#01Oct 22, 2026
math.SG
SympFormer: Inertial Attention Blocks from Damped Hamiltonian Dynamics on Spaces of Probability Measures
Viktor Stein, Levin Maier, Wuchen Li and 1 more
The attention mechanism underpins modern large language models, yet their dynamics remain only partially understood. Recent perspectives interpret attention blocks as interacting particle systems, whose mean-field limits correspond to gradient flows of energy functionals on spaces of probability densities on $\mathbb R^d$ equipped with Wasserstein-type metrics. We extend this viewpoint by deriving accelerated attention blocks from inertial, Nesterov-type damped Hamiltonian dynamics on these density spaces, developing the geometry and differential calculus needed to make the construction rigorous. In particular, we introduce a local-mobility geometry for the Otto-Wasserstein and softmax-Wasserstein settings, and a quotient Stein geometry that resolves the degeneracy of the linear-attention kernel to yield a well-defined metric and differential calculus. We provide rigorous foundations when softmax attention admits a gradient-flow formulation and when its inertial counterpart admits a damped Hamiltonian on explicitly characterized domains. For linear attention, we obtain a finite-dimensional solution theory with sharp conditions for global existence versus finite-time second-moment blow-up. We further prove a quantitative link between continuous moment blow-up and the discrete-layer depth at which token norms cross a prescribed threshold. Finally, motivated by the finite-token geometry, we propose geometric corrections to token and momentum updates. In decoder-only, nanoGPT-style experiments on TinyStories and OpenWebText, these corrections lower validation negative log-likelihood on next-token prediction relative to momentum-only controls.
#02Oct 22, 2026
cs.CL
Would an LLM Match into a Residency? Uncertainty, Reasoning and Image Use of Open Language Models on the Spanish MIR Examination
Miguel Fernandez-de-Retana, Unai Zulaika, Ruben Sanchez-Corcuera and 2 more
Accuracy on medical examinations is common evidence offered for the clinical use of language models, yet a health system that intends to run an open model on its own hardware needs to know considerably more: whether the model recognizes when it is likely to be wrong, whether medical fine-tuning improves this ability, whether its correct answers rest on sound clinical reasoning, or whether it actually uses the images it is given. We examine these questions for four open models of 8 to 27 billion parameters on the Spanish MIR residency examination, scoring 1,552 questions from eight editions under the exam's own rules. Answering every question, the three larger models would rank among roughly the first 600 to 3,000 candidates, with 41 to 43 specialties still open to them. Abstention, however, never pays, because even their least confident answers are right more often than the 25% break-even. Medical fine-tuning reduced overconfidence at the cost of accuracy, without improving how well confidence separates right from wrong answers. Moreover, 13-38% of correct answers rejected a distractor for a clinically unsound reason that confidence did not reflect, and removing or swapping the provided contextual images left accuracy unchanged.
#03Oct 22, 2026
cs.AI
Direct Pass@$k$ Optimization for Thought Pruning: Balancing Quality and Diversity in Test-Time Reasoning
Mahdi Salmani, Adel Javanmard, Devansh Gupta and 1 more
Test-time scaling has emerged as a key inference strategy for reasoning models, where a model generates a large number of candidate solutions in the hope of finding at least one correct solution. Running all candidate solutions to completion can be computationally demanding, so recent work prunes the candidate pool after generating only a short prefix, committing further computation only to the surviving candidates. Existing pruning strategies divide broadly into two categories: Quality-based methods assign a score to each partial solution and discard those predicted to ultimately lead to an incorrect final solution, while diversity-based methods select partial solutions that follow different reasoning trajectories and are therefore likely to cover different final solutions. The tension between these two objectives is an instance of exploration versus exploitation, yet little to no prior work on pruning exposes this trade-off directly, let alone optimizes it. In this paper, we introduce a pruning procedure that addresses this trade-off directly by optimizing pass@$k$ over the selected solution pool. In particular, it chooses the pool to maximize the probability that at least one of its partial solutions, when continued to completion, yields a correct answer. Because this objective simultaneously rewards high-quality partial solutions and penalizes redundancy among them, quality and diversity are naturally balanced by the criterion itself rather than through separate ad-hoc design choices. On eight open-weight reasoning models, our method outperforms existing quality-based and diversity-based pruning strategies. The gains are particularly pronounced when the pruning procedure keeps a moderate number of partial solutions for continued generation, highlighting the value of directly balancing quality and diversity during pruning.
#04Oct 22, 2026
math.NA
Phase Transition from Random Features to Fully Trainable Networks
Xin He, Huan Lao, Xiliang Lu and 1 more
Constructing accurate and computationally tractable approximation spaces is a fundamental task in scientific computing. Partially trainable neural networks bridge fixed random features and fully adaptive representations. This paper quantifies a phase transition pattern in approximation bounds and observes analogous behavior in training errors: a rapid but smooth initial improvement followed by diminishing returns, with a sharper transition under greater initialization mismatch. Under polynomial spectral decay with exponent $α>1$ and leverage compatibility, we establish a squared $L_2$ approximation rate $\mathcal{O}((κ_P\log(m/δ)/m)^α)$ for two-layer networks of width $m$ and trainable proportion $P$, where $κ_P=(P/κ_1^*+(1-P)/κ_0^*)^{-1}$. The bound holds uniformly over the unit ball of a reproducing kernel Hilbert space (RKHS) with probability at least $1-δ$. The constants $κ_0^*$ and $κ_1^*$ quantify the fixed and trainable groups' sampling mismatches relative to the target RKHS. For spherical biased-ReLU features in dimension $d$, we verify these assumptions with $α=(d+2)/(d-1)$ and $κ_0^*=4\sqrt{2}τ(d+1)/d$ for initialization scale $τ\ge1$, and $κ_1^*=1$ under ridge leverage sampling. For $d=5$ and $τ=10$, a trainable proportion of approximately $4.1\%$ captures $90\%$ of the reduction in the approximation bound from $P=0$ to $P=1$. A structured three-layer example illustrates how a few trainable first-layer neurons exploit fixed downstream blocks through nonlinear composition. Numerical experiments across diverse settings, including networks with up to 15 layers, exhibit similar phase transitions in training errors.
#05Oct 22, 2026
cs.LG
OP-Print: Tracing Membership Imprints in On-Policy Self-Distillation
Qi Li, Xinchao Wang
On-policy distillation has been widely adopted for alleviating the distribution mismatch of off-policy distillation. Self-distillation further removes the need for an external teacher by using the student's own outputs, enhanced with extra context or demonstrations, as supervision. This teacher-free design makes on-policy self-distillation practical, yet its privacy risks remain largely unexplored. In particular, using two coupled views of the same training example may strengthen its memorization. Motivated by this, we present the first study of membership inference attacks against on-policy self-distillation. We investigate both gray-box and black-box threat models and propose a unified attack framework, named OP-Print. OP-Print builds on our finding that the supervision imprint in on-policy self-distillation is sparse and structured, concentrating on uncertainty-sensitive correction sites rather than across the entire rollout. Exploiting this property, we construct attack signals tailored to the two access levels: in the gray-box setting, OP-Print directly aggregates the privileged teacher-student residual correction along the same rollout; in the black-box setting, it measures restart-based output-basin stability as an observable surrogate for the hidden correction process. Across diverse models and distillation schemes, all baseline attacks remain essentially at random guessing even under gray-box access, whereas OP-Print is broadly validated to be effective in both gray-box and black-box settings.