Reinforcement learning (RL), particularly RL with Verifiable Rewards (RLVR), has recently emerged as a central paradigm for enhancing large language models' (LLMs) reasoning abilities, demonstrating remarkable effectiveness across reasoning tasks. Recent studies suggest that high-entropy tokens play an exceptionally important role in model training, since training with only the highest 20% entropy tokens yields significant performance gains. However, why such high-entropy tokens are beneficial remains insufficiently understood. In this work, we find that although high-entropy tokens within one answer tend to correlate with large gradient magnitude, entropy alone fails to consistently reflect token importance across different answers, considering the variations in the answer-level reward signals. Based on this observation, we introduce the Gradient Magnitude-based Token Selection (GMTS) method to quantify token importance, which leverages the entropy-gradient connection to approximate gradient-magnitude rankings for token selection. We find that training on the top 20% tokens ranked by GMTS consistently outperforms entropy-based token selection across three reasoning domains and various model sizes, suggesting that GMTS provides a more fine-grained estimate of token contribution for RLVR training.
Large Language Models (LLMs) trained on extensive scientific research are increasingly integrated as assistants for scientific discovery. However, most research papers omit the fine-grained cognitive process of examining constraints, failed alternatives, and iterative decisions required to achieve the desired goal. Such cognitive processes are vital for real-world scientists working toward specific goals under constraints. In this paper, we show that LLMs, when trained to produce such cognitive traces, perform better as scientific discovery assistants than when trained solely on scientific literature. We propose COGTRL, a trajectory-level reinforcement learning framework that trains LLMs to emulate cognitively grounded reasoning by jointly optimizing cognitive traces and the scientific steps produced in an interleaved manner. Across two 3B-parameter models and two scientific domains (AI and Materials Science), COGTRL improves method quality by an average of 7.85 points over comparable 3B model baselines and achieves competitive performance relative to 70B parameter models. Moreover, analysis by domain experts shows a preference for methods generated by COGTRL over the baselines.
Synthetic data generation has become a cornerstone for advancing large language models. However, the lack of the quantitative analysis for error tolerance became a critical bottleneck. Consequently, current filtering strategies fluctuate between two extremes: they are either overly aggressive, risking the exclusion of potentially valuable samples, or overly permissive, failing to eliminate erroneous samples effectively. To bridge this gap, this paper introduces Atomic Tree Operation Modeling (ATOM), a framework that decomposes data into functional units ($f(x)\rightarrow y$). ATOM distinguishes benign Operand $x$ perturbations from fatal Operator $f$ perturbations. The former are needlessly discarded by aggressive filtering, while the latter slip through permissive filtering. Our experiments reveal a double dissociation: models are robust to operand perturbations but collapse under operator perturbations. By prioritizing operator over aggressive operand precision, our ATOM-synthesized data outperforms rigorous baselines (e.g., +3.1% gain over LIMA), suggesting that operator diversity matters more than operand precision. Our code is available at https://github.com/Lut-hub/ATOM.
Zewen Ding, Zezhong Wu, Zhou Tao +5cs.LG cs.AI cs.CL
On-policy self-distillation (OPSD) improves reasoning by training a problem-only student on its own rollouts using dense token-level supervision from a privileged teacher that also sees a reference solution. However, standard OPSD treats the teacher distribution as a fixed target along the student's rollout and updates only the student %, although -- even though privileged conditioning does not guarantee that the teacher always provides the most appropriate target for problem-only reasoning. This one-way supervision can therefore misdirect the student when the teacher distribution is misaligned with valid student reasoning. We therefore introduce Verifier-Informed Student-to-Teacher Adaptation (VISTA), which preserves the standard OPSD student update while using outcome-verified rollouts to adapt the teacher toward the student distribution. Within each verified rollout, VISTA further restricts this adaptation to the top-$k$ positions with the largest teacher--student KL divergence. Notably, VISTA reuses the rollout and loss function from standard OPSD, introducing no additional sampling or separate reward objective. Across AIME24, AIME25, and HMMT25 with Qwen3 models at 1.7B, 4B, and 8B, VISTA achieves the highest Avg@12 at every scale, improving over OPSD by $0.6$, $0.7$, and $2.1$ points, respectively. These results demonstrate the value of student supervision from outcome-verified rollouts and highlight student-to-teacher adaptation as a promising direction for OPSD.
Specializing large language models to vertical domains improves domain-specific behavior but often degrades general capabilities such as reasoning, coding, instruction following, and creative writing. We study this domain--general trade-off in Multi-Teacher On-Policy Distillation (MOPD), where a specialized student is supervised on its own sampled trajectories by domain and general teachers. Standard MOPD faces two limitations: ordinary on-policy sampling rarely exposes tokens with large positive teacher--student advantages, while the advantage sign alone does not establish whether the resulting update direction is reliable. We propose uncertainty-calibrated MOPD to address these limitations. Dual-temperature sampling broadens the candidate trajectory pool, and positive-advantage-density filtering selects trajectories with stronger positive learning signals. Centered log-likelihood (CLL) filtering then computes an entropy-calibrated teacher-endorsement score and probabilistically retains token updates according to direction--endorsement consistency. Experiments on role-playing and medical-domain specialization show that our method improves the general-capability average over standard MOPD by $4.73\%$ and $10.84\%$, respectively, while maintaining vertical-domain performance. Ablations and diagnostic analyses further confirm that the gains do not merely result from a larger rollout budget and that the proposed trajectory- and token-level mechanisms address their intended failure modes.
Self-evolving language models have recently emerged as a promising path toward superintelligence, with the advantage of reducing the cost of human supervision. While considerable progress has been made in verifiable domains, self-evolution in unverifiable domains remains substantially less explored. We propose Judge co-adaptation from Zero data (J-Zero), a unified Challenger--Solver--Judge co-evolution framework that supports self-improvement across both domains. The Challenger and Solver co-evolve through an adversarial interaction: the Challenger generates increasingly difficult tasks, while the Solver learns to produce higher-quality responses to them. In parallel, the Judge co-adapts using preference pairs whose ordering is known in advance from how each response was produced, i.e., the Solver's answer over the Challenger's, and its decomposed-and-recombined answer over its one-shot answer, rather than from the Judge's own scores. J-Zero outperforms the baselines by an average of 4.2 points on verifiable and 8.0 points on unverifiable domains, and continues to improve through at least ten iterations, whereas the baselines degrade after two.
Mathematical reasoning has seen rapid progress in large language models (LLMs), yet existing methods optimize predominantly for final-answer correctness, raising the question whether models truly internalize mathematical concepts or merely memorize solution patterns. In human mathematics education, example-based reasoning such as constructing counterexamples to test theorem boundaries reflects deep conceptual understanding, but remains underdeveloped in current LLMs. Enhancing this capability through preference optimization presents two key challenges: (1) the model's limited example-based reasoning ability makes constructing effective preference pairs inherently difficult; and (2) capability acquisition is progressive, as the model must first learn to adopt this strategy before learning to apply it correctly. Therefore we propose INSPIRE, an Internalize-Then-Improve approach combining Reference-Guided Student Internalization (RGSI), which produces high-quality preference candidates under the policy model's own distribution, with a stage-wise rubric preference training strategy that decomposes learning into method-oriented and correctness-oriented stages. Experiments across multiple model scales and families demonstrate consistent improvements, even surpassing larger open-source models, while evaluations on out-of-distribution benchmarks confirm no degradation in general mathematical reasoning ability.
On-policy self-distillation (OPSD) uses a privileged copy of the student model to provide dense supervision without an external teacher. OPSD keeps this privileged teacher fixed, even though the student distribution and output style change during training. We propose DualOPSD, an asymmetric alternating framework that adapts both policies. The student first learns from the privileged teacher. The teacher then moves toward the updated student distribution on the same student trajectory. This update makes later supervision responsive to the learner and does not require another rollout. On Qwen3-8B in non-thinking mode, DualOPSD improves avg@12 over OPSD by 23.61, 13.89, and 10.00 points on AIME 2024, AIME 2025, and HMMT 2025. Results at 1.7B and 4B show that the accuracy gain depends on model scale. Across all three scales, DualOPSD reduces truncation. The 4B diagnostic also shows lower KL in both directions between the teacher and student.
Pre-norm is the standard normalization placement in modern Transformers because it facilitates joint optimization of full-depth models. We ask whether this preference persists when depth is introduced through a curriculum. In curriculum depth growth, each appended block receives the boundary representation produced by a trained prefix, making normalization placement relevant to forward conditioning. We therefore test whether placement and training curriculum interact. In a controlled distillation study with a Qwen3-8B teacher and a nine-layer student, pre-norm and post-norm are indistinguishable under joint training, differing by $0.0004$ validation CE, while post-norm improves over pre-norm by $0.0328$ under curriculum growth, an order of magnitude larger. A post-joint control matched by student active-layer tokens remains worse than post-grow, which rules out compute as the sole explanation. The ranking crosses over during the curriculum: post-norm takes the lead once blocks are appended. Single-block and freeze controls localize the ranking change to block appending rather than shallow-block quality or retraining. Boundary diagnostics associate post-norm with stable residual scales and pre-norm with structural-token scale drift; on a fixed batch, the final pre-grow block is also nearly identity-mapped. Together with the phase-wise crossover, these observations are consistent with boundary-scale conditioning after new blocks are appended. The results motivate treating normalization placement and training curriculum as coupled design choices in this distillation setting.
On-policy distillation (OPD) trains a student on its own trajectories under dense token-level supervision from a teacher. Reward-extrapolation methods such as ExOPD amplify the teacher-reference log-likelihood ratio to move beyond direct imitation, but apply a single global coefficient $λ$ to every token. This can drive the student to fit extreme peaks in the implicit reward, causing reward hacking and unstable training, and the optimal $λ$ varies across domains, requiring costly sweeps. We propose REOPD, a reliability-adaptive reward extrapolation framework for OPD. REOPD combines a token-level compatibility weight with a batch-level adaptive budget, yielding a token-wise coefficient $λ_{b,t}=1+γ_b q_t$ that preserves teacher alignment while selectively extrapolating along reliable teacher-reference directions. It requires no verifier, reward model, value model, or extra rollout beyond standard OPD. REOPD outperforms G-OPD on single-teacher mathematics and on both domains in the multi-teacher setting, while matching G-OPD on single-teacher code, demonstrating effective fine-grained reliability adaptation across domains and teacher configurations.
Vu Duc Anh, Nhat M. Hoang, Do Xuan Long +3cs.CL cs.AI
Achieving effective self-correction, where models verify and correct their own mistakes, remains a fundamental challenge for large language models (LLMs). In this work, we propose Self-Fix Step-DPO (SFS-DPO), a reinforcement learning based, two-stage framework for step-level self-verification and self-correction. The first stage strengthens step-level reasoning via step-level preference optimization, while the second stage explicitly trains models to self-verify and self-correct. We further introduce a teacher-assisted variant, SFS-DPO-R, which incorporates explanatory rationales for error verification to provide stronger corrective signals. Comprehensive in-domain and out-of-domain evaluations across multiple LLMs demonstrate that SFS-DPO and SFS-DPO-R consistently outperform prior step-level training baselines. Our analysis further reveals improvements in self-correction frequency and effectiveness, highlighting the importance of strengthening step-level reasoning for robust performance.
Yongkang Yang, Zhezheng Hao, Hong Zhang +8cs.AI cs.LG
On-policy self-distillation (OPSD) improves the reasoning abilities of LLMs by internalizing privileged context into model parameters through self-distillation. Two recent research lines promote vanilla OPSD by choosing which tokens to learn from and by controlling how much privileged information the teacher receives, respectively. However, we show that each line optimizes one variable while holding the other fixed, which leads to a suboptimal solution. We argue that the two variables are coupled through the student's learning capacity: the privileged information sets the per-token divergence the teacher prescribes, while token weighting selects which of these the student must absorb. We formalize the two lines of work into a unified optimization framework, which maximizes the aggregate teacher--student divergence, subject to a budget on the aggregate learning difficulty the student can absorb. Under this modelling, we propose Unified On-Policy Self-Distillation (USD), a lightweight online algorithm to solve the Lagrangian. USD reveals that a single dual variable governs both decisions: at one price for learning difficulty, it simultaneously sets the token-selection threshold and the direction of privileged-information adjustment, keeping supervision matched to the student's evolving capacity. Through extensive experiments, USD consistently demonstrates superior performance over OPSD and token- and PI-side baselines across various model scales on various reasoning benchmarks. Code is available at https://github.com/lauvlalala/USD.
Large language models (LLMs) have increasingly supported response generation grounded in user-provided knowledge spanning heterogeneous structures. However, existing benchmarks provide limited assessment of whether LLMs can faithfully perform multi-hop reasoning chains across such knowledge contexts while remaining robust to variations in their input order. We introduce TKFQA, a factuality consistency benchmark comprising 10,130 question-answering (QA) pairs grounded in tables, texts, and knowledge graphs (KGs). Each example is constructed from an explicit counterfactual reasoning chain, enabling the joint evaluation of answer correctness, reasoning-chain accuracy, and robustness to different input-order. An extensive evaluation of 14 open- and closed-source LLMs reveals that state-of-the-art models exhibit limited reasoning-chain accuracy and remain sensitive to variations in the input order of heterogeneous knowledge contexts. To address these limitations, we propose ORLF, an LLM-agnostic training framework that models cross-context topological relations through knowledge-specific latent vectors. ORLF integrates context-wise position encoding, a latent-bridge attention mask, and topological knowledge bias to preserve knowledge-specific bias and encode topological semantics. Experiments across four LLM backbones show that ORLF outperforms competitive training-free and LoRA-based baselines, improving average Exact Match and Reasoning-Chain Accuracy by 2.15% and 4.29%, respectively, while reducing order-induced performance standard deviation by 0.04% to 3.01%.
Preference alignment often makes large language models (LLMs) overconfident and poorly calibrated. Traditional post-hoc temperature scaling is inherently domain-dependent: a temperature fitted on one domain does not generalize across domains. This motivates us to modify model parameters during training to improve calibration. We propose maximizing the entropy of predictive distributions as the calibration objective, which directly targets overconfidence by discouraging overly concentrated predictions. Inspired by temperature scaling, we realize this through a bilevel optimization formulation, where the lower level trains the model under a parametric loss and the upper level selects loss hyperparameters to maximize entropy. To make the framework practical at LLM scale, we adopt an efficient first-order approximation that avoids explicit second-order computation. Across both multiple-choice and open-ended generative question answering, experiments demonstrate that our method yields well-calibrated LLMs with particular advantages in out-of-domain generalization.
On-Policy distillation (OPD) transfers teacher capabilities by supervising student-sampled trajectories with dense token-level teacher signals. Recent selective OPD methods improve this process by prioritizing signals that are confident, informative, or learnable. However, the assumptions overlook a fundamental failure mode of language models: their token-level judgments can be driven by input-agnostic language priors, formatting conventions, or stereotyped reasoning templates rather than task-specific evidence. We refer to such optimization-relevant but weakly input-grounded supervision as spurious signals in OPD, which may produce large gradients while contributing little task-improving direction. To mitigate this issue, we propose SA-OPD, a Spurious-Signal-Aware On-Policy Distillation framework that identifies and filters misleading token-level supervision based on input-groundedness and optimization impact. SA-OPD introduces a lightweight input-groundedness proxy estimating whether a token-level distillation signal truly depends on the input. It then filters only tokens that simultaneously exhibit low input-groundedness and extreme distillation divergence, thereby removing high-impact spurious updates and achieving fine-grained OPD optimization. Extensive experiments on both large language model (LLM) and vision-language model (VLM) settings demonstrate that SA-OPD consistently outperforms Vanilla OPD and competitive selective methods. These results establish input-groundedness as a key dimension for OPD supervision selection and offer a simple, effective strategy for mitigating spurious updates.
On-policy self-distillation (OPSD) improves reasoning by using a privileged view of a model conditioned on reference solutions to supervise a student view that observes only the question. However, the teacher-provided token-level targets may depend on reference-specific information unavailable at inference time. We propose Problem-Space-Guided OPSD (PS-OPSD), which replaces the complete solution with trajectory-grounded guidance describing the initial state, goal conditions, constraints, and a selected state-transition path. The student rollout and OPSD objective remain unchanged. Across three mathematical reasoning benchmarks and model scales ranging from 1.7B to 8B, PS-OPSD achieves the highest aggregate question-only accuracy among the compared methods. Controlled experiments further indicate that guidance relevance and path coherence contribute to these gains, highlighting the representation of privileged information as an important design choice in OPSD.
On-policy knowledge distillation transfers reasoning from large teachers to compact students, but existing approaches suffer three compounding failure modes: (i) cold-start collapse, where a fresh student assigns near-zero mass to teacher-preferred tokens; (ii) state-agnostic divergence scheduling, where time-only forward/reverse-KL interpolation ignores the student's coverage state; and (iii) binary reward sparsity, where pass/fail signals discard information from partially correct traces. We present CADENCE, a unified framework with a targeted fix for each. Its DRIFT mechanism schedules a per-token convex mixture of forward-KL and reverse-KL surrogate objectives on student-sampled trajectories (per-token surrogates, not sequence-level KL gradient estimators). Six components extend it: (A) COVA, a coverage-adaptive $β$ schedule accelerating the forward-to-reverse transition; (B) FTB, a forking-token boost concentrating gradient at high-entropy positions via a globally-normalized entropy reference; (C) CCD, a dense reward adding numerical-proximity partial credit for incorrect-but-close traces; (D) LAP, brevity-preferential correct-rollout reinforcement; (E) EMR, an entropy-matching calibration regularizer; (F) BSD, a bootstrapped self-distillation phase. On GSM8K and MATH-500 (corrected 512-token protocol, 5 seeds, reported std), CADENCE distills a 0.5B student from a 1.5B teacher to 69.8 $\pm$ 0.5% GSM8K pass@1 (from 48.7% pretrained; 63.2% of the teacher gap closed) and to 72.1 $\pm$ 0.4% with a 3B teacher (76.2% closed), beating the strongest matched-compute label-using baseline (DRIFT+binary reward) by +4.4 $\pm$ 0.7 points. All experiments run on a single Apple Mac Studio (M-series, 64GB unified memory), showing principled distillation reaches strong reasoning quality without datacenter-scale hardware.
Zhanming Shen, Jintao Tong, Shaotian Yan +9cs.AI cs.LG
On-policy self-distillation (OPSD) has emerged as a promising paradigm for improving LLM reasoning, where a privileged teacher with access to reference solutions provides token-level supervision on the student's own generated trajectories. However, we find that OPSD consistently fails on long chain-of-thought (long-CoT) reasoning models, yielding at best marginal gains while destabilizing the reflective reasoning capability these models depend on. Through a novel decomposition of the teacher's supervision signal, we identify the root cause: the teacher's supervision is dominated by a reference-induced component that drives rote memorization of reference-specific shortcuts, while the question-conditioned, inference-transferable component is ignored or actively opposed. Based on this diagnosis, we propose a two-step solution. First, we construct a reference-only teacher (the same model conditioned on the reference without the question) to isolate the non-transferable component of the supervision signal; the residual after subtracting this component captures the question-conditioned, inference-transferable correction. Second, we use pointwise mutual information (PMI) as the mechanism to transform this residual into a well-formed PMI target distribution that the student can directly distill from, filtering out the reference-induced shortcut. Experiments on four long-CoT models across two datasets demonstrate consistent improvements over both the base model and standard OPSD, while preserving the models' natural epistemic behavior throughout training.
In Large Language Model (LLM) training, data mixing plays a pivotal role in determining model performance. Recent methods optimize mixture weights via proxy models, but they rely on the assumption of static data distributions. As a result, when the underlying data pool shifts, these methods require costly retraining from scratch. This limitation restricts their ability to scale seamlessly from small settings to larger data pools and model sizes. In this paper, we propose CausalMix to address this limitation by casting data mixture optimization as a causal inference problem. We formulate the statistical features of the data pool as covariates and the domain mixture as the treatment. After fitting a causal model on 512 runs of Qwen2.5-0.5B to estimate the Conditional Average Treatment Effect (CATE), we extrapolate the optimal mixture for an 800K data pool and apply it to train a 7B model. Furthermore, we successfully generalize the framework to long chain-of-thought data on Qwen3-4B-Base. By leveraging causal modeling to isolate confounding biases, CausalMix dynamically infers state-dependent optimal data mixtures. Extensive experiments show that the mixture guided by CausalMix consistently improves performance across multiple downstream tasks, outperforming RegMix and other baselines. In addition, we use the CATE Interpreter to provide visual analysis of the learned mixing strategy. Overall, CausalMix offers a causal and interpretable framework for optimizing LLM data mixtures.
On-policy self-distillation (OPSD) teaches large language models new skills through a teacher that shares the student's backbone and supervises its own rollouts. Existing teachers either inject privileged context at the input -- inducing post-hoc rationalization -- or fine-tune weights, accumulating drift and forgetting across tasks. We propose \method, whose teacher differs from the student only by a learnable soft prompt: trained on $(x, y_\text{gold})$ pairs with the backbone frozen, the prompt yields a task-specific teacher that preserves the student's exact representational geometry. \method\ extends naturally to multi-task settings by routing each example in a merged corpus to its corresponding soft-prompt teacher, allowing a single student to absorb knowledge from $K$ teachers in parallel; at inference, all prompts are discarded. On Qwen3-1.7B-Base and Phi-4-mini-instruct across four tasks (Science, Tool Use, Biology, Math), the single-task variant (OPD with a PT teacher) matches or exceeds full fine-tuning while training orders of magnitude fewer parameters, and the multi-task variant achieves the best overall average ($56.2$ on Qwen3-1.7B-Base) while preserving general-capability benchmarks -- in contrast to sequential SFT, which degrades both.
On-policy distillation (OPD) offers superior capacity transfer by supervising student-sampled trajectories with dense token-level signals. To furnish high-quality supervision sources and thereby elevate the performance frontier of distillation, an intuitive direction is to infuse privileged information to either teacher or student itself. However, this additional input induces a potential failure mode we dub privilege illusion: a pattern that conflates the transferable capability gap that students are meant to close, and the information asymmetry gap that can only be mimicked but never replicated. This issue is further amplified by the inherent non-uniformity of token-level supervision, where only a small subset of tokens carries pivotal capability-bearing signals. To this end, we propose DOPD, an advantage-aware dual distillation paradigm that dynamically routes token-level supervision between privileged teacher and privileged student policies based on their advantage gap and relative probabilities. Each token receives supervision of different strength, objective, and strategy from either teacher or student itself, which transfers credible capability while simultaneously receiving auxiliary signals, to alleviate privilege illusion. Extensive experiments on both large language model (LLM) and vision-language model (VLM) settings demonstrate that DOPD consistently outperforms Vanilla OPD and other counterparts. Further results on stability, robustness, continual learning, and out-of-distribution tasks validate its superiority.
On-policy distillation (OPD) has a property absent in offline distillation and RL: teacher supervision quality depends on student competence. Incoherent rollouts yield noisy gradients; already-mastered tokens yield redundant ones. This creates waste at three scales (tokens, training phases, and prompts) yet existing methods supervise uniformly. We introduce SEAD, which uses entropy as a unified probe of this competence-dependent degradation at three scales: (1) joint teacher-student entropy partitions tokens into zones receiving tailored divergences or zero gradient (approx. 50% skipped); (2) a cosine schedule anneals from forward to reverse KL as competence grows; (3) a competence-gated curriculum introduces prompts easy-to-hard. These components are symbiotically necessary: token selection requires coherent rollouts (curriculum), annealing requires monotonic improvement (also curriculum). On OLMo-3 (7B to 32B), SEAD achieves +4.8 avg accuracy over vanilla OPD across six math benchmarks, with ablations confirming super-additive interactions.
Recent research has explored the integration of knowledge graphs (KGs) with large language models (LLMs) to enhance their performance on downstream knowledge-intensive tasks, particularly knowledge graph question answering (KGQA). Existing approaches primarily combine LLMs with KGs through retrieval-augmented generation (RAG)-based, agent-based, and SPARQL-based methods. Although these methods have achieved notable success, they still suffer from several limitations, including structural information loss, unfaithful reasoning, and limited flexibility and generalization. To address these challenges, this paper proposes KG2Code, a novel approach that transforms knowledge graphs into a code-based representation, preserving structural semantics while naturally aligning with the code-aware pretraining of modern LLMs. Based on KG2Code, KG2Code-QA is further introduced as a KGQA framework that formulates KGQA as a code generation task. This formulation enables the generation of verifiable reasoning traces and executable code, thereby substantially mitigating the impact of hallucinations. In addition, an automated pipeline is developed to construct a large-scale, high-quality code corpus for effectively training open-source LLMs on KG2Code-QA. After training, LLMs are able to perform KGQA in zero-shot scenarios. Extensive experiments demonstrate that the proposed approach significantly outperforms existing KG-enhanced LLM methods for KGQA, while exhibiting strong generalization to unseen KGs. The code and data are available at Github.
Zhuo Zuo, Li Yue, Wenhao Zheng +2cs.CL cs.AI cs.CE cs.LG
Despite their strong general capabilities, large language models (LLMs) often remain unreliable when outputs must be numerically precise. A key reason is the training objective: standard cross-entropy treats numeric tokens as unstructured categories and ignores the metric structure of their values. We address this mismatch with Smooth Maximum Mean Discrepancy (SMMD), which builds on the classic MMD by incorporating value-distance kernels over numeric tokens and graph-based smoothness. With this kernel defined over a numeric sub-vocabulary, SMMD aligns the predicted numeric distribution to the target via kernel matching and smooths the prediction-target residual over the induced kernel graph to encourage local consistency. We evaluate SMMD on four numeric-target tasks: mathematical reasoning, arithmetic calculation, clock-time recognition, and chart question answering, across multiple open-weight LLM and VLM backbones. SMMD consistently improves accuracy over both cross-entropy and recent numeric-target losses; analyses show complementary effects between MMD and smoothness and underscore the importance of distance-based kernel design. Code is available at https://github.com/Zuozhuo/smmd-loss.
Sequential learning is order-dependent: from Pile-style next-token domain adaptation to instruction-SFT and DPO, N candidate sources induce N! possible curricula. We show that the local order effect is governed by a computable geometric quantity, the Lie-bracket commutator of gradient update fields, yielding a pairwise score for whether A->B or B->A is better for a target domain. The pairwise bracket primitive also defines a Lie-Bracket Tournament: with a shared theta_0 target-gradient reference, Hessian symmetry gives Borda/row-sum scores from one Hessian-vector product per source, O(N) dot products, and an O(N log N) sort, without materializing the O(N^2) edge matrix. Empirically, the planner reaches 98.1%/98.9% pairwise accuracy at k=1 for instruction-SFT/DPO, remains at 73.1%/72.2% at k=20, and preserves the original pretraining-domain evidence with 82.4-92.0% accuracy across four LLMs and 91.1% on diffusion. At curriculum scale, it recovers the best of all 3! schedules in 87.5% of trials, ranks 85 Stack programming-language source domains for a Python target in the 99th sampled percentile, and reaches the 99.0-99.6th sampled percentile on 56 MMLU subjects, sharply above the reported descending gradient-norm baseline. These results reframe sequential learning as a geometric tournament problem: commutators provide both local pairwise order information and a scalable primitive for many-domain schedules.
Knowledge injection via synthetic data is crucial for enhancing Large Language Models (LLMs). However, current synthesis methods simply stop at preset token counts or fixed data ratios, lacking awareness of knowledge distribution. This results in some domains being sparse while others are redundant, limiting LLM knowledge boundaries. We revisit knowledge injection from a distribution perspective and hypothesize that an optimal knowledge distribution exists to maximize knowledge boundary expansion. We propose KDoS (Knowledge Distribution-optimized Synthesis), a framework that introduces knowledge density to drive synthesis through a three-stage feedback mechanism, shifting from blind generation to distribution-optimized synthesis. We construct Wikipedia-based synthetic data with varying knowledge distributions and conduct experiments on models from 0.6B to 16B (Qwen, Ling, LLaMA) and data scales from 1B to 5B tokens. Our key findings are: (1) an optimal knowledge distribution consistently maximizes boundary expansion; (2) this distribution is stable across backbones and scales; (3) KDoS outperforms baselines across six knowledge benchmarks. Our work offers a new perspective and practical framework for synthetic data-driven knowledge injection.
Because large language models (LLMs) are impressively successful in predicting text, it appears that they must have access to a 'world model' representing causal and definitional structure. However, the dominant formalisms of modern causal inference -- Judea Pearl's interventionist approach and the Neyman-Rubin potential outcomes framework -- struggle to illuminate how LLMs learn causal structure. I resolve this puzzle by arguing that LLMs employ a specific inductive approach based on a difference-making logic -- sometimes called variational induction. I demonstrate how central aspects of this logic are realized during training, where LLMs require enormous amounts of text data from a wide range of contexts to identify difference- and indifference-makers within word sequences. Furthermore, I analyze specific architectural features of LLMs -- such as token embeddings and self-attention -- to determine their roles in variational induction. The difference-making logic of LLMs fundamentally parallels the experimental method, where causal relations are derived by systematically varying individual circumstances to determine their influence on a phenomenon.
As Large Language Model (LLM) datasets scale to trillions of tokens, data selection has emerged as a critical frontier to filter out uninformative noise and construct adaptive learning trajectories. Beyond static heuristic filtering, advanced data selection methods for LLM training largely follow two paradigms, each with fundamental limitations. Influence-based methods provide principled bi-level objectives but require intractable inverse-Hessian computations, while excess-loss methods are computationally efficient but rely on a static reference model that becomes misaligned with the evolving proxy model during training. We propose BLADE (Bi-Level Adaptive Data sElection), a Hessian-free framework for data selection. BLADE reformulates the bi-level optimization problem underlying influence-based methods as a penalized single-level objective via Lagrange multipliers, avoiding inverse-Hessian computation while revealing a principled connection to excess-loss based data selection. The resulting objective recovers an excess-loss form but replaces the static reference model with a dynamic one that stays synchronized with training. Theoretically, we prove that this penalized formulation guarantees first-order convergence. For efficient online batch selection, we instantiate BLADE as a memoryless randomized block-coordinate Frank-Wolfe algorithm. Extensive experiments show that BLADE consistently outperforms state-of-the-art data selection baselines, providing a practical recipe for LLM training.
With the growth of LLMs' (Large Language Models) capabilities, there has been an increasing push to curate high quality datasets by filtering samples in the training data. In general, Data Attribution (DA) methods aim to estimate how individual samples in a training dataset can precondition a model to generate certain outputs. As an example, one might be interested in which samples in the data could be the source of toxic behavior after training the LLM. Many methods quantify this conditioning through the paradigm of influence functions. While methods of this family are effective in its function, they lack the necessary processing speed and storage compactness to be practically implemented on large datasets. We propose a method, Influcoder, as a quick and cost-effective approach to influence-based Data Attribution at scale.
Recent work has shown that on-policy distillation can internalize privileged context, such as system prompts or task hints, into a student model so that the context is no longer needed at inference time. Although this approach successfully improves the student's no-context performance, we identify an interesting and previously unstudied phenomenon: in many settings, reintroducing the original privileged context to the distilled student actually degrades its performance, even on instances it already solves correctly without context. We term this context-induced degradation and argue that robust internalization demands not only matching the teacher's context-conditioned behavior, but also remaining stable when the context is reintroduced, a property we call context removability. Motivated by this observation, we propose a lightweight consistency regularizer that first anchors the student's no-context output via stop-gradient, then penalizes the context-conditioned output for deviating from it via forward KL divergence. This simple addition requires only one extra forward pass per training step, yet it effectively mitigates context-induced degradation and, in many cases, even improves no-context performance. Across 12 configurations spanning diverse domains and model families, our method improves context-conditioned accuracy in the majority of settings, reduces context-induced harm in 11 out of 12 settings, and effectively eliminates response-length inflation. A mechanistic case study further confirms that context removability is achieved at the representation level, with hidden states remaining nearly identical regardless of whether the context is present.