Sharp minimax bounds for DRO with optimal transport discrepancy

Determine whether distributionally robust optimization (DRO) estimators with uncertainty sets defined via optimal transport discrepancy (e.g., Wasserstein distance) achieve sharp minimax bounds for the excess worst-case loss sup_{Q in B_δ(P_*)} E_Q[ℓ(θ̂_{n,δ}, ξ)] − inf_{θ∈Θ} sup_{Q in B_δ(P_*)} E_Q[ℓ(θ, ξ)], analogous to the results established for φ-divergence-based DRO estimators with fixed radius.

Background

The paper reviews statistical optimality results for DRO estimators. For φ-divergence-based uncertainty sets with fixed radius, prior work has established sharp minimax bounds for the worst-case excess risk, demonstrating uniform performance guarantees.

The authors note that such worst-case losses may be conservative and explicitly highlight that an analogous sharp minimax theory for other uncertainty structures—specifically optimal transport (Wasserstein) discrepancy—has not been established. This identifies a concrete gap in current theory regarding whether the same type of sharp minimax guarantees hold in the optimal transport setting.