Chapter 5 · Book pp. 141–173 · August 9, 2026 edition
Faster Low-Accuracy Samplers
Study randomized midpoint, Hamiltonian, and underdamped methods that improve low-accuracy complexity.
Begin with 5.1 Open this chapter in the canonical August 9 source ↗Chapter route
This chapter develops Hamiltonian flow, momentum refreshment, underdamped Langevin, hypocoercivity. Its main destination is to connect the definitions below to the results that later chapters consume.
Core definitions
- Randomized midpoint uses an internal random evaluation time to reduce discretization bias.
- Hamiltonian Monte Carlo alternates approximate Hamiltonian flow with momentum refreshment.
- Underdamped Langevin evolves position and momentum with friction and noise.
- Hypocoercive metrics couple position and momentum errors.
Main results
- Randomized midpoint improves the low-accuracy dimension dependence over basic Euler discretization.
- Hamiltonian trajectories exploit second-order phase-space motion before refreshment.
- Underdamped Langevin contracts in a suitable mixed metric under strong log-concavity.
- Discretized accelerated dynamics produce faster low-accuracy sampling guarantees under stated smoothness and step-size regimes.
Contents
Why is this chapter route valid?
Analytic contracts
- Invariant phase-space laws require both position and momentum normalization.
- Hypocoercive estimates mix position and velocity norms; coercivity is not pointwise in the position coordinate alone.
- Exact Hamiltonian flow and numerical integrators must not be conflated.
Open boundaries
- Phase-space invariant-law theorem
- Hypocoercive convergence packet
- HMC flow and refreshment kernel
View Lean formalization
These mappings are evidence links, not a claim that the entire chapter is formalized.
No declaration-level source block is mapped for this chapter yet.