Yifan Chen, Xiaoou Cheng, Jonathan Niles-Weed +1 more
Some of the fastest ways to draw samples from a complicated probability distribution share a known flaw: they are biased, meaning they systematically miss the true target by a little. The usual fix, a Metropolis-Hastings accept-reject step, removes the bias but forces tiny step sizes, which makes everything slow. So it helps to know when you can just live with the bias.
This paper extends a reassuring result to two popular unadjusted samplers. The rough message: the bias is 'delocalized', spread thinly across all the dimensions, so if you only care about a handful of variables at a time, the error you actually feel stays small. Concretely, controlling the error on any K of the variables takes only about the square root of K integration steps, as long as the variables interact weakly or sparsely.
This is a plain gloss of a technical abstract, so the paper holds the precise assumptions and bounds.
Unadjusted samplers such as unadjusted Hamiltonian Monte Carlo and underdamped Langevin are well-known to be biased. Metropolis--Hastings adjustment has been conventionally incorporated into Hamiltonian Monte Carlo to eliminate the bias. However, this adjustment can significantly increase the iteration complexity due to the small step size required for reasonable Metropolis acceptance rates. In this work, we extend the \emph{delocalization of bias} phenomenon, previously…
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