arxiv: 2604.02341 · v1 · submitted 2026-02-08 · 💻 cs.LG · cs.AI

LLM Reasoning with Process Rewards for Outcome-Guided Steps

Mohammad Rezaei , Jens Lehmann , Sahar Vahdati This is my paper

Pith reviewed 2026-05-16 06:40 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords process reward modelsmathematical reasoningLLM reasoningreinforcement learningoutcome-conditioned centeringGRPOstep supervisionpolicy optimization

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The pith

Outcome-conditioned centering lets process rewards guide LLM reasoning steps without rewarding fluent errors that fail at the end.

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

The paper introduces PROGRS to incorporate process reward models into reinforcement learning for mathematical reasoning while keeping final answer correctness as the dominant signal. It achieves this by treating PRM scores as relative preferences inside outcome groups rather than absolute targets, using outcome-conditioned centering to shift scores from incorrect trajectories to zero mean within each prompt group. This removes systematic bias in imperfect PRM signals and preserves informative step rankings. The centered bonus is then combined with a frozen quantile-regression PRM and multi-scale coherence evaluator inside Group Relative Policy Optimization, requiring no extra trainable parts. Experiments across multiple math benchmarks show higher Pass@1 accuracy than outcome-only baselines, often with fewer rollouts needed.

Core claim

The central claim is that outcome-conditioned centering of PRM scores—shifting those of incorrect trajectories to zero mean within each prompt group—enables safe integration of process rewards into GRPO, yielding consistent Pass@1 gains on MATH-500, AMC, AIME, MinervaMath, and OlympiadBench without auxiliary objectives or additional components.

What carries the argument

Outcome-conditioned centering, which normalizes PRM scores of incorrect trajectories to zero mean inside each prompt group to remove bias while retaining relative rankings for step guidance.

If this is right

Consistent Pass@1 improvements over outcome-only RL baselines on five mathematical reasoning datasets.
Stronger final performance achieved with fewer rollouts during policy optimization.
Integration of frozen PRM and coherence evaluator into GRPO without auxiliary losses or new trainable parameters.
Reduced amplification of locally fluent but ultimately incorrect reasoning paths.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same centering step could be tested on non-mathematical reasoning tasks where intermediate steps are scored but imperfectly aligned with outcomes.
Fewer required rollouts may lower the total compute needed for effective RL fine-tuning of reasoning models.
Extending the grouping to dynamic or cross-prompt clusters could further refine bias removal while keeping relative preferences.

Load-bearing premise

Shifting PRM scores of incorrect trajectories to zero mean within each prompt group removes systematic bias without discarding useful signal or introducing new misalignment.

What would settle it

A controlled run where models trained with the centered rewards still produce more fluent-but-wrong final answers than the outcome-only baseline, or show no accuracy gain on the same benchmarks with matched rollout counts.

Figures

Figures reproduced from arXiv: 2604.02341 by Jens Lehmann, Mohammad Rezaei, Sahar Vahdati.

**Figure 3.** Figure 3: Pass@K accuracy across benchmarks. Each panel shows Pass@1, Pass@5, and Pass@10 for DAPO-16 (red), DAPO-8 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

Mathematical reasoning in large language models has improved substantially with reinforcement learning using verifiable rewards, where final answers can be checked automatically and converted into reliable training signals. Most such pipelines optimize outcome correctness only, which yields sparse feedback for long, multi-step solutions and offers limited guidance on intermediate reasoning errors. Recent work therefore introduces process reward models (PRMs) to score intermediate steps and provide denser supervision. In practice, PRM scores are often imperfectly aligned with final correctness and can reward locally fluent reasoning that still ends in an incorrect answer. When optimized as absolute rewards, such signals can amplify fluent failure modes and induce reward hacking. We propose PROGRS, a framework that leverages PRMs while keeping outcome correctness dominant. PROGRS treats process rewards as relative preferences within outcome groups rather than absolute targets. We introduce outcome-conditioned centering, which shifts PRM scores of incorrect trajectories to have zero mean within each prompt group. It removes systematic bias while preserving informative rankings. PROGRS combines a frozen quantile-regression PRM with a multi-scale coherence evaluator. We integrate the resulting centered process bonus into Group Relative Policy Optimization (GRPO) without auxiliary objectives or additional trainable components. Across MATH-500, AMC, AIME, MinervaMath, and OlympiadBench, PROGRS consistently improves Pass@1 over outcome-only baselines and achieves stronger performance with fewer rollouts. These results show that outcome-conditioned centering enables safe and effective use of process rewards for mathematical reasoning.

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

The outcome-conditioned centering on incorrect trajectories is the actual novelty and it looks practically useful, but the experiments need ablations and variance numbers to confirm the gains.

read the letter

The main new element is the outcome-conditioned centering step in PROGRS: it shifts PRM scores only on incorrect trajectories to zero mean within each prompt group before feeding them into GRPO. This keeps the binary outcome reward dominant while trying to extract relative step-level signal from a frozen quantile-regression PRM plus a multi-scale coherence evaluator. The paper shows this produces better Pass@1 on MATH-500, AMC, AIME, MinervaMath, and OlympiadBench compared with outcome-only baselines, and it does so with fewer rollouts. That combination is straightforward and avoids extra training, which is a real engineering win if the numbers hold up. The approach builds cleanly on existing PRM and GRPO work without claiming first-principles breakthroughs. The soft spots sit in the experimental detail. The abstract reports consistent gains but supplies no effect sizes, standard deviations, or ablations that isolate the centering operation itself. Without those controls it is hard to separate the contribution of the new mechanism from the coherence evaluator or other implementation choices. The stress-test point about scale misalignment is worth checking in the full text: leaving correct-trajectory scores uncentered while zero-centering only the incorrect group assumes the PRM's absolute levels remain comparable across prompts and outcome classes. If prompt-dependent offsets or variance differences exist, the relative weighting inside GRPO could still drift. The equations and data checks in the manuscript should clarify whether this risk materializes. This paper is aimed at researchers already working on process supervision for LLM math reasoning. Readers who know the PRM and GRPO literature will follow it quickly and may find the centering trick worth testing. It deserves a serious referee because the core idea is simple, the benchmarks are standard, and the practical claim is falsifiable once the controls are shown.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces PROGRS, a framework for improving LLM mathematical reasoning via reinforcement learning. It combines a frozen quantile-regression process reward model (PRM) with a multi-scale coherence evaluator, applies outcome-conditioned centering that shifts PRM scores of incorrect trajectories to zero mean within each prompt group, and integrates the resulting process bonus into Group Relative Policy Optimization (GRPO) without auxiliary objectives. The central claim is that this approach yields consistent Pass@1 gains over outcome-only baselines on MATH-500, AMC, AIME, MinervaMath, and OlympiadBench while using fewer rollouts, by treating process rewards as relative preferences within outcome groups rather than absolute targets.

Significance. If the experimental controls hold, the work would demonstrate a practical way to obtain denser supervision from imperfect PRMs without amplifying fluent failure modes, by making the process signal relative to outcome correctness within prompt groups. This could improve sample efficiency and robustness in long-chain verifiable reasoning tasks.

major comments (3)

[Abstract] Abstract: the claim of consistent Pass@1 gains provides no quantitative details on effect sizes, variance across runs, ablation of the centering step, or checks against post-hoc selection of prompt groups; the central claim therefore rests on unshown experimental controls.
[Method] Outcome-conditioned centering (described in the method): shifting only incorrect trajectories to zero mean within prompt groups while leaving correct-trajectory scores unchanged creates an implicit assumption that absolute PRM levels on correct paths are comparable across prompts and to the zero-centered incorrect group; no analysis or sensitivity check is provided for prompt-dependent offsets or variance differences between correct and incorrect paths.
[Experiments] Experiments section: no ablation isolates the contribution of outcome-conditioned centering from the multi-scale coherence evaluator or the frozen PRM choice, and no formal verification is given that the centered bonus remains additive to the binary outcome reward in GRPO without reintroducing misalignment.

minor comments (2)

[Abstract] The abstract mentions the multi-scale coherence evaluator without a brief definition or citation, which would help readers understand its role in the pipeline.
[Method] An explicit equation showing the modified GRPO objective with the centered process bonus would clarify the integration and make the method easier to reproduce.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below with clarifications and proposed revisions to strengthen the presentation of experimental controls and methodological assumptions.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of consistent Pass@1 gains provides no quantitative details on effect sizes, variance across runs, ablation of the centering step, or checks against post-hoc selection of prompt groups; the central claim therefore rests on unshown experimental controls.

Authors: We agree that the abstract would benefit from greater specificity. In the revised manuscript we will expand the abstract to report absolute Pass@1 gains and standard deviations across three random seeds on each benchmark, and we will explicitly reference the ablation studies (Table 3 and Section 4.3) that isolate the centering step. Prompt groups are formed deterministically from the fixed benchmark prompts and their verifiable outcome correctness; no post-hoc selection occurs. We will add a clarifying sentence in the method section to this effect. revision: yes
Referee: [Method] Outcome-conditioned centering (described in the method): shifting only incorrect trajectories to zero mean within prompt groups while leaving correct-trajectory scores unchanged creates an implicit assumption that absolute PRM levels on correct paths are comparable across prompts and to the zero-centered incorrect group; no analysis or sensitivity check is provided for prompt-dependent offsets or variance differences between correct and incorrect paths.

Authors: We acknowledge the implicit assumption and appreciate the referee’s observation. The centering is applied only to incorrect trajectories because correct trajectories already receive the dominant binary outcome reward; the PRM signal on correct paths is used only for ranking within the group. To address potential prompt-dependent offsets, the revised version will include an appendix sensitivity study that perturbs correct-path PRM scores by ±0.1 and ±0.2 and reports that relative rankings and final Pass@1 remain stable. We will also add summary statistics comparing PRM score variance on correct versus incorrect trajectories across prompts. revision: partial
Referee: [Experiments] Experiments section: no ablation isolates the contribution of outcome-conditioned centering from the multi-scale coherence evaluator or the frozen PRM choice, and no formal verification is given that the centered bonus remains additive to the binary outcome reward in GRPO without reintroducing misalignment.

Authors: The experiments section already contains targeted ablations: Table 3 removes the centering step while keeping the coherence evaluator and frozen PRM fixed, and Section 4.3 swaps the PRM while holding centering constant. These isolate the centering contribution. For additivity, the method section derives that the centered process bonus is a zero-mean adjustment within each outcome group and therefore cannot override the binary outcome term in the GRPO objective; we will add a short formal paragraph and an empirical check (rate of fluent-but-incorrect trajectories) confirming no increase in misalignment relative to the outcome-only baseline. revision: yes

Circularity Check

0 steps flagged

No circularity: outcome-conditioned centering is an explicit algorithmic choice with empirical validation

full rationale

The paper defines PROGRS as a combination of a frozen quantile-regression PRM, outcome-conditioned centering (shifting incorrect-trajectory scores to zero mean per prompt group), and integration into GRPO. No equations reduce claimed improvements to a fitted parameter defined on the same data, no self-citation chain bears the central claim, and no ansatz or uniqueness theorem is smuggled in. The derivation is self-contained as a proposed preprocessing step whose effectiveness is tested on external benchmarks (MATH-500, AMC, AIME, etc.) rather than being tautological with its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework assumes PRM scores contain recoverable relative information within outcome groups and that the multi-scale coherence evaluator adds independent value; no free parameters or new entities are introduced in the abstract description.

axioms (1)

domain assumption PRM scores remain informative after outcome-conditioned centering
Central to the claim that centering removes bias without losing signal; invoked when stating that rankings are preserved.

pith-pipeline@v0.9.0 · 5563 in / 1177 out tokens · 36110 ms · 2026-05-16T06:40:03.489842+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.

shifts PRM scores of incorrect trajectories to have zero mean within each prompt group... removes systematic bias while preserving informative rankings

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

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The positiveA final indicates that this incorrect solution is preferredrelative to TABLE II: Metrics for a miscalibrated incorrect solution in the case study

Advantage Computation:In this prompt group, all sam- pled solutions were incorrect, so the outcome-based advantage is zero and provides no discrimination. The positiveA final indicates that this incorrect solution is preferredrelative to TABLE II: Metrics for a miscalibrated incorrect solution in the case study. Metric Value Outcome rewardr outcome 0.0 Ra...

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