Incremental Without Replacement Sampling in Nonconvex Optimization

TL;DR

This paper provides convergence guarantees for incremental gradient methods without replacement in nonconvex optimization, offering explicit complexity estimates and analysis for both smooth and nonsmooth settings.

Contribution

It introduces a versatile incremental gradient scheme with convergence guarantees for nonconvex problems, extending analysis to without-replacement sampling.

Findings

Explicit complexity estimates in smooth settings.

Sequence attracted to solutions in nonsmooth settings.

Convergence guarantees for incremental without-replacement sampling.

Abstract

Minibatch decomposition methods for empirical risk minimization are commonly analysed in a stochastic approximation setting, also known as sampling with replacement. On the other hands modern implementations of such techniques are incremental: they rely on sampling without replacement, for which available analysis are much scarcer. We provide convergence guaranties for the latter variant by analysing a versatile incremental gradient scheme. For this scheme, we consider constant, decreasing or adaptive step sizes. In the smooth setting we obtain explicit complexity estimates in terms of epoch counter. In the nonsmooth setting we prove that the sequence is attracted by solutions of optimality conditions of the problem.