Random directions stochastic approximation with deterministic perturbations

TL;DR

This paper develops deterministic perturbation schemes for random directions stochastic approximation, introducing new first- and second-order algorithms with proven convergence and superior convergence rates in convex quadratic optimization.

Contribution

It presents the first second-order deterministic perturbation algorithms for RDSA, with theoretical convergence guarantees and improved convergence rates.

Findings

Algorithms are asymptotically unbiased

Proven convergence of the algorithms

Numerical experiments validate theoretical results

Abstract

We introduce deterministic perturbation schemes for the recently proposed random directions stochastic approximation (RDSA) [17], and propose new first-order and second-order algorithms. In the latter case, these are the first second-order algorithms to incorporate deterministic perturbations. We show that the gradient and/or Hessian estimates in the resulting algorithms with deterministic perturbations are asymptotically unbiased, so that the algorithms are provably convergent. Furthermore, we derive convergence rates to establish the superiority of the first-order and second-order algorithms, for the special case of a convex and quadratic optimization problem, respectively. Numerical experiments are used to validate the theoretical results.