The Sample Complexity of Meta Sparse Regression

This paper addresses the meta-learning problem in sparse linear regression with infinite tasks. We assume that the learner can access several similar tasks. The goal of the learner is to transfer knowledge from the prior tasks to a similar but novel task. For $p$ parameters, size of the support set $k$, and $l$ samples per task, we show that $T \in O((k \log (p-k)) / l)$ tasks are sufficient in order to recover the common support of all tasks. With the recovered support, we can greatly reduce the sample complexity for estimating the parameter of the novel task, i.e., $l \in O(1)$ with respect to $T$ and $p$. We also prove that our rates are minimax optimal. A key difference between meta-learning and the classical multi-task learning, is that meta-learning focuses only on the recovery of the parameters of the novel task, while multi-task learning estimates the parameter of all tasks, which requires $l$ to grow with $T$. Instead, our efficient meta-learning estimator allows for $l$ to be constant with respect to $T$ (i.e., few-shot learning).