Hamiltonian Monte Carlo for efficient Gaussian sampling: long and random steps
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Hamiltonian Monte Carlo (HMC) is a Markov chain algorithm for sampling from a high-dimensional distribution with density e^-f(x), given access to the gradient of f. A particular case of interest is that of a d-dimensional Gaussian distribution with covariance matrix Σ, in which case f(x) = x^⊤Σ^-1 x. We show that HMC can sample from a distribution that is ε-close in total variation distance using O(√(κ) d^1/4log(1/ε)) gradient queries, where κ is the condition number of Σ. Our algorithm uses long and random integration times for the Hamiltonian dynamics. This contrasts with (and was motivated by) recent results that give an Ω(κ d^1/2) query lower bound for HMC with fixed integration times, even for the Gaussian case.
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