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Mixing of the No-U-Turn Sampler and the Geometry of Gaussian Concentration
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abstract
We prove that the mixing time of the No-U-Turn Sampler (NUTS), when initialized in the concentration region of the canonical Gaussian measure, scales as $d^{1/4}$, up to logarithmic factors, where $d$ is the dimension. This scaling is expected to be sharp. This result is based on a coupling argument that leverages the geometric structure of the target distribution. Specifically, concentration of measure results in a striking uniformity in NUTS' locally adapted transitions, which holds with high probability. This uniformity is formalized by interpreting NUTS as an accept/reject Markov chain, where the mixing properties for the more uniform accept chain are analytically tractable. Additionally, our analysis uncovers a previously unnoticed issue with the path length adaptation procedure of NUTS, specifically related to looping behavior, which we address in detail.
Forward citations
Cited by 2 Pith papers
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A Profile-Separation Framework for Quantitative Convergence of No-U-Turn Samplers
Under a new profile-separation condition, multinomial and biased-progressive NUTS mix in O~(1 + a*^2 kappa^2(1+gamma)^{4/3}) and O~(1 + a*^4 kappa^3(1+gamma)^2) transitions.
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On Accelerated Mixing of the No-U-turn Sampler
In Gaussian targets, NUTS is shown to select critical orbit lengths (and hence mix in O(1) transitions) exactly in a parameter phase A, while outside A there are step sizes for which it selects short orbits and mixes ...
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