Flow-based Distributionally Robust Optimization

We present a computationally efficient framework, called \texttt{FlowDRO}, for solving flow-based distributionally robust optimization (DRO) problems with Wasserstein uncertainty sets, when requiring the worst-case distribution (also called the Least Favorable Distribution, LFD) to be continuous so that the algorithm can be scalable to problems with larger sample sizes and achieve better generalization capability for the induced robust algorithms. To tackle the computationally challenging infinitely dimensional optimization problem, we leverage flow-based models, continuous-time invertible transport maps between the data distribution and the target distribution, and develop a Wasserstein proximal gradient flow type of algorithm. In practice, we parameterize the transport maps by a sequence of neural networks progressively trained in blocks by gradient descent. Our computational framework is general, can handle high-dimensional data with large sample sizes, and can be useful for various applications. We demonstrate its usage in adversarial learning, distributionally robust hypothesis testing, and a new mechanism for data-driven distribution perturbation differential privacy, where the proposed method gives strong empirical performance on real high-dimensional data.