Bayesian Optimization of Risk Measures

TL;DR

This paper develops Bayesian optimization algorithms tailored for risk measures like VaR and CVaR, improving decision-making under uncertainty by efficiently modeling the underlying functions with Gaussian processes.

Contribution

It introduces novel Bayesian optimization algorithms that leverage the structure of risk measure objectives, enhancing sampling efficiency for expensive black-box functions.

Findings

Algorithms outperform standard methods in numerical experiments.

Significant reduction in number of evaluations needed.

Effective in portfolio optimization and robust system design.

Abstract

We consider Bayesian optimization of objective functions of the form $\rho[ F(x, W) ]$, where $F$ is a black-box expensive-to-evaluate function and $\rho$ denotes either the VaR or CVaR risk measure, computed with respect to the randomness induced by the environmental random variable $W$. Such problems arise in decision making under uncertainty, such as in portfolio optimization and robust systems design. We propose a family of novel Bayesian optimization algorithms that exploit the structure of the objective function to substantially improve sampling efficiency. Instead of modeling the objective function directly as is typical in Bayesian optimization, these algorithms model $F$ as a Gaussian process, and use the implied posterior on the objective function to decide which points to evaluate. We demonstrate the effectiveness of our approach in a variety of numerical experiments.