NysAct: A Scalable Preconditioned Gradient Descent using Nystrom Approximation

Adaptive gradient methods are computationally efficient and converge quickly, but they often suffer from poor generalization. In contrast, second-order methods enhance convergence and generalization but typically incur high computational and memory costs. In this work, we introduce NysAct, a scalable first-order gradient preconditioning method that strikes a balance between state-of-the-art first-order and second-order optimization methods. NysAct leverages an eigenvalue-shifted Nystrom method to approximate the activation covariance matrix, which is used as a preconditioning matrix, significantly reducing time and memory complexities with minimal impact on test accuracy. Our experiments show that NysAct not only achieves improved test accuracy compared to both first-order and second-order methods but also demands considerably less computational resources than existing second-order methods. Code is available at https://github.com/hseung88/nysact.

MAC: An Efficient Gradient Preconditioning using Mean Activation Approximated Curvature

Second-order optimization methods for training neural networks, such as KFAC, exhibit superior convergence by utilizing curvature information of loss landscape. However, it comes at the expense of high computational burden. In this work, we analyze the two components that constitute the layer-wise Fisher information matrix (FIM) used in KFAC: the Kronecker factors related to activations and pre-activation gradients. Based on empirical observations on their eigenspectra, we propose efficient approximations for them, resulting in a computationally efficient optimization method called MAC. To the best of our knowledge, MAC is the first algorithm to apply the Kronecker factorization to the FIM of attention layers used in transformers and explicitly integrate attention scores into the preconditioning. We also study the convergence property of MAC on nonlinear neural networks and provide two conditions under which it converges to global minima. Our extensive evaluations on various network architectures and datasets show that the proposed method outperforms KFAC and other state-of-the-art methods in terms of accuracy, end-to-end training time, and memory usage. Code is available at https://github.com/hseung88/mac.

An Adaptive Method Stabilizing Activations for Enhanced Generalization

We introduce AdaAct, a novel optimization algorithm that adjusts learning rates according to activation variance. Our method enhances the stability of neuron outputs by incorporating neuron-wise adaptivity during the training process, which subsequently leads to better generalization -- a complementary approach to conventional activation regularization methods. Experimental results demonstrate AdaAct's competitive performance across standard image classification benchmarks. We evaluate AdaAct on CIFAR and ImageNet, comparing it with other state-of-the-art methods. Importantly, AdaAct effectively bridges the gap between the convergence speed of Adam and the strong generalization capabilities of SGD, all while maintaining competitive execution times. Code is available at https://github.com/hseung88/adaact.