DC-LA: Difference-of-Convex Langevin Algorithm
Hoang Phuc Hau Luu, Zhongjian Wang
TL;DR
The paper addresses sampling from a nonconvex, nonsmooth DC-regularized Gibbs distribution $\pi \propto e^{-V}$ with $V=f+(r_1-r_2)$, where $f$ is Lipschitz-smooth and $r_i$ are convex. It proposes DC-LA, a forward-backward Langevin algorithm that uses Moreau envelopes $r_1^{\lambda}$ and $r_2^{\lambda}$ to form an augmented potential $V_{\lambda}$ and a practical update, with an unrolled form linking $Y_k$ and $X_k$ through proximal steps. Theoretical contributions establish convergence in $q$-Wasserstein distance to $\pi$ (up to discretization and smoothing) under distant dissipativity, for both nonsmooth and smooth variants of $r_2$, and for a range of regularity regimes. Empirically, DC-LA yields faithful posterior distributions and meaningful uncertainty quantification in synthetic experiments and in CT reconstruction with data-driven DC priors, outperforming baselines on density fidelity and variance visualization. Overall, the work extends non-log-concave sampling to non-weakly-convex DC regularizers, providing a robust, theoretically-grounded tool for high-dimensional Bayesian inference and imaging applications.
Abstract
We study a sampling problem whose target distribution is $π\propto \exp(-f-r)$ where the data fidelity term $f$ is Lipschitz smooth while the regularizer term $r=r_1-r_2$ is a non-smooth difference-of-convex (DC) function, i.e., $r_1,r_2$ are convex. By leveraging the DC structure of $r$, we can smooth out $r$ by applying Moreau envelopes to $r_1$ and $r_2$ separately. In line of DC programming, we then redistribute the concave part of the regularizer to the data fidelity and study its corresponding proximal Langevin algorithm (termed DC-LA). We establish convergence of DC-LA to the target distribution $π$, up to discretization and smoothing errors, in the $q$-Wasserstein distance for all $q \in \mathbb{N}^*$, under the assumption that $V$ is distant dissipative. Our results improve previous work on non-log-concave sampling in terms of a more general framework and assumptions. Numerical experiments show that DC-LA produces accurate distributions in synthetic settings and reliably provides uncertainty quantification in a real-world Computed Tomography application.