Tweedie deals with conditions under which the convergence of langevin diffusion is exponentially fast, and whether it extends to higher moments.
The Unadjusted Langevin Algorithm (ULA) is a discrete-time Markov chain $\mathbf{U}n$ that approximates the continuous Langevin diffusion $\mathbf{L}t$. It uses a first-order Gaussian approximation to the transition distribution over a grid of size $h$, constructing the update as: $$ \mathbf{U}n \mid \mathbf{U}{n-1} \sim N\left(\mathbf{U}{n-1} + \frac{1}{2} h \nabla \log \pi(\mathbf{U}{n-1}), h I_k\right), $$ as introduced by Parisi (1981).
Under the assumption that $\nabla \log \pi(\mathbf{x})$ is continuous, the ULA chain is $\mu^{\text{Leb}}$-irreducible and weak Feller. By Theorem 15.0.1 of Meyn and Tweedie (1993a), geometric ergodicity can be established if there exists a function $V \geq 1$ such that: $$ \int P(\mathbf{x}, \mathrm{d}\mathbf{y}) V(\mathbf{y}) \leq \lambda V(\mathbf{x}) + b \mathbf{1}_C(\mathbf{x}), $$ for some $\lambda < 1$, $b < \infty$, and a compact set $C$. This condition corresponds to the discrete analogue of the drift condition for the Langevin diffusion.
However, the ULA algorithm has significant limitations. Since it only approximates the true Langevin diffusion, it generally converges to a stationary distribution that differs from the target $\pi$. Consequently, its convergence behavior can vary: The chain may fail to exhibit geometric ergodicity even if the continuous diffusion is exponentially ergodic. In some cases, the ULA chain may even be transient, despite $\mathbf{L}_t$ having a well-behaved invariant distribution.
ULA does not converge reliably, but it is still used sometimes due to the computational simplicity.
Theorem 1 (Non-explosivity and Invariance of $\pi$) Assume, $\nabla \log \pi(\mathbf{x})$ is continuously differentiable and there exist constants $a, b < \infty$ and $N > 0$ such that: $$ \nabla \log \pi(\mathbf{x}) \cdot \mathbf{x} \leq a|\mathbf{x}|^2 + b, \quad |\mathbf{x}| > N. $$
Then, the Langevin diffusion $\mathbf{L}_t$ satisfies:
- It is non-explosive, $\mu^{\text{Leb}}$-irreducible, aperiodic, strong Feller, and all compact sets are small.
- The measure $\pi$ is invariant, and for all $\mathbf{x}$: $$ |P_t^{\mathrm{L}}(\mathbf{x}, \cdot) - \pi| \to 0 \quad \text{as } t \to \infty, $$ where $P_t^{\mathrm{L}}(\mathbf{x}, A)$ is the transition probability of $\mathbf{L}_t$
Non-explosivity is shown by comparing $|\mathbf{L}t|$ with an Ornstein-Uhlenbeck process. Local boundedness of the drift ensures $\mu^{\text{Leb}}$-irreducibility and strong Feller properties (via extensions of Bhattacharya’s results). Aperiodicity follows from irreducibility of the skeleton chain. $\pi$-invariance derives from properties of the generator: $$ \mathscr{A}{\mathrm{L}} f(\mathbf{x}) = \nabla \log \pi(\mathbf{x}) \cdot \nabla f(\mathbf{x}) + \nabla^2 f(\mathbf{x}), $$ for any $f \in C^2$, and the total variation convergence follows from Meyn and Tweedie’s results.
Theorem 2 ($V$-Uniform Ergodicity) Suppose $\mathbf{L}t$ satisfies the conditions of Theorem 1, and there exists a function $V \geq 1$ such that: $$ \mathscr{A}{\mathrm{L}} V(\mathbf{x}) \leq -c V(\mathbf{x}) + b \mathbf{1}_C(\mathbf{x}), $$ where $c, b > 0$ and $C \subset \mathbb{R}^k$ is compact. Then:
- $\mathbf{L}_t$ is $V$-uniformly ergodic: $$ |P_t^{\mathrm{L}}(\mathbf{x}, \cdot) - \pi|_V \leq V(\mathbf{x}) R \rho^t, \quad t \geq 0, $$ for some $R < \infty, \rho < 1$.
- If $\mathbf{L}t$ is exponentially ergodic, then there exist $\kappa > 1, \delta > 0$ such that: $$ \sup{\mathbf{x} \in C} \mathbb{E}\left[\kappa^{\tau_C^\delta}\right] < \infty, $$ where $\tau_C^\delta = \inf{t \geq \delta : \mathbf{L}_t \in C}$.
Use Meyn and Tweedie’s drift condition results to show $V$-uniform ergodicity. Compactness of $C$ and strong Feller properties imply smallness of $C$, ensuring the hitting time result.
Theorem 3 (Sufficient Conditions for Exponential Ergodicity) Suppose $|\pi(\mathbf{x})|$ is bounded for $|\mathbf{x}| \geq S > 0$. A sufficient condition for exponential ergodicity is: $$ \liminf_{|\mathbf{x}| \to \infty} \left[(1-d)|\nabla \log \pi(\mathbf{x})|^2 + \nabla^2 \log \pi(\mathbf{x})\right] > 0, $$ for some $0 < d < 1$.
Choose $V = \pi^{-d}$ as a Lyapunov function. Verify the drift condition $\mathscr{A}_{\mathrm{L}} V \leq -c V + b \mathbf{1}_C$ under the given assumptions. Apply Meyn and Tweedie’s criteria for geometric ergodicity.