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Chopthin-Consensus Power Sampling: A Diversity-Preserving Approach to LLM Decoding

Minoo Ahmadi, Seyedarmin Azizi, Erfan Baghaei Potraghloo, Mehdi Kamal, Massoud Pedram

arXiv:2609.12243v1cs.CLcs.AIstat.ML

TL;DR

Equal-weight resampling in SMC can discard low-weight reasoning paths that may reach correct answers, reducing genealogical diversity. CCPS replaces it with Chopthin and semantic-majority selection; across the reported settings, it raises oracle coverage in 13 of 15 and matches or improves Power-SMC accuracy in 14 of 15.

  • Problem

    Equal-weight SMC resampling can discard low-weight trajectories that may contain correct reasoning paths and reduce genealogical diversity.

    Source: §3.2 · p. 5

  • Method

    CCPS carries unequal weights forward with Chopthin and selects answers by merging exact duplicate trajectories, clustering semantically equivalent answers, and choosing the largest coalition of distinct trajectories.

    Source: §1 · pp. 2–3

  • Results

    Across three open models and five benchmarks, CCPS matches or improves Power-SMC final-answer accuracy in 14 of 15 settings, while Chopthin raises oracle coverage in 13 of 15.

    Source: §8 · p. 11

  • Takeaways & Limitations

    The reported findings identify diversity-preserving resampling and diversity-aware selection as complementary mechanisms for training-free LLM reasoning.

    Source: §1 · p. 3

    Evaluating Chopthin combinations with other SMC-based decoding methods remains future work.

    Source: §8 · p. 11

Abstract

from arXiv · show

Inference-time power sampling via Sequential Monte Carlo (SMC) can substantially improve large language model (LLM) reasoning without requiring post-training. However, many existing SMC approaches rely on equal-weight resampling, which can aggressively prune low-weight trajectories, discarding potentially correct reasoning paths and degrading the genealogical diversity of the search space. To address this, we introduce Chopthin-Consensus Power Sampling (CCPS). Our method applies the Chopthin resampler to LLM decoding: rather than equalizing weights and forcing unnecessary particle duplication, it enforces an upper bound on the ratio between the largest and smallest weights and carries the unequal weights forward. This targeted intervention preserves a richer set of distinct reasoning paths, keeps the weighted SMC approximation unchanged in conditional expectation, and guarantees a lower bound on the post-resampling effective sample size (ESS). To fully exploit this enriched population, we employ a semantic-majority selection mechanism that merges token-identical final trajectories, clusters semantically equivalent answers, and returns the answer supported by the largest number of distinct trajectories. Evaluating across three open-weight models and five reasoning benchmarks, we show that Chopthin increases oracle coverage in 13 of 15 settings. Combined with semantic-majority selection, CCPS matches or exceeds the final-answer accuracy of the Power-SMC baseline in 14 of 15 settings, delivering absolute gains of up to 10.6 percentage points. These findings demonstrate that diversity-preserving resampling and diversity-aware selection are complementary mechanisms for training-free LLM reasoning. Code is available at github.com/MinooAhmadii/chopthin-consensus-power-sampling.

1 Introduction

Power sampling seeks reasoning gains at inference time, but equal-weight resampling can erase distinct paths, including potentially correct ones. CCPS pairs Chopthin resampling with semantic-majority selection; across the reported settings, it improves oracle coverage in 13 of 15 and matches or improves Power-SMC accuracy in 14 of 15.

1 Introduction

  • Power sampling sharpens the base model’s complete-sequence distribution without training, while SMC approximates it by incrementally decoding and weighting parallel partial trajectories.RL post-training requires a dedicated pipeline, reward signals or verifiers, and substantial computational resources.Source: §1 · pp. 1–2
  • Equal-weight resampling can delete low-weight particles that contain correct reasoning paths, while restored weight-based ESS can mask the resulting loss of genealogical diversity.Particle weights reflect relative prefix mass, not whether an unfinished trajectory is logically sound or will yield the right answer.Source: §1 · p. 2
  • Chopthin bounds the ratio between output weights and carries unequal weights forward, using targeted thinning and chopping to preserve more low-weight trajectories.The method is designed to retain unbiasedness, exact particle count, and weight conservation.Source: §1 · p. 2
  • CCPS merges token-identical final trajectories, clusters semantically equivalent answers, and returns the answer supported by the largest coalition of distinct trajectories.For code, clustering can use functional execution on self-generated test inputs.Source: §1 · p. 3
  • Across three open models and five reasoning benchmarks, Chopthin raises oracle coverage in 13 of 15 settings, while CCPS matches or improves final-answer accuracy over Power-SMC in 14 of 15.Source: §1 · p. 3

2 Background: Power sampling as sequential Monte Carlo

The background defines a sequence-level power target and describes how SMC approximates it with a population of weighted prefixes. It then specifies ESS monitoring and the resampling interface used to manage weight concentration.

2.1 The sequence-level power distribution

  • The sequence-level power distribution raises each complete sequence’s base-model probability to an exponent greater than 1, but exact sampling is intractable and tokenwise temperature scaling does not induce that distribution.The normalizer sums over an exponentially large space of complete sequences.Source: §2.1 · p. 4

2.2 Sequential Monte Carlo over prefixes

  • SMC decodes N prefixes together, sampling each next token from a proposal and updating cumulative importance weights toward intermediate targets with an exponent ramp from 1 to α.The ramp reaches α over the first T_ramp tokens and mitigates early weight degeneracy; after the ramp, the weighted population targets the power distribution over reachable sequences.Source: §2.2 · p. 4

2.3 Effective sample size and the resampling interface

  • At block boundaries, the sampler triggers resampling when ESS_t < κN, and the resampler returns N particles with ancestor indices and output weights.Standard resampling methods equalize output weights to 1/N; this method changes that rule while leaving the rest of the SMC construction unchanged.Source: §2.3 · pp. 4–5

3 Motivation: weight balance is not trajectory diversity

ESS measures weight balance, not the number of distinct trajectories or lineages that survive. Equal-weight resampling can delete potentially correct paths and collapse lineages, motivating a rule that controls weight concentration without requiring exact equality.

3.1 Weight balance and trajectory diversity are different objectives

  • ESS tracks weight balance, while surviving root lineages and token-distinct complete trajectories measure different aspects of trajectory diversity.Distinct lineages can converge to the same final trajectory, and descendants of one lineage can diverge.Source: §3.1 · p. 5

3.2 Stochastic deletion and lineage collapse

  • Systematic resampling deletes particles with Nw_i < 1 with probability 1 − Nw_i, although their current weight does not reliably predict whether they could reach a correct answer.This stochastic deletion can remove trajectories that otherwise would have reached a correct solution.Source: §3.2 · p. 5
  • Repeated equalizing events can reduce the number of independent lineages as high-weight particles produce copies of the same prefix.Equal-weight resampling also converts continuous particle weights into integer offspring counts instead of carrying weight information forward.Source: §3.2 · p. 5

3.3 Design objective

  • A bound on the largest-to-smallest output-weight ratio can guarantee an ESS floor without exact equality or perturbing the entire population.The design objective also retains the weighted target and particle budget while avoiding erasure of accumulated weight information.Source: §3.3 · p. 5
  • Weight-based selection can rarely return a surviving correct trajectory when its final weight is low, motivating a selector that uses the whole population.Source: §3.3 · p. 6

4 Chopthin-Consensus Power Sampling

CCPS replaces equal-weight resampling with Chopthin, which bounds weight unevenness and carries unequal weights forward. It then selects a consensus answer by merging duplicate trajectories, clustering equivalent answers, and voting across distinct trajectories.

4.1 Chopthin preliminaries: bounded-weight resampling

  • A bounded output-weight ratio controls weight concentration and guarantees an ESS floor, while allowing Chopthin to retain unequal weights rather than force equality.Particles in the keep band pass through unchanged, and equal input weights leave every particle unchanged.Source: §4.1 · p. 6
  • Chopthin returns exactly N particles, conserves total weight, and is unbiased, so the weighted target is unchanged in expectation.A light particle survives with probability w_i/a, exactly its survival probability under systematic resampling, so it is no more likely to be deleted under Chopthin.Source: §4.1 · p. 6

4.2 Carried weights and integration

  • At each event, CCPS carries Chopthin’s unequal output weights forward as cumulative weights, which later decoding increments multiply.The weights at the end of decoding therefore retain information from before the last resampling event.Source: §4.2 · pp. 6–7

4.3 Semantic-majority selection

  • Semantic-majority selection merges token-identical trajectories, clusters answers judged equivalent, and returns the cluster supported by the most distinct trajectories.Pooled weights break ties among duplicate trajectories and, subsequently, ties between clusters; the output is a consensus point estimate, not a sample from the power distribution.Source: §4.3 · p. 7

5 Experiments

The experiments compare CCPS with Power-SMC and other decoding baselines across three Qwen models and five benchmarks, finding that Chopthin usually raises oracle coverage and CCPS usually matches or improves final-answer accuracy. The discussion distinguishes the roles of resampling and selection, while ablations probe parameter sensitivity.

5.1 Setup

  • Across three Qwen models and five benchmarks without fine-tuning, generation uses matched N=32, α=2, τ=0.5, κ=0.5 settings, while Power-SMC draws weights and CCPS uses semantic majority.The compared models are Qwen2.5-Math-7B, Qwen2.5-7B, and Qwen3-4B; the benchmarks are MATH500, GSM8K, AIME 2022–2024, GPQA Diamond, and HumanEval.Source: §5.1 · p. 7
  • Oracle coverage is the at-least-one-correct rate in a single fixed-compute correlated population, not an i.i.d. pass@N estimate.It is a ceiling for a selector that must return one of the final answers.Source: §5.1 · p. 8

5.2 Comparison study

  • Chopthin raises oracle coverage over Power-SMC in 13 of 15 cells, ties one, and dips one by at most 1.5 points; CCPS matches or improves accuracy in 14 of 15, by up to 10.6 points.The accuracy gain is on Qwen3-4B on GPQA; the sole exception is Qwen3-4B on HumanEval (−0.6).Source: §5.2 · p. 8, Figure 2 · p. 8, Table 2 · p. 8
  • Semantic majority matches or improves on weight draw under both resamplers in 28 of 30 comparisons; with majority held fixed, Chopthin wins 9 cells, ties 2, and loses 4.The selector exceptions are Qwen3-4B on HumanEval under systematic resampling and on AIME under Chopthin.Source: §5.2 · pp. 8–9, Table 1 · p. 8
  • On Qwen2.5-Math-7B/MATH500, Figure 3 reports accuracy variation within 1.4 points and coverage within 1.6 across η, alongside a uniform-reset diagnostic hybrid.The hybrid applies Chopthin’s allocation and then resets each weight to 1/N; the caption identifies it as a diagnostic, not a valid resampler.Source: Figure 3 · p. 9

5.3 Discussion

  • Higher oracle coverage does not guarantee chosen-answer accuracy: on Qwen3-4B/MATH500, CCPS has 84.8 coverage but 81.8 accuracy, below baseline decoding’s 82.0.The discussion attributes the mismatch to correct trajectories that survive but are not returned by the current selector.Source: §5.3 · p. 9

6 Ablations

The ablations find that CCPS is robust across the tested ratio bounds and that its default bound performs best for majority accuracy. They also separate the effects of Chopthin’s offspring allocation from carrying its unequal output weights forward.

6.1 Sensitivity to the ratio bound η

  • The ratio-bound sweep varies η induced by target ESS fraction ρ with trigger κ=0.5; looser bounds perturb the population less and preserve more distinct trajectories.All arms are identical to CCPS except η.Source: §6.1 · pp. 9–10
  • Every tested η keeps oracle coverage above the systematic baseline of 84.6, by +0.8 to +2.4; the trigger-matched default ρ=0.5 gives the best majority accuracy.The authors report no further tuning of η.Source: §6.1 · pp. 9–10

6.2 Carried weights versus uniform reset

  • The uniform-reset hybrid is not a valid resampler: resetting weights to 1/N without adjusting offspring counts breaks Chopthin’s unbiasedness for the product of count and output weight.The diagnostic separates which particles survive from what weight information the survivors carry.Source: §6.2 · p. 10
  • In the diagnostic, the hybrid and carried-weight Chopthin preserve the same mean distinct trajectories (15.2), versus 13.0 under systematic resampling; carrying weights adds +0.8 majority accuracy and +1.2 oracle coverage.The reported gains are relative to the hybrid that resets output weights to 1/N.Source: §6.2 · p. 10

7 Related work

Related work on particle-based LLM inference commonly re-equalizes weights, while diversity-preserving alternatives modify particle preferences or restore diversity around equalizing resampling. The paper positions its resampler and selector as complementary to existing sampling approaches.

7 Related work

  • Prior particle-based LLM inference methods re-equalize weights at resampling; the authors say theirs is the first, to their knowledge, to bound post-resampling weight unevenness and guarantee post-resampling ESS.Other diversity efforts temper or augment weights, enlarge the eligible pool, or rejuvenate particles after equalizing resampling; CCPS instead changes how much the population is perturbed, without extra rollouts, learned potentials, or reward models.Source: §7 · p. 10
  • Unlike marginal-sharpening approaches that alter the target distribution over answers, CCPS keeps Power-SMC’s sequence-level target and changes the resampler and selector.Its selector adapts clustering and agreement ideas to a resampled population by merging duplicates before voting.Source: §7 · p. 10

8 Conclusion

Replacing systematic resampling with Chopthin raises oracle coverage in 13 of 15 settings, and pairing it with semantic-majority selection matches or improves Power-SMC accuracy in 14 of 15. The appendix details Chopthin’s threshold-based thinning and chopping, target-preservation properties, and ESS-bound parameter choice.

8 Conclusion

  • 13 of 15 settings show higher oracle coverage with Chopthin, and CCPS matches or improves Power-SMC accuracy in 14 of 15, by up to 10.6 percentage points.The conclusion says the resampling change may combine with other SMC decoding methods; evaluating such combinations remains future work.Source: §8 · pp. 10–11

A Details of the Chopthin resampler

  • The appendix presents the Chopthin resampler’s interface, three-band behavior, and guarantees as the implementation details behind the main-text method.The main text uses ancestor indices and output weights, the three-band behavior, and Propositions 1 and 2.Source: A Details of the Chopthin resampler · p. 13

A.1 Expected-offspring function and threshold

  • The threshold partitions particles into light ones that are thinned and heavy ones that are kept or chopped; the upper-endpoint convention keeps equal-weight particles unchanged.Particles in the keep band pass through with expected offspring count one.Source: §A.1 · p. 14

A.2 Thinning and chopping

  • Light particles survive with probability wi/a and receive weight a, while heavy-particle weight corrections preserve unbiasedness and keep output weights within [a, ηa].The correction conserves total weight in exact arithmetic and to numerical tolerance in implementation.Source: §A.2 · p. 14

A.3 Pseudocode

  • The appendix’s pseudocode adapts Gandy and Lau’s Chopthin algorithm and explicitly guards the case where every heavy expected count is an integer.In that case, NU = 0 and ζ is set to zero.Source: §A.3 · p. 14

A.4 Why η ≥4

  • η ≥4 is required so every admissible split keeps its pieces within [a, ηa].The constraint follows by considering weights just below 2a.Source: §A.4 · p. 14

A.5 Target preservation

  • Chopthin leaves the weighted empirical approximation unchanged in conditional expectation while changing resampling variance and genealogy.A finite population remains an approximation of the target, not an exact draw.Source: §A.5 · pp. 14–15

A.6 Our choice of η

  • At N = 32, the chosen η gives a post-resampling ESS floor of 13.17 (0.412N), approaching 0.5N as N grows.The appendix targets ρ = 0.5, yielding η ≈5.83.Source: §A.6 · p. 15

A.7 Solver and integration into Power-SMC

The experiments integrate Chopthin into Power-SMC under shared decoding settings and paired randomness. The appendix pseudocode carries Chopthin’s unequal weights forward, while the α-ramp updates importance weights toward the tempered target.

A.7 Solver and integration into Power-SMC

  • The two resampling arms share the model, proposal, decoding configuration, and paired random streams, differing in the resampler.The evaluation uses three Qwen models across five benchmarks without additional fine-tuning; the decoding setup uses α = 2, τ = 0.5, top-p = 0.9, and ESS trigger κ = 0.5.Source: §A.7 · p. 16

B CCPS pseudocode

  • CCPS changes Power-SMC at resampling and selection: it carries Chopthin’s output weights forward, then deduplicates trajectories and selects the largest semantic answer cluster.Cluster ties are broken by pooled weight.Source: B CCPS pseudocode · pp. 16–17

C Power-SMC incremental weights and the α-ramp

  • During the α-ramp, incremental weights reweight the prefix toward the tempered target; after the ramp, the update reduces to pθ(yt | x, y<t)^α/q(yt | x, y<t).The identity holds between resampling events, and stopping during the ramp does not distort the final target.Source: C Power-SMC incremental weights and… · p. 17

D Selector definitions

The selector definitions distinguish weight-based baselines from semantic majority, which votes over distinct final trajectories and uses pooled weights only for ties. On HumanEval, the table reports exact correct counts for four resampler-selector combinations across three models.

D Selector definitions

  • The weight-draw rule samples by normalized final weights, ARGMAX returns the highest-weight particle’s answer, and ORACLE measures whether any population answer is correct.ORACLE is a population ceiling rather than a selector available without the gold answer.Source: D Selector definitions · p. 18
  • Semantic majority deduplicates token-identical trajectories, clusters equivalent answers, and votes once per distinct trajectory, using pooled weight only to break ties.For symbolic tasks the representative is the first answer that creates a cluster; for HumanEval it is the cluster’s highest-pooled-weight program.Source: D Selector definitions · p. 18
  • Clustering compares candidate answers with one another, not with the gold answer, and uses benchmark-specific equivalence checks and greedy match-to-representative assignment.Unparseable answers are excluded from symbolic votes; HumanEval uses behavioral signatures on fixed generated test inputs.Source: D Selector definitions · p. 18
  • On HumanEval, Chopthin plus behavioral majority yields 101, 126, and 116 correct out of 164, versus 100, 125, and 116 for systematic resampling plus behavioral majority.The model order is Qwen2.5-Math-7B, Qwen2.5-7B, then Qwen3-4B.Source: D Selector definitions · p. 18

E.1 Oracle coverage values

Oracle coverage exceeds majority-selector accuracy in every cell, leaving selector headroom after resampling. The reported GRPO comparison is contextual rather than controlled because evaluation protocols are unmatched.

E.1 Oracle coverage values

  • Coverage exceeds majority-selector accuracy in every cell, with a median gap of 4.2 points and a smallest gap of +1.5 for Qwen2.5-7B on AIME under Chopthin.Table 3 reports oracle coverage, the fraction of problems whose final population contains at least one correct trajectory; it is a ceiling for a selector, not i.i.d. pass@N.Source: §E.1 · p. 18, Table 3 · p. 19

E.2 Reported GRPO reference values

  • The reported GRPO references show CCPS competitive on MATH500 and stronger on HumanEval, while GRPO leads on GPQA for both models.The table’s prompts, checkpoints, graders, and budgets are not matched to the CCPS evaluation, so the values are context rather than a controlled comparison.Source: §E.2 · p. 19, Table 4 · p. 19

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