Quasi-Monte Carlo Graph Random Features
This work addresses the need for more efficient and accurate kernel approximations on graphs, which is incremental as it builds on existing GRFs but introduces a novel correlation mechanism.
The paper tackles the problem of improving the accuracy of graph random features (GRFs) by introducing a quasi-Monte Carlo method that induces negative correlations between random walk lengths through antithetic termination, resulting in lower-variance estimators for the 2-regularised Laplacian kernel with empirical accuracy gains on tasks like graph diffusion approximation.
We present a novel mechanism to improve the accuracy of the recently-introduced class of graph random features (GRFs). 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.