Stochastic Average Model Methods

TL;DR

This paper introduces stochastic average model (SAM) methods for efficiently solving finite-sum minimization problems by sampling component functions to reduce computational cost, with promising numerical results in derivative-free optimization.

Contribution

The paper proposes SAM methods that sample component functions based on variance-minimizing distributions, extending trust-region methods with a derivative-free solver called SAM-POUNDERS.

Findings

SAM methods reduce computational cost in finite-sum problems.

Numerical results show effectiveness of SAM-POUNDERS in derivative-free optimization.

Sampling strategies improve variance control in stochastic models.

Abstract

We consider the solution of finite-sum minimization problems, such as those appearing in nonlinear least-squares or general empirical risk minimization problems. We are motivated by problems in which the summand functions are computationally expensive and evaluating all summands on every iteration of an optimization method may be undesirable. We present the idea of stochastic average model (SAM) methods, inspired by stochastic average gradient methods. SAM methods sample component functions on each iteration of a trust-region method according to a discrete probability distribution on component functions; the distribution is designed to minimize an upper bound on the variance of the resulting stochastic model. We present promising numerical results concerning an implemented variant extending the derivative-free model-based trust-region solver POUNDERS, which we name SAM-POUNDERS.