Gradient-Descent Based Convergence and Error Analysis of Flow Matching in Deep ReLU Networks

Flow matching has proven to be a highly efficient technique for generative modeling of complex data, and works by learning the constructed velocity fields of a continuous normalizing flow via deep least-squares regression. However, most existing theoretical investigations on the accuracy of flow models, while significant, assume that the velocity function has been approximated to a certain accuracy, and then use this a priori bound to control the error of generation. We provide a quantitative understanding of the whole generation process, from training to sampling. More precisely, we provide a non-asymptotic convergence analysis of flow matching under gradient descent for a general class of deep ReLU neural networks, which elucidates how to design the training and sampling process for effective generation. For instance, our theory indicates the explicit effect of increasing the depth and width of the network, as well as the gradient-descent hyperparameters, on the final target accuracy. By propagating the convergence error across the generation process, we derive end-to-end error bounds on the Wasserstein-2 distance between the generated and target distributions.