The Langevin Algorithm in the Non Smooth Log-Concave Case

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Proves quantitative convergence for the unadjusted Langevin discretization on log‑concave but possibly non‑smooth potentials, showing explicit step‑size and step‑count prescriptions: take δ of order ε^2 and run O(ε^{-2}) steps (with polynomial dependence on curvature/regularity constants) to reach ε accuracy in standard metrics. Analysis uses smoothing via Moreau envelopes, Girsanov/change‑of‑measure and functional inequalities, compares total variation vs relative entropy through Pinsker inequality, and invokes log‑Sobolev/Poincaré when available.

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1. 0:00Langevin algorithms basics
2. 4:24Langevin diffusion basics
3. 10:05Pinsker inequality bounds
4. 17:01Convergence under log-concavity
5. 24:19Sobolev inequalities role
6. 31:13Wasserstein distance applications
7. 38:04Convex bodies techniques
8. 44:14Projection methods
9. 50:10Convexity and diffusion
10. 55:27Summary and open questions