Gradient Descent with Low-Rank Objective Functions

Romain Cosson, Ali Jadbabaie, Anuran Makur, Amirhossein Reisizadeh, Devavrat Shah

Abstract: Several recent empirical studies demonstrate that important machine learning tasks, e.g., training deep neural networks, exhibit low-rank structure, where the loss function varies significantly in only a few directions of the input space. In this paper, we leverage such low-rank structure to reduce the high computational cost of canonical gradient-based methods such as gradient descent (GD). Our proposed Low-Rank Gradient Descent (LRGD) algorithm finds an $\epsilon$ -minimizer of a p-dimensional function by first identifying $r\leq p$ significant directions, and then estimating the true p-dimensional gradient at every iteration by computing directional derivatives only along those $r$ directions. We establish that the “directional oracle complexity” of LRGD for strongly convex objective functions is $\mathrm{O}(r\log(1/\epsilon)+rp)$ . Therefore, when $r\ll p$ , LRGD provides significant improvement over the known complexity of $\mathcal{O}(p\log(1/\epsilon))$ of GD in the strongly convex setting. Furthermore, using real and synthetic data, we empirically find that LRGD provides significant gains over GD when the data has low-rank structure, and in the absence of such structure, LRGD does not degrade performance compared to GD.