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arXiv2026

Difficulty-Adaptive Tree-Structured Policy Optimization for Expanding Reasoning Coverage in RLVR

Reinforcement Learning with Verifiable Rewards (RLVR) has been central to the recent success of Large Reasoning Models. However, while RLVR significantly improves single-sample accuracy, it often fails to expand the model's intrinsic reasoning coverage (pass@k) due to limited exploration during training. To address this, we optimize the structural design of train-time rollouts to enhance pass@k. Our analysis identifies three key design principles: (1) difficulty-adaptive rollout can play an important role in expanding pass@k, beyond serving as an efficiency heuristic; (2) tree-based rollout outperforms parallel sampling in discovering correct answers; and (3) sentence-entropy-guided forking overcomes the localization phenomenon of token-level branching to maximize semantic diversity. Building on these insights, we propose DATPO (Difficulty-Adaptive Sentence-entropy-guided Tree-structured Policy Optimization). DATPO integrates difficulty-adaptive tree search with a sibling-diversity advantage term, explicitly promoting semantic diversity to expand reasoning coverage during training. Experiments on mathematical reasoning benchmarks demonstrate that DATPO outperforms baselines especially in pass@k, which directly translates to superior test-time scaling performance.

cs.LG↗

arXiv2026

Charts Are Beyond Pixels: Probing for Layer-Wise Chart Understanding and Editing

Charts are structured visual compositions whose elements have distinct functional roles, semantic correspondences, and visibility relations. This structural view motivates evaluating whether models can understand and manipulate charts at the layer level. Existing chart benchmarks, however, primarily assess the correctness or fidelity of final outputs and do not directly evaluate these layer-wise behaviors. We present LayerWiseBench, a benchmark organized around three core concepts, layer attribution, layer binding, and visibility ordering, that structure its chart-understanding and chart-editing evaluations. Generated from executable chart programs, LayerWiseBench pairs each rendered chart with spatially aligned per-layer RGBA assets and construction-derived labels for functional roles, semantic bindings, and visibility relations. From this layer-wise representation, we derive controlled understanding questions, editing targets, reference images, and evaluation regions. It contains 2,800 source charts across 14 chart paradigms, from which we derive 7,329 layer-wise understanding questions and 53,791 instruction-guided editing variants. Among the evaluated VLMs, Qwen3.5-27B, which achieves the highest QA macro-average, obtains 93.04% accuracy on layer attribution and 97.46% on layer binding, but only 61.46% on visibility ordering. Across the four evaluated image editors, overall mIoU ranges from 1.49% to 4.93%, and visibility-constrained edits have the lowest mIoU for every editor, ranging from 0.37% to 2.00%. Taken together, these results identify tasks involving front-to-back relations between overlapping components as a recurring challenge across understanding and editing, motivating more explicit modeling of component identity and visibility relations.

cs.CV↗

arXiv2026

Learning to build covering structures with continuous adjustments

Robotic construction offers the potential to use materials more efficiently and create complex geometries, but current methods rely on rigid, high-precision plans that cannot accommodate the tolerances, inaccuracies, and unexpected changes inherent in physical fabrication. In this work, we introduce a reinforcement learning approach that forgoes predefined plans entirely, instead generating construction sequences adaptively as the structure is built. Our method operates on graph-structured state representations and a mixed (parameterized) action space, requiring both discrete block selection and continuous placement parameters. Because the stability simulation of a structure is computationally heavy, we develop an efficient exploration strategy by incorporating unilateral edges into graph neural networks, extending soft actor-critic (SAC) to this hybrid setting. We evaluate our algorithm, HSAC, against the prior method hybrid-PPO (HPPO), demonstrating significantly higher asymptotic performance and good sample efficiency. We also demonstrate HSAC's robustness to hyperparameter choices and its exploration capability, handling up to 10 discrete actions without performance degradation. Finally, we validate our approach on a physical two-robot setup, successfully building a spanning arch with 3D-printed blocks in closed-loop execution, confirming that policies trained in simulation transfer to real hardware.

cs.RO↗

arXiv2026

Optimal estimation for Functional Linear Regression with Noisy Discretized Data

In this paper, we consider the scalar-on-function linear regression model under a realistic sampling scheme in which the functional covariates are observed on a regular grid and contaminated by additive noise. We propose a two-step estimation procedure: first, the underlying curves are reconstructed from the discrete noisy observations using a Fourier-based projection method; second, the slope function is estimated by a penalized least-squares criterion over finite-dimensional trigonometric spaces, with data-driven selection of the model dimension. We establish oracle-type inequalities for the prediction error, both with respect to the reconstructed curves and to the true latent curves. Under regularity assumptions on the slope function and polynomial decay of the eigenvalues of the covariate, we derive convergence rates for the prediction error and show that our estimator attains the minimax rate when the number of grid points is sufficiently large. Finally, the proposed method is illustrated on simulated data and on a real meteorological dataset.

stat.ML↗

arXiv2026

Neither Adversarial Training Nor Purification: Emergent Adversarial Robustness from Oscillatory Predictive Learning

Adversarial robustness in computer vision is still largely achieved through adversarial training or test-time adversarial purification, both of which introduce significant computational overhead by generating adversarial examples during training or performing iterative denoising at test time. We study whether empirical robustness can instead emerge from architectural and representation-learning inductive biases. We introduce Oscillatory Predictive Learning (OPL), a two-stage framework that combines Artificial Kuramoto Oscillatory Neurons (AKOrN) with predictive self-supervised pretraining using X-PhiNet. Because our default checkpoint uses randomized initial oscillator states, we compare it with other randomized adversarial defense methods that provide precise, reproducible, and strong attack protocols. Experiments on CIFAR-10 and CIFAR-100, with additional corruption evaluation on CIFAR-10-C, demonstrate that our method achieves competitive results under the AutoAttack-rand evaluation protocol. On CIFAR-10 and CIFAR-100, OPL attains 76.63$\pm$0.76$\%$ and 50.44$\%$ robust accuracy, respectively, under $\ell_\infty$, $ε=8/255$, AutoAttack-rand with EoT $K=20$.

cs.LG↗

arXiv2026

HOPE: Heterophily-Aware Open-Set Node Classification with Pseudo-Extrapolation

Standard open-set node classification methods rely on the homophily assumption, where connected nodes share labels. However, real-world graphs are often heterophilic, exposing the limitations of current methods and posing new challenges to open-set node classification. On the one hand, cross-class connectivity causes representations from different known or unknown classes to become intertwined after aggregation, undermining their discriminative capacity. On the other hand, structural mixture invalidates threshold-based open-set methods and cross-class feature interpolation, leading to unreliable unknown-class rejection. To address these challenges, we propose HOPE, a Heterophily-aware Open-set node classification method with Pseudo-Extrapolation. To adapt open-set graph neural networks (GNNs) to heterophilic scenarios, HOPE uses a structure-augmented feature initialization layer to capture multi-hop structural patterns. Meanwhile, we design a trustworthy neighborhood aggregation mechanism for standard GNNs to dynamically filter noisy cross-class neighbors. To enhance unknown-class rejection, we introduce a heterophily-guided pseudo-extrapolation strategy. It dynamically maintains known-class centers and extrapolates along cross-class neighborhood displacement directions, synthesizing pseudo-unknown proxies near structurally ambiguous regions. Finally, we optimize the network with joint classification and logit margin regularization, routing synthetic proxies into a dedicated rejection slot without imposing geometric margin constraints in the representation space. Extensive experiments on multiple datasets show that HOPE consistently outperforms state-of-the-art models, validating its effectiveness, robustness, and efficiency.

cs.LG↗

arXiv2026

Hyperparameter Scaling Laws Across MoE Sparsity

Mixture-of-Experts (MoE) models expand model capacity without a proportional increase in training compute, but increasing sparsity makes reliable hyperparameter transfer challenging. In this work, we show that conventional hyperparameter scaling laws are insufficient for ultra-sparse MoEs: the optimal learning rate and batch size vary with activation ratio, and these shifts cannot be explained by either total or activated parameter count alone. To characterize this dependence, we conduct 1,800 pre-training runs spanning six activated-parameter scales and models with up to 6B total non-embedding parameters, processing approximately 20 trillion tokens at a cost of 200,000 equivalent H800 GPU-hours. Our results reconcile conflicting findings in prior work by revealing two scaling regimes. At fixed sparsity, the optimal batch size follows a power-law relationship with training tokens $D$, whereas the optimal learning rate scales with training compute $C$ and remains robust to the allocation between model size and data. Across sparsity levels, the activation ratio $A$ enters both relationships as an additional multiplicative power-law factor. These observations lead to unified hyperparameter scaling laws that transfer across MoE sparsity levels. Large-scale evaluation shows that the scaling form outperforms alternative functional forms. On a held-out ultra-sparse MoE with 12B total parameters and only 1/64 of its experts activated, the predicted hyperparameters remain close to the observed optima, supporting joint extrapolation across model scale and sparsity. Further experiments demonstrate transfer across expert granularities and isolate the effect of activation ratio from that of total expert count.

cs.LG↗

arXiv2026

Chimaera: A Mixture-of-Graph-Experts Architecture for Cross-Task and Cross-Dataset Graph Learning

Designing foundation models for graphs is challenging due to the irregular structure of graphs and the different sizes and characteristics of embeddings. Chimaera integrates mixture-of-experts with graph foundation models (GFM). It integrates different GFM architectures, such as graph prompts and linear GNN models. Large language models are used to generate embeddings, and experts can be trained and combined following different strategies, GFMs, embeddings, etc. Furthermore, Chimaera extends existing linear GNNs to support link-level and graph-level tasks in addition to node-level tasks. Empirical analyses are performed on same-task and cross-task experiments with node, link, and graph classification tasks using six benchmark text-attributed graph datasets. The experiments demonstrate the effectiveness of Chimaera and its capabilities for transfer across tasks and datasets. Further insights include the need to use both large and small language models to generate embeddings for the experts, a strong cross-task transferability of simple but effective linear GNNs, and using few samples only to provide strong results.

cs.LG↗

arXiv2026

BAFF: Bid-Aware Filter Family for Mitigating Training Data Interference in RTB A/B Tests

In online A/B tests for real-time bidding (RTB), control and treatment models are typically trained on a shared serving log that includes data generated by the counterpart model. This shared-log training biases each model's training data through two channels: the counterpart model may have selected a different ad from the ad-candidate pool (ad-ranking disagreement) and may have bid a different price (bid-pricing disagreement), potentially distorting the A/B test outcome. Log-splitting eliminates the bias but sacrifices training data; log-sharing retains all data but leaves the bias unaddressed. We formalize the Bid-Aware Filter Family (BAFF), a class of (k,l)-parameterized hard filters that controls tolerance to each channel independently, providing a structured search space between these two extremes. We further propose a three-stage online measurement protocol that enables evaluating data-sharing strategies by their deviation from an interference-free reference model in production. In offline simulation, a (k,l) sweep surfaces operating points with smaller deviation from the interference-free reference model than both log-sharing and log-splitting. In a live RTB deployment on a demand-side platform (DSP), filter-based variants preserve the reference model's business metrics (e.g., CPC, CTR) more closely than both baselines. The best operating point is setting-dependent, underscoring the practical value of the search space itself.

cs.LG↗

arXiv2026

PAC-Bayesian Bounds for Learning Partially Observed Stochastic Linear Time-Invariant State-Space Systems with Inputs and Sub-Gaussian Noise

In this paper we derive a Probably Approximately Correct (PAC)-Bayesian error bound for partially observed linear time-invariant (LTI) stochastic dynamical systems in state-space form with inputs and sub-Gaussian noise. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. The bound derived in this paper relates the expectation of prediction errors with the prediction error generated by the model on the data used for learning. In addition, we show that it can also be used to derive bounds for the parameter estimation error. In turn, this allows us to provide finite-sample error bounds for the prediction error and parameter estimation error for a wide class of system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

cs.LG↗

arXiv2026

A Note on Scaling in Randomly Rotated Quantization and Its Connection to the CDEF +1 Pythagorean Relation

Quantization schemes based on randomized rotations have recently received renewed attention, including the roles of MMSE and unbiased reconstruction scalings. In this note, we point out the connection to classical results in statistical signal processing and communication theory. Specifically, the two reconstruction scales used in the EDEN line of work admit a natural interpretation as finite-dimensional, realization-dependent counterparts of the Wiener and unbiased coefficients in the classical CDEF formulation. At finite blocklength, the CDEF +1 relation holds pointwise for each rotation realization as an exact geometric (Pythagorean) identity, but does not hold after averaging the distortions over the rotation. The classical SNR relation $\sf{SNR}_{\rm MMSE}=\sf{SNR}_{\rm MMSE,U}+1$ is recovered as $d\to\infty$: once the overall scale is handled separately, the empirical coordinate statistics of a randomly rotated vector approach their i.i.d. Gaussian counterparts, and the rotation-dependent quantities concentrate. Importantly, EDEN goes beyond this classical correspondence: for every finite $d$, its Haar-rotation formulation guarantees exact conditional unbiasedness, a stronger property than the second-order notion of unbiasedness in CDEF. We further comment on two distinct roles random rotations play in quantization: one is approximate Gaussianization of the coordinates; the other is decorrelation of reconstruction errors across quantization branches.

cs.IT↗

arXiv2026

Eliciting Weak-to-Strong Generalization with On-Policy Reverse Distillation

Weak-to-strong generalization asks whether stronger models can learn from weaker supervisors and surpass them. This question is particularly important for successive model generations and multi-domain consolidation, where repeating frontier-scale post-training from scratch can be prohibitively expensive. Yet conventional distillation treats the weak teacher as an optimization target, potentially imposing its capacity ceiling on the student. We introduce On-Policy Reverse Distillation (OPRD), which evaluates the teacher's policy shift relative to its reference policy on student rollouts and amplifies the component of the student's verifier-driven policy gradient along that direction. By rescaling only verifier-supported updates, OPRD preserves the stationary points of policy optimization while accelerating learning beyond the teacher. In both successive model transfer and multi-teacher distillation, OPRD achieves higher performance with fewer student updates than existing RL and distillation approaches. Response-style analysis shows that OPRD students remain closer to models trained with verifier-based RL alone than to their weak teachers, suggesting that teacher guidance accelerates rather than redirects the student's own optimization. Results in conventional strong-to-weak distillation further demonstrate that OPRD effectively combines verifier-driven policy optimization with teacher guidance regardless of capacity ordering.

cs.LG↗

arXiv2026

Ostrich: Taking Large Strides Through Stiff Contact in Differentiable Dynamics

Three properties determine whether a differentiable simulator can drive gradient-based optimization through contact: simulation accuracy, gradient reliability, and per-iteration cost. Tape-based engines such as MJX and Newton Semi-Implicit require timesteps small enough to keep contacts numerically tractable, and their backpropagation memory grows linearly with the number of timesteps T. Surrogate models bound memory by approximating contact away, but the resulting gradients lose the geometry the optimization depends on. We present Ostrich, a GPU-accelerated rigid-body simulator that resolves hard contacts and friction with non-smooth Newton iteration at large timesteps (h ~ 0.1 s), and differentiates the converged residual via the implicit function theorem, reusing the forward Schur complement to compute the adjoint at O(1) memory per timestep. On real-robot trajectories over a pallet obstacle, Ostrich holds MuJoCo's sim-to-real accuracy up to a 50x larger timestep. Its gradients converge from random initializations where MJX descends slowly and Newton Semi-Implicit stalls; a warm iteration runs 211x faster than MJX's and 4.7x faster than Semi-Implicit's. On the same scene Ostrich differentiates 8,192 parallel worlds on a single 24 GB GPU, sustaining 29x checkpointed MJX's optimization throughput; without checkpointing both baselines exhaust memory at far fewer worlds. We close with a gradient-based trajectory optimization demonstration over triangle-mesh terrain across a 10 s horizon, a setting where prior engines either restrict to primitive geometry or face the convergence and memory limits shown above.

cs.RO↗

arXiv2026

A Closed-Form Estimator and Diagnostic Battery for Anchor-Judge Error Correlation, Under a Single-Common-Factor Model

When an external reference set (an anchor) is used to decompose an LLM-judge panel's error into a quality signal and a shared common-mode error, standard practice assumes the anchor is uncontaminated: its error uncorrelated with the judges' shared error. We study when that assumption can be dropped and replaced by an estimate. Under a single-common-factor model, >=2 judges and >=2 anchors point-identify the quality variance, the common-mode variance, and each anchor's contamination correlation rho_k in closed form, with an exact per-anchor-pair failure boundary; a designated clean-anchor estimator, by contrast, reports a contaminated companion anchor as fully clean once its trusted anchor is itself contaminated. Because the single-common-factor assumption is itself untestable, the estimator ships gated behind a calibrated diagnostic battery (judge-covariance dispersion; over-identification; a family-block test from judge metadata, with a family-blocked estimator that removes family-level shared-residual bias exactly), bootstrap confidence intervals with measured coverage, and a weak-identification screen. A proposition maps which violations bias rho_k, in which direction, and which evade detection. For ordinal scores we show an identification hierarchy: with all variables ordinal, rho_k is not identified at any number of anchors; with ordinal judges and >=3 continuous anchors it is, and we give an estimator for that case. On real data the validation is asymmetric, and we say so plainly: the diagnostics are validated in the rejecting direction (both real panels we test are correctly rejected by the model-adequacy pre-test), while the estimator is validated in simulation and stress-tested semi-synthetically under oracle calibration; no real panel has yet passed the pre-test, and the pre-test exists precisely to say so. All results replay offline from shipped, checksummed artifacts.

stat.ME↗

arXiv2026

Length Generalization for Transformers via Compression

Recent advancements in transformer length generalization theory enable us to reliably predict when a transformer can learn to solve a task. In particular, the C-RASP hypothesis (a formalized version of the so-called RASP-l conjecture) posits that transformers length-generalize on a task if and only if a solution is expressible in the C-RASP language. While this hypothesis has strong empirical validation, theoretical problems arise from the fact that no computable length generalization bounds exist for C-RASP, alongside the discovery of seemingly contradictory experiments. To address these problems, we refine the C-RASP hypothesis utilizing the recently-proposed fragments C-RASP+ and C-RASP1. These fragments have computable length generalization bounds, though in the worst case requiring an extremely large (double exponential) sample size. It is an open question whether these sample size bounds are tight. In this paper, we resolve this open question by providing an exponentially tighter bound. In doing so, we show a polynomial length generalization bound for transformers if we adopt compressed strings, via a novel connection to power words. As an application, we show how this yields a fine-grained analysis of the C-RASP conjecture that resolves contradicting experimental evidence against it.

cs.LG↗

arXiv2026

Earth System World Model for What-If Simulations: A Case Study for Terrestrial Ecosystems

Machine learning emulators have become essential for accelerating expensive Earth-system simulations, but most existing approaches remain passive forecasters: they reproduce simulator trajectories under prescribed forcings without an explicit interaction mechanism for user-specified interventions. This limits their use in interactive scientific workflows and Earth-system digital twins, where users often need to explore how a system would respond if selected state components were changed. We propose an action-conditioned world-modeling framework for Earth-system emulation that reformulates simulator trajectories as supervision for controllable state-transition learning. The key idea is transition-action pretraining: naturally observed state changes are treated as label-free action supervision, allowing the model to learn both prescribed dynamics and action-conditioned responses without manually annotated interventions. We further introduce masked response learning to infer unobserved variables under partial state edits and learn coupled system dependencies. We test this framework on ecosystem dynamics across six global regions and multiple stand ages. Experiments show that the model preserves competitive long-horizon emulation accuracy while enabling controllable structural interventions and coherent responses in coupled ecosystem-cycle variables. These results suggest a practical route from passive Earth-system emulators toward interactive, intervention-aware scientific surrogates.

cs.LG↗

arXiv2026

High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression

Sparsity is a powerful structural resource in optimization and statistics. We develop frameworks for leveraging sparsity in sampling problems over the Hamming slice $\mathcal{X}_k^d:=\{\mathbf{x}\in\{\pm 1\}^d:|\{i:\mathbf{x}_i=1\}|=k\}$, in high-dimensional regimes where $k\ll d$ (i.e., where $\mathcal{X}_k^d$ is \emph{highly magnetized}). We use our frameworks to design improved samplers for canonical problems in the study of \emph{Ising models} and \emph{Bayesian sparse linear regression}. Our first main result considers the \emph{Sherrington--Kirkpatrick} (SK) model restricted to fixed-magnetization slices $\mathcal{X}_k^d$. We give a polynomial-time sampler for fixed-magnetization SK models at any inverse temperature $β>0$, under arbitrary external fields, provided that $k\le c_βd$ for an appropriate constant $c_β$. By combining this result with an annealing strategy for estimating normalizing constants, we obtain polynomial-time samplers for the SK model at arbitrarily low temperatures under a sufficiently strong external field of strength $h$. In the large-$β$ limit, our framework permits sampling at field strengths within constant factors of the \emph{Almeida--Thouless line} delineating the replica-symmetric and replica-symmetry-breaking regions ([dAT78]), improving polynomially over the field strength $h(β)$ required by the recent work of [BAR26]. Our second main result concerns the measurement complexity of polynomial-time Bayesian sparse linear regression. Recent work by [KSTZ25] shows how to sample from the canonical \emph{Gaussian spike-and-slab posterior} with expected sparsity $k$, at any signal-to-noise ratio, given $n\gtrsim k^3\log^3 d$ Gaussian measurements. We improve this requirement to $n\gtrsim k^{3/2}\log^2 d+k\log^3 d$, using a common sparsity-aware framework underlying both our results.

cs.DS↗

arXiv2026

ONE CYLinder: A Benchmark for Graph-Based Surrogate Modeling of Unsteady Bluff-Body Flows

Graph-based surrogate models offer a promising route to accelerate computational fluid dynamics (CFD) simulations on unstructured meshes. However, their development is limited by the scarcity of benchmark datasets spanning multiple flow regimes and standardized protocols for long-horizon autoregressive prediction. We introduce ONECYL (ONE CYLinder), a new benchmark for unsteady flow past a circular cylinder across laminar, transitional, and high-Reynolds-number regimes. The benchmark comprises 450 high-fidelity Variational Multiscale finite-element simulations (270,000 flow snapshots) with randomized cylinder geometries, providing time-resolved velocity and pressure fields together with mesh connectivity, geometric descriptors, Reynolds numbers, and integrated aerodynamic quantities. Beyond the dataset, ONECYL establishes a unified evaluation framework combining full-field rollout errors, virtual probes, and drag and lift predictions to assess numerical accuracy and physical fidelity. To accompany the benchmark, we develop a Graph Transformer as a reference baseline predicting velocity and pressure fields autoregressively on unstructured meshes. Using ONECYL, we investigate geometric representations and physics-based regularization across the three Reynolds-number regimes. The results show that explicitly encoding the cylinder geometry through a level-set representation consistently improves long-horizon prediction accuracy and generalization to unseen geometries, while divergence-based regularization becomes increasingly beneficial as flow complexity increases. The ONECYL benchmark and its Graph Transformer baseline provide a reproducible framework for evaluating graph-based surrogate models and establish a foundation for future research on long-horizon prediction of unsteady bluff-body flows.

physics.flu-dyn↗