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We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with log-concave and smooth densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of $d$-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most $varepsilon > 0$ in Wasserstein distance from the target distribution in $Oleft(frac{d^{1/4}}{ varepsilon^{1/2}} right)$ steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with $alpha$-th order smoothness, we prove that the mixing time scales as $O left(frac{d^{1/4}}{varepsilon^{1/2}} + frac{d^{1/2}}{varepsilon^{1/(alpha - 1)}} right)$.
We consider the problem of sampling from a density of the form $p(x) propto exp(-f(x)- g(x))$, where $f: mathbb{R}^d rightarrow mathbb{R}$ is a smooth and strongly convex function and $g: mathbb{R}^d rightarrow mathbb{R}$ is a convex and Lipschitz fu
We study the problem of sampling from the power posterior distribution in Bayesian Gaussian mixture models, a robust version of the classical posterior. This power posterior is known to be non-log-concave and multi-modal, which leads to exponential m
What is the information leakage of an iterative learning algorithm about its training data, when the internal state of the algorithm is emph{not} observable? How much is the contribution of each specific training epoch to the final leakage? We study
A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and in Wasserstein distance is presented. All these are obtained in the context of sampling from a target distribution $pi$ that has a density $hat{pi}$ on
We give an approximation algorithm for MaxCut and provide guarantees on the average fraction of edges cut on $d$-regular graphs of girth $geq 2k$. For every $d geq 3$ and $k geq 4$, our approximation guarantees are better than those of all other clas