",
+        "bbox": [
+            174,
+            364,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Recall the state value function (1) at state $x _ { 0 }$ given primal policy $\\pi$ and adversarial policy $g$ , we can rewrite the state value function as follows, ",
+        "bbox": [
+            171,
+            426,
+            823,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9dc28286115132af8485c5bbb0f2e2f2b71d95de514a4c479f9ab887fb165c17.jpg",
+        "text": "$$\n\\begin{array} { r l } & { C _ { \\gamma } ^ { \\pi , g } ( x _ { 0 } ) = \\mathbb { E } _ { x _ { 0 } } ^ { \\pi , g } [ \\Sigma _ { n = 0 } ^ { \\infty } \\gamma ^ { n } c ( x _ { n } , a _ { n } ) ] } \\\\ & { \\quad \\quad \\quad \\quad = \\mathbb { E } _ { x _ { 0 } } ^ { a _ { 0 } \\sim \\pi , q _ { 0 } } [ c ( x _ { 0 } , a _ { 0 } ) + \\mathbb { E } _ { x _ { 1 } \\sim q _ { 0 } ( \\cdot \\vert x _ { 0 } , a _ { 0 } ) } ^ { ( 1 ) } [ \\Sigma _ { n = 1 } ^ { \\infty } \\gamma ^ { n } c ( x _ { n } , a _ { n } ) ] ] } \\\\ & { \\quad \\quad \\quad \\quad = \\mathbb { E } _ { x _ { 0 } } ^ { a _ { 0 } \\sim \\pi , q _ { 0 } } [ c ( x _ { 0 } , a _ { 0 } ) + \\gamma \\int _ { x _ { 1 } \\in \\mathcal { X } } q _ { 0 } ( d x _ { 1 } \\vert x _ { 0 } , a _ { 0 } ) C _ { \\gamma } ^ { ( 1 ) \\pi , ( ^ { 1 } ) g } ( x _ { 1 } ) ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            457,
+            738,
+            539
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where ${ } ^ { ( 1 ) } \\pi = ( \\pi _ { 1 } , \\pi _ { 2 } , . . . )$ and ${ } ^ { ( 1 ) } g = ( g _ { 1 } , g _ { 2 } , . . . )$ are the shift policies. Since $c$ is continuous and bounded, the value function is actually continuous in $\\mathcal { X }$ and belongs to $[ 0 , \\frac { \\bar { c } } { 1 - \\gamma } ]$ . ",
+        "bbox": [
+            171,
+            540,
+            823,
+            571
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let $u : \\mathcal { X }  \\mathbb { R }$ be a measurable, upper semi-continuous function with $u \\in [ 0 , \\frac { \\bar { c } } { 1 - \\gamma } ]$ , and let $\\mathbb { U }$ denote the set of all such functions. For state $x \\in \\mathcal { X }$ and action $a \\in A ( x )$ . Consider the following operator $H ^ { a }$ defined on $\\mathbb { U }$ : ",
+        "bbox": [
+            173,
+            579,
+            825,
+            625
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d14594b86d213897541686005ba09e7cfdd25f7ace6e567a82e7005c3edcd389.jpg",
+        "text": "$$\n( H ^ { a } u ) ( x ) : = c ( x , a ) + \\operatorname* { s u p } _ { Q \\in { \\mathcal Q } } \\gamma \\int _ { y \\in { \\mathcal X } } Q ( d y | x , a ) u ( y ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            627,
+            676,
+            662
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Applying Lagrangian method and the strong duality property brought by Wasserstein distance (Blanchet $\\&$ Murthy, 2019), we reformulate (5) to the following form: ",
+        "bbox": [
+            173,
+            671,
+            826,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/723990a91db7589e23e74610b61bd2de28b8f8c80eecd43e586a4c960cd12ed3.jpg",
+        "text": "$$\n( H ^ { a } u ) ( x ) = \\operatorname* { i n f } _ { \\lambda \\geq 0 } c ( x , a ) + \\gamma \\lambda \\delta + \\gamma \\int _ { y \\in \\mathcal { X } } P ( d y | x , a ) [ \\operatorname* { s u p } _ { z \\in \\mathcal { X } } ( u ( z ) - \\lambda \\kappa ( z , y ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            715,
+            756,
+            751
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The significance of this strong dual representation lies on the fact that the operator $\\mathrm { s u p } _ { \\mathfrak { Q } }$ in eq. (5) is replaced by $\\operatorname { s u p } _ { z \\in { \\mathcal { X } } }$ in eq. (6), which leads a much easier optimization algorithm. The righthand side of eq. (6) is a normal iterated-risk function. That is, it reduces the infinite-dimensional probability-searching problem (5) into an ordinary finite-dimensional optimization problem (6). ",
+        "bbox": [
+            174,
+            757,
+            823,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It is easy to verify that $H ^ { a }$ maps $\\mathbb { U }$ to $\\mathbb { U }$ . Thus, given a state $x \\in \\mathcal { X }$ and agent policy $\\pi$ , we have the following expected Bellman-form operator: ",
+        "bbox": [
+            176,
+            819,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1da98fc985b74ed4751e4d97aa9bed36e4d5651960a91edd9e5962c2712af019.jpg",
+        "text": "$$\n\\begin{array} { l } { { ( H ^ { \\pi } u ) ( x ) : = \\displaystyle \\int _ { a \\in A ( x ) } \\pi ( d a | x ) H ^ { a } u ( x ) } } \\\\ { { \\mathrm { ~ } = \\displaystyle \\operatorname* { i n f } _ { \\lambda \\geq 0 } \\gamma \\lambda \\delta + \\displaystyle \\int _ { a \\in A ( x ) } \\pi ( d a | x ) [ c ( x , a ) + \\gamma \\displaystyle \\int _ { \\mathcal X } P ( d y | x , a ) [ \\underset { z \\in \\mathcal X } { \\operatorname* { s u p } } ( u ( z ) - \\lambda \\kappa ( z , y ) ) ] ] . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            851,
+            810,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Similarly, $H ^ { \\pi }$ maps $\\mathbb { U }$ to $\\mathbb { U }$ as well. Under the following Assumption 1, we are able to define the optimal iteration operator and show its contraction property. ",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1. $\\mathcal { X }$ is a compact metric space. For any $x \\in \\mathcal { X }$ , $A ( x )$ is compact and $H ^ { a }$ is lower semi-continuous on $a \\in A ( x )$ . ",
+        "bbox": [
+            173,
+            135,
+            823,
+            165
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Then, given an initial state $x \\in \\mathcal { X }$ , the following optimal operator over $\\mathbb { U }$ is well-defined. ",
+        "bbox": [
+            169,
+            174,
+            756,
+            189
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d7f876969a5599a6c517f8bb55d3931b6de7d487df4ac950bc2fe9113fef9362.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle ( H u ) ( x ) : = \\operatorname* { i n f } _ { a \\in A ( x ) } H ^ { a } u ( x ) \\ } \\\\ { \\displaystyle \\qquad = \\operatorname* { i n f } _ { a \\in A ( x ) , \\lambda \\geq 0 } c ( x , a ) + \\gamma \\lambda \\delta + \\gamma \\int _ { \\mathcal X } P ( d y | x , a ) [ \\underset { z \\in \\mathcal X } { \\operatorname* { s u p } } ( u ( z ) - \\lambda \\kappa ( z , y ) ) ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            191,
+            766,
+            253
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "It is simple to verify that $H$ maps $\\mathbb { U }$ to $\\mathbb { U }$ . The contraction property of $H$ is shown in Lemma 1. We put the proof in the appendix. ",
+        "bbox": [
+            173,
+            263,
+            825,
+            292
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. $H$ is a contraction operator in $\\mathbb { U }$ under $L _ { \\infty }$ norm. There exists an unique element in $\\mathbb { U }$ , denoted as $u ^ { * }$ , satisfying $H u ^ { * } = u ^ { * }$ . ",
+        "bbox": [
+            173,
+            295,
+            823,
+            324
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For any $u _ { 0 } \\in \\mathbb { U }$ , $u _ { n } : = H u _ { n - 1 } = H ^ { n } u _ { 0 } .$ . Due to the contraction, we have ",
+        "bbox": [
+            176,
+            334,
+            660,
+            349
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a6900ff669841ad13c37a1f849e2897b9580da1b8ebe02776c4cfb441b28c3fd.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { n \\to \\infty } u _ { n } = \\operatorname* { l i m } _ { n \\to \\infty } H ^ { n } u _ { 0 } = u ^ { * } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            353,
+            599,
+            376
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which indicates an iterative procedure of finding the optimal value function. Based on this optimal value function, we can demonstrate the existence of optimal policies, and single out an optimal policy who is deterministic, Markov and stationary, as shown in Theorem 1. We put the proof in the appendix. ",
+        "bbox": [
+            174,
+            378,
+            825,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. There exists a deterministic Markov stationary policy $f \\in \\mathbb { F }$ that satisfies ",
+        "bbox": [
+            173,
+            438,
+            741,
+            454
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7fc77ef5a6cea8aa3f20b8faef43a5c501c340d4934bb8a0f9cb91fe628937b3.jpg",
+        "text": "$$\nH ^ { f } u ^ { * } = H u ^ { * } = u ^ { * } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            457,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now obtain the existence of an unique robust optimal value function, as well as a robust optimal policies, which is deterministic, Markov and stationary. Through an iterative procedure as (9), we can design corresponding algorithms for robust Reinforcement Learning. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 SENSITIVITY ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            546,
+            382,
+            560
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Before going to the algorithm design, we present a sensitivity analysis for the optimal value function w.r.t. the radius $\\delta$ and the Wasserstein order $p$ . Let $\\lambda ^ { * }$ and $z ^ { * } ( y , \\lambda ^ { * } ) = \\mathrm { \\bar { \\ a r g m a x } } _ { z \\in \\mathcal { X } } ( u ( z ) -$ $\\lambda ^ { * } \\kappa ( z , y ) )$ be a solution of equation (8), and $\\lambda ^ { * }$ is non-negative. ",
+        "bbox": [
+            174,
+            571,
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+            614
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "If $\\lambda ^ { * } = 0$ , which means the worst transition kernel is within our fixed $\\epsilon$ -Wasserstein ball, equation (8) can be reduced to an ordinary problem: ",
+        "bbox": [
+            174,
+            621,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/97fb110c5a73cca9bf96e8fa9814f910c37778676930ae54934ba2bc1f2ee7b0.jpg",
+        "text": "$$\n( H u ) ( x ) = \\operatorname* { i n f } _ { a \\in A ( x ) } c ( x , a ) + \\gamma \\operatorname* { s u p } _ { z \\in \\mathcal { X } } u ( z ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            652,
+            635,
+            679
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Thus $u ^ { * }$ has nothing to do with $\\delta$ or $p$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "If $\\lambda ^ { * } > 0$ , via the envelop theorem, the gradient of optimal value function w.r.t. $\\delta$ can be calculated as follows. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c5f8b5e3b2e1475f6bd38d4ec9631d36771e4d11e5a199683df7840a36d5ca9f.jpg",
+        "text": "$$\n\\frac { \\partial { u ^ { * } } ( x ) } { \\partial { \\delta } } = \\gamma \\lambda ^ { * } > 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This gradient remains positive. That is, the optimal value function increases as the volume of Wasserstein ball increases (remember that $\\begin{array} { r } { \\delta = \\frac { 1 } { p } { \\epsilon } ^ { p } } \\end{array}$ and the value function represents the discounted cost). Similarly, via the envelop theorem, the gradient w.r.t. $p$ can be calculated as follows. ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e3f27a2625b45c02ed7222ab74f297b8189ba945657f9ed0bce22c46f89387aa.jpg",
+        "text": "$$\n\\frac { \\partial u ^ { * } ( x ) } { \\partial p } = - \\gamma \\lambda ^ { * } \\int _ { \\mathcal { X } } P ( d y | x , a ) ( \\log \\parallel z ^ { * } ( y , \\lambda ^ { * } ) - y \\parallel _ { 2 } - \\frac { 1 } { p } ) \\frac { \\parallel z ^ { * } ( y , \\lambda ^ { * } ) - y \\parallel _ { 2 } ^ { p } } { p } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since $\\lambda ^ { * } > 0$ , the worst transition kernel $Q ^ { * }$ satisfies $W _ { p } ( Q ^ { * } , P ) = \\epsilon$ , i.e. $D _ { \\kappa } ( Q ^ { * } , P ) = \\delta .$ .1 Notice that calculating $z ^ { \\ast } ( y , \\lambda ^ { \\ast } )$ for $y$ is actually trying to find an optimal transport map $\\dot { T } _ { p } : \\mathcal { X }  \\mathcal { X }$ , which substantially perturbs $P$ to $Q ^ { * }$ . Recall that $u ^ { * }$ is upper semi-continuous and its domain is compact, and then we can actually regard $u ^ { * }$ as the Kantorovich potential (Villani, 2008) for a transport cost function $\\lambda ^ { * } \\kappa$ in the transport from $P$ to $Q ^ { * }$ . For $p > 1$ , $\\lambda ^ { * } \\kappa$ is strictly convex. Through theorem 1.17 in Santambrogio (2015), we can write the optimal transport map in an explicit way, as well as the gradient over $p$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/4c59393166fb1e34222b62efd7c6b852b92baaaabf93f32748bcf70019f1ad86.jpg",
+        "text": "$$\n\\begin{array} { c } { { z ^ { * } ( y , \\lambda ^ { * } ) = T _ { p } ( y ) = y - ( \\lambda ^ { * } ) ^ { - \\frac { 1 } { p - 1 } } \\parallel \\nabla _ { y } u ^ { * } ( y ) \\parallel ^ { - \\frac { p - 2 } { p - 1 } } \\nabla _ { y } u ^ { * } ( y ) , p > 1 . } } \\\\ { { \\displaystyle \\frac { u ^ { * } ( x ) } { \\partial p } = - \\frac { \\gamma \\lambda ^ { * } } { p ( p - 1 ) } \\int _ { \\mathcal X } P ( d y | x , a ) ( \\log \\parallel \\frac { \\nabla _ { y } u ^ { * } ( y ) } { \\lambda ^ { * } } \\parallel _ { 2 } - \\frac { p - 1 } { p } ) \\cdot \\parallel \\frac { \\nabla _ { y } u ^ { * } ( y ) } { \\lambda ^ { * } } \\parallel _ { 2 } ^ { \\frac { p } { p - 1 } } , p > 1 . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Thus when $\\begin{array} { r } { \\frac { 1 } { p } \\leq 1 - \\log \\parallel \\frac { \\nabla _ { y } u ^ { * } ( y ) } { \\lambda ^ { * } } \\parallel _ { 2 } } \\end{array}$ ∇yu∗(y)λ∗ k2 for all y ∈ X , the gradient over p is non-negative. Larger $\\lambda ^ { * }$ makes non-negativity more likely to happen. Remember that $\\lambda ^ { * }$ actually reflect the extent of robustness, i.e., larger $\\lambda ^ { * }$ coincides with smaller radius $\\epsilon$ . Intuitively, when the volume of Wasserstein ball is very small, the extent of perturbation at each point is small with high probability, making the gradient (11) positive. Thus in such situation, smaller $p$ is preferred. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 WASSERSTEIN ROBUST ADVANTANGE ACTOR-CRITIC ALGORITHMS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In reinforcement learning, the agent does not know the precise environment dynamics, i.e., the transition kernel and immediate cost function are unknown. Some researchers leverage an adversarial agent to inject perturbations into environmental parameters during training procedures (Pinto et al., 2017). However, such methods have to work with pre-defined environmental parameters, and are lack of quantified robustness toward transition kernels. Other researchers borrow the idea of adversarial examples and disturb observed states in a heuristic way (Nguyen et al., 2015). They also lose the explanation of robustness towards system dynamics. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Following the analysis in Section 3, we develop a robust Advantage Actor-Critic algorithm: a critic neural network with parameters $w$ , denoted by $u _ { w }$ , is employed to estimate value function; and an actor neural network with parameters $\\theta$ , denoted by $\\pi _ { \\theta }$ , is designed as the primal policy. Rewrite equation (8): ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ce3007911cc72df0127970bc095e86bb898ad80fdcb392564120d18207b31cbd.jpg",
+        "text": "$$\n' H u _ { w } ) ( x ) = \\operatorname* { i n f } _ { \\theta , \\lambda \\geq 0 } \\int _ { a \\in A ( x ) } \\pi _ { \\theta } ( d a | x ) [ c ( x , a ) + \\gamma \\lambda \\delta + \\gamma \\int _ { \\mathcal { X } } P ( d y | x , a ) [ \\underset { z \\in \\mathcal { X } } { \\operatorname* { s u p } } ( u _ { w } ( z ) - \\lambda \\kappa ( z , y ) ) ] ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Update for $z$ and $\\lambda { : }$ Let $f _ { w } ( z ; y , \\lambda ) : = u _ { w } ( z ) - \\lambda \\kappa ( z , y )$ where $\\begin{array} { r } { \\kappa ( z , y ) = \\frac { 1 } { p } \\parallel z - y \\parallel ^ { p } , p \\ge 1 } \\end{array}$ Given $y \\in \\mathcal X$ and $\\lambda \\in [ 0 , \\infty )$ , denote ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/700fabd03f6b4848dee25685ecd1bd5d866eaf683f6db14fe5c92e04dd629966.jpg",
+        "text": "$$\nz _ { y , \\lambda } : = \\arg \\operatorname* { m a x } _ { z \\in \\mathcal { X } } f _ { w } ( z ; y , \\lambda ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Initially, $z _ { y , \\lambda }$ can be treated as the maximum perturbation to state $y \\in \\mathcal { X }$ , given the penalty $\\lambda$ . The gradient of $f _ { w }$ over $z$ is: $\\nabla _ { z } f _ { w } = \\nabla _ { z } u _ { w } ( z ) - \\lambda | | z - y | | ^ { p - 2 } ( z - y )$ . Let $G _ { w } ( \\lambda ; x , a ) : =$ $\\begin{array} { r } { \\lambda \\delta + \\int _ { \\mathcal { X } } P ( d y | x , a ) [ \\mathrm { s u p } _ { z \\in \\mathcal { X } } ( u _ { w } ( z ) - \\lambda \\kappa ( z , y ) ) ] . } \\end{array}$ and we get ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/169b4e0b88793b3d8cf994a56fac78299db2c56b905340f127b17dce76b21ec4.jpg",
+        "text": "$$\n\\lambda _ { x , a } : = \\arg \\operatorname* { m i n } _ { \\lambda } G _ { w } ( \\lambda ; x , a ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Combining the envelope theorem, we can obtain the gradient of $G _ { w }$ w.r.t. $\\lambda \\colon \\nabla _ { \\lambda } G _ { w } \\ = \\ \\delta -$ $\\begin{array} { r } { \\int _ { \\mathcal { X } } P ( d y | x , a ) \\kappa ( z _ { y , \\lambda } , y ) } \\end{array}$ . The expectation in the gradient can be approximated by Monte Carlo: take action $a$ at state $x$ for $n$ times; under the reference transition kernel $P$ , observe the next $y ^ { j }$ . $( x , a , c , y ^ { j } )$ , $j = 1 , 2 , \\cdots , n$ ; and then we can approximate $\\nabla _ { \\lambda } G _ { w } \\approx$ $\\begin{array} { r } { \\delta - \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } \\kappa \\bar { ( z _ { y ^ { j } , \\lambda } , \\dot { y } ^ { j } ) } } \\end{array}$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Critic Update Rule: Given state $x \\in \\mathcal { X }$ and policy $\\pi _ { \\theta }$ , let ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/02fe7e80c056c5510c22109c4f7cfee76213846f69ef894a9bd0ea2f9deb36ff.jpg",
+        "text": "$$\nJ ( \\theta , w , x ) : = \\int _ { a \\in A ( x ) } \\pi _ { \\theta } ( d a | x ) [ c ( x , a ) + \\gamma G _ { w } ( \\lambda _ { x , a } ; x , a ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To calculate $J$ , similarly, we leverage Monte Carlo, take actions $a _ { i } \\sim \\pi _ { \\theta } ( \\cdot | x )$ , $i = 1 , 2 , \\cdots , m$ at the same state $x$ for $m$ times, observe $m$ “state-action” pairs $( x , a _ { i } )$ , $i = 1 , 2 , \\cdots , m$ , and then approximate $\\begin{array} { r } { J ( \\theta , w , x ) \\approx \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } [ c ( x , a _ { i } ) + \\gamma G _ { w } ( \\lambda _ { x , a } ; \\overset { . } { x } , a _ { i } ) ] } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Let $e ( x , a _ { i } ) : = c ( x , a _ { i } ) + \\gamma G _ { w } ( \\lambda _ { x , a } ; x , a _ { i } ) - u _ { w } ( x )$ , and $e ( x )$ denote the difference between the observed cost and the critic network: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e0ce3e917b3126dcb414809dc70d18287537f81bd42118a0eb99d685002ee76e.jpg",
+        "text": "$$\ne ( x ) : = J ( \\theta , w , x ) - u _ { w } ( x ) \\approx \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } e ( x , a _ { i } ) = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } [ c ( x , a _ { i } ) + \\gamma G _ { w } ( \\lambda _ { x , a } ; x , a _ { i } ) - u _ { w } ( x ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Through the envelope theorem, we can obtain the following gradient of $e ( x )$ w.r.t. $w$ : ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/71bc3cb2e2a7ee856db8ff377aa52c2ebc80feb5edd6ee181527132d20bd8f8c.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\nabla _ { w } e ( x ) = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\gamma \\int _ { \\mathcal { X } } P ( d y | x , a _ { i } ) \\nabla _ { w } u _ { w } ( z _ { y , \\lambda _ { x , a } } ) - \\nabla _ { w } u _ { w } ( x ) } \\\\ { \\displaystyle \\approx \\frac { \\gamma } { m n } \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } \\nabla _ { w } u _ { w } ( z _ { y _ { i } ^ { j } , \\lambda _ { x , a _ { i } } } ) - \\nabla _ { w } u _ { w } ( x ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Notice that we should actually update the critic network via minimizing ${ \\textstyle \\frac { 1 } { 2 } } e ( x ) ^ { 2 }$ , and the gradient is ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a36c2e1c141b2f0b2fff62e707b07cc48d86a18433f5af81ee1e16c3d43a0377.jpg",
+        "text": "$$\n\\nabla _ { w } \\frac { 1 } { 2 } e ( x ) ^ { 2 } = e ( x ) \\cdot \\nabla _ { w } e ( x ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In practice, we usually can let $m = n = 1$ to obtain faster convergence. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Actor Update Rule: In classical AC algorithms, directly minimizing “state-action” value function $J ( \\theta , w , x )$ may cause large variance and slow convergence, and optimizing the advantage function is a better choice instead. The advantage function is ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9aafa386794b8e9adf78c7994f132bab7ea3943abade046b0b6466f381f16df4.jpg",
+        "text": "$$\nA ( x , a ) : = c ( x , a ) + \\gamma G _ { w } ( \\lambda _ { x , a } ; x , a ) - u _ { w } ( x ) = e ( x , a ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Thus we can find the optimal $\\theta$ via minimizing the expected advantage function $A ( x , \\theta ) \\ =$ $\\begin{array} { r } { \\int _ { a \\in A ( x ) } \\pi _ { \\theta } ( d a | x ) e ( x , a ) } \\end{array}$ . Similarly, we can approximate the gradient of $A$ w.r.t. $\\theta$ as follows: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/832053c0d247f29a3ba8db810097503f6b0baaf71d61a9e06d7e42b5484b2ebf.jpg",
+        "text": "$$\n\\nabla _ { \\theta } A ( x , \\theta ) = \\int _ { a \\in A ( x ) } \\pi _ { \\theta } ( d a | x ) \\nabla _ { \\theta } \\log \\pi _ { \\theta } ( d a | x ) e ( x , a ) \\approx { \\frac { 1 } { m } } \\sum _ { i = 1 } ^ { m } \\nabla _ { \\theta } \\log \\pi _ { \\theta } ( x , a _ { i } ) e ( x , a _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finally, we obtain a corresponding Robust Advantage Actor-Critic algorithms. We name it Wasserstein Robust Advantage Actor-Critic algorithm with order $p$ , described in Algorithm 1 and Algorithm 2. Algorithm 1 is actually an inner loop that certifies the extent of perturbations, while Algorithm 2 finds the optimal policy in a normal way. Let the learning rates satisfy the RobbinsMonro condition (Robbins & Monro, 1951), and $\\beta _ { 1 } \\doteq o ( \\beta _ { 2 } ) , \\beta _ { 2 } = o ( \\tilde { \\beta _ { 3 } } ) , \\beta _ { 3 } = o ( \\tilde { \\beta } _ { 4 } )$ , and via the multi-time-scales theory (Borkar, 2008), the convergence to a local minimum can be guaranteed. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we will verify WRAAC algorithm in Cart-Pole environment 2. State space has four dimensions, including cart position, cart velocity, pole angle and pole velocity at tip. There are only two admissible actions: left or right. The target is to prevent the pole from falling over. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our baseline includes the ordinary Advantage Actor-Critic algorithm. Policies are learnt under the default environment for WRAAC and the baseline. Then, we test the performances of these two policies under different environmental dynamics. We change the simulated environmental parameters such as gravity or pole-length to emulate different test dynamics. Note that the unit change on gravity and pole-length will result in different extents of the dynamic’s robustness. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We apply WRAAC algorithm of order 2, and fix the degree of dynamical robustness at $\\delta = 1 0$ . For each quadruple $( x , a , r , y )$ , if $y$ is not the last state of the trajectory, we set initial $\\lambda$ be 0 and initial $z$ be $\\begin{array} { r } { y + \\delta \\times ( 0 , \\frac { 1 } { \\sqrt { 2 6 } } , 0 , \\frac { 5 } { \\sqrt { 2 6 } } ) } \\end{array}$ (designed according to the simulated dynamics of Cart-Pole). If $y$ is the last state, we set $\\lambda \\equiv 0$ and $z \\equiv y$ . The baseline policy and WRAAC are tested in environments with different gravity or different pole-length, shown in Figure 1 and Figure 2. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Remember that different parameters in the Cart-Pole environment have different effects to the dynamic’s robustness. We can see that our robust algorithm changes smoothly as parameter changes, ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Calculating Perturbations. ",
+        "text_level": 1,
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Input: $x \\in \\mathcal { X }$ , $w$ , $a \\in A ( x )$ , $\\delta \\geq 0$ , $\\lambda \\geq 0$ , $e = 0$ , $g _ { e } = 0$ , discount factor $\\alpha$ , order $p \\geq 1$ , $\\kappa = 0$ ,   \nlearning rates $\\beta _ { 1 }$ , $\\beta _ { 2 }$ .   \nfor $j = 1 , 2 , \\cdots , n$ do collect roll-out $( x , a , c ^ { j } , y ^ { j } )$ . $z ^ { j } \\gets y ^ { j }$ . $z$ update: $\\begin{array} { r l } & { g _ { z }  \\nabla _ { z } u _ { w } ( z ) - \\lambda ( | | z ^ { j } - y ^ { j } | | ^ { p - 2 } ) ( z ^ { j } - y ^ { j } ) , } \\\\ & { z ^ { j }  z ^ { j } + \\beta _ { 1 } \\cdot g _ { z } , } \\\\ & { e  e + c ^ { j } + \\alpha [ \\lambda \\delta + [ u _ { w } ( z ^ { j } ) - \\lambda \\frac { 1 } { p } | | z ^ { j } - y ^ { j } | | ^ { p } ] ] - u _ { w } ( x ) } \\\\ & { g _ { e }  g _ { e } + \\alpha \\nabla _ { w } u _ { w } ( z ) - \\nabla _ { w } u _ { w } ( x ) } \\\\ & { \\kappa  \\kappa + \\frac { 1 } { p } | | z - y ^ { j } | | ^ { p } , } \\end{array}$   \nend for $\\lambda$ update:   \ngλ ← δ − 1n κ,   \nλ ← λ + β2 · gλ ,   \ne = 1n ege = 1n ge   \nInput: $\\overline { { x \\in \\mathcal { X } , \\theta , w , \\delta \\geq 0 } }$ , discount factor $\\gamma$ , order $p \\geq 1$ , learning rates $\\beta _ { 3 }$ , $\\beta _ { 4 }$   \nfor each step do $E = 0$ , $g _ { E } = 0$ . for $i = 1 , 2 , \\cdots , m$ do sample $a _ { i } \\sim \\pi _ { \\theta } ( \\cdot | x )$ ; use Algorithm 1 and obtain $e , g _ { e }$ . $\\begin{array} { l } { { e _ { i }  e } } \\\\ { { E  E + e } } \\\\ { { g _ { E }  g _ { E } + g _ { e } } } \\end{array}$ end for $w$ update: $\\begin{array} { r l } & { w \\gets w - \\beta _ { 3 } \\cdot ( \\frac { 1 } { m } E ) \\cdot ( \\frac { 1 } { m } g _ { E } ) } \\\\ & { \\theta \\mathop { \\bf u p d a t e : } } \\\\ & { g _ { \\theta } = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\nabla _ { \\theta } \\log \\pi _ { \\theta } ( x , a _ { i } ) e _ { i } } \\\\ & { \\theta \\gets \\theta - \\beta _ { 4 } \\cdot g _ { \\theta } } \\end{array}$ state update: choose $\\bar { a } \\sim \\pi _ { \\theta } ( \\cdot | x )$ , and collect roll-out $( x , a , c , y )$ . $x \\gets y$   \nend for   \nOutput: $\\theta , w$ . ",
+        "bbox": [
+            176,
+            117,
+            826,
+            367
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            407,
+            815,
+            690
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "while the baseline plunges. When the perturbation of parameter reaches some level (related with the fixed $\\delta = 1 0$ ), our robust policy keeps the pole from falling over for a longer time, which indicates that our algorithm does learn some level of robustness, compared with baseline. If the perturbation of parameter is small, the baseline performs better, due to the fact that the perturbed environment is close to the default environment. ",
+        "bbox": [
+            173,
+            722,
+            825,
+            791
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            819,
+            328,
+            835
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we investigate the robust Reinforcement Learning with Wasserstein constraint. The derived theoretical framework can be reformulated into a tractable iterated-risk aware problem and the theoretical guarantee is then obtained by building connection between robustness to transition probabilities and robustness to states. Subsequently, we demonstrate the existence of optimal policies, provide a sensitivity analysis to reveal the effects of uncertainty set, and design a proper two-stage learning algorithm WRAAC. The experimental results on the Cart-Pole environment verified the effectiveness and robustness of our proposed approaches. ",
+        "bbox": [
+            174,
+            853,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/a598949459870ee8e6009da7b178e94976fd301af0448a983b43b37b839d7a67.jpg",
+        "image_caption": [
+            "Figure 1: Robustness to gravity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            223,
+            111,
+            470,
+            253
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/c5afc9d31e8c354fc4e577dc764b40a109c0fed76c22f56146f29a123ed93b56.jpg",
+        "image_caption": [
+            "Figure 2: Robustness to length. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            519,
+            111,
+            766,
+            252
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            168,
+            290,
+            823,
+            319
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Future works may favor a complete study for the effects of the radius of Wasserstein ball in our WRAAC algorithm. We are also interested in studying robust policy improvement in a data-driven situation where we only have access to the set of collected trajectories. ",
+        "bbox": [
+            174,
+            325,
+            825,
+            367
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            299,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The trajectory space $( \\Omega , { \\mathcal { F } } )$ , where $\\mathcal { F }$ is the $\\sigma$ -algebra of $\\Omega$ , satisfies ",
+        "bbox": [
+            173,
+            133,
+            622,
+            148
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ae7e84aac59695d12008c1b376cc2ce861d97ce0279449d2cf83bebc7b72c93b.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\bullet } & { \\mathbb { P } _ { x } ^ { \\pi , g } ( X _ { 0 } = x ) = 1 , } \\\\ { \\bullet } & { \\mathbb { P } _ { x } ^ { \\pi , g } ( d a | \\omega _ { n } ) = \\pi _ { n } ( d a | \\omega _ { n } ) , } \\\\ { \\bullet } & { \\mathbb { P } _ { x } ^ { \\pi , g } ( d q | \\tilde { \\omega } _ { n } ) = \\mathbb { 1 } ( g _ { n } ( \\tilde { \\omega } _ { n } ) \\in d q ) , } \\\\ { \\bullet } & { \\mathbb { P } _ { x } ^ { \\pi , g } ( X _ { n + 1 } \\in d x | \\omega _ { n } , a _ { n } , q _ { n } ) = q _ { n } ( X _ { n + 1 } \\in d x | \\omega _ { n } , a _ { n } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            157,
+            606,
+            234
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof of Lemma 1: ",
+        "bbox": [
+            174,
+            243,
+            300,
+            257
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof. (1) First, for $\\{ u _ { 1 } , u _ { 2 } \\} \\subset \\mathbb { U }$ , if $u _ { 1 } \\geq u _ { 2 }$ , it’s easy to have $H u _ { 1 } \\geq H u _ { 2 }$ , i.e., the operator $H$ is monotone about $u$ .   \n(2) For any real constant $C$ and $u \\in \\mathbb { U }$ , we can verify that $H ( u + C ) = H u + \\gamma C$ .   \n(3) For any $u _ { 1 } \\in \\mathbb { U }$ , $u _ { 2 } \\in \\mathbb { U }$ , there is $u _ { 1 } \\leq u _ { 2 } + | | u _ { 1 } - u _ { 2 } | | _ { \\infty }$ . Combining (1) and (2), we have $H u _ { 1 } \\leq H u _ { 2 } + \\gamma | | u _ { 1 } - u _ { 2 } | | _ { \\infty }$ , i.e., $H u _ { 1 } - H u _ { 2 } \\leq \\gamma | | u _ { 1 } - u _ { 2 } | | _ { \\infty }$ . Thus $| | H u _ { 1 } - H u _ { 2 } | | _ { \\infty } \\leq$ $\\gamma | | u _ { 1 } - u _ { 2 } | | _ { \\infty }$ . Furthermore, since $\\gamma \\in ( 0 , 1 )$ , the operator $H$ has the contract property in $\\mathbb { U }$ under $L _ { \\infty }$ norm. ",
+        "bbox": [
+            173,
+            272,
+            826,
+            372
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(4) Via Banach fixed-point theorem, there exist an unique $u ^ { * } \\in \\mathbb { U }$ satisfying $H u ^ { * } = u ^ { * }$ ",
+        "bbox": [
+            173,
+            371,
+            746,
+            386
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof of Theorem 1: ",
+        "bbox": [
+            174,
+            400,
+            308,
+            415
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof. Due to Assumption 1, for any $u \\in \\mathbb { U }$ , it is a measurable function on $\\mathbb { K } _ { A }$ , an $( H ^ { a } u ) ( x )$ is lower semi-continuous w.r.t. $a$ . Based on the measurable selection theorem (see Lemma 8.3.8 in (Hernandez-Lerma & Lasserre, 2012b)), there is a deterministic Markov stationary policy ´ $f \\in \\mathbb { F }$ , satisfying $H ^ { f } u ^ { * } = H u ^ { * } = u ^ { * }$ . □ ",
+        "bbox": [
+            174,
+            430,
+            825,
+            486
+        ],
+        "page_idx": 10
+    }
+]

parse/train/SJGCiw5gl/SJGCiw5gl.md

@@ -0,0 +1,387 @@
+# PRUNING CONVOLUTIONAL NEURAL NETWORKS FOR RESOURCE EFFICIENT INFERENCE
+
+Pavlo Molchanov, Stephen Tyree, Tero Karras, Timo Aila, Jan Kautz NVIDIA {pmolchanov, styree, tkarras, taila, jkautz}@nvidia.com
+
+# ABSTRACT
+
+We propose a new formulation for pruning convolutional kernels in neural networks to enable efficient inference. We interleave greedy criteria-based pruning with finetuning by backpropagation—a computationally efficient procedure that maintains good generalization in the pruned network. We propose a new criterion based on Taylor expansion that approximates the change in the cost function induced by pruning network parameters. We focus on transfer learning, where large pretrained networks are adapted to specialized tasks. The proposed criterion demonstrates superior performance compared to other criteria, e.g. the norm of kernel weights or feature map activation, for pruning large CNNs after adaptation to fine-grained classification tasks (Birds-200 and Flowers-102) relaying only on the first order gradient information. We also show that pruning can lead to more than $1 0 \times$ theoretical reduction in adapted 3D-convolutional filters with a small drop in accuracy in a recurrent gesture classifier. Finally, we show results for the largescale ImageNet dataset to emphasize the flexibility of our approach.
+
+# 1 INTRODUCTION
+
+Convolutional neural networks (CNN) are used extensively in computer vision applications, including object classification and localization, pedestrian and car detection, and video classification. Many problems like these focus on specialized domains for which there are only small amounts of carefully curated training data. In these cases, accuracy may be improved by fine-tuning an existing deep network previously trained on a much larger labeled vision dataset, such as images from ImageNet (Russakovsky et al., 2015) or videos from Sports-1M (Karpathy et al., 2014). While transfer learning of this form supports state of the art accuracy, inference is expensive due to the time, power, and memory demanded by the heavyweight architecture of the fine-tuned network.
+
+While modern deep CNNs are composed of a variety of layer types, runtime during prediction is dominated by the evaluation of convolutional layers. With the goal of speeding up inference, we prune entire feature maps so the resulting networks may be run efficiently even on embedded devices. We interleave greedy criteria-based pruning with fine-tuning by backpropagation, a computationally efficient procedure that maintains good generalization in the pruned network.
+
+Neural network pruning was pioneered in the early development of neural networks (Reed, 1993). Optimal Brain Damage (LeCun et al., 1990) and Optimal Brain Surgeon (Hassibi & Stork, 1993) leverage a second-order Taylor expansion to select parameters for deletion, using pruning as regularization to improve training and generalization. This method requires computation of the Hessian matrix partially or completely, which adds memory and computation costs to standard fine-tuning.
+
+In line with our work, Anwar et al. (2015) describe structured pruning in convolutional layers at the level of feature maps and kernels, as well as strided sparsity to prune with regularity within kernels. Pruning is accomplished by particle filtering wherein configurations are weighted by misclassification rate. The method demonstrates good results on small CNNs, but larger CNNs are not addressed.
+
+Han et al. (2015) introduce a simpler approach by fine-tuning with a strong $\ell _ { 2 }$ regularization term and dropping parameters with values below a predefined threshold. Such unstructured pruning is very effective for network compression, and this approach demonstrates good performance for intra-kernel pruning. But compression may not translate directly to faster inference since modern hardware exploits regularities in computation for high throughput. So specialized hardware may be needed for efficient inference of a network with intra-kernel sparsity (Han et al., 2016). This approach also requires long fine-tuning times that may exceed the original network training by a factor of 3 or larger. Group sparsity based regularization of network parameters was proposed to penalize unimportant parameters (Wen et al., 2016; Zhou et al., 2016; Alvarez & Salzmann, 2016; Lebedev & Lempitsky, 2016). Regularization-based pruning techniques require per layer sensitivity analysis which adds extra computations. In contrast, our approach relies on global rescaling of criteria for all layers and does not require sensitivity estimation. Moreover, our approach is faster as we directly prune unimportant parameters instead of waiting for their values to be made sufficiently small by optimization under regularization.
+
+Other approaches include combining parameters with correlated weights (Srinivas & Babu, 2015), reducing precision (Gupta et al., 2015; Rastegari et al., 2016) or tensor decomposition (Kim et al., 2015). These approaches usually require a separate training procedure or significant fine-tuning, but potentially may be combined with our method for additional speedups.
+
+# 2 METHOD
+
+The proposed method for pruning consists of the following steps: 1) Fine-tune the network until convergence on the target task; 2) Alternate iterations of pruning and further fine-tuning; 3) Stop pruning after reaching the target trade-off between accuracy and pruning objective, e.g. floating point operations (FLOPs) or memory utilization.
+
+The procedure is simple, but its success hinges on employing the right pruning criterion. In this section, we introduce several efficient pruning criteria and related technical considerations.
+
+![](images/5e0ef15f39588156fc82e686d0f6339a4e5abfa37f201046a638c44d125fb561.jpg)  
+Figure 1: Network pruning as a backward filter.
+
+Consider a set of training examples $\begin{array} { r l r l } { { \mathcal { D } } } & { { } = { } } & { \{ \mathcal { X } }  & { { } = } \end{array}$ $\left\{ { \bf x } _ { 0 } , { \bf x } _ { 1 } , . . . , { \bf x } _ { N } \} , \mathcal { V } = \left\{ y _ { 0 } , y _ { 1 } , . . . , y _ { N } \right\} \right\}$ , where $\mathbf { x }$ and $y$ represent an inparameters1 $\mathcal { W } = \{ ( \mathbf { w } _ { 1 } ^ { 1 } , b _ { 1 } ^ { \bar { 1 } } ) , ( \mathbf { w } _ { 1 } ^ { 2 } , b _ { 1 } ^ { 2 } ) , . . . ( \mathbf { w } _ { L } ^ { C _ { \ell } } , b _ { L } ^ { \bar { C } _ { \ell } } ) \}$ he network’sare optimized $\mathcal { C } ( \mathcal { D } | \mathcal { W } )$   
+a cost function $\mathcal { C } ( \cdot )$ is a negative log-likelihood function. A cost function is selected independently of pruning and depends only on the task to be solved by the original network. In the case of transfer learning, we adapt a large network initialized with parameters ${ \mathcal { W } } _ { 0 }$ pretrained on a related but distinct dataset.   
+During pruning, we refine a subset of parameters which preserves   
+the accuracy of the adapted network, $\bar { \mathcal { C } } ( \mathcal { D } | \mathcal { W } ^ { \prime } ) \approx \mathcal { C } ( \mathcal { D } | \bar { \mathcal { W } } )$ . This corresponds to a combinatorial   
+optimization:
+
+$$
+\operatorname* { m i n } _ { W ^ { \prime } } \left| \mathcal { C } ( \mathcal { D } | \mathcal { W } ^ { \prime } ) - \mathcal { C } ( \mathcal { D } | \mathcal { W } ) \right| \quad \mathrm { s . t . } \quad | | \mathcal { W } ^ { \prime } | | _ { 0 } \leq B ,
+$$
+
+where the $\ell _ { 0 }$ norm in $| | \mathcal { W } ^ { \prime } | | _ { 0 }$ bounds the number of non-zero parameters $B$ in $W ^ { \prime }$ . Intuitively, if $\mathcal { W } ^ { \prime } = \mathcal { W }$ we reach the global minimum of the error function, however $| | \mathcal { W } ^ { \prime } | | _ { 0 }$ will also have its maximum.
+
+Finding a good subset of parameters while maintaining a cost value as close as possible to the original is a combinatorial problem. It will require $2 ^ { | \mathcal { W } | }$ evaluations of the cost function for a selected subset of data. For current networks it would be impossible to compute: for example, VGG-16 has $| \mathcal { W } | = 4 2 2 4$ convolutional feature maps. While it is impossible to solve this optimization exactly for networks of any reasonable size, in this work we investigate a class of greedy methods. Starting with a full set of parameters $\mathcal { W }$ , we iteratively identify and remove the least important parameters, as illustrated in Figure 1. By removing parameters at each iteration, we ensure the eventual satisfaction of the $\ell _ { 0 }$ bound on $\mathcal { W } ^ { \prime }$ .
+
+Since we focus our analysis on pruning feature maps from convolutional layers, let us denote a set of image feature maps by $\mathbf { z } _ { \ell } \doteq \mathbb { R } ^ { H _ { \ell } ^ { \smile } \times W _ { \ell } \times C _ { \ell } }$ with dimensionality $H _ { \ell } \times W _ { \ell }$ and $C _ { \ell }$ individual maps (or channels).2 The feature maps can either be the input to the network, $\mathbf { z } _ { 0 }$ , or the output from a convolutional layer, $\mathbf { z } _ { \ell }$ with $\ell \in [ 1 , 2 , . . . , L ]$ . Individual feature maps are denoted $\mathbf { z } _ { \ell } ^ { ( k ) }$ for $k \in [ 1 , 2 , . . . , C _ { \ell } ]$ . A convolutional layer $\ell$ applies the convolution operation $( * )$ to a set of input feature maps $\mathbf { z } _ { \ell - 1 }$ with kernels parameterized by $\mathbf { w } _ { \ell } ^ { ( k ) } \in \mathbb { R } ^ { C _ { \ell - 1 } \times p \times p }$ :
+
+$$
+\begin{array} { r } { \mathbf { z } _ { \ell } ^ { ( k ) } = \mathbf { g } _ { \ell } ^ { ( k ) } \mathcal { R } \big ( \mathbf { z } _ { \ell - 1 } \ast \mathbf { w } _ { \ell } ^ { ( k ) } + b _ { \ell } ^ { ( k ) } \big ) , } \end{array}
+$$
+
+where $\mathbf { z } _ { \ell } ^ { ( k ) } \in \mathbb { R } ^ { H _ { \ell } \times W _ { \ell } }$ is the result of convolving each of $C _ { \ell - 1 }$ kernels of size $p \times p$ with its respective input feature map and adding bias $b _ { \ell } ^ { ( k ) }$ . We introduce a pruning gate $\mathbf { g } _ { l } \in \{ 0 , 1 \} ^ { C _ { l } }$ , an external switch which determines if a particular feature map is included or pruned during feed-forward propagation, such that when $\mathbf { g }$ is vectorized: $w ^ { \prime } = \mathbf { g } \mathcal { W }$ .
+
+# 2.1 ORACLE PRUNING
+
+Minimizing the difference in accuracy between the full and pruned models depends on the criterion for identifying the “least important” parameters, called saliency, at each step. The best criterion would be an exact empirical evaluation of each parameter, which we denote the oracle criterion, accomplished by ablating each non-zero parameter $w \in \mathcal { W } ^ { \prime }$ in turn and recording the cost’s difference.
+
+We distinguish two ways of using this oracle estimation of importance: 1) oracle-loss quantifies importance as the signed change in loss, $\mathcal { C } ( \mathcal { D } | \mathcal { W } ^ { \prime } ) - \mathcal { C } ( \mathcal { D } | \mathcal { W } )$ , and 2) oracle-abs adopts the absolute difference, $| \mathcal { C } ( \mathcal { D } | \mathcal { W } ^ { \prime } ) - \mathcal { C } ( \mathcal { D } | \mathcal { W } ) |$ . While both discourage pruning which increases the loss, the oracle-loss version encourages pruning which may decrease the loss, while oracle-abs penalizes any pruning in proportion to its change in loss, regardless of the direction of change.
+
+While the oracle is optimal for this greedy procedure, it is prohibitively costly to compute, requiring $| | W ^ { \prime } | | _ { 0 }$ evaluations on a training dataset, one evaluation for each remaining non-zero parameter. Since estimation of parameter importance is key to both the accuracy and the efficiency of this pruning approach, we propose and evaluate several criteria in terms of performance and estimation cost.
+
+# 2.2 CRITERIA FOR PRUNING
+
+There are many heuristic criteria which are much more computationally efficient than the oracle. For the specific case of evaluating the importance of a feature map (and implicitly the set of convolutional kernels from which it is computed), reasonable criteria include: the combined $\ell _ { 2 }$ -norm of the kernel weights, the mean, standard deviation or percentage of the feature map’s activation, and mutual information between activations and predictions. We describe these criteria in the following paragraphs and propose a new criterion which is based on the Taylor expansion.
+
+Minimum weight. Pruning by magnitude of kernel weights is perhaps the simplest possible criterion, and it does not require any additional computation during the fine-tuning process. In case of pruning according to the norm of a set of weights, the criterion is evaluated as: $\begin{array} { r } { \bar { \Theta } _ { M W } : \mathbb { R } ^ { C _ { \ell - 1 } \times p \times p } \stackrel { \cdot } {  } \mathbb { R } } \end{array}$ , with $\begin{array} { r } { \Theta _ { M W } ( \mathbf { \bar { w } } ) = \frac { 1 } { | \mathbf { w } | } \sum _ { i } w _ { i } ^ { 2 } } \end{array}$ , where $| \mathbf { w } |$ is dimensionality of the set of weights after vectorization. The motivation to apply this type of pruning is that a convolutional kernel with low $\ell _ { 2 }$ norm detects less important features than those with a high norm. This can be aided during training by applying $\ell _ { 1 }$ or $\ell _ { 2 }$ regularization, which will push unimportant kernels to have smaller values.
+
+Activation. One of the reasons for the popularity of the ReLU activation is the sparsity in activation that is induced, allowing convolutional layers to act as feature detectors. Therefore it is reasonable to assume that if an activation value (an output feature map) is small then this feature detector is not important for prediction task at hand. We may evaluate this by mean activation, $\Theta _ { M A }$ : $\mathbb { R } ^ { H _ { l } \times W _ { \ell } \times C _ { \ell } } \to \mathbb { R }$ , with $\begin{array} { r } { \Theta _ { M A } ( { \bf a } ) = \frac { 1 } { \lvert { \bf a } \rvert } \sum _ { i } a _ { i } } \end{array}$ for activation $\mathbf { a } = \mathbf { z } _ { l } ^ { ( k ) }$ , or by the standard deviation of the activation, $\begin{array} { r } { \Theta _ { M A _ { - } s t d } ( \mathbf { a } ) = \sqrt { \frac { 1 } { | \mathbf { a } | } \sum _ { i } ( a _ { i } - \mu _ { \mathbf { a } } ) ^ { 2 } } , } \end{array}$ .
+
+Mutual information. Mutual information (MI) is a measure of how much information is present in one variable about another variable. We apply MI as a criterion for pruning, $\Theta _ { M I } : \mathbb { R } ^ { H _ { l } \times W _ { \ell } \times C _ { \ell } }  \mathbb { R }$ , with $\Theta _ { M I } ( \mathbf { a } ) = M I ( \mathbf { a } , y )$ , where $y$ is the target of neural network. MI is defined for continuous variables, so to simplify computation, we exchange it with information gain (IG), which is defined for quantized variables $I G ( y | x ) = H ( x ) + H ( y ) - H ( x , y )$ , where $H ( x )$ is the entropy of variable $x$ . We accumulate statistics on activations and ground truth for a number of updates, then quantize the values and compute IG.
+
+Taylor expansion. We phrase pruning as an optimization problem, trying to find $\mathcal { W } ^ { \prime }$ with bounded number of non-zero elements that minimize $\left| \Delta \dot { C } ( h _ { i } ) \right| = \left| \dot { \mathcal { C } ( \mathcal { D } | \mathcal { W } ^ { \prime } ) } - \dot { \mathcal { C } } ( \bar { \mathcal { D } } | \mathcal { W } ) \right|$ . With this approach based on the Taylor expansion, we directly approximate change in the loss function from removing a particular parameter. Let $h _ { i }$ be the output produced from parameter $i$ . In the case of feature maps, $h = \{ z _ { 0 } ^ { ( 1 ) } , z _ { 0 } ^ { ( 2 ) } , . . . , z _ { L } ^ { ( C _ { \ell } ) } \}$ z(C\`)L }. For notational convenience, we consider the cost function equally dependent on parameters and outputs computed from parameters: $\mathcal { C } ( D | h _ { i } ) = \mathcal { C } ( \mathcal { D } | ( \mathbf { w } , b ) _ { i } )$ . Assuming independence of parameters, we have:
+
+$$
+\big | \Delta \mathcal { C } ( h _ { i } ) \big | = \big | \mathcal { C } ( \mathcal { D } , h _ { i } = 0 ) - \mathcal { C } ( \mathcal { D } , h _ { i } ) \big | ,
+$$
+
+where $\mathcal { C } ( \mathcal { D } , h _ { i } = 0 )$ is a cost value if output $h _ { i }$ is pruned, while $\mathcal { C } ( \mathcal { D } , h _ { i } )$ is the cost if it is not pruned. While parameters are in reality inter-dependent, we already make an independence assumption at each gradient step during training.
+
+To approximate $\Delta \mathcal { C } ( h _ { i } )$ , we use the first-degree Taylor polynomial. For a function $f ( x )$ , the Taylor expansion at point $x = a$ is
+
+$$
+f ( x ) = \sum _ { p = 0 } ^ { P } \frac { f ^ { ( p ) } ( a ) } { p ! } ( x - a ) ^ { p } + R _ { p } ( x ) ,
+$$
+
+where $f ^ { ( p ) } ( a )$ is the $p$ -th derivative of $f$ evaluated at point $a$ , and $R _ { p } ( x )$ is the $p$ -th order remainder. Approximating $\mathcal { C } ( \mathcal { D } , h _ { i } = 0 )$ ) with a first-order Taylor polynomial near $h _ { i } = 0$ , we have:
+
+$$
+\mathcal { C } ( \mathcal { D } , h _ { i } = 0 ) \ = \ \mathcal { C } ( \mathcal { D } , h _ { i } ) - \frac { \delta \mathcal { C } } { \delta h _ { i } } h _ { i } + R _ { 1 } ( h _ { i } = 0 ) .
+$$
+
+The remainder $R _ { 1 } ( h _ { i } = 0 )$ can be calculated through the Lagrange form:
+
+$$
+R _ { 1 } ( h _ { i } = 0 ) = \frac { \delta ^ { 2 } \mathcal { C } } { \delta ( h _ { i } ^ { 2 } = \xi ) } \frac { h _ { i } ^ { 2 } } { 2 } ,
+$$
+
+where $\xi$ is a real number between 0 and $h _ { i }$ . However, we neglect this first-order remainder, largely due to the significant calculation required, but also in part because the widely-used ReLU activation function encourages a smaller second order term. Finally, by substituting Eq. (5) into Eq. (3) and ignoring the remainder, we have $\Theta _ { T E } : \mathbb { R } ^ { H _ { l } \times W _ { l } \times C _ { l } }  \bar { \mathbb { R } ^ { + } }$ , with
+
+$$
+\Theta _ { T E } ( h _ { i } ) = \left| \Delta \mathcal { C } ( h _ { i } ) \right| = \left| \mathcal { C } ( \mathcal { D } , h _ { i } ) - \frac { \delta \mathcal { C } } { \delta h _ { i } } h _ { i } - \mathcal { C } ( \mathcal { D } , h _ { i } ) \right| = \left| \frac { \delta \mathcal { C } } { \delta h _ { i } } h _ { i } \right| .
+$$
+
+Intuitively, this criterion prunes parameters that have an almost flat gradient of the cost function w.r.t. feature map $h _ { i }$ . This approach requires accumulation of the product of the activation and the gradient of the cost function w.r.t. to the activation, which is easily computed from the same computations for back-propagation. $\Theta _ { T E }$ is computed for a multi-variate output, such as a feature map, by
+
+$$
+\Theta _ { T E } ( z _ { l } ^ { ( k ) } ) = \bigg | \frac { 1 } { M } \sum _ { m } \frac { \delta C } { \delta z _ { l , m } ^ { ( k ) } } z _ { l , m } ^ { ( k ) } \bigg | ,
+$$
+
+where $M$ is length of vectorized feature map. For a minibatch with $T > 1$ examples, the criterion is computed for each example separately and averaged over $T$ .
+
+Independently of our work, Figurnov et al. (2016) came up with similar metric based on the Taylor expansion, called impact, to evaluate importance of spatial cells in a convolutional layer. It shows that the same metric can be applied to evaluate importance of different groups of parameters.
+
+Relation to Optimal Brain Damage. The Taylor criterion proposed above relies on approximating the change in loss caused by removing a feature map. The core idea is the same as in Optimal Brain Damage (OBD) (LeCun et al., 1990). Here we consider the differences more carefully.
+
+$\begin{array} { r } { y = \frac { \delta \mathcal { C } } { \delta h } h } \end{array}$ ry difference is thfor cost function erm tends to zero: $\mathcal { C }$ nt ofddeand rder tertivation . At fac $h$ of the. Aftvalue aylor expansion, in our notation sufficient training epochs, theoffers little useful information, $\frac { \delta { \mathcal { C } } } { \delta h }  0$ $\mathbb { E } ( y ) = 0$ $y$ hence OBD regards the term as zero and focuses on the second-order term.
+
+However, the variance of $y$ is non-zero and correlates with the stability of the local function w.r.t. activation $h$ . By considering the absolute change in the cost3 induced by pruning (as in Eq. 3), we use the absolute value of the first-order term, $| y |$ . Under assumption that samples come from independent and identical distribution, $\mathbb { E } ( | y | ) = \sigma { \sqrt { 2 } } / { \sqrt { \pi } }$ where $\sigma$ is the standard deviation of $y$ , known as the expected value of the half-normal distribution. So, while $y$ tends to zero, the expectation of $| y |$ is proportional to the variance of $y$ , a value which is empirically more informative as a pruning criterion.
+
+As an additional benefit, we avoid the computation of the second-order Taylor expansion term, or its simplification - diagonal of the Hessian, as required in OBD.
+
+We found important to compare proposed Taylor criteria to OBD. As described in the original papers (LeCun et al., 1990; 1998), OBD can be efficiently implemented similarly to standard back propagation algorithm doubling backward propagation time and memory usage when used together with standard fine-tuning. Efficient implementation of the original OBD algorithm might require significant changes to the framework based on automatic differentiation like Theano to efficiently compute only diagonal of the Hessian instead of the full matrix. Several researchers tried to tackle this problem with approximation techniques (Martens, 2010; Martens et al., 2012). In our implementation, we use efficient way of computing Hessian-vector product (Pearlmutter, 1994) and matrix diagonal approximation proposed by (Bekas et al., 2007), please refer to more details in appendix. With current implementation, OBD is 30 times slower than Taylor technique for saliency estimation, and 3 times slower for iterative pruning, however with different implementation can only be $50 \%$ slower as mentioned in the original paper.
+
+Average Percentage of Zeros (APoZ). Hu et al. (2016) proposed to explore sparsity in activations for network pruning. ReLU activation function imposes sparsity during inference, and average percentage of positive activations at the output can determine importance of the neuron. Intuitively, it is a good criteria, however feature maps at the first layers have similar APoZ regardless of the network’s target as they learn to be Gabor like filters. We will use APoZ to estimate saliency of feature maps.
+
+# 2.3 NORMALIZATION
+
+Some criteria return “raw” values, whose scale varies with the depth of the parameter’s layer in the network. A simple layer-wise $\ell _ { 2 }$ -normalization can achieve adequate rescaling across layers:
+
+$$
+\hat { \Theta } ( \mathbf { z } _ { l } ^ { ( k ) } ) = \frac { \Theta ( \mathbf { z } _ { l } ^ { ( k ) } ) } { \sqrt { \sum _ { j } \left( \Theta ( \mathbf { z } _ { l } ^ { ( j ) } ) \right) ^ { 2 } } } .
+$$
+
+# 2.4 FLOPS REGULARIZED PRUNING
+
+One of the main reasons to apply pruning is to reduce number of operations in the network. Feature maps from different layers require different amounts of computation due the number and sizes of input feature maps and convolution kernels. To take this into account we introduce FLOPs regularization:
+
+$$
+\Theta ( \mathbf { z } _ { l } ^ { ( k ) } ) = \Theta ( \mathbf { z } _ { l } ^ { ( k ) } ) - \lambda \Theta _ { l } ^ { f l o p s } ,
+$$
+
+where $\lambda$ controls the amount of regularization. For our experiments, we use $\lambda = 1 0 ^ { - 3 }$ . $\Theta ^ { f l o p s }$ is computed under the assumption that convolution is implemented as a sliding window (see Appendix). Other regularization conditions may be applied, e.g. storage size, kernel sizes, or memory footprint.
+
+![](images/7fe7129df240cf9ae8d98709d3020505acef56bbe5ea0edf667a6f729cb30b60.jpg)  
+Figure 2: Global statistics of oracle ranking, shown by layer for Birds-200 transfer learning.
+
+![](images/26f057cd82b064948e11ae7322c6ce4962c9162ccb788834c1a5d0e8caf66437.jpg)  
+Figure 3: Pruning without fine-tuning using oracle ranking for Birds-200 transfer learning.
+
+# 3 RESULTS
+
+We empirically study the pruning criteria and procedure detailed in the previous section for a variety of problems. We focus many experiments on transfer learning problems, a setting where pruning seems to excel. We also present results for pruning large networks on their original tasks for more direct comparison with the existing pruning literature. Experiments are performed within Theano (Theano Development Team, 2016). Training and pruning are performed on the respective training sets for each problem, while results are reported on appropriate holdout sets, unless otherwise indicated. For all experiments we prune a single feature map at every pruning iteration, allowing fine-tuning and re-evaluation of the criterion to account for dependency between parameters.
+
+# 3.1 CHARACTERIZING THE ORACLE RANKING
+
+We begin by explicitly computing the oracle for a single pruning iteration of a visual transfer learning problem. We fine-tune the VGG-16 network (Simonyan & Zisserman, 2014) for classification of bird species using the Caltech-UCSD Birds 200-2011 dataset (Wah et al., 2011). The dataset consists of nearly 6000 training images and 5700 test images, covering 200 species. We fine-tune VGG-16 for 60 epochs with learning rate 0.0001 to achieve a test accuracy of $7 2 . 2 \%$ using uncropped images.
+
+To compute the oracle, we evaluate the change in loss caused by removing each individual feature map from the fine-tuned VGG-16 network. (See Appendix A.3 for additional analysis.) We rank feature maps by their contributions to the loss, where rank 1 indicates the most important feature map—removing it results in the highest increase in loss—and rank 4224 indicates the least important. Statistics of global ranks are shown in Fig. 2 grouped by convolutional layer. We observe: (1) Median global importance tends to decrease with depth. (2) Layers with max-pooling tend to be more important than those without. (VGG-16 has pooling after layers 2, 4, 7, 10, and 13.) However, (3) maximum and minimum ranks show that every layer has some feature maps that are globally important and others that are globally less important. Taken together with the results of subsequent experiments, we opt for encouraging a balanced pruning that distributes selection across all layers.
+
+Next, we iteratively prune the network using pre-computed oracle ranking. In this experiment, we do not update the parameters of the network or the oracle ranking between iterations. Training accuracy is illustrated in Fig. 3 over many pruning iterations. Surprisingly, pruning by smallest absolute change in loss (Oracle-abs) yields higher accuracy than pruning by the net effect on loss (Oracle-loss). Even though the oracle indicates that removing some feature maps individually may decrease loss, instability accumulates due the large absolute changes that are induced. These results support pruning by absolute difference in cost, as constructed in Eq. 1.
+
+# 3.2 EVALUATING PROPOSED CRITERIA VERSUS THE ORACLE
+
+To evaluate computationally efficient criteria as substitutes for the oracle, we compute Spearman’s rank correlation, an estimate of how well two predictors provide monotonically related outputs,
+
+Table 1: Spearman’s rank correlation of criteria vs. oracle for convolutional feature maps of VGG-16 and AlexNet fine-tuned on Birds-200 and Flowers-102 datasets, and AlexNet trained on ImageNet.   
+
+<table><tr><td rowspan="3"></td><td colspan="6">AlexNet /Flowers-102</td><td colspan="6">VGG-16/Birds-200</td></tr><tr><td rowspan="2">Weight</td><td rowspan="2">Mean</td><td colspan="2">Activation S.d.</td><td rowspan="2">OBD</td><td rowspan="2">Taylor</td><td rowspan="2">Weight</td><td colspan="2">Activation</td><td rowspan="2">OBD</td><td rowspan="2">Taylor</td><td rowspan="2">Mutual Info.</td></tr><tr><td>APoZ</td><td></td><td>Mean S.d.</td><td>APoZ</td></tr><tr><td>Per layer</td><td>0.17</td><td>0.65</td><td>0.67</td><td>0.54</td><td>0.64 0.77</td><td></td><td>0.27 0.56</td><td>0.57</td><td>0.35</td><td>0.59</td><td>0.73</td><td>0.28</td></tr><tr><td>All layers</td><td>0.28</td><td>0.51</td><td>0.53</td><td>0.68</td><td>0.37</td><td></td><td>0.34</td><td>0.35 0.30</td><td>0.43</td><td>0.65</td><td>0.14</td><td>0.35</td></tr><tr><td>(w/ l2-norm)</td><td>0.13</td><td>0.63</td><td>0.61</td><td>0.41 0.60</td><td></td><td>0.75</td><td>0.33 0.64</td><td>0.66</td><td>0.51</td><td>1</td><td>0.73</td><td>0.47</td></tr><tr><td colspan="5">AlexNet/Birds-200</td><td colspan="6">VGG-16/Flowers-102</td><td rowspan="4"></td></tr><tr><td>Per layer</td><td>0.36</td><td>0.57</td><td>0.65</td><td>0.42 0.54</td><td>0.81</td><td></td><td>0.51</td><td>0.47</td><td>0.36</td><td>0.21</td><td>0.6</td></tr><tr><td>All layers</td><td>0.32</td><td>0.37</td><td>0.51</td><td></td><td></td><td></td><td>0.19</td><td></td><td></td><td></td><td></td></tr><tr><td>(w/ l2-norm)</td><td>0.23</td><td>0.54 0.57</td><td>0.28 0.49</td><td>0.61 1</td><td>0.37 0.78</td><td>0.35 0.28</td><td>0.53 0.66</td><td>0.45 0.65</td><td>0.61 0.61</td><td>0.28 1</td><td>0.02 0.7</td></tr><tr><td colspan="9">AlexNet/ImageNet</td><td></td><td></td><td></td></tr><tr><td>Per layer</td><td>0.57</td><td>0.09</td><td>0.19</td><td>-0.06</td><td>0.58</td><td>0.58</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>All layers</td><td>0.67</td><td></td><td></td><td></td><td></td><td>0.11</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(w/ lz-norm)</td><td>0.44</td><td>0.00 0.10</td><td>0.13 0.19</td><td>-0.08 0.19</td><td>0.72 -</td><td>0.55</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+![](images/3619e15867c1012f68a8762ade1c9b939f48ebbfa25334d28cfbc56f9050e842.jpg)  
+Figure 4: Pruning of feature maps in VGG-16 fine-tuned on the Birds-200 dataset.
+
+even if their relationship is not linear. Given the difference between oracle4 and criterion ranks $d _ { i } = r a n k ( \Theta _ { o r a c l e } ( i ) ) - r a n k ( \Theta _ { c r i t e r i o n } ( i ) )$ for each parameter $i$ , the rank correlation is computed:
+
+$$
+\mathcal { S } = 1 - \frac { 6 } { N ( N ^ { 2 } - 1 ) } \sum _ { i = 1 } ^ { N } { d _ { i } } ^ { 2 } ,
+$$
+
+where $N$ is the number of parameters (and the highest rank). This correlation coefficient takes values in $[ - 1 , 1 ]$ , where $- 1$ implies full negative correlation, 0 no correlation, and 1 full positive correlation.
+
+We show Spearman’s correlation in Table 1 to compare the oracle-abs ranking to rankings by different criteria on a set of networks/datasets some of which are going to be introduced later. Data-dependent criteria (all except weight magnitude) are computed on training data during the fine-tuning before or between pruning iterations. As a sanity check, we evaluate random ranking and observe 0.0 correlation across all layers. “Per layer” analysis shows ranking within each convolutional layer, while “All layers” describes ranking across layers. While several criteria do not scale well across layers with raw values, a layer-wise $\ell _ { 2 }$ -normalization significantly improves performance. The Taylor criterion has the highest correlation among the criteria, both within layers and across layers (with $\ell _ { 2 }$ normalization). OBD shows the best correlation across layers when no normalization used; it also shows best results for correlation on ImageNet dataset. (See Appendix A.2 for further analysis.)
+
+# 3.3 PRUNING FINE-TUNED IMAGENET NETWORKS
+
+We now evaluate the full iterative pruning procedure on two transfer learning problems. We focus on reducing the number of convolutional feature maps and the total estimated floating point operations (FLOPs). Fine-grained recognition is difficult for relatively small datasets without relying on transfer learning. Branson et al. (2014) show that training CNN from scratch on the Birds-200 dataset achieves test accuracy of only $1 0 . 9 \%$ . We compare results to training a randomly initialized CNN with half the number of parameters per layer, denoted "from scratch".
+
+![](images/822f4ff9f919bb1d8f8e9b83b6da949a22e0ff915fd8c12fd1bba68c34172521.jpg)  
+Figure 5: Pruning of feature maps in AlexNet on fine-tuned on Flowers-102.
+
+Fig. 4 shows pruning of VGG-16 after fine-tuning on the Birds-200 dataset (as described previously). At each pruning iteration, we remove a single feature map and then perform 30 minibatch SGD updates with batch-size 32, momentum 0.9, learning rate $1 \dot { 0 } ^ { - 4 }$ , and weight decay $1 0 ^ { - 4 }$ . The figure depicts accuracy relative to the pruning rate (left) and estimated GFLOPs (right). The Taylor criterion shows the highest accuracy for nearly the entire range of pruning ratios, and with FLOPs regularization demonstrates the best performance relative to the number of operations. OBD shows slightly worse performance of pruning in terms of parameters, however significantly worse in terms of FLOPs.
+
+In Fig. 5, we show pruning of the CaffeNet implementation of AlexNet (Krizhevsky et al., 2012) after adapting to the Oxford Flowers 102 dataset (Nilsback & Zisserman, 2008), with 2040 training and 6129 test images from 102 species of flowers. Criteria correlation with oracle-abs is summarized in Table 1. We initially fine-tune the network for 20 epochs using a learning rate of 0.001, achieving a final test accuracy of $8 0 . 1 \%$ . Then pruning procedes as previously described for Birds-200, except with only 10 mini-batch updates between pruning iterations. We observe the superior performance of the Taylor and OBD criteria in both number of parameters and GFLOPs.
+
+We observed that Taylor criterion shows the best performance which is closely followed by OBD with a bit lower Spearman’s rank correlation coefficient. Implementing OBD takes more effort because of computation of diagonal of the Hessian and it is $50 \%$ to $300 \%$ slower than Taylor criteria that relies on first order gradient only.
+
+Fig. 6 shows pruning with the Taylor technique and a varying number of fine-tuning updates between pruning iterations. Increasing the number of updates results in higher accuracy, but at the cost of additional runtime of the pruning procedure.
+
+During pruning we observe a small drop in accuracy. One of the reasons is fine-tuning between pruning iterations. Accuracy of the initial network can be improved with longer fine tunning and search of better optimization parameters. For example accuracy of unpruned VGG16 network on Birds-200 goes up to $7 5 \%$ after extra $1 2 8 \mathrm { k }$ updates. And AlexNet on Flowers-102 goes up to $8 2 . 9 \%$ after $1 3 0 \mathrm { k }$ updates. It should be noted that with farther fine-tuning of pruned networks we can achieve higher accuracy as well, therefore the one-to-one comparison of accuracies is rough.
+
+# 3.4 PRUNING A RECURRENT 3D-CNN NETWORK FOR HAND GESTURE RECOGNITION
+
+Molchanov et al. (2016) learn to recognize 25 dynamic hand gestures in streaming video with a large recurrent neural network. The network is constructed by adding recurrent connections to a 3D-CNN pretrained on the Sports-1M video dataset (Karpathy et al., 2014) and fine tuning on a gesture dataset. The full network achieves an accuracy of $8 0 . { \bar { 7 } } \%$ when trained on the depth modality, but a single inference requires an estimated 37.8 GFLOPs, too much for deployment on an embedded GPU. After several iterations of pruning with the Taylor criterion with learning rate 0.0003, momentum 0.9, FLOPs regularization $1 0 ^ { - 3 }$ , we reduce inference to 3.0 GFLOPs, as shown in Fig. 7. While pruning increases classification error by nearly $6 \%$ , additional fine-tuning restores much of the lost accuracy, yielding a final pruned network with a $1 2 . 6 \times$ reduction in GFLOPs and only a $2 . 5 \%$ loss in accuracy.
+
+![](images/038373025e262cf10fd623e13c94664ea2dd66069597f4da6a9719238ddc69b8.jpg)  
+Figure 6: Varying the number of minibatch updates between pruning iterations with AlexNet/Flowers-102 and the Taylor criterion.
+
+![](images/cb229bfde9ed46387d76a6d50f8a2d64269f3a0b7a80bcbb9dd4fbc7c43bf0ea.jpg)  
+Figure 7: Pruning of a recurrent 3D-CNN for dynamic hand gesture recognition (Molchanov et al., 2016).
+
+![](images/cb7b49f5685ff7bd9bdeeedba7cb02bece13d8eb6d67d12bfe3ef712dab5b3b0.jpg)  
+Figure 8: Pruning of AlexNet on Imagenet with varying number of updates between pruning iterations.
+
+![](images/d601c52809af81845b0c530255babe65cfaaf1767a11d40d517b1740701ae618.jpg)
+
+# 3.5 PRUNING NETWORKS FOR IMAGENET
+
+We also test our pruning scheme on the largescale ImageNet classification task. In the first experiment, we begin with a trained CaffeNet implementation of AlexNet with $7 9 . 2 \%$ top-5 validation accuracy. Between pruning iterations, we fine-tune with learning rate $1 0 ^ { - 4 }$ , momen$\mathrm { t u m } 0 . 9$ , weight decay $1 0 ^ { - 4 }$ , batch size 32, and drop-out $5 0 \%$ . Using a subset of 5000 training images, we compute oracle-abs and Spearman’s rank correlation with the criteria, as shown in Table 1. Pruning traces are illustrated in Fig. 8. We observe: 1) Taylor performs better than random or minimum weight pruning when 100 updates are used between pruning iterations. When results are displayed w.r.t. FLOPs, the difference with random pruning is only $0 \% - 4 \%$ , but the difference is higher, $1 \% - \mathrm { i } 0 \%$ , when plotted with the number of feature maps pruned. 2) Increasing the number of updates from 100 to 1000 improves performance of pruning significantly for both the Taylor criterion and random pruning.
+
+![](images/cffc122da067354b60cec34a5cc11ec4923edf0e2b2704848395409dc29722ab.jpg)  
+Figure 9: Pruning of the VGG-16 network on ImageNet, with additional following fine-tuning at 11.5 and 8 GFLOPs.
+
+Table 2: Actual speed up of networks pruned by Taylor criterion for various hardware setup. All measurements were performed with PyTorch with cuDNN v5.1.0, except R3DCNN which was implemented in $\mathrm { C } { + } { + }$ with cuDNN v4.0.4). Results for ImageNet dataset are reported as top-5 accuracy on validation set. Results on AlexNet / Flowers-102 are reported for pruning with 1000 updates between iterations and no fine-tuning after pruning.   
+
+<table><tr><td>Hardware</td><td>Batch</td><td>Accuracy</td><td>Time, ms</td><td>Accuracy</td><td>Time (speed up)</td><td>Accuracy Time (speed up)</td></tr><tr><td>AlexNet/Flowers-102,1.46 GFLOPs</td><td></td><td></td><td></td><td>41% feature maps,0.4GFLOPs</td><td></td><td>19.5% feature maps, 0.2 GFLOPs</td></tr><tr><td>CPU: Intel Core i7-5930K</td><td>16</td><td>80.1%</td><td>226.4</td><td>79.8%(-0.3%)</td><td>121.4 (1.9x) 74.1%(-6.0%)</td><td>87.0 (2.6x)</td></tr><tr><td>GPU: GeForce GTX TITAN X(Pascal)</td><td>16</td><td></td><td>4.8</td><td></td><td>2.4 (2.0x)</td><td>1.9 (2.5x)</td></tr><tr><td>GPU:GeForce GTX TITANX(Pascal)</td><td>512</td><td></td><td>88.3</td><td></td><td>36.6 (2.4x)</td><td>27.4 (3.2x)</td></tr><tr><td>GPU: NVIDIA Jetson TX1</td><td>32</td><td></td><td>169.2</td><td></td><td>73.6 (2.3x)</td><td>58.6 (2.9x)</td></tr><tr><td>VGG-16/ImageNet,30.96 GFLOPs</td><td></td><td></td><td></td><td>66% feature maps,11.5 GFLOPs</td><td></td><td>52% feature maps, 8.0 GFLOPs</td></tr><tr><td>CPU:Intel Core i7-5930K</td><td>16</td><td>89.3%</td><td>2564.7</td><td>87.0% (-2.3%)</td><td>1483.3 (1.7x) 84.5% (-4.8%)</td><td>1218.4 (2.1x)</td></tr><tr><td>GPU: GeForce GTX TITAN X(Pascal)</td><td>16</td><td></td><td>68.3</td><td></td><td>31.0 (2.2x)</td><td>20.2 (3.4x)</td></tr><tr><td>GPU: NVIDIA Jetson TX1</td><td>4</td><td></td><td>456.6</td><td></td><td>182.5 (2.5x)</td><td>138.2 (3.3x)</td></tr><tr><td>R3DCNN/nvGesture,37.8 GFLOPs</td><td></td><td></td><td></td><td>25% feature maps,3GFLOPs</td><td></td><td></td></tr><tr><td>GPU:GeForce GT730M</td><td>1</td><td>80.7%</td><td>438.0</td><td>78.2% (-2.5%)</td><td>85.0 (5.2x)</td><td></td></tr></table>
+
+For a second experiment, we prune a trained VGG-16 network with the same parameters as before, except enabling FLOPs regularization. We stop pruning at two points, 11.5 and 8.0 GFLOPs, and fine-tune both models for an additional five epochs with learning rate $1 0 ^ { - 4 }$ . Fine-tuning after pruning significantly improves results: the network pruned to 11.5 GFLOPs improves from $8 3 \%$ to $8 7 \%$ top-5 validation accuracy, and the network pruned to 8.0 GFLOPs improves from $7 7 . 8 \%$ to $8 4 . 5 \%$ .
+
+# 3.6 SPEED UP MEASUREMENTS
+
+During pruning we were measuring reduction in computations by FLOPs, which is a common practice (Han et al., 2015; Lavin, 2015a;b). Improvements in FLOPs result in monotonically decreasing inference time of the networks because of removing entire feature map from the layer. However, time consumed by inference dependents on particular implementation of convolution operator, parallelization algorithm, hardware, scheduling, memory transfer rate etc. Therefore we measure improvement in the inference time for selected networks to see real speed up compared to unpruned networks in Table 2. We observe significant speed ups by proposed pruning scheme.
+
+# 4 CONCLUSIONS
+
+We propose a new scheme for iteratively pruning deep convolutional neural networks. We find: 1) CNNs may be successfully pruned by iteratively removing the least important parameters—feature maps in this case—according to heuristic selection criteria; 2) a Taylor expansion-based criterion demonstrates significant improvement over other criteria; 3) per-layer normalization of the criterion is important to obtain global scaling.
+
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+
+# A APPENDIX
+
+# A.1 FLOPS COMPUTATION
+
+To compute the number of floating-point operations (FLOPs), we assume convolution is implemented as a sliding window and that the nonlinearity function is computed for free. For convolutional kernels we have:
+
+$$
+\mathrm { F L O P s } = 2 H W ( C _ { i n } K ^ { 2 } + 1 ) C _ { o u t } ,
+$$
+
+where $H$ , $W$ and $C _ { i n }$ are height, width and number of channels of the input feature map, $K$ is the kernel width (assumed to be symmetric), and $C _ { o u t }$ is the number of output channels.
+
+For fully connected layers we compute FLOPs as:
+
+$$
+\mathrm { { F L O P s } } = ( 2 I - 1 ) O ,
+$$
+
+where $I$ is the input dimensionality and $O$ is the output dimensionality.
+
+We apply FLOPs regularization during pruning to prune neurons with higher FLOPs first. FLOPs per convolutional neuron in every layer:
+
+VGG16: $\Theta ^ { f l o p s } = [ 3 . 1 , 5 7 . 8 , 1 4 . 1 , 2 8 . 9 , 7 . 0 , 1 4 . 5 , 1 4 . 5 , 3 . 5 , 7 . 2 , 7 . 2 , 1 . 8 , 1 . 8 , 1 . 8 , 1 . 8 , ]$ , 1.8]   
+AlexNet: $\Theta ^ { f l o p s } = [ 2 . 3 , 1 . 7 , 0 . 8 , 0 . 6 , 0 . 6 ]$   
+R3DCNN: $\Theta ^ { f l o p s } = [ 5 . 6 , 8 6 . 9 , 2 1 . 7 , 4 3 . 4 , 5 . 4 , 1 0 . 8 , 1 . 4 , 1 . 4 ]$
+
+# A.2 NORMALIZATION ACROSS LAYERS
+
+Scaling a criterion across layers is very important for pruning. If the criterion is not properly scaled, then a hand-tuned multiplier would need to be selected for each layer. Statistics of feature map ranking by different criteria are shown in Fig. 10. Without normalization (Fig. 14a–14d), the weight magnitude criterion tends to rank feature maps from the first layers more important than last layers; the activation criterion ranks middle layers more important; and Taylor ranks first layers higher. After $\ell _ { 2 }$ normalization (Fig. 10d–10f), all criteria have a shape more similar to the oracle, where each layer has some feature maps which are highly important and others which are unimportant.
+
+![](images/0bd0840462f9fae0679e0d449d6b8d7c088a8ca9bf27dfccac2664d69fb279a0.jpg)  
+Figure 10: Statistics of feature map ranking by raw criteria values (top) and by criteria values after $\ell _ { 2 }$ normalization (bottom).
+
+<table><tr><td rowspan="2"></td><td rowspan="2">MI</td><td rowspan="2">Weight</td><td colspan="3">Activation</td><td rowspan="2">OBD</td><td rowspan="2">Taylor</td></tr><tr><td>Mean</td><td>S.d.</td><td>APoZ</td></tr><tr><td>Per layer</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Layer 1</td><td>0.41</td><td>0.40</td><td>0.65</td><td>0.78</td><td>0.36</td><td>0.54</td><td>0.95</td></tr><tr><td>Layer 2</td><td>0.23</td><td>0.57</td><td>0.56</td><td>0.59</td><td>0.33</td><td>0.78</td><td>0.90</td></tr><tr><td>Layer 3</td><td>0.14</td><td>0.55</td><td>0.48</td><td>0.45</td><td>0.51</td><td>0.66</td><td>0.74</td></tr><tr><td>Layer 4</td><td>0.26</td><td>0.23</td><td>0.58</td><td>0.42</td><td>0.10</td><td>0.36</td><td>0.80</td></tr><tr><td>Layer 5</td><td>0.17</td><td>0.28</td><td>0.49</td><td>0.52</td><td>0.15</td><td>0.54</td><td>0.69</td></tr><tr><td>Layer 6</td><td>0.21</td><td>0.18</td><td>0.41</td><td>0.48</td><td>0.16</td><td>0.49</td><td>0.63</td></tr><tr><td>Layer 7</td><td>0.12</td><td>0.19</td><td>0.54</td><td>0.49</td><td>0.38</td><td>0.55</td><td>0.71</td></tr><tr><td>Layer8</td><td>0.18</td><td>0.23</td><td>0.43</td><td>0.42</td><td>0.30</td><td>0.50</td><td>0.54</td></tr><tr><td>Layer 9</td><td>0.21</td><td>0.18</td><td>0.50</td><td>0.55</td><td>0.35</td><td>0.53</td><td>0.61</td></tr><tr><td>Layer 10</td><td>0.26</td><td>0.15</td><td>0.59</td><td>0.60</td><td>0.45</td><td>0.61</td><td>0.66</td></tr><tr><td>Layer 11</td><td>0.41</td><td>0.12</td><td>0.61</td><td>0.65</td><td>0.45</td><td>0.64</td><td>0.72</td></tr><tr><td>Layer 12</td><td>0.47</td><td>0.15</td><td>0.60</td><td>0.66</td><td>0.39</td><td>0.66</td><td>0.72</td></tr><tr><td>Layer 13</td><td>0.61</td><td>0.21</td><td>0.77</td><td>0.76</td><td>0.65</td><td>0.76</td><td>0.77</td></tr><tr><td>Mean</td><td>0.28</td><td>0.27</td><td>0.56</td><td>0.57</td><td>0.35</td><td>0.59</td><td>0.73</td></tr><tr><td>All layers</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>No normalization</td><td>0.35</td><td>0.34</td><td>0.35</td><td>0.30</td><td>0.43</td><td>0.65</td><td>0.14</td></tr><tr><td>l1 normalization</td><td>0.47</td><td>0.37</td><td>0.63</td><td>0.63</td><td>0.52</td><td>0.65</td><td>0.71</td></tr><tr><td>l2 normalization</td><td>0.47</td><td>0.33</td><td>0.64</td><td>0.66</td><td>0.51</td><td>0.60</td><td>0.73</td></tr><tr><td>Min-max normalization</td><td>0.27</td><td>0.17</td><td>0.52</td><td>0.57</td><td>0.42</td><td>0.54</td><td>0.67</td></tr></table>
+
+Table 3: Spearman’s rank correlation of criteria vs oracle-abs in VGG-16 fine-tuned on Birds 200.
+
+# A.3 ORACLE COMPUTATION FOR VGG-16 ON BIRDS-200
+
+We compute the change in the loss caused by removing individual feature maps from the VGG-16 network, after fine-tuning on the Birds-200 dataset. Results are illustrated in Fig. 11a-11b for each feature map in layers 1 and 13, respectively. To compute the oracle estimate for a feature map, we remove the feature map and compute the network prediction for each image in the training set using the central crop with no data augmentation or dropout. We draw the following conclusions:
+
+• The contribution of feature maps range from positive (above the red line) to slightly negative (below the red line), implying the existence of some feature maps which decrease the training cost when removed.   
+• There are many feature maps with little contribution to the network output, indicated by almost zero change in loss when removed.   
+• Both layers contain a small number of feature maps which induce a significant increase in the loss when removed.
+
+![](images/331248b1ce09f6104dc08ad70fa6d25b399860365c037d079ffcd5e1c72162b8.jpg)  
+Figure 11: Change in training loss as a function of the removal of a single feature map from the VGG-16 network after fine-tuning on Birds-200. Results are plotted for two convolutional layers w.r.t. the index of the removed feature map index. The loss with all feature maps, 0.00461, is indicated with a red horizontal line.
+
+![](images/fe5c5536861fe96a4283fc23d0bab5d4d42c6223e63c40a29225584a4450ed59.jpg)  
+Figure 12: Comparison of our iterative pruning with pruning by regularization
+
+Table 3 contains a layer-by-layer listing of Spearman’s rank correlation of several criteria with the ranking of oracle-abs. In this more detailed comparison, we see the Taylor criterion shows higher correlation for all individual layers. For several methods including Taylor, the worst correlations are observed for the middle of the network, layers 5-10. We also evaluate several techniques for normalization of the raw criteria values for comparison across layers. The table shows the best performance is obtained by $\ell _ { 2 }$ normalization, hence we select it for our method.
+
+# A.4 COMPARISON WITH WEIGHT REGULARIZATION
+
+Han et al. (2015) find that fine-tuning with high $\ell _ { 1 }$ or $\ell _ { 2 }$ regularization causes unimportant connections to be suppressed. Connections with energy lower than some threshold can be removed on the assumption that they do not contribute much to subsequent layers. The same work also finds that thresholds must be set separately for each layer depending on its sensitivity to pruning. The procedure to evaluate sensitivity is time-consuming as it requires pruning layers independently during evaluation.
+
+The idea of pruning with high regularization can be extended to removing the kernels for an entire feature map if the $\ell _ { 2 }$ norm of those kernels is below a predefined threshold. We compare our approach with this regularization-based pruning for the task of pruning the last convolutional layer of VGG-16 fine-tuned for Birds-200. By considering only a single layer, we avoid the need to compute layerwise sensitivity. Parameters for optimization during fine-tuning are the same as other experiments with the Birds-200 dataset. For the regularization technique, the pruning threshold is set to $\sigma = 1 0 ^ { - 5 }$ while we vary the regularization coefficient $\gamma$ of the $\ell _ { 2 }$ norm on each feature map kernel.5 We prune only kernel weights, while keeping the bias to maintain the same expected output.
+
+A comparison between pruning based on regularization and our greedy scheme is illustrated in Fig. 12. We observe that our approach has higher test accuracy for the same number of remaining unpruned feature maps, when pruning $8 5 \%$ or more of the feature maps. We observe that with high regularization all weights tend to zero, not only unimportant weights as Han et al. (2015) observe in the case of ImageNet networks. The intuition here is that with regularization we push all weights down and potentially can affect important connections for transfer learning, whereas in our iterative procedure we only remove unimportant parameters leaving others untouched.
+
+# A.5 COMBINATION OF CRITERIA
+
+One of the possibilities to improve saliency estimation is to combine several criteria together. One of the straight forward combinations is Taylor and mean activation of the neuron. We compute the joint criteria as $\Theta _ { j o i n t } ( \mathbf { z } _ { l } ^ { ( k ) } ) = ( 1 - \lambda ) \hat { \Theta } _ { T a y l o r } ( \mathbf { z } _ { l } ^ { ( k ) } ) + \lambda \hat { \Theta } _ { A c t i v a t i o n } ( \mathbf { z } _ { l } ^ { ( k ) } )$ and perform a grid search of parameter $\lambda$ in Fig.13. The highest correlation value for each dataset is marked with with vertical bar with $\lambda$ and gain. We observe that the gain of linearly combining criteria is negligibly small (see $\Delta$ ’s in the figure).
+
+![](images/7ac80e5c32922b43e5033947e7c3653a5f67c9778c4a8649833a70f3b6e2dc9f.jpg)  
+Figure 13: Spearman rank correlation for linear combination of criteria. The per layer metric is used. Each $\Delta$ indicates the gain in correlation for one experiment.
+
+# A.6 OPTIMAL BRAIN DAMAGE IMPLEMENTATION
+
+OBD computes saliency of a parameter by computing a product of the squared magnitude of the parameter and the corresponding element on the diagonal of the Hessian. For many deep learning frameworks, an efficient implementation of the diagonal evaluation is not straightforward and approximation techniques must be applied. Our implementation of Hessian diagonal computation was inspired by Dauphin et al. (2015) work, where the technique proposed by Bekas et al. (2007) was used to evaluate SGD preconditioned with the Jacobi preconditioner. It was shown that diagonal of the Hessian can be approximated as:
+
+$$
+\mathrm { d i a g } ( \mathbf { H } ) = \mathbb { E } [ \mathbf { v } \odot \mathbf { H } \mathbf { v } ] = \mathbb { E } [ \mathbf { v } \odot \nabla ( \nabla \mathcal { C } \cdot \mathbf { v } ) ] ,
+$$
+
+where $\odot$ is the element-wise product, $\mathbf { v }$ are random vectors with entries $\pm 1$ , and $\nabla$ is the gradient operator. To compute saliency with OBD, we randomly draw $\mathbf { v }$ and compute the diagonal over 10 iterations for a single minibatch for 1000 mini batches. We found that this number of mini batches is required to compute close approximation of the Hessian’s diagonal (which we verified). Computing saliency this way is computationally expensive for iterative pruning, and we use a slightly different but more efficient procedure. Before the first pruning iteration, saliency is initialized from values computed off-line with 1000 minibatches and 10 iterations, as described above. Then, at every minibatch we compute the OBD criteria with only one iteration and apply an exponential moving averaging with a coefficient of 0.99. We verified that this computes a close approximation to the Hessian’s diagonal.
+
+# A.7 CORRELATION OF TAYLOR CRITERION WITH GRADIENT AND ACTIVATION
+
+The Taylor criterion is composed of both an activation term and a gradient term. In Figure 14, we depict the correlation between the Taylor criterion and each constituent part. We consider expected absolute value of the gradient instead of the mean, because otherwise it tends to zero. The plots are computed from pruning criteria for an unpruned VGG network fine-tuned for the Birds-200 dataset. (Values are shown after layer-wise normalization). Figure 14(a-b) depict the Taylor criterion in the y-axis for all neurons w.r.t. the gradient and activation components, respectively. The bottom $1 0 \%$ of neurons (lowest Taylor criterion, most likely to be pruned) are depicted in red, while the top $1 0 \%$ are shown in green. Considering all neurons, both gradient and activation components demonstrate a linear trend with the Taylor criterion. However, for the bottom $1 0 \%$ of neurons, as shown in Figure 14(c-d), the activation criterion shows much stronger correlation, with lower activations indicating lower Taylor scores.
+
+![](images/1c248872de3c5daef6857d28123867f99e9e6e2807ee078a0d1e49c1ec54a75e.jpg)  
+Figure 14: Correlation of Taylor criterion with gradient and activation (after layer-wise $\ell _ { 2 }$ normalization) for all neurons (a-b) and bottom $1 0 \%$ of neurons (c-d) for unpruned VGG after fine-tuning on Birds-200.

parse/train/SJGCiw5gl/SJGCiw5gl_content_list.json

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+    {
+        "type": "text",
+        "text": "PRUNING CONVOLUTIONAL NEURAL NETWORKS FOR RESOURCE EFFICIENT INFERENCE ",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Pavlo Molchanov, Stephen Tyree, Tero Karras, Timo Aila, Jan Kautz NVIDIA {pmolchanov, styree, tkarras, taila, jkautz}@nvidia.com ",
+        "bbox": [
+            183,
+            169,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a new formulation for pruning convolutional kernels in neural networks to enable efficient inference. We interleave greedy criteria-based pruning with finetuning by backpropagation—a computationally efficient procedure that maintains good generalization in the pruned network. We propose a new criterion based on Taylor expansion that approximates the change in the cost function induced by pruning network parameters. We focus on transfer learning, where large pretrained networks are adapted to specialized tasks. The proposed criterion demonstrates superior performance compared to other criteria, e.g. the norm of kernel weights or feature map activation, for pruning large CNNs after adaptation to fine-grained classification tasks (Birds-200 and Flowers-102) relaying only on the first order gradient information. We also show that pruning can lead to more than $1 0 \\times$ theoretical reduction in adapted 3D-convolutional filters with a small drop in accuracy in a recurrent gesture classifier. Finally, we show results for the largescale ImageNet dataset to emphasize the flexibility of our approach. ",
+        "bbox": [
+            233,
+            281,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            502,
+            336,
+            517
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolutional neural networks (CNN) are used extensively in computer vision applications, including object classification and localization, pedestrian and car detection, and video classification. Many problems like these focus on specialized domains for which there are only small amounts of carefully curated training data. In these cases, accuracy may be improved by fine-tuning an existing deep network previously trained on a much larger labeled vision dataset, such as images from ImageNet (Russakovsky et al., 2015) or videos from Sports-1M (Karpathy et al., 2014). While transfer learning of this form supports state of the art accuracy, inference is expensive due to the time, power, and memory demanded by the heavyweight architecture of the fine-tuned network. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While modern deep CNNs are composed of a variety of layer types, runtime during prediction is dominated by the evaluation of convolutional layers. With the goal of speeding up inference, we prune entire feature maps so the resulting networks may be run efficiently even on embedded devices. We interleave greedy criteria-based pruning with fine-tuning by backpropagation, a computationally efficient procedure that maintains good generalization in the pruned network. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural network pruning was pioneered in the early development of neural networks (Reed, 1993). Optimal Brain Damage (LeCun et al., 1990) and Optimal Brain Surgeon (Hassibi & Stork, 1993) leverage a second-order Taylor expansion to select parameters for deletion, using pruning as regularization to improve training and generalization. This method requires computation of the Hessian matrix partially or completely, which adds memory and computation costs to standard fine-tuning. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In line with our work, Anwar et al. (2015) describe structured pruning in convolutional layers at the level of feature maps and kernels, as well as strided sparsity to prune with regularity within kernels. Pruning is accomplished by particle filtering wherein configurations are weighted by misclassification rate. The method demonstrates good results on small CNNs, but larger CNNs are not addressed. ",
+        "bbox": [
+            176,
+            805,
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+            861
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Han et al. (2015) introduce a simpler approach by fine-tuning with a strong $\\ell _ { 2 }$ regularization term and dropping parameters with values below a predefined threshold. Such unstructured pruning is very effective for network compression, and this approach demonstrates good performance for intra-kernel pruning. But compression may not translate directly to faster inference since modern hardware exploits regularities in computation for high throughput. So specialized hardware may be needed for efficient inference of a network with intra-kernel sparsity (Han et al., 2016). This approach also requires long fine-tuning times that may exceed the original network training by a factor of 3 or larger. Group sparsity based regularization of network parameters was proposed to penalize unimportant parameters (Wen et al., 2016; Zhou et al., 2016; Alvarez & Salzmann, 2016; Lebedev & Lempitsky, 2016). Regularization-based pruning techniques require per layer sensitivity analysis which adds extra computations. In contrast, our approach relies on global rescaling of criteria for all layers and does not require sensitivity estimation. Moreover, our approach is faster as we directly prune unimportant parameters instead of waiting for their values to be made sufficiently small by optimization under regularization. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Other approaches include combining parameters with correlated weights (Srinivas & Babu, 2015), reducing precision (Gupta et al., 2015; Rastegari et al., 2016) or tensor decomposition (Kim et al., 2015). These approaches usually require a separate training procedure or significant fine-tuning, but potentially may be combined with our method for additional speedups. ",
+        "bbox": [
+            174,
+            250,
+            826,
+            306
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            325,
+            281,
+            342
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The proposed method for pruning consists of the following steps: 1) Fine-tune the network until convergence on the target task; 2) Alternate iterations of pruning and further fine-tuning; 3) Stop pruning after reaching the target trade-off between accuracy and pruning objective, e.g. floating point operations (FLOPs) or memory utilization. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The procedure is simple, but its success hinges on employing the right pruning criterion. In this section, we introduce several efficient pruning criteria and related technical considerations. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/5e0ef15f39588156fc82e686d0f6339a4e5abfa37f201046a638c44d125fb561.jpg",
+        "image_caption": [
+            "Figure 1: Network pruning as a backward filter. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            645,
+            367,
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+            589
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Consider a set of training examples $\\begin{array} { r l r l } { { \\mathcal { D } } } & { { } = { } } & { \\{ \\mathcal { X } }  & { { } = } \\end{array}$ $\\left\\{ { \\bf x } _ { 0 } , { \\bf x } _ { 1 } , . . . , { \\bf x } _ { N } \\} , \\mathcal { V } = \\left\\{ y _ { 0 } , y _ { 1 } , . . . , y _ { N } \\right\\} \\right\\}$ , where $\\mathbf { x }$ and $y$ represent an inparameters1 $\\mathcal { W } = \\{ ( \\mathbf { w } _ { 1 } ^ { 1 } , b _ { 1 } ^ { \\bar { 1 } } ) , ( \\mathbf { w } _ { 1 } ^ { 2 } , b _ { 1 } ^ { 2 } ) , . . . ( \\mathbf { w } _ { L } ^ { C _ { \\ell } } , b _ { L } ^ { \\bar { C } _ { \\ell } } ) \\}$ he network’sare optimized $\\mathcal { C } ( \\mathcal { D } | \\mathcal { W } )$   \na cost function $\\mathcal { C } ( \\cdot )$ is a negative log-likelihood function. A cost function is selected independently of pruning and depends only on the task to be solved by the original network. In the case of transfer learning, we adapt a large network initialized with parameters ${ \\mathcal { W } } _ { 0 }$ pretrained on a related but distinct dataset.   \nDuring pruning, we refine a subset of parameters which preserves   \nthe accuracy of the adapted network, $\\bar { \\mathcal { C } } ( \\mathcal { D } | \\mathcal { W } ^ { \\prime } ) \\approx \\mathcal { C } ( \\mathcal { D } | \\bar { \\mathcal { W } } )$ . This corresponds to a combinatorial   \noptimization: ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/9aa18199b837a1d5ef12fce3d5964bd1d7cdce9b22fdc58d33b37816e24449fb.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { W ^ { \\prime } } \\left| \\mathcal { C } ( \\mathcal { D } | \\mathcal { W } ^ { \\prime } ) - \\mathcal { C } ( \\mathcal { D } | \\mathcal { W } ) \\right| \\quad \\mathrm { s . t . } \\quad | | \\mathcal { W } ^ { \\prime } | | _ { 0 } \\leq B ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            688,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where the $\\ell _ { 0 }$ norm in $| | \\mathcal { W } ^ { \\prime } | | _ { 0 }$ bounds the number of non-zero parameters $B$ in $W ^ { \\prime }$ . Intuitively, if $\\mathcal { W } ^ { \\prime } = \\mathcal { W }$ we reach the global minimum of the error function, however $| | \\mathcal { W } ^ { \\prime } | | _ { 0 }$ will also have its maximum. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Finding a good subset of parameters while maintaining a cost value as close as possible to the original is a combinatorial problem. It will require $2 ^ { | \\mathcal { W } | }$ evaluations of the cost function for a selected subset of data. For current networks it would be impossible to compute: for example, VGG-16 has $| \\mathcal { W } | = 4 2 2 4$ convolutional feature maps. While it is impossible to solve this optimization exactly for networks of any reasonable size, in this work we investigate a class of greedy methods. Starting with a full set of parameters $\\mathcal { W }$ , we iteratively identify and remove the least important parameters, as illustrated in Figure 1. By removing parameters at each iteration, we ensure the eventual satisfaction of the $\\ell _ { 0 }$ bound on $\\mathcal { W } ^ { \\prime }$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Since we focus our analysis on pruning feature maps from convolutional layers, let us denote a set of image feature maps by $\\mathbf { z } _ { \\ell } \\doteq \\mathbb { R } ^ { H _ { \\ell } ^ { \\smile } \\times W _ { \\ell } \\times C _ { \\ell } }$ with dimensionality $H _ { \\ell } \\times W _ { \\ell }$ and $C _ { \\ell }$ individual maps (or channels).2 The feature maps can either be the input to the network, $\\mathbf { z } _ { 0 }$ , or the output from a convolutional layer, $\\mathbf { z } _ { \\ell }$ with $\\ell \\in [ 1 , 2 , . . . , L ]$ . Individual feature maps are denoted $\\mathbf { z } _ { \\ell } ^ { ( k ) }$ for $k \\in [ 1 , 2 , . . . , C _ { \\ell } ]$ . A convolutional layer $\\ell$ applies the convolution operation $( * )$ to a set of input feature maps $\\mathbf { z } _ { \\ell - 1 }$ with kernels parameterized by $\\mathbf { w } _ { \\ell } ^ { ( k ) } \\in \\mathbb { R } ^ { C _ { \\ell - 1 } \\times p \\times p }$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7b2fc427ecb7720fcbb16aa2bc6a1d2f9600c607e0657c1e7f015025b311a52f.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { z } _ { \\ell } ^ { ( k ) } = \\mathbf { g } _ { \\ell } ^ { ( k ) } \\mathcal { R } \\big ( \\mathbf { z } _ { \\ell - 1 } \\ast \\mathbf { w } _ { \\ell } ^ { ( k ) } + b _ { \\ell } ^ { ( k ) } \\big ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { z } _ { \\ell } ^ { ( k ) } \\in \\mathbb { R } ^ { H _ { \\ell } \\times W _ { \\ell } }$ is the result of convolving each of $C _ { \\ell - 1 }$ kernels of size $p \\times p$ with its respective input feature map and adding bias $b _ { \\ell } ^ { ( k ) }$ . We introduce a pruning gate $\\mathbf { g } _ { l } \\in \\{ 0 , 1 \\} ^ { C _ { l } }$ , an external switch which determines if a particular feature map is included or pruned during feed-forward propagation, such that when $\\mathbf { g }$ is vectorized: $w ^ { \\prime } = \\mathbf { g } \\mathcal { W }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 ORACLE PRUNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Minimizing the difference in accuracy between the full and pruned models depends on the criterion for identifying the “least important” parameters, called saliency, at each step. The best criterion would be an exact empirical evaluation of each parameter, which we denote the oracle criterion, accomplished by ablating each non-zero parameter $w \\in \\mathcal { W } ^ { \\prime }$ in turn and recording the cost’s difference. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We distinguish two ways of using this oracle estimation of importance: 1) oracle-loss quantifies importance as the signed change in loss, $\\mathcal { C } ( \\mathcal { D } | \\mathcal { W } ^ { \\prime } ) - \\mathcal { C } ( \\mathcal { D } | \\mathcal { W } )$ , and 2) oracle-abs adopts the absolute difference, $| \\mathcal { C } ( \\mathcal { D } | \\mathcal { W } ^ { \\prime } ) - \\mathcal { C } ( \\mathcal { D } | \\mathcal { W } ) |$ . While both discourage pruning which increases the loss, the oracle-loss version encourages pruning which may decrease the loss, while oracle-abs penalizes any pruning in proportion to its change in loss, regardless of the direction of change. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While the oracle is optimal for this greedy procedure, it is prohibitively costly to compute, requiring $| | W ^ { \\prime } | | _ { 0 }$ evaluations on a training dataset, one evaluation for each remaining non-zero parameter. Since estimation of parameter importance is key to both the accuracy and the efficiency of this pruning approach, we propose and evaluate several criteria in terms of performance and estimation cost. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.2 CRITERIA FOR PRUNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There are many heuristic criteria which are much more computationally efficient than the oracle. For the specific case of evaluating the importance of a feature map (and implicitly the set of convolutional kernels from which it is computed), reasonable criteria include: the combined $\\ell _ { 2 }$ -norm of the kernel weights, the mean, standard deviation or percentage of the feature map’s activation, and mutual information between activations and predictions. We describe these criteria in the following paragraphs and propose a new criterion which is based on the Taylor expansion. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Minimum weight. Pruning by magnitude of kernel weights is perhaps the simplest possible criterion, and it does not require any additional computation during the fine-tuning process. In case of pruning according to the norm of a set of weights, the criterion is evaluated as: $\\begin{array} { r } { \\bar { \\Theta } _ { M W } : \\mathbb { R } ^ { C _ { \\ell - 1 } \\times p \\times p } \\stackrel { \\cdot } {  } \\mathbb { R } } \\end{array}$ , with $\\begin{array} { r } { \\Theta _ { M W } ( \\mathbf { \\bar { w } } ) = \\frac { 1 } { | \\mathbf { w } | } \\sum _ { i } w _ { i } ^ { 2 } } \\end{array}$ , where $| \\mathbf { w } |$ is dimensionality of the set of weights after vectorization. The motivation to apply this type of pruning is that a convolutional kernel with low $\\ell _ { 2 }$ norm detects less important features than those with a high norm. This can be aided during training by applying $\\ell _ { 1 }$ or $\\ell _ { 2 }$ regularization, which will push unimportant kernels to have smaller values. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Activation. One of the reasons for the popularity of the ReLU activation is the sparsity in activation that is induced, allowing convolutional layers to act as feature detectors. Therefore it is reasonable to assume that if an activation value (an output feature map) is small then this feature detector is not important for prediction task at hand. We may evaluate this by mean activation, $\\Theta _ { M A }$ : $\\mathbb { R } ^ { H _ { l } \\times W _ { \\ell } \\times C _ { \\ell } } \\to \\mathbb { R }$ , with $\\begin{array} { r } { \\Theta _ { M A } ( { \\bf a } ) = \\frac { 1 } { \\lvert { \\bf a } \\rvert } \\sum _ { i } a _ { i } } \\end{array}$ for activation $\\mathbf { a } = \\mathbf { z } _ { l } ^ { ( k ) }$ , or by the standard deviation of the activation, $\\begin{array} { r } { \\Theta _ { M A _ { - } s t d } ( \\mathbf { a } ) = \\sqrt { \\frac { 1 } { | \\mathbf { a } | } \\sum _ { i } ( a _ { i } - \\mu _ { \\mathbf { a } } ) ^ { 2 } } , } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Mutual information. Mutual information (MI) is a measure of how much information is present in one variable about another variable. We apply MI as a criterion for pruning, $\\Theta _ { M I } : \\mathbb { R } ^ { H _ { l } \\times W _ { \\ell } \\times C _ { \\ell } }  \\mathbb { R }$ , with $\\Theta _ { M I } ( \\mathbf { a } ) = M I ( \\mathbf { a } , y )$ , where $y$ is the target of neural network. MI is defined for continuous variables, so to simplify computation, we exchange it with information gain (IG), which is defined for quantized variables $I G ( y | x ) = H ( x ) + H ( y ) - H ( x , y )$ , where $H ( x )$ is the entropy of variable $x$ . We accumulate statistics on activations and ground truth for a number of updates, then quantize the values and compute IG. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Taylor expansion. We phrase pruning as an optimization problem, trying to find $\\mathcal { W } ^ { \\prime }$ with bounded number of non-zero elements that minimize $\\left| \\Delta \\dot { C } ( h _ { i } ) \\right| = \\left| \\dot { \\mathcal { C } ( \\mathcal { D } | \\mathcal { W } ^ { \\prime } ) } - \\dot { \\mathcal { C } } ( \\bar { \\mathcal { D } } | \\mathcal { W } ) \\right|$ . With this approach based on the Taylor expansion, we directly approximate change in the loss function from removing a particular parameter. Let $h _ { i }$ be the output produced from parameter $i$ . In the case of feature maps, $h = \\{ z _ { 0 } ^ { ( 1 ) } , z _ { 0 } ^ { ( 2 ) } , . . . , z _ { L } ^ { ( C _ { \\ell } ) } \\}$ z(C\\`)L }. For notational convenience, we consider the cost function equally dependent on parameters and outputs computed from parameters: $\\mathcal { C } ( D | h _ { i } ) = \\mathcal { C } ( \\mathcal { D } | ( \\mathbf { w } , b ) _ { i } )$ . Assuming independence of parameters, we have: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/df5a20ca0e75ac0e09d576e0fe3adfa07290dfa4f150a62a4aca0e27da88e2e7.jpg",
+        "text": "$$\n\\big | \\Delta \\mathcal { C } ( h _ { i } ) \\big | = \\big | \\mathcal { C } ( \\mathcal { D } , h _ { i } = 0 ) - \\mathcal { C } ( \\mathcal { D } , h _ { i } ) \\big | ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { C } ( \\mathcal { D } , h _ { i } = 0 )$ is a cost value if output $h _ { i }$ is pruned, while $\\mathcal { C } ( \\mathcal { D } , h _ { i } )$ is the cost if it is not pruned. While parameters are in reality inter-dependent, we already make an independence assumption at each gradient step during training. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To approximate $\\Delta \\mathcal { C } ( h _ { i } )$ , we use the first-degree Taylor polynomial. For a function $f ( x )$ , the Taylor expansion at point $x = a$ is ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/54a820da9ca76fdc58c74e526634e485e5ddb1537ce5dc72e89523fbf5ad4a78.jpg",
+        "text": "$$\nf ( x ) = \\sum _ { p = 0 } ^ { P } \\frac { f ^ { ( p ) } ( a ) } { p ! } ( x - a ) ^ { p } + R _ { p } ( x ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $f ^ { ( p ) } ( a )$ is the $p$ -th derivative of $f$ evaluated at point $a$ , and $R _ { p } ( x )$ is the $p$ -th order remainder. Approximating $\\mathcal { C } ( \\mathcal { D } , h _ { i } = 0 )$ ) with a first-order Taylor polynomial near $h _ { i } = 0$ , we have: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9614916549f7dc3de2fa537c658a3656f63e12fe153267d21ac766ea37b4ca75.jpg",
+        "text": "$$\n\\mathcal { C } ( \\mathcal { D } , h _ { i } = 0 ) \\ = \\ \\mathcal { C } ( \\mathcal { D } , h _ { i } ) - \\frac { \\delta \\mathcal { C } } { \\delta h _ { i } } h _ { i } + R _ { 1 } ( h _ { i } = 0 ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The remainder $R _ { 1 } ( h _ { i } = 0 )$ can be calculated through the Lagrange form: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2639c3193a63d3fdb73a63fdde66acd4a61ad0e2374f1c154ee2dadf775257a3.jpg",
+        "text": "$$\nR _ { 1 } ( h _ { i } = 0 ) = \\frac { \\delta ^ { 2 } \\mathcal { C } } { \\delta ( h _ { i } ^ { 2 } = \\xi ) } \\frac { h _ { i } ^ { 2 } } { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            584,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\xi$ is a real number between 0 and $h _ { i }$ . However, we neglect this first-order remainder, largely due to the significant calculation required, but also in part because the widely-used ReLU activation function encourages a smaller second order term. Finally, by substituting Eq. (5) into Eq. (3) and ignoring the remainder, we have $\\Theta _ { T E } : \\mathbb { R } ^ { H _ { l } \\times W _ { l } \\times C _ { l } }  \\bar { \\mathbb { R } ^ { + } }$ , with ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f4a3dd58fcce434b47110a9be563cd95ac28e2c62edda66009bc0593f2a7b9f.jpg",
+        "text": "$$\n\\Theta _ { T E } ( h _ { i } ) = \\left| \\Delta \\mathcal { C } ( h _ { i } ) \\right| = \\left| \\mathcal { C } ( \\mathcal { D } , h _ { i } ) - \\frac { \\delta \\mathcal { C } } { \\delta h _ { i } } h _ { i } - \\mathcal { C } ( \\mathcal { D } , h _ { i } ) \\right| = \\left| \\frac { \\delta \\mathcal { C } } { \\delta h _ { i } } h _ { i } \\right| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            271,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, this criterion prunes parameters that have an almost flat gradient of the cost function w.r.t. feature map $h _ { i }$ . This approach requires accumulation of the product of the activation and the gradient of the cost function w.r.t. to the activation, which is easily computed from the same computations for back-propagation. $\\Theta _ { T E }$ is computed for a multi-variate output, such as a feature map, by ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e6835651681e9f1de885af1ac4e3feac4717ce7d7dc2c464b14a28d52025051c.jpg",
+        "text": "$$\n\\Theta _ { T E } ( z _ { l } ^ { ( k ) } ) = \\bigg | \\frac { 1 } { M } \\sum _ { m } \\frac { \\delta C } { \\delta z _ { l , m } ^ { ( k ) } } z _ { l , m } ^ { ( k ) } \\bigg | ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            800,
+            614,
+            840
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $M$ is length of vectorized feature map. For a minibatch with $T > 1$ examples, the criterion is computed for each example separately and averaged over $T$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Independently of our work, Figurnov et al. (2016) came up with similar metric based on the Taylor expansion, called impact, to evaluate importance of spatial cells in a convolutional layer. It shows that the same metric can be applied to evaluate importance of different groups of parameters. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Relation to Optimal Brain Damage. The Taylor criterion proposed above relies on approximating the change in loss caused by removing a feature map. The core idea is the same as in Optimal Brain Damage (OBD) (LeCun et al., 1990). Here we consider the differences more carefully. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$\\begin{array} { r } { y = \\frac { \\delta \\mathcal { C } } { \\delta h } h } \\end{array}$ ry difference is thfor cost function erm tends to zero: $\\mathcal { C }$ nt ofddeand rder tertivation . At fac $h$ of the. Aftvalue aylor expansion, in our notation sufficient training epochs, theoffers little useful information, $\\frac { \\delta { \\mathcal { C } } } { \\delta h }  0$ $\\mathbb { E } ( y ) = 0$ $y$ hence OBD regards the term as zero and focuses on the second-order term. ",
+        "bbox": [
+            174,
+            152,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "However, the variance of $y$ is non-zero and correlates with the stability of the local function w.r.t. activation $h$ . By considering the absolute change in the cost3 induced by pruning (as in Eq. 3), we use the absolute value of the first-order term, $| y |$ . Under assumption that samples come from independent and identical distribution, $\\mathbb { E } ( | y | ) = \\sigma { \\sqrt { 2 } } / { \\sqrt { \\pi } }$ where $\\sigma$ is the standard deviation of $y$ , known as the expected value of the half-normal distribution. So, while $y$ tends to zero, the expectation of $| y |$ is proportional to the variance of $y$ , a value which is empirically more informative as a pruning criterion. ",
+        "bbox": [
+            174,
+            218,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As an additional benefit, we avoid the computation of the second-order Taylor expansion term, or its simplification - diagonal of the Hessian, as required in OBD. ",
+        "bbox": [
+            176,
+            310,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We found important to compare proposed Taylor criteria to OBD. As described in the original papers (LeCun et al., 1990; 1998), OBD can be efficiently implemented similarly to standard back propagation algorithm doubling backward propagation time and memory usage when used together with standard fine-tuning. Efficient implementation of the original OBD algorithm might require significant changes to the framework based on automatic differentiation like Theano to efficiently compute only diagonal of the Hessian instead of the full matrix. Several researchers tried to tackle this problem with approximation techniques (Martens, 2010; Martens et al., 2012). In our implementation, we use efficient way of computing Hessian-vector product (Pearlmutter, 1994) and matrix diagonal approximation proposed by (Bekas et al., 2007), please refer to more details in appendix. With current implementation, OBD is 30 times slower than Taylor technique for saliency estimation, and 3 times slower for iterative pruning, however with different implementation can only be $50 \\%$ slower as mentioned in the original paper. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Average Percentage of Zeros (APoZ). Hu et al. (2016) proposed to explore sparsity in activations for network pruning. ReLU activation function imposes sparsity during inference, and average percentage of positive activations at the output can determine importance of the neuron. Intuitively, it is a good criteria, however feature maps at the first layers have similar APoZ regardless of the network’s target as they learn to be Gabor like filters. We will use APoZ to estimate saliency of feature maps. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.3 NORMALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            627,
+            336,
+            642
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Some criteria return “raw” values, whose scale varies with the depth of the parameter’s layer in the network. A simple layer-wise $\\ell _ { 2 }$ -normalization can achieve adequate rescaling across layers: ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6402959440a281d008c5d297c801ad503860482804a2a7b027547c836a8bcba1.jpg",
+        "text": "$$\n\\hat { \\Theta } ( \\mathbf { z } _ { l } ^ { ( k ) } ) = \\frac { \\Theta ( \\mathbf { z } _ { l } ^ { ( k ) } ) } { \\sqrt { \\sum _ { j } \\left( \\Theta ( \\mathbf { z } _ { l } ^ { ( j ) } ) \\right) ^ { 2 } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.4 FLOPS REGULARIZED PRUNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            747,
+            437,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One of the main reasons to apply pruning is to reduce number of operations in the network. Feature maps from different layers require different amounts of computation due the number and sizes of input feature maps and convolution kernels. To take this into account we introduce FLOPs regularization: ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e10f1a4103fb9b85f28f05314e22f32e8147b93febd8b045d012e1ba2b5336e.jpg",
+        "text": "$$\n\\Theta ( \\mathbf { z } _ { l } ^ { ( k ) } ) = \\Theta ( \\mathbf { z } _ { l } ^ { ( k ) } ) - \\lambda \\Theta _ { l } ^ { f l o p s } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            820,
+            602,
+            842
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda$ controls the amount of regularization. For our experiments, we use $\\lambda = 1 0 ^ { - 3 }$ . $\\Theta ^ { f l o p s }$ is computed under the assumption that convolution is implemented as a sliding window (see Appendix). Other regularization conditions may be applied, e.g. storage size, kernel sizes, or memory footprint. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/7fe7129df240cf9ae8d98709d3020505acef56bbe5ea0edf667a6f729cb30b60.jpg",
+        "image_caption": [
+            "Figure 2: Global statistics of oracle ranking, shown by layer for Birds-200 transfer learning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            101,
+            470,
+            267
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/26f057cd82b064948e11ae7322c6ce4962c9162ccb788834c1a5d0e8caf66437.jpg",
+        "image_caption": [
+            "Figure 3: Pruning without fine-tuning using oracle ranking for Birds-200 transfer learning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3 RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            325,
+            281,
+            342
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We empirically study the pruning criteria and procedure detailed in the previous section for a variety of problems. We focus many experiments on transfer learning problems, a setting where pruning seems to excel. We also present results for pruning large networks on their original tasks for more direct comparison with the existing pruning literature. Experiments are performed within Theano (Theano Development Team, 2016). Training and pruning are performed on the respective training sets for each problem, while results are reported on appropriate holdout sets, unless otherwise indicated. For all experiments we prune a single feature map at every pruning iteration, allowing fine-tuning and re-evaluation of the criterion to account for dependency between parameters. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.1 CHARACTERIZING THE ORACLE RANKING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    {
+        "type": "text",
+        "text": "We begin by explicitly computing the oracle for a single pruning iteration of a visual transfer learning problem. We fine-tune the VGG-16 network (Simonyan & Zisserman, 2014) for classification of bird species using the Caltech-UCSD Birds 200-2011 dataset (Wah et al., 2011). The dataset consists of nearly 6000 training images and 5700 test images, covering 200 species. We fine-tune VGG-16 for 60 epochs with learning rate 0.0001 to achieve a test accuracy of $7 2 . 2 \\%$ using uncropped images. ",
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+        "type": "text",
+        "text": "To compute the oracle, we evaluate the change in loss caused by removing each individual feature map from the fine-tuned VGG-16 network. (See Appendix A.3 for additional analysis.) We rank feature maps by their contributions to the loss, where rank 1 indicates the most important feature map—removing it results in the highest increase in loss—and rank 4224 indicates the least important. Statistics of global ranks are shown in Fig. 2 grouped by convolutional layer. We observe: (1) Median global importance tends to decrease with depth. (2) Layers with max-pooling tend to be more important than those without. (VGG-16 has pooling after layers 2, 4, 7, 10, and 13.) However, (3) maximum and minimum ranks show that every layer has some feature maps that are globally important and others that are globally less important. Taken together with the results of subsequent experiments, we opt for encouraging a balanced pruning that distributes selection across all layers. ",
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+    {
+        "type": "text",
+        "text": "Next, we iteratively prune the network using pre-computed oracle ranking. In this experiment, we do not update the parameters of the network or the oracle ranking between iterations. Training accuracy is illustrated in Fig. 3 over many pruning iterations. Surprisingly, pruning by smallest absolute change in loss (Oracle-abs) yields higher accuracy than pruning by the net effect on loss (Oracle-loss). Even though the oracle indicates that removing some feature maps individually may decrease loss, instability accumulates due the large absolute changes that are induced. These results support pruning by absolute difference in cost, as constructed in Eq. 1. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.2 EVALUATING PROPOSED CRITERIA VERSUS THE ORACLE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To evaluate computationally efficient criteria as substitutes for the oracle, we compute Spearman’s rank correlation, an estimate of how well two predictors provide monotonically related outputs, ",
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+    {
+        "type": "table",
+        "img_path": "images/f579c5d0c9f130d32bf93859d2e52eea8f07c58a9a2cf4d60c89c97776274633.jpg",
+        "table_caption": [
+            "Table 1: Spearman’s rank correlation of criteria vs. oracle for convolutional feature maps of VGG-16 and AlexNet fine-tuned on Birds-200 and Flowers-102 datasets, and AlexNet trained on ImageNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td colspan=\"6\">AlexNet /Flowers-102</td><td colspan=\"6\">VGG-16/Birds-200</td></tr><tr><td rowspan=\"2\">Weight</td><td rowspan=\"2\">Mean</td><td colspan=\"2\">Activation S.d.</td><td rowspan=\"2\">OBD</td><td rowspan=\"2\">Taylor</td><td rowspan=\"2\">Weight</td><td colspan=\"2\">Activation</td><td rowspan=\"2\">OBD</td><td rowspan=\"2\">Taylor</td><td rowspan=\"2\">Mutual Info.</td></tr><tr><td>APoZ</td><td></td><td>Mean S.d.</td><td>APoZ</td></tr><tr><td>Per layer</td><td>0.17</td><td>0.65</td><td>0.67</td><td>0.54</td><td>0.64 0.77</td><td></td><td>0.27 0.56</td><td>0.57</td><td>0.35</td><td>0.59</td><td>0.73</td><td>0.28</td></tr><tr><td>All layers</td><td>0.28</td><td>0.51</td><td>0.53</td><td>0.68</td><td>0.37</td><td></td><td>0.34</td><td>0.35 0.30</td><td>0.43</td><td>0.65</td><td>0.14</td><td>0.35</td></tr><tr><td>(w/ l2-norm)</td><td>0.13</td><td>0.63</td><td>0.61</td><td>0.41 0.60</td><td></td><td>0.75</td><td>0.33 0.64</td><td>0.66</td><td>0.51</td><td>1</td><td>0.73</td><td>0.47</td></tr><tr><td colspan=\"5\">AlexNet/Birds-200</td><td colspan=\"6\">VGG-16/Flowers-102</td><td rowspan=\"4\"></td></tr><tr><td>Per layer</td><td>0.36</td><td>0.57</td><td>0.65</td><td>0.42 0.54</td><td>0.81</td><td></td><td>0.51</td><td>0.47</td><td>0.36</td><td>0.21</td><td>0.6</td></tr><tr><td>All layers</td><td>0.32</td><td>0.37</td><td>0.51</td><td></td><td></td><td></td><td>0.19</td><td></td><td></td><td></td><td></td></tr><tr><td>(w/ l2-norm)</td><td>0.23</td><td>0.54 0.57</td><td>0.28 0.49</td><td>0.61 1</td><td>0.37 0.78</td><td>0.35 0.28</td><td>0.53 0.66</td><td>0.45 0.65</td><td>0.61 0.61</td><td>0.28 1</td><td>0.02 0.7</td></tr><tr><td colspan=\"9\">AlexNet/ImageNet</td><td></td><td></td><td></td></tr><tr><td>Per layer</td><td>0.57</td><td>0.09</td><td>0.19</td><td>-0.06</td><td>0.58</td><td>0.58</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>All layers</td><td>0.67</td><td></td><td></td><td></td><td></td><td>0.11</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(w/ lz-norm)</td><td>0.44</td><td>0.00 0.10</td><td>0.13 0.19</td><td>-0.08 0.19</td><td>0.72 -</td><td>0.55</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+        "type": "image",
+        "img_path": "images/3619e15867c1012f68a8762ade1c9b939f48ebbfa25334d28cfbc56f9050e842.jpg",
+        "image_caption": [
+            "Figure 4: Pruning of feature maps in VGG-16 fine-tuned on the Birds-200 dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "even if their relationship is not linear. Given the difference between oracle4 and criterion ranks $d _ { i } = r a n k ( \\Theta _ { o r a c l e } ( i ) ) - r a n k ( \\Theta _ { c r i t e r i o n } ( i ) )$ for each parameter $i$ , the rank correlation is computed: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f15736383a3f137daf6d261483f632860e98529edf7031323ab62a29cc198635.jpg",
+        "text": "$$\n\\mathcal { S } = 1 - \\frac { 6 } { N ( N ^ { 2 } - 1 ) } \\sum _ { i = 1 } ^ { N } { d _ { i } } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $N$ is the number of parameters (and the highest rank). This correlation coefficient takes values in $[ - 1 , 1 ]$ , where $- 1$ implies full negative correlation, 0 no correlation, and 1 full positive correlation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We show Spearman’s correlation in Table 1 to compare the oracle-abs ranking to rankings by different criteria on a set of networks/datasets some of which are going to be introduced later. Data-dependent criteria (all except weight magnitude) are computed on training data during the fine-tuning before or between pruning iterations. As a sanity check, we evaluate random ranking and observe 0.0 correlation across all layers. “Per layer” analysis shows ranking within each convolutional layer, while “All layers” describes ranking across layers. While several criteria do not scale well across layers with raw values, a layer-wise $\\ell _ { 2 }$ -normalization significantly improves performance. The Taylor criterion has the highest correlation among the criteria, both within layers and across layers (with $\\ell _ { 2 }$ normalization). OBD shows the best correlation across layers when no normalization used; it also shows best results for correlation on ImageNet dataset. (See Appendix A.2 for further analysis.) ",
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+    {
+        "type": "text",
+        "text": "3.3 PRUNING FINE-TUNED IMAGENET NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now evaluate the full iterative pruning procedure on two transfer learning problems. We focus on reducing the number of convolutional feature maps and the total estimated floating point operations (FLOPs). Fine-grained recognition is difficult for relatively small datasets without relying on transfer learning. Branson et al. (2014) show that training CNN from scratch on the Birds-200 dataset achieves test accuracy of only $1 0 . 9 \\%$ . We compare results to training a randomly initialized CNN with half the number of parameters per layer, denoted \"from scratch\". ",
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+        "img_path": "images/822f4ff9f919bb1d8f8e9b83b6da949a22e0ff915fd8c12fd1bba68c34172521.jpg",
+        "image_caption": [
+            "Figure 5: Pruning of feature maps in AlexNet on fine-tuned on Flowers-102. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Fig. 4 shows pruning of VGG-16 after fine-tuning on the Birds-200 dataset (as described previously). At each pruning iteration, we remove a single feature map and then perform 30 minibatch SGD updates with batch-size 32, momentum 0.9, learning rate $1 \\dot { 0 } ^ { - 4 }$ , and weight decay $1 0 ^ { - 4 }$ . The figure depicts accuracy relative to the pruning rate (left) and estimated GFLOPs (right). The Taylor criterion shows the highest accuracy for nearly the entire range of pruning ratios, and with FLOPs regularization demonstrates the best performance relative to the number of operations. OBD shows slightly worse performance of pruning in terms of parameters, however significantly worse in terms of FLOPs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Fig. 5, we show pruning of the CaffeNet implementation of AlexNet (Krizhevsky et al., 2012) after adapting to the Oxford Flowers 102 dataset (Nilsback & Zisserman, 2008), with 2040 training and 6129 test images from 102 species of flowers. Criteria correlation with oracle-abs is summarized in Table 1. We initially fine-tune the network for 20 epochs using a learning rate of 0.001, achieving a final test accuracy of $8 0 . 1 \\%$ . Then pruning procedes as previously described for Birds-200, except with only 10 mini-batch updates between pruning iterations. We observe the superior performance of the Taylor and OBD criteria in both number of parameters and GFLOPs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We observed that Taylor criterion shows the best performance which is closely followed by OBD with a bit lower Spearman’s rank correlation coefficient. Implementing OBD takes more effort because of computation of diagonal of the Hessian and it is $50 \\%$ to $300 \\%$ slower than Taylor criteria that relies on first order gradient only. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Fig. 6 shows pruning with the Taylor technique and a varying number of fine-tuning updates between pruning iterations. Increasing the number of updates results in higher accuracy, but at the cost of additional runtime of the pruning procedure. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "During pruning we observe a small drop in accuracy. One of the reasons is fine-tuning between pruning iterations. Accuracy of the initial network can be improved with longer fine tunning and search of better optimization parameters. For example accuracy of unpruned VGG16 network on Birds-200 goes up to $7 5 \\%$ after extra $1 2 8 \\mathrm { k }$ updates. And AlexNet on Flowers-102 goes up to $8 2 . 9 \\%$ after $1 3 0 \\mathrm { k }$ updates. It should be noted that with farther fine-tuning of pruned networks we can achieve higher accuracy as well, therefore the one-to-one comparison of accuracies is rough. ",
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+    {
+        "type": "text",
+        "text": "3.4 PRUNING A RECURRENT 3D-CNN NETWORK FOR HAND GESTURE RECOGNITION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Molchanov et al. (2016) learn to recognize 25 dynamic hand gestures in streaming video with a large recurrent neural network. The network is constructed by adding recurrent connections to a 3D-CNN pretrained on the Sports-1M video dataset (Karpathy et al., 2014) and fine tuning on a gesture dataset. The full network achieves an accuracy of $8 0 . { \\bar { 7 } } \\%$ when trained on the depth modality, but a single inference requires an estimated 37.8 GFLOPs, too much for deployment on an embedded GPU. After several iterations of pruning with the Taylor criterion with learning rate 0.0003, momentum 0.9, FLOPs regularization $1 0 ^ { - 3 }$ , we reduce inference to 3.0 GFLOPs, as shown in Fig. 7. While pruning increases classification error by nearly $6 \\%$ , additional fine-tuning restores much of the lost accuracy, yielding a final pruned network with a $1 2 . 6 \\times$ reduction in GFLOPs and only a $2 . 5 \\%$ loss in accuracy. ",
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+        "img_path": "images/038373025e262cf10fd623e13c94664ea2dd66069597f4da6a9719238ddc69b8.jpg",
+        "image_caption": [
+            "Figure 6: Varying the number of minibatch updates between pruning iterations with AlexNet/Flowers-102 and the Taylor criterion. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "image_caption": [
+            "Figure 7: Pruning of a recurrent 3D-CNN for dynamic hand gesture recognition (Molchanov et al., 2016). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/cb7b49f5685ff7bd9bdeeedba7cb02bece13d8eb6d67d12bfe3ef712dab5b3b0.jpg",
+        "image_caption": [
+            "Figure 8: Pruning of AlexNet on Imagenet with varying number of updates between pruning iterations. "
+        ],
+        "image_footnote": [],
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+        "image_caption": [],
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+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "3.5 PRUNING NETWORKS FOR IMAGENET ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We also test our pruning scheme on the largescale ImageNet classification task. In the first experiment, we begin with a trained CaffeNet implementation of AlexNet with $7 9 . 2 \\%$ top-5 validation accuracy. Between pruning iterations, we fine-tune with learning rate $1 0 ^ { - 4 }$ , momen$\\mathrm { t u m } 0 . 9$ , weight decay $1 0 ^ { - 4 }$ , batch size 32, and drop-out $5 0 \\%$ . Using a subset of 5000 training images, we compute oracle-abs and Spearman’s rank correlation with the criteria, as shown in Table 1. Pruning traces are illustrated in Fig. 8. We observe: 1) Taylor performs better than random or minimum weight pruning when 100 updates are used between pruning iterations. When results are displayed w.r.t. FLOPs, the difference with random pruning is only $0 \\% - 4 \\%$ , but the difference is higher, $1 \\% - \\mathrm { i } 0 \\%$ , when plotted with the number of feature maps pruned. 2) Increasing the number of updates from 100 to 1000 improves performance of pruning significantly for both the Taylor criterion and random pruning. ",
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+    {
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+        "img_path": "images/cffc122da067354b60cec34a5cc11ec4923edf0e2b2704848395409dc29722ab.jpg",
+        "image_caption": [
+            "Figure 9: Pruning of the VGG-16 network on ImageNet, with additional following fine-tuning at 11.5 and 8 GFLOPs. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "table",
+        "img_path": "images/129c2ac9893ac1b0a68de6921bd3a933a657028d7bba29ea6f6f438b4ba672a4.jpg",
+        "table_caption": [
+            "Table 2: Actual speed up of networks pruned by Taylor criterion for various hardware setup. All measurements were performed with PyTorch with cuDNN v5.1.0, except R3DCNN which was implemented in $\\mathrm { C } { + } { + }$ with cuDNN v4.0.4). Results for ImageNet dataset are reported as top-5 accuracy on validation set. Results on AlexNet / Flowers-102 are reported for pruning with 1000 updates between iterations and no fine-tuning after pruning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hardware</td><td>Batch</td><td>Accuracy</td><td>Time, ms</td><td>Accuracy</td><td>Time (speed up)</td><td>Accuracy Time (speed up)</td></tr><tr><td>AlexNet/Flowers-102,1.46 GFLOPs</td><td></td><td></td><td></td><td>41% feature maps,0.4GFLOPs</td><td></td><td>19.5% feature maps, 0.2 GFLOPs</td></tr><tr><td>CPU: Intel Core i7-5930K</td><td>16</td><td>80.1%</td><td>226.4</td><td>79.8%(-0.3%)</td><td>121.4 (1.9x) 74.1%(-6.0%)</td><td>87.0 (2.6x)</td></tr><tr><td>GPU: GeForce GTX TITAN X(Pascal)</td><td>16</td><td></td><td>4.8</td><td></td><td>2.4 (2.0x)</td><td>1.9 (2.5x)</td></tr><tr><td>GPU:GeForce GTX TITANX(Pascal)</td><td>512</td><td></td><td>88.3</td><td></td><td>36.6 (2.4x)</td><td>27.4 (3.2x)</td></tr><tr><td>GPU: NVIDIA Jetson TX1</td><td>32</td><td></td><td>169.2</td><td></td><td>73.6 (2.3x)</td><td>58.6 (2.9x)</td></tr><tr><td>VGG-16/ImageNet,30.96 GFLOPs</td><td></td><td></td><td></td><td>66% feature maps,11.5 GFLOPs</td><td></td><td>52% feature maps, 8.0 GFLOPs</td></tr><tr><td>CPU:Intel Core i7-5930K</td><td>16</td><td>89.3%</td><td>2564.7</td><td>87.0% (-2.3%)</td><td>1483.3 (1.7x) 84.5% (-4.8%)</td><td>1218.4 (2.1x)</td></tr><tr><td>GPU: GeForce GTX TITAN X(Pascal)</td><td>16</td><td></td><td>68.3</td><td></td><td>31.0 (2.2x)</td><td>20.2 (3.4x)</td></tr><tr><td>GPU: NVIDIA Jetson TX1</td><td>4</td><td></td><td>456.6</td><td></td><td>182.5 (2.5x)</td><td>138.2 (3.3x)</td></tr><tr><td>R3DCNN/nvGesture,37.8 GFLOPs</td><td></td><td></td><td></td><td>25% feature maps,3GFLOPs</td><td></td><td></td></tr><tr><td>GPU:GeForce GT730M</td><td>1</td><td>80.7%</td><td>438.0</td><td>78.2% (-2.5%)</td><td>85.0 (5.2x)</td><td></td></tr></table>",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "For a second experiment, we prune a trained VGG-16 network with the same parameters as before, except enabling FLOPs regularization. We stop pruning at two points, 11.5 and 8.0 GFLOPs, and fine-tune both models for an additional five epochs with learning rate $1 0 ^ { - 4 }$ . Fine-tuning after pruning significantly improves results: the network pruned to 11.5 GFLOPs improves from $8 3 \\%$ to $8 7 \\%$ top-5 validation accuracy, and the network pruned to 8.0 GFLOPs improves from $7 7 . 8 \\%$ to $8 4 . 5 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "3.6 SPEED UP MEASUREMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "During pruning we were measuring reduction in computations by FLOPs, which is a common practice (Han et al., 2015; Lavin, 2015a;b). Improvements in FLOPs result in monotonically decreasing inference time of the networks because of removing entire feature map from the layer. However, time consumed by inference dependents on particular implementation of convolution operator, parallelization algorithm, hardware, scheduling, memory transfer rate etc. Therefore we measure improvement in the inference time for selected networks to see real speed up compared to unpruned networks in Table 2. We observe significant speed ups by proposed pruning scheme. ",
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "4 CONCLUSIONS ",
+        "text_level": 1,
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "We propose a new scheme for iteratively pruning deep convolutional neural networks. We find: 1) CNNs may be successfully pruned by iteratively removing the least important parameters—feature maps in this case—according to heuristic selection criteria; 2) a Taylor expansion-based criterion demonstrates significant improvement over other criteria; 3) per-layer normalization of the criterion is important to obtain global scaling. ",
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+        "page_idx": 9
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            299,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 FLOPS COMPUTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            135,
+            375,
+            150
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To compute the number of floating-point operations (FLOPs), we assume convolution is implemented as a sliding window and that the nonlinearity function is computed for free. For convolutional kernels we have: ",
+        "bbox": [
+            173,
+            162,
+            825,
+            205
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e0a3c0134f52b137d60502e36e2bba0650da626853e73098b5d5244a134aec9.jpg",
+        "text": "$$\n\\mathrm { F L O P s } = 2 H W ( C _ { i n } K ^ { 2 } + 1 ) C _ { o u t } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            208,
+            617,
+            227
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $H$ , $W$ and $C _ { i n }$ are height, width and number of channels of the input feature map, $K$ is the kernel width (assumed to be symmetric), and $C _ { o u t }$ is the number of output channels. ",
+        "bbox": [
+            174,
+            234,
+            828,
+            265
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For fully connected layers we compute FLOPs as: ",
+        "bbox": [
+            173,
+            270,
+            503,
+            285
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fee2ee704f961d1a59e9e942ce10c4193b4daf41c42d9757c06bef53db3b2737.jpg",
+        "text": "$$\n\\mathrm { { F L O P s } } = ( 2 I - 1 ) O ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            424,
+            295,
+            571,
+            313
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $I$ is the input dimensionality and $O$ is the output dimensionality. ",
+        "bbox": [
+            174,
+            321,
+            637,
+            338
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We apply FLOPs regularization during pruning to prune neurons with higher FLOPs first. FLOPs per convolutional neuron in every layer: ",
+        "bbox": [
+            173,
+            344,
+            823,
+            373
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "VGG16: $\\Theta ^ { f l o p s } = [ 3 . 1 , 5 7 . 8 , 1 4 . 1 , 2 8 . 9 , 7 . 0 , 1 4 . 5 , 1 4 . 5 , 3 . 5 , 7 . 2 , 7 . 2 , 1 . 8 , 1 . 8 , 1 . 8 , 1 . 8 , ]$ , 1.8]   \nAlexNet: $\\Theta ^ { f l o p s } = [ 2 . 3 , 1 . 7 , 0 . 8 , 0 . 6 , 0 . 6 ]$   \nR3DCNN: $\\Theta ^ { f l o p s } = [ 5 . 6 , 8 6 . 9 , 2 1 . 7 , 4 3 . 4 , 5 . 4 , 1 0 . 8 , 1 . 4 , 1 . 4 ]$ ",
+        "bbox": [
+            209,
+            383,
+            787,
+            444
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 NORMALIZATION ACROSS LAYERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            463,
+            457,
+            478
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Scaling a criterion across layers is very important for pruning. If the criterion is not properly scaled, then a hand-tuned multiplier would need to be selected for each layer. Statistics of feature map ranking by different criteria are shown in Fig. 10. Without normalization (Fig. 14a–14d), the weight magnitude criterion tends to rank feature maps from the first layers more important than last layers; the activation criterion ranks middle layers more important; and Taylor ranks first layers higher. After $\\ell _ { 2 }$ normalization (Fig. 10d–10f), all criteria have a shape more similar to the oracle, where each layer has some feature maps which are highly important and others which are unimportant. ",
+        "bbox": [
+            173,
+            491,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/0bd0840462f9fae0679e0d449d6b8d7c088a8ca9bf27dfccac2664d69fb279a0.jpg",
+        "image_caption": [
+            "Figure 10: Statistics of feature map ranking by raw criteria values (top) and by criteria values after $\\ell _ { 2 }$ normalization (bottom). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            611,
+            808,
+            886
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/c478294b512edb157a0452023f937f8f33b26890affded397e5f00c49e243031.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">MI</td><td rowspan=\"2\">Weight</td><td colspan=\"3\">Activation</td><td rowspan=\"2\">OBD</td><td rowspan=\"2\">Taylor</td></tr><tr><td>Mean</td><td>S.d.</td><td>APoZ</td></tr><tr><td>Per layer</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Layer 1</td><td>0.41</td><td>0.40</td><td>0.65</td><td>0.78</td><td>0.36</td><td>0.54</td><td>0.95</td></tr><tr><td>Layer 2</td><td>0.23</td><td>0.57</td><td>0.56</td><td>0.59</td><td>0.33</td><td>0.78</td><td>0.90</td></tr><tr><td>Layer 3</td><td>0.14</td><td>0.55</td><td>0.48</td><td>0.45</td><td>0.51</td><td>0.66</td><td>0.74</td></tr><tr><td>Layer 4</td><td>0.26</td><td>0.23</td><td>0.58</td><td>0.42</td><td>0.10</td><td>0.36</td><td>0.80</td></tr><tr><td>Layer 5</td><td>0.17</td><td>0.28</td><td>0.49</td><td>0.52</td><td>0.15</td><td>0.54</td><td>0.69</td></tr><tr><td>Layer 6</td><td>0.21</td><td>0.18</td><td>0.41</td><td>0.48</td><td>0.16</td><td>0.49</td><td>0.63</td></tr><tr><td>Layer 7</td><td>0.12</td><td>0.19</td><td>0.54</td><td>0.49</td><td>0.38</td><td>0.55</td><td>0.71</td></tr><tr><td>Layer8</td><td>0.18</td><td>0.23</td><td>0.43</td><td>0.42</td><td>0.30</td><td>0.50</td><td>0.54</td></tr><tr><td>Layer 9</td><td>0.21</td><td>0.18</td><td>0.50</td><td>0.55</td><td>0.35</td><td>0.53</td><td>0.61</td></tr><tr><td>Layer 10</td><td>0.26</td><td>0.15</td><td>0.59</td><td>0.60</td><td>0.45</td><td>0.61</td><td>0.66</td></tr><tr><td>Layer 11</td><td>0.41</td><td>0.12</td><td>0.61</td><td>0.65</td><td>0.45</td><td>0.64</td><td>0.72</td></tr><tr><td>Layer 12</td><td>0.47</td><td>0.15</td><td>0.60</td><td>0.66</td><td>0.39</td><td>0.66</td><td>0.72</td></tr><tr><td>Layer 13</td><td>0.61</td><td>0.21</td><td>0.77</td><td>0.76</td><td>0.65</td><td>0.76</td><td>0.77</td></tr><tr><td>Mean</td><td>0.28</td><td>0.27</td><td>0.56</td><td>0.57</td><td>0.35</td><td>0.59</td><td>0.73</td></tr><tr><td>All layers</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>No normalization</td><td>0.35</td><td>0.34</td><td>0.35</td><td>0.30</td><td>0.43</td><td>0.65</td><td>0.14</td></tr><tr><td>l1 normalization</td><td>0.47</td><td>0.37</td><td>0.63</td><td>0.63</td><td>0.52</td><td>0.65</td><td>0.71</td></tr><tr><td>l2 normalization</td><td>0.47</td><td>0.33</td><td>0.64</td><td>0.66</td><td>0.51</td><td>0.60</td><td>0.73</td></tr><tr><td>Min-max normalization</td><td>0.27</td><td>0.17</td><td>0.52</td><td>0.57</td><td>0.42</td><td>0.54</td><td>0.67</td></tr></table>",
+        "bbox": [
+            256,
+            101,
+            741,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Spearman’s rank correlation of criteria vs oracle-abs in VGG-16 fine-tuned on Birds 200. ",
+        "bbox": [
+            176,
+            383,
+            820,
+            397
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3 ORACLE COMPUTATION FOR VGG-16 ON BIRDS-200 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            421,
+            586,
+            436
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We compute the change in the loss caused by removing individual feature maps from the VGG-16 network, after fine-tuning on the Birds-200 dataset. Results are illustrated in Fig. 11a-11b for each feature map in layers 1 and 13, respectively. To compute the oracle estimate for a feature map, we remove the feature map and compute the network prediction for each image in the training set using the central crop with no data augmentation or dropout. We draw the following conclusions: ",
+        "bbox": [
+            173,
+            450,
+            825,
+            520
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "• The contribution of feature maps range from positive (above the red line) to slightly negative (below the red line), implying the existence of some feature maps which decrease the training cost when removed.   \n• There are many feature maps with little contribution to the network output, indicated by almost zero change in loss when removed.   \n• Both layers contain a small number of feature maps which induce a significant increase in the loss when removed. ",
+        "bbox": [
+            210,
+            535,
+            825,
+            659
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/331248b1ce09f6104dc08ad70fa6d25b399860365c037d079ffcd5e1c72162b8.jpg",
+        "image_caption": [
+            "Figure 11: Change in training loss as a function of the removal of a single feature map from the VGG-16 network after fine-tuning on Birds-200. Results are plotted for two convolutional layers w.r.t. the index of the removed feature map index. The loss with all feature maps, 0.00461, is indicated with a red horizontal line. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            684,
+            790,
+            857
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/fe5c5536861fe96a4283fc23d0bab5d4d42c6223e63c40a29225584a4450ed59.jpg",
+        "image_caption": [
+            "Figure 12: Comparison of our iterative pruning with pruning by regularization "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            104,
+            784,
+            270
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 3 contains a layer-by-layer listing of Spearman’s rank correlation of several criteria with the ranking of oracle-abs. In this more detailed comparison, we see the Taylor criterion shows higher correlation for all individual layers. For several methods including Taylor, the worst correlations are observed for the middle of the network, layers 5-10. We also evaluate several techniques for normalization of the raw criteria values for comparison across layers. The table shows the best performance is obtained by $\\ell _ { 2 }$ normalization, hence we select it for our method. ",
+        "bbox": [
+            174,
+            327,
+            825,
+            411
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.4 COMPARISON WITH WEIGHT REGULARIZATION",
+        "text_level": 1,
+        "bbox": [
+            176,
+            434,
+            542,
+            446
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Han et al. (2015) find that fine-tuning with high $\\ell _ { 1 }$ or $\\ell _ { 2 }$ regularization causes unimportant connections to be suppressed. Connections with energy lower than some threshold can be removed on the assumption that they do not contribute much to subsequent layers. The same work also finds that thresholds must be set separately for each layer depending on its sensitivity to pruning. The procedure to evaluate sensitivity is time-consuming as it requires pruning layers independently during evaluation. ",
+        "bbox": [
+            174,
+            460,
+            825,
+            531
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The idea of pruning with high regularization can be extended to removing the kernels for an entire feature map if the $\\ell _ { 2 }$ norm of those kernels is below a predefined threshold. We compare our approach with this regularization-based pruning for the task of pruning the last convolutional layer of VGG-16 fine-tuned for Birds-200. By considering only a single layer, we avoid the need to compute layerwise sensitivity. Parameters for optimization during fine-tuning are the same as other experiments with the Birds-200 dataset. For the regularization technique, the pruning threshold is set to $\\sigma = 1 0 ^ { - 5 }$ while we vary the regularization coefficient $\\gamma$ of the $\\ell _ { 2 }$ norm on each feature map kernel.5 We prune only kernel weights, while keeping the bias to maintain the same expected output. ",
+        "bbox": [
+            174,
+            537,
+            825,
+            650
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A comparison between pruning based on regularization and our greedy scheme is illustrated in Fig. 12. We observe that our approach has higher test accuracy for the same number of remaining unpruned feature maps, when pruning $8 5 \\%$ or more of the feature maps. We observe that with high regularization all weights tend to zero, not only unimportant weights as Han et al. (2015) observe in the case of ImageNet networks. The intuition here is that with regularization we push all weights down and potentially can affect important connections for transfer learning, whereas in our iterative procedure we only remove unimportant parameters leaving others untouched. ",
+        "bbox": [
+            173,
+            656,
+            825,
+            753
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.5 COMBINATION OF CRITERIA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            776,
+            413,
+            790
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "One of the possibilities to improve saliency estimation is to combine several criteria together. One of the straight forward combinations is Taylor and mean activation of the neuron. We compute the joint criteria as $\\Theta _ { j o i n t } ( \\mathbf { z } _ { l } ^ { ( k ) } ) = ( 1 - \\lambda ) \\hat { \\Theta } _ { T a y l o r } ( \\mathbf { z } _ { l } ^ { ( k ) } ) + \\lambda \\hat { \\Theta } _ { A c t i v a t i o n } ( \\mathbf { z } _ { l } ^ { ( k ) } )$ and perform a grid search of parameter $\\lambda$ in Fig.13. The highest correlation value for each dataset is marked with with vertical bar with $\\lambda$ and gain. We observe that the gain of linearly combining criteria is negligibly small (see $\\Delta$ ’s in the figure). ",
+        "bbox": [
+            174,
+            803,
+            825,
+            891
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/7ac80e5c32922b43e5033947e7c3653a5f67c9778c4a8649833a70f3b6e2dc9f.jpg",
+        "image_caption": [
+            "Figure 13: Spearman rank correlation for linear combination of criteria. The per layer metric is used. Each $\\Delta$ indicates the gain in correlation for one experiment. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            354,
+            103,
+            643,
+            314
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.6 OPTIMAL BRAIN DAMAGE IMPLEMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            383,
+            532,
+            397
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "OBD computes saliency of a parameter by computing a product of the squared magnitude of the parameter and the corresponding element on the diagonal of the Hessian. For many deep learning frameworks, an efficient implementation of the diagonal evaluation is not straightforward and approximation techniques must be applied. Our implementation of Hessian diagonal computation was inspired by Dauphin et al. (2015) work, where the technique proposed by Bekas et al. (2007) was used to evaluate SGD preconditioned with the Jacobi preconditioner. It was shown that diagonal of the Hessian can be approximated as: ",
+        "bbox": [
+            174,
+            410,
+            825,
+            506
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0789f40c79bfa1fdee711cf349c690aa21e0b9b1c636287cba2ba20718e4c918.jpg",
+        "text": "$$\n\\mathrm { d i a g } ( \\mathbf { H } ) = \\mathbb { E } [ \\mathbf { v } \\odot \\mathbf { H } \\mathbf { v } ] = \\mathbb { E } [ \\mathbf { v } \\odot \\nabla ( \\nabla \\mathcal { C } \\cdot \\mathbf { v } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            513,
+            643,
+            531
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ is the element-wise product, $\\mathbf { v }$ are random vectors with entries $\\pm 1$ , and $\\nabla$ is the gradient operator. To compute saliency with OBD, we randomly draw $\\mathbf { v }$ and compute the diagonal over 10 iterations for a single minibatch for 1000 mini batches. We found that this number of mini batches is required to compute close approximation of the Hessian’s diagonal (which we verified). Computing saliency this way is computationally expensive for iterative pruning, and we use a slightly different but more efficient procedure. Before the first pruning iteration, saliency is initialized from values computed off-line with 1000 minibatches and 10 iterations, as described above. Then, at every minibatch we compute the OBD criteria with only one iteration and apply an exponential moving averaging with a coefficient of 0.99. We verified that this computes a close approximation to the Hessian’s diagonal. ",
+        "bbox": [
+            173,
+            537,
+            825,
+            676
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.7 CORRELATION OF TAYLOR CRITERION WITH GRADIENT AND ACTIVATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            694,
+            725,
+            708
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The Taylor criterion is composed of both an activation term and a gradient term. In Figure 14, we depict the correlation between the Taylor criterion and each constituent part. We consider expected absolute value of the gradient instead of the mean, because otherwise it tends to zero. The plots are computed from pruning criteria for an unpruned VGG network fine-tuned for the Birds-200 dataset. (Values are shown after layer-wise normalization). Figure 14(a-b) depict the Taylor criterion in the y-axis for all neurons w.r.t. the gradient and activation components, respectively. The bottom $1 0 \\%$ of neurons (lowest Taylor criterion, most likely to be pruned) are depicted in red, while the top $1 0 \\%$ are shown in green. Considering all neurons, both gradient and activation components demonstrate a linear trend with the Taylor criterion. However, for the bottom $1 0 \\%$ of neurons, as shown in Figure 14(c-d), the activation criterion shows much stronger correlation, with lower activations indicating lower Taylor scores. ",
+        "bbox": [
+            173,
+            719,
+            826,
+            872
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/1c248872de3c5daef6857d28123867f99e9e6e2807ee078a0d1e49c1ec54a75e.jpg",
+        "image_caption": [
+            "Figure 14: Correlation of Taylor criterion with gradient and activation (after layer-wise $\\ell _ { 2 }$ normalization) for all neurons (a-b) and bottom $1 0 \\%$ of neurons (c-d) for unpruned VGG after fine-tuning on Birds-200. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            246,
+            325,
+            753,
+            648
+        ],
+        "page_idx": 16
+    }
+]

parse/train/Sy-tszZRZ/Sy-tszZRZ.md

@@ -0,0 +1,655 @@
+# BOUNDING AND COUNTING LINEAR REGIONS OF DEEP NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+In this paper, we study the representational power of deep neural networks (DNN) that belong to the family of piecewise-linear (PWL) functions, based on PWL activation units such as rectifier or maxout. We investigate the complexity of such networks by studying the number of linear regions of the PWL function. Typically, a PWL function from a DNN can be seen as a large family of linear functions acting on millions of such regions. We directly build upon the work of Montufar et al. (2014), Mont ´ ufar (2017), and Raghu et al. (2017) by refining the ´ upper and lower bounds on the number of linear regions for rectified and maxout networks. In addition to achieving tighter bounds, we also develop a novel method to perform exact enumeration or counting of the number of linear regions with a mixed-integer linear formulation that maps the input space to output. We use this new capability to visualize how the number of linear regions change while training DNNs.
+
+# 1 INTRODUCTION
+
+We have witnessed an unprecedented success of deep learning algorithms in computer vision, speech, and other domains (Krizhevsky et al., 2012; Ciresan et al., 2012; Goodfellow et al., 2013; Hinton et al., 2012). While the popular deep learning architectures such as AlexNet (Krizhevsky et al., 2012), GoogleNet (Szegedy et al., 2015), and residual networks (He et al., 2016) have shown record beating performance on various image recognition tasks, empirical results still govern the design of network architecture in terms of depth and activation functions. Two important practical considerations that are part of most successful architectures are greater depth and the use of PWL activation functions such as rectified linear units (ReLUs). Due to the large gap between theory and practice, many researchers have been looking at the theoretical modeling of the representational power of DNNs (Cybenko, 1989; Anthony & Bartlett, 1999; Pascanu et al., 2014; Montufar et al., ´ 2014; Bianchini & Scarselli, 2014; Eldan & Shamir, 2016; Telgarsky, 2015; Mhaskar et al., 2016; Raghu et al., 2017; Montufar, 2017). ´
+
+Any continuous function can be approximated to arbitrary accuracy using a single hidden layer of sigmoid activation functions (Cybenko, 1989). This does not imply that shallow networks are sufficient to model all problems in practice. Typically, shallow networks require exponentially more number of neurons to model functions that can be modeled using much fewer activation functions in deeper ones (Delalleau & Bengio, 2011). There have been a wide variety of activation functions such as threshold $( f ( z ) = ( z > 0 ) )$ , logistic $( f ( z ) = 1 / ( 1 + \exp ( - e ) ) )$ , hyperbolic tangent $( f ( z ) =$ $\operatorname { t a n h } ( z ) )$ , rectified linear units (ReLUs $f ( z ) = \operatorname* { m a x } \{ 0 , z \} )$ , and maxouts $( f ( z _ { 1 } , z _ { 2 } , \ldots , z _ { k } ) =$ $\operatorname* { m a x } \{ z _ { 1 } , z _ { 2 } , \dots , z _ { k } \} )$ . The activation functions offer different modeling capabilities. For example, sigmoid networks are shown to be more expressive than similar-sized threshold networks (Maass et al., 1994). It was recently shown that ReLUs are more expressive than similar-sized threshold networks by deriving transformations from one network to another (Pan & Srikumar, 2016).
+
+The complexity of neural networks belonging to the family of PWL functions can be analyzed by looking at how the network can partition the input space to an exponential number of linear response regions (Pascanu et al., 2014; Montufar et al., 2014). The basic idea of a PWL function is simple: ´ we can divide the input space into several regions and we have individual linear functions for each of these regions. Functions partitioning the input space to a larger number of linear regions are considered to be more complex ones, or in other words, possess better representational power. In the case of ReLUs, it was shown that deep networks separate their input space into exponentially more linear response regions than their shallow counterparts despite using the same number of activation functions (Pascanu et al., 2014). The results were later extended and improved (Montufar et al., ´ 2014; Raghu et al., 2017; Montufar, 2017; Arora et al., 2016). In particular, Mont ´ ufar et al. (2014) ´ shows both upper and lower bounds on the maximal number of linear regions for a ReLU DNN and a single layer maxout network, and a lower bound for a maxout DNN. Furthermore, Raghu et al. (2017) and Montufar (2017) improve the upper bound for a ReLU DNN. This upper bound ´ asymptotically matches the lower bound from Montufar et al. (2014) when the number of layers ´ and input dimension are constant and all layers have the same width. Finally, Arora et al. (2016) improves the lower bound by providing a family of ReLU DNNS with an exponential number of regions given fixed size and depth.
+
+In this work, we directly improve on the results of Montufar et al. (Pascanu et al., 2014; Mont ´ ufar ´ et al., 2014; Montufar, 2017) and Raghu et al. (Raghu et al., 2017) in better understanding the ´ representational power of DNNs employing PWL activation functions.
+
+# 2 NOTATIONS AND BACKGROUND
+
+We will only consider feedforward neural networks in this paper. Let us assume that the network has $n _ { 0 }$ input variables given by $\textbf { x } = \{ x _ { 1 } , x _ { 2 } , \ldots , x _ { n _ { 0 } } \}$ , and $m$ output variables given by ${ \textbf { y } } =$ $\{ y _ { 1 } , y _ { 2 } , . . . , y _ { m } \}$ . Each hidden layer $l = \{ 1 , 2 , \ldots , L \}$ has $n _ { l }$ hidden neurons whose activations are given by $\mathbf { h } ^ { l } = \left\{ h _ { 1 } ^ { l } , h _ { 2 } ^ { l } , \dots , h _ { n _ { l } } ^ { l } \right\}$ . Let $W ^ { \tilde { l } }$ be the $n _ { l } \times n _ { l - 1 }$ matrix where each row corresponds to the weights of a neuron of layer $l$ . Let $\mathbf { b } ^ { l }$ be the bias vector used to obtain the activation functions of neurons in layer $l$ . Based on the ${ \mathrm { R e L U } } ( x ) = \operatorname* { m a x } \{ 0 , x \}$ activation function, the activations of the hidden neurons and the outputs are given below:
+
+$$
+\begin{array} { r c l } { \mathbf { h } ^ { 1 } } & { = } & { \operatorname* { m a x } \{ 0 , W ^ { 1 } \mathbf { x } + b ^ { 1 } \} } \\ { \mathbf { h } ^ { l } } & { = } & { \operatorname* { m a x } \{ 0 , W ^ { l } \mathbf { h } ^ { l - 1 } + b ^ { l } \} } \\ { \mathbf { y } } & { = } & { W ^ { L + 1 } \mathbf { h } ^ { \mathbf { L } } } \end{array}
+$$
+
+As considered in Pascanu et al. (2014), the output layer is a linear layer that computes the linear combination of the activations from the previous layer without any ReLUs.
+
+We can treat the DNN as a piecewise linear (PWL) function $F : \mathbb { R } ^ { n _ { 0 } }  \mathbb { R } ^ { m }$ that maps the input $\mathbf { x }$ in $\mathbb { R } ^ { n _ { 0 } }$ to $\mathbf { y }$ in $\mathbb { R } ^ { m }$ . This paper primarily deals with investigating the bounds on the linear regions of this PWL function. There are two subtly different definitions for linear regions in the literature and we will formally define them.
+
+Definition 1. Given a PWL function $F : \mathbb { R } ^ { n _ { 0 } }  \mathbb { R } ^ { m }$ , a linear region is defined as a maximal connected subset of the input space $\mathbb { R } ^ { n _ { 0 } }$ , on which $F$ is linear (Pascanu et al., 2014; Montufar ´ et al., 2014).
+
+Activation Pattern: Let us consider an input vector $\mathbf x = \{ x _ { 1 } , x _ { 2 } , \dots , x _ { n _ { 0 } } \}$ . For every layer $l$ we define an activation set $S ^ { l } \subseteq \{ 1 , 2 , \dotsc , n _ { l } \}$ such that $e \in S ^ { l }$ if and only if the ReLU $e$ is active, that is, $h _ { e } ^ { l } > 0$ . We aggregate these activation sets into a set $\pmb { S } = ( S ^ { 1 } , \overleftarrow { \ldots } , S ^ { l } )$ , which we call an activation pattern. Note that we may consider activation patterns up to a layer $l \leq L$ . Activation patterns were previously defined in terms of strings (Raghu et al., 2017; Montufar, 2017). ´
+
+We say that an input $\mathbf { x }$ corresponds to an activation pattern $s$ in a DNN if feeding $\mathbf { x }$ to the DNN results in the activations in $s$ .
+
+Definition 2. Given a PWL function $F : \mathbb { R } ^ { n _ { 0 } }  \mathbb { R } ^ { m }$ represented by a DNN, a linear region is the set of input vectors $\mathbf { x }$ that corresponds to an activation pattern $s$ in the DNN.
+
+We prefer to look at linear regions as activation patterns and we interchangeably refer to $s$ as an activation pattern or a region. Definitions 1 and 2 are essentially the same, except in a few degenerate cases. There could be scenarios where two different activation patterns may correspond to two adjacent regions with the same linear function. In this case, Definition 1 will produce only one linear region whereas Definition 2 will yield two linear regions. This has no effect on the bounds that we derive in this paper.
+
+In Fig. 1(a) we show a simple ReLU DNN with two inputs $\{ x _ { 1 } , x _ { 2 } \}$ and 3 hidden layers.
+
+![](images/07e07782af901a7f744422d434ba86436d8e0f158821fe2e12e70e55addd6872.jpg)  
+Figure 1: (a) Simple DNN with two inputs and three hidden layers with 2 activation units each. (b), (c), and (d) Visualization of the hyperplanes from the first, second, and third hidden layers respectively partitioning the input space into several linear regions. The arrows indicate the directions in which the corresponding neurons are activated. (e), $( f )$ , and (g) Visualization of the hyperplanes from the first, second, and third hidden layers in the space given by the outputs of their respective previous layers.
+
+The activation units $\{ a , b , c , d , e , f \}$ in the hidden layers can be thought of as hyperplanes that each divide the space in two. On one side of the hyperplane, the unit outputs a positive value. For all points on the other side of the hyperplane including itself, the unit outputs 0.
+
+One may wonder: into how many regions do $n$ hyperplanes split a space? Zaslavsky (1975) shows that an arrangement of $n$ hyperplanes divides a $d$ -dimensional space into at most $\textstyle \sum _ { s = 0 } ^ { d } { \binom { n } { s } }$ regions, a bound that is attained when they are in general position. The term general position basically means that a small perturbation of the hyperplanes does not change the number of regions. This corresponds to the exact maximal number of regions of a single layer DNN with $n$ ReLUs and input dimension $d$ .
+
+In Figs. $1 ( \mathbf { b } ) \mathbf { - } ( \mathbf { g } )$ , we provide a visualization of how ReLUs partition the input space. Figs. 1(e), (f), and (g) show the hyperplanes corresponding to the ReLUs at layers $l = 1 , 2$ , and 3 respectively. Figs. 1(b), (c), and (d) consider these same hyperplanes in the input space $x$ . In Fig. 1(b), as per Zaslavsky (1975), the 2D input space is partitioned into 4 regions $( \bar { \binom { 2 } { 0 } } + \bar { \binom { 2 } { 1 } } + \binom { 2 } { 2 } = 4 )$ . In Figs. 1(c) and (d), we add the hyperplanes from the second and third layers respectively, which are affected by the transformations applied in the earlier hidden layers. The regions are further partitioned as we consider additional layers.
+
+Fig. 1 also highlights that activation boundaries behave like hyperplanes when inside a region and may bend whenever they intersect with a boundary from a previous layer. This has also been pointed out by Raghu et al. (2017). In particular, they cannot appear twice in the same region as they are defined by a single hyperplane if we fix the region. Moreover, these boundaries do not need to be connected, as illustrated in Fig. 2.
+
+# Main Contributions
+
+We summarize the main contributions of this paper below:
+
+• We achieve tighter upper and lower bounds on the maximal number of linear regions of the PWL function corresponding to a DNN that employs ReLUs. As a special case, we present the exact maximal number of regions when the input dimension is one. We additionally provide the first upper bound on the number of linear regions for multi-layer maxout networks (See Sections 3 and 4).
+
+![](images/b52127ccbebe4c0c9a8ff662d52e35193b9101774a7dd2d983e95ba390ce54ab.jpg)  
+Figure 2: (a) $A$ network with one input $x _ { 1 }$ and three activation units $a , b ,$ , and $c$ . $( b )$ We show the hyperplanes $x _ { 1 } = 0$ and $- x _ { 1 } + 1 = 0$ corresponding to the two activation units in the first hidden layer. In other words, the activation units are given by $h _ { a } = \operatorname* { m a x } \{ 0 , x _ { 1 } \}$ and $h _ { b } = \operatorname* { m a x } \{ 0 , - x _ { 1 } +$ $1 \}$ . (c) The activation unit in the third layer is given by $h _ { c } = \operatorname* { m a x } \{ 0 , 4 h _ { a } + 2 h _ { b } - 3 \}$ . (d) The activation boundary for neuron $c$ is disconnected.
+
+• We show for ReLUs that the exact maximal number of linear regions of shallow networks is larger than that of deep networks if the input dimension exceeds the number of neurons. This result is particularly interesting, since it cannot be inferred from the bounds derived in prior work.
+
+• We use a mixed-integer linear formulation to show that exact counting of the linear regions is indeed possible. For the first time, we show the exact counting of the number of linear regions for several small-sized DNNs during the training process. This new capability can be used to evaluate the tightness of the bounds and potentially analyze the correlation between validation accuracy and the number of linear regions. It also provides new insights as to how the linear regions vary during the training process (See Section 5 and 6).
+
+# 3 TIGHTER BOUNDS FOR RECTIFIER NETWORKS
+
+Montufar et al. (2014) derive an upper bound of ´ $2 ^ { N }$ for $N$ hidden units, which can be obtained by mapping linear regions to activation patterns. Raghu et al. (2017) improves this result by deriving an asymptotic upper bound of $O ( n ^ { L n _ { 0 } } )$ to the maximal number of regions, assuming $n _ { l } = n$ for all layers $l$ and $n _ { 0 } = O ( 1 )$ . Montufar (2017) further tightens the upper bound to ´ $\begin{array} { r } { \prod _ { l = 1 } ^ { L } \sum _ { j = 0 } ^ { d _ { l } } { \binom { n _ { l } } { j } } } \end{array}$ where $d _ { l } = \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } , . . . , n _ { l } \}$ .
+
+Moreover, Montufar et al. (2014) prove a lower bound of ´ $\left( \prod _ { l = 1 } ^ { L - 1 } \lfloor n _ { l } / n _ { 0 } \rfloor ^ { n _ { 0 } } \right) \sum _ { j = 0 } ^ { n _ { 0 } } { \binom { n _ { L } } { j } }$ when $n \geq n _ { 0 }$ , or asymptotically $\Omega ( ( n / n _ { 0 } ) ^ { ( L - 1 ) n _ { 0 } } n ^ { n _ { 0 } } )$ . Arora et al. (2016) present a lower bound of $\textstyle 2 \sum _ { j = 0 } ^ { n _ { 0 } - 1 } { \binom { m - 1 } { j } } w ^ { L - 1 }$ where $2 m = n _ { 1 }$ and $w = n _ { l }$ for all $l = 2 , \ldots , L$ . By choosing $m$ and $w$ appropriately, this lower bound is $\Omega ( s ^ { n _ { 0 } } )$ where $s$ is the total size of the network. We derive both upper and lower bounds that improve upon these previous results.
+
+# 3.1 AN UPPER BOUND ON THE NUMBER OF LINEAR REGIONS
+
+In this section, we prove the following upper bound on the number of regions.
+
+Theorem 1. Consider a deep rectifier network with $L$ layers, $n _ { l }$ rectified linear units at each layer $l$ , and an input of dimension $n _ { 0 }$ . The maximal number of regions of this neural network is at most
+
+$$
+\sum _ { ( j _ { 1 } , \ldots , j _ { L } ) \in J } \prod _ { l = 1 } ^ { L } { \binom { n _ { l } } { j _ { l } } }
+$$
+
+where J = {(j1, . . . , jL) ∈ ZL : 0 ≤ jl ≤ min{n0, n1 − j1, . . . , nl−1 − jl−1, nl} ∀l = 1, . . . , L}.   
+This bound is tight when $L = 1$ .
+
+Note that this is a stronger upper bound than the one that appeared in Montufar (2017), which can ´ be derived from this bound by relaxing the terms $n _ { l } - j _ { l }$ to $n _ { l }$ and factoring the expression. When $n _ { 0 } = O ( 1 )$ and all layers have the same width $n$ , this expression has the same best known asymptotic bound $\dot { O ( n ^ { L n _ { 0 } } ) }$ first presented in Raghu et al. (2017).
+
+Two insights can be extracted from the above expression:
+
+1. Bottleneck effect. The bound is sensitive to the positioning of layers that are small relative to the others, a property we call the bottleneck effect. If we subtract a neuron from one of two layers with the same width, choosing the one closer to the input layer will lead to a larger (or equal) decrease in the bound. This occurs because each index $j _ { l }$ is essentially limited by the widths of the current and previous layers, $n _ { 0 } , n _ { 1 } , \ldots , n _ { l }$ . In other words, smaller widths in the first few layers of the network imply a bottleneck on the bound.
+
+In particular for a 2-layer network, we show in Appendix A that if the input dimension is sufficiently large to not create its own bottleneck, then moving a neuron from the first layer to the second layer strictly decreases the bound, as it tightens a bottleneck.
+
+Figure 3a illustrates this behavior. For the solid line, we keep the total size of the network the same but shift from a small-to-large network (i.e., smaller width near the input layer and larger width near the output layer) to a large-to-small network in terms of width. We see that the bound monotonically increases as we reduce the bottleneck. If we add a layer of constant width at the end, represented by the dashed line, the bound decreases when the layers before the last become too small and create a bottleneck for the last layer.
+
+While this is a property of the upper bound rather than one of the exact maximal number of regions, we observe in Section 6 that empirical results for the number of regions of a trained network exhibit a behavior that resembles the bound as the width of the layers vary.
+
+2. Deep vs shallow for large input dimensions. In several applications such as imaging, the input dimension can be very large. Montufar et al. (2014) show that if the input dimension ´ $n _ { 0 }$ is constant, then the number of regions of deep networks is asymptotically larger than that of shallow (single-layer) networks. We complement this picture by establishing that if the input dimension is large, then shallow networks can attain more regions than deep networks.
+
+More precisely, we compare a deep network with $L$ layers of equal width $n$ and a shallow network with one layer of width $L n$ . In Appendix A, we show using Theorem 1 that if the input dimension $n _ { 0 }$ exceeds the size of the network $L n$ , then the ratio between the exact maximal number of regions of the deep and of the shallow network goes to zero as $L$ approaches infinity.
+
+We also show in Appendix A that in a 2-layer network, if the input dimension $n _ { 0 }$ is larger than both widths $n _ { 1 }$ and $n _ { 2 }$ , then turning it into a shallow network with a layer of $n _ { 1 } + n _ { 2 }$ ReLUs increases the exact maximal number of regions.
+
+Figure 3b illustrates this behavior. As we increase the number of layers while keeping the total size of the network constant, the bound plateaus at a value lower than the exact maximal number of regions for shallow networks. Moreover, the number of layers that yields the highest bound decreases as we increase the input dimension $n _ { 0 }$ .
+
+It is important to note that this property cannot be inferred from previous upper bounds derived in prior work, since they are at least $2 ^ { N }$ when $n _ { 0 } \ge \operatorname* { m a x } \{ n _ { 1 } , . . . , n _ { L } \}$ , where $N$ is the total number of neurons.
+
+We remark that asymptotically both deep and shallow networks can attain exponentially many regions when the input dimension is at least $n$ (see Appendix B).
+
+![](images/0f4019d6e5c2faf0ece041df2fc6d28f6e6f9572f0d1dbe4f2a870bd91f3f2b2.jpg)  
+Figure 3: Bounds from Theorem 1: (a) is in semilog scale, has input dimension $n _ { 0 } = 3 2 { \mathrm { ~ } }$ , and the width of the first five layers is $1 6 - 2 k , 1 6 - k , 1 6 , 1 6 + k , 1 6 + 2 k$ ; (b) is in linear scale, evenly distributes 60 neurons in 1 to 6 layers (the single-layer case is exact), and the input dimension varies.
+
+We now build towards the proof of Theorem 1. For a given activation set $S ^ { l }$ and a matrix $W$ with $n _ { l }$ rows, let $\sigma _ { S ^ { l } } ( W )$ be the operation that zeroes out the rows of $W$ that are inactive according to $S ^ { l }$ . This represents the effect of the ReLUs. For a region $s$ at layer $l - 1$ , define $\bar { W } _ { S } ^ { l } : = \bar { \Psi }$ $W ^ { l } \sigma _ { S ^ { l - 1 } } ( W ^ { l - 1 } ) \cdot \cdot \cdot \sigma _ { S ^ { 1 } } ( W ^ { 1 } )$ .
+
+Each region $s$ at layer $l - 1$ may be partitioned by a set of hyperplanes defined by the neurons of layer $l$ . When viewed in the input space, these hyperplanes are the rows of $\bar { W } _ { S } ^ { l } x + \bar { b } = 0$ for some $b$ . To verify this, note that, if we recursively substitute out the hidden variables $h _ { l - 1 } , \ldots , h _ { 1 }$ from the original hyperplane $W ^ { l } h _ { l - 1 } + b _ { l } = 0$ following $s$ , the resulting weight matrix applied to $x$ is ${ \bar { W } } _ { S } ^ { l }$ .
+
+Finally, we define the dimension of a region $s$ at layer $l \_ 1 \_ 1$ as $\begin{array} { r l } { \dim ( S ) } & { { } : = } \end{array}$ $\operatorname { r a n k } ( \overleftarrow { \boldsymbol { \sigma } } _ { S ^ { l - 1 } } ( W ^ { l - 1 } ) \cdot \cdot \cdot \boldsymbol { \sigma } _ { S ^ { 1 } } ( W ^ { 1 } ) )$ . This can be interpreted as the dimension of the space corresponding to $s$ that $W ^ { l }$ effectively partitions.
+
+The proof of Theorem 1 focuses on the dimension of each region $s$ . A key observation is that once it falls to a certain value, the regions contained in $s$ cannot recover to a higher dimension.
+
+Zaslavsky (1975) showed that the maximal number of regions in $\mathbb { R } ^ { d }$ induced by an arrangement of $m$ hyperplanes is at most $\textstyle \sum _ { j = 0 } ^ { d } { \binom { m } { j } }$ . Moreover, this value is attained if and only if the hyperplanes are in general position. The lemma below tightens this bound for a special case where the hyperplanes may not be in general position.
+
+Lemma 2. Consider m hyperplanes in regions induced by the hyperplanes is a $\mathbb { R } ^ { d }$ deost s of  . $W x + b = 0$ . Then the number of $\sum _ { j = 0 } ^ { \mathrm { r a n k } ( W ) } \binom { m } { j }$
+
+The proof is given in Appendix C. Its key idea is that it suffices to count regions within the row space of $W$ . The next lemma brings Lemma 2 into our context.
+
+Lemma 3. The number of regions induced by the $n _ { l }$ neurons at layer l within a certain region $s$ is at most Pmin{j=0 $\sum _ { j = 0 } ^ { \operatorname* { m i n } \left\{ n _ { l } , \dim ( { \cal S } ) \right\} } \binom { n _ { l } } { j }$ .
+
+Proof. The hyperplanes in a region $s$ of the input space are given by the rows of $\bar { W } _ { S } ^ { l } x \ +$ $ { \boldsymbol { b } } \quad = \quad 0$ for some $b$ . By the definition of ${ \bar { W } } _ { S } ^ { l }$ , the rank of ${ \bar { W } } _ { S } ^ { l }$ is upper bounded by $\begin{array} { r l } { \operatorname* { m i n } _ { - } \{ \mathrm { r a n k } ( W ^ { l } ) , \mathrm { r a n k } ( \sigma _ { S ^ { l - 1 } } ( W ^ { l - 1 } ) \cdot \cdot \cdot \sigma _ { S ^ { 1 } } ( W ^ { 1 } ) ) \} } & { = \ : \operatorname* { m i n } \{ \mathrm { r a n k } ( W ^ { l } ) , \dim ( S ) \} } \end{array}$ . That is, $\mathrm { r a n k } ( \bar { W } _ { \mathcal { S } } ^ { \bar { l } } ) \leq \operatorname* { m i n } \{ n _ { l } , \dim ( \tilde { \mathcal { S } } ) \}$ . Applying Lemma 2 yields the result. □
+
+In the next lemma, we show that the dimension of a region $s$ can be bounded recursively in terms of the dimension of the region containing $s$ and the number of activated neurons defining $s$ .
+
+Lemma 4. Let $s$ be a region at layer $l$ and $S ^ { \prime }$ be the region at layer $l - 1$ that contains it. Then $\mathrm { d i m } ( S ) \leq \operatorname* { m i n } \{ | S ^ { l } | , \mathrm { d i m } ( S ^ { \prime } ) \}$ .
+
+Proof. $\dim ( { \mathcal S } ) \ = \ \operatorname { r a n k } ( \sigma _ { S ^ { l } } ( W ^ { l } ) \cdot \cdot \cdot \sigma _ { S ^ { 1 } } ( W ^ { 1 } ) ) \ \leq \ \operatorname* { m i n } \{ \operatorname { r a n k } ( \sigma _ { S ^ { l } } ( W ^ { l } ) ) , \operatorname { r a n k } ( \sigma _ { S ^ { l - 1 } } ( W ^ { l - 1 } ) \cdot \cdot \cdot \sigma _ { S ^ { 1 } } ( W ^ { 1 } ) ) \} \ = \ \operatorname* { m i n } \{ \operatorname { r a n k } ( \sigma _ { S ^ { l } } ( W ^ { l } ) ) , \operatorname { r a n k } ( \sigma _ { S ^ { l - 1 } } ( W ^ { 1 } ) \cdot \cdot \cdot \sigma _ { S ^ { 1 } } ( W ^ { 1 } ) ) \}$ $\sigma _ { S ^ { 1 } } ( W ^ { 1 } ) ) \leq \operatorname* { m i n } \{ | S ^ { l } | , \dim ( \bar { S } ^ { \prime } ) \}$ . The last inequality comes from the fact that the zeroed out rows do not count towards the rank of the matrix. □
+
+In the remainder of the proof of Theorem 1, we combine Lemmas 3 and 4 to construct a recurrence $R ( l , d )$ that bounds the number of regions within a given region of dimension $d$ . Simplifying this recurrence yields the expression in Theorem 1. We formalize this idea and complete the proof of Theorem 1 in Appendix D.
+
+As a side note, Theorem 1 can be further tightened if the weight matrices are known to have small rank. The bound from Lemma 3 can be rewritten as j=0 Pmin{rank(W l),dim(S)} nl
 if we do not relax $\mathrm { r a n k } ( W ^ { l } )$ to $n _ { l }$ in the proof. The term $\mathrm { r a n k } ( W ^ { l } )$ follows through the proof of Theorem 1 and the index set $J$ in the theorem becomes $\{ ( j _ { 1 } , \dotsc , j _ { L } ) \in \mathbb { Z } ^ { L } : 0 \leq j _ { l } \leq \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } - j _ { 1 } , \dotsc , n _ { l - 1 } -$ $j _ { l - 1 } , \mathrm { r a n k } ( W ^ { l } ) \} \forall l \geq 1 \}$ .
+
+A key insight from Lemmas 3 and 4 is that the dimensions of the regions are non-increasing as we move through the layers partitioning it. In other words, if at any layer the dimension of a region becomes small, then that region will not be able to be further partitioned into a large number of regions. For instance, if the dimension of a region falls to zero, then that region will never be further partitioned. This suggests that if we want to have many regions, we need to keep dimensions high. We use this idea in the next section to construct a DNN with many regions.
+
+# 3.2 THE CASE OF DIMENSION ONE
+
+If the input dimension $n _ { 0 }$ is equal to 1 and $n _ { l } = n$ for all layers $l$ , the upper bound presented in the previous section reduces to $( n + 1 ) ^ { L }$ . On the other hand, the lower bound given by Montufar et al. ´ (2014) becomes $n ^ { L - 1 } ( n + 1 )$ . It is then natural to ask: are either of these bounds tight? The answer is that the upper bound is tight in the case of $n _ { 0 } = 1$ , assuming there are sufficiently many neurons.
+
+Theorem 5. Consider a deep rectifier network with $L$ layers, $n \geq 3$ rectified linear units at each layer $l$ , and an input of dimension $^ { l }$ . The maximal number of regions of this neural network is exactly $\textstyle \prod _ { l = 1 } ^ { L } ( n _ { l } + 1 )$ .
+
+The expression above is a simplified form of the upper bound from Theorem 1 in the case $n _ { 0 } = 1$
+
+The proof of this theorem in Appendix E has a construction with $n + 1$ regions that replicate themselves as we add layers, instead of $n$ as in Montufar et al. (2014). That is motivated by an insight ´ from the previous section: in order to obtain more regions, we want the dimension of every region to be as large as possible. When $n _ { 0 } = 1$ , we want all regions to have dimension one. This intuition leads to a new construction with one additional region that can be replicated with other strategies.
+
+# 3.3 A LOWER BOUND ON THE MAXIMAL NUMBER OF LINEAR REGIONS
+
+Both the lower bound from Montufar et al. (2014) and from Arora et al. (2016) can be slightly ´ improved, since their approaches are based on extending a 1-dimensional construction similar to the one in Section 3.2. We do both since they are not directly comparable: the former bound is in terms of the number of neurons in each layer and the latter is in terms of the total size of the network.
+
+Theorem 6. The maximal number of linear regions induced by a rectifier network with $n _ { 0 }$ input units and $L$ hidden layers with $n _ { l } \ge 3 n _ { 0 }$ for all $l$ is lower bounded by
+
+$$
+\left( \prod _ { l = 1 } ^ { L - 1 } \left( \left\lfloor { \frac { n _ { l } } { n _ { 0 } } } \right\rfloor + 1 \right) ^ { n _ { 0 } } \right) \sum _ { j = 0 } ^ { n _ { 0 } } { \binom { n _ { L } } { j } } .
+$$
+
+The proof of this theorem is in Appendix F. For comparison, the differences between the lower bound theorem (Theorem 5) from Montufar et al. (2014) and the above theorem is the replacement´ of the condition $n _ { l } \ge n _ { 0 }$ by the more restrictive $n _ { l } \ge 3 n _ { 0 }$ , and of $\lfloor n _ { l } / n _ { 0 } \rfloor$ by $\lfloor n _ { l } / n _ { 0 } \rfloor + { \bar { 1 } }$ .
+
+Theand em 7. For any value hidden layers of size $m \geq 1$ $w \geq 2$ , the has ith lin $n _ { 0 }$ input units regions. $L$ $2 m + w ( L - 1 )$ $\textstyle 2 \sum _ { j = 0 } ^ { n _ { 0 } - 1 } { \binom { m - 1 } { j } } ( w + 1 ) ^ { L - 1 }$
+
+The proof of this theorem is in Appendix G. The differences between Theorem 2.11(i) from Arora et al. (2016) and the above theorem is the replacement of $w$ by $w + 1$ . They construct a $2 m$ -width layer with many regions and use a one-dimensional construction for the remaining layers.
+
+# 4 AN UPPER BOUND ON THE NUMBER OF LINEAR REGIONS FOR MAXOUT NETWORKS
+
+We now consider a deep neural network composed of maxout units. Given weights $W _ { j } ^ { l }$ for $j =$ $1 , \ldots , k$ , the output of a rank- $k$ maxout layer $l$ is given by
+
+$$
+\begin{array} { r c l } { \mathbf { h } ^ { l } } & { = } & { \operatorname* { m a x } \{ W _ { 1 } ^ { l } \mathbf { h } ^ { l - 1 } + b _ { 1 } ^ { l } , \dots , W _ { k } ^ { l } \mathbf { h } ^ { l - 1 } + b _ { k } ^ { l } \} } \end{array}
+$$
+
+In terms of bounding number of regions, a major difference between the next result for maxout units and the previous one for ReLUs is that reductions in dimensionality due to inactive neurons with zeroed output become a particular case now. Nevertheless, using techniques similar to the ones from Section 3.1, the following theorem can be shown (see Appendix H for the proof).
+
+Theorem 8. Consider a deep neural network with $L$ layers, $n _ { l }$ rank- $k$ maxout units at each layer $l$ , and an input of dimension $n _ { 0 }$ . The maximal number of regions of this neural network is at most
+
+$$
+\prod _ { l = 1 } ^ { L } \sum _ { j = 0 } ^ { d _ { l } } { \binom { k ( k - 1 ) } { j } } n _ { l } \rangle
+$$
+
+where $d _ { l } = \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } , . . . , n _ { l } \}$
+
+Asymptotically, if $n _ { l } = n$ for all $l = 1 , \ldots , L , n \geq n _ { 0 }$ , and $n _ { 0 } = O ( 1 )$ , then the maximal number of regions is at most $O ( ( k ^ { 2 } n ) ^ { L n _ { 0 } } )$ .
+
+# 5 EXACT COUNTING OF LINEAR REGIONS
+
+If the input space $\mathbf { x } \in \mathbb { R } ^ { n _ { 0 } }$ is bounded by minimum and maximum values along each dimension, or else if $\mathbf { x }$ corresponds to a polytope more generally, then we can define a mixed-integer linear formulation mapping polyhedral regions of $\mathbf { x }$ to the output space $\mathbf { y } \in \mathbb { R } ^ { m }$ . The assumption that $\mathbf { x }$ is bounded and polyhedral is natural in most applications, where each value $x _ { i }$ has known lower and upper bounds (e.g., the value can vary from 0 to 1 for image pixels). Among other things, we can use this formulation to count the number of linear regions.
+
+In the formulation that follows, we use continuous variables to represent the input $\mathbf { x }$ , which we can also denote as $\mathbf { h } ^ { 0 }$ , the output of each neuron $i$ in layer $l$ as $\hat { h } _ { i } ^ { l }$ , and the output $\mathbf { y }$ as $\mathbf { h } ^ { L + 1 }$ . To simplify the representation, we lift this formulation to a space that also contains the output of a complementary set of neurons, each of which is active when the corresponding neuron is not. Namely, for each neuron $i$ in layer $l$ we also have a variable $\overline { { h } } _ { i } ^ { l } : = \operatorname* { m a x } \{ 0 , - W _ { i } ^ { l } h ^ { l - 1 } - b _ { i } ^ { l } \}$ . We use binary variables of the form $z _ { i } ^ { l }$ to denote if each neuron $i$ in layer $l$ is active or else if the complement of such neuron is. Finally, we assume $M$ to be a sufficiently large constant.
+
+For a given neuron $i$ in layer $l$ , the following set of constraints maps the input to the output:
+
+$$
+W _ { i } ^ { l } h ^ { l - 1 } + b _ { i } ^ { l } = h _ { i } ^ { l } - \overline { { h } } _ { i } ^ { l } , h _ { i } ^ { l } \leq M z _ { i } ^ { l } , \overline { { h } } _ { i } ^ { l } \leq M ( 1 - z _ { i } ^ { l } ) , h _ { i } ^ { l } \geq 0 , \overline { { h } } _ { i } ^ { l } \geq 0 , z _ { i } ^ { l } \in \{ 0 , 1 \}
+$$
+
+Theorem 9. Provided that $| w _ { i } ^ { l } h _ { \ j } ^ { l - 1 } + b _ { i } ^ { l } | \leq M$ for any possible value of $h ^ { l - 1 }$ , a formulation with the set of constraints (1) for each neuron of a rectifier network is such that a feasible solution with $a$ fixed value for $x$ yields the output $y$ of the neural network.
+
+The proof for the statement above is given in Appendix I. More details on the procedure for exact counting are in Appendix J. In addition, we show the theory for unrestricted inputs and a mixedinteger formulation for maxout networks in Appendices $\mathrm { K }$ and L, respectively.
+
+These results have important consequences. First, they allow us to tap into the literature of mixedinteger representability (Jeroslow, 1987) and disjunctive programming (Balas, 1979) to understand what can be modeled on rectifier networks with a finite number of neurons and layers. To the best of our knowledge, that has not been discussed before. Second, they imply that we can use mixedinteger optimization solvers to analyze the $\displaystyle ( \mathbf { x } , \mathbf { y } )$ mapping of a trained neural network. For example, Cheng et al. (2017) use another mixed-integer formulation to generate adversarial examples of a DNN. That is technically feasible due to the linear proportion between the size of the neural network and that of the mixed-integer formulation. Compared to Cheng et al. (2017), we show in Appendix I that formulation (1) can be implemented with further refinements on the value of the $M$ constants.
+
+# 6 EXPERIMENTS
+
+We perform two different experiments for region counting using small-sized networks with ReLU activation units on the MNIST benchmark dataset (LeCun et al., 1998). In the first experiment, we generate rectifier networks with 1 to 4 hidden layers having 10 neurons each, with final test error between 6 and $8 \%$ . The training was carried out for 20 epochs or training steps, and we count the number of linear regions during each training step. For those networks, we count the number of linear regions within $0 \leq x \leq 1$ in which a single neuron is active in the output layer, hence partitioning these regions in terms of the digits that they classify. In Fig. 4, we show how the number of regions classifying each digit progresses during training. Some digits have zero linear regions in the beginning, which explains why they begin later in the plot. The total number of such regions per training step is presented in Fig. 5(a) and error measures are found in Appendix M. Overall, we observe that the number of linear regions jumps orders of magnitude are varies more widely for each added layer. Furthermore, there is an initial jump in the number of linear regions classifying each digit that seems proportional to the number of layers.
+
+![](images/b0fd727e33064b769f81904c9e203a73da45e91e71934a3a4c40a50ff8890256.jpg)  
+Figure 4: Total number of regions classifying each digit (different colors for 0-9) of MNIST alone as training progresses, each plot corresponding to a different number of hidden layers.
+
+![](images/cb3cec966f5be5cf494d43b3ffc73b31635f2d71ff8f9380ed851fbb73976ccd.jpg)  
+Figure 5: (a) Total number of linear regions classifying a single digit of MNIST as training progresses, each plot corresponding to a different number of hidden layers. (b) Comparison of upper bounds from Montufar et al. (2014), Mont ´ ufar (2017), and from Theorem 1 with the total number of ´ linear regions of a network with two hidden layers totaling 22 neurons.
+
+In the second experiment, we train rectifier networks with two hidden layers summing up to 22 neurons. We train a network for each width configuration under the same conditions as above, with the test error in half of them ranging from 5 to $6 \%$ . In this case, we count all linear regions within $0 \leq x \leq 1$ , hence not restricting by activation in output layer as before. The number of linear regions of these networks are plotted in Fig. 5(b), along with the upper bound from Theorem 1 and the upper bounds from Montufar et al. (2014) and Mont ´ ufar (2017). Error measures of both experiments can ´ be found in Appendix M and runtimes for counting the linear regions in Appendix N.
+
+# 7 DISCUSSION
+
+The representational power of a DNN can be studied by observing the number of linear regions of the PWL function that the DNN represents. In this work, we improve on the upper and lower bounds on the linear regions for rectified networks derived in prior work (Montufar et al., 2014; Raghu et al., ´ 2017; Montufar, 2017; Arora et al., 2016) and introduce a first upper bound for multi-layer maxout ´ networks. We obtain several valuable insights from our extensions.
+
+Our ReLU upper bound indicates that small widths in early layers cause a bottleneck effect on the number of regions. If we reduce the width of an early layer, the dimensions of the linear regions become irrecoverably smaller throughout the network and the regions will not be able to be partitioned as much. Moreover, the dimensions of the linear regions are not only driven by width, but also the number of activated ReLUs corresponding to the region. This intuition allowed us to create a 1-dimensional construction with the maximal number of regions by eliminating a zero-dimensional bottleneck. An unexpected and useful consequence of our result is that shallow networks can attain more linear regions when the input dimensions exceed the number of neurons of the DNN.
+
+In addition to achieving tighter bounds, we use a mixed-integer linear formulation that maps the input space to the output to show the exact counting of the number of linear regions for several small-sized DNNs during the training process. In the first experiment, we observed that the number of linear regions correctly classifying each digit of the MNIST benchmark increases and vary in proportion to the depth of the network during the first training epochs. In the second experiment, we count the total number of linear regions as we vary the width of two layers with a fixed number of neurons, and we experimentally validate the bottleneck effect by observing that the results follow a similar pattern to the upper bound that we show.
+
+Our current results suggest new avenues for future research. First, we believe that the study of linear regions may eventually lead to insights in how to design better DNNs in practice, for example by further validating the bottleneck effect found in this study. Other properties of the bounds may turn into actionable insights if confirmed as these bounds get sufficiently close to the actual number of regions. For example, the plots in Appendix O show that there are particular network depths that maximize our ReLU upper bound for a given input dimension and number of neurons. In a sense, the number of neurons is a proxy to the computational resources available. We also believe that analyzing the shape of the linear regions is a promising idea for future work, which could provide further insight in how to design DNNs. Another important line of research is to understand the exact relation between the number of linear regions and accuracy, which may also involve the potential for overfitting. We conjecture that the network training is not likely to generalize well if there are so many regions that each point can be singled out in a different region, in particular if regions with similar labels are unlikely to be compositionally related. Second, applying exact counting to larger networks would depend on more efficient algorithms or on using approximations instead. In any case, the exact counting at a smaller scale can assess the quality of the current bounds and possibly derive insights for tighter bounds in future work, hence leading to insights that could be scaled up.
+
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+R.G. Jeroslow. Representability in mixed integer programming, I: Characterization results. Discrete Applied Mathematics, 17(3):223 – 243, 1987.   
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+Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   
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+H. Mhaskar, Q. Liao, and T. A. Poggio. Learning real and boolean functions: When is deep better than shallow. CoRR, abs/1603.00988, 2016.   
+G. Montufar. Notes on the number of linear regions of deep neural networks. In ´ SampTA, 2017.   
+G. Montufar, R. Pascanu, K. Cho, and Y. Bengio. On the number of linear regions of deep neural ´ networks. In NIPS, 2014.   
+X. Pan and V. Srikumar. Expressiveness of rectifier networks. In ICML, 2016.   
+R. Pascanu, G. Montufar, and Y. Bengio. On the number of response regions of deep feedforward ´ networks with piecewise linear activations. In ICLR, 2014.   
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+M. Telgarsky. Representation benefits of deep feedforward networks. CoRR, abs/1509.08101, 2015.   
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+
+# Appendices
+
+Most of the proofs for theorems and lemmas associated with the upper and lower bounds on the linear regions are provided below. The theory for mixed-integer formulation for exact counting in the case of maxouts and unrestricted inputs are also provided below.
+
+# A ANALYSIS OF THE BOUND FROM THEOREM 1
+
+In this section, we present properties of the upper bound for the number of regions of a rectifier network from Theorem 1. Denote the bound by $B ( n _ { 0 } , n _ { 1 } , . . . , n _ { L } )$ , where $n _ { 0 }$ is the input dimension and $n _ { 1 } , \ldots , n _ { L }$ are the widths of layers 1 through $L$ of the network. That is,
+
+$$
+B ( n _ { 0 } , n _ { 1 } , \ldots , n _ { L } ) : = \sum _ { ( j _ { 1 } , \ldots , j _ { L } ) \in J } \prod _ { l = 1 } ^ { L } { \binom { n _ { l } } { j _ { l } } }
+$$
+
+Instead of expressing $J$ as in Theorem 1, we rearrange it to a more convenient form for the proofs in this section:
+
+$$
+\begin{array} { c } { { J = \{ ( j _ { 1 } , \dots , j _ { L } ) \in \mathbb { Z } ^ { L } : j _ { l } + j _ { k } \leq n _ { k } \forall k = 1 , \dots , l - 1 \forall l = 2 , \dots , L } } \\ { { j _ { l } \leq n _ { 0 } \forall l = 1 , \dots , L } } \\  { 0 \leq j _ { l } \leq n _ { l } \forall l = 1 , \dots , L \} . } \end{array}
+$$
+
+Note that whenever we assume ${ n } _ { 0 } \geq \operatorname* { m a x } \{ n _ { 1 } , . . . , n _ { l } \}$ , then the bound inequality for $n _ { 0 }$ becomes redundant and can be removed.
+
+Some of the results have implications in terms of the exact maximal number of regions. We denote it by $R ( n _ { 0 } , n _ { 1 } , . . . , n _ { L } )$ , following the same notation above.
+
+Moreover, the following lemma is useful throughout the section.
+
+Lemma 10.
+
+$$
+\sum _ { j = 0 } ^ { k } { \binom { n _ { 1 } + \ldots + n _ { L } } { j } } = \sum _ { \stackrel { j _ { 1 } + \ldots + j _ { L } \leq k } { 0 \leq j _ { l } \leq n _ { l } \overline { { \forall l } } } } { \binom { n _ { 1 } } { j _ { 1 } } } { \binom { n _ { 2 } } { j _ { 2 } } } \cdot \cdot \cdot { \binom { n _ { L } } { j _ { L } } } .
+$$
+
+Proof. The result comes from taking a generalization of Vandermonde’s identity and adding the summation of $j$ from 0 to $k$ as above. □
+
+We first examine some properties related to 2-layer networks. The proposition below characterizes the bound when $L = 2$ for large input dimensions.
+
+Proposition 11. Consider a 2-layer network with widths $n _ { 1 } , n _ { 2 }$ and input dimension $n _ { 0 } \geq n _ { 1 }$ and $n _ { 0 } \geq n _ { 2 }$ . Then
+
+$$
+B ( n _ { 0 } , n _ { 1 } , n _ { 2 } ) = \sum _ { j = 0 } ^ { n _ { 1 } } { \binom { n _ { 1 } + n _ { 2 } } { j } }
+$$
+
+If $\dot { n } _ { 0 } < n _ { 1 }$ or $n _ { 0 } < n _ { 2 }$ , the above holds with inequality: $\begin{array} { r } { B ( n _ { 0 } , n _ { 1 } , n _ { 2 } ) \leq \sum _ { j = 0 } ^ { n _ { 1 } } { \binom { n _ { 1 } + n _ { 2 } } { j } } . } \end{array}$
+
+Proof. If $n _ { 0 } \geq n _ { 1 }$ and $n _ { 0 } \geq n _ { 2 }$ , the bound inequalities for $n _ { 0 }$ in the index set $J$ become redundant. By applying Lemma 10, we obtain
+
+$$
+B ( n _ { 0 } , n _ { 1 } , n _ { 2 } ) = \sum _ { 0 \leq j _ { 1 } + j _ { 2 } \leq n _ { 1 } } { \binom { n _ { 1 } } { j _ { 1 } } } { \binom { n _ { 2 } } { j _ { 2 } } } = \sum _ { j = 0 } ^ { n _ { 1 } } { \binom { n _ { 1 } + n _ { 2 } } { j } } .
+$$
+
+If $n _ { 0 } < n _ { 1 }$ or $n _ { 0 } < n _ { 2 }$ , then its index set $J$ is contained by the one above, and thus the first equal sign above becomes a less-or-equal sign. □
+
+Recall that the expression on the right-hand side of Proposition 11 is equal to the maximal number of regions of a single-layer network with $n _ { 1 } + n _ { 2 }$ ReLUs and input dimension $n _ { 1 }$ , as discussed in Section 2. Hence, the proposition implies that for large input dimensions, a two-layer network has no more regions than a single-layer network with the same number of neurons, as formalized below.
+
+Corollary 12. Consider a 2-layer network with widths $n _ { 1 } , n _ { 2 } \geq 1$ and input dimension $n _ { 0 } \geq n _ { 1 }$ and $n _ { 0 } \geq n _ { 2 }$ . Then $R ( n _ { 0 } , n _ { 1 } , n _ { 2 } ) \leq R ( n _ { 0 } , n _ { 1 } + n _ { 2 } )$ .
+
+Moreover, this inequality is strict when $n _ { 0 } > n _ { 1 }$
+
+Proof. This is a direct consequence of Proposition 11:
+
+$$
+R ( n _ { 0 } , n _ { 1 } , n _ { 2 } ) \leq B ( n _ { 0 } , n _ { 1 } , n _ { 2 } ) = \sum _ { j = 0 } ^ { n _ { 1 } } { \binom { n _ { 1 } + n _ { 2 } } { j } } \leq \sum _ { j = 0 } ^ { n _ { 0 } } { \binom { n _ { 1 } + n _ { 2 } } { j } } = R ( n _ { 0 } , n _ { 1 } + n _ { 2 } ) .
+$$
+
+Note that if $n _ { 0 } > n _ { 1 }$ , then the second inequality can be turned into a strict inequality.
+
+The next corollary illustrates the bottleneck effect for two layers. It states that for large input dimensions, moving a neuron from the second layer to the first strictly increases the bound.
+
+Corollary 13. Consider a 2-layer network with widths $n _ { 1 } , n _ { 2 }$ and input dimension $n _ { 0 } \geq n _ { 1 } + 1$ and $n _ { 0 } \geq n _ { 2 } + 1$ . Then $B ( n _ { 0 } , n _ { 1 } + 1 , n _ { 2 } ) > B ( n _ { 0 } , n _ { 1 } , n _ { 2 } + 1 )$ .
+
+Proof. By Proposition 11,
+
+$$
+B ( n _ { 0 } , n _ { 1 } + 1 , n _ { 2 } ) = \sum _ { j = 0 } ^ { n _ { 1 } + 1 } { \binom { ( n _ { 1 } + 1 ) + n _ { 2 } } { j } } > \sum _ { j = 0 } ^ { n _ { 1 } } { \binom { n _ { 1 } + ( n _ { 2 } + 1 ) } { j } } = B ( n _ { 0 } , n _ { 1 } , n _ { 2 } + 1 ) .
+$$
+
+The assumption that $n _ { 0 }$ must be large is required for the above proposition; otherwise, the input itself may create a bottleneck with respect to the second layer as we decrease its size. Note that the bottleneck affects all subsequent layers, not only the layer immediately after it.
+
+However, it is not true that moving neurons to earlier layers always increases the bound. For instance, with three layers, $B ( 4 , 3 , 2 , 1 ) = 4 7 > 4 6 = B ( 4 , 4 , 1 , 1 )$ .
+
+In the remainder of this section, we consider deep networks of equal widths $n$ . The next proposition can be viewed as an extension of Proposition 11 for multiple layers. It states that for a network with widths and input dimension $n$ and at least 4 layers, if we halve the number of layers and redistribute the neurons so that the widths become $2 n$ , then the bound increases. In other words, if we assume the bound to be close to the maximal number of regions, it suggests that making a deep network shallower allows for more regions when the input dimension is equal to the width.
+
+Proposition 14. Consider a $2 L$ -layer network with equal widths $n$ and input dimension $n _ { 0 } = n$ . Then
+
+$$
+B ( n , \underbrace { n , \ldots , n } _ { 2 L { t i m e s } } ) \leq B ( n , \underbrace { 2 n , \ldots , 2 n } _ { L { t i m e s } } ) .
+$$
+
+This inequality is met with equality when $L = 1$ and strict inequality when $L \geq 2$ .
+
+Proof. When $n _ { 0 } = n$ , the inequalities $j _ { l } \le \operatorname* { m i n } \{ n _ { 0 } , 2 n - j _ { 1 } , . . . , 2 n - j _ { l - 1 } , 2 n \}$ appearing in $J$ (in the form presented in Theorem 1) can be simplified to $j _ { l } \le n$ . Therefore, using Lemma 10, the bound on the right-hand side becomes
+
+$$
+\begin{array} { r l } & { 3 ( n , \underbrace { 2 n , \ldots , 2 n } _ { L \mathrm { ~ t i m e s } } ) = \displaystyle \sum _ { j _ { 1 } = 0 } ^ { n } \sum _ { j _ { 2 } = 0 } ^ { n } \cdots \sum _ { j _ { L } = 0 } ^ { n } \prod _ { l = 1 } ^ { L } \left( \ O \sum _ { j _ { l } } ^ { 2 n } \right) = \left( \displaystyle \sum _ { j _ { 2 } = 0 } ^ { n } \binom { 2 n } { j } \right) ^ { L } = \left( \displaystyle \sum _ { j _ { 1 } = 0 } ^ { n } \sum _ { j _ { 2 } = 0 } ^ { n - j _ { 1 } } \binom { n } { j _ { 1 } } \binom { n } { j _ { 2 } } \right) ^ { L } } \\ & { \qquad \ge \displaystyle \sum _ { ( j _ { 1 } , \ldots , j _ { 2 } , L ) \in J } \prod _ { l = 1 } ^ { 2 L } \binom { n } { j _ { l } } = B ( n , \underbrace { n , \ldots , n } _ { 2 L \mathrm { ~ t i m e s } } ) . } \end{array}
+$$
+
+where $J$ above is the index set from Theorem 1 applied to $n _ { 0 } = n _ { l } = n$ for all $l = 1 , \ldots , 2 L$ . Note that we can turn the inequality into equality when $L = 1$ (also becoming a consequence of Proposition 11) and into strict inequality when $L \geq 2$ . □
+
+Next, we provide an upper bound that is independent of $n _ { 0 }$ .
+
+Proposition 15. Consider an $L$ -layer network with equal widths n and any input dimension $n _ { 0 } \geq 0$ .
+
+$$
+B ( n _ { 0 } , n , \ldots , n ) \leq 2 ^ { L n } \left( { \frac { 1 } { 2 } } + { \frac { 1 } { 2 { \sqrt { \pi n } } } } \right) ^ { L / 2 } { \sqrt { 2 } }
+$$
+
+Proof. Since we are deriving an upper bound, we can assume $n _ { 0 } \geq n$ , as the bound is nondecreasing on $n _ { 0 }$ . We first assume that $L$ is even. We relax some of the constraints of the index set $J$ from Theorem 1 and apply Vandermonde’s identity on each pair:
+
+$$
+\begin{array} { r l } & { B ( n _ { 0 } , n , \ldots , n ) \le \displaystyle \sum _ { j _ { 1 } = 0 } ^ { n } \sum _ { j _ { 2 } = 0 } ^ { n - j _ { 1 } } { \binom { n } { j _ { 1 } } } { \binom { n } { j _ { 2 } } } \displaystyle \sum _ { j _ { 3 } = 0 } ^ { n } \sum _ { j _ { 4 } = 0 } ^ { n - j _ { 3 } } { \binom { n } { j _ { 3 } } } { \binom { n } { j _ { 4 } } } \cdots \displaystyle \sum _ { j _ { L - 1 } = 0 } ^ { n } \sum _ { j _ { L } = 0 } ^ { n - j _ { L - 1 } } { \binom { n } { j _ { L - 1 } } } { \binom { n } { j _ { L } } } } \\ & { \qquad = \displaystyle \left( \sum _ { j = 0 } ^ { n } { \binom { 2 n } { j } } \right) ^ { L / 2 } = \left( \frac { 2 ^ { 2 n } + { \binom { 2 n } { n } } } { 2 } \right) ^ { L / 2 } \le \left( \frac { 2 ^ { 2 n } + \frac { 2 ^ { 2 n } } { \sqrt { \pi n } } } { 2 } \right) ^ { L / 2 } } \\ & { \qquad = 2 ^ { L n } \left( \frac { 1 } { 2 } + \frac { 1 } { 2 \sqrt { \pi n } } \right) ^ { L / 2 } . } \end{array}
+$$
+
+The bound on $\binom { 2 n } { n }$ is a direct application of Stirling’s approximation (Stirling, 1730). If $L$ is odd, then we can write
+
+$$
+\begin{array} { l } { \displaystyle B ( n _ { 0 } , n , \dots , n ) \leq \left( \sum _ { j = 0 } ^ { n } \binom { 2 n } { j } \right) ^ { ( L - 1 ) / 2 } \left( \sum _ { j = 0 } ^ { n } \binom { n } { j } \right) = \left( \sum _ { j = 0 } ^ { n } \binom { 2 n } { j } \right) ^ { L / 2 } \frac { 2 ^ { n } } { \left( \sum _ { j = 0 } ^ { n } \binom { 2 n } { j } \right) ^ { 1 / 2 } } } \\ { \leq \left( \sum _ { j = 0 } ^ { n } \binom { 2 n } { j } \right) ^ { L / 2 } \frac { 2 ^ { n } } { ( 2 ^ { 2 n } / 2 ) ^ { 1 / 2 } } = \left( \sum _ { j = 0 } ^ { n } \binom { 2 n } { j } \right) ^ { L / 2 } \sqrt { 2 } } \\ { \leq 2 ^ { L n } \left( \frac 1 2 + \frac { 1 } { 2 \sqrt { \pi n } } \right) ^ { L / 2 } \sqrt { 2 } } \end{array}
+$$
+
+where the last inequality is analogous to the even case. Hence, the result follows.
+
+Corollary 16. Consider an $L$ -layer network with equal widths $n$ and any input dimension $n _ { 0 } \geq 0$ .
+
+$$
+\operatorname* { l i m } _ { L \to \infty } { \frac { R ( n _ { 0 } , n , \ldots , n ) } { 2 ^ { L n } } } = 0
+$$
+
+Proof. By Proposition 15 and Theorem 1, the ratio between $R ( n _ { 0 } , n , \ldots , n )$ and $2 ^ { L n }$ is at most ${ \sqrt { { \frac { 1 } { 2 } } + { \frac { 1 } { 2 { \sqrt { \pi n } } } } } } ^ { L } { \sqrt { 2 } }$ . Since the base of the first term is less than 1 for all $n \geq 1$ and $\sqrt { 2 }$ is a constant, the ratio goes to 0 as $L$ goes to infinity.
+
+In particular, Corollary 16 implies that if $n _ { 0 }$ exceeds the total size of the network, that is, $n _ { 0 } \geq L n$ , then limL→∞ $\begin{array} { r } { \operatorname* { l i m } _ { L  \infty } \frac { R ( n _ { 0 } , n , \dots , n ) } { R ( n _ { 0 } , L n ) } = 0 } \end{array}$ R(n0,n,...,n) = 0. In other words, the ratio between the maximal number of regions of a deep network and a shallow network goes to zero as $L$ goes to infinity.
+
+B EXPONENTIAL MAXIMAL NUMBER OF REGIONS WHEN INPUT DIMENSION IS LARGE
+
+Proposition 17. Consider an $L$ -layer rectifier network with equal widths $n$ and input dimension $n _ { 0 } \geq n / 3$ . Then the maximal number of regions is $\Omega ( 2 ^ { \frac { 2 } { 3 } L n } )$ .
+
+Proof. It suffices to show that a lower bound such as the one from Theorem 6 grows exponentially large. For simplicity, we consider the lower bound $( \prod _ { l = 1 } ^ { L } ( \lfloor n _ { l } / n _ { 0 } \rfloor + 1 ) ) ^ { n _ { 0 } }$ , which is the bound obtained before the last tightening step in the proof of Theorem 6 (see Appendix F).
+
+Note that replacing $n _ { 0 }$ in the above expression by a value $n _ { 0 } ^ { \prime }$ smaller than the input dimension still yields a valid lower bound. This holds because increasing the input dimension of a network from $n _ { 0 } ^ { \prime }$ to $n _ { 0 }$ cannot decrease its maximal number of regions.
+
+Choose $n _ { 0 } ^ { \prime } ~ = ~ \lfloor n / 3 \rfloor$ , which satisfies $n _ { 0 } ^ { \prime } ~ \le ~ n _ { 0 }$ and the condition $n \geq 3 n _ { 0 } ^ { \prime }$ of Theorem 6. The lower bound can be expressed as $( \lfloor n / \lfloor n / 3 \rfloor \rfloor + 1 ) ^ { L \lfloor n / 3 \rfloor } \ge 4 ^ { L \lfloor n / 3 \rfloor }$ . This implies that the maximal number of regions is $\Omega ( 2 ^ { { \frac { 2 } { 3 } } L n } )$ . □
+
+# C PROOF OF LEMMA 2
+
+Lemma 2. Consider m hyperplanes in regions induced by the hyperplanes is a $\mathbb { R } ^ { d }$ deost s of  . $W x + b = 0$ . Then the number of $\sum _ { j = 0 } ^ { \mathrm { r a n k } ( W ) } \binom { m } { j }$
+
+Proof. Consider the row space $\mathcal { R } ( W )$ of $W$ , which is a subspace of $\mathbb { R } ^ { d }$ of dimension rank $( W )$ . We show that the number of regions $N _ { \mathbb { R } ^ { d } }$ in $\mathbb { R } ^ { d }$ is equal to the number of regions $N _ { \mathcal { R } ( W ) }$ in $\mathcal { R } ( W )$ induced by $W x + b = 0$ restricted to $\mathcal { R } ( W )$ . This suffices to prove the lemma since $\mathcal { R } ( W )$ has at most Prank(j=0 $\sum _ { j = 0 } ^ { \mathrm { r a n k } ( W ) } \binom { m } { j }$ regions according to Zaslavsky’s theorem.
+
+Since $\mathcal { R } ( W )$ is a subspace of $\mathbb { R } ^ { d }$ , it directly follows that $N _ { \mathcal { R } ( W ) } \leq N _ { \mathbb { R } ^ { d } }$ . To show the converse, we apply the orthogonal decomposition theorem from linear algebra: any point $\bar { x } \in \mathbb { R } ^ { d }$ can be expressed uniquely as $\bar { x } = \hat { x } + y$ , where ${ \hat { x } } \in { \mathcal { R } } ( W )$ and $y \in \mathcal { R } ( \overline { { W } } ) ^ { \perp }$ . Here, ${ \mathcal { R } } ( W ) ^ { \perp } = \operatorname { K e r } ( W ) : = \{ y \in$ $\mathbb { R } ^ { d } : W \boldsymbol { y } = 0 \}$ , and thus $W { \bar { x } } = W { \hat { x } } + W y = W { \hat { x } }$ . This means $\bar { x }$ and $\hat { x }$ lie on the same side of each hyperplane of $W x = b$ and thus belong to the same region. In other words, given any $\bar { x } \in \mathbb { R } ^ { d }$ , its region is the same one that ${ \hat { x } } \in { \mathcal { R } } ( W )$ lies in. Therefore, $N _ { \mathbb { R } ^ { d } } \leq N _ { \mathcal { R } ( W ) }$ . Hence, $N _ { \mathbb { R } ^ { d } } = N _ { \mathscr { R } ( W ) }$ and the result follows.
+
+# D PROOF OF THEOREM 1
+
+Theorem 1. Consider a deep rectifier network with $L$ layers, $n _ { l }$ rectified linear units at each layer $l$ , and an input of dimension $n _ { 0 }$ . The maximal number of regions of this neural network is at most
+
+$$
+\sum _ { ( j _ { 1 } , \ldots , j _ { L } ) \in J } \prod _ { l = 1 } ^ { L } { \binom { n _ { l } } { j _ { l } } }
+$$
+
+where $J = \{ ( j _ { 1 } , \ldots , j _ { L } ) \in \mathbb { Z } ^ { L } : 0 \leq j _ { l } \leq \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } - j _ { 1 } , \ldots , n _ { l - 1 } - j _ { l - 1 } , n _ { l } \} \forall l = 1 , \ldots , L \} .$ This bound is tight when $L = 1$ .
+
+Proof. As illustrated in Figure 1, the partitioning can be viewed as a sequential process: at each layer, we partition the regions obtained from the previous layer. When viewed in the input space, each region $s$ obtained at layer $l - 1$ is potentially partitioned by $n _ { l }$ hyperplanes given by the rows of $\bar { W } _ { S } ^ { l } + b = 0$ for some bias $b$ . Some of these hyperplanes may fall outside the interior of $s$ and do not partition the region.
+
+With this process in mind, we recursively bound the number of subregions within a region. More precisely, we construct a recurrence $R ( l , d )$ to be an upper bound to the maximal number of regions obtained from partitioning a region of dimension $d$ with layers $l , l + 1 , \ldots , L$ . The base case of the recurrence is given by Lemma 3: $\begin{array} { r } { R ( L , d ) = \sum _ { j = 0 } ^ { \operatorname* { m i n } \{ n _ { L } , d \} } \binom { n _ { L } } { j } } \end{array}$ . Based on Lemma 4, we can write the recurrence by grouping together regions with the same activation set size $\begin{array} { r } { R ( l , d ) = \sum _ { j = 0 } ^ { n _ { l } } { N _ { n _ { l } , d , j } \bar { R } ( l ^ { ' } + 1 , \bar { \operatorname* { m i n } } \{ j , d \} ) } } \end{array}$ for all $l = 1 , \ldots , L - 1$ . Here, $N _ { n _ { l } , d , j }$ $| S ^ { l } |$ represents the , as follows: maximum number of regions with $| S ^ { l } | = j$ obtained by partitioning a space of dimension $d$ with $n _ { l }$ hyperplanes. We bound this value next.
+
+For each $j$ , there are at most $\binom { n _ { l } } { j }$ regions with $| S ^ { l } | = j$ , as they can be viewed as subsets of $n _ { l }$ neurons of size $j$ . In total, Lemma 3 states that there are at most Pmin{nl,d}j=0 nlj 
 regions. If we allow these regions to have the highest $| S ^ { l } |$ possible, for each $j$ from 0 to $\operatorname* { m i n } \{ n _ { l } , d \}$ we have at most $\textstyle { \binom { n _ { l } } { n _ { l } - j } } = { \binom { n _ { l } } { j } }$ regions with $\vert S ^ { l } \vert = n _ { l } - j$ .
+
+Therefore, we can write the recurrence as
+
+$$
+R ( l , d ) = \left\{ \begin{array} { l l } { \displaystyle \sum _ { j = 0 } ^ { \operatorname* { m i n } \{ n _ { l } , d \} } \binom { n _ { l } } { j } R ( l + 1 , \operatorname* { m i n } \{ n _ { l } - j , d \} ) } & { \mathrm { i f ~ } 1 \leq l \leq L - 1 , } \\ { \displaystyle \sum _ { j = 0 } ^ { \operatorname* { m i n } \{ n _ { L } , d \} } \binom { n _ { L } } { j } } & { \mathrm { i f ~ } l = L . } \end{array} \right.
+$$
+
+The recurrence $R ( 1 , n _ { 0 } )$ can be unpacked to
+
+$$
+\sum _ { j _ { 1 } = 0 } ^ { \operatorname* { m i n } \{ n _ { 1 } , d _ { 1 } \} } { \binom { n _ { 1 } } { j _ { 1 } } } \sum _ { j _ { 2 } = 0 } ^ { \operatorname* { m i n } \{ n _ { 2 } , d _ { 2 } \} } { \binom { n _ { 2 } } { j _ { 2 } } } \cdot \cdot \cdot \sum _ { j _ { L } = 0 } ^ { \operatorname* { m i n } \{ n _ { L } , d _ { L } \} } { \binom { n _ { L } } { j _ { L } } }
+$$
+
+where $d _ { l } = \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } - j _ { 1 } , \dots , n _ { l - 1 } - j _ { l - 1 } \}$ . This can be made more compact, resulting in the final expression.
+
+The bound is tight when L = 1 since it becomes Pmin{j=0 $\textstyle \sum _ { j = 0 } ^ { \operatorname* { m i n } \left\{ n _ { 0 } , n _ { 1 } \right\} } { \binom { n _ { 1 } } { j } }$ , which is the maximal number of regions of a single-layer network. □
+
+# E PROOF OF THEOREM 5
+
+Theorem 5. Consider a deep rectifier network with $L$ layers, $n \geq 3$ rectified linear units at each layer $l _ { i }$ , and an input of dimension $^ { l }$ . The maximal number of regions of this neural network is exactly $\textstyle \prod _ { l = 1 } ^ { L } ( n _ { l } + 1 )$ .
+
+Proof. Section 3 provides us with a helpful insight to construct an example with a large number of regions. It tells us that we want regions to have large dimension in general. In particular, regions of dimension zero cannot be further partitioned. This suggests that the one-dimensional construction from Montufar et al. (2014) can be improved, as it contains ´ $n$ regions of dimension one and 1 region of dimension zero. This is because all ReLUs point to the same direction as depicted in Fig. 6, leaving one region with an empty activation pattern.
+
+Our construction essentially increases the dimension of this region from zero to one. This is done by shifting the neurons forward and flipping the direction of the third neuron, as illustrated in Fig. 6. We assume $n \geq 3$ .
+
+We review the intuition behind the construction strategy from Montufar et al. (2014). They construct ´ a linear function ${ \tilde { h } } : \mathbb { R }  \mathbb { R }$ with a zigzag pattern from $[ 0 , 1 ]$ to $[ 0 , 1 ]$ that is composed of $n$ ReLUs. More precisely, $\tilde { h } ( x ) = ( 1 , - 1 , 1 , \ldots , \pm 1 ) ^ { \top } ( h _ { 1 } ( x ) , h _ { 2 } ( x ) , \ldots , h _ { n } ( x ) )$ , where $h _ { i } ( x )$ for $i = 1 , \ldots , n$ are ReLUs. This linear function can be absorbed in the preactivation function of the next layer.
+
+The zigzag pattern allows it to replicate in each slope a scaled copy of the function in the domain $[ 0 , 1 ]$ . Fig. 7 shows an example of this effect. Essentially, when we compose $\tilde { h }$ with itself, each linear piece in $[ t _ { 1 } , t _ { 2 } ]$ such that $\tilde { h } ( t _ { 1 } ) = 0$ and $\tilde { h } ( t _ { 2 } ) = 1$ maps the entire function $\tilde { h }$ to the interval $[ t _ { 1 } , t _ { 2 } ]$ , and each piece such that $\tilde { h } ( t _ { 1 } ) = 1$ and $\tilde { h } ( t _ { 2 } ) = 2$ does the same in a backward manner.
+
+![](images/0114c7270652a587769ce91eb6d5ef38b6526dd45b117bbbf03b3ef4af89e5bc.jpg)  
+Figure 6: (a) The 1D construction from Montufar et al. (2014). All units point to the right, leaving ´ a region with dimension zero before the origin. (b) The 1D construction described in this section. Within the interval [0, 1] there are five regions instead of the four in (a).
+
+![](images/5003997364e094431777a365528f787a5b98122dfd0c9a1b0ed8c8bd05486c5a.jpg)  
+Figure 7: A function with a zigzag pattern composed with itself. Note that the entire function is replicated within each linear region, up to a scaling factor.
+
+In our construction, we want to use $n$ ReLUs to create $n + 1$ regions instead of $n$ . In other words, we want the construct this zigzag pattern with $n + 1$ slopes. In order to do that, we take two steps to give ourselves more freedom. First, observe that we only need each linear piece to go from zero to one or one to zero; that is, the construction works independently of the length of each piece. Therefore, we turn the breakpoints into parameters $t _ { 1 } , t _ { 2 } , \ldots , t _ { n }$ , where $0 < t _ { 1 } < t _ { 2 } <$ $\ldots < t _ { n } < 1 .$ . Second, we add sign and bias parameters to the function $\tilde { h }$ . That is, $\tilde { h } ( x ) =$ $( s _ { 1 } , s _ { 2 } , \ldots , s _ { n } ) ^ { \top } ( h _ { 1 } ( x ) , h _ { 2 } ( x ) , \ldots , \bar { h } _ { n } ( x ) ) + d$ , where $s _ { i } \in \{ - 1 , + 1 \}$ and $d$ are parameters to be set. Here, $h _ { i } ( x ) = \operatorname* { m a x } \{ 0 , \tilde { w } _ { i } x + \tilde { b } _ { i } \}$ since it is a ReLU.
+
+We define $w _ { i } = s _ { i } \tilde { w } _ { i }$ and $b _ { i } = s _ { i } { \tilde { b } } _ { i }$ , which are the weights and biases we seek in each interval to form the zigzag pattern. The parameters $s _ { i }$ are needed because the signs of $\tilde { w } _ { i }$ cannot be arbitrary: it must match the directions the ReLUs point towards. In particular, we need a positive slope $( \tilde { w } _ { i } > 0 )$ ) if we want $i$ to point right, and a negative slope $( \tilde { w } _ { i } < 0 )$ ) if we want $i$ to point left. Hence, without loss of generality, we do not need to consider the $s _ { i }$ ’s any further since they will be directly defined from the signs of the $w _ { i }$ ’s and the directions. More precisely, $s _ { i } ~ = ~ 1$ if $w _ { i } ~ \geq ~ 0$ and $s _ { i } ~ = ~ - 1$ otherwise for $i = 1 , 2 , 4 , \dots , n$ , and $s _ { 3 } = - 1$ if $w _ { 3 } \geq 0$ and $s _ { 3 } = 1$ otherwise.
+
+To summarize, our parameters are the weights $w _ { i }$ and biases $b _ { i }$ for each ReLU, a global bias $d$ , and the breakpoints $0 < t _ { 1 } < . . . < t _ { n } < 1$ . Our goal is to find values for these parameters such that each piece in the function $\tilde { h }$ with domain in $[ 0 , 1 ]$ is linear from zero to one or one to zero.
+
+More precisely, if the domain is $[ s , t ]$ , we want each linear piece to be either $\scriptstyle { \frac { 1 } { t - s } } x - { \frac { s } { t - s } }$ or $- { \frac { 1 } { t - s } } x +$ $\frac { t } { t - s }$ , which define linear functions from zero to one and from one to zero respectively. Since we want a zigzag pattern, the former should happen for the interval $[ t _ { i } , t _ { i - 1 } ]$ when $i$ is odd and the latter should happen when $i$ is even.
+
+There is one more set of parameters that we will fix. Each ReLU corresponds to a hyperplane, or a point in dimension one. In fact, these points are the breakpoints $t _ { 1 } , \ldots , t _ { n }$ . They have directions that define for which inputs the neuron is activated. For instance, if a neuron $h _ { i }$ points to the right, then the neuron $h _ { i } ( x )$ outputs zero if $x \leq t _ { i }$ and the linear function $w _ { i } x + b _ { i }$ if $x > t _ { i }$ .
+
+As previously discussed, in our construction all neurons point right except for the third neuron $h _ { 3 }$ , which points left. This is to ensure that the region before $t _ { 1 }$ has one activated neuron instead of zero, which would happen if all neurons pointed left. However, although ensuring every region has dimension one is necessary to reach the bound, not every set of directions yields valid weights. These directions are chosen so that they admit valid weights.
+
+The directions of the neurons tells us which neurons are activated in each region. From left to right, we start with $h _ { 3 }$ activated, then we activate $h _ { 1 }$ and $h _ { 2 }$ as we move forward, we deactivate $h _ { 3 }$ , and finally we activate $h _ { 4 } , \ldots , h _ { n }$ in sequence. This yields the following system of equations, where $t _ { n + 1 }$ is defined as 1 for simplicity:
+
+$$
+\begin{array} { c c } { { w _ { 3 } x + ( b _ { 3 } + d ) = \displaystyle \frac { 1 } { t _ { 1 } } x } } & { { ( R _ { 1 } ) } } \\ { { \displaystyle ( w _ { 1 } + w _ { 3 } ) x + ( b _ { 1 } + b _ { 3 } + d ) = - \displaystyle \frac { 1 } { t _ { 2 } - t _ { 1 } } x + \displaystyle \frac { t _ { 2 } } { t _ { 2 } - t _ { 1 } } } } & { { ( R _ { 2 } ) } } \\ { { \displaystyle ( w _ { 1 } + w _ { 2 } + w _ { 3 } ) x + ( b _ { 1 } + b _ { 2 } + b _ { 3 } + d ) = \displaystyle \frac { 1 } { t _ { 3 } - t _ { 2 } } x - \displaystyle \frac { t _ { 2 } } { t _ { 3 } - t _ { 2 } } } } & { { ( R _ { 3 } ) } } \\ { { \displaystyle ( w _ { 1 } + w _ { 2 } ) x + ( b _ { 1 } + w _ { 2 } ) x + ( b _ { 1 } + b _ { 2 } + d ) = - \displaystyle \frac { 1 } { t _ { 4 } - t _ { 3 } } x + \displaystyle \frac { t _ { 4 } } { t _ { 4 } - t _ { 3 } } } } & { { ( R _ { 4 } ) } } \\ { { \displaystyle \left( w _ { 1 } + w _ { 2 } + \displaystyle \sum _ { j = 4 } ^ { i - 1 } w _ { j } \right) x + \displaystyle \left( b _ { 1 } + b _ { 2 } + \displaystyle \sum _ { j = 4 } ^ { i - 1 } b _ { j } + d \right) = \displaystyle \left\{ \frac { t _ { i } } { t _ { - } t _ { i - 1 } } x - \displaystyle \frac { t _ { i - 1 } } { t _ { i - 1 } } \right. } } & { { \mathrm { i f ~ } i \mathrm { ~ s ~ o d d } } } \\ { { \displaystyle \left. - \displaystyle \frac { 1 } { t _ { 2 } - t _ { i - 1 } } x + \displaystyle \frac { t _ { i - 1 } } { t _ { i } - t _ { i - 1 } } \right. } } & { { \mathrm { i f ~ } } i \mathrm { ~ i s ~ e v e n } } \end{array}
+$$
+
+for all $i = 5 , \dots , n + 1$
+
+It is left to show that there exists a solution to this system of linear equations such that $0 < t _ { 1 } <$ $\ldots < t _ { n } < 1$ .
+
+First, note that all of the biases $b _ { 1 } , \ldots , b _ { n } , d$ can be written in terms of $t _ { 1 } , \ldots , t _ { n }$ . Note that if we subtract $( R _ { 4 } )$ from $( R _ { 3 } )$ , we can express $b _ { 3 }$ in terms of the $t _ { i }$ variables. The remaining equations become triangular, and therefore given any values for $t _ { i }$ ’s we can back-substitute the remaining bias variables.
+
+The same subtraction yields $w _ { 3 }$ in terms of $t _ { i }$ ’s. However, both $( R _ { 1 } )$ and $\left( R _ { 3 } \right) - \left( R _ { 4 } \right)$ define $w _ { 3 }$ in terms of the $t _ { i }$ variables, so they must be the same:
+
+$$
+{ \frac { 1 } { t _ { 1 } } } = { \frac { 1 } { t _ { 3 } - t _ { 2 } } } + { \frac { 1 } { t _ { 4 } - t _ { 3 } } } .
+$$
+
+If we find values for $t _ { i }$ ’s satisfying this equation and $0 < t _ { 1 } < . . . < t _ { n } < 1$ , all other weights can be obtained by back-substitution since eliminating $w _ { 3 }$ yields a triangular set of equations.
+
+In particular, the following values are valid: $\textstyle t _ { 1 } = { \frac { 1 } { 2 n + 1 } }$ and $\begin{array} { r } { t _ { i } = \frac { 2 i - 1 } { 2 n + 1 } } \end{array}$ 2i−1 for all i = 2, . . . , n. The remaining weights and biases can be obtained as described above, which completes the desired construction.
+
+As an example, a construction with four units is depicted in Fig. 6. Its breakpoints are $\textstyle t _ { 1 } ~ = ~ { \frac { 1 } { 9 } }$ , $t _ { 2 } ~ = ~ \frac { 3 } { 9 } , ~ t _ { 3 } ~ = ~ \frac { 5 } { 9 }$ , and $t _ { 4 } ~ = ~ \frac { 7 } { 9 }$ . Its ReLUs are $h _ { 1 } ( x ) \ = \ \operatorname* { m a x } \{ 0 , - { \frac { 2 7 } { 2 } } x \ + \ { \frac { 3 } { 2 } } \}$ , $h _ { 2 } ( x ) \ =$ $\mathrm { m a x } \{ 0 , 9 x \textrm { - } 3 \}$ , $h _ { 3 } ( x ) \ = \ \operatorname* { m a x } \{ 0 , 9 x \ - \ 5 \} .$ , and $h _ { 4 } ( x ) ~ = ~ \operatorname* { m a x } \{ 0 , 9 x \}$ . Finally, $\tilde { h } ( x ) \ =$ $( - 1 , 1 , - 1 , 1 ) ^ { \top } ( h _ { 1 } ( x ) , h _ { 2 } ( x ) , h _ { 3 } ( x ) , h _ { 4 } ( x ) ) + 5$ .
+
+# F PROOF OF THEOREM 6
+
+Theorem 6. The maximal number of linear regions induced by a rectifier network with $n _ { 0 }$ input units and $L$ hidden layers with $n _ { l } \ge 3 n _ { 0 }$ for all $l$ is lower bounded by
+
+$$
+{ \binom { L - 1 } { l = 1 } } ( \lfloor { \frac { n _ { l } } { n _ { 0 } } } \rfloor + 1 ) ^ { n _ { 0 } } ) \sum _ { j = 0 } ^ { n _ { 0 } } { \binom { n _ { L } } { j } } .
+$$
+
+Proof. We follow the proof of Theorem 5 from (Montufar et al., 2014) except that we use a different ´ 1-dimensional construction. The main idea of the proof is to organize the network into $n _ { 0 }$ independent networks with input dimension 1 each and apply the 1-dimensional construction to each individual network. In particular, for each layer $l$ we assign $\lfloor n _ { l } / n _ { 0 } \rfloor$ ReLUs to each network, ignoring any remainder units. In (Montufar et al., 2014), each of these networks have at least ´ $\textstyle \prod _ { l = 1 } ^ { L } \lfloor n _ { l } / n _ { 0 } \rfloor$ regions. We instead use Theorem 5 to attain $\textstyle \prod _ { l = 1 } ^ { L } ( \lfloor n _ { l } / n _ { 0 } \rfloor + 1 )$ regions in each network.
+
+Since the networks are independent from each other, the number of activation patterns of the compound network is the product of the number of activation patterns of each of the $n _ { 0 }$ networks. Hence, the same holds for the number of regions. Therefore, the number of regions of this network is at least $( \prod _ { l = 1 } ^ { L } ( \lfloor n _ { l } / n _ { 0 } \rfloor + 1 ) ) ^ { n _ { 0 } }$ .
+
+In addition, we can replace the last layer by a function representing an arrangement of $n _ { L }$ hyperplanes in general position that partitions $( 0 , 1 ) ^ { n _ { 0 } }$ into $\textstyle \sum _ { j = 0 } ^ { \hat { n } _ { 0 } } { \binom { n _ { L } } { j } }$ regions. This yields the lower bound of $\begin{array} { r } { \prod _ { l = 1 } ^ { L - 1 } ( \lfloor n _ { l } / n _ { 0 } \rfloor + 1 ) ^ { n _ { 0 } } \sum _ { j = 0 } ^ { n _ { 0 } } { \binom { n _ { L } } { j } } } \end{array}$ .
+
+# G PROOF OF THEOREM 7
+
+Theorem 7. For any values of $m \geq 1$ and $w \geq 2$ , there exists a rectifier network with $n _ { 0 }$ input units and $L$ hidden layers of size $2 m + w ( L - 1 )$ that has $\textstyle 2 \sum _ { j = 0 } ^ { n _ { 0 } - 1 } { \binom { m - 1 } { j } } ( w + 1 ) ^ { L - 1 }$ linear regions.
+
+Proof. Theorem 6.1 and Lemma 6.2 in Arora et al. (2016) imply that for any $m \geq 1$ , we can construct a layer representing a function from $\mathbb { R } ^ { n }$ to $\mathbb { R }$ with $2 m$ ReLUs that has $2 \textstyle \sum _ { j = 0 } ^ { n _ { 0 } - 1 } { \binom { m - 1 } { j } }$ regions. Consider the network where this layer is the first one and the remaining layers are the onedimensional layers from Theorem 5, each of size $w$ . Then this network has size $2 m + w ( L - 1 )$ and $\textstyle 2 \sum _ { j = 0 } ^ { n _ { 0 } - 1 } { \binom { m - 1 } { j } } ( w + 1 ) ^ { L - 1 }$ regions. □
+
+# H PROOF OF THEOREM 8
+
+Theorem 8. Consider a deep neural network with $L$ layers, $n _ { l }$ rank- $k$ maxout units at each layer $l$ , and an input of dimension $n _ { 0 }$ . The maximal number of regions of this neural network is at most
+
+$$
+\prod _ { l = 1 } ^ { L } \sum _ { j = 0 } ^ { d _ { l } } { \binom { k ( k - 1 ) } { 2 } } n _ { l } \rangle
+$$
+
+where $d _ { l } = \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } , . . . , n _ { l } \}$
+
+Asymptotically, if $n _ { l } = n$ for all $l = 1 , \ldots , L , n \geq n _ { 0 }$ , and $n _ { 0 } = O ( 1 )$ , then the maximal number of regions is at most $O ( ( k ^ { 2 } n ) ^ { L n _ { 0 } } )$ .
+
+Proof. We denote by $W _ { j } ^ { l }$ the $n _ { l } \times n _ { l - 1 }$ matrix where the rows are given by the $j$ -th weight vectors of each rank- $k$ maxout unit at layer $l$ , for $j = 1 , \dots , k$ . Similarly, $b _ { j } ^ { l }$ is the vector composed of the $j$ -th biases at layer $l$ .
+
+In the case of maxout, an activation pattern $\pmb { \mathcal { S } } = ( S ^ { 1 } , \dots , S ^ { l } )$ is such that $S ^ { l }$ is a vector that maps from layer- $\mathbf { \xi } _ { l }$ neurons to $\{ 1 , \ldots , k \}$ . We say that the activation of a neuron is $j$ if $w _ { j } x + b _ { j }$ attains the maximum among all of its functions; that is, $w _ { j } x + b _ { j } \geq w _ { j ^ { \prime } } x + b _ { j ^ { \prime } }$ for all $j ^ { \prime } = 1 , \dotsc , j$ . In the case of ties, we assume the function with lowest index is considered as its activation.
+
+Similarly to the ReLU case, denote by $\phi _ { S ^ { l } } : \mathbb { R } ^ { n _ { l } \times n _ { l - 1 } \times k }  \mathbb { R } ^ { n _ { l } \times n _ { l - 1 } }$ the operator that selects the rows of $W _ { 1 } ^ { l } , \ldots , W _ { k } ^ { l }$ that correspond to the activations in $S ^ { l }$ . More precisely, $\phi _ { S ^ { l } } ( W _ { 1 } ^ { l } , \dots , W _ { k } ^ { l } )$ is a matrix $W$ such that its $i$ -th row is the $i$ -th row of $W _ { j } ^ { l }$ , where $j$ is the neuron $i$ ’s activation in $S ^ { l }$ . This essentially applies the maxout effect on the weight matrices given an activation pattern.
+
+Montufar et al. (2014) provides an upper bound of ´ $\textstyle \sum _ { j = 0 } ^ { n _ { 0 } } { \binom { k ^ { 2 } n } { j } }$ for the number of regions for a single rank- maxout layer with $n$ neurons. The reasoning is as follows. For a single maxout unit, there is one region per linear function. The boundaries between the regions are composed by pieces that are each contained in a hyperplane. Each piece is part of the boundary of at least two regions and conversely each pair of regions corresponds to at most one piece. Extending these pieces into hyperplanes cannot decrease the number of regions. Therefore, if we now consider $n$ maxout units in a single layer, we can have at most the number of regions of an arrangement of $k ^ { 2 } n$ hyperplanes. In the results below we replace $k ^ { 2 }$ by $\binom { k } { 2 }$ , as only pairs of distinct functions need to be considered.
+
+We need to define more precisely these ${ \binom { k } { 2 } } n$ hyperplanes in order to apply a strategy similar to the one from the Section 3.1. In a single layer setting, they are given by $w _ { j } x + b _ { j } = w _ { j ^ { \prime } } + b _ { j ^ { \prime } }$ for each distinct pair ${ j , j ^ { \prime } }$ within a neuron. In order to extend this to multiple layers, consider a ${ \binom { k } { 2 } } n _ { l } \times n _ { l - 1 }$ matrix $\hat { W } _ { l }$ where its rows are given by $w _ { j } - w _ { j ^ { \prime } }$ for every distinct pair ${ j , j ^ { \prime } }$ within a neuron $i$ and for every neuron $i = 1 , \dots , n _ { l }$ . Given a region $s$ , we can now write the weight matrix corresponding to the hyperplanes described above: $\hat { W } _ { S } ^ { l } : = \hat { W } ^ { l } ~ \phi _ { S ^ { l - 1 } } ( W _ { 1 } ^ { l - 1 } , \ldots , W _ { k } ^ { l - 1 } ) \cdot \cdot \cdot \phi _ { S ^ { 1 } } ( W _ { 1 } ^ { 1 } , \ldots , W _ { k } ^ { 1 } )$ . In other words, the hyperplanes that extend the boundary pieces within region $s$ are given by the rows of $\hat { W } _ { S } ^ { l } x + b = 0$ for some bias $b$ .
+
+A main difference between the maxout case and the ReLU case is that the maxout operator $\phi$ does not guarantee reductions in rank, unlike the ReLU operator $\sigma$ . We show the analogous of Lemma 3 for the maxout case. However, we fully relax the rank.
+
+Lemma 18. The number of regions induced by the $n _ { l }$ neurons at layer $l$ within a certain region $s$ is at most $\textstyle \sum _ { j = 0 } ^ { d _ { l } } { \binom { { \frac { k ( k - 1 ) } { 2 } } n _ { l } } { j } }$ 1) nl
, where dl = min{n0, n1, . . . , nl}.
+
+Proof. For a fixed region $s$ , an upper bound is given by the number of regions of the hyperplane arrangement corresponding to $\hat { W } _ { S } ^ { l } x + b = 0$ for some bias $b$ . The rank of ${ \hat { W } } _ { S } ^ { l }$ is upper bounded by
+
+$$
+\begin{array} { r l } & { \mathrm { r a n k } ( \hat { W } _ { \mathcal { S } } ^ { l } ) = \mathrm { r a n k } ( \hat { W } ^ { l } \ \phi _ { S ^ { l - 1 } } ( W _ { 1 } ^ { l - 1 } , \ldots , W _ { k } ^ { l - 1 } ) \cdots \phi _ { S ^ { 1 } } ( W _ { 1 } ^ { 1 } , \ldots , W _ { k } ^ { 1 } ) ) } \\ & { \qquad \leq \operatorname* { m i n } \{ \mathrm { r a n k } ( \hat { W } ^ { l } ) , \mathrm { r a n k } ( \phi _ { S ^ { l - 1 } } ( W _ { 1 } ^ { l - 1 } , \ldots , W _ { k } ^ { l - 1 } ) ) , \ldots , \mathrm { r a n k } ( \phi _ { S ^ { 1 } } ( W _ { 1 } ^ { 1 } , \ldots , W _ { k } ^ { 1 } ) ) \} } \\ & { \qquad \leq \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } , \ldots , n _ { l } \} . } \end{array}
+$$
+
+Applying Lemma 2 yields the result.
+
+Since we can consider the partitioning of regions independently from each other, Lemma 18 implies that thwhere ons of a rank-. $k$ maxout network is at most $\begin{array} { r } { \prod _ { l = 1 } ^ { L } \sum _ { j = 0 } ^ { d _ { l } } \binom { \frac { k ( k - 1 ) } { 2 } n _ { l } } { j } } \end{array}$ $d _ { l } = \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } , . . . , n _ { l } \}$
+
+# I PROOF OF THEOREM 9
+
+Theorem 9. Provided that $| w _ { i } ^ { l } h _ { j } ^ { l - 1 } + b _ { i } ^ { l } | \leq M$ for any possible value of $h ^ { l - 1 }$ , $a$ formulation with the set of constraints (1) for each neuron of a rectifier network is such that a feasible solution with a fixed value for $x$ yields the output y of the neural network.
+
+Proof. For ease of explanation, we expand the set of constraints (1) as follows:
+
+$$
+\begin{array} { r } { \bar { \boldsymbol { h } } _ { i } ^ { l } + \boldsymbol { b } _ { i } ^ { l } = \boldsymbol { h } _ { i } ^ { l } - \overline { { \boldsymbol { h } } } _ { i } ^ { l } } \\ { \boldsymbol { h } _ { i } ^ { l } \le M \boldsymbol { z } _ { i } ^ { l } } \\ { \overline { { \boldsymbol { h } } } _ { i } ^ { l } \le M ( 1 - \boldsymbol { z } _ { i } ^ { l } ) } \\ { \boldsymbol { h } _ { i } ^ { l } \ge 0 } \\ { \overline { { \boldsymbol { h } } } _ { i } ^ { l } \ge 0 } \\ { \overline { { \boldsymbol { h } } } _ { i } ^ { l } \ge 0 } \\ { \boldsymbol { z } _ { i } ^ { l } \in \{ 0 , 1 \} } \end{array}
+$$
+
+It suffices to prove that the constraints for each neuron map the input to the output in the same way that the neural network would. If $W _ { i } ^ { l } \mathbf { h } ^ { l - 1 } + b _ { i } ^ { l } > 0$ , it follows that $h _ { i } ^ { l } - \overline { { h } } _ { i } ^ { l } > 0$ according to (2). Since both variables are non-negative due to (5) and (6) whereas one is non-positive due to (3), (4), and (7), then $z _ { i } ^ { l } = 1$ and $h _ { i } ^ { l } = \operatorname* { m a x } \left\{ 0 , W _ { i } ^ { l } \mathbf { h } ^ { l - 1 } + b _ { i } ^ { l } \right\}$ . If ${ \cal W } _ { i } ^ { l } { \bf h } ^ { l - 1 } + b _ { i } ^ { l } < 0$ , then it similarly follows that $h _ { i } ^ { l } - \overline { { h } } _ { i } ^ { l } < 0$ , $z _ { i } ^ { l } = 0$ , and thus $\overline { { h } } _ { i } ^ { l } = \operatorname* { m i n } \left\{ 0 , W _ { i } ^ { l } \mathbf { h } _ { j } ^ { l - 1 } + b _ { i } ^ { l } \right\}$ . If $\begin{array} { r } { \sum _ { j } W _ { i } ^ { l } \mathbf { h } _ { j } ^ { l - 1 } + b _ { i } ^ { l } = 0 } \end{array}$ , then either $h _ { i } ^ { l } = 0$ or $\overline { { h } } _ { i } ^ { l } = 0$ due to constraints (5) to (7) whereas (2) implies that $\overline { { h } } _ { i } ^ { l } = 0$ or $h _ { i } ^ { l } = 0$ , respectively. In this case, the value of $z _ { i } ^ { l }$ is arbitrary but irrelevant. □
+
+# J EXACT COUNTING FOR RECTIFIER NETWORKS USING A MIXED-INTEGER FORMULATION
+
+A systematic method to count these solutions is the one-tree approach (Danna et al., 2007), which resumes the search after an optimal solution has been found using the same branch-and-bound tree. That method can also be applied to near-optimal solutions by revisiting nodes pruned when solving for an optimal solution. Note that in constraints (1), the variables $z _ { i } ^ { l }$ can be either 0 or 1 when they lie on the activation boundary, whereas we want to consider a neuron active only when its output is strictly positive. This discrepancy may cause double-counting when activation boundaries overlap. We can address that by defining an objective function that maximizes the minimum output $f$ of an active neuron, which is positive in non-degenerate cases. The formulation is as follows:
+
+max $f$ s.t. (1) for each neuron i in layer $l$ $\begin{array} { l } { f \leq h _ { i } ^ { l } + ( 1 - z _ { i } ^ { l } ) M } \\ { x \in X } \end{array}$ for each neuron i in layer $l$
+
+Corollary 19. The number of $z$ assignments of (8) yielding a positive objective function value corresponds to the number of linear regions of the neural network.
+
+Proof. Implicit in the discussion above.
+
+Corollary 20. If the input $X$ is a polytope, then $( x , y )$ is mixed-integer representable.
+
+Proof. Immediate from the existence of a mixed-integer formulation mapping $x$ to $y$ , which is correct as long as the input is bounded and therefore a sufficiently large $M$ exists. □
+
+In practice, the value of constant $M$ should be chosen to be as small as possible, which also implies choosing different values on different places to make the formulation tighter and more stable numerically (Camm et al., 1990). For the constraints set (1), it suffices to choose $M$ to be as large as either $h _ { i } ^ { \bar { l } }$ or $\bar { h } _ { i } ^ { l }$ can be given the bounds on the input. Hence, we can respectively replace $M$ with $H _ { i } ^ { l }$ and $\bar { H } _ { i } ^ { l }$ in the constraints involving those variables. If we are given lower and upper bounds for $X$ , which we can use for $H ^ { 0 }$ and $\bar { H } ^ { 0 }$ , then we can define subsequent bounds as follows:
+
+$$
+\begin{array} { l } { { \displaystyle { \cal H } _ { i } ^ { l } = \operatorname* { m a x } \left\{ 0 , \sum _ { j } \operatorname* { m a x } \left\{ 0 , w _ { i j } ^ { l } { \cal H } _ { j } ^ { l - 1 } \right\} + b _ { i } ^ { l } \right\} } } \\ { ~ } \\ { { \displaystyle { \overline { { { \cal H } } } _ { i } ^ { l } = \operatorname* { m a x } \left\{ 0 , \sum _ { j } \operatorname* { m a x } \left\{ 0 , - w _ { i j } ^ { l } { \cal H } _ { j } ^ { l - 1 } \right\} - b _ { i } ^ { l } \right\} } } } \end{array}
+$$
+
+For the constraint involving $f$ in formulation (8), we should choose a slightly larger value than $H _ { i } ^ { l }$ for correctness because some neurons may never be active within the input bounds.
+
+# K COUNTING LINEAR REGIONS OF RELUS WITH UNRESTRICTED INPUTS
+
+More generally, we can represent linear regions as a disjunctive program (Balas, 1979), which consist of a union of polyhedra. Disjunctive programs are used in the integer programming literature to generate cutting planes by lift-and-project (Balas et al., 1993). In what follows, we assume that a neuron can be either active or inactive when the output lies on the activation hyperplane.
+
+For each active neuron, we can use the following constraints to map input to output:
+
+$$
+\begin{array} { r } { w _ { i } ^ { l } h ^ { l - 1 } + b _ { i } ^ { l } = h _ { i } ^ { l } } \\ { h _ { i } ^ { l } \ge 0 } \end{array}
+$$
+
+For each inactive neuron, we use the following constraint:
+
+$$
+\begin{array} { r } { w _ { i } ^ { l } h ^ { l - 1 } + b _ { i } ^ { l } \leq 0 } \\ { h _ { i } ^ { l } = 0 } \end{array}
+$$
+
+Theorem 21. The set of linear regions of a rectifier network is a union of polyhedra.
+
+Proof. First, the activation set $S ^ { l }$ for each level $l$ defines the following mapping:
+
+$$
+\bigcup _ { S ^ { l } \subseteq \{ 1 , \dotsc , n _ { l } \} , l \in \{ 1 , \dotsc , L + 1 \} } \left\{ ( h ^ { 0 } , h ^ { 1 } , \dotsc , h ^ { L + 1 } ) \mid ( 9 ) - ( 1 0 ) { \mathrm { i f ~ } } i \in S ^ { l } ; ( 1 1 ) - ( 1 2 ) { \mathrm { ~ o t h e r w i s e ~ } } \right\}
+$$
+
+Consequently, we can project the variables sets $h ^ { 1 } , \ldots , h ^ { L + 1 }$ out of each of those terms by FourierMotzkin elimination (Fourier, 1826), thereby yielding a polyhedron for each combination of active sets across the layers. □
+
+Note that the result above is similar in essence to Theorem 2 of Raghu et al. (2017).
+
+Corollary 22. If $X$ is unrestricted, then the number of linear regions can be counted using (8) if M is large enough.
+
+Proof. To count regions, we only need one point $x$ from each linear region. Since the number of linear regions is finite, then it suffices if $M$ is large enough to correctly map a single point in each region. Conversely, each infeasible linear region either corresponds to empty sets of (13) or else to a polyhedron $P$ such that $\{ ( h ^ { 1 } , \ldots , h ^ { L + 1 } ) \stackrel { \smile } { \in } P \mid h _ { i } ^ { l } > 0 \forall l \stackrel { \cdot } { \in } \{ 1 , \ldots , L \stackrel { \cdot } { + } 1 \} , i \in S ^ { l } \}$ is empty, and neither case would yield a solution for the $z$ -projection of (8). □
+
+# L MIXED-INTEGER REPRESENTABILITY OF MAXOUT UNITS
+
+In what follows, we assume that we are given a neuron $i$ in level $l$ with output $h _ { i } ^ { l }$ . For that neuron, we denote the vector of weights as $w _ { 1 } ^ { l i } , \ldots , w _ { k } ^ { l i }$ . Thus, the neuron output corresponds to
+
+$$
+h _ { i } ^ { l } : = \operatorname* { m a x } \left\{ w _ { 1 } ^ { l i } h ^ { l - 1 } + b _ { 1 } , \ldots , w _ { k } ^ { l i } h ^ { l - 1 } + b _ { k } \right\}
+$$
+
+Hence, we can connect inputs to outputs for that given neuron as follows:
+
+$$
+\begin{array} { r l } { w _ { j } ^ { l i } h _ { j } ^ { l - 1 } + b _ { j } ^ { l i } = g _ { j } ^ { l i } , } & { \quad j = 1 , \dots , k } \\ { h _ { i } ^ { l } \geq g _ { j } ^ { l i } , } & { \quad j = 1 , \dots , k } \\ { h _ { i } ^ { l } \leq g _ { j } ^ { l i } + M ( 1 - z _ { j } ^ { l i } ) } & { \quad j = 1 , \dots , k } \\ { z _ { j } ^ { l i } \in \{ 0 , 1 \} , } & { \quad j = 1 , \dots , k } \\ { \displaystyle \sum _ { j = 1 } ^ { k } z _ { j } ^ { l i } = 1 } & { } \end{array}
+$$
+
+The formulation above generalizes that for ReLUs with some small modifications. First, we are computing the output of each term with constraint (14). The output of the neuron is lower bounded by that of each term with constraint (15). Finally, we have a binary variable $z _ { m } ^ { l i }$ per term of each neuron, which denotes which neuron is active. Constraint (18) enforces that only one variable is at one per neuron, whereas constraint (16) equates the output of the neuron with the active term. Each constant $M$ should be chosen in a way that the other terms can vary freely, hence effectively disabling the constraint when the corresponding binary variable is at zero.
+
+# M ACCURACY AND ERROR MEASURES OF THE SAMPLE NETWORKS
+
+Figure 8 shows the error during training for different configurations in the first experiment. Figure 9 shows the errors after training for different configurations in the second experiment. In both, we observe some relation between accuracy and the order of magnitude of the linear regions, which suggest that linear regions represent a reasonable proxy to the representational power of DNNs.
+
+![](images/1475ee431808e1b02b0b609167be526c0f05e48a3d6a9ea7a7b392481477773e.jpg)  
+Figure 8: Contrast of cross-entropy along training with number of regions identifying a single digit in the first experiment: (a) shows training error in green; (b) shows validation error in purple.
+
+![](images/6295ed94fcc87454197b5b50d7409bcf1dbd5ba673ce83729c2040d5805ed1b5.jpg)  
+Figure 9: Contrast of final errors with number of regions and bound in the second experiment: (a) shows training error in green and validation error in purple; (b) shows accuracy in red.
+
+# N RUNTIMES FOR COUNTING THE LINEAR REGIONS
+
+Table 1 reports the runtimes to count different configurations of networks on each experiment.   
+
+<table><tr><td></td><td>ExperimentNetwork widths</td><td>Runtime (s)</td></tr><tr><td>1</td><td>1×10</td><td>6.0×10-2</td></tr><tr><td rowspan="8">2</td><td>2×10</td><td>1.1 × 10²</td></tr><tr><td>3×10</td><td>1.8 ×103</td></tr><tr><td>4×10</td><td>5.2 ×104</td></tr><tr><td>1; 21; 10</td><td>1.0 × 10-2</td></tr><tr><td>2; 20; 10</td><td>4.5 × 10-1</td></tr><tr><td>3;19; 10</td><td>1.9 ×100</td></tr><tr><td>4; 18; 10</td><td>3.8 × 101</td></tr><tr><td>5; 17; 10</td><td>2.0×10²</td></tr><tr><td></td><td>6;16; 10</td><td>4.1× 102</td></tr><tr><td></td><td>7;15;10</td><td>1.2 × 103</td></tr><tr><td></td><td>9; 13;10</td><td>7.5× 103</td></tr><tr><td></td><td>10; 12; 10</td><td>1.5× 104</td></tr><tr><td></td><td>11; 11; 10</td><td>3.3× 104</td></tr><tr><td></td><td>12; 10; 10</td><td>4.4× 104</td></tr><tr><td></td><td>13; 9; 10</td><td>5.8×104</td></tr><tr><td></td><td></td><td>6.6 ×104</td></tr><tr><td></td><td>14; 8;10</td><td>7.5 ×104</td></tr><tr><td></td><td>15; 7; 10 16; 6; 10</td><td>3.0 ×105</td></tr><tr><td></td><td>17; 5;10</td><td>2.8×105</td></tr><tr><td></td><td></td><td>2.3×105</td></tr><tr><td></td><td>18;4;10</td><td></td></tr><tr><td></td><td>19;3;10</td><td>2.7 × 105</td></tr><tr><td></td><td>20; 2; 10</td><td>1.1 × 105</td></tr><tr><td></td><td>21; 1; 10</td><td>4.0 ×104</td></tr></table>
+
+Table 1: Runtimes for counting the trained networks for each configuration used in the experiments.
+
+# O UPPER BOUND BY VARYING THE TOTAL NUMBER OF NEURONS
+
+Figure 10a shows that the upper bound from Theorem 1 can only be maximized if more layers are added as the number of neurons increase. In contrast, Figure 10b shows that the smallest depth preserving such growth is better because there is a secondary, although still exponential, effect that starts shrinks the bound if the number of layers is too large for the total number of neurons.
+
+![](images/b6c3c57f1f50dfd1ecb4894ebe4b4fdd89ec5f289b6827ee6080c130b53e51e6.jpg)  
+Figure 10: Bounds from Theorem 1 in semilog scale for $n _ { 0 } = 6 0 $ as the total number of neurons increase by evenly distributing such neurons in 1 to 4 layers: (a) actual values showing overall impact of more depth; and (b) ratio by sum over all layers showing local impact of particular depths.

parse/train/ilVv1LO0Ew/ilVv1LO0Ew.md

@@ -0,0 +1,234 @@
+# Exploring Cross-Video and Cross-Modality Signals for Weakly-Supervised Audio-Visual Video Parsing
+
+Yan-Bo Lin1,2 Hung-Yu Tseng3 Hsin-Ying Lee4 Yen-Yu Lin1 Ming-Hsuan Yang3,5,6
+
+1National Yang Ming Chiao Tung University 2UNC Chapel Hill 3UC Merc 4Snap Research 5Google Research 6Yonsei University yblin@unc.edu htseng6@ucmerced.edu hlee5@snap.com lin@cs.nctu.edu.tw mhyang@ucmerced.edu
+
+# Abstract
+
+The audio-visual video parsing task aims to temporally parse a video into audio or visual event categories. However, it is labor-intensive to temporally annotate audio and visual events and thus hampers the learning of a parsing model. To this end, we propose to explore additional cross-video and cross-modality supervisory signals to facilitate weakly-supervised audio-visual video parsing. The proposed method exploits both the common and diverse event semantics across videos to identify audio or visual events. In addition, our method explores event co-occurrence across audio, visual, and audio-visual streams. We leverage the explored cross-modality co-occurrence to localize segments of target events while excluding irrelevant ones. The discovered supervisory signals across different videos and modalities can greatly facilitate the training with only video-level annotations. Quantitative and qualitative results demonstrate that the proposed method performs favorably against existing methods on weakly-supervised audio-visual video parsing.
+
+# 1 Introduction
+
+Humans perceive multisensory signals via seeing, hearing, touching, etc., and obtain multimodal information while exploring the surrounding environments. Visual and audio signals, the most common modalities, motivate researchers to jointly comprehend audio-visual events (e.g., see people singing and hear their sounds) [1, 2, 3, 4, 5, 6, 7]. Events visible in images while hearable in audio are referred to as audio-visual events. However, learning-based models tend to recognize a particular audio-visual event by using the data from the dominant modality with richer information and overlook clues from either audio only or visual only events which still contribute to holistic video understanding. Therefore, the resultant models can generalize well on audio-visual events only instead of comprehensively understanding all kinds of video events. To address this issue, we target at audio-visual video parsing [4, 6] where predictions for audio, visual, and audio-visual events with temporal boundaries are all required but separately evaluated.
+
+The time-consuming and labor-intensive annotation process poses a major challenge for the audiovisual video parsing task. To address this issue, Tian et al. [4] handle this task in a weakly-supervised manner given only video-level labels, which indicate events of presence without temporal boundaries and detailed modalities. They develop an audio-visual co-attention mechanism to assemble discriminative multimodal representations and use multiple instance learning to aggregate frame-level predictions into video-level ones. However, video-level labels alone cannot identify which modality events are from. Wu et al. [6] then propose to perform label refinement by swapping the audio and visual tracks of different videos to estimate and remove irrelevant event categories for each modality. They further adopt temporal contrastive learning to align audio and visual representations from the same frame. However, the contrastive learning is based on the assumption that audio and visual signals are synchronized, which may not hold in practical scenarios with complex events. Furthermore, these methods [4, 6] only consider audio and visual tracks of a single video without exploiting the relationship across categories and videos, which also provide rich shared semantics regarding event categories.
+
+In this work, we propose to leverage audio and visual data across different videos to explore shared information of each category. For example, videos with singing events may have similar patterns whatever in an audio or a visual modality. By observing all videos in a training batch, we can not only explore the shared semantics among audio-visual data but also exclude unrelated events. In addition to the relationship across different videos, we exploit the dependency between event categories. For example, when people are singing, there is usually a music accompaniment. Therefore, we propose to treat audio, visual, and audio-visual streams separately and adopt an audio-visual class co-occurrence module that jointly explores the relationship of different categories among all streams. By measuring the similarity of event categories from audio, visual, and audio-visual events, the correlated events are more likely to be correctly determined as the presence or absence of event categories. Such a strategy can robustly learn the correlation of categories within/across modality and fully exploit video data. The proposed strategy can be applied to existing methods on video parsing.
+
+We evaluate the proposed method on the LLP [4] dataset. Videos are parsed into audio, visual, and audio-visual events under both segment and event levels, and evaluated with F-scores metrics. Both qualitative and quantitative results demonstrate the effectiveness of the proposed method on the audio-visual video parsing task. The main contributions of this work are summarized as follows:
+
+• We leverage audio and visual data across different videos and tracks, which can learn common semantics of the same events and discern unrelated clues. • We develop an audio-visual event co-occurrence module that jointly considers the relationship of categories in audio, visual, and audio-visual modalities, which can prevent models from differentiating the representations of the related events. • Qualitative and quantitative experimental results on the benchmark dataset demonstrate that the proposed method performs favorably against the state-of-the-arts in various settings.
+
+# 2 Related Work
+
+Audio-Visual Representation Learning. Implicit correlation between audio and visual data from videos provides rich information for audio-visual representation learning. First, the audio-visual pairs from the same video clip [8, 9, 10, 11, 12, 13, 14, 15, 16, 17] are strongly correlated based on the assumption that audio and visual data from a video are synchronized and highly correlated. Moreover, features extracted from unpaired video clips tend to be more diverse than those from the same clips. Second, by exploring audio-visual temporal synchronization [18, 19], temporal information can be served as a training guidance. Given a video sequence, existing methods [18, 19] distinguish audio and visual features from different frames while correlating features from the same frames. Such an idea enhances robust audio-visual representation learning that is essential to several tasks such as audio-visual event localization/parsing/recognition [1, 2, 3, 4, 5, 6, 7, 20], sound separation [21, 22, 23, 24, 25, 26, 27, 28, 29, 30], audio spatialization [31, 32, 33, 34, 35, 36, 37, 38], and sound localization [39, 40, 41, 42, 43, 44]. Instead of random sampling sound and images, our method selects both related and irreverent videos to explore common semantics and discern dissimilar events.
+
+Audio-Visual Video Event Localization and Parsing. Audio-visual video parsing aims to detect events in videos and identify audio, visual, and audio-visual events (e.g., seeing the event and hearing its sound) and activities. Videos can be parsed with event categories and boundaries in both audio and visual modalities. Early researches [5, 7, 2, 3] aim to jointly derive audiovisual information in each local segment of the input video for audio-visual event localization, which emphasizes to detect only audio-visual events. However, due to the inconsistent information observed from audio and visual signals, data from either modality with insufficient clues may degrade the performance of prediction. Therefore, the work [7] focuses on audio/visual data with relevant categorical events to tackle this issue. Although methods of this category present favorable results, they are applicable to audio-visual event localization, which considers only synchronous audio-visual events or not. Recently, multi-modal multiple instance learning (MMIL) based methods with hybrid attention [4] carry out weakly-supervised audio-visual video parsing. These methods aggregate segment-level predictions into video-level ones, with which optimizing a model by using video-level or weak labels is enabled. Since video-level labels are typically insufficient to identify either audio or visual events, Wu et al. [6] generate pseudo labels for each modality by exchanging audio and visual tracks between unrelated videos. However, we notice that videos with replaced sounds or images may share some common semantics. Our method can exploit videos in a training batch to extract their common semantics for a categorical event and discern unrelated clues. Furthermore, we can leverage the relationship between event classes to find out related events (e.g., singing may accompany music).
+
+![](images/270ce2aeb0adebd19f77c3fe3a37657e6dbf1b204e12e9a0401ee9485ffe46c2.jpg)  
+Figure 1: Algorithmic overview. Our framework consists of a visual feature extractor, an audio feature extractor, a feature aggregation module, MMIL pooling, shared cross-modality semantics, and an cross-modality co-occurrence module. Given $n$ videos of $T$ seconds, the visual and audio feature extractors compute their visual and audio features. The feature aggregation module [4] conducts self- and cross-modality attention to aggregate segment-wise audio $\bar { \hat { \mathbf { f } ^ { a } } }$ and visual $\hat { \mathbf { f } ^ { v } }$ representations. We map segment-wise aggregated features to class-specific features by exploring cross-modality co-occurrence. By performing self- and cross-modality attention for class features, we identify within and cross modalities relationship between classes for event predictions. Note that $\otimes$ denotes matrix multiplication with the softmax operation performing on each row, and the green block only shows the example for segment-wise visual prediction at time $t$ . We also leverage the aggregated features of all $n$ videos to figure out common semantics regrading events by maximizing the similarities between related videos while minimizing those between unrelated videos with Eq. 8. The MMIL Pooling [4] is an attention-based pooling function that aggregates segment-wise results to produce video-level ones, which are optimized by the binary cross entropy loss described in Eq. 3 and Eq. 6.
+
+# 3 Proposed Method
+
+In this paper, we propose a novel framework for weakly-supervised audio-visual video parsing. In order to explore common semantics across videos and dependency across event categories, the proposed model leverages all audio and visual signals across videos in a training batch and the correlation between classes for each training instance. In Section 3.1, we first define the notations and settings considered in this paper and revisit the common backbone [4, 6] for weakly-supervised audio-visual video parsing, which consists of feature aggregation and multi-modal multiple instance learning (MMIL) pooling. Then in Section 3.2 and Section 3.3, we detail the modules we propose to capture dependency across different events and information across different videos, respectively.
+
+# 3.1 Preliminaries
+
+Problem Formulation and Notations. Given a video sequence $S$ with $T$ seconds long, we obtain $T$ non-overlapping audio and visual segments where each segment is one-second long. Models are aiming to predict the event labels for each segment, which may contain several or no events. At time $t$ , there are three targets for audio, visual, and audio-visual events: $\mathbf { y } _ { t } ^ { a } \in \mathbb { R } ^ { 1 \times C } , \mathbf { y } _ { t } ^ { v } \in \mathbb { R } ^ { 1 \times C }$ and $\mathbf { y } _ { t } ^ { a v } \in \mathbb { R } ^ { 1 \times C }$ are multi-class event label with $C$ event categories. $\mathbf { y } _ { t } ^ { a } , \mathbf { y } _ { t } ^ { v }$ , and ${ \bf y } _ { t } ^ { a v }$ denote audio, visual, and audio-visual event labels, respectively. We note that detailed annotations (e.g., $\mathbf { y } _ { t } ^ { a }$ , $\mathbf { y } _ { t } ^ { a }$ , and ${ \bf y } _ { t } ^ { a v }$ ) are not accessible during training and only available during evaluation. As for training, only video-level annotations are available during training. Video-level annotations only contain action event categories without indicating specific times slots or modalities (e.g., audio and visual event).
+
+Revisit of Weakly-Supervised Audio-Visual Video Parsing. The previous method [4] presents promising results with feature aggregation based on transformers and multimodal multiple instance learning (MMIL) pooling. Given a video sequence $S$ of $T$ frames, we denote its audio and visual feature sets by ${ \bf F } ^ { a ^ { * } } = \{ { \bf f } _ { 1 } ^ { a ^ { * } } , . . . , { \bf f } _ { T } ^ { a } \} \in \mathbb { R } ^ { T \times d }$ and $\mathbf { F } ^ { v } = \{ \mathbf { f } _ { 1 } ^ { v } , . . . , \mathbf { f } _ { T } ^ { v } \} \in \mathbb { R } ^ { T \times d }$ , respectively, where $d$ is the feature dimension. The transformer encoder [45] is employed to aggregate both within-modality and cross-modality information using multi-head attention blocks:
+
+$$
+\begin{array} { l } { { \phi _ { s e l f } ( { \bf f } _ { t } ^ { a } , { \bf F } ^ { a } , { \bf F } ^ { a } ) = \mathrm { S o f t m a x } ( \frac { { \bf f } _ { t } ^ { a } { \bf F } ^ { a } ^ { \top } } { \sqrt { d } } ) { \bf F } ^ { a } , } } \\ { { \phi _ { c r o s s } ( { \bf f } _ { t } ^ { a } , { \bf F } ^ { v } , { \bf F } ^ { v } ) = \mathrm { S o f t m a x } ( \frac { { \bf f } _ { t } ^ { a } { \bf F } ^ { v } ^ { \top } } { \sqrt { d } } ) { \bf F } ^ { v } , } } \end{array}
+$$
+
+where $\phi _ { s e l f } ( \cdot )$ and $\phi _ { c r o s s } ( \cdot )$ are self-attention and cross-modality attention functions respectively. They perform dot-product on features across time stamps by using non-shared MLPs. Then the jointly aggregated representations are described as follows:
+
+$$
+\begin{array} { r } { \hat { \mathbf { f } } _ { t } ^ { a } = \mathbf { f } _ { t } ^ { a } + \phi _ { s e l f } ( \mathbf { f } _ { t } ^ { a } , \mathbf { F } ^ { a } , \mathbf { F } ^ { a } ) + \phi _ { c r o s s } ( \mathbf { f } _ { t } ^ { a } , \mathbf { F } ^ { v } , \mathbf { F } ^ { v } ) , } \\ { \hat { \mathbf { f } } _ { t } ^ { v } = \mathbf { f } _ { t } ^ { v } + \phi _ { s e l f } ( \mathbf { f } _ { t } ^ { v } , \mathbf { F } ^ { v } , \mathbf { F } ^ { v } ) + \phi _ { c r o s s } ( \mathbf { f } _ { t } ^ { v } , \mathbf { F } ^ { a } , \mathbf { F } ^ { a } ) , } \end{array}
+$$
+
+With the aggregated audio and visual features $\hat { \mathbf { f } } _ { t } ^ { a }$ and $\hat { \mathbf { f } } _ { t } ^ { v }$ , we can obtain the frame-wise event prediction $\hat { \mathbf { p } } _ { t } ^ { a } \in \mathbb { R } ^ { 1 \times \widetilde { C } }$ and $\hat { \mathbf { p } } _ { t } ^ { v } \in \mathbb { R } ^ { 1 \times C }$ , and the attention weights computed by MLPs and normalized by a softmax function for audio, visual, and audio-visual streams (i.e., $\mathbf { w } _ { t } ^ { a } \in \mathbb { R } ^ { 1 \times C }$ , $\mathbf { w } _ { t } ^ { v } \in \mathbb { R } ^ { 1 \times C }$ , and $\mathbf { w } _ { t } ^ { a v } \in \mathbb { R } ^ { 2 \times C } ,$ ). Then the video-level prediction is gathered with the MMIL pooling:
+
+$$
+\bar { \mathbf { p } } ^ { a } = \sum _ { t = 1 } ^ { T } \mathbf { w } _ { t } ^ { a } \hat { \mathbf { p } } _ { t } ^ { a } , \bar { \mathbf { p } } ^ { v } = \sum _ { t = 1 } ^ { T } \mathbf { w } _ { t } ^ { v } \hat { \mathbf { p } } _ { t } ^ { v } , \mathrm { a n d } \bar { \mathbf { p } } ^ { a v } = \sum _ { t = 1 } ^ { T } \mathbf { w } _ { t } ^ { a v } [ 0 ] \mathbf { w } _ { t } ^ { a } \hat { \mathbf { p } } _ { t } ^ { a } + \mathbf { w } _ { t } ^ { a v } [ 1 ] \mathbf { w } _ { t } ^ { v } \hat { \mathbf { p } } _ { t } ^ { v } .
+$$
+
+The model can then be optimized using the binary cross-entropy loss function between $\bar { \bf p }$ and a video-level weak label $\bar { \mathbf { y } } \in \mathbb { R } ^ { 1 \times C }$ , which does not indicate time boundaries and modalities for events.
+
+# 3.2 Cross-Modality Co-Occurrence
+
+Videos with multi-label events contain rich information among event categories because the related events are likely to present at the same time. The correlation is useful for models to robustly predict the presence or absence of events.
+
+Similar to [46], to explicitly model the relationship between event categories in different modalities, we first obtain the representations for each class and then measure the correlation. We note that the class relationships may be different in audio and visual modalities. That is why the work [46] cannot be directly applied to audio-visual video parsing since audio or visual events can be partially or jointly presented at a single frame. Thus, jointly understanding the class relationship within a modality and across two modalities can benefit the audio-visual video parsing task.
+
+In order to map the frame-wise audio and visual features into class-level ones, the nonlinear transformation with MLPs is formulated as follows:
+
+$$
+\begin{array} { r } { \mathbf { a } _ { t , c } = \operatorname { R e L U } ( \hat { \mathbf { f } } _ { t } ^ { a } \mathbf { M } _ { c } ^ { a } + \mathbf { b } _ { c } ^ { a } ) , } \\ { \mathbf { v } _ { t , c } = \operatorname { R e L U } ( \hat { \mathbf { f } } _ { t } ^ { v } \mathbf { M } _ { c } ^ { v } + \mathbf { b } _ { c } ^ { v } ) , } \end{array}
+$$
+
+where $\mathbf { a } _ { t , c }$ and $\mathbf { v } _ { t , c }$ are audio and visual class-level features for class $c$ at time $t$ with dimension $1 \times d _ { c }$ , respectively. The weights and biases for class $c$ for audio and visual features are denoted as $\mathbf { M } _ { c } ^ { a }$ $\mathbf { \Psi } _ { : } ^ { i } , \mathbf { M } _ { c } ^ { i } \in \mathbb { R } ^ { d \times } \mathbf { \tilde { { d } } } _ { c }$ and ${ \bf b } _ { c } ^ { a }$ $\mathbf { \bar { b } } _ { c } ^ { v } \in \mathbb { R } ^ { 1 \times d _ { c } }$ . With class-level representations, we can further model the relationship between event categories within and across modalities by self-attention and crossmodality co-attention mechanism:
+
+$$
+\begin{array} { r } { \hat { \mathbf { a } } _ { t , c } = \mathbf { a } _ { t , c } + \phi _ { s e l f } ( \mathbf { a } _ { t , c } , \mathbf { A } _ { t } , \mathbf { A } _ { t } ) + \phi _ { c r o s s } ( \mathbf { a } _ { t , c } , \mathbf { V } _ { t } , \mathbf { V } _ { t } ) , } \\ { \hat { \mathbf { v } } _ { t , c } = \mathbf { v } _ { t , c } + \phi _ { s e l f } ( \mathbf { v } _ { t , c } , \mathbf { V } _ { t } , \mathbf { V } _ { t } ) + \phi _ { c r o s s } ( \mathbf { v } _ { t , c } , \mathbf { A } _ { t } , \mathbf { A } _ { t } ) , } \end{array}
+$$
+
+where $\mathbf { A } _ { t } = \{ \mathbf { a } _ { t , 1 } , \dotsc , \mathbf { a } _ { t , C } \}$ and $\mathbf { V } _ { t } = \{ \mathbf { v } _ { t , 1 } , \dots , \mathbf { v } _ { t , C } \}$ are sets of audio and visual class features at time $t$ . $\hat { \mathbf { a } } _ { t , c }$ and $\hat { \mathbf { v } } _ { t , c }$ are now co-occcurence features that consider the relationships between categories within and across modalities. We can then predict the probability for each event at time $t$ by MLPs and aggregate every segment-wise predictions into video-level ones i.e.,
+
+$$
+\begin{array} { r l } & { \hat { \mathbf { p } } _ { t } ^ { a } = \sigma ( \mathrm { M L P } _ { a } ( \{ \hat { \mathbf { a } } _ { t , 1 } , \dots , \hat { \mathbf { a } } _ { t , C } \} ) ) , \quad \hat { \mathbf { p } } _ { t } ^ { v } = \sigma ( \mathrm { M L P } _ { v } ( \{ \hat { \mathbf { v } } _ { t , 1 } , \dots , \hat { \mathbf { v } } _ { t , C } \} ) ) , } \\ & { \bar { \mathbf { p } } ^ { a } , \bar { \mathbf { p } } ^ { v } , \bar { \mathbf { p } } ^ { a v } = \mathrm { M M I L } ( \{ \hat { \mathbf { p } } _ { 1 } ^ { a } , \dots , \hat { \mathbf { p } } _ { T } ^ { a } \} , \{ \hat { \mathbf { p } } _ { 1 } ^ { v } , \dots , \hat { \mathbf { p } } _ { T } ^ { v } \} ) } \end{array}
+$$
+
+where $\sigma$ is the sigmoid function, and $\mathrm { M M L } ( \cdot )$ is the multi-modal multiple instance learning pooling described in Eq. 3 taking all segment-wise predictions as inputs. The video-level prediction can be optimized by the binary cross-entropy loss function with a video-level weak label $\bar { \mathbf { y } }$ .
+
+# 3.3 Shared Cross-Modality Semantics across Videos
+
+The information across different videos provides rich supervisory signals that benefit the training of weakly-supervised audio-visual video parsing. By observing videos in a training batch, we can discover both the common and diverse event semantics. With video-level labels, we can initially associate related and irrelevant videos. In order to obtain a discriminative categorical representation, we would like to encourage audio and visual representations from related events to be similar and differentiate those from irrelevant videos. However, targeting at segment-wise representations with specific events is difficult due to the lack of temporal annotations. Therefore, we seek event-related frames through the weights from MMIL pooling in Eq. 3:
+
+$$
+\tilde { \mathbf { f } } ^ { a } = \sum _ { t = 1 } ^ { T } \Big [ \frac { \exp \bigl ( g \bigl ( \bar { \mathbf { y } } \odot \mathbf { w } _ { t } ^ { a } \bigr ) \bigr ) } { \sum _ { t ^ { \prime } = 1 } ^ { T } \exp \bigl ( g \bigl ( \bar { \mathbf { y } } \odot \mathbf { w } _ { t ^ { \prime } } ^ { a } \bigr ) \bigr ) } \hat { \mathbf { f } } _ { t } ^ { a } \Big ] , \quad \tilde { \mathbf { f } } ^ { v } = \sum _ { t = 1 } ^ { T } \Big [ \frac { \exp \bigl ( g \bigl ( \bar { \mathbf { y } } \odot \mathbf { w } _ { t } ^ { v } \bigr ) \bigr ) } { \sum _ { t ^ { \prime } = 1 } ^ { T } \exp \bigl ( g \bigl ( \bar { \mathbf { y } } \odot \mathbf { w } _ { t ^ { \prime } } ^ { v } \bigr ) \bigr ) } \hat { \mathbf { f } } _ { t } ^ { v } \Big ] ,
+$$
+
+where $\odot$ and $g ( . )$ are element-wise dot product and summation function over all elements respectively.
+
+With video-level labels and features ${ \tilde { \mathbf { f } } } ^ { a }$ and $\tilde { \mathbf { f } } ^ { v }$ ), we adopt contrastive learning [47, 48, 49] to encourage features across modalities with the same event category (at least one) to be close and those with different events to be far away from each other. We leverage all $n$ videos in a batch to explore diverse semantics, where the sets of audio and visual features are denoted as $\{ \widetilde { \bf f } _ { ( 0 ) } ^ { a } , . . . , \widetilde { \bf f } _ { ( n ) } ^ { a } \}$ and $\{ \tilde { \mathbf { f } } _ { ( 0 ) } ^ { v } , . . . , \tilde { \mathbf { f } } _ { ( n ) } ^ { v } \}$ respectively with video-level labels $\left\{ \bar { \mathbf { y } } _ { ( 0 ) } , . . . , \bar { \mathbf { y } } _ { ( n ) } \right\}$ . The relationship across videos can be optimized by the proposed training objective as follows:
+
+$$
+\mathcal { L } _ { \mathrm { c o n t r a s t } } = - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \Big [ \log \frac { \sum _ { j = 1 } ^ { n } f ( \bar { \bf y } _ { i } \cdot \bar { \bf y } _ { j } ) \exp ( \tilde { \bf f } _ { ( i ) } ^ { a } \cdot \tilde { \bf f } _ { ( j ) } ^ { v } / \tau ) } { \sum _ { j = 1 } ^ { n } \exp ( \tilde { \bf f } _ { ( i ) } ^ { a } \cdot \tilde { \bf f } _ { ( j ) } ^ { v } / \tau ) } \Big ] ,
+$$
+
+where $f ( \cdot )$ is a clipping function that clips values over 1, and $\tau$ denotes a hyper-parameter controlling the temperature. Thus, the proposed method can be optimized by joint the binary cross-entropy loss mentioned in Section 3.1 and the contrastive learning loss in Eq. 8. Our training strategy can exploit cross-modality information across videos and event categories to understand common semantics while ignoring irrelevant ones.
+
+# 4 Experimental Results
+
+Datasets. We use the Look, Listen and Parse (LLP) Dataset [4] for all experiments. The LLP dataset consists of 11, 849 10-seconds video clips annotated with 25 event categories. It covers various real-life scenes such as speech, music performances, car, cheering, dog, etc. Particularly, there are 7202 video clips labeled with more than one event category. We use the 10000 video clips with only video-level event annotations for model training. The detailed annotations (e.g., individual audio and visual events per second) are available for the remaining 1849 validation and test videos. For all experiments, we use the official data splits from the LLP dataset.
+
+Evaluation Metrics. Following previous work [4, 6], we adopt F-scores as the evaluation metrics. Note that all types of events (audio, visual, and audio-visual) are measured under both segmentlevel and event-level metrics. The segment-level metrics can evaluate snippet-wise prediction results. As for the event-level metrics, the clips are extracted by concatenating positive consecutive segments in the same events. Then, we compute the event-level F-scores with $\mathrm { m I o U } = 0 . 5$ as the threshold. Furthermore, the overall Type $\ @ \mathbf { A V }$ performance on audio-visual scene is also considered by computing the averaged audio, visual, and audio-visual event evaluation results. Instead of directly averaging results from different event types, Event@AV considers all audio and visual event categories for each sample.
+
+Table 1: Quantitative results of weakly-supervised audio-visual video parsing. We evaluate all methods on the LLP dataset [4] with F-scores in five different event types and two kinds of segments. The first row indicates five different event types (audio, visual, audio-visual, Type@AV, and Event@AV). In the second row, two kinds of segments are shown: Seg. and Event are segmentlevel and event-level; and $^ *$ indicates only label refinement is utilized for fair comparisons.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Audio</td><td colspan="2">Visual</td><td colspan="2">Audio-visual</td><td colspan="2">Type@AV</td><td colspan="2">Event@AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>AVE [5]</td><td>47.2</td><td>40.4</td><td>37.1</td><td>34.7</td><td>35.4</td><td>31.6</td><td>39.9</td><td>35.5</td><td>41.6</td><td>36.5</td></tr><tr><td>AVSDN [2]</td><td>47.8</td><td>34.1</td><td>52.0</td><td>46.3</td><td>37.1</td><td>26.5</td><td>45.7</td><td>35.6</td><td>50.8</td><td>37.7</td></tr><tr><td>AVSDN + Ours</td><td>48.3</td><td>41.2</td><td>52.4</td><td>48.5</td><td>46.9</td><td>40.0</td><td>49.2</td><td>43.2</td><td>53.2</td><td>40.1</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + Ours</td><td>59.2</td><td>51.3</td><td>59.9</td><td>55.5</td><td>53.4</td><td>46.2</td><td>57.5</td><td>51.0</td><td>58.1</td><td>49.7</td></tr><tr><td>MA [6]</td><td>60.3</td><td>53.6</td><td>60.0</td><td>56.4</td><td>55.1</td><td>49.0</td><td>58.9</td><td>53.0</td><td>57.9</td><td>50.6</td></tr><tr><td>MA*</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA*+Ours</td><td>60.8</td><td>53.8</td><td>63.5</td><td>58.9</td><td>57.0</td><td>49.5</td><td>60.5</td><td>54.0</td><td>59.5</td><td>52.1</td></tr></table>
+
+Implementation Details. We implement the proposed method using PyTorch [50], and conduct the training and evaluation processes on a single NVIDIA GTX 1080 Ti GPU with 11 GB memory. Following [4, 6], we use the same visual and audio encoders for fair comparisons. We adopt both ResNet-152 [51] pre-trained on ImageNet [52] and 3D ResNet [53] pre-trained on Kinetics-400 [54] as visual feature extractors. Visual frames are sampled at 8 fps and their 2D and 3D visual features are extracted. The 2D and 3D visual features are concatenated and then processed by an MLP as the segment-wise representations. As for audio data, we utilize VGGish [55] pre-trained on AudioSet [56] to extract 128-dimensional audio features. The code and models are publicly available.
+
+Evaluated methods. We compare the proposed method based on several baselines to the following weakly-unsupervised approaches to the audio-visual video parsing task:
+
+• AVE [5] consists of an audio-guided co-attention mechanism to adaptively learn the sounding regions. We note that AVE [5] deals with the audio-visual event localization task. Thus, we follow [4] and add additional audio and visual parsing branches for the weakly-supervised audio-visual video parsing task as a baseline.   
+• AVSDN [2] is a sequence-to-sequence-based model to integrate global audio and visual features to local ones. Since AVSDN [2] also deals with the audio-visual event localization task, we make the same modifications to AVSDN as those to AVE.   
+• HAN [4] is a multi-modal multiple instance learning-based method with a hybrid attention network.   
+• MA [6] reports the state-of-the-art performance on the weakly-supervised audio-visual video parsing task. It is a method based on HAN with the label refinement and the audio-visual contrastive learning differentiating temporal segments.
+
+# 4.1 Quantitative Evaluation
+
+Table 1 shows the quantitative comparisons on the LLP dataset [4]. The proposed method performs favorably against the competing approaches on the weakly-supervised audio-visual video parsing task. Since our method can be easily extended to existing methods, we extend the proposed on three baselines. The third, fifth, and last rows in Table 1 indicate that the proposed method generally benefits three baselines on several metrics of the audio-visual video parsing task by a large margin. We note that $\mathbf { M A } ^ { * }$ [6] only utilizes label refinement to refine labels for each modality, and temporal difference audio-visual contrastive learning [6] is not implemented.
+
+Table 2: Ablation study. We investigate the effect of using different design components in the proposed method. We show how proposed cross-modality co-occurrence (CM-Co) in Section 3.3 and shared cross-modality semantics across videos (CM-S) module in Section 3.2 improve the baselines.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Audio</td><td colspan="2">Visual</td><td colspan="2">Audio-visual</td><td colspan="2">Type@AV</td><td colspan="2">Event@AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + CM-S</td><td>58.1</td><td>49.6</td><td>58.3</td><td>53.6</td><td>53.2</td><td>46.3</td><td>56.5</td><td>49.8</td><td>55.9</td><td>47.5</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>51.4</td><td>57.4</td><td>52.4</td><td>51.9</td><td>44.2</td><td>56.3</td><td>49.3</td><td>57.4</td><td>48.5</td></tr><tr><td>HAN + Ours</td><td>59.2</td><td>51.3</td><td>59.9</td><td>55.5</td><td>53.4</td><td>46.2</td><td>57.5</td><td>51.0</td><td>58.1</td><td>49.7</td></tr><tr><td>MA [28]</td><td>60.3</td><td>53.6</td><td>60.0</td><td>56.4</td><td>55.1</td><td>49.0</td><td>58.9</td><td>53.0</td><td>57.9</td><td>50.6</td></tr><tr><td>MA + CM-Co</td><td>61.1</td><td>53.3</td><td>61.7</td><td>57.3</td><td>56.3</td><td>49.0</td><td>59.7</td><td>53.0</td><td>58.9</td><td>51.2</td></tr><tr><td>MA*</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA* + CM-S</td><td>60.4</td><td>53.5</td><td>60.7</td><td>56.5</td><td>55.8</td><td>47.5</td><td>58.9</td><td>52.5</td><td>58.6</td><td>51.0</td></tr><tr><td>MA* + CM-Co</td><td>60.5</td><td>53.6</td><td>61.3</td><td>56.5</td><td>54.9</td><td>46.7</td><td>58.9</td><td>52.3</td><td>59.1</td><td>51.4</td></tr><tr><td>MA* +Ours</td><td>60.8</td><td>53.8</td><td>63.5</td><td>58.9</td><td>57.0</td><td>49.5</td><td>60.5</td><td>54.0</td><td>59.5</td><td>52.1</td></tr></table>
+
+We notice that our method significantly improves baselines in the metrics of visual, audio-visual, Type $@ \mathrm { A V } ,$ and Event $@$ AV. By observing the class distribution of training sets, we find that $3 1 \%$ , $7 \%$ , and $9 \%$ training videos contain speech, singing, and violin events. These events are more likely to present in the audio modality. Therefore, the video-level labels would limit the performance regarding visual events. The proposed method can leverage additional cross-video and cross-modality supervisory signals to explore common semantics, which can improve results in vision-related metrics.
+
+# 4.2 Ablation Study
+
+Cross-Modality Co-Occurrence and Semantics across Video. We conduct the ablation study to analyze the individual impact of each developed component in the proposed method. The results are presented in Table 2. CM-Co represents the usage of the cross-modality co-occurrence module described in Section 3.2, which leverages the relationship between categories within and cross modalities. CM-S indicates the shared cross-modality semantics across videos module described in Section 3.3, which considers all audio and visual information across videos in a batch.
+
+In Table 2, we note that both CM-S and CM-Co can improve baselines in several metrics. By exploring common semantics among training videos (CM-S), we improve the performance on visual and audio-visual evaluation by a large margin. Such a strategy can exploit additional information from videos to address the potential drawback of video-level labels described in Section 4.1. Furthermore, the proposed cross-modality co-occurrence module (CM-Co) also presents favorable results. We note that the significant improvement in Event@AV evaluation with the usage of CM-Co can verify the efficacy of considering the relationship between categories within and across modalities. Since Event $@$ AV considers all audio and visual events for the F-score (e.g., truth positive from both audio and visual events), the improvement of Event@AV indicates our cross-modality co-occurrence can perform well on video parsing when events present in an audio or a visual modality.
+
+In the second group of the evaluated methods in Table 2, we verify if the proposed CM-S works better than the contrastive learning method in MA. We perform our CM-S on the MA model. The CM-S exploits information across different videos to address the issue that audio and visual tracks may not be synchronized. Instead, the contrastive learning method in MA is developed based on the assumption of synchronization to associate the audio-visual representation in a single video. Since our CM-S learns diverse and common semantics, it is effective and complementary to the contrastive learning approach in MA performing on a single video. We note that our CM-S generally improves the performance over all segment-level metrics, which supports our claim.
+
+Self-attention and Cross-Modality Co-attention in Co-Occurrence. Since our cross-modality co-occurrence module exploits self-attention among class-level features in the same modality and cross-modality co-attention on cross-modality class-level representations to model the relationship between categories in the same and different modalities. Taking class-level audio features in Eq. 5 as an example, the class-level self-attention and cross-modality co-attention are $\mathrm { a t t n } ( \mathbf { a } _ { t , c } , \mathbf { A } _ { t } , \mathbf { A } _ { t } )$ and $\operatorname { a t t n } ( \mathbf { a } _ { t , c } , \mathbf { V } _ { t } , \mathbf { V } _ { t } ) .$ , respectively.
+
+Table 3: Ablation study. We investigate the effect of different developed mechanisms in the proposed cross-modality co-occurrence (CM-Co) module in Section 3.2. In Eq. 5, class-level features are processed by self-attention and cross-modality co-attention mechanisms. A Only and $\mathbf { V }$ Only indicate only self-attention performs for individual audio and visual events respectively. AV denotes performing self-attention for audio and visual events. CM-Co is the proposed method that considers relationship between categories within and cross modalities by both self-attention and cross-modality co-attention mechanisms.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Audio</td><td colspan="2">Visual</td><td colspan="2">Audio-visual</td><td colspan="2">Type@AV</td><td colspan="2">Event@ AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + A Only</td><td>60.5</td><td>52.3</td><td>49.8</td><td>43.9</td><td>45.6</td><td>38.3</td><td>52.0</td><td>44.8</td><td>55.7</td><td>45.9</td></tr><tr><td>HAN + V Only</td><td>56.1</td><td>44.5</td><td>56.8</td><td>53.2</td><td>49.7</td><td>40.7</td><td>54.2</td><td>46.1</td><td>54.1</td><td>44.6</td></tr><tr><td>HAN + AV</td><td>59.5</td><td>50.3</td><td>55.1</td><td>50.5</td><td>48.6</td><td>40.3</td><td>54.4</td><td>47.0</td><td>56.0</td><td>47.4</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>51.4</td><td>57.4</td><td>52.4</td><td>51.9</td><td>44.2</td><td>56.3</td><td>49.3</td><td>57.4</td><td>48.5</td></tr><tr><td>MA*[6]</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA*+ A Only</td><td>60.7</td><td>52.7</td><td>53.9</td><td>47.9</td><td>50.1</td><td>42.2</td><td>54.9</td><td>47.6</td><td>57.0</td><td>47.1</td></tr><tr><td>MA* + V Only</td><td>46.8</td><td>34.4</td><td>60.8</td><td>57.0</td><td>42.8</td><td>31.1</td><td>50.1</td><td>40.9</td><td>52.6</td><td>40.4</td></tr><tr><td>MA*+ AV</td><td>58.3</td><td>50.4</td><td>59.4</td><td>55.2</td><td>53.9</td><td>46.9</td><td>57.2</td><td>50.8</td><td>56.7</td><td>48.5</td></tr><tr><td>MA*+ CM-Co</td><td>60.5</td><td>53.6</td><td>61.3</td><td>56.5</td><td>54.9</td><td>46.7</td><td>58.9</td><td>52.3</td><td>59.1</td><td>51.4</td></tr></table>
+
+Table 4: Ablation study. We evaluate the proposed method in accuracy, efficiency, and model sizes. We show the numbers of parameters and FLOPs for the proposed cross-modality co-occurrence (CM-Co) and HAN [4] with a few layers.   
+Note that the results are all in the segment level.   
+
+<table><tr><td>Method</td><td>Audio</td><td>Visual</td><td>Audio-visual</td><td>Type@AV</td><td>Event@AV</td><td>GFLOPs</td><td>Params</td></tr><tr><td>HAN 1 Layer</td><td>60.1</td><td>52.9</td><td>48.9</td><td>54.0</td><td>55.4</td><td>6.63</td><td>2.4M</td></tr><tr><td>HAN 2 Layers</td><td>58.2</td><td>55.4</td><td>50.6</td><td>54.7</td><td>54.9</td><td>7.28</td><td>2.9M</td></tr><tr><td>HAN 3 Layers</td><td>58.1</td><td>55.2</td><td>50.3</td><td>54.5</td><td>54.6</td><td>7.97</td><td>3.5M</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>57.4</td><td>51.9</td><td>56.3</td><td>57.4</td><td>6.99</td><td>2.8M</td></tr></table>
+
+Table 3 presents the results in various modifications of the cross-modality co-occurrence module. We note that the design of co-occurrence in the same and cross modalities can generally improve the results in several metrics. We also evaluate the co-occurrence module in a single modality. The results are shown in the second, third, seventh, and eighth rows in Table 3, where A Only and $\mathbf { V }$ Only indicate the co-occurrence module only leverages the relationship between categories in audio or visual data respectively. As the results shown in the second and seventh rows, training with co-occurrence in audio events only (i.e., A Only) can slightly improve the performance on audio events. Similarly, considering visual event only (i.e., V Only) can benefit the results regarding visual events. Furthermore, the co-occurrence for both audio and visual categories (AV) in the fourth and ninth rows can contribute to the results in general metrics such as Type $@$ AV and Event $@ \mathrm { A V } .$ . We then further consider the correlation between events across modalities. That is the cross-modality co-occurrence module (CM-Co) in the fifth and tenth rows. The results can confirm the efficacy of the proposed cross-modality co-occurrence module in all metrics except segment-level audio events caused by similar reasons discussed in Section 4.1.
+
+Model Capacity. Since our cross-modality co-occurrence module leverages class-level representations, it would increase the capability of models on capturing information. For fair comparisons, we add extra parameters to HAN [4] to analyze whether more parameters can contribute to performance gain. Specifically, we increase the number of layers in its transformer-based feature aggregation to 2 and 3, respectively.
+
+In Table 4, we report the results in accuracy, computational costs, and model sizes. The first three rows show the performance of HAN with different numbers of layers. We note that HAN with one extra layer has more parameters than the proposed co-occurrence module. However, the results of HAN with extra layers indicate that using more parameters/layers for HAN does not improve the performance. The proposed cross-modality co-occurrence module enhances HAN more effectively.
+
+![](images/3a3e5bddfd4d0b8b3df955da6588c994004175b863c809999ccff152b083248e.jpg)  
+Figure 2: Qualitative comparisons. We compare the proposed method with the state-of-the-art weakly-supervised audio-visual video parsing method on the LLP dataset [4]. The frame-wise annotations are shown in gray and purple bars. The gray bar denotes visual events, and the purple bar represents audio events. GT_V and GT_A are the ground-truth visual and audio events respectively. Our results are shown in the green block, and the results by the competing method, MA [6], are present in the blue block.
+
+![](images/21cead53b19ede061abdc05068f5d07b4594cffe449abe292ef47d1a62306ab2.jpg)  
+Figure 3: Audio feature distribution by using t-SNE. The upper figure shows the distribution by our method. The lower figure presents that by $\mathbf { M A } ^ { * }$ . The legend lists all event combinations.
+
+# 4.3 Qualitative Evaluation
+
+Qualitative Results. We present the qualitative results of the evaluated methods in Figure 2. GT_V and GT_A show the ground-truth annotations for visual and audio events, respectively. Pred_V and Pred_A present the predictions made by our method and the state-of-the-art competing method, MA [6], respectively. Our results are shown in the green block, while the results of MA are present in the blue block. In general, our method presents more accurate predictions in both audio and visual events than MA. We note that the whole violin is shown after 7 seconds. That would hamper models for understanding visual events e.g., MA predicts wrong results on violin visual events before 6 seconds. Since our method leverages the relationship between categories, it can still predict correct temporal boundaries for guitar events by jointly considering cello events in the videos.
+
+![](images/ed423064930570c6abeaee1db5cadd585aab212a9182498adeec79bb58aa61f7.jpg)  
+Figure 4: Visual feature distribution by using t-SNE. The upper figure shows the distribution by our method. The lower figure presents that by MA∗. The legend lists all event combinations.
+
+Feature Distribution Visualized by t-SNE. We apply t-SNE to the aggregated audio and visual features from each segment described in Eq. 2. The visualization results are present in Figure 3 and Figure 4, respectively. The legends list all the combinations of multiple labels. For example, in Figure 3, audio events of singing are present as blue spots, and the mixed sounds of singing and violin are shown as purple spots. We note that the related events including multiple events are shown in similar colors. In Figure 4, the proposed method achieves better performance in the sense that similar color spots are closer than the spots in $\mathbf { M A } ^ { * }$ .
+
+# 5 Conclusions
+
+In this paper, we present a novel audio-visual video parsing framework in a weakly-supervised manner that can be applied to existing methods. We propose two modules to exploit the relationship across videos, modalities, and event categories, and explore additional supervisory signals that can benefit audio-visual video parsing. The shared cross-modality semantics module leverages common and diverse event semantics across videos to learn robust cross-modality representations that facilitate models to identify audio, visual, and audio-visual events. Furthermore, the cross-modality co-occurrence module aims to learn the relationship between event categories. It helps localize segments of target events and can exclude irrelevant ones by performing self-attention and crossmodality co-attention on class-wise features, Extensive experimental results show that our approach substantially improves several baselines and performs favorably against the state-of-the-art methods.
+
+Acknowledgments. This work was supported in part by the Ministry of Science and Technology under grants 109- 2221-E-009-113-MY3, 110-2628-E-A49-008, and 110-2634-F007-015. It was also funded in part by Qualcomm through a Taiwan University Research Collaboration Project, the Higher Education Sprout Project of the National Yang Ming Chiao Tung University, and Ministry of Education.
+
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parse/train/ilVv1LO0Ew/ilVv1LO0Ew_content_list.json

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+    {
+        "type": "text",
+        "text": "Exploring Cross-Video and Cross-Modality Signals for Weakly-Supervised Audio-Visual Video Parsing ",
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+    {
+        "type": "text",
+        "text": "Yan-Bo Lin1,2 Hung-Yu Tseng3 Hsin-Ying Lee4 Yen-Yu Lin1 Ming-Hsuan Yang3,5,6 ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1National Yang Ming Chiao Tung University 2UNC Chapel Hill 3UC Merc 4Snap Research 5Google Research 6Yonsei University yblin@unc.edu htseng6@ucmerced.edu hlee5@snap.com lin@cs.nctu.edu.tw mhyang@ucmerced.edu ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "The audio-visual video parsing task aims to temporally parse a video into audio or visual event categories. However, it is labor-intensive to temporally annotate audio and visual events and thus hampers the learning of a parsing model. To this end, we propose to explore additional cross-video and cross-modality supervisory signals to facilitate weakly-supervised audio-visual video parsing. The proposed method exploits both the common and diverse event semantics across videos to identify audio or visual events. In addition, our method explores event co-occurrence across audio, visual, and audio-visual streams. We leverage the explored cross-modality co-occurrence to localize segments of target events while excluding irrelevant ones. The discovered supervisory signals across different videos and modalities can greatly facilitate the training with only video-level annotations. Quantitative and qualitative results demonstrate that the proposed method performs favorably against existing methods on weakly-supervised audio-visual video parsing. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Humans perceive multisensory signals via seeing, hearing, touching, etc., and obtain multimodal information while exploring the surrounding environments. Visual and audio signals, the most common modalities, motivate researchers to jointly comprehend audio-visual events (e.g., see people singing and hear their sounds) [1, 2, 3, 4, 5, 6, 7]. Events visible in images while hearable in audio are referred to as audio-visual events. However, learning-based models tend to recognize a particular audio-visual event by using the data from the dominant modality with richer information and overlook clues from either audio only or visual only events which still contribute to holistic video understanding. Therefore, the resultant models can generalize well on audio-visual events only instead of comprehensively understanding all kinds of video events. To address this issue, we target at audio-visual video parsing [4, 6] where predictions for audio, visual, and audio-visual events with temporal boundaries are all required but separately evaluated. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The time-consuming and labor-intensive annotation process poses a major challenge for the audiovisual video parsing task. To address this issue, Tian et al. [4] handle this task in a weakly-supervised manner given only video-level labels, which indicate events of presence without temporal boundaries and detailed modalities. They develop an audio-visual co-attention mechanism to assemble discriminative multimodal representations and use multiple instance learning to aggregate frame-level predictions into video-level ones. However, video-level labels alone cannot identify which modality events are from. Wu et al. [6] then propose to perform label refinement by swapping the audio and visual tracks of different videos to estimate and remove irrelevant event categories for each modality. They further adopt temporal contrastive learning to align audio and visual representations from the same frame. However, the contrastive learning is based on the assumption that audio and visual signals are synchronized, which may not hold in practical scenarios with complex events. Furthermore, these methods [4, 6] only consider audio and visual tracks of a single video without exploiting the relationship across categories and videos, which also provide rich shared semantics regarding event categories. ",
+        "bbox": [
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+    {
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+    {
+        "type": "text",
+        "text": "In this work, we propose to leverage audio and visual data across different videos to explore shared information of each category. For example, videos with singing events may have similar patterns whatever in an audio or a visual modality. By observing all videos in a training batch, we can not only explore the shared semantics among audio-visual data but also exclude unrelated events. In addition to the relationship across different videos, we exploit the dependency between event categories. For example, when people are singing, there is usually a music accompaniment. Therefore, we propose to treat audio, visual, and audio-visual streams separately and adopt an audio-visual class co-occurrence module that jointly explores the relationship of different categories among all streams. By measuring the similarity of event categories from audio, visual, and audio-visual events, the correlated events are more likely to be correctly determined as the presence or absence of event categories. Such a strategy can robustly learn the correlation of categories within/across modality and fully exploit video data. The proposed strategy can be applied to existing methods on video parsing. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate the proposed method on the LLP [4] dataset. Videos are parsed into audio, visual, and audio-visual events under both segment and event levels, and evaluated with F-scores metrics. Both qualitative and quantitative results demonstrate the effectiveness of the proposed method on the audio-visual video parsing task. The main contributions of this work are summarized as follows: ",
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+    {
+        "type": "text",
+        "text": "• We leverage audio and visual data across different videos and tracks, which can learn common semantics of the same events and discern unrelated clues. • We develop an audio-visual event co-occurrence module that jointly considers the relationship of categories in audio, visual, and audio-visual modalities, which can prevent models from differentiating the representations of the related events. • Qualitative and quantitative experimental results on the benchmark dataset demonstrate that the proposed method performs favorably against the state-of-the-arts in various settings. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Audio-Visual Representation Learning. Implicit correlation between audio and visual data from videos provides rich information for audio-visual representation learning. First, the audio-visual pairs from the same video clip [8, 9, 10, 11, 12, 13, 14, 15, 16, 17] are strongly correlated based on the assumption that audio and visual data from a video are synchronized and highly correlated. Moreover, features extracted from unpaired video clips tend to be more diverse than those from the same clips. Second, by exploring audio-visual temporal synchronization [18, 19], temporal information can be served as a training guidance. Given a video sequence, existing methods [18, 19] distinguish audio and visual features from different frames while correlating features from the same frames. Such an idea enhances robust audio-visual representation learning that is essential to several tasks such as audio-visual event localization/parsing/recognition [1, 2, 3, 4, 5, 6, 7, 20], sound separation [21, 22, 23, 24, 25, 26, 27, 28, 29, 30], audio spatialization [31, 32, 33, 34, 35, 36, 37, 38], and sound localization [39, 40, 41, 42, 43, 44]. Instead of random sampling sound and images, our method selects both related and irreverent videos to explore common semantics and discern dissimilar events. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Audio-Visual Video Event Localization and Parsing. Audio-visual video parsing aims to detect events in videos and identify audio, visual, and audio-visual events (e.g., seeing the event and hearing its sound) and activities. Videos can be parsed with event categories and boundaries in both audio and visual modalities. Early researches [5, 7, 2, 3] aim to jointly derive audiovisual information in each local segment of the input video for audio-visual event localization, which emphasizes to detect only audio-visual events. However, due to the inconsistent information observed from audio and visual signals, data from either modality with insufficient clues may degrade the performance of prediction. Therefore, the work [7] focuses on audio/visual data with relevant categorical events to tackle this issue. Although methods of this category present favorable results, they are applicable to audio-visual event localization, which considers only synchronous audio-visual events or not. Recently, multi-modal multiple instance learning (MMIL) based methods with hybrid attention [4] carry out weakly-supervised audio-visual video parsing. These methods aggregate segment-level predictions into video-level ones, with which optimizing a model by using video-level or weak labels is enabled. Since video-level labels are typically insufficient to identify either audio or visual events, Wu et al. [6] generate pseudo labels for each modality by exchanging audio and visual tracks between unrelated videos. However, we notice that videos with replaced sounds or images may share some common semantics. Our method can exploit videos in a training batch to extract their common semantics for a categorical event and discern unrelated clues. Furthermore, we can leverage the relationship between event classes to find out related events (e.g., singing may accompany music). ",
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+    {
+        "type": "image",
+        "img_path": "images/270ce2aeb0adebd19f77c3fe3a37657e6dbf1b204e12e9a0401ee9485ffe46c2.jpg",
+        "image_caption": [
+            "Figure 1: Algorithmic overview. Our framework consists of a visual feature extractor, an audio feature extractor, a feature aggregation module, MMIL pooling, shared cross-modality semantics, and an cross-modality co-occurrence module. Given $n$ videos of $T$ seconds, the visual and audio feature extractors compute their visual and audio features. The feature aggregation module [4] conducts self- and cross-modality attention to aggregate segment-wise audio $\\bar { \\hat { \\mathbf { f } ^ { a } } }$ and visual $\\hat { \\mathbf { f } ^ { v } }$ representations. We map segment-wise aggregated features to class-specific features by exploring cross-modality co-occurrence. By performing self- and cross-modality attention for class features, we identify within and cross modalities relationship between classes for event predictions. Note that $\\otimes$ denotes matrix multiplication with the softmax operation performing on each row, and the green block only shows the example for segment-wise visual prediction at time $t$ . We also leverage the aggregated features of all $n$ videos to figure out common semantics regrading events by maximizing the similarities between related videos while minimizing those between unrelated videos with Eq. 8. The MMIL Pooling [4] is an attention-based pooling function that aggregates segment-wise results to produce video-level ones, which are optimized by the binary cross entropy loss described in Eq. 3 and Eq. 6. "
+        ],
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+    {
+        "type": "text",
+        "text": "3 Proposed Method ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a novel framework for weakly-supervised audio-visual video parsing. In order to explore common semantics across videos and dependency across event categories, the proposed model leverages all audio and visual signals across videos in a training batch and the correlation between classes for each training instance. In Section 3.1, we first define the notations and settings considered in this paper and revisit the common backbone [4, 6] for weakly-supervised audio-visual video parsing, which consists of feature aggregation and multi-modal multiple instance learning (MMIL) pooling. Then in Section 3.2 and Section 3.3, we detail the modules we propose to capture dependency across different events and information across different videos, respectively. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.1 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Problem Formulation and Notations. Given a video sequence $S$ with $T$ seconds long, we obtain $T$ non-overlapping audio and visual segments where each segment is one-second long. Models are aiming to predict the event labels for each segment, which may contain several or no events. At time $t$ , there are three targets for audio, visual, and audio-visual events: $\\mathbf { y } _ { t } ^ { a } \\in \\mathbb { R } ^ { 1 \\times C } , \\mathbf { y } _ { t } ^ { v } \\in \\mathbb { R } ^ { 1 \\times C }$ and $\\mathbf { y } _ { t } ^ { a v } \\in \\mathbb { R } ^ { 1 \\times C }$ are multi-class event label with $C$ event categories. $\\mathbf { y } _ { t } ^ { a } , \\mathbf { y } _ { t } ^ { v }$ , and ${ \\bf y } _ { t } ^ { a v }$ denote audio, visual, and audio-visual event labels, respectively. We note that detailed annotations (e.g., $\\mathbf { y } _ { t } ^ { a }$ , $\\mathbf { y } _ { t } ^ { a }$ , and ${ \\bf y } _ { t } ^ { a v }$ ) are not accessible during training and only available during evaluation. As for training, only video-level annotations are available during training. Video-level annotations only contain action event categories without indicating specific times slots or modalities (e.g., audio and visual event). ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Revisit of Weakly-Supervised Audio-Visual Video Parsing. The previous method [4] presents promising results with feature aggregation based on transformers and multimodal multiple instance learning (MMIL) pooling. Given a video sequence $S$ of $T$ frames, we denote its audio and visual feature sets by ${ \\bf F } ^ { a ^ { * } } = \\{ { \\bf f } _ { 1 } ^ { a ^ { * } } , . . . , { \\bf f } _ { T } ^ { a } \\} \\in \\mathbb { R } ^ { T \\times d }$ and $\\mathbf { F } ^ { v } = \\{ \\mathbf { f } _ { 1 } ^ { v } , . . . , \\mathbf { f } _ { T } ^ { v } \\} \\in \\mathbb { R } ^ { T \\times d }$ , respectively, where $d$ is the feature dimension. The transformer encoder [45] is employed to aggregate both within-modality and cross-modality information using multi-head attention blocks: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/df014a1b954a2bcc0c32855513113f4fe555449e7a17c022439841acbd5fd03e.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\phi _ { s e l f } ( { \\bf f } _ { t } ^ { a } , { \\bf F } ^ { a } , { \\bf F } ^ { a } ) = \\mathrm { S o f t m a x } ( \\frac { { \\bf f } _ { t } ^ { a } { \\bf F } ^ { a } ^ { \\top } } { \\sqrt { d } } ) { \\bf F } ^ { a } , } } \\\\ { { \\phi _ { c r o s s } ( { \\bf f } _ { t } ^ { a } , { \\bf F } ^ { v } , { \\bf F } ^ { v } ) = \\mathrm { S o f t m a x } ( \\frac { { \\bf f } _ { t } ^ { a } { \\bf F } ^ { v } ^ { \\top } } { \\sqrt { d } } ) { \\bf F } ^ { v } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            263,
+            645,
+            339
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\phi _ { s e l f } ( \\cdot )$ and $\\phi _ { c r o s s } ( \\cdot )$ are self-attention and cross-modality attention functions respectively. They perform dot-product on features across time stamps by using non-shared MLPs. Then the jointly aggregated representations are described as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e6cff82b396fe130840dfd8f000dea5c20b6fe84a3191f893872089e8a3c5a7d.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\mathbf { f } } _ { t } ^ { a } = \\mathbf { f } _ { t } ^ { a } + \\phi _ { s e l f } ( \\mathbf { f } _ { t } ^ { a } , \\mathbf { F } ^ { a } , \\mathbf { F } ^ { a } ) + \\phi _ { c r o s s } ( \\mathbf { f } _ { t } ^ { a } , \\mathbf { F } ^ { v } , \\mathbf { F } ^ { v } ) , } \\\\ { \\hat { \\mathbf { f } } _ { t } ^ { v } = \\mathbf { f } _ { t } ^ { v } + \\phi _ { s e l f } ( \\mathbf { f } _ { t } ^ { v } , \\mathbf { F } ^ { v } , \\mathbf { F } ^ { v } ) + \\phi _ { c r o s s } ( \\mathbf { f } _ { t } ^ { v } , \\mathbf { F } ^ { a } , \\mathbf { F } ^ { a } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "With the aggregated audio and visual features $\\hat { \\mathbf { f } } _ { t } ^ { a }$ and $\\hat { \\mathbf { f } } _ { t } ^ { v }$ , we can obtain the frame-wise event prediction $\\hat { \\mathbf { p } } _ { t } ^ { a } \\in \\mathbb { R } ^ { 1 \\times \\widetilde { C } }$ and $\\hat { \\mathbf { p } } _ { t } ^ { v } \\in \\mathbb { R } ^ { 1 \\times C }$ , and the attention weights computed by MLPs and normalized by a softmax function for audio, visual, and audio-visual streams (i.e., $\\mathbf { w } _ { t } ^ { a } \\in \\mathbb { R } ^ { 1 \\times C }$ , $\\mathbf { w } _ { t } ^ { v } \\in \\mathbb { R } ^ { 1 \\times C }$ , and $\\mathbf { w } _ { t } ^ { a v } \\in \\mathbb { R } ^ { 2 \\times C } ,$ ). Then the video-level prediction is gathered with the MMIL pooling: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/55d25e95859dcdbc0e18dcaa2f5b054e82fd68aa4b4e701a080a2cb7163a0878.jpg",
+        "text": "$$\n\\bar { \\mathbf { p } } ^ { a } = \\sum _ { t = 1 } ^ { T } \\mathbf { w } _ { t } ^ { a } \\hat { \\mathbf { p } } _ { t } ^ { a } , \\bar { \\mathbf { p } } ^ { v } = \\sum _ { t = 1 } ^ { T } \\mathbf { w } _ { t } ^ { v } \\hat { \\mathbf { p } } _ { t } ^ { v } , \\mathrm { a n d } \\bar { \\mathbf { p } } ^ { a v } = \\sum _ { t = 1 } ^ { T } \\mathbf { w } _ { t } ^ { a v } [ 0 ] \\mathbf { w } _ { t } ^ { a } \\hat { \\mathbf { p } } _ { t } ^ { a } + \\mathbf { w } _ { t } ^ { a v } [ 1 ] \\mathbf { w } _ { t } ^ { v } \\hat { \\mathbf { p } } _ { t } ^ { v } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "The model can then be optimized using the binary cross-entropy loss function between $\\bar { \\bf p }$ and a video-level weak label $\\bar { \\mathbf { y } } \\in \\mathbb { R } ^ { 1 \\times C }$ , which does not indicate time boundaries and modalities for events. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Cross-Modality Co-Occurrence ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Videos with multi-label events contain rich information among event categories because the related events are likely to present at the same time. The correlation is useful for models to robustly predict the presence or absence of events. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Similar to [46], to explicitly model the relationship between event categories in different modalities, we first obtain the representations for each class and then measure the correlation. We note that the class relationships may be different in audio and visual modalities. That is why the work [46] cannot be directly applied to audio-visual video parsing since audio or visual events can be partially or jointly presented at a single frame. Thus, jointly understanding the class relationship within a modality and across two modalities can benefit the audio-visual video parsing task. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In order to map the frame-wise audio and visual features into class-level ones, the nonlinear transformation with MLPs is formulated as follows: ",
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7d313b9172ca9b6250573624cb928ff08260ac3770f3f987b57f8e82d08c0916.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { a } _ { t , c } = \\operatorname { R e L U } ( \\hat { \\mathbf { f } } _ { t } ^ { a } \\mathbf { M } _ { c } ^ { a } + \\mathbf { b } _ { c } ^ { a } ) , } \\\\ { \\mathbf { v } _ { t , c } = \\operatorname { R e L U } ( \\hat { \\mathbf { f } } _ { t } ^ { v } \\mathbf { M } _ { c } ^ { v } + \\mathbf { b } _ { c } ^ { v } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { a } _ { t , c }$ and $\\mathbf { v } _ { t , c }$ are audio and visual class-level features for class $c$ at time $t$ with dimension $1 \\times d _ { c }$ , respectively. The weights and biases for class $c$ for audio and visual features are denoted as $\\mathbf { M } _ { c } ^ { a }$ $\\mathbf { \\Psi } _ { : } ^ { i } , \\mathbf { M } _ { c } ^ { i } \\in \\mathbb { R } ^ { d \\times } \\mathbf { \\tilde { { d } } } _ { c }$ and ${ \\bf b } _ { c } ^ { a }$ $\\mathbf { \\bar { b } } _ { c } ^ { v } \\in \\mathbb { R } ^ { 1 \\times d _ { c } }$ . With class-level representations, we can further model the relationship between event categories within and across modalities by self-attention and crossmodality co-attention mechanism: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/72881a6d61d9ee642027dbd9fe83a19d454f3be0d7ee0d0382843a0d8348f6b1.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\mathbf { a } } _ { t , c } = \\mathbf { a } _ { t , c } + \\phi _ { s e l f } ( \\mathbf { a } _ { t , c } , \\mathbf { A } _ { t } , \\mathbf { A } _ { t } ) + \\phi _ { c r o s s } ( \\mathbf { a } _ { t , c } , \\mathbf { V } _ { t } , \\mathbf { V } _ { t } ) , } \\\\ { \\hat { \\mathbf { v } } _ { t , c } = \\mathbf { v } _ { t , c } + \\phi _ { s e l f } ( \\mathbf { v } _ { t , c } , \\mathbf { V } _ { t } , \\mathbf { V } _ { t } ) + \\phi _ { c r o s s } ( \\mathbf { v } _ { t , c } , \\mathbf { A } _ { t } , \\mathbf { A } _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { A } _ { t } = \\{ \\mathbf { a } _ { t , 1 } , \\dotsc , \\mathbf { a } _ { t , C } \\}$ and $\\mathbf { V } _ { t } = \\{ \\mathbf { v } _ { t , 1 } , \\dots , \\mathbf { v } _ { t , C } \\}$ are sets of audio and visual class features at time $t$ . $\\hat { \\mathbf { a } } _ { t , c }$ and $\\hat { \\mathbf { v } } _ { t , c }$ are now co-occcurence features that consider the relationships between categories within and across modalities. We can then predict the probability for each event at time $t$ by MLPs and aggregate every segment-wise predictions into video-level ones i.e., ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c29410692dc2863506c070ce822f56ae4879f19b55b5ec100c21bc53504c3d84.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\hat { \\mathbf { p } } _ { t } ^ { a } = \\sigma ( \\mathrm { M L P } _ { a } ( \\{ \\hat { \\mathbf { a } } _ { t , 1 } , \\dots , \\hat { \\mathbf { a } } _ { t , C } \\} ) ) , \\quad \\hat { \\mathbf { p } } _ { t } ^ { v } = \\sigma ( \\mathrm { M L P } _ { v } ( \\{ \\hat { \\mathbf { v } } _ { t , 1 } , \\dots , \\hat { \\mathbf { v } } _ { t , C } \\} ) ) , } \\\\ & { \\bar { \\mathbf { p } } ^ { a } , \\bar { \\mathbf { p } } ^ { v } , \\bar { \\mathbf { p } } ^ { a v } = \\mathrm { M M I L } ( \\{ \\hat { \\mathbf { p } } _ { 1 } ^ { a } , \\dots , \\hat { \\mathbf { p } } _ { T } ^ { a } \\} , \\{ \\hat { \\mathbf { p } } _ { 1 } ^ { v } , \\dots , \\hat { \\mathbf { p } } _ { T } ^ { v } \\} ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            185,
+            740,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma$ is the sigmoid function, and $\\mathrm { M M L } ( \\cdot )$ is the multi-modal multiple instance learning pooling described in Eq. 3 taking all segment-wise predictions as inputs. The video-level prediction can be optimized by the binary cross-entropy loss function with a video-level weak label $\\bar { \\mathbf { y } }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Shared Cross-Modality Semantics across Videos ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The information across different videos provides rich supervisory signals that benefit the training of weakly-supervised audio-visual video parsing. By observing videos in a training batch, we can discover both the common and diverse event semantics. With video-level labels, we can initially associate related and irrelevant videos. In order to obtain a discriminative categorical representation, we would like to encourage audio and visual representations from related events to be similar and differentiate those from irrelevant videos. However, targeting at segment-wise representations with specific events is difficult due to the lack of temporal annotations. Therefore, we seek event-related frames through the weights from MMIL pooling in Eq. 3: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b75402bbc144e55bc1f513cac29fe6ded89ed203a5d73acdcd8a4f394e79fba.jpg",
+        "text": "$$\n\\tilde { \\mathbf { f } } ^ { a } = \\sum _ { t = 1 } ^ { T } \\Big [ \\frac { \\exp \\bigl ( g \\bigl ( \\bar { \\mathbf { y } } \\odot \\mathbf { w } _ { t } ^ { a } \\bigr ) \\bigr ) } { \\sum _ { t ^ { \\prime } = 1 } ^ { T } \\exp \\bigl ( g \\bigl ( \\bar { \\mathbf { y } } \\odot \\mathbf { w } _ { t ^ { \\prime } } ^ { a } \\bigr ) \\bigr ) } \\hat { \\mathbf { f } } _ { t } ^ { a } \\Big ] , \\quad \\tilde { \\mathbf { f } } ^ { v } = \\sum _ { t = 1 } ^ { T } \\Big [ \\frac { \\exp \\bigl ( g \\bigl ( \\bar { \\mathbf { y } } \\odot \\mathbf { w } _ { t } ^ { v } \\bigr ) \\bigr ) } { \\sum _ { t ^ { \\prime } = 1 } ^ { T } \\exp \\bigl ( g \\bigl ( \\bar { \\mathbf { y } } \\odot \\mathbf { w } _ { t ^ { \\prime } } ^ { v } \\bigr ) \\bigr ) } \\hat { \\mathbf { f } } _ { t } ^ { v } \\Big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            223,
+            424,
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+            468
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ and $g ( . )$ are element-wise dot product and summation function over all elements respectively. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With video-level labels and features ${ \\tilde { \\mathbf { f } } } ^ { a }$ and $\\tilde { \\mathbf { f } } ^ { v }$ ), we adopt contrastive learning [47, 48, 49] to encourage features across modalities with the same event category (at least one) to be close and those with different events to be far away from each other. We leverage all $n$ videos in a batch to explore diverse semantics, where the sets of audio and visual features are denoted as $\\{ \\widetilde { \\bf f } _ { ( 0 ) } ^ { a } , . . . , \\widetilde { \\bf f } _ { ( n ) } ^ { a } \\}$ and $\\{ \\tilde { \\mathbf { f } } _ { ( 0 ) } ^ { v } , . . . , \\tilde { \\mathbf { f } } _ { ( n ) } ^ { v } \\}$ respectively with video-level labels $\\left\\{ \\bar { \\mathbf { y } } _ { ( 0 ) } , . . . , \\bar { \\mathbf { y } } _ { ( n ) } \\right\\}$ . The relationship across videos can be optimized by the proposed training objective as follows: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/575eda778d80d968a5746712c2c81b94feb4fdca31b45c4f7807df8f1f5230ba.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { c o n t r a s t } } = - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\Big [ \\log \\frac { \\sum _ { j = 1 } ^ { n } f ( \\bar { \\bf y } _ { i } \\cdot \\bar { \\bf y } _ { j } ) \\exp ( \\tilde { \\bf f } _ { ( i ) } ^ { a } \\cdot \\tilde { \\bf f } _ { ( j ) } ^ { v } / \\tau ) } { \\sum _ { j = 1 } ^ { n } \\exp ( \\tilde { \\bf f } _ { ( i ) } ^ { a } \\cdot \\tilde { \\bf f } _ { ( j ) } ^ { v } / \\tau ) } \\Big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            592,
+            704,
+            636
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $f ( \\cdot )$ is a clipping function that clips values over 1, and $\\tau$ denotes a hyper-parameter controlling the temperature. Thus, the proposed method can be optimized by joint the binary cross-entropy loss mentioned in Section 3.1 and the contrastive learning loss in Eq. 8. Our training strategy can exploit cross-modality information across videos and event categories to understand common semantics while ignoring irrelevant ones. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Experimental Results ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Datasets. We use the Look, Listen and Parse (LLP) Dataset [4] for all experiments. The LLP dataset consists of 11, 849 10-seconds video clips annotated with 25 event categories. It covers various real-life scenes such as speech, music performances, car, cheering, dog, etc. Particularly, there are 7202 video clips labeled with more than one event category. We use the 10000 video clips with only video-level event annotations for model training. The detailed annotations (e.g., individual audio and visual events per second) are available for the remaining 1849 validation and test videos. For all experiments, we use the official data splits from the LLP dataset. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Evaluation Metrics. Following previous work [4, 6], we adopt F-scores as the evaluation metrics. Note that all types of events (audio, visual, and audio-visual) are measured under both segmentlevel and event-level metrics. The segment-level metrics can evaluate snippet-wise prediction results. As for the event-level metrics, the clips are extracted by concatenating positive consecutive segments in the same events. Then, we compute the event-level F-scores with $\\mathrm { m I o U } = 0 . 5$ as the threshold. Furthermore, the overall Type $\\ @ \\mathbf { A V }$ performance on audio-visual scene is also considered by computing the averaged audio, visual, and audio-visual event evaluation results. Instead of directly averaging results from different event types, Event@AV considers all audio and visual event categories for each sample. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/084b18971d504f27393e74bd41c53c9cc07fa98ff1b6441b9862e1db77699897.jpg",
+        "table_caption": [
+            "Table 1: Quantitative results of weakly-supervised audio-visual video parsing. We evaluate all methods on the LLP dataset [4] with F-scores in five different event types and two kinds of segments. The first row indicates five different event types (audio, visual, audio-visual, Type@AV, and Event@AV). In the second row, two kinds of segments are shown: Seg. and Event are segmentlevel and event-level; and $^ *$ indicates only label refinement is utilized for fair comparisons. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Audio</td><td colspan=\"2\">Visual</td><td colspan=\"2\">Audio-visual</td><td colspan=\"2\">Type@AV</td><td colspan=\"2\">Event@AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>AVE [5]</td><td>47.2</td><td>40.4</td><td>37.1</td><td>34.7</td><td>35.4</td><td>31.6</td><td>39.9</td><td>35.5</td><td>41.6</td><td>36.5</td></tr><tr><td>AVSDN [2]</td><td>47.8</td><td>34.1</td><td>52.0</td><td>46.3</td><td>37.1</td><td>26.5</td><td>45.7</td><td>35.6</td><td>50.8</td><td>37.7</td></tr><tr><td>AVSDN + Ours</td><td>48.3</td><td>41.2</td><td>52.4</td><td>48.5</td><td>46.9</td><td>40.0</td><td>49.2</td><td>43.2</td><td>53.2</td><td>40.1</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + Ours</td><td>59.2</td><td>51.3</td><td>59.9</td><td>55.5</td><td>53.4</td><td>46.2</td><td>57.5</td><td>51.0</td><td>58.1</td><td>49.7</td></tr><tr><td>MA [6]</td><td>60.3</td><td>53.6</td><td>60.0</td><td>56.4</td><td>55.1</td><td>49.0</td><td>58.9</td><td>53.0</td><td>57.9</td><td>50.6</td></tr><tr><td>MA*</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA*+Ours</td><td>60.8</td><td>53.8</td><td>63.5</td><td>58.9</td><td>57.0</td><td>49.5</td><td>60.5</td><td>54.0</td><td>59.5</td><td>52.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Implementation Details. We implement the proposed method using PyTorch [50], and conduct the training and evaluation processes on a single NVIDIA GTX 1080 Ti GPU with 11 GB memory. Following [4, 6], we use the same visual and audio encoders for fair comparisons. We adopt both ResNet-152 [51] pre-trained on ImageNet [52] and 3D ResNet [53] pre-trained on Kinetics-400 [54] as visual feature extractors. Visual frames are sampled at 8 fps and their 2D and 3D visual features are extracted. The 2D and 3D visual features are concatenated and then processed by an MLP as the segment-wise representations. As for audio data, we utilize VGGish [55] pre-trained on AudioSet [56] to extract 128-dimensional audio features. The code and models are publicly available. ",
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+        "type": "text",
+        "text": "Evaluated methods. We compare the proposed method based on several baselines to the following weakly-unsupervised approaches to the audio-visual video parsing task: ",
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+        "text": "• AVE [5] consists of an audio-guided co-attention mechanism to adaptively learn the sounding regions. We note that AVE [5] deals with the audio-visual event localization task. Thus, we follow [4] and add additional audio and visual parsing branches for the weakly-supervised audio-visual video parsing task as a baseline.   \n• AVSDN [2] is a sequence-to-sequence-based model to integrate global audio and visual features to local ones. Since AVSDN [2] also deals with the audio-visual event localization task, we make the same modifications to AVSDN as those to AVE.   \n• HAN [4] is a multi-modal multiple instance learning-based method with a hybrid attention network.   \n• MA [6] reports the state-of-the-art performance on the weakly-supervised audio-visual video parsing task. It is a method based on HAN with the label refinement and the audio-visual contrastive learning differentiating temporal segments. ",
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+    {
+        "type": "text",
+        "text": "4.1 Quantitative Evaluation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 1 shows the quantitative comparisons on the LLP dataset [4]. The proposed method performs favorably against the competing approaches on the weakly-supervised audio-visual video parsing task. Since our method can be easily extended to existing methods, we extend the proposed on three baselines. The third, fifth, and last rows in Table 1 indicate that the proposed method generally benefits three baselines on several metrics of the audio-visual video parsing task by a large margin. We note that $\\mathbf { M A } ^ { * }$ [6] only utilizes label refinement to refine labels for each modality, and temporal difference audio-visual contrastive learning [6] is not implemented. ",
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+    {
+        "type": "table",
+        "img_path": "images/3c22ac25c5061d220d82ddeb6f86b8e8cce2d27d122739cee7c6f25d9429a171.jpg",
+        "table_caption": [
+            "Table 2: Ablation study. We investigate the effect of using different design components in the proposed method. We show how proposed cross-modality co-occurrence (CM-Co) in Section 3.3 and shared cross-modality semantics across videos (CM-S) module in Section 3.2 improve the baselines. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Audio</td><td colspan=\"2\">Visual</td><td colspan=\"2\">Audio-visual</td><td colspan=\"2\">Type@AV</td><td colspan=\"2\">Event@AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + CM-S</td><td>58.1</td><td>49.6</td><td>58.3</td><td>53.6</td><td>53.2</td><td>46.3</td><td>56.5</td><td>49.8</td><td>55.9</td><td>47.5</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>51.4</td><td>57.4</td><td>52.4</td><td>51.9</td><td>44.2</td><td>56.3</td><td>49.3</td><td>57.4</td><td>48.5</td></tr><tr><td>HAN + Ours</td><td>59.2</td><td>51.3</td><td>59.9</td><td>55.5</td><td>53.4</td><td>46.2</td><td>57.5</td><td>51.0</td><td>58.1</td><td>49.7</td></tr><tr><td>MA [28]</td><td>60.3</td><td>53.6</td><td>60.0</td><td>56.4</td><td>55.1</td><td>49.0</td><td>58.9</td><td>53.0</td><td>57.9</td><td>50.6</td></tr><tr><td>MA + CM-Co</td><td>61.1</td><td>53.3</td><td>61.7</td><td>57.3</td><td>56.3</td><td>49.0</td><td>59.7</td><td>53.0</td><td>58.9</td><td>51.2</td></tr><tr><td>MA*</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA* + CM-S</td><td>60.4</td><td>53.5</td><td>60.7</td><td>56.5</td><td>55.8</td><td>47.5</td><td>58.9</td><td>52.5</td><td>58.6</td><td>51.0</td></tr><tr><td>MA* + CM-Co</td><td>60.5</td><td>53.6</td><td>61.3</td><td>56.5</td><td>54.9</td><td>46.7</td><td>58.9</td><td>52.3</td><td>59.1</td><td>51.4</td></tr><tr><td>MA* +Ours</td><td>60.8</td><td>53.8</td><td>63.5</td><td>58.9</td><td>57.0</td><td>49.5</td><td>60.5</td><td>54.0</td><td>59.5</td><td>52.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "We notice that our method significantly improves baselines in the metrics of visual, audio-visual, Type $@ \\mathrm { A V } ,$ and Event $@$ AV. By observing the class distribution of training sets, we find that $3 1 \\%$ , $7 \\%$ , and $9 \\%$ training videos contain speech, singing, and violin events. These events are more likely to present in the audio modality. Therefore, the video-level labels would limit the performance regarding visual events. The proposed method can leverage additional cross-video and cross-modality supervisory signals to explore common semantics, which can improve results in vision-related metrics. ",
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+        "type": "text",
+        "text": "4.2 Ablation Study ",
+        "text_level": 1,
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+        "text": "Cross-Modality Co-Occurrence and Semantics across Video. We conduct the ablation study to analyze the individual impact of each developed component in the proposed method. The results are presented in Table 2. CM-Co represents the usage of the cross-modality co-occurrence module described in Section 3.2, which leverages the relationship between categories within and cross modalities. CM-S indicates the shared cross-modality semantics across videos module described in Section 3.3, which considers all audio and visual information across videos in a batch. ",
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+        "type": "text",
+        "text": "In Table 2, we note that both CM-S and CM-Co can improve baselines in several metrics. By exploring common semantics among training videos (CM-S), we improve the performance on visual and audio-visual evaluation by a large margin. Such a strategy can exploit additional information from videos to address the potential drawback of video-level labels described in Section 4.1. Furthermore, the proposed cross-modality co-occurrence module (CM-Co) also presents favorable results. We note that the significant improvement in Event@AV evaluation with the usage of CM-Co can verify the efficacy of considering the relationship between categories within and across modalities. Since Event $@$ AV considers all audio and visual events for the F-score (e.g., truth positive from both audio and visual events), the improvement of Event@AV indicates our cross-modality co-occurrence can perform well on video parsing when events present in an audio or a visual modality. ",
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+    {
+        "type": "text",
+        "text": "In the second group of the evaluated methods in Table 2, we verify if the proposed CM-S works better than the contrastive learning method in MA. We perform our CM-S on the MA model. The CM-S exploits information across different videos to address the issue that audio and visual tracks may not be synchronized. Instead, the contrastive learning method in MA is developed based on the assumption of synchronization to associate the audio-visual representation in a single video. Since our CM-S learns diverse and common semantics, it is effective and complementary to the contrastive learning approach in MA performing on a single video. We note that our CM-S generally improves the performance over all segment-level metrics, which supports our claim. ",
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+        "type": "text",
+        "text": "Self-attention and Cross-Modality Co-attention in Co-Occurrence. Since our cross-modality co-occurrence module exploits self-attention among class-level features in the same modality and cross-modality co-attention on cross-modality class-level representations to model the relationship between categories in the same and different modalities. Taking class-level audio features in Eq. 5 as an example, the class-level self-attention and cross-modality co-attention are $\\mathrm { a t t n } ( \\mathbf { a } _ { t , c } , \\mathbf { A } _ { t } , \\mathbf { A } _ { t } )$ and $\\operatorname { a t t n } ( \\mathbf { a } _ { t , c } , \\mathbf { V } _ { t } , \\mathbf { V } _ { t } ) .$ , respectively. ",
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+    {
+        "type": "table",
+        "img_path": "images/6d2690df8819bd3c6c2c8effa4b228db879b7c72a522eedf405aa11e230fcf3e.jpg",
+        "table_caption": [
+            "Table 3: Ablation study. We investigate the effect of different developed mechanisms in the proposed cross-modality co-occurrence (CM-Co) module in Section 3.2. In Eq. 5, class-level features are processed by self-attention and cross-modality co-attention mechanisms. A Only and $\\mathbf { V }$ Only indicate only self-attention performs for individual audio and visual events respectively. AV denotes performing self-attention for audio and visual events. CM-Co is the proposed method that considers relationship between categories within and cross modalities by both self-attention and cross-modality co-attention mechanisms. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Audio</td><td colspan=\"2\">Visual</td><td colspan=\"2\">Audio-visual</td><td colspan=\"2\">Type@AV</td><td colspan=\"2\">Event@ AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + A Only</td><td>60.5</td><td>52.3</td><td>49.8</td><td>43.9</td><td>45.6</td><td>38.3</td><td>52.0</td><td>44.8</td><td>55.7</td><td>45.9</td></tr><tr><td>HAN + V Only</td><td>56.1</td><td>44.5</td><td>56.8</td><td>53.2</td><td>49.7</td><td>40.7</td><td>54.2</td><td>46.1</td><td>54.1</td><td>44.6</td></tr><tr><td>HAN + AV</td><td>59.5</td><td>50.3</td><td>55.1</td><td>50.5</td><td>48.6</td><td>40.3</td><td>54.4</td><td>47.0</td><td>56.0</td><td>47.4</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>51.4</td><td>57.4</td><td>52.4</td><td>51.9</td><td>44.2</td><td>56.3</td><td>49.3</td><td>57.4</td><td>48.5</td></tr><tr><td>MA*[6]</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA*+ A Only</td><td>60.7</td><td>52.7</td><td>53.9</td><td>47.9</td><td>50.1</td><td>42.2</td><td>54.9</td><td>47.6</td><td>57.0</td><td>47.1</td></tr><tr><td>MA* + V Only</td><td>46.8</td><td>34.4</td><td>60.8</td><td>57.0</td><td>42.8</td><td>31.1</td><td>50.1</td><td>40.9</td><td>52.6</td><td>40.4</td></tr><tr><td>MA*+ AV</td><td>58.3</td><td>50.4</td><td>59.4</td><td>55.2</td><td>53.9</td><td>46.9</td><td>57.2</td><td>50.8</td><td>56.7</td><td>48.5</td></tr><tr><td>MA*+ CM-Co</td><td>60.5</td><td>53.6</td><td>61.3</td><td>56.5</td><td>54.9</td><td>46.7</td><td>58.9</td><td>52.3</td><td>59.1</td><td>51.4</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/89f0e2785faa97f098bed9fc6e71d1f81760b2cf7c930cd9333bc2b3bd34ab9b.jpg",
+        "table_caption": [
+            "Table 4: Ablation study. We evaluate the proposed method in accuracy, efficiency, and model sizes. We show the numbers of parameters and FLOPs for the proposed cross-modality co-occurrence (CM-Co) and HAN [4] with a few layers. ",
+            "Note that the results are all in the segment level. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Audio</td><td>Visual</td><td>Audio-visual</td><td>Type@AV</td><td>Event@AV</td><td>GFLOPs</td><td>Params</td></tr><tr><td>HAN 1 Layer</td><td>60.1</td><td>52.9</td><td>48.9</td><td>54.0</td><td>55.4</td><td>6.63</td><td>2.4M</td></tr><tr><td>HAN 2 Layers</td><td>58.2</td><td>55.4</td><td>50.6</td><td>54.7</td><td>54.9</td><td>7.28</td><td>2.9M</td></tr><tr><td>HAN 3 Layers</td><td>58.1</td><td>55.2</td><td>50.3</td><td>54.5</td><td>54.6</td><td>7.97</td><td>3.5M</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>57.4</td><td>51.9</td><td>56.3</td><td>57.4</td><td>6.99</td><td>2.8M</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 3 presents the results in various modifications of the cross-modality co-occurrence module. We note that the design of co-occurrence in the same and cross modalities can generally improve the results in several metrics. We also evaluate the co-occurrence module in a single modality. The results are shown in the second, third, seventh, and eighth rows in Table 3, where A Only and $\\mathbf { V }$ Only indicate the co-occurrence module only leverages the relationship between categories in audio or visual data respectively. As the results shown in the second and seventh rows, training with co-occurrence in audio events only (i.e., A Only) can slightly improve the performance on audio events. Similarly, considering visual event only (i.e., V Only) can benefit the results regarding visual events. Furthermore, the co-occurrence for both audio and visual categories (AV) in the fourth and ninth rows can contribute to the results in general metrics such as Type $@$ AV and Event $@ \\mathrm { A V } .$ . We then further consider the correlation between events across modalities. That is the cross-modality co-occurrence module (CM-Co) in the fifth and tenth rows. The results can confirm the efficacy of the proposed cross-modality co-occurrence module in all metrics except segment-level audio events caused by similar reasons discussed in Section 4.1. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Model Capacity. Since our cross-modality co-occurrence module leverages class-level representations, it would increase the capability of models on capturing information. For fair comparisons, we add extra parameters to HAN [4] to analyze whether more parameters can contribute to performance gain. Specifically, we increase the number of layers in its transformer-based feature aggregation to 2 and 3, respectively. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Table 4, we report the results in accuracy, computational costs, and model sizes. The first three rows show the performance of HAN with different numbers of layers. We note that HAN with one extra layer has more parameters than the proposed co-occurrence module. However, the results of HAN with extra layers indicate that using more parameters/layers for HAN does not improve the performance. The proposed cross-modality co-occurrence module enhances HAN more effectively. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/3a3e5bddfd4d0b8b3df955da6588c994004175b863c809999ccff152b083248e.jpg",
+        "image_caption": [
+            "Figure 2: Qualitative comparisons. We compare the proposed method with the state-of-the-art weakly-supervised audio-visual video parsing method on the LLP dataset [4]. The frame-wise annotations are shown in gray and purple bars. The gray bar denotes visual events, and the purple bar represents audio events. GT_V and GT_A are the ground-truth visual and audio events respectively. Our results are shown in the green block, and the results by the competing method, MA [6], are present in the blue block. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/21cead53b19ede061abdc05068f5d07b4594cffe449abe292ef47d1a62306ab2.jpg",
+        "image_caption": [
+            "Figure 3: Audio feature distribution by using t-SNE. The upper figure shows the distribution by our method. The lower figure presents that by $\\mathbf { M A } ^ { * }$ . The legend lists all event combinations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.3 Qualitative Evaluation ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Qualitative Results. We present the qualitative results of the evaluated methods in Figure 2. GT_V and GT_A show the ground-truth annotations for visual and audio events, respectively. Pred_V and Pred_A present the predictions made by our method and the state-of-the-art competing method, MA [6], respectively. Our results are shown in the green block, while the results of MA are present in the blue block. In general, our method presents more accurate predictions in both audio and visual events than MA. We note that the whole violin is shown after 7 seconds. That would hamper models for understanding visual events e.g., MA predicts wrong results on violin visual events before 6 seconds. Since our method leverages the relationship between categories, it can still predict correct temporal boundaries for guitar events by jointly considering cello events in the videos. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/ed423064930570c6abeaee1db5cadd585aab212a9182498adeec79bb58aa61f7.jpg",
+        "image_caption": [
+            "Figure 4: Visual feature distribution by using t-SNE. The upper figure shows the distribution by our method. The lower figure presents that by MA∗. The legend lists all event combinations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
+            90,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Feature Distribution Visualized by t-SNE. We apply t-SNE to the aggregated audio and visual features from each segment described in Eq. 2. The visualization results are present in Figure 3 and Figure 4, respectively. The legends list all the combinations of multiple labels. For example, in Figure 3, audio events of singing are present as blue spots, and the mixed sounds of singing and violin are shown as purple spots. We note that the related events including multiple events are shown in similar colors. In Figure 4, the proposed method achieves better performance in the sense that similar color spots are closer than the spots in $\\mathbf { M A } ^ { * }$ . ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusions ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a novel audio-visual video parsing framework in a weakly-supervised manner that can be applied to existing methods. We propose two modules to exploit the relationship across videos, modalities, and event categories, and explore additional supervisory signals that can benefit audio-visual video parsing. The shared cross-modality semantics module leverages common and diverse event semantics across videos to learn robust cross-modality representations that facilitate models to identify audio, visual, and audio-visual events. Furthermore, the cross-modality co-occurrence module aims to learn the relationship between event categories. It helps localize segments of target events and can exclude irrelevant ones by performing self-attention and crossmodality co-attention on class-wise features, Extensive experimental results show that our approach substantially improves several baselines and performs favorably against the state-of-the-art methods. ",
+        "bbox": [
+            174,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments. This work was supported in part by the Ministry of Science and Technology under grants 109- 2221-E-009-113-MY3, 110-2628-E-A49-008, and 110-2634-F007-015. It was also funded in part by Qualcomm through a Taiwan University Research Collaboration Project, the Higher Education Sprout Project of the National Yang Ming Chiao Tung University, and Ministry of Education. ",
+        "bbox": [
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