Unadjusted Langevin algorithm (ULA)

Assume AC (The Axiom of Choice). Let $d\ge 1$, let $U:{\mathbb{R}}^d\to \mathbb{R}$ be Borel measurable in the sense of Borel measurable and Lebesgue measurable functions on ${\mathbb{R}}^n$ and continuously differentiable with $0<Z_U:={\int }_{{\mathbb{R}}^d}{e}^{-U(x)}dx<\mathrm{\infty },$ and define the target probability ${\pi }_U(dx):=Z_U^{-1}{e}^{-U(x)}dx\mathrm{.}$ This is the measure with density $Z_U^{-1}{e}^{-U}$ relative to Lebesgue measure (The measure with density $f$ relative to $\mu$ and Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$); the probability-measure convention is Probability measures and probability spaces, and the normalizer makes its total mass one. Given a step size $\gamma >0$, an initial state $X_0\in {\mathbb{R}}^d$, and an IID sequence $Z_1,Z_2,\dots \sim N_d(0,I_d)$, using the multivariate-normal law of Multivariate normal law, including singular covariance and standard-normal convention of Standard normal and normal laws, repeat for $k\ge 0$:

1. Compute the Euclidean gradient $\mathrm{\nabla }U(X_k)$ using the convention of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case.
2. Take the fixed-step gradient update ${\tilde{X}}_{k+1}=X_k-\gamma \mathrm{\nabla }U(X_k)$, the fixed-step branch of Gradient descent.
3. Set $X_{k+1}={\tilde{X}}_{k+1}+\sqrt{2\gamma }{Z}_{k+1}\mathrm{.}$

The one-step transition law is $Q_{\gamma }(x,\cdot )=N_d(x-\gamma \mathrm{\nabla }U(x),2\gamma I_d)\mathrm{.}$ The procedure applies the proposal at every step and has no accept/reject correction. The probability-kernel and density properties of $Q_{\gamma }$ are proved in The ULA Gaussian proposal is a probability kernel with a positive density ↗.

The definition only asserts a well-defined Gaussian update. It asserts no convergence rate, invariant-law existence, or equality of the invariant law with ${\pi }_U$; those are separate mathematical claims.