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.

Learning to Discover Iterative Spectral Algorithms

We introduce AutoSpec, a neural network framework for discovering iterative spectral algorithms for large-scale numerical linear algebra and numerical optimization. Our self-supervised models adapt to input operators using coarse spectral information (e.g., eigenvalue estimates and residual norms), and they predict recurrence coefficients for computing or applying a matrix polynomial tailored to a downstream task. The effectiveness of AutoSpec relies on three ingredients: an architecture whose inference pass implements short, executable numerical linear algebra recurrences; efficient training on small synthetic problems with transfer to large-scale real-world operators; and task-defined objectives that enforce the desired approximation or preconditioning behavior across the range of spectral profiles represented in the training set. We apply AutoSpec to discovering algorithms for representative numerical linear algebra tasks: accelerating matrix-function approximation; accelerating sparse linear solvers; and spectral filtering/preconditioning for eigenvalue computations. On real-world matrices, the learned procedures deliver orders-of-magnitude improvements in accuracy and/or reductions in iteration count, relative to basic baselines. We also find clear connections to classical theory: the induced polynomials often exhibit near-equiripple, near-minimax behavior characteristic of Chebyshev polynomials.

Landscaper: Understanding Loss Landscapes Through Multi-Dimensional Topological Analysis

Loss landscapes are a powerful tool for understanding neural network optimization and generalization, yet traditional low-dimensional analyses often miss complex topological features. We present Landscaper, an open-source Python package for arbitrary-dimensional loss landscape analysis. Landscaper combines Hessian-based subspace construction with topological data analysis to reveal geometric structures such as basin hierarchy and connectivity. A key component is the Saddle-Minimum Average Distance (SMAD) for quantifying landscape smoothness. We demonstrate Landscaper's effectiveness across various architectures and tasks, including those involving pre-trained language models, showing that SMAD captures training transitions, such as landscape simplification, that conventional metrics miss. We also illustrate Landscaper's performance in challenging chemical property prediction tasks, where SMAD can serve as a metric for out-of-distribution generalization, offering valuable insights for model diagnostics and architecture design in data-scarce scientific machine learning scenarios.

Depth, Not Data: An Analysis of Hessian Spectral Bifurcation

The eigenvalue distribution of the Hessian matrix plays a crucial role in understanding the optimization landscape of deep neural networks. Prior work has attributed the well-documented ``bulk-and-spike'' spectral structure, where a few dominant eigenvalues are separated from a bulk of smaller ones, to the imbalance in the data covariance matrix. In this work, we challenge this view by demonstrating that such spectral Bifurcation can arise purely from the network architecture, independent of data imbalance. Specifically, we analyze a deep linear network setup and prove that, even when the data covariance is perfectly balanced, the Hessian still exhibits a Bifurcation eigenvalue structure: a dominant cluster and a bulk cluster. Crucially, we establish that the ratio between dominant and bulk eigenvalues scales linearly with the network depth. This reveals that the spectral gap is strongly affected by the network architecture rather than solely by data distribution. Our results suggest that both model architecture and data characteristics should be considered when designing optimization algorithms for deep networks.

Suspicious Alignment of SGD: A Fine-Grained Step Size Condition Analysis

This paper explores the suspicious alignment phenomenon in stochastic gradient descent (SGD) under ill-conditioned optimization, where the Hessian spectrum splits into dominant and bulk subspaces. This phenomenon describes the behavior of gradient alignment in SGD updates. Specifically, during the initial phase of SGD updates, the alignment between the gradient and the dominant subspace tends to decrease. Subsequently, it enters a rising phase and eventually stabilizes in a high-alignment phase. The alignment is considered ``suspicious'' because, paradoxically, the projected gradient update along this highly-aligned dominant subspace proves ineffective at reducing the loss. The focus of this work is to give a fine-grained analysis in a high-dimensional quadratic setup about how step size selection produces this phenomenon. Our main contribution can be summarized as follows: We propose a step-size condition revealing that in low-alignment regimes, an adaptive critical step size $η_t^*$ separates alignment-decreasing ($η_t < η_t^*$) from alignment-increasing ($η_t > η_t^*$) regimes, whereas in high-alignment regimes, the alignment is self-correcting and decreases regardless of the step size. We further show that under sufficient ill-conditioning, a step size interval exists where projecting the SGD updates to the bulk space decreases the loss while projecting them to the dominant space increases the loss, which explains a recent empirical observation that projecting gradient updates to the dominant subspace is ineffective. Finally, based on this adaptive step-size theory, we prove that for a constant step size and large initialization, SGD exhibits this distinct two-phase behavior: an initial alignment-decreasing phase, followed by stabilization at high alignment.

KCES: Training-Free Defense for Robust Graph Neural Networks via Kernel Complexity

Graph Neural Networks (GNNs) have achieved impressive success across a wide range of graph-based tasks, yet they remain highly vulnerable to small, imperceptible perturbations and adversarial attacks. Although numerous defense methods have been proposed to address these vulnerabilities, many rely on heuristic metrics, overfit to specific attack patterns, and suffer from high computational complexity. In this paper, we propose Kernel Complexity-Based Edge Sanitization (KCES), a training-free, model-agnostic defense framework. KCES leverages Graph Kernel Complexity (GKC), a novel metric derived from the graph's Gram matrix that characterizes GNN generalization via its test error bound. Building on GKC, we define a KC score for each edge, measuring the change in GKC when the edge is removed. Edges with high KC scores, typically introduced by adversarial perturbations, are pruned to mitigate their harmful effects, thereby enhancing GNNs' robustness. KCES can also be seamlessly integrated with existing defense strategies as a plug-and-play module without requiring training. Theoretical analysis and extensive experiments demonstrate that KCES consistently enhances GNN robustness, outperforms state-of-the-art baselines, and amplifies the effectiveness of existing defenses, offering a principled and efficient solution for securing GNNs.

Eigenspectrum Analysis of Neural Networks without Aspect Ratio Bias

Diagnosing deep neural networks (DNNs) through the eigenspectrum of weight matrices has been an active area of research in recent years. At a high level, eigenspectrum analysis of DNNs involves measuring the heavytailness of the empirical spectral densities (ESD) of weight matrices. It provides insight into how well a model is trained and can guide decisions on assigning better layer-wise training hyperparameters. In this paper, we address a challenge associated with such eigenspectrum methods: the impact of the aspect ratio of weight matrices on estimated heavytailness metrics. We demonstrate that matrices of varying sizes (and aspect ratios) introduce a non-negligible bias in estimating heavytailness metrics, leading to inaccurate model diagnosis and layer-wise hyperparameter assignment. To overcome this challenge, we propose FARMS (Fixed-Aspect-Ratio Matrix Subsampling), a method that normalizes the weight matrices by subsampling submatrices with a fixed aspect ratio. Instead of measuring the heavytailness of the original ESD, we measure the average ESD of these subsampled submatrices. We show that measuring the heavytailness of these submatrices with the fixed aspect ratio can effectively mitigate the aspect ratio bias. We validate our approach across various optimization techniques and application domains that involve eigenspectrum analysis of weights, including image classification in computer vision (CV) models, scientific machine learning (SciML) model training, and large language model (LLM) pruning. Our results show that despite its simplicity, FARMS uniformly improves the accuracy of eigenspectrum analysis while enabling more effective layer-wise hyperparameter assignment in these application domains. In one of the LLM pruning experiments, FARMS reduces the perplexity of the LLaMA-7B model by 17.3% when compared with the state-of-the-art method.

Why LLM Safety Guardrails Collapse After Fine-tuning: A Similarity Analysis Between Alignment and Fine-tuning Datasets

Recent advancements in large language models (LLMs) have underscored their vulnerability to safety alignment jailbreaks, particularly when subjected to downstream fine-tuning. However, existing mitigation strategies primarily focus on reactively addressing jailbreak incidents after safety guardrails have been compromised, removing harmful gradients during fine-tuning, or continuously reinforcing safety alignment throughout fine-tuning. As such, they tend to overlook a critical upstream factor: the role of the original safety-alignment data. This paper therefore investigates the degradation of safety guardrails through the lens of representation similarity between upstream alignment datasets and downstream fine-tuning tasks. Our experiments demonstrate that high similarity between these datasets significantly weakens safety guardrails, making models more susceptible to jailbreaks. Conversely, low similarity between these two types of datasets yields substantially more robust models and thus reduces harmfulness score by up to 10.33%. By highlighting the importance of upstream dataset design in the building of durable safety guardrails and reducing real-world vulnerability to jailbreak attacks, these findings offer actionable insights for fine-tuning service providers.

A Model Zoo on Phase Transitions in Neural Networks

Using the weights of trained Neural Network (NN) models as data modality has recently gained traction as a research field - dubbed Weight Space Learning (WSL). Multiple recent works propose WSL methods to analyze models, evaluate methods, or synthesize weights. Weight space learning methods require populations of trained models as datasets for development and evaluation. However, existing collections of models - called `model zoos' - are unstructured or follow a rudimentary definition of diversity. In parallel, work rooted in statistical physics has identified phases and phase transitions in NN models. Models are homogeneous within the same phase but qualitatively differ from one phase to another. We combine the idea of `model zoos' with phase information to create a controlled notion of diversity in populations. We introduce 12 large-scale zoos that systematically cover known phases and vary over model architecture, size, and datasets. These datasets cover different modalities, such as computer vision, natural language processing, and scientific ML. For every model, we compute loss landscape metrics and validate full coverage of the phases. With this dataset, we provide the community with a resource with a wide range of potential applications for WSL and beyond. Evidence suggests the loss landscape phase plays a role in applications such as model training, analysis, or sparsification. We demonstrate this in an exploratory study of the downstream methods like transfer learning or model weights averaging.