Gradient-based optimisation of the conditional-value-at-risk using the multi-level Monte Carlo method

TL;DR

This paper develops a gradient-based optimization method for the Conditional-Value-at-Risk (CVaR) using an enhanced Multi-Level Monte Carlo approach, enabling efficient risk minimization in complex stochastic models.

Contribution

It introduces modifications to the MLMC estimator and combines it with an inexact gradient descent algorithm, providing proven exponential convergence for CVaR optimization.

Findings

Demonstrates optimal asymptotic cost-tolerance behavior in numerical examples.

Proves exponential convergence under strong convexity and Lipschitz conditions.

Ensures prescribed accuracy in CVaR and sensitivity estimation.

Abstract

In this work, we tackle the problem of minimising the Conditional-Value-at-Risk (CVaR) of output quantities of complex differential models with random input data, using gradient-based approaches in combination with the Multi-Level Monte Carlo (MLMC) method. In particular, we consider the framework of multi-level Monte Carlo for parametric expectations and propose modifications of the MLMC estimator, error estimation procedure, and adaptive MLMC parameter selection to ensure the estimation of the CVaR and sensitivities for a given design with a prescribed accuracy. We then propose combining the MLMC framework with an alternating inexact minimisation-gradient descent algorithm, for which we prove exponential convergence in the optimisation iterations under the assumptions of strong convexity and Lipschitz continuity of the gradient of the objective function. We demonstrate the performance of our approach on two numerical examples of practical relevance, which evidence the same optimal asymptotic cost-tolerance behaviour as standard MLMC methods for fixed design computations of output expectations.