Adaptive Self-Consistency: From Black-Box Sampling to Distribution-Valued Feedback

Jingkai Huang; Yunfan Zhang; Will Ma; Weihua Zhou; Zhengyuan Zhou

cs.LGSep 30, 2026

Adaptive Self-Consistency: From Black-Box Sampling to Distribution-Valued Feedback

Authors: Jingkai Huang, Yunfan Zhang, Will Ma, Weihua Zhou, Zhengyuan Zhou

Organizations: Stern School of Business New York University · School of Management Zhejiang University

Abstract

Self-consistency samples many reasoning trajectories and aggregates their final answers, treating the LLM as a black box that returns one answer per trajectory. Yet the final answer of each trajectory is sampled from a softmax vector that is available from the model's log-probabilities. We refer to this as the grey-box setting in which each trajectory reveals this answer distribution rather than a single draw from it. We formulate efficient inference in this setting as sequential mode identification with distribution-valued observations: sample trajectories one at a time and stop as soon as the LLM's modal answer is identified at a prescribed confidence level. We characterize the asymptotic stopping rate of mode identification with distribution-valued observations exactly and show that it is never worse than the black-box rate. We then propose the ASC-D algorithm, a betting stopping rule that attains this asymptotic stopping rate. On MMLU-Redux, ASC-D uses $46.4$ -- $95.6\%$ fewer trajectories than answer-only adaptive self-consistency baselines and achieves the highest fixed-budget correct-certification rate across three open-source models.

Figures & tables

Panel of Figure 1 — Figure 1: Black-box and grey-box feedback from the same reasoning process.

$c$	$0.01$	$0.1$	$1$	$2$	$3$	$5$	$10$	$20$	$50$	$100$	$1000$
$T_{\rm cw}(F)$	$139.52$	$139.50$	$138.22$	$125.66$	$109.15$	$84.99$	$55.76$	$33.70$	$15.62$	$11.85$	$10.76$
$T_{\rm gb}(F)$	$138.15$	$126.95$	$70.33$	$47.24$	$35.70$	$24.17$	$15.01$	$12.34$	$11.16$	$10.82$	$10.52$
$T_{\rm bb}(F)/(c+1)$	$138.14$	$126.84$	$69.76$	$46.51$	$34.88$	$23.25$	$12.68$	$6.64$	$2.74$	$1.38$	$0.14$

Table 1: Characteristic times for

F=\mathrm{Dir}(c\boldsymbol{\pi})

with

\boldsymbol{\pi}=(0.4,0.3,0.2,0.1)

. For all

c

T_{\rm bb}(F)=139.52

and

1/\log(1+\Delta)=10.49

. The values of

T_{\rm cw}(F)

and

T_{\rm gb}(F)

are computed numerically through simulation. The last row is the fixed-

c

, small-gap approximation in ( 7 ).

		ASC-D		ASC-D (one-hot)		PPR-1v1		CITE-4
$T_{a}$	$\Delta_{\mathrm{geo}}$	Num. Gen.	Ans. Acc.	Num. Gen.	Ans. Acc.	Num. Gen.	Ans. Acc.	Num. Gen.	Ans. Acc.
4	0.61	24.67	72.51%	41.82	71.83%	46.05	71.65%	58.19	71.70%
8	0.42	27.82	72.40%	90.73	70.97%	99.21	70.92%	113.92	70.94%
16	0.21	42.17	72.36%	346.55	70.78%	380.76	70.77%	403.87	70.78%
32	0.10	76.87	72.16%	1593.49	70.73%	1732.97	70.74%	1711.74	70.74%

Table 3: Number of trajectory generations required for

95\%

confidence and the answer accuracy on the 90 common questions. Here,

\Delta_{\rm geo}

denotes the geometric mean of the modal gap.

Model	Method	$1-\delta=0.95$	$1-\delta=0.975$	$1-\delta=0.99$
Llama-3.2-1B	ASC-D	56.57%	53.38%	49.83%
	CITE-4	37.51%	35.21%	32.93%
	PPR-1v1	36.11%	34.14%	31.94%
Llama-3.1-8B	ASC-D	80.62%	78.84%	76.44%
	CITE-4	71.69%	69.48%	66.94%
	PPR-1v1	70.34%	68.34%	66.01%

Table 4: Correct-certification rate within

N=128

trajectories on the 90 common questions.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Option token	Token ID	$\ell_{k}$	$q_{k}$	$\theta_{k}$
A	362	$-1.3715$	$0.2537$	$0.2635$
B	426	$-0.8715$	$0.4183$	$0.4344$
C	356	$-1.6215$	$0.1976$	$0.2052$
D	423	$-2.3715$	$0.0933$	$0.0969$
Sum			$0.9630$	$1.0000$

Appendix

Table 5: A single real-world grey-box observation. The raw probabilities

q_{k}

are the full-vocabulary next-token probabilities assigned to the four allowed label tokens. Their total mass is

0.9630

and renormalizing over A–D at

T_{a}=1

gives the observed vector

\boldsymbol{\theta}=(0.2635,0.4344,0.2052,0.0969)

Figure 2: Characteristic-time scaling across the feedback family

F_{\rho}

Figure 3: Mean stopping time versus Dirichlet concentration.

	ASC-D	ASC-D (one-hot)	PPR-1v1	CITE-4
Median $\widehat{\beta}$	0.766	1.969	1.966	1.854
95% CI	$[0.692,0.810]$	$[1.953,2.001]$	$[1.949,1.990]$	$[1.841,1.861]$

Appendix

Table 6: Gap-scaling exponents on the 90 common questions.

Figure 4: Required trajectory budget versus inverse empirical modal gap on the 90 common questions. Points show geometric means, error bars show 95% paired question-bootstrap confidence intervals, and reference lines have slopes one and two on the log–log scale.

Model	Method	$1-\delta=0.95$	$1-\delta=0.975$	$1-\delta=0.99$
Llama-3.2-1B	ASC-D	74.48%	71.99%	69.06%
	CITE-4	55.59%	53.40%	51.20%
	PPR-1v1	54.25%	52.43%	50.30%
Qwen3-1.7B	ASC-D	97.48%	97.27%	96.94%
	CITE-4	97.07%	96.72%	96.20%
	PPR-1v1	96.86%	96.55%	95.97%

Appendix

Table 7: Correct-certification rate within

N=128

trajectories on 100 ARC-Challenge questions. Each entry aggregates 200 paired replays per question.

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Self-consistency (SC) is a widely used test-time inference technique for improving performance in chain-of-thought reasoning. It consists of generating multiple responses, or ``samples", from a large language model (LLM) and selecting the most frequent answer. This procedure can naturally be viewed as a majority vote or empirical mode estimation. Despite its effectiveness, self-consistency is prohibitively expensive at scale when naively applied to datasets, and it lacks a unified theoretical understanding of sample efficiency and scaling behavior. In this paper, we provide the first comprehensive analysis of SC's scaling behavior and its variants, drawing on mode estimation and voting theory. We derive and empirically validate power law scaling for self-consistency across datasets, and analyze the sample efficiency for fixed-allocation and dynamic-allocation sampling schemes. From these insights, we introduce Blend-ASC, a novel variant of self-consistency that dynamically allocates samples to questions during inference, achieving state-of-the-art sample efficiency. Our approach uses 4.8 times fewer samples than vanilla SC on average, outperforming both fixed- and dynamic-allocation SC baselines, thereby demonstrating the superiority of our approach in terms of efficiency. In contrast to existing variants, we note that Blend-ASC is hyperparameter-free, supports batching, and can fit any budget of samples, ensuring it can be easily applied to any self-consistency application.

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