We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via the Poisson equation allowing us to consider a very broad class of subexponentially ergodic processes. These results add to existing concentration inequalities for additive functionals of diffusion processes which have so far been only available for either bounded functions or for unbounded functions of processes from a significantly smaller class. We demonstrate the power of these exponential inequalities by two examples of very different areas. Considering a possibly high-dimensional parametric nonlinear drift model under sparsity constraints, we apply the continuous-time concentration results to validate the restricted eigenvalue condition for Lasso estimation, which is fundamental for the derivation of oracle inequalities. The results for discrete additive functionals are used to investigate the unadjusted Langevin MCMC algorithm for sampling of moderately heavy-tailed densities $\pi$. In particular, we provide PAC bounds for the sample Monte Carlo estimator of integrals $\pi(f)$ for polynomially growing functions $f$ that quantify sufficient sample and step sizes for approximation within a prescribed margin with high probability.
翻译:我们的分析依赖于通过 Poisson 方程式进行的一种方法。 我们的分析依赖于通过 Poisson 方程式进行的一种方法,让我们能够考虑一种非常广泛的亚爆炸性过激过程。 这些结果加剧了扩散过程的添加性功能的现有集中性不平等,迄今为止,只有约束性功能或相当小的等级的无约束性功能才有这种扩散过程的无约束性功能。我们通过两个非常不同的领域的例子来展示这些指数性不平等的力量。考虑到在宽度限制下可能具有高度的参数性非线性漂移模型,我们应用连续时间集中结果来验证Lasso估算的有限机能值条件,这是衍生或末端不平等的根本。离性添加功能的结果被用来调查用于对中度重成型的密度取样的未经调整的兰氏MC算法。我们特别通过两个不同领域的例子来展示这些指数性不平等的威力。我们使用连续时间集中度浓度结果来验证Lasso的限值条件,而Lasso 值对于测算值至关重要,而这种测算结果用于在一定的基度范围内以美元为一定的基数的基数的基数的基数的基度内度。