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We study deterministic and randomized midpoint discretizations of Langevin dynamics for a target $\pi \propto e^{-V}$, where $0 \prec \alpha I\preceq\nabla^2V\preceq\beta I$ and $\kappa=\beta/\alpha$. To achieve $\sqrt\alpha\,W_2\leqslant\varepsilon$, we show that deterministic Heun uses at most $\widetilde O(\kappa^{4...
We study the problem of sampling from a continuous density $\pi\propto \exp(-V)$ on $\mathbb R^d$, where $V\in C^2(\mathbb R^d)$ has a $\beta$-Lipschitz gradient and $\pi$ satisfies a logarithmic Sobolev inequality with constant $\alpha^{-1}$, and write $\kappa = \beta/\alpha$. We introduce the Gaussian cloud sampler,...
We prove two dimension-free estimates in Gaussian space. The first is an optimal Meyer-type inequality for the Gaussian divergence: for $p\geq2$, \[ \|\delta V\|_{L^p(\gamma_d)} \leq \sqrt p\,|\mathbb E V| +Cp\,\|DV\|_{L^p(\gamma_d;\mathrm{HS}_d)}\, . \] Here $\mathrm{HS}_d$ is the space of $d\times d$ matrices with it...
We develop a new low-accuracy sampler, called smoothed Picard Hamiltonian Monte Carlo, which combines Gaussian smoothing, Picard iteration, and higher-order discretization. For a log-concave target $\pi \propto \exp(-V)$ in dimension $d$ satisfying $0 \prec \alpha I \preceq \nabla^2 V \preceq \beta I$, with condition n...
The Proximal Bouncy Particle Sampler is a new sampler that combines ideas from the proximal sampler and the bouncy particle sampler that returns a sample whose law is $\varepsilon$-close to $\mu$ in total variation distance using gradient queries in expectation.
Abstract. We establish a log-Sobolev inequality for the stationary distribution of mean-field Langevin dynamics with a constant that is independent of the number of particles [Formula: see text]. Our proof proceeds by establishing the existence of a Lipschitz transport map from the standard Gaussian measure via the re...
This work develops a generative counterpart to the theory of benign overfitting and algorithmic regularization for overparameterized neural networks in the supervised lazy-training regime by studying denoising score matching in a vector-valued reproducing kernel Hilbert space with an inner-product kernel.
We study exact simulation of diffusions via rejection sampling on path space using unbiased estimators of the density ratio obtained from Girsanov's theorem. When applied to the underdamped Langevin diffusion, it yields an algorithm for sampling from a strongly log-concave and log-smooth distribution with condition num...
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