Accelerated Distributed Stochastic Nonconvex Optimization Over Time-Varying Directed Networks

Distributed stochastic nonconvex optimization problems have recently received attention due to the growing interest of signal processing, computer vision, and natural language processing communities in applications deployed over distributed learning systems (e.g., federated learning). We study the setting where the data is distributed across the nodes of a time-varying directed network, a topology suitable for modeling dynamic networks experiencing communication delays and straggler effects. The network nodes, which can access only their local objectives and query a stochastic first-order oracle to obtain gradient estimates, collaborate to minimize a global objective function by exchanging messages with their neighbors. We propose an algorithm, novel to this setting, that leverages stochastic gradient descent with momentum and gradient tracking to solve distributed nonconvex optimization problems over time-varying networks. To analyze the algorithm, we tackle the challenges that arise when analyzing dynamic network systems that communicate gradient acceleration components. We prove that the algorithm's oracle complexity is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(1/\epsilon ^{1.5})$</tex-math></inline-formula>, and that under Polyak-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{L}$</tex-math></inline-formula>ojasiewicz condition the algorithm converges linearly to a steady error state. The proposed scheme is tested on several learning tasks: a nonconvex logistic regression experiment on the MNIST dataset, an image classification task on the CIFAR-10 dataset, and an NLP classification test on the IMDB dataset. We further present numerical simulations with an objective that satisfies the PL condition. The results demonstrate superior performance of the proposed framework compared to the existing related methods.