Faster Optimization on Sparse Graphs via Neural Reparametrization
This method accelerates optimization for problems on sparse graphs, such as heat diffusion and synchronization, but is incremental as it builds on existing GNN and optimization techniques.
The paper tackles the computational expense of second-order Newton's methods in optimization by proposing neural reparametrization, which uses graph neural networks to implement an efficient Quasi-Newton method on sparse graphs, achieving speed-ups of 10-100x.
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.