Optimal Self-Consistency for Efficient Reasoning with Large Language Models

TL;DR

This work reframes Self-Consistency (SC) as a mode-estimation problem and develops a theory linking per-question margins to exponential error decay, then extends the analysis to dataset-scale scaling by connecting margin distributions to power-law performance. It introduces Blend-ASC, a hyperparameter-free adaptive SC method that combines a theoretically optimal dynamic allocation with practical ASC signals, achieving significant sample-efficiency gains (up to 6.8× fewer samples) across diverse models and benchmarks. The results provide both per-question and dataset-level scaling laws, along with asymptotically optimal stopping guarantees for dynamic allocation, offering a principled path to scalable, cost-effective test-time reasoning in LLMs. The approach has practical impact by enabling efficient inference with no extra hyperparameters, potentially benefiting any self-consistency-based reasoning system and related verifier-driven inference methods.

Abstract

Self-consistency (SC) is a widely used test-time inference technique for improving performance in chain-of-thought reasoning. It involves 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, SC is prohibitively expensive at scale when naively applied to datasets, and it lacks a unified theoretical treatment 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 6.8x 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, Blend-ASC is hyperparameter-free and can fit an arbitrary sample budget, ensuring it can be easily applied to any self-consistency application.

Figures (13)

• Figure 1: (Left) Blend-ASC outperforms SC, ASC, Fixed-Allocation SC, and asymptotically-optimal PPR-1v1, by converging to the limiting answer the fastest on aligned questions. (Right) SC exhibits scaling laws across free-response datasets, with power-law convergence to its limiting error.
• Figure 2: Example of SC.
• Figure 3: Margin correlates with decay rate across several model and dataset combinations, where decay is fit for $x\geq 16$ for $\epsilon$ to have negligible impact on the bound.
• Figure 4: Large dataset sizes induce power-law scaling. (Left) Margin distribution for $\mathcal{D}_1-\mathcal{D}_3$ with $n=1$. (Middle) Error scaling $\mathcal{D}_1-\mathcal{D}_3$, with $\mathcal{D}_3$ having the fastest convergence. (Right) Margin distribution from sampling 100 points from each dataset and applying KDE.
• Figure 5: Scaling behavior of Self-Consistency on aligned (left), misaligned (middle), and full (right) datasets for free-response (top) and multiple-choice (bottom) benchmarks.

Theorems & Definitions (4)

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