Ensuring both accuracy and robustness in time series prediction is critical to many applications, ranging from urban planning to pandemic management. With sufficient training data where all spatiotemporal patterns are well-represented, existing deep-learning models can make reasonably accurate predictions. However, existing methods fail when the training data are drawn from different circumstances (e.g., traffic patterns on regular days) compared to test data (e.g., traffic patterns after a natural disaster). Such challenges are usually classified under domain generalization. In this work, we show that one way to address this challenge in the context of spatiotemporal prediction is by incorporating domain differential equations into Graph Convolutional Networks (GCNs). We theoretically derive conditions where GCNs incorporating such domain differential equations are robust to mismatched training and testing data compared to baseline domain agnostic models. To support our theory, we propose two domain-differential-equation-informed networks called Reaction-Diffusion Graph Convolutional Network (RDGCN), which incorporates differential equations for traffic speed evolution, and Susceptible-Infectious-Recovered Graph Convolutional Network (SIRGCN), which incorporates a disease propagation model. Both RDGCN and SIRGCN are based on reliable and interpretable domain differential equations that allow the models to generalize to unseen patterns. We experimentally show that RDGCN and SIRGCN are more robust with mismatched testing data than the state-of-the-art deep learning methods.

The challenge of communication-efficient distributed optimization has attracted attention in recent years. In this paper, a communication efficient algorithm, called ordering-based alternating direction method of multipliers (OADMM) is devised in a general fully decentralized network setting where a worker can only exchange messages with neighbors. Compared to the classical ADMM, a key feature of OADMM is that transmissions are ordered among workers at each iteration such that a worker with the most informative data broadcasts its local variable to neighbors first, and neighbors who have not transmitted yet can update their local variables based on that received transmission. In OADMM, we prohibit workers from transmitting if their current local variables are not sufficiently different from their previously transmitted value. A variant of OADMM, called SOADMM, is proposed where transmissions are ordered but transmissions are never stopped for each node at each iteration. Numerical results demonstrate that given a targeted accuracy, OADMM can significantly reduce the number of communications compared to existing algorithms including ADMM. We also show numerically that SOADMM can accelerate convergence, resulting in communication savings compared to the classical ADMM.

A very large number of communications are typically required to solve distributed learning tasks, and this critically limits scalability and convergence speed in wireless communications applications. In this paper, we devise a Gradient Descent method with Sparsification and Error Correction (GD-SEC) to improve the communications efficiency in a general worker-server architecture. Motivated by a variety of wireless communications learning scenarios, GD-SEC reduces the number of bits per communication from worker to server with no degradation in the order of the convergence rate. This enables larger-scale model learning without sacrificing convergence or accuracy. At each iteration of GD-SEC, instead of directly transmitting the entire gradient vector, each worker computes the difference between its current gradient and a linear combination of its previously transmitted gradients, and then transmits the sparsified gradient difference to the server. A key feature of GD-SEC is that any given component of the gradient difference vector will not be transmitted if its magnitude is not sufficiently large. An error correction technique is used at each worker to compensate for the error resulting from sparsification. We prove that GD-SEC is guaranteed to converge for strongly convex, convex, and nonconvex optimization problems with the same order of convergence rate as GD. Furthermore, if the objective function is strongly convex, GD-SEC has a fast linear convergence rate. Numerical results not only validate the convergence rate of GD-SEC but also explore the communication bit savings it provides. Given a target accuracy, GD-SEC can significantly reduce the communications load compared to the best existing algorithms without slowing down the optimization process.