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arxiv: 2606.08815 · v1 · pith:NVTOTHPGnew · submitted 2026-06-07 · 💻 cs.AI · cs.CL· cs.LG

Momentum for Reasoning: Dense Intrinsic Signals in Policy Optimization

Hao Chen , Zhanming Shen , Liyao Li , Yanyu Chen , Xuhang Zhu , Xiaomeng Hu , Qi Zhang , Ru Peng
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Xiaoyu Shen Haobo Wang Junbo Zhao
This is my paper

Pith reviewed 2026-06-27 18:24 UTC · model grok-4.3

classification 💻 cs.AI cs.CLcs.LG
keywords reinforcement learningpolicy optimizationlarge language modelsmathematical reasoningintrinsic rewardszero-advantage collapsehallucinated certaintyRLVR
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The pith

Intrinsic signals from a policy's own probabilities can densify sparse outcome rewards and reduce two structural failure modes in LLM reasoning training.

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

The paper identifies that binary outcome rewards in group relative policy optimization create zero-advantage collapse, where entire groups of rollouts yield identical results and gradients vanish, plus hallucinated certainty, where models grow overconfident on incorrect paths. It proposes ISPO to address both by adding intrinsic signals computed only from the policy's conditional probabilities: a sequence-level measure of how informative a thinking trajectory is for the final answer, and a token-level hinge that penalizes confidently wrong predictions at critical tokens. These signals are tested across three base models and five mathematical reasoning benchmarks, producing consistent gains that are largest on the hardest tasks. Training diagnostics show measurable drops in both failure modes. A reader would care because the method supplies denser learning signals without external verifiers or additional models.

Core claim

ISPO densifies the binary outcome reward by combining a sequence-level signal that measures how informative the thinking trajectory is for the final answer with a token-level directional reward whose hallucinated-certainty hinge penalizes confidently-wrong predictions at critical decision tokens; across three base models and five mathematical reasoning benchmarks this yields consistent outperformance over baselines, largest gains on the hardest benchmarks, and confirmed reductions in both zero-advantage collapse and hallucinated certainty.

What carries the argument

ISPO (Intrinsic Signal Policy Optimization), which adds sequence-level trajectory informativeness and token-level hallucinated-certainty hinge signals derived solely from the policy's conditional probabilities.

If this is right

  • Performance gains are largest on the hardest benchmarks, exactly where zero-advantage collapse is most common.
  • Training-dynamics measurements show both zero-advantage collapse and hallucinated certainty decrease under ISPO.
  • The method delivers consistent improvements across three different base models and five benchmarks.
  • Both failure modes are mitigated without requiring external verification or additional reward models.

Where Pith is reading between the lines

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

  • The same internal-probability signals might extend to non-mathematical domains where outcome verification is costly.
  • If the signals truly capture trajectory quality, they could reduce reliance on verifiable-reward setups altogether.
  • Combining the token-level hinge with other forms of self-supervised shaping could further stabilize long-horizon training.
  • The approach implies that policy entropy and conditional likelihoods already contain usable information about reasoning correctness.

Load-bearing premise

Signals computed only from the policy's own conditional probabilities give a reliable measure of trajectory informativeness and can penalize hallucinated certainty without introducing new biases.

What would settle it

A run on the same benchmarks where the frequency of zero-advantage groups stays the same or rises, or where average on incorrect tokens fails to drop, would show the intrinsic signals do not address the claimed failure modes.

Figures

Figures reproduced from arXiv: 2606.08815 by Haobo Wang, Hao Chen, Junbo Zhao, Liyao Li, Qi Zhang, Ru Peng, Xiaomeng Hu, Xiaoyu Shen, Xuhang Zhu, Yanyu Chen, Zhanming Shen.

Figure 1
Figure 1. Figure 1: ISPO outperforms competitive base￾lines across all five benchmarks. Pass@1 accu￾racy on AIME24, AIME25, AMC23, MATH-500, and Olympiad with Qwen2.5-Math-7B. Numbers above the ISPO bars are absolute Pass@1. ISPO’s largest gains arise on the hardest AIME-level benchmarks, where zero-advantage collapse is most frequent ( [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Three empirical signals missed by sparse outcomes. (a) In rollout groups where every trajectory shares the same outcome (σoutcome = 0), the within-group Conditional IFD distribution is clearly non-degenerate: all-correct groups (blue) sit visibly right of all-wrong groups (red). (b) The cumulative gradient norm follows a Lorenz-like curve: the top 5% of tokens carry ∼50% of the gradient and the top 10% car… view at source ↗
Figure 3
Figure 3. Figure 3: ISPO framework. For each prompt Q, the policy πθ samples G rollouts τi = (Q, Ti , Ai). ISPO augments the sparse outcome reward with two intrinsic signals computed from πθ’s own log-probabilities: a sequence-level Conditional IFD measuring how much the thinking trajectory Ti sharpens the answer distribution, and a token-level directional reward at critical tokens that is asymmetric by outcome. The composite… view at source ↗
Figure 4
Figure 4. Figure 4: shows two diagnostic curves that directly target the failure modes of Section 3.1: the zero￾advantage rate over training (left) and the entropy at critical tokens on incorrect rollouts (right). The 0 200 400 600 800 1000 Training step 0.0 0.2 0.4 0.6 Zero-adv.\ rate Vanilla GRPO Dr.\,GRPO ISPO (ours) 0 200 400 600 800 1000 Training step 0.5 1.0 1.5 2.0 E ntro p y o n ( ) hinge threshold H Vanilla GRPO Dr.\… view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the Conditional IFD distribution across training on Qwen2.5-Math-7B, split by rollout outcome. At step 0 (base policy), correct and incorrect distributions overlap around zero, indicating no learned thinking–answer dependence. As training progresses, the correct-rollout distribution shifts rightward (mean ∼0 → ∼0.3 with a heavy positive tail) while the incorrect-rollout distribution stays comp… view at source ↗
Figure 6
Figure 6. Figure 6: Fate of zero-advantage prompts. Among prompts whose rollout group is all-wrong (σoutcome = 0) at an early checkpoint (exactly the batches GRPO discards), the cumulative fraction subsequently solved (Pass@1) over continued training. GRPO improves on them only slowly and indirectly; ISPO keeps a non-zero gradient and converts a substantial fraction into solved prompts. icy later solves [PITH_FULL_IMAGE:figu… view at source ↗
read the original abstract

Reinforcement learning with verifiable rewards (RLVR) has emerged as a powerful paradigm for eliciting long-chain reasoning in large language models. However, existing methods based on Group Relative Policy Optimization (GRPO) rely on a binary outcome reward, which induces two structural failure modes: Zero-Advantage Collapse, in which all rollouts in a group share the same outcome and the gradient vanishes, and Hallucinated Certainty, in which the model becomes increasingly confident on incorrect rollouts late in training. We address both modes by densifying the reward with intrinsic signals computed entirely from the policy's own conditional probabilities, and propose ISPO (Intrinsic Signal Policy Optimization, which combines a sequence-level signal measuring how informative the thinking trajectory is for the final answer, with a token-level directional reward whose hallucinated-certainty hinge penalizes confidently-wrong predictions at critical decision tokens. Across three base models and five mathematical reasoning benchmarks, ISPO consistently outperforms competitive baselines, with the largest gains on the hardest benchmarks where zero-advantage collapse is most frequent, and training-dynamics diagnostics confirm that both failure modes are decreased.

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.

Referee Report

3 major / 2 minor

Summary. The paper introduces ISPO (Intrinsic Signal Policy Optimization) to address two failure modes in Group Relative Policy Optimization (GRPO) for RLVR on mathematical reasoning tasks in LLMs: zero-advantage collapse (vanishing gradients when all group rollouts share the same outcome) and hallucinated certainty (increasing confidence on incorrect trajectories). It densifies the binary outcome reward using two intrinsic signals computed solely from the policy's conditional probabilities—a sequence-level informativeness signal for thinking trajectories and a token-level hallucinated-certainty hinge that penalizes confident errors at critical tokens—claiming consistent outperformance over baselines across three base models and five benchmarks (largest gains on hardest tasks), with diagnostics showing reduced incidence of both failure modes.

Significance. If the intrinsic signals prove reliable without external anchors, the approach could stabilize and improve RL-based reasoning elicitation in LLMs by mitigating common collapse issues in sparse-reward settings, offering a parameter-free densification method that scales to harder benchmarks. The emphasis on training-dynamics diagnostics is a positive step toward falsifiable validation of the claimed failure-mode reductions.

major comments (3)
  1. [Abstract, §3] Abstract and §3 (method): The central claim that the sequence-level informativeness signal and token-level hallucinated-certainty hinge (both derived exclusively from the policy's own conditional probabilities) reliably measure trajectory usefulness and penalize errors is load-bearing, yet the manuscript provides no external verification, ground-truth comparison, or ablation against oracle signals; this leaves open the risk that the signals reinforce existing policy errors when probability mass concentrates on incorrect reasoning paths.
  2. [§4] §4 (experiments): The reported outperformance and largest gains on hardest benchmarks are presented without error bars, statistical significance tests, or per-benchmark breakdown of zero-advantage frequency before/after ISPO; without these, it is unclear whether the gains are attributable to reduced collapse or to other factors.
  3. [§5] §5 (diagnostics): The training-dynamics diagnostics confirming decreased zero-advantage collapse and hallucinated certainty rely on the same intrinsic signals whose validity is under question; an independent metric (e.g., external verifier agreement or human judgment of reasoning steps) is needed to establish that the observed reductions are not artifacts of the signal definitions.
minor comments (2)
  1. [Abstract] The abstract would benefit from a one-sentence definition or equation sketch of the two intrinsic signals to allow readers to assess their non-circularity immediately.
  2. [§3] Notation for the hinge function and informativeness measure should be introduced with explicit dependence on the policy π_θ to clarify that no external model or reward is involved.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback, which highlights important aspects of validation and statistical rigor. We address each major comment below, proposing revisions where the manuscript can be strengthened without altering its core claims.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and §3 (method): The central claim that the sequence-level informativeness signal and token-level hallucinated-certainty hinge (both derived exclusively from the policy's own conditional probabilities) reliably measure trajectory usefulness and penalize errors is load-bearing, yet the manuscript provides no external verification, ground-truth comparison, or ablation against oracle signals; this leaves open the risk that the signals reinforce existing policy errors when probability mass concentrates on incorrect reasoning paths.

    Authors: We agree that direct external verification against oracle signals would provide stronger evidence. The design choice for fully intrinsic signals (computed only from policy conditional probabilities) is deliberate to ensure the method remains parameter-free and applicable without external verifiers or ground-truth reasoning annotations, which are often unavailable at scale. The empirical gains on harder benchmarks, where collapse is prevalent, offer indirect support, but we acknowledge the risk of reinforcing errors. In revision, we will add a targeted ablation on a subset of MATH problems with available step-level annotations, comparing intrinsic signals to an oracle informativeness measure, and include a limitations paragraph discussing potential error reinforcement. revision: partial

  2. Referee: [§4] §4 (experiments): The reported outperformance and largest gains on hardest benchmarks are presented without error bars, statistical significance tests, or per-benchmark breakdown of zero-advantage frequency before/after ISPO; without these, it is unclear whether the gains are attributable to reduced collapse or to other factors.

    Authors: This is a valid point on reporting standards. The current results aggregate across runs without variance measures. In the revised manuscript, we will rerun experiments with at least three random seeds per setting, report mean and standard deviation, apply paired t-tests for significance on key comparisons, and add a table breaking down zero-advantage frequency per benchmark before and after ISPO to directly link gains to collapse reduction. revision: yes

  3. Referee: [§5] §5 (diagnostics): The training-dynamics diagnostics confirming decreased zero-advantage collapse and hallucinated certainty rely on the same intrinsic signals whose validity is under question; an independent metric (e.g., external verifier agreement or human judgment of reasoning steps) is needed to establish that the observed reductions are not artifacts of the signal definitions.

    Authors: We recognize that the diagnostics are defined using the proposed signals, creating potential circularity. The performance improvements on held-out benchmarks provide an external check that overall reasoning quality improves, but this does not directly validate the per-trajectory signal reductions. In revision, we will expand §5 to explicitly state this limitation, add a qualitative example analysis of trajectories where signals diverge from final outcome, and discuss the value of future human or verifier-based validation while noting that such metrics were outside the scope of the current intrinsic-signal focus. revision: partial

Circularity Check

0 steps flagged

No circularity: signals defined from policy probabilities in standard RL manner

full rationale

The paper defines sequence-level informativeness and token-level hallucinated-certainty signals directly from the policy's own conditional probabilities to densify the binary outcome reward. This is a self-contained construction within the RL objective and does not reduce any claimed prediction or result to a fitted parameter or prior self-citation by construction. No equations or steps in the abstract or description exhibit self-definitional equivalence, uniqueness imported via self-citation, or renaming of known results. Empirical gains on benchmarks are presented as external validation rather than forced by the input definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to identify free parameters, axioms, or invented entities; the approach relies on standard policy probabilities.

pith-pipeline@v0.9.1-grok · 5755 in / 1002 out tokens · 19826 ms · 2026-06-27T18:24:33.589565+00:00 · methodology

discussion (0)

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Reference graph

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