Contaminated Multi-task Learning with Heterogeneity: Fundamental Limits and Optimal Algorithms

Integrating information across related tasks can improve estimation and prediction in transfer, multi-task, and federated learning, but contamination and heterogeneity make robust borrowing challenging. We study a contaminated multi-task empirical risk minimization (ERM) framework in which an $\epsilon$ fraction of $K$ tasks, each with sample size $n$, may be arbitrarily contaminated while the remaining tasks are heterogeneous. Our goal is to estimate both the global minimizer of the average risk and the clean task-specific minimizers, thereby combining robustness and personalization. In the Gaussian mean model, we show that several common paradigms, including adaptive and robust regularization around a shared center, global matrix regularization, decomposition-based regularization, and score-based outlier-task detection, all suffer from a worst-case contamination error of order $\epsilon\sqrt{d/n}$, which is suboptimal compared to the lower bound $\epsilon/\sqrt{n}$. This identifies a dimension-dependent barrier for these approaches. We then establish minimax lower bounds for a general heterogeneous ERM setting and propose a computationally efficient filtering-based robust multi-task gradient descent method. Under local strong convexity, smoothness, and sub-Gaussian gradient assumptions, the proposed method attains high-probability upper bounds matching the minimax rates up to logarithmic factors over a broad regime. In particular, it removes the extra $\sqrt{d}$ contamination dependence of many regularization-based methods and score-based outlier detection, while achieving personalization to local tasks under strong heterogeneity. Simulations and a real-data analysis demonstrate strong robustness and personalization relative to a broad range of benchmark methods.