Learning with Stochastic Orders

Carles Domingo-Enrich, Yair Schiff, Youssef Mroueh

Keywords: optimal transport, stochastic order, Choquet order, convex function, input convex neural network, integral probability metric, image generation, statistical rates

TL;DR: We propose and study discrepancies and distances between probability measures that arise from the convex or Choquet order, which capture dominance constraints and are useful in applications like image generation.

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, exploiting the relation between convex orders and optimal transport, 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. 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. Finally, our ICMNs class of convex functions and its derived Rademacher Complexity are of independent interest beyond their application in convex orders. Code to reproduce experimental results is available at https://github.com/yair-schiff/stochastic-orders-ICMN.

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