Stochastic Optimization under Hidden Convexity

TL;DR

This paper studies stochastic optimization problems with hidden convexity, proposing methods with proven convergence guarantees even when the convex reformulation map is unavailable, applicable to various non-convex real-world problems.

Contribution

It introduces the first sample complexity guarantees for stochastic gradient methods solving hidden convexity problems, including last iterate convergence with momentum.

Findings

Provided the first convergence guarantees for such problems.

Achieved last iterate convergence results in smooth settings.

Applicable to diverse non-convex applications like control and reinforcement learning.

Abstract

In this work, we consider constrained stochastic optimization problems under hidden convexity, i.e., those that admit a convex reformulation via non-linear (but invertible) map $c(\cdot)$. A number of non-convex problems ranging from optimal control, revenue and inventory management, to convex reinforcement learning all admit such a hidden convex structure. Unfortunately, in the majority of applications considered, the map $c(\cdot)$ is unavailable or implicit; therefore, directly solving the convex reformulation is not possible. On the other hand, the stochastic gradients with respect to the original variable are often easy to obtain. Motivated by these observations, we examine the basic projected stochastic (sub-) gradient methods for solving such problems under hidden convexity. We provide the first sample complexity guarantees for global convergence in smooth and non-smooth settings. Additionally, in the smooth setting, we improve our results to the last iterate convergence in terms of function value gap using the momentum variant of projected stochastic gradient descent.