Nonparametric Density Estimation & Convergence Rates for GANs under Besov IPM Losses

Along line ofwork has established convergence rates ofthe empirical distribution tothe true distribution in spaces as general as unbounded metric spaces [54, 25, 45]). In the Euclidean setting, this is well understood [14,2,18], although, to the best of our knowledge, minimax lower bounds have been proven only recently [45]; this setting intersects with our work in the caseσd = 1,σg = 0, pd =,matchingourminimaxrateofn 1/D+n 1/2.