Incremental Without Replacement Sampling in Nonconvex Optimization
This work addresses a gap in theoretical analysis for widely used incremental optimization methods, offering insights for machine learning practitioners dealing with nonconvex problems.
The paper tackles the problem of analyzing incremental gradient methods with sampling without replacement for nonconvex optimization, providing convergence guarantees and explicit complexity estimates for smooth cases and attraction to optimality conditions for nonsmooth cases.
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.