We consider the constrained sampling problem where the goal is to sample from a distribution $\pi(x)\propto e^{-f(x)}$ and $x$ is constrained on a convex body $\mathcal{C}\subset \mathbb{R}^d$. Motivated by penalty methods from optimization, we propose penalized Langevin Dynamics (PLD) and penalized Hamiltonian Monte Carlo (PHMC) that convert the constrained sampling problem into an unconstrained one by introducing a penalty function for constraint violations. When $f$ is smooth and the gradient is available, we show $\tilde{\mathcal{O}}(d/\varepsilon^{10})$ iteration complexity for PLD to sample the target up to an $\varepsilon$-error where the error is measured in terms of the total variation distance and $\tilde{\mathcal{O}}(\cdot)$ hides some logarithmic factors. For PHMC, we improve this result to $\tilde{\mathcal{O}}(\sqrt{d}/\varepsilon^{7})$ when the Hessian of $f$ is Lipschitz and the boundary of $\mathcal{C}$ is sufficiently smooth. To our knowledge, these are the first convergence rate results for Hamiltonian Monte Carlo methods in the constrained sampling setting that can handle non-convex $f$ and can provide guarantees with the best dimension dependency among existing methods with deterministic gradients. We then consider the setting where unbiased stochastic gradients are available. We propose PSGLD and PSGHMC that can handle stochastic gradients without Metropolis-Hasting correction steps. When $f$ is strongly convex and smooth, we obtain an iteration complexity of $\tilde{\mathcal{O}}(d/\varepsilon^{18})$ and $\tilde{\mathcal{O}}(d\sqrt{d}/\varepsilon^{39})$ respectively in the 2-Wasserstein distance. For the more general case, when $f$ is smooth and non-convex, we also provide finite-time performance bounds and iteration complexity results. Finally, we test our algorithms on Bayesian LASSO regression and Bayesian constrained deep learning problems.

翻译：我们考虑限制的取样问题, 目标来自分配 $\ pi( x)\ proto e\ { f( x)} $ 和 $x 的样本, 在一个螺旋体上 $\ mathcal{ C\ subb{ R\ d$。 受到来自优化的惩罚方法的驱动, 我们提议惩罚 Langevin Dynalive (PLD) 和 Hamilton Monte Carlo (PHMC), 将限制的取样问题转换成一个不受约束的样本, 通过引入限制违反的处罚功能。 当美元平滑和梯度可用时, 我们展示的是 美元 美元 美元 美元 = mathcalalal { O} (d/ v) 美元 美元 的样本复杂性, PLD 来将目标采样到 $\ valeplational $ 美元 美元, 当我们用整个变差距离来测量错误时, 能够隐藏一些对调调值 。 对于 PHMIC, 我们改进的结果是 美元 美元= 美元 美元 lisax ral deal deal deal ral ral ral ration 和 ration 。