Reviews: Nonparametric Density Estimation under Adversarial Losses

Overview: This paper looks at nonparametric density estimation under certain classes of metrics, which the authors call "adversarial losses". To define adversarial losses, assume we have two probability distributions P and Q, and suppose X P and Y Q. Consider the supremum of E[f(X)] - E[f(Y)], when f ranges over a class F_d of functions. This is the "adversarial loss with respect to class F_d", or simply the "F_d-metric", and generalizes several metrics, for instance the L_1 metric. Now, assume P is an unkown distribution belonging to some known class F_G, and we have n i.i.d. How small can we make this distance?