Log-Concave Sampling

We establish near-linear accuracy bounds for the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA). The target is $π\propto e^{-f-g}$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with Lipschitz gradient and $g$ is convex and globally Lipschitz. Under an explicit parameter-dependent step-size condition, we bound the invariant-measure bias relative to the Moreau-smoothed target by $\widetilde O(h)$, with only logarithmic dependence on the inverse smoothing parameter in the error coefficient. Combining this estimate with the Moreau approximation bias and Wasserstein contraction gives $\widetilde O(\varepsilon^{-1})$ iterations to make the $N$th-iterate law $μ_N$ satisfy $\sqrt m\,W_2(μ_N,π)\le\varepsilon$, for fixed model parameters and initialization. We bound the stationary error directly, without assuming third derivatives or a Lipschitz Hessian. Each iteration uses one gradient evaluation and one exact proximal evaluation. The key idea in our analysis is to convert a second-order stationary residual into a Wasserstein bound using a Poisson-based estimate.

We study the convergence of Wasserstein-Fisher-Rao (WFR) gradient flows for sampling from probability distributions known up to a normalisation constant. By combining Wasserstein transport with Fisher-Rao birth-death dynamics, WFR flows balance exploration and selection. These flows have been recognised as a promising mechanism to accelerate convergence beyond Langevin dynamics. We show that for a class of strongly log-concave target distributions satisfying additional curvature conditions, WFR flows preserve strong log-concavity, in contrast to Wasserstein flows which enjoy this property only in the Gaussian setting. Exploiting this result, we derive explicit non-asymptotic convergence rates for the symmetrised Kullback-Leibler divergence, without requiring a warm-start as required in current estimates. In particular, we show that the convergence rate decomposes additively into Wasserstein and Fisher-Rao contributions, thereby confirming a recent conjecture within this setting. These results provide refined convergence guarantees and further develop the theoretical foundations of WFR gradient flows for sampling and Bayesian inference.

We show that logconcave probability measures along the Gaussian cooling path have thin-shell stability, generalizing the thin-shell theorem. This result leads to improved complexity for the fundamental problem of sampling an arbitrary logconcave distribution from a cold start. For (near-)isotropic logconcave distributions, the complexity is nearly $n^{2.5}$, improving the previous bound of $n^{2.75}$, and matching the complexity of the abstract Speedy walk.

We investigate the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in any fixed Euclidean dimension. The potential is $\mu$-strongly convex and $L$-smooth, with an unknown mode in the ball of radius $\mu^{-1/2}$ about the origin. We have access to unbiased stochastic oracles with the variance at most $\sigma^2$. For every $\sigma^2\ge0$ and total variation (TV) accuracy $0<\varepsilon\le1/10$, we prove that the tight complexity of sampling a distribution within $\epsilon$-TV distance from the target distribution is $$N^\star_{\text{TV}}=\Theta\!\left(\log(1+\kappa)+ \frac{\sigma^2}{\mu\epsilon}\right),$$ where $\kappa:=\frac L\mu$ is the condition number. Note that this complexity bound is simultaneously tight for the condition number $\kappa$ and accuracy $\epsilon$. Besides, our tight complexity bound is adaptive to noiseless setting $\sigma=0$, which is $N^\star_{\text{TV}}=\Theta\!\left(\log(1+\kappa)\right)$.

We study the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA) for $\pi(\,\mathrm{d} x)\propto e^{-f(x)-g(x)}\,\mathrm{d} x$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with $L_f$-Lipschitz gradient and $g:\mathbb{R}^d\to\mathbb{R}$ is convex and globally $G$-Lipschitz. For the Moreau-smoothed target $\pi_\lambda$ and the MYULA invariant law $\widehat\pi_{\lambda,h}$, we prove $$\sqrt m\,W_2(\pi_\lambda,\widehat\pi_{\lambda,h}) =O(h)+\widetilde O(h^{3/4})$$ under $0<h(L_f+\lambda^{-1})\le c$, with only logarithmic dependence on $\lambda^{-1}$ in the error coefficients. Combining this estimate with the Moreau approximation bias yields $\widetilde O(\varepsilon^{-4/3})$ iterations to achieve $\sqrt m\,W_2(\mu_N,\pi)\le\varepsilon$, for fixed model parameters and initialization. The proof combines a discrete Poisson corrector with active-trace estimates and a shared-noise bound for the exact--Euler two-point curvature.

We provide new estimates in Wasserstein distance for the asymptotic bias of the unadjusted Langevin algorithm, in the classical setting of log-smooth strongly log-concave measures. Our bound implies a Wasserstein mixing time of order $κ\sqrt{d}/\varepsilon$, where $κ$ is the condition number, $d$ is the dimension, and $\varepsilon$ is the target precision: this improves by a factor of $\sqrt{d}/\varepsilon$ over the previous state-of-the-art results.

We show the Randomized Hamiltonian Monte Carlo (RHMC) algorithm has accelerated mixing time guarantees for sampling from log-concave probability distributions. RHMC proceeds by repeatedly simulating the continuous-time Hamiltonian dynamics for some random integration times, and resetting the velocity to be an independent Gaussian random variable between each simulation. We show that when the target distribution is log-concave and satisfies an $α$-Talagrand inequality (for example, if the target distribution is $α$-strongly log-concave), if we use a random integration time from either the triangular or the exponential distribution with mean $Θ(α^{-1/2})$, then RHMC converges exponentially fast in KL divergence, and the total integration time to reach error $\varepsilon$ in KL divergence scales as $O(α^{-1/2} \log(\varepsilon^{-1}))$. We also show that when the target distribution is log-concave, if we use a sequence of random integration times from the triangular distribution with exponentially increasing means, then the total integration time to reach error $\varepsilon$ in KL divergence scales as $O(\varepsilon^{-1/2})$. Our analysis relies on a bound on the average KL divergence along Hamiltonian dynamics, which is inspired by an analogous result on accelerated optimization methods based on Hamiltonian dynamics.

We give a simple, unified, and nearly tight bound for sampling arbitrary logconcave distributions from a warm start using the In-and-Out algorithm along with exponential lifting. The main new ingredient in the analysis is an improved bound on the Poincaré constant of a lifted distribution. As a consequence, the resulting convergence rate is nearly tight for both constrained settings (e.g., Gaussian restricted to a convex body) and well-conditioned settings (e.g., strongly logconcave and smooth densities).

We establish improved nonasymptotic bounds for Langevin Monte Carlo in the strongly log-concave setting, when the error is measured by the Wasserstein distance. The main result shows that the discretization error is governed by an average coordinate-wise smoothness constant, rather than by the usual global smoothness constant. The proof is short and probabilistic, and relies on a refined use of the synchronous coupling. We further show that the same ideas lead to improved bounds for variable step sizes, for potentials whose Laplacian is Lipschitz-continuous, and for finite-sum problems sampled by stochastic-gradient Langevin dynamics with fixed point control variates. In the Laplacian-smooth case, the usual Hessian-Lipschitz contribution is replaced by a weaker trace-type third-order smoothness quantity. In the finite-sum setting, the resulting SGLD bound improves the dependence on the root mean square smoothness of the component functions. Applications to generalized linear models with Gaussian design show that these refinements can yield substantial, dimension-dependent improvements over previously known bounds, especially for correlated covariates.

We study the query complexity of obtaining a relative Fisher information guarantee for sampling from a log-smooth non-log-concave distribution; this is a sampling analog of finding an approximate stationary point in optimization. Our algorithm is based on the proximal sampler, which is an implicit discretization of the Langevin diffusion, and requires an implementation of the backward step known as the restricted Gaussian oracle (RGO). We show that by leveraging the recent results for log-concave sampling with high-accuracy guarantees in Rényi divergence, we can obtain an approximate RGO implementation that -- when used with the proximal sampler -- yields a complexity guarantee in relative Fisher information that inherits the same dimension dependence as log-concave sampling, and improves upon prior work for non-log-concave sampling. We also show a converse reduction that any improvement in the dimension dependence in relative Fisher information for non-log-concave sampling will yield an improved dimension dependence for high-accuracy log-concave sampling.

We propose an algorithm to sample from composite log-concave distributions over $\mathbb{R}^d$, i.e., densities of the form $π\propto e^{-f-g}$, assuming access to gradient evaluations of $f$ and a restricted Gaussian oracle (RGO) for $g$. The latter requirement means that we can easily sample from the density $\text{RGO}_{g,h,y}(x) \propto \exp(-g(x) -\frac{1}{2h}||y-x||^2)$, which is the sampling analogue of the proximal operator for $g$. If $f + g$ is $α$-strongly convex and $f$ is $β$-smooth, our sampler achieves $\varepsilon$ error in total variation distance in $\widetilde{\mathcal O}(κ\sqrt d \log^4(1/\varepsilon))$ iterations where $κ:= β/α$, which matches prior state-of-the-art results for the case $g=0$. We further extend our results to cases where (1) $π$ is non-log-concave but satisfies a Poincaré or log-Sobolev inequality, and (2) $f$ is non-smooth but Lipschitz.

We propose Decentralized Proximal Stochastic Gradient Langevin Dynamics (DE-PSGLD), a decentralized Markov chain Monte Carlo (MCMC) algorithm for sampling from a log-concave probability distribution constrained to a convex domain. Constraints are enforced through a shared proximal regularization based on the Moreau-Yosida envelope, enabling unconstrained updates while preserving consistency with the target constrained posterior. We establish non-asymptotic convergence guarantees in the 2-Wasserstein distance for both individual agent iterates and their network averages. Our analysis shows that DE-PSGLD converges to a regularized Gibbs distribution and quantifies the bias introduced by the proximal approximation. We evaluate DE-PSGLD for different sampling problems on synthetic and real datasets. As the first decentralized approach for constrained domains, our algorithm exhibits fast posterior concentration and high predictive accuracy.

Sampling based on score diffusions has led to striking empirical results, and has attracted considerable attention from various research communities. It depends on availability of (approximate) Stein score functions for various levels of additive noise. We show how in some generality, the availability of scores allows the general problem to be ``reduced'' to sampling from an adaptively constructed sequence of $K$ strongly log-concave (SLC) sub-problems. The reduction is simple, constructive and algorithm-independent, so that any SLC sampler can be used as a subroutine. Various bounds on score-based sampling complexity follow directly: for instance, high-accuracy SLC samplers yield $\tilde{\mathcal{O}}(K \sqrt{d} \operatorname{polylog}(1/\varepsilon))$ guarantees for accuracy $\varepsilon$ in dimension $d$, where randomized midpoint SLC schemes yield $\tilde{\mathcal{O}}(K d^{1/3} \operatorname{poly}(1/\varepsilon))$ guarantees. When the original distribution itself is SLC, we prove that $K \leq 1 + \log_2(κ)$, thereby obtaining the first efficient procedure with logarithmic dependence on condition number $κ$; for general distributions, the quantity $K$ depends on the geometry of score Hessian across the trajectory. Our analysis is direct and simple, involving techniques and insights complementary to those in standard analyses of discretized diffusions.