LLM 理论进展

Chain-of-Thought reasoning has enabled large language models to achieve substantial performance gains on complex tasks. However, these gains come at the cost of dramatically increased token consumption. This raises a fundamental question: is every token in the reasoning trace equally valuable? We present a diagnostic and optimization framework grounded in a key empirical finding: the value of tokens within a CoT reasoning sequence is highly non-uniform, and this non-uniformity can be effectively characterized by token-level log probability signals. We show that normalized log probability helps distinguish core tokens, which carry structural and decisive reasoning content, from redundant tokens, which are exploratory, low-confidence filler that contributes less directly to the final answer. Building on these findings, we formulate the TokenProbe framework around two empirical findings and one claim: findings identify token value inequality first and then establish TokenProbe as a core-token proxy, and the claim introduces an efficient GRPO objective positing that selectively compressing redundant tokens can yield Pareto improvements in the accuracy-token efficiency space. Empirically, our method preserves reasoning quality while reducing the token usage by 76% of the baseline. Under matched reasoning-length budgets, we show that it can even outperform strong flagship baselines like Gemini-3.1-Pro. Homepage: https://runjia.tech/tokenprobe/.

People typically assume that LMs stably mature from pattern-matching parrots to generalizable intelligence during pre-training. We build a toy eval suite and show this mental model is wrong: throughout pre-training, LMs frequently and suddenly hop between parrot-like and intelligence-like computations. We call this mode-hopping. Across our suite, LMs suddenly latch onto memorized or in-context patterns instead of in-context learning, use System 1 instead of System 2 thinking, pick up what sounds true instead of what is true, fail at multi-hop persona QA, out-of-context reasoning, and emergent misalignment — then just as suddenly revert and generalize. Mode-hopping is not explained by standard optimization dynamics: it is locally stable and cannot be fixed by checkpoint averaging. We instead think of it as a capacity allocation problem: in a capacity-bounded model, generalizable circuits must compete with the shallow ones learned early in training, and the data in each pre-training window may decide which circuits win. Our suite provides a new efficient lens on generalization. We demonstrate two concrete applications: (i) select intermediate pre-training checkpoints that strongly generalize reasoning and alignment, better than the final pre- or mid-training checkpoints, and (ii) select pre-training data that controls and stabilizes generalization dynamics.

Large language models can memorize private or harmful information, motivating machine unlearning methods that remove targeted knowledge while preserving other capabilities. However, existing evaluations rely primarily on behavioral benchmarks, which assess whether a model appears to forget but provide limited insight into how unlearning changes the model or affects related knowledge. We introduce TRIAGE (Tripartite Representation-internal Introspection for Adjacency Gap Evaluation), a benchmark-agnostic evaluation framework for characterizing these changes. TRIAGE uses diagonal approximations of the Fisher information and Hessian to measure changes in parameter sensitivity and local curvature, and utilizes a Forget / Adjacent-Retain / Generic-Retain partition to quantify an adjacency gap in semantically related knowledge. Based on the magnitude and distribution of these changes, TRIAGE further classifies each algorithm's update as no-op, partially localized, collateral dominant, or globally destructive. Across 12 unlearning methods, four language models, and the WMDP, TOFU, and MUSE benchmarks, we find that methods with similar behavioral forgetting can produce substantially different internal changes and patterns of collateral damage. These signatures also vary across models and benchmarks, indicating that the effects of unlearning are not determined solely by the unlearning algorithm. TRIAGE can be applied alongside existing unlearning benchmarks to complement behavioral evaluation with a model-internal view of how unlearning reshapes the model's parameter space and affects retained knowledge.

Transformers process information strictly forward through depth, preventing deeper computation from revisiting and refining earlier representations. To augment the standard forward pass, existing approaches either re-execute depth, incurring additional computation, or modify the residual stream using predefined directions, limiting their instance-level adaptation. Recently, inference-time feedback offers a direct mechanism for recycling endogenously produced computation by writing deeper residual states back to earlier layers, yet what should be fed back remains unclear. We argue that the depth increment Delta, capturing newly accumulated computation between two layers, provides a more effective, composable, and scalable feedback signal than the full state. Building on this observation, we introduce ReFlux, a learnable feedback graph that dynamically selects and composes increment-carrying routes. ReFlux supports synchronous feedback to the same token and streaming feedback to subsequent tokens. Extensive experiments across various models, corpora, and benchmarks show that synchronous ReFlux consistently reduces perplexity across ten language-modeling corpora, and improves accuracy by 2.1-2.3 points, with gains reaching 4.7 points on multi-hop reasoning. Streaming ReFlux further retains most of these gains while preserving the base model's 1x theoretical backbone FLOPs. These results establish ReFlux as an efficient paradigm for unlocking the latent computational potential of LLMs, allowing them to revisit the same tokens with fresh representations. Code implementation can be found at https://github.com/gooogleshanghai/reflux.

In tri-modal visual question answering (VQA), auxiliary text is commonly used to complement visual and textual inputs, yet its reliability is often uncontrolled. While prior work studies modality conflicts in general, it remains unclear how different types of unreliable auxiliary text affect answer selection under fixed image-question-option contexts. In this work, we show that the most harmful auxiliary text is not necessarily the most factually incorrect, but the one that aligns with the question while contradicting the image and favoring a specific distractor, leading to systematic redirection of model predictions. To isolate this effect, we introduce the Textual Reliability Ladder, a controlled diagnostic protocol that decomposes auxiliary text along three axes: image consistency, question relevance, and option support. Across multiple datasets (ScienceQA, VCR, A-OKVQA, Causal-VidQA) and recent VLMs, we find that such distractor-supporting text induces the largest accuracy drops (up to 53.1%) and concentrates errors on specific incorrect options. To mitigate this failure mode, we propose a training-free inference-time intervention that explicitly counteracts this redirection effect via noise-stability steering and dynamic grounding, reducing redirected errors while largely preserving performance under faithful text. Our results highlight that auxiliary-text reliability must be understood at the decision level, rather than solely through factual correctness, and provide a practical pathway toward more robust tri-modal reasoning.

The ability of vision-language models (VLMs) to abstain from unanswerable questions is as important as their ability to answer answerable ones accurately. Recently, several benchmarks have emerged to evaluate and improve VLM abstention, but they have substantial limitations. First, samples often contain shortcut cues in images or questions that reveal answerability, while an explicit "unanswerable" option further prevents accurate assessment of spontaneous abstention. Second, as training data, they generally provide only binary labels without fine-grained explanations for deeper supervision. To address these limitations, we introduce Visual Answerability Diagnosis with Rationales (VAD-R), a benchmark constructed through a two-stage pipeline of shortcut filtering and quality verification to prevent answerability leakage. Each example is annotated with step-by-step rationales and causal evidence-gap labels. Evaluation of state-of-the-art open- and closed-source VLMs on VAD-R reveals limited spontaneous abstention, with average recall rates of only 11.4% and 16.3%, respectively. Probing analyses show that hidden-state representations in certain layers can effectively distinguish answerability, yet this distinction fails to manifest in final responses. Motivated by this observation, we introduce Rep2Act, a representation-to-action alignment method that translates latent answerability awareness into explicit abstention decisions. Rep2Act improves action accuracy on VAD-R from 56.67% to 86.33% for Qwen2.5-VL-3B and from 59.33% to 88.67% for Qwen2.5-VL-7B. On the out-of-distribution TUBench, Rep2Act achieves an average F1 score of 53.3% with only a 3B model, surpassing the closed-source GPT-4 Turbo and GPT-4o by 16.2% and 1.1%, respectively.

Persona prompts ask language models to answer as particular kinds of people. We test whether relationships learned from these effects predict responses to new questions and remain useful across models and prompts. Across 57 attributes, three behavioral domains, and seven pairs of open 7 to 9B checkpoints, persona effects can be predictable without being portable. Separate attribute and task gains improve prediction beyond shared scaling significantly in OLMo-3 and Qwen2.5, with the most robust evidence in OLMo-3. In that model, target refitting significantly outperforms gains borrowed from each of the other six pairs. Across model transfers, borrowed gains with one amplitude underperform shared scaling in most directions; allowing two target parameters removes the significant losses but yields no significant benefit over target shared scaling. After rewording, refitting significantly outperforms reuse with one amplitude in all six tested pairs, while changes of examples or country context often preserve reuse value. In the tested prompt transfers, regularized updates outperform both reuse strategies in median at 64 target questions per attribute. A separate survey comparison finds that selecting the more responsive checkpoint can worsen human fit; responsiveness is confounded with training status, and temperature calibration largely removes this cost but not errors in group ordering. Within the tested gain representation, apparent transfer can come from target calibration; source relationships must add predictive value beyond calibration and regularization. Code and data are available at https://github.com/thzva/persona-gain

Model merging efficiently combines specialized large language models (LLMs) without joint retraining, but can substantially alter expert routing in Mixture-of-Experts (MoE) models. Such routing drift is often interpreted as routing failure, raising a fundamental question that remains unclear: does routing drift after MoE merging actually indicate routing failure, and what evidence should justify repair? We investigate these questions across DeepSeekMoE, OLMoE, and Qwen3-MoE proposing a routing analysis toolkit for controlled counterfactual interventions and token-level analysis. By crossing source and merged router inputs and parameters, we attribute most expert reassignments to input shifts rather than parameter changes at the same layer. However, source-relative routing differences poorly predict next-token likelihood gains from source-route restoration, and different expert selections can produce directionally similar mixture outputs. We therefore operationalize routing failure as task loss recoverable under a specified routing intervention, with non-routing parameters fixed. These tests detect recoverable loss under deliberate router corruption, whereas source-route restoration does not establish reliable task benefits in the evaluated merged models. Motivated by these, we propose Selective Router Repair (SRR) as a case study, and find that source-specialist token-likelihood advantages do not reliably identify beneficial local corrections. Together, these findings show that routing drift alone is insufficient evidence of routing failure: source-informed corrections must be judged by their task-level intervention effects. The analysis toolkit and SRR code are released.

Repeated sampling can reveal a correct numerical answer without yielding either a reliable system output or a supported derivation. We present Coverage, Realization, and Validity Evidence (CRV), an evaluation protocol for sampled large language model (LLM) reasoning over formal geometry states. Coverage is answer availability, realization is readout accuracy on the frozen candidate pool, and validity evidence is a label-blinded critic judgment of derivational support rather than a proof certificate. CRV freezes each candidate pool before comparing readouts and analyzes covered failures by correct-answer multiplicity and within-problem discrimination. On HardShift441, a 441-problem set for which a reference solver leaves 406 problems unsolved, a LoRA-adapted Qwen2.5-7B generator obtains 24.2% average single-sample accuracy and 68.9% pass@16, whereas verifier-weighted self-consistency (WSC) reaches 38.0%. Readout accuracy is particularly low when the correct answer occurs only once or twice in the pool. In a separate constructed audit of 195 covered problems, the critic labels 12 correct-answer representatives as supported, 181 as refuted, and two as uncertain. These results show that coverage, realization, and validity evidence from the critic are distinct quantities and should be reported separately.

Transformer-based language models perform well on symbolic tasks, yet it remains unclear whether they learn generalizable rules or rely on statistical shortcuts. Mechanistic studies link algorithmic behavior to structured internal representations, motivating the hypothesis that robust reasoning benefits from separating values from the types that control their manipulation. Can making this separation an architectural primitive improve the learnability and generalization of logical mechanisms? We introduce STRAT (STratified Registers And Types), which partitions the residual stream into orthogonal Data and Type subspaces and uses Type-based attention and gating to govern Data transformations. Controlled arithmetic ablations identify three failure modes associated with data-control interference: the Linear Trap, Gradient Wall, and Open Gate Trap. Mechanistic analysis reveals interpretable logical structure, and in arithmetic, STRAT reduces median OOD error 35-fold relative to a Transformer baseline. On each of 11 datasets spanning 10 tasks, STRAT outperforms the Transformer baseline in mean accuracy, by 26 percentage points on average, with both models trained from 10 base examples per dataset using identical task-specific augmentation where applicable. Under distribution shift, STRAT's mean accuracy drops by only 2.39 percentage points, compared with 11.75 for the Transformer.

A two-hop question makes a language model retrieve twice: once to produce a bridge entity, and once to retrieve with it. Transformers resolve that entity in their early layers. Hybrid models replace most of the attention with a recurrent state, so where the key becomes usable, and where it is spent on an answer, is not known. Answering "Where is the ball?" from "the ball belongs to Alice" and "Alice is in the garden" turns on Alice, a name the question does not mention. A model could reach garden through Alice, the key it computed, or through where the fact sits in the prompt. Across twelve models, dense and hybrid, we move a hidden state from one story into another where the two routes lead to different places, and read off which place the model gives. An attention layer converts the key in every model we tested, and crossing that layer removes its usable effect, all of it where no attention follows. In sequential hybrids this makes the answer a handoff: recurrent layers carry the key forward, and attention spends it. Retrieval does not always stop there. Writing a different fact into the memory of the recurrent layers that follow the last attention layer can move the answer toward that fact, multiplying its odds by 1.3 to 2.7, and which hybrids do this is not settled by their architecture or training. That read is addressed by the key, not by position: recurrent state in a hybrid is not only a carrier, but a memory that later layers can query.

Language models (LMs) often hallucinate by committing to confident answers rather than abstaining, even when they do not have enough information to answer reliably. A large body of existing work mitigates hallucination through detection or abstention mechanisms, but leaves open how models internally arrive at the decision to commit or abstain in the first place. We study this decision through mechanistic analysis, framing hallucination as unsupported commitment: the model commits despite exhibiting signals of unanswerability. Using causal gating, we identify a Commit-Abstain Circuit (CAC), a sparse, causally localised subset of attention heads and MLP sublayers underlying this decision. Across ten LMs (3B-14B) from five families and three benchmarks, the CAC exhibits a recurring accumulate-yet-undercorrect pattern: commitment-promoting components build up commitment in earlier layers, while abstention-promoting components act later as corrective signals that are often insufficient to overturn the accumulated commitment. Building on this finding, a lightweight policy trained on CAC activations improves decision accuracy by 12.2 points over the model's intrinsic commit-abstain margin, reduces false abstentions by 2.5 times, transfers to unseen benchmarks, and extends to larger models (27B-35B). The CAC is both diagnostic, clarifying how models overcommit, and practical, enabling improved abstention decisions.

Multi-step agents are trained on flat action streams: SFT and RLVR weight every token uniformly and ignore the sub-procedures that recur across tasks, the hierarchy that lets humans plan top-down from reusable routines. This structure sits unused, and flat training uses each scarce trajectory less fully than its content allows. Recent agents do use that structure, but only as LLM-written skills in context, never in the weights, so their gains do not generalize beyond retrieval. We instead recover this hierarchy from the data itself and train on it, with no LLM calls. Following text tokenizers, which build a vocabulary by counting alone, we score action spans by reusability and merge canonicalized actions into a reusable eXperience tree (X-Tree). Each X-Tree node captures how a frequent and success-bearing skill is composed from sub-skills, guiding efficient generalization. We integrate X-Tree into three training settings: offline RL, with each node as a training instance; online RLVR, with an adaptive skill bonus; and on-policy self-distillation, with X-Tree as the self-teacher's privileged context. Across WebArena, ScienceWorld, and WebShop at three model scales, X-Tree improves over standard recipes at matched data and budget by up to 4.5% SR on WebArena, 5.8% SR on ScienceWorld and 4.1% success on WebShop. Matched analyses attribute the gains to the X-Tree structure and the three integrations.

Distillation enables student language models to acquire new capabilities from expert teachers. However, integrating knowledge from multi-teacher, multi-domain demonstrations into a single student remains challenging. We study supervised fine-tuning (SFT) in this setting, where students must acquire diverse capabilities while maintaining generalization beyond the training tasks. Our experiments reveal varying trade-offs between in-distribution learning and out-of-distribution generalization across SFT methods, motivating more explicit control over this balance. To this end, we propose soft-target fine-tuning (SoFT) to balance learning from teacher demonstrations with retaining the Base model's existing capabilities. SoFT sets a minimum target probability for each demonstrated token while making the smallest KL change to the Base distribution. The resulting objective couples learning from demonstrations with adaptively weighted regularization toward the Base model. We further use domain-specific gradient budgets to control this balance and determine a probability threshold for each trajectory. Experiments on mixed-domain reasoning and agentic tasks show that SoFT achieves the best overall performance among the compared methods, with improvements in both in-distribution capability acquisition and out-of-distribution generalization.

On-policy distillation (OPD) is a promising approach to language-model adaptation, aligning teacher supervision with the student's own generated trajectories. When adaptation prompts are distributed across clients, can this process benefit from federated collaboration? We study federated OPD and find that substantial collaboration gains can be obscured by learning-rate sensitivity: FedAvg can perform no better than independent local training at a small learning rate, yet recover a clear advantage at a larger rate. We explain this phenomenon through the student's dual role as learner and generator of future training data. An aggregation-induced optimization lag can delay access to useful teacher supervision, which in turn slows subsequent learning. Our theory establishes this aggregation--rollout feedback in a solvable model with a common optimum and stable updates, and identifies two coupled roles of learning rate: learning from current supervision and reaching future supervision. Guided by this analysis, we propose FedTOPS (Federated Teacher-guided On-Policy Scaling), which reuses teacher feedback on current trajectories to adapt the FedAvg update magnitude under clientwise predictive-change constraints. Across six mathematical reasoning benchmarks, FedTOPS improves macro Avg@8 over FedAvg by 4.56--14.57 percentage points across the evaluated student models and local learning rates.

*Reinforcement learning (RL)* can induce substantial reasoning capabilities in large language models (LLMs), but how much of this capability transfers across model scales, and how quickly, remains unclear. We study the scaling properties of *on-policy distillation (OPD)* across *weak-to-strong*, *same-base*, and *strong-to-weak* teacher--student setups. We find that early OPD training dynamics uniformly exhibit a regular *useful-transfer* regime, in which held-out accuracy (the *gold score*, $G$) rises approximately linearly in $d=\sqrt{\mathrm{KL}(π_θ\Vert π_{\mathrm{ref}})}$, the square root of token-level reverse KL divergence from the student initialization. In every observed weak-to-strong pair, the student's peak gold score exceeds its teacher's own, so a compact RL expert can transfer capability to a much larger student via OPD. To estimate OPD outcomes, we fit *power laws* for how $G_{\mathrm{peak}}$ and the slope of the useful-transfer regime scale with student and teacher parameter counts and with teacher gold score. These laws show that peak gold score improves with teacher scale only up to roughly the student's scale, and that at a matched gold score smaller teachers transfer better, so a teacher's score alone does not define its supervision value. We also study the scaling effects of two OPD variants, bootstrapping weak-to-strong OPD, and the degree of on-policy supervision.

LLM agents require reliable step-level evaluation to compare candidate branches and allocate computation effectively. However, lightweight evaluation remains challenging. External verifiers introduce additional inference cost, while agent-produced confidence or self-evaluation scores can be miscalibrated, especially when candidates are generated by heterogeneous agents. We propose Generalized Layer-wise Intrinsic Distributional Evaluation (GLIDE) for LLM agents. \textsc{GLIDE} derives intrinsic step evidence from layer-wise residual coherence, which measures whether local residual updates consistently support the global residual change induced by a candidate step. It calibrates this evidence against the recent score distribution of the generating agent and converts it into a pessimistic reward that jointly accounts for absolute residual evidence and agent-relative standing. The reward provides a cross-agent value signal for MCTS branch selection, while normalized predictive uncertainty guides adaptive branching. Experiments on multi-hop reasoning, sequential decision making, and symbolic logic show that \textsc{GLIDE} improves task performance, step-level ranking quality, and computational efficiency without external verifiers or task-specific supervision.

Large-momentum effective theory (LaMET) provides a first-principles framework for computing the $x$ dependence of light-cone parton distributions from lattice QCD. Over the past decade, theoretical and numerical advances have established a mature multi-stage workflow for systematic calculation of parton physics, although its implementation still requires expert judgment and substantial repeated effort. We present lamet-agent, an open-source large language model (LLM) agent framework that organizes this workflow into an executable, reproducible, and inspectable analysis pipeline. The present release supports collinear quark distributions and implements correlator analysis, renormalization, Fourier transformation, perturbative matching, continuum, physical pion mass and infinite-momentum extrapolations, and automated result review. We validate it on four end-to-end analyses: pion parton distribution functions in the gauge-invariant and Coulomb-gauge formulations, and pion and kaon distribution amplitudes, obtaining results consistent with the published calculations. Extensions to transverse-momentum-dependent distributions, generalized transverse-momentum-dependent distributions, and gluonic distribution functions are planned for subsequent releases.

Multi-agent systems built on large language models (LLMs) are largely homogeneous, as their agents behave alike even across distinct LLMs. We show that when such agents act concurrently without communication, they collide on targets they must split and diverge on targets they must take together, a double failure we term the symmetry trap. Theory of Mind (ToM), widely used for coordination without communication, cannot escape this trap, since homogeneous agents form the same prediction of one another and respond to it in the same way. We propose Theory of Scene (ToS), a training-free reasoning schema in which each agent reads its public role, the only difference between the agents, and the task context they all observe. Homogeneous agents thereby derive one division of labor, each taking the part its role fixes, which turns homogeneity from the cause of the trap into the cure. ToS reads the role together with the scene through role gating, which determines whether ownership overlaps or is already divided, and the task context through task coupling, which infers whether the team must converge on each target, divide it, or take its stages in turn. We evaluate on DivvyBench, a controlled environment we introduce, whose target types make an episode Competitive, Cooperative, or Mixed across Tabletop, Airspace, and Household scenarios, and on two established agentic benchmarks, GovSim and Overcooked. ToS outperforms all six baselines on every benchmark, and each baseline falls far behind it in at least one setting. Against ToM given the same inputs, ToS raises the DivvyBench success rate from 71.1% to 99.6%, the GovSim total gain from 207 to 400, and the Overcooked level-normalized throughput from 1.41 to 1.67.

LLM-based multi-agent systems (MASs) have shown strong potential for solving complex tasks, but most assume that task requirements are sufficiently specified before execution. In practice, user requests are often incomplete, and additional requirements may only become clear during reasoning, tool use, or execution. We refer to such problems as progressively specified tasks. To systematically study this setting, we introduce ProgSpec, a benchmark that evaluates final outputs against requirements explicitly stated in the initial request and additional requirements supported by the available task evidence. We further propose RepoMAS, an issue-driven multi-agent framework inspired by open-source project management. RepoMAS records newly discovered requirements, conflicts, and failures as structured Issues and uses them to revise the task specification and execution structure during problem solving. Across ProgSpec and five existing benchmarks, RepoMAS achieves the best performance. Further analyses show that its issue-driven revision and repository maintenance mechanisms consistently contribute to performance. These results highlight the importance of allowing MASs to revise not only how a task is solved, but also revise their explicit representation of task requirements during execution.

The memory usage and decoding latency of LLM inference grow rapidly with context length. To reduce these costs, key-value (KV) cache compression methods selectively retain cached states based on token importance or differences in attention patterns across heads. However, we discover that retrieval capability varies substantially with relative distance, even within the same attention head. To exploit this structure, we introduce Distance-KV, which learns a static KV retention pattern over the joint space of layers, attention heads, and relative distances. The pattern is learned offline with the language model frozen and reused across inputs to prune and compact the KV cache without online importance scoring. Across three backbone models and four long-context benchmarks, Distance-KV consistently achieves the best overall performance among competing KV cache compression methods, exceeding the strongest compression baseline by up to 9.3 points on RULER at 128K. On Llama-3.1-8B-Instruct at 128K, Distance-KV reduces KV cache memory by 65.4% and achieves a $1.66\times$ decoding speedup relative to Dense. Together, these results identify relative distance as an important structural dimension for understanding how LLMs retrieve information over long contexts and for designing more efficient inference methods.

Model-internal features can be studied through both their ability to identify a specified concept and their causal effect when manipulated, e.g., through steering or weight editing. A prominent approach to feature learning is Sparse Autoencoders (SAEs), which learn broad feature dictionaries whose relation to particular concepts is typically identified post hoc. However, many interpretability questions are instead hypothesis-driven and concern a concept specified in advance. We study this setting as targeted feature learning, where a single feature is constructed for such a predefined concept. We present a controlled comparison across three model signals (activation values, activation gradients, and parameter gradients) and two estimators (contrastive mean and a learned one-dimensional encoder-decoder), yielding six targeted methods, with CAA and GRADIEND as existing instances and four new methods covering the remaining combinations. We compare these methods against pretrained SAEs across 15 tasks and three language models, evaluating both detection and causal intervention. Across models, the strongest detection performance is achieved by contrastive activation value methods, whereas the strongest intervention performance is achieved by gradient-based methods. Overall, our results show that targeted feature quality depends jointly on the model signal and estimator, with detection and intervention capturing complementary properties.

On-policy reasoning models can learn from task rewards or teacher signals, but these sources differ in form and can favor conflicting updates, leaving unclear which should guide a given reasoning action. We introduce Expected Reasoning-Step Return (ERSR), which treats semantic reasoning steps as macro-actions and uses Monte Carlo student-policy rollouts to estimate the expected final task reward of student-generated and teacher-proposed actions in a common return space for step-level comparison. ERSR analysis reveals an outcome-dependent asymmetry: student actions are more beneficial than teacher replacements on successful trajectories, whereas teacher replacements become more beneficial on failed trajectories. We further show that student answer-probe gains track student-step ERSR utility and distinguish beneficial from harmful reasoning steps. Based on these findings, we propose Return-Referenced On-Policy Learning (R$^2$OPL), which reinforces student reasoning on successful trajectories and distills teacher signals on failed ones, while using group success rate for difficulty scaling and student-probe gains for step-level modulation. Experiments across reasoning benchmarks and teacher--student configurations show that R$^2$OPL consistently outperforms strong baselines. ERSR training dynamics further show that R$^2$OPL jointly exploits substantial utility from both reward- and teacher-side signals, whereas existing hybrids often leave substantial residual utility in one branch.

Large Language Model (LLM) pretraining performance is jointly shaped by three components of the training triplet: the optimizer, model architecture, and training data stream. However, how these components influence performance in distinct ways remains unclear. We take a first step toward isolating their effects by studying relative generalization. We introduce Relative Generalization Invariance (RGI), the invariance of the validation-loss difference between any two tokens across models. We show that RGI approximately holds across a wide range of optimizers and moderate architectural variations, suggesting that these choices induce an approximately uniform shift in token-wise losses. In contrast, changing the training data stream can substantially alter relative generalization. We further show that RGI cannot be explained by the neural tangent kernel or mean-field regimes alone and prove that it can emerge in an overparameterized quadratic model. Overall, our work identifies RGI as a new phenomenon in LLM pretraining that helps distinguish the effects of optimizers and architectures from those of training data.

We investigate how generalization and memorization develop along the learning trajectory of neural language models. Using controlled synthetic grammars, we examine both the geometry of representation space and model behaviour over training. We find that continuous regions of representation space not occupied by observed tokens become systematically structured from the earliest stages of learning, forming category-level geometric organization that supports generalization to unattested combinations. Generalization therefore emerges from the beginning of learning rather than only after extensive memorization. With prolonged training, larger models increasingly distinguish observed from unobserved grammatical combinations. At the same time, the continuous geometric structure supporting category-level generalization is gradually destructed. Together, these results suggest that neural language models initially learn through categorization-based generalization, followed by a gradual transition toward more exemplar-specific memorization.

We investigate the effectiveness of artificial intelligences (AI)-specifically large language models (LLMs)-relative to human scientists at high-level cognitive tasks in social science such as theory formulation, predictions of novel empirical results, and theory revision in response to new evidence. The research domain was academic discourse regarding gender and race inequality. Our findings, comparing 25 LLMs with 13 senior researchers and 60 doctoral scholars, reveal that the AIs outperformed most humans individually on most of the present tasks, while human theories were more diverse and exhibited greater gains in predictive accuracy from aggregation. AI-generated theories were more extensively elaborated, involving additional theoretical paths and latent variables, and were rated as higher quality than human theories by independent raters blinded to source. However, this theoretical complexity was in part ornamental, in that it was not associated with more accurate predictions about empirical patterns in data; in contrast, human scientists achieved greater predictive efficiency with simpler theories. The AIs were significantly more likely than human scientists to revise their theories to incorporate new evidence; human scientists updated their beliefs in a selective way that is sensitive to prior prediction errors. We speculate that the superior processing capacity of artificial intelligences makes them especially well-suited to tasks requiring grappling with complexity, but that the greater diversity of human ideas is essential to wise crowds and collective creativity.

Pre-training data poisoning of large language models is usually studied using targeted backdoors and their survival through safety post-training, which leaves open a more basic question: how does a model's clean data performance degrade as the poison rate $\varepsilon$ grows? Motivated by our controlled pre-training runs of OLMo-style models, in which the relative clean data validation perplexity increase $Δ$ between poisoned and clean models matched in architecture, token budget, and optimization schedule is well fit by a power law $Δ\approx C\varepsilon^{a}$ with a non-integer exponent, we ask what such a law requires theoretically. We first prove an analyticity barrier: whenever the contaminated objective depends analytically on $\varepsilon$ around a nondegenerate clean data optimum, $Δ$ is generically quadratic in $\varepsilon$, so a generic non-integer exponent is a signature of genuinely singular structure. We then supply that structure in solvable truncated ridge regression with heavy-tailed covariates, controlled by $q_\star$, and a label-shift poisoning. Our central result is that the excess risk scaling exponent depending on the order of limits: in the higher dimensional proportional regime it is $ε^{q_\star/(q_\star+2)}$, whereas taking the ample-data limit first gives $ε^{2-2/q_\star}$, and the limits do not commute. We confirm this prediction through several numerical simulations. Finally, we argue that finite training time plays the role of truncation on the curvature spectrum in local LLM pre-training, deriving the observed scaling law under heavy tailed inverse curvature spectrum as a modeling hypothesis.

Transformers and state-space models (SSMs) are two prominent sequential learning architectures, yet their comparison remains largely empirical and existing theoretical analyses are typically task-specific or architecturally restricted. In this paper, we develop belief geometry, a unified analytical framework for comparing the representational capabilities of broad classes of attention and SSMs. Starting from a generalized formulation of in-context linear regression and using cumulative Bayes regret as our measure, we abstract three capabilities required by many sequential learning problems in our belief geometry: evidence assembly, belief maintenance, and addressing. We then study three cases of our formulation that isolate these capabilities and yield sharp architectural lessons: For belief maintenance, SSMs attain the optimal regret over stationary aggregation kernels; for positional assembly, SSMs have a memory advantage; and for content addressing, softmax attention has an exponential width advantage over sigmoid-selective SSMs. Experiments with LLaMA-type Transformers and Mamba-2 show that these architectural insights extend beyond our analytically tractable classes and linear-regression testbed.

The neural scaling law relating longer training to better performance through a power law is central to today's large language models (LLMs), yet its origin remains debated. One recent proposal is that power laws can emerge from the strong non-linearity of a single softmax head learning peaked distributions. What happens with multiple softmax functions, as in LLMs, is unclear. Here, we show through toy models that any softmax learning peaked distributions, regardless of its position in the model, can develop logit magnitudes that grow in a power law with exponent $1/3$, becoming a training bottleneck whose loss contribution decays as a power law with the same exponent $1/3$. The overall loss therefore obeys $1/3$ scaling whenever at least one softmax learns peaked distributions. We confirm that many softmax functions in LLMs learn peaked distributions and that LLM loss scaling matches this $1/3$ prediction. Moreover, logit growth dynamics reveal that attention heads, rather than the language modeling head, are the bottleneck likely driving the $1/3$ loss scaling in LLMs. Attention trying to concentrate on specific information, which is the heart of Transformers, may therefore also be the heart of the neural scaling law of training.

Prompt engineering has become an indispensable tool for using large language models (LLMs), turning LLMs into task-specific experts without changing their weights. Despite notable theoretical advances in prompt engineering, the theory for the more practical hard or discrete prompts is largely open. In this paper, we try to fill this gap either by providing a complete solution or by making substantial progress on the three core theoretical questions regarding hard prompts. First, we show that determining the existence of a hard prompt for a transformer to solve a downstream task is NP-complete and that finding an optimal hard prompt is NP-hard, which is the first computational complexity result for hard prompting, as far as we know. Second, we show that, unlike soft or continuous prompts, hard prompts have essential limitations: hard prompts are not complete; short hard prompts do not significantly enhance the ability of transformers; and long hard prompts exhibit the "prompt dominating answer phenomenon," meaning that, with high probability, the same answer is given for all queries of the same length. On the other hand, linear hard prompts do not have the limitations of short or long prompts. Third, we provide a tight bound on the size of the task in terms of the prompt length for the performance of prompts on the finite task to generalize to the entire data distribution, leading to a necessary and sufficient condition for generalizability. This is the first result on generalization for prompting, as far as we know. Our findings not only offer the first theoretical insights into hard prompts but also provide provably reliable practical guidance for real-world LLM usage.