Calibration-Free Early Stopping for Chain-of-Thought Reasoning via E-processes

Dong-Jae Lee; Gyojin Han; Junmo Kim

cs.CLOct 18, 2026

Calibration-Free Early Stopping for Chain-of-Thought Reasoning via E-processes

Authors: Dong-Jae Lee, Gyojin Han, Junmo Kim

Organizations: KAIST

Abstract

Reasoning large language models (LLMs) benefit from test-time scaling, motivating adaptive policies that spend tokens when they help and stop early otherwise. Existing rules halt when a confidence or stability signal crosses a threshold, but such thresholds rarely transfer across models and datasets, and validation-calibrated rules lose their guarantee under shifts in test-time horizon or signal dynamics. We reframe adaptive stopping as sequential evidence accumulation along the reasoning trajectory rather than threshold calibration on held-out data. Our framework runs two e-processes on a shared per-chunk signal: a commit stream tracks the signal's position relative to its slow trend, and an abstain stream tracks its signed velocity, making the two anti-correlated by construction. Under signal-level nulls on the per-chunk increments, Ville's inequality yields an anytime-valid signal-level false-alarm bound at user-specified significance levels, with no held-out calibration. Across various model--dataset combinations, the framework matches or exceeds calibration-based stopping on the token--accuracy Pareto front using zero calibration data, and the same significance level transfers unchanged across cells where calibration becomes infeasible.

Figures & tables

Figure 1: Token-use vs. accuracy Pareto curves aggregated over all 12 model–dataset combinations. The left panel shows dual stopping, where early commit and early abstain are both active so the model can either commit to an intermediate answer or defer to a fallback policy before the trajectory ends. The right panel shows upper-only stopping, where only early commit is applied and the model falls back to the terminal answer otherwise.

	Dual stopping						Upper-only stopping
TokU	CT-Conf	CT-EAT	CT-Stab	Ours-Conf	Ours-EAT	Ours-Stab	CT-Conf	CT-EAT	CT-Stab	Ours-Conf	Ours-EAT	Ours-Stab
$\leq 0.2$	0.214	0.286	0.302	0.131	0.158	0.337	0.275	0.286	0.310	—	—	0.317
$\leq 0.3$	0.243	0.347	0.352	0.180	0.235	0.441	0.310	0.364	0.408	—	—	0.408
$\leq 0.4$	0.280	0.398	0.397	0.245	0.358	0.517	0.363	0.453	0.485	—	—	0.491
$\leq 0.5$	0.357	0.441	0.415	0.364	0.444	0.574	0.408	0.545	0.538	—	—	0.567
$\leq 0.6$	0.394	0.445	0.415	0.434	0.530	0.616	0.502	0.595	0.614	—	0.508	0.637

Table 1: Accuracy at each token budget, averaged over 40 splits and 12 cells. CT methods use

n_{\mathrm{val}}=50

while Ours uses zero calibration data. CT-Stab applies CT’s calibration rule to the same stability signal used by Ours. —: no Pareto operating point exists at or below that budget (the e-process commit threshold was not crossed at any

\alpha

in the sweep). Bold : highest value in each half-row.

Panel of Figure 2 — Figure 2: Token-use vs. accuracy Pareto under cross-combination deployment with Qwen3-8B/AIME as the fixed calibration source. CT same-cell and the proposed method are evaluated on the same target cells as reference envelopes. CT cross-cell performance degrades substantially when the calibration distribution does not match the target cell, whereas the proposed method operates identically without re-calibration.

Panel of Figure 3 — Figure 3: E-process dynamics. (a) Three canonical trajectory outcomes at $\alpha{=}0.1$ : $\log E^{+}_{t}$ rises and crosses the threshold for a correct early prediction ( commit ); neither stream crosses for a hard instance that exhausts the full token budget ( full-run ); $\log E^{-}_{t}$ crosses first, correctly flagging a trajectory that would have produced a wrong terminal answer ( abstain ). (b) Realized commit FP rate and abstain FN rate as $\alpha$ is swept across all 12 model–dataset combinations; both decrease monotonically, confirming that $(\alpha_{+},\alpha_{-})$ provide calibration-free monotone knobs on the two error types.

Figure 4: Stopping outcome distribution for Qwen3-8B on MATH-500 under two operating points. Each bar shows the fraction of trajectories in each outcome category, further decomposed into correct and incorrect predictions, broken down by problem solvability.

Figure 5: Four choices of the upper evidence increment

\phi^{+}

are compared by measuring mean token use as a function of

\alpha_{+}

(log scale), with the lower stream fixed to

\phi^{-}=\mathrm{sign}(-\Delta s_{t})

and results aggregated over all 12 model–dataset combinations.

Figure 6: Five

\lambda

update strategies are compared on the token use–accuracy Pareto curve, with

\phi^{+}

fixed to EMA 0.05 and results aggregated over all 12 model–dataset combinations.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A: Per-combination Pareto envelopes (upper-only, left; dual, right). Best viewed zoomed in.

Figure B: Token-use vs. accuracy Pareto curves under cross-cell deployment (dual stopping), averaged over all

12\times 11=132

source

\to

target pairs. CT-EAT same-cell uses each target cell’s own calibration data as a reference; CT-EAT cross-cell transfers the source-calibrated threshold to a different target cell. The proposed method (Ours-Stab) requires no calibration data and operates identically across all target cells.

Method	$\leq 0.2$	$\leq 0.4$	$\leq 0.6$	$\leq 0.8$
Truncation	0.385	0.447	0.503	0.568
CT-Conf	0.399	0.439	0.490	0.554
CT-EAT	0.407	0.456	0.489	0.506
CT-Stab	0.423	0.465	0.473	0.473
CES-Conf	0.112	0.209	0.375	0.511
CES-EAT	–	0.367	0.538	0.616

Appendix

Table A: Dual-stopping accuracy on GPQA-Diamond, aggregated over five models.

Method	$\leq 0.2$	$\leq 0.4$	$\leq 0.6$	$\leq 0.8$
Truncation	0.292	0.368	0.450	0.486
CT-Conf	0.286	0.348	0.425	0.469
CT-EAT	0.284	0.314	0.317	0.317
CT-Stab	0.339	0.339	0.339	0.339
CES-Conf	–	0.084	0.283	0.381
CES-EAT	–	0.251	0.424	0.488

Appendix

Table B: Dual-stopping accuracy on CRUXEval-O, aggregated over two models.

Method	$\leq 0.2$	$\leq 0.4$	$\leq 0.6$	$\leq 0.8$
Truncation	0.141	0.261	0.390	0.485
CT-Conf	0.203	0.298	0.400	0.480
CT-EAT	0.266	0.334	0.347	0.347
CT-Stab	0.261	0.340	0.364	0.364
CES-Conf	0.142	0.262	0.390	0.466
CES-EAT	0.173	0.346	0.474	0.538

Appendix

Table C: Dual-stopping accuracy for DeepSeek-R1-Distill-Llama-8B, aggregated over five datasets. Full-run accuracy across cells ranges from 0.475 to 0.744.

Method	$\leq 0.2$	$\leq 0.4$	$\leq 0.6$	$\leq 0.8$
Truncation	0.236	0.350	0.503	0.620
CT-Conf	0.297	0.403	0.510	0.597
CT-EAT	0.355	0.438	0.447	0.452
CT-Stab	0.363	0.437	0.442	0.442
CES-Conf	–	0.274	0.478	0.598
CES-EAT	0.206	0.450	0.571	0.651

Appendix

Table D: Dual-stopping accuracy for DeepSeek-R1-Distill-Llama-70B, aggregated over five datasets. Full-run accuracy across cells ranges from 0.604 to 0.828; AIME full-run accuracy is 0.700.

Method	$\leq 0.3$	$\leq 0.5$	$\leq 0.7$
Truncation	0.277	0.435	0.548
CT-Conf (probe-free)	0.311	0.434	0.547
CT-EAT	0.323	0.390	0.430
CT-Stab	0.343	0.390	0.408
CES-Stab	0.353	0.478	0.566

Appendix

Table E: Dual-stopping accuracy at normalized FLOPs budgets, aggregated over the 12 main-paper model–dataset cells.

Model	Dataset	Final Acc.	$\alpha_{+}=0.1$		$\alpha_{+}=0.01$		$c_{+}$	$c_{-}$
			FP / All	FP / Final-Correct	FP / All	FP / Final-Correct
Qwen3-8B	AIME	0.659	0.354	0.538	0.165	0.251	246	6.90
Qwen3-8B	MATH-500	0.858	0.168	0.196	0.068	0.079	451	0.80
Qwen3-8B	OlympiadBench	0.669	0.230	0.344	0.096	0.144	569	0.76
Qwen3-8B	DeepScaler	0.643	0.212	0.329	0.105	0.163	622	2.03
Qwen3-30B-A3B	AIME	0.777	0.361	0.464	0.188	0.242	419	119.90

Appendix

Table F: Per-cell accuracy-relevant commit errors under two denominator conventions, together with empirical relaxed-null constants

c_{+}

and

c_{-}

. “FP / All” divides the accuracy-relevant early-commit count by all trajectories in the cell, whereas “FP / Final-Correct” conditions the denominator on trajectories whose full-budget terminal answer is correct.

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The field of Language Reasoning Models (LRMs) has been very active over the past few years with advances in training and inference techniques enabling LRMs to reason longer, and more accurately. However, a growing body of studies show that LRMs are still inefficient, over-generating verification and reflection steps. Additionally, the high-level role of each reasoning step and how different step types contribute to the generation of correct answers, is largely underexplored. To address this challenge, we introduce TRACES (Tagging of the Reasoning steps enabling Adaptive Cost-Efficient early-Stopping), a lightweight framework that tags reasoning steps in real-time, and enable adaptive, cost-efficient early stopping of large-language-model inferences. Building on this framework we monitor reasoning behaviors during inferences, and we find that LRMs tend to shift their reasoning behavior after reaching a correct answer. We demonstrate that the monitoring of the specific type of steps can produce effective interpretable early stopping criteria. We evaluate the TRACES framework on three mathematical reasoning benchmarks, namely, MATH500, GSM8K, AIME and two knowledge and reasoning benchmarks, MMLU and GPQA respectively. We achieve 20 to 50% token reduction while maintaining comparable accuracy to standard generation.

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