SHARPER RATES AND FLEXIBLE FRAMEWORK FOR NONCONVEX SGD WITH CLIENT AND DATA SAMPLING Anonymous authors Paper under double-blind review

Abstract

We revisit the classical problem of finding an approximately stationary point of the average of n smooth and possibly nonconvex functions. The optimal complexity of stochastic first-order methods in terms of the number of gradient evaluations of individual functions is O n + n 1/2 ε -1 , attained by the optimal SGD methods SPIDER (Fang et al., 2018) and PAGE (Li et al., 2021), for example, where ε is the error tolerance. However, i) the big-O notation hides crucial dependencies on the smoothness constants associated with the functions, and ii) the rates and theory in these methods assume simplistic sampling mechanisms that do not offer any flexibility. In this work we remedy the situation. First, we generalize the PAGE algorithm so that it can provably work with virtually any (unbiased) sampling mechanism. This is particularly useful in federated learning, as it allows us to construct and better understand the impact of various combinations of client and data sampling strategies. Second, our analysis is sharper as we make explicit use of certain novel inequalities that capture the intricate interplay between the smoothness constants and the sampling procedure. Indeed, our analysis is better even for the simple sampling procedure analyzed in the PAGE paper. However, this already improved bound can be further sharpened by a different sampling scheme which we propose. In summary, we provide the most general and most accurate analysis of optimal SGD in the smooth nonconvex regime. Finally, our theoretical findings are supposed with carefully designed experiments.

1. INTRODUCTION

In this paper, we consider the minimization of the average of n smooth functions (1) in the nonconvex setting in the regime when the number of functions n is very large. In this regime, calculation of the exact gradient can be infeasible and the classical gradient descent method (GD) (Nesterov, 2018) can not be applied. The structure of the problem is generic, and such problems arise in many applications, including machine learning (Bishop & Nasrabadi, 2006) and computer vision (Goodfellow et al., 2016) . Problems of this form are the basis of empirical risk minimization (ERM), which is the prevalent paradigm for training supervised machine learning models.

1.1. FINITE-SUM OPTIMIZATION IN THE SMOOTH NONCONVEX REGIME

We consider the finite-sum optimization problem min x∈R d f (x) := 1 n n i=1 f i (x) , where f i : R d → R is a smooth (and possibly nonconvex) function for all i ∈ [n] := {1, . . . , n}. We are interested in randomized algorithms that find an ε-stationary point of (1) by returning a random point x such that E ∇f ( x) 2 ≤ ε. The main efficiency metric of gradient-based algorithms for finding such a point is the (expected) number of gradient evaluations ∇f i ; we will refer to it as the complexity of an algorithm.

