langevin dynamics
**Langevin Dynamics** is a stochastic sampling algorithm that generates samples from a target probability distribution p(x) by simulating a continuous-time stochastic differential equation whose stationary distribution equals the target, using only the score function ∇_x log p(x) and injected Gaussian noise. In the discrete-time implementation (Langevin Monte Carlo), iterates follow: x_{t+1} = x_t + (ε/2)·∇_x log p(x_t) + √ε · z_t, where z_t ~ N(0,I) and ε is the step size.
**Why Langevin Dynamics Matters in AI/ML:**
Langevin dynamics provides the **fundamental sampling mechanism** for score-based generative models, converting a learned score function into a practical sample generator through iterative gradient-guided denoising with stochastic perturbation.
• **Score-driven sampling** — The gradient ∇_x log p(x) pushes samples toward high-probability regions while the noise term √ε·z prevents collapse to the mode and ensures the samples eventually cover the full distribution rather than concentrating at a single point
• **Continuous-time SDE** — The continuous formulation dx = (1/2)∇_x log p(x)dt + dW_t (overdamped Langevin equation) has p(x) as its unique stationary distribution; the discrete-time version converges as ε → 0 with corrections for finite step size
• **Annealed Langevin dynamics** — For multi-modal distributions, standard Langevin dynamics mixes slowly between modes; annealing the noise level from large σ₁ to small σ_L uses the corresponding score estimates s_θ(x, σ_l) at each level, enabling mode-hopping at high noise and refinement at low noise
• **Predictor-corrector sampling** — In score-based generative models, Langevin dynamics serves as the "corrector" step that refines samples within each noise level after a "predictor" step that transitions between noise levels, combining numerical ODE/SDE solutions with score-based refinement
• **Underdamped Langevin** — Adding momentum variables (like HMC) creates underdamped Langevin dynamics: dv = -γv dt + ∇_x log p(x)dt + √(2γ)dW; this reduces to HMC in the undamped limit and provides faster mixing than overdamped Langevin
| Parameter | Role | Typical Value |
|-----------|------|---------------|
| Step Size (ε) | Controls update magnitude | 10⁻⁴ to 10⁻² |
| Noise Scale | √ε · N(0,I) | Proportional to √step size |
| Score Function | ∇_x log p(x) | Learned neural network |
| Iterations | Steps to convergence | 100-10,000 |
| Annealing Levels | Noise schedule stages | 10-1000 |
| Convergence | To stationary distribution | As ε→0, iterations→∞ |
**Langevin dynamics is the fundamental bridge between score function estimation and sample generation, providing the iterative, gradient-guided stochastic process that converts learned scores into samples from the target distribution, serving as the core sampling engine for all score-based and diffusion generative models.**