Logsmooth Gradient Concentration and Tighter Runtimes for Metropolized Hamiltonian Monte Carlo
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We show that the gradient norm ∇ f(x) for x ∼(-f(x)), where f is strongly convex and smooth, concentrates tightly around its mean. This removes a barrier in the prior state-of-the-art analysis for the well-studied Metropolized Hamiltonian Monte Carlo (HMC) algorithm for sampling from a strongly logconcave distribution <cit.>. We correspondingly demonstrate that Metropolized HMC mixes in Õ(κ d) iterations[%s], improving upon the Õ(κ^1.5√(d) + κ d) runtime of <cit.> by a factor (κ/d)^1/2 when the condition number κ is large. Our mixing time analysis introduces several techniques which to our knowledge have not appeared in the literature and may be of independent interest, including restrictions to a nonconvex set with good conductance behavior, and a new reduction technique for boosting a constant-accuracy total variation guarantee under weak warmness assumptions. This is the first mixing time result for logconcave distributions using only first-order function information which achieves linear dependence on κ; we also give evidence that this dependence is likely to be necessary for standard Metropolized first-order methods.
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