Stochastic version of second-order (Newton-Raphson) optimization using only function measurements

Consider the problem of loss function minimization when only (possibly noisy) measurements of the loss function are available. In particular, no measurements of the gradient of the loss function are assumed available (as required in the steepest descent or Newton-Raphson algorithms). Stochastic approximation (SA) algorithms of the multivariate Kiefer-Wolfowitz (finite-difference) form have long been considered for such problems, but with only limited success. The simultaneous perturbation SA (SPSA) algorithm has successfully addressed one of the major shortcomings of those finite-difference SA algorithms by significantly reducing the number of measurements required in many multivariate problems of practical interest. This SPSA algorithm displays the classic behaviour of 1st-order search algorithms by typically exhibiting a steep initial decline in the loss function followed by a slow decline to the optimum. This paper presents a 2nd-order SPSA algorithm that is based on estimating both the loss function gradient and inverse Hessian matrix at each iteration. The aim of this approach is to emulate the acceleration properties associated with deterministic algorithms of Newton-Raphson form, particularly in the terminal phase where the 1st-order SPSA algorithm slows down in its convergence. This 2nd-order SPSA algorithm requires only three loss function measurements at each iteration, independent of the problem dimension. This paper includes a formal convergence result for this 2nd-order approach.