See Theorem 2 in Section 3.1 formoredetail.

This perspectiveparallels an earlier phenomenon inthe much better understood field of optimization where convexity has played a preponderant role for both theoretical and methodological advances [Nes04; Bub15].

Given a convex function f: Rd R, the problem of sampling from a distribution e f(x) is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc.

Importance sampling is a Monte Carlo method which designs estimators of expectations under a target distribution using weighted samples from a proposal distribution. When the target distribution is complex, such as multimodal distributions in highdimensional spaces, the efficiency of importance sampling critically depends on the choice of the proposal distribution. In this paper, we propose a novel adaptive scheme for the construction of efficient proposal distributions. Our algorithm promotes efficient exploration of the target distribution by combining global sampling mechanisms with a delayed weighting procedure. The proposed weighting mechanism plays a key role by enabling rapid resampling in regions where the proposal distribution is poorly adapted to the target. Our sampling algorithm is shown to be geometrically convergent under mild assumptions and is illustrated through various numerical experiments.

We establish sample complexity guarantees for estimating the covariance matrix of strongly log-concave smooth distributions using the unadjusted Langevin algorithm (ULA). We quantitatively compare our complexity estimates on single-chain ULA with embarrassingly parallel ULA and derive that the sample complexity of the single-chain approach is smaller than that of embarrassingly parallel ULA by a logarithmic factor in the dimension and the reciprocal of the prescribed precision, with the difference arising from effective bias reduction through burn-in. The key technical contribution is a concentration bound for the sample covariance matrix around its expectation, derived via a log-Sobolev inequality for the joint distribution of ULA iterates.

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The performance of the proposed algorithm is evaluated on a wide variety of machine learning tasks.

Motivated by applications in deep learning, where the global Lipschitz continuity condition is often not satisfied, we examine the problem of sampling from distributions with super-linearly growing log-gradients. We propose a novel tamed Langevin dynamics-based algorithm, called kTULA, to solve the aforementioned sampling problem, and provide a theoretical guarantee for its performance. More precisely, we establish a non-asymptotic convergence bound in Kullback-Leibler (KL) divergence with the best-known rate of convergence equal to $2-\overlineε$, $\overlineε>0$, which significantly improves relevant results in existing literature. This enables us to obtain an improved non-asymptotic error bound in Wasserstein-2 distance, which can be used to further derive a non-asymptotic guarantee for kTULA to solve the associated optimization problems. To illustrate the applicability of kTULA, we apply the proposed algorithm to the problem of sampling from a high-dimensional double-well potential distribution and to an optimization problem involving a neural network. We show that our main results can be used to provide theoretical guarantees for the performance of kTULA.

In this article, we study the problem of sampling from distributions whose densities are not necessarily smooth nor log-concave. We propose a simple Langevin-based algorithm that does not rely on popular but computationally challenging techniques, such as the Moreau Yosida envelope or Gaussian smoothing. We derive non-asymptotic guarantees for the convergence of the algorithm to the target distribution in Wasserstein distances. Non asymptotic bounds are also provided for the performance of the algorithm as an optimizer, specifically for the solution of associated excess risk optimization problems.