The perturbed prox-preconditioned spider algorithm: non-asymptotic convergence bounds

TL;DR

The paper introduces 3P-SPIDER, a novel stochastic optimization algorithm that combines preconditioning and variance reduction, achieving near-optimal convergence bounds even with approximate gradient estimations.

Contribution

It proposes the 3P-SPIDER algorithm, extending SPIDER with preconditioning and handling approximate gradients, providing non-asymptotic convergence guarantees for nonconvex, nonsmooth problems.

Findings

Achieves near-optimal oracle inequality O(n^(1/2) / epsilon).

Handles approximate gradient estimations via Monte Carlo methods.

Demonstrates effectiveness on penalized empirical loss minimization.

Abstract

A novel algorithm named Perturbed Prox-Preconditioned SPIDER (3P-SPIDER) is introduced. It is a stochastic variancereduced proximal-gradient type algorithm built on Stochastic Path Integral Differential EstimatoR (SPIDER), an algorithm known to achieve near-optimal first-order oracle inequality for nonconvex and nonsmooth optimization. Compared to the vanilla prox-SPIDER, 3P-SPIDER uses preconditioned gradient estimators. Preconditioning can either be applied "explicitly" to a gradient estimator or be introduced "implicitly" as in applications to the EM algorithm. 3P-SPIDER also assumes that the preconditioned gradients may (possibly) be not known in closed analytical form and therefore must be approximated which adds an additional degree of perturbation. Studying the convergence in expectation, we show that 3P-SPIDER achieves a near-optimal oracle inequality O(n^(1/2) /epsilon) where n is the number of observations and epsilon the target precision even when the gradient is estimated by Monte Carlo methods. We illustrate the algorithm on an application to the minimization of a penalized empirical loss.