PETS: A Principled Framework Towards Optimal Trajectory Allocation for Efficient Test-Time Self-Consistency

Zhangyi Liu; Huaizhi Qu; Xiaowei Yin; He Sun; Yanjun Han; Tianlong Chen; Zhun Deng

cs.LGFeb 18, 2026

PETS: A Principled Framework Towards Optimal Trajectory Allocation for Efficient Test-Time Self-Consistency

Authors: Zhangyi Liu, Huaizhi Qu, Xiaowei Yin, He Sun, Yanjun Han, Tianlong Chen, Zhun Deng

Organizations: Stanford University · UNC at Chapel Hill · Yale University · New York University

Abstract

Test-time scaling can improve model performance by aggregating stochastic reasoning trajectories. However, achieving sample-efficient test-time self-consistency under a limited budget remains an open challenge. We introduce PETS (Principled and Efficient Test-TimeSelf-Consistency), which initiates a principled study of trajectory allocation through an optimization framework. Central to our approach is the self-consistency rate, a new measure defined as agreement with the infinite-budget majority vote. This formulation makes sample-efficient test-time allocation theoretically grounded and amenable to rigorous analysis. We study both offline and online settings. In the offline regime, where all questions are known in advance, we connect trajectory allocation to crowdsourcing, a classic and well-developed area, by modeling reasoning traces as workers. This perspective allows us to leverage rich existing theory, yielding theoretical guarantees and an efficient majority-voting-based allocation algorithm. In the online streaming regime, where questions arrive sequentially and allocations must be made on the fly, we propose a novel method inspired by the offline framework. Our approach adapts budgets to question difficulty while preserving strong theoretical guarantees and computational efficiency. Experiments show that PETS consistently outperforms uniform allocation. On GPQA, PETS achieves perfect self-consistency in both settings while reducing the sampling budget by up to 75% (offline) and 55% (online) relative to uniform allocation. Code is available at https://github.com/ZDCSlab/PETS.

Figures & tables

Figure 1: In this paper, we study how to allocate an LLM’s sampling budget across questions to best match the full-budget outcome under self-consistency. Our results show that PETS substantially reduces the required budget while maintaining accuracy.

Crowdsourcing	Test-time self-consistency
Item $i$ , Worker label $Y_{i}^{(b)}$	Question $q_{i}$ , Trace answer $Y_{i}^{(b)}$
Adaptive worker assignment to items	Adaptive trajectory sampling for questions
Posterior over labels and worker reliability	Posterior over answers and confidence weights
Worker-specific reliability	Trace confidence without persistent worker ID
Goal: infer the true label	Goal: infer the population majority answer $y_{i}^{\infty}$ (or $y_{i}^{W,\infty}$ )

Table 1: Connection to crowdsourcing.

Figure 2: Budget allocation plan of the offline and online settings on 9 simulated binary choice questions,

\mathcal{Y}=\{1,2\}

. Each question is associated with a

\theta=\max(\theta_{1},\theta_{2})

, and larger theta indicates easier questions.

		Qwen3-4B			Qwen3-30B			Qwen-Long			GPT-20B			GPT-120B
Dataset	Method	# traces $\downarrow$	Con. $\uparrow$	Acc. $\uparrow$	# traces	Con.	Acc.	# traces	Con.	Acc.	# traces	Con.	Acc.	# traces	Con.	Acc.
	PETS-Off.	2780	1.00	69.7	2607	1.00	71.8	2513	1.00	76.3	3180	1.00	7 5.7	2580	1.00	82.5
	PETS-Off. (conf)	3 667	0 .99	70.3	3 687	0 .99	7 2.1	2 887	0 .99	77.0	3 693	0 .99	76.2	3 540	0 .99	8 2.1
	Conf. guided	10652	0.96	68.5	9702	0.96	71.2	8158	0.96	76.1	10058	0.96	75.4	8474	0.97	81.6
	Uniform	11013	0.96	68.9	10367	0.96	71.5	9973	0.96	7 6.5	10853	0.96	75.1	10393	0.97	81.5
GPQA	Uniform (conf)	11453	0.95	6 9.9	11607	0.95	72.2	11400	0.95	76.4	11193	0.96	75.6	10500	0.96	80.8

Table 2: PETS-Offline Results. We compare PETS against confidence-guided and uniform sample budget allocation. “(conf)” denotes the trace confidence-weighted variant. # traces reports the number of sampled traces required to reach full consistency (consistency

=1

); Con. and Acc. report each method’s achieved consistency and accuracy evaluated at the trace count where PETS-Off. attains consistency 1.

Figure 3: Budget allocation curve in the offline setting. “(conf)” denotes the trace confidence-weighted variant. Consistency is computed within each matched-variant comparison group: PETS-Offline vs. Uniform, and PETS-Offline (conf) vs. Uniform (conf).

		Qwen3-4B			Qwen3-30B			Qwen-Long			GPT-20B			GPT-120B
Dataset	Method	# traces $\downarrow$	Con. $\uparrow$	Acc. $\uparrow$	# traces	Con.	Acc.	# traces	Con.	Acc.	# traces	Con.	Acc.	# traces	Con.	Acc.
	PETS-On.	4662	1.00	68.8	3826	1.00	71.6	4 852	1.00	76.3	6826	1.00	75.9	4172	1.00	82.4
	PETS-On. (conf)	4 888	1.00	69.6	4 357	0 .99	7 1.8	4146	1.00	76.9	7 269	0 .99	7 6.1	5 094	0 .99	81.6
	Conf. guided	7623	0 .97	68.2	7361	0.98	70.5	7782	0 .98	75.4	9355	0.98	74.3	7162	0.97	80.5
	Uniform	9520	0 .97	68.1	8456	0.97	71.2	9195	0.97	76.0	9699	0.98	7 6.1	8109	0.98	8 1.8
GPQA	Uniform (conf)	9856	0 .97	6 9.4	9257	0.96	72.0	9526	0.97	7 6.4	9895	0.98	76.3	9593	0.97	81.4

Table 3: PETS-Online Results. We compare PETS against confidence-guided and uniform sample budget allocation. “(conf)” denotes the trace confidence-weighted variant. # traces reports the number of sampled traces required to reach full consistency (consistency

=1

); Con. and Acc. report each method’s achieved consistency and accuracy evaluated at the trace count where PETS-On. attains consistency 1.

Figure 4: Budget allocation curve in the online setting. “(conf)” denotes the trace confidence-weighted variant. Consistency is computed within each matched-variant comparison group: PETS-Online vs. Uniform, and PETS-Online (conf) vs. Uniform (conf). Oracle variant assumes access to the latent parameter

\bm{\theta}

, while in the online setting,

\bm{\theta}

is learnt from a training dataset.

Dataset	Model	$k{=}2$	$k{=}3$	$k{=}5$	$k{=}8$
GPQA	Qwen3-4B	0.6905	0.6905	0.6964	0.6905
GPQA	GPT-120B	0.8036	0.8036	0.8214	0.8274
AIME 2024	Qwen3-4B	0.9500	0.9000	0.9000	0.9000
AIME 2024	GPT-120B	1.0000	1.0000	1.0000	1.0000

Table 4: Effect of Oracle KMeans cluster number

k

on Oracle accuracy. The Oracle uses KMeans clustering on ground-truth

(a,b)

parameters to partition questions into

k

difficulty bins. Predictor and Baseline results are unaffected by this parameter.

Dataset	Model	$k{=}2$	$k{=}3$	$k{=}4$	$k{=}5$
GPQA	Qwen3-4B	0.6845	0.6845	0.6845	0.6786
GPQA	GPT-120B	0.8214	0.8095	0.8095	0.8095
AIME 2024	Qwen3-4B	0.9500	0.9000	0.9000	0.9000
AIME 2024	GPT-120B	1.0000	1.0000	1.0000	1.0000

Table 5: Effect of Predictor difficulty bin number

k

on Predictor accuracy. The Predictor assigns each question to one of

k

bins based on the number of distinct answers among

K_{0}{=}4

warm-up.

Dataset	Model	Online Predictor	Online Uniform	Offline OKG
GPQA	Qwen3-4B	$4.73$	$<0.01$	$212.2$
GPQA	GPT-120B	$4.68$	$<0.01$	$204.1$
AIME 2024	Qwen3-4B	$1.55$	$0.08$	$36.0$
AIME 2024	GPT-120B	$1.19$	$0.03$	$39.0$

Table 6: Wall-clock runtime (seconds) of different budget allocation methods. Online Predictor trains bucket statistics on 30 questions and then assigns each test question a budget via

K_{0}{=}4

warm-up samples. Online Uniform applies a fixed per-question budget with no training. Offline OKG performs sequential Monte Carlo allocation (

B{=}16

per question,

n_{\mathrm{samples}}{=}500

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

R_{i}^{+}(S^{t})\ \approx\ \max_{m\in[M]}\Bigl(U(\mathrm{Upd}(S_{i}^{t};m,\beta_{i,m}^{t}))-U(S_{i}^{t})\Bigr)

Appendix

Algorithm 2 Optimistic Knowledge Gradient (confidence-weighted Bayesian setting)

Figure 5: Probability that multinomial majority voting selects the true best option as a function of budget. We plot

\mathbb{P}\!\left(Y^{\mathrm{Maj}}(B)=\arg\max_{y\in\mathcal{Y}}\theta_{y}\right)

versus

\textsc{Budget}\in\{1,\ldots,64\}

for different ground-truth preference vectors

\bm{\theta}=(\theta_{1},\ldots,\theta_{M})

. Gray curves ( Exact ) are computed from the true

\bm{\theta}

(exact for

M=2,4

; Monte Carlo estimates for

M=10

), while green dotted curves ( Probit ) are produced by fitting a two-parameter probit model and evaluating the fitted model across budgets. The fitted probit curves closely track the exact/MC curves in all regimes. Panels correspond to

M\in\{2,4,10\}

. For

M=4

, we use: dominate

[0.8,0.1,0.1,0]

, head-heavy

[0.6,0.3,0.05,0.05]

, linear

[0.4,0.3,0.2,0.1]

, flat

[0.3,0.25,0.25,0.2]

, and uniform

[0.25,0.25,0.25,0.25]

. For

M=10

, we use: dominate

[0.8,0.15,0.05,0,\ldots,0]

, head-heavy

[0.25,0.15,0.10,0.0714,\ldots,0.0714]

, linear

[10,9,\ldots,1]/55

, flat

\mathrm{normalize}([10,9,\ldots,1]^{0.4})

, and uniform

[0.1,\ldots,0.1]

Figure 6: GPQA offline

Figure 7: AIME 25 offline

Figure 8: AIME 24 offline

Figure 9: HMMT offline

Figure 10: BRUMO offline

Figure 11: GPQA online

Figure 12: AIME 25 online

Figure 13: AIME 24 online

Figure 14: HMMT online

Figure 15: BRUMO online

Figure 16: Summarization of the detailed offline comparison across datasets and models. Pass@1 denotes single-pass prediction accuracy (a lower-bound reference), while MV@128 denotes majority voting over 128 samples, used as a finite proxy of infinite-budget performance (an upper-bound reference).

Figure 17: Summarization of the detailed online comparison across datasets and models. Pass@1 denotes single-pass prediction accuracy (a lower-bound reference), while MV@128 denotes majority voting over 128 samples, used as a finite proxy of infinite-budget performance (an upper-bound reference).

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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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