Sampling Crash Course (continued)
summary
Shows that reflection coupling yields contraction only after applying a carefully chosen concave WF distance f with explicit constants depending on L, k, R, giving dimension-free bounds when R is fixed; proves L2 relaxation time equals (about) one half the Poincaré constant and entropy relaxation equals one half the Log‑Sobolev constant. Analyzes the unadjusted Langevin bias: for Gaussian targets the invariant‑measure error scales ~¼ h sqrt(d), so h must be O(d^{-1/2} ε^{-1}), and kinetic (underdamped) dynamics reach optimal rate ~1/max σi versus overdamped ~1/max σi^2.
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chapters (18)
- 0:00Start
- 8:12Total Variation Bounds
- 11:22References
- 12:20Chi-Square Divergence
- 14:27Relative Entropy
- 15:45Mixing Time
- 22:40Time Derivative of the Relative Entropy
- 26:14Sprut Perturbation Lemma
- 29:15Euler Discretization
- 31:36Invariant Measure
- 33:13Synchronous Coupling
- 39:04Metropolis Adjusted Longitude Algorithm
- 42:13Second-Order Methods
- 44:46Dynamics
- 45:16Boltzmann Gibbs Distribution
- 46:24Mixing Properties
- 47:49Wasserstein Contraction Coefficient
- 57:01Hamiltonian Monte Carlo
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