A minimax approach to duality for linear distributional sensitivity testing

Abstract

We consider the dual formulation of the problem of finding the maximum of 𝔼𝜈⁡[𝑓⁡(𝑋)], where ν is allowed to vary over all the probability measures on a Polish space 𝒳 for which 𝑑𝑐⁡(𝜇,𝜈)≤𝑟, with 𝑑𝑐 an optimal transport distance, f a real-valued function on 𝒳 satisfying some regularity, μ a ‘baseline’ measure and 𝑟≥ 0. Whereas some derivations of the dual rely on Fenchel duality, applied on a vector space of functions in duality with a vector space of measures, we impose compactness on 𝒳 to allow the use of the minimax theorem of Ky Fan, which does not require vector space structure.