TY - JOUR

T1 - Decomposition algorithm for distributionally robust optimization using Wasserstein metric with an application to a class of regression models

AU - Luo, Fengqiao

AU - Mehrotra, Sanjay

N1 - Funding Information:
We are grateful to Changhyeok Lee and Liwei Zeng for a valuable discussion0. We especially thank one anonymous reviewer for helpful technical comments that resulted in a refinement of the ε-optimality statement as provided in the proof of Theorem 3.2 . We also acknowledge NSF grants CMMI-1362003 and CMMI-1100868 that supported this research.
Funding Information:
We are grateful to Changhyeok Lee and Liwei Zeng for a valuable discussion0. We especially thank one anonymous reviewer for helpful technical comments that resulted in a refinement of the ε-optimality statement as provided in the proof of Theorem 3.2. We also acknowledge NSF grants CMMI-1362003 and CMMI-1100868 that supported this research.
Publisher Copyright:
© 2019

PY - 2019/10/1

Y1 - 2019/10/1

N2 - We study distributionally robust optimization (DRO) problems where the ambiguity set is defined using the Wasserstein metric and can account for a bounded support. We show that this class of DRO problems can be reformulated as decomposable semi-infinite programs. We use a cutting-surface method to solve the reformulated problem for the general nonlinear model, assuming that we have a separation oracle. As examples, we consider the problems arising from the machine learning models where variables couple with data in a bilinear form in the loss function. We present a branch-and-bound algorithm to solve the separation problem for this case using an iterative piece-wise linear approximation scheme. We use a distributionally robust generalization of the logistic regression model to test our algorithm. We also show that it is possible to approximate the logistic-loss function with significantly less linear pieces than those needed for a general loss function to achieve a given accuracy when generating a cutting surface. Numerical experiments on the distributionally robust logistic regression models show that the number of oracle calls are typically 20–50 to achieve 5-digit precision. The solution found by the model has better predicting power than classical logistic regression when the sample size is small.

AB - We study distributionally robust optimization (DRO) problems where the ambiguity set is defined using the Wasserstein metric and can account for a bounded support. We show that this class of DRO problems can be reformulated as decomposable semi-infinite programs. We use a cutting-surface method to solve the reformulated problem for the general nonlinear model, assuming that we have a separation oracle. As examples, we consider the problems arising from the machine learning models where variables couple with data in a bilinear form in the loss function. We present a branch-and-bound algorithm to solve the separation problem for this case using an iterative piece-wise linear approximation scheme. We use a distributionally robust generalization of the logistic regression model to test our algorithm. We also show that it is possible to approximate the logistic-loss function with significantly less linear pieces than those needed for a general loss function to achieve a given accuracy when generating a cutting surface. Numerical experiments on the distributionally robust logistic regression models show that the number of oracle calls are typically 20–50 to achieve 5-digit precision. The solution found by the model has better predicting power than classical logistic regression when the sample size is small.

KW - Distributionally robust optimization

KW - Logistic regression

KW - Robustness and sensitivity analysis

KW - Semi-infinite programming

KW - Wasserstein metric

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U2 - 10.1016/j.ejor.2019.03.008

DO - 10.1016/j.ejor.2019.03.008

M3 - Article

AN - SCOPUS:85063958018

VL - 278

SP - 20

EP - 35

JO - European Journal of Operational Research

JF - European Journal of Operational Research

SN - 0377-2217

IS - 1

ER -