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The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithmthat incorporates the gradient of the logarithm of the target density in itsproposal distribution. In an earlier joint work \cite{pill:stu:12}, the authorhad extended the seminal work of \cite{Robe:Rose:98} and showed that instationarity, MALA applied to an $N$-dimensional approximation of the targetwill take ${\cal O}(N^{\frac13})$ steps to explore its target measure. It wasalso shown in \cite{Robe:Rose:98,pill:stu:12} that, as a consequence of thediffusion limit, the MALA algorithm is optimized at an average acceptanceprobability of $0.574$. In \cite{pere:16}, Pereyra introduced the proximal MALAalgorithm where the gradient of the log target density is replaced by theproximal function (mainly aimed at implementing MALA non-differentiable targetdensities). In this paper, we show that for a wide class of twicedifferentiable target densities, the proximal MALA enjoys the same optimalscaling as that of MALA in high dimensions and also has an average optimalacceptance probability of $0.574$. The results of this paper thus give thefollowing practically useful guideline: for smooth target densities where it isexpensive to compute the gradient while implementing MALA, users may replacethe gradient with the corresponding proximal function (that can be oftencomputed relatively cheaply via convex optimization) \emph{without} losing anyefficiency. This confirms some of the empirical observations made in\cite{pere:16}.

Optimal Scaling for the Proximal Langevin Algorithm in High Dimensions