The performance of trained neural networks is robust to harsh levels of pruning. Coupled with the ever-growing size of deep learning models, this observation has motivated extensive research on learning sparse models. In this work, we focus on the task of controlling the level of sparsity when performing sparse learning. Existing methods based on sparsity-inducing penalties involve expensive trial-anderror tuning of the penalty factor, thus lacking direct control of the resulting model sparsity. In response, we adopt a constrained formulation: using the gate mechanism proposed by Louizos et al. [31], we formulate a constrained optimization problem where sparsification is guided by the training objective and the desired sparsity target in an end-to-end fashion. Experiments on CIFAR-{10, 100}, TinyImageNet, and ImageNet using WideResNet and ResNet{18, 50} models validate the effectiveness of our proposal and demonstrate that we can reliably achieve pre-determined sparsity targets without compromising on predictive performance.

It is well-known that given a smooth, bounded-from-below, and possibly nonconvex function, standard gradient-based methods can find -stationary points (with gradient norm less than) in O(1/ 2) iterations. However, many important nonconvex optimization problems, such as those associated with training modern neural networks, are inherently not smooth, making these results inapplicable. In this paper, we study nonsmooth nonconvex optimization from an oracle complexity viewpoint, where the algorithm is assumed to be given access only to local information about the function at various points. We provide two main results (under mild assumptions): First, we consider the problem of getting near -stationary points. This is perhaps the most natural relaxation of finding -stationary points, which is impossible in the nonsmooth nonconvex case. We prove that this relaxed goal cannot be achieved efficiently, for any distance and smaller than some constants. Our second result deals with the possibility of tackling nonsmooth nonconvex optimization by reduction to smooth optimization: Namely, applying smooth optimization methods on a smooth approximation of the objective function. For this approach, we prove an inherent trade-off between oracle complexity and smoothness: On the one hand, smoothing a nonsmooth nonconvex function can be done very efficiently (e.g., by randomized smoothing), but with dimension-dependent factors in the smoothness parameter, which can strongly affect iteration complexity when plugging into standard smooth optimization methods. On the other hand, these dimension factors can be eliminated with suitable smoothing methods, but only by making the oracle complexity of the smoothing process exponentially large.

In this appendix, we discuss a technique to optimize over the counterfactuals found by counterfactual explanation methods, such as [6]. We restate lemma 3.1 and provide a proof. Lemma 3.1 Assuming the counterfactual algorithm A (x) follows the form of the objective in equation 1, @@xcf G(x,A (x)) = 0, and m is the number of parameters in the model, we can write the derivative of counterfactual algorithm A with respect to model parameters as the Jacobian, @ @ A (x)= @2G(x,A (x)) @x2cf 1 G(x,xcf) (7) This problem is identical to a well-studied class of bi-level optimization problems in deep learning. In these problems, we must compute the derivative of a function with respect to some parameter (here) that includes an inner argmin, which itself depends on the parameter. We follow [44] to complete the proof.

Stochastic and adversarial data are two widely studied settings in online learning. But many optimization tasks are neither i.i.d.

We consider unconstrained minimization of smooth convex functions. We propose a novel variational perspective using forced Euler-Lagrange equation that allows for studying high-resolution ODEs. Through this, we obtain a faster convergence rate for gradient norm minimization using Nesterov's accelerated gradient method. Additionally, we show that Nesterov's method can be interpreted as a ratematching discretization of an appropriately chosen high-resolution ODE. Finally, using the results from the new variational perspective, we propose a stochastic method for noisy gradients.