Diversity Combining for Multi-Path LLM Reasoning

Guangsheng Yu; Litianyi Zhang; Qin Wang; Xu Wang; Mingyuan Li; Shaoxiong Ji; Ren Ping Liu; Massimo Piccardi

cs.AISep 30, 2026

Diversity Combining for Multi-Path LLM Reasoning

Authors: Guangsheng Yu, Litianyi Zhang, Qin Wang, Xu Wang, Mingyuan Li, Shaoxiong Ji, Ren Ping Liu, Massimo Piccardi

Organizations: University of Technology Sydney · The University of Sydney · CSIRO · ELLIS Institute Finland · University of Turku

Abstract

Multi-path reasoning methods such as self-consistency (SC) sample $K$ reasoning paths and choose the most frequent answer. However, their gains quickly plateau as $K$ increases, and existing methods do not predict when this saturation will occur. We formalize multi-path LLM reasoning as a diversity combining problem from wireless communications: each path is a noisy channel observation, and the pairwise correlation of path correctness caps the design-effect effective sample size of the vote at a finite ceiling. Generalized least squares (GLS) analysis shows that, under exchangeability, the optimal symmetric linear combiner of latent embeddings is uniform, supporting majority vote as the natural default in standard SC while leaving room for weighting or pruning under heterogeneous prompt-template branches. Across 5 models and 12 benchmarks, prompt-template diversity reduces path correlation in $55$ of $57$ valid cells, with the strongest effect on open-ended QA. We derive an Adaptive-K rule that uses a four-path pilot to select $K^*$ , retaining $96$ -- $103\%$ of MV@ $K{=}32$ accuracy across Math, QA, and NLU.

Figures & tables

Fig. 1: Diversity combining for multi-path LLM reasoning. The LLM generates

K

correlated paths

\mathbf{z}_{k}=h_{k}\mathbf{z}(x)+\mathbf{n}_{k}

(

h_{k}

: reasoning quality;

\mathbf{n}_{k}

: errors;

\rho

: pairwise correlation), collapsed to correctness indicators

Y_{k}

. Aggregation : under exchangeability, the GLS-optimal linear combiner is uniform (Corollary IV.5 ), supporting uniform majority vote as the natural default. Diagnostic :

K_{\text{eff}}^{\text{vote}}=K/(1+(K{-}1)c)

saturates at

1/c

; Adaptive-K estimates

K^{*}

from a

K{=}4

pilot.

$K$	Agree	$\hat{c}$	$K_{\text{eff}}^{\text{vote}}$	Ceiling	% Ceil.	$\bar{p}$ (%) $\uparrow$	Tokens $\downarrow$
1	–	–	1.0	–	–	78.0 $\pm$ 2.5	323
4	0.868	0.604	1.42	1.66	86%	79.0 $\pm$ 1.3	1293
8	0.866	0.593	1.55	1.69	92%	79.2 $\pm$ 0.8	2592
16	0.862	0.579	1.65	1.73	96%	79.4 $\pm$ 0.8	5179
32	0.864	0.586	1.67	1.71	98%	79.2 $\pm$ 0.4	10327

TABLE I: Effective diversity analysis for Qwen2.5-7B on GSM8K (5 seeds

\times

100 instances = 500 pooled). Agree is the mean pairwise agreement

\binom{K}{2}^{-1}\sum_{j<k}\Pr(Y^{(j)}{=}Y^{(k)})

. Correctness correlation

\hat{c}

is computed via ( 3 ).

K_{\text{eff}}^{\text{vote}}

is the design-effect ( 5 ); Ceiling

=1/\hat{c}

\bar{p}

is the mean per-path accuracy, reported as per-seed mean

\pm

std over 5 seeds; it is not the majority-vote accuracy MV@

K

of Table II . The

K{=}1

row is a single sampled path.

Model	$\hat{c}_{K{=}4}$	4	8	16	32	MV@32	BB	Binom
		$K_{\text{eff}}^{\text{vote}}$ at $K{=}$					Predicted MV@32
Qwen2.5-7B	0.60	1.42	1.55	1.65	1.67	81.9%	81.1%	100.0%
Llama-3.1-8B	0.53	1.55	1.72	1.86	1.94	79.3%	77.8%	99.9%
Mistral-7B	0.45	1.71	1.89	2.00	2.13	42.4%	41.3%	20.6%

TABLE II: Saturation and beta-binomial calibration on GSM8K (Qwen2.5-7B, Llama-3.1-8B, Mistral-7B, 5 seeds

\times

100 instances = 500 pooled per cell).

K_{\text{eff}}^{\text{vote}}=K/(1+(K{-}1)\hat{c})

; all models reach

96

–

98\%

of the ceiling

1/\hat{c}

K{=}32

. MV@

K

denotes the binary majority-vote accuracy at

K

paths (§ V-A ), which is distinct from the mean per-path accuracy

\bar{p}

of Table I . BB (beta-binomial) and Binom (independence-binomial) columns are in-sample MV@32 predictions; both estimators are formally defined in the next subsection (Eq. ( 9 )).

Domain	Benchmark	Answer space	$n$	Mean $\Delta\rho$ (%) $\downarrow$	Mean $\Delta K_{\text{eff}}^{\text{vote}}$ $\uparrow$	Mean $\hat{\rho}_{\text{SC}}$	Mean $\hat{\rho}_{\text{PT}}$
QA	TriviaQA	open text	5	$-$ 71.5	$+$ 2.1	0.68	0.19
QA	HotpotQA	open text	5	$-$ 62.5	$+$ 2.0	0.55	0.17
Science/MC	ARC-C	bounded (4)	5	$-$ 48.7	$+$ 1.1	0.60	0.28
Science/MC	MMLU	bounded (4)	5	$-$ 34.6	$+$ 0.7	0.48	0.31
Commonsense	HellaSwag	bounded (4)	5	$-$ 36.0	$+$ 0.9	0.45	0.25
Commonsense	WinoGrande	binary (2)	5	$-$ 36.7	$+$ 0.7	0.60	0.35

TABLE III: Prompt-template diversity across 12 benchmarks (

K{=}8

, pooled over 5 seeds,

n{=}250

per cell except six cells with

100

–

235

instances in one arm: MATH on Qwen-7B, Qwen-32B and Llama-8B, MMLU on Qwen-7B and Qwen-32B, and MBPP on Qwen-7B). Five models evaluated: Qwen2.5-0.5B/7B/32B, Llama-3.1-8B, Mistral-7B. The Answer space column gives the effective evaluated output space (post-extraction value compared against gold), used as the relevant axis for the correlation analysis.

\Delta\rho

is the relative change in pairwise correlation,

(\hat{\rho}_{\text{PT}}-\hat{\rho}_{\text{SC}})/\hat{\rho}_{\text{SC}}

, expressed as a percentage (so

-71.5

on TriviaQA means correlation drops by

71.5\%

of its SC value, not by 71.5 percentage points). Cells with base accuracy

<2\%

excluded;

n

= number of valid models per benchmark. Each row reports the mean across

n

models.

† Retention above $100\%$ is consistent with finite-sample variance: the paired bootstrap
Task	Model	$\hat{c}_{K{=}4}$	$K^{*}$	MV@ $K^{*}$	MV@32	Retained	Net cost
GSM8K (Math)	Qwen-7B	0.60	6	81.6%	81.9%	100%	31%
GSM8K (Math)	Llama-8B	0.53	8	78.1%	79.3%	98%	38%
GSM8K (Math)	Mistral-7B	0.45	10	43.6%	42.4%	103% †	44%
HotpotQA (QA)	Llama-8B	0.61	6	50.2%	52.2%	96%	31%
BoolQ (NLU)	Llama-8B	0.79	4	80.2%	80.3%	100%	25%

TABLE IV: Adaptive-K compute savings across three domains.

\hat{c}

is estimated from

K{=}4

SC (5 seeds

\times

n{=}100

= 500 pooled for GSM8K, 3 seeds

\times

n{=}100

= 300 for HotpotQA and BoolQ). Retained

=

MV accuracy at

K^{*}

as a percentage of MV at

K{=}32

. Net cost

=(K^{*}+4)/32

accounts for the

K{=}4

pilot in addition to the operating-point paths.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
$x\in\mathcal{A}$	Ground-truth answer in discrete answer space.
$\mathbf{z}(x)\in\mathbb{R}^{D}$	Latent embedding of answer $x$ .
$K$	Number of reasoning paths.
$h_{k}$	Channel gain of path $k$ ( $h_{k}>0$ ).
$\mathbf{n}_{k}$	Additive noise vector of path $k$ .
$\rho\in[0,1]$	Latent channel gain correlation (in $\mathbf{R}$ ).

Appendix

TABLE V: Notation used throughout the paper.

Result	Assumptions used	Observable
$K_{\text{eff}}^{\text{vote}}$ , Eq. ( 5 )	Equicorrelated correctness indicators (design effect [ 10 ] )	✓
Theorem IV.2	i.i.d. Bernoulli correctness (independence reference)	✓
Beta-binomial, Eq. ( 9 )	$\Theta\sim\text{Beta}(\alpha,\beta)$ , $Y_{k}\mid\Theta$ i.i.d. Bernoulli $(\Theta)$ [ 25 ]	✓
Adaptive-K, Eq. ( 12 )	$K_{\text{eff}}^{\text{vote}}$ and a marginal-gain threshold $\varepsilon$	✓
Theorem IV.1 , latent rank	Eq. ( 1 ) with Assumption III.1 and the equicorrelated matrix ( 2 )	✗
Proposition III.4	Gaussian decision surrogate (Assumption III.3 )	✗

Appendix

TABLE VI: Assumption dependencies of each result. “Observable” marks results computed from output-level correctness alone.

Study	Formal saturation bound	Aggregation optimality analysis	Predicts $K^{*}$	Cross-task characterization	Answer-space analysis
SC [ 1 ]	✗	✗	✗	✗	✗
CISC [ 2 ]	✗	✗	✗	✗	✗
Entropy Voting [ 22 ]	✗	✗	✗	✗	✗
Optimal Agg. [ 23 ]	✗	$\boldsymbol{\sim}$	✗	✗	✗
SoftCoT [ 4 , 11 ]	✗	✗	✗	✗	✗
Test-time scaling [ 15 ]	✗	✗	$\boldsymbol{\sim}$	✗	✗

Appendix

TABLE VII: Analytical coverage of multi-path reasoning studies. Existing methods focus on aggregation strategies, online stopping, or empirical scaling laws; this work provides an upstream correlation-based diagnostic framework. ✓ = yes, ✗ = no,

\boldsymbol{\sim}

= partial. Superscripts indicate the predictive mechanism: online stops sampling per query based on observed agreement/quality; difficulty predicts

K

from a query-difficulty mixture model fit to scaling curves; correlation predicts

K^{*}

from a closed-form ceiling derived from inter-path correctness correlation. Compound-inference scaling [ 6 ] fits query-difficulty heterogeneity as a mixture over scaling curves. Our

\hat{c}

equals the corrected between-instance variance of per-instance accuracy divided by

\bar{p}(1-\bar{p})

(§ III-B ), so it summarizes the same heterogeneity in one vote-level statistic without decomposing its sources.

Domain	$n$	$\mathrm{MV}_{\text{SC}}$	$\mathrm{MV}_{\text{PT}}$	$\Delta\mathrm{Acc}$ (pp)
QA	10	6.9%	6.5%	$-0.4\pm 1.8$
Sci/MC	10	60.7%	59.3%	$-1.5\pm 5.4$
Commonsense	10	63.6%	63.9%	$+0.3\pm 8.7$
NLU	7	50.2%	52.5%	$+2.3\pm 2.1$
Code	10	13.1%	9.3%	$-3.9\pm 11.2$
Math	10	46.7%	45.4%	$-1.3\pm 3.8$

Appendix

TABLE VIII: Per-domain MV accuracy under SC and PT (

K{=}8

, 5 seeds

\times

n{=}50

pooled).

n

= number of valid model-benchmark cells (after

\bar{p}<2\%

exclusion).

\pm

denotes within-domain standard deviation across cells.

$K$	Observed MV	Beta-binomial error	Independence error
4	95.8%	0.8	2.8
8	96.8%	1.2	3.1
16	96.8%	1.3	3.2
32	96.8%	1.7	3.2

Appendix

TABLE IX: Qwen3.5-9B (thinking) on GSM8K, 3 seeds

\times

100 instances. Observed binary MV and held-out absolute prediction errors (pp), averaged over the two instance halves.

Model	$n$	$\hat{c}$ mean	$\hat{c}$ std	$K^{*}$ mean	$K^{*}$ std
Qwen-7B	25	0.571	0.122	7.9	6.77
	50	0.589	0.083	6.9	1.69
	100	0.596	0.055	6.6	1.02
	200	0.605	0.039	6.5	0.72
Llama-8B	25	0.512	0.105	8.7	3.14
	50	0.522	0.068	8.1	1.53

Appendix

TABLE X: Pilot size sensitivity on GSM8K (

K{=}4

, 500 bootstrap resamples of instances from pooled 5-seed data).

Model	$\varepsilon$	$K^{*}$	MV@ $K^{*}$	Retained
Qwen-7B	0.01	10	82.4%	101%
	0.025	6	81.6%	100%
	0.05	5	81.4%	99%
	0.1	3	79.8%	97%
Llama-8B	0.01	13	78.8%	99%
	0.025	8	78.1%	98%

Appendix

TABLE XI: Sensitivity of Adaptive-K to threshold

\varepsilon

on GSM8K (5 seeds

\times

n{=}100

= 500 pooled). Retained = MV@

K^{*}

as a percentage of MV@

K{=}32

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