Metalearning and multitask learning are two frameworks for solving a group of related learning tasks more efficiently than we could hope to solve each of the individual tasks on their own. In multitask learning, we are given a fixed set of related learning tasks and need to output one accurate model per task, whereas in metalearning we are given tasks that are drawn i.i.d. from a metadistribution and need to output some common information that can be easily specialized to new, previously unseen tasks from the metadistribution. In this work, we consider a binary classification setting where tasks are related by a shared representation, that is, every task $P$ of interest can be solved by a classifier of the form $f_{P} \circ h$ where $h \in H$ is a map from features to some representation space that is shared across tasks, and $f_{P} \in F$ is a task-specific classifier from the representation space to labels. The main question we ask in this work is how much data do we need to metalearn a good representation? Here, the amount of data is measured in terms of both the number of tasks $t$ that we need to see and the number of samples $n$ per task. We focus on the regime where the number of samples per task is extremely small. Our main result shows that, in a distribution-free setting where the feature vectors are in $\mathbb{R}^d$, the representation is a linear map from $\mathbb{R}^d \to \mathbb{R}^k$, and the task-specific classifiers are halfspaces in $\mathbb{R}^k$, we can metalearn a representation with error $\varepsilon$ using just $n = k+2$ samples per task, and $d \cdot (1/\varepsilon)^{O(k)}$ tasks. Learning with so few samples per task is remarkable because metalearning would be impossible with $k+1$ samples per task, and because we cannot even hope to learn an accurate task-specific classifier with just $k+2$ samples per task.

翻译：元学习和多任务学习是两种框架，旨在比单独解决每个任务更高效地处理一组相关学习任务。在多任务学习中，给定一组固定的相关学习任务，需要为每个任务输出一个准确的模型；而在元学习中，任务是从一个元分布中独立同分布地抽取的，需要输出一些通用信息，这些信息可以轻松特化到该元分布中新的、未见过的任务。本研究考虑一个二分类场景，其中任务通过共享表示相关联，即每个相关任务 $P$ 可通过形如 $f_{P} \circ h$ 的分类器求解，其中 $h \in H$ 是从特征到共享表示空间的映射，$f_{P} \in F$ 是从表示空间到标签的任务特定分类器。本研究探讨的核心问题是：学习一个良好表示需要多少数据？这里，数据量通过需要观察的任务数量 $t$ 和每个任务的样本数量 $n$ 来衡量。我们聚焦于每个任务样本量极小的情形。主要结果表明，在特征向量位于 $\mathbb{R}^d$、表示是从 $\mathbb{R}^d$ 到 $\mathbb{R}^k$ 的线性映射、且任务特定分类器为 $\mathbb{R}^k$ 中半空间的无分布场景下，我们只需每个任务提供 $n = k+2$ 个样本和 $d \cdot (1/\varepsilon)^{O(k)}$ 个任务，即可元学习到误差为 $\varepsilon$ 的表示。每个任务仅需如此少的样本具有显著意义，因为当每个任务样本量为 $k+1$ 时元学习将不可能实现，甚至无法期望仅用 $k+2$ 个样本就学习到准确的任务特定分类器。