Stochastic optimization methods for the simultaneous control of parameter-dependent systems

TL;DR

This paper compares stochastic optimization methods like SGD and CSG with classical algorithms for controlling parameter-dependent systems, demonstrating their efficiency in reducing computational costs through numerical experiments.

Contribution

It introduces the application of stochastic gradient methods to control problems and compares their performance with traditional algorithms, highlighting advantages in high-parameter scenarios.

Findings

SGD and CSG reduce computational complexity for large parameter problems.

Stochastic methods outperform classical algorithms in high-dimensional control tasks.

Numerical experiments confirm efficiency gains of stochastic approaches.

Abstract

We address the application of stochastic optimization methods for the simultaneous control of parameter-dependent systems. In particular, we focus on the classical Stochastic Gradient Descent (SGD) approach of Robbins and Monro, and on the recently developed Continuous Stochastic Gradient (CSG) algorithm. We consider the problem of computing simultaneous controls through the minimization of a cost functional defined as the superposition of individual costs for each realization of the system. We compare the performances of these stochastic approaches, in terms of their computational complexity, with those of the more classical Gradient Descent (GD) and Conjugate Gradient (CG) algorithms, and we discuss the advantages and disadvantages of each methodology. In agreement with well-established results in the machine learning context, we show how the SGD and CSG algorithms can significantly reduce the computational burden when treating control problems depending on a large amount of parameters. This is corroborated by numerical experiments.