Faster Optimization on Sparse Graphs via Neural Reparametrization

TL;DR

This paper introduces a neural reparametrization technique using graph neural networks to efficiently implement a Quasi-Newton method on sparse graphs, significantly accelerating optimization tasks.

Contribution

It presents a novel GNN-based approach that leverages second-order information for faster optimization, compatible with existing optimizers and applicable to various scientific problems.

Findings

Achieves 10-100x speedup in optimization on sparse graphs.

Effectively utilizes second-order information via GNNs.

Applicable to heat diffusion, synchronization, and persistent homology.

Abstract

In mathematical optimization, second-order Newton's methods generally converge faster than first-order methods, but they require the inverse of the Hessian, hence are computationally expensive. However, we discover that on sparse graphs, graph neural networks (GNN) can implement an efficient Quasi-Newton method that can speed up optimization by a factor of 10-100x. Our method, neural reparametrization, modifies the optimization parameters as the output of a GNN to reshape the optimization landscape. Using a precomputed Hessian as the propagation rule, the GNN can effectively utilize the second-order information, reaching a similar effect as adaptive gradient methods. As our method solves optimization through architecture design, it can be used in conjunction with any optimizers such as Adam and RMSProp. We show the application of our method on scientifically relevant problems including heat diffusion, synchronization and persistent homology.