Stochastic (Approximate) Proximal Point Methods: Convergence, Optimality, and Adaptivity

TL;DR

This paper introduces the aProx family of model-based stochastic convex optimization methods, which improve convergence, robustness, and adaptivity over classical approaches with minimal additional computational cost.

Contribution

It proposes the aProx family of methods that unify and extend stochastic subgradient, proximal point, and bundle methods with stronger theoretical guarantees and practical advantages.

Findings

Models converge with probability 1.

Methods achieve optimal asymptotic normality.

Experimental results show advantages over standard subgradient methods.

Abstract

We develop model-based methods for solving stochastic convex optimization problems, introducing the approximate-proximal point, or aProx, family, which includes stochastic subgradient, proximal point, and bundle methods. When the modeling approaches we propose are appropriately accurate, the methods enjoy stronger convergence and robustness guarantees than classical approaches, even though the model-based methods typically add little to no computational overhead over stochastic subgradient methods. For example, we show that improved models converge with probability 1 and enjoy optimal asymptotic normality results under weak assumptions; these methods are also adaptive to a natural class of what we term easy optimization problems, achieving linear convergence under appropriate strong growth conditions on the objective. Our substantial experimental investigation shows the advantages of more accurate modeling over standard subgradient methods across many smooth and non-smooth optimization problems.