proximal sampler

At the Columbia workshop last week, Andre Wibisono presented work related with a recent arXival on the exponentially fast convergence of both unadjusted Langevin and proximal sampler algorithms under strong [definitely strong] log-concavity assumptions. The idea behind the proximal sampler is to target the demarginalised density

$g(x,y) \propto \exp\{\log f(x) - ||x-y||^2/2\eta\}\quad\eta>0$

by introducing an auxiliary Gaussian vector y, which preserves f(x) as the marginal distribution on the first component vector X. While the auxiliary Y is (obviously) conditionally Gaussian, the conditional of X is at least as challenging as simulating from f. Unless η is chosen small enough to regularize log g(x) into a strongly log-concave function, since

$\log g(x,y) \le \log g(x^\star,y) -\beta||x-x^\star||^2$

when x*=x*(y) is the maximiser of log g(x,y) (for a given value y) and β>0 is the appropriate log-concavity constant. This inequality means that an accept-reject can be implemented to simulate from the conditional of X given Y but it requires both the factor β and the derivation of x*(y), hence a pretty good understanding and a rather high regularity of the actual target f(x). Besides, the regularization term ||x-y||² means that y is approximately the previous value of the (sub)chain X, hence it creates a rappel force that slows down the exploration of the target.

Since the arXival does not contain numerical comparisons, I attempted one using the (2D) banana shaped distribution,

target=function(x,sig,B,mu)-x[1]^2/2/sig-(x[2]+B*x[1]^2-mu)^2/2

with μ=σ=B=10. Comparing with a vanilla random walk Metropolis with three potential scales, chosen randomly at each iteration. Since I did not want to check whether or not the target was log-concave (and derive the corresponding β), I used the Normal distribution centred at proposal x*(y) of a Metropolis step, again with several scales. The following is the representation of the samples (sienna for MCMC, navy blue for proximal with β=50, dark green for β=5), with a lesser rate of tail exploration for the proximal samplers. It is thus unclear to me the theoretical characterisations of the method translate into practical efficiency beyond the most regular cases.

This entry was posted on April 28, 2025 at 12:25 am and is filed under Books, pictures, R, Statistics, Travel, University life with tags banana, Bayesian optimization, Columbia University, log-concave functions, Metropolis-Hastings algorithm, New York, New York city, proximal sampler, random walk Metropolis algorithm, Statistical learning, ULA, unadjusted Langevin algorithm, workshop. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “proximal sampler”

Radford Neal Says:
May 8, 2025 at 5:16 pm

I’d suggest using a higher-dimensional distribution for this test. In 2D, the best Metropolis proposal distribution will have a large scale (and hence very low acceptance probability), and its behaviour will be not at all typical of higher dimensional distributions (to be precise, distributions in which there are at least three highly-constrained dimensions, rather than one as in your example).

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