Incorporating Linear Dependencies into Graph Gaussian Processes

Abstract

Graph Gaussian processes are an important technique for learning unknown functions on graphs while quantifying uncertainty. These processes encode prior information by using kernels that reflect the structure of the graph, allowing function values at nearby nodes to be correlated. However, there are limited choices for kernels on graphs, and most existing graph kernels can be shown to rely on the graph Laplacian and behave in a manner that resembles Euclidean radial basis functions. In many applications, additional prior information which goes beyond the graph structure encoded in Laplacian is available: in this work, we study the case where the dependencies between nodes in the target function are known as linear, possibly up to some noise. We propose a type of kernel for graph Gaussian processes that incorporate linear dependencies between nodes, based on an inter-domain-type construction. We show that this construction results in kernels that can encode directed information, and are robust under misspecified linear dependencies. We also show that the graph Matérn kernel, one of the commonly used Laplacian-based kernels, can be obtained as a special case of this construction. We illustrate the properties of these kernels on a set of synthetic examples. We then evaluate these kernels in a real-world traffic speed prediction task, and show that they easily out-perform the baseline kernels. We also use these kernels to learn offline reinforcement learning policies in maze environments. We show that they are significantly more stable and data-efficient than strong baselines, and they can incorporate prior information to generalize to unseen tasks.