---
title: Hessian Riemannian Langevin Monte Carlo (HRLMC)
url: https://www.emergentmind.com/topics/hessian-riemannian-langevin-monte-carlo-hrlmc-algorithm
type: topic
---

# Hessian Riemannian Langevin Monte Carlo (HRLMC)

The Hessian Riemannian Langevin Monte Carlo (HRLMC) algorithm is a geometric extension of Langevin Monte Carlo for efficient sampling from high-dimensional probability densities, particularly those lacking global regularity in the standard Euclidean setting. HRLMC leverages a Hessian manifold structure and mirror-descent techniques to model non-flat or constrained geometries, providing non-asymptotic convergence bounds in Wasserstein and Kullback–Leibler distances. This framework unifies Riemannian stochastic differential equations (SDEs), interior-point methods, and optimal transport theory, resolving difficulties in the analysis and computation of Riemannian discretizations by introducing relative smoothness, self-concordance, and explicit metric-dependent contraction rates [2002.04363], [2204.10818].

## 1. Hessian Manifold Structure and Riemannian Metrics

Let $X \subset \mathbb{R}^p$ be open and contractible, and let $\varphi : X \to \mathbb{R}$ be a $C^2$ Legendre-type convex function (often called the "mirror potential" or barrier function). The Hessian manifold is endowed with the metric
$$
g_x(u, v) = u^\top [D^2 \varphi(x)]\, v
$$
where $D^2 \varphi(x)$ is positive definite for all $x \in X$. This construction allows dual coordinates $y = \nabla \varphi(x)$, in which the metric becomes Euclidean and Riemannian quantities push forward via the mirror map. Important classes of $\varphi$ include the Euclidean case $\varphi(x) = \frac{1}{2} \|x\|^2$ and the log-barrier $\varphi(x) = -\sum_i \log(a_i^\top x - b_i)$ for polytope constraints, which yield $g(x) = \nabla^2 \varphi(x)$ [2002.04363], [2204.10818].

## 2. Riemannian Langevin Diffusion: SDE Formulation

Let $\pi$ be a target density $d\pi(x) = e^{-f(x)} dx$, $f \in C^3(X)$. The Riemannian Langevin diffusion (RLD) on $(X, g)$ is the Itô SDE:
$$
dX_t = \Big[\theta(X_t) - D^2\varphi(X_t)^{-1} \nabla f(X_t)\Big]dt + \sqrt{2}\; D^2\varphi(X_t)^{-1/2} dB_t
$$
where
$$
\theta(x) = -D^2\varphi(x)^{-1} \, \mathrm{Tr}[ D^3 \varphi(x) D^2\varphi(x)^{-1} ]
$$
The invariant measure of this diffusion is $\pi$ by Fokker–Planck analysis. In coordinates $y = \nabla\varphi(x)$ the SDE simplifies, and the diffusion is natural Brownian motion on the manifold, with drift $g^{-1}\nabla (\log \nu)$ and diffusion $g^{-1/2}$ [2002.04363], [2204.10818].

## 3. Discrete HRLMC Scheme and Mirror Descent Analogy

The discretization leading to HRLMC is performed in the dual space and mapped back via the Legendre conjugate. With step-size $h_{k+1}$ and i.i.d. $\xi_{k+1} \sim \mathcal{N}(0, I_p)$:
\[
\begin{align*}
\text{Initialize } x_0 \in X \\
y_k = \nabla \varphi(x_k) \\
y_{k+1} = y_k - h_{k+1} \nabla f(x_k) + \sqrt{2 h_{k+1} D^2\varphi(x_k)}\, \xi_{k+1} \\
x_{k+1} = \nabla \varphi^*(y_{k+1})
\end{align*}
\]
HRLMC is the sampling analogue of mirror descent: setting $\xi = 0$ yields the mirror descent step for $f$ with mirror map $\varphi$. The stochastic perturbation ensures the correct stationary distribution $\pi$. In the Euclidean case, HRLMC recovers Unadjusted Langevin Algorithm (ULA) [2002.04363]. For constrained domains (e.g., polytopes), ULA in local charts uses the metric and steps:
\[
x_{k+1} = x_k - \eta\; g(x_k)^{-1} \nabla f(x_k) + \sqrt{2\eta}\; g(x_k)^{-1/2} \xi_k
\]
[2204.10818].

## 4. Assumptions, Metric Properties, and Self-Concordance

Non-asymptotic convergence proofs require the following conditions:
- **A1 (Self-concordance-like control):** $\exists \kappa \geq 0$ such that $\|D^2\varphi(x)^{1/2} - D^2\varphi(x')^{1/2}\|_F \leq \kappa \|\nabla\varphi(x) - \nabla\varphi(x')\|_2$.
- **A2 (Moment bound):** $R = \mathbb{E}_{X \sim \pi}[\|D^2\varphi(X)\|_2] < \infty$.
- **A3 (Relative strong convexity of $f$):** $\exists m \geq 0$ with $m\|\nabla\varphi(x) - \nabla\varphi(x')\|_2^2 \leq \langle\nabla f(x)-\nabla f(x'), \nabla\varphi(x)-\nabla\varphi(x')\rangle$.
- **A4 (Relative gradient Lipschitzness):** $\exists M > 0$ such that $\|\nabla f(x) - \nabla f(x')\|_2 \leq M \|\nabla\varphi(x) - \nabla\varphi(x')\|_2$.
- **A5 (Commutator bound):** $ \|\left[D^{2}\varphi(x)^{-1}, D^2f(x)\right]\|_2 \leq \delta $.

Self-concordance is extended to second-order conditions: bounds on derivatives of the metric and its inverse, controlling the Jacobian growth and ensuring discretization errors remain manageable [2204.10818].

## 5. Non-Asymptotic Convergence Bounds and Wasserstein Geometry

HRLMC delivers explicit, non-asymptotic contraction rates in the Wasserstein ground metric:
\[
W_{2,\varphi}^2(\mu, \nu) = \inf_{X \sim \mu, X' \sim \nu} \mathbb{E} \left[\|\nabla\varphi(X) - \nabla\varphi(X')\|_2^2\right]
\]
With step-size $h_{k+1}$ and contraction factor $\rho_{k+1}$, for all $k$,
\[
W_{2,\varphi}(\mu_{k+1}, \pi) \leq \rho_{k+1}\, W_{2,\varphi}(\mu_k, \pi) + h_{k+1}\, p^{1/2}\, \beta_1(R,\kappa) + h_{k+1}^{3/2} p^{1/2} \beta_2(R,M,\kappa)
\]
where $\tilde{\kappa} = \sqrt{\kappa^2 + \delta (4M+\delta)/2(m+M)}$ and explicit constants $\beta_1, \beta_2$ are supplied as functions of dimension, moment bounds, and regularity parameters. For constant step-size, iterates contract to an explicit Wasserstein ball around $\pi$ and remain inside, with radius proportional to $\kappa$ and local geometry [2002.04363].

For second-order self-concordant metrics, the convergence in Kullback–Leibler divergence is:
\[
H_\nu(\rho_k) \leq e^{-c\,\alpha\,\epsilon\, k} H_\nu(\rho_0) + O(n^{5/2} L_2 + \sqrt{n} L_2^2 + n L_3) \frac{\epsilon}{\alpha}
\]
where $H_\nu$ denotes KL divergence relative to $\nu$ and $L_2$, $L_3$ are Lipschitz constants of $F$ and its Hessian [2204.10818].

## 6. Specializations and Applications

**Euclidean LMC:** Setting $\varphi(x) = \frac{1}{2}\|x\|^2$ yields $g = I$, $\kappa = \delta = 0$, recovers classical LMC with optimal Wasserstein bounds.

**Constrained Sampling:** For polytopes $\mathcal{M} = \{x: Ax \geq b\}$, using the log-barrier $\varphi(x) = -\sum_i \log(a_i^\top x - b_i)$ yields $g(x) = \nabla^2 \varphi(x)$ with self-concordant constants $\gamma_1 = 2, \gamma_2 = 4, \gamma_3 = 6$. HRLMC enables polynomial-time sampling of isoperimetric densities given gradient oracles and avoids global smoothness requirements.

**Other Metrics:** Adapting $\varphi$ to the geometry of $f$ allows sampling from densities in non-Euclidean geometries, e.g., Dirichlet or gamma posteriors on the simplex [2002.04363].

## 7. Key Analytical Tools and Proofs

HRLMC analysis relies on:
- **Extended Baillon–Haddad Inequality:** For relative smoothness, establishes cocoercivity bounds between gradients of $f$ and the mirror map.
- **Itô’s Isometry and Minkowski Integral Inequality:** Manage stochastic error terms and bias in diffusion discretization.
- **Chung’s Lemma:** Demonstrates contraction to a Wasserstein ball under vanishing step-sizes.
- **Bismut–Elworthy–Li Formula:** Controls higher-order terms in stochastic flows.
- **Self-Concordance Theory:** Manages metric derivatives and stability (Nesterov–Nemirovskii).

These tools together establish dimension-dependent rates, validate non-asymptotic guarantees, and ensure robustness to variance in local geometry [2002.04363], [2204.10818].

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HRLMC thus constitutes an advanced geometric Monte Carlo method tailored for high-dimensional, non-Euclidean, and constrained sampling problems, bridging stochastic analysis, convex optimization, and differential geometry. The rigorous contraction rates and metric-dependent bounds enable its application to non-smooth log-concave target distributions and provide theoretical foundations for sampling in complex domains.

Source: https://www.emergentmind.com/topics/hessian-riemannian-langevin-monte-carlo-hrlmc-algorithm