We consider a collaborative learning setting where the goal of each agent is to improve their own model by leveraging the expertise of collaborators, in addition to their own training data. To facilitate the exchange of expertise among agents, we propose a distillation-based method leveraging shared unlabeled auxiliary data, which is pseudo-labeled by the collective. Central to our method is a trust weighting scheme that serves to adaptively weigh the influence of each collaborator on the pseudo-labels until a consensus on how to label the auxiliary data is reached. We demonstrate empirically that our collaboration scheme is able to significantly boost individual models' performance in the target domain from which the auxiliary data is sampled. At the same time, it can provably mitigate the negative impact of bad models on the collective. By design, our method adeptly accommodates heterogeneity in model architectures and substantially reduces communication overhead compared to typical collaborative learning methods.

Still, the limited certified robustness that is currently achievable has been a bottleneck for their practical adoption. Gowal et al. and Wang et al. have shown

Experiments demonstrate the advantage of our framework.

We thank the reviewers for their insightful feedback. Can you evaluate on additional in-distribution dataset? We will definitely include these results in the final version. How to distinguish methodological differences w.r .t. prior work? The idea of using the energy score for OOD detection is novel and theoretically motivated.

Empirical evaluation is conducted on the challenging CLRS Algorithmic Reasoning Benchmark, which consists of 30 diverse algorithmic tasks.

At the same time, it can provably mitigate the negative impact of bad models on the collective.

Large language models (LLMs) have achieved remarkable success across widespread tasks, yet their application in low-resource domains remains a significant challenge due to data scarcity and the high risk of overfitting. While in-domain data is limited, there exist vast amounts of similar general-domain data, and our initial findings reveal that they could potentially serve as auxiliary supervision for domain enhancement. This observation leads us to our central research question: how to effectively select the most valuable auxiliary data to maximize domain-specific performance, particularly when traditional methods are inapplicable due to a lack of large in-domain data pools or validation sets. To address this, we propose NTK-Selector, a principled and efficient framework for selecting general-domain auxiliary data to enhance domain-specific performance via neural tangent kernels (NTK). Our method tackles two challenges of directly applying NTK to LLMs, theoretical assumptions and prohibitive computational cost, by empirically demonstrating a stable NTK-like behavior in LLMs during LoRA fine-tuning and proposing a Jacobian-free approximation method. Extensive experiments across four low-resource domains (medical, financial, legal, and psychological) demonstrate that NTK-Selector consistently improves downstream performance. Specifically, fine-tuning on 1,000 in-domain samples alone only yielded +0.8 points for Llama3-8B-Instruct and +0.9 points for Qwen3-8B. In contrast, enriching with 9,000 auxiliary samples selected by NTK-Selector led to substantial gains of +8.7 and +5.1 points, which corresponds to a 10.9x and 5.7x improvement over the domain-only setting. Each task is augmented with 9K auxiliary samples selected by Random, LESS, and NTK-Selector from Cot Collection based on 1K domain samples. The emergence of large language models (LLMs) has led to remarkable advancements across a wide spectrum of natural language processing tasks (Touvron et al., 2023; Chowdhery et al., 2023; Y ang et al., 2025). However, their formidable capabilities are predominantly anchored in the availability of immense, high-quality pre-training and instruction-tuning datasets.

Multimodal Large Language Models (MLLMs) hold great promise for advanced reasoning at the intersection of text and images, yet they have not fully realized this potential. MLLMs typically integrate an LLM, a vision encoder, and a connector that maps the vision encoder's embeddings into the LLM's text embedding space. Although each component is pretrained on massive datasets with billions of samples, the entire multimodal model is typically trained on only thousands (or a few million) samples, which can result in weak performance on complex reasoning tasks. To address these shortcomings, instead of relying on extensive labeled datasets for fine-tuning, we propose a Test-Time Warmup method that adapts the MLLM per test instance by leveraging data from weakly supervised auxiliary tasks. With our approach, we observe a relative performance improvement of 4.03% on MMMU, 5.28% on VQA-Rad, and 1.63% on GQA on the Llama-Vision-Instruct model. Our method demonstrates that 'warming up' before inference can enhance MLLMs' robustness across diverse reasoning tasks.

Still, the limited certified robustness that is currently achievable has been a bottleneck for their practical adoption. Gowal et al. and Wang et al. have shown