Tensor Normal Training for Deep Learning Models

Yi Ren, Donald Goldfarb

Despite the predominant use of first-order methods for training deep learning models, second-order methods, and in particular, natural gradient methods, remain of interest because of their potential for accelerating training through the use of curvature information. Several methods with non-diagonal preconditioning matrices, including KFAC and Shampoo, have been proposed and shown to be effective. Based on the so-called tensor normal (TN) distribution, we propose and analyze a brand new approximate natural gradient method, Tensor Normal Training (TNT), which like Shampoo, only requires knowledge on the shape of the training parameters. By approximating the probabilistically based Fisher matrix, as opposed to the empirical Fisher matrix, our method uses the layer-wise covariance of the sampling based gradient as the pre-conditioning matrix. Moreover, the assumption that the sampling-based (tensor) gradient follows a TN distribution, ensures that its covariance has a Kronecker separable structure, which leads to a tractable approximation to the Fisher matrix. Consequently, TNT's memory requirements and per-iteration computational costs are only slightly higher than those for first-order methods. In our experiments, TNT exhibited superior optimization performance to KFAC and Shampoo, and to state-of-the-art first-order methods. Moreover, TNT demonstrated its ability to generalize as well as these first-order methods, using fewer epochs.

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