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.
Self-play, a type of training algorithm that enables a model to self-improve, has recently shown promising empirical results in the context of formal theorem proving using Large Language Models (LLMs). (Dong & Ma, 2025) instantiate self-play with two cooperating agents: a prover, which proves theorems, and a conjecturer, which generates new theorems as a curriculum to the prover. In this paper, we provide a theoretical framework for understanding the self-improvement capabilities of self-play algorithms for theorem proving. First, we formalize the set of theorems as a graph, with nodes as theorems and edges between pairs of theorems with similar semantics. We introduce a set of primitive assumptions that characterize the guarantees of a trained prover and how a conjecturer can access the structure of the graph. Second, we show that if the underlying graph of theorems is well-connected, then a prover-conjecturer system, where the conjecturing algorithm is based on a reversible random walk, is sufficient to grow the set of proved theorems exponentially. Third, motivated by an issue encountered empirically by self-play algorithms, where the conjecturer tends to generate artificially complex and non-fundamental theorems, we propose a diversity measure for a training distribution of theorems generated by a conjecturer and an improved conjecturing algorithm that locally maximizes this diversity measure, by computing the diffusion similarity between neighboring theorems in the theorem graph. Finally, we describe a method to compute the diffusion similarity by using contrastive learning to embed nodes into Euclidean space and then computing the inner-product between embeddings.
Self-play can train language models without external supervision. However, existing methods require rule-checkable answers, leaving open-ended tasks dependent on curated prompts or frontier-model judges. We introduce SCOPE, a data-free self-play framework for open-ended tasks that co-evolves two policies: a Challenger that generates document-grounded tasks, and a Solver that answers them through multi-turn retrieval. A frozen copy of the initial model serves as the self-judge, which writes task-specific rubrics from the source document and grades Solver responses against them. Across three 7-8B instruction-tuned models (Qwen2.5, Qwen3, OLMo-3), SCOPE improves open-ended performance by up to +10.4 points on eight benchmarks and matches or exceeds GRPO_data trained on ~9K curated prompts. Although trained only on open-ended tasks, SCOPE also improves held-out short-form QA by up to +13.8 points on seven held-out benchmarks, surpassing GRPO_data on all three models. Ablations show that co-evolving the Challenger is necessary to keep tasks near the Solver's frontier, that gains arise from improvements in both retrieval and synthesis with the relative contribution varying by task, and that rubric generation quality is the bottleneck for self-judging.
Wai-Chung Kwan, Aryo Pradipta Gema, Joshua Ong Jun Leang +1
We introduce PopuLoRA, a population-based asymmetric self-play framework for reinforcement learning with verifiable rewards (RLVR) post-training of LLMs. Teachers and students are specialised LoRA adapters on a shared frozen base: teachers propose problems, matched students solve them under a programmatic verifier, and cross-evaluation between sub-populations replaces the self-calibration that limits single-agent self-play. A family of LoRA weight-space evolution operators (mutations and crossovers that produce same-rank population members in seconds) serves as the replacement step of a population-based training loop at 7B scale. We instantiate PopuLoRA on top of Absolute Zero Reasoner and compare it against a per-adapter compute-matched single-agent baseline. Where the single agent self-calibrates to generating easy problems it can reliably solve, the population enters a co-evolutionary arms race: teachers produce increasingly complex problems, student solve rates oscillate, and problem-space coverage keeps expanding throughout training. Despite lower training-time reward, the population mean outperforms the baseline on three code benchmarks (HumanEval+, MBPP+, LiveCodeBench) and seven math benchmarks (AIME 24/25, AMC 23, MATH-500, Minerva, GSM8K, OlympiadBench), and even the weakest member of the population beats the baseline on aggregate.
Roger Creus Castanyer, Geoffrey Bradway, Lorenz Wolf +3