Matthew Ng Cheng En | Searcharxiv

arXiv2026

Robust SGLD algorithm for solving non-convex distributionally robust optimisation problems

In this paper we develop a Stochastic Gradient Langevin Dynamics (SGLD) algorithm tailored for solving a certain class of non-convex distributionally robust optimisation (DRO) problems. By deriving non-asymptotic convergence bounds, we build an algorithm which for any prescribed accuracy $\varepsilon>0$ outputs an estimator whose expected excess risk is at most $\varepsilon$. As a concrete application, we consider the problem of identifying the best non-linear estimator of a given regression model involving a neural network using adversarially corrupted samples. We formulate this problem as a DRO problem and demonstrate both theoretically and numerically the applicability of the proposed robust SGLD algorithm. Moreover, numerical experiments show that the robust SGLD estimator outperforms the estimator obtained using vanilla SGLD in terms of test accuracy, which highlights the advantage of incorporating model uncertainty when optimising with perturbed samples.

math.OC↗

arXiv2024

Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting

We consider the problem of sampling from a high-dimensional target distribution $π_β$ on $\mathbb{R}^d$ with density proportional to $θ\mapsto e^{-βU(θ)}$ using explicit numerical schemes based on discretising the Langevin stochastic differential equation (SDE). In recent literature, taming has been proposed and studied as a method for ensuring stability of Langevin-based numerical schemes in the case of super-linearly growing drift coefficients for the Langevin SDE. In particular, the Tamed Unadjusted Langevin Algorithm (TULA) was proposed in [Bro+19] to sample from such target distributions with the gradient of the potential $U$ being super-linearly growing. However, theoretical guarantees in Wasserstein distances for Langevin-based algorithms have traditionally been derived assuming strong convexity of the potential $U$. In this paper, we propose a novel taming factor and derive, under a setting with possibly non-convex potential $U$ and super-linearly growing gradient of $U$, non-asymptotic theoretical bounds in Wasserstein-1 and Wasserstein-2 distances between the law of our algorithm, which we name the modified Tamed Unadjusted Langevin Algorithm (mTULA), and the target distribution $π_β$. We obtain respective rates of convergence $\mathcal{O}(λ)$ and $\mathcal{O}(λ^{1/2})$ in Wasserstein-1 and Wasserstein-2 distances for the discretisation error of mTULA in step size $λ$. High-dimensional numerical simulations which support our theoretical findings are presented to showcase the applicability of our algorithm.

math.PR↗