Distributionally Robust Optimization with Markovian Data

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LI, Mengmeng, Tobias SUTTER, Daniel KUHN, 2021. Distributionally Robust Optimization with Markovian Data. 38th International Conference on Machine Learning (Virtual), Jul 18, 2021 - Jul 24, 2021. In: MEILA, Marina, ed., Tong ZHANG, ed.. Proceedings of the 38th International Conference on Machine Learning, pp. 6493-6503. ISSN 2640-3498

@inproceedings{Li2021Distr-55737, title={Distributionally Robust Optimization with Markovian Data}, url={https://proceedings.mlr.press/v139/li21t.html}, year={2021}, number={139}, issn={2640-3498}, series={Proceedings of Machine Learning Research}, booktitle={Proceedings of the 38th International Conference on Machine Learning}, pages={6493--6503}, editor={Meila, Marina and Zhang, Tong}, author={Li, Mengmeng and Sutter, Tobias and Kuhn, Daniel} }

We study a stochastic program where the probability distribution of the uncertain problem parameters is unknown and only indirectly observed via finitely many correlated samples generated by an unknown Markov chain with d states. We propose a data-driven distributionally robust optimization model to estimate the problem’s objective function and optimal solution. By leveraging results from large deviations theory, we derive statistical guarantees on the quality of these estimators. The underlying worst-case expectation problem is nonconvex and involves O(d<sup>2</sup>) decision variables. Thus, it cannot be solved efficiently for large d. By exploiting the structure of this problem, we devise a customized Frank-Wolfe algorithm with convex direction-finding subproblems of size O(d). We prove that this algorithm finds a stationary point efficiently under mild conditions. The efficiency of the method is predicated on a dimensionality reduction enabled by a dual reformulation. Numerical experiments indicate that our approach has better computational and statistical properties than the state-of-the-art methods. Kuhn, Daniel Kuhn, Daniel 2021 terms-of-use 2021-12-02T12:37:11Z eng Sutter, Tobias 2021-12-02T12:37:11Z Li, Mengmeng Sutter, Tobias Li, Mengmeng Distributionally Robust Optimization with Markovian Data

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