# Optimal Scaling for the Proximal Langevin Algorithm in High Dimensions

arXiv: https://arxiv.org/abs/2204.10793 · HTML: https://mlsanity.com/paper/2204.10793/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 20 references that are in the map.
- First posted on arXiv 21 Apr 2022.

## Builds on (most cited first, 2 of 2)

- [Proximal Markov chain Monte Carlo algorithms](https://mlsanity.com/paper/1306.0187/) (2013)
- [Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning](https://mlsanity.com/paper/2012.12137/) (2020)

## Built on by (most cited first, 2 of 2)

- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [Optimal scaling of MCMC algorithms: the Hamiltonian approach](https://mlsanity.com/paper/2607.00586/) (2026)

## Abstract (arXiv)

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 citetpill:stu:12, the author had extended the seminal work of citeRobe:Rose:98 and showed that in stationarity, MALA applied to an N-dimensional approximation of the target will take cal O(Nfrac13) 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 citetpere: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) without losing any efficiency gains from optimal scaling. This confirms some of the empirical observations made in citepere:16.