Optimizing tail risks using an importance sampling based extrapolation for heavy-tailed objectives

TL;DR

This paper introduces a novel importance sampling-based extrapolation method to efficiently approximate CVaR objectives in heavy-tailed distributions, reducing data requirements and enabling practical tail risk optimization.

Contribution

It develops a new CVaR approximation formula leveraging heavy-tailed distribution self-similarity, improving data efficiency over existing methods.

Findings

The proposed method is statistically consistent.

It significantly reduces data needs for tail risk estimation.

Numerical experiments show superior performance and ease of implementation.

Abstract

Motivated by the prominence of Conditional Value-at-Risk (CVaR) as a measure for tail risk in settings affected by uncertainty, we develop a new formula for approximating CVaR based optimization objectives and their gradients from limited samples. A key difficulty that limits the widespread practical use of these optimization formulations is the large amount of data required by the state-of-the-art sample average approximation schemes to approximate the CVaR objective with high fidelity. Unlike the state-of-the-art sample average approximations which require impractically large amounts of data in tail probability regions, the proposed approximation scheme exploits the self-similarity of heavy-tailed distributions to extrapolate data from suitable lower quantiles. The resulting approximations are shown to be statistically consistent and are amenable for optimization by means of conventional gradient descent. The approximation is guided by means of a systematic importance-sampling scheme whose asymptotic variance reduction properties are rigorously examined. Numerical experiments demonstrate the superiority of the proposed approximations and the ease of implementation points to the versatility of settings to which the approximation scheme can be applied.