Any-scale Balanced Samplers for Discrete Space

The locally balanced informed proposal has proved to be highly effective for sampling from discrete spaces. However, its success relies on the "local'' factor, which ensures that whenever the proposal distribution is restricted to be near the current state, the locally balanced weight functions are asymptotically optimal and the gradient approximations are accurate. In seeking a more efficient sampling algorithm, many recent works have considered increasing the scale of the proposal distributions, but this causes the "local'' factor to no longer hold. Instead, we propose any-scale balanced samplers to repair the gap in non-local proposals. In particular, we substitute the locally balanced function with an any-scale balanced function that can self-adjust to achieve better efficiency for proposal distributions at any scale. We also use quadratic approximations to capture curvature of the target distribution and reduce the error in the gradient approximation, while employing a Gaussian integral trick with a special estimated diagonal to efficiently sample from the quadratic proposal distribution. On various synthetic and real distributions, the proposed sampler substantially outperforms existing approaches.