论文标题
具有标准分裂方案的高维MCMC，用于阻尼不足的Langevin扩散
High-dimensional MCMC with a standard splitting scheme for the underdamped Langevin diffusion
论文作者
论文摘要
研究了基于阻尼不足的Langevin扩散的Markov采样器的效率，以具有凸和光滑电势的高维靶标。我们考虑了一个经典的二阶集成器，该集成符仅需要一个迭代中的一个梯度计算。与以前在类似采样器上的作品相反，对于离散的时间链本身，证明了瓦斯恒星距离的无维度收缩和总方差距离的收敛速率。对于大都市调整和未调整的链，都获得了非反应的瓦斯汀和总变异效率界限和浓度不平等。 \ nv {特别是对于未调整的链条，就尺寸$ d $和所需的准确性$ \ varepsilon $而言，Wasserstein效率的界限是$ \ sqrt d / \ sqrt d / \ varepsilon $在一般情况下，在一般情况下，$ \ sqrt {d / \ sqrt {d / \ vareps $ $ d^{1/4}/\ sqrt \ varepsilon $在可分离目标的情况下，根据其他动力学Langevin或HMC方案的已知结果。
The efficiency of a Markov sampler based on the underdamped Langevin diffusion is studied for high dimensional targets with convex and smooth potentials. We consider a classical second-order integrator which requires only one gradient computation per iteration. Contrary to previous works on similar samplers, a dimension-free contraction of Wasserstein distances and convergence rate for the total variance distance are proven for the discrete time chain itself. Non-asymptotic Wasserstein and total variation efficiency bounds and concentration inequalities are obtained for both the Metropolis adjusted and unadjusted chains. \nv{In particular, for the unadjusted chain,} in terms of the dimension $d$ and the desired accuracy $\varepsilon$, the Wasserstein efficiency bounds are of order $\sqrt d / \varepsilon$ in the general case, $\sqrt{d/\varepsilon}$ if the Hessian of the potential is Lipschitz, and $d^{1/4}/\sqrt\varepsilon$ in the case of a separable target, in accordance with known results for other kinetic Langevin or HMC schemes.
