arXiv subjects

Di He

Publications and source records attributed to Di He.

At least 19 recordsLinked to original sources

arXiv2026

Your LR Schedule May Not Be Principled: Effective Learning Rate Scheduling in Weight-Normalized Training

Weight-normalized (WN) training is emerging as a compelling alternative to conventional weight decay (WD), with recent studies reporting substantial pretraining loss gains. We show that these gains are driven primarily by the induced effective learning rate (ELR) trajectory rather than by normalization alone. Matching ELR trajectories largely collapses the training loss dynamics of WN and WD. Schedule design should therefore target ELR rather than the nominal learning rate (LR) alone. Building on this principle, we propose AIR (Affine Inverse-square-Root decay), which combines a theoretically derived $1/\sqrt{t}$ decay with an affine adjustment to satisfy prescribed peak and terminal LRs over a finite training horizon. We further identify late-stage ELR scheduling as a key source of its gains, because a shared late schedule attenuates the effect of earlier LR choices. Experiments on 128 B-token Mixture-of-Experts pretraining demonstrate that AIR outperforms cosine and other candidate schedules in training loss, downstream performance, and hyperparameter robustness, reducing terminal training loss by approximately 0.01 relative to cosine at matched cumulative LR. Applying the same ELR trajectory to WD reproduces the corresponding training dynamics, showing that the benefits extend beyond fixed-norm training and establishing ELR as the principled scheduling object.

cs.LG↗

arXiv2026

BizAgentBench: Unveiling Hidden Challenges in Deploying Long-Horizon Agents for Real-World Business

Existing long-horizon agent benchmarks often emphasize final task success while providing limited assessment of intermediate execution validity. Yet real-world business operations require agents to navigate extensive, potentially outdated or conflicting policies while satisfying critical constraints throughout execution. These requirements are particularly consequential in high-stakes domains such as payments, where intermediate violations can incur financial losses or regulatory consequences regardless of the final outcome. To bridge this evaluation gap, we introduce BizAgentBench, a multi-turn benchmark grounded in real-world procedures from a leading payments company. Its executable environment integrates 789 knowledge documents and 601 discoverable business tools, with consistent policies, states, and tool behavior. The benchmark comprises 50 tasks involving resource constraints, temporal dependencies, prior commitments, and conflicting or outdated guidance. Each task is validated through reference execution, while deterministic checks assess both goal attainment and execution validity without prescribing a single solution path. Experiments reveal substantial limitations even among frontier models: Claude Opus 5 and GPT-6 Sol achieve strict success rates of only 64.0% and 61.3%, respectively. For Qwen3.8-27B, goal attainment reaches 42.0%, but strict success drops to 32.7%, demonstrating how outcome-based evaluation can mask invalid execution. Beyond success rates, DeepSeek-V4.1-Flash averages 191.3 tool calls per run, approximately 70% more than reference executions, revealing substantial execution overhead. Overall, BizAgentBench establishes a testbed for evaluating and advancing long-horizon agents beyond task completion, toward reliable and constraint-compliant execution in complex business environments.

cs.CE↗

arXiv2026

Scaling to Tens of Thousands of Test-Time Iterations with Loop-Native Attention Residuals

In this paper, we argue that Looped Transformers need their own residual connections to prevent performance degradation as the number of iterations grows. We first introduce an interesting phenomenon we observed in Looped Transformers, called latent overthinking: noisy state updates overwrite correct intermediate deductions and even undo completed solutions. This leaves subsequent iterations to recover lost information from an already degraded representation: once an error arises in an earlier loop, often as a result of long-range propagation through the recurrence, later loops find it difficult to correct. In this paper, we introduce InfiLoop, a loop-native residual connection that learns which past computations to retain and how much to accept from each new update. InfiLoop combines content-based weighting with learned temporal decay to maintain a running summary of recurrent states. An exact streaming recurrence keeps its persistent aggregation memory constant as the loop count grows. The resulting adaptive update suppresses unreliable proposals and preserves useful intermediate states. Across extensive reasoning tasks, a 7M-parameter InfiLoop model outperforms existing recursive architectures, reaching 97.9% exact accuracy on Sudoku-Extreme, and 13.6% pass@2 on ARC-AGI-2. Notably, on Sudoku-Extreme, InfiLoop continues to improve with test-time looping beyond 20,000 effective steps, showing that added depth translates directly into stronger reasoning. Our code is available at https://github.com/pixeli99/InfiLoop.

cs.AI↗

arXiv2026

L2G-SR: Learning to Guide LLM-based Symbolic Regression

Symbolic regression searches for an equation that explains observed data. Numerical error is commonly used to guide the search, but it alone can favor an accurate surrogate over a candidate with the right structure. We propose L2G-SR, a framework for LLM-based symbolic regression that guides search with a learned structural verifier. We construct structural supervision signal using tree edit distance (TED), accounting for commutative operators and alternative expression forms. A verifier first aligns numerical observations with the formulas, then learns to score a candidate's structural similarity to the target formula. At search time, the target formula is hidden and the verifier complements numerical error in parent selection. An analyst then uses verifier feedback and diagnostic tools to propose revisions, which a generator converts into refined expressions. On LLM-SRBench, L2G-SR achieves state-of-the-art numerical fit, equation level accuracy and TED across the evaluated LLM backbones and domains. L2G-SR demonstrates the value of learned structural guidance for LLM-based symbolic regression.

cs.AI↗

arXiv2026

Designing Reinforcement Learning for Diffusion Models: A Unified Path-Space View

Reinforcement learning (RL) post-training provides a direct way to align diffusion models with human preferences and task-specific rewards. However, current RL algorithms for diffusion models remain fragmented: reverse-trajectory methods rely on discretized likelihood ratios, whereas forward-matching methods train on reward-labeled noising versions of the rollout samples. This paper shows that these seemingly different losses arise from a single path-space principle. Starting from the regularized diffusion-RL objective, we use importance sampling between sampling SDEs to obtain an explicit policy-gradient estimator on trajectory space. The estimator contains the stochastic Itô integral underlying Flow-GRPO-type updates; we derive an equivalent variance-reduced value-gradient form that recovers the forward-matching structure of AWM and DiffusionNFT. This identifies the empirical gap between these method families as a variance-reduction effect rather than a difference in RL principle. The derivation yields a unified design space organized by value-gradient estimation, weight functions, and sampling choices. Within this space, we propose a multi-sample KDE value-gradient estimator that reuses rollout groups, together with scale-bounded weight families that retain stable existing recipes while excluding singular ones. Experiments on SD3.5-M and Qwen-Image models validate the variance-reduction explanation and show that the resulting recipe improves over prior diffusion-RL baselines.

cs.LG↗

arXiv2026

Looping Beyond Twice: A Scalable Recipe for Looped Mixture-of-Experts

Looped Transformers introduce recurrent depth as a new scaling axis for LLMs: by repeatedly applying shared Transformer blocks, they increase effective depth without increasing parameter count. However, the benefits of looping remain unclear for large MoE LLMs under FLOPs-matched comparisons. The main reason is that the gains from additional iterations diminish quickly and can even turn into degradation, so the extra FLOPs spent on looping yield little substantial improvement. Consequently, prior work typically settles on two loops. We identify two main obstacles to scaling looped MoE. First, looping inherits and amplifies the curse of depth: hidden-state variance grows with each iteration as residual updates accumulate, which destabilizes deep recurrence and causes representations to drift. Second, looped MoE suffers from expert selection collapse: routers repeatedly select the same experts across loops, so extra iterations add computation without adding computational diversity. Guided by this diagnosis, we propose LOOM, built on a single principle: each loop should contribute new computation while keeping the recurrent state stable. LOOM stabilizes recurrence by scaling residual updates to bound variance growth and re-injecting the input embedding at every loop, and diversifies it through per-loop routers that engage different experts and a Looping Residual that carries earlier outputs forward. Experiments across 100M-1.7B models show stable scaling to 9-12 loops. Under near-iso-FLOP, the 700M model performs best at 5 loops, reducing perplexity from 18.36 to 16.54 and improving average zero-shot accuracy from 38.84% to 39.53% over the non-looped baseline. Without FLOP matching, the 1.7B model trained on 60B tokens peaks at 9 loops, reducing perplexity from 9.62 to 7.77 and improving average zero-shot accuracy from 42.4% to 47.7%. Code is available https://github.com/hed-ucas/LOOM.

cs.LG↗

arXiv2026

Lossless Anti-Distillation Sampling

Frontier commercial generative models face a growing threat from distillation, whereby a distiller harvests generated responses and trains a competing model at drastically lower cost. Existing defenses either modify the generation to degrade distillation performance, sacrificing response quality, or rely on behavioral detection mechanisms that can be readily bypassed through multi-account querying. In this work, we propose Lossless Anti-Distillation Sampling (LADS), which leaves the generation itself unchanged while substantially reducing the effectiveness of distillation. Concretely, LADS controls the latent randomness underlying inference through a coupling mechanism that preserves within-account generation independence while inducing cross-account dependence. By construction, each benign user, who typically holds only a single account, receives the same experience under LADS as they would without any defense, thereby enjoying a lossless experience. However, for a multi-account task-specific distiller, semantically similar queries submitted across different accounts are assigned coupled randomness, inducing dependence in the harvested data and thereby degrading the generalization performance of the distilled model. Using uniform convergence theory, we show that LADS provably degrades the distiller's generalization gap relative to standard i.i.d. sampling. Experiments on image generation, mathematical reasoning, and code generation confirm that LADS substantially degrades the performance of distilled students while preserving exact statistical fidelity for individual users.

cs.LG↗

arXiv2026

Towards Solving the Gilbert-Pollak Conjecture via Large Language Models

The Gilbert-Pollak Conjecture \citep{gilbert1968steiner}, also known as the Steiner Ratio Conjecture, states that for any finite point set in the Euclidean plane, the Steiner minimum tree has length at least $\sqrt{3}/2 \approx 0.866$ times that of the Euclidean minimum spanning tree (the Steiner ratio). A sequence of improvements through the 1980s culminated in a lower bound of $0.824$, with no substantial progress reported over the past three decades. Recent advances in LLMs have demonstrated strong performance on contest-level mathematical problems, yet their potential for addressing open, research-level questions remains largely unexplored. In this work, we present a novel AI system for obtaining tighter lower bounds on the Steiner ratio. Rather than directly prompting LLMs to solve the conjecture, we task them with generating rule-constrained geometric lemmas implemented as executable code. These lemmas are then used to construct a collection of specialized functions, which we call verification functions, that yield theoretically certified lower bounds of the Steiner ratio. Through progressive lemma refinement driven by reflection, the system establishes a new certified lower bound of 0.8559 for the Steiner ratio. The entire research effort involves only thousands of LLM calls, demonstrating the strong potential of LLM-based systems for advanced mathematical research.

cs.DM↗

arXiv2026

AREX: Towards a Recursively Self-Improving Agent for Deep Research

Deep research requires agents to find answers that jointly satisfy multiple constraints. Discovering such answers is costly, whereas verifying a candidate can often be decomposed into tractable constraint-wise checks. This discovery--verification asymmetry suggests that a research agent should do more than simply search longer: it should recursively improve its current answer by verifying intermediate results and using the partially verified state to guide subsequent refinement. We introduce AREX, a family of Recursively Self-Improving (RSI) deep research agents. AREX alternates between an inner research loop that gathers evidence and constructs a provisional answer, and an outer self-improvement loop that audits the answer constraint-wise, identifies unresolved claims, and launches targeted follow-up research. To sustain RSI over long horizons, AREX learns an autonomous context-update tool that compresses growing interaction history into a compact improvement state preserving verified evidence and unresolved constraints, without relying on an external model. We train AREX on verified synthetic tasks and high-quality trajectories through agentic mid-training and long-horizon reinforcement learning. To mitigate sparse final rewards during long horizon learning, we emphasize key steps where decisive evidence is acquired or erroneous research directions are corrected. We instantiate a dense 4B model and a 122B-A10B Mixture-of-Experts model. Across BrowseComp, WideSearch, DeepSearchQA, Humanity's Last Exam (HLE), and other reasoning and tool-use benchmarks, AREX substantially outperforms comparable-scale baselines and remains competitive with models using substantially more activated parameters.

cs.AI↗

arXiv2026

Test-Time Scaling for Scientific Equation Discovery

Test-time scaling (TTS) improves language model reasoning by allocating additional test-time compute, but prior work mainly studies closed-ended tasks such as math and coding. We study TTS for automated equation discovery, an open-ended setting where models search over candidate equations and rely on observed datapoints for feedback. We formulate LLM-driven equation discovery as an iterative search process that unifies Best-of-N, sequential refinement, tree search, and evolution-style methods under a common compute-allocation view. To isolate allocation effects from prompt engineering and other heuristics, we compare minimal parallel controllers under fixed budgets. On LLM-SRBench equation-discovery tasks, we find that search width is the dominant allocation parameter: the best width in our sweep generally increases with the compute budget, while the population--branching split and controller choice matter less. Appropriate width selection also improves wall-clock efficiency by increasing parallelism. These results suggest that, given an informative verifier, controlling exploration and exploitation is central to scaling LLM-based equation discovery.

cs.CL↗

arXiv2026

Ouroboros-Spatial: Closing the Data-Model Loop for Spatial Reasoning

Spatial reasoning remains a persistent challenge for multimodal large language models (MLLMs). Existing approaches largely rely on large-scale, statically curated datasets, where all training samples are treated uniformly regardless of the model's evolving capabilities. This static paradigm is inherently data-inefficient: training capacity is often spent on samples that are either trivial or overly difficult for the model at its current stage. To address this limitation, we propose Ouroboros-Spatial, a self-evolving training framework in which the model plays dual roles as a proposer and a solver. In each iteration, a frozen proposer generates spatial question-answer (QA) pairs from 3D scene metadata and raw video frames, together with executable code for deriving reliable ground truth. A learnable solver is then fine-tuned on the accepted samples, and its per-sample prediction confidence is used as a difficulty signal. This signal is fed back to the proposer in the next iteration, guiding it to generate questions better matched to the solver's current capabilities. Through this closed-loop design, the training distribution co-evolves with model ability, reducing redundant trivial examples while filtering out ambiguous or uninformative samples with limited learning value. Across six spatial reasoning benchmarks, Ouroboros-Spatial substantially improves Qwen3-VL-4B and Qwen3-VL-8B while using an order of magnitude fewer training examples than recent large-scale curated datasets. On VSI-Bench, it yields absolute gains of 9.9 and 6.8 points for the 4B and 8B models, respectively, enabling both to outperform a wide range of strong open-source and proprietary baselines.

cs.CV↗

arXiv2026

Structured Scaling of AI Discovery Across Diverse Scientific Domains

Scientific discovery often requires many cycles of proposing, testing, and refining candidate solutions. Language models can increasingly participate in these loops, but simply generating more attempts does not ensure progress: parallel searches may duplicate one another and iterative refinement may become trapped in poor directions. The central challenge is therefore not only to scale AI-driven discovery, but to structure that scaling so that evaluation signals compound over time. Here we introduce SimpleTES (Simple Test-time Evaluation-driven Scaling), a framework that focuses on the structured scaling of AI discovery loops, organizing evaluator queries across independent trajectories, iterative refinement, local candidate selection, and the selective reuse of evaluated histories. Drawing on structural features of scientific communities, SimpleTES uses a single open-source GPT-OSS model to establish new state-of-the-art solutions across 28 open-ended problems in diverse scientific domains ranging from quantum physics and astronomy to biology, AI, and mathematics. These include a 24.5% reduction in quantum circuit compilation overhead, up to 23% lower propulsive cost for deep-space trajectories, a 2.17x faster lasso-path solver, an 8.5% lower-error whole-brain neural-activity predictor, the fastest reported TriMul kernel, and new mathematical constructions beyond prior human or AI records. We further post-train the model for long-horizon discovery by assigning each attempt the final outcome of the trajectory it helped produce. This improves performance on both training and held-out mathematics problems, further advancing the frontier. Together, these results establish structured scaling as a general mechanism for advancing AI scientific discovery.

cs.LG↗

arXiv2026

Muse: Representation Geometry of Muon Beyond Normalized Momentum

Muon-style optimizers apply a polar map to matrix momentum, but their updates also depend on the representation of each parameter block before orthogonalization. We study this representation choice as a form of optimizer geometry and introduce {\method}, a family of Muon-style optimizers that shares the same momentum rule and Newton--Schulz backend across native, nearest-square, skinny, and vector representations. Each Frobenius-isometric representation induces a distinct polar steepest-descent geometry, in which the shorter matrix dimension determines the number of supported singular channels, the pullback scaling, and the constants in stochastic nonconvex convergence bounds. In a teacher--student model, curvature collapse and an isotropic Marchenko--Pastur spectral profile connect early-stage dissipation to the represented nuclear-to-squared-Frobenius norm ratio. Pretraining experiments on LLaMA2-130M and LLaMA2-600M, together with fixed-momentum diagnostics, show that balanced non-native representations can match the performance of the native representation, whereas reducing the shorter dimension weakens the scaling and singular-channel support, leading to behavior that increasingly resembles normalized momentum.

cs.LG↗

arXiv2026

SPARKLING: Balancing Signal Preservation and Symmetry Breaking for Width-Progressive Learning

Progressive Learning (PL) reduces pre-training computational overhead by gradually increasing model scale. While prior work has extensively explored depth expansion, width expansion remains significantly understudied, with the few existing methods limited to the early stages of training. However, expanding width during the mid-stage is essential for maximizing computational savings, yet it remains a formidable challenge due to severe training instabilities. Empirically, we show that naive initialization at this stage disrupts activation statistics, triggering loss spikes, while copy-based initialization introduces gradient symmetry that hinders feature diversity. To address these issues, we propose SPARKLING (balancing {S}ignal {P}reservation {A}nd symmet{R}y brea{K}ing for width-progressive {L}earn{ING}), a novel framework for mid-stage width expansion. Our method achieves signal preservation via RMS-scale consistency, stabilizing activation statistics during expansion. Symmetry breaking is ensured through asymmetric optimizer state reset and asymmetric learning rate re-warmup. Extensive experiments on dense and Mixture-of-Experts (MoE) models demonstrate that, across multiple width axes and optimizer families, SPARKLING consistently outperforms training from scratch and reduces training cost by up to 35% under $2\times$ width expansion.

cs.LG↗

arXiv2026

Parallel Rollout Approximation for Pixel-Space Autoregressive Image Generation

Pixel-space continuous-token autoregressive (AR) generation directly models images as sequences of raw pixel patches, avoiding discrete tokenization or a separately pretrained tokenizer. However, it faces coupled challenges: high-dimensional patch generation causes large single-step errors, and teacher-forced training creates a train--inference gap that makes these errors accumulate across AR steps. Existing fixes such as $x$-prediction and input noise injection only partially mitigate these issues. Exact rollout training better matches inference-time conditions, but is impractical due to prohibitively slow sequential sampling. We propose \emph{Parallel Rollout Approximation} (PRA), a scalable framework that addresses both challenges jointly. PRA generates low-dimensional intermediate states instead of high-dimensional pixel patches, then maps them back to pixel-space tokens with a pixel decoder, preserving a pixel-in, pixel-out AR interface. It also constructs inference-like pixel inputs through the same intermediate-state-to-pixel path used at inference, independently across positions, approximating the pixel-feedback interface encountered during inference-time rollout while retaining parallel teacher-forced training. On class-conditional ImageNet-1K generation at $256\times256$ resolution, PRA-S with 135M parameters achieves an FID of 2.58, surpassing the previous billion-scale pixel-space AR result of 3.60. Scaling to PRA-L with 511M parameters further improves FID to 1.94, establishing a new state of the art among pixel-space AR models. Beyond generation, PRA achieves higher ImageNet classification probing accuracy than other AR and diffusion baselines, suggesting its potential for unified pixel-space image generation and understanding.

cs.CV↗

arXiv2026

One LR Doesn't Fit All: Heavy-Tail Guided Layerwise Learning Rates for LLMs

Learning rate configuration is a fundamental aspect of modern deep learning. The prevailing practice of applying a uniform learning rate across all layers overlooks the structural heterogeneity of Transformers, potentially limiting their effectiveness as the backbone of Large Language Models (LLMs). In this paper, we introduce Layerwise Learning Rate (LLR), an adaptive scheme that assigns distinct learning rates to individual Transformer layers. Our method is grounded in Heavy-Tailed Self-Regularization (HT-SR) theory, which characterizes the empirical spectral density (ESD) of weight correlation matrices to quantify heavy-tailedness. Layers with weaker heavy-tailedness are assigned larger learning rates to accelerate training, while layers with stronger heavy-tailedness receive smaller learning rates. By tailoring learning rates in this manner, LLR promotes more balanced training across layers, leading to faster convergence and improved generalization. Extensive experiments across architectures ranging from LLaMA to GPT-nano, optimizers including AdamW and Muon, and model scales from 60M to 3B parameters with up to 100B training tokens demonstrate the effectiveness of LLR. LLR achieves up to 1.5x training speedup and consistently outperforms uniform-learning-rate baselines. In particular, it improves the average zero-shot accuracy of 1B models from 47.09% to 49.02%, and that of 3B models from 48.58% to 50.61%. A key advantage of LLR is its low tuning overhead: it can transfer nearly optimal learning-rate settings directly from the uniform baseline. Code is available at https://github.com/hed-ucas/Layer-wise-Learning-Rate.

cs.LG↗

arXiv2026

Quotient-Space Diffusion Models

Diffusion-based generative models have reformed generative AI, and also enabled new capabilities in the science domain, e.g., fast generation of 3D structures of molecules. In such tasks, there is often a symmetry in the system, identifying elements that can be converted by certain transformations as equivalent. Equivariant diffusion models guarantee a symmetric distribution, but miss the opportunity to make learning easier, while alignment-based simplification attempts fail to preserve the target distribution. In this work, we develop quotient-space diffusion models, a principled generative framework to fully handle and leverage symmetry. By viewing the intrinsic generation process on the quotient space, the exact construction that removes symmetry redundancy, the framework simplifies learning by allowing model output to have an arbitrary intra-equivalence-class movement, while generating the correct symmetric target distribution with guarantee. We instantiate the framework for molecular structure generation which follows $\mathrm{SE}(3)$ (rigid-body movement) symmetry. It improves the performance over equivariant diffusion models and outperforms alignment-based methods universally for small molecules and proteins, representing a new framework that surpasses previous symmetry treatments in generative models.

cs.LG↗

arXiv2026

Diagnosing and Improving Diffusion Models by Estimating the Optimal Loss Value

Diffusion models have achieved remarkable success in generative modeling. Despite more stable training, the loss of diffusion models is not indicative of absolute data-fitting quality, since its optimal value is typically not zero but unknown, leading to confusion between large optimal loss and insufficient model capacity. In this work, we advocate the need to estimate the optimal loss value for diagnosing and improving diffusion models. We first derive the optimal loss in closed form under a unified formulation of diffusion models, and develop effective estimators for it, including a stochastic variant scalable to large datasets with proper control of variance and bias. With this tool, we unlock the inherent metric for diagnosing the training quality of mainstream diffusion model variants, and develop a more performant training schedule based on the optimal loss. Moreover, using models with 120M to 1.5B parameters, we find that the power law is better demonstrated after subtracting the optimal loss from the actual training loss, suggesting a more principled setting for investigating the scaling law for diffusion models.

cs.LG↗