Adaptive Gradient Methods

Shampoo and its efficient variant, SOAP, employ structured second-moment estimations and have shown strong performance for training neural networks (NNs). In practice, however, Shampoo typically requires step-size grafting with Adam to be competitive, and SOAP mitigates this by applying Adam in Shampoo's eigenbasis -- at the cost of additional memory overhead from Adam in both methods. Prior analyses have largely relied on the Frobenius norm to motivate these estimation schemes. We instead recast their estimation procedures as covariance estimation under Kullback-Leibler (KL) divergence minimization, revealing a previously overlooked theoretical limitation and motivating principled redesigns. Building on this perspective, we develop $\textbf{KL-Shampoo}$ and $\textbf{KL-SOAP}$, practical schemes that match or exceed the performance of Shampoo and SOAP in NN pre-training while achieving SOAP-level per-iteration runtime. Notably, KL-Shampoo does not rely on Adam to attain competitive performance, eliminating the memory overhead introduced by Adam. Across our experiments, KL-Shampoo consistently outperforms SOAP, Shampoo, and even KL-SOAP, establishing the KL-based approach as a promising foundation for designing structured methods in NN optimization. An implementation of KL-Shampoo/KL-SOAP is available at https://github.com/yorkerlin/KL-Methods

As the default optimizer for training large language models, AdamW has achieved remarkable success in deep learning. However, its convergence behavior is not theoretically well-understood. This paper establishes the convergence rate $\frac{1}{K}\sum_{k=1}^KE\left[||\nabla f(x^k)||_1\right]\leq O(\frac{\sqrt{d}C}{K^{1/4}})$ for AdamW measured by $\ell_1$ norm, where $K$ represents the iteration number, $d$ denotes the model dimension, and $C$ matches the constant in the optimal convergence rate of SGD. Theoretically, we have $||\nabla f(x)||_2\ll ||\nabla f(x)||_1\leq \sqrt{d}||\nabla f(x)||_2$ for any high-dimensional vector $x$ and $E\left[||\nabla f(x)||_1\right]\geq\sqrt{\frac{2d}π}E\left[||\nabla f(x)||_2\right]$ when each element of $\nabla f(x)$ is generated from Gaussian distribution $\mathcal N(0,1)$. Empirically, our experimental results on real-world deep learning tasks reveal $||\nabla f(x)||_1=\varTheta(\sqrt{d})||\nabla f(x)||_2$. Both support that our convergence rate can be considered to be analogous to the optimal $\frac{1}{K}\sum_{k=1}^KE\left[||\nabla f(x^k)||_2\right]\leq O(\frac{C}{K^{1/4}})$ convergence rate of SGD in the ideal case. We also extend our result to NAdamW, an AdamW variant that employs a double-momentum mechanism, and demonstrate that it maintains the same convergence rate.

Client-drift compensation methods mitigate the impact of data heterogeneity in federated learning (FL). While drift correction is well studied for stochastic gradient descent (SGD)-based FL, its interaction with adaptive client optimizers remains less understood. Adaptive optimizers such as Adam build each update from running averages of past gradients, so correcting only the final update direction leaves those averages, and the drift they contain, unchanged. We propose Federated Adaptive Gradient Correction ({\tt FA-GC}), which corrects each gradient before it enters these running averages, and compare it with Federated Adaptive Naive Tracking ({\tt FA-NT}), which adds the correction to the final direction. We show that, in an idealized setting without gradient noise, the direction FA-GC follows vanishes only at stationary points of the global objective, whereas the direction FA-NT follows can vanish elsewhere when the optimizer is nonlinear. We also prove convergence of FA-GC for Adam-family methods, gradient normalization, and Muon, which orthogonalizes matrix updates. Unlike our bound for FA-NT, these bounds contain no term for client heterogeneity. On seven image and language tasks, FA-GC with Adam needs the fewest rounds on average to reach the target accuracy, and on the three image tasks it also uses less total communication than uncorrected local Adam and FA-NT. On the QNLI question-entailment task with SOAP, which applies Adam in a rotated basis, and Muon under strong heterogeneity, FA-GC gives the highest accuracy, while FA-NT falls below the uncorrected optimizers.

Long-term time-series forecasting supports a wide range of applications, including weather prediction and electricity demand planning. Frequency-domain methods address this task by decomposing observations into components that describe temporal variations at different scales. However, separate representations do not by themselves provide an explicit mechanism for adjusting the training emphasis across components. Under a shared forecasting objective, the frequency-specific networks can retain different levels of coefficient prediction error, motivating an error-dependent adjustment to their gradients. To this end, we propose BEAT (Balanced frEquency Adaptive Tuning), a framework that combines frequency-specific error monitoring with adaptive gradient modulation. We design a Frequency-Specific Monitor that compares predicted and target wavelet coefficients in a common normalized space and expresses each discrepancy relative to a reference error computed from the detail components. We further introduce a Dynamical Gradient Balancer that converts these ratios into positive, bounded coefficients. Components with higher relative errors receive larger gradient weights, whereas those with lower relative errors receive smaller weights. A shared modulation-strength parameter controls the departure from unmodulated training, and the monitoring and balancing operations are used only during training. Experiments on seven real-world datasets show that BEAT achieves competitive performance against state-of-the-art forecasting methods.

The standard stochastic gradient descent (SGD) optimization method, as well as adaptive methods such as the Adam optimizer fail to converge if the learning rates do not converge to zero (particularly, in the situation of constant learning rates). In practice, human-tuned deterministic learning rate schedules or small constant learning rates are often used, and implementations in machine learning frameworks like Tensorflow and Pytorch typically employ constant learning rates. We propose a learning-rate-adaptive approach for SGD methods, adjusting the learning rate based on empirical estimates for the objective function values. Specifically, we propose a learning-rate-adaptive variant of the Adam optimizer and implement it for several machine learning problems, including deep learning methods for partial differential equations such as deep Kolmogorov methods, physics-informed neural networks, and deep Ritz methods. We refer to https://github.com/deeplearningmethods/adaptive-learning-rate for the Python source codes for the numerical simulations in this work. Our results show that the proposed adaptive Adam variant achieves faster reductions of the objective function value compared to Adam with default learning rates. For certain quadratic minimization problems, we rigorously prove that an adaptive SGD variant converges to the global minimizer. This proof uses properties of invariant measures of the SGD dynamics and a generalized convergence analysis for SGD with random predictable learning rates which we develop in this work.

One of the primary reasons behind the success of neural networks has been the emergence of an array of new, highly-successful optimizers, perhaps most importantly the Adam optimizer. It is widely used for training neural networks, yet notoriously hard to interpret. Lacking a clear physical intuition, Adam is difficult to generalize to manifolds. Some attempts have been made to directly apply parts of the Adam algorithm to manifolds or to find an underlying structure, but a full generalization has remained elusive. In this work a new approach is presented that leverages the special structure of the manifolds which are relevant for optimization of neural networks, such as the Stiefel manifold, the symplectic Stiefel manifold and the Grassmann manifold: all of these are homogeneous spaces and as such admit a global tangent space representation. This is a common vector space, often called the Lie subspace, that makes the generalization of all steps in the Adam optimizer (as well as other optimizers) possible. It is thus possible to extend the Adam optimizer to manifolds without a projection step, something that was not possible before. The resulting algorithm is then applied to train transformers and a symplectic autoencoder for which orthogonality constraints are enforced up to machine precision and we conclusively demonstrate the advantage of the proposed optimizer over existing methods.

Training logistic regression over encrypted data has emerged as a prominent approach to addressing security concerns in recent years. In this paper, we introduce an efficient gradient variant, termed the quadratic gradient, which is specifically designed for privacy-preserving logistic regression while remaining equally effective in plaintext optimization. By incorporating this quadratic gradient, we enhance Nesterov's Accelerated Gradient (NAG), Adaptive Gradient (AdaGrad), and Adam algorithms. We evaluate these enhanced algorithms across various datasets, with experimental results demonstrating state-of-the-art convergence rates that significantly outperform traditional first-order gradient methods. Furthermore, we apply the enhanced NAG method to implement homomorphic logistic regression training, achieving comparable performance within only four iterations. The proposed quadratic-gradient approach offers a unified framework that synergizes the advantages of first-order gradient methods and second-order Newton-type methods, suggesting broad applicability to diverse numerical optimization tasks.