We consider the convergence rates of loss and uncertainty-based active learning algorithms under various assumptions. Firstly, we establish a set of conditions that ensure convergence rates when applied to linear classifiers and linearly separable datasets. This includes demonstrating convergence rate guarantees for loss-based sampling with various loss functions. Secondly, we introduce a framework that allows us to derive convergence rate bounds for loss-based sampling by leveraging known convergence rate bounds for stochastic gradient descent algorithms. Lastly, we propose a new algorithm that combines point sampling and stochastic Polyak's step size. We establish a condition on the sampling process, ensuring a convergence rate guarantee for this algorithm, particularly in the case of smooth convex loss functions. Our numerical results showcase the efficiency of the proposed algorithm.

翻译：我们研究了基于损失与不确定性的主动学习算法在不同假设条件下的收敛速率。首先，我们建立了一组确保线性分类器在可线性分离数据集上实现收敛的条件，其中包括证明基于损失的采样方法在使用不同损失函数时的收敛速率保证。其次，我们提出一个分析框架，通过利用随机梯度下降算法已知的收敛速率界，推导出基于损失采样的收敛速率边界。最后，我们设计了一种结合点采样与随机Polyak步长的新型算法，证明了该算法在光滑凸损失函数情形下的收敛速率保证，并建立了采样过程所需满足的条件。数值实验结果验证了所提算法的有效性。