Accelerated Stochastic Matrix Inversion: General Theory and Speeding up BFGS Rules for Faster Second-Order Optimization
For optimization and machine learning practitioners, this provides faster second-order optimization methods via accelerated matrix inversion and quasi-Newton updates.
The paper presents the first accelerated randomized algorithm for matrix inversion, which can invert positive definite matrices while maintaining positive definiteness of iterates. This leads to accelerated quasi-Newton updates that outperform classical rules in numerical experiments and accelerate training of machine learning models.
We present the first accelerated randomized algorithm for solving linear systems in Euclidean spaces. One essential problem of this type is the matrix inversion problem. In particular, our algorithm can be specialized to invert positive definite matrices in such a way that all iterates (approximate solutions) generated by the algorithm are positive definite matrices themselves. This opens the way for many applications in the field of optimization and machine learning. As an application of our general theory, we develop the {\em first accelerated (deterministic and stochastic) quasi-Newton updates}. Our updates lead to provably more aggressive approximations of the inverse Hessian, and lead to speed-ups over classical non-accelerated rules in numerical experiments. Experiments with empirical risk minimization show that our rules can accelerate training of machine learning models.