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Optimal Self-Consistency for Efficient Reasoning with Large Language Models

ICML 2026 · Large language models › Reasoning and chain of thought · approach: Routing, pruning and distillation for reasoning · 1 citations (OpenAlex)

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Analyzes self-consistency scaling via mode estimation and voting theory, derives power-law dataset scaling and fixed/dynamic allocation bounds, and proposes Blend-ASC, a hyperparameter-free adaptive SC variant using 4.8x fewer samples than vanilla SC.

Problem

Self-consistency (majority-vote sampling for CoT reasoning) is expensive at dataset scale and lacks a principled understanding of its sample efficiency and scaling behavior; adaptive variants are underexplored and often require hyperparameter tuning.

Approach

Views self-consistency as empirical mode estimation, derives a margin-based exponential per-question error bound (margin = (sqrt(p1)-sqrt(p2))^2) that yields power-law dataset scaling, and combines ASC's early confidence with PPR-1v1's asymptotically optimal stopping rule via a linearly blended ranking to allocate samples dynamically without hyperparameters.

Key results

• Blend-ASC reduces samples by 4.8x on average relative to vanilla SC (average improvement factor over 6 model-dataset pairs).
• At 64-sample SC budget, Blend-ASC achieves 5.33x average sample improvement vs 4.09x for fixed allocation and 4.36x for ASC.
• Per-question margin correlates with empirical error decay rate across models and benchmarks (high Pearson correlation reported).

Surprising findings

• Fixed-allocation SC with oracle access to the margin is only competitive under an unrealistic oracle assumption, and ASC often stalls at large sample counts.
• Dataset-level scaling for multiple-choice benchmarks is non-monotonic and hard to predict theoretically.

Limitations

• Theory assumes majority-vote-style aggregation and does not apply to open-ended generation.
• Dataset-level bounds for datasets dominated by incorrect answers are not derived.
• Experiments use margins estimated from 100 samples per question and sampled generations rather than live inference.

Experimental setup

Datasets: GSM8K MATH MMLU GPQA-Diamond

Benchmarks: GSM8K MATH GPQA-Diamond

Metrics: reducible error sample efficiency (samples needed to match SC error) average improvement factor

Baselines: vanilla self-consistency Adaptive Self-Consistency (ASC) PPR-1v1 Fixed-Allocation SC Early-Stopping Self-Consistency (ESC)

Builds on: self-consistency adaptive self-consistency PPR-1v1

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