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

JOURNAL OF MACHINE LEARNING RESEARCH（2022）

摘要

As a popular meta-learning approach, the model-agnostic meta-learning (MAML) algo-rithm has been widely used due to its simplicity and effectiveness. However, the conver-gence 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 algo-rithm 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.

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关键词

Computational complexity, convergence rate, finite-sum, meta-learning, multi-step MAML, nonconvex, resampling

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