Constrained Dikin-Langevin diffusion for polyhedra

TL;DR

The paper addresses constrained sampling and optimization on polyhedral domains by leveraging interior-point (Dikin) geometry. It introduces a barrier-aware Dikin--Langevin diffusion with drift and diffusion premultiplied by the inverse log-barrier Hessian, and proves a boundary no-flux property ensuring feasibility; a discretize-then-correct scheme using Euler--Maruyama proposals and Metropolis--Hastings corrections targets the constrained Boltzmann density $\rho_\beta(x) \propto \mathbf{1}_U(x)\exp(-f(x)/\beta)$. The main contributions are (i) a continuous-time process that preserves the interior and (ii) a practical, MH-corrected algorithm that avoids reflections and exhibits improved convergence diagnostics and cross-well mobility on anisotropic box constraints and multimodal landscapes. The approach yields a robust, reflection-free method for constrained sampling and optimization with strong performance near faces and corners, and the authors outline several extensions to smoother constraints, nonreversible variants, and scalable linear-algebra techniques for large-scale barrier computations.

Abstract

Interior-point geometry offers a straightforward approach to constrained sampling and optimization on polyhedra, eliminating reflections and ad hoc projections. We exploit the Dikin log-barrier to define a Dikin--Langevin diffusion whose drift and noise are modulated by the inverse barrier Hessian. In continuous time, we establish a boundary no-flux property; trajectories started in the interior remain in $U$ almost surely, so feasibility is maintained by construction. For computation, we adopt a discretize-then-correct design: an Euler--Maruyama proposal with state-dependent covariance, followed by a Metropolis--Hastings correction that targets the exact constrained law and reduces to a Dikin random walk when $f$ is constant. Numerically, the unadjusted diffusion exhibits the expected first-order step size bias, while the MH-adjusted variant delivers strong convergence diagnostics on anisotropic, box-constrained Gaussians (rank-normalized split-$\hat{R}$ concentrated near $1$) and higher inter-well transition counts on a bimodal target, indicating superior cross-well mobility. Taken together, these results demonstrate that coupling calibrated stochasticity with interior-point preconditioning provides a practical, reflection-free approach to sampling and optimization over polyhedral domains, offering clear advantages near faces, corners, and in nonconvex landscapes.

Figures (3)

• Figure 1: Discretization bias of the Dikin--Langevin SDE for various time steps, $\textrm{dt}$, when sampling from a 20-dimensional Gaussian distribution truncated to a ball of radius one, centered at the origin. The error is computed as ${|}{E}[\|x\|]-\frac{1}{N}\sum_{n=1}^N\|X_{n}\|{|}$, where $X_n$ are samples obtained after integrating the SDE for $t=0.1n$ using an Euler--Maruyama discretization of the SDE.
• Figure 2: Convergence of the rolling-mean estimator on a log scale. For each algorithm, the black curve is the median of $| {\|x\|}^2_{1:t}-\mu^\ast |$ over $200$ independent runs, where ${\|x\|}_{1:t}^2=t^{-1}\sum_{i=1}^t \|x_i\|^2$ and $\mu^\ast$ is the ground-truth expectation. The red band marks the interdecile range (10th-90th percentiles).
• Figure 3: Distribution of inter-well transitions per chain for two samplers on a bimodal distribution. Out of $200$ independent runs, the bars indicate the number of chains that achieved a given number of transitions over $20\,000$ iterations.

Theorems & Definitions (4)

• Remark 1
• Theorem 1
• proof
• Remark 2