标题： 阻尼Langevin MCMC具有三阶收敛性 显示英文标题

标题： Underdamped Langevin MCMC with third order convergence

摘要： 在本文中，我们提出了一种新的数值方法用于欠阻尼朗之万扩散（ULD），并在目标分布$d$维且$p(x)\propto e^{-f(x)}$是强对数凹且具有不同平滑程度的情况下，给出了其采样误差在 2-Wasserstein 距离下的非渐近分析。精确地说，在梯度和 Hessian 的假设下$f$是 Lipschitz 连续的，我们的算法分别在$\mathcal{O}(\sqrt{d}/\varepsilon)$和$\mathcal{O}(\sqrt{d}/\sqrt{\varepsilon})$步骤中达到 2-Wasserstein 误差$\varepsilon$。因此，在匹配假设下，我们的算法与其他流行的朗之万 MCMC 算法具有相似的复杂度。 然而，如果我们进一步假设$f$的三阶导数是 Lipschitz 连续的，那么我们的算法在$\mathcal{O}(\sqrt{d}/\varepsilon^{\frac{1}{3}})$步骤内达到$\varepsilon$的 2-Wasserstein 误差。据我们所知，这是首个具有三阶收敛性的 ULD 梯度仅方法。为了支持我们的理论，我们在一系列真实数据集上进行了贝叶斯逻辑回归，其中我们的算法与现有的欠阻尼 Langevin MCMC 算法和流行的 No U-Turn Sampler (NUTS) 相比表现出有竞争力的性能。 显示英文摘要

摘要： In this paper, we propose a new numerical method for the underdamped Langevin diffusion (ULD) and present a non-asymptotic analysis of its sampling error in the 2-Wasserstein distance when the $d$-dimensional target distribution $p(x)\propto e^{-f(x)}$ is strongly log-concave and has varying degrees of smoothness. Precisely, under the assumptions that the gradient and Hessian of $f$ are Lipschitz continuous, our algorithm achieves a 2-Wasserstein error of $\varepsilon$ in $\mathcal{O}(\sqrt{d}/\varepsilon)$ and $\mathcal{O}(\sqrt{d}/\sqrt{\varepsilon})$ steps respectively. Therefore, our algorithm has a similar complexity as other popular Langevin MCMC algorithms under matching assumptions. However, if we additionally assume that the third derivative of $f$ is Lipschitz continuous, then our algorithm achieves a 2-Wasserstein error of $\varepsilon$ in $\mathcal{O}(\sqrt{d}/\varepsilon^{\frac{1}{3}})$ steps. To the best of our knowledge, this is the first gradient-only method for ULD with third order convergence. To support our theory, we perform Bayesian logistic regression across a range of real-world datasets, where our algorithm achieves competitive performance compared to an existing underdamped Langevin MCMC algorithm and the popular No U-Turn Sampler (NUTS).