Zeroth-Order Langevin Dynamics for Stochastic Nonconvex Optimization: Curvature Regularization and Non-Asymptotic Guarantees

Stefano Bruno, Dong-Young Lim

Zeroth-order methods provide a way to optimize stochastic nonconvex objectives using only noisy function evaluations. However, the smoothing inherent in their gradient estimators effectively modifies the objective being optimized. In particular, Gaussian smoothing with smoothing parameter $η>0$ induces, to leading order, an implicit Hessian-trace regularization, which motivates the study of the curvature-regularized objective of the form $f(x)+(η^2/2) tr(\nabla^2 f(x))$. We develop a non-asymptotic theory in the nonconvex setting for a class of algorithms called stochastic zeroth-order Langevin dynamics (ZOLD), which use only noisy function evaluations and do not have access to first- or higher-order derivative information, and derive excess-risk guarantees for the curvature-regularized objective. We first control the finite-time distribution of the ZOLD iterates in Wasserstein distance with respect to the Gibbs measure associated with the Gaussian-smoothed objective. An explicit coupling between the smoothing and inverse temperature parameters then quantifies the proximity of this measure to the Gibbs measure associated with the corresponding Hessian-trace-regularized objective.…