First-order ANIL provably learns representations despite overparametrization

TL;DR

This paper proves that first-order ANIL, a meta-learning method, can theoretically learn shared linear representations even with overparametrization, ensuring effective adaptation to new tasks.

Contribution

It provides the first theoretical proof that first-order ANIL learns shared representations under overparametrization in a linear setting.

Findings

First-order ANIL learns low-rank shared representations.

Overparametrization leads to asymptotically low-rank solutions.

Shared representations enable effective one-step adaptation.

Abstract

Due to its empirical success in few-shot classification and reinforcement learning, meta-learning has recently received significant interest. Meta-learning methods leverage data from previous tasks to learn a new task in a sample-efficient manner. In particular, model-agnostic methods look for initialization points from which gradient descent quickly adapts to any new task. Although it has been empirically suggested that such methods perform well by learning shared representations during pretraining, there is limited theoretical evidence of such behavior. More importantly, it has not been shown that these methods still learn a shared structure, despite architectural misspecifications. In this direction, this work shows, in the limit of an infinite number of tasks, that first-order ANIL with a linear two-layer network architecture successfully learns linear shared representations. This result even holds with overparametrization; having a width larger than the dimension of the shared representations results in an asymptotically low-rank solution. The learned solution then yields a good adaptation performance on any new task after a single gradient step. Overall, this illustrates how well model-agnostic methods such as first-order ANIL can learn shared representations.