Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes
Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes
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DOI：
10.1287/moor.2021.1178
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发表时间：
2018-10
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影响因子：
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通讯作者：
J. Blanchet;Karthyek Murthy;Fan Zhang
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文献类型：
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作者：
J. Blanchet;Karthyek Murthy;Fan Zhang
考虑了具有局部强凸运输代价函数和仿射决策规则的基于运输的分布鲁棒优化问题。在传统的凸性假设的基础上的损失函数，我们得到的价值函数，最优策略，最坏情况下的最优运输对抗模型的结构结果。这些结果揭示了嵌入在DRO问题中的丰富结构（例如，强凸性，即使非DRO问题不是强凸的，DRO约束的拉格朗日的适当缩放等，这对于有效算法的设计是至关重要的）。作为这些结果的结果，人们可以开发出有效的优化程序，具有相同的样本和迭代复杂度作为一个自然的非DRO基准算法，如随机梯度下降。
We consider optimal transport-based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about the value function, the optimal policy, and the worst-case optimal transport adversarial model. These results expose a rich structure embedded in the DRO problem (e.g., strong convexity even if the non-DRO problem is not strongly convex, a suitable scaling of the Lagrangian for the DRO constraint, etc., which are crucial for the design of efficient algorithms). As a consequence of these results, one can develop efficient optimization procedures that have the same sample and iteration complexity as a natural non-DRO benchmark algorithm, such as stochastic gradient descent.