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. These distributional estimates in Wasserstein distance lead to excess-risk bounds for the curvature-regularized objective. Our analysis accommodates assumptions under which the stochastic function evaluations are allowed to have non-uniform dependence on the data stream. In addition, the constants in the non-asymptotic bounds do not rely on log-Sobolev or Poincaré inequalities as in prior zeroth-order Langevin analyses. To the best of our knowledge, this is the first non-asymptotic analysis of a zeroth-order Langevin method that simultaneously establishes excess-risk guarantees for a curvature-regularized objective and Wasserstein guarantees for the associated Gibbs measure in the nonconvex setting.

翻译：暂无翻译