Near-optimal and Efficient First-Order Algorithm for Multi-Task Learning with Shared Linear Representation
It provides the first efficient and near-optimal first-order method for likelihood-based multi-task learning, addressing a key bottleneck in the field.
The paper proposes a first-order algorithm for multi-task learning with shared linear representation that converges in Õ(1) iterations and achieves near-optimal estimation error of Õ(dk/(TN)), improving over existing likelihood-based methods by a factor of k.
Multi-task learning (MTL) has emerged as a pivotal paradigm in machine learning by leveraging shared structures across multiple related tasks. Despite its empirical success, the development of likelihood-based efficiently solvable algorithms--even for shared linear representations--remains largely underdeveloped, primarily due to the non-convex structure intrinsic to matrix factorization. This paper introduces a first-order algorithm that jointly learns a shared representation and task-specific parameters, with guaranteed efficiency. Notably, it converges in $\widetilde{\mathcal{O}}(1)$ iterations and attains a \emph{near-optimal} estimation error of $\widetilde{\mathcal{O}}(dk/(TN))$, \emph{improving} over existing likelihood-based methods by a factor of $k$, where $d$, $k$, $T$, $N$ denote input dimension, representation dimension, task count, and samples per task, respectively. Our results justify that likelihood-based first-order methods can efficiently solve the MTL problem.