Self-Consistency in LLM Inference

• Self-consistency (SC) is a probabilistic inference technique that aggregates multiple independent completions via empirical mode estimation.
• It uses statistical voting and chain-of-thought reasoning to improve accuracy, with theoretical guarantees on exponential error decay.
• Dynamic methods like ASC and Blend-ASC optimize sample allocation, reducing sample usage by up to 6.8× while preserving accuracy.

Self-consistency (SC) is a probabilistic test-time inference technique that enhances the reliability and accuracy of LLMs in complex reasoning tasks. It is primarily implemented by generating multiple independent completions—each possibly with an explicit chain-of-thought (CoT)—and selecting the most frequent final answer via empirical mode estimation, i.e., majority voting. Recent research has established a rigorous theoretical framework for SC, analyzed its scaling characteristics, and proposed efficient dynamic allocation, stopping criteria, and variants to address sample inefficiency and adaptivity in both per-instance and dataset-level regimes (Feng et al., 15 Nov 2025).

1. Mathematical Formalism and Theoretical Framework

Formally, for a question $q$, an LLM induces a discrete distribution $\mu(r \mid q)$ over possible answers $r$. Given $n$ independent samples,

$r_1, \dots, r_n \sim \mu(\cdot \mid q),$

self-consistency returns the empirical mode,

$$\hat{a}_n = \arg\max_{a \in A} \sum_{i=1}^n \mathbf{1}[r_i = a],$$

which, under the language of voting theory, is a plurality rule over $n$ "votes."

Theoretical guarantees establish that, for "aligned" questions where the true answer is the mode ($p_1 = \mu(a^*\mid q)$, runner-up $p_2$), the error rate decays exponentially: $$\Pr[\hat{a}_n \ne a^*] \leq \exp\left(-n(m + \epsilon_n)\right), \quad m = (\sqrt{p_1} - \sqrt{p_2})^2,$$ with $$\epsilon_n = O\left(\frac{\log n}{n}\right) \rightarrow 0$$. This implies that to guarantee misclassification probability $\le \delta$, it suffices to draw

$$n \gtrsim \frac{\ln(1/\delta)}{(\sqrt{p_1} - \sqrt{p_2})^2}$$

samples. The proof relies on refined bounds for multi-class majority vote in finite-alphabet settings (Feng et al., 15 Nov 2025).

Empirically, when applied across datasets, the average error $\mathrm{Err}(n)$ exhibits power-law decay,

$\mathrm{Err}(n) \approx C n^{-\alpha},$

with empirical exponents $\alpha \approx 0.4$–$0.5$ on free-response tasks (e.g., GSM8K, MATH). Multiple-choice settings exhibit weaker or non-monotonic scaling due to error being concentrated among a small number of distractor answers.

2. Sample Allocation and Efficiency

Fixed-Allocation SC

The classical SC baseline fixes $n$ samples per question regardless of difficulty. In a hypothetical oracle setting with known per-question margin $m$, the real-valued optimal allocation under margin density $p(m)\propto m^{-r}$ yields error scaling as $\bar{x}^{-1}$ for typical $r=1/2$.

Dynamic-Allocation SC

To address sample inefficiency and adaptivity, dynamic stopping rules have been formalized:

• ASC (Adaptive Self-Consistency): Tracks the leading ($n_1$) and runner-up ($n_2$) answer counts, assuming a Beta$(1,1)$ prior on $p_1/(p_1+p_2)$. Sampling stops when

$$\Pr[p_1 < p_2] = \int_0^{1/2} \mathrm{Beta}(n_1+1, n_2+1) < \tau,$$

for a threshold $\tau$.

• PPR-1v1: Implements a martingale confidence sequence over the two leading answers (out of $K$ seen), stopping when

$\mathrm{Beta}(n_1+1, n_2+1) \le \delta/(K-1).$

This approach matches the information-theoretic lower bound on samples needed to statistically distinguish the mode (Feng et al., 15 Nov 2025).

3. Blend-ASC: A Hyperparameter-Free, Budgeted Variant

Blend-ASC is a recent hyperparameter-free algorithm that integrates the benefits of ASC (sample efficiency at small budgets) with the asymptotic guarantees of PPR-1v1 (optimal exponential decay). At each sampling step $t$ (out of total $B$ budget for $Q$ questions), every question $q$ is assigned a blended score via a weighted combination of ASC and PPR-1v1 confidence metrics, with the blend parameter $(1-t/B)$. The procedure:

• Selects the question $q^*$ with minimum blended score,
• Draws an additional sample,
• Updates vote counts and confidence metrics,
• Caps per-question allocations at $16 \times (t/Q)$ to prevent runaway sampling.

After $B$ samples, the empirical mode per question is reported. Blend-ASC matches or exceeds SC performance while using $6.8\times$ fewer samples on average (e.g., for LLaMA-3.2-3B, Qwen-2.5-MATH-7B, and Qwen-2.5-32B on GSM8K, MATH, GPQA, and MMLU) (Feng et al., 15 Nov 2025).

Method	Avg. Samples (SC=64)	Efficiency Gain	Accuracy (vs SC)
Fixed-Alloc SC	15–27	$\sim$4.6x	Match
ASC	6–33	$\sim$5.9x	Match
Blend-ASC	6–26	$\sim$6.8x	Match or Sup.

4. Empirical Validation and Scaling Behavior

Large-scale experiments were conducted on:

• Models: LLaMA-3.2-3B, Qwen2.5-MATH-7B, Qwen2.5-32B,
• Tasks: Free-response (GSM8K, MATH, GPQA-Diamond) and multiple-choice (MMLU).

Key findings:

• SC@64 establishes a robust error baseline.
• Blend-ASC consistently outperforms vanilla SC, fixed-allocation, and ASC on sample efficiency without compromising accuracy.
• Average error continues to decay exponentially with sample count under Blend-ASC across datasets and models.
• Power law exponents and scaling coefficients for error can be estimated via pilot runs at $n \in \{16, 32, 64\}$ and log-linear regression.

5. Practical Recommendations and Planning

• Sample Budgeting: Fit the observed error $\mathrm{Err}(n)\approx C n^{-\alpha}$ on a held-out pilot set to estimate $(C,\alpha)$. To achieve target error $\epsilon$, set

$n\approx (C/\epsilon)^{1/\alpha}.$

• Per-Question Difficulty: When per-question margin $m$ is unknown (the general case), prefer dynamic schemes—specifically Blend-ASC for any fixed total budget $B$.
• Target Error Control: For a target error $\delta$, set $B$ via $\delta\approx C B^{-\alpha}$.
• Runaway Sampling: Cap per-question samples to $\sim16\times$ the running average (as implemented in Blend-ASC).

Blend-ASC is hyperparameter-free, precisely budgeted, and thus suitable for large-scale or resource-constrained settings (Feng et al., 15 Nov 2025).

6. Context within the Literature and Future Directions

SC is now a central component in test-time scaling for LLM inference. Its statistical behavior, sample complexity, and scaling dynamics are now formally characterized using tools from mode estimation and voting theory (Feng et al., 15 Nov 2025). The presented dynamic allocation strategies, especially Blend-ASC, set new efficiency baselines without requiring parameter tuning.

Open directions include: improved margin estimation under model uncertainty, further integration with post-training objectives that sharpen answer distributions, and robust scaling to open-ended generative outputs where mode identification is non-trivial or requires semantic equivalence.

• "Optimal Self-Consistency for Efficient Reasoning with LLMs" (Feng et al., 15 Nov 2025)