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CITE: Anytime Valid Statistical Inference in LLM Self-Consistency
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Proposes CITE, an anytime-valid sequential test that certifies a prespecified LLM answer as the unique mode of its response distribution over an unknown, possibly infinite category set, with stopping-time and minimax guarantees and a confidence-weighted extension.
Problem
Deciding when to stop sampling LLM outputs in self-consistency while controlling the false-certification probability at a prescribed level under data-dependent stopping, with an unknown and possibly infinite answer set.
Approach
Combine three time-uniform components via an intersection-union test: pairwise e-processes against the empirical runner-up, a time-uniform lower confidence bound on the target probability, and a per-category unseen-mass upper bound; stop when the pairwise e-value and LCB-versus-unseen conditions both hold.
Key results
- Under the null, CITE shows zero false certifications in simulations, while KR and MMC show nonzero rates up to 0.002 and 0.014 respectively.
- On Qwen3-30B/ER-REASON, Bonferroni saturates at certification rate 0.111 for N>=256, whereas CITE reaches >=0.98 certification at mean stopping time about 12.
- Max false-certification rate when target is the empirical runner-up is 0.052 (gpt-oss-20b/ER-REASON), within Monte Carlo error of the nominal 0.05.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Pairwise e-process plus LCB and unseen-bound stopping rule (CITE) vs fixed-sample Bonferroni | CITE certification on diffuse simulation Setting 1 · Simulation Setting 1 (diffuse) | certification rate at N=64: 0.188 vs 0.100 | Helped |
| CITE vs Bonferroni in easy concentrated setting | CITE on Setting 2 · Simulation Setting 2 (concentrated) | certification rate at N=64: 0.890 vs 0.928 | Hurt |
| Fixed-target KR sample-split test vs CITE under null | KR baseline · Simulation Settings 3-4 (null) | false certification rate: up to 0.002 | Hurt |
| MMC sequential leader tracking vs CITE on LLM-like diffuse setting | MMC baseline · Simulation Setting 5 (LLM-like) | certification rate: at most 0.002 through N=2048 vs CITE 0.932 at N=512 | Hurt |
| CITE vs fixed-sample Bonferroni on ER-REASON with Qwen3-30B | CITE · ER-REASON (Qwen3-30B) | certification rate: at least 0.98 at mean stopping time ~12 vs Bonferroni 0.111 at N>=256 | Helped |
| CITE vs KR baseline on FrontierScience-Olympiad (Qwen3-30B) | CITE · FrontierScience-Olympiad (Qwen3-30B) | certification rate: lower by at most 0.087 | Hurt |
| CITE vs MMC at N=1024 on FrontierScience-Olympiad (gpt-oss-20b) | CITE · FrontierScience-Olympiad (gpt-oss-20b) | certification rate: approximately 2x MMC | Helped |
| CITE vs MMC on ER-REASON | CITE · ER-REASON | certification rate difference: differ by at most 0.054 | No clear effect |
| Empirical false certification with runner-up target across methods and problems | CITE and baselines · AIME 2026 / FrontierScience / ER-REASON (runner-up target) | max false certification rate: 0.052 max on gpt-oss-20b/ER-REASON, at most 0.012 elsewhere | No clear effect |
Surprising findings
- KR often has higher finite-sample power than CITE, though it is not anytime-valid.
- MMC remains at certification rate at most 0.002 through N=2048 in a moderate LLM-like simulation setting.
Limitations
- Certifies a fixed prespecified target only; data-adaptive top-category certification requires an open correction.
- In the very small p_r regime the pairwise term upper bound is weaker and the lower bound gap is open.
- Certifies mode of the response distribution, not answer correctness.
Experimental setup
Datasets: AIME 2026 FrontierScience-Olympiad ER-REASON
Benchmarks: AIME 2026 FrontierScience-Olympiad ER-REASON
Metrics: Type-I error (false certification rate) certification rate mean stopping time accuracy vs sample budget
Baselines: Bonferroni KR MMC
Builds on: testing by betting Ville's inequality self-consistency Martingale Majority Certificate
Closest papers by meaning
- Sample Smart, Not Hard: Correctness-First Decoding for Better Reasoning in LLMs (ICLR 2026 · similarity 0.91)
- Optimal Self-Consistency for Efficient Reasoning with Large Language Models (ICML 2026 · similarity 0.90)
- A Theoretical Study on Bridging Internal Probability and Self-Consistency for LLM Reasoning (NeurIPS 2025 · similarity 0.90)
- Latent Self-Consistency for Reliable Majority-Set Selection in Short- and Long-Answer Reasoning (AAAI 2026 · similarity 0.90)
- Incentivizing LLMs to Self-Verify Their Answers (NeurIPS 2025 · similarity 0.89)
- Logical Consistency of Large Language Models in Fact-Checking (ICLR 2025 · similarity 0.89)
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