Quasi-Monte Carlo Graph Random Features

We present a novel mechanism to improve the accuracy of the recently-introduced class of graph random features (GRFs) [Choromanski, 2023]. Our method induces negative correlations between the lengths of the algorithm's random walks by imposing antithetic termination: a procedure to sample more diverse random walks which may be of independent interest. It has a trivial drop-in implementation. We derive strong theoretical guarantees on the properties of these quasi-Monte Carlo GRFs (q-GRFs), proving that they yield lower-variance estimators of the 2-regularised Laplacian kernel under mild conditions. Remarkably, our results hold for any graph topology. We demonstrate empirical accuracy improvements on a variety of tasks including a new practical application: time-efficient approximation of the graph diffusion process. To our knowledge, q-GRFs constitute the first rigorously studied quasi-Monte Carlo scheme for kernels defined on combinatorial objects, inviting new research on correlations between graph random walks.(1)