Matrix-Free Preconditioning in Online Learning
This work addresses computational efficiency in online learning for practitioners, though it is incremental with mild improvements in logarithmic factors.
The paper tackles the problem of online convex optimization by introducing an algorithm that achieves regret bounds between those of optimal full-matrix preconditioning and diagonal preconditioning, with computational efficiency matching online gradient descent. It demonstrates performance improvements in synthetic and deep learning benchmarks.
We provide an online convex optimization algorithm with regret that interpolates between the regret of an algorithm using an optimal preconditioning matrix and one using a diagonal preconditioning matrix. Our regret bound is never worse than that obtained by diagonal preconditioning, and in certain setting even surpasses that of algorithms with full-matrix preconditioning. Importantly, our algorithm runs in the same time and space complexity as online gradient descent. Along the way we incorporate new techniques that mildly streamline and improve logarithmic factors in prior regret analyses. We conclude by benchmarking our algorithm on synthetic data and deep learning tasks.