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arxiv: 2605.00380 · v2 · submitted 2026-05-01 · 💻 cs.LG · cs.CL

Recognition: 2 theorem links

· Lean Theorem

ResRL: Boosting LLM Reasoning via Negative Sample Projection Residual Reinforcement Learning

Guojun Yin, Jiajun Chai, Jie Cao, Li Wang, Ran He, Wei Lin, Xiaodong Lu, Xiaohan Wang, Zihan Lin

Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:01 UTC · model grok-4.3

classification 💻 cs.LG cs.CL
keywords LLM reasoningreinforcement learningnegative sample reinforcementprojection residualSVD low-rank subspacediversity preservationmathematical reasoninggradient modulation
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The pith

ResRL improves LLM reasoning by projecting negative-token representations onto a low-rank positive subspace and modulating gradients with the residuals.

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

This paper establishes that negative sample reinforcement can suppress shared semantic information between positive and negative responses, limiting diversity in LLM outputs. ResRL counters this by using SVD to find a low-rank positive subspace, projecting negative hidden states onto it, and applying the residuals to adjust negative gradients conservatively. A reader would care because effective reasoning in LLMs requires both accurate answers and the capacity to generate varied solution attempts rather than converging to limited patterns. The approach is shown to yield gains on diverse tasks from math to agent interactions.

Core claim

ResRL projects negative-token hidden representations onto an SVD-based low-rank positive subspace and uses projection residuals to modulate negative gradients. This decouples similar semantic distributions between positive and negative responses, improving reasoning while preserving diversity and outperforming strong baselines on average across twelve benchmarks spanning Mathematics, Code, Agent Tasks, and Function Calling.

What carries the argument

The projection of negative-token hidden representations onto an SVD-derived low-rank positive subspace, with the resulting residuals used to reweight negative advantages and reduce head-gradient interference.

If this is right

  • Outperforms prior methods including NSR on average across twelve benchmarks in mathematics, code, agent tasks, and function calling.
  • Surpasses NSR specifically on mathematical reasoning by 9.4% in Avg@16 and 7.0% in Pass@128.
  • Preserves generation diversity by avoiding suppression of shared semantic distributions.
  • Uses a single-forward proxy upper-bounding representation alignment to guide the conservative reweighting.

Where Pith is reading between the lines

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

  • Similar projection-based decoupling could be explored in other LLM alignment techniques that balance positive and negative feedback.
  • Representation-level interventions like this may help stabilize training in broader reinforcement learning from human feedback settings.
  • Scalability tests on models larger than those used here would clarify whether the SVD computation remains efficient at scale.

Load-bearing premise

The SVD-based projection successfully isolates the components unique to negative responses from the shared semantic information that both positive and negative responses need for effective reasoning.

What would settle it

If the method is applied and either reasoning accuracy does not increase or diversity metrics decline compared to NSR on the reported benchmarks, or if the proxy for representation alignment does not decrease after projection.

Figures

Figures reproduced from arXiv: 2605.00380 by Guojun Yin, Jiajun Chai, Jie Cao, Li Wang, Ran He, Wei Lin, Xiaodong Lu, Xiaohan Wang, Zihan Lin.

Figure 1
Figure 1. Figure 1: ResRL overview. Overlapping positive/negative seman￾tic distributions (S.D.) can cause GRPO/NSR to penalize shared valid tokens. ResRL utilizes negative projection residuals Ri,t to reweight gradients, reducing shared semantic penalties. increased negative weighting. Consequently, while NSR effectively improves Pass@k, it may demonstrate limited efficacy in boosting Pass@1. This motivates a central questio… view at source ↗
Figure 2
Figure 2. Figure 2: Pass@k performance on AIME24/25 and AMC23 using Qwen3-4B. ResRL consistently dominates the high-k regime, outperforming the base model and diversity-oriented baselines such as NSR and FlowRL, indicating a widened capability frontier [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pass@k performance on AIME24/25 and AMC23 using Qwen3-1.7B. ResRL demonstrates consistent superiority on the challenging AIME datasets across all sampling budgets (k = 20 to 2 7 ). On AMC23, ResRL leads in low-sample regimes and converges with baselines at high k due to task saturation [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Impact of rank k on model performance and optimization stability. (a) AIME2024 and (b) AIME2025 accuracy (Avg@16) curves across different ranks (k = 8, k = 64, k = 128, k = 256), demonstrating the protection-discrimination tradeoff. (c) Actor gradient norm highlighting the stability of updates, with larger ranks showing bursty gradients indicative of high variance. actor KL and entropy, indicating broader … view at source ↗
Figure 5
Figure 5. Figure 5: Pass@k performance on AIME24/25 and AMC23 using Qwen3-8B. ResRL dominates practical low-to-mid regimes (k≤2 6 ) and remains competitive at high compute (k=27 ). This confirms ResRL optimizes the precision–diversity trade-off, securing reliable reasoning without relying on the high-variance “brute-force” exploration of unconstrained methods. 0 50 100 150 200 Training Step 0.2 0.3 0.4 0.5 AIME2024 Accuracy 0… view at source ↗
Figure 6
Figure 6. Figure 6: Ablation of the KL penalty using Qwen3-8B. Removing the explicit KL term (red) significantly boosts accuracy on (a) AIME2024 and (b) AIME2025 (Avg@16) compared to the standard configuration (purple). (c) The rising KL divergence reflects an expanded exploration horizon enabled by ResRL. Crucially, training remains stable despite this drift, confirming that ResRL’s projection￾based weighting acts as a suffi… view at source ↗
Figure 7
Figure 7. Figure 7: Impact of hidden layer selection on reasoning performance. Utilizing the penultimate hidden layer (purple) yields significantly superior accuracy on (a) AIME2024 and (b) AIME2025 compared to the final hidden layer (green). (c) The optimization dynamics, characterized by elevated KL divergence and actor entropy, indicate that the penultimate layer facilitates more sufficient exploration. Crucially, this con… view at source ↗
Figure 8
Figure 8. Figure 8: Sensitivity analysis of the quantile hyperparameter. Lower quantile thresholds (0.1 and 0.2) accelerate convergence and yield superior accuracy on (a) AIME2024 and (b) AIME2025 compared to the more permissive 0.3 level (green). (c) The elevated KL divergence at lower quantiles indicates that stricter residual penalization drives more aggressive exploration. Crucially, the 0.1 configuration maintains optimi… view at source ↗
Figure 9
Figure 9. Figure 9: Long-horizon training stability of ResRL (Qwen3-8B) without explicit KL regularization. We extend the training to 800 steps to verify asymptotic stability. Despite the removal of the KL penalty, the model exhibits (Top) continuous performance gains on AIME 2024/2025 and a natural, bounded rise in KL divergence indicative of effective exploration; and (Bottom) stable dynamics in response length, actor entro… view at source ↗
Figure 10
Figure 10. Figure 10: Effect of the SVD subspace budget Mmax on learning dynamics. Ablations over Mmax∈ {2048, 4096, 6144, 8192} (rank k fixed) under group rollouts (G=4) and max response length 4096. We report AIME2024/2025 accuracy (top row, left/middle), policy drift measured by actor KL loss (top row, right), optimization stability via actor gradient norm (bottom row, left), exploration via actor entropy (bottom row, middl… view at source ↗
Figure 11
Figure 11. Figure 11: Impact of representation normalization on performance and stability. Removing the LayerNorm and centering mechanism (green) results in degraded performance on (a) AIME2024 and (b) AIME2025 compared to the standard ResRL configuration (purple). (c) The optimization dynamics reveal that the unnormalized configuration suffers from severe instability, evidenced by high-variance spikes in actor gradient norms.… view at source ↗
Figure 12
Figure 12. Figure 12: Responses generated by the ResRL-trained Qwen3-8B model (Rollout 1, no think mode) on the OlympiadBench test set. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Response generated by the ResRL-trained Qwen3-8B model (Rollout 2, no think mode) on the OlympiadBench test set. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Response generated by the ResRL-trained Qwen3-8B model (Rollout 3, no think mode) on the OlympiadBench test set. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Response generated by the ResRL-trained Qwen3-8B model (Rollout 4, no think mode) on the OlympiadBench test set. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Response generated by the ResRL-trained Qwen3-8B model (Rollout 1, no think mode) on the Math500 test set. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Response generated by the ResRL-trained Qwen3-8B model (Rollout 2, no think mode) on the Math500 test set. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Response generated by the ResRL-trained Qwen3-8B model (Rollout 3, no think mode) on the Math500 test set. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Response generated by the ResRL-trained Qwen3-8B model (Rollout 4, no think mode) on the Math500 test set. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Response generated by the ResRL-trained Qwen3-4B model (Rollout 1, think mode) on the Humaneval+ code test set (using Brute Force method). 41 [PITH_FULL_IMAGE:figures/full_fig_p041_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Response generated by the ResRL-trained Qwen3-4B model (Rollout 2, think mode) on the Humaneval+ code test set (using Sorting & Scanning method). 42 [PITH_FULL_IMAGE:figures/full_fig_p042_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Response generated by the ResRL-trained Qwen3-4B model (Rollout 1, think mode) on the Codeforces test set. 43 [PITH_FULL_IMAGE:figures/full_fig_p043_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Response generated by the ResRL-trained Qwen3-4B model (Rollout 2, think mode) on the Codeforces test set. 44 [PITH_FULL_IMAGE:figures/full_fig_p044_23.png] view at source ↗
read the original abstract

Reinforcement Learning with Verifiable Rewards (RLVR) enhances reasoning of Large Language Models (LLMs) but usually exhibits limited generation diversity due to the over-incentivization of positive rewards. Although methods like Negative Sample Reinforcement (NSR) mitigate this issue by upweighting penalty from negative samples, they may suppress the semantic distributions shared between positive and negative responses. To boost reasoning ability without losing diversity, this paper proposes negative sample projection Residual Reinforcement Learning (ResRL) that decouples similar semantic distributions among positive and negative responses. We theoretically link Lazy Likelihood Displacement (LLD) to negative-positive head-gradient interference and derive a single-forward proxy that upper-bounds representation alignment to guide conservative advantage reweighting. ResRL then projects negative-token hidden representations onto an SVD-based low-rank positive subspace and uses projection residuals to modulate negative gradients, improving reasoning while preserving diversity and outperforming strong baselines on average across twelve benchmarks spanning Mathematics, Code, Agent Tasks, and Function Calling. Notably, ResRL surpasses NSR on mathematical reasoning by 9.4\% in Avg@16 and 7.0\% in Pass@128. Code is available at https://github.com/1229095296/ResRL.git.

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 manuscript proposes ResRL, a reinforcement learning method for LLMs that projects negative-token hidden representations onto an SVD-derived low-rank positive subspace and modulates negative gradients via the resulting residuals. It claims this decouples shared semantic distributions between positive and negative samples (addressing limitations in NSR), theoretically links Lazy Likelihood Displacement to head-gradient interference via a single-forward alignment proxy, and empirically yields gains in reasoning diversity and performance, outperforming baselines on average across twelve benchmarks in mathematics, code, agent tasks, and function calling (notably +9.4% Avg@16 and +7.0% Pass@128 over NSR on math).

Significance. If the central mechanism holds, ResRL could meaningfully advance RLVR techniques by balancing reward optimization against diversity loss without post-hoc suppression of useful negative information. The explicit code release supports reproducibility, and the gradient-interference framing provides a potentially useful lens for future work on negative sampling in LLM alignment. However, the current support for both the theoretical proxy and the empirical robustness remains limited.

major comments (3)
  1. [Theoretical Analysis] Theoretical section (derivation of LLD-to-gradient-interference link and single-forward proxy): the abstract asserts an upper bound on representation alignment that guides conservative advantage reweighting, but no explicit steps, assumptions, or proof sketch are provided; this is load-bearing for the claim that the SVD projection is theoretically justified rather than heuristic.
  2. [Experiments] Experimental section (benchmark results and SVD rank): the reported 9.4% Avg@16 and 7.0% Pass@128 gains on mathematical reasoning are presented as direct outcomes, yet the manuscript does not specify the exact twelve benchmarks, the chosen SVD subspace rank (a free parameter), its selection criterion, or any ablation confirming no post-hoc tuning; this undermines verification that the projection residual truly preserves diversity without suppressing shared reasoning signals.
  3. [Method] Method section (projection residual modulation): the weakest assumption—that the low-rank positive subspace successfully decouples similar semantic distributions without losing useful shared information—is stated but not supported by any quantitative analysis of information loss or gradient interference reduction; this is central to the diversity-preservation claim.
minor comments (2)
  1. [Experiments] A table explicitly listing the twelve benchmarks, their categories, and the exact evaluation metrics (Avg@16, Pass@128, etc.) would improve clarity and allow direct comparison with NSR and other baselines.
  2. [Method] Notation for the SVD-based subspace and projection residual should be defined once with consistent symbols across equations and text to avoid ambiguity in the gradient modulation step.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the potential of ResRL to advance RLVR methods. We address each major comment point by point below, providing clarifications and committing to revisions that strengthen the manuscript without misrepresenting our current results.

read point-by-point responses

  1. Referee: [Theoretical Analysis] Theoretical section (derivation of LLD-to-gradient-interference link and single-forward proxy): the abstract asserts an upper bound on representation alignment that guides conservative advantage reweighting, but no explicit steps, assumptions, or proof sketch are provided; this is load-bearing for the claim that the SVD projection is theoretically justified rather than heuristic.

    Authors: We acknowledge that the current manuscript presents the link between Lazy Likelihood Displacement and head-gradient interference along with the single-forward proxy but does not include a full step-by-step derivation or explicit list of assumptions. In the revised version, we will expand the theoretical section with a detailed proof sketch, including all assumptions and the mathematical steps deriving the upper bound on representation alignment. This will clarify the theoretical justification for the SVD projection. revision: yes

  2. Referee: [Experiments] Experimental section (benchmark results and SVD rank): the reported 9.4% Avg@16 and 7.0% Pass@128 gains on mathematical reasoning are presented as direct outcomes, yet the manuscript does not specify the exact twelve benchmarks, the chosen SVD subspace rank (a free parameter), its selection criterion, or any ablation confirming no post-hoc tuning; this undermines verification that the projection residual truly preserves diversity without suppressing shared reasoning signals.

    Authors: We agree that explicit details are required for reproducibility and verification. The revised manuscript will list all twelve benchmarks by name and category. We will also specify the SVD rank employed, describe its selection criterion, and add an ablation study across multiple ranks to demonstrate robustness and confirm that performance improvements are not the result of post-hoc tuning while preserving diversity. revision: yes

  3. Referee: [Method] Method section (projection residual modulation): the weakest assumption—that the low-rank positive subspace successfully decouples similar semantic distributions without losing useful shared information—is stated but not supported by any quantitative analysis of information loss or gradient interference reduction; this is central to the diversity-preservation claim.

    Authors: We recognize that the central assumption would be strengthened by quantitative evidence. In the revision, we will add an analysis subsection reporting metrics such as projection error norms and gradient interference reduction (via pre- and post-modulation comparisons) to demonstrate that shared semantic information is largely retained while decoupling is achieved. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins with a theoretical connection between Lazy Likelihood Displacement (LLD) and negative-positive head-gradient interference, followed by derivation of a single-forward alignment proxy. This proxy then motivates an SVD-based low-rank projection of negative-token hidden states onto the positive subspace, with residuals used to modulate gradients. No step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the projection and residual modulation are introduced as new mechanisms grounded in the stated theory rather than tautological re-use of prior NSR quantities. Empirical results on the twelve benchmarks are reported as direct outcomes of this construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the effectiveness of the introduced projection residual technique and the theoretical proxy for representation alignment, which are new to this work; limited details available from abstract prevent exhaustive enumeration of all free parameters or axioms.

free parameters (1)
  • SVD subspace rank
    The low-rank dimension for the positive subspace is a hyperparameter that must be selected, though its specific value or selection process is not stated in the abstract.
axioms (2)
  • domain assumption Lazy Likelihood Displacement links to negative-positive head-gradient interference
    The paper states a theoretical connection used to derive the proxy, but the full justification is not provided in the abstract.
  • domain assumption Single-forward proxy upper-bounds representation alignment
    This proxy is derived to guide conservative advantage reweighting, with details absent from the abstract.

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