Optimal Scaling for the Proximal Langevin Algorithm in High Dimensions
Natesh S. Pillai
TL;DR
This work establishes that proximal MALA, which replaces the gradient of the log-target with a proximal operator, shares the same optimal high-dimensional scaling as standard MALA for a wide class of smooth targets, achieving an asymptotic acceptance probability of $0.574$. By embedding the analysis in an infinite-dimensional Hilbert-space framework with a Gaussian reference measure and convex, differentiable $\Psi$, the authors derive a diffusion limit for the proximal MALA chain with proposal variance scaling $\delta = \ell N^{-1/3}$, and identify the limiting speed function $h(\ell) = \ell \mathbb{E}[1 \wedge e^{Z_\ell}]$ where $Z_\ell \sim N(-\ell^3/4,\ell^3/2)$. The core technique hinges on a Gaussian approximation of the log-acceptance, allowing a drift-martingale decomposition that transfers to a SPDE-driven diffusion in the limit; this yields a principled guideline for using proximal updates when gradients are costly. The results thus justify gradient-free proximal updates as practically efficient alternatives to MALA in high-dimensional settings, including Bayesian inverse problems and data assimilation, while preserving optimal scaling and acceptance properties.
Abstract
The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm that incorporates the gradient of the logarithm of the target density in its proposal distribution. In an earlier joint work \citet{pill:stu:12}, the author had extended the seminal work of \cite{Robe:Rose:98} and showed that in stationarity, MALA applied to an $N-$dimensional approximation of the target will take ${\cal O}(N^{\frac13})$ steps to explore its target measure. It was also shown that the MALA algorithm is optimized at an average acceptance probability of $0.574$. In \citet{pere:16}, the author introduced the proximal MALA algorithm where the gradient of the log target density is replaced by the proximal function. In this paper, we show that for a wide class of twice differentiable target densities, the proximal MALA enjoys the same optimal scaling as that of MALA in high dimensions and also has an average optimal acceptance probability of $0.574$. The results of this paper thus give the following practically useful guideline: for smooth target densities where it is expensive to compute the gradient while implementing MALA, users may replace the gradient with the corresponding proximal function (that can be often computed relatively cheaply via convex optimization) \emph{without} losing any efficiency gains from optimal scaling. This confirms some of the empirical observations made in \cite{pere:16}.