Accelerating adaptive online learning by matrix approximation

Abstract

Adaptive subgradient methods are able to leverage the second-order information of functions to improve the regret and have become popular for online learning and optimization. According to the amount of information used, these methods can be divided into diagonal-matrix version (ADA-DIAG) and full-matrix version (ADA-FULL). In practice, ADA-DIAG is the most commonly adopted instead of ADA-FULL, because ADA-FULL is computationally intractable in high dimensions though it has smaller regret when gradients are correlated. In this paper, we propose to employ techniques of matrix approximation to accelerate ADA-FULL and develop two methods based on random projections. Compared with ADA-FULL, at each iteration, our methods reduce the space complexity from $$O(d^2)$$ to $$O(\tau d)$$ and the time complexity from $$O(d^3)$$ to $$O(\tau ^2 d)$$ where d is the dimensionality of the data and $$\tau \ll d$$ is the number of random projections. Experimental results about online convex optimization and training convolutional neural networks show that our methods are comparable to ADA-FULL and outperform other state-of-the-art algorithms including ADA-DIAG.