Representation learning based on multi-task pretraining has become a powerful approach in many domains. In particular, task-aware representation learning aims to learn an optimal representation for a specific target task by sampling data from a set of source tasks, while task-agnostic representation learning seeks to learn a universal representation for a class of tasks. In this paper, we propose a general and versatile algorithmic and theoretic framework for \textit{active representation learning}, where the learner optimally chooses which source tasks to sample from. This framework, along with a tractable meta algorithm, allows most arbitrary target and source task spaces (from discrete to continuous), covers both task-aware and task-agnostic settings, and is compatible with deep representation learning practices. We provide several instantiations under this framework, from bilinear and feature-based nonlinear to general nonlinear cases. In the bilinear case, by leveraging the non-uniform spectrum of the task representation and the calibrated source-target relevance, we prove that the sample complexity to achieve $\varepsilon$-excess risk on target scales with $ (k^*)^2 \|v^*\|_2^2 \varepsilon^{-2}$ where $k^*$ is the effective dimension of the target and $\|v^*\|_2^2 \in (0,1]$ represents the connection between source and target space. Compared to the passive one, this can save up to $\frac{1}{d_W}$ of sample complexity, where $d_W$ is the task space dimension. Finally, we demonstrate different instantiations of our meta algorithm in synthetic datasets and robotics problems, from pendulum simulations to real-world drone flight datasets. On average, our algorithms outperform baselines by $20\%-70\%$.

翻译：基于多任务预训练的表示学习已成为众多领域中的强大方法。具体而言，任务感知表示学习旨在通过从源任务集合中采样数据，学习针对特定目标任务的最优表示；而任务无关表示学习则致力于为某一类任务学习通用表示。本文提出一种通用且灵活的算法与理论框架——**主动表示学习**，其中学习器能够最优地选择从哪些源任务进行采样。该框架结合可处理的元算法，可容纳大部分任意类型的目标与源任务空间（从离散到连续），覆盖任务感知与任务无关两种设置，并与深度表示学习实践兼容。我们在此框架下提供了多种实例化方案，涵盖双线性、基于特征的非线性以及一般非线性情形。在双线性情形中，通过利用任务表示的非均匀谱以及经校准的源-目标相关性，我们证明了达到目标$\varepsilon$-超额风险所需的样本复杂度为$(k^*)^2 \|v^*\|_2^2 \varepsilon^{-2}$，其中$k^*$为目标的有效维度，$\|v^*\|_2^2 \in (0,1]$表示源空间与目标空间的关联程度。与被动方法相比，这最多可节省$\frac{1}{d_W}$的样本复杂度，其中$d_W$为任务空间维度。最后，我们在合成数据集与机器人问题（从摆锤仿真到真实无人机飞行数据集）中展示了元算法的多种实例化效果。平均而言，我们的算法相比基线方法性能提升$20\%-70\%$。