论文标题
多步型元学习的理论融合
Theoretical Convergence of Multi-Step Model-Agnostic Meta-Learning
论文作者
论文摘要
作为一种流行的元学习方法，由于其简单性和有效性，模型不足的元学习（MAML）算法已被广泛使用。但是，一般多步MAML的收敛仍然未开发。在本文中，我们开发了一个新的理论框架，以为实践中感兴趣的两种目标功能提供这种融合保证：（a）重新采样案例（例如，加强学习），其中损失函数在期望中以算法运行为预期和新数据； （b）有限的案例（例如，有监督的学习），其中损失功能采用有限的和给定样品的形式。在这两种情况下，我们都表征了收敛速率和计算复杂性，以在一般的非convex设置中用于多步MAML的$ε$可准确解决方案。特别是，我们的结果表明，需要与内部阶段的数字$ n $成反比，以使$ n $ -Step MAML呈现，以保证收敛。从技术角度来看，我们开发了新的技术来处理多步MAML的元梯度的嵌套结构，这可能具有独立的兴趣。
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 $ε$-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.
