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Found 23 papers, 3 papers with code
Statistical optimal transport
We present an introduction to the field of statistical optimal transport, based on lectures given at \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour XLIX.
Sampling from the Mean-Field Stationary Distribution
We study the complexity of sampling from the stationary distribution of a mean-field SDE, or equivalently, the complexity of minimizing a functional over the space of probability measures which includes an interaction term.
Fast parallel sampling under isoperimetry
For our main application, we show how to combine the TV distance guarantees of our algorithms with prior works and obtain RNC sampling-to-counting reductions for families of discrete distribution on the hypercube $\{\pm 1\}^n$ that are closed under exponential tilts and have bounded covariance.
Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space
We develop a theory of finite-dimensional polyhedral subsets over the Wasserstein space and optimization of functionals over them via first-order methods.
Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space
Of key interest in statistics and machine learning is Gaussian VI, which approximates $\pi$ by minimizing the Kullback-Leibler (KL) divergence to $\pi$ over the space of Gaussians.
Query lower bounds for log-concave sampling
Log-concave sampling has witnessed remarkable algorithmic advances in recent years, but the corresponding problem of proving lower bounds for this task has remained elusive, with lower bounds previously known only in dimension one.
Faster high-accuracy log-concave sampling via algorithmic warm starts
Understanding the complexity of sampling from a strongly log-concave and log-smooth distribution $\pi$ on $\mathbb{R}^d$ to high accuracy is a fundamental problem, both from a practical and theoretical standpoint.
Improved Discretization Analysis for Underdamped Langevin Monte Carlo
As a byproduct, we also obtain the first KL divergence guarantees for ULMC without Hessian smoothness under strong log-concavity, which is based on a new result on the log-Sobolev constant along the underdamped Langevin diffusion.
Learning threshold neurons via the "edge of stability"
For these models, we provably establish the edge of stability phenomenon and discover a sharp phase transition for the step size below which the neural network fails to learn "threshold-like" neurons (i. e., neurons with a non-zero first-layer bias).
Fisher information lower bounds for sampling
We prove two lower bounds for the complexity of non-log-concave sampling within the framework of Balasubramanian et al. (2022), who introduced the use of Fisher information (FI) bounds as a notion of approximate first-order stationarity in sampling.
Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions
We provide theoretical convergence guarantees for score-based generative models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which constitute the backbone of large-scale real-world generative models such as DALL$\cdot$E 2.
Variational inference via Wasserstein gradient flows
Along with Markov chain Monte Carlo (MCMC) methods, variational inference (VI) has emerged as a central computational approach to large-scale Bayesian inference.
Improved analysis for a proximal algorithm for sampling
We study the proximal sampler of Lee, Shen, and Tian (2021) and obtain new convergence guarantees under weaker assumptions than strong log-concavity: namely, our results hold for (1) weakly log-concave targets, and (2) targets satisfying isoperimetric assumptions which allow for non-log-concavity.
Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo
For the task of sampling from a density $\pi \propto \exp(-V)$ on $\mathbb{R}^d$, where $V$ is possibly non-convex but $L$-gradient Lipschitz, we prove that averaged Langevin Monte Carlo outputs a sample with $\varepsilon$-relative Fisher information after $O( L^2 d^2/\varepsilon^2)$ iterations.
Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev
Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution $\pi$ under the sole assumption that $\pi$ satisfies a Poincar\'e inequality.
The entropic barrier is $n$-self-concordant
For any convex body $K \subseteq \mathbb R^n$, S. Bubeck and R. Eldan introduced the entropic barrier on $K$ and showed that it is a $(1+o(1)) \, n$-self-concordant barrier.
Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent
We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric.
Rejection sampling from shape-constrained distributions in sublinear time
We consider the task of generating exact samples from a target distribution, known up to normalization, over a finite alphabet.
The query complexity of sampling from strongly log-concave distributions in one dimension
We establish the first tight lower bound of $\Omega(\log\log\kappa)$ on the query complexity of sampling from the class of strongly log-concave and log-smooth distributions with condition number $\kappa$ in one dimension.
Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm
Conventional wisdom in the sampling literature, backed by a popular diffusion scaling limit, suggests that the mixing time of the Metropolis-Adjusted Langevin Algorithm (MALA) scales as $O(d^{1/3})$, where $d$ is the dimension.
Efficient constrained sampling via the mirror-Langevin algorithm
We propose a new discretization of the mirror-Langevin diffusion and give a crisp proof of its convergence.
SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence
Stein Variational Gradient Descent (SVGD), a popular sampling algorithm, is often described as the kernelized gradient flow for the Kullback-Leibler divergence in the geometry of optimal transport.
Exponential ergodicity of mirror-Langevin diffusions
Motivated by the problem of sampling from ill-conditioned log-concave distributions, we give a clean non-asymptotic convergence analysis of mirror-Langevin diffusions as introduced in Zhang et al. (2020).
