Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networks
This work addresses a computational bottleneck in optimal transport and probability optimization, offering a method with potential applications in machine learning and scientific domains, though it appears incremental as it builds on existing ICNN and JKO frameworks.
The authors tackled the challenge of optimizing functionals over probability spaces using gradient flows, which is difficult in high dimensions due to the need to optimize over convex functions. They proposed using input-convex neural networks to approximate the JKO scheme, demonstrating feasibility in low-dimensional PDEs and viability in high-dimensional molecular generation.
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.