On the convergence of loss and uncertainty-based active learning algorithms

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.