We then investigate in detail two adaptivelearning rates-anidealized exactlinesearch andAdaGrad-Norm -on the least squares problem.

This paper introduces a new problem-dependent regret measure for online convex optimization with smooth losses. The notion, which we call the $G^\star$ regret, depends on the cumulative squared gradient norm evaluated at the decision in hindsight $\sum_{t=1}^T \|\nabla \ell(x^\star)\|^2$. We show that the $G^\star$ regret strictly refines the existing $L^\star$ (small loss) regret, and that it can be arbitrarily sharper when the losses have vanishing curvature around the hindsight decision. We establish upper and lower bounds on the $G^\star$ regret and extend our results to dynamic regret and bandit settings. As a byproduct, we refine the existing convergence analysis of stochastic optimization algorithms in the interpolation regime. Some experiments validate our theoretical findings.

We develop a framework for analyzing the training and learning rate dynamics on a large class of high-dimensional optimization problems, which we call the high line, trained using one-pass stochastic gradient descent (SGD) with adaptive learning rates. We give exact expressions for the risk and learning rate curves in terms of a deterministic solution to a system of ODEs. We then investigate in detail two adaptive learning rates -- an idealized exact line search and AdaGrad-Norm -- on the least squares problem. When the data covariance matrix has strictly positive eigenvalues, this idealized exact line search strategy can exhibit arbitrarily slower convergence when compared to the optimal fixed learning rate with SGD. Moreover we exactly characterize the limiting learning rate (as time goes to infinity) for line search in the setting where the data covariance has only two distinct eigenvalues. For noiseless targets, we further demonstrate that the AdaGrad-Norm learning rate converges to a deterministic constant inversely proportional to the average eigenvalue of the data covariance matrix, and identify a phase transition when the covariance density of eigenvalues follows a power law distribution.

We give exact expressions for the risk and learning rate curves in terms of a deterministic solution to a system of ODEs.

We propose a conversion scheme that turns regret minimizing algorithms into fixed point iterations, with convergence guarantees following from regret bounds. The resulting iterations can be seen as a grand extension of the classical Krasnoselskii--Mann iterations, as the latter are recovered by converting the Online Gradient Descent algorithm. This approach yields new simple iterations for finding fixed points of non-self operators. We also focus on converting algorithms from the AdaGrad family of regret minimizers, and thus obtain fixed point iterations with adaptive guarantees of a new kind. Numerical experiments on various problems demonstrate faster convergence of AdaGrad-based fixed point iterations over Krasnoselskii--Mann iterations.

We develop a framework for analyzing the training and learning rate dynamics on a large class of high-dimensional optimization problems, which we call the high line, trained using one-pass stochastic gradient descent (SGD) with adaptive learning rates. We give exact expressions for the risk and learning rate curves in terms of a deterministic solution to a system of ODEs. We then investigate in detail two adaptive learning rates -- an idealized exact line search and AdaGrad-Norm -- on the least squares problem. When the data covariance matrix has strictly positive eigenvalues, this idealized exact line search strategy can exhibit arbitrarily slower convergence when compared to the optimal fixed learning rate with SGD. Moreover we exactly characterize the limiting learning rate (as time goes to infinity) for line search in the setting where the data covariance has only two distinct eigenvalues.

Recent results in non-convex stochastic optimization demonstrate the convergence of popular adaptive algorithms (e.g., AdaGrad) under the $(L_0, L_1)$-smoothness condition, but the rate of convergence is a higher-order polynomial in terms of problem parameters like the smoothness constants. The complexity guaranteed by such algorithms to find an $ε$-stationary point may be significantly larger than the optimal complexity of $Θ\left( ΔL σ^2 ε^{-4} \right)$ achieved by SGD in the $L$-smooth setting, where $Δ$ is the initial optimality gap, $σ^2$ is the variance of stochastic gradient. However, it is currently not known whether these higher-order dependencies can be tightened. To answer this question, we investigate complexity lower bounds for several adaptive optimization algorithms in the $(L_0, L_1)$-smooth setting, with a focus on the dependence in terms of problem parameters $Δ, L_0, L_1$. We provide complexity bounds for three variations of AdaGrad, which show at least a quadratic dependence on problem parameters $Δ, L_0, L_1$. Notably, we show that the decorrelated variant of AdaGrad-Norm requires at least $Ω\left( Δ^2 L_1^2 σ^2 ε^{-4} \right)$ stochastic gradient queries to find an $ε$-stationary point. We also provide a lower bound for SGD with a broad class of adaptive stepsizes. Our results show that, for certain adaptive algorithms, the $(L_0, L_1)$-smooth setting is fundamentally more difficult than the standard smooth setting, in terms of the initial optimality gap and the smoothness constants.

We present a novel unified analysis for a broad class of adaptive optimization algorithms with structured (e.g., layerwise, diagonal, and kronecker-factored) preconditioners for both online regret minimization and offline convex optimization. Our analysis not only provides matching rate to several important structured preconditioned algorithms including diagonal AdaGrad, full-matrix AdaGrad, and AdaGrad-Norm, but also gives an improved convergence rate for a one-sided variant of Shampoo over that of original Shampoo. Interestingly, more structured preconditioners (e.g., diagonal Adagrad, AdaGrad-Norm which use less space and compute) are often presented as computationally efficient approximations to full-matrix Adagrad, aiming for improved optimization performance through better approximations. Our unified analysis challenges this prevailing view and reveals, perhaps surprisingly, that more structured preconditioners, despite using less space and computation per step, can outperform their less structured counterparts. To demonstrate this, we show that one-sided Shampoo, which is relatively much cheaper than full-matrix AdaGrad could outperform it both theoretically and experimentally.

Adaptive optimizers have emerged as powerful tools in deep learning, dynamically adjusting the learning rate based on iterative gradients. These adaptive methods have significantly succeeded in various deep learning tasks, outperforming stochastic gradient descent (SGD). However, although AdaGrad is a cornerstone adaptive optimizer, its theoretical analysis is inadequate in addressing asymptotic convergence and non-asymptotic convergence rates on non-convex optimization. This study aims to provide a comprehensive analysis and complete picture of AdaGrad. We first introduce a novel stopping time technique from probabilistic theory to establish stability for the norm version of AdaGrad under milder conditions. We further derive two forms of asymptotic convergence: almost sure and mean-square. Furthermore, we demonstrate the near-optimal non-asymptotic convergence rate measured by the average-squared gradients in expectation, which is rarely explored and stronger than the existing high-probability results, under the mild assumptions. The techniques developed in this work are potentially independent of interest for future research on other adaptive stochastic algorithms.

Adaptive gradient methods are arguably the most successful optimization algorithms for neural network training. While it is well-known that adaptive gradient methods can achieve better dimensional dependence than stochastic gradient descent (SGD) under favorable geometry for stochastic convex optimization, the theoretical justification for their success in stochastic non-convex optimization remains elusive. In this paper, we aim to close this gap by analyzing the convergence rates of AdaGrad measured by the $\ell_1$-norm of the gradient. Specifically, when the objective has $L$-Lipschitz gradient and the stochastic gradient variance is bounded by $\sigma^2$, we prove a worst-case convergence rate of $\tilde{\mathcal{O}}(\frac{\sqrt{d}L}{\sqrt{T}} + \frac{\sqrt{d} \sigma}{T^{1/4}})$, where $d$ is the dimension of the problem.We also present a lower bound of ${\Omega}(\frac{\sqrt{d}}{\sqrt{T}})$ for minimizing the gradient $\ell_1$-norm in the deterministic setting, showing the tightness of our upper bound in the noiseless case. Moreover, under more fine-grained assumptions on the smoothness structure of the objective and the gradient noise and under favorable gradient $\ell_1/\ell_2$ geometry, we show that AdaGrad can potentially shave a factor of $\sqrt{d}$ compared to SGD. To the best of our knowledge, this is the first result for adaptive gradient methods that demonstrates a provable gain over SGD in the non-convex setting.