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GENERALIZED PROXIMAL LANGEVIN ALGORITHMS VIA BACKWARD DIFFERENTIATION FORMULA
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DOI：10.3934/cpaa.2026047.png)
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Abstract
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Sampling from complex probability distributions is a cornerstone of statistical inference and machine learning. Classical algorithms such as the unadjusted Langevin algorithm (ULA) and the Metropolis-adjusted Langevin algorithm (MALA) provide theoretical mixing guarantees under certain conditions. However, their practical performance often deteriorates in high-dimensional or ill-conditioned settings due to instability and slow convergence. Recent advances-including higher-order discretization (e.g., Runge-Kutta (RK) methods) and the proximal Langevin algorithm (PLA)-have demonstrated potential in mitigating these challenges, offering improved stability and efficiency. Proximal algorithms, in particular, reformulate computationally challenging implicit updates as optimization problems, thereby enabling the leveraging of advances in gradient-based optimization algorithms, e.g., momentum methods. In this paper, we propose a class of higher-order proximal algorithms, referred to as Prox-BDF, which are derived from the proximal formulation of the backward differentiation formula (BDF). These algorithms synergistically combine the accuracy benefits of higher-order temporal discretization with the stability advantages of implicit proximal schemes. Notably, Prox-BDF algorithms exhibit superior stability when applied to stiff or ill-conditioned potentials and achieve higher temporal accuracy than explicit methods such as ULA, existing proximal sampling approaches, and RK-based schemes. Through numerical experiments, we show that Prox-BDF algorithms substantially reduce discretization bias, enable using larger step sizes, and outperform the baseline Langevin methods in terms of both accuracy and robustness.
Keywords:
Sampling
high-order methods
proximal algorithms
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