Minimax Optimality for Factored Optimal-Transport Distributionally-Robust Policy Learning in Contextual Bandits

Offline contextual-bandit policies may be deployed under simultaneous shifts in the context distribution and reward laws. We study policy learning for a factored distributionally robust optimization (DRO) objective that models these shifts through separate ambiguity sets. Optimal transport (OT) ambiguity sets are particularly suitable for this purpose, allowing the transport cost to encode application-specific severity of a shift. We develop a pessimistic OT-DRO learner based on sharp confidence bounds for robust rewards and analyze context estimation using a distribution-dependent concentration bound. Requiring knowledge of neither the behavior policy nor the context distribution, our learner achieves a regret of order $D{C}_{\mathrm{rob}}^{\star }\sqrt{\log (|\mathcal{X}||\mathcal{A}|/\delta )/n}$. Our robust optimal-policy coverage coefficient $C_{\mathrm{rob}}^{\star }$ is a natural generalization of the single-policy coverage coefficient in prior non-robust pessimism work, capturing the additional statistical difficulty caused by worst-case context shift. A matching lower bound shows that this rate, including its dependence on $C_{\mathrm{rob}}^{\star }$, is minimax-optimal up to constants.