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.

翻译：遗憾是算法决策中不确定性的代价。量化遗憾通常需要通过样本平均近似（SAA）进行计算成本高昂的模拟，其复杂度为$\mathcal{O}(Bn^{2}d^{3})$，其中$B$为场景数，$n$为变量数，$d$为约束数。本文证明任一随机优化问题中的期望遗憾均具有精确分解形式\mathrm{Regret}(c) = \mathrm{Cov}(c,\,π^{*}(c)) + R(c)，其中$c$为不确定参数向量，$π^{*}(c)$为最优决策，$R(c)$为残差项，我们给出了其在Lipschitz、光滑和强凸条件下的显式界限。对于线性规划和无约束二次规划（包括经典的Markowitz投资组合问题），我们证明$R(c)=0$严格成立，从而\mathrm{Regret}(c) = \mathrm{Cov}(c,π^{*}(c))无需近似即成立。当历史成本-决策对$\{(c_i, π^*(c_i))\}$可用时，协方差可在$\mathcal{O}(nd^{2})$时间内估计完成，比SAA快数个数量级，且该估计仅需单次数据遍历。我们推导了浓度界、中心极限定理和渐近无偏的残差估计量，并在合成LP、QP和整数规划实例以及使用十年CRSP权益数据的滚动窗口投资组合实验上验证了所有结果。