Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networks

Abstract

Gradient flows are a powerful tool for optimizing functionals in general metric spaces, including the space of probabilities endowed with the Wasserstein metric. A typical approach to solving this optimization problem relies on its connection to the dynamic formulation of optimal transport and the celebrated Jordan-Kinderlehrer-Otto (JKO) scheme. However, this formulation involves optimization over convex functions, which is challenging, especially in high dimensions. In this work, we propose an approach that relies on the recently introduced input-convex neural networks (ICNN) to parametrize the space of convex functions in order to approximate the JKO scheme, as well as in designing functionals over measures that enjoy convergence guarantees. We derive a computationally efficient implementation of this JKO-ICNN framework and experimentally demonstrate its feasibility and validity in approximating solutions of low-dimensional partial differential equations with known solutions. We also demonstrate its viability in high-dimensional applications through an experiment in controlled generation for molecular discovery.

References in corpus (8)

• PyTorch: An Imperative Style, High-Performance Deep Learning Library
• Variational Inference with Normalizing Flows
• Input Convex Neural Networks
• Residual Flows for Invertible Generative Modeling
• Approximate inference with Wasserstein gradient flows
• Discretization of functionals involving the Monge-Ampère operator
• Large-Scale Wasserstein Gradient Flows
• Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport

Cited by in corpus (4)

• Real-Time Machine-Learning-Based Optimization Using Input Convex Long Short-Term Memory Network
• Convergence of flow-based generative models via proximal gradient descent in Wasserstein space
• Large-Scale Wasserstein Gradient Flows
• Input Convex Gradient Networks