Online Non-Convex Learning: Following the Perturbed Leader is Optimal

We study the problem of online learning with non-convex losses, where the learner has access to an offline optimization oracle. We show that the classical Follow the Perturbed Leader (FTPL) algorithm achieves optimal regret rate of $O(T^{-1/2})$ in this setting. This improves upon the previous best-known regret rate of $O(T^{-1/3})$ for FTPL. We further show that an optimistic variant of FTPL achieves better regret bounds when the sequence of losses encountered by the learner is `predictable'.

SGD without Replacement: Sharper Rates for General Smooth Convex Functions

We study stochastic gradient descent {\em without replacement} (\sgdwor) for smooth convex functions. \sgdwor is widely observed to converge faster than true \sgd where each sample is drawn independently {\em with replacement}~\cite{bottou2009curiously} and hence, is more popular in practice. But it's convergence properties are not well understood as sampling without replacement leads to coupling between iterates and gradients. By using method of exchangeable pairs to bound Wasserstein distance, we provide the first non-asymptotic results for \sgdwor when applied to {\em general smooth, strongly-convex} functions. In particular, we show that \sgdwor converges at a rate of $O(1/K^2)$ while \sgd~is known to converge at $O(1/K)$ rate, where $K$ denotes the number of passes over data and is required to be {\em large enough}. Existing results for \sgdwor in this setting require additional {\em Hessian Lipschitz assumption}~\cite{gurbuzbalaban2015random,haochen2018random}. For {\em small} $K$, we show \sgdwor can achieve same convergence rate as \sgd for {\em general smooth strongly-convex} functions. Existing results in this setting require $K=1$ and hold only for generalized linear models \cite{shamir2016without}. In addition, by careful analysis of the coupling, for both large and small $K$, we obtain better dependence on problem dependent parameters like condition number.

Stochastic Gradient Descent Escapes Saddle Points Efficiently

This paper considers the perturbed stochastic gradient descent algorithm and shows that it finds $\epsilon$-second order stationary points ($\left\|\nabla f(x)\right\|\leq \epsilon$ and $\nabla^2 f(x) \succeq -\sqrt{\epsilon} \mathbf{I}$) in $\tilde{O}(d/\epsilon^4)$ iterations, giving the first result that has linear dependence on dimension for this setting. For the special case, where stochastic gradients are Lipschitz, the dependence on dimension reduces to polylogarithmic. In addition to giving new results, this paper also presents a simplified proof strategy that gives a shorter and more elegant proof of previously known results (Jin et al. 2017) on perturbed gradient descent algorithm.

A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm

In this note, we derive concentration inequalities for random vectors with subGaussian norm (a generalization of both subGaussian random vectors and norm bounded random vectors), which are tight up to logarithmic factors.

Minmax Optimization: Stable Limit Points of Gradient Descent Ascent are Locally Optimal

Minmax optimization, especially in its general nonconvex-nonconcave formulation, has found extensive applications in modern machine learning frameworks such as generative adversarial networks (GAN), adversarial training and multi-agent reinforcement learning. Gradient-based algorithms, in particular gradient descent ascent (GDA), are widely used in practice to solve these problems. Despite the practical popularity of GDA, however, its theoretical behavior has been considered highly undesirable. Indeed, apart from possiblity of non-convergence, recent results (Daskalakis and Panageas, 2018; Mazumdar and Ratliff, 2018; Adolphs et al., 2018) show that even when GDA converges, its stable limit points can be points that are not local Nash equilibria, thus not game-theoretically meaningful. In this paper, we initiate a discussion on the proper optimality measures for minmax optimization, and introduce a new notion of local optimality---local minmax---as a more suitable alternative to the notion of local Nash equilibrium. We establish favorable properties of local minmax points, and show, most importantly, that as the ratio of the ascent step size to the descent step size goes to infinity, stable limit points of GDA are exactly local minmax points up to degenerate points, demonstrating that all stable limit points of GDA have a game-theoretic meaning for minmax problems.

On the insufficiency of existing momentum schemes for Stochastic Optimization

Momentum based stochastic gradient methods such as heavy ball (HB) and Nesterov's accelerated gradient descent (NAG) method are widely used in practice for training deep networks and other supervised learning models, as they often provide significant improvements over stochastic gradient descent (SGD). Rigorously speaking, "fast gradient" methods have provable improvements over gradient descent only for the deterministic case, where the gradients are exact. In the stochastic case, the popular explanations for their wide applicability is that when these fast gradient methods are applied in the stochastic case, they partially mimic their exact gradient counterparts, resulting in some practical gain. This work provides a counterpoint to this belief by proving that there exist simple problem instances where these methods cannot outperform SGD despite the best setting of its parameters. These negative problem instances are, in an informal sense, generic; they do not look like carefully constructed pathological instances. These results suggest (along with empirical evidence) that HB or NAG's practical performance gains are a by-product of mini-batching. Furthermore, this work provides a viable (and provable) alternative, which, on the same set of problem instances, significantly improves over HB, NAG, and SGD's performance. This algorithm, referred to as Accelerated Stochastic Gradient Descent (ASGD), is a simple to implement stochastic algorithm, based on a relatively less popular variant of Nesterov's Acceleration. Extensive empirical results in this paper show that ASGD has performance gains over HB, NAG, and SGD.

Accelerating Stochastic Gradient Descent For Least Squares Regression

There is widespread sentiment that it is not possible to effectively utilize fast gradient methods (e.g. Nesterov's acceleration, conjugate gradient, heavy ball) for the purposes of stochastic optimization due to their instability and error accumulation, a notion made precise in d'Aspremont 2008 and Devolder, Glineur, and Nesterov 2014. This work considers these issues for the special case of stochastic approximation for the least squares regression problem, and our main result refutes the conventional wisdom by showing that acceleration can be made robust to statistical errors. In particular, this work introduces an accelerated stochastic gradient method that provably achieves the minimax optimal statistical risk faster than stochastic gradient descent. Critical to the analysis is a sharp characterization of accelerated stochastic gradient descent as a stochastic process. We hope this characterization gives insights towards the broader question of designing simple and effective accelerated stochastic methods for more general convex and non-convex optimization problems.

Parallelizing Stochastic Gradient Descent for Least Squares Regression: mini-batching, averaging, and model misspecification

This work characterizes the benefits of averaging schemes widely used in conjunction with stochastic gradient descent (SGD). In particular, this work provides a sharp analysis of: (1) mini-batching, a method of averaging many samples of a stochastic gradient to both reduce the variance of the stochastic gradient estimate and for parallelizing SGD and (2) tail-averaging, a method involving averaging the final few iterates of SGD to decrease the variance in SGD's final iterate. This work presents non-asymptotic excess risk bounds for these schemes for the stochastic approximation problem of least squares regression. Furthermore, this work establishes a precise problem-dependent extent to which mini-batch SGD yields provable near-linear parallelization speedups over SGD with batch size one. This allows for understanding learning rate versus batch size tradeoffs for the final iterate of an SGD method. These results are then utilized in providing a highly parallelizable SGD method that obtains the minimax risk with nearly the same number of serial updates as batch gradient descent, improving significantly over existing SGD methods. A non-asymptotic analysis of communication efficient parallelization schemes such as model-averaging/parameter mixing methods is then provided. Finally, this work sheds light on some fundamental differences in SGD's behavior when dealing with agnostic noise in the (non-realizable) least squares regression problem. In particular, the work shows that the stepsizes that ensure minimax risk for the agnostic case must be a function of the noise properties. This paper builds on the operator view of analyzing SGD methods, introduced by Defossez and Bach (2015), followed by developing a novel analysis in bounding these operators to characterize the excess risk. These techniques are of broader interest in analyzing computational aspects of stochastic approximation.

A Markov Chain Theory Approach to Characterizing the Minimax Optimality of Stochastic Gradient Descent (for Least Squares)

This work provides a simplified proof of the statistical minimax optimality of (iterate averaged) stochastic gradient descent (SGD), for the special case of least squares. This result is obtained by analyzing SGD as a stochastic process and by sharply characterizing the stationary covariance matrix of this process. The finite rate optimality characterization captures the constant factors and addresses model mis-specification.

Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form

Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer--Monteiro factorization approach for solving SDPs. We show that all approximate local optima are global optima for the penalty formulation of appropriately rank-constrained SDPs as long as the number of constraints scales sub-quadratically with the desired rank of the optimal solution. Our result is based on a simple penalty function formulation of the rank-constrained SDP along with a smoothed analysis to avoid worst-case cost matrices. We particularize our results to two applications, namely, Max-Cut and matrix completion.