arxiv: 2605.09853 · v1 · submitted 2026-05-11 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Exploration-Driven Optimization for Test-Time Large Language Model Reasoning

Changhao Li , Yuchen Zhuang , Chenxiao Gao , Haotian Sun , Rushi Qiang , Chao Zhang , Bo Dai

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:28 UTC · model grok-4.3

classification 💻 cs.LG

keywords Exploration-Driven OptimizationLLM reasoningpost-trainingreinforcement learningtest-time scalingsolution diversityDirect Preference Optimization

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The pith

Exploration-Driven Optimization adds diversity-promoting objectives to LLM RL post-training to enhance reasoning with test-time scaling.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to address the tension between RL post-training that sharpens LLM probability distributions and inference-time methods that require diverse sampling. It introduces Exploration-Driven Optimization (EDO) by extending reward-biasing exploration into iterative post-training and combining it with standard RL objectives. This produces two variants, ED-iDPO and ED-GRPO, that generate more diverse solutions. A sympathetic reader would care because the approach leads to better reasoning performance especially when using test-time computation like self-consistency, while keeping training stable.

Core claim

EDO extends reward-biasing style exploration objectives to iterative post-training and integrates them into RL objectives, encouraging greater diversity in sampled solutions while facilitating more effective inference-time computation. When incorporated into iDPO and GRPO, the resulting methods show greater solution diversity and improved reasoning abilities. On three in-distribution reasoning benchmarks, EDO achieves 1.0-1.3% improvement over strongest baselines, with an additional 1.5% average gain on five out-of-distribution tasks. It also preserves model entropy and stabilizes RL training dynamics.

What carries the argument

Exploration-Driven Optimization (EDO), which extends reward-biasing exploration objectives to iterative post-training and integrates them into standard RL objectives.

Load-bearing premise

That adding reward-biasing exploration objectives to iterative post-training will reliably boost solution diversity and reasoning performance without new instabilities or loss of core model capabilities.

What would settle it

A direct comparison showing that ED-iDPO and ED-GRPO produce no more diverse solutions or no accuracy improvements over standard iDPO and GRPO on the same benchmarks and test-time methods would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.09853 by Bo Dai, Changhao Li, Chao Zhang, Chenxiao Gao, Haotian Sun, Rushi Qiang, Yuchen Zhuang.

**Figure 1.** Figure 1: EDO exhibits a more flattened probability distribution, encouraging greater solution diversity and enhancing reasoning performance when integrated with inference-time computation techniques. indicates the trainable parameters. Recent advances have shown the increasing effectiveness of post-training techniques for enhancing the reasoning capabilities of large language models (LLMs). State-of-the-art mod… view at source ↗

**Figure 2.** Figure 2: Accuracy vs. Epochs on Math and GSM8K across different optimization methods. SelfConsistency (SC) is applied at inference, and results are averaged over three runs. 88 90 92 94 ED-iDPO GSM8K 74 76 78 80 MATH 38 40 42 44 46 s1K 88 90 92 94 ED-GRPO 74 76 78 80 38 40 42 44 46 Self-Consistency Skywork-PRM-1.5B Qwen2.5-Math-PRM-7B [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 4.** Figure 4: Evolution of policy entropy during training on GSM8K and Math datasets. Base ETO ED-iDPO 0.364 0.365 0.366 0.367 Math 0.484 0.486 0.488 0.490 0.492 GSM8K 0.38 0.39 0.40 S1K Base GRPO DAPO ED-GRPO 0.20 0.25 0.30 0.35 0.40 Math 0.3 0.4 0.5 GSM8K 0.3 0.4 0.5 0.6 S1K [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 6.** Figure 6: Case study of EDO on the MATH dataset. For the given question, the iterative online DPO-trained model consistently produces homogeneous solutions, leading to incorrect answers even when combined with test-time scaling methods. In contrast, the model trained with ED-iDPO generates significantly more diverse solutions, ultimately yielding the correct answer. For visualization purposes, we display only 3 cand… view at source ↗

read the original abstract

Post-training techniques combined with inference-time scaling significantly enhance the reasoning and alignment capabilities of large language models (LLMs). However, a fundamental tension arises: inference-time methods benefit from diverse sampling from a relatively flattened probability distribution, whereas reinforcement learning (RL)-based post-training inherently sharpens these distributions. To address this, we propose Exploration-Driven Optimization (EDO), which extends reward-biasing style exploration objectives to iterative post-training and integrates them into standard RL objectives, encouraging greater diversity in sampled solutions while facilitating more effective inference-time computation. We incorporate EDO into iterative Direct Preference Optimization (iDPO) and Group Relative Policy Optimization (GRPO), resulting in two variants: ED-iDPO and ED-GRPO. Extensive experiments demonstrate that both ED-iDPO and ED-GRPO exhibit greater solution diversity and improved reasoning abilities, particularly when combined with test-time computation techniques like self-consistency. Across three in-distribution reasoning benchmarks, EDO achieves a 1.0-1.3\% improvement over the strongest baselines, and delivers an additional 1.5\% average gain on five out-of-distribution tasks. Beyond accuracy, EDO preserves model entropy and stabilizes RL training dynamics, highlighting its effectiveness in preventing over-optimization collapse. Taken together, these results establish EDO as a practical framework for balancing exploration and exploitation in LLM reasoning, especially in settings that rely on test-time scaling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

EDO adds a reward-biasing exploration term to iterative post-training so models stay diverse enough for test-time scaling to help, with small accuracy lifts on reasoning benchmarks.

read the letter

EDO is a practical fix for the mismatch between RL post-training, which narrows distributions, and test-time methods that need variety. The authors fold exploration objectives into iDPO and GRPO to produce ED-iDPO and ED-GRPO, then show these keep entropy higher and stabilize training while delivering 1.0-1.3% gains on in-distribution reasoning tasks and an extra 1.5% on out-of-distribution ones when self-consistency is used at inference.

Referee Report

1 major / 3 minor

Summary. The paper introduces Exploration-Driven Optimization (EDO), which extends reward-biasing exploration objectives into iterative post-training by modifying iDPO and GRPO into ED-iDPO and ED-GRPO variants. The central claim is that this increases solution diversity, yields 1.0-1.3% gains over strong baselines on three in-distribution reasoning benchmarks and an additional 1.5% average on five out-of-distribution tasks when paired with test-time scaling (e.g., self-consistency), while preserving model entropy and stabilizing RL training dynamics to avoid over-optimization collapse.

Significance. If the empirical results hold under the reported controls, the work is significant for resolving the tension between RL post-training (which sharpens distributions) and inference-time methods (which benefit from diversity). The inclusion of training curves, benchmark tables, and explicit modified objectives provides direct evidence for the stability and diversity claims, strengthening the practical contribution to LLM reasoning pipelines.

major comments (1)

§4.1-4.2 (ED-iDPO and ED-GRPO objectives): the exploration term is added to the base losses, but the manuscript does not report the specific coefficient values used across experiments or include an ablation on its sensitivity; this is load-bearing for the claim that EDO reliably increases diversity without new instabilities, as the gains could be sensitive to this choice.

minor comments (3)

Table 1 and Table 2: report the number of random seeds and standard deviations for the accuracy numbers to allow readers to assess whether the 1.0-1.5% gains are statistically distinguishable from baseline variance.
§5.3 (OOD tasks): clarify the exact definition of 'out-of-distribution' (e.g., domain shift vs. task shift) and whether the same base models and training data were used, to strengthen the generalization claim.
Figure 2 (entropy curves): add a direct comparison panel showing entropy under standard iDPO/GRPO versus the EDO variants on the same plot for easier visual assessment of the preservation claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and recommendation for minor revision. The feedback correctly identifies a gap in the presentation of the exploration coefficient that affects reproducibility and the strength of our stability claims. We address this point directly below.

read point-by-point responses

Referee: [—] §4.1-4.2 (ED-iDPO and ED-GRPO objectives): the exploration term is added to the base losses, but the manuscript does not report the specific coefficient values used across experiments or include an ablation on its sensitivity; this is load-bearing for the claim that EDO reliably increases diversity without new instabilities, as the gains could be sensitive to this choice.

Authors: We agree that the manuscript does not explicitly report the coefficient values for the exploration term or provide a sensitivity ablation, and that this information is important for supporting the claims of reliable diversity gains without new instabilities. In the revised version we will state the exact coefficient values used in all reported experiments (chosen via validation-set tuning to balance the objectives) directly in Sections 4.1 and 4.2. We will also add an ablation study in the appendix that varies the coefficient over a reasonable range and reports the resulting effects on solution diversity, benchmark accuracy, entropy preservation, and training stability. These additions will make the hyperparameter choice transparent and will directly substantiate that the reported benefits are not overly sensitive to this choice. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical claims independent of derivations

full rationale

The paper proposes Exploration-Driven Optimization (EDO) by extending reward-biasing exploration objectives into iterative post-training and integrating them with iDPO and GRPO to form ED-iDPO and ED-GRPO. It reports empirical gains (1.0-1.3% in-distribution, 1.5% OOD) plus entropy preservation from benchmark experiments. No equations, derivations, or self-referential definitions appear in the provided text that would reduce predictions to inputs by construction, rename fitted parameters as predictions, or rely on load-bearing self-citations. The central claims rest on direct experimental comparisons rather than any self-definitional or fitted-input circular chain, rendering the argument self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only view yields no explicit free parameters, axioms, or invented entities; the method implicitly assumes that reward-biasing exploration can be stably integrated into existing RL objectives without side effects.

pith-pipeline@v0.9.0 · 5563 in / 1141 out tokens · 25237 ms · 2026-05-12T04:28:06.012959+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We propose Exploration-Driven Optimization (EDO), which extends reward-biasing style exploration objectives to iterative post-training and integrates them into standard RL objectives... LED-iDPO = LiDPO − αJ∗(r(t))
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

EDO preserves model entropy and stabilizes RL training dynamics

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

We were told that Mohamed is currently twice 30 years old, so he is currently 30 *2=60 years old

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[2]

That means that four years ago he must have been 60 - 4 = 56 years old

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[3]

Four years ago, Kody was half as old as Mohamed, so Kody must have been 56 / 2 = 28 years old then

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[4]

Since Kody was 28 years old four years ago, she must now be 28 + 4 = 32 years old

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[5]

Q: Carla bought 2 bags of mini peanut butter cups on clearance

So the answer is 32. Q: Carla bought 2 bags of mini peanut butter cups on clearance. Each bag was $6.00 but was 75% off. How much did she spend on 2 bags of candy? A: Let’s think step by step

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[6]

Each bag was $6.00 but was 75% off

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[7]

So each bag cost $6.00 * (1 - 0.75) = $6.00 * 0.25 = $1.50

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[8]

So she spent $1.50 * 2 = $3.00

Carla bought 2 bags. So she spent $1.50 * 2 = $3.00

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[9]

Q: If Pam is currently twice as young as Rena is, and in 10 years Rena will be 5 years older than her, how old is Pam now? A: Let’s think step by step

So the answer is 3. Q: If Pam is currently twice as young as Rena is, and in 10 years Rena will be 5 years older than her, how old is Pam now? A: Let’s think step by step

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[10]

Since Rena will be 5 years older than Pam in 10 years, she must be 5 years older than Pam now as well

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[11]

If Pam is currently twice as young as Rena, that means that Rena is currently twice as old as Pam is

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[12]

So if P stands for Pam’s age now and R stands for Rena’s age now, then we know that R = 2 * P And since Rena is 5 years older than Pam now, we know that R = P + 5

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[13]

By substitution, we have P + 5 = 2 * P, which means that P = 5

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[14]

Q: Cappuccinos cost $2, iced teas cost $3, cafe lattes cost $1.5 and espressos cost $1 each

So the answer is 5. Q: Cappuccinos cost $2, iced teas cost $3, cafe lattes cost $1.5 and espressos cost $1 each. Sandy orders some drinks for herself and some friends. She orders three cappuccinos, two iced teas, two cafe lattes, and two espressos. How much change does she receive back for a twenty-dollar bill? A: Let’s think step by step

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[15]

Sandy ordered three cappuccinos, which cost $2 each, so she spent $2 * 3 = $6 on cappuccinos

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[16]

She ordered two iced teas, which cost $3 each, so she spent $3 * 2 = $6 dollars on ice teas

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[17]

She ordered two cafe lattes, which cost $1.5 each, so she spent $1.5 * 2 = $3 on cafe lattes

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[18]

She ordered two espressos, which cost $1 each, so she spent $1 * 2 = $2 on espressos

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[19]

So altogether, Sandy spent $6 + $6 + $3 + $2 = $17 on drinks, which means that sandy will get $20 - $17 = $3 as change

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[20]

[END OF EXAMPLE] Please answer the following question: Q:{Question} A: Let’s think step by step

So the answer is 3. [END OF EXAMPLE] Please answer the following question: Q:{Question} A: Let’s think step by step. You MUST write the final answer only as an integer after the phrase ’So the answer is’. G.2 Math For Math, we use a zero-shot prompt to guide the policy model, which also remains consistent across both training and evaluation: 26 Math Promp...

work page 2024
[21]

2) Sequential Scaling: It focuses on extending Chain-of-thoughts (Wei et al., 2022) by incorporating reflective reasoning processes

are typically used for aggregating the generated candidates. 2) Sequential Scaling: It focuses on extending Chain-of-thoughts (Wei et al., 2022) by incorporating reflective reasoning processes. A prominent example is Deepseek-R1 (Guo et al., 2025), which enhances reasoning abilities by training with GRPO (Shao et al., 2024) and extending the reasoning tra...

work page 2022