Zeroth-Order Optimization Finds Flat Minima

TL;DR

This paper demonstrates that zeroth-order optimization algorithms tend to converge to flat minima characterized by small Hessian trace, supported by theoretical analysis and experiments in machine learning tasks.

Contribution

It reveals the implicit bias of zeroth-order methods towards flat minima and provides convergence rates for finding such solutions in convex, smooth functions.

Findings

Zeroth-order methods favor solutions with small Hessian trace.

Convergence rates to flat minima are established for convex functions.

Experiments confirm theoretical predictions in classification and language modeling.

Abstract

Zeroth-order methods are extensively used in machine learning applications where gradients are infeasible or expensive to compute, such as black-box attacks, reinforcement learning, and language model fine-tuning. Existing optimization theory focuses on convergence to an arbitrary stationary point, but less is known on the implicit regularization that provides a fine-grained characterization on which particular solutions are finally reached. We show that zeroth-order optimization with the standard two-point estimator favors solutions with small trace of Hessian, which is widely used in previous work to distinguish between sharp and flat minima. We further provide convergence rates of zeroth-order optimization to approximate flat minima for convex and sufficiently smooth functions, where flat minima are defined as the minimizers that achieve the smallest trace of Hessian among all optimal solutions. Experiments on binary classification tasks with convex losses and language model fine-tuning support our theoretical findings.