Traditional models of active learning assume a learner can directly manipulate or query a covariate $X$ in order to study its relationship with a response $Y$. However, if $X$ is a feature of a complex system, it may be possible only to indirectly influence $X$ by manipulating a control variable $Z$, a scenario we refer to as Indirect Active Learning. Under a nonparametric model of Indirect Active Learning with a fixed budget, we study minimax convergence rates for estimating the relationship between $X$ and $Y$ locally at a point, obtaining different rates depending on the complexities and noise levels of the relationships between $Z$ and $X$ and between $X$ and $Y$. We also identify minimax rates for passive learning under comparable assumptions. In many cases, our results show that, while there is an asymptotic benefit to active learning, this benefit is fully realized by a simple two-stage learner that runs two passive experiments in sequence.
翻译:传统的主动学习模式假定学习者可以直接操纵或询问共同变换美元,以便研究与应答美元的关系。然而,如果美元是复杂系统的一个特点,则可能只能通过操纵控制变量Z美元(我们称之为间接积极学习)间接影响X美元(我们称之为间接积极学习的假想)。根据非对称模式,以固定预算间接积极学习,我们研究一个点估算X美元与当地美元之间关系的微量峰值趋同率,根据Z美元与X美元之间的关系的复杂性和噪音程度以及X美元与Y美元之间的关系获得不同的利率。我们还确定了在类似假设下被动学习的微量成交率。在许多情况下,我们的结果显示,虽然积极学习有一个无损的收益,但通过一个按顺序进行两次被动实验的简单两阶段学习者完全实现了这一效益。