Distributionally Robust Optimization (DRO) accounts for uncertainty in data distributions by optimizing the model performance against the worst possible distribution within an ambiguity set. In this paper, we propose a DRO framework that relies on a new distance inspired by Unbalanced Optimal Transport (UOT). The proposed UOT distance employs a soft penalization term instead of hard constraints, enabling the construction of an ambiguity set that is more resilient to outliers. Under smoothness conditions, we establish strong duality of the proposed DRO problem. Moreover, we introduce a computationally efficient Lagrangian penalty formulation for which we show that strong duality also holds. Finally, we provide empirical results that demonstrate that our method offers improved robustness to outliers and is computationally less demanding for regression and classification tasks.

We consider an online stochastic game with risk-averse agents whose goal is to learn optimal decisions that minimize the risk of incurring significantly high costs. Specifically, we use the Conditional Value at Risk (CVaR) as a risk measure that the agents can estimate using bandit feedback in the form of the cost values of only their selected actions. Since the distributions of the cost functions depend on the actions of all agents that are generally unobservable, they are themselves unknown and, therefore, the CVaR values of the costs are difficult to compute. To address this challenge, we propose a new online risk-averse learning algorithm that relies on one-point zeroth-order estimation of the CVaR gradients computed using CVaR values that are estimated by appropriately sampling the cost functions. We show that this algorithm achieves sub-linear regret with high probability. We also propose two variants of this algorithm that improve performance. The first variant relies on a new sampling strategy that uses samples from the previous iteration to improve the estimation accuracy of the CVaR values. The second variant employs residual feedback that uses CVaR values from the previous iteration to reduce the variance of the CVaR gradient estimates. We theoretically analyze the convergence properties of these variants and illustrate their performance on an online market problem that we model as a Cournot game.