从频率派到贝叶斯的情境优化

中文摘要 AI 辅助

在数据驱动的情境随机优化中，现有方法 predominantly 是频率派的：它们承诺单一预测模型并将其视为真实情况，从而产生对小样本和中样本情况下模型不确定性和采样变异性脆弱的处方。我们提出贝叶斯情境优化（BCO），一个在参数空间上维护 Gibbs 后验的框架。这种决策聚焦的后验根据候选模型的实证决策质量而非统计拟合度进行加权，从而避免承诺于可能错误设定的似然，同时编码与数据一致的所有合理模型的完整分布。然后通过在后验预测分布下最小化期望决策成本来推导贝叶斯情境策略，通过在后验上聚合来对冲模型不确定性。我们建立了三个理论保证：（i）Gibbs 后验以指数速度集中于频率派最佳类别参数集；（ii）当模型不确定性不可忽略时，BCO 可以严格优于频率派最佳类别策略；（iii）BCO 达到相对于所有概率测度的预言机的 $O(n^{-1/2})$ 超额风险率，直至一个错误设定项和一个预言机聚合差距。计算上，我们定制了一个变分推断方案，其每次迭代成本与频率派替代方案相同，以及一个处理不可微问题的无梯度 Metropolis-Hastings 算法。在两阶段运输规划、情境报童和收益约束投资组合问题的数值实验中，确认 BCO 相对于核估计器和决策聚焦基线持续降低样本外成本和方差，在显著模型不确定性下的小样本和中样本情况下收益最为显著。

英文摘要

In data-driven contextual stochastic optimization, existing approaches are predominantly frequentist: they commit to a single predictive model and treat it as ground truth, yielding prescriptions that are fragile to model uncertainty and sampling variability in small- and moderate-sample regimes. We propose Bayesian contextual optimization (BCO), a framework that maintains a Gibbs posterior over the parameter space. This decision-focused posterior weights candidate models by their empirical decision quality rather than statistical fit, thereby avoiding commitment to a potentially misspecified likelihood while encoding the full distribution of plausible models consistent with the data. A Bayesian contextual policy is then derived by minimizing the expected decision cost under the posterior predictive distribution, hedging prescriptions against model uncertainty by aggregating over the posterior. We establish three theoretical guarantees: (i) the Gibbs posterior concentrates exponentially fast around the frequentist best-in-class parameter set; (ii) BCO can strictly improve over the frequentist best-in-class policy when model uncertainty is non-negligible; and (iii) BCO attains an $O(n^{-1/2})$ excess risk rate against the oracle over all probability measures up to a misspecification term and an oracle aggregate gap. Computationally, we tailor a variational inference scheme that has the same per-iteration cost as frequentist alternatives and a gradient-free Metropolis-Hastings algorithm that handles nondifferentiable problems. Numerical experiments on two-stage shipment planning, contextual newsvendor, and return-constrained portfolio problems confirm that BCO consistently reduces out-of-sample cost and variance relative to kernel estimators and decision-focused baselines, with the most pronounced gains in small- and moderate-sample regimes under substantial model uncertainty.

发表机构

• Laboratoire Génie Industriel, CentraleSupélec, Université Paris-Saclay（巴黎萨克雷大学中央理工学院工业工程实验室）

机构由 AI 辅助整理，请以论文原文为准。