arxiv: 2604.11056 · v1 · submitted 2026-04-13 · 💻 cs.LG · cs.AI

Recognition: 2 theorem links

· Lean Theorem

Rethinking Token-Level Credit Assignment in RLVR: A Polarity-Entropy Analysis

Hange Zhou, Haodong Wu, Hongyu Ge, Keyi Wu, Qihong Lin, Siyi Liu, Yongqi Zhang, Yuhang He, Zhuo Zheng, Zixin Zhong

Pith reviewed 2026-05-10 15:29 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords RLVRcredit assignmenttoken entropyconditional mutual informationpolicy optimizationLLM reasoningreinforcement learning

0 comments

The pith

The credit a token can carry in RLVR is upper-bounded by its entropy.

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

The paper examines the credit assignment problem that arises when sparse outcome rewards are used to improve reasoning in large language models via reinforcement learning. It introduces a Four Quadrant Decomposition that separates token updates according to reward polarity and token entropy, then shows through controlled ablations that reasoning gains concentrate in the high-entropy quadrants. Adapting conditional mutual information to the autoregressive setting yields a proof that the credit any token can receive is limited by its entropy. The same analysis reveals how uniform reward broadcasting in GRPO dilutes signals at uncertain positions and over-credits deterministic ones. These observations motivate Entropy-Aware Policy Optimization, which modulates per-token updates and improves performance across model families.

Core claim

In the autoregressive RLVR setting, the credit a token can carry is upper-bounded by its entropy, as shown by adapting conditional mutual information. The Four Quadrant Decomposition isolates that reasoning improvements concentrate in high-entropy quadrants, while gradient analysis of GRPO shows signal dilution at high-entropy positions and over-crediting of low-entropy tokens. This view supplies testable predictions about the distinct roles of positive and negative updates and grounds the design of Entropy-Aware Policy Optimization.

What carries the argument

The Four Quadrant Decomposition, which partitions token updates by reward polarity and entropy, together with the adaptation of Conditional Mutual Information that proves token credit is upper-bounded by entropy.

If this is right

Reasoning improvements arise primarily from high-entropy tokens, with positive and negative updates playing distinct roles there.
Uniform reward broadcast dilutes the learning signal at high-entropy positions while over-crediting deterministic low-entropy tokens.
Modulating token-level learning rates according to entropy, as done in EAPO, produces measurable gains over standard RLVR methods.
The entropy bound supplies concrete predictions about where future reward-shaping techniques should focus their adjustments.

Where Pith is reading between the lines

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

The same entropy-credit relation may help diagnose why RLVR succeeds on some reasoning tasks but stalls on others that contain fewer uncertain steps.
Training loops could monitor token entropy on the fly and selectively amplify gradients only at high-entropy positions without changing the reward function.
If the bound is tight, deliberately increasing entropy at selected reasoning steps before applying RLVR might enlarge the effective credit window for those tokens.

Load-bearing premise

The adaptation of conditional mutual information to the autoregressive RLVR setting correctly captures the credit a token can carry.

What would settle it

An experiment in which low-entropy tokens produce larger reasoning gains than high-entropy tokens, or in which measured token credit exceeds the entropy bound during RLVR training.

Figures

Figures reproduced from arXiv: 2604.11056 by Hange Zhou, Haodong Wu, Hongyu Ge, Keyi Wu, Qihong Lin, Siyi Liu, Yongqi Zhang, Yuhang He, Zhuo Zheng, Zixin Zhong.

**Figure 1.** Figure 1: Tokens in Bold exhibit high entropy. High-entropy and low-entropy tokens hold different credit upper bounds in an information-theoretic view. We decompose token updates in RLVR into four quadrants to study the token-level credit assignment mechanism. See Appendix I for full entropy examples. Our framework confirms the predictions. Useful credit is concentrated at high-entropy positions rather than routine… view at source ↗

**Figure 2.** Figure 2: The training dynamics and evaluation of four quadrant decomposition. (a) Training solve rate. (b) Training [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Performance–efficiency trade-off for 1.5B [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Token entropy distribution. While lowentropy distributions remain aligned, a sharp divergence emerges between positive and negative samples in the high-entropy quadrant [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

**Figure 5.** Figure 5: Test time compute performance of four-quadrant training objectives. The x-axis represents varying [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗

**Figure 6.** Figure 6: Entropy dynamics analysis of 7B models. GRPO suffers from entropy collapse, which constrains exploratory capacity, while EntroReg tends to over-flatten the output distribution. We further analyze the change in the four quadrants by tracking absolute entropy shifts. EAPO predominantly modulates the high-entropy quadrants (PHR and NHR), whereas the low-entropy quadrants remain comparatively stable, confirm… view at source ↗

**Figure 7.** Figure 7: Hyperparameter analysis. We evaluate the impact of [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

**Figure 8.** Figure 8: Advantage value distributions. Comparison of [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗

**Figure 9.** Figure 9: High entropy tokens word cloud [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗

**Figure 10.** Figure 10: Low entropy tokens word cloud classification of tokens into high and low entropy categories. We are curious whether entropy levels represent consistent model behaviors, regardless of correctness. For example, do low-entropy tokens usually indicate overconfident mistakes or robust reasoning steps in incorrect responses? To this end, we examine several examples and visualize two representative examples in … view at source ↗

**Figure 11.** Figure 11: Visualization of token entropy in incorrect and correct responses. [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗

read the original abstract

Reinforcement Learning with Verifiable Rewards (RLVR) has substantially improved the reasoning ability of Large Language Models (LLMs). However, its sparse outcome-based rewards pose a fundamental credit assignment problem. We analyze this problem through the joint lens of reward polarity and token entropy. Our diagnostic tool, the Four Quadrant Decomposition, isolates token updates by polarity and entropy, and controlled ablations show that reasoning improvements concentrate in the high-entropy quadrants. To justify this observation theoretically, we adapt Conditional Mutual Information to the autoregressive RLVR setting and prove that the credit a token can carry is upper-bounded by its entropy. This view yields testable predictions that reasoning gains arise primarily from high-entropy tokens, with unique roles for positive and negative updates. A gradient analysis of GRPO further reveals how uniform reward broadcast dilutes signal at high-entropy positions while over-crediting deterministic tokens. Grounded in these insights, we propose Entropy-Aware Policy Optimization (EAPO) that modulates token-level learning signals accordingly. Extensive experiments demonstrate that EAPO outperforms strong baselines across two model families.

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 quadrant decomposition and EAPO deliver measurable gains on RLVR reasoning tasks, but the CMI bound's connection to actual policy gradients is the part that needs checking.

read the letter

The main takeaway is that this paper gives a practical way to think about token credit assignment in reinforcement learning with verifiable rewards. By breaking tokens into four quadrants based on reward polarity and entropy, they show that improvements cluster in high-entropy tokens. They support this with a theoretical bound using conditional mutual information and introduce Entropy-Aware Policy Optimization, which outperforms baselines. The new parts are the quadrant analysis and the specific EAPO modulation. The ablations are controlled and the results hold across model families, which is a plus. The gradient analysis of how uniform rewards affect high-entropy positions is also useful. On the downside, the proof adapts CMI but I am not fully convinced it captures the actual credit in the RLVR objective without additional assumptions about how the reward factors. The stress test note raises a fair point about whether the bound constrains the policy gradient term directly. Also, the paper might benefit from more details on variance in advantage estimation. Overall, this is for the RL-for-LLMs crowd. It has enough substance to go to peer review, where the theory can be checked more carefully while the empirical side is already promising.

Referee Report

3 major / 2 minor

Summary. The paper addresses token-level credit assignment in Reinforcement Learning with Verifiable Rewards (RLVR) for LLM reasoning. It introduces the Four Quadrant Decomposition tool based on reward polarity and token entropy, uses controlled ablations to show that reasoning improvements concentrate in high-entropy quadrants, adapts Conditional Mutual Information (CMI) to the autoregressive RLVR setting to prove that token credit is upper-bounded by entropy, analyzes how GRPO dilutes signals at high-entropy positions, and proposes Entropy-Aware Policy Optimization (EAPO) that outperforms baselines across two model families.

Significance. If the CMI adaptation is shown to map rigorously onto the actual policy-gradient signals and the empirical concentration result is robust, the work provides a useful diagnostic lens and practical method for credit assignment under sparse outcome rewards. The combination of quadrant analysis, theoretical bound, GRPO gradient insights, and the EAPO algorithm could guide more targeted updates in RLVR, with the experiments across model families strengthening applicability.

major comments (3)

The central theoretical claim adapts CMI to prove that the credit a token can carry is upper-bounded by its entropy. While I(token_i; R | tokens_<i) ≤ H(token_i | tokens_<i) holds by definition, the manuscript must derive the explicit correspondence between this MI quantity and the magnitude of the token-level learning signal (advantage-weighted log-prob term) under the sparse verifiable reward and GRPO-style objective; without this step the bound does not necessarily constrain the policy update.
Four Quadrant Decomposition: the isolation of token updates by polarity and entropy relies on quadrant thresholds that appear post-hoc. The claim that reasoning improvements concentrate in high-entropy quadrants is load-bearing for the overall narrative; the paper should demonstrate robustness to alternative threshold choices or non-quadrant partitions.
Gradient analysis of GRPO: the argument that uniform reward broadcast dilutes signal at high-entropy positions while over-crediting deterministic tokens requires a more precise accounting of how entropy interacts with advantage estimation variance and the autoregressive factorization of the trajectory.

minor comments (2)

Abstract: the claim of 'extensive experiments' would benefit from naming the specific baselines and reporting key performance deltas to allow readers to assess the practical impact immediately.
Notation: ensure that the definitions of conditional mutual information, entropy, and polarity are introduced with consistent symbols before their first use in the theoretical sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and detailed report. We address each major comment below with clarifications and commit to specific revisions that strengthen the theoretical and empirical components of the manuscript.

read point-by-point responses

Referee: The central theoretical claim adapts CMI to prove that the credit a token can carry is upper-bounded by its entropy. While I(token_i; R | tokens_<i) ≤ H(token_i | tokens_<i) holds by definition, the manuscript must derive the explicit correspondence between this MI quantity and the magnitude of the token-level learning signal (advantage-weighted log-prob term) under the sparse verifiable reward and GRPO-style objective; without this step the bound does not necessarily constrain the policy update.

Authors: We agree that the current presentation of the CMI adaptation stops short of an explicit mapping to the GRPO policy-gradient term. In the revision we will insert a new lemma in Section 3.2 that decomposes the expected advantage-weighted log-probability under the sparse outcome reward into an information-theoretic component, showing that its magnitude is bounded above by I(token_i; R | tokens_<i) and hence by H(token_i | tokens_<i). The derivation will explicitly use the autoregressive factorization and the fact that the verifiable reward is a deterministic function of the full trajectory. revision: yes
Referee: Four Quadrant Decomposition: the isolation of token updates by polarity and entropy relies on quadrant thresholds that appear post-hoc. The claim that reasoning improvements concentrate in high-entropy quadrants is load-bearing for the overall narrative; the paper should demonstrate robustness to alternative threshold choices or non-quadrant partitions.

Authors: The referee correctly notes that the quadrant thresholds were selected after initial inspection. We will add an appendix containing sensitivity analyses that vary the entropy threshold across the 25th, 50th, and 75th percentiles of the observed entropy distribution, as well as results obtained by replacing the quadrant partition with k-means clustering on the (polarity, entropy) plane. These additional experiments confirm that the concentration of reasoning gains in high-entropy regions is stable under these alternatives. revision: yes
Referee: Gradient analysis of GRPO: the argument that uniform reward broadcast dilutes signal at high-entropy positions while over-crediting deterministic tokens requires a more precise accounting of how entropy interacts with advantage estimation variance and the autoregressive factorization of the trajectory.

Authors: We accept that the current gradient analysis would benefit from greater precision on variance and autoregressive effects. In the revised manuscript we will expand the analysis to include (i) a conditional-variance decomposition of the advantage estimator given token entropy and (ii) an explicit accounting of how the autoregressive product of conditional probabilities propagates the single scalar reward through high- versus low-entropy positions. The updated section will contain the corresponding mathematical steps and a small-scale illustrative simulation. revision: yes

Circularity Check

1 steps flagged

CMI adaptation yields definitional entropy bound on 'credit' rather than RL-specific derivation

specific steps

self definitional [Abstract (theoretical justification paragraph) and Section 3 (CMI adaptation)]
"To justify this observation theoretically, we adapt Conditional Mutual Information to the autoregressive RLVR setting and prove that the credit a token can carry is upper-bounded by its entropy."

The statement equates 'credit' with the conditional mutual information I(token_i; R | tokens_<i) and then invokes the universal inequality I ≤ H. Because this inequality is true by definition for any joint distribution, the 'proof' adds no new constraint derived from the RLVR reward structure, trajectory sparsity, or policy-gradient update; the upper bound is forced once the identification is made.

full rationale

The paper's central theoretical claim adapts conditional mutual information to RLVR and 'proves' an entropy upper bound on token credit. This inequality holds identically for any random variables by the definition of mutual information (I(X;Y|Z) ≤ H(X|Z)), so the bound is tautological once credit is identified with the adapted CMI term. No additional derivation from the sparse verifiable reward, GRPO objective, or advantage-weighted log-prob is required or supplied to reach the bound; the result therefore reduces to its definitional input. The subsequent claim that reasoning gains concentrate in high-entropy tokens inherits this circularity. Other elements (Four Quadrant Decomposition, EAPO proposal, experiments) are independent and do not rely on the bound, limiting the overall circularity to partial.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the CMI adaptation is treated as a domain assumption whose validity cannot be audited without the full derivation.

pith-pipeline@v0.9.0 · 5522 in / 1146 out tokens · 27649 ms · 2026-05-10T15:29:49.448633+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
I(o_i,t; r_i | s_t) = H(o_i,t | s_t) - H(o_i,t | r_i, s_t) ≤ H_i,t (Proposition 1)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear
Four Quadrant Decomposition (PHR/PLR/NHR/NLR) isolating updates by polarity and entropy

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

From Instance Selection to Fixed-Pool Data Recipe Search for Supervised Fine-Tuning
cs.LG 2026-05 conditional novelty 7.0

AutoSelection discovers data recipes from a 90K instruction pool that outperform full-data training and other selectors on reasoning tasks for SFT across multiple models.
Not All Tokens Learn Alike: Attention Entropy Reveals Heterogeneous Signals in RL Reasoning
cs.CL 2026-05 unverdicted novelty 7.0

Attention entropy splits RL training tokens into stable anchors and volatile explorers, and entropy-aware reweighting improves held-out reasoning performance.

Reference graph

Works this paper leans on

9 extracted references · 7 canonical work pages · cited by 2 Pith papers · 4 internal anchors

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[8]

When Hi,t ≈0 , the model has committed to a near-deterministic choice (e.g., syntax or for- mula fragments), so I(o i,t;r i|st)≈0 regardless of the outcome

Low-entropy tokens carry near-zero credit. When Hi,t ≈0 , the model has committed to a near-deterministic choice (e.g., syntax or for- mula fragments), so I(o i,t;r i|st)≈0 regardless of the outcome. GRPO’s uniform ˆAi,t thus sys- tematically overestimates their contribution. 2.High-entropy tokens have maximal informa- tion budget.At positions where Hi,t ...
[9]

what should I have chosen, knowing the outcome?

The gap H(o i,t|ri, st) captures polarity- dependent structure.In positive samples, the posterior πhs(v|st, ri = 1) may sharpen around the correct token, lowering H(o i,t|ri = 1, s t) and increasing the realized CMI. In negative samples, the posterior may diffuse across mul- tiple plausible alternatives, yielding a different credit profile. This motivates...

work page arXiv 2021