Low-Rank Gradient Descent

Several recent empirical studies demonstrate that important machine learning tasks such as training deep neural networks, exhibit a low-rank structure, where most of the variation in the loss function occurs only in a few directions of the input space. In this article, 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 $\epsilon$ -approximate stationary point 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 complexities” of LRGD for strongly convex and non-convex objective functions are ${\mathcal {O}}(r \log (1/\epsilon) + rp)$ and ${\mathcal {O}}(r/\epsilon ^{2} + rp)$ , respectively. Therefore, when $r \ll p$ , LRGD provides significant improvement over the known complexities of ${\mathcal {O}}(p \log (1/\epsilon))$ and ${\mathcal {O}}(p/\epsilon ^{2})$ of GD in the strongly convex and non-convex settings, respectively. Furthermore, we formally characterize the classes of exactly and approximately low-rank functions. Empirically, using real and synthetic data, 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 . This suggests that LRGD could be used in practice in any setting in place of GD .