Large Batch Analysis for Adagrad Under Anisotropic Smoothness

Adaptive gradient algorithms have been widely adopted in training large-scale deep neural networks, especially large foundation models. Despite their huge success in practice, their theoretical advantages over stochastic gradient descent (SGD) have not been fully understood, especially in the large batch-size setting commonly used in practice. This is because the only theoretical result that can demonstrate the benefit of Adagrad over SGD was obtained in the original paper of Adagrad for nonsmooth objective functions. However, for nonsmooth objective functions, there can be a linear slowdown of convergence when batch size increases, and thus a convergence analysis based on nonsmooth assumption cannot be used for large batch algorithms. In this work, we resolve this gap between theory and practice by providing a new analysis of Adagrad on both convex and nonconvex smooth objectives suitable for the large batch setting. It is shown that under the anisotropic smoothness and noise conditions, increased batch size does not slow down convergence for Adagrad, and thus it can still achieve a faster convergence guarantee over SGD even in the large batch setting. We present detailed comparisons between SGD and Adagrad to provide a better understanding of the benefits of adaptive gradient methods. Experiments in logistic regression and instruction following fine-tuning tasks provide strong evidence to support our theoretical analysis.

On the Complexity of Finite-Sum Smooth Optimization under the Polyak-Łojasiewicz Condition

This paper considers the optimization problem of the form $\min_{{\bf x}\in{\mathbb R}^d} f({\bf x})\triangleq \frac{1}{n}\sum_{i=1}^n f_i({\bf x})$, where $f(\cdot)$ satisfies the Polyak--{\L}ojasiewicz (PL) condition with parameter $\mu$ and $\{f_i(\cdot)\}_{i=1}^n$ is $L$-mean-squared smooth. We show that any gradient method requires at least $\Omega(n+\kappa\sqrt{n}\log(1/\epsilon))$ incremental first-order oracle (IFO) calls to find an $\epsilon$-suboptimal solution, where $\kappa\triangleq L/\mu$ is the condition number of the problem. This result nearly matches upper bounds of IFO complexity for best-known first-order methods. We also study the problem of minimizing the PL function in the distributed setting such that the individuals $f_1(\cdot),\dots,f_n(\cdot)$ are located on a connected network of $n$ agents. We provide lower bounds of $\Omega(\kappa/\sqrt{\gamma}\,\log(1/\epsilon))$, $\Omega((\kappa+\tau\kappa/\sqrt{\gamma}\,)\log(1/\epsilon))$ and $\Omega\big(n+\kappa\sqrt{n}\log(1/\epsilon)\big)$ for communication rounds, time cost and local first-order oracle calls respectively, where $\gamma\in(0,1]$ is the spectral gap of the mixing matrix associated with the network and~$\tau>0$ is the time cost of per communication round. Furthermore, we propose a decentralized first-order method that nearly matches above lower bounds in expectation.

Accelerated Convergence of Stochastic Heavy Ball Method under Anisotropic Gradient Noise

Heavy-ball momentum with decaying learning rates is widely used with SGD for optimizing deep learning models. In contrast to its empirical popularity, the understanding of its theoretical property is still quite limited, especially under the standard anisotropic gradient noise condition for quadratic regression problems. Although it is widely conjectured that heavy-ball momentum method can provide accelerated convergence and should work well in large batch settings, there is no rigorous theoretical analysis. In this paper, we fill this theoretical gap by establishing a non-asymptotic convergence bound for stochastic heavy-ball methods with step decay scheduler on quadratic objectives, under the anisotropic gradient noise condition. As a direct implication, we show that heavy-ball momentum can provide $\tilde{\mathcal{O}}(\sqrt{\kappa})$ accelerated convergence of the bias term of SGD while still achieving near-optimal convergence rate with respect to the stochastic variance term. The combined effect implies an overall convergence rate within log factors from the statistical minimax rate. This means SGD with heavy-ball momentum is useful in the large-batch settings such as distributed machine learning or federated learning, where a smaller number of iterations can significantly reduce the number of communication rounds, leading to acceleration in practice.