Spectral-factorized Positive-definite Curvature Learning for NN Training

Many training methods, such as Adam(W) and Shampoo, learn a positive-definite curvature matrix and apply an inverse root before preconditioning. Recently, non-diagonal training methods, such as Shampoo, have gained significant attention; however, they remain computationally inefficient and are limited to specific types of curvature information due to the costly matrix root computation via matrix decomposition. To address this, we propose a Riemannian optimization approach that dynamically adapts spectral-factorized positive-definite curvature estimates, enabling the efficient application of arbitrary matrix roots and generic curvature learning. We demonstrate the efficacy and versatility of our approach in positive-definite matrix optimization and covariance adaptation for gradient-free optimization, as well as its efficiency in curvature learning for neural net training.