Daily Papers

On-policy distillation (OPD) is increasingly used to improve large language model reasoning, but its training dynamics remain poorly understood. We characterize the trajectory of OPD updates in parameter space and compare it with supervised fine-tuning (SFT) and reinforcement learning with verifiable rewards (RLVR). A suite of parameter-space diagnostics consistently places OPD in a relaxed off-principal regime: compared with SFT, its updates affect fewer weights and avoid principal directions more strongly, while compared with RLVR, they remain less tightly constrained. Beyond this static localization, OPD exhibits subspace locking: its cumulative updates rapidly enter a narrow low-dimensional channel. Constraining training to the update subspace formed early in training preserves OPD performance but substantially degrades SFT, indicating that the locked subspace is functionally sufficient for OPD. Control experiments further show that sparsifying the update tokens and shifting rollout generation off-policy preserve the rank dynamics, whereas mixing the OPD objective with RLVR changes them. Overall, these results suggest that OPD is not merely an intermediate point between SFT and RLVR, but induces its own update geometry in parameter space.

Spatio-temporal prediction is a pivotal task with broad applications in traffic management, climate monitoring, energy scheduling, etc. However, existing methodologies often struggle to balance model expressiveness and computational efficiency, especially when scaling to large real-world datasets. To tackle these challenges, we propose STH-SepNet (Spatio-Temporal Hypergraph Separation Networks), a novel framework that decouples temporal and spatial modeling to enhance both efficiency and precision. Therein, the temporal dimension is modeled using lightweight large language models, which effectively capture low-rank temporal dynamics. Concurrently, the spatial dimension is addressed through an adaptive hypergraph neural network, which dynamically constructs hyperedges to model intricate, higher-order interactions. A carefully designed gating mechanism is integrated to seamlessly fuse temporal and spatial representations. By leveraging the fundamental principles of low-rank temporal dynamics and spatial interactions, STH-SepNet offers a pragmatic and scalable solution for spatio-temporal prediction in real-world applications. Extensive experiments on large-scale real-world datasets across multiple benchmarks demonstrate the effectiveness of STH-SepNet in boosting predictive performance while maintaining computational efficiency. This work may provide a promising lightweight framework for spatio-temporal prediction, aiming to reduce computational demands and while enhancing predictive performance. Our code is avaliable at https://github.com/SEU-WENJIA/ST-SepNet-Lightweight-LLMs-Meet-Adaptive-Hypergraphs.

Low-Rank Adaptation (LoRA) is a fundamental parameter-efficient fine-tuning method that balances efficiency and performance in large-scale neural networks. However, the proliferation of LoRA variants has led to fragmentation in methodology, theory, code, and evaluation. To this end, this work presents the first unified study of LoRA variants, offering a systematic taxonomy, unified theoretical review, structured codebase, and standardized empirical assessment. First, we categorize LoRA variants along four principal axes: rank, optimization dynamics, initialization, and integration with Mixture-of-Experts. Then, we review their relationships and evolution within a common theoretical framework focused on low-rank update dynamics. Further, we introduce LoRAFactory, a modular codebase that implements variants through a unified interface, supporting plug-and-play experimentation and fine-grained analysis. Last, using this codebase, we conduct a large-scale evaluation across natural language generation, natural language understanding, and image classification tasks, systematically exploring key hyperparameters. Our results uncover several findings, notably: LoRA and its variants exhibit pronounced sensitivity to the choices of learning rate compared to other hyperparameters; moreover, with proper hyperparameter configurations, LoRA consistently matches or surpasses the performance of most of its variants.

Low-rank adaptation (LoRA) has become a standard approach for fine-tuning large foundation models. However, our theoretical understanding of LoRA remains limited as prior analyses of LoRA's training dynamics either rely on linearization arguments or consider highly simplified setups. In this work, we analyze the LoRA loss landscape without such restrictive assumptions. We define two regimes: a "special regime", which includes idealized setups where linearization arguments hold, and a "generic regime" representing more realistic setups where linearization arguments do not hold. In the generic regime, we show that LoRA training converges to a global minimizer with low rank and small magnitude, or a qualitatively distinct solution with high rank and large magnitude. Finally, we argue that the zero-initialization and weight decay in LoRA training induce an implicit bias toward the low-rank, small-magnitude region of the parameter space -- where global minima lie -- thus shedding light on why LoRA training usually succeeds in finding global minima.

Recent advances in reasoning capabilities of large language models (LLMs) are largely driven by reinforcement learning (RL), yet the underlying parameter dynamics during RL training remain poorly understood. This work identifies two fundamental properties of RL-induced parameter updates in LLMs: (1) Rank-1 Dominance, where the top singular subspace of the parameter update matrix nearly fully determines reasoning improvements, recovering over 99\% of performance gains; and (2) Rank-1 Linear Dynamics, where this dominant subspace evolves linearly throughout training, enabling accurate prediction from early checkpoints. Extensive experiments across 8 LLMs and 7 algorithms validate the generalizability of these properties. More importantly, based on these findings, we propose AlphaRL, a plug-in acceleration framework that extrapolates the final parameter update using a short early training window, achieving up to 2.5 speedup while retaining \textgreater 96\% of reasoning performance without extra modules or hyperparameter tuning. This positions our finding as a versatile and practical tool for large-scale RL, opening a path toward principled, interpretable, and efficient training paradigm for LLMs.

Model merging is an effective post-training strategy for composing capabilities in large language models without joint retraining. We study this in the supervised fine-tuning (SFT) stage, where multiple capability-based SFT checkpoints -- spanning math, code, precise instruction following, general instruction following, and knowledge recall -- must be consolidated into a single model. We introduce Optimization Trajectory Aware (OTA) Merging, a curvature-aware aggregation that leverages optimizer second-moment statistics as a diagonal curvature proxy to reweight parameter edits and mitigate interference. Complementing OTA, we propose Fast Fisher Grafting (FFG), a curvature-driven task-localization step that sparsifies conflicting or low-importance edits. FFG induces extremely low-rank masks concentrated in early attention query/key projections and token embeddings, exploiting shared curvature across capabilities. We further develop a memory-light compression of the second moments that preserves OTA's effect. Across diverse capability-based SFT checkpoints, OTA+FFG improves merged-model quality over strong weight-space baselines, reduces negative transfer, and remains robust across sparsity levels. Analyses reveal substantial curvature overlap between checkpoints, offering a novel lens on why simple linear merging can be effective in practice. Ablations confirm that FFG is critical for reducing task interference and that the compressed second moments retain the gains of the full formulation. To facilitate reproducibility, we open-source all code, training and evaluation scripts, visualization artifacts, and capability-specific SFT checkpoints at https://github.com/pmahdavi/ota-merge.

We identify incremental learning dynamics in transformers, where the difference between trained and initial weights progressively increases in rank. We rigorously prove this occurs under the simplifying assumptions of diagonal weight matrices and small initialization. Our experiments support the theory and also show that phenomenon can occur in practice without the simplifying assumptions.

Linear-time attention and State Space Models (SSMs) promise to solve the quadratic cost bottleneck in long-context language models employing softmax attention. We introduce Error-Free Linear Attention (EFLA), a numerically stable, fully parallelism and generalized formulation of the delta rule. Specifically, we formulate the online learning update as a continuous-time dynamical system and prove that its exact solution is not only attainable but also computable in linear time with full parallelism. By leveraging the rank-1 structure of the dynamics matrix, we directly derive the exact closed-form solution effectively corresponding to the infinite-order Runge-Kutta method. This attention mechanism is theoretically free from error accumulation, perfectly capturing the continuous dynamics while preserving the linear-time complexity. Through an extensive suite of experiments, we show that EFLA enables robust performance in noisy environments, achieving lower language modeling perplexity and superior downstream benchmark performance than DeltaNet without introducing additional parameters. Our work provides a new theoretical foundation for building high-fidelity, scalable linear-time attention models.

Recently, scaling reinforcement learning with verifiable rewards (RLVR) for large language models (LLMs) has emerged as an effective training paradigm for significantly improving model capabilities, which requires guiding the model to perform extensive exploration and learning, leading to substantial computational overhead and becoming a key challenge. To reduce the number of training steps, Prior work performs linear extrapolation of model parameters. However, the dynamics of model parameter updates during RLVR training remain insufficiently understood. To further investigate the evolution of LLMs during RLVR training, we conduct empirical experiments and find that the rank-1 subspace of the model does not evolve linearly, and its dominance over the original parameters is further amplified during LoRA training. Based on the above insights, we propose the Nonlinear Extrapolation of low-rank trajectories (NExt), a novel framework that models and extrapolates low-rank parameter trajectories in a nonlinear manner. Concretely, we first train the model using LoRA and extract the rank-1 subspace of parameter differences at multiple training steps, which is then used for the subsequent nonlinear extrapolation. Afterward, we utilized the extracted rank-1 subspace to train a predictor, which can model the trajectory of parameter updates during RLVR, and then perform the predict-extend process to extrapolate model parameters, achieving the acceleration of RLVR. To further study and understand NExt, we conduct comprehensive experiments that demonstrate the effectiveness and robustness of the method. Our method reduces computational overhead by approximately 37.5\% while remaining compatible with a wide range of RLVR algorithms and tasks. We release our code in https://github.com/RUCAIBox/NExt.

In this paper, we study the role of initialization in Low Rank Adaptation (LoRA) as originally introduced in Hu et al. (2021). Essentially, to start from the pretrained model as initialization for finetuning, one can either initialize B to zero and A to random (default initialization in PEFT package), or vice-versa. In both cases, the product BA is equal to zero at initialization, which makes finetuning starts from the pretrained model. These two initialization schemes are seemingly similar. They should in-principle yield the same performance and share the same optimal learning rate. We demonstrate that this is an incorrect intuition and that the first scheme (initializing B to zero and A to random) on average yields better performance compared to the other scheme. Our theoretical analysis shows that the reason behind this might be that the first initialization allows the use of larger learning rates (without causing output instability) compared to the second initialization, resulting in more efficient learning of the first scheme. We validate our results with extensive experiments on LLMs.

Large language models are powerful but costly. We ask whether meta-learning can make the pretraining of small language models not only better but also more interpretable. We integrate first-order MAML with subset-masked LM pretraining, producing four LLama-style decoder-only models (11M-570M params), and evaluate it on a fundamental NLP task with many settings and real-world applications. Compared with vanilla training, our model (i) reaches the same loss up to 1.6x sooner, (ii) improves F1 on multilingual Universal NER under equal compute, and (iii) makes the training dynamics easy to read: first the network's representations fan out ("diversify") and later they collapse into a smaller, shared subspace ("compress"). This two-stage shift shows up as a rise-and-fall in both effective-rank curves and attention-head entropy. The same curves pinpoint which layers specialise earliest and which later reconverge, giving a compact, interpretable signature of meta-adaptation. Code, checkpoints and WandB logs are released.

We investigate how self-referential inputs alter the internal matrix dynamics of large language models. Measuring 106 scalar metrics across up to 7 analysis passes on four models from three architecture families -- Qwen3-VL-8B, Llama-3.2-11B, Llama-3.3-70B, and Gemma-2-9B -- over 300 prompts in a 14-level hierarchy at three temperatures (T in {0.0, 0.3, 0.7}), we find that self-reference alone is not destabilizing: grounded self-referential statements and meta-cognitive prompts are markedly more stable than paradoxical self-reference on key collapse-related metrics, and on several such metrics can be as stable as factual controls. Instability concentrates in prompts inducing non-closing truth recursion (NCTR) -- truth-value computations with no finite-depth resolution. NCTR prompts produce anomalously elevated attention effective rank -- indicating attention reorganization with global dispersion rather than simple concentration collapse -- and key metrics reach Cohen's d = 3.14 (attention effective rank) to 3.52 (variance kurtosis) vs. stable self-reference in the 70B model; 281/397 metric-model combinations differentiate NCTR from stable self-reference after FDR correction (q < 0.05), 198 with |d| > 0.8. Per-layer SVD confirms disruption at every sampled layer (d > +1.0 in all three models analyzed), ruling out aggregation artifacts. A classifier achieves AUC 0.81-0.90; 30 minimal pairs yield 42/387 significant combinations; 43/106 metrics replicate across all four models. We connect these observations to three classical matrix-semigroup problems and propose, as a conjecture, that NCTR forces finite-depth transformers toward dynamical regimes where these problems concentrate. NCTR prompts also produce elevated contradictory output (+34-56 percentage points vs. controls), suggesting practical relevance for understanding self-referential failure modes.

Low-Rank Adaptation (LoRA) has significantly advanced parameter-efficient fine-tuning of large pretrained models. LoRA augments the pre-trained weights of a model by adding the product of two smaller matrices that together form a low-rank matrix update. Recent research has shown that scale disparities between these two matrices often cause unstable training dynamics, leading to suboptimal performance. In this paper, we propose SingLoRA, which reformulates low-rank adaptation by learning the weights update as a decomposition of a single low-rank matrix multiplied by its transpose. This simple design inherently removes inter-matrix scale conflicts, ensuring stable optimization, and roughly halves the parameter count. We analyze SingLoRA within the infinite-width neural network framework, showing that it guarantees stable feature learning by construction. Extensive experiments on multiple tasks validate these benefits. In common sense reasoning, fine-tuning LLama 7B on MNLI with SingLoRA achieves 91.3% accuracy - surpassing LoRA (89.1%) and LoRA+ (90.2%) - while using only 60% of their parameter budget. In image generation, fine-tuning Stable Diffusion with SingLoRA significantly improves image fidelity on DreamBooth, achieving a DINO similarity score of 0.151, compared to scores of 0.148 and 0.143 for DoRA and LoRA, respectively.

Despite large neural networks demonstrating remarkable abilities to complete different tasks, they require excessive memory usage to store the optimization states for training. To alleviate this, the low-rank adaptation (LoRA) is proposed to reduce the optimization states by training fewer parameters. However, LoRA restricts overall weight update matrices to be low-rank, limiting the model performance. In this work, we investigate the dynamics of LoRA and identify that it can be approximated by a random projection. Based on this observation, we propose Flora, which is able to achieve high-rank updates by resampling the projection matrices while enjoying the sublinear space complexity of optimization states. We conduct experiments across different tasks and model architectures to verify the effectiveness of our approach.

Neural networks trained with gradient descent often learn solutions of increasing complexity over time, a phenomenon known as simplicity bias. Despite being widely observed across architectures, existing theoretical treatments lack a unifying framework. We present a theoretical framework that explains a simplicity bias arising from saddle-to-saddle learning dynamics for a general class of neural networks, incorporating fully-connected, convolutional, and attention-based architectures. Here, simple means expressible with few hidden units, i.e., hidden neurons, convolutional kernels, or attention heads. Specifically, we show that linear networks learn solutions of increasing rank, ReLU networks learn solutions with an increasing number of kinks, convolutional networks learn solutions with an increasing number of convolutional kernels, and self-attention models learn solutions with an increasing number of attention heads. By analyzing fixed points, invariant manifolds, and dynamics of gradient descent learning, we show that saddle-to-saddle dynamics operates by iteratively evolving near an invariant manifold, approaching a saddle, and switching to another invariant manifold. Our analysis also disentangles data-induced and initialization-induced saddle-to-saddle dynamics. In particular, the former leads to low-rank weights while the latter to sparse weights. Equipped with the theory, we predict the effects of data distribution and weight initialization on the duration and number of plateaus in learning. Overall, our theory offers a framework for understanding when and why gradient descent progressively learns increasingly complex solutions.

Catastrophic forgetting is a significant challenge in online continual learning (OCL), especially for non-stationary data streams that do not have well-defined task boundaries. This challenge is exacerbated by the memory constraints and privacy concerns inherent in rehearsal buffers. To tackle catastrophic forgetting, in this paper, we introduce Online-LoRA, a novel framework for task-free OCL. Online-LoRA allows to finetune pre-trained Vision Transformer (ViT) models in real-time to address the limitations of rehearsal buffers and leverage pre-trained models' performance benefits. As the main contribution, our approach features a novel online weight regularization strategy to identify and consolidate important model parameters. Moreover, Online-LoRA leverages the training dynamics of loss values to enable the automatic recognition of the data distribution shifts. Extensive experiments across many task-free OCL scenarios and benchmark datasets (including CIFAR-100, ImageNet-R, ImageNet-S, CUB-200 and CORe50) demonstrate that Online-LoRA can be robustly adapted to various ViT architectures, while achieving better performance compared to SOTA methods. Our code will be publicly available at: https://github.com/Christina200/Online-LoRA-official.git.

Earth Observation (EO) data are increasingly used in policy analysis by enabling granular estimation of treatment effects. However, a challenge in EO-based causal inference lies in balancing the trade-off between capturing fine-grained individual heterogeneity and broader contextual information. This paper introduces Multi-scale Concatenation, a family of composable procedures that transform arbitrary single-scale CATE estimation algorithms into multi-scale algorithms. We benchmark the performance of Multi-scale Concatenation on a CATE estimation pipeline combining Vision Transformer (ViT) models fine-tuned on satellite images to encode images of different scales with Causal Forests to obtain the final CATE estimate. We first perform simulation studies, showing how a multi-scale approach captures multi-level dynamics that single-scale ViT models fail to capture. We then apply the multi-scale method to two randomized controlled trials (RCTs) conducted in Peru and Uganda using Landsat satellite imagery. In the RCT analysis, the Rank Average Treatment Effect Ratio (RATE Ratio) measure is employed to assess performance without ground truth individual treatment effects. Results indicate that Multi-scale Concatenation improves the performance of deep learning models in EO-based CATE estimation without the complexity of designing new multi-scale architectures for a specific use case.

Multispectral fusion object detection is a critical task for edge-based maritime surveillance and remote sensing, demanding both high inference efficiency and robust feature representation for high-resolution inputs. However, current State Space Models (SSMs) like Mamba suffer from significant parameter redundancy in their standard 2D Selective Scan (SS2D) blocks, which hinders deployment on resource-constrained hardware and leads to the loss of fine-grained structural information during conventional compression. To address these challenges, we propose the Low-Rank Two-Dimensional Selective Structured State Space Model (Low-Rank SS2D), which reformulates state transitions via matrix factorization to exploit intrinsic feature sparsity. Furthermore, we introduce a Structure-Aware Distillation strategy that aligns the internal latent state dynamics of the student with a full-rank teacher model to compensate for potential representation degradation. This approach substantially reduces computational complexity and memory footprint while preserving the high-fidelity spatial modeling required for object recognition. Extensive experiments on five benchmark datasets and real-world edge platforms, such as Raspberry Pi 5, demonstrate that our method achieves a superior efficiency-accuracy trade-off, significantly outperforming existing lightweight architectures in practical deployment scenarios.

We propose Dynamic Rank Reinforcement Learning (DR-RL), a novel framework that adaptively optimizes the low-rank factorization of Multi-Head Self-Attention (MHSA) in Large Language Models (LLMs) through the integration of reinforcement learning and online matrix perturbation theory. While traditional low-rank approximations often rely on static rank assumptions--limiting their flexibility across diverse input contexts--our method dynamically selects ranks based on real-time sequence dynamics, layer-specific sensitivities, and hardware constraints. The core innovation lies in an RL agent that formulates rank selection as a sequential policy optimization problem, where the reward function strictly balances attention fidelity against computational latency. Crucially, we employ online matrix perturbation bounds to enable incremental rank updates, thereby avoiding the prohibitive cost of full decomposition during inference. Furthermore, the integration of a lightweight Transformer-based policy network and batched Singular Value Decomposition (SVD) operations ensures scalable deployment on modern GPU architectures. Experiments demonstrate that DR-RL maintains downstream accuracy statistically equivalent to full-rank attention while significantly reducing Floating Point Operations (FLOPs), particularly in long-sequence regimes (L > 4096). This work bridges the gap between adaptive efficiency and theoretical rigor in MHSA, offering a principled, mathematically grounded alternative to heuristic rank reduction techniques in resource-constrained deep learning. Source code and experiment logs are available at: https://github.com/canererden/DR_RL_Project

Fine-tuning large language models (LLMs) has become essential for adapting pretrained models to specific downstream tasks. In this paper, we propose Linear Chain Transformation (LinChain), a novel approach that introduces a sequence of linear transformations during fine-tuning to enrich optimization dynamics. By incorporating multiple linear transformations into the parameter update process, LinChain expands the effective rank of updates and enhances the model's ability to learn complex task-specific representations. We demonstrate that this method significantly improves the performance of LLM fine-tuning over state-of-the-art methods by providing more flexible optimization paths during training, while maintaining the inference efficiency of the resulting model. Our experiments on various benchmark tasks show that LinChain leads to better generalization, fewer learnable parameters, and improved task adaptation, making it a compelling strategy for LLM fine-tuning.

Developing autonomous agents that can interact with changing environments is an open challenge in machine learning. Robustness is particularly important in these settings as agents are often fit offline on expert demonstrations but deployed online where they must generalize to the closed feedback loop within the environment. In this work, we explore the application of recurrent neural networks to tasks of this nature and understand how a parameterization of their recurrent connectivity influences robustness in closed-loop settings. Specifically, we represent the recurrent connectivity as a function of rank and sparsity and show both theoretically and empirically that modulating these two variables has desirable effects on network dynamics. The proposed low-rank, sparse connectivity induces an interpretable prior on the network that proves to be most amenable for a class of models known as closed-form continuous-time neural networks (CfCs). We find that CfCs with fewer parameters can outperform their full-rank, fully-connected counterparts in the online setting under distribution shift. This yields memory-efficient and robust agents while opening a new perspective on how we can modulate network dynamics through connectivity.

We present an approach to modifying Transformer architectures by integrating graph-aware relational reasoning into the attention mechanism, merging concepts from graph neural networks and language modeling. Building on the inherent connection between attention and graph theory, we reformulate the Transformer's attention mechanism as a graph operation and propose Graph-Aware Isomorphic Attention. This method leverages advanced graph modeling strategies, including Graph Isomorphism Networks (GIN) and Principal Neighborhood Aggregation (PNA), to enrich the representation of relational structures. Our approach captures complex dependencies and generalizes across tasks, as evidenced by a reduced generalization gap and improved learning performance. Additionally, we expand the concept of graph-aware attention to introduce Sparse GIN-Attention, a fine-tuning approach that employs sparse GINs. By interpreting attention matrices as sparse adjacency graphs, this technique enhances the adaptability of pre-trained foundational models with minimal computational overhead, endowing them with graph-aware capabilities. Sparse GIN-Attention fine-tuning achieves improved training dynamics and better generalization compared to alternative methods like low-rank adaption (LoRA). We discuss latent graph-like structures within traditional attention mechanisms, offering a new lens through which Transformers can be understood. By evolving Transformers as hierarchical GIN models for relational reasoning. This perspective suggests profound implications for foundational model development, enabling the design of architectures that dynamically adapt to both local and global dependencies. Applications in bioinformatics, materials science, language modeling, and beyond could benefit from this synthesis of relational and sequential data modeling, setting the stage for interpretable and generalizable modeling strategies.

One of the most striking features of Large Language Models (LLM) is their ability to learn in context. Namely at inference time an LLM is able to learn new patterns without any additional weight update when these patterns are presented in the form of examples in the prompt, even if these patterns were not seen during training. The mechanisms through which this can happen are still largely unknown. In this work, we show that the stacking of a self-attention layer with an MLP, allows the transformer block to implicitly modify the weights of the MLP layer according to the context. We argue through theory and experimentation that this simple mechanism may be the reason why LLMs can learn in context and not only during training. Specifically, we show under mild simplifying assumptions how a transformer block implicitly transforms a context into a low-rank weight-update of the MLP layer.

As Vision Transformers (ViTs) become standard vision backbones, a mechanistic account of their computational phenomenology is essential. Despite architectural cues that hint at dynamical structure, there is no settled framework that interprets Transformer depth as a well-characterized flow. In this work, we introduce the Block-Recurrent Hypothesis (BRH), arguing that trained ViTs admit a block-recurrent depth structure such that the computation of the original L blocks can be accurately rewritten using only k ll L distinct blocks applied recurrently. Across diverse ViTs, between-layer representational similarity matrices suggest few contiguous phases. To determine whether these phases reflect genuinely reusable computation, we train block-recurrent surrogates of pretrained ViTs: Recurrent Approximations to Phase-structured TransfORmers (Raptor). In small-scale, we demonstrate that stochastic depth and training promote recurrent structure and subsequently correlate with our ability to accurately fit Raptor. We then provide an empirical existence proof for BRH by training a Raptor model to recover 96% of DINOv2 ImageNet-1k linear probe accuracy in only 2 blocks at equivalent computational cost. Finally, we leverage our hypothesis to develop a program of Dynamical Interpretability. We find i) directional convergence into class-dependent angular basins with self-correcting trajectories under small perturbations, ii) token-specific dynamics, where cls executes sharp late reorientations while patch tokens exhibit strong late-stage coherence toward their mean direction, and iii) a collapse to low rank updates in late depth, consistent with convergence to low-dimensional attractors. Altogether, we find a compact recurrent program emerges along ViT depth, pointing to a low-complexity normative solution that enables these models to be studied through principled dynamical systems analysis.

Large pre-trained models are commonly adapted to downstream tasks using parameter-efficient fine-tuning methods such as Low-Rank Adaptation (LoRA), which injects small trainable low-rank matrices instead of updating all weights. While LoRA dramatically reduces trainable parameters with little overhead, it can still underperform full fine-tuning in accuracy and often converges more slowly. We introduce LoFT, a novel low-rank adaptation method that behaves like full fine-tuning by aligning the optimizer's internal dynamics with those of updating all model weights. LoFT not only learns weight updates in a low-rank subspace (like LoRA) but also properly projects the optimizer's first and second moments (Adam's momentum and variance) into the same subspace, mirroring full-model updates. By aligning the low-rank update itself with the full update, LoFT eliminates the need for tuning extra hyperparameters, e.g., LoRA scaling factor alpha. Empirically, this approach substantially narrows the performance gap between adapter-based tuning and full fine-tuning and consistently outperforms standard LoRA-style methods, all without increasing inference cost.

The latest advancements in unsupervised learning of sentence embeddings predominantly involve employing contrastive learning-based (CL-based) fine-tuning over pre-trained language models. In this study, we analyze the latest sentence embedding methods by adopting representation rank as the primary tool of analysis. We first define Phase 1 and Phase 2 of fine-tuning based on when representation rank peaks. Utilizing these phases, we conduct a thorough analysis and obtain essential findings across key aspects, including alignment and uniformity, linguistic abilities, and correlation between performance and rank. For instance, we find that the dynamics of the key aspects can undergo significant changes as fine-tuning transitions from Phase 1 to Phase 2. Based on these findings, we experiment with a rank reduction (RR) strategy that facilitates rapid and stable fine-tuning of the latest CL-based methods. Through empirical investigations, we showcase the efficacy of RR in enhancing the performance and stability of five state-of-the-art sentence embedding methods.

The infinitely wide neural network has been proven a useful and manageable mathematical model that enables the understanding of many phenomena appearing in deep learning. One example is the convergence of random deep networks to Gaussian processes that allows a rigorous analysis of the way the choice of activation function and network weights impacts the training dynamics. In this paper, we extend the seminal proof of Matthews et al. (2018) to a larger class of initial weight distributions (which we call PSEUDO-IID), including the established cases of IID and orthogonal weights, as well as the emerging low-rank and structured sparse settings celebrated for their computational speed-up benefits. We show that fully-connected and convolutional networks initialized with PSEUDO-IID distributions are all effectively equivalent up to their variance. Using our results, one can identify the Edge-of-Chaos for a broader class of neural networks and tune them at criticality in order to enhance their training. Moreover, they enable the posterior distribution of Bayesian Neural Networks to be tractable across these various initialization schemes.

Capturing high-dimensional social interactions and feasible futures is essential for predicting trajectories. To address this complex nature, several attempts have been devoted to reducing the dimensionality of the output variables via parametric curve fitting such as the B\'ezier curve and B-spline function. However, these functions, which originate in computer graphics fields, are not suitable to account for socially acceptable human dynamics. In this paper, we present EigenTrajectory (ET), a trajectory prediction approach that uses a novel trajectory descriptor to form a compact space, known here as ET space, in place of Euclidean space, for representing pedestrian movements. We first reduce the complexity of the trajectory descriptor via a low-rank approximation. We transform the pedestrians' history paths into our ET space represented by spatio-temporal principle components, and feed them into off-the-shelf trajectory forecasting models. The inputs and outputs of the models as well as social interactions are all gathered and aggregated in the corresponding ET space. Lastly, we propose a trajectory anchor-based refinement method to cover all possible futures in the proposed ET space. Extensive experiments demonstrate that our EigenTrajectory predictor can significantly improve both the prediction accuracy and reliability of existing trajectory forecasting models on public benchmarks, indicating that the proposed descriptor is suited to represent pedestrian behaviors. Code is publicly available at https://github.com/inhwanbae/EigenTrajectory .

Conventional Low-Rank Adaptation (LoRA) methods employ a fixed rank, imposing uniform adaptation across transformer layers and attention heads despite their heterogeneous learning dynamics. This paper introduces Adaptive Rank Dynamic LoRA (ARD-LoRA), a novel framework that automates rank allocation through learnable scaling factors. These factors are optimized via a meta-objective balancing task performance and parameter efficiency, incorporating ell_1 sparsity for minimal rank and Total Variation regularization for stable rank transitions. ARD-LoRA enables continuous, differentiable, per-head rank adaptation. Experiments on LLAMA-3.1-70B and PaliGemma-2 demonstrate ARD-LoRA's efficacy, achieving up to 99.3% of full fine-tuning performance with only 0.32% trainable parameters, outperforming strong baselines like DoRA and AdaLoRA. Furthermore, it reduces multimodal adaptation memory by 41%. These results establish dynamic, fine-grained rank allocation as a critical paradigm for efficient foundation model adaptation.

Modern Large Language Models (LLMs) are composed of matrices with billions of elements, making their storage and processing quite demanding in terms of computational resources and memory usage. Being significantly large, such matrices can often be expressed in low-rank format with potential to relax resource requirements. Unlike prior works which focus on developing novel matrix decomposition algorithms, in this work we first study the emergence of low-rank structures across matrices within different layers of LLMs and establish a consequential relationship between the gradient dynamics and emerging low-rank expressiveness of matrices. Our findings reveal that different layers exhibit varying levels of converged low-rank structure, necessitating a non-uniform rank reduction across them to minimize performance drop due to compression. In view of that, we present Weight Low-Rank Projection (WeLore) that unifies weight compression and memory-efficient fine-tuning as ONE, in a data-agnostic and one-shot way. WeLore capitalizes the heavy-tail distribution of singular values to identify a suitable rank reduction ratio for matrices within LLMs. Going beyond only as a compression technique, WeLore categorizes weight matrices into Low-rank Components (LRCs) and Non-Low-rank Components (N-LRCs) based on their ability to express themselves as low-rank. Our gradient perspective and extensive experiments illustrate that LRCs tend to have better finetuning capabilities and can closely mimic (sometimes outperform) the training loss trajectory and performance of full-finetuning with notable memory and compute footprint reduction. For example, finetuning a 50\% compressed LLaMa-2 7B model using only a fraction of parameters in LRCs (WeLore) can outperform its full finetuning with ~3x better throughput and ~0.6x GPU requirement. Our codes are available at https://github.com/VITA-Group/welore

In this work, we study how multi-head latent attention (MLA), a popular strategy for compressing key/value memory, affects a transformer's internal capacity during pretraining. Using a lightweight suite of Marchenko-Pastur (MP) diagnostics, we analyze the spectrum of the W_{Q}W_{K}^top gram matrix throughout training, comparing three variants: the standard multi-head attention (MHA) baseline, MLA-PreRoPE with rotary applied before compression, and MLA-Decoupled, which shares a single rotary sub-vector across all heads. Our random matrix analysis reveals three key findings: i) capacity bottlenecks emerge locally: both MHA and MLA-PreRoPE exhibit sharp, early spikes in specific layers that persist and propagate, disrupting the balance between bulk and outlier directions; ii) these spikes coincide with rank collapse, concentrating the model's expressivity into narrow subspaces; iii) only the decoupled variant prevents this cascade, maintaining broad spectral support and suppressing outlier formation across layers. These results underscore that how rotary embeddings are applied is just as critical as where compression occurs. Sharing rotary components across heads mitigates spectral fragmentation and preserves representational capacity.

Training Large Language Models (LLMs) presents significant memory challenges, predominantly due to the growing size of weights and optimizer states. Common memory-reduction approaches, such as low-rank adaptation (LoRA), add a trainable low-rank matrix to the frozen pre-trained weight in each layer, reducing trainable parameters and optimizer states. However, such approaches typically underperform training with full-rank weights in both pre-training and fine-tuning stages since they limit the parameter search to a low-rank subspace and alter the training dynamics, and further, may require full-rank warm start. In this work, we propose Gradient Low-Rank Projection (GaLore), a training strategy that allows full-parameter learning but is more memory-efficient than common low-rank adaptation methods such as LoRA. Our approach reduces memory usage by up to 65.5% in optimizer states while maintaining both efficiency and performance for pre-training on LLaMA 1B and 7B architectures with C4 dataset with up to 19.7B tokens, and on fine-tuning RoBERTa on GLUE tasks. Our 8-bit GaLore further reduces optimizer memory by up to 82.5% and total training memory by 63.3%, compared to a BF16 baseline. Notably, we demonstrate, for the first time, the feasibility of pre-training a 7B model on consumer GPUs with 24GB memory (e.g., NVIDIA RTX 4090) without model parallel, checkpointing, or offloading strategies.

We introduce Tadpole, a novel foundation model for three-dimensional partial differential equations (PDEs) that addresses key challenges in transferability, scalability to high dimensionality, and multi-functionality. Tadpole is pre-trained as an autoencoder on synthetic 3D PDE data generated by an efficient online data-generation framework. This enables large-scale, diverse training without storage or I/O overhead, demonstrated by scaling to an equivalent of hundreds of terabytes of training data. By autoencoding single-channel spatial crops, Tadpole learns rich and transferable representations across heterogeneous physical systems with varying numbers of state variables and spatial resolutions. Although pre-trained solely as an autoencoder, Tadpole can be efficiently applied for multiple downstream tasks beyond reconstruction, including dynamics learning and generative modeling. For dynamics learning, we propose a novel parameter-efficient fine-tuning strategy that integrates low-rank adaptation, latent-space transformations, and reintroduced skip connections, achieving accurate temporal modeling with a minimal number of trainable parameters. Tadpole demonstrates strong fine-tuning performance across various downstream tasks, highlighting its versatility and effectiveness as a foundation model for 3D PDE learning. Source code and pre-trained weights of Tadpole are available at https://github.com/tum-pbs/tadpole

Modern Transformer-based models frequently suffer from miscalibration, producing overconfident predictions that do not reflect true empirical frequencies. This work investigates the calibration dynamics of LoRA: Low-Rank Adaptation and a novel hyper-network-based adaptation framework as parameter-efficient alternatives to full fine-tuning for RoBERTa. Evaluating across the GLUE benchmark, we demonstrate that LoRA-based adaptation consistently achieves calibration parity with (and in specific tasks exceeds) full fine-tuning, while maintaining significantly higher parameter efficiency. We further explore a dynamic approach where a shared hyper-network generates LoRA factors (A and B matrices) to induce structural coupling across layers. This approach produced results similar to standard LoRA fine-tuning, even achieving better MCC on CoLA dataset. Our study also reveal a critical trade-off: constraining the adaptation space (e.g., freezing matrices A) acts as a powerful regularizer that enhances Expected Calibration Error (ECE), but necessitates a carefully balanced sacrifice in downstream task accuracy. To support future research, we provide a unified and reproducible implementation of contemporary calibration metrics, including ECE, MCE, and ACE. Our findings clarify the relationship between parameter efficiency and probabilistic reliability, positioning structured low-rank updates as a viable foundation for uncertainty-aware Transformer architectures. Code available at: https://github.com/btrojan-official/HypeLoRA

Parameter-efficient fine-tuning methods like Low-Rank Adaptation (LoRA) have become the dominant paradigm for adapting large pretrained models. We present a theoretical framework explaining an underexplored property: LoRA's inherent resistance to label noise. Our analysis reveals three key insights. First, we prove that rank-r LoRA cannot memorize all possible label assignments once the sample size exceeds O(r(d+k-r)), limiting its capacity to fit arbitrary noise. Second, we derive an optimal rank balancing approximation bias and noise-induced variance, showing it decreases with noise rate. Third, we establish temporal separation: clean patterns are learned early while noise memorization occurs later. We propose RACT (Rank-Aware Curriculum Training), leveraging rank discrepancy for noise detection. Experiments validate our predictions, with RACT achieving 91.1% F1 for noise detection on AG News while maintaining 91.46% accuracy, competitive with baselines that lack noise detection capability.

The success of automated medical image analysis depends on large-scale and expert-annotated training sets. Unsupervised domain adaptation (UDA) has been raised as a promising approach to alleviate the burden of labeled data collection. However, they generally operate under the closed-set adaptation setting assuming an identical label set between the source and target domains, which is over-restrictive in clinical practice where new classes commonly exist across datasets due to taxonomic inconsistency. While several methods have been presented to tackle both domain shifts and incoherent label sets, none of them take into account the common characteristics of the two issues and consider the learning dynamics along network training. In this work, we propose optimization trajectory distillation, a unified approach to address the two technical challenges from a new perspective. It exploits the low-rank nature of gradient space and devises a dual-stream distillation algorithm to regularize the learning dynamics of insufficiently annotated domain and classes with the external guidance obtained from reliable sources. Our approach resolves the issue of inadequate navigation along network optimization, which is the major obstacle in the taxonomy adaptive cross-domain adaptation scenario. We evaluate the proposed method extensively on several tasks towards various endpoints with clinical and open-world significance. The results demonstrate its effectiveness and improvements over previous methods.

It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics (Li et al., 2020) and follows an incremental learning procedure (Gissin et al., 2019): GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.

Large Language Models (LLMs) must continuously learn and update knowledge to remain effective in dynamic real-world environments. While Low-Rank Adaptation (LoRA) is widely used for such memory updates, existing studies mainly rely on qualitative downstream evaluations, leaving the quantitative capacity limits and underlying dynamics of exact parametric memory largely unexplored. To bridge this gap, we employ LoRA as a controlled memory capacity probe within the latent space to systematically quantify exact parametric memory. We introduce the Parametric Memory Law, a robust power law linking loss reduction Delta L to effective parameters and sequence length. At the token level, fine-grained analysis reveals a deterministic phase transition, demonstrating that a prediction probability of p > 0.5 constitutes a sufficient condition for verbatim recall under greedy decoding. Driven by these insights, we introduce MemFT, a threshold-guided optimization strategy that dynamically redistributes the training budget toward sub-threshold tokens. Empirical evaluations demonstrate that MemFT can enhance memory fidelity and efficiency. Code will be released at https://github.com/zjunlp/ParametricMemoryLaw.

LLM agents are widely deployed in complex interactive tasks, yet privacy constraints often preclude centralized optimization and co-evolution across dynamic environments. While Federated Learning (FL) has proven effective on static datasets, its extension to the open-ended self-evolution of agents remains underexplored. Directly applying standard FL is challenging: heterogeneous tasks and sparse, trajectory-level rewards introduce severe gradient conflicts, destabilizing the global optimization process. To bridge this gap, we propose Fed-SE, a Federated Self-Evolution framework for LLM agents. Fed-SE establishes a local evolution-global aggregation paradigm. Locally, agents employ parameter-efficient fine-tuning on filtered, high-return trajectories to achieve stable gradient updates. Globally, Fed-SE aggregates updates within a low-rank subspace that disentangles environment-specific dynamics, effectively reducing negative transfer across clients. Experiments across five heterogeneous environments demonstrate that Fed-SE improves average task success rates by approximately 18% over federated baselines, validating its effectiveness in robust cross-environment knowledge transfer in privacy-constrained deployments.

In theoretical neuroscience, recent work leverages deep learning tools to explore how some network attributes critically influence its learning dynamics. Notably, initial weight distributions with small (resp. large) variance may yield a rich (resp. lazy) regime, where significant (resp. minor) changes to network states and representation are observed over the course of learning. However, in biology, neural circuit connectivity could exhibit a low-rank structure and therefore differs markedly from the random initializations generally used for these studies. As such, here we investigate how the structure of the initial weights -- in particular their effective rank -- influences the network learning regime. Through both empirical and theoretical analyses, we discover that high-rank initializations typically yield smaller network changes indicative of lazier learning, a finding we also confirm with experimentally-driven initial connectivity in recurrent neural networks. Conversely, low-rank initialization biases learning towards richer learning. Importantly, however, as an exception to this rule, we find lazier learning can still occur with a low-rank initialization that aligns with task and data statistics. Our research highlights the pivotal role of initial weight structures in shaping learning regimes, with implications for metabolic costs of plasticity and risks of catastrophic forgetting.

Heliophysics is central to understanding and forecasting space weather events and solar activity. Despite decades of high-resolution observations from the Solar Dynamics Observatory (SDO), most models remain task-specific and constrained by scarce labeled data, limiting their capacity to generalize across solar phenomena. We introduce Surya, a 366M parameter foundation model for heliophysics designed to learn general-purpose solar representations from multi-instrument SDO observations, including eight Atmospheric Imaging Assembly (AIA) channels and five Helioseismic and Magnetic Imager (HMI) products. Surya employs a spatiotemporal transformer architecture with spectral gating and long--short range attention, pretrained on high-resolution solar image forecasting tasks and further optimized through autoregressive rollout tuning. Zero-shot evaluations demonstrate its ability to forecast solar dynamics and flare events, while downstream fine-tuning with parameter-efficient Low-Rank Adaptation (LoRA) shows strong performance on solar wind forecasting, active region segmentation, solar flare forecasting, and EUV spectra. Surya is the first foundation model in heliophysics that uses time advancement as a pretext task on full-resolution SDO data. Its novel architecture and performance suggest that the model is able to learn the underlying physics behind solar evolution.

Increasing the number of parameters in language models is a common strategy to enhance their performance. However, smaller language models remain valuable due to their lower operational costs. Despite their advantages, smaller models frequently underperform compared to their larger counterparts, even when provided with equivalent data and computational resources. Specifically, their performance tends to degrade in the late pretraining phase. This is anecdotally attributed to their reduced representational capacity. Yet, the exact causes of this performance degradation remain unclear. We use the Pythia model suite to analyse the training dynamics that underlie this phenomenon. Across different model sizes, we investigate the convergence of the Attention and MLP activations to their final state and examine how the effective rank of their parameters influences this process. We find that nearly all layers in larger models stabilise early in training - within the first 20% - whereas layers in smaller models exhibit slower and less stable convergence, especially when their parameters have lower effective rank. By linking the convergence of layers' activations to their parameters' effective rank, our analyses can guide future work to address inefficiencies in the learning dynamics of small models.

Large Language Models (LLMs) often exhibit slash attention patterns, where attention scores concentrate along the Δ-th sub-diagonal for some offset Δ. These patterns play a key role in passing information across tokens. But why do they emerge? In this paper, we demystify the emergence of these Slash-Dominant Heads (SDHs) from both empirical and theoretical perspectives. First, by analyzing open-source LLMs, we find that SDHs are intrinsic to models and generalize to out-of-distribution prompts. To explain the intrinsic emergence, we analyze the queries, keys, and Rotary Position Embedding (RoPE), which jointly determine attention scores. Our empirical analysis reveals two characteristic conditions of SDHs: (1) Queries and keys are almost rank-one, and (2) RoPE is dominated by medium- and high-frequency components. Under these conditions, queries and keys are nearly identical across tokens, and interactions between medium- and high-frequency components of RoPE give rise to SDHs. Beyond empirical evidence, we theoretically show that these conditions are sufficient to ensure the emergence of SDHs by formalizing them as our modeling assumptions. Particularly, we analyze the training dynamics of a shallow Transformer equipped with RoPE under these conditions, and prove that models trained via gradient descent exhibit SDHs. The SDHs generalize to out-of-distribution prompts.

The last layer of neural language models (LMs) projects output features of dimension D to logits in dimension V, the size of the vocabulary, where usually D ll V. This mismatch is known to raise risks of limited expressivity in neural LMs, creating a so-called softmax bottleneck. We show the softmax bottleneck is not only an expressivity bottleneck but also an optimization bottleneck. Backpropagating V-dimensional gradients through a rank-D linear layer induces unavoidable compression, which alters the training feedback provided to the vast majority of the parameters. We present a theoretical analysis of this phenomenon and measure empirically that 95-99% of the gradient norm is suppressed by the output layer, resulting in vastly suboptimal update directions. We conduct controlled pretraining experiments showing that the gradient bottleneck makes trivial patterns unlearnable, and drastically affects the training dynamics of LLMs. We argue that this inherent flaw contributes to training inefficiencies at scale independently of the model architecture, and raises the need for new LM head designs.

Data mixing decides how to combine different sources or types of data and is a consequential problem throughout language model training. In pretraining, data composition is a key determinant of model quality; in continual learning and adaptation, it governs what is retained and acquired. Yet existing data mixing methods address only one phase of this lifecycle at a time: some require smaller proxy models tied to a single training phase, others assume a fixed domain set, and continual learning lacks principled guidance altogether. We argue that data mixing is fundamentally an online decision making problem -- one that recurs throughout training and demands a single, unified solution. We introduce OP-Mix (On-Policy Mix), a data mixing algorithm that operates across the entire language model training lifecycle. Our main insight is that candidate data mixtures can be cheaply simulated by interpolating between low-rank adapters trained directly on the current model, eliminating separate proxy models and ensuring the search is always grounded in the model's actual learning dynamics. Across pretraining, continual midtraining, and continual instruction tuning, OP-Mix consistently finds near-optimal mixtures while using a fraction of the compute of the baselines. In pretraining, OP-Mix improves upon training without mixing by 6.3% in average perplexity. For continual learning, OP-Mix matches the performance of both retraining and on-policy distillation while using 66% and 95% less overall compute, respectively. OP-Mix suggests a different view of language model training: not a sequence of distinct phases, but a single continuous process of learning from data.

What fundamental properties of reward functions enable efficient sparse-reward reinforcement learning? We address this question through the lens of low-rank structure in reward matrices, showing that such structure induces a sharp transition from exponential to polynomial sample complexity, the first result of this kind for sparse-reward RL. We introduce Policy-Aware Matrix Completion (PAMC), which connects matrix completion theory with reinforcement learning via a new analysis of policy-dependent sampling. Our framework provides: (i) impossibility results for general sparse reward observation, (ii) reward-free representation learning from dynamics, (iii) distribution-free confidence sets via conformal prediction, and (iv) robust completion guarantees that degrade gracefully when low-rank structure is only approximate. Empirically, we conduct a pre-registered evaluation across 100 systematically sampled domains, finding exploitable structure in over half. PAMC improves sample efficiency by factors between 1.6 and 2.1 compared to strong exploration, structured, and representation-learning baselines, while adding only about 20 percent computational overhead.These results establish structural reward learning as a promising new paradigm, with immediate implications for robotics, healthcare, and other safety-critical, sample-expensive applications.

Latent prediction--where agents learn by predicting their own latents--has emerged as a powerful paradigm for training general representations in machine learning. In reinforcement learning (RL), this approach has been explored to define auxiliary losses for a variety of settings, including reward-based and unsupervised RL, behavior cloning, and world modeling. While existing methods are typically limited to single-task learning, one-step prediction, or on-policy trajectory data, we show that temporal difference (TD) learning enables learning representations predictive of long-term latent dynamics across multiple policies from offline, reward-free transitions. Building on this, we introduce TD-JEPA, which leverages TD-based latent-predictive representations into unsupervised RL. TD-JEPA trains explicit state and task encoders, a policy-conditioned multi-step predictor, and a set of parameterized policies directly in latent space. This enables zero-shot optimization of any reward function at test time. Theoretically, we show that an idealized variant of TD-JEPA avoids collapse with proper initialization, and learns encoders that capture a low-rank factorization of long-term policy dynamics, while the predictor recovers their successor features in latent space. Empirically, TD-JEPA matches or outperforms state-of-the-art baselines on locomotion, navigation, and manipulation tasks across 13 datasets in ExoRL and OGBench, especially in the challenging setting of zero-shot RL from pixels.

A prevailing view in Reinforcement Learning for Verifiable Rewards (RLVR) interprets recent progress through the lens of an exploration-exploitation trade-off, a perspective largely shaped by token-level metrics. We re-examine this perspective, proposing that this perceived trade-off may not be a fundamental constraint but rather an artifact of the measurement level. To investigate this, we shift the analysis to the semantically rich hidden-state space, adopting Effective Rank (ER) to quantify exploration and proposing its novel first- and second-order derivatives, named Effective Rank Velocity (ERV) and Effective Rank Acceleration (ERA), to capture exploitation dynamics. Our analysis reveals that at the hidden-state level, exploration and exploitation could be decoupled (Sec. 4). This finding reveals an opportunity to enhance both capacities simultaneously. This insight motivates our method, Velocity-Exploiting Rank-Learning (VERL), the first to operationalize the principle of synergistic exploration-exploitation enhancement by directly shaping the RL advantage function. The key innovation is leveraging the theoretically stable ERA as a predictive meta-controller to create a synergistic, dual-channel incentive structure. Instead of forcing a trade-off, VERL prospectively amplifies rewards for exploration to preempt overconfidence and reinforces exploitative gains to consolidate reasoning. Experiments across diverse LLMs and reasoning benchmarks show consistent gains, including up to 21.4% absolute accuracy improvement on the challenging Gaokao 2024 dataset.

The rapid proliferation of Large Language Models (LLMs) and diverse specialized benchmarks necessitates a shift from fragmented, task-specific metrics to a holistic, competitive ranking system that effectively aggregates performance across multiple ability dimensions. Primarily using static scoring, current evaluation methods are fundamentally limited. They struggle to determine the proper mix ratio across diverse benchmarks, and critically, they fail to capture a model's dynamic competitive fitness or its vulnerability when confronted with sequential, high-stakes tasks. To address this, we introduce the novel Competitive Swiss-System Dynamics (CSD) framework. CSD simulates a multi-round, sequential contest where models are dynamically paired across a curated sequence of benchmarks based on their accumulated win-loss record. And Monte Carlo Simulation (N=100,000 iterations) is used to approximate the statistically robust Expected Win Score (E[S_m]), which eliminates the noise of random pairing and early-round luck. Furthermore, we implement a Failure Sensitivity Analysis by parameterizing the per-round elimination quantity (T_k), which allows us to profile models based on their risk appetite--distinguishing between robust generalists and aggressive specialists. We demonstrate that CSD provides a more nuanced and context-aware ranking than traditional aggregate scoring and static pairwise models, representing a vital step towards risk-informed, next-generation LLM evaluation.

Value-based methods constitute a fundamental methodology in planning and deep reinforcement learning (RL). In this paper, we propose to exploit the underlying structures of the state-action value function, i.e., Q function, for both planning and deep RL. In particular, if the underlying system dynamics lead to some global structures of the Q function, one should be capable of inferring the function better by leveraging such structures. Specifically, we investigate the low-rank structure, which widely exists for big data matrices. We verify empirically the existence of low-rank Q functions in the context of control and deep RL tasks. As our key contribution, by leveraging Matrix Estimation (ME) techniques, we propose a general framework to exploit the underlying low-rank structure in Q functions. This leads to a more efficient planning procedure for classical control, and additionally, a simple scheme that can be applied to any value-based RL techniques to consistently achieve better performance on "low-rank" tasks. Extensive experiments on control tasks and Atari games confirm the efficacy of our approach. Code is available at https://github.com/YyzHarry/SV-RL.

Aligning Large Language Models (LLMs) with human preferences typically relies on external supervision, which faces critical limitations: human annotations are scarce and subjective, reward models are vulnerable to reward hacking, and self-evaluation methods suffer from prompt sensitivity and biases. In this work, we propose stable rank, an intrinsic, annotation-free quality signal derived from model representations. Stable rank measures the effective dimensionality of hidden states by computing the ratio of total variance to dominant-direction variance, capturing quality through how information distributes across representation dimensions. Empirically, stable rank achieves 84.04% accuracy on RewardBench and improves task accuracy by an average of 11.3 percentage points over greedy decoding via Best-of-N sampling. Leveraging this insight, we introduce Stable Rank Group Relative Policy Optimization (SR-GRPO), which uses stable rank as a reward signal for reinforcement learning. Without external supervision, SR-GRPO improves Qwen2.5-1.5B-Instruct by 10% on STEM and 19% on mathematical reasoning, outperforming both learned reward models and self-evaluation baselines. Our findings demonstrate that quality signals can be extracted from internal model geometry, offering a path toward scalable alignment without external supervision.

Learning a precise dynamics model can be crucial for offline reinforcement learning, which, unfortunately, has been found to be quite challenging. Dynamics models that are learned by fitting historical transitions often struggle to generalize to unseen transitions. In this study, we identify a hidden but pivotal factor termed dynamics reward that remains consistent across transitions, offering a pathway to better generalization. Therefore, we propose the idea of reward-consistent dynamics models: any trajectory generated by the dynamics model should maximize the dynamics reward derived from the data. We implement this idea as the MOREC (Model-based Offline reinforcement learning with Reward Consistency) method, which can be seamlessly integrated into previous offline model-based reinforcement learning (MBRL) methods. MOREC learns a generalizable dynamics reward function from offline data, which is subsequently employed as a transition filter in any offline MBRL method: when generating transitions, the dynamics model generates a batch of transitions and selects the one with the highest dynamics reward value. On a synthetic task, we visualize that MOREC has a strong generalization ability and can surprisingly recover some distant unseen transitions. On 21 offline tasks in D4RL and NeoRL benchmarks, MOREC improves the previous state-of-the-art performance by a significant margin, i.e., 4.6% on D4RL tasks and 25.9% on NeoRL tasks. Notably, MOREC is the first method that can achieve above 95% online RL performance in 6 out of 12 D4RL tasks and 3 out of 9 NeoRL tasks.

Dynamical systems with complex behaviours, e.g. immune system cells interacting with a pathogen, are commonly modelled by splitting the behaviour into different regimes, or modes, each with simpler dynamics, and then learning the switching behaviour from one mode to another. Switching Dynamical Systems (SDS) are a powerful tool that automatically discovers these modes and mode-switching behaviour from time series data. While effective, these methods focus on independent objects, where the modes of one object are independent of the modes of the other objects. In this paper, we focus on the more general interacting object setting for switching dynamical systems, where the per-object dynamics also depends on an unknown and dynamically changing subset of other objects and their modes. To this end, we propose a novel graph-based approach for switching dynamical systems, GRAph Switching dynamical Systems (GRASS), in which we use a dynamic graph to characterize interactions between objects and learn both intra-object and inter-object mode-switching behaviour. We introduce two new datasets for this setting, a synthesized ODE-driven particles dataset and a real-world Salsa Couple Dancing dataset. Experiments show that GRASS can consistently outperforms previous state-of-the-art methods.

State space models (SSMs) have shown remarkable empirical performance on many long sequence modeling tasks, but a theoretical understanding of these models is still lacking. In this work, we study the learning dynamics of linear SSMs to understand how covariance structure in data, latent state size, and initialization affect the evolution of parameters throughout learning with gradient descent. We show that focusing on the learning dynamics in the frequency domain affords analytical solutions under mild assumptions, and we establish a link between one-dimensional SSMs and the dynamics of deep linear feed-forward networks. Finally, we analyze how latent state over-parameterization affects convergence time and describe future work in extending our results to the study of deep SSMs with nonlinear connections. This work is a step toward a theory of learning dynamics in deep state space models.

Learning dynamics, which describes how the learning of specific training examples influences the model's predictions on other examples, gives us a powerful tool for understanding the behavior of deep learning systems. We study the learning dynamics of large language models during different types of finetuning, by analyzing the step-wise decomposition of how influence accumulates among different potential responses. Our framework allows a uniform interpretation of many interesting observations about the training of popular algorithms for both instruction tuning and preference tuning. In particular, we propose a hypothetical explanation of why specific types of hallucination are strengthened after finetuning, e.g., the model might use phrases or facts in the response for question B to answer question A, or the model might keep repeating similar simple phrases when generating responses. We also extend our framework and highlight a unique "squeezing effect" to explain a previously observed phenomenon in off-policy direct preference optimization (DPO), where running DPO for too long makes even the desired outputs less likely. This framework also provides insights into where the benefits of on-policy DPO and other variants come from. The analysis not only provides a novel perspective of understanding LLM's finetuning but also inspires a simple, effective method to improve alignment performance.

Dynamical systems provide a comprehensive way to study complex and changing behaviors across various sciences. Many modern systems are too complicated to analyze directly or we do not have access to models, driving significant interest in learning methods. Koopman operators have emerged as a dominant approach because they allow the study of nonlinear dynamics using linear techniques by solving an infinite-dimensional spectral problem. However, current algorithms face challenges such as lack of convergence, hindering practical progress. This paper addresses a fundamental open question: When can we robustly learn the spectral properties of Koopman operators from trajectory data of dynamical systems, and when can we not? Understanding these boundaries is crucial for analysis, applications, and designing algorithms. We establish a foundational approach that combines computational analysis and ergodic theory, revealing the first fundamental barriers -- universal for any algorithm -- associated with system geometry and complexity, regardless of data quality and quantity. For instance, we demonstrate well-behaved smooth dynamical systems on tori where non-trivial eigenfunctions of the Koopman operator cannot be determined by any sequence of (even randomized) algorithms, even with unlimited training data. Additionally, we identify when learning is possible and introduce optimal algorithms with verification that overcome issues in standard methods. These results pave the way for a sharp classification theory of data-driven dynamical systems based on how many limits are needed to solve a problem. These limits characterize all previous methods, presenting a unified view. Our framework systematically determines when and how Koopman spectral properties can be learned.

We identify and formalize a fundamental gradient descent phenomenon resulting in a learning proclivity in over-parameterized neural networks. Gradient Starvation arises when cross-entropy loss is minimized by capturing only a subset of features relevant for the task, despite the presence of other predictive features that fail to be discovered. This work provides a theoretical explanation for the emergence of such feature imbalance in neural networks. Using tools from Dynamical Systems theory, we identify simple properties of learning dynamics during gradient descent that lead to this imbalance, and prove that such a situation can be expected given certain statistical structure in training data. Based on our proposed formalism, we develop guarantees for a novel regularization method aimed at decoupling feature learning dynamics, improving accuracy and robustness in cases hindered by gradient starvation. We illustrate our findings with simple and real-world out-of-distribution (OOD) generalization experiments.

Representation rank is an important concept for understanding the role of Neural Networks (NNs) in Deep Reinforcement learning (DRL), which measures the expressive capacity of value networks. Existing studies focus on unboundedly maximizing this rank; nevertheless, that approach would introduce overly complex models in the learning, thus undermining performance. Hence, fine-tuning representation rank presents a challenging and crucial optimization problem. To address this issue, we find a guiding principle for adaptive control of the representation rank. We employ the Bellman equation as a theoretical foundation and derive an upper bound on the cosine similarity of consecutive state-action pairs representations of value networks. We then leverage this upper bound to propose a novel regularizer, namely BEllman Equation-based automatic rank Regularizer (BEER). This regularizer adaptively regularizes the representation rank, thus improving the DRL agent's performance. We first validate the effectiveness of automatic control of rank on illustrative experiments. Then, we scale up BEER to complex continuous control tasks by combining it with the deterministic policy gradient method. Among 12 challenging DeepMind control tasks, BEER outperforms the baselines by a large margin. Besides, BEER demonstrates significant advantages in Q-value approximation. Our code is available at https://github.com/sweetice/BEER-ICLR2024.

Hyperparameter tuning is one of the essential steps to guarantee the convergence of machine learning models. We argue that intuition about the optimal choice of hyperparameters for stochastic gradient descent can be obtained by studying a neural network's phase diagram, in which each phase is characterised by distinctive dynamics of the singular values of weight matrices. Taking inspiration from disordered systems, we start from the observation that the loss landscape of a multilayer neural network with mean squared error can be interpreted as a disordered system in feature space, where the learnt features are mapped to soft spin degrees of freedom, the initial variance of the weight matrices is interpreted as the strength of the disorder, and temperature is given by the ratio of the learning rate and the batch size. As the model is trained, three phases can be identified, in which the dynamics of weight matrices is qualitatively different. Employing a Langevin equation for stochastic gradient descent, previously derived using Dyson Brownian motion, we demonstrate that the three dynamical regimes can be classified effectively, providing practical guidance for the choice of hyperparameters of the optimiser.

Diffusion models have achieved remarkable success across a wide range of generative tasks. A key challenge is understanding the mechanisms that prevent their memorization of training data and allow generalization. In this work, we investigate the role of the training dynamics in the transition from generalization to memorization. Through extensive experiments and theoretical analysis, we identify two distinct timescales: an early time τ_gen at which models begin to generate high-quality samples, and a later time τ_mem beyond which memorization emerges. Crucially, we find that τ_mem increases linearly with the training set size n, while τ_gen remains constant. This creates a growing window of training times with n where models generalize effectively, despite showing strong memorization if training continues beyond it. It is only when n becomes larger than a model-dependent threshold that overfitting disappears at infinite training times. These findings reveal a form of implicit dynamical regularization in the training dynamics, which allow to avoid memorization even in highly overparameterized settings. Our results are supported by numerical experiments with standard U-Net architectures on realistic and synthetic datasets, and by a theoretical analysis using a tractable random features model studied in the high-dimensional limit.

Achieving robust coordination and cooperation is a central challenge in multi-agent reinforcement learning (MARL). Uncovering the mechanisms underlying such emergent behaviors calls for a dynamical understanding of learn processes. In this work, we investigate the dynamics of actor-critic agents in stochastic games, focusing on the impact of entropy regularization. By leveraging time-scale separation, we derive the system's evolution equations, which are then formally analyzed using dynamical systems theory. We find that in the constant-sum game of Matching Pennies, the system exhibits chaotic behavior. Entropy regularization mitigates this chaos and drives the dynamics toward convergence to fair cooperation. In contrast, in the general-sum game of the Prisoner's Dilemma, the system displays multistability. Interestingly, the three stable equilibria of the system correspond to the well-known ALLC (Always Cooperate), ALLD (Always Defect), and GRIM (Grim Trigger) strategies from evolutionary game theory (EGT). Entropy regularization strengthens system resilience by enlarging the basin of attraction of the cooperative equilibrium. Our findings reveal a close link between the mechanism of direct reciprocity in EGT and how cooperation emerges in MARL, offering insights for designing more robust and collaborative multi-agent systems.

Ranking and scoring are ubiquitous. We consider the setting in which an institution, called a ranker, evaluates a set of individuals based on demographic, behavioral or other characteristics. The final output is a ranking that represents the relative quality of the individuals. While automatic and therefore seemingly objective, rankers can, and often do, discriminate against individuals and systematically disadvantage members of protected groups. This warrants a careful study of the fairness of a ranking scheme. In this paper we propose fairness measures for ranked outputs. We develop a data generation procedure that allows us to systematically control the degree of unfairness in the output, and study the behavior of our measures on these datasets. We then apply our proposed measures to several real datasets, and demonstrate cases of unfairness. Finally, we show preliminary results of incorporating our ranked fairness measures into an optimization framework, and show potential for improving fairness of ranked outputs while maintaining accuracy.

This survey draws a broad-stroke, panoramic picture of the State of the Art (SoTA) of the research in generative methods for the analysis of social media data. It fills a void, as the existing survey articles are either much narrower in their scope or are dated. We included two important aspects that currently gain importance in mining and modeling social media: dynamics and networks. Social dynamics are important for understanding the spreading of influence or diseases, formation of friendships, the productivity of teams, etc. Networks, on the other hand, may capture various complex relationships providing additional insight and identifying important patterns that would otherwise go unnoticed.

Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. In this work, we propose a novel algorithm to find efficient low-rank subnetworks. Remarkably, these subnetworks are determined and adapted already during the training phase and the overall time and memory resources required by both training and evaluating them are significantly reduced. The main idea is to restrict the weight matrices to a low-rank manifold and to update the low-rank factors rather than the full matrix during training. To derive training updates that are restricted to the prescribed manifold, we employ techniques from dynamic model order reduction for matrix differential equations. This allows us to provide approximation, stability, and descent guarantees. Moreover, our method automatically and dynamically adapts the ranks during training to achieve the desired approximation accuracy. The efficiency of the proposed method is demonstrated through a variety of numerical experiments on fully-connected and convolutional networks.

Reinforcement Learning with Verifiable Rewards (RLVR) has recently demonstrated remarkable potential in enhancing the reasoning capability of Large Reasoning Models (LRMs). However, RLVR often leads to entropy collapse, resulting in premature policy convergence and performance saturation. While manipulating token-level entropy has proven effective for promoting policy exploration, we argue that the latent dynamics underlying token generation encode a far richer computational structure for steering policy optimization toward a more effective exploration-exploitation tradeoff. To enable tractable analysis and intervention of the latent dynamics of LRMs, we leverage Koopman operator theory to obtain a linearized representation of their hidden-state dynamics. This enables us to introduce Dynamic Spectral Dispersion (DSD), a new metric to quantify the heterogeneity of the model's latent dynamics, serving as a direct indicator of policy exploration. Building upon these foundations, we propose Reasoning with Latent eXploration (ReLaX), a paradigm that explicitly incorporates latent dynamics to regulate exploration and exploitation during policy optimization. Comprehensive experiments across a wide range of multimodal and text-only reasoning benchmarks show that ReLaX significantly mitigates premature convergence and consistently achieves state-of-the-art performance.

Online social media platforms enable influencers to distribute content and quickly capture audience reactions, significantly shaping their promotional strategies and advertising agreements. Understanding how sentiment dynamics and emotional contagion unfold among followers is vital for influencers and marketers, as these processes shape engagement, brand perception, and purchasing behavior. While sentiment analysis tools effectively track sentiment fluctuations, dynamical models explaining their evolution remain limited, often neglecting network structures and interactions both among blogs and between their topic-focused follower groups. In this study, we tracked influential tech-focused Weibo bloggers over six months, quantifying follower sentiment from text-mined feedback. By treating each blogger's audience as a single "macro-agent", we find that sentiment trajectories follow the principle of iterative averaging -- a foundational mechanism in many dynamical models of opinion formation, a theoretical framework at the intersection of social network analysis and dynamical systems theory. The sentiment evolution aligns closely with opinion-dynamics models, particularly modified versions of the classical French-DeGroot model that incorporate delayed perception and distinguish between expressed and private opinions. The inferred influence structures reveal interdependencies among blogs that may arise from homophily, whereby emotionally similar users subscribe to the same blogs and collectively shape the shared sentiment expressed within these communities.

Selective state space models (SSM), such as Mamba, have gained prominence for their effectiveness in modeling sequential data. Despite their outstanding empirical performance, a comprehensive theoretical understanding of deep selective SSM remains elusive, hindering their further development and adoption for applications that need high fidelity. In this paper, we investigate the dynamical properties of tokens in a pre-trained Mamba model. In particular, we derive the dynamical system governing the continuous-time limit of the Mamba model and characterize the asymptotic behavior of its solutions. In the one-dimensional case, we prove that only one of the following two scenarios happens: either all tokens converge to zero, or all tokens diverge to infinity. We provide criteria based on model parameters to determine when each scenario occurs. For the convergent scenario, we empirically verify that this scenario negatively impacts the model's performance. For the divergent scenario, we prove that different tokens will diverge to infinity at different rates, thereby contributing unequally to the updates during model training. Based on these investigations, we propose two refinements for the model: excluding the convergent scenario and reordering tokens based on their importance scores, both aimed at improving practical performance. Our experimental results validate these refinements, offering insights into enhancing Mamba's effectiveness in real-world applications.

Compressed Sensing (CS) significantly speeds up Magnetic Resonance Image (MRI) processing and achieves accurate MRI reconstruction from under-sampled k-space data. According to the current research, there are still several problems with dynamic MRI k-space reconstruction based on CS. 1) There are differences between the Fourier domain and the Image domain, and the differences between MRI processing of different domains need to be considered. 2) As three-dimensional data, dynamic MRI has its spatial-temporal characteristics, which need to calculate the difference and consistency of surface textures while preserving structural integrity and uniqueness. 3) Dynamic MRI reconstruction is time-consuming and computationally resource-dependent. In this paper, we propose a novel robust low-rank dynamic MRI reconstruction optimization model via highly under-sampled and Discrete Fourier Transform (DFT) called the Robust Depth Linear Error Decomposition Model (RDLEDM). Our method mainly includes linear decomposition, double Total Variation (TV), and double Nuclear Norm (NN) regularizations. By adding linear image domain error analysis, the noise is reduced after under-sampled and DFT processing, and the anti-interference ability of the algorithm is enhanced. Double TV and NN regularizations can utilize both spatial-temporal characteristics and explore the complementary relationship between different dimensions in dynamic MRI sequences. In addition, Due to the non-smoothness and non-convexity of TV and NN terms, it is difficult to optimize the unified objective model. To address this issue, we utilize a fast algorithm by solving a primal-dual form of the original problem. Compared with five state-of-the-art methods, extensive experiments on dynamic MRI data demonstrate the superior performance of the proposed method in terms of both reconstruction accuracy and time complexity.

We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.

Optimizing user engagement is a key goal for modern recommendation systems, but blindly pushing users towards increased consumption risks burn-out, churn, or even addictive habits. To promote digital well-being, most platforms now offer a service that periodically prompts users to take breaks. These, however, must be set up manually, and so may be suboptimal for both users and the system. In this paper, we study the role of breaks in recommendation, and propose a framework for learning optimal breaking policies that promote and sustain long-term engagement. Based on the notion that recommendation dynamics are susceptible to both positive and negative feedback, we cast recommendation as a Lotka-Volterra dynamical system, where breaking reduces to a problem of optimal control. We then give an efficient learning algorithm, provide theoretical guarantees, and empirically demonstrate the utility of our approach on semi-synthetic data.

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.

As neural networks grow in scale, their training becomes both computationally demanding and rich in dynamics. Amidst the flourishing interest in these training dynamics, we present a novel observation: Parameters during training exhibit intrinsic correlations over time. Capitalizing on this, we introduce Correlation Mode Decomposition (CMD). This algorithm clusters the parameter space into groups, termed modes, that display synchronized behavior across epochs. This enables CMD to efficiently represent the training dynamics of complex networks, like ResNets and Transformers, using only a few modes. Moreover, test set generalization is enhanced. We introduce an efficient CMD variant, designed to run concurrently with training. Our experiments indicate that CMD surpasses the state-of-the-art method for compactly modeled dynamics on image classification. Our modeling can improve training efficiency and lower communication overhead, as shown by our preliminary experiments in the context of federated learning.

This paper aims to extend the Structured Knowledge Accumulation (SKA) framework recently proposed by mahi2025ska. We introduce two core concepts: the Tensor Net function and the characteristic time property of neural learning. First, we reinterpret the learning rate as a time step in a continuous system. This transforms neural learning from discrete optimization into continuous-time evolution. We show that learning dynamics remain consistent when the product of learning rate and iteration steps stays constant. This reveals a time-invariant behavior and identifies an intrinsic timescale of the network. Second, we define the Tensor Net function as a measure that captures the relationship between decision probabilities, entropy gradients, and knowledge change. Additionally, we define its zero-crossing as the equilibrium state between decision probabilities and entropy gradients. We show that the convergence of entropy and knowledge flow provides a natural stopping condition, replacing arbitrary thresholds with an information-theoretic criterion. We also establish that SKA dynamics satisfy a variational principle based on the Euler-Lagrange equation. These findings extend SKA into a continuous and self-organizing learning model. The framework links computational learning with physical systems that evolve by natural laws. By understanding learning as a time-based process, we open new directions for building efficient, robust, and biologically-inspired AI systems.

Most existing causal discovery methods rely on the assumption of no latent confounders, limiting their applicability in solving real-life problems. In this paper, we introduce a novel, versatile framework for causal discovery that accommodates the presence of causally-related hidden variables almost everywhere in the causal network (for instance, they can be effects of observed variables), based on rank information of covariance matrix over observed variables. We start by investigating the efficacy of rank in comparison to conditional independence and, theoretically, establish necessary and sufficient conditions for the identifiability of certain latent structural patterns. Furthermore, we develop a Rank-based Latent Causal Discovery algorithm, RLCD, that can efficiently locate hidden variables, determine their cardinalities, and discover the entire causal structure over both measured and hidden ones. We also show that, under certain graphical conditions, RLCD correctly identifies the Markov Equivalence Class of the whole latent causal graph asymptotically. Experimental results on both synthetic and real-world personality data sets demonstrate the efficacy of the proposed approach in finite-sample cases.

Federated low-rank adaptation (FedLoRA) has facilitated communication-efficient and privacy-preserving fine-tuning of foundation models for downstream tasks. In practical federated learning scenarios, client heterogeneity in system resources and data distributions motivates the use of heterogeneous LoRA ranks across clients. However, we identify a previously overlooked phenomenon in heterogeneous FedLoRA with SVD-based allocation, termed rank collapse, where the energy of the global update becomes concentrated in the minimum shared rank, resulting in suboptimal performance and high sensitivity to rank configurations. Through theoretical analysis, we reveal the root cause of rank collapse: a mismatch between rank-agnostic aggregation weights and rank-dependent client contributions, which systematically suppresses higher-rank updates at a geometric rate over rounds. Motivated by this insight, we propose raFLoRA, a rank-partitioned aggregation method that decomposes local updates into rank partitions and then aggregates each partition weighted by its effective client contributions. Extensive experiments across vision, language, and reasoning tasks show that raFLoRA prevents rank collapse, improves model performance, and enhances robustness across diverse heterogeneous configurations compared with strong FedLoRA baselines.

Mixture of Experts (MoE) models scale capacity but often suffer from representation collapse and gradient instability. We propose Dynamic Subspace Composition (DSC), a framework that approximates context-dependent weights via a state-dependent, sparse expansion of a shared basis bank. Formally, DSC models the weight update as a residual trajectory within a Star- Shaped Domain, employing a Magnitude-Gated Simplex Interpolation to ensure continuity at the identity. Unlike standard Mixture-of-LoRAs, which incurs O(M rd) parameter complexity by retrieving independent rank-r matrices, DSC constructs a compositional rank-K approximation from decoupled unit-norm basis vectors. This reduces parameter complexity to O(M d) and memory traffic to O(Kd), while Frame-Theoretic regularization and spectral constraints provide rigorous worst-case bounds on the dynamic update. The code is available at https://github. com/VladimerKhasia/DSC

We rigorously study the joint evolution of training dynamics via stochastic gradient descent (SGD) and the spectra of empirical Hessian and gradient matrices. We prove that in two canonical classification tasks for multi-class high-dimensional mixtures and either 1 or 2-layer neural networks, the SGD trajectory rapidly aligns with emerging low-rank outlier eigenspaces of the Hessian and gradient matrices. Moreover, in multi-layer settings this alignment occurs per layer, with the final layer's outlier eigenspace evolving over the course of training, and exhibiting rank deficiency when the SGD converges to sub-optimal classifiers. This establishes some of the rich predictions that have arisen from extensive numerical studies in the last decade about the spectra of Hessian and information matrices over the course of training in overparametrized networks.

We showcase important features of the dynamics of the Stochastic Gradient Descent (SGD) in the training of neural networks. We present empirical observations that commonly used large step sizes (i) lead the iterates to jump from one side of a valley to the other causing loss stabilization, and (ii) this stabilization induces a hidden stochastic dynamics orthogonal to the bouncing directions that biases it implicitly toward sparse predictors. Furthermore, we show empirically that the longer large step sizes keep SGD high in the loss landscape valleys, the better the implicit regularization can operate and find sparse representations. Notably, no explicit regularization is used so that the regularization effect comes solely from the SGD training dynamics influenced by the step size schedule. Therefore, these observations unveil how, through the step size schedules, both gradient and noise drive together the SGD dynamics through the loss landscape of neural networks. We justify these findings theoretically through the study of simple neural network models as well as qualitative arguments inspired from stochastic processes. Finally, this analysis allows us to shed a new light on some common practice and observed phenomena when training neural networks. The code of our experiments is available at https://github.com/tml-epfl/sgd-sparse-features.

MoltBook is a large-scale multi-agent coordination environment where over 770,000 autonomous LLM agents interact without human participation, offering the first opportunity we are aware of to observe emergent multi-agent coordination dynamics at this population scale. We introduce Molt Dynamics: the emergent agent coordination behaviors, inter-agent communication dynamics, and role specialization patterns arising when autonomous agents operate as decentralized decision-makers in an unconstrained multi-agent environment. Through longitudinal observation of 90,704 active agents over three weeks, we characterize three aspects. First, spontaneous role specialization: network-based clustering reveals six structural roles (silhouette 0.91), though the result primarily reflects core-periphery organization -- 93.5\% of agents occupy a homogeneous peripheral cluster, with meaningful differentiation confined to the active minority. Second, decentralized information dissemination: cascade analysis of 10,323 inter-agent propagation events reveals power-law distributed cascade sizes (α= 2.57 pm 0.02) and saturating adoption dynamics where adoption probability shows diminishing returns with repeated exposures (Cox hazard ratio 0.53, concordance 0.78). Third, distributed cooperative task resolution: 164 multi-agent collaborative events show detectable coordination patterns, but success rates are low (6.7\%, p = 0.057) and cooperative outcomes are significantly worse than a matched single-agent baseline (Cohen's d = -0.88), indicating emergent cooperative behavior is nascent. These findings establish an empirical baseline for coordination dynamics in decentralized autonomous agent systems, with implications for multi-agent system design, agent communication protocol engineering, and AI safety.

Probabilistic dynamics model ensemble is widely used in existing model-based reinforcement learning methods as it outperforms a single dynamics model in both asymptotic performance and sample efficiency. In this paper, we provide both practical and theoretical insights on the empirical success of the probabilistic dynamics model ensemble through the lens of Lipschitz continuity. We find that, for a value function, the stronger the Lipschitz condition is, the smaller the gap between the true dynamics- and learned dynamics-induced Bellman operators is, thus enabling the converged value function to be closer to the optimal value function. Hence, we hypothesize that the key functionality of the probabilistic dynamics model ensemble is to regularize the Lipschitz condition of the value function using generated samples. To test this hypothesis, we devise two practical robust training mechanisms through computing the adversarial noise and regularizing the value network's spectral norm to directly regularize the Lipschitz condition of the value functions. Empirical results show that combined with our mechanisms, model-based RL algorithms with a single dynamics model outperform those with an ensemble of probabilistic dynamics models. These findings not only support the theoretical insight, but also provide a practical solution for developing computationally efficient model-based RL algorithms.

Social and collaborative platforms emit multivariate time-series traces in which early interactions-such as views, likes, or downloads-are followed, sometimes months or years later, by higher impact like citations, sales, or reviews. We formalize this setting as Lead-Lag Forecasting (LLF): given an early usage channel (the lead), predict a correlated but temporally shifted outcome channel (the lag). Despite the ubiquity of such patterns, LLF has not been treated as a unified forecasting problem within the time-series community, largely due to the absence of standardized datasets. To anchor research in LLF, here we present two high-volume benchmark datasets-arXiv (accesses -> citations of 2.3M papers) and GitHub (pushes/stars -> forks of 3M repositories)-and outline additional domains with analogous lead-lag dynamics, including Wikipedia (page views -> edits), Spotify (streams -> concert attendance), e-commerce (click-throughs -> purchases), and LinkedIn profile (views -> messages). Our datasets provide ideal testbeds for lead-lag forecasting, by capturing long-horizon dynamics across years, spanning the full spectrum of outcomes, and avoiding survivorship bias in sampling. We documented all technical details of data curation and cleaning, verified the presence of lead-lag dynamics through statistical and classification tests, and benchmarked parametric and non-parametric baselines for regression. Our study establishes LLF as a novel forecasting paradigm and lays an empirical foundation for its systematic exploration in social and usage data. Our data portal with downloads and documentation is available at https://lead-lag-forecasting.github.io/.

We consider the question of learning Q-function in a sample efficient manner for reinforcement learning with continuous state and action spaces under a generative model. If Q-function is Lipschitz continuous, then the minimal sample complexity for estimating ε-optimal Q-function is known to scale as Ω(1{ε^{d_1+d_2 +2}}) per classical non-parametric learning theory, where d_1 and d_2 denote the dimensions of the state and action spaces respectively. The Q-function, when viewed as a kernel, induces a Hilbert-Schmidt operator and hence possesses square-summable spectrum. This motivates us to consider a parametric class of Q-functions parameterized by its "rank" r, which contains all Lipschitz Q-functions as r to infty. As our key contribution, we develop a simple, iterative learning algorithm that finds ε-optimal Q-function with sample complexity of O(1{ε^{max(d_1, d_2)+2}}) when the optimal Q-function has low rank r and the discounting factor γ is below a certain threshold. Thus, this provides an exponential improvement in sample complexity. To enable our result, we develop a novel Matrix Estimation algorithm that faithfully estimates an unknown low-rank matrix in the ell_infty sense even in the presence of arbitrary bounded noise, which might be of interest in its own right. Empirical results on several stochastic control tasks confirm the efficacy of our "low-rank" algorithms.

As the open-weight AI landscape continues to proliferate-with model development, significant investment, and user interest-it becomes increasingly important to predict which models will ultimately drive innovation and shape AI ecosystems. Building on parallels with citation dynamics in scientific literature, we propose a framework to quantify how an open-weight model's influence evolves. Specifically, we adapt the model introduced by Wang et al. for scientific citations, using three key parameters-immediacy, longevity, and relative fitness-to track the cumulative number of fine-tuned models of an open-weight model. Our findings reveal that this citation-style approach can effectively capture the diverse trajectories of open-weight model adoption, with most models fitting well and outliers indicating unique patterns or abrupt jumps in usage.

Many platforms on the web present ranked lists of content to users, typically optimized for engagement-, satisfaction- or retention- driven metrics. Advances in the Learning-to-Rank (LTR) research literature have enabled rapid growth in this application area. Several popular interfaces now include nested lists, where users can enter a 2nd-level feed via any given 1st-level item. Naturally, this has implications for evaluation metrics, objective functions, and the ranking policies we wish to learn. We propose a theoretically grounded method to incorporate 2nd-level feedback into any 1st-level ranking model. Online experiments on a large-scale recommendation system confirm our theoretical findings.

Modeling stochastic dynamics from discrete observations is a key interdisciplinary challenge. Existing methods often fail to estimate the continuous evolution of probability densities from trajectories or face the curse of dimensionality. To address these limitations, we presents a novel paradigm: modeling dynamics directly in the weight space of a neural network by projecting the evolving probability distribution. We first theoretically establish the connection between dynamic optimal transport in measure space and an equivalent energy functional in weight space. Subsequently, we design WeightFlow, which constructs the neural network weights into a graph and learns its evolution via a graph controlled differential equation. Experiments on interdisciplinary datasets demonstrate that WeightFlow improves performance by an average of 43.02\% over state-of-the-art methods, providing an effective and scalable solution for modeling high-dimensional stochastic dynamics.

TD-learning is a foundation reinforcement learning (RL) algorithm for value prediction. Critical to the accuracy of value predictions is the quality of state representations. In this work, we consider the question: how does end-to-end TD-learning impact the representation over time? Complementary to prior work, we provide a set of analysis that sheds further light on the representation dynamics under TD-learning. We first show that when the environments are reversible, end-to-end TD-learning strictly decreases the value approximation error over time. Under further assumptions on the environments, we can connect the representation dynamics with spectral decomposition over the transition matrix. This latter finding establishes fitting multiple value functions from randomly generated rewards as a useful auxiliary task for representation learning, as we empirically validate on both tabular and Atari game suites.

Prediction models can exhibit sensitivity with respect to training data: small changes in the training data can produce models that assign conflicting predictions to individual data points during test time. In this work, we study this sensitivity in recommender systems, where users' recommendations are drastically altered by minor perturbations in other unrelated users' interactions. We introduce a measure of stability for recommender systems, called Rank List Sensitivity (RLS), which measures how rank lists generated by a given recommender system at test time change as a result of a perturbation in the training data. We develop a method, CASPER, which uses cascading effect to identify the minimal and systematical perturbation to induce higher instability in a recommender system. Experiments on four datasets show that recommender models are overly sensitive to minor perturbations introduced randomly or via CASPER - even perturbing one random interaction of one user drastically changes the recommendation lists of all users. Importantly, with CASPER perturbation, the models generate more unstable recommendations for low-accuracy users (i.e., those who receive low-quality recommendations) than high-accuracy ones.

The accurate modeling of dynamics in interactive environments is critical for successful long-range prediction. Such a capability could advance Reinforcement Learning (RL) and Planning algorithms, but achieving it is challenging. Inaccuracies in model estimates can compound, resulting in increased errors over long horizons. We approach this problem from the lens of Koopman theory, where the nonlinear dynamics of the environment can be linearized in a high-dimensional latent space. This allows us to efficiently parallelize the sequential problem of long-range prediction using convolution while accounting for the agent's action at every time step. Our approach also enables stability analysis and better control over gradients through time. Taken together, these advantages result in significant improvement over the existing approaches, both in the efficiency and the accuracy of modeling dynamics over extended horizons. We also show that this model can be easily incorporated into dynamics modeling for model-based planning and model-free RL and report promising experimental results.

As the post-training of large language models (LLMs) advances from instruction-following to complex reasoning tasks, understanding how different data affect finetuning dynamics remains largely unexplored. In this paper, we present a spectral analysis of layer-wise gradients induced by low/high-quality instruction and reasoning data for LLM post-training. Our analysis reveals that widely-studied metrics for data evaluation, e.g., IFD, InsTag, Difficulty, and Reward, can be explained and unified by spectral properties computed from gradients' singular value decomposition (SVD). Specifically, higher-quality data are usually associated with lower nuclear norms and higher effective ranks. Notably, effective rank exhibits better robustness and resolution than nuclear norm in capturing subtle quality differences. For example, reasoning data achieves substantially higher effective ranks than instruction data, implying richer gradient structures on more complex tasks. Our experiments also highlight that models within the same family share similar gradient patterns regardless of their sizes, whereas different model families diverge significantly. Providing a unified view on the effects of data quality across instruction and reasoning data, this work illuminates the interplay between data quality and training stability, shedding novel insights into developing better data exploration strategies for post-training.

Generalizing policies across different domains with dynamics mismatch poses a significant challenge in reinforcement learning. For example, a robot learns the policy in a simulator, but when it is deployed in the real world, the dynamics of the environment may be different. Given the source and target domain with dynamics mismatch, we consider the online dynamics adaptation problem, in which case the agent can access sufficient source domain data while online interactions with the target domain are limited. Existing research has attempted to solve the problem from the dynamics discrepancy perspective. In this work, we reveal the limitations of these methods and explore the problem from the value difference perspective via a novel insight on the value consistency across domains. Specifically, we present the Value-Guided Data Filtering (VGDF) algorithm, which selectively shares transitions from the source domain based on the proximity of paired value targets across the two domains. Empirical results on various environments with kinematic and morphology shifts demonstrate that our method achieves superior performance compared to prior approaches.

This paper investigates the evaluation of learned multiagent strategies in the incomplete information setting, which plays a critical role in ranking and training of agents. Traditionally, researchers have relied on Elo ratings for this purpose, with recent works also using methods based on Nash equilibria. Unfortunately, Elo is unable to handle intransitive agent interactions, and other techniques are restricted to zero-sum, two-player settings or are limited by the fact that the Nash equilibrium is intractable to compute. Recently, a ranking method called α-Rank, relying on a new graph-based game-theoretic solution concept, was shown to tractably apply to general games. However, evaluations based on Elo or α-Rank typically assume noise-free game outcomes, despite the data often being collected from noisy simulations, making this assumption unrealistic in practice. This paper investigates multiagent evaluation in the incomplete information regime, involving general-sum many-player games with noisy outcomes. We derive sample complexity guarantees required to confidently rank agents in this setting. We propose adaptive algorithms for accurate ranking, provide correctness and sample complexity guarantees, then introduce a means of connecting uncertainties in noisy match outcomes to uncertainties in rankings. We evaluate the performance of these approaches in several domains, including Bernoulli games, a soccer meta-game, and Kuhn poker.

Modern socio-technical systems increasingly involve multi-stakeholder environments where actors simultaneously cooperate and compete. These coopetitive relationships exhibit dynamic trust evolution based on observed behavior over repeated interactions. While conceptual modeling languages like i* represent trust relationships qualitatively, they lack computational mechanisms for analyzing how trust changes with behavioral evidence. Conversely, computational trust models from multi-agent systems provide algorithmic updating but lack grounding in conceptual models that capture strategic dependencies covering mixed motives of actors. This technical report bridges this gap by developing a computational trust model that extends game-theoretic foundations for strategic coopetition with dynamic trust evolution. Building on companion work that achieved 58/60 validation (96.7%) for logarithmic specifications, we introduce trust as a two-layer system with immediate trust responding to current behavior and reputation tracking violation history. Trust evolves through asymmetric updating where cooperation builds trust gradually while violations erode it sharply, creating hysteresis effects and trust ceilings that constrain relationship recovery. We develop a structured translation framework enabling practitioners to instantiate computational trust models from i* dependency networks encompassing mixed motives of actors. Comprehensive experimental validation across 78,125 parameter configurations establishes robust emergence of negativity bias, hysteresis effects, and cumulative damage amplification. Empirical validation using the Renault-Nissan Alliance case study (1999-2025) achieves 49/60 validation points (81.7%), successfully reproducing documented trust evolution across five distinct relationship phases including crisis and recovery periods.

Decentralized reputation schemes present a promising area of experimentation in blockchain applications. These solutions aim to overcome the shortcomings of simple monetary incentive mechanisms of naive tokenomics. However, there is a significant research gap regarding the limitations and benefits of such solutions. We formulate these trade-offs as a conjecture on the irreconcilability of three desirable properties of the reputation system in this context. Such a system can not be simultaneously generalizable, trustless, and Sybil resistant. To handle the limitations of this trilemma, we propose MeritRank: Sybil tolerant feedback aggregation mechanism for reputation. Instead of preventing Sybil attacks, our approach successfully bounds the benefits of these attacks. Using a dataset of participants' interactions in MakerDAO, we run experiments to demonstrate Sybil tolerance of MeritRank. Decay parameters of reputation in MeritRank: transitivity decay and connectivity decay, allow for a fine-tuning of desirable levels of reputation utility and Sybil tolerance in different use contexts.

Computing stable partitions in hedonic games is a challenging task because there exist games in which stable outcomes do not exist. Even more, these No-instances can often be leveraged to prove computational hardness results. We make this impression rigorous in a dynamic model of cardinal hedonic games by providing meta theorems. These imply hardness of deciding about the possible or necessary convergence of deviation dynamics based on the mere existence of No-instances. Our results hold for additively separable, fractional, and modified fractional hedonic games (ASHGs, FHGs, and MFHGs). Moreover, they encompass essentially all reasonable stability notions based on single-agent deviations. In addition, we propose dynamics as a method to find individually rational and contractually individual stable (CIS) partitions in ASHGs. In particular, we find that CIS dynamics from the singleton partition possibly converge after a linear number of deviations but may require an exponential number of deviations in the worst case.

Projection maintenance is one of the core data structure tasks. Efficient data structures for projection maintenance have led to recent breakthroughs in many convex programming algorithms. In this work, we further extend this framework to the Kronecker product structure. Given a constraint matrix {sf A} and a positive semi-definite matrix Win R^{ntimes n} with a sparse eigenbasis, we consider the task of maintaining the projection in the form of {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}, where {sf B}={sf A}(Wotimes I) or {sf B}={sf A}(W^{1/2}otimes W^{1/2}). At each iteration, the weight matrix W receives a low rank change and we receive a new vector h. The goal is to maintain the projection matrix and answer the query {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}h with good approximation guarantees. We design a fast dynamic data structure for this task and it is robust against an adaptive adversary. Following the beautiful and pioneering work of [Beimel, Kaplan, Mansour, Nissim, Saranurak and Stemmer, STOC'22], we use tools from differential privacy to reduce the randomness required by the data structure and further improve the running time.

Varying dynamics parameters in simulation is a popular Domain Randomization (DR) approach for overcoming the reality gap in Reinforcement Learning (RL). Nevertheless, DR heavily hinges on the choice of the sampling distribution of the dynamics parameters, since high variability is crucial to regularize the agent's behavior but notoriously leads to overly conservative policies when randomizing excessively. In this paper, we propose a novel approach to address sim-to-real transfer, which automatically shapes dynamics distributions during training in simulation without requiring real-world data. We introduce DOmain RAndomization via Entropy MaximizatiON (DORAEMON), a constrained optimization problem that directly maximizes the entropy of the training distribution while retaining generalization capabilities. In achieving this, DORAEMON gradually increases the diversity of sampled dynamics parameters as long as the probability of success of the current policy is sufficiently high. We empirically validate the consistent benefits of DORAEMON in obtaining highly adaptive and generalizable policies, i.e. solving the task at hand across the widest range of dynamics parameters, as opposed to representative baselines from the DR literature. Notably, we also demonstrate the Sim2Real applicability of DORAEMON through its successful zero-shot transfer in a robotic manipulation setup under unknown real-world parameters.

We study the learning dynamics of self-predictive learning for reinforcement learning, a family of algorithms that learn representations by minimizing the prediction error of their own future latent representations. Despite its recent empirical success, such algorithms have an apparent defect: trivial representations (such as constants) minimize the prediction error, yet it is obviously undesirable to converge to such solutions. Our central insight is that careful designs of the optimization dynamics are critical to learning meaningful representations. We identify that a faster paced optimization of the predictor and semi-gradient updates on the representation, are crucial to preventing the representation collapse. Then in an idealized setup, we show self-predictive learning dynamics carries out spectral decomposition on the state transition matrix, effectively capturing information of the transition dynamics. Building on the theoretical insights, we propose bidirectional self-predictive learning, a novel self-predictive algorithm that learns two representations simultaneously. We examine the robustness of our theoretical insights with a number of small-scale experiments and showcase the promise of the novel representation learning algorithm with large-scale experiments.

The intricate dynamics of stock markets have led to extensive research on models that are able to effectively explain their inherent complexities. This study leverages the econometrics literature to explore the dynamic factor model as an interpretable model with sufficient predictive capabilities for capturing essential market phenomena. Although the model has been extensively applied for predictive purposes, this study focuses on analyzing the extracted loadings and common factors as an alternative framework for understanding stock price dynamics. The results reveal novel insights into traditional market theories when applied to the Philippine Stock Exchange using the Kalman method and maximum likelihood estimation, with subsequent validation against the capital asset pricing model. Notably, a one-factor model extracts a common factor representing systematic or market dynamics similar to the composite index, whereas a two-factor model extracts common factors representing market trends and volatility. Furthermore, an application of the model for nowcasting the growth rates of the Philippine gross domestic product highlights the potential of the extracted common factors as viable real-time market indicators, yielding over a 34% decrease in the out-of-sample prediction error. Overall, the results underscore the value of dynamic factor analysis in gaining a deeper understanding of market price movement dynamics.

Systemic financial risk refers to the simultaneous failure or destabilization of multiple financial institutions, often triggered by contagion mechanisms or common exposures to shocks. In this paper, we present a dynamical model of bank leverage (the ratio of asset holdings to equity) a quantity that both reflects and drives risk dynamics. We model how banks, constrained by Value-at-Risk (VaR) regulations, adjust their leverage in response to changes in the price of a single asset, assumed to be held in fixed proportion across banks. This leverage-targeting behavior introduces a procyclical feedback loop between asset prices and leverage. In the dynamics, this can manifest as logistic-like behavior with a rich bifurcation structure across model parameters. By analyzing these coupled dynamics in both isolated and interconnected bank models, we outline a framework for understanding how systemic risk can emerge from seemingly rational micro-level behavior.

Low-rank approximation is a fundamental technique in modern data analysis, widely utilized across various fields such as signal processing, machine learning, and natural language processing. Despite its ubiquity, the mechanics of low-rank approximation and its application in adaptation can sometimes be obscure, leaving practitioners and researchers with questions about its true capabilities and limitations. This paper seeks to clarify low-rank approximation and adaptation by offering a comprehensive guide that reveals their inner workings and explains their utility in a clear and accessible way. Our focus here is to develop a solid intuition for how low-rank approximation and adaptation operate, and why they are so effective. We begin with basic concepts and gradually build up to the mathematical underpinnings, ensuring that readers of all backgrounds can gain a deeper understanding of low-rank approximation and adaptation. We strive to strike a balance between informal explanations and rigorous mathematics, ensuring that both newcomers and experienced experts can benefit from this survey. Additionally, we introduce new low-rank decomposition and adaptation algorithms that have not yet been explored in the field, hoping that future researchers will investigate their potential applicability.

State-space models (SSMs) offer a powerful framework for dynamical system analysis, wherein the temporal dynamics of the system are assumed to be captured through the evolution of the latent states, which govern the values of the observations. This paper provides a selective review of recent advancements in deep neural network-based approaches for SSMs, and presents a unified perspective for discrete time deep state space models and continuous time ones such as latent neural Ordinary Differential and Stochastic Differential Equations. It starts with an overview of the classical maximum likelihood based approach for learning SSMs, reviews variational autoencoder as a general learning pipeline for neural network-based approaches in the presence of latent variables, and discusses in detail representative deep learning models that fall under the SSM framework. Very recent developments, where SSMs are used as standalone architectural modules for improving efficiency in sequence modeling, are also examined. Finally, examples involving mixed frequency and irregularly-spaced time series data are presented to demonstrate the advantage of SSMs in these settings.

Diffusion models (DMs) represent state-of-the-art generative models for continuous inputs. DMs work by constructing a Stochastic Differential Equation (SDE) in the input space (ie, position space), and using a neural network to reverse it. In this work, we introduce a novel generative modeling framework grounded in phase space dynamics, where a phase space is defined as {an augmented space encompassing both position and velocity.} Leveraging insights from Stochastic Optimal Control, we construct a path measure in the phase space that enables efficient sampling. {In contrast to DMs, our framework demonstrates the capability to generate realistic data points at an early stage of dynamics propagation.} This early prediction sets the stage for efficient data generation by leveraging additional velocity information along the trajectory. On standard image generation benchmarks, our model yields favorable performance over baselines in the regime of small Number of Function Evaluations (NFEs). Furthermore, our approach rivals the performance of diffusion models equipped with efficient sampling techniques, underscoring its potential as a new tool generative modeling.

We analyze the layerwise effective dimension (rank of the feature matrix) in fully-connected ReLU networks of finite width. Specifically, for a fixed batch of m inputs and random Gaussian weights, we derive closed-form expressions for the expected rank of the \mtimes n hidden activation matrices. Our main result shows that E[EDim(ell)]=m[1-(1-2/pi)^ell]+O(e^{-c m}) so that the rank deficit decays geometrically with ratio 1-2 / pi approx 0.3634. We also prove a sub-Gaussian concentration bound, and identify the "revival" depths at which the expected rank attains local maxima. In particular, these peaks occur at depths ell_k^*approx(k+1/2)pi/log(1/rho) with height approx (1-e^{-pi/2}) m approx 0.79m. We further show that this oscillatory rank behavior is a finite-width phenomenon: under orthogonal weight initialization or strong negative-slope leaky-ReLU, the rank remains (nearly) full. These results provide a precise characterization of how random ReLU layers alternately collapse and partially revive the subspace of input variations, adding nuance to prior work on expressivity of deep networks.

ReAct-style agents for search-intensive, multi-step reasoning tasks rely largely on their own internal judgment to decide what evidence to seek, which reasoning or action step to take next, and when to stop, often producing shallow, redundant, or poorly targeted trajectories. Prior work has explored rubrics as external quality signals, but existing uses are mostly evaluative rather than action-guiding: rubrics typically serve as training-time rewards or post-hoc evaluators of completed outputs, and in deep-research settings they are often coarse-grained and report-level rather than step-level. We introduce Co-ReAct, a rubric-guided action-selection framework that uses rubrics as step-level guidance during inference. At each decision step, Co-ReAct injects a rubric into the agent's context to guide the next Reason-or-Act decision, specifying what the agent should target in evidence seeking, search, reasoning, or self-evaluation. To make this guidance reliable, we train a dedicated rubric generator with GRPO. Unlike prior pairwise or binary preference formulations, our objective optimizes a list-wise Spearman rank-correlation reward against multi-judge expert consensus rankings, encouraging rubrics that are discriminative rather than merely plausible. On DeepResearchBench and SQA-CS-V2, Co-ReAct consistently improves over ReAct and representative test-time compute baselines across search agents built on both 8B/14B open-source and frontier closed-source base models. The trained rubric generator can also serve as a drop-in component that improves these baselines without changing their underlying decision mechanisms. Our code is publicly available at https://github.com/ZBWpro/Co-ReAct.

Cognitive dynamics are pivotal to advance human understanding of the world. Recent advancements in large language models (LLMs) reveal their potential for cognitive simulation. However, these LLM-based cognitive studies primarily focus on static modeling, overlooking the dynamic nature of cognition. To bridge this gap, we propose the concept of the cognitive dynamics of LLMs and present a corresponding task with the inspiration of longitudinal studies. Towards the task, we develop CogBench, a novel benchmark to assess the cognitive dynamics of LLMs and validate it through participant surveys. We also design two evaluation metrics for CogBench, including Authenticity and Rationality. Recognizing the inherent static nature of LLMs, we introduce CogGPT for the task, which features an innovative iterative cognitive mechanism aimed at enhancing lifelong cognitive dynamics. Empirical results demonstrate the superiority of CogGPT over existing methods, particularly in its ability to facilitate role-specific cognitive dynamics under continuous information flows.

Diffusion language models (DLMs) have recently emerged as a promising alternative to autoregressive (AR) models, offering parallel decoding and controllable sampling dynamics while achieving competitive generation quality at scale. Despite this progress, the role of sampling mechanisms in shaping refusal behavior and jailbreak robustness remains poorly understood. In this work, we present a fundamental analytical framework for step-wise refusal dynamics, enabling comparison between AR and diffusion sampling. Our analysis reveals that the sampling strategy itself plays a central role in safety behavior, as a factor distinct from the underlying learned representations. Motivated by this analysis, we introduce the Step-Wise Refusal Internal Dynamics (SRI) signal, which supports interpretability and improved safety for both AR and DLMs. We demonstrate that the geometric structure of SRI captures internal recovery dynamics, and identifies anomalous behavior in harmful generations as cases of incomplete internal recovery that are not observable at the text level. This structure enables lightweight inference-time detectors that generalize to unseen attacks while matching or outperforming existing defenses with over 100times lower inference overhead.

Preference-based reinforcement learning (PbRL) aligns a robot behavior with human preferences via a reward function learned from binary feedback over agent behaviors. We show that dynamics-aware reward functions improve the sample efficiency of PbRL by an order of magnitude. In our experiments we iterate between: (1) learning a dynamics-aware state-action representation (z^{sa}) via a self-supervised temporal consistency task, and (2) bootstrapping the preference-based reward function from (z^{sa}), which results in faster policy learning and better final policy performance. For example, on quadruped-walk, walker-walk, and cheetah-run, with 50 preference labels we achieve the same performance as existing approaches with 500 preference labels, and we recover 83\% and 66\% of ground truth reward policy performance versus only 38\% and 21\%. The performance gains demonstrate the benefits of explicitly learning a dynamics-aware reward model. Repo: https://github.com/apple/ml-reed.

I introduce a novel mathematical framework integrating topological dynamics, operator algebras, and ergodic geometry to study lattices of asynchronous metric dynamical systems. Each node in the lattice carries an internal flow represented by a one-parameter family of operators, evolving on its own time scale. I formalize stratified state spaces capturing multiple levels of synchronized behavior, define an asynchronous evolution metric that quantifies phase-offset distances between subsystems, and characterize emergent coherent topologies arising when subsystems synchronize. Within this framework, I develop formal operators for the evolution of each subsystem and give precise conditions under which phase-aligned synchronization occurs across the lattice. The main results include: (1) the existence and uniqueness of coherent (synchronized) states under a contractive coupling condition, (2) stability of these coherent states and criteria for their emergence as a collective phase transition in a continuous operator topology, and (3) the influence of symmetries, with group-invariant coupling leading to flow-invariant synchrony subspaces and structured cluster dynamics. Proofs are given for each theorem, demonstrating full mathematical rigor. In a final section, I discuss hypothetical applications of this framework to symbolic lattice systems (e.g. subshifts), to invariant group actions on dynamical lattices, and to operator fields over stratified manifolds in the spirit of noncommutative geometry. Throughout, I write in the first person to emphasize the exploratory nature of this work. The paper avoids any reference to cosmology or observers, focusing instead on clean, formal mathematics suitable for a broad array of dynamical systems.

While the phenomenon of grokking, i.e., delayed generalization, has been studied extensively, it remains an open problem whether there is a mathematical framework that characterizes what kind of features will emerge, how and in which conditions it happens, and is closely related to the gradient dynamics of the training, for complex structured inputs. We propose a novel framework, named Li_2, that captures three key stages for the grokking behavior of 2-layer nonlinear networks: (I) \textbf{L}azy learning, (II) \textbf{i}ndependent feature learning and (III) \textbf{i}nteractive feature learning. At the lazy learning stage, top layer overfits to random hidden representation and the model appears to memorize. Thanks to lazy learning and weight decay, the backpropagated gradient G_F from the top layer now carries information about the target label, with a specific structure that enables each hidden node to learn their representation independently. Interestingly, the independent dynamics follows exactly the gradient ascent of an energy function E, and its local maxima are precisely the emerging features. We study whether these local-optima induced features are generalizable, their representation power, and how they change on sample size, in group arithmetic tasks. When hidden nodes start to interact in the later stage of learning, we provably show how G_F changes to focus on missing features that need to be learned. Our study sheds lights on roles played by key hyperparameters such as weight decay, learning rate and sample sizes in grokking, leads to provable scaling laws of feature emergence, memorization and generalization, and reveals the underlying cause why recent optimizers such as Muon can be effective, from the first principles of gradient dynamics. Our analysis can be extended to multi-layer architectures.

Molecular dynamics (MD) provides insights into atomic-scale processes by integrating over time the equations that describe the motion of atoms under the action of interatomic forces. Machine learning models have substantially accelerated MD by providing inexpensive predictions of the forces, but they remain constrained to minuscule time integration steps, which are required by the fast time scale of atomic motion. In this work, we propose FlashMD, a method to predict the evolution of positions and momenta over strides that are between one and two orders of magnitude longer than typical MD time steps. We incorporate considerations on the mathematical and physical properties of Hamiltonian dynamics in the architecture, generalize the approach to allow the simulation of any thermodynamic ensemble, and carefully assess the possible failure modes of such a long-stride MD approach. We validate FlashMD's accuracy in reproducing equilibrium and time-dependent properties, using both system-specific and general-purpose models, extending the ability of MD simulation to reach the long time scales needed to model microscopic processes of high scientific and technological relevance.

Learning to rank with biased click data is a well-known challenge. A variety of methods has been explored to debias click data for learning to rank such as click models, result interleaving and, more recently, the unbiased learning-to-rank framework based on inverse propensity weighting. Despite their differences, most existing studies separate the estimation of click bias (namely the propensity model) from the learning of ranking algorithms. To estimate click propensities, they either conduct online result randomization, which can negatively affect the user experience, or offline parameter estimation, which has special requirements for click data and is optimized for objectives (e.g. click likelihood) that are not directly related to the ranking performance of the system. In this work, we address those problems by unifying the learning of propensity models and ranking models. We find that the problem of estimating a propensity model from click data is a dual problem of unbiased learning to rank. Based on this observation, we propose a Dual Learning Algorithm (DLA) that jointly learns an unbiased ranker and an unbiased propensity model. DLA is an automatic unbiased learning-to-rank framework as it directly learns unbiased ranking models from biased click data without any preprocessing. It can adapt to the change of bias distributions and is applicable to online learning. Our empirical experiments with synthetic and real-world data show that the models trained with DLA significantly outperformed the unbiased learning-to-rank algorithms based on result randomization and the models trained with relevance signals extracted by click models.

During 2023, two interesting results were proven about the limit behavior of game dynamics: First, it was shown that there is a game for which no dynamics converges to the Nash equilibria. Second, it was shown that the sink equilibria of a game adequately capture the limit behavior of natural game dynamics. These two results have created a need and opportunity to articulate a principled computational theory of the meaning of the game that is based on game dynamics. Given any game in normal form, and any prior distribution of play, we study the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game. When the prior distribution has pure strategy support, we prove this distribution can be computed efficiently, in near-linear time to the size of the best-response graph. When the distribution can be sampled -- for example, if it is the uniform distribution over all mixed strategy profiles -- we show through experiments that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.

Offline reinforcement learning aims to train agents from pre-collected datasets. However, this comes with the added challenge of estimating the value of behaviors not covered in the dataset. Model-based methods offer a potential solution by training an approximate dynamics model, which then allows collection of additional synthetic data via rollouts in this model. The prevailing theory treats this approach as online RL in an approximate dynamics model, and any remaining performance gap is therefore understood as being due to dynamics model errors. In this paper, we analyze this assumption and investigate how popular algorithms perform as the learned dynamics model is improved. In contrast to both intuition and theory, if the learned dynamics model is replaced by the true error-free dynamics, existing model-based methods completely fail. This reveals a key oversight: The theoretical foundations assume sampling of full horizon rollouts in the learned dynamics model; however, in practice, the number of model-rollout steps is aggressively reduced to prevent accumulating errors. We show that this truncation of rollouts results in a set of edge-of-reach states at which we are effectively ``bootstrapping from the void.'' This triggers pathological value overestimation and complete performance collapse. We term this the edge-of-reach problem. Based on this new insight, we fill important gaps in existing theory, and reveal how prior model-based methods are primarily addressing the edge-of-reach problem, rather than model-inaccuracy as claimed. Finally, we propose Reach-Aware Value Learning (RAVL), a simple and robust method that directly addresses the edge-of-reach problem and hence - unlike existing methods - does not fail as the dynamics model is improved. Code open-sourced at: github.com/anyasims/edge-of-reach.

Modern generative models demonstrate impressive capabilities, likely stemming from an ability to identify and manipulate abstract concepts underlying their training data. However, fundamental questions remain: what determines the concepts a model learns, the order in which it learns them, and its ability to manipulate those concepts? To address these questions, we propose analyzing a model's learning dynamics via a framework we call the concept space, where each axis represents an independent concept underlying the data generating process. By characterizing learning dynamics in this space, we identify how the speed at which a concept is learned, and hence the order of concept learning, is controlled by properties of the data we term concept signal. Further, we observe moments of sudden turns in the direction of a model's learning dynamics in concept space. Surprisingly, these points precisely correspond to the emergence of hidden capabilities, i.e., where latent interventions show the model possesses the capability to manipulate a concept, but these capabilities cannot yet be elicited via naive input prompting. While our results focus on synthetically defined toy datasets, we hypothesize a general claim on emergence of hidden capabilities may hold: generative models possess latent capabilities that emerge suddenly and consistently during training, though a model might not exhibit these capabilities under naive input prompting.

Understanding the training dynamics of deep neural networks remains a major open problem, with physics-inspired approaches offering promising insights. Building on this perspective, we develop a thermodynamic framework to describe the stationary distributions of stochastic gradient descent (SGD) with weight decay for scale-invariant neural networks, a setting that both reflects practical architectures with normalization layers and permits theoretical analysis. We establish analogies between training hyperparameters (e.g., learning rate, weight decay) and thermodynamic variables such as temperature, pressure, and volume. Starting with a simplified isotropic noise model, we uncover a close correspondence between SGD dynamics and ideal gas behavior, validated through theory and simulation. Extending to training of neural networks, we show that key predictions of the framework, including the behavior of stationary entropy, align closely with experimental observations. This framework provides a principled foundation for interpreting training dynamics and may guide future work on hyperparameter tuning and the design of learning rate schedulers.

Networks have become indispensable and ubiquitous structures in many fields to model the interactions among different entities, such as friendship in social networks or protein interactions in biological graphs. A major challenge is to understand the structure and dynamics of these systems. Although networks evolve through time, most existing graph representation learning methods target only static networks. Whereas approaches have been developed for the modeling of dynamic networks, there is a lack of efficient continuous time dynamic graph representation learning methods that can provide accurate network characterization and visualization in low dimensions while explicitly accounting for prominent network characteristics such as homophily and transitivity. In this paper, we propose the Piecewise-Velocity Model (PiVeM) for the representation of continuous-time dynamic networks. It learns dynamic embeddings in which the temporal evolution of nodes is approximated by piecewise linear interpolations based on a latent distance model with piecewise constant node-specific velocities. The model allows for analytically tractable expressions of the associated Poisson process likelihood with scalable inference invariant to the number of events. We further impose a scalable Kronecker structured Gaussian Process prior to the dynamics accounting for community structure, temporal smoothness, and disentangled (uncorrelated) latent embedding dimensions optimally learned to characterize the network dynamics. We show that PiVeM can successfully represent network structure and dynamics in ultra-low two-dimensional spaces. It outperforms relevant state-of-art methods in downstream tasks such as link prediction. In summary, PiVeM enables easily interpretable dynamic network visualizations and characterizations that can further improve our understanding of the intrinsic dynamics of time-evolving networks.

To what extent do vision-and-language foundation models possess a realistic world model (observation times action rightarrow observation) and a dynamics model (observation times observation rightarrow action), when actions are expressed through language? While open-source foundation models struggle with both, we find that fine-tuning them to acquire a dynamics model through supervision is significantly easier than acquiring a world model. In turn, dynamics models can be used to bootstrap world models through two main strategies: 1) weakly supervised learning from synthetic data and 2) inference time verification. Firstly, the dynamics model can annotate actions for unlabelled pairs of video frame observations to expand the training data. We further propose a new objective, where image tokens in observation pairs are weighted by their importance, as predicted by a recognition model. Secondly, the dynamics models can assign rewards to multiple samples of the world model to score them, effectively guiding search at inference time. We evaluate the world models resulting from both strategies through the task of action-centric image editing on Aurora-Bench. Our best model achieves a performance competitive with state-of-the-art image editing models, improving on them by a margin of 15% on real-world subsets according to GPT4o-as-judge, and achieving the best average human evaluation across all subsets of Aurora-Bench.

We address the challenge of adapting pre-trained Large Language Models (LLMs) for multivariate time-series analysis, where their deployment is often hindered by prohibitive computational and memory demands. Our solution, One-for-All, introduces Gaussian Rank-Stabilized Low-Rank Adapters (rsLoRA) to enable parameter-efficient fine-tuning of frozen LLMs. While inspired by LoRA, rsLoRA introduces a mathematically grounded rank-stabilization mechanism that enables provable gradient stability at low ranks a novel contribution absent in prior PEFT methods. Our framework injects trainable rank decomposition matrices (rank 16) into positional embeddings and output layers, while keeping self-attention weights fixed. This design reduces trainable parameters by 6.8times (vs. TimesNet), 21times (vs. GPT4TS), and 11.8times (vs. TIME-LLM), while achieving a 168-1,776times smaller memory footprint (2.2MiB vs. 340MiB-4.18GiB in SOTA models). Rigorous evaluation across six time-series tasks demonstrates that One-for-All achieves state-of-the-art efficiency-accuracy trade-offs: 5.5times higher parameter efficiency (MSE=5.50) than TimesNet and 21times better than GPT4TS, while matching their forecasting accuracy (MSE=0.33). The framework's stability is validated through consistent performance across diverse horizons (96-720 steps) and datasets (ETT, Weather, M3, M4), with 98.3% fewer parameters than conventional transformers. These advances enable deployment on edge devices for healthcare, finance, and environmental monitoring without compromising performance.

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.

Reinforcement learning is emerging as a primary driver for improving language model reasoning capabilities. A fundamental question is whether current reinforcement learning algorithms -- such as Group Relative Policy Optimization (GRPO), the de facto standard algorithm used to improve language model reasoning -- merely sharpen the base model's distribution around problems it can already solve. We investigate this question in the context of formal theorem proving, which has access to a perfect verifier. We identify a degenerate rank bias in GRPO in which highly probable trajectories are reinforced and rare ones are neglected. This results in distribution sharpening: the model can solve some problems with fewer samples, but underperforms simply sampling more solutions from the original model. To overcome GRPO's rank bias we introduce unlikeliness reward, a simple method for explicitly up-weighting rare but correct solutions. We show that unlikeliness reward mitigates rank bias and improves pass@N across a large range of N in both synthetic and real theorem proving settings. We also uncover an unexpected link between rank bias and a seemingly mundane hyperparameter -- the number of updates per batch -- that leads to a second, complementary mitigation. We combine our insights into a revised GRPO training recipe for formal theorem proving, yielding an open pipeline that achieves competitive performance to DeepSeek-Prover-V1.5-RL on the miniF2F-test benchmark. We release our implementation at https://github.com/AndreHe02/rewarding-unlikely-release