Abstract
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.
