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Distributionally robust optimization (DRO) has been shown to offer a principled way to regularize learning models. In this paper, we find that Tikhonov regularization is distributionally robust in an optimal transport sense (i.e. if an adversary chooses distributions in a suitable optimal transport neighborhood of the empirical measure), provided that suitable martingale constraints are also imposed. Further, we introduce a relaxation of the martingale constraints which not only provide a unified viewpoint to a class of existing robust methods but also lead to new regularization tools. To realize these novel tools, provably efficient computational algorithms are proposed. As a byproduct, the strong duality theorem proved in this paper can be potentially applied to other problems of independent interest.

We study the problem of transfer learning, observing that previous efforts to understand its information-theoretic limits do not fully exploit the geometric structure of the source and target domains. In contrast, our study first illustrates the benefits of incorporating a natural geometric structure within a linear regression model, which corresponds to the generalized eigenvalue problem formed by the Gram matrices of both domains. We next establish a finite-sample minimax lower bound, propose a refined model interpolation estimator that enjoys a matching upper bound, and then extend our framework to multiple source domains and generalized linear models. Surprisingly, as long as information is available on the distance between the source and target parameters, negative-transfer does not occur. Simulation studies show that our proposed interpolation estimator outperforms state-of-the-art transfer learning methods in both moderate- and high-dimensional settings.

Reinforcement learning (RL) has demonstrated remarkable achievements in simulated environments. However, carrying this success to real environments requires the important attribute of robustness, which the existing RL algorithms often lack as they assume that the future deployment environment is the same as the training environment (i.e. simulator) in which the policy is learned. This assumption often does not hold due to the discrepancy between the simulator and the real environment and, as a result, and hence renders the learned policy fragile when deployed. In this paper, we propose a novel distributionally robust Q-learning algorithm that learns the best policy in the worst distributional perturbation of the environment. Our algorithm first transforms the infinite-dimensional learning problem (since the environment MDP perturbation lies in an infinite-dimensional space) into a finite-dimensional dual problem and subsequently uses a multi-level Monte-Carlo scheme to approximate the dual value using samples from the simulator. Despite the complexity, we show that the resulting distributionally robust Q-learning algorithm asymptotically converges to optimal worst-case policy, thus making it robust to future environment changes. Simulation results further demonstrate its strong empirical robustness.

We revisit Markowitz’s mean-variance portfolio selection model by considering a distributionally robust version, in which the region of distributional uncertainty is around the empirical measure and the discrepancy between probability measures is dictated by the Wasserstein distance. We reduce this problem into an empirical variance minimization problem with an additional regularization term. Moreover, we extend the recently developed inference methodology to our setting in order to select the size of the distributional uncertainty as well as the associated robust target return rate in a data-driven way. Finally, we report extensive back-testing results on S&P 500 that compare the performance of our model with those of several well-known models including the Fama–French and Black–Litterman models. This paper was accepted by David Simchi-Levi, finance.

Le, Tam; Nguyen, Truyen; Yamada, Makoto; Blanchet, Jose, and Nguyen, Viet Anh. Adversarial Regression with Doubly Non-negative Weighting Matrices. Advances in Neural Information Processing Systems. M. Ranzato and A. Beygelzimer and Y. Dauphin and P.S. Liang and J. Wortman Vaughan, editors. Vol. 34, (2021). pp. 16964--16976. https://proceedings.neurips.cc/paper/2021/file/8cfef17bee2b7a75a3ce09d40b497f6b-Paper.pdf

Summary Estimators based on Wasserstein distributionally robust optimization are obtained as solutions of min-max problems in which the statistician selects a parameter minimizing the worst-case loss among all probability models within a certain distance from the underlying empirical measure in a Wasserstein sense. While motivated by the need to identify optimal model parameters or decision choices that are robust to model misspecification, these distributionally robust estimators recover a wide range of regularized estimators, including square-root lasso and support vector machines, among others. This paper studies the asymptotic normality of these distributionally robust estimators as well as the properties of an optimal confidence region induced by the Wasserstein distributionally robust optimization formulation. In addition, key properties of min-max distributionally robust optimization problems are also studied; for example, we show that distributionally robust estimators regularize the loss based on its derivative, and we also derive general sufficient conditions which show the equivalence between the min-max distributionally robust optimization problem and the corresponding max-min formulation.