Learning with Stochastic Orders

TL;DR

This paper introduces new methods for learning high-dimensional probability distributions with stochastic order constraints, using optimal transport-based distances and neural network surrogates, validated on synthetic and image data.

Contribution

It proposes the Choquet-Toland distance and Variational Dominance Criterion for stochastic order learning, along with input convex maxout networks for scalable approximation.

Findings

The proposed methods effectively learn distributions with stochastic order constraints.

Surrogates via ICMNs achieve parametric rates despite high dimensionality.

Experimental results show promising performance on synthetic and image data.

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.