A kernel-based stochastic approximation framework for contextual optimization

TL;DR

This paper introduces a kernel-based stochastic approximation framework for solving a broad class of contextual stochastic optimization problems, demonstrating strong convergence and finite-time performance bounds.

Contribution

The paper proposes a novel KBSA framework that handles various contextual measures and includes variants with improved convergence and multi-conditioning capabilities.

Findings

Proves strong convergence of KBSA under certain conditions.

Provides finite-time mean squared error bounds for the iterates.

Demonstrates effectiveness through simulation experiments.

Abstract

We present a kernel-based stochastic approximation (KBSA) framework for solving contextual stochastic optimization problems with differentiable objective functions. The framework only relies on system output estimates and can be applied to address a large class of contextual measures, including conditional expectations, conditional quantiles, CoVaR, and conditional expected shortfalls.Under appropriate conditions, we show the strong convergence of KBSA and characterize its finite-time performance in terms of bounds on the mean squared errors of the sequences of iterates produced. In addition, we discuss variants of the framework, including a version based on high-order kernels for further enhancing the convergence rate of the method and an extension of KBSA for handling contextual measures involving multiple conditioning events.Simulation experiments are also carried out to illustrate the framework.