Finding Low-Rank Matrix Weights in DNNs via Riemannian Optimization: RAdaGrad and RAdamW

Finding low-rank matrix weights is a key technique for addressing the high memory usage and computational demands of large models. Most existing algorithms rely on the factorization of the low-rank matrix weights, which is non-unique and redundant. Their convergence is slow especially when the target low-rank matrices are ill-conditioned, because the convergence rate depends on the condition number of the Jacobian operator for the factorization and the Hessian of the loss function with respect to the weight matrix. To address this challenge, we adopt the Riemannian gradient descent (RGD) algorithm on the Riemannian manifold of fixed-rank matrices to update the entire low-rank weight matrix. This algorithm completely avoids the factorization, thereby eliminating the negative impact of the Jacobian condition number.