Regret Equals Covariance: A Closed-Form Characterization for Stochastic Optimization

2026-05-13

AI Summary

This work addresses the computational burden of sample average approximation in quantifying regret due to uncertainty in stochastic optimization. It establishes, for the first time, an exact equality linking regret to the covariance between uncertain parameters and optimal decisions, proving that the residual term vanishes under specific conditions. This yields closed-form, approximation-free expressions for linear programs and unconstrained quadratic programs. By integrating covariance estimation, smoothness analysis, and concentration inequalities, the proposed method requires only a single pass over the data, reducing computational complexity from 𝒪(Bn²d³) to 𝒪(nd²). Empirical validation on synthetic LP/QP instances, integer programs, and a decade-long rolling portfolio optimization task using CRSP data demonstrates both high efficiency and accuracy.

Abstract

Regret is the cost of uncertainty in algorithmic decision-making. Quantifying regret typically requires computationally expensive simulation via Sample Average Approximation (SAA), with complexity $\mathcal{O}(Bn^{2}d^{3})$ in the number of scenarios $B$, variables $n$, and constraints $d$. % This paper proves that expected regret in any stochastic optimization problem admits the exact decomposition % \begin{equation*} \mathrm{Regret}(c) = \mathrm{Cov}(c,\,π^{*}(c)) + R(c), \end{equation*} % where $c$ is the vector of uncertain parameters, $π^{*}(c)$ is the optimal decision, and $R(c)$ is a residual whose magnitude we bound explicitly under Lipschitz, smooth, and strongly convex conditions. % For linear programs and unconstrained quadratic programs, including the classical Markowitz portfolio problem, we prove $R(c)=0$ exactly, so that $\mathrm{Regret}(c) = \mathrm{Cov}(c,π^{*}(c))$ holds without approximation. % When historical cost-decision pairs $\{(c_i, π^*(c_i))\}$ are available, the covariance can be estimated in $\mathcal{O}(nd^{2})$ time, which is orders of magnitude faster than SAA. The estimation is performed by a single pass through the data. % We derive concentration bounds, a central limit theorem, and an asymptotically unbiased residual estimator, and we validate all results on synthetic LP, QP, and integer programming instances and on a rolling-window portfolio experiment using ten years of CRSP equity data.

Problem

• regret
• stochastic optimization
• covariance
• sample average approximation
• uncertainty quantification

Innovation

• regret decomposition
• covariance characterization
• stochastic optimization
• computational efficiency
• Markowitz portfolio