New Lower Bounds for Stochastic Non-Convex Optimization through Divergence Decomposition

TL;DR

This paper establishes tight lower bounds on the number of noisy gradient queries needed for stochastic non-convex optimization in various settings, using divergence decomposition to identify fundamental limits.

Contribution

It introduces new tight lower bounds for stochastic non-convex optimization across multiple function classes, employing divergence decomposition and a novel function identification approach.

Findings

Lower bounds are tight up to logarithmic factors.

A specialized 1D algorithm achieves faster convergence rates.

Certain dimensional thresholds are intrinsic to the problem complexity.

Abstract

We study fundamental limits of first-order stochastic optimization in a range of nonconvex settings, including L-smooth functions satisfying Quasar-Convexity (QC), Quadratic Growth (QG), and Restricted Secant Inequalities (RSI). While the convergence properties of standard algorithms are well-understood in deterministic regimes, significantly fewer results address the stochastic case, where only unbiased and noisy gradients are available. We establish new lower bounds on the number of noisy gradient queries to minimize these classes of functions, also showing that they are tight (up to a logarithmic factor) in all the relevant quantities characterizing each class. Our approach reformulates the optimization task as a function identification problem, leveraging divergence decomposition arguments to construct a challenging subclass that leads to sharp lower bounds. Furthermore, we present a specialized algorithm in the one-dimensional setting that achieves faster rates, suggesting that certain dimensional thresholds are intrinsic to the complexity of non-convex stochastic optimization.