Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration
share
A novel randomized time integrator is suggested for unadjusted Hamiltonian Monte Carlo (uHMC) in place of the usual Verlet integrator; namely, a stratified Monte Carlo (sMC) integrator which involves a minor modification to Verlet, and hence, is easy to implement. For target distributions of the form μ(dx) ∝ e^-U(x) dx where U: ℝ^d →ℝ_≥ 0 is both K-strongly convex and L-gradient Lipschitz, and initial distributions ν with finite second moment, coupling proofs reveal that an ε-accurate approximation of the target distribution μ in L^2-Wasserstein distance 𝒲^2 can be achieved by the uHMC algorithm with sMC time integration using O((d/K)^1/3 (L/K)^5/3ε^-2/3log( 𝒲^2(μ, ν) / ε)^+) gradient evaluations; whereas without additional assumptions the corresponding complexity of the uHMC algorithm with Verlet time integration is in general O((d/K)^1/2 (L/K)^2 ε^-1log( 𝒲^2(μ, ν) / ε)^+ ). Duration randomization, which has a similar effect as partial momentum refreshment, is also treated. In this case, without additional assumptions on the target distribution, the complexity of duration-randomized uHMC with sMC time integration improves to O(max((d/K)^1/4 (L/K)^3/2ε^-1/2,(d/K)^1/3 (L/K)^4/3ε^-2/3) ) up to logarithmic factors. The improvement due to duration randomization turns out to be analogous to that of time integrator randomization.
READ FULL TEXT