Nonlinear Stochastic Approximation Procedures for $L_p$ Loss Functions

Abstract

The classical stochastic approximation problem can be regarded as choosing design points so that the responses are close to some target level in the expected squared distance. Motivated by different loss criteria, a family of stochastic approximation algorithms is proposed. This family has the same simplicity as the classical Robbins-Monro procedure does and contains the latter as a special case. Using appropriate representations and martingale limit theorems, we establish asymptotic properties for this family. Using the semiparametric formulation, lower bounds are obtained for estimating the desired parameters under any adaptive design, showing that the proposed algorithms with appropriate scaling are asymptotically efficient.