Weak Supervision

Where do learning signals come from when there is no ground truth in post-training? We show that inference compute itself can serve as supervision. By generating parallel rollouts and converting them into reference estimates, models can learn without human labels—critically, even in non-verifiable domains like healthcare guidance where no programmatic checker exists. We call this framework *Compute as Teacher (CaT)* and it turns inference-time compute from parallel rollouts into supervision for RL training. The framework has two components: (1) reference estimation which aggregates rollouts into a pseudo-reference answer, and (2) reward derivation which converts that pseudo-reference into RL rewards. For (1), we explore a simple method we call *synthesis*, but the framework admits any aggregator. For (2), we introduce self-proposed rubrics for non-verifiable domains. These are binary, auditable criteria generated from the pseudo-reference and scored by an LLM judge. On HealthBench, models trained with CaT match or exceed inference-time aggregation quality while using 9× less test-time compute. Here, CaT also competes with learning from expert physician annotations, yielding up to +30% relative improvement over the initial policy. The framework extends naturally to verifiable rewards, matching the best existing baselines on MATH-500 in test-time RL and demonstrating 'drop-in' versatility across both types of domains.

Weak-to-strong generalization (W2SG) is the phenomenon in which a powerful student model, trained on labels produced by a weaker teacher, ultimately outperforms the teacher on the target task. In this work, we theoretically investigate how W2SG can arise via a generalized bias–variance decomposition under Bregman divergence. We show that the expected population risk gap between the student and the teacher is characterized by the expected misfit between the two models. Unlike earlier misfit-based analyses, our theory removes several restrictive assumptions, e.g., it does not require the student hypothesis class to be convex. Our results indicate that W2SG is more likely when the student effectively approximates the teacher's posterior mean. Specializing to squared loss, we provide a sufficient condition (illustrated through a concrete example) under which the student converges to its posterior mean teacher; in particular, increasing the student model size can ensure this convergence. For cross-entropy loss, our analysis further suggests that lowering the entropy of the student's predictive distribution can promote W2SG. We also find that the reverse cross-entropy, unlike the standard forward cross-entropy, is less sensitive to the teacher's predictive uncertainty. Finally, we verify these theoretical insights empirically and demonstrate that incorporating reverse cross-entropy consistently improves student performance.

Antibody expression ranking is a critical task in antibody design, yet its modelling is severely hindered by the scarcity of labeled experimental data. To address this, we propose a unified preference-based learning framework that integrates scarce quantitative expression data with large-scale weak positive supervision from immunization data. We adapt Direct Preference Optimization (DPO) to protein language models by introducing a union-masked log-likelihood approximation and IMGT-based alignment, enabling efficient training on variable-length sequences. Evaluating on a diverse internal dataset of 1254 labeled sequences and 4 million unlabeled camelid-derived antibodies, we show that our method consistently outperforms baselines on most metrics. Our results demonstrate that preference learning can effectively learn from weak supervision, providing a scalable solution for antibody expressibility optimization in data-constrained settings. Project page: https://kisoji-biotechnology-inc.github.io/Preference-Expression-Ranking/.

Ensuring safe behavior in reinforcement learning (RL) is challenging when safety constraints are implicit and cannot be densely measured. In many settings, supervision is limited to coarse approvals or rejections of whole trajectories (e.g., whether a rollout remained within an unknown safety threshold). We propose **TraCeS** (Trajectory-based Constraint Estimation for Safety), a method for learning **per-timestep violation credit** from such sparse trajectory-level labels. **TraCeS** trains a sequential violation estimator whose per-step credits factorize the predicted probability that a trajectory has **not yet violated** the constraint, and integrates this learned signal into constrained policy optimization. The method requires neither a known cost function nor a known threshold, and remains compatible with standard continuous-control algorithms. We provide a theoretical analysis of the approximation gap introduced by the learning objective, and demonstrate empirically that **TraCeS** improves constraint satisfaction and feedback efficiency over baselines across multiple continuous-control benchmarks, including long-horizon tasks and settings with noisy or inconsistent labels.

Can a diffusion model trained on bedrooms recover human faces? Diffusion models are widely used as priors for inverse problems, but standard approaches usually assume a high-fidelity model trained on data that closely match the unknown signal. In practice, one often must use a mismatched or low-fidelity diffusion prior. Surprisingly, these weak priors often perform nearly as well as full-strength, in-domain baselines. We study when and why inverse solvers are robust to weak diffusion priors. Through extensive experiments, we find that weak priors succeed when measurements are highly informative (e.g., many observed pixels), and we identify regimes where they fail. To explain this behavior, we combine Bayesian-consistency theory with local-correlation analysis: the theory gives conditions under which high-dimensional measurements make the posterior concentrate near the true signal, while the correlation analysis shows that weak and stronger natural-image priors can share similar local spatial structure. These results provide a principled justification on when weak diffusion priors can be used reliably. Code is available at https://github.com/jjia131/weak-diffusion-priors-inverse-problem.

Standard preference alignment relies on a binary forced-choice paradigm, assuming definitive preferences for all pairs. However, we find that indistinguishable pairs are prevalent even in standard benchmarks, where quality differences of two responses often fall below the labeler's discriminative resolution limit. Forcing a choice in such cases could inject significant noise that undermines policy optimization. In this work, we propose a silent-aware framework that introduces a principled way to allow annotators to stay silent (i.e., express ties) and then explicitly model these ties during optimization. Our findings reveal a compelling phenomenon: when ties are properly modeled, supervision from small models yields alignment surpassing that of forced-choice LLMs or human experts. This discovery highlights a cost-effective path for alignment: respecting a labeler’s resolution limit is more critical than increasing its capability, while simultaneously unlocking the latent value in existing benchmarks by properly modeling inherent tie signals without requiring any re-labeling effort. To leverage these signals, we propose several optimization objectives to drive the policy toward high-reward regions while mitigating unreliable updates that lead to arbitrary distribution shifts. Our approaches significantly surpass conventional alignment performance, consistently outperforming the strongest available baselines across diverse benchmarks.

Multimodal Process Reward Models (MPRMs) are central to step-level supervision for visual reasoning in MLLMs. Training MPRMs typically requires large-scale Monte Carlo (MC)-annotated corpora, incurring substantial training cost. This paper studies the data efficiency for MPRM training. Our preliminary experiments reveal that MPRM training quickly saturates under random subsampling of the training data, indicating substantial redundancy within existing MC-annotated corpora. To explain this, we formalize a theoretical framework and reveal that informative gradient updates depend on two factors: label mixtures of positive/negative steps and label reliability (average MC scores of positive steps). Guided by these insights, we propose the Balanced-Information Score (BIS), which prioritizes both mixture and reliability based on existing MC signals at the rollout level, without incurring any additional cost. Across two backbones (InternVL2.5-8B and Qwen2.5-VL-7B) on VisualProcessBench, BIS-selected subsets consistently match and even surpass the full-data performance at small fractions. Notably, the BIS subset reaches full-data performance using only 10% of the training data, improving over random subsampling by a relative 4.1%.

Interpreting residual architectures as discretizations of Ordinary Differential Equations characterizes the forward pass as a continuous latent flow. Although this framework defines the mechanics of inference, conventional training paradigms primarily constrain the terminal state, leaving the intermediate evolution unregulated. In this work, we formalize the forward pass as a Conditional Discriminative Flow and investigate its intrinsic kinematic laws. Using Tuned Lens analysis, we discover that the accumulation of semantic evidence follows a consistent exponential schedule. This finding confirms that deep models naturally require an extended phase of feature construction prior to a rapid transition toward categorical certainty in the terminal layers. Standard training ignores this latent progression. To resolve this impedance mismatch, we propose Deep Trajectory Supervision, a framework that aligns auxiliary supervision with this intrinsic exponential bias. By rectifying the trajectory of the inference flow, DTS functions as a critical physical inductive bias. Empirical evaluations on ImageNet-1K and various benchmarks demonstrate that DTS significantly accelerates convergence and improves terminal performance.

Due to the difficulty of obtaining ground-truth data for 4D radar scene flow estimation, previous methods typically rely on either self-supervised losses or cross-modal supervision using 3D LiDAR data, 2D images, and odometry. However, self-supervised approaches often yield suboptimal results due to radar's inherently low-fidelity measurements, while existing cross-modal supervised methods introduce complex multi-task architecture and require costly LiDAR sensors to generate pseudo radar scene flow labels from pretrained 3D tracking models. To overcome these limitations, we propose a task-specific iterative framework for weakly supervised radar scene flow learning, using only images and odometry for auxiliary supervision during training. Specially, we establish two novel instance-aware self-supervised losses by exploiting off-the-shelf 2D tracking and segmentation algorithms to obtain tracked instance masks, which are back-projected into 3D space to provide instance-level semantic guidance; for static regions, we integrate vehicle odometry with radar's intrinsic motion cues to construct a rigid static loss. Extensive experiments on the real-world View-of-Delft (VoD) dataset demonstrate that our method not only surpasses state-of-the-art cross-modal supervised approaches that rely on 3D multi-object tracking on dense LiDAR point clouds but also outperforms existing fully supervised scene flow estimation methods. The code is open-sourced at \href{https://github.com/FuJingyun/IterFlow}{https://github.com/FuJingyun/IterFlow}.

In scientific reasoning tasks, the veracity of the reasoning process is as critical as the final outcome. While Process Reward Models (PRMs) offer a solution to the coarse-grained supervision problems inherent in Outcome Reward Models (ORMs), their deployment is hindered by the prohibitive cost of obtaining expert-verified step-wise labels. This paper addresses the challenge of training reliable PRMs using abundant but noisy "weak" supervision. We argue that existing Weak-to-Strong Generalization (W2SG) theories lack prescriptive guidelines for selecting high-quality training signals from noisy data. To bridge this gap, we introduce the Dual-Consensus Weak-to-Strong (DC-W2S) framework. By intersecting Self-Consensus (SC) metrics among weak supervisors with Neighborhood-Consensus (NC) metrics in the embedding space, we stratify supervision signals into distinct reliability regimes. We then employ a curriculum of instance-level balanced sampling and label-level reliability-aware masking to guide the training process. We demonstrate that DC-W2S enables the training of robust PRMs for complex reasoning without exhaustive expert annotation, proving that strategic data curation is more effective than indiscriminate training on large-scale noisy datasets.

Group Relative Policy Optimization (GRPO) is effective for training language models on complex reasoning. However, since the objective is defined relative to a group of sampled trajectories, extended deliberation can create more chances to realize relative gains, leading to inefficient reasoning and overthinking, and complicating the trade-off between correctness and rollout efficiency. Controlling this behavior is difficult in practice, considering (i) Length penalties are hard to calibrate because longer rollouts may reflect harder problems that require longer reasoning, penalizing tokens risks truncating useful reasoning along with redundant continuation; and (ii) supervision that directly indicates when to continue or stop is typically unavailable beyond final answer correctness. We propose Weakly Supervised GRPO (WS-GRPO), which improves rollout efficiency by converting terminal rewards into correctness-aware guidance over partial trajectories. Unlike global length penalties that are hard to calibrate, WS-GRPO trains a preference model from outcome-only correctness to produce prefix-level signals that indicate when additional continuation is beneficial. Thus, WS-GRPO supplies outcome-derived continue/stop guidance, reducing redundant deliberation while maintaining accuracy. We provide theoretical results and empirically show on reasoning benchmarks that WS-GRPO substantially reduces rollout length while remaining competitive with GRPO baselines.

As AI systems begin to automate complex tasks, supervision increasingly relies on weaker models or limited human oversight that cannot fully verify output quality. A model more capable than its supervisors could exploit this gap through sandbagging, producing work that appears acceptable but falls short of its true abilities. Can training elicit a model's best work even without reliable verification? We study this using model organisms trained to sandbag, testing elicitation techniques on problem-solving math, graduate-level science, and competitive coding tasks. We find that training with weak supervision can reliably elicit sandbagging models when supervised fine-tuning (SFT) and reinforcement learning (RL) are combined: SFT on weak demonstrations breaks the sandbagging behavior, enabling RL to then fully elicit performance. Neither method succeeds reliably alone—RL without SFT almost always leads to reward hacking rather than genuine improvement, and SFT without RL fails to elicit full performance when the supervisor is much weaker than the untrusted model. Critically, this relies on training being indistinguishable from deployment; when models can distinguish between training and deployment, they can perform well during training while continuing to sandbag afterward. Our results provide initial evidence that training is a viable mitigation against sandbagging, while highlighting the importance of making training indistinguishable from deployment.

Many learning problems require uncovering a hidden ordering that reveals structure in unordered data, such as monotonicity in sorting or spatial continuity in jigsaw reconstruction. In these settings, permutations can be learned as latent operators by optimizing objectives defined directly on the reordered output, often without access to ground-truth orderings. Differentiable relaxations such as Gumbel–Sinkhorn make this approach practical by approximating permutation matrices with doubly stochastic matrices. However, learning from structure without supervision induces a non-uniform uncertainty: some assignments become confident early, while others remain ambiguous. Existing methods control this process using a single global temperature, forcing all assignments to sharpen or diffuse simultaneously and leading to instability at scale. We introduce an entropy-adaptive formulation of Gumbel–Sinkhorn that locally modulates temperature based on assignment uncertainty. This allows confident assignments to discretize early while preserving exploration where uncertainty remains. Across sorting and jigsaw reconstruction tasks and in routing-style settings, adaptive entropy control improves training stability and final permutation quality relative to fixed-temperature baselines, particularly as problem size and assignment ambiguity increase.

While deep learning has revolutionized financial forecasting through sophisticated architectures, the design of the supervision signal itself is rarely scrutinized. We challenge the canonical assumption that training labels must strictly mirror inference targets, uncovering the Label Horizon Paradox: the optimal supervision signal often deviates from the prediction goal, shifting across intermediate horizons governed by market dynamics. We theoretically ground this phenomenon in a dynamic signal-noise trade-off, demonstrating that generalization hinges on the competition between marginal signal realization and noise accumulation. To operationalize this insight, we propose a bi-level optimization framework that autonomously identifies the optimal proxy label within a single training run. Extensive experiments on large-scale financial datasets demonstrate consistent improvements over conventional baselines, thereby opening new avenues for label-centric research in financial forecasting.

Weak-to-strong generalization studies how to improve a strong student using supervision from a weaker teacher when reliable labels are scarce. We view this primarily as a data selection problem, where the key challenge is to identify which weak labels are reliable enough to serve as a training signal. To address this, we introduce trust functions that assign each weak label a scalar trust score and use these scores to filter weak supervision. Across several domains, including world knowledge, quantitative reasoning, decision making, trust filtering yields students that match and sometimes surpass ground-truth supervision, achieving near-lossless weak-to-strong generalization. Moreover, trust functions enable an iterative weak-to-strong chain that compounds gains by training a student and reusing it as the next teacher, producing the strongest final model. Our analyses suggest that neural trust functions improve learning through more than label error reduction. They induce an implicit easy-first curriculum, recover near-optimal alternatives where ground truth labels are incomplete, and produce more coherent gradient updates, offering a mechanistic account of the stability and efficiency of trust-filtered weak-to-strong generalization.