#01Oct 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.
#02Oct 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.
#03Oct 22, 2026
cs.LG
Rethinking Sequence Compression with Test-time Regression
Sihan Yang, Jie Huang, Hanxuan Li and 4 more
Sequence compression is a fundamental operation and a critical information bottleneck in efficient language and multimodal models, and yet prevailing approaches based on pooling or learned projections remain underperforming. We revisit sequence compression through the lens of test-time regression: input tokens define a regression problem whose fitted coefficients form the compressed representation. Importantly, the regression problem is learned end-to-end via backpropagation, while its coefficients are computed efficiently in closed form during the forward pass. This leads to TTR-Compressor, a general alternative to conventional pooling and projection-based compression. We evaluate TTR-Compressor as the compressor for block-wise sparse attention and the visual token merger for multimodal large language models. On block-wise sparse attention, TTR-Compressor improves RULER performance over attention-based compressors by a large margin. For visual token merging, it consistently outperforms the standard MLP merger and matches its performance with 55.6% fewer visual tokens. These results establish test-time regression as an effective primitive for sequence compression in efficient language and multimodal models.
#04Oct 22, 2026
cs.CV
Beyond Best-of-$k$: Prior-Consistent DAPS for Ambiguous Nonlinear Inverse Problems
Merve Kocabas, Benjamin A. Burns, Andreas Geiger and 2 more
Diffusion models have become strong priors for solving inverse problems. However, nonlinear inverse problems such as phase retrieval remain challenging when multiple measurement-consistent solutions are distinguishable only by the prior; we call such measurements ambiguous. Existing diffusion-based samplers rely on local approximations during reconstruction. For an ambiguous measurement, these approximations can allow measurement guidance to favor a measurement-consistent mode that is close to a prior-supported mode but is itself not supported by the prior. Evaluation often masks this type of failure by reporting the best-of-$k$ runs, where the best is selected by comparison to the ground truth. We introduce Prior-Consistent DAPS (PC-DAPS), a variant of Decoupled Annealing Posterior Sampling (DAPS) designed to address this failure case by maintaining multiple candidate solutions within a single sampling run. As annealing noise decreases, PC-DAPS prunes and replaces weaker candidates according to their prior consistency, without any ground-truth access. PC-DAPS reduces the incidence of prior-inconsistent samples and improves sample quality on nonlinear inverse problems while preserving sample quality and diversity on linear inverse problems.
#05Oct 22, 2026
cs.LG
On the Token Collapse Phenomenon in Textual Graph Soft Prompting
Andrea Giuseppe Di Francesco, Rishabh Jain, Franco Maria Nardini and 3 more
Recent methods for injecting graph structure into LLM prompts include graph soft prompting, in which a graph neural network (GNN) learns graph-specific tokens that capture relational information among entities in textual prompts. G-Retriever combines graph soft prompts with a verbalized description of the same graph to strengthen the textual prompt signal. We conduct an in-depth analysis of G-Retriever and uncover a pathological behavior we term token collapse. Across three benchmarks, three graph encoders, two LLM families, and both frozen and LoRA-adapted models, G-Retriever's graph tokens become almost indistinguishable across inputs. We provide evidence that the same failure is inherited by subsequent methods, even with multi-token pooling. Under the standard setup, swapping a token with the dataset mean, a token from another example, or one derived from a corrupted graph produces no resolved answer-level effect in accuracy. Moreover, a learned constant prompt that never accesses the graph matches or surpasses G-Retriever on all three benchmarks. These findings indicate that, rather than conveying graph content as intended, the collapsed token functions primarily as a generic soft prompt. We also examine conditions linked to this collapse and propose a two-stage procedure that yields non-collapsed tokens while maintaining competitive performance in most cases. However, token variability alone does not prove that the model uses graph information: via interventions and decoding probes, we show that depending on the configuration, a token can reflect the question, an answer prior, or graph-derived content that the LLM may or may not use. Overall, our results offer a practical approach to evaluating graph-token alignment rather than inferring it from task accuracy alone. Code to reproduce our experiments is available at https://github.com/difra100/Token_Collapse_KGQA.