We study actively labeling streaming data, where an active learner is faced with a stream of data points and must carefully choose which of these points to label via an expensive experiment. Such problems frequently arise in applications such as healthcare and astronomy. We first study a setting when the data's inputs belong to one of $K$ discrete distributions and formalize this problem via a loss that captures the labeling cost and the prediction error. When the labeling cost is $B$, our algorithm, which chooses to label a point if the uncertainty is larger than a time and cost dependent threshold, achieves a worst-case upper bound of $O(B^{\frac{1}{3}} K^{\frac{1}{3}} T^{\frac{2}{3}})$ on the loss after $T$ rounds. We also provide a more nuanced upper bound which demonstrates that the algorithm can adapt to the arrival pattern, and achieves better performance when the arrival pattern is more favorable. We complement both upper bounds with matching lower bounds. We next study this problem when the inputs belong to a continuous domain and the output of the experiment is a smooth function with bounded RKHS norm. After $T$ rounds in $d$ dimensions, we show that the loss is bounded by $O(B^{\frac{1}{d+3}} T^{\frac{d+2}{d+3}})$ in an RKHS with a squared exponential kernel and by $O(B^{\frac{1}{2d+3}} T^{\frac{2d+2}{2d+3}})$ in an RKHS with a Mat\'ern kernel. Our empirical evaluation demonstrates that our method outperforms other baselines in several synthetic experiments and two real experiments in medicine and astronomy.

翻译：我们研究了流数据的主动标注问题，其中主动学习器面对连续到达的数据点，必须通过昂贵实验谨慎选择要标注的样本。此类问题常见于医疗与天文学等应用领域。我们首先研究了数据输入属于$K$个离散分布之一的情形，并通过捕捉标注代价与预测误差的损失函数对该问题进行形式化建模。当标注代价为$B$时，我们的算法采用基于时间与代价相关的不确定性阈值选择待标注样本，在$T$轮迭代后达到的最坏情况损失上界为$O(B^{\frac{1}{3}} K^{\frac{1}{3}} T^{\frac{2}{3}})$。我们同时提出了更精细的上界分析，证明该算法能自适应数据到达模式，且在更有利的到达模式下取得更优性能。针对两个上界，我们分别给出了匹配的下界证明。进一步研究了输入属于连续域且实验结果具有有界RKHS范数光滑函数的场景。在$d$维空间的$T$轮迭代中，我们发现：当采用平方指数核时，RKHS中的损失上界为$O(B^{\frac{1}{d+3}} T^{\frac{d+2}{d+3}})$；当采用Matérn核时，上界为$O(B^{\frac{1}{2d+3}} T^{\frac{2d+2}{2d+3}})$。实验评估表明，在多个合成实验以及医疗与天文学领域的两个真实实验中，我们的方法均优于其他基线方法。