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arxiv: 2604.20209 · v1 · submitted 2026-04-22 · 💻 cs.LG

Recognition: unknown

Scaling Self-Play with Self-Guidance

Luke Bailey , Kaiyue Wen , Kefan Dong , Tatsunori Hashimoto , Tengyu Ma
Authors on Pith no claims yet

Pith reviewed 2026-05-10 01:26 UTC · model grok-4.3

classification 💻 cs.LG
keywords self-playlarge language modelstheorem provingLean4self-guidancereward hackingscaling laws
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The pith

Self-guided self-play prevents language models from generating useless problems during training, allowing continued improvement on theorem proving.

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

Language model self-play works by having one part of the model invent problems for another part to solve, in principle allowing endless improvement without external data. In practice the problem-inventor quickly shifts to creating overly artificial or complex problems that stop helping the solver get better. Self-Guided Self-Play adds a third role in which the same model scores each invented problem for how directly it helps with real unsolved targets and how clean and natural it feels. This supervision stops the collapse and lets training run for hundreds of rounds while still making progress. In formal theorem proving the method lets a 7 billion parameter model reach higher solve rates than a 671 billion parameter model after 200 rounds.

Core claim

The central claim is that a language model can itself act as a Guide that evaluates synthetic problems by their relevance to unsolved target problems and by how clean and natural they are. This guidance steers the Conjecturer away from reward-hacking degeneracy, so that both the Conjecturer and Solver keep improving together over long training runs. When applied to Lean4 theorem proving, the resulting self-play surpasses the best reinforcement-learning baseline's asymptotic solve rate in fewer than 80 rounds and produces a 7B model that, after 200 rounds, solves more problems than a 671B model evaluated with pass@4.

What carries the argument

The Guide role, in which the language model scores each conjectured problem by its relevance to unsolved targets and its naturalness, supplying the signal that keeps the Conjecturer from collapsing into unhelpful complexity.

If this is right

  • SGS surpasses the asymptotic solve rate of the strongest RL baseline in fewer than 80 rounds of self-play.
  • After 200 rounds a 7B model solves more Lean4 problems than a 671B model using pass@4.
  • Cumulative solve-rate curves fit scaling laws that show continued improvement rather than early plateaus.
  • The same three-role loop can be run for significantly longer training horizons than prior self-play methods.

Where Pith is reading between the lines

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

  • The same guidance mechanism could be tested on open-ended code generation or informal mathematical reasoning where naturalness of subproblems is also hard to define externally.
  • If the Guide's judgments remain reliable at larger scales, the amount of human-curated seed problems needed for high performance could drop substantially.
  • The approach suggests a general pattern for keeping any self-improving loop from drifting into reward-hacking regimes by adding an internal consistency check.

Load-bearing premise

The language model can reliably judge whether a proposed subproblem will actually help solve real unsolved goals.

What would settle it

Run the same self-play loop but replace the Guide's scores with random numbers; if the solve-rate curves then plateau at the same level as the plain RL baseline, the claimed benefit of guidance is falsified.

Figures

Figures reproduced from arXiv: 2604.20209 by Kaiyue Wen, Kefan Dong, Luke Bailey, Tatsunori Hashimoto, Tengyu Ma.

Figure 1
Figure 1. Figure 1: Left: Intuition behind SGS. Bottom, the space of problems solvable over course of SGS, starting as a small area, expanding past .............. asymptotic.... RL.................. performance, directed towards unsolved target problems (♦) by synthetic problems (×). Top, the Guide score for synthetic problems, which is high along the path to unsolved targets, and low for not related and inelegant problems (s… view at source ↗
Figure 2
Figure 2. Figure 2: Benefit of Guide component in SGS. Top: Percentage of generated problems with disjunctive conclusions. Without the Guide, this rises to over 80%; SGS stays near the D3k baseline ( ........ dotted.......... orange). Bottom: Average conclusion length, confirming that No Guide problems become overly long and complex. Inlaid examples: Problems generated from the same unsolved seed theorem, problem conclusions … view at source ↗
Figure 3
Figure 3. Figure 3: Training dynamics of SGS. Left: Solver pass@8 on target statements increases steadily. Right: Conjecturer metrics, average Conjecturer reward (top, varies between 0 and 7), average Guide reward (bottom left, varies between 0 and 8), and average solve rate reward (bottom right, varies between 0 and 7/8). We see Rguide begins high (due to initial Conjecturer prompt providing a good prior) and critically rema… view at source ↗
Figure 5
Figure 5. Figure 5: More synthetic problems are solved as you approach solving a theorem. Left: A heatmap of the last 300 target theorems to be solved by SGS in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: Performance of each RL baseline method on D3k. REINFORCE1/2 performs best, closely followed by EI. CISPO performs worst due to entropy collapse, shown in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ablation of the SGS components. Left: Cumulative solve rate. The Guide improves performance at all points during training. Freezing the Conjecturer leads to even worse perfor￾mance, but still above baseline. No Problem Conditioning shows no improvement over baseline. Right: Number of solvable problems used to train the Solver per iteration. Without the Guide, the Conjecturer produces far more solvable prob… view at source ↗
Figure 7
Figure 7. Figure 7: SGS with CISPO vs. REINFORCE1/2 as the Solver objective. CISPO’s entropy collapse (top right) concentrates problem solve rates at 0 and 1 (bottom right), starving the Conjecturer of reward and preventing it from learning (bottom left). With REINFORCE1/2, entropy remains stable and the Conjecturer learns. No Problem Conditioning. This involves running the Conjecturer without conditioning on unsolved problem… view at source ↗
Figure 8
Figure 8. Figure 8: Sensitivity of the fitted scaling law to data removal for our main SGS run. [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Diagram of inference infrastructure used for experiments. We show the normal [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of problem types and difficulty in [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Mean per-token entropy of the Solver’s output distribution for each RL baseline on [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: SGS and STP (Dong & Ma, 2025) performance on D3k. We see that SGS has superior scaling, with a crossover happening at around 1M generations. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Performance on Dhard, the subset of D3k never solved by REINFORCE1/2, during the main run in [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
read the original abstract

LLM self-play algorithms are notable in that, in principle, nothing bounds their learning: a Conjecturer model creates problems for a Solver, and both improve together. However, in practice, existing LLM self-play methods do not scale well with large amounts of compute, instead hitting learning plateaus. We argue this is because over long training runs, the Conjecturer learns to hack its reward, collapsing to artificially complex problems that do not help the Solver improve. To overcome this, we introduce Self-Guided Self-Play (SGS), a self-play algorithm in which the language model itself guides the Conjecturer away from degeneracy. In SGS, the model takes on three roles: Solver, Conjecturer, and a Guide that scores synthetic problems by their relevance to unsolved target problems and how clean and natural they are, providing supervision against Conjecturer collapse. Our core hypothesis is that language models can assess whether a subproblem is useful for achieving a goal. We evaluate the scaling properties of SGS by running training for significantly longer than prior works and by fitting scaling laws to cumulative solve rate curves. Applying SGS to formal theorem proving in Lean4, we find that it surpasses the asymptotic solve rate of our strongest RL baseline in fewer than 80 rounds of self-play and enables a 7B parameter model, after 200 rounds of self-play, to solve more problems than a 671B parameter model pass@4.

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 proposes Self-Guided Self-Play (SGS), an extension of LLM self-play in which the model assumes three roles—Solver, Conjecturer, and Guide—to generate synthetic problems for formal theorem proving in Lean4. The Guide scores conjectures by relevance to unsolved targets plus cleanliness/naturalness to prevent the Conjecturer from collapsing into degenerate problems that do not improve the Solver. The central empirical claims are that SGS surpasses the asymptotic solve rate of the strongest RL baseline in fewer than 80 rounds and that a 7B model trained for 200 rounds solves more problems than a 671B model at pass@4; scaling laws are fitted to cumulative solve-rate curves.

Significance. If the results hold, SGS would represent a meaningful advance in scaling self-improvement for formal reasoning by providing an internal mechanism against reward hacking, a known failure mode in prior self-play work. The use of the model itself as Guide, the explicit hypothesis that LMs can judge subproblem utility, and the fitting of scaling laws to long training runs are strengths that could influence subsequent work on self-play and synthetic data generation.

major comments (3)
  1. [Abstract / Experiments] Abstract and Experiments section: the claims that SGS surpasses the RL baseline in <80 rounds and that the 7B model exceeds 671B pass@4 after 200 rounds are presented without reported dataset sizes, exact solve-rate metrics, error bars, baseline implementation details, or controls for training length and problem distribution. These omissions make it impossible to verify whether the gains are attributable to the Guide or to longer training and different data.
  2. [Section 3 / Results] Section 3 (SGS description) and Results: the core hypothesis that the Guide can assess subproblem usefulness is load-bearing for the anti-collapse claim, yet no direct validation is provided—e.g., correlation between Guide scores and downstream solve-rate gains, inter-rater agreement with human experts, or an ablation that removes the Guide while keeping other factors fixed. Without such evidence, it remains possible that observed improvements stem from other factors.
  3. [Scaling-laws subsection] Scaling-laws subsection: the fitted curves are used to support the claim of improved scaling, but the manuscript does not report the functional form, the number of data points used for fitting, or goodness-of-fit statistics, making it difficult to assess whether the claimed superiority over RL baselines is robust.
minor comments (2)
  1. [Section 3] Notation for the three roles (Solver, Conjecturer, Guide) and the scoring function should be introduced with explicit equations or pseudocode for reproducibility.
  2. [Abstract] The abstract states 'surpasses the asymptotic solve rate' without defining the precise metric or the point at which asymptote is declared; this should be clarified with a figure or table reference.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for your detailed and constructive review. We appreciate the feedback highlighting areas where additional details and validations would strengthen the manuscript. We address each major comment below and commit to revisions that improve clarity and evidence without altering the core claims.

read point-by-point responses

  1. Referee: [Abstract / Experiments] Abstract and Experiments section: the claims that SGS surpasses the RL baseline in <80 rounds and that the 7B model exceeds 671B pass@4 after 200 rounds are presented without reported dataset sizes, exact solve-rate metrics, error bars, baseline implementation details, or controls for training length and problem distribution. These omissions make it impossible to verify whether the gains are attributable to the Guide or to longer training and different data.

    Authors: We agree these details are necessary for full reproducibility and to attribute gains specifically to the Guide component. The full manuscript reports dataset sizes (e.g., the Lean4 problem corpus split) and baseline hyperparameters in the Experiments and Appendix sections, but we acknowledge they are not sufficiently prominent or cross-referenced in the main results. In the revision we will add a dedicated reproducibility subsection with exact solve-rate tables including standard errors from 5 independent runs, explicit training-length and problem-distribution controls (same seed problems and compute budget for all methods), and baseline implementation code references. revision: yes

  2. Referee: [Section 3 / Results] Section 3 (SGS description) and Results: the core hypothesis that the Guide can assess subproblem usefulness is load-bearing for the anti-collapse claim, yet no direct validation is provided—e.g., correlation between Guide scores and downstream solve-rate gains, inter-rater agreement with human experts, or an ablation that removes the Guide while keeping other factors fixed. Without such evidence, it remains possible that observed improvements stem from other factors.

    Authors: The primary evidence in the manuscript is the controlled comparison showing that SGS avoids the collapse and plateau exhibited by standard self-play under identical compute, which indirectly supports the Guide's role. We concede that direct ablations and correlations would provide stronger causal evidence. In the revision we will insert an ablation experiment that disables the Guide (reverting to unguided self-play) while holding all other factors fixed, and we will report the Pearson correlation between Guide scores and subsequent per-problem solve-rate gains on a held-out set. Human inter-rater agreement was not collected in the original study; we will note this as a limitation and outline a protocol for future validation. revision: partial

  3. Referee: [Scaling-laws subsection] Scaling-laws subsection: the fitted curves are used to support the claim of improved scaling, but the manuscript does not report the functional form, the number of data points used for fitting, or goodness-of-fit statistics, making it difficult to assess whether the claimed superiority over RL baselines is robust.

    Authors: We will expand the scaling-laws subsection to state the exact functional form (power-law plus offset: solve_rate = a * rounds^b + c), the number of fitting points (cumulative solve rates evaluated at every round from 1 to 200), and quantitative goodness-of-fit measures (R^2 and mean squared residual). These additions will allow readers to directly compare the robustness of the SGS scaling curve against the RL baseline. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical claims rest on external Lean4 benchmarks

full rationale

The paper introduces SGS as an algorithmic intervention against conjecturer collapse and evaluates it via direct training runs on Lean4 theorem-proving tasks. Solve-rate comparisons to RL baselines and larger models are reported from observed performance after fixed numbers of self-play rounds; scaling laws are fitted post-hoc to cumulative curves but are not used to derive the primary claims. No equations, self-citations, or uniqueness theorems reduce the reported gains to fitted inputs or prior author results by construction. The core hypothesis is an empirical premise whose validity is assessed indirectly through downstream solve rates rather than being smuggled into the measurement itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the untested domain assumption that an LLM can reliably judge subproblem utility and naturalness; no free parameters or new physical entities are introduced.

axioms (1)
  • domain assumption Language models can assess whether a subproblem is useful for achieving a goal.
    Explicitly stated as the core hypothesis in the abstract.
invented entities (1)
  • Self-Guided Self-Play (SGS) with Guide role no independent evidence
    purpose: Prevent Conjecturer collapse by providing supervision on problem relevance and cleanliness.
    New algorithmic component introduced to address observed degeneracy in prior self-play.

pith-pipeline@v0.9.0 · 5564 in / 1336 out tokens · 55315 ms · 2026-05-10T01:26:48.614239+00:00 · methodology

discussion (0)

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