Conservation laws for neural network training: two papers at @NeurIPS23 & @ICML24

We study conservation laws during the (euclidean or not) gradient or momentum flow of neural networks.

Keep the Momentum: Conservation Laws beyond Euclidean Gradient Flows, accepted at ICML24

1/ We define the concept of conservation laws for momentum flows and show how to extend the framework from our previous paper (Abide by the Law and Follow the Flaw: Conservation Laws for Gradient Flows, oral @NeurIPS23) for non-Euclidean gradient flow (GF) and momentum flow (MF) settings. In stark contrast to the case of GF, conservation laws for MF exhibit temporal dependence.

2/ We discover new conservation laws for linear networks in the Euclidean momentum case, and these new laws are complete. In contrast, there is no conservation law for ReLU networks in the Euclidean momentum case

3/ In a non-Euclidean context, such as in NMF or for ICNN implemented with two-layer ReLU networks, we discover new conservation laws for gradient flows and find none in the momentum case. We obtain new conservation laws in the Natural Gradient Flow case.

4/ We shed light on a quasi-systematic loss of conservation when transitioning from the GF to the MF setting.