We also study the more general case of an additional entropy regularizer.

We consider the penalized distributionally robust optimization (DRO) problem with a closed, convex uncertainty set, a setting that encompasses learning using f -DRO and spectral/ L-risk minimization.

We identify a natural and necessary condition called the "Optimal

GPand JPCaresupportedbythe Simons Foundation, McKnight Foundation, Grossman Centerforthe Statisticsof Mind, and Gatsby Charitable Trust.

We study whether and how can we model a joint distributionp(x,z)using two conditional modelsp(x|z) and q(z|x) that form a cycle.

Work done during an internship at Microsoft Research Asia.