Theoretical Convergence of Multi-Step Model-Agnostic Meta-Learning

TL;DR

This paper provides the first theoretical convergence analysis for multi-step MAML, establishing convergence rates and complexity bounds for both resampling and finite-sum cases in nonconvex settings.

Contribution

It introduces a new theoretical framework that guarantees convergence of multi-step MAML in practical scenarios, addressing both resampling and finite-sum cases.

Findings

Inner stepsize should be inversely proportional to the number of inner steps for convergence.

Characterized convergence rate and complexity for multi-step MAML in nonconvex settings.

Developed novel techniques for nested meta gradient analysis.

Abstract

As a popular meta-learning approach, the model-agnostic meta-learning (MAML) algorithm has been widely used due to its simplicity and effectiveness. However, the convergence of the general multi-step MAML still remains unexplored. In this paper, we develop a new theoretical framework to provide such convergence guarantee for two types of objective functions that are of interest in practice: (a) resampling case (e.g., reinforcement learning), where loss functions take the form in expectation and new data are sampled as the algorithm runs; and (b) finite-sum case (e.g., supervised learning), where loss functions take the finite-sum form with given samples. For both cases, we characterize the convergence rate and the computational complexity to attain an $\epsilon$-accurate solution for multi-step MAML in the general nonconvex setting. In particular, our results suggest that an inner-stage stepsize needs to be chosen inversely proportional to the number $N$ of inner-stage steps in order for $N$-step MAML to have guaranteed convergence. From the technical perspective, we develop novel techniques to deal with the nested structure of the meta gradient for multi-step MAML, which can be of independent interest.