Drift Optimization of Regulated Stochastic Models Using Sample Average Approximation

TL;DR

This paper develops a Sample Average Approximation method for optimizing the drift in regulated stochastic models, enabling practical solutions for complex infinite-dimensional problems in operations research, machine learning, and statistics.

Contribution

It introduces a novel SAA framework incorporating path and function-space discretization with Monte Carlo sampling for drift optimization in regulated stochastic processes.

Findings

Proposes a pathwise directional derivative construction for the SAA method.

Provides consistency and complexity analysis of the proposed approach.

Guides on balancing computational effort between optimization and discretization steps.

Abstract

This paper introduces a drift optimization model of stochastic optimization problems driven by regulated stochastic processes. A broad range of problems across operations research, machine learning, and statistics can be viewed as optimizing the "drift" associated with a process by minimizing a cost functional, while respecting path constraints imposed by a Lipschitz continuous regulator. Towards an implementable solution to such infinite-dimensional problems, we develop the fundamentals of a Sample Average Approximation (SAA) method that incorporates (i) path discretization, (ii) function-space discretization, and (iii) Monte Carlo sampling, and that is solved using an optimization recursion such as mirror descent. We start by constructing pathwise directional derivatives for use within the SAA method, followed by consistency and complexity calculations. The characterized complexity is expressed as a function of the number of optimization steps, and the computational effort involved in (i)--(iii), leading to guidance on how to trade-off the computational effort allocated to optimization steps versus the "dimension reduction" steps in (i)--(iii).