**Abstract**: Learning high-dimensional distributions is often done with explicit likelihood modeling or implicit modeling via minimizing integral probability metrics (IPMs). In this paper, we expand this learning paradigm to stochastic orders, namely, the convex or Choquet order between probability measures. Towards this end, we introduce the Choquet-Toland distance between probability measures, that can be used as a drop-in replacement for IPMs. We also introduce the Variational Dominance Criterion (VDC) to learn probability measures with dominance constraints, that encode the desired stochastic order between the learned measure and a known baseline. We analyze both quantities and show that they suffer from the curse of dimensionality and propose surrogates via input convex maxout networks (ICMNs), that enjoy parametric rates. Finally, we provide a min-max framework for learning with stochastic orders and validate it experimentally on synthetic and high-dimensional image generation, with promising results. The code is available at https://github.com/yair-schiff/stochastic-orders-ICMN.

Joint work with Carles Domingo Enrich and Yair Schiff.

**Bio**: Youssef Mroueh is a research staff member in IBM since April 2015. He received his PhD in computer science in February 2015 from MIT, CSAIL, where he was advised by Professor Tomaso Poggio. In 2011, he obtained his engineering diploma from Ecole Polytechnique Paris France, and a Master of Science in Applied Maths from Ecole des Mines de Paris. He is interested in Deep Learning, Machine Learning, Optimal transport, multimodal learning, Statistical Learning Theory, Computer Vision and Artificial Intelligence.