SMART: A Spectral Transfer Approach to Multi-Task Learning

We study the multi-task linear regression problem in the presence of contaminated tasks. We address the setting where the unknown parameters of a majority of tasks are close in the $\ell_2$-norm, while a fraction of tasks are arbitrary outliers. Existing theoretical frameworks for this problem rely heavily on the assumption that the empirical second moment of each task has a minimum eigenvalue bounded away from zero (order $Ω(1)$). Crucially, this assumption fails in many high-dimensional scenarios, rendering prior guarantees vacuous. To overcome this limitation, we propose an estimator based on matrix-weighted norm regularization. We also introduce a relative balancedness condition, quantified by a balancedness constant, that compares each task's second moment with the average inlier geometry and relaxes the need for taskwise second-moment lower bounds. In favorable regimes with moderate balancedness, our prediction MSE bounds match the rate of Duan and Wang (2023) under substantially weaker spectral assumptions; the resulting task-overall MSE is minimax optimal up to logarithmic factors. Furthermore, we demonstrate that our estimator enjoys a safety guarantee: when the relevant balancedness constant is large or infinite, or when tasks are unrelated, the method performs no worse than independent task learning.

Most existing multitask learning approaches are limited by their reliance on task-specific loss functions tailored to the scale and type of each outcome. When outcomes differ across tasks, these losses are generally not directly comparable, which makes it difficult to formulate a unified objective and may limit information sharing across tasks. We propose a multitask transformation framework in which task-specific responses may differ through unknown monotone transformations. Motivated by high-dimensional biological applications in which the predictor dimension may diverge with the sample size while only a common subset of predictors is informative, we consider shared sparsity across tasks. Under this framework, we estimate the target functions and identify important predictors by optimizing a smoothed rank-based criterion with a group-Lasso penalty, implemented through a multitask deep neural network with a shared first layer. We establish the nonasymptotic excess-risk bounds, and variable-selection consistency for the proposed estimator. Simulation studies show that the proposed method achieves competitive prediction and variable-selection performance compared with competing approaches. Analyses of gene-expression studies with continuous, binary, and mixed outcomes further illustrate that the proposed method improves prediction and identifies biologically meaningful shared predictors.

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 $ε$ 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 $ε\sqrt{d/n}$, which is suboptimal compared to the lower bound $ε/\sqrt{n}$. 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.