Trust Functions: Near-Lossless Weak-to-Strong Generalization by Learning When to Trust the Weak Teacher

As large language models become stronger, weak supervisors may fail to provide reliable labels, preferences, or final judgments for complex outputs, limiting both weak-to-strong generalization and scalable oversight. We study a more tractable form of weak supervision: using a weak model as a critic rather than as a labeler or judge. Instead of solving the task or selecting the correct answer, the weak critic only needs to provide a non-misleading revision direction that helps the strong model better use its own knowledge. We call this setting weak-critic strong oversight. We first show that weak critiques can improve frozen strong models at inference time, and that critique quality is key to this improvement. We then propose progressive on-policy critique distillation (OPCD), which filters high-quality critiques and distills critic-guided behavior into the strong model through adaptive self-teacher signals. Experiments on reasoning and alignment benchmarks show that our method improves strong models over training epochs, suggesting an effective path for scalable oversight with weak supervision.

Weak-to-strong generalization is a phenomenon in post-training whereby a strong student model, when finetuned solely with feedback from a weaker teacher, can not only surpass the teacher, but can improve upon its own capabilities. Recent work of Burns et al. (2023) demonstrated that this can occur in the setting of frontier language models, and subsequently there has been a flurry of both empirical work trying to exploit this phenomenon, as well as theoretical work attempting to understand it. In this work, we demonstrate that weak-to-strong generalization occurs in standard linear logistic regression, under mild distributional assumptions on the data. In fact, we show that this happens for most student-teacher pairs, suggesting that weak-to-strong generalization is in fact \emph{almost inevitable}, even in this basic setting. Notably, our setting does not require the student to be more expressive or have more model capacity in any way compared to the teacher, which runs contrary to the prevailing theoretical belief that a mismatch in model capacity is a central mechanism to weak-to-strong generalization.

Multi-teacher on-policy distillation allows a student to learn from complementary specialists on its own trajectories. Domain-routed approaches, however, select one teacher per example and keep it fixed throughout the response. This design both depends on labels that mixed training corpora often lack and cannot adapt teacher selection when the expertise required changes within a trajectory. We propose \textbf{TrustMOPD}, which replaces example-level teacher selection with label-free, token-level supervision allocation. At each student-generated prefix, TrustMOPD uses each specialist's RL-induced displacement from a shared pre-RL reference as a proxy for local reliability, calibrates these scores across teachers, and constructs a weighted distillation target. Across mathematics, code, and instruction following, TrustMOPD outperforms the strongest label-free baseline, increasing the recovery ratio from $54.4\%$ to $91.5\%$ on \textsc{SingleCap} and from $54.5\%$ to $98.0\%$ on \textsc{MultiCap}, while approaching label-based MOPD on \textsc{SingleCap}. Randomizing token-level weights independently of the student-generated prefix performs no better than uniform weighting, supporting the importance of conditioning supervision on the evolving generation context.