Regret Equals Covariance: A Closed-Form Characterization for Stochastic Optimization
Quantifying regret typically involves repeatedly re-solving a stochastic optimization problem across simulated scenarios. Such an approach is costly: the cost scales with the number of scenarios and the cost of each solve. We propose that regret can instead be measured using the joint moments of the cost vector $c$ and the optimal decision $π^*(c)$. Working with the optimal concave and Lipschitz under only a feasible region value function $v(c) = \min_{z\in Z} c^\top z$, we show that regret always decomposes as $\mathrm{Regret} = -\mathrm{Cov}(c,π^*(c)) - R(c)$ for a residual $R(c)$. We further show that $R(c)=0$ when $π^*$ is affine in $c$, which holds for unconstrained and equality-cons...
| Authors | Irene Aldridge |
|---|---|
| Primary category | econ.EM |
| Published | 13 May 2026 |
| Last updated at source | 27 September 2026 |
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