Chapter 10 · Book pp. 249–271 · August 9, 2026 edition
Structured Sampling
Adapt Langevin analysis to stochastic gradients, coordinate updates, and mirror geometry.
Begin with 10.1 Open this chapter in the canonical August 9 source ↗Chapter route
This chapter develops stochastic gradient, coordinate method, mirror map, relative smoothness. Its main destination is to connect the definitions below to the results that later chapters consume.
Core definitions
- A stochastic-gradient oracle returns an unbiased or controlled-bias gradient estimate with a variance contract.
- Coordinate methods update selected components according to a sampling schedule.
- Mirror Langevin replaces Euclidean geometry by a convex mirror map and its dual coordinates.
Main results
- Stochastic-gradient Langevin bounds separate oracle noise from discretization and mixing error.
- Coordinate samplers exploit anisotropic smoothness through nonuniform update frequencies.
- Mirror Langevin transfers continuous and discrete analysis to non-Euclidean geometry.
- Each structured method exposes an oracle- or geometry-specific complexity bound.
Contents
Why is this chapter route valid?
Analytic contracts
- Stochastic-gradient unbiasedness and variance bounds are conditional statements with a specified filtration or kernel.
- Coordinate schedules, coordinate-dependent step sizes, and anisotropic norms must be measurable and retained in constants.
- Mirror maps need an open effective domain, invertible gradient map, and boundary/nonexplosion control for the transformed diffusion.
- Oracle error, discretization error, and continuous-time convergence remain separate terms.
Open boundaries
- Stochastic-gradient kernel and conditional-moment packet
- Coordinate-update kernel and anisotropic smoothness chain
- Mirror-map domain and transformed-SDE regularity
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.