We study the selective learning problem introduced by Qiao and Valiant (2019), in which the learner observes $n$ labeled data points one at a time. At a time of its choosing, the learner selects a window length $w$ and a model $\hat\ell$ from the model class $\mathcal{L}$, and then labels the next $w$ data points using $\hat\ell$. The excess risk incurred by the learner is defined as the difference between the average loss of $\hat\ell$ over those $w$ data points and the smallest possible average loss among all models in $\mathcal{L}$ over those $w$ data points. We give an improved algorithm, termed the hybrid exponential weights algorithm, that achieves an expected excess risk of $O((\log\log|\mathcal{L}| + \log\log n)/\log n)$. This result gives a doubly exponential improvement in the dependence on $|\mathcal{L}|$ over the best known bound of $O(\sqrt{|\mathcal{L}|/\log n})$. We complement the positive result with an almost matching lower bound, which suggests the worst-case optimality of the algorithm. We also study a more restrictive family of learning algorithms that are bounded-recall in the sense that when a prediction window of length $w$ is chosen, the learner's decision only depends on the most recent $w$ data points. We analyze an exponential weights variant of the ERM algorithm in Qiao and Valiant (2019). This new algorithm achieves an expected excess risk of $O(\sqrt{\log |\mathcal{L}|/\log n})$, which is shown to be nearly optimal among all bounded-recall learners. Our analysis builds on a generalized version of the selective mean prediction problem in Drucker (2013); Qiao and Valiant (2019), which may be of independent interest.
翻译:我们研究Qiao 和 Valiant (2019年) 引入的选择性学习问题, 学习者在其中一次观察标注的美元数据点。 在选择的时候, 学习者选择了一个窗口长度 $w$, 模型$\ mathcal{L} 美元, 然后用 $\ hat\ ell 来标注下一个 $w$ 的数据点。 学习者面临的超额风险被定义为 美元相对于这些 $w$ 数据点的平均损失与所有模型之间最小的平均损失之间的差别。 在选择的时候, 学习者选择了一个窗口长度 $ 美元 标标标标标注的模型 $w; 学习者选择了一个模型 $( log\ log\ logmacal {L} + log\ log n n. 。 学习者对这个模型的预期超额风险可能是 = 美元 美元 listalalal_ listalal_ dal_ legal ligal) 。