arXiv subjects

Samy Jelassi

Publications and source records attributed to Samy Jelassi.

30 records · Page 2Linked to original sources

arXiv2022

Dissecting adaptive methods in GANs

Adaptive methods are a crucial component widely used for training generative adversarial networks (GANs). While there has been some work to pinpoint the "marginal value of adaptive methods" in standard tasks, it remains unclear why they are still critical for GAN training. In this paper, we formally study how adaptive methods help train GANs; inspired by the grafting method proposed in arXiv:2002.11803 [cs.LG], we separate the magnitude and direction components of the Adam updates, and graft them to the direction and magnitude of SGDA updates respectively. By considering an update rule with the magnitude of the Adam update and the normalized direction of SGD, we empirically show that the adaptive magnitude of Adam is key for GAN training. This motivates us to have a closer look at the class of normalized stochastic gradient descent ascent (nSGDA) methods in the context of GAN training. We propose a synthetic theoretical framework to compare the performance of nSGDA and SGDA for GAN training with neural networks. We prove that in that setting, GANs trained with nSGDA recover all the modes of the true distribution, whereas the same networks trained with SGDA (and any learning rate configuration) suffer from mode collapse. The critical insight in our analysis is that normalizing the gradients forces the discriminator and generator to be updated at the same pace. We also experimentally show that for several datasets, Adam's performance can be recovered with nSGDA methods.

cs.LG↗

arXiv2022

Towards understanding how momentum improves generalization in deep learning

Stochastic gradient descent (SGD) with momentum is widely used for training modern deep learning architectures. While it is well-understood that using momentum can lead to faster convergence rate in various settings, it has also been observed that momentum yields higher generalization. Prior work argue that momentum stabilizes the SGD noise during training and this leads to higher generalization. In this paper, we adopt another perspective and first empirically show that gradient descent with momentum (GD+M) significantly improves generalization compared to gradient descent (GD) in some deep learning problems. From this observation, we formally study how momentum improves generalization. We devise a binary classification setting where a one-hidden layer (over-parameterized) convolutional neural network trained with GD+M provably generalizes better than the same network trained with GD, when both algorithms are similarly initialized. The key insight in our analysis is that momentum is beneficial in datasets where the examples share some feature but differ in their margin. Contrary to GD that memorizes the small margin data, GD+M still learns the feature in these data thanks to its historical gradients. Lastly, we empirically validate our theoretical findings.

cs.LG↗

arXiv2021

A Permutation-Equivariant Neural Network Architecture For Auction Design

Designing an incentive compatible auction that maximizes expected revenue is a central problem in Auction Design. Theoretical approaches to the problem have hit some limits in the past decades and analytical solutions are known for only a few simple settings. Computational approaches to the problem through the use of LPs have their own set of limitations. Building on the success of deep learning, a new approach was recently proposed by Duetting et al. (2019) in which the auction is modeled by a feed-forward neural network and the design problem is framed as a learning problem. The neural architectures used in that work are general purpose and do not take advantage of any of the symmetries the problem could present, such as permutation equivariance. In this work, we consider auction design problems that have permutation-equivariant symmetry and construct a neural architecture that is capable of perfectly recovering the permutation-equivariant optimal mechanism, which we show is not possible with the previous architecture. We demonstrate that permutation-equivariant architectures are not only capable of recovering previous results, they also have better generalization properties.

cs.GT↗

arXiv2021

Auction learning as a two-player game

Designing an incentive compatible auction that maximizes expected revenue is a central problem in Auction Design. While theoretical approaches to the problem have hit some limits, a recent research direction initiated by Duetting et al. (2019) consists in building neural network architectures to find optimal auctions. We propose two conceptual deviations from their approach which result in enhanced performance. First, we use recent results in theoretical auction design (Rubinstein and Weinberg, 2018) to introduce a time-independent Lagrangian. This not only circumvents the need for an expensive hyper-parameter search (as in prior work), but also provides a principled metric to compare the performance of two auctions (absent from prior work). Second, the optimization procedure in previous work uses an inner maximization loop to compute optimal misreports. We amortize this process through the introduction of an additional neural network. We demonstrate the effectiveness of our approach by learning competitive or strictly improved auctions compared to prior work. Both results together further imply a novel formulation of Auction Design as a two-player game with stationary utility functions.

cs.GT↗

arXiv2021

Depth separation beyond radial functions

High-dimensional depth separation results for neural networks show that certain functions can be efficiently approximated by two-hidden-layer networks but not by one-hidden-layer ones in high-dimensions $d$. Existing results of this type mainly focus on functions with an underlying radial or one-dimensional structure, which are usually not encountered in practice. The first contribution of this paper is to extend such results to a more general class of functions, namely functions with piece-wise oscillatory structure, by building on the proof strategy of (Eldan and Shamir, 2016). We complement these results by showing that, if the domain radius and the rate of oscillation of the objective function are constant, then approximation by one-hidden-layer networks holds at a $\mathrm{poly}(d)$ rate for any fixed error threshold. A common theme in the proofs of depth-separation results is the fact that one-hidden-layer networks fail to approximate high-energy functions whose Fourier representation is spread in the domain. On the other hand, existing approximation results of a function by one-hidden-layer neural networks rely on the function having a sparse Fourier representation. The choice of the domain also represents a source of gaps between upper and lower approximation bounds. Focusing on a fixed approximation domain, namely the sphere $\mathbb{S}^{d-1}$ in dimension $d$, we provide a characterisation of both functions which are efficiently approximable by one-hidden-layer networks and of functions which are provably not, in terms of their Fourier expansion.

cs.LG↗

arXiv2021

Adaptivity without Compromise: A Momentumized, Adaptive, Dual Averaged Gradient Method for Stochastic Optimization

We introduce MADGRAD, a novel optimization method in the family of AdaGrad adaptive gradient methods. MADGRAD shows excellent performance on deep learning optimization problems from multiple fields, including classification and image-to-image tasks in vision, and recurrent and bidirectionally-masked models in natural language processing. For each of these tasks, MADGRAD matches or outperforms both SGD and ADAM in test set performance, even on problems for which adaptive methods normally perform poorly.

cs.LG↗

arXiv2021

Towards closing the gap between the theory and practice of SVRG

Among the very first variance reduced stochastic methods for solving the empirical risk minimization problem was the SVRG method (Johnson & Zhang 2013). SVRG is an inner-outer loop based method, where in the outer loop a reference full gradient is evaluated, after which $m \in \mathbb{N}$ steps of an inner loop are executed where the reference gradient is used to build a variance reduced estimate of the current gradient. The simplicity of the SVRG method and its analysis have led to multiple extensions and variants for even non-convex optimization. We provide a more general analysis of SVRG than had been previously done by using arbitrary sampling, which allows us to analyse virtually all forms of mini-batching through a single theorem. Furthermore, our analysis is focused on more practical variants of SVRG including a new variant of the loopless SVRG (Hofman et al 2015, Kovalev et al 2019, Kulunchakov and Mairal 2019) and a variant of k-SVRG (Raj and Stich 2018) where $m=n$ and where $n$ is the number of data points. Since our setup and analysis reflect what is done in practice, we are able to set the parameters such as the mini-batch size and step size using our theory in such a way that produces a more efficient algorithm in practice, as we show in extensive numerical experiments.

math.OC↗

arXiv2021

A mean-field analysis of two-player zero-sum games

Finding Nash equilibria in two-player zero-sum continuous games is a central problem in machine learning, e.g. for training both GANs and robust models. The existence of pure Nash equilibria requires strong conditions which are not typically met in practice. Mixed Nash equilibria exist in greater generality and may be found using mirror descent. Yet this approach does not scale to high dimensions. To address this limitation, we parametrize mixed strategies as mixtures of particles, whose positions and weights are updated using gradient descent-ascent. We study this dynamics as an interacting gradient flow over measure spaces endowed with the Wasserstein-Fisher-Rao metric. We establish global convergence to an approximate equilibrium for the related Langevin gradient-ascent dynamic. We prove a law of large numbers that relates particle dynamics to mean-field dynamics. Our method identifies mixed equilibria in high dimensions and is demonstrably effective for training mixtures of GANs.

cs.LG↗

arXiv2020

Dual Averaging is Surprisingly Effective for Deep Learning Optimization

First-order stochastic optimization methods are currently the most widely used class of methods for training deep neural networks. However, the choice of the optimizer has become an ad-hoc rule that can significantly affect the performance. For instance, SGD with momentum (SGD+M) is typically used in computer vision (CV) and Adam is used for training transformer models for Natural Language Processing (NLP). Using the wrong method can lead to significant performance degradation. Inspired by the dual averaging algorithm, we propose Modernized Dual Averaging (MDA), an optimizer that is able to perform as well as SGD+M in CV and as Adam in NLP. Our method is not adaptive and is significantly simpler than Adam. We show that MDA induces a decaying uncentered $L_2$-regularization compared to vanilla SGD+M and hypothesize that this may explain why it works on NLP problems where SGD+M fails.

cs.LG↗

arXiv2020

Extragradient with player sampling for faster Nash equilibrium finding

Data-driven modeling increasingly requires to find a Nash equilibrium in multi-player games, e.g. when training GANs. In this paper, we analyse a new extra-gradient method for Nash equilibrium finding, that performs gradient extrapolations and updates on a random subset of players at each iteration. This approach provably exhibits a better rate of convergence than full extra-gradient for non-smooth convex games with noisy gradient oracle. We propose an additional variance reduction mechanism to obtain speed-ups in smooth convex games. Our approach makes extrapolation amenable to massive multiplayer settings, and brings empirical speed-ups, in particular when using a heuristic cyclic sampling scheme. Most importantly, it allows to train faster and better GANs and mixtures of GANs.

stat.ML↗

arXiv2019

Global convergence of neuron birth-death dynamics

Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions. In this work, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We implement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean-field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of parameters. We illustrate our algorithms with two models to provide intuition for the mechanism through which convergence is accelerated.

stat.ML↗

arXiv2018

Smoothed analysis of the low-rank approach for smooth semidefinite programs

We consider semidefinite programs (SDPs) of size n with equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size $n$ by $k$ such that $X = YY^*$ is the SDP variable. The advantages of such formulation are twofold: the dimension of the optimization variable is reduced and positive semidefiniteness is naturally enforced. However, the problem in Y is non-convex. In prior work, it has been shown that, when the constraints on the factorized variable regularly define a smooth manifold, provided k is large enough, for almost all cost matrices, all second-order stationary points (SOSPs) are optimal. Importantly, in practice, one can only compute points which approximately satisfy necessary optimality conditions, leading to the question: are such points also approximately optimal? To this end, and under similar assumptions, we use smoothed analysis to show that approximate SOSPs for a randomly perturbed objective function are approximate global optima, with k scaling like the square root of the number of constraints (up to log factors). Moreover, we bound the optimality gap at the approximate solution of the perturbed problem with respect to the original problem. We particularize our results to an SDP relaxation of phase retrieval.

stat.ML↗