We thank all the reviewers for their insightful and encouraging comments. Per your suggestion, we will update the appendix by adding more explanations about the proof ideas. Similarly, we can extend convergence results in Theorem 4 in Appendix from FS to IFS. We expect that the approach developed in this paper will fuel this future investigation. We will update this into the revision.

Multi-task learning (MTL) has shown effectiveness in exploiting shared information across tasks to improve generalization. MTL assumes tasks share similarities that can improve performance. In addition, boosting algorithms have demonstrated exceptional performance across diverse learning problems, primarily due to their ability to focus on hard-to-learn instances and iteratively reduce residual errors. This makes them a promising approach for learning multi-task problems. However, real-world MTL scenarios often involve tasks that are not well-aligned (known as outlier or adversarial tasks), which do not share beneficial similarities with others and can, in fact, deteriorate the performance of the overall model. To overcome this challenge, we propose Robust-Multi-Task Gradient Boosting (R-MTGB), a novel boosting framework that explicitly models and adapts to task heterogeneity during training. R-MTGB structures the learning process into three sequential blocks: (1) learning shared patterns, (2) partitioning tasks into outliers and non-outliers with regularized parameters, and (3) fine-tuning task-specific predictors. This architecture enables R-MTGB to automatically detect and penalize outlier tasks while promoting effective knowledge transfer among related tasks. Our method integrates these mechanisms seamlessly within gradient boosting, allowing robust handling of noisy or adversarial tasks without sacrificing accuracy. Extensive experiments on both synthetic benchmarks and real-world datasets demonstrate that our approach successfully isolates outliers, transfers knowledge, and consistently reduces prediction errors for each task individually, and achieves overall performance gains across all tasks. These results highlight robustness, adaptability, and reliable convergence of R-MTGB in challenging MTL environments.

Large language model (LLM) query routers are critical to modern AI platforms as they seek to improve efficiency by assigning inference queries to accurate, yet low-cost models. Parametric routers typically use trained neural networks for LLM selection but suffer from retraining and maintenance overheads. Nonparametric routers are training-free, instead estimating LLM accuracy and cost via similarity between encodings of the input query and training set queries. However, like their parametric counterparts, nonparametric routers struggle to generalize to outlier queries, an issue exacerbated by limited diversity in training sets which are costly to expand and difficult to keep current with ever-evolving use cases. We propose ProxRouter, which applies an exponentially tilted aggregation mechanism to balance bias and variance in nonparametric routers, improving their robustness to outliers. Experiments show ProxRouter enhances outlier routing while preserving inlier performance with minimal overhead.

We thank all the reviewers for their insightful and encouraging comments. Per your suggestion, we will update the appendix by adding more explanations about the proof ideas. Similarly, we can extend convergence results in Theorem 4 in Appendix from FS to IFS. We expect that the approach developed in this paper will fuel this future investigation. We will update this into the revision.

In the age of large and heterogeneous datasets, the integration of information from diverse sources is essential to improve parameter estimation. Multi-task learning offers a powerful approach by enabling simultaneous learning across related tasks. In this work, we introduce a late fusion framework for multi-task learning with semiparametric models that involve infinite-dimensional nuisance parameters, focusing on applications such as heterogeneous treatment effect estimation across multiple data sources, including electronic health records from different hospitals or clinical trial data. Our framework is two-step: first, initial double machine-learning estimators are obtained through individual task learning; second, these estimators are adaptively aggregated to exploit task similarities while remaining robust to task-specific differences. In particular, the framework avoids individual level data sharing, preserving privacy. Additionally, we propose a novel multi-task learning method for nuisance parameter estimation, which further enhances parameter estimation when nuisance parameters exhibit similarity across tasks. We establish theoretical guarantees for the method, demonstrating faster convergence rates compared to individual task learning when tasks share similar parametric components. Extensive simulations and real data applications complement the theoretical findings of our work while highlight the effectiveness of our framework even in moderate sample sizes.

Meta-learning is a general approach to equip machine learning models with the ability to handle few-shot scenarios when dealing with many tasks. Most existing meta-learning methods work based on the assumption that all tasks are of equal importance. However, real-world applications often present heterogeneous tasks characterized by varying difficulty levels, noise in training samples, or being distinctively different from most other tasks. In this paper, we introduce a novel meta-learning method designed to effectively manage such heterogeneous tasks by employing rank-based task-level learning objectives, Heterogeneous Tasks Robust Meta-learning (HeTRoM). HeTRoM is proficient in handling heterogeneous tasks, and it prevents easy tasks from overwhelming the meta-learner. The approach allows for an efficient iterative optimization algorithm based on bi-level optimization, which is then improved by integrating statistical guidance. Our experimental results demonstrate that our method provides flexibility, enabling users to adapt to diverse task settings and enhancing the meta-learner's overall performance.

Multi-task learning (MTL) aims to improve estimation and prediction performance by sharing common information among related tasks. One natural assumption in MTL is that tasks are classified into clusters based on their characteristics. However, existing MTL methods based on this assumption often ignore outlier tasks that have large task-specific components or no relation to other tasks. To address this issue, we propose a novel MTL method called Multi-Task Learning via Robust Regularized Clustering (MTLRRC). MTLRRC incorporates robust regularization terms inspired by robust convex clustering, which is further extended to handle non-convex and group-sparse penalties. The extension allows MTLRRC to simultaneously perform robust task clustering and outlier task detection. The connection between the extended robust clustering and the multivariate M-estimator is also established. This provides an interpretation of the robustness of MTLRRC against outlier tasks. An efficient algorithm based on a modified alternating direction method of multipliers is developed for the estimation of the parameters. The effectiveness of MTLRRC is demonstrated through simulation studies and application to real data.

Unsupervised learning has been widely used in many real-world applications. One of the simplest and most important unsupervised learning models is the Gaussian mixture model (GMM). In this work, we study the multi-task learning problem on GMMs, which aims to leverage potentially similar GMM parameter structures among tasks to obtain improved learning performance compared to single-task learning. We propose a multi-task GMM learning procedure based on the EM algorithm that not only can effectively utilize unknown similarity between related tasks but is also robust against a fraction of outlier tasks from arbitrary distributions. The proposed procedure is shown to achieve minimax optimal rate of convergence for both parameter estimation error and the excess mis-clustering error, in a wide range of regimes. Moreover, we generalize our approach to tackle the problem of transfer learning for GMMs, where similar theoretical results are derived. Finally, we demonstrate the effectiveness of our methods through simulations and real data examples. To the best of our knowledge, this is the first work studying multi-task and transfer learning on GMMs with theoretical guarantees.

Unsupervised Federated Learning: A Federated Gradient EM Algorithm for Heterogeneous Mixture Models with Robustness against Adversarial AttacksYe Tian Haolei Weng Yang Feng Columbia University Michigan State University New York University Abstract While supervised federated learning approaches have enjoyed significant success, the domain of unsupervised federated learning remains relatively underexplored. In this paper, we introduce a novel federated gradient EM algorithm designed for the unsupervised learning of mixture models with heterogeneous mixture proportions across tasks. We begin with a comprehensive finite-sample theory that holds for general mixture models, then apply this general theory on Gaussian Mixture Models (GMMs) and Mixture of Regressions (MoRs) to characterize the explicit estimation error of model parameters and mixture proportions. Our proposed federated gradient EM algorithm demonstrates several key advantages: adaptability to unknown task similarity, resilience against adversarial attacks on a small fraction of data sources, protection of local data privacy, and computational and communication efficiency. 1 INTRODUCTION Federated learning (FDL) is a machine learning paradigm that allows the training of statistical models by leveraging data from various local tasks, while ensuring the data remains decentralized to protect privacy (Li et al., 2020a). Introduced a few years ago, notably by Google (Koneˇ cn` y et al., 2016; McMahan et al., 2017), FDL has witnessed remarkable success in a diverse range of applications, including smartphones (Hard et al., 2018), healthcare (Antunes et al., 2022), and the internet of things (Nguyen et al., 2021). In this paper, we delve into the realm of unsupervised FDL, a scenario in which each task involves a mixture of distributions. Before proceeding further, we summarize mathematical notations used in this paper here. For two positive sequences { a n} and { b n}, a n b n means a n/b n 0, a n b n or a n = O (b n) means a n/b n C <, and a n b n means a n/b n, b n/a n C < . For a random variable x n and a positive sequence a n, x n = O p(a n) means that for any ϵ > 0, there exists M > 0 such that sup n P( |x n/a n| > M) ϵ . For a vector x R d, x 2 represents its Euclidean norm. For any positive integer K, 1: K and [ K ] stand for the set { 1, 2, .. . And for any set S, |S | denotes its cardinality and S c denotes its complement.

Representation multi-task learning (MTL) and transfer learning (TL) have achieved tremendous success in practice. However, the theoretical understanding of these methods is still lacking. Most existing theoretical works focus on cases where all tasks share the same representation, and claim that MTL and TL almost always improve performance. However, as the number of tasks grows, assuming all tasks share the same representation is unrealistic. Also, this does not always match empirical findings, which suggest that a shared representation may not necessarily improve single-task or target-only learning performance. In this paper, we aim to understand how to learn from tasks with \textit{similar but not exactly the same} linear representations, while dealing with outlier tasks. With a known intrinsic dimension, we propose two algorithms that are \textit{adaptive} to the similarity structure and \textit{robust} to outlier tasks under both MTL and TL settings. Our algorithms outperform single-task or target-only learning when representations across tasks are sufficiently similar and the fraction of outlier tasks is small. Furthermore, they always perform no worse than single-task learning or target-only learning, even when the representations are dissimilar. We provide information-theoretic lower bounds to show that our algorithms are nearly \textit{minimax} optimal in a large regime. We also propose an algorithm to adapt to the unknown intrinsic dimension. We conduct two simulation studies to verify our theoretical results.