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Recent research across mathematical problem solving, proof assistant programming and multimodal jailbreaking documents a striking finding: when (multimodal) language model tackle a suite of tasks with multiple attempts per task -- succeeding if any attempt is correct -- then the negative log of the average success rate scales a power law in the number of attempts.In this work, we identify an apparent puzzle: a simple mathematical calculation predicts that on each problem, the failure rate should fall exponentially with the number of attempts.We confirm this prediction empirically, raising a question: from where does aggregate polynomial scaling emerge?We then answer this question by demonstrating per-problem exponential scaling can be made consistent with aggregate polynomial scaling if the distribution of single-attempt success probabilities is heavy tailed such that a small fraction of tasks with extremely low success probabilities collectively warp the aggregate success trend into a power law - even as each problem scales exponentially on its own.We further demonstrate that this distributional perspective explains previously observed deviations from power law scaling, and provides a simple method for forecasting the power law exponent with an order of magnitude lower relative error, or equivalently, ∼2−4 orders of magnitude less inference compute.Overall, our work contributes to a better understanding of how neural language model performance improves with scaling inference compute and the development of scaling-predictable evaluations of (multimodal) language models.
From extracting features to generating text, the outputs of large language models (LLMs) typically rely on their final layers, following the conventional wisdom that earlier layers capture only low-level cues. However, our analysis shows that intermediate layers can encode even richer representations, often improving performance on a wide range of downstream tasks. To explain and quantify these hidden-layer properties, we propose a unified framework of representation quality metrics based on information theory, geometry, and invariance to input perturbations. Our framework highlights how each model layer balances information compression and signal preservation, revealing why mid-depth embeddings can exceed the last layer’s performance. Through extensive experiments on 32 text-embedding tasks across various architectures (transformers, state-space models) and domains (language, vision), we demonstrate that intermediate layers consistently provide stronger features, challenging the standard view on final-layer embeddings and opening new directions on using mid-layer representations for more robust and accurate representations.
We propose an online adaptive sampling algorithm for solving stochastic nonsmooth difference-of-convex (DC) problems under time-varying distributions. At each iteration, the algorithm relies solely on data generated from the current distribution and employs distinct adaptive sampling rates for the convex and concave components of the DC function, a novel design guided by our theoretical analysis. We show that, under proper conditions on the convergence of distributions, the algorithm converges subsequentially to DC critical points almost surely. Furthermore, the sample size requirement of our proposed algorithm matches the results achieved in the smooth case or when a measurable subgradient selector is available, both under static distributions. A key element of this analysis is the derivation of a novel O(√p/n) pointwise convergence rate (modulo logarithmic factors) for the sample average approximation of subdifferential mappings, where p is the dimension of the variable and n is the sample size -- a result of independent interest. Numerical experiments confirm that the proposed algorithm is both efficient and effective for addressing stochastic nonsmooth problems.
The Multimodal Large Language Model (MLLM) is currently experiencing rapid growth, driven by the advanced capabilities of language-based LLMs. Unlike their specialist predecessors, existing MLLMs are evolving towards a Multimodal Generalist paradigm. Initially limited to understanding multiple modalities, these models have advanced to not only comprehend but also generate across modalities. Their capabilities have expanded from coarse-grained to fine-grained multimodal understanding and from supporting singular modalities to accommodating a wide array of or even arbitrary modalities. To assess the capabilities of various MLLMs, a diverse array of benchmark test sets has been proposed. This leads to a critical question: *Can we simply assume that higher performance across tasks indicates a stronger MLLM capability, bringing us closer to human-level AI?*We argue that the answer is not as straightforward as it seems. In this project, we introduce an evaluation framework to delineate the capabilities and behaviors of current multimodal generalists. This framework, named **General-Level**, establishes 5-scale levels of MLLM performance and generality, offering a methodology to compare MLLMs and gauge the progress of existing systems towards more robust multimodal generalists and, ultimately, towards AGI (Artificial General Intelligence). Central to our framework is the use of **Synergy** as the evaluative criterion, categorizing capabilities based on whether MLLMs preserve synergy across comprehension and generation, as well as across multimodal interactions.To evaluate the comprehensive abilities of various generalists, we present a massive multimodal benchmark, **General-Bench**, which encompasses a broader spectrum of skills, modalities, formats, and capabilities, including over 700 tasks and 325,800 instances. The evaluation results that involve over 100 existing state-of-the-art MLLMs uncover the capability rankings of generalists, highlighting the challenges in reaching genuine AI. We expect this project to pave the way for future research on next-generation multimodal foundation models, providing a robust infrastructure to accelerate the realization of AGI.Project Page: https://generalist.top/,Leaderboard: https://generalist.top/leaderboard/,Benchmark: https://huggingface.co/General-Level/.
We consider the basic statistical challenge of designing an "all-purpose" mean estimation algorithm that is recommendable across a variety of settings and models.Recent work by [Lee and Valiant 2022] introduced the first 1-d mean estimator whose error in the standard finite-variance+i.i.d. setting is optimal even in its constant factors; experimental demonstration of its good performance was shown by [Gobet et al. 2022].Yet, unlike for classic (but not necessarily practical) estimators such as median-of-means and trimmed mean, this new algorithm lacked proven robustness guarantees in other settings, including the settings of adversarial data corruption and heavy-tailed distributions with infinite variance.Such robustness is important for practical use cases.This raises a research question: is it possible to have a mean estimator that is robust, *without* sacrificing provably optimal performance in the standard i.i.d. setting?In this work, we show that Lee and Valiant's estimator is in fact an "all-purpose" mean estimator by proving:(A) It is robust to an η-fraction of data corruption, even in the strong contamination model; it has optimal estimation error O(σ√η) for distributions with variance σ2.(B) For distributions with finite zth moment, for z∈(1,2), it has optimal estimation error, matching the lower bounds of [Devroye et al. 2016] up to constants.We further show (C) that outlier robustness for 1-d mean estimators in fact implies neighborhood optimality, a notion of beyond worst-case and distribution-dependent optimality recently introduced by [Dang et al. 2023].Previously, such an optimality guarantee was only known for median-of-means, but now it holds also for all estimators that are simultaneously *robust* and *sub-Gaussian*, including Lee and Valiant's, resolving a question raised by Dang et al.Lastly, we show (D) the asymptotic normality and efficiency of Lee and Valiant's estimator, as further evidence for its performance across many settings.
The peer review process in major artificial intelligence (AI) conferences faces unprecedented challenges with the surge of paper submissions (exceeding 10,000 submissions per venue), accompanied by growing concerns over review quality and reviewer responsibility. This position paper argues for **the need to transform the traditional one-way review system into a bi-directional feedback loop where authors evaluate review quality and reviewers earn formal accreditation, creating an accountability framework that promotes a sustainable, high-quality peer review system.** The current review system can be viewed as an interaction between three parties: the authors, reviewers, and system (i.e., conference), where we posit that all three parties share responsibility for the current problems. However, issues with authors can only be addressed through policy enforcement and detection tools, and ethical concerns can only be corrected through self-reflection. As such, this paper focuses on reforming reviewer accountability with systematic rewards through two key mechanisms: (1) a two-stage bi-directional review system that allows authors to evaluate reviews while minimizing retaliatory behavior, (2) a systematic reviewer reward system that incentivizes quality reviewing. We ask for the community's strong interest in these problems and the reforms that are needed to enhance the peer review process.
Inspired by the principle of deliberate practice in human learning, we propose Deliberate Practice for Synthetic Data Generation (DP), a novel framework that improves sample efficiency through dynamic synthetic data generation. Prior work has shown that scaling synthetic data is inherently challenging, as naively adding new data leads to diminishing returns. To address this, pruning has been identified as a key mechanism for improving scaling, enabling models to focus on the most informative synthetic samples. Rather than generating a large dataset and pruning it afterward, DP efficiently approximates the direct generation of informative samples. We theoretically show how training on challenging, informative examples improves scaling laws and empirically validate that DP achieves better scaling performance with significantly fewer training samples and iterations. On ImageNet-100, DP generates 3.4x fewer samples and requires six times fewer iterations, while on ImageNet-1k, it generates 8x fewer samples with a 30% reduction in iterations, all while achieving superior performance compared to prior work.
Machine learning is increasingly used in government programs to identify and support the most vulnerable individuals, prioritizing assistance for those at greatest risk over optimizing aggregate outcomes. This paper examines the welfare impacts of prediction in equity-driven contexts, and how they compare to other policy levers, such as expanding bureaucratic capacity. Through mathematical models and a real-world case study on long-term unemployment amongst German residents, we develop a comprehensive understanding of the relative effectiveness of prediction in surfacing the worst-off. Our findings provide clear analytical frameworks and practical, data-driven tools that empower policymakers to make principled decisions when designing these systems.
Normalizing Flows (NFs) are likelihood-based models for continuous inputs. They have demonstrated promising results on both density estimation and generative modeling tasks, but have received relatively little attention in recent years. In this work, we demonstrate that NFs are more powerful than previously believed. We present TarFlow: a simple and scalable architecture that enables highly performant NF models. TarFlow can be thought of as a Transformer-based variant of Masked Autoregressive Flows (MAFs): it consists of a stack of autoregressive Transformer blocks on image patches, alternating the autoregression direction between layers. TarFlow is straightforward to train end-to-end, and capable of directly modeling and generating pixels. We also propose three key techniques to improve sample quality: Gaussian noise augmentation during training, a post training denoising procedure, and an effective guidance method for both class-conditional and unconditional settings. Putting these together, TarFlow sets new state-of-the-art results on likelihood estimation for images, beating the previous best methods by a large margin, and generates samples with quality and diversity comparable to diffusion models, for the first time with a stand-alone NF model. We make our code available at https://github.com/apple/ml-tarflow.
Neural networks trained to solve modular arithmetic tasks exhibit grokking, a phenomenon where the test accuracy starts improving long after the model achieves 100% training accuracy in the training process. It is often taken as an example of "emergence", where model ability manifests sharply through a phase transition. In this work, we show that the phenomenon of grokking is not specific to neural networks nor to gradient descent-based optimization. Specifically, we show that this phenomenon occurs when learning modular arithmetic with Recursive Feature Machines (RFM), an iterative algorithm that uses the Average Gradient Outer Product (AGOP) to enable task-specific feature learning with general machine learning models. When used in conjunction with kernel machines, iterating RFM results in a fast transition from random, near zero, test accuracy to perfect test accuracy. This transition cannot be predicted from the training loss, which is identically zero, nor from the test loss, which remains constant in initial iterations. Instead, as we show, the transition is completely determined by feature learning: RFM gradually learns block-circulant features to solve modular arithmetic. Paralleling the results for RFM, we show that neural networks that solve modular arithmetic also learn block-circulant features. Furthermore, we present theoretical evidence that RFM uses such block-circulant features to implement the Fourier Multiplication Algorithm, which prior work posited as the generalizing solution neural networks learn on these tasks. Our results demonstrate that emergence can result purely from learning task-relevant features and is not specific to neural architectures nor gradient descent-based optimization methods. Furthermore, our work provides more evidence for AGOP as a key mechanism for feature learning in neural networks.
Even when massively overparameterized, deep neural networks show a remarkable ability to generalize. Research on this phenomenon has focused on generalization within distribution, via smooth interpolation. Yet in some settings neural networks also learn to extrapolate to data far beyond the bounds of the original training set, sometimes even allowing for infinite generalization, implying that an algorithm capable of solving the task has been learned. Here we undertake a case study of the learning dynamics of recurrent neural networks trained on the streaming parity task in order to develop an effective theory of algorithm development. The streaming parity task is a simple but nonlinear task defined on sequences up to arbitrary length. We show that, with sufficient finite training experience, RNNs exhibit a phase transition to perfect infinite generalization. Using an effective theory for the representational dynamics, we find an implicit representational merger effect which can be interpreted as the construction of a finite automaton that reproduces the task. Overall, our results disclose one mechanism by which neural networks can generalize infinitely from finite training experience.
As multimodal large language models (MLLMs) advance, MLLM-based virtual agents have demonstrated remarkable performance. However, existing benchmarks face significant limitations, including uncontrollable task complexity, extensive manual annotation, and a lack of multidimensional evaluation. In response to these challenges, we introduce OmniBench, a self-generating, graph-based benchmark with an automated pipeline for synthesizing tasks of controllable complexity through subtask composition. To evaluate the diverse capabilities of virtual agents on the graph, we further present OmniEval, a multidimensional evaluation framework that includes subtask-level evaluation, graph-based metrics, and comprehensive tests across 10 capabilities. Our synthesized dataset contains 36k graph-structured tasks across 20 scenarios, achieving a 91% human acceptance rate. Training on our graph-structured data shows that it improves generalization across environments. We conduct multidimensional evaluations for virtual agents, revealing their performance across various capabilities and paving the way for future advancements. Our project is available at https://omni-bench.github.io.
We present rStar-Math to demonstrate that small language models (SLMs) can rival or even surpass the math reasoning capability of OpenAI o1, without distillation from superior models. rStar-Math achieves this by exercising ``deep thinking'' through Monte Carlo Tree Search (MCTS), where a math policy SLM performs test-time search guided by an SLM-based process reward model. rStar-Math introduces three innovations to tackle the challenges in training the two SLMs: (1) a novel code-augmented CoT data synthesis method, which performs extensive MCTS rollouts to generate step-by-step verified reasoning trajectories used to train the policy SLM; (2) a novel process reward model training method that avoids na\"ive step-level score annotation, yielding a more effective process preference model (PPM); (3) a self-evolution recipe in which the policy SLM and PPM are built from scratch and iteratively evolved to improve reasoning capabilities. Through 4 rounds of self-evolution with millions of synthesized solutions for 747k math problems, rStar-Math boosts SLMs' math reasoning to state-of-the-art levels. On MATH benchmark, it improves Qwen2.5-Math-7B from 58.8% to 90.0%, surpassing o1-preview by +4.5%. On the USA Math Olympiad (AIME), rStar-Math solves an average of 53.3% (8/15) of problems, ranking among the top 20% of the brightest high school math students. Code and data are available at https://github.com/microsoft/rStar.
Classic ensembles generalize better than any single component model. In contrast, recent empirical studies find that modern ensembles of (overparameterized) neural networks may not provide any inherent generalization advantage over single but larger neural networks. This paper clarifies how modern overparameterized ensembles differ from their classic underparameterized counterparts, using ensembles of random feature (RF) regressors as a basis for developing theory. In contrast to the underparameterized regime, where ensembling typically induces regularization and increases generalization, we prove with minimal assumptions that infinite ensembles of overparameterized RF regressors become pointwise equivalent to (single) infinite-width RF regressors, and finite width ensembles rapidly converge to single models with the same parameter budget. These results, which are exact for ridgeless models and approximate for small ridge penalties, imply that overparameterized ensembles and single large models exhibit nearly identical generalization. We further characterize the predictive variance amongst ensemble members, demonstrating that it quantifies the expected effects of increasing capacity rather than capturing any conventional notion of uncertainty. Our results challenge common assumptions about the advantages of ensembles in overparameterized settings, prompting a reconsideration of how well intuitions from underparameterized ensembles transfer to deep ensembles and the overparameterized regime.
Leveraging Multi-modal Large Language Models (MLLMs) to create embodied agents offers a promising avenue for tackling real-world tasks. While language-centric embodied agents have garnered substantial attention, MLLM-based embodied agents remain underexplored due to the lack of comprehensive evaluation frameworks. To bridge this gap, we introduce EmbodiedBench, an extensive benchmark designed to evaluate vision-driven embodied agents.EmbodiedBench features: (1) a diverse set of 1,128 testing tasks across four environments, ranging from high-level semantic tasks (e.g., household) to low-level tasks involving atomic actions (e.g., navigation and manipulation); and (2) six meticulously curated subsets evaluating essential agent capabilities like commonsense reasoning, complex instruction understanding, spatial awareness, visual perception, and long-term planning.Through extensive experiments, we evaluated 24 leading proprietary and open-source MLLMs within EmbodiedBench. Our findings reveal that: MLLMs excel at high-level tasks but struggle with low-level manipulation, with the best model, GPT-4o, scoring only 28.9% on average. EmbodiedBench provides a multifaceted standardized evaluation platform that not only highlights existing challenges but also offers valuable insights to advance MLLM-based embodied agents. Our code and dataset are available at [https://embodiedbench.github.io](https://embodiedbench.github.io).
In recent years, masked diffusion models (MDMs) have emerged as a promising alternative approach for generative modeling over discrete domains. Compared to autoregressive models (ARMs), MDMs trade off complexity at training time with flexibility at inference time. At training time, they must learn to solve an exponentially large number of infilling problems, but at inference time, they can decode tokens in essentially arbitrary order. In this work we closely examine these two competing effects. On the training front, we theoretically and empirically demonstrate that MDMs indeed train on computationally intractable subproblems compared to their autoregressive counterparts. On the inference front, we show that a suitable strategy for adaptively choosing the token decoding order significantly enhances the capabilities of MDMs, allowing them to sidestep hard subproblems. On logic puzzles like Sudoku, we show that adaptive inference can boost solving accuracy in pretrained MDMs from <7\% to ≈90\%, even outperforming ARMs that were explicitly trained via teacher forcing to learn the right order of decoding.
Large Language Models are typically trained with next-turn rewards, limiting their ability to optimize for long-term interaction. As a result, they often respond passively to ambiguous or open-ended user requests, failing to help users reach their ultimate intents and leading to inefficient conversations. To address these limitations, we introduce CollabLLM, a novel and general training framework that enhances multiturn human-LLM collaboration. Its key innovation is a collaborative simulation that estimates the long-term contribution of responsesusing Multiturn-aware Rewards. By reinforcement fine-tuning these rewards, CollabLLM goes beyond responding to user requests, and actively uncovers user intent and offers insightful suggestions—a key step towards more human-centered AI. We also devise a multiturn interaction benchmark with three challenging tasks such as document creation. CollabLLM significantly outperforms our baselines with averages of 18.5% higher task performance and 46.3% improved interactivity by LLM judges. Finally, we conduct a large user study with 201 judges, where CollabLLM increases user satisfaction by 17.6% and reduces user spent time by 10.4%.
A key task in certain democratic processes is to produce a concise slate of statements that proportionally represents the full spectrum of user opinions. This task is similar to committee elections, but unlike traditional settings, the candidate set comprises all possible statements of varying lengths, and so it can only be accessed through specific queries. Combining social choice and large language models, prior work has approached this challenge through a framework of generative social choice. We extend the framework in two fundamental ways, providing theoretical guarantees even in the face of approximately optimal queries and a budget limit on the overall length of the slate. Using GPT-4o to implement queries, we showcase our approach on datasets related to city improvement measures and drug reviews, demonstrating its effectiveness in generating representative slates from unstructured user opinions.
Optimal transport (OT) has enjoyed great success in machine learning as a principled way to align datasets via a least-cost correspondence, driven in large part by the runtime efficiency of the Sinkhorn algorithm (Cuturi, 2013). However, Sinkhorn has quadratic space complexity in the number of points, limiting scalability to larger datasets. Low-rank OT achieves linear-space complexity, but by definition, cannot compute a one-to-one correspondence between points. When the optimal transport problem is an assignment problem between datasets then an optimal mapping, known as the _Monge map_, is guaranteed to be a bijection. In this setting, we show that the factors of an optimal low-rank coupling co-cluster each point with its image under the Monge map. We leverage this invariant to derive an algorithm, _Hierarchical Refinement_ (`HiRef`), that dynamically constructs a multiscale partition of each dataset using low-rank OT subproblems, culminating in a bijective coupling. Hierarchical Refinement uses linear space and has log-linear runtime, retaining the space advantage of low-rank OT while overcoming its limited resolution. We demonstrate the advantages of Hierarchical Refinement on several datasets, including ones containing over a million points, scaling full-rank OT to problems previously beyond Sinkhorn's reach.
Multimodal retrieval-augmented generation (RAG) plays a crucial role in domains such as knowledge-based visual question answering (KB-VQA), where models should effectively integrate additional knowledge to generate a response. However, existing vision and language models (VLMs) are not inherently designed for context-augmented generation, limiting their effectiveness in such tasks. While synthetic data generation has recently gained attention for training large VLMs, its application for context-augmented generation remains underexplored. To address this gap, we introduce SKVQA, a large-scale synthetic multimodal dataset containing over 2 million visual question-answer pairs, each associated with external knowledge sources to determine the final answer. Compared to previous datasets, SKVQA exhibits 11× more unique questions, greater domain diversity, and a broader spectrum of image sources. Through human evaluations, we confirm the high quality of the generated question-answer pairs and their contextual relevance. Extensive experiments show that SKVQA serves both as a challenging benchmark for knowledge-based VQA and as an effective training resource for adapting generative multimodal models to context-augmented generation. Our results further indicate that models trained on SKVQA demonstrate enhanced generalization in both context-aware VQA and multimodal RAG settings.
We introduce in-context denoising, a task that refines the connection between attention-based architectures and dense associative memory (DAM) networks, also known as modern Hopfield networks. Using a Bayesian framework, we show theoretically and empirically that certain restricted denoising problems can be solved optimally even by a single-layer transformer. We demonstrate that a trained attention layer processes each denoising prompt by performing a single gradient descent update on a context-aware DAM energy landscape, where context tokens serve as associative memories and the query token acts as an initial state. This one-step update yields better solutions than exact retrieval of either a context token or a spurious local minimum, providing a concrete example of DAM networks extending beyond the standard retrieval paradigm. Overall, this work solidifies the link between associative memory and attention mechanisms first identified by Ramsauer et al., and demonstrates the relevance of associative memory models in the study of in-context learning.
Understanding neural network training dynamics at scale is an important open problem. Although realistic model architectures, optimizers, and data interact in complex ways that make predictive theory challenging, we show that compute-optimally trained models exhibit remarkably precise collective regularities. Specifically, loss curves from models of varying sizes collapse onto a single universal curve when training compute and loss are normalized to unity at the end of training. With learning rate decay, discrepancies between normalized curves fall below the noise floor of individual models' loss curves across random seeds, yielding an exceptionally tight collapse we term "supercollapse." We observe supercollapse across learning rate schedules, datasets, and architectures, including transformers trained on next-token prediction. This collapse breaks down when hyperparameters are scaled suboptimally, providing a practical indicator of proper scaling. We explain these phenomena by connecting collapse to the power-law structure in typical neural scaling laws, and analyzing a simple but effective model of SGD noise dynamics that accurately captures how learning rate schedules deform loss curves away from power laws while preserving universality, and why learning rate decay suppresses variance to enable supercollapse.
Detecting AI-generated images (AIGIs), such as natural images or face images, has become increasingly important yet challenging. In this paper, we start from a new perspective to excavate the reason behind the failure generalization in AIGI detection, named the asymmetry phenomenon, where a naively trained detector tends to favor overfitting to the limited and monotonous fake patterns, causing the feature space to become highly constrained and low-ranked, which is proved seriously limiting the expressivity and generalization. One potential remedy is incorporating the pre-trained knowledge within the vision foundation models (higher-ranked) to expand the feature space, alleviating the model's overfitting to fake. To this end, we employ Singular Value Decomposition (SVD) to decompose the original feature space into two orthogonal subspaces. By freezing the principal components and adapting only the remained components, we preserve the pre-trained knowledge while learning fake patterns. Compared to existing full-parameters and LoRA-based tuning methods, we explicitly ensure orthogonality, enabling the higher rank of the whole feature space, effectively minimizing overfitting and enhancing generalization. We finally identify a crucial insight: our method implicitly learns a vital prior that fakes are actually derived from the real, indicating a hierarchical relationship rather than independence. Modeling this prior, we believe, is essential for achieving superior generalization. Our codes are publicly available at https://github.com/YZY-stack/Effort-AIGI-Detection.
Many applications in machine learning involve data represented as probability distributions. The emergence of such data requires radically novel techniques to design tractable gradient flows on probability distributions over this type of (infinite-dimensional) objects. For instance, being able to flow labeled datasets is a core task for applications ranging from domain adaptation to transfer learning or dataset distillation. In this setting, we propose to represent each class by the associated conditional distribution of features, and to model the dataset as a mixture distribution supported on these classes (which are themselves probability distributions), meaning that labeled datasets can be seen as probability distributions over probability distributions. We endow this space with a metric structure from optimal transport, namely the Wasserstein over Wasserstein (WoW) distance, derive a differential structure on this space, and define WoW gradient flows. The latter enables to design dynamics over this space that decrease a given objective functional. We apply our framework to transfer learning and dataset distillation tasks, leveraging our gradient flow construction as well as novel tractable functionals that take the form of Maximum Mean Discrepancies with Sliced-Wasserstein based kernels between probability distributions.
Recurrent neural networks (RNNs) are powerful models used widely in both machine learning and neuroscience to learn tasks with temporal dependencies and to model neural dynamics. However, despite significant advancements in the theory of RNNs, there is still limited understanding of their learning process and the impact of the temporal structure of data. Here, we bridge this gap by analyzing the learning dynamics of linear RNNs (LRNNs) analytically, enabled by a novel framework that accounts for task dynamics. Our mathematical result reveals four key properties of LRNNs: (1) Learning of data singular values is ordered by both scale and temporal precedence, such that singular values that are larger and occur later are learned faster. (2) Task dynamics impact solution stability and extrapolation ability. (3) The loss function contains an effective regularization term that incentivizes small weights and mediates a tradeoff between recurrent and feedforward computation. (4) Recurrence encourages feature learning, as shown through a novel derivation of the neural tangent kernel for finite-width LRNNs. As a final proof-of-concept, we apply our theoretical framework to explain the behavior of LRNNs performing sensory integration tasks. Our work provides a first analytical treatment of the relationship between the temporal dependencies in tasks and learning dynamics in LRNNs, building a foundation for understanding how complex dynamic behavior emerges in cognitive models.