Teacher-Free Self-Training Amplifies but Does Not Compound: A Pass@K Crossover on a Free-Verifier Domain

Can post-trained large language models (LLMs) further improve themselves using only unlabeled prompts, without external teachers or feedback from tools? We study this setting starting only from unlabeled seed questions with no ground-truth solutions, across three reasoning domains: math, science, and coding. We propose Self-Verified Distillation, a simple post-training refinement algorithm in which the model generates candidate solutions to these seed questions, filters them using prompt-based self-verification, and trains on the resulting self-curated dataset. Inspired by the UQ benchmark's use of multiple validators to screen candidate answers to hard unsolved questions, we adapt this validation-based filtering idea to self-training: the model filters its own generated solutions through a three-stage cascade of cycle-consistency, factuality, and correctness checks, accepting a solution only if it passes all stages with unanimous judge votes. We find that sampling more candidate generations and using a larger verification budget during training data construction produces higher-quality self-curated data and, in turn, better reasoning models. We then train Qwen3 models at multiple scales with Self-Verified Distillation and obtain gains across all three domains. For Qwen3-4B, our method improves aggregate held-out pass@1 by +16.7 points in math (AIME26 and HMMT), +11.1 points in science (GPQA Diamond and HLE), and +8.3 points in coding (LCBv5 and LCBv6), with gains also extending to 0.6B and 8B models. Compared to our test-time-only baseline (UQ-TTC), which improves performance by spending extra compute at inference time, Self-Verified Distillation achieves better performance in most settings while requiring only a single inference call at test time.

Can a language model improve from plain text sampled from itself, with no prompts, no teacher, no verifier, and no reward model? Yes, but only when the synthetic corpus is compatible with the student, a relational property of the source-student pair rather than an intrinsic property of the data. We call this the latent capability resurfacing hypothesis: weak self-training can amplify capabilities already present in the pretrained model, but only under this compatibility condition. We study this in the minimal setting of prompt-free unconditional self-training, where base language models are fine-tuned on text generated from the BOS token alone, with no task specification or external supervision. We report three findings. First, synthetic utility is relational rather than intrinsic: self-generated data is the most effective source, same-lineage transfer outperforms stronger but differently trained sources, and cross-family transfer is substantially weaker. Second, common intrinsic proxies fail: neither benchmark-level semantic similarity nor average per-token likelihood under the student predicts which corpora help. Third, this regime produces a surprising byproduct. In controlled Pythia experiments, capability and verbatim memorization decouple: benchmark utility is preserved or improved while held-out exact-match extraction drops by over 95 percent, with no forget set, privacy objective, or targeted unlearning. Together, these results suggest that prompt-free self-training works by amplifying what the student already knows, not by importing structure from the data. They also reveal a regime in which capability and verbatim memorization can be separated without any explicit unlearning objective.

Mechanisms for continued self-improvement of language models without external supervision remain an open challenge. We propose Peer-Predictive Self-Training (PST), a label-free fine-tuning framework in which multiple language models improve collaboratively by using a cross-model aggregate response as an internal training signal. Given a prompt, models generate responses sequentially; the final aggregated answer, which is often more reliable than individual responses in practice, serves as an internal reference for learning. We measure how informative each intermediate response is about the aggregate using pointwise mutual information (PMI), and use this signal to scale self-training updates: responses already aligned with the aggregate receive smaller updates, while less informative or misaligned responses receive larger ones. On mathematical reasoning benchmarks, including SimulEq, MATH-500-Numeric, and MultiArith, PST improves exact-match accuracy by 2.2--4.3 percentage points across Gemma-2-2B, LLaMA-3.2-1B, and Qwen2.5-1.5B, and reduces the average generator--verifier gap (GV-Gap) by 26--40%, while requiring no external supervision, no teacher--student hierarchy, and only cross-model interactions. These results suggest that peer-predictive feedback from cross-model generations can provide an effective mechanism for self-supervised language-model improvement.