Training deep neural networks (DNNs) is a structured optimization problem, because the parameters are naturally represented by matrices and tensors rather than simple vectors. Under this structural representation, it has been widely observed that gradients are low-rank and Hessians are approximately block-wise diagonal. These structured properties are crucial for designing efficient optimization algorithms but may not be utilized by current popular optimizers like Adam. In this paper, we present a novel optimization algorithm ASGO that capitalizes on these properties by employing a preconditioner that is adaptively updated using structured gradients. By fine-grained theoretical analysis, ASGO is proven to achieve superior convergence rates compared to existing structured gradient methods. Based on the convergence theory, we further demonstrate that ASGO can benefit from the low-rank and block-wise diagonal properties. We also discuss practical modifications of ASGO and empirically verify the effectiveness of the algorithm on language model tasks.

Deep Neural Networks (DNNs) are currently predominantly trained using first-order methods. Some of these methods (e.g., Adam, AdaGrad, and RMSprop, and their variants) incorporate a small amount of curvature information by using a diagonal matrix to precondition the stochastic gradient. Recently, effective second-order methods, such as KFAC, K-BFGS, Shampoo, and TNT, have been developed for training DNNs, by preconditioning the stochastic gradient by layer-wise block-diagonal matrices. Here we propose a “mini-block Fisher (MBF)” preconditioned stochastic gradient method, that lies in between these two classes of methods. Specifically, our method uses a block-diagonal approximation to the empirical Fisher matrix, where for each layer in the DNN, whether it is convolutional or feed-forward and fully connected, the associated diagonal block is itself block-diagonal and is composed of a large number of mini-blocks of modest size. Our novel approach utilizes the parallelism of GPUs to efficiently perform computations on the large number of matrices in each layer. Consequently, MBF’s per-iteration computational cost is only slightly higher than it is for first-order methods. The performance of MBF is compared to that of several baseline methods, on Autoencoder, Convolutional Neural Network (CNN), and Graph Convolutional Network (GCN) problems, to validate its effectiveness both in terms of time efficiency and generalization power. Finally, it is proved that an idealized version of MBF converges linearly.

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, Shampoo, and K-BFGS, 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 of the shape of the training parameters. By approximating the probabilistically based Fisher matrix, as opposed to the empirical Fisher matrix, our method uses the block-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 state-of-the-art first-order methods, and comparable optimization performance to the state-of-the-art second-order methods KFAC and Shampoo. Moreover, TNT demonstrated its ability to generalize as well as first-order methods, while using fewer epochs.

We consider the development of practical stochastic quasi-Newton, and in particular Kronecker-factored block diagonal BFGS and L-BFGS methods, for training deep neural networks (DNNs). In DNN training, the number of variables and components of the gradient n is often of the order of tens of millions and the Hessian has n^2 elements. Consequently, computing and storing a full n times n BFGS approximation or storing a modest number of (step, change in gradient) vector pairs for use in an L-BFGS implementation is out of the question. In our proposed methods, we approximate the Hessian by a block-diagonal matrix and use the structure of the gradient and Hessian to further approximate these blocks, each of which corresponds to a layer, as the Kronecker product of two much smaller matrices. This is analogous to the approach in KFAC , which computes a Kronecker-factored block diagonal approximation to the Fisher matrix in a stochastic natural gradient method. Because the indefinite and highly variable nature of the Hessian in a DNN, we also propose a new damping approach to keep the upper as well as the lower bounds of the BFGS and L-BFGS approximations bounded. In tests on autoencoder feed-forward network models with either nine or thirteen layers applied to three datasets, our methods outperformed or performed comparably to KFAC and state-of-the-art first-order stochastic methods.