Without-Replacement Sampling for Stochastic Gradient Methods: Convergence Results and Application to Distributed Optimization

TL;DR

This paper analyzes the convergence of stochastic gradient methods using without-replacement sampling, providing guarantees and applications to distributed optimization, which is more practical and often more effective than traditional with-replacement sampling.

Contribution

It offers the first competitive convergence guarantees for without-replacement sampling across multiple algorithms and introduces a nearly-optimal distributed regularized least squares algorithm.

Findings

Provides convergence guarantees for without-replacement sampling

Develops a distributed algorithm for regularized least squares

Achieves near-optimal communication and runtime complexity

Abstract

Stochastic gradient methods for machine learning and optimization problems are usually analyzed assuming data points are sampled \emph{with} replacement. In practice, however, sampling \emph{without} replacement is very common, easier to implement in many cases, and often performs better. In this paper, we provide competitive convergence guarantees for without-replacement sampling, under various scenarios, for three types of algorithms: Any algorithm with online regret guarantees, stochastic gradient descent, and SVRG. A useful application of our SVRG analysis is a nearly-optimal algorithm for regularized least squares in a distributed setting, in terms of both communication complexity and runtime complexity, when the data is randomly partitioned and the condition number can be as large as the data size per machine (up to logarithmic factors). Our proof techniques combine ideas from stochastic optimization, adversarial online learning, and transductive learning theory, and can potentially be applied to other stochastic optimization and learning problems.