We study convergence rates of loss and uncertainty-based active learning algorithms under various assumptions. First, we provide a set of conditions under which a convergence rate guarantee holds, and use this for linear classifiers and linearly separable datasets to show convergence rate guarantees for loss-based sampling and different loss functions. Second, we provide a framework that allows us to derive convergence rate bounds for loss-based sampling by deploying known convergence rate bounds for stochastic gradient descent algorithms. Third, and last, we propose an active learning algorithm that combines sampling of points and stochastic Polyak's step size. We show a condition on the sampling that ensures a convergence rate guarantee for this algorithm for smooth convex loss functions. Our numerical results demonstrate efficiency of our proposed algorithm.

翻译：我们研究了在各种假设下基于损失和不确定性的主动学习算法的收敛速率。首先，我们提供了一组保证收敛速率的条件，并将其用于线性分类器和线性可分数据集，以展示基于损失采样和不同损失函数的收敛速率保证。其次，我们提出一个框架，通过利用已知的随机梯度下降算法的收敛速率界，来推导基于损失采样的收敛速率界。最后，我们提出了一种结合点采样和随机Polyak步长的主动学习算法。我们证明了采样条件可确保该算法在光滑凸损失函数下的收敛速率保证。数值结果展示了所提算法的效率。