Generalized Simultaneous Perturbation-based Gradient Search with Reduced Estimator Bias

TL;DR

This paper introduces a family of generalized gradient estimators using noisy function measurements, reducing bias through increased measurements, and analyzes their convergence with experimental validation on benchmark functions.

Contribution

It proposes new generalized simultaneous perturbation estimators with balanced and unbalanced variants, providing theoretical analysis and experimental validation.

Findings

More function measurements lead to lower estimator bias.

The estimators converge asymptotically and non-asymptotically.

Experimental results validate theoretical bias reduction and convergence.

Abstract

We present in this paper a family of generalized simultaneous perturbation-based gradient search (GSPGS) estimators that use noisy function measurements. The number of function measurements required by each estimator is guided by the desired level of accuracy. We first present in detail unbalanced generalized simultaneous perturbation stochastic approximation (GSPSA) estimators and later present the balanced versions (B-GSPSA) of these. We extend this idea further and present the generalized smoothed functional (GSF) and generalized random directions stochastic approximation (GRDSA) estimators, respectively, as well as their balanced variants. We show that estimators within any specified class requiring more number of function measurements result in lower estimator bias. We present a detailed analysis of both the asymptotic and non-asymptotic convergence of the resulting stochastic approximation schemes. We further present a series of experimental results with the various GSPGS estimators on the Rastrigin and quadratic function objectives. Our experiments are seen to validate our theoretical findings.