Best-Of-N

Top-two algorithms are simple and effective for fixed-confidence best-arm identification, but their sharp non-asymptotic behavior is still not well understood. We study this problem for Bernoulli bandits through $β$-EB-TCI, the empirical-best top-two rule of Jourdan et al., whose challenger is chosen using a Bernoulli transportation cost with a logarithmic count penalty. We prove that, after the empirical leader has become the true best arm and its sampling fraction stays close to $β$, the stopping time is $T_β^{\star}(μ)\log(1/δ)$ up to lower-order concentration terms. We also show that, in this regime, every challenger is sampled linearly often. Thus, for the original algorithm without forced exploration, the main remaining difficulty is to control when the empirical leader becomes permanently correct. These results imply a non-asymptotic high-probability bound for all Bernoulli instances with a unique best arm. If the algorithm satisfies a finite-mean sufficient-exploration condition, the bound further yields the sharp expected sample complexity. In particular, this gives the sharp expectation result for the unguarded Bernoulli rule when all arm means are pairwise distinct, using the sufficient-exploration result of Jourdan et al. Finally, if we add a mild forced-exploration rule that contributes only $O(\sqrt{Kt})$ pulls up to time $t$, we obtain a self-contained expected sample-complexity theorem for any number of arms under the unique-best-arm assumption. We also identify a limitation of proof strategies that try to handle equal suboptimal means through a single index-comparison argument.

A self-check defense asks the target model to assess a request before answering it; SAGE, the strongest published instance, reports an average 99% defense success rate. We show it can be breached by composing two attacks that are individually harmless against it: an established code-completion encoding and an established best-of-N search, neither of which exceeds 4.7% of behaviors alone. Composed, with the search budget spent on the encoding, they reach 67/22/15% across three open targets, and the effect persists on a 70B target. We then explain the composition rather than only reporting it. First, a self-check defense borrows its strength from the target: SAGE does not detect the attack, it asks the model to, and the four targets convert that request into an explicit refusal between 32% and 97% of the time, which orders the spread in defended coverage even though undefended reach is near-identical. Second, which attack survives is decided by the type of defense, and it inverts: against transform defenses the code encoding retains far more of its undefended reach than the character search, while against gate defenses the ordering flips. We account for this with the number of independent probes an attack delivers to a defense's decision boundary. Finally, we report a validity defect we found and repaired in our own pipeline, a deterministic attack under greedy decoding has no best-of-N variation channel at all, and give the one-line diagnostic that detects it. All claims rest on 310,000 generations scored by a human-validated judge.

BoN improves model outputs by sampling several candidates and selecting one with a proxy score, but it assumes that complete candidates can be evaluated reliably. Many vision-language tasks instead provide only partial verification: a finding, span, value, region, or relation may be checkable even when no dependable whole-response verifier exists. Moreover, the same claim may recur across candidates with opposing stances, allowing one observation to support part of the pool and contradict another. We introduce Best-of-Evidence (BoE), an inference-time selection framework that keeps the BoN candidate pool fixed, represents reusable claims with a signed candidate--factor graph, and allocates a limited budget to evidence actions that can change the final choice. BoE formalizes selection under partial verification and provides a practical score-based controller, with the zero-budget case recovering the underlying BoN decision. Theoretically, we show that residual evidence capacity limits any evidence-driven improvement and that shared factor queries can achieve an O(log K) versus Θ(K) query separation in a factor-code model. Common-ledger experiments on four medical VQA settings show that BoE can improve fixed-pool selection and rescue some BoN failures when evidence is reliable, contrastive, and decision-relevant, while also revealing the channel-quality and candidate-generation limits that prevent universal gains.

We study the coupled objective J_K^WOR = E_{S ~ PL-WOR_K}[max_{i in S} R_i]: the expected maximum reward of a size-K Plackett-Luce draw without replacement, the law of Gumbel-Top-K / Stochastic Beam Search decoding. This estimand differs from the conventional i.i.d. objective J_K^iid = E[max_{i<=K} R_i] targeted by existing sample-reuse Max@K estimators, and reusing their i.i.d. weights under the coupled sampler is provably biased (a closed-form three-item instance gives E[g_iid] = (4/5) grad J_K^WOR exactly; pass@K under the coupled sampler is the binary-reward special case). Generic joint-score REINFORCE is already unbiased for J_K^WOR; what it lacks is sample reuse. Our contribution is to instantiate standard rank-conditioned Horvitz-Thompson estimation for the J_K^WOR subset total: from one Gumbel-Top-n pool (n>K) and its observed priority threshold we build an estimator that reuses all C(n,K) embedded K-subsets, unbiased with an unbiased exact score-function surrogate gradient, plus a reward-sorted Max-specific dynamic program that collapses the C(n,K)-term subset sum (with K!-cost set probabilities) exactly to a one-dimensional integral. A fixed-Q quadrature evaluation costs O(n log n + nKQ) arithmetic and is numerically, not algebraically, exact; no epsilon-approximation rate is certified. Each nonzero degree-K Horvitz-Thompson term has finite second moment exactly when n >= 2K; under the same assumptions the full surrogate gradient has finite second moment whenever n >= 2K (sharpness there is open). At K=1 the construction recovers classical priority sampling. All quantities require only the values and differentiable computation graphs of the n+1 drawn items' probabilities, so finite structured sequence policies sampled by exact SBS are covered. A certified finite-Q quadrature bound and countably infinite support remain open. Validation code is included as ancillary files.

Inference-time alignment methods, such as Best-of-$N$, offer a flexible alternative to training-based alignment by using reward models to select high-quality responses generated by a reference LLM. However, the efficacy of these methods is inherently limited by the response quality: if the reference LLM assigns negligible probability to high-reward responses, no selection strategy will succeed in finding aligned outputs. In this work, we propose Best-of-Better-$N$ (BoBN), an in context learning-based generation framework to address this challenge. Our method utilizes retrieval from high-reward examples relevant to the input query and task. Crucially, we introduce a restyling step where retrieved responses are rewritten by the reference LLM to align with the target task's format and style. These restyled examples are used in-context to shift the sampling distribution toward the high-reward region. We analytically characterize how in-context learning shifts the output distribution of pretrained transformers toward the high-reward region, resulting in provable benefits on the target task. We then evaluate BoBN on safety alignment and mathematical reasoning benchmarks across several reference LLMs. BoBN's higher-quality responses enable better performance to be achieved when the number of responses $N$ is fixed, and smaller $N$ required to achieve a target performance.

In longitudinal clinical practice, every chest X-ray is read in the context of the patients prior exam, and much of what the radiologist communicates is the change from one visit to the next. To the best of our knowledge, we present the first training-free best-of-N sampling scheme for pre-trained chest X-ray report generators that is explicitly aware of this longitudinal prior to current transition. We call it transition-aware best-of-N sampling, each report is split into sentences and embedded into an unordered set in Rd; each (prior, current) pair is reduced to a fixed-dim directional vector via a set-to-set distance designed to encode the change between the two sets; and candidates are scored by cosine distance from their candidate transition vector to a cached bank of ground-truth training transition vectors, aggregated as min or kNN. We instantiate the framework with four directional set distances (mean-shift, novelty residual, directed-Hausdorff anchor, and cost-weighted optimal transport) and evaluate on a multi-visit AP-PA cohort, running inference under three prompts on three vision-language generators. Transition-aware best-of-N outperforms random selection across the board, with the largest relative gains on the Impression section.

Best-of-$N$ inference scaling (drawing $N$ candidate answers from a language model and returning the one a reward model ranks highest) improves accuracy by an amount that varies across models, but predicting that amount in advance currently requires running the procedure end-to-end. Prior work links cheap statistics of a model's sampled outputs and validation-set correctness (how often samples agree, how diverse they are, how confident the model is, and where correct samples appear) to model behavior, but does not isolate which of these form a stable, compact predictor of best-of-$N$ gain. We fit ridge predictors on features computed from a single labeled validation-set sampling pass, use bootstrap-Lasso as a stability analysis of the candidate feature set, and give a concentration analysis with an explicit linear-approximation residual. Across three base-model families, six post-training methods, and math and reasoning task domains, the stability analysis identifies a strict three-feature core spanning prompt-level agreement spread, label-assisted first-correct-sample position, and completion-length variance; a compact ridge predictor built from this core plus an entropy add-on reaches Spearman $ρ= 0.90$ with actual best-of-$N$ gain under a reward-model verifier. The intended use is labeled validation-set screening of candidate configurations before paying the full reward-model scoring cost.

Test-time scaling for vision-language-action (VLA) policies, methods such as RoboMonkey, SEAL, MG-Select, and V-GPS, samples K candidate action chunks at inference and executes the verifier-best. When all K candidates are unsafe, the system executes a violating action with no warning. We propose BOKBO, the first conformal abstention layer for K-sample VLA inference, providing finite-sample distribution-free guarantees on executed-violation rate. We provide both global and per-task (Mondrian) variants, with the per-task variant closing the conditional gap on the hardest tasks. Our analysis exposes a structural failure of policy-internal nonconformity scores under perturbation-based K-sampling: the base-policy confidence proxy and K-sample disagreement correlate at 0.98 with the action-noise hyperparameter $σ$, while correlating at the noise floor with actual safety violations. We test the failure's scope by replicating the analysis under token-level temperature sampling and find the failure is mechanism-specific and partially mitigated under policy-stochasticity-based sampling. A learned violation predictor conditioned on semantic visual features and task identity supports tight calibration: at $ε$ = 0.05 on libero_object_temp_x0.1 with OpenVLA-OFT, the conditional CRC bound holds on 86% of bootstrap splits with 78% coverage and 70% net task success. Mondrian-BOKBO raises the minimum per-task conditional hold fraction from 0.71 to 0.93. Results are stable across 5 training seeds, replicate within bootstrap noise on $π_0$-FAST, hold on libero_spatial_temp_x0.1 as a co-equal benchmark, and survive four within-suite distribution shifts. We additionally identify and correct a methodological pitfall: globally-set force thresholds well below expert-typical manipulation forces conflate unsafe behavior with normal manipulation, inflating violation rates by $5\times$.

Best-of-$N$ sampling is widely used to construct pairwise preference data: $N$ candidates are drawn from a base distribution, and the best is paired with a rejected response. Despite its widespread use, what Bradley--Terry (BT) reward learning extracts from such data, and how to choose $N$ and the base distribution, remain unclear. We specialize a recent analysis of preference data via its induced conditional distribution to Best-of-$N$. For independent-reference variants, we derive closed-form reward targets as explicit functions of $N$ and the base distribution, and show that they preserve the latent reward ranking. For the practical Best-vs-Random and Best-vs-Worst variants, chosen and rejected responses are coupled through the same candidate set, so exact BT representability generally fails; nevertheless, bounded-class minimizers approach the reference targets as $N$ grows. Although margin and connectivity are known to govern sample efficiency in pairwise preference learning, Best-of-$N$ couples them through $N$ in opposing directions: larger $N$ widens pairwise margins but reduces connectivity. This trade-off yields two design principles: use larger $N$ when preference labels are the bottleneck, smaller $N$ when generation is the bottleneck; and shape the base distribution to place mass between the responses whose comparison matters most at test time. Experiments on synthetic and real preference data support the predicted dependence on sample size and base-distribution shape.

On-policy distillation (OPD), which supervises a student on its own sampled trajectories, has emerged as a data-efficient post-training method for improving reasoning while avoiding the reward dependence of reinforcement learning and the catastrophic forgetting often observed in standard supervised fine-tuning. However, standard OPD typically computes teacher supervision under noisy student-generated contexts and often relies on a single stochastic teacher rollout per prompt. As a result, the supervision signal can be high-variance: the sampled teacher trajectory can be incorrect, uninformative, or poorly matched to the student's current reasoning behavior. To address this limitation, we propose BRTS, a Best-of-N Rollout Teacher Selection framework for on-policy distillation. BRTS augments standard student-context OPD with a teacher-context supervision branch constructed from the curated teacher trajectory. Rather than distilling from the first sampled teacher rollout, BRTS samples a small pool of teacher trajectories and selects the auxiliary trajectory using a simple priority rule: correctness first, student alignment second. When multiple correct teacher trajectories are available, BRTS chooses the one most aligned with the student's current behavior; when unconditioned teacher samples fail on harder prompts, it invokes a ground-truth-conditioned recovery step to elicit a natural derivation. The selected trajectory is then used to provide reliable teacher-context supervision inside the OPD loop, augmented with an auxiliary loss on the teacher trajectory. Experiments on AIME 2024, AIME 2025, and AMC 2023 show that BRTS improves over standard OPD on challenging reasoning benchmarks, with the largest gains on harder datasets. Our code is available at https://github.com/BWGZK-keke/BRTS.

Traditional change detection identifies where changes occur, but does not explain what changed in natural language. Existing remote sensing change captioning datasets typically describe overall image-level differences, leaving fine-grained localized semantic reasoning largely unexplored. To close this gap, we present RSRCC, a new benchmark for remote sensing change question-answering containing 126k questions, split into 87k training, 17.1k validation, and 22k test instances. Unlike prior datasets, RSRCC is built around localized, change-specific questions that require reasoning about a particular semantic change. To the best of our knowledge, this is the first remote sensing change question-answering benchmark designed explicitly for such fine-grained reasoning-based supervision. To construct RSRCC, we introduce a hierarchical semi-supervised curation pipeline that uses Best-of-N ranking as a critical final ambiguity-resolution stage. First, candidate change regions are extracted from semantic segmentation masks, then initially screened using an image-text embedding model, and finally validated through retrieval-augmented vision-language curation with Best-of-N ranking. This process enables scalable filtering of noisy and ambiguous candidates while preserving semantically meaningful changes. The dataset is available at https://huggingface.co/datasets/google/RSRCC.

This paper proposes a novel method that incorporates empowerment when reasoning actions in reinforcement learning (RL), thereby achieving the flexibility of exploration-exploitation dilemma (EED). In previous methods, empowerment for promoting exploration has been provided as a bonus term to the task-specific reward function as an intrinsically-motivated RL. However, this approach introduces a delay until the policy that accounts for empowerment is learned, making it difficult to adjust the emphasis on exploration as needed. On the other hand, a trick devised for fine-tuning recent foundation models at reasoning, so-called best-of-N (BoN) sampling, allows for the implicit acquisition of modified policies without explicitly learning them. It is expected that applying this trick to exploration-promoting terms, such as empowerment, will enable more flexible adjustment of EED. Therefore, this paper investigates BoN sampling for empowerment. Furthermore, to adjust the degree of policy modification in a generalizable manner while maintaining computational cost, this paper proposes a novel BoN sampling method extended by Tsalis statistics. Through toy problems, the proposed method's cability to balance EED is verified. In addition, it is demonstrated that the proposed method improves RL performance to solve complex locomotion tasks.

Test-time scaling (TTS) improves large language models (LLMs) by allocating additional compute at inference time. In practice, TTS is often achieved through parallel scaling: generating multiple candidate responses and selecting the best via a Best-of-N (BoN) strategy. Its effectiveness therefore hinges on the scoring function. Learned scorers such as process reward models (PRMs) can be strong, but they are expensive to train and run. Lightweight confidence heuristics based on token log-probabilities are much cheaper, yet we find that they often perform substantially worse. To improve on lightweight confidence heuristics without incurring the full cost of stronger learned scorers, we introduce SCATR, a simple and efficient BoN ranking method that learns a lightweight scorer from a small calibration set using hidden representations from the base model. Across coding and mathematical reasoning benchmarks, SCATR improves over prior confidence-based baselines by up to 9%. Relative to LoRA fine-tuning on the same calibration data, it achieves comparable accuracy with up to 8000x fewer trainable parameters and much lower compute, reducing training and inference latency by up to 150x and 1000x, respectively. SCATR is also competitive with strong PRM baselines, and in several settings improves accuracy by up to 7.8% on math and 4.2% on coding while enabling up to 1000x faster inference. Overall, SCATR offers a strong accuracy-efficiency trade-off for scalable test-time selection.