Forward Tangent-Centered Sinkhorn DRO: Exact Nominal Recovery from Optimal Transport to KL

Abstract

Fixed-reference entropy-regularized transport yields smooth distributionally robust optimization (DRO) objectives, but its value as a function of the adversarial marginal is generally minimized at a kernel-smoothed law rather than at the nominal distribution. Consequently, the smallest raw ambiguity set need not recover nominal risk. Building on prior information-geometric plan-KL orientations, we study the forward tangent-centered discrepancy obtained by subtracting the value and first variation at the nominal law. This is the unique affine recentering, up to the simplex gauge; for symmetric costs, its plan-KL representation coincides with the forward candidate noted by Amari et al. (2018). The representation shows that the discrepancy dominates marginal KL, is uniquely zero at the nominal law, and is strongly convex in total variation. The resulting DRO problem retains a scalar log-sum-exp dual and closed-form conditional adversarial weights. On finite supports with symmetric zero-diagonal costs, the discrepancy converges uniformly to the unregularized transport cost at low temperature; for any finite cost, its rescaled value converges uniformly to marginal KL at high temperature. We derive its exact local Hessian, an explicit finite-sample containment radius, and a generalized chi-square calibration for multinomial data. Controlled simulations verify nominal centering, both limiting regimes, and practical plug-in coverage.