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

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.

The importance of better models in stochastic optimization

“…To make our approach a bit more concrete, we identify several models that fit into our framework. These have appeared [15,8,2], but we believe a self-contained presentation beneficial. Each of these models satisfies our conditions (C.i)-(C.iii).…”

“…Proximal point methods: In the convex setting [4,28,2], the stochastic proximal point method uses the model f x (y; s) := f (y; s); in the weakly convex setting, we regularize and use…”

“…Most optimization methods iterate by making an approximation-a model-of the objective near the current iterate, then minimizing this model and re-approximating. Stochastic (sub)gradient methods [31,27] instantiate this approach using a linear approximation; following initial work of our own and others [2,15,8], we study the modeling approach in more depth for stochastic optimization. Thus, the aProx algorithms we develop [2] iterate as follows: for k = 1, 2, .…”

“…In our initial study of stability in optimization [2], we defined an algorithm as stable if its iterates remain bounded, then showed several consequences of this in convex optimization (which we review presently). A weakness, however, was that the only algorithms whose stability we were able to demonstrate were very close approximations to stochastic proximal point methods.…”

“…Assumption A2 is relatively weak, and is equivalent to assuming that E[ f (x; S) 2 2 ] is bounded on compacta; it allows arbitrary growth-exponential, super-exponential-in the norm of the subgradients, just requiring that the second moment exists. .…”