Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach
Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach
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DOI：
10.1287/moor.2020.1085
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发表时间：
2016-10
期刊：
影响因子：
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通讯作者：
John C. Duchi;P. Glynn;Hongseok Namkoong
中科院分区：
文献类型：
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作者：
John C. Duchi;P. Glynn;Hongseok Namkoong
我们研究随机优化问题的统计推断和分布鲁棒解方法，重点是最优值的置信区间和渐近实现精确覆盖的解决方案。我们开发了一个广义的经验似然框架的基础上构造的非参数f-发散球的分布不确定性集阿达玛可微泛函，特别是随机优化问题。作为这一理论的后果，我们提供了一个原则性的方法来选择分布不确定性区域的大小，以提供单侧和双侧的置信区间，实现精确的覆盖。我们还给出了我们的分布鲁棒制剂的渐近展开，显示如何鲁棒正则化问题的方差。最后，我们表明，我们研究的分布鲁棒配方的优化享受（基本上）相同的一致性性质，在经典的样本平均近似。我们的一般方法适用于快速混合平稳序列，包括几何遍历哈里斯经常马尔可夫链。
We study statistical inference and distributionally robust solution methods for stochastic optimization problems, focusing on confidence intervals for optimal values and solutions that achieve exact coverage asymptotically. We develop a generalized empirical likelihood framework—based on distributional uncertainty sets constructed from nonparametric f-divergence balls—for Hadamard differentiable functionals, and in particular, stochastic optimization problems. As consequences of this theory, we provide a principled method for choosing the size of distributional uncertainty regions to provide one- and two-sided confidence intervals that achieve exact coverage. We also give an asymptotic expansion for our distributionally robust formulation, showing how robustification regularizes problems by their variance. Finally, we show that optimizers of the distributionally robust formulations we study enjoy (essentially) the same consistency properties as those in classical sample average approximations. Our general approach applies to quickly mixing stationary sequences, including geometrically ergodic Harris recurrent Markov chains.