Conservative Hamiltonian Monte Carlo
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We introduce a new class of Hamiltonian Monte Carlo (HMC) algorithm called Conservative Hamiltonian Monte Carlo (CHMC), where energy-preserving integrators, derived from the Discrete Multiplier Method, are used instead of symplectic integrators. Due to the volume being no longer preserved under such a proposal map, a correction involving the determinant of the Jacobian of the proposal map is introduced within the acceptance probability of HMC. For a p-th order accurate energy-preserving integrator using a time step size τ, we show that CHMC satisfies stationarity without detailed balance. Moreover, we show that CHMC satisfies approximate stationarity with an error of 𝒪(τ^(m+1)p) if the determinant of the Jacobian is truncated to its first m+1 terms of its Taylor polynomial in τ^p. We also establish a lower bound on the acceptance probability of CHMC which depends only on the desired tolerance δ for the energy error and approximate determinant. In particular, a cost-effective and gradient-free version of CHMC is obtained by approximating the determinant of the Jacobian as unity, leading to an 𝒪(τ^p) error to the stationary distribution and a lower bound on the acceptance probability depending only on δ. Furthermore, numerical experiments show increased performance in acceptance probability and convergence to the stationary distribution for the Gradient-free CHMC over HMC in high dimensional problems.
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