With the motive of training all the parameters of a neural network, we study why and when one can achieve this by iteratively creating, training, and combining randomly selected subnetworks. Such scenarios have either implicitly or explicitly emerged in the recent literature: see e.g., the Dropout family of regularization techniques, or some distributed ML training protocols that reduce communication/computation complexities, such as the Independent Subnet Training protocol. While these methods are studied empirically and utilized in practice, they often enjoy partial or no theoretical support, especially when applied on neural network-based objectives. In this manuscript, our focus is on overparameterized single hidden layer neural networks with ReLU activations in the lazy training regime. By carefully analyzing $i)$ the subnetworks' neural tangent kernel, $ii)$ the surrogate functions' gradient, and $iii)$ how we sample and combine the surrogate functions, we prove linear convergence rate of the training error --up to a neighborhood around the optimal point-- for an overparameterized single-hidden layer perceptron with a regression loss. Our analysis reveals a dependency of the size of the neighborhood around the optimal point on the number of surrogate models and the number of local training steps for each selected subnetwork. Moreover, the considered framework generalizes and provides new insights on dropout training, multi-sample dropout training, as well as Independent Subnet Training; for each case, we provide convergence results as corollaries of our main theorem.