RMNP: Row-Momentum Normalized Preconditioning for Scalable Matrix-Based Optimization

Preconditioned adaptive methods have gained significant attention for training deep neural networks, as they capture rich curvature information of the loss landscape . The central challenge in this field lies in balancing preconditioning effectiveness with computational efficiency of implementing the preconditioner. Among recent advances, \textsc{Muon} stands out by using Newton-Schulz iteration to obtain preconditioned updates without explicitly constructing the preconditioning matrix. Despite its advantages, the efficiency of \textsc{Muon} still leaves room for further improvement. In this paper, we introduce \textsc{RMNP} (Row Momentum Normalized Preconditioning), an optimizer that replaces Newton-Schulz iteration with a simple row-wise $\ell_2$ normalization operation, motivated by the empirically observed diagonal block structure of the Transformer layerwise Hessian. This substitution reduces the per-iteration computational complexity from $\mathcal{O}(mn\cdot\min(m,n))$ to $\mathcal{O}(mn)$ for an $m\times n$ weight matrix while maintaining comparable optimization performance. Theoretically, we establish convergence guarantees for \textsc{RMNP} in the non-convex setting that match recent results for \textsc{Muon} optimizers, achieving the information-theoretic minimax optimal complexity. Extensive experiments on large language model pretraining show that \textsc{RMNP} delivers competitive optimization performance compared with \textsc{Muon} while substantially reducing preconditioning wall-clock time. Our code is available at \href{https://anonymous.4open.science/r/RMNP-E8E1/}{this link}.

HTMuon: Improving Muon via Heavy-Tailed Spectral Correction

Muon has recently shown promising results in LLM training. In this work, we study how to further improve Muon. We argue that Muon's orthogonalized update rule suppresses the emergence of heavy-tailed weight spectra and over-emphasizes the training along noise-dominated directions. Motivated by the Heavy-Tailed Self-Regularization (HT-SR) theory, we propose HTMuon. HTMuon preserves Muon's ability to capture parameter interdependencies while producing heavier-tailed updates and inducing heavier-tailed weight spectra. Experiments on LLM pretraining and image classification show that HTMuon consistently improves performance over state-of-the-art baselines and can also serve as a plug-in on top of existing Muon variants. For example, on LLaMA pretraining on the C4 dataset, HTMuon reduces perplexity by up to $0.98$ compared to Muon. We further theoretically show that HTMuon corresponds to steepest descent under the Schatten-$q$ norm constraint and provide convergence analysis in smooth non-convex settings. The implementation of HTMuon is available at https://github.com/TDCSZ327/HTmuon.

Distributed Sign Momentum with Local Steps for Training Transformers

Pre-training Transformer models is resource-intensive, and recent studies have shown that sign momentum is an efficient technique for training large-scale deep learning models, particularly Transformers. However, its application in distributed training or federated learning remains underexplored. This paper investigates a novel communication-efficient distributed sign momentum method with local updates. Our proposed method allows for a broad class of base optimizers for local updates, and uses sign momentum in global updates, where momentum is generated from differences accumulated during local steps. We evaluate our method on the pre-training of various GPT-2 models, and the empirical results show significant improvement compared to other distributed methods with local updates. Furthermore, by approximating the sign operator with a randomized version that acts as a continuous analog in expectation, we present an $O(1/\sqrt{T})$ convergence for one instance of the proposed method for nonconvex smooth functions.

Large Deviations and Improved Mean-squared Error Rates of Nonlinear SGD: Heavy-tailed Noise and Power of Symmetry

We study large deviations and mean-squared error (MSE) guarantees of a general framework of nonlinear stochastic gradient methods in the online setting, in the presence of heavy-tailed noise. Unlike existing works that rely on the closed form of a nonlinearity (typically clipping), our framework treats the nonlinearity in a black-box manner, allowing us to provide unified guarantees for a broad class of bounded nonlinearities, including many popular ones, like sign, quantization, normalization, as well as component-wise and joint clipping. We provide several strong results for a broad range of step-sizes in the presence of heavy-tailed noise with symmetric probability density function, positive in a neighbourhood of zero and potentially unbounded moments. In particular, for non-convex costs we provide a large deviation upper bound for the minimum norm-squared of gradients, showing an asymptotic tail decay on an exponential scale, at a rate $\sqrt{t} / \log(t)$. We establish the accompanying rate function, showing an explicit dependence on the choice of step-size, nonlinearity, noise and problem parameters. Next, for non-convex costs and the minimum norm-squared of gradients, we derive the optimal MSE rate $\widetilde{\mathcal{O}}(t^{-1/2})$. Moreover, for strongly convex costs and the last iterate, we provide an MSE rate that can be made arbitrarily close to the optimal rate $\mathcal{O}(t^{-1})$, improving on the state-of-the-art results in the presence of heavy-tailed noise. Finally, we establish almost sure convergence of the minimum norm-squared of gradients, providing an explicit rate, which can be made arbitrarily close to $o(t^{-1/4})$.

Nonlinear Stochastic Gradient Descent and Heavy-tailed Noise: A Unified Framework and High-probability Guarantees

We study high-probability convergence in online learning, in the presence of heavy-tailed noise. To combat the heavy tails, a general framework of nonlinear SGD methods is considered, subsuming several popular nonlinearities like sign, quantization, component-wise and joint clipping. In our work the nonlinearity is treated in a black-box manner, allowing us to establish unified guarantees for a broad range of nonlinear methods. For symmetric noise and non-convex costs we establish convergence of gradient norm-squared, at a rate $\widetilde{\mathcal{O}}(t^{-1/4})$, while for the last iterate of strongly convex costs we establish convergence to the population optima, at a rate $\mathcal{O}(t^{-\zeta})$, where $\zeta \in (0,1)$ depends on noise and problem parameters. Further, if the noise is a (biased) mixture of symmetric and non-symmetric components, we show convergence to a neighbourhood of stationarity, whose size depends on the mixture coefficient, nonlinearity and noise. Compared to state-of-the-art, who only consider clipping and require unbiased noise with bounded $p$-th moments, $p \in (1,2]$, we provide guarantees for a broad class of nonlinearities, without any assumptions on noise moments. While the rate exponents in state-of-the-art depend on noise moments and vanish as $p \rightarrow 1$, our exponents are constant and strictly better whenever $p < 6/5$ for non-convex and $p < 8/7$ for strongly convex costs. Experiments validate our theory, demonstrating noise symmetry in real-life settings and showing that clipping is not always the optimal nonlinearity, further underlining the value of a general framework.

Revisiting Image Classifier Training for Improved Certified Robust Defense against Adversarial Patches

Certifiably robust defenses against adversarial patches for image classifiers ensure correct prediction against any changes to a constrained neighborhood of pixels. PatchCleanser arXiv:2108.09135 [cs.CV], the state-of-the-art certified defense, uses a double-masking strategy for robust classification. The success of this strategy relies heavily on the model's invariance to image pixel masking. In this paper, we take a closer look at model training schemes to improve this invariance. Instead of using Random Cutout arXiv:1708.04552v2 [cs.CV] augmentations like PatchCleanser, we introduce the notion of worst-case masking, i.e., selecting masked images which maximize classification loss. However, finding worst-case masks requires an exhaustive search, which might be prohibitively expensive to do on-the-fly during training. To solve this problem, we propose a two-round greedy masking strategy (Greedy Cutout) which finds an approximate worst-case mask location with much less compute. We show that the models trained with our Greedy Cutout improves certified robust accuracy over Random Cutout in PatchCleanser across a range of datasets and architectures. Certified robust accuracy on ImageNet with a ViT-B16-224 model increases from 58.1\% to 62.3\% against a 3\% square patch applied anywhere on the image.