arXiv subjects

Samuel Horváth

Publications and source records attributed to Samuel Horváth.

At least 19 recordsLinked to original sources

arXiv2026

High-Probability Convergence of Clipped SGD under Heavy-Tailed Noise and $(L_0,L_1)$-Smoothness

Gradient clipping is widely used in language-model training to control heavy-tailed gradient noise and can improve convergence guarantees over stochastic gradient descent (SGD) under $(L_0,L_1)$-smoothness. Under these joint conditions, a central challenge is to obtain high-probability guarantees without exponential dependence on $L_1R_0$, where $R_0$ bounds the initial distance to a minimizer. We resolve this challenge for convex objectives, establishing, to the best of our knowledge, the first such guarantees for standard Clip-SGD. We assume unbiased stochastic gradients with bounded central $α$-th moment, $α\in(1,2]$. Our bounds have only polylogarithmic dependence on the inverse failure probability and recover known deterministic generalized-smoothness rates when the noise vanishes, as well as the standard high-probability rate under heavy-tailed noise in the classical $L$-smooth setting. The convex rate is attained by a computable output that averages iterates whose sampled stochastic gradients are not clipped, requiring neither function values nor extra oracle calls. We establish a lower bound for Clip-SGD with any fixed stepsize and clipping level, showing that, under the stated iteration conditions, our convex stochastic rate is optimal up to logarithmic factors in the iteration budget. Our upper-bound analysis uses a directional clipping-bias bound to absorb part of the bias into the progress generated by the clipped population gradient, avoiding an exponentially large local smoothness constant. We also obtain nonconvex high-probability guarantees that recover the noiseless rate, match the classical-smooth stochastic iteration rate when $L_1=0$, and have no explicit $L_1$ dependence in the asymptotically dominant stochastic term.

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arXiv2026

LionMuon: Alternating Spectral and Sign Descent for Efficient Training

Pretraining a language model takes enormous compute, and the right optimizer can save a good part of it. Muon's spectral step gives a stronger direction than a sign step, but it is expensive. Every step runs Newton-Schulz iterations on the full matrix and, in distributed training, an extra all-reduce. Sign steps, as in Lion and Signum, are cheap and stay local to each device. We propose LionMuon, which takes one Muon step every $P$ iterations and Lion steps in between, with a single dual-EMA momentum buffer shared by both. Muon's compute and communication are paid once per $P$ steps, and the optimizer state is half of AdamW's. A single-EMA variant, SignMuon, already improves on Muon. We prove complexity bounds under heavy-tailed noise in which the period sets an interpolation between Muon's and Lion's smoothness and noise constants, and which say when LionMuon is faster than both. On 124M and 355M models trained on FineWeb, LionMuon with $P=2$ and $P=5$ reaches a lower loss than Muon, AdamW, Lion and Signum at the same number of tokens. Under 4-GPU data-parallel training it reaches Muon's final loss with a third less wall-clock on PCIe, and it beats the communication-efficient Muon variants Dion and MuonBP on loss at no more exposed communication, while keeping the exact gradient. Code: https://github.com/brain-lab-research/lion-muon

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arXiv2026

LionMuon: Alternating Spectral and Sign Descent for Efficient Training

In large-scale optimization, the cheapness and effectiveness of update steps are the most crucial factors for a successful optimizer. Sign-based optimizers like Lion or Signum produce cheap per-step updates, whereas Muon's spectral matrix-sign update gives a much stronger direction at a substantially higher per-step cost. In this work, we propose LionMuon, which retains the effectiveness of Muon steps while considerably cutting the averaged iteration cost, similar to sign-based methods. It alternates between Lion's and Muon's updates on a fixed period P, sharing a single dual-EMA momentum buffer between them. The optimizer state memory therefore matches Lion and is exactly half of AdamW's. A simpler single-EMA variant, SignMuon, by itself already outperforms pure Muon. At P = 2, LionMuon Pareto-dominates Muon, Lion, Signum, and AdamW on every dataset and architecture we tested at 124M model size, reaching lower validation loss at lower compute, and the same advantage persists at 355M and 720M scale. On the theory side, we prove sharp complexity bounds under heavy-tailed noise which are governed by period-averaged smoothness and noise that interpolate between Muon's and Lion's constants. These bounds predict the compute-optimal period and the conditions under which LionMuon outruns Muon and Lion. Code: https://github.com/brain-lab-research/lion-muon

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arXiv2026

One-Step Gradient Delay is Not a Barrier for Large-Scale Asynchronous Pipeline Parallel LLM Pretraining

Modern large-scale LLM pretraining benefits from utilizing Pipeline Parallelism; however, synchronous implementations leave GPUs idle during pipeline bubbles, wasting computational resources. Asynchronous Pipeline Parallelism eliminates these bubbles, maximizing throughput at the cost of gradient staleness. Among asynchronous schedules, PipeDream-2BW is particularly appealing: unlike the original PipeDream schedule, it ensures a constant one-step gradient delay regardless of pipeline depth. However, its adoption remains limited due to the common belief that optimizing under staleness is fundamentally unstable. In this work, we challenge this assumption, demonstrating that degradation under one-step delay depends strongly on optimizer choice rather than being an intrinsic limitation. We provide the first comprehensive empirical analysis showing that while AdamW, the predominant optimizer at the time when PipeDream-2BW was introduced, indeed suffers from severe degradation, recent methods like Muon exhibit strong robustness under a one-step delay. We introduce an optimizer-agnostic Error Feedback-inspired correction to further mitigate delay effects. We provide supporting theoretical analysis demonstrating convergence for Muon with and without this correction. Extensive evaluation on models up to 10B parameters confirms that our strategies bridge the performance gap with synchronous training, highlighting the practical potential of asynchronous pipeline parallelism at scale.

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arXiv2026

CacheMuon: Using Temporal Preconditioning To Approximate Polar Factor

Muon is an optimizer that computes updates using the polar factor of the momentum matrix and has shown strong empirical performance across a range of training settings. A key component of Muon is the Newton-Schulz iteration used to compute this polar factor. Although this avoids the cost of an exact singular value decomposition, it remains expensive in practice because it is applied at every optimization step. At the same time, the momentum matrix changes smoothly over training, suggesting strong temporal correlation in the corresponding polar factors. In this paper, we exploit this structure and propose CacheMuon, a temporal preconditioning method that reuses information from previous optimization steps to approximate the polar factor at the current step. This reduces redundant orthogonalization computation across iterations. We analyze CacheMuon as an inexact Muon update, with error controlled by fresh-solver error and cache staleness. Empirically, CacheMuon provides a controllable quality-efficiency frontier: conservative thresholds closely match fresh Muon on language-model and vision training while reducing orthogonalization FLOPs, whereas more aggressive thresholds yield larger arithmetic savings at the cost of modest validation-quality degradation.

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arXiv2026

FlexRank: Nested Low-Rank Knowledge Decomposition for Adaptive Model Deployment

The growing scale of deep neural networks, encompassing large language models (LLMs) and vision transformers (ViTs), has made training from scratch prohibitively expensive and deployment increasingly costly. These models are often used as computational monoliths with fixed cost, hindering adaptive deployment across different cost budgets. We argue that nested components, ordered by importance, can be extracted from pretrained models and selectively activated within the available computational budget. To this end, our proposed FlexRank method leverages low-rank weight decomposition with nested, importance-based consolidation to extract submodels of increasing capabilities. Our approach enables a "train-once, deploy-everywhere" paradigm offering a graceful trade-off between cost and performance without training from scratch for each budget - advancing practical deployment of large models.

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arXiv2026

Modulated learning for private and distributed regression with just a single sample per client device

This work focuses on the question of learning from a large number of devices with each device holding only a single sample of data. Several real-world applications exist to this one sample per client setup up including learning from fitness trackers, data/app usage aggregators, body-worn sensing devices, and daily event monitors to name a few. When a client has only one sample, the standard federated learning paradigm breaks down as a local update based on that single point is far from being useful, especially in the earlier rounds for estimation of the model coefficients. This utility is further weakened by the privacy-inducing noise applied at every round. This work caters to this problem to enable such clients to collaboratively contribute to effectively learn a global model without leaking the privacy of their data. The proposed approach injects a single, carefully calibrated noisy perturbation to transform the sample at each client, followed by a post-processed representation which is shared with the server. These representations aggregated at the server are processed to obtain an unbiased gradient update that in expectation matches the non-private centralized gradient while preserving data privacy. This approach is different than traditional private federated learning, where the communication payloads involve model coefficients as opposed to privately transformed data samples. This method enables devices with extremely limited data to collaborate and learn accurate, privacy-preserving models without requiring large local datasets or sacrificing individual privacy.

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arXiv2026

Preconditioned Norms: A Unified Framework for Steepest Descent, Quasi-Newton and Adaptive Methods

Optimization lies at the core of modern deep learning, yet existing methods often face a fundamental trade-off between adapting to problem geometry and leveraging curvature utilization. Steepest descent algorithms adapt to different geometries through norm choices but remain strictly first-order, whereas quasi-Newton and adaptive optimizers incorporate curvature information but are restricted to Frobenius geometry, limiting their applicability across diverse architectures. In this work, we propose a unified framework generalizing steepest descent, quasi-Newton methods, and adaptive methods through the novel notion of preconditioned matrix norms. This abstraction reveals that widely used optimizers such as SGD and Adam, as well as more advanced approaches like Muon and KL-Shampoo, and recent hybrids including SOAP and SPlus, all emerge as special cases of the same principle. Within this framework, we provide the first systematic treatment of affine and scale invariance in the matrix-parameterized setting, establishing necessary and sufficient conditions under generalized norms. Building on this foundation, we introduce two new methods, $\texttt{MuAdam}$ and $\texttt{MuAdam-SANIA}$, which combine the spectral geometry of Muon with Adam-style preconditioning. Our experiments demonstrate that these optimizers are competitive with, and in some cases outperform, existing state-of-the-art methods. Our code is available at https://github.com/brain-lab-research/LIB/tree/quasi_descent

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arXiv2026

Who to Trust? Aggregating Client Predictions in Federated Distillation

Under data heterogeneity (e.g., $\textit{class mismatch}$), clients may produce unreliable predictions for instances belonging to unfamiliar classes. An equally weighted combination of such predictions can corrupt the teacher signal used for distillation. In this paper, we provide a theoretical analysis of Federated Distillation and show that aggregating client predictions on a shared public dataset converges to a neighborhood of the optimum, where the neighborhood size is governed by the aggregation quality. We further propose two uncertainty-aware aggregation methods, $\mathbf{UWA}$ and $\mathbf{sUWA}$, which leverage density-based uncertainty estimates to down-weight unreliable client predictions. Experiments on image and text classification benchmarks demonstrate that our methods are particularly effective under high data heterogeneity, while matching standard averaging when heterogeneity is low.

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arXiv2026

Byzantine-Robust Optimization under $(L_0, L_1)$-Smoothness

We consider distributed optimization under Byzantine attacks in the presence of $(L_0,L_1)$-smoothness, a generalization of standard $L$-smoothness that captures functions with state-dependent gradient Lipschitz constants. We propose Byz-NSGDM, a normalized stochastic gradient descent method with momentum that achieves robustness against Byzantine workers while maintaining convergence guarantees. Our algorithm combines momentum normalization with Byzantine-robust aggregation enhanced by Nearest Neighbor Mixing (NNM) to handle both the challenges posed by $(L_0,L_1)$-smoothness and Byzantine adversaries. We prove that Byz-NSGDM achieves a convergence rate of $O(K^{-1/4})$ up to a Byzantine bias floor proportional to the robustness coefficient and gradient heterogeneity. Experimental validation on heterogeneous MNIST classification, synthetic $(L_0,L_1)$-smooth optimization, and character-level language modeling with a small GPT model demonstrates the effectiveness of our approach against various Byzantine attack strategies. An ablation study further shows that Byz-NSGDM is robust across a wide range of momentum and learning rate choices.

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arXiv2026

Learning in the Null Space: Small Singular Values for Continual Learning

Alleviating catastrophic forgetting while enabling further learning is a primary challenge in continual learning (CL). Orthogonal-based training methods have gained attention for their efficiency and strong theoretical properties, and many existing approaches enforce orthogonality through gradient projection. In this paper, we revisit orthogonality and exploit the fact that small singular values correspond to directions that are nearly orthogonal to the input space of previous tasks. Building on this principle, we introduce NESS (Null-space Estimated from Small Singular values), a CL method that applies orthogonality directly in the weight space rather than through gradient manipulation. Specifically, NESS constructs an approximate null space using the smallest singular values of each layer's input representation and parameterizes task-specific updates via a compact low-rank adaptation (LoRA-style) formulation constrained to this subspace. The subspace basis is fixed to preserve the null-space constraint, and only a single trainable matrix is learned for each task. This design ensures that the resulting updates remain approximately in the null space of previous inputs while enabling adaptation to new tasks. Our theoretical analysis and experiments on three benchmark datasets demonstrate competitive performance, low forgetting, and stable accuracy across tasks, highlighting the role of small singular values in continual learning. The code is available at https://github.com/pacman-ctm/NESS.

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arXiv2026

Beyond SGD, Without SVD: Proximal Subspace Iteration LoRA with Diagonal Fractional K-FAC

Low-Rank Adaptation (LoRA) fine-tunes large models by learning low-rank updates on top of frozen weights, dramatically reducing trainable parameters and memory. In this work, we address the gap between training with full steps with low-rank projections (SVDLoRA) and LoRA fine-tuning. We propose LoRSum, a memory-efficient subroutine that closes this gap for gradient descent by casting LoRA optimization as a proximal sub-problem and solving it efficiently with alternating least squares updates, which we prove to be an implicit block power method. We recover several recently proposed preconditioning methods for LoRA as special cases, and show that LoRSum can also be used for updating a low-rank momentum. In order to address full steps with preconditioned gradient descent, we propose a scaled variant of LoRSum that uses structured metrics such as K-FAC and Shampoo, and we show that storing the diagonal of these metrics still allows them to perform well while remaining memory-efficient. Experiments on a synthetic task, CIFAR-100, and language-model fine-tuning on GLUE, SQuAD v2, and WikiText-103, show that our method can match or improve LoRA baselines given modest compute overhead, while avoiding full-matrix SVD projections and retaining LoRA-style parameter efficiency.

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arXiv2026

What Scalable Second-Order Information Knows for Pruning at Initialization

Pruning remains an effective strategy for reducing both the costs and environmental impact associated with deploying large neural networks (NNs) while maintaining performance. Classical methods, such as OBD (LeCun et al., 1989) and OBS (Hassibi et al., 1992), demonstrate that utilizing curvature information can significantly enhance the balance between network complexity and performance. However, the computation and storage of the Hessian matrix make it impractical for modern NNs, motivating the use of approximations. Recent research (Gur et al., 2018; Karakida et al., 2019) suggests that the top eigenvalues guide optimization in a small subspace, are identifiable early, and remain consistent during training. Motivated by these findings, we revisit pruning at initialization (PaI) to evaluate scalable, unbiased second-order approximations, such as the Empirical Fisher and Hutchinson diagonals. Our experiments show that these methods capture sufficient curvature information to improve the identification of critical parameters compared to first-order baselines, while maintaining linear complexity. Additionally, we empirically demonstrate that updating batch normalization statistics as a warmup phase improves the performance of data-dependent criteria and mitigates the issue of layer collapse. Notably, Hutchinson-based criteria consistently outperformed or matched existing PaI algorithms across various models (including VGG, ResNet, and ViT) and datasets (such as CIFAR-10/100, TinyImageNet, and ImageNet). Our findings suggest that scalable second-order approximations strike an effective balance between computational efficiency and accuracy, making them a valuable addition to the pruning toolkit. We make our code available.

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arXiv2025

MT-DAO: Multi-Timescale Distributed Adaptive Optimizers with Local Updates

Training large models with distributed data parallelism (DDP) requires frequent communication of gradients across workers, which can saturate bandwidth. Infrequent communication strategies (e.g., Local SGD) reduce this overhead but, when applied to adaptive optimizers, often suffer a performance gap relative to fully synchronous DDP. We trace this gap to a time-scale mismatch: the optimizer's fast-moving momentum, tuned for frequent updates, decays too quickly to smooth gradients over long intervals, leading to noise-dominated optimization. To address this, we propose MT-DAO, a family of optimizers that employs multiple slow- and fast-moving first momenta or the gradient to track update dynamics across different time scales, for which we provide the first convergence guarantees. Empirically, for language-model pre-training, this eliminates the performance gap with DDP, outperforming infrequent-communication baselines in perplexity and reducing iso-token wall-clock time by 6-27% on Ethernet interconnects. At the 720M scale, MT-DAO reaches a target perplexity in 24% fewer steps and 35% less time than the single-momentum DDP baseline. MT-DAO enables effective cross-datacenter training and training over wide geographic areas.

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arXiv2025

Loss-Transformation Invariance in the Damped Newton Method

The Newton method is a powerful optimization algorithm, valued for its rapid local convergence and elegant geometric properties. However, its theoretical guarantees are usually limited to convex problems. In this work, we ask whether convexity is truly necessary. We introduce the concept of loss-transformation invariance, showing that damped Newton methods are unaffected by monotone transformations of the loss - apart from a simple rescaling of the step size. This insight allows difficult losses to be replaced with easier transformed versions, enabling convexification of many nonconvex problems while preserving the same sequence of iterates. Our analysis also explains the effectiveness of unconventional stepsizes in Newton's method, including values greater than one and even negative steps.

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arXiv2025

Differentially Private Clipped-SGD: High-Probability Convergence with Arbitrary Clipping Level

Gradient clipping is a fundamental tool in Deep Learning, improving the high-probability convergence of stochastic first-order methods like SGD, AdaGrad, and Adam under heavy-tailed noise, which is common in training large language models. It is also a crucial component of Differential Privacy (DP) mechanisms. However, existing high-probability convergence analyses typically require the clipping threshold to increase with the number of optimization steps, which is incompatible with standard DP mechanisms like the Gaussian mechanism. In this work, we close this gap by providing the first high-probability convergence analysis for DP-Clipped-SGD with a fixed clipping level, applicable to both convex and non-convex smooth optimization under heavy-tailed noise, characterized by a bounded central $α$-th moment assumption, $α\in (1,2]$. Our results show that, with a fixed clipping level, the method converges to a neighborhood of the optimal solution with a faster rate than the existing ones. The neighborhood can be balanced against the noise introduced by DP, providing a refined trade-off between convergence speed and privacy guarantees.

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arXiv2025

Faster Than SVD, Smarter Than SGD: The OPLoRA Alternating Update

Low-Rank Adaptation (LoRA) fine-tunes large models by learning low-rank updates on top of frozen weights, dramatically reducing trainable parameters and memory. However, there is still a gap between full training with low-rank projections (SVDLoRA) and LoRA fine-tuning, indicating that LoRA steps can be further improved. In this study, we propose OPLoRA, a memory-efficient optimizer that closes this gap by casting LoRA optimization as an interpretable sub-problem and solving it efficiently with alternating least squares updates, where 1-2 alternating steps are empirically found to be sufficient to closely match truncated SVD without ever forming the full matrix. We also retrieve the recently proposed preconditioning methods for LoRA as a special case. OPLoRA supports momentum by maintaining a low-rank estimate using the same subroutine (LoRSum) for computing the step, with a memory budget of 3 times the number of LoRA parameters (i.e., same as Adam). We also propose an experimental scaled variant that uses the K-FAC metric, which could be of interest. Across a linear task, MNIST, CIFAR-100, and RoBERTa-base (MNLI), OPLoRA consistently approaches SVDLoRA's performance using significantly less memory.

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arXiv2025

Simple Stepsize for Quasi-Newton Methods with Global Convergence Guarantees

Quasi-Newton methods are widely used for solving convex optimization problems due to their ease of implementation, practical efficiency, and strong local convergence guarantees. However, their global convergence is typically established only under specific line search strategies and the assumption of strong convexity. In this work, we extend the theoretical understanding of Quasi-Newton methods by introducing a simple stepsize schedule that guarantees a global convergence rate of ${O}(1/k)$ for the convex functions. Furthermore, we show that when the inexactness of the Hessian approximation is controlled within a prescribed relative accuracy, the method attains an accelerated convergence rate of ${O}(1/k^2)$ -- matching the best-known rates of both Nesterov's accelerated gradient method and cubically regularized Newton methods. We validate our theoretical findings through empirical comparisons, demonstrating clear improvements over standard Quasi-Newton baselines. To further enhance robustness, we develop an adaptive variant that adjusts to the function's curvature while retaining the global convergence guarantees of the non-adaptive algorithm.

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