A Normal Map-Based Proximal Stochastic Gradient Method: Convergence and Identification Properties

TL;DR

This paper introduces a normal map-based stochastic gradient method that guarantees convergence and finite-time identification of active substructures in nonconvex stochastic optimization problems.

Contribution

It proposes a novel NSGD algorithm that overcomes limitations of existing PSGD methods by ensuring convergence and finite-time manifold identification without convexity assumptions.

Findings

Global convergence to stationary points

Finite-time identification of active manifolds

Complexity bounds matching existing PSGD results

Abstract

The proximal stochastic gradient method (PSGD) is one of the state-of-the-art approaches for stochastic composite-type problems. In contrast to its deterministic counterpart, PSGD has been found to have difficulties with the correct identification of underlying substructures (such as supports, low rank patterns, or active constraints) and it does not possess a finite-time manifold identification property. Existing solutions rely on convexity assumptions or on the additional usage of variance reduction techniques. In this paper, we address these limitations and present a simple variant of PSGD based on Robinson's normal map. The proposed normal map-based proximal stochastic gradient method (NSGD) is shown to converge globally, i.e., accumulation points of the generated iterates correspond to stationary points almost surely. In addition, we establish complexity bounds for NSGD that match the known results for PSGD and we prove that NSGD can almost surely identify active manifolds in finite-time in a general nonconvex setting. Our derivations are built on almost sure iterate convergence guarantees and utilize analysis techniques based on the Kurdyka-Lojasiewicz inequality.