Daily Papers

This paper considers the problem of variable selection allowing for parameter instability. It distinguishes between signal and pseudo-signal variables that are correlated with the target variable, and noise variables that are not, and investigate the asymptotic properties of the One Covariate at a Time Multiple Testing (OCMT) method proposed by Chudik et al. (2018) under parameter insatiability. It is established that OCMT continues to asymptotically select an approximating model that includes all the signals and none of the noise variables. Properties of post selection regressions are also investigated, and in-sample fit of the selected regression is shown to have the oracle property. The theoretical results support the use of unweighted observations at the selection stage of OCMT, whilst applying down-weighting of observations only at the forecasting stage. Monte Carlo and empirical applications show that OCMT without down-weighting at the selection stage yields smaller mean squared forecast errors compared to Lasso, Adaptive Lasso, and boosting.

Reliable evaluation is fundamental to the progress of Large Language Models (LLMs), yet the evaluation process during pre-training is plagued by significant instability that obscures true learning dynamics. In this work, we systematically diagnose this instability, attributing it to two distinct sources: Parameter Instability from training stochasticity and Evaluation Instability from noisy measurement protocols. To counteract both sources of noise, we introduce MaP, a dual-pronged framework that synergistically integrates checkpoint Merging and the Pass@k metric. Checkpoint merging smooths the parameter space by averaging recent model weights, while Pass@k provides a robust, low-variance statistical estimate of model capability. Extensive experiments show that MaP yields significantly smoother performance curves, reduces inter-run variance, and ensures more consistent model rankings. Ultimately, MaP provides a more reliable and faithful lens for observing LLM training dynamics, laying a crucial empirical foundation for LLM research.

Zero-shot anomaly detection (ZSAD) requires detecting and localizing anomalies without access to target-class anomaly samples. Mainstream methods rely on vision-language models (VLMs) such as CLIP: they build hand-crafted or learned prompt sets for normal and abnormal semantics, then compute image-text similarities for open-set discrimination. While effective, this paradigm depends on a text encoder and cross-modal alignment, which can lead to training instability and parameter redundancy. This work revisits the necessity of the text branch in ZSAD and presents VisualAD, a purely visual framework built on Vision Transformers. We introduce two learnable tokens within a frozen backbone to directly encode normality and abnormality. Through multi-layer self-attention, these tokens interact with patch tokens, gradually acquiring high-level notions of normality and anomaly while guiding patches to highlight anomaly-related cues. Additionally, we incorporate a Spatial-Aware Cross-Attention (SCA) module and a lightweight Self-Alignment Function (SAF): SCA injects fine-grained spatial information into the tokens, and SAF recalibrates patch features before anomaly scoring. VisualAD achieves state-of-the-art performance on 13 zero-shot anomaly detection benchmarks spanning industrial and medical domains, and adapts seamlessly to pretrained vision backbones such as the CLIP image encoder and DINOv2. Code: https://github.com/7HHHHH/VisualAD

Robust Fine-Tuning (RFT) is a low-cost strategy to obtain adversarial robustness in downstream applications, without requiring a lot of computational resources and collecting significant amounts of data. This paper uncovers an issue with the existing RFT, where optimizing both adversarial and natural objectives through the feature extractor (FE) yields significantly divergent gradient directions. This divergence introduces instability in the optimization process, thereby hindering the attainment of adversarial robustness and rendering RFT highly sensitive to hyperparameters. To mitigate this issue, we propose a low-rank (LoRa) branch that disentangles RFT into two distinct components: optimizing natural objectives via the LoRa branch and adversarial objectives via the FE. Besides, we introduce heuristic strategies for automating the scheduling of the learning rate and the scalars of loss terms. Extensive empirical evaluations demonstrate that our proposed automated RFT disentangled via the LoRa branch (AutoLoRa) achieves new state-of-the-art results across a range of downstream tasks. AutoLoRa holds significant practical utility, as it automatically converts a pre-trained FE into an adversarially robust model for downstream tasks without the need for searching hyperparameters.

Training instability remains a critical challenge in large language model (LLM) pretraining, often manifesting as sudden gradient explosions that waste significant computational resources. We study training failures in a 5M-parameter NanoGPT model scaled via μP, identifying two key phenomena preceding collapse: (1) rapid decline in weight matrix stable rank (ratio of squared Frobenius norm to squared spectral norm), and (2) increasing alignment between adjacent layer Jacobians. We prove theoretically that these two conditions jointly cause exponential gradient norm growth with network depth. To break this instability mechanism, we propose MSign, a new optimizer that periodically applies matrix sign operations to restore stable rank. Experiments on models from 5M to 3B parameters demonstrate that MSign effectively prevents training failures with a computational overhead of less than 7.0%.

Machine unlearning in foundation models (e.g., language and vision transformers) is essential for privacy and safety; however, existing approaches are unstable and unreliable. A widely used strategy, the gradient difference method, applies gradient descent to retained data while performing gradient ascent on forgotten data. When combined with cross-entropy, this procedure can trigger the unbounded growth of weights and gradients, degrading both forgetting and retention. We provide a theoretical framework that explains this failure by showing how ascent destabilizes optimization in transformer feedforward MLP layers. Guided by this insight, we propose *Bounded Parameter-Efficient Unlearning*, which stabilizes LoRA-based fine-tuning by applying bounded functions to MLP adapters. This controls the weight dynamics during ascent and enables reliable convergence. We validate the approach on Vision Transformer class deletion on CIFAR-100, where GD+Sine is the only evaluated method to achieve both high forget quality and model utility across ViT-B/16, ViT-L/14, and DeiT-S architectures, and demonstrate generality on language-model benchmarks (TOFU, TDEC, MUSE) across architectures from 22M to 8B parameters, achieving improved forgetting while preserving utility.

Neural scaling laws approximate a language model's loss as a power-law function of parameter count N and token count D. Following Chinchilla-style compute-optimal training, many studies fit scaling laws from runs performed under a fixed tokens-per-parameter (TPP) ratio k and set D = kN. We show that this collinear design, combined with the empirically common near-equality of the exponents governing N and D, induces an inherent ill-conditioning in the Gauss-Newton least-squares problem: the condition number of the design grows as the inverse square of the gap between the N and D-exponents. The scale coefficients become practically unidentifiable, with confidence intervals inflating by an order of magnitude or more, yielding a ``sloppy'' model whose extrapolations degrade sharply off the training ray. We prove this for four scaling-law formalisms and derive a closed-form TPP-diversity threshold that is necessary and sufficient for well-conditioned estimation. Empirically, non-collinear designs outperform collinear ones on held-out splits with a 97.3\% win rate across four laws, five corpora, multiple floating point precision modes. We further show the degeneracy is rooted in Jacobian geometry and is not an artifact of the loss function: any smooth estimation objective whose curvature involves the Jacobian inherits the same ill-conditioning.

We present a theory for the previously unexplained divergent behavior noticed in the training of large language models. We argue that the phenomenon is an artifact of the dominant optimization algorithm used for training, called Adam. We observe that Adam can enter a state in which the parameter update vector has a relatively large norm and is essentially uncorrelated with the direction of descent on the training loss landscape, leading to divergence. This artifact is more likely to be observed in the training of a deep model with a large batch size, which is the typical setting of large-scale language model training. To argue the theory, we present observations from the training runs of the language models of different scales: 7 billion, 30 billion, 65 billion, and 546 billion parameters.

Recent advancements in state-space models (SSMs) have showcased effective performance in modeling long-range dependencies with subquadratic complexity. However, pure SSM-based models still face challenges related to stability and achieving optimal performance on computer vision tasks. Our paper addresses the challenges of scaling SSM-based models for computer vision, particularly the instability and inefficiency of large model sizes. To address this, we introduce a Modulated Group Mamba layer which divides the input channels into four groups and applies our proposed SSM-based efficient Visual Single Selective Scanning (VSSS) block independently to each group, with each VSSS block scanning in one of the four spatial directions. The Modulated Group Mamba layer also wraps the four VSSS blocks into a channel modulation operator to improve cross-channel communication. Furthermore, we introduce a distillation-based training objective to stabilize the training of large models, leading to consistent performance gains. Our comprehensive experiments demonstrate the merits of the proposed contributions, leading to superior performance over existing methods for image classification on ImageNet-1K, object detection, instance segmentation on MS-COCO, and semantic segmentation on ADE20K. Our tiny variant with 23M parameters achieves state-of-the-art performance with a classification top-1 accuracy of 83.3% on ImageNet-1K, while being 26% efficient in terms of parameters, compared to the best existing Mamba design of same model size. Our code and models are available at: https://github.com/Amshaker/GroupMamba.

Parameter-efficient fine-tuning (PEFT) is a standard way to adapt multimodal large language models, yet extremely low-rank settings -- especially rank-1 LoRA -- are often unstable. We show that this instability is not solely due to limited capacity: in the rank-1 regime, optimization is highly sensitive to the update direction. Concretely, pretrained vision and text features form mismatched anisotropic regions, yielding a dominant "gap" direction that acts like a translation component and disproportionately steers early gradients under rank-1 constraints. Analyzing pretrained representations, we identify a modality-gap axis that dominates early gradient flow, while a random rank-1 initialization is unlikely to align with it, leading to weak gradients and training collapse. We propose Gap-Init, a geometry-aware initialization that aligns the rank-1 LoRA direction with an estimated modality-gap vector from a small calibration set, while keeping the initial LoRA update zero. Across multiple vision-language tasks and backbones, Gap-Init consistently stabilizes rank-1 training and can match or outperform strong rank-8 baselines. Our results suggest that at the extreme low-rank limit, initial alignment can matter as much as rank itself.

To improve detection robustness in adverse conditions (e.g., haze and low light), image restoration is commonly applied as a pre-processing step to enhance image quality for the detector. However, the functional mismatch between restoration and detection networks can introduce instability and hinder effective integration -- an issue that remains underexplored. We revisit this limitation through the lens of Lipschitz continuity, analyzing the functional differences between restoration and detection networks in both the input space and the parameter space. Our analysis shows that restoration networks perform smooth, continuous transformations, while object detectors operate with discontinuous decision boundaries, making them highly sensitive to minor perturbations. This mismatch introduces instability in traditional cascade frameworks, where even imperceptible noise from restoration is amplified during detection, disrupting gradient flow and hindering optimization. To address this, we propose Lipschitz-regularized object detection (LROD), a simple yet effective framework that integrates image restoration directly into the detector's feature learning, harmonizing the Lipschitz continuity of both tasks during training. We implement this framework as Lipschitz-regularized YOLO (LR-YOLO), extending seamlessly to existing YOLO detectors. Extensive experiments on haze and low-light benchmarks demonstrate that LR-YOLO consistently improves detection stability, optimization smoothness, and overall accuracy.

In deep learning, models typically reuse the same parameters for all inputs. Mixture of Experts (MoE) defies this and instead selects different parameters for each incoming example. The result is a sparsely-activated model -- with outrageous numbers of parameters -- but a constant computational cost. However, despite several notable successes of MoE, widespread adoption has been hindered by complexity, communication costs and training instability -- we address these with the Switch Transformer. We simplify the MoE routing algorithm and design intuitive improved models with reduced communication and computational costs. Our proposed training techniques help wrangle the instabilities and we show large sparse models may be trained, for the first time, with lower precision (bfloat16) formats. We design models based off T5-Base and T5-Large to obtain up to 7x increases in pre-training speed with the same computational resources. These improvements extend into multilingual settings where we measure gains over the mT5-Base version across all 101 languages. Finally, we advance the current scale of language models by pre-training up to trillion parameter models on the "Colossal Clean Crawled Corpus" and achieve a 4x speedup over the T5-XXL model.

While fine-tuning of pre-trained language models generally helps to overcome the lack of labelled training samples, it also displays model performance instability. This instability mainly originates from randomness in initialisation or data shuffling. To address this, researchers either modify the training process or augment the available samples, which typically results in increased computational costs. We propose a new mitigation strategy, called Delayed Ensemble with Noisy Interpolation (DENI), that leverages the strengths of ensembling, noise regularisation and model interpolation, while retaining computational efficiency. We compare DENI with 9 representative mitigation strategies across 3 models, 4 tuning strategies and 7 text classification datasets. We show that: 1) DENI outperforms the best performing mitigation strategy (Ensemble), while using only a fraction of its cost; 2) the mitigation strategies are beneficial for parameter-efficient fine-tuning (PEFT) methods, outperforming full fine-tuning in specific cases; and 3) combining DENI with data augmentation often leads to even more effective instability mitigation.

We present Gazal-R1, a 32-billion-parameter language model that achieves state-of-the-art performance in medical reasoning while providing transparent, step-by-step explanations for clinical decision-making. Built upon Qwen3 32B, our model demonstrates that strategic training can enable mid-sized models to outperform significantly larger counterparts in specialized domains. We developed a novel two-stage training pipeline: first, supervised fine-tuning on a carefully curated dataset of 107,033 synthetic medical reasoning examples that teaches structured clinical thinking, enhanced by advanced parameter-efficient techniques including Weight-Decomposed Low-Rank Adaptation (DoRA) and Rank-Stabilized LoRA (rsLoRA); second, reinforcement learning using Group Relative Policy Optimization (GRPO) with a sophisticated multi-component reward system that refines accuracy, format adherence, and reasoning quality. Gazal-R1 achieves exceptional performance across medical benchmarks, scoring 87.1% on MedQA, 81.6% on MMLU Pro (Medical), and 79.6% on PubMedQA, surpassing models up to 12x larger. Beyond its strong empirical results, this work provides detailed insights into the challenges of training reasoning-capable models in specialized domains, including issues with reward hacking, training instability, and the fundamental tension between factual recall and detailed reasoning. Our methodology offers a reproducible framework for developing high-capability, domain-specific language models that balance performance, efficiency, and explainability.

DARTS is a popular algorithm for neural architecture search (NAS). Despite its great advantage in search efficiency, DARTS often suffers weak stability, which reflects in the large variation among individual trials as well as the sensitivity to the hyper-parameters of the search process. This paper owes such instability to an optimization gap between the super-network and its sub-networks, namely, improving the validation accuracy of the super-network does not necessarily lead to a higher expectation on the performance of the sampled sub-networks. Then, we point out that the gap is due to the inaccurate estimation of the architectural gradients, based on which we propose an amended estimation method. Mathematically, our method guarantees a bounded error from the true gradients while the original estimation does not. Our approach bridges the gap from two aspects, namely, amending the estimation on the architectural gradients, and unifying the hyper-parameter settings in the search and re-training stages. Experiments on CIFAR10 and ImageNet demonstrate that our approach largely improves search stability and, more importantly, enables DARTS-based approaches to explore much larger search spaces that have not been investigated before.

The success of Hyper-Connections (HC) in neural networks (NN) has also highlighted issues related to its training instability and restricted scalability. The Manifold-Constrained Hyper-Connections (mHC) mitigate these challenges by projecting the residual connection space onto a Birkhoff polytope, however, it faces two issues: 1) its iterative Sinkhorn-Knopp (SK) algorithm does not always yield exact doubly stochastic residual matrices; 2) mHC incurs a prohibitive O(n^3C) parameter complexity with n as the width of the residual stream and C as the feature dimension. The recently proposed mHC-lite reparametrizes the residual matrix via the Birkhoff-von-Neumann theorem to guarantee double stochasticity, but also faces a factorial explosion in its parameter complexity, O left( nC cdot n! right). To address both challenges, we propose KromHC, which uses the Kronecker products of smaller doubly stochastic matrices to parametrize the residual matrix in mHC. By enforcing manifold constraints across the factor residual matrices along each mode of the tensorized residual stream, KromHC guarantees exact double stochasticity of the residual matrices while reducing parameter complexity to O(n^2C). Comprehensive experiments demonstrate that KromHC matches or even outperforms state-of-the-art (SOTA) mHC variants, while requiring significantly fewer trainable parameters. The code is available at https://github.com/wz1119/KromHC.

We study the Cooper instability in jellium model in the controlled regime of small to intermediate values of the Coulomb parameter r_s leq 2. We confirm that superconductivity naturally emerges from purely repulsive interactions described by the Kukkonen-Overhauser vertex function. By employing the implicit renormalization approach we reveal that even in the small-r_s limit, the dominant mechanism behind Cooper instability is based on dynamic screening of the Coulomb interaction--accurately captured by the random phase approximation, whereas the Kohn-Luttinger contribution is negligibly small and, thus, not relevant.

Function-calling language models are essential for agentic AI systems that translate natural language into executable structured actions, yet existing models exhibit severe structural instability when applied to Arabic. We present AISA-AR-FunctionCall, a production-oriented Arabic function-calling framework built on a 270M-parameter FunctionGemma backbone and trained through systematic dataset auditing, schema repair, tool-aware prompt restructuring, and full-parameter supervised fine-tuning. On a held-out test set, fine-tuning reduces parse failures from 87\% to below 1\%, improves function name accuracy by more than eightfold, and substantially enhances argument alignment across dialects and domains. Error analysis reveals a transition from structural collapse to semantic misalignment, suggesting that serialization stability and decision-level reasoning are separable challenges. We further explore a reasoning-augmented LoRA variant that introduces explicit intermediate reasoning prior to tool invocation. All datasets and models are publicly released under the AISA framework.

Recent advancements in Multimodal Large Language Models (MLLMs) pursue omni-perception capabilities, yet integrating robust sensory grounding with complex reasoning remains a challenge, particularly for underrepresented regions. In this report, we introduce the research preview of MERaLiON2-Omni (Alpha), a 10B-parameter multilingual omni-perception tailored for Southeast Asia (SEA). We present a progressive training pipeline that explicitly decouples and then integrates "System 1" (Perception) and "System 2" (Reasoning) capabilities. First, we establish a robust Perception Backbone by aligning region-specific audio-visual cues (e.g., Singlish code-switching, local cultural landmarks) with a multilingual LLM through orthogonal modality adaptation. Second, to inject cognitive capabilities without large-scale supervision, we propose a cost-effective Generate-Judge-Refine pipeline. By utilizing a Super-LLM to filter hallucinations and resolve conflicts via a consensus mechanism, we synthesize high-quality silver data that transfers textual Chain-of-Thought reasoning to multimodal scenarios. Comprehensive evaluation on our newly introduced SEA-Omni Benchmark Suite reveals an Efficiency-Stability Paradox: while reasoning acts as a non-linear amplifier for abstract tasks (boosting mathematical and instruction-following performance significantly), it introduces instability in low-level sensory processing. Specifically, we identify Temporal Drift in long-context audio, where extended reasoning desynchronizes the model from acoustic timestamps, and Visual Over-interpretation, where logic overrides pixel-level reality. This report details the architecture, the data-efficient training recipe, and a diagnostic analysis of the trade-offs between robust perception and structured reasoning.

Building deep reinforcement learning (RL) agents that find a good policy with few samples has proven notoriously challenging. To achieve sample efficiency, recent work has explored updating neural networks with large numbers of gradient steps for every new sample. While such high update-to-data (UTD) ratios have shown strong empirical performance, they also introduce instability to the training process. Previous approaches need to rely on periodic neural network parameter resets to address this instability, but restarting the training process is infeasible in many real-world applications and requires tuning the resetting interval. In this paper, we focus on one of the core difficulties of stable training with limited samples: the inability of learned value functions to generalize to unobserved on-policy actions. We mitigate this issue directly by augmenting the off-policy RL training process with a small amount of data generated from a learned world model. Our method, Model-Augmented Data for TD Learning (MAD-TD), uses small amounts of generated data to stabilize high UTD training and achieve competitive performance on the most challenging tasks in the DeepMind control suite. Our experiments further highlight the importance of employing a good model to generate data, MAD-TD's ability to combat value overestimation, and its practical stability gains for continued learning.

Achieving a balance between high-fidelity visual quality and low-latency streaming remains a formidable challenge in audio-driven portrait generation. Existing large-scale models often suffer from prohibitive computational costs, while lightweight alternatives typically compromise on holistic facial representations and temporal stability. In this paper, we propose SoulX-FlashHead, a unified 1.3B-parameter framework designed for real-time, infinite-length, and high-fidelity streaming video generation. To address the instability of audio features in streaming scenarios, we introduce Streaming-Aware Spatiotemporal Pre-training equipped with a Temporal Audio Context Cache mechanism, which ensures robust feature extraction from short audio fragments. Furthermore, to mitigate the error accumulation and identity drift inherent in long-sequence autoregressive generation, we propose Oracle-Guided Bidirectional Distillation, leveraging ground-truth motion priors to provide precise physical guidance. We also present VividHead, a large-scale, high-quality dataset containing 782 hours of strictly aligned footage to support robust training. Extensive experiments demonstrate that SoulX-FlashHead achieves state-of-the-art performance on HDTF and VFHQ benchmarks. Notably, our Lite variant achieves an inference speed of 96 FPS on a single NVIDIA RTX 4090, facilitating ultra-fast interaction without sacrificing visual coherence.

Diffusion models have demonstrated impressive image synthesis performance, yet many UNet-based models are trained at certain fixed resolutions. Their quality tends to degrade when generating images at out-of-training resolutions. We trace this issue to resolution-dependent parameter behaviors, where weights that function well at the default resolution can become adverse when spatial scales shift, weakening semantic alignment and causing structural instability in the UNet architecture. Based on this analysis, this paper introduces CR-Diff, a novel method that improves the cross-resolution visual consistency by pruning some parameters of the diffusion model. Specifically, CR-Diff has two stages. It first performs block-wise pruning to selectively eliminate adverse weights. Then, a pruned output amplification is conducted to further purify the pruned predictions. Empirically, extensive experiments suggest that CR-Diff can improve perceptual fidelity and semantic coherence across various diffusion backbones and unseen resolutions, while largely preserving the performance at default resolutions. Additionally, CR-Diff supports prompt-specific refinement, enabling quality enhancement on demand.

Large Language Models (LLMs) are highly resource-intensive to fine-tune due to their enormous size. While low-rank adaptation is a prominent parameter-efficient fine-tuning approach, it suffers from sensitivity to hyperparameter choices, leading to instability in model performance on fine-tuning downstream tasks. This paper highlights the importance of effective parameterization in low-rank fine-tuning to reduce estimator variance and enhance the stability of final model outputs. We propose MonteCLoRA, an efficient fine-tuning technique that employs Monte Carlo estimation to learn an unbiased posterior estimation of low-rank parameters with low expected variance, stabilizing fine-tuned LLMs with only O(r) additional parameters, for a given rank r. MonteCLoRA shows 0.5% and 1.6% improvements in accuracy and robustness over unregularized low-rank adaptation method on natural language understanding tasks with pre-trained RoBERTa-base. Furthermore, in generative tasks with pre-trained LLaMA-1-7B and LLaMA-3.2-3B-Instruct, MonteCLoRA demonstrates robust performance with 50% and 62% lower spreads respectively than the contemporary efficient fine-tuning methods. The theoretical and empirical results presented in the paper underscore how parameterization and hyperpriors balance exploration-exploitation in the low-rank parametric space, therefore leading to more optimal and robust parameter estimation during efficient fine-tuning.

Ad-hoc instruction fine-tuning of large language models (LLMs) is widely adopted for domain-specific adaptation. While domain-specific supervised fine-tuning (SFT) is effective and efficient, it often weakens cross-domain generalization and struggles with noisy training data. To address these challenges, we propose DONOD, a lightweight model-intrinsic data pruning method. Our approach evaluates data using two model-parameter-based metrics: Delta of Norm (DON), which captures the cumulative influence on model weights, and Norm of Delta (NOD), which quantifies weight instability. Moreover, by employing the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) algorithm, we effectively filter noisy, unlearnable, and generalization-harming samples without relying on auxiliary models during the SFT process. Experiments on mathematical tasks demonstrate that data selected by DONOD achieve superior fine-tuning efficiency and improved robustness against noisy data. By filtering out 70% of the full dataset, we improve target-domain accuracy by 14.90% and cross-domain accuracy by 5.67%. Meanwhile, our selected data present superior cross-architecture generalization. Data pruned by smaller models (e.g., Llama 3.1-8B) generalize effectively on larger models (e.g., Llama 2-13B). Compared to existing related methodologies, DONOD demonstrates comparable or superior performance while remaining dataset-agnostic, enabling broader applicability.

Real-world multivariate time series can exhibit intricate multi-scale structures, including global trends, local periodicities, and non-stationary regimes, which makes long-horizon forecasting challenging. Although sparse Mixture-of-Experts (MoE) approaches improve scalability and specialization, they typically rely on homogeneous MLP experts that poorly capture the diverse temporal dynamics of time series data. We address these limitations with MoHETS, an encoder-only Transformer that integrates sparse Mixture-of-Heterogeneous-Experts (MoHE) layers. MoHE routes temporal patches to a small subset of expert networks, combining a shared depthwise-convolution expert for sequence-level continuity with routed Fourier-based experts for patch-level periodic structures. MoHETS further improves robustness to non-stationary dynamics by incorporating exogenous information via cross-attention over covariate patch embeddings. Finally, we replace parameter-heavy linear projection heads with a lightweight convolutional patch decoder, improving parameter efficiency, reducing training instability, and allowing a single model to generalize across arbitrary forecast horizons. We validate across seven multivariate benchmarks and multiple horizons, with MoHETS consistently achieving state-of-the-art performance, reducing the average MSE by 12% compared to strong recent baselines, demonstrating effective heterogeneous specialization for long-term forecasting.

We present a model for chi^{(2)} waveguides accounting for three modes, two of which make an avoided crossing at the second harmonic wavelength. We introduce two linearly coupled pure modes and adjust the coupling to replicate the waveguide dispersion near the avoided crossing. Analysis of the nonlinear system reveals continuous wave (CW) solutions across much of the parameter-space and prevalence of its modulational instability. We also predict the existence of the avoided-crossing solitons, and study peculiarities of their dynamics and spectral properties, which include formation of a pedestal in the pulse tails and associated pronounced spectral peaks. Mapping these solitons onto the linear dispersion diagrams, we make connections between their existence and CW existence and stability. We also simulate the two-color soliton generation from a single frequency pump pulse to back up its formation and stability properties.

Training large language models (LLMs) poses challenges due to their massive scale and heterogeneous architectures. While adaptive optimizers like AdamW help address gradient variations, they still struggle with efficient and effective parameter-wise learning rate estimation, resulting in training instability, slow convergence, and poor compatibility with parameter-efficient fine-tuning (PEFT) techniques. This work introduces Scaling with Gradient Grouping (SGG), an optimizer wrapper that improves adaptive learning rate estimation by dynamic grouping and group-specific scaling. SGG first groups gradient statistics in each layer into clusters and then applies cluster-specific scaling to calibrate learning rates for each parameter, thus imposing collective group-wise constraints while maintaining precise per-parameter adaptation. Experiments on diverse (M)LLM benchmarks show that SGG integrates seamlessly with existing optimizers, and offers consistent gains and faster convergence over baselines, with various model sizes. Its stability across varying batch sizes and learning rates establishes SGG as a robust choice for LLM optimization.

The push for efficient text to image synthesis has moved the field toward one step sampling, yet existing methods still face a three way tradeoff among fidelity, inference speed, and training efficiency. Approaches that rely on external discriminators can sharpen one step performance, but they often introduce training instability, high GPU memory overhead, and slow convergence, which complicates scaling and parameter efficient tuning. In contrast, regression based distillation and consistency objectives are easier to optimize, but they typically lose fine details when constrained to a single step. We present APEX, built on a key theoretical insight: adversarial correction signals can be extracted endogenously from a flow model through condition shifting. Using a transformation creates a shifted condition branch whose velocity field serves as an independent estimator of the model's current generation distribution, yielding a gradient that is provably GAN aligned, replacing the sample dependent discriminator terms that cause gradient vanishing. This discriminator free design is architecture preserving, making APEX a plug and play framework compatible with both full parameter and LoRA based tuning. Empirically, our 0.6B model surpasses FLUX-Schnell 12B (20times more parameters) in one step quality. With LoRA tuning on Qwen-Image 20B, APEX reaches a GenEval score of 0.89 at NFE=1 in 6 hours, surpassing the original 50-step teacher (0.87) and providing a 15.33times inference speedup. Code is available https://github.com/LINs-lab/APEX.

We present Ring-1T, the first open-source, state-of-the-art thinking model with a trillion-scale parameter. It features 1 trillion total parameters and activates approximately 50 billion per token. Training such models at a trillion-parameter scale introduces unprecedented challenges, including train-inference misalignment, inefficiencies in rollout processing, and bottlenecks in the RL system. To address these, we pioneer three interconnected innovations: (1) IcePop stabilizes RL training via token-level discrepancy masking and clipping, resolving instability from training-inference mismatches; (2) C3PO++ improves resource utilization for long rollouts under a token budget by dynamically partitioning them, thereby obtaining high time efficiency; and (3) ASystem, a high-performance RL framework designed to overcome the systemic bottlenecks that impede trillion-parameter model training. Ring-1T delivers breakthrough results across critical benchmarks: 93.4 on AIME-2025, 86.72 on HMMT-2025, 2088 on CodeForces, and 55.94 on ARC-AGI-v1. Notably, it attains a silver medal-level result on the IMO-2025, underscoring its exceptional reasoning capabilities. By releasing the complete 1T parameter MoE model to the community, we provide the research community with direct access to cutting-edge reasoning capabilities. This contribution marks a significant milestone in democratizing large-scale reasoning intelligence and establishes a new baseline for open-source model performance.

We present a continuous geometric framework that models the discrete algebraic operations of the Transformer architecture as an integro-differential equation (IDE) on a semantic fiber bundle calE = calM times R^d. Beginning from a single geometric axiom -- that the token sequence forms a discrete 1-manifold equipped with a canonical measure lattice -- we translate every core component of the modern Transformer (RMSNorm, RoPE, Softmax Attention, FFN, Residual Stream, SGD, Weight Decay) into a cohesive vocabulary of differential geometry, measure theory, and stochastic calculus. The resulting framework yields quantitative predictions spanning entropic optimal transport (Attention as a Schrödinger bridge) and non-equilibrium thermodynamics (SGD as Itô diffusion violating detailed balance). We conduct a six-part experimental campaign across five architectures (Qwen3, LLaMA\nobreakdash-3.1, Gemma\nobreakdash-3, GPT-2, Mistral) spanning 124M to 8B parameters. The empirical observables are quantitatively consistent with the geometric predictions: the ε^{-1/2} Lipschitz scaling calibration at machine precision (R^2 = 1.000), the Lie--Trotter operator-splitting torsion, the symmetric ablation instability confirming the Dual-Law of Topological Stability, the calO(1/k) thermodynamic suppression of Poincaré recurrence on the RoPE torus, the thermodynamic context-limit phase transition, and the Non-Equilibrium Steady State parameter vortex -- verified across two optimizers (AdamW and Pure SGD) to exclude momentum artifacts. The results demonstrate that analyzing Transformers through the lens of continuous stochastic differential geometry provides a predictive descriptive vocabulary for the stability limits, context bounds, and optimization dynamics of Large Language Models.

We prove stability for a class of heterogeneous catalysis models in the L_p-setting. We consider a setting in a finite three-dimensional pore of cylinder-like geometry, with the lateral walls acting as a catalytic surface. Under a reasonable condition on the involved parameters, we show that given equilibria are normally stable, i.e. solutions are attracted at an exponential rate. The potential incidence of instability is discussed as well.

Reasoning failures in large language models (LLMs) are typically measured only at the end of a generation, yet many failures manifest as a process-level breakdown: the model "loses the thread" mid-reasoning. We study whether such breakdowns are detectable from inference-time observables available in standard APIs (token log probabilities), without any training or fine-tuning. We define a simple instability signal that combines consecutive-step distributional shift (JSD) and uncertainty (entropy), summarize each trace by its peak instability strength, and show that this signal reliably predicts failure. Across GSM8K and HotpotQA, instability strength predicts wrong answers with above-chance AUC and yields monotonic bucket-level accuracy decline at scale across model sizes. Crucially, we show that instability is not uniformly harmful: early instability can reflect subsequent stabilization and a correct final answer (corrective instability), whereas late instability is more often followed by failure (destructive instability), even at comparable peak magnitudes, indicating that recoverability depends not only on how strongly the distribution changes but also on when such changes occur relative to the remaining decoding horizon. The method is model-agnostic, training-free, and reproducible, and is presented as a diagnostic lens rather than a corrective or control mechanism.

Recent work suggests that (stochastic) gradient descent self-organizes near an instability boundary, shaping both optimization and the solutions found. Momentum and mini-batch gradients are widely used in practical deep learning optimization, but it remains unclear whether they operate in a comparable regime of instability. We demonstrate that SGD with momentum exhibits an Edge of Stochastic Stability (EoSS)-like regime with batch-size-dependent behavior that cannot be explained by a single momentum-adjusted stability threshold. Batch Sharpness (the expected directional mini-batch curvature) stabilizes in two distinct regimes: at small batch sizes it converges to a lower plateau 2(1-β)/η, reflecting amplification of stochastic fluctuations by momentum and favoring flatter regions than vanilla SGD; at large batch sizes it converges to a higher plateau 2(1+β)/η, where momentum recovers its classical stabilizing effect and favors sharper regions consistent with full-batch dynamics. We further show that this aligns with linear stability thresholds and discuss the implications for hyperparameter tuning and coupling.

Teams that have trained large Transformer-based models have reported training instabilities at large scale that did not appear when training with the same hyperparameters at smaller scales. Although the causes of such instabilities are of scientific interest, the amount of resources required to reproduce them has made investigation difficult. In this work, we seek ways to reproduce and study training stability and instability at smaller scales. First, we focus on two sources of training instability described in previous work: the growth of logits in attention layers (Dehghani et al., 2023) and divergence of the output logits from the log probabilities (Chowdhery et al., 2022). By measuring the relationship between learning rate and loss across scales, we show that these instabilities also appear in small models when training at high learning rates, and that mitigations previously employed at large scales are equally effective in this regime. This prompts us to investigate the extent to which other known optimizer and model interventions influence the sensitivity of the final loss to changes in the learning rate. To this end, we study methods such as warm-up, weight decay, and the muParam (Yang et al., 2022), and combine techniques to train small models that achieve similar losses across orders of magnitude of learning rate variation. Finally, to conclude our exploration we study two cases where instabilities can be predicted before they emerge by examining the scaling behavior of model activation and gradient norms.

Advances in endoscopy use in surgeries face challenges like inadequate lighting. Deep learning, notably the Denoising Diffusion Probabilistic Model (DDPM), holds promise for low-light image enhancement in the medical field. However, DDPMs are computationally demanding and slow, limiting their practical medical applications. To bridge this gap, we propose a lightweight DDPM, dubbed LighTDiff. It adopts a T-shape model architecture to capture global structural information using low-resolution images and gradually recover the details in subsequent denoising steps. We further prone the model to significantly reduce the model size while retaining performance. While discarding certain downsampling operations to save parameters leads to instability and low efficiency in convergence during the training, we introduce a Temporal Light Unit (TLU), a plug-and-play module, for more stable training and better performance. TLU associates time steps with denoised image features, establishing temporal dependencies of the denoising steps and improving denoising outcomes. Moreover, while recovering images using the diffusion model, potential spectral shifts were noted. We further introduce a Chroma Balancer (CB) to mitigate this issue. Our LighTDiff outperforms many competitive LLIE methods with exceptional computational efficiency.

Exact firing rate models, also known as next-generation neural mass models (NG-NMMs), provide a rigorous description of the dynamics of neural populations. While in its simplest form a single population only displays fixed-point activity, multi-population models may display a range of different behaviors. In this work, we study the dynamics of all-excitatory or all-inhibitory NG-NMMs coupled through sparse random networks with row-normalized network topology. Linear stability analysis of the homogeneous states of the system, representing asynchronous neural activity, provides a dispersion relation linking the emergence of spatiotemporal dynamics to the spectra of the connectivity matrix. Using bounds from random matrix theory, we identify the parameter regions where instabilities occur. In undirected networks, only inhibitory systems produce heterogeneous stationary patterns, corresponding to a winner-takes-all mechanism. In directed networks, exotic rhythmic states with high frequencies emerge in both, excitatory and inhibitory systems. Numerical simulations reveal that these hectic oscillatory states correspond to high-dimensional chaos with extensive properties.

Multi-agent large language model (LLM) systems often rely on verifier and critic agents to suppress hallucinations, but verification is delayed. During this delay, false claims can propagate through the agent network. We model this process as delayed consensus on a graph with grounded corrector nodes. Spectral decomposition by the grounded Laplacian yields a closed-form stability threshold for the verification dose: correction that is too strong or too delayed can turn consensus into oscillation. The most unstable regime occurs when the communication and verification delays coincide; for delay two, the threshold is the inverse golden ratio. The same framework gives a supermodular placement objective and a greedy (1-1/e)-approximation rule for assigning a limited corrector budget to influential nodes. Experiments across five open models confirm the predicted dose-delay oscillations. By contrast, grounded factual answering makes truth an absorbing boundary and eliminates the effect, suggesting that the instability is specific to signed-belief tasks while grounded verification remains stabilizing

Benchmarks measure whether a model is correct. They do not measure whether a model is reliable. This distinction is largely academic for single-shot inference, but becomes critical for agentic AI systems, where a single rephrased prompt can trigger cascading failures in multi-step execution. Yet this form of instability is not captured by existing evaluations. Hallucinations live in variance: they arise when semantically equivalent prompts activate inconsistent internal pathways, producing divergent outputs. Consistent but incorrect outputs reflect bias or missing knowledge; confident guessing reflects calibration failure. Neither constitutes hallucination under this definition. When error is variance-dominated, reducing redundant pathways improves reliability without adding knowledge. We formalize this through Semantic Stability (SS), measured via Paraphrase Consistency (PC@k): generate k paraphrases, greedy decode each, compute mode agreement. SS is a diagnostic for variance-driven unreliability, not a method for improving correctness. We show that a dense Qwen3-0.6B agrees with itself only 23.8% of the time; at 32% sparsity, agreement jumps to 55.9%. A phase diagram reveals the sweet spot where variance reduction outpaces bias accumulation, and regimes where stability collapses onto wrong answers.

There is an emerging interest in generating robust counterfactual explanations that would remain valid if the model is updated or changed even slightly. Towards finding robust counterfactuals, existing literature often assumes that the original model m and the new model M are bounded in the parameter space, i.e., |Params(M){-}Params(m)|{<}Delta. However, models can often change significantly in the parameter space with little to no change in their predictions or accuracy on the given dataset. In this work, we introduce a mathematical abstraction termed naturally-occurring model change, which allows for arbitrary changes in the parameter space such that the change in predictions on points that lie on the data manifold is limited. Next, we propose a measure -- that we call Stability -- to quantify the robustness of counterfactuals to potential model changes for differentiable models, e.g., neural networks. Our main contribution is to show that counterfactuals with sufficiently high value of Stability as defined by our measure will remain valid after potential ``naturally-occurring'' model changes with high probability (leveraging concentration bounds for Lipschitz function of independent Gaussians). Since our quantification depends on the local Lipschitz constant around a data point which is not always available, we also examine practical relaxations of our proposed measure and demonstrate experimentally how they can be incorporated to find robust counterfactuals for neural networks that are close, realistic, and remain valid after potential model changes.

We introduce a novel approach for analyzing the training dynamics of ReLU networks by examining the characteristic activation boundaries of individual ReLU neurons. Our proposed analysis reveals a critical instability in common neural network parameterizations and normalizations during stochastic optimization, which impedes fast convergence and hurts generalization performance. Addressing this, we propose Geometric Parameterization (GmP), a novel neural network parameterization technique that effectively separates the radial and angular components of weights in the hyperspherical coordinate system. We show theoretically that GmP resolves the aforementioned instability issue. We report empirical results on various models and benchmarks to verify GmP's theoretical advantages of optimization stability, convergence speed and generalization performance.

Latent Diffusion Models (LDMs) achieve high-fidelity synthesis but suffer from latent space brittleness, causing discontinuous semantic jumps during editing. We introduce a Riemannian framework to diagnose this instability by analyzing the generative Jacobian, decomposing geometry into Local Scaling (capacity) and Local Complexity (curvature). Our study uncovers a ``Geometric Decoupling": while curvature in normal generation functionally encodes image detail, OOD generation exhibits a functional decoupling where extreme curvature is wasted on unstable semantic boundaries rather than perceptible details. This geometric misallocation identifies ``Geometric Hotspots" as the structural root of instability, providing a robust intrinsic metric for diagnosing generative reliability.

Current LLM unlearning methods face a critical security vulnerability that undermines their fundamental purpose: while they appear to successfully remove sensitive or harmful knowledge, this ``forgotten" information remains precariously recoverable through relearning attacks. We identify that the root cause is that conventional methods optimizing the forgetting loss at individual data points will drive model parameters toward sharp minima in the loss landscape. In these unstable regions, even minimal parameter perturbations can drastically alter the model's behaviors. Consequently, relearning attacks exploit this vulnerability by using just a few fine-tuning samples to navigate the steep gradients surrounding these unstable regions, thereby rapidly recovering knowledge that was supposedly erased. This exposes a critical robustness gap between apparent unlearning and actual knowledge removal. To address this issue, we propose StableUN, a bi-level feedback-guided optimization framework that explicitly seeks more stable parameter regions via neighborhood-aware optimization. It integrates forgetting feedback, which uses adversarial perturbations to probe parameter neighborhoods, with remembering feedback to preserve model utility, aligning the two objectives through gradient projection. Experiments on WMDP and MUSE benchmarks demonstrate that our method is significantly more robust against both relearning and jailbreaking attacks while maintaining competitive utility performance.

Frontier large language model training consumes massive accelerator fleets and long wall-clock computation, making stability failures costly when they occur. After a numerical or a hyperparameter fault has already destabilized the training dynamics, it may continue for thousands of steps while loss and gradient norms still appear normal. We study mechanism-driven detection of training instability by deriving internal monitors from the functional role of each critical module and from the earliest computational sites where failures are expected to produce measurable signatures. For low-precision flash attention, we monitor the spectral entropy of a QK bilinear decomposition, whose first-order term becomes abnormal before the loss fully collapses. For MoE routers, we derive indicators from their role in expert selection. Our fault-injection experiments on low-precision attention, large learning-rate, and combined faults show that these signals provide distinct signatures for different failures, triggering thousands of steps before loss divergence.

Differentiable matching layers and residual connection paradigms, often implemented via entropy-regularized Optimal Transport (OT), serve as critical mechanisms in structural prediction and architectural scaling. However, recovering discrete permutations or maintaining identity mappings via annealing εto 0 is notoriously unstable. In this work, we identify a fundamental mechanism for this failure: Premature Mode Collapse. By analyzing the non-normal dynamics of the Sinkhorn fixed-point map, we reveal a theoretical thermodynamic speed limit: standard exponential cooling outpaces the contraction rate of the inference operator, which degrades as O(1/ε). To address this, we propose Efficient Piecewise Hybrid Adaptive Stability Control (EPH-ASC), an adaptive scheduling algorithm that monitors the stability of the inference process. We demonstrate that EPH-ASC is essential for stabilizing Manifold-Constrained Hyper-Connections (mHC) during large-scale training on the FineWeb-Edu dataset, effectively preventing late-stage gradient explosions by enforcing a linear stability law.

Preference learning in large language models relies on reward models as proxies for human judgment. However, these models frequently exhibit preference instability, producing contradictory preference assignments in response to subtle, meaning-preserving input variations. We analyze this instability at the representation level under three semantic-preserving perturbation types: paraphrasing, pattern injection, and backdoor triggers. We attribute this instability to over-reliance on predictive yet brittle features, which we term unstable features, and isolate them via Sparse Autoencoders (SAEs) in a sparse latent space where benign and perturbed inputs activate distinctly separable patterns. Building on this separability, we propose two SAE-based instability mitigation strategies: SAE Feature Steering, which identifies and suppresses anomalously activated features at inference, and SAE Residual Correction, which learns adaptive adjustments over SAE features to restore correct preferences. Our methods substantially reduce incorrect preference assignments on harmlessness and hallucination benchmarks while preserving benign performance and general utility on other tasks, without retraining the reward model. Our code and data are available in https://github.com/shunchang-liu/pisa.

Safety alignment of Large Language Models (LLMs) is extremely fragile, as fine-tuning on a small number of benign samples can erase safety behaviors learned from millions of preference examples. Existing studies attempt to explain this phenomenon by comparing parameters and hidden states before and after fine-tuning, but overlook their dynamic evolution during fine-tuning. In this paper, we uncover a critical mechanism underlying safety degradation by analyzing parameter dynamics, where benign fine-tuning causes parameters to cumulatively drift toward danger-aligned directions, progressively undermining the model's safety. This finding suggests that samples contributing more to this drift has greater fine-tuning risks. Based on this insight, we propose a method of Sample-Level Quantification of Safety Degradation (SQSD), which quantifies the influence of each training sample on safety degradation. Specifically, SQSD computes continuous risk scores to samples by measuring their induced parameter updates' projection difference between danger and safety directions. Extensive experiments across multiple models and datasets demonstrate that SQSD effectively quantifies sample-level fine-tuning risks and exhibits strong transferability across model architectures, parameter scales, and parameter-efficient methods.

Although deep learning (DL) has received much attention in accelerated magnetic resonance imaging (MRI), recent studies show that tiny input perturbations may lead to instabilities of DL-based MRI reconstruction models. However, the approaches of robustifying these models are underdeveloped. Compared to image classification, it could be much more challenging to achieve a robust MRI image reconstruction network considering its regression-based learning objective, limited amount of training data, and lack of efficient robustness metrics. To circumvent the above limitations, our work revisits the problem of DL-based image reconstruction through the lens of robust machine learning. We find a new instability source of MRI image reconstruction, i.e., the lack of reconstruction robustness against spatial transformations of an input, e.g., rotation and cutout. Inspired by this new robustness metric, we develop a robustness-aware image reconstruction method that can defend against both pixel-wise adversarial perturbations as well as spatial transformations. Extensive experiments are also conducted to demonstrate the effectiveness of our proposed approaches.

Scaling limits, such as infinite-width limits, serve as promising theoretical tools to study large-scale models. However, it is widely believed that existing infinite-width theory does not faithfully explain the behavior of practical networks, especially those trained in standard parameterization (SP) meaning He initialization with a global learning rate. For instance, existing theory for SP predicts instability at large learning rates and vanishing feature learning at stable ones. In practice, however, optimal learning rates decay slower than theoretically predicted and networks exhibit both stable training and non-trivial feature learning, even at very large widths. Here, we show that this discrepancy is not fully explained by finite-width phenomena. Instead, we find a resolution through a finer-grained analysis of the regime previously considered unstable and therefore uninteresting. In particular, we show that, under cross-entropy (CE) loss, the unstable regime comprises two distinct sub-regimes: a catastrophically unstable regime and a more benign controlled divergence regime, where logits diverge but gradients and activations remain stable. Moreover, under large learning rates at the edge of the controlled divergence regime, there exists a well-defined infinite width limit where features continue to evolve in all the hidden layers. In experiments across optimizers, architectures, and data modalities, we validate that neural networks operate in this controlled divergence regime under CE loss but not under MSE loss. Our empirical evidence suggests that width-scaling considerations are surprisingly useful for predicting empirically maximal stable learning rate exponents which provide useful guidance on optimal learning rate exponents. Finally, our analysis clarifies the effectiveness and limitations of recently proposed layerwise learning rate scaling for standard initialization.

Large language models (LLMs) are increasingly deployed in settings where reasoning, such as multi-step problem solving and chain-of-thought, is essential. Yet, current evaluation practices overwhelmingly report single-run accuracy while ignoring the intrinsic uncertainty that naturally arises from stochastic decoding. This omission creates a blind spot because practitioners cannot reliably assess whether a method's reported performance is stable, reproducible, or cost-consistent. We introduce ReasonBENCH, the first benchmark designed to quantify the underlying instability in LLM reasoning. ReasonBENCH provides (i) a modular evaluation library that standardizes reasoning frameworks, models, and tasks, (ii) a multi-run protocol that reports statistically reliable metrics for both quality and cost, and (iii) a public leaderboard to encourage variance-aware reporting. Across tasks from different domains, we find that the vast majority of reasoning strategies and models exhibit high instability. Notably, even strategies with similar average performance can display confidence intervals up to four times wider, and the top-performing methods often incur higher and less stable costs. Such instability compromises reproducibility across runs and, consequently, the reliability of reported performance. To better understand these dynamics, we further analyze the impact of prompts, model families, and scale on the trade-off between solve rate and stability. Our results highlight reproducibility as a critical dimension for reliable LLM reasoning and provide a foundation for future reasoning methods and uncertainty quantification techniques. ReasonBENCH is publicly available at https://github.com/au-clan/ReasonBench .

We present an open-source Python library for simulating two-dimensional incompressible Kelvin-Helmholtz instabilities in stratified shear flows. The solver employs a fractional-step projection method with spectral Poisson solution via Fast Sine Transform, achieving second-order spatial accuracy. Implementation leverages NumPy, SciPy, and Numba JIT compilation for efficient computation. Four canonical test cases explore Reynolds numbers 1000--5000 and Richardson numbers 0.1--0.3: classical shear layer, double shear configuration, rotating flow, and forced turbulence. Statistical analysis using Shannon entropy and complexity indices reveals that double shear layers achieve 2.8times higher mixing rates than forced turbulence despite lower Reynolds numbers. The solver runs efficiently on standard desktop hardware, with 384times192 grid simulations completing in approximately 31 minutes. Results demonstrate that mixing efficiency depends on instability generation pathways rather than intensity measures alone, challenging Richardson number-based parameterizations and suggesting refinements for subgrid-scale representation in climate models.

This paper introduces a continuous-time stochastic dynamical framework for understanding how large language models (LLMs) may self-amplify latent biases or toxicity through their own chain-of-thought reasoning. The model posits an instantaneous "severity" variable x(t) in [0,1] evolving under a stochastic differential equation (SDE) with a drift term μ(x) and diffusion σ(x). Crucially, such a process can be consistently analyzed via the Fokker--Planck approach if each incremental step behaves nearly Markovian in severity space. The analysis investigates critical phenomena, showing that certain parameter regimes create phase transitions from subcritical (self-correcting) to supercritical (runaway severity). The paper derives stationary distributions, first-passage times to harmful thresholds, and scaling laws near critical points. Finally, it highlights implications for agents and extended LLM reasoning models: in principle, these equations might serve as a basis for formal verification of whether a model remains stable or propagates bias over repeated inferences.

Fine-tuning pre-trained models has been ubiquitously proven to be effective in a wide range of NLP tasks. However, fine-tuning the whole model is parameter inefficient as it always yields an entirely new model for each task. Currently, many research works propose to only fine-tune a small portion of the parameters while keeping most of the parameters shared across different tasks. These methods achieve surprisingly good performance and are shown to be more stable than their corresponding fully fine-tuned counterparts. However, such kind of methods is still not well understood. Some natural questions arise: How does the parameter sparsity lead to promising performance? Why is the model more stable than the fully fine-tuned models? How to choose the tunable parameters? In this paper, we first categorize the existing methods into random approaches, rule-based approaches, and projection-based approaches based on how they choose which parameters to tune. Then, we show that all of the methods are actually sparse fine-tuned models and conduct a novel theoretical analysis of them. We indicate that the sparsity is actually imposing a regularization on the original model by controlling the upper bound of the stability. Such stability leads to better generalization capability which has been empirically observed in a lot of recent research works. Despite the effectiveness of sparsity grounded by our theory, it still remains an open problem of how to choose the tunable parameters. To better choose the tunable parameters, we propose a novel Second-order Approximation Method (SAM) which approximates the original problem with an analytically solvable optimization function. The tunable parameters are determined by directly optimizing the approximation function. The experimental results show that our proposed SAM model outperforms many strong baseline models and it also verifies our theoretical analysis.

Stochastic Gradient Descent (SGD) stands as a cornerstone optimization algorithm with proven real-world empirical successes but relatively limited theoretical understanding. Recent research has illuminated a key factor contributing to its practical efficacy: the implicit regularization it instigates. Several studies have investigated the linear stability property of SGD in the vicinity of a stationary point as a predictive proxy for sharpness and generalization error in overparameterized neural networks (Wu et al., 2022; Jastrzebski et al., 2019; Cohen et al., 2021). In this paper, we delve deeper into the relationship between linear stability and sharpness. More specifically, we meticulously delineate the necessary and sufficient conditions for linear stability, contingent on hyperparameters of SGD and the sharpness at the optimum. Towards this end, we introduce a novel coherence measure of the loss Hessian that encapsulates pertinent geometric properties of the loss function that are relevant to the linear stability of SGD. It enables us to provide a simplified sufficient condition for identifying linear instability at an optimum. Notably, compared to previous works, our analysis relies on significantly milder assumptions and is applicable for a broader class of loss functions than known before, encompassing not only mean-squared error but also cross-entropy loss.

The transition of an impulsively excited kink oscillation of a solar coronal loop to an oscillation with a stationary amplitude, i.e., the damping pattern, is determined using the low-dimensional self-oscillation model. In the model, the decayless kink oscillations are sustained by the interaction of the oscillating loop with an external quasi-steady flow. The analytical solution is based on the assumption that the combined effect of the effective dissipation, for example, by resonant absorption, and interaction with an external flow, is weak. The effect is characterised by a dimensionless coupling parameter. The damping pattern is found to depend upon the initial amplitude and the coupling parameter. The approximate expression shows a good agreement with a numerical solution of the self-oscillation equation. The plausibility of the established damping pattern is demonstrated by an observational example. Notably, the damping pattern is not exponential, and the characteristic decay time is different from the time determined by the traditionally used exponential damping fit. Implications of this finding for seismology of the solar coronal plasmas are discussed. In particular, it is suggested that a very rapid, in less than the oscillation period, decay of the oscillation to the stationary level, achieved for larger values of the coupling parameter, can explain the relative rareness of the kink oscillation events.

Under the standard split, Muon gets hidden matrices and AdamW embeddings/output head. Muon groks modular addition faster, but its solutions do not hold. All nine configurations on (a+b) bmod 113 grok and later lose generalization. Across five seeds the selected AdamW reference falls below threshold on four, reaching 27.59%. Instability persists across two moduli, two widths, two training fractions, subtraction, and depth. The failure arises at the representation-readout interface, identified only jointly up to an invertible map unselected by the loss. After solving the training set, the gradient falls to order 10^{-6} and the optimizers respond differently: step-size elasticity is -0.03 for Muon versus +1.5 for AdamW, and the Muon group moves 8.0 times faster per parameter. From bit-identical states, freezing either group prevents failure. Freezing embeddings/readout removes it in five runs over 451,400 post-grokking steps and five paired seeds: unfrozen arms record 137-321 sub-threshold evaluations, frozen arms none. Removing Muon's normalization and orthogonalization is no substitute: it collapses representation from 326 effective conjugate pairs to 4, shows no recurrent collapse, and fails terminally. Fourier filtering separates circuit failure from masking. Across 43 checkpoints over five seeds and three regimes, the task-aligned family reaches exactly 100% alone. In circuit failure it no longer solves the task; in masking it remains perfect while the full model reaches 45.85%, giving a positive margin on every example, including errors, but being outvoted by a near-equal adversarial remainder. Rescaling it restores 99.9%; grokking is the same condition resolving upward. The task selects the family, swapping (k,k) for (k,-k) under subtraction. Across an abrupt collapse, standard Fourier support is unchanged and the power-distribution cosine remains 0.9899.

One-step generative modeling has emerged as a leading approach for amortizing the inference cost of diffusion and flow-matching models. Among distillation-free methods, MeanFlow training is notoriously unstable, with non-decreasing loss and unbounded gradient variance. In this work, we establish a theory that attributes this pathology to a misuse of the conditional velocity field. We show that the conditional velocity plays two distinct statistical roles in the loss: both as an unbiased regression target and as a Monte Carlo control variate in a Jacobi-vector product, with the original MeanFlow loss assigning the wrong coefficient to the latter. We derive the optimal coefficient in closed form and show that a family of fixes in concurrent works corresponds to different practical realizations of the same optimum. A controlled sweep of this coefficient on two-dimensional benchmarks and on a latent Diffusion Transformer recovers the predicted bias-variance ordering. Our DiT experiment also reveals a quantitative FID-MSE landscape mismatch. Specifically, although the gradient-MSE is minimized at an interior coefficient value near β!=!0.94, the coefficient that minimizes FID prefers to use conditional velocity directly at the unbiased corner. Our analysis therefore explains why MeanFlow is unstable and unifies its concurrent remedies, and shows that the variance-optimal coefficient need not coincide with the quality-optimal one.

In this work, we consider the generalized Benjamin-Bona-Mahony equation $partial_t u+partial_x u+partial_x( |u|^pu)-partial_t partial_x^{2}u=0, quad(t,x) in R times R, with p>4. This equation has the traveling wave solutions \phi_{c}(x-ct), for any frequency c>1. It has been proved by Souganidis and Strauss Strauss-1990 that, there exists a number c_{0}(p)>1, such that solitary waves \phi_{c}(x-ct) with 1<c<c_{0}(p) is orbitally unstable, while for c>c_{0}(p), \phi_{c}(x-ct) is orbitally stable. The linear exponential instability in the former case was further proved by Pego and Weinstein Pego-1991-eigenvalue. In this paper, we prove the orbital instability in the critical case c=c_{0}(p)$.

Scaling laws for large language models depend critically on the optimizer and parameterization. Existing hyperparameter transfer laws are mainly developed for first-order optimizers, and they do not structurally prevent training instability at scale. Recent hypersphere optimization methods constrain weight matrices to a fixed-norm hypersphere, offering a promising alternative for more stable scaling. We introduce HyperP (Hypersphere Parameterization), the first framework for transferring optimal learning rates across model width, depth, training tokens, and Mixture-of-Experts (MoE) granularity under the Frobenius-sphere constraint with the Muon optimizer. We prove that weight decay is a first-order no-op on the Frobenius sphere, show that Depth-μP remains necessary, and find that the optimal learning rate follows the same data-scaling power law with the "magic exponent" 0.32 previously observed for AdamW. A single base learning rate tuned at the smallest scale transfers across all compute budgets under HyperP, yielding 1.58times compute efficiency over a strong Muon baseline at 6times10^{21} FLOPs. Moreover, HyperP delivers transferable stability: all monitored instability indicators, including Z-values, output RMS, and activation outliers, remain bounded and non-increasing under training FLOPs scaling. We also propose SqrtGate, an MoE gating mechanism derived from the hypersphere constraint that preserves output RMS across MoE granularities for improved granularity scaling, and show that hypersphere optimization enables substantially larger auxiliary load-balancing weights, yielding both strong performance and good expert balance. We release our training codebase at https://github.com/microsoft/ArchScale.

Reinforcement learning is structurally harder than supervised learning because the policy changes the data distribution it learns from. The resulting fragility is especially visible in large-model training, where the training and rollout systems differ in numerical precision, sampling, and other implementation details. Existing methods manage this fragility by adding hyper-parameters to the training objective, which makes the algorithm more sensitive to its configuration and requires retuning whenever the task, model scale, or distribution mismatch changes. This fragility traces to two concerns that current objectives entangle through hyper-parameters set before training begins: a trust-region concern, that updates should not move the policy too far from its current value, and an off-policy concern, that data from older or different behavior policies should influence the update only to the extent that it remains reliable. Neither concern is a constant to set in advance, and their severity is reflected in the policy-ratio distribution of the current batch. We present a simple yet effective batch-adaptive objective that replaces fixed clipping with the normalized effective sample size of the policy ratios. The same statistic caps the score-function weight and sets the strength of an off-policy regularizer, so the update stays close to the usual on-policy score-function update when ratios are nearly uniform, and tightens automatically when stale or mismatched data cause ratio concentration, while retaining a nonzero learning signal on high-ratio tokens. Experiments across a wide range of settings show that our method matches or exceeds tuned baselines, introducing no new objective hyper-parameters and removing several existing ones. The code is available at https://github.com/FeynRL-project/FeynRL.

Global stability of differentially rotating plasma is investigated using a generalized effective potential. We first, for a current-free system, obtain a general form of an effective potential in terms of the free energies of global curvature and gradients of rotation for non-axisymmetric disturbances. We then examine the stability of differentially rotating disks for several rotation profiles and present the associated effective potential for the onset of these instabilities in the MHD regime. In particular, results for global axisymmetric magnetorotational instability (MRI) as well as local and global non-axisymmetric modes are presented. The latter constitute two distinct non-axisymmetric modes, a high frequency local MRI and a global low-frequency non-axisymmetric mode (the magneto-curvature mode, introduced in Ebrahimi&Pharr, ApJ 2022), confined either between two Alfv\'enic resonances or an Alfv\'enic resonance and a boundary.

Recent progress in multi-turn reinforcement learning (RL) has significantly improved reasoning LLMs' performances on complex interactive tasks. Despite advances in stabilization techniques such as fine-grained credit assignment and trajectory filtering, instability remains pervasive and often leads to training collapse. We argue that this instability stems from inefficient exploration in multi-turn settings, where policies continue to generate low-information actions that neither reduce uncertainty nor advance task progress. To address this issue, we propose Token- and Turn-level Policy Optimization (T^2PO), an uncertainty-aware framework that explicitly controls exploration at fine-grained levels. At the token level, T^2PO monitors uncertainty dynamics and triggers a thinking intervention once the marginal uncertainty change falls below a threshold. At the turn level, T^2PO identifies interactions with negligible exploration progress and dynamically resamples such turns to avoid wasted rollouts. We evaluate T^2PO in diverse environments, including WebShop, ALFWorld, and Search QA, demonstrating substantial gains in training stability and performance improvements with better exploration efficiency. Code is available at: https://github.com/WillDreamer/T2PO.

Data-parallel (DP) training with synchronous all-reduce is a dominant paradigm for full-parameter fine-tuning of large language models (LLMs). While parameter synchronization guarantees numerical equivalence of model weights after each iteration, it does not necessarily imply alignment of worker-level optimization dynamics before gradient aggregation. This paper identifies and studies this latent mismatch, termed silent inconsistency, where cross-worker divergence in losses and gradients can remain invisible under conventional aggregated monitoring signals. We propose a lightweight, model-agnostic diagnostic framework that quantifies worker-level consistency using training signals readily available in standard pipelines. Specifically, we introduce three complementary metrics: loss dispersion, gradient-norm dispersion, and gradient-direction consistency measured by inter-worker cosine similarity. The proposed metrics incur negligible overhead and require no modification to model architecture, synchronization mechanisms, or optimization algorithms. We validate the framework by fully fine-tuning the 1B-parameter openPangu-Embedded-1B-V1.1 model on the tatsu-lab/alpaca dataset using an 8-NPU DP setup, under controlled perturbations of cross-rank stochasticity. Experimental results show that progressively desynchronized data shuffling and random seeds lead to substantial increases in loss/gradient dispersion and reduced directional alignment, despite smooth globally averaged loss curves. These findings demonstrate that the proposed indicators provide actionable visibility into hidden instability modes in large-scale DP fine-tuning, enabling more reliable diagnosis and configuration assessment.

Nowadays, software systems tend to include Artificial Intelligence (AI) components. Changes in the operational environment have been known to negatively impact the stability of AI-enabled software systems by causing unintended changes in behavior. However, how an environment configuration impacts the behavior of such systems has yet to be explored. Understanding and quantifying the degree of instability caused by different environment settings can help practitioners decide the best environment configuration for the most stable AI systems. To achieve this goal, we performed experiments with eight different combinations of three key environment variables (operating system, Python version, and CPU architecture) on 30 open-source AI-enabled systems using the Travis CI platform. We determine the existence and the degree of instability introduced by each configuration using three metrics: the output of an AI component of the system (model performance), the time required to build and run the system (processing time), and the cost associated with building and running the system (expense). Our results indicate that changes in environment configurations lead to instability across all three metrics; however, it is observed more frequently with respect to processing time and expense rather than model performance. For example, between Linux and MacOS, instability is observed in 23\%, 96.67\%, and 100\% of the studied projects in model performance, processing time, and expense, respectively. Our findings underscore the importance of identifying the optimal combination of configuration settings to mitigate drops in model performance and reduce the processing time and expense before deploying an AI-enabled system.

Prospect Theory (PT) models human decision-making behaviour under uncertainty, among which linguistic uncertainty is commonly adopted in real-world scenarios. Although recent studies have developed some frameworks to test PT parameters for Large Language Models (LLMs), few have considered the fitness of PT itself on LLMs. Moreover, whether PT is robust under linguistic uncertainty perturbations, especially epistemic markers (e.g. "likely"), remains highly under-explored. To address these gaps, we design a three-stage workflow based on a classic behavioural economics experimental setup. We first estimate PT parameters with economics questions and evaluate PT's fitness with performance metrics. We then derive probability mappings for epistemic markers in the same context, and inject these mappings into the prompt to investigate the stability of PT parameters. Our findings suggest that modelling LLMs' decision-making with PT is not consistently reliable across models, and applying Prospect Theory to LLMs is likely not robust to epistemic uncertainty. The findings caution against the deployment of PT-based frameworks in real-world applications where epistemic ambiguity is prevalent, giving valuable insights in behaviour interpretation and future alignment direction for LLM decision-making.

Group Relative Policy Optimization (GRPO) is highly effective for post-training autoregressive (AR) language models, yet its direct application to diffusion large language models (dLLMs) often triggers reward collapse. We identify two sources of incompatibility. First, GRPO relies on importance ratios defined by sequence probabilities, which are intractable in dLLMs and must be estimated (e.g., via ELBO-based or mean-field likelihood proxies), yielding inherently noisy ratios. Second, standard GRPO's formulation is not designed for estimated ratios: its conditional clipping can be anomalously bypassed by model-agnostic estimation noise, producing gradient spikes, while its fixed group-size normalization amplifies gradient-magnitude fluctuations under high-variance ratio estimates. We show these effects form a self-reinforcing instability loop that drives policy drift and further increases ratio variance. To break this loop, we propose StableDRL, a reformulation of GRPO tailored for dLLMs that uses (i) unconditional clipping to suppress outlier-induced spikes and (ii) self-normalization to constrain updates within the convex hull of per-sample gradients. We further extend StableDRL to block-wise diffusion models via a staircase attention mechanism.

Ensemble Kalman Inversion (EKI) is a derivative-free, ensemble-based method for inverse and optimization problems. Its continuous-time formulation can be interpreted as an interacting particle system driven by a Kalman-type preconditioned descent direction. A well-known limitation of this dynamics is the possible premature collapse of the covariance of the ensemble, which makes the method sensitive to the initial ensemble. We introduce a second-order particle system in which the particles evolve according to an inertial dynamics. The model combines a Kalman-type relaxation force with damping, attraction towards the ensemble mean, and a short-range repulsive interaction designed to counteract ensemble collapse. The resulting dynamics can be interpreted as a heavy-ball reformulation of continuous-time EKI enriched by competing attractive and repulsive mechanisms. For linear inverse problems, we analyze the induced mean and fluctuation dynamics and identify a parameter regime in which fully collapsed configurations are linearly unstable. We further characterize asymptotic equilibria through a constrained optimality condition on the subspace retained by the limiting ensemble covariance and derive an exponential decay estimate. Numerical experiments illustrate the effect of inertia and repulsion on the ensemble dynamics and compare the proposed second-order method with first-order EKI-type

While locate-then-edit knowledge editing efficiently updates knowledge encoded within Large Language Models (LLMs), a critical generalization failure mode emerges in the practical same-subject knowledge editing scenario: models fail to recall the updated knowledge when following user instructions, despite successfully recalling it in the original edited form. This paper identifies the geometric root of this generalization collapse as a fundamental conflict where the inner activation drifts induced by prompt variations exceed the model's geometric tolerance for generalization after editing. We attribute this instability to a dual pathology: (1) The joint optimization with orthogonal gradients collapses solutions into sharp minima with narrow stability, and (2) the standard covariance constraint paradoxically acts as a Covariance Trap that amplifies input perturbations. To resolve this, we introduce RoSE (Robust Same-subject Editing), which employs Isotropic Geometric Alignment to minimize representational deviation and Hierarchical Knowledge Integration to smooth the optimization landscape. Extensive experiments demonstrate that RoSE significantly improves instruction-following capabilities, laying the foundation for robust interactive parametric memory of LLM agents.

Differentiable programming techniques are widely used in the community and are responsible for the machine learning renaissance of the past several decades. While these methods are powerful, they have limits. In this short report, we discuss a common chaos based failure mode which appears in a variety of differentiable circumstances, ranging from recurrent neural networks and numerical physics simulation to training learned optimizers. We trace this failure to the spectrum of the Jacobian of the system under study, and provide criteria for when a practitioner might expect this failure to spoil their differentiation based optimization algorithms.

Recent studies show that Large Language Models (LLMs) with safety alignment can be jail-broken by fine-tuning on a dataset mixed with harmful data. First time in the literature, we show that the jail-broken effect can be mitigated by separating states in the finetuning stage to optimize the alignment and user datasets. Unfortunately, our subsequent study shows that this simple Bi-State Optimization (BSO) solution experiences convergence instability when steps invested in its alignment state is too small, leading to downgraded alignment performance. By statistical analysis, we show that the excess drift towards consensus could be a probable reason for the instability. To remedy this issue, we propose Lazy(i) safety alignment (Lisa), which introduces a proximal term to constraint the drift of each state. Theoretically, the benefit of the proximal term is supported by the convergence analysis, wherein we show that a sufficient large proximal factor is necessary to guarantee Lisa's convergence. Empirically, our results on four downstream finetuning tasks show that Lisa with a proximal term can significantly increase alignment performance while maintaining the LLM's accuracy on the user tasks. Code is available at https://github.com/git-disl/Lisa.

Pretraining of large language models is not only expensive but also prone to certain training instabilities. A specific instability that often occurs at the end of training is output logit divergence. The most widely used mitigation strategies, z-loss and logit soft-capping, merely address the symptoms rather than the underlying cause of the problem. In this paper, we analyze the instability from the perspective of the output embeddings' geometry and identify anisotropic embeddings as its source. Based on this, we propose output embedding centering (OEC) as a new mitigation strategy, and demonstrate that it suppresses output logit divergence. OEC can be implemented in two different ways: as a deterministic operation called μ-centering, or a regularization method called μ-loss. Our experiments show that both variants outperform z-loss in terms of training stability, while being on par with logit soft-capping. This holds true both in the presence and the absence of weight tying. As a secondary result, we find that μ-loss is significantly less sensitive to regularization hyperparameter tuning than z-loss.

Consistency models (CMs) are a powerful class of diffusion-based generative models optimized for fast sampling. Most existing CMs are trained using discretized timesteps, which introduce additional hyperparameters and are prone to discretization errors. While continuous-time formulations can mitigate these issues, their success has been limited by training instability. To address this, we propose a simplified theoretical framework that unifies previous parameterizations of diffusion models and CMs, identifying the root causes of instability. Based on this analysis, we introduce key improvements in diffusion process parameterization, network architecture, and training objectives. These changes enable us to train continuous-time CMs at an unprecedented scale, reaching 1.5B parameters on ImageNet 512x512. Our proposed training algorithm, using only two sampling steps, achieves FID scores of 2.06 on CIFAR-10, 1.48 on ImageNet 64x64, and 1.88 on ImageNet 512x512, narrowing the gap in FID scores with the best existing diffusion models to within 10%.

Electroencephalography (EEG) is a cornerstone of brain-computer interfaces and clinical neuroscience, yet deep learning models are typically trained and evaluated under a single, unreported preprocessing pipeline. We formalize preprocessing choices as a counterfactual intervention space and show that EEG predictions are surprisingly unstable under this space: across six datasets spanning four paradigms, up to 42% of trial-level predictions flip when only the preprocessing changes, a variability that standard uncertainty methods do not explicitly quantify because they condition on a fixed preprocessing pipeline. We provide three tools to make this instability measurable, decomposable, and reducible. First, a Walsh-Hadamard decomposition of the 2^7 pipeline space reveals that sensitivity is near-additive in practice under the binary intervention design, enabling efficient step-by-step optimization. Second, we introduce Preprocessing Uncertainty (PU), a per-trial diagnostic that captures a dimension of instability complementary to model-based confidence. Third, we study Normalized Adaptive PGI (NA-PGI), a graph-structured regularizer that exploits the compositional structure of preprocessing interventions as one mitigation strategy with clear scope conditions.

This paper establishes Lipschitz stability for the simultaneous recovery of a variable density coefficient and the initial displacement in a damped biharmonic wave equation. The data consist of the boundary Cauchy data for the Laplacian of the solution, \(Δu |_{\partial Ω}\) and \( \partial_{n}(Δu)|_{\partial Ω}.\) We first prove that the associated system operator generates a contraction semigroup, which ensures the well-posedness of the forward problem. A key observability inequality is then derived via multiplier techniques. Building on this foundation, explicit stability estimates for the inverse problem are obtained. These estimates demonstrate that the biharmonic structure inherently enhances the stability of parameter identification, with the stability constants exhibiting an explicit dependence on the damping coefficient via the factor \( (1 + γ)^{1/2} \). This work provides a rigorous theoretical basis for applications in non-destructive testing and dynamic inversion.

Capturing aleatoric uncertainty is a critical part of many machine learning systems. In deep learning, a common approach to this end is to train a neural network to estimate the parameters of a heteroscedastic Gaussian distribution by maximizing the logarithm of the likelihood function under the observed data. In this work, we examine this approach and identify potential hazards associated with the use of log-likelihood in conjunction with gradient-based optimizers. First, we present a synthetic example illustrating how this approach can lead to very poor but stable parameter estimates. Second, we identify the culprit to be the log-likelihood loss, along with certain conditions that exacerbate the issue. Third, we present an alternative formulation, termed β-NLL, in which each data point's contribution to the loss is weighted by the β-exponentiated variance estimate. We show that using an appropriate β largely mitigates the issue in our illustrative example. Fourth, we evaluate this approach on a range of domains and tasks and show that it achieves considerable improvements and performs more robustly concerning hyperparameters, both in predictive RMSE and log-likelihood criteria.

In the era of proliferation of large language and image generation models, the phenomenon of "model collapse" refers to the situation whereby as a model is trained recursively on data generated from previous generations of itself over time, its performance degrades until the model eventually becomes completely useless, i.e the model collapses. In this work, we study this phenomenon in the setting of high-dimensional regression and obtain analytic formulae which quantitatively outline this phenomenon in a broad range of regimes. In the special case of polynomial decaying spectral and source conditions, we obtain modified scaling laws which exhibit new crossover phenomena from fast to slow rates. We also propose a simple strategy based on adaptive regularization to mitigate model collapse. Our theoretical results are validated with experiments.

Traditional fixed-depth architectures scale quality by increasing training FLOPs, typically through increased parameterization, at the expense of a higher memory footprint, or data. A potential alternative is looped architectures, which instead increase FLOPs by sending activations through a block of layers in a loop. While promising, existing recipes for training looped architectures can be unstable, suffering from residual explosion and loss spikes. We address these challenges by recasting looping as a nonlinear time-variant dynamical system over the residual stream. Via a linear approximation to this system, we find that instability occurs in existing looped architectures as a result of large spectral norms in their injection parameters. To address these instability issues, we propose Parcae, a novel stable, looped architecture that constrains the spectral norm of the injection parameters via discretization of a negative diagonal parameterization. As a result, Parcae achieves up to 6.3% lower validation perplexity over prior large-scale looped models. Using our stable looped architecture, we investigate the scaling properties of looping as a medium to improve quality by increasing FLOPs in training and test-time. For training, we derive predictable power laws to scale FLOPs while keeping parameter count fixed. Our initial scaling laws suggest that looping and data should be increased in tandem, given a fixed FLOP budget. At test-time, we find that Parcae can use looping to scale compute, following a predictable, saturating exponential decay. When scaled up to 1.3B parameters, we find that Parcae improves CORE and Core-Extended quality by 2.99 and 1.18 points when compared to strong Transformer baselines under a fixed parameter and data budget, achieving a relative quality of up to 87.5% a Transformer twice the size.

Despite significant progress in model editing methods, their application in real-world scenarios remains challenging as they often cause large language models (LLMs) to collapse. Among them, ROME is particularly concerning, as it could disrupt LLMs with only a single edit. In this paper, we study the root causes of such collapse. Through extensive analysis, we identify two primary factors that contribute to the collapse: i) inconsistent handling of prefixed and unprefixed keys in the parameter update equation may result in very small denominators, causing excessively large parameter updates; ii) the subject of collapse cases is usually the first token, whose unprefixed key distribution significantly differs from the prefixed key distribution in autoregressive transformers, causing the aforementioned issue to materialize. To validate our findings, we propose a simple yet effective approach: uniformly using prefixed keys during editing phase and adding prefixes during testing phase to ensure the consistency between training and testing. The experimental results show that the proposed solution can prevent model collapse while maintaining the effectiveness of the edits.

Hybrid reasoning systems that combine learned components with model-based inference are increasingly deployed in tool-augmented decision loops, yet their runtime behavior under partial observability and sustained evidence mismatch remains poorly understood. In practice, failures often arise as gradual divergence of internal reasoning dynamics rather than as isolated prediction errors. This work studies runtime stability in hybrid reasoning systems from a Kalman-inspired perspective. We model reasoning as a stochastic inference process driven by an internal innovation signal and introduce cognitive drift as a measurable runtime phenomenon. Stability is defined in terms of detectability, bounded divergence, and recoverability rather than task-level correctness. We propose a runtime stability framework that monitors innovation statistics, detects emerging instability, and triggers recovery-aware control mechanisms. Experiments on multi-step, tool-augmented reasoning tasks demonstrate reliable instability detection prior to task failure and show that recovery, when feasible, re-establishes bounded internal behavior within finite time. These results emphasize runtime stability as a system-level requirement for reliable reasoning under uncertainty.

Large Language Models (LLMs) can significantly improve their reasoning capabilities by interacting with external tools, a paradigm known as Tool-Integrated Reasoning (TIR). However, extending TIR to multi-turn scenarios using Reinforcement Learning (RL) is often hindered by training instability and performance collapse. We identify that such instability is primarily caused by a distributional drift from external tool feedback, leading to the generation of low-probability tokens. This issue compounds over successive turns, causing catastrophic gradient norm explosions that derail the training process. To address this challenge, we introduce SimpleTIR , a plug-and-play algorithm that stabilizes multi-turn TIR training. Its core strategy is to identify and filter out trajectories containing void turns, i.e., turns that yield neither a code block nor a final answer. By removing these problematic trajectories from the policy update, SimpleTIR effectively blocks the harmful, high-magnitude gradients, thus stabilizing the learning dynamics. Extensive experiments show that SimpleTIR achieves state-of-the-art performance on challenging math reasoning benchmarks, notably elevating the AIME24 score from a text-only baseline of 22.1 to 50.5 when starting from the Qwen2.5-7B base model. Furthermore, by avoiding the constraints of supervised fine-tuning, SimpleTIR encourages the model to discover diverse and sophisticated reasoning patterns, such as self-correction and cross-validation.

DDEBIFTOOL is a collection of Matlab routines for numerical bifurcation analysis of systems of delay differential equations with discrete constant and state-dependent delays. The package supports continuation and stability analysis of steady state solutions and periodic solutions. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. To analyse the stability of steady state solutions, approximations are computed to the rightmost, stability-determining roots of the characteristic equation which can subsequently be used as starting values in a Newton procedure. For periodic solutions, approximations to the Floquet multipliers are computed. The manual describes the structure of the package, its routines, and its data and method parameter structures.

Power transforms are popular parametric techniques for making data more Gaussian-like, and are widely used as preprocessing steps in statistical analysis and machine learning. However, we find that direct implementations of power transforms suffer from severe numerical instabilities, which can lead to incorrect results or even crashes. In this paper, we provide a comprehensive analysis of the sources of these instabilities and propose effective remedies. We further extend power transforms to the federated learning setting, addressing both numerical and distributional challenges that arise in this context. Experiments on real-world datasets demonstrate that our methods are both effective and robust, substantially improving stability compared to existing approaches.

Scaling model performance typically requires increasing model size. Looped Transformer offers a compelling alternative by iteratively reusing the same Transformer blocks, trading additional computation for improved performance without increasing parameter count or context length. Because the number of loop iterations can be adjusted at inference, it also provides a natural mechanism for balancing performance and test-time compute. However, Looped Transformer still suffers from training instability when the number of loop iterations increases. Our analysis reveals that this instability stems from two sources: gradient oscillation and residual explosion. To address these two problems, we propose the Fully Looped Transformer, which introduces two parameter-free modifications: (1) Fully Looped Architecture, which distributes inter-loop signals across all layers to mitigate residual explosion; (2) Attention Injection, which reuses the existing attention block to suppress gradient oscillation. These modifications stabilize training dynamics, enabling the Fully Looped Transformer to be trained stably up to 12 loop iterations, whereas other baseline looped models collapse in this regime. In milder settings where Looped Transformer does not collapse, Fully Looped Transformer still improves average downstream-task performance by up to 13.2\%. Overall, our experiments demonstrate that Fully Looped Transformer improves training stability, enhances downstream performance, and provides preliminary adaptability under different test-time compute budgets by varying loop iterations at inference.

We introduce a new predator-prey model by replacing the growth and predation constant by a square matrix, and the population density as a population vector. The classical Lotka-Volterra model describes a population that either modulates or converges. Stability analysis of such models have been extensively studied by the works of Merdan (https://doi.org/10.1016/j.chaos.2007.06.062). The new model adds complexity by introducing an age group structure where the population of each age group evolves as prescribed by the Leslie matrix. The added complexity changes the behavior of the model such that the population either displays roughly an exponential growth or decay. We first provide an exact equation that describes a time evolution and use analytic techniques to obtain an approximate growth factor. We also discuss the variants of the Leslie model, i.e., the complex value predator-prey model and the competitive model. We then prove the Last Species Standing theorem that determines the dominant population in the large time limit. The recursive structure of the model denies the application of simple regression. We discuss a machine learning scheme that allows an admissible fit for the population evolution of Paramecium Aurelia and Paramecium Caudatum. Another potential avenue to simplify the computation is to use the machinery of quantum operators. We demonstrate the potential of this approach by computing the Hamiltonian of a simple Leslie system.

Water looks unchanged as it warms, then at a critical point it boils. We ask whether long-horizon language agents show an analogous transition in their implicit world models. In some parameter settings, changing state load by a small amount, or adding a single step of horizon, leaves behavior nearly unchanged; near a critical boundary, the same small change causes a sudden world collapse. We study this effect in a deterministic task family with exact per-step gold state. A large grid search over state cardinality, dependency density, horizon, branching, observation mode, and mutation rate reveals a phase diagram: a solved plateau, a narrow transition band, and a collapse floor. Per-step traces show the mechanism: world-state fidelity fails before action validity, so the agent is not merely choosing a bad action; it is acting from a corrupted world. Stronger models translate the critical boundary but do not remove the qualitative transition. These results make world-model collapse a measurable bottleneck for long-horizon agents.

Training stability remains a critical bottleneck for Group Relative Policy Optimization (GRPO), often manifesting as a trade-off between reasoning plasticity and general capability retention. We identify a root cause as the geometric conflict between plasticity and stability gradients, which leads to destructive interference. Crucially, we argue that deterministic projection methods are suboptimal for GRPO as they overlook the intrinsic stochasticity of group-based gradient estimates. To address this, we propose Probabilistic Conflict Resolution (PCR), a Bayesian framework that models gradients as random variables. PCR dynamically arbitrates conflicts via an uncertainty-aware ``soft projection'' mechanism, optimizing the signal-to-noise ratio. Extensive experiments demonstrate that PCR significantly smooths the training trajectory and achieves superior performance in various reasoning tasks.

In diffusion models, UNet is the most popular network backbone, since its long skip connects (LSCs) to connect distant network blocks can aggregate long-distant information and alleviate vanishing gradient. Unfortunately, UNet often suffers from unstable training in diffusion models which can be alleviated by scaling its LSC coefficients smaller. However, theoretical understandings of the instability of UNet in diffusion models and also the performance improvement of LSC scaling remain absent yet. To solve this issue, we theoretically show that the coefficients of LSCs in UNet have big effects on the stableness of the forward and backward propagation and robustness of UNet. Specifically, the hidden feature and gradient of UNet at any layer can oscillate and their oscillation ranges are actually large which explains the instability of UNet training. Moreover, UNet is also provably sensitive to perturbed input, and predicts an output distant from the desired output, yielding oscillatory loss and thus oscillatory gradient. Besides, we also observe the theoretical benefits of the LSC coefficient scaling of UNet in the stableness of hidden features and gradient and also robustness. Finally, inspired by our theory, we propose an effective coefficient scaling framework ScaleLong that scales the coefficients of LSC in UNet and better improves the training stability of UNet. Experimental results on four famous datasets show that our methods are superior to stabilize training and yield about 1.5x training acceleration on different diffusion models with UNet or UViT backbones. Code: https://github.com/sail-sg/ScaleLong

In this article, we consider a newly proposed parameterization of the viscosity coefficient zeta, specifically zeta=zeta_0 {Omega^s_m} H , where zeta_0 = zeta_0{{Omega^s_{m_0}}} within the coincident f(Q) gravity formalism. We consider a non-linear function f(Q)= -Q +alpha Q^n, where alpha and n are arbitrary model parameters, which is a power-law correction to the STEGR scenario. We find an autonomous system by invoking the dimensionless density parameters as the governing phase-space variables. We discuss the physical significance of the model corresponding to the parameter choices n=-1 and n=2 along with the exponent choices s=0, 0.5, and 1.05. We find that model I shows the stable de-Sitter type or stable phantom type (depending on the choice of exponent s) behavior with no transition epoch, whereas model II shows the evolutionary phase from the radiation epoch to the accelerated de-Sitter epoch via passing through the matter-dominated epoch. Hence, we conclude that model I provides a good description of the late-time cosmology but fails to describe the transition epoch, whereas model II modifies the description in the context of the early universe and provides a good description of the matter and radiation era along with the transition phase.

The solution of linear inverse problems arising, for example, in signal and image processing is a challenging problem since the ill-conditioning amplifies, in the solution, the noise present in the data. Recently introduced algorithms based on deep learning overwhelm the more traditional model-based approaches in performance, but they typically suffer from instability with respect to data perturbation. In this paper, we theoretically analyze the trade-off between stability and accuracy of neural networks, when used to solve linear imaging inverse problems for not under-determined cases. Moreover, we propose different supervised and unsupervised solutions to increase the network stability and maintain a good accuracy, by means of regularization properties inherited from a model-based iterative scheme during the network training and pre-processing stabilizing operator in the neural networks. Extensive numerical experiments on image deblurring confirm the theoretical results and the effectiveness of the proposed deep learning-based approaches to handle noise on the data.

Regulatory institutions (from content moderation platforms to financial supervisors) observe, deliberate, and intervene only after a characteristic delay. We ask whether this processing lag alone can destabilize a multi-agent system that would otherwise remain stable, without exogenous shocks, coordination among agents, or malicious actors. We study this question in two stages. First, we analyze a delayed replicator equation in which autonomous agents receive a benefit from radical behavior but face punishment based on a lagged institutional alarm signal. We derive a closed-form critical delay threshold beyond which the unique interior equilibrium loses stability through a Hopf bifurcation, and prove via center manifold reduction that the bifurcation is supercritical (producing bounded oscillations, not explosive growth) for the entire sigmoid response-function family. Second, we embed N=240 agents on a network and equip them with reinforcement learning (tabular Q-learning), comparing three decision architectures in a factorial design: non-reactive agents (fixed policy), reactive agents (threshold heuristic without memory), and Q-learning agents (adaptive with cumulative value estimates). The results reveal a hierarchy opposite to the naive expectation that learning amplifies instability: non-reactive agents are immune to delay (0% runaway across all tested values), reactive agents collapse catastrophically (96% runaway by delay geq 8 steps), and Q-learning agents achieve partial resilience (66% runaway at delay = 20). The destabilizing ingredient is reactivity to delayed signals: agents that immediately exploit low-alarm windows trigger oscillatory feedback loops. Learning buffers this through implicit punishment memory encoded in Q-values

Synchronization in one dimension displays generic scale invariance with universal properties previously observed in surface kinetic roughening and the wider context of the Kardar-Parisi-Zhang (KPZ) universality class. This has been established for phase oscillators and also for some limit-cycle oscillators, both in the presence of columnar (quenched) disorder and of time-dependent noise, by extensive numerical simulations, and has been analytically motivated by continuum approximations in the strong oscillator coupling limit. The robustness and the precise boundaries in parameter space for such critical behavior remain unclear, however, which may preclude further developments, including the extension of these results to higher dimensions and the experimental observation of nonequilibrium criticality in synchronizing (e.g.~electronic or chemical) oscillators. We here present complete numerical phase diagrams of one-dimensional synchronization, including saturation times and values, but, most importantly, also dynamical features giving insight into the gradual emergence of synchronous dynamics, based on systems of phase oscillators with either type of randomness. In the absence of synchronization, the dynamics evolves as expected for random deposition (for time-dependent noise) or linear growth (for columnar disorder), while a crossover from Edwards-Wilkinson to Kardar-Parisi-Zhang behavior (with the corresponding type of randomness) is observed as the randomness strength, or the nonoddity of the coupling among oscillators, is increased in the synchronous region -- their combined effect being partially captured by the so-called KPZ coupling. The distortion of scaling due to phase slips near the desynchronization boundary, a feature that is likely to play a role in experimental contexts, is also discussed.

The phase evolution of gravitational waves encodes critical information about the orbital dynamics of binary systems. In this work, we test the robustness of parameterized tests against unmodeled deviations from general relativity. We demonstrate that these parameterized tests are flexible and sensitive in detecting generic deviations in the waveform using the Cutler-Vallisneri bias formalism. This universality arises from examining the inherent geometry of the waveform signal and understanding how biases manifest. We show how Bayes factors are governed by the intrinsic geometry of the waveform signal manifold when parameterized tests are used to approximate generic violations of GR. We use the singular value decomposition to propose templates that are orthogonal to parameterized tests, identifying degeneracies and enhancing the detection of potential deviations. More broadly, the geometric framework developed here clarifies -- at a fundamental level -- how subtle waveform effects (including orbital eccentricity, spin precession, waveform systematics, and instrumental glitches) can mimic one another in data, and when they are intrinsically distinguishable.

We introduce rd-spiral, an open-source Python library for simulating 2D reaction-diffusion systems using pseudo-spectral methods. The framework combines FFT-based spatial discretization with adaptive Dormand-Prince time integration, achieving exponential convergence while maintaining pedagogical clarity. We analyze three dynamical regimes: stable spirals, spatiotemporal chaos, and pattern decay, revealing extreme non-Gaussian statistics (kurtosis >96) in stable states. Information-theoretic metrics show 10.7% reduction in activator-inhibitor coupling during turbulence versus 6.5% in stable regimes. The solver handles stiffness ratios >6:1 with features including automated equilibrium classification and checkpointing. Effect sizes (delta=0.37--0.78) distinguish regimes, with asymmetric field sensitivities to perturbations. By balancing computational rigor with educational transparency, rd-spiral bridges theoretical and practical nonlinear dynamics.

Training modern neural networks is increasingly fragile, with rare but severe destabilizing updates often causing irreversible divergence or silent performance degradation. Existing optimization methods primarily rely on preventive mechanisms embedded within the optimizer, offering limited ability to detect and recover from instability once it occurs. We introduce a supervisory runtime stability framework that treats optimization as a controlled stochastic process. By isolating an innovation signal derived from secondary measurements, such as validation probes, the framework enables automatic detection and recovery from destabilizing updates without modifying the underlying optimizer. We provide theoretical runtime safety guarantees that formalize bounded degradation and recovery. Our implementation incurs minimal overhead and is compatible with memory-constrained training settings.

Hyperparameter (HP) tuning in deep learning is an expensive process, prohibitively so for neural networks (NNs) with billions of parameters. We show that, in the recently discovered Maximal Update Parametrization (muP), many optimal HPs remain stable even as model size changes. This leads to a new HP tuning paradigm we call muTransfer: parametrize the target model in muP, tune the HP indirectly on a smaller model, and zero-shot transfer them to the full-sized model, i.e., without directly tuning the latter at all. We verify muTransfer on Transformer and ResNet. For example, 1) by transferring pretraining HPs from a model of 13M parameters, we outperform published numbers of BERT-large (350M parameters), with a total tuning cost equivalent to pretraining BERT-large once; 2) by transferring from 40M parameters, we outperform published numbers of the 6.7B GPT-3 model, with tuning cost only 7% of total pretraining cost. A Pytorch implementation of our technique can be found at github.com/microsoft/mup and installable via `pip install mup`.

Neural models are usually adapted through changes in parameters shared among model components via fine-tuning, alignment-based training, and reinforcement learning. These changes have been found effective in short-term optimization. However, they result in long-term alterations in the model's base behavior. In this study, we introduce the concept of structural irreversibility as a characteristic of shared-parameter model adaptation. This concept refers to the intertwining of task-specific objectives with the representational identity of the model. We show that when parameters are directly mutated, the resulting model behaves divergently from the original model. This divergence cannot be reversed deterministically without an explicit parameter snapshot. We introduce reversible behavioral learning, in which model behaviors are structurally dissociated from identity parameters and can be deterministically unloaded through an explicit unload process. We also introduce the Recoverability Factor as a normalized measure of behavioral recoverability and provide additional diagnostics based on model divergence. Experiments show that reversible model adaptation achieves rollback within numerical precision, whereas shared-parameter mutation exhibits persistent post-reset divergence.

The alignment of Small Language Models (SLMs) in the 70--500M parameter range using reinforcement learning is often considered unstable, though the underlying failure mechanisms have not been systematically investigated. In the State-of-the-Art (SOTA) research, fifteen (model, corpus) configurations were trained using Proximal Policy Optimization (PPO). The experiments included Pythia-70M, 160M, 410M and SmolLM2-135M, 360M on the TinyStories, CNN/DailyMail, and Wikitext-103 corpora. Three reproducible failure modes were identified in small-scale language models: silent LoRA parameter freezing in standard PEFT/TRL pipelines, numerical overflow in importance ratios when using bfloat16, and catastrophic policy collapse due to reward-model error. These issues were addressed using a merge-and-reinitialize adapter technique, float32 precision during PPO updates, and a three-layer safety mechanism comprising reward whitening, importance-ratio guarding, and weight rollback. In this paper, a capacity-headroom hypothesis is proposed, which states that PPO performance at the SLM scale depends on both a fluent supervised model (PPL<20) and a discriminative reward signal, rather than on the number of model parameters. The proposed system converged stably in all experiments and improved preference win rate over the SFT baseline in configurations with a fluent prior and an informative reward signal. Furthermore, it outperformed instruction-tuned baselines while requiring significantly less training data. All checkpoints, preference datasets, and training scripts are publicly released^§.

We present dla-ideal-solver, a high-performance framework for simulating two-dimensional Diffusion-Limited Aggregation (DLA) using Numba-accelerated Python. By leveraging just-in-time (JIT) compilation, we achieve computational throughput comparable to legacy static implementations while retaining high-level flexibility. We investigate the Laplacian growth instability across varying injection geometries and walker concentrations. Our analysis confirms the robustness of the standard fractal dimension D_f approx 1.71 for dilute regimes, consistent with the Witten-Sander universality class. However, we report a distinct crossover to Eden-like compact growth (D_f approx 1.87) in high-density environments, attributed to the saturation of the screening length. Beyond standard mass-radius scaling, we employ generalized Rényi dimensions and lacunarity metrics to quantify the monofractal character and spatial heterogeneity of the aggregates. This work establishes a reproducible, open-source testbed for exploring phase transitions in non-equilibrium statistical mechanics.

This paper is a study of fine-tuning of BERT contextual representations, with focus on commonly observed instabilities in few-sample scenarios. We identify several factors that cause this instability: the common use of a non-standard optimization method with biased gradient estimation; the limited applicability of significant parts of the BERT network for down-stream tasks; and the prevalent practice of using a pre-determined, and small number of training iterations. We empirically test the impact of these factors, and identify alternative practices that resolve the commonly observed instability of the process. In light of these observations, we re-visit recently proposed methods to improve few-sample fine-tuning with BERT and re-evaluate their effectiveness. Generally, we observe the impact of these methods diminishes significantly with our modified process.

Designing sampling algorithms for multimodal targets that remain stable under refinement of the finite-dimensional approximation of an underlying function-space problem is a central challenge. Annealed Langevin dynamics (ALD) is a natural alternative to classical Langevin in this context, since it is often observed to improve exploration across modes. Yet a gap remains between its empirical success and existing theory: under which conditions can ALD be guaranteed to remain stable across dimensions? In this paper, we bridge this gap by providing a uniform-in-dimension analysis of continuous-time ALD for Gaussian-mixture targets. Along an explicit annealing path obtained by gradually removing Gaussian smoothing from the target, we identify spectral conditions linking the smoothing covariance to the component covariances under which ALD achieves a prescribed accuracy in Kullback-Leibler divergence within a dimension-uniform time horizon. We then establish stability in a perturbative regime with imperfect initialization and approximate scores. Under a misspecified-mixture score model, we show that preconditioning ALD with an operator whose spectrum decays sufficiently fast prevents error terms from accumulating across coordinates and thereby preserves dimension-uniform control.

Design of reliable systems must guarantee stability against input perturbations. In machine learning, such guarantee entails preventing overfitting and ensuring robustness of models against corruption of input data. In order to maximize stability, we analyze and develop a computationally efficient implementation of Jacobian regularization that increases classification margins of neural networks. The stabilizing effect of the Jacobian regularizer leads to significant improvements in robustness, as measured against both random and adversarial input perturbations, without severely degrading generalization properties on clean data.

Neural network training is inherently sensitive to initialization and the randomness induced by stochastic gradient descent. However, it is unclear to what extent such effects lead to meaningfully different networks, either in terms of the models' weights or the underlying functions that were learned. In this work, we show that during the initial "chaotic" phase of training, even extremely small perturbations reliably causes otherwise identical training trajectories to diverge-an effect that diminishes rapidly over training time. We quantify this divergence through (i) L^2 distance between parameters, (ii) the loss barrier when interpolating between networks, (iii) L^2 and barrier between parameters after permutation alignment, and (iv) representational similarity between intermediate activations; revealing how perturbations across different hyperparameter or fine-tuning settings drive training trajectories toward distinct loss minima. Our findings provide insights into neural network training stability, with practical implications for fine-tuning, model merging, and diversity of model ensembles.

In this work, we study the evolution of the loss Hessian across many classification tasks in order to understand the effect the curvature of the loss has on the training dynamics. Whereas prior work has focused on how different learning rates affect the loss Hessian observed during training, we also analyze the effects of model initialization, architectural choices, and common training heuristics such as gradient clipping and learning rate warmup. Our results demonstrate that successful model and hyperparameter choices allow the early optimization trajectory to either avoid -- or navigate out of -- regions of high curvature and into flatter regions that tolerate a higher learning rate. Our results suggest a unifying perspective on how disparate mitigation strategies for training instability ultimately address the same underlying failure mode of neural network optimization, namely poor conditioning. Inspired by the conditioning perspective, we show that learning rate warmup can improve training stability just as much as batch normalization, layer normalization, MetaInit, GradInit, and Fixup initialization.

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.

Designing effective positional encodings for graphs is key to building powerful graph transformers and enhancing message-passing graph neural networks. Although widespread, using Laplacian eigenvectors as positional encodings faces two fundamental challenges: (1) Non-uniqueness: there are many different eigendecompositions of the same Laplacian, and (2) Instability: small perturbations to the Laplacian could result in completely different eigenspaces, leading to unpredictable changes in positional encoding. Despite many attempts to address non-uniqueness, most methods overlook stability, leading to poor generalization on unseen graph structures. We identify the cause of instability to be a "hard partition" of eigenspaces. Hence, we introduce Stable and Expressive Positional Encodings (SPE), an architecture for processing eigenvectors that uses eigenvalues to "softly partition" eigenspaces. SPE is the first architecture that is (1) provably stable, and (2) universally expressive for basis invariant functions whilst respecting all symmetries of eigenvectors. Besides guaranteed stability, we prove that SPE is at least as expressive as existing methods, and highly capable of counting graph structures. Finally, we evaluate the effectiveness of our method on molecular property prediction, and out-of-distribution generalization tasks, finding improved generalization compared to existing positional encoding methods.

We identity a by-far-unrecognized problem of Adam-style optimizers which results from unnecessary coupling between momentum and adaptivity. The coupling leads to instability and divergence when the momentum and adaptivity parameters are mismatched. In this work, we propose a method, Laprop, which decouples momentum and adaptivity in the Adam-style methods. We show that the decoupling leads to greater flexibility in the hyperparameters and allows for a straightforward interpolation between the signed gradient methods and the adaptive gradient methods. We experimentally show that Laprop has consistently improved speed and stability over Adam on a variety of tasks. We also bound the regret of Laprop on a convex problem and show that our bound differs from that of Adam by a key factor, which demonstrates its advantage.

The capacity to precisely edit genomes has outpaced our ability to predict the consequences. A cell can be genetically perfect and therapeutically useless: edited exactly as intended, yet unstable, drifting toward unintended fates, or selected for properties that compromise safety. This paradox reflects a deeper gap in how we evaluate biological intervention. Current frameworks excel at measuring what was done to a cell but remain blind to what the cell has become. We argue that this blindness stems from treating cells as collections of independent variables rather than as dynamical systems occupying positions on high-dimensional state manifolds. Drawing on Waddington's epigenetic landscape, we propose geometric stability as a missing axis of evaluation: the directional coherence of cellular responses to perturbation. This metric distinguishes interventions that guide cells coherently toward stable states from those that scatter them across the state manifold. Validation across diverse perturbation datasets reveals that geometric stability captures regulatory architecture invisible to conventional metrics, discriminating pleiotropic master regulators from lineage-specific factors without prior biological annotation. As precision medicine increasingly relies on cellular reprogramming, the question shifts from ``did the intervention occur?'' to ``is the resulting state stable?'' Geometric stability provides a framework for answering.

Optical Chemical Structure Recognition (OCSR) is critical for converting 2D molecular diagrams from printed literature into machine-readable formats. While Vision-Language Models have shown promise in end-to-end OCR tasks, their direct application to OCSR remains challenging, and direct full-parameter supervised fine-tuning often fails. In this work, we adapt DeepSeek-OCR-2 for molecular optical recognition by formulating the task as image-conditioned SMILES generation. To overcome training instabilities, we propose a two-stage progressive supervised fine-tuning strategy: starting with parameter-efficient LoRA and transitioning to selective full-parameter fine-tuning with split learning rates. We train our model on a large-scale corpus combining synthetic renderings from PubChem and realistic patent images from USPTO-MOL to improve coverage and robustness. Our fine-tuned model, MolSeek-OCR, demonstrates competitive capabilities, achieving exact matching accuracies comparable to the best-performing image-to-sequence model. However, it remains inferior to state-of-the-art image-to-graph modelS. Furthermore, we explore reinforcement-style post-training and data-curation-based refinement, finding that they fail to improve the strict sequence-level fidelity required for exact SMILES matching.

Agentic reinforcement learning (ARL) has rapidly gained attention as a promising paradigm for training agents to solve complex, multi-step interactive tasks. Despite encouraging early results, ARL remains highly unstable, often leading to training collapse. This instability limits scalability to larger environments and longer interaction horizons, and constrains systematic exploration of algorithmic design choices. In this paper, we first propose ARLArena, a stable training recipe and systematic analysis framework that examines training stability in a controlled and reproducible setting. ARLArena first constructs a clean and standardized testbed. Then, we decompose policy gradient into four core design dimensions and assess the performance and stability of each dimension. Through this fine-grained analysis, we distill a unified perspective on ARL and propose SAMPO, a stable agentic policy optimization method designed to mitigate the dominant sources of instability in ARL. Empirically, SAMPO achieves consistently stable training and strong performance across diverse agentic tasks. Overall, this study provides a unifying policy gradient perspective for ARL and offers practical guidance for building stable and reproducible LLM-based agent training pipelines.

The Information Bottleneck (IB) method frequently suffers from unstable optimization, characterized by abrupt representation shifts near critical points of the IB trade-off parameter, beta. In this paper, I introduce a novel approach to achieve stable and convex IB optimization through symbolic continuation and entropy-regularized trajectories. I analytically prove convexity and uniqueness of the IB solution path when an entropy regularization term is included, and demonstrate how this stabilizes representation learning across a wide range of eta values. Additionally, I provide extensive sensitivity analyses around critical points (beta) with statistically robust uncertainty quantification (95% confidence intervals). The open-source implementation, experimental results, and reproducibility framework included in this work offer a clear path for practical deployment and future extension of my proposed method.

Knowledge distillation (KD) is a widely adopted technique for transferring knowledge from large language models to smaller student models; however, conventional supervised KD often suffers from a distribution mismatch between training and inference. While on-policy KD approaches attempt to mitigate this issue by learning directly from student-generated outputs, they frequently encounter training instabilities because the distributional gap between the novice student and the expert teacher is often too wide to bridge directly. These challenges manifest as pathological gradients in forward KL objectives or diversity collapse in reverse KL regimes. To address these limitations, we propose Veto, an objective-level reformulation that constructs a geometric bridge in the logit space. Unlike prior methods that mix data samples, Veto creates an intermediate target distribution that promotes alignment between the teacher and the student. By introducing a tunable parameter beta, Veto serves as an Adaptive Gradient Veto that stabilizes optimization by suppressing harmful gradients on low-confidence tokens, while simultaneously acting as a Decisiveness Knob to balance reward-driven performance with output diversity. Extensive experiments across various reasoning and generation tasks demonstrate that Veto consistently outperforms supervised fine-tuning and existing on-policy baselines.

Straight-through estimator (STE), which enables the gradient flow over the non-differentiable function via approximation, has been favored in studies related to quantization-aware training (QAT). However, STE incurs unstable convergence during QAT, resulting in notable quality degradation in low precision. Recently, pseudoquantization training has been proposed as an alternative approach to updating the learnable parameters using the pseudo-quantization noise instead of STE. In this study, we propose a novel noise proxy-based integrated pseudoquantization (NIPQ) that enables unified support of pseudoquantization for both activation and weight by integrating the idea of truncation on the pseudo-quantization framework. NIPQ updates all of the quantization parameters (e.g., bit-width and truncation boundary) as well as the network parameters via gradient descent without STE instability. According to our extensive experiments, NIPQ outperforms existing quantization algorithms in various vision and language applications by a large margin.

Whether physics foundation models can be usefully deployed on laboratory experiments remains an open question for scientific machine learning (ML). We test this question on the Rayleigh-Taylor instability (RTI), a ubiquitous and demanding fluid instability seen from tabletop flows to supernova explosions, in which small perturbations at a density interface grow into chaotic, multiscale mixing as a lighter fluid accelerates into a heavier one. Standard ML models struggle with RTI, and despite over a century of theoretical, numerical, and experimental work, it carries an unresolved discrepancy between simulation and experiment: the late-time mixing growth rate, α, measured in most laboratory experiments (sim 0.06-0.07), is roughly three times the value from idealized direct numerical simulations (DNS, sim 0.02). The gap's origin remains debated. These properties make RTI a stringent test for a question that matters well beyond RTI: can foundation models trained only on simulations generalise to sparse, messy, and noisy laboratory settings? We finetune Walrus, a foundation model for continuum dynamics, on three or fewer DNS realizations and recover key RTI physics over long rollouts. Applied zero-shot to sliding-barrier laboratory data, the finetuned model leaves the DNS-like regime and enters the observed growth band, having never seen a single experimental sample. These results provide independent, data-driven evidence that initial conditions play a crucial role in the longstanding sim-experiment gap in α. The model also generalises zero-shot to stable stratification, a buoyancy regime absent from training, correctly slowing mixing-layer growth. Together, our results show that foundation models can generalise well beyond their training data, predicting laboratory behavior and unseen physical regimes, opening new ways to probe longstanding simulation-experiment gaps.

We discover that common diffusion noise schedules do not enforce the last timestep to have zero signal-to-noise ratio (SNR), and some implementations of diffusion samplers do not start from the last timestep. Such designs are flawed and do not reflect the fact that the model is given pure Gaussian noise at inference, creating a discrepancy between training and inference. We show that the flawed design causes real problems in existing implementations. In Stable Diffusion, it severely limits the model to only generate images with medium brightness and prevents it from generating very bright and dark samples. We propose a few simple fixes: (1) rescale the noise schedule to enforce zero terminal SNR; (2) train the model with v prediction; (3) change the sampler to always start from the last timestep; (4) rescale classifier-free guidance to prevent over-exposure. These simple changes ensure the diffusion process is congruent between training and inference and allow the model to generate samples more faithful to the original data distribution.

Recurrent models are a popular choice for video enhancement tasks such as video denoising or super-resolution. In this work, we focus on their stability as dynamical systems and show that they tend to fail catastrophically at inference time on long video sequences. To address this issue, we (1) introduce a diagnostic tool which produces input sequences optimized to trigger instabilities and that can be interpreted as visualizations of temporal receptive fields, and (2) propose two approaches to enforce the stability of a model during training: constraining the spectral norm or constraining the stable rank of its convolutional layers. We then introduce Stable Rank Normalization for Convolutional layers (SRN-C), a new algorithm that enforces these constraints. Our experimental results suggest that SRN-C successfully enforces stability in recurrent video processing models without a significant performance loss.

Reinforcement learning with verifiable rewards (RLVR) improves language-model reasoning, but GRPO-style optimization remains prone to collapse. We analyse this instability through token-level gradient dynamics, deriving a taxonomy that predicts how updates affect next-token probabilities and entropy. The taxonomy shows that stability depends jointly on the advantage sign and token distribution under the current policy. Motivated by this finding, we propose Winner Advantage Policy Optimization (WAPO), a simple online clipped policy-gradient objective that updates only on positive-advantage completions. Across mathematical reasoning and multi-hop QA benchmarks, WAPO improves training stability and matches or outperforms baselines across multiple model families. Full code can be found at https://github.com/layer6ai-labs/wapo.

Federated Learning with Low-Rank Adaptation (LoRA) faces three critical challenges under client heterogeneity: (1) Initialization-Induced Instability due to random initialization misaligning client subspaces; (2) Rank Incompatibility and Aggregation Error when averaging LoRA parameters of different ranks, which biases the global model; and (3) exacerbated Client Drift under Non-IID Data, impairing generalization. To address these challenges, we propose ILoRA, a unified framework that integrates three core innovations: a QR-based orthonormal initialization to ensure all clients start in a coherent subspace; a Concatenated QR Aggregation mechanism that fuses heterogeneous-rank updates via concatenation and decomposition, preserving information while maintaining dimension alignment; and an AdamW optimizer with rank-aware control variates to correct local updates and mitigate client drift. Supported by theoretical convergence guarantees, extensive experiments on vision and NLP benchmarks demonstrate that ILoRA consistently achieves superior accuracy and convergence stability compared to existing federated LoRA methods.

Existing benchmarks measure capability -- whether a model succeeds on a single attempt -- but production deployments require reliability -- consistent success across repeated attempts on tasks of varying duration. We show these properties diverge systematically as task duration grows, and that pass@1 on short tasks is structurally blind to this divergence. We introduce a reliability science framework for long-horizon LLM agents with four metrics: Reliability Decay Curve (RDC), Variance Amplification Factor (VAF), Graceful Degradation Score (GDS), and Meltdown Onset Point (MOP). We evaluate 10 models across 23,392 episodes on a 396-task benchmark spanning four duration buckets and three domains. Key findings: (1) reliability decay is domain-stratified -- SE GDS drops from 0.90 to 0.44 while document processing is nearly flat (0.74 to 0.71); (2) VAF bifurcates by capability tier -- high VAF is a capability signature, not an instability signal; (3) capability and reliability rankings diverge substantially, with multi-rank inversions at long horizons; (4) frontier models have the highest meltdown rates (up to 19%) because they attempt ambitious multi-step strategies that sometimes spiral; and (5) memory scaffolds universally hurt long-horizon performance across all 10 models. These results motivate reliability as a first-class evaluation dimension alongside capability.

We study the parameter placement problem: given a fixed budget of k trainable entries within the B matrix of a LoRA adapter (A frozen), does the choice of which k matter? Under supervised fine-tuning, random and informed subsets achieve comparable performance. Under GRPO on base models, random placement fails to improve over the base model, while gradient-informed placement recovers standard LoRA accuracy. This regime dependence traces to gradient structure: SFT gradients are low-rank and directionally stable, so any subset accumulates coherent updates; GRPO gradients are high-rank and near-orthogonal across steps, so only elements with consistently signed gradients retain the learning signal. Our scoring procedure identifies these critical parameters in under 10 seconds at less than 0.5% of training cost. Selected parameters concentrate on residual-stream-writing projections (V, O, Down), stable across model families and scales (1.5B - 8B).

PPO has been widely adopted for training large language models (LLMs) at the token level in multi-turn dialogue and reasoning tasks. However, its performance is often unstable and prone to collapse. Through empirical analysis, we identify two main sources of instability in this setting: (1)~token-level importance sampling, which is misaligned with the natural granularity of multi-turn environments that have distinct turn-level stages, and (2) inaccurate advantage estimates from off-policy samples, where the critic has not learned to evaluate certain state-action pairs, resulting in high-variance gradients and unstable updates. To address these challenges, we introduce two complementary stabilization techniques: (1) turn-level importance sampling, which aligns optimization with the natural structure of multi-turn reasoning, and (2) clipping-bias correction, which normalizes gradients by downweighting unreliable, highly off-policy samples. Depending on how these components are combined, we obtain three variants: Turn-PPO (turn-level sampling only), S-PPO (clipping-bias correction applied to token-level PPO), and ST-PPO (turn-level sampling combined with clipping-bias correction). In our experiments, we primarily study ST-PPO and S-PPO, which together demonstrate how the two stabilization mechanisms address complementary sources of instability. Experiments on multi-turn search tasks across general QA, multi-hop QA, and medical multiple-choice QA benchmarks show that ST-PPO and S-PPO consistently prevent the performance collapses observed in large-model training, maintain lower clipping ratios throughout optimization, and achieve higher task performance than standard token-level PPO. These results demonstrate that combining turn-level importance sampling with clipping-bias correction provides a practical and scalable solution for stabilizing multi-turn LLM agent training.

Effective LLM training depends on predictable scaling of key quantities -- such as final loss and optimal hyperparameters -- with model and dataset size. Qiu et al. (2025) recently showed that this predictability can extend beyond scalars: whole training loss curves can *collapse* onto a universal trajectory after a simple normalization. What remains unclear is whether this phenomenon persists for LLM families trained under *practical scaling recipes*, where width, depth, learning rate, batch size, and weight decay are scaled jointly. We show that it does: loss curves collapse across scales precisely when optimization hyperparameters are set optimally for the given data budget, in accordance with recent empirical scaling laws. Collapse therefore emerges as a signature of compute-efficient training. We demonstrate two applications at scale: (1) deviation-from-collapse provides a sensitive, early diagnostic of training pathologies, and (2) predictability of collapsed curves enables early stopping in large-scale hyperparameter tuning. Finally, we train a competitive LLM family, *Celerity*, using these insights, establishing collapse as an effective tool for developing efficient LLMs.