Wasserstein Distributionally Robust Risk-Sensitive Estimation via Conditional Value-at-Risk
2026-04-20 • Machine Learning
Machine Learning
AI summaryⓘ
The authors introduce a new way to estimate an unknown signal from observed data when the exact relationship between them is uncertain. They assume the true probabilities come from a set of possible distributions near a known one, and measure errors using a risk-focused metric called conditional value-at-risk (CVaR). They focus on simple linear estimators and show that the best ones for this worst-case risk can be found efficiently using a type of optimization called semidefinite programming. They test their method on forecasting electricity prices and find it reduces risky prediction errors compared to other approaches.
distributionally robust estimationconditional value-at-risk (CVaR)Wasserstein distanceambiguity setrandom vectorsaffine estimatorsemidefinite programmingworst-case riskelectricity price forecastingout-of-sample evaluation
Authors
Feras Al Taha, Eilyan Bitar
Abstract
We propose a distributionally robust approach to risk-sensitive estimation of an unknown signal x from an observed signal y. The unknown signal and observation are modeled as random vectors whose joint probability distribution is unknown, but assumed to belong to a given type-2 Wasserstein ball of distributions, termed the ambiguity set. The performance of an estimator is measured according to the conditional value-at-risk (CVaR) of the squared estimation error. Within this framework, we study the problem of computing affine estimators that minimize the worst-case CVaR over all distributions in the given ambiguity set. As our main result, we show that, when the nominal distribution at the center of the Wasserstein ball is finitely supported, such estimators can be exactly computed by solving a tractable semidefinite program. We evaluate the proposed estimators on a wholesale electricity price forecasting task using real market data and show that they deliver lower out-of-sample CVaR of squared error compared to existing methods.