Block stochastic gradient descent for linear ill-posed problems in Hilbert spaces under the power-type source condition

In this paper, we consider linear ill-posed inverse problems in Hilbert spaces under the classical power-type source condition. We propose a block stochastic gradient descent (Block SGD) algorithm, which generalizes the classical SGD method by allowing updates on data blocks rather than single indices. Under the power-type source condition, we derive a novel mean squared error bound between the true solution and the regularized solution, in terms of the step size, the batch size, and the block number. In particular, our estimate shows that the stochastic error decreases as the batch size increases, which is confirmed by numerical experiments.