Asymptotic Behavior of Multi--Task Learning: Implicit Regularization and Double Descent Effects

We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator maps, we derive near-optimal upper bounds for approximation and statistical generalization. On the lower-bound side, we establish a curse of parametric complexity and prove corresponding minimax rates. Together, these results show that shared representations across tasks do not increase the overall cost: multi-task operator learning follows the same scaling laws as single operator learning. We also compare MNO with a multi-task extension of DeepONet based on concatenated task inputs and show that, from a worst-case approximation-complexity perspective, both architectures satisfy essentially the same asymptotic rates.

Model merging has emerged as a promising paradigm for enabling multi-task capabilities without additional training. However, traditional basic merging methods often experience performance degradation due to parameter conflicts, even when applied to similar tasks. While recent personalized merging frameworks successfully preserve task-specific information to maintain performance, they typically incur storage overhead. In this paper, we propose Decomposition, Thresholding, and Scaling (DTS), an approximation-based personalized merging framework that pushes task-specific storage efficiency. DTS first applies singular value decomposition to the task-specific information and retains only a small subset of singular values and vectors. It then introduces a novel thresholding strategy that partitions singular vector elements into groups and assigns a scaling factor to each group. To enable generalization to unseen tasks, we further extend DTS with a variant that fuses task-specific information in a data-free manner based on the semantic similarity of task characteristics. Extensive experiments demonstrate that DTS consistently outperforms state-of-the-art baselines while requiring only 1% extra storage per task. Furthermore, experiments on unseen tasks show that the DTS variant achieves significantly better generalization performance. Our code is available at https://github.com/krumpguo/DTS.

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.