Teacher-Student Learning

Knowledge distillation transfers reasoning capabilities from large teachers to efficient students. However, token-level on-policy distillation (OPD) constrains student exploration and requires teacher token probabilities, precluding distillation from black-box teachers that provide only text outputs. We introduce On-policy Verbal Distillation (OVD), a framework that uses verbal scores from black-box teachers to rank student-generated sub-trajectories, retaining high-scoring ones and replacing low-scoring ones with teacher-generated continuations. We analyze when ranking induced by verbal scores can guide distribution approximation: under a density-ratio calibration condition on acceptance probabilities and bounded teacher-replacement error, we bound the approximation error between the resulting mixed trajectory distribution and a teacher-preferred target. On Web Q&A, OVD achieves 41.09% average EM with teacher feedback at inference, exceeding the strongest evaluated baseline by 5.89 percentage points. On AMC23, OVD-FR improves accuracy over RLVR by 10.0 percentage points (52.5% to 62.5%) after 600 training steps on 128 problems. Further experiments suggest that retaining student-generated prefixes helps preserve exploration and mitigate trajectory-level entropy collapse. OVD also improves training efficiency: resampling selected suffixes rather than entire responses reduces mean per-step training time by 10.2% in the 128-problem setting. Project page: https://menik1126.github.io/ovd-project-page/.

RL methods for scaling large reasoning models stall on datasets with low initial success rates, and thus little training signal. We investigate a fundamental question: Can a pretrained LLM leverage latent knowledge to generate an automated curriculum for problems it cannot solve? We explore this with SOAR: An asymmetric self-play framework that uses meta-RL to surface these pedagogical signals. A teacher model proposes synthetic problems for a student model, and is rewarded with its improvement on a subset of hard problems, thus grounding the curriculum in real student progress rather than intrinsic proxy rewards. Our study on the hardest subsets of math benchmarks (0/128 success) reveals three core findings. First, it is possible to realize bilevel meta-RL that unlocks learning under sparse, binary rewards by sharpening a latent capacity of pretrained models to generate useful problems. Second, grounded rewards outperform intrinsic learnability rewards used in prior LLM self-play, reliably avoiding typical instability and diversity collapse modes. Third, the structure and well-posedness of questions are more critical for learning progress than solution correctness. Our results suggest that the ability to generate useful stepping stones does not require the preexisting ability to solve the hard problems, paving a principled path to escape reasoning plateaus without additional curated data

Knowledge Distillation (KD) is a central paradigm for transferring knowledge from a large teacher network to a typically smaller student model, often by leveraging soft probabilistic outputs. While KD has shown strong empirical success in numerous applications, its theoretical underpinnings remain only partially understood. In this work, we adopt a Bayesian perspective on KD to rigorously analyze the convergence behavior of students trained with Stochastic Gradient Descent (SGD). We study two regimes: $(i)$ when the teacher provides the exact Bayes Class Probabilities (BCPs); and $(ii)$ supervision with noisy approximations of the BCPs. Our analysis shows that learning from BCPs yields variance reduction and removes neighborhood terms in the convergence bounds compared to one-hot supervision. We further characterize how the level of noise affects generalization and accuracy. Motivated by these insights, we advocate the use of Bayesian deep learning models, which typically provide improved estimates of the BCPs, as teachers in KD. Consistent with our analysis, we experimentally demonstrate that students distilled from Bayesian teachers not only achieve higher accuracies (up to +4.27%), but also exhibit more stable convergence (up to 30% less noise), compared to students distilled from deterministic teachers.

Step distillation accelerates diffusion sampling by training a few-step student to imitate a many-step teacher, but distillation itself remains expensive. Typically, this requires thousands of GPU-hours and a large pre-generated trajectory dataset. We introduce ReDiF, which casts step distillation as terminal-reward policy optimization rather than step-wise regression. The student is optimized against a reward computed on the terminal sample, measuring alignment with the teacher's output, instead of matching the teacher's intermediate trajectory under a reconstruction or consistency loss. Because the reward need not be differentiable or trajectory-aligned, ReDiF admits non-differentiable objectives, multi-objective combinations, and preferences the teacher does not express, while exploration lets the student find sampling paths matched to its own step schedule. ReDiF converges in 400 policy updates with 3200 rollouts on a single A100 GPU with 1,000 noise-class pairs and no paired dataset: about 4 GPU-hours, against roughly 336 A100-hours reported for DMD2 on the same EDM teacher. At 8 steps on ImageNet-64, it achieves an FID 3.64 points better than the strongest retrained distillation baseline in the low-training regime under the same single-GPU budget. The formulation is also orthogonal to existing distillation objectives: added to the DMD2 loss, it further improves DMD2's FID by 4.4.

Distilling the capabilities from a large reasoning model (LRM) to a smaller student model often involves training on substantial amounts of reasoning data. However, knowledge distillation (KD) over lengthy sequences with prompt (P), chain-of-thought (CoT), and answer (A) sections makes the process computationally expensive. In this work, we investigate how the allocation of supervision across different sections (P, CoT, A) affects student performance. Our analysis shows that selective KD over only the CoT tokens can be effective when the prompt and answer information is encompassed by it. Building on this insight, we establish a truncation protocol to quantify computation-quality tradeoffs as a function of sequence length. We observe that beyond a specific length, longer training sequences provide marginal returns for downstream performance but require substantially higher memory and FLOPs. To this end, training on only the first $50\%$ of tokens of every training sequence can retain, on average, $\approx91\%$ of full-sequence performance on math benchmarks while reducing training time, memory usage, and FLOPs by about $50\%$ each. Codes are available at https://github.com/weiruichen01/distilling-the-essence.

An adaptive structural learning method of Restricted Boltzmann Machine (RBM) and Deep Belief Network (DBN) has been developed as one of prominent deep learning models. The neuron generation-annihilation algorithm in RBM and layer generation algorithm in DBN make an optimal network structure for given input during the learning. In this paper, our model is applied to an automatic recognition method of road network system, called RoadTracer. RoadTracer can generate a road map on the ground surface from aerial photograph data. A novel method of RoadTracer using the Teacher-Student based ensemble learning model of Adaptive DBN is proposed, since the road maps contain many complicated features so that a model with high representation power to detect should be required. The experimental results showed the detection accuracy of the proposed model was improved from 40.0% to 89.0% on average in the seven major cities among the test dataset. In addition, we challenged to apply our method to the detection of available roads when landslide by natural disaster is occurred, in order to rapidly obtain a way of transportation. For fast inference, a small size of the trained model was implemented on a small embedded edge device as lightweight deep learning. We reported the detection results for the satellite image before and after the rainfall disaster in Japan. This version of the article was improved the search algorithm at the border around image.

Training student models on synthetic data generated by strong teacher models is a promising way to distilling the capabilities of teachers. However, recent studies show that stronger models are not always optimal teachers, revealing a mismatch between teacher outputs and student learnability. To address this issue, we propose PerSyn (Personalized data Synthesis), a novel synthesis strategy that operates under a new Route then Generate'' paradigm to create data tailored to each student model, enabling it to learn more effectively. Specifically, PerSyn first assigns each prompt to its optimal teacher via a query-level router that jointly considers student learnability and teacher response quality. Each teacher then synthesizes data only for its assigned prompts, making the process more efficient than the conventional Generate then Select'' paradigm, where all teachers must generate parallel responses for the entire prompt set before constructing the final dataset. Extensive experiments across different model families and scales demonstrate that PerSyn consistently achieves superior or comparable performance to all baselines in instruct tuning and math reasoning settings. Further analysis verifies the effectiveness of PerSyn and offers extra insights to propel future research.

Distilling reasoning traces from strong teacher models has become the standard recipe for building capable small language models. Yet reasoning traces are 5-20$\times$ longer than standard instruction fine-tuning (IFT) outputs, meaning every practitioner who chooses reasoning distillation implicitly forgoes training a larger IFT model on the same compute budget. Whether this trade-off is worthwhile remains unaddressed. We study it with a controlled experiment: a single teacher generates paired IFT and reasoning outputs for identical prompts by toggling only its reasoning mode, isolating supervision format as the sole variable. Training students at five scales (0.5B to 14B) and evaluating on 18 benchmarks, we find that at matched FLOPs, IFT lies on or near the Pareto frontier across the majority of configurations. Reasoning reaches the Pareto frontier only on open-ended tasks at 7B and above. Even there, a sequential curriculum mixing just 25-50% reasoning data with IFT captures most of the accuracy benefit at far lower compute cost.

Supervised deep learning achieves strong performance in biomedical image segmentation but relies on costly pixel-wise annotations, motivating semi-supervised approaches that exploit unlabeled data. We introduce a diffusion-based teacher--student framework in which segmentation predictions are used to condition image denoising, encouraging the production of more informative pseudo-labels. The teacher is first pretrained through an unsupervised reconstruction task using diffusion-style corruption, timestep conditioning, and denoising. Starting from a corrupted empty mask, the model predicts an intermediate segmentation that conditions image denoising, encouraging the predicted mask to capture structural information useful for recovering the original image. The resulting teacher is then co-trained with a student using supervised segmentation on labeled samples and cross pseudo-supervision on unlabeled data. We further introduce a multi-round extension during co-training, in which the teacher generates multiple stochastic image reconstructions and corresponding segmentation predictions, providing additional reconstruction and alignment signals to improve its pseudo-labels. We evaluate the proposed framework on three public 2D biomedical segmentation datasets and a 3D left atrial segmentation benchmark. Across several labeling regimes, our method achieves competitive or superior performance compared with state-of-the-art semi-supervised approaches, with the largest gains observed under severe label scarcity.

Amortized inference is the task of training a parametric model, such as a neural network, to approximate a distribution with a given unnormalized density where exact sampling is intractable. When sampling is implemented as a sequential decision-making process, reinforcement learning (RL) methods, such as generative flow networks, can be used to train the sampling policy. Off-policy RL training facilitates the discovery of diverse, high-reward candidates, but existing methods still face challenges in efficient exploration. We propose to use an adaptive training distribution (the \teacher) to guide the training of the primary amortized sampler (the \student). The \teacher, an auxiliary behavior model, is trained to sample high-loss regions of the \student and can generalize across unexplored modes, thereby enhancing mode coverage by providing an efficient training curriculum. We validate the effectiveness of this approach in a synthetic environment designed to present an exploration challenge, two diffusion-based sampling tasks, and four biochemical discovery tasks demonstrating its ability to improve sample efficiency and mode coverage. Source code is available at https://github.com/alstn12088/adaptive-teacher.

Knowledge distillation transfers knowledge from large teacher models to more compact student networks. The standard approach minimizes the Kullback-Leibler (KL) divergence between the probabilistic outputs of the teacher and student, aligning predictions but neglecting the structural relationships encoded within the teacher's internal representations. Recent advances have adopted contrastive learning objectives to address this limitation; however, such instance-discrimination-based methods induce a "class collision problem", in which semantically related samples are inappropriately pushed apart despite belonging to similar classes. To overcome this, we propose Relational Representation Distillation (RRD) that preserves the relative relationships among instances rather than enforcing absolute separation. Our method introduces separate temperature parameters for teacher and student distributions, with a sharper teacher (low $τ_t$) emphasizing primary relationships and a softer student (high $τ_s$) maintaining secondary similarities. This dual-temperature formulation creates an implicit information bottleneck that preserves fine-grained relational structure while avoiding the over-separation characteristic of contrastive losses. We establish theoretical connections showing that InfoNCE emerges as a limiting case of our objective when $τ_t \rightarrow 0$, and empirically demonstrate that this relaxed formulation yields superior relational alignment and generalization across classification and detection tasks.

While on-policy distillation (OPD) reduces exposure bias by training student language models on their own rollouts, early student errors in long-horizon agentic scenarios can lead to contexts unfamiliar to the teacher. To improve trajectory quality, recent work on agentic OPD introduces teacher intervention into training rollouts by switching the executor between the student and the teacher. However, existing methods determine how much teacher intervention is needed---but not when. To address this limitation, we propose DASH-OPD (Discrepancy-Aware Switching with Hysteresis for OPD), the first agentic OPD method to perform adaptive, bidirectional executor switching. At each turn, DASH-OPD measures teacher--student discrepancy using a mean log-probability ratio over action tokens. Student-to-teacher ratios on student turns serve as drift signals, while teacher-to-student ratios on teacher turns serve as recovery signals. These signals are accumulated over multiple turns to form drift and recovery evidence, respectively. DASH-OPD switches executors when either type of evidence exceeds its corresponding switching threshold, introducing hysteresis that prevents rapid switching triggered by transient discrepancy fluctuations. Across three benchmarks and two student model sizes, DASH-OPD outperforms five baselines in all 14 task performance comparisons, while requiring the fewest interaction turns in nine of ten efficiency comparisons. Code, models, and training logs are available at https://github.com/Lucian1115/DASH-OPD