Multi-Level Stochastic Gradient Methods for Nested Composition Optimization

TL;DR

This paper introduces multi-level stochastic gradient methods for complex nested optimization problems, achieving improved convergence rates especially when component functions are smooth, with applications in risk-averse optimization.

Contribution

It develops a multi-level stochastic gradient framework with accelerated methods and provides convergence guarantees for both convex and nonconvex cases.

Findings

Convergence rate of $O(n^{-1/2^T})$ for basic multi-level methods.

Accelerated methods improve convergence to $O(n^{-4/(7+T)})$ for smooth functions.

Validated effectiveness through numerical experiments.

Abstract

Stochastic gradient methods are scalable for solving large-scale optimization problems that involve empirical expectations of loss functions. Existing results mainly apply to optimization problems where the objectives are one- or two-level expectations. In this paper, we consider the multi-level compositional optimization problem that involves compositions of multi-level component functions and nested expectations over a random path. It finds applications in risk-averse optimization and sequential planning. We propose a class of multi-level stochastic gradient methods that are motivated from the method of multi-timescale stochastic approximation. First we propose a basic $T$-level stochastic compositional gradient algorithm, establish its almost sure convergence and obtain an $n$-iteration error bound $O (n^{-1/2^T})$. Then we develop accelerated multi-level stochastic gradient methods by using an extrapolation-interpolation scheme to take advantage of the smoothness of individual component functions. When all component functions are smooth, we show that the convergence rate improves to $O(n^{-4/(7+T)})$ for general objectives and $O (n^{-4/(3+T)})$ for strongly convex objectives. We also provide almost sure convergence and rate of convergence results for nonconvex problems. The proposed methods and theoretical results are validated using numerical experiments.