Geometric Breakthroughs for Neural Network Stability

Training large neural networks demands healthy tensors to prevent numerical instability and improve learning. This post introduces a compelling approach: constraining weight matrices to submanifolds, enabling co-designed optimization algorithms. It details manifold optimization, particularly using the Stiefel manifold for weights, and presents the Manifold Muon optimizer, which outperformed AdamW in small experiments. The concept extends to "Modular Manifolds," an abstraction for principled, layer-wise learning rate budgeting based on Lipschitz sensitivity. This framework promises more robust and automatic neural network training, opening various research avenues.