Gradient_Descent
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The Paradox of Outcome Optimization: A Causal Information-Theoretic Bound on Reasoning Shortcuts in LLMs
2026-05-30 11:06:13Zihan Chen, Yiming Zhang, Wenxiang Geng, Zenghui Ding, Yining SunAbstract
Large Language Models (LLMs) aligned via outcome-based Reinforcement Learning (RL) frequently exhibit a critical failure mode: they achieve high performance on in-distribution benchmarks while demonstrating brittle reasoning capabilities on out-of-distribution (OOD) tasks. We term this phenomenon Reward-Induced Manifold Collapse. We establish a theoretical framework bridging Structural Causal Models (SCM) and the Information Bottleneck (IB) principle to explain this paradox. We define reasoning as a high-complexity causal process and shortcut learning as the exploitation of low-complexity spurious correlations. Under the implicit inductive bias of Stochastic Gradient Descent (SGD), models optimized for outcome rewards are biased toward shortcut solutions whenever the training distribution allows for a ``Markovian Screening'' of the true causal mechanism. We derive a new generalization bound based on Semantic Coverage Measure ($\eta$) rather than sample size, showing why data scaling on homogeneous distributions may fail to correct reasoning flaws. We also show that Process Reward Models (PRMs) function as Topological Filters, enforcing step-wise mutual information constraints that render the low-complexity shortcut manifold inadmissible. These results provide a mathematical grounding for the role of process supervision beyond simple credit assignment.
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URL
https://arxiv.org/abs/2606.00674
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Consolidating Rewarded Perturbations for LLM Post-Training
2026-05-29 16:16:13Zheyu Zhang, Shuo Yang, Gjergji KasneciAbstract
Post-training of language models is commonly framed as a sample-score-update loop implemented by gradient descent. A recent line of work, exemplified by RandOpt, relocates this loop to weight space, sampling Gaussian perturbations around a pretrained model and ensembling the top-K rewarded specialists at inference. While competitive with PPO and GRPO under matched training compute, this prediction-level ensemble incurs K forward passes per test example and does not extend cleanly to free-form generation. We ask whether the rewarded population can instead be folded into a single deployable model, replacing the inference-time ensemble with one consolidated update. A split-half analysis over 25 model-task pairs reveals reproducible low-rank structure in every case. We turn this geometry into CoRP (Consolidating Rewarded Perturbations), a gradient-free operator that combines reward-weighted aggregation, compatibility-aware reweighting, and a held-out validation gate, with no gradient flowing through the language model. Across five language models from 0.5B to 8B and five tasks covering math, code, and creative writing, CoRP improves the base model by 8.1 points on average. Using one tenth of RandOpt's perturbation budget, CoRP exceeds single-inference RandOpt by 6.5 points and recovers more than half of the gain of the 50-pass majority-vote ensemble, at one forward pass per test example.
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URL
https://arxiv.org/abs/2605.31494
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Gradient Descent with Large Step Size Restores Symmetry in Deep Linear Networks with Multi-Pathway
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Recent analyses of multi-pathway Deep Linear Networks use Gradient Flow to predict a "winner-takes-all" specialization in which path symmetry breaks and each feature concentrates in a single pathway. In this work, we show that discrete Gradient Descent (GD) with a large step size tells a different story. We prove that single-path solutions are sharp minima, whereas distributing signals across pathways reduces sharpness by a factor that decreases with both the number of pathways and depth. Consequently, while early training reproduces the depth-driven symmetry breaking predicted by GF, oscillations at the Edge of Stability subsequently override this tendency and drive the network into a re-balancing phase, where signals redistribute across pathways. Together, these results clarify how depth shapes pathway competition and explain why large-step GD favors shared representations rather than persistent single-pathway dominance.
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URL
https://arxiv.org/abs/2606.05219
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Building Generalization Into Behavior Generation Via Adaptive Compositions of Regularities
2026-05-29 10:21:35Aravind Battaje, Malte Bernhard, Vito Mengers, Oliver BrockAbstract
Generalization in robotics requires prior knowledge about how the world is structured, yet this structure changes from one situation to the next. This paper investigates the proposition that generalization arises from adaptively composing regularities -- predictable relationships within the robot-environment system -- into situation-appropriate structures for behavior generation. We examine this proposition by analyzing the mechanism in AICON (Active InterCONnect), a framework representing regularities as interacting processes in a differentiable network, where sensory feedback realizes composition and gradient descent generates behavior. To isolate adaptive composition as the key mechanism, we study a simple simulated problem in which all relevant regularities can be identified. We expose the resulting model to a wide range of novel conditions not considered during design, and we find that it generates context-appropriate behavior in all but one case, where encoded regularities are provably insufficient. Ablations reveal that the network automatically modulates which regularities influence behavior based on their informativeness. These results suggest that adaptive composition of regularities constitutes a powerful inductive bias for building generalization into behavior generation.
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https://arxiv.org/abs/2605.31110
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Improving Adversarial Robustness of Attribution via Implicit Regularization
2026-05-28 14:19:39Amir Mehrpanah, Matteo Gamba, Hossein AzizpourAbstract
The adversarial robustness of attributions is a fundamental requirement for reliable explainability in deep learning, yet existing approaches typically rely on computationally expensive explicit regularization. In this work, we show that attribution robustness can arise implicitly from the learning dynamics of standard stochastic gradient descent. We theoretically motivate this effect through connections between parameter-space and input-space curvature, and validate it across architectures, datasets, and attribution methods, with negligible computational overhead. In contrast, we prove that such robustness gains often does not transfer to attention-based attribution under softmax normalization, due to inherent entropy constraints, and we validate this limitation experimentally. Finally, we show that replacing softmax attention with kernel-based attention restores the robustness gains in transformer models. Our results highlight learning dynamics as a principled and practical mechanism for robust explainability, and reveal fundamental limitations of attention-based attribution under normalization.
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URL
https://arxiv.org/abs/2605.29983
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On the Optimizer Dependence of Neural Scaling Laws
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The scaling exponent $\alpha$ in neural scaling laws $L(N) \propto N^{-\alpha}$ is commonly treated as a fixed constant set by architecture and data. We present evidence that $\alpha$ depends systematically on the optimizer. In controlled random-feature regression experiments -- the canonical theoretical framework for neural scaling -- we measure $\alpha$ across five optimizer variants and six spectral conditions. Preconditioned optimizers consistently yield steeper scaling (larger $\alpha$), with the $\alpha$-shift increasing across most of the tested spectral range, peaking near $s = 1.5$, and remaining large at $s = 2.0$. At $s \approx 1.0$ (characteristic of natural language), the full natural gradient achieves $\alpha \approx 0.31$ versus $\alpha \approx 0.12$ for gradient descent -- a $2.6\times$ larger fitted exponent that, within the random-feature model, compounds with each model-size doubling. Whether and how this exponent shift transfers to large-scale LLM training -- where recent evidence suggests the advantage may attenuate with scale -- remains an important open question. Our results imply that scaling-law forecasts should account for optimizer choice, and we provide a spectral diagnostic predicting when advanced optimizers will pay off.
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URL
https://arxiv.org/abs/2605.29387
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Stochastic Gradient Descent with Momentum is Algorithmically Stable
2026-05-27 14:17:38Yunwen Lei, Zimeng Wang, Xiaoming YuanAbstract
Stochastic gradient descent with momentum (SGDM) is one of the most widely used optimization algorithms in machine learning. While optimization properties of SGDM have been extensively studied in the literature, it remains insufficiently understood whether and when SGDM can generalize well to unseen data. In particular, it has been conjectured that while momentum accelerates training, it may degrade generalization. In this paper, we close this gap by developing a comprehensive generalization analysis of SGDM through the lens of algorithmic stability. More specifically, we introduce a generalized SGDM framework that encompasses both Polyak's and Nesterov's momentum schemes, and establish tight on-average model stability bounds for smooth and convex problems. Notably, the obtained bounds exploit small optimization error bounds along the trajectory, apply to any momentum parameter in the interval $[0, 1)$, and do not require the commonly assumed Lipschitzness of loss functions. We further derive optimization error bounds for the generalized SGDM, and combine them with our generalization analyses to obtain optimal excess population risk bounds for SGDM with both Polyak's and Nesterov's momentum.
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URL
https://arxiv.org/abs/2605.28517
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Where Does Toxicity Live? Mechanistic Localization and Targeted Suppression in Language Models
2026-05-27 05:41:19Himanshu Beniwal, Mayank SinghAbstract
Large language models frequently generate toxic, hateful, or harmful content, yet existing mitigation methods rely on costly retraining or output-level filtering with no mechanistic insight into where toxicity originates internally. We introduce Meow2X and TRNE, two complementary retraining-free frameworks that localize toxicity to specific layers and neurons by analyzing activation differentials between toxic and neutral prompts, then suppress them via inference-time scaling or minimal rank-one weight edits -- without any gradient descent. Evaluations across five LMs, two benchmarks, and 90 configurations using dual safety evaluators demonstrate consistent toxicity reduction while preserving language modeling quality. Our analysis reveals that toxicity is disproportionately encoded in early MLP layers, varies across architectures, and is systematically underestimated by single-evaluator setups -- underscoring the need for multi-evaluator safety assessment. By bridging mechanistic interpretability with practical detoxification, our framework offers a principled path toward safer, more transparent language models.
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URL
https://arxiv.org/abs/2605.27997
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SkillGrad: Optimizing Agent Skills Like Gradient Descent
2026-05-26 23:18:43Hanyu Wang, Yifan Lan, Bochuan Cao, Lu Lin, Jinghui ChenAbstract
Agent skills provide a lightweight way to adapt LLM agents to specialized domains by storing reusable procedural knowledge in structured files. However, whether downloaded from third parties or self-generated, these skills are often unreliable, incomplete, or outdated. Existing skill-evolution methods often address these deficiencies through heuristic reflections without an explicit optimization formulation. In this paper, we propose SkillGrad, a gradient-descent-inspired framework for optimizing agent skills. SkillGrad treats the skill package as a structured parameter to optimize in a gradient descent fashion: task executions provide trajectory-level loss evidence, automatic diagnoses then provide text-based gradients that indicate the correction directions. To stabilize optimization across iterations, a momentum agent accumulates recurring diagnostic patterns into a persistent memory overlay. Finally, an LLM-based patcher executes the parameter update by applying layer-aware edits to the skill package. Evaluated on SpreadsheetBench Verified and WikiTableQuestions, SkillGrad consistently outperforms training-based skill evolution baselines across two backbone LLMs, improving over the strongest training-based baseline by $6.7$ percentage points on average. Ablations further show that momentum and contrastive diagnosis both contribute to the final skill quality.
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URL
https://arxiv.org/abs/2605.27760
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Aligning Few-Step Generative Models by Amortizing Sample-based Variational Inference
2026-05-26 05:02:49Jaewoo Lee, Hyeongyu Kang, Dohyun Kim, Kyuil Sim, Woocheol Shin, Minsu Kim, Taeyoung Yun, Jeongjae Lee, Sanghyeok Choi, Tabitha Edith Lee, Jongchul Ye, Jinkyoo ParkAbstract
Aligning a few-step generative model is challenging, since existing alignment frameworks typically rely on restrictive assumptions: a tractable likelihood, a specific ODE/SDE solver, or a particular model family. We introduce FAV, Few-step Generative Models Alignment via Sample-based Variational Inference, a general alignment framework that requires only sample access to the generator and the reference distribution. We cast alignment as sampling from a reward-tilted distribution anchored to a reference distribution. We leverage Stein Variational Gradient Descent as a sample-based variational inference scheme and amortize its particle updates into the generator parameters via fixed-point regression. We evaluate FAV on two domains: robotics manipulation and image generator alignment. On generative policy alignment for robotic manipulation, FAV outperforms prevailing policy extraction baselines across 56 offline and 30 offline-to-online RL tasks. For image generator alignment, FAV fine-tunes diverse few-step backbones, including GAN, drifting model, consistency models, and flow maps, scaling from ImageNet-$256$ to 1024$^2$ text-to-image synthesis. Code is available at this https URL.
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URL
https://arxiv.org/abs/2605.26552
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In-Context Optimization for Retrieval-Augmented Generation: A Gradient-Descent Perspective
2026-05-25 22:04:54Mingchen Li, Jiatan Huang, Chuxu Zhang, Liang Zhao, Hong YuAbstract
In-context learning has recently been linked to implicit gradient descent in linear self-attention models, suggesting that context can induce a forward-pass update. Retrieval-augmented generation (RAG) also relies on context, but retrieved documents are usually treated as static evidence rather than signals for adaptation. We study RAG as an in-context optimization process. First, we show that one linear self-attention layer can implement one gradient-descent step on a unified linearized RAG objective covering both projection-based and dot-product retrieval interfaces. This gives an exact regime where retrieval-augmented prediction and in-context optimization coincide. We use this result not as a literal model of LLM computation, but as a guide for adapting the interaction between queries and retrieved evidence. We then test the boundary of this correspondence: it remains stable under controlled linear extensions, but becomes feature-distribution dependent under nonlinear architectures. Finally, we turn this view into a lightweight method for frozen RAG LLMs. The method keeps the retriever and backbone fixed, and predicts a context-conditioned update to a generator-side evidence-use interface. Across seven QA benchmarks, two retrievers, and two frozen LLM backbones, this forward-only update improves a shared-interface baseline, transfers to held-out tasks, and approaches test-time gradient adaptation at much lower per-query cost.
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URL
https://arxiv.org/abs/2605.26356
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Planning Neural Dynamics with Lie Group Embedding through Supervised Projective Manifold Learning
2026-05-24 22:47:25Tianwei Wang, Bryan Chen, Qian Zuo, Qiyue Xia, Xin Li, Wei PangAbstract
We propose Lie group embedded dynamical neural networks (LieEDNN) and the corresponding learning algorithms based on gradient descent and metric projection on smooth manifold, where we treat Lie group as an intrinsic representation for continuous symmetry of manifold geometry. Thereby we achieve learnable and stable dynamics on the underlying manifold for general Lie group, and we are able to utilize the powerful representation capability of Lie group such as SO(3) and SE(3) to solve real world engineering problems in areas such as robotics, graphics, and control. Two core challenges are: (i) General Lie groups are incompatible with addition arithmetic, which is necessary for neural network interactions. (ii) The dynamics evolve in the nonlinear representation space of special algebra rather than the normal Euclidean space, which violates the paradigm of common neural ODEs. To address these two challenges, we firstly introduce adjoint Lie group action on the Lie algebra, which induces a linear mapping and transfer to the block-wise structure of weight matrices, such that addition could operate on the Lie algebra as a vector space. Then we parameterize the Lie algebra and the adjoint action as linear transformation so that the architecture is aligned with neural network perceptrons. Explicitly, this embedding appears as block-wise manifold constraints on weights, and we develop algorithms to learn the equilibrium with stability guarantees of the temporal neural network dynamics. Experiments are implemented on a specific Lie group SE(3), with the application scenario of telescopic manipulators.
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URL
https://arxiv.org/abs/2605.26167
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Cultivating Machine Intelligence: The OMEGA Shift from Top-Down Optimization to Autopoietic Cognitive Ecologies
2026-05-24 13:13:16Ata G. ZareAbstract
The dominant artificial intelligence paradigm trains neural architectures via gradient descent against proxy objectives and reinforcement learning from human feedback. While remarkably capable, this top-down optimization inherently generates structural failure modes, including hallucination, sycophancy, reward hacking, and alignment fragility, which represent paradigmatic limitations rather than mere engineering defects. In response, we introduce RECLAIM (Recursive, Ecological, Cognitive, Lifelike, Adaptive, Intelligent Machine), a theoretical framework for cultivating intelligence through computational ecology rather than engineering it through strict optimization. The model is supported by four interlocking theoretical pillars. General Darwinism replaces gradients with blind variation and selective retention, while non-agentic emergence substitutes evaluative rewards with environmental physics to structurally prevent specification gaming against human intent. Concurrently, the Polya-Hebbian bridge applies Polya urn dynamics to Hebbian reinforcement for path-dependent specialization, and the free energy principle is integrated as environmental thermodynamics rather than as an agent objective. The architecture situates autopoietic units, bounded by Markov blankets and competing for finite computational energy, within a data ecology shaped by cognitive food chains and Red Queen arms races. This framework suggests the spontaneous emergence of dual-process cognition, sensory specialization, analogical reasoning, and intrinsic motivation as natural consequences of evolution under resource constraints. We conceptualize this paradigm transition as the OMEGA shift, representing a move from optimization and maximization to emergence through generative autopoiesis.
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URL
https://arxiv.org/abs/2605.25062
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How to Build Marcus's Algebraic Mind: Algebro-Deterministic Substrate over Galois Fields
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In The Algebraic Mind, Gary Marcus identified three components essential for any adequate cognitive architecture: operations over variables, recursively structured representations, and a distinction between mental representations of individuals and kinds. He argued that standard multilayer perceptrons supported none of these, acknowledging that a neural implementation using registers and treelets, constructed via developmental programs rather than gradient descent, remained a programmatic conjecture. Twenty-five years later, the required substrate is now available. Our newly developed PyVaCoAl/VaCoAl is a hyperdimensional computing architecture organized end-to-end around a single algebraic primitive: XOR-and-shift over GF(2), implemented by primitive-polynomial linear-feedback shift registers. The architecture supports reversible variable binding via Bind(R,F) = R XOR shift(F), non-commutative compositional bundling that distinguishes "the dog bites the man" from "the man bites the dog," and address-space individual/kind separation under the same algebra. A companion perspective argues that the dentate gyrus-CA3 circuit is a biological homologue of this same engine, with developmentally specified mossy-fiber targeting supplying the innate microcircuitry Marcus anticipated. In this paper, we map the correspondence between Marcus's three pillars and the operational commitments of PyVaCoAl/VaCoAl. We reinterpret the treelet as an algebraic register set indexed by a primitive generator polynomial, arguing that this architecture provides a functional neural substrate meeting Marcus's specifications far more closely than the tensor products, circular convolution, or temporal synchrony available in 2001. We also demonstrate how this substrate naturally extends to Pearl's rung-3 counterfactual reasoning, a capability the original treelet program did not directly target.
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URL
https://arxiv.org/abs/2605.21379
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The Double Dilemma in Multi-Task Radiology Report Generation: A Gradient Dynamics Analysis and Solution
2026-05-21 15:40:13Erjian Zhang, Yatong Hao, Liejun Wang, Zhiqing GuoAbstract
While multi-task learning based automatic radiology report generation (RRG) is widely adopted to ensure clinical consistency, most focus on architectural designs yet remain limited to coarse linear scalarization strategies. These strategies cannot effectively balance the hard constraints of discriminative clinical supervision with the smoothness requirements of report generation. To address these problems, we analyze the failure mechanism of linear scalarization from the perspective of gradient dynamics, utilizing the stochastic differential equation (SDE) framework to characterize it as a "Double Dilemma" of drift term deviation and diffusion term decay. Based on this, we propose a backbone-agnostic optimizer named Conflict-Averse Magnitude-Enhanced Gradient Descent (CAME-Grad). Through conflict-averse direction rectification and magnitude-enhanced energy injection, the algorithm not only ensures geometric validity, but also avoids local optimal solutions. Then, the adaptive gradient fusion mechanism is used to establish a dynamic balance between the theoretical optimal direction and the task-specific inductive bias. Experiments show that as a universal plug-and-play optimizer, CAME-Grad brings substantial and consistent improvements across eight diverse RRG methods, elevating overall clinical efficacy performance by an average of 2.3\% on MIMIC-CXR and 1.9\% on IU X-Ray. Our code is available at this https URL.
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URL
https://arxiv.org/abs/2605.22635
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Large-Step Training Dynamics of a Two-Factor Linear Transformer Model
Abstract
Gradient-flow analyses show that simplified linear transformers can learn the in-context linear-regression algorithm, but they do not explain the finite-step behavior of gradient descent at large learning rates. Motivated by empirical work on high-learning-rate transformer instabilities and by the cubic-map phase diagram for quadratic regression, we study an exactly reducible one-prompt linear-transformer training problem. After normalization, the dynamics reduce to a two-factor product map with an effective step-size parameter \(\mu\). On the balanced slice, this map recovers the known scalar cubic transition from monotone convergence to catapult convergence, periodic and chaotic bounded nonconvergence, and divergence. We then analyze the full two-dimensional system and show that, for \(0<\mu<2\), it has an explicit invariant Chebyshev ellipse separating forward-invariant regions; this ellipse carries off-balanced chaotic dynamics but is transversely repelling, while balanced scalar attractors can be transversely attracting. These results show that large constant learning rates can change the training attractor of the learned transformer rather than merely accelerating convergence: beyond sharp stability thresholds, finite-step training may settle into cycles, bounded chaos, or divergence instead of a single in-context linear-regression solution. We also discuss the consequences for mini-batch gradient descent based training methods.
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URL
https://arxiv.org/abs/2605.21292
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Training Neural Networks with Optimal Double-Bayesian Learning
2026-05-19 15:39:36Vy Bui, Hang Yu, Karthik Kantipudi, Ziv Yaniv, Stefan JaegerAbstract
Backpropagation with gradient descent is a common optimization strategy employed by most neural network architectures in machine learning. However, finding optimal hyperparameters to guide training has proven challenging. While it is widely acknowledged that selecting appropriate parameters is crucial for avoiding overfitting and achieving unbiased outcomes, this choice remains largely based on empirical experiments and experience. This paper presents a new probabilistic framework for the learning rate, a key parameter in stochastic gradient descent. The framework develops classic Bayesian statistics into a double-Bayesian decision mechanism involving two antagonistic Bayesian processes. A theoretically optimal learning rate can be derived from these two processes and used for stochastic gradient descent. Experiments across various classification, segmentation, and detection tasks corroborate the practical significance of the theoretically derived learning rate. The paper also discusses the ramifications of the proposed double-Bayesian framework for network training and model performance.
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URL
https://arxiv.org/abs/2605.20009
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Spectral Unforgetting: Post-Hoc Recovery of Damaged Capabilities Without Retraining
2026-05-19 11:01:39Aarash Abro, Muhammad TahirAbstract
Fine-tuning a language model for a target task routinely degrades capabilities the training data never explicitly threatened. We study this phenomenon, known as catastrophic forgetting, and propose a post-hoc repair solution that uses only the pretrained checkpoint $W_{\mathrm{base}}$ and its fine-tuned descendant $W_{\mathrm{ft}}$. The goal is not merely to revert the model toward the base checkpoint, but to recover capabilities damaged by fine-tuning while preserving both the target-task gains and any beneficial held-out improvements. We introduce DG-Hard, a checkpoint-only spectral repair method for the fine-tuning update $\Delta = W_{\mathrm{ft}} - W_{\mathrm{base}}$. DG-Hard treats $\Delta$ as a low-rank task-aligned signal embedded in an IID-like noise residual that gradient descent has no incentive to remove, and applies the Donoho-Gavish hard singular-value threshold to each weight-delta matrix, keeping the structured high-energy part of the update and removing the spectral bulk. This reduces repair to a closed-form SVD filtering step requiring no data-dependent tuning. A central difficulty is evaluation: average accuracy hides per-benchmark failures, while naive recovery scores reward models that simply revert toward the base. We therefore introduce a partition-conditional metric that separately tracks healing, preservation, non-damage, and target-task retention. Across $14$ (model, task) settings and nine cross-domain held-out benchmarks, DG-Hard achieves the strongest balanced repair among post-hoc baselines. DG-Hard also restores safety alignment degraded by benign fine-tuning on three independent safety axes, despite using no alignment data. These results suggest that part of fine-tuning-induced capability loss is not an unavoidable consequence of specialization, but a removable spectral residue in the weight update itself.
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URL
https://arxiv.org/abs/2605.20296
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Efficient Bilevel Optimization for Meta Label Correction in Noisy Label Learning
2026-05-18 04:12:14Ba Hoang Anh Nguyen, Viet Cuong TaAbstract
Training a deep neural network with noisy labels could reduce data annotation cost but may introduce noise into the learned model. In meta label correction approaches, an additional meta model besides the main model is trained with a small, clean dataset to correct the large, noisy dataset. However, the update of the meta model requires the computation of hypergradients at the inner step of the main model which signif- icantly increases the computational cost. To improve the training efficiency, we first introduce the dynamic barrier gradient descent into standard meta label correction. While this naive extenstion is able to speed up the training process to approximately first- order complexity, it lacks mechanisms to prevent the leakage of noisy signals to the main model and to stabilize the learning of the meta model. Based on this observation, we propose the EBOMLC method, which is designed with three key improvements including one-step inner loop update, mixture upper loss and alignment- aware dynamic barrier. Empirical results on CIFAR-10 and CIFAR-100 demonstrate that EBOMLC consistently outperforms other baselines, especially under high noise rate settings, while reducing training time of the meta label correction approach.
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URL
https://arxiv.org/abs/2605.17833
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DynMuon: A Dynamic Spectral Shaping View of Muon
2026-05-16 18:30:11Fangzhou Wu, Rikhav Shah, Sandeep Silwal, Qiuyi ZhangAbstract
In recent years, Muon has emerged as the dominant method for training large language models, and transformers more broadly. The essential difference, when compared to standard gradient descent methods, is to replace the usual update matrix $M=U\Sigma V^\top$ with its polar factor $UV^\top$. In this work, we consider a class of Muon-like updates, where we replace the update $M$ with $U\Sigma^p V^\top$ for some parameter $p$. We call this a "spectral-shaping" operation, and develop a theory of how to pick $p$ which depends on (a) local curvature of the loss function, (b) noise stemming from stochastic gradients and label noise, and (c) training stage. Our theory and experimentation reveal a previously overlooked behavior: positive $p$ helps early by emphasizing high-curvature directions and accelerating signal contraction, while mildly negative $p$ helps later by reallocating update strength toward low-curvature directions that still contain useful training signals. Building on the insight, we propose DynMuon, an efficient dynamic spectral shaping method that schedules $p$ from positive to mildly negative over training. Extensive experiments across model sizes, architectures, and training settings show that DynMuon consistently achieves lower validation loss than Muon, while requiring 10.6-26.5% fewer steps to reach the same target loss.
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URL
https://arxiv.org/abs/2605.17109
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