We develop a contraction-based framework for proving mixing-time bounds for Markov chain Monte Carlo algorithms. The framework is built around global and local contraction coefficients of Markov kernels under the $\mathsf E_γ$-divergence with $γ\ge1$. For projected Langevin Monte Carlo on a compact convex domain, we show that Gaussian smoothing yields an explicit global contraction coefficient for the $\mathsf E_γ$-divergence. This gives a direct proof of exponential convergence to the discretized stationary distribution for general smooth, possibly non-convex potentials. The rate is explicit, accommodates arbitrary random-batch sampling schemes, and yields convergence guarantees for several divergences, including KL, $χ^2$, and Rényi divergences. For independent Metropolis--Hastings with target $π$, proposal $q$, and unbounded importance weight $w=dπ/dq$, global contraction coefficients are typically trivial. We therefore introduce a local contraction coefficient on the core $C_R=\{w\le R\}$ and prove that it controls the rejection profile on the core. This yields warm-start convergence bounds governed by the local contraction coefficient and the tail profile $H_R=π(w>R)$, recovering sharp existing moment-based convergence rates when $\mathbb E_q[w^p]<\infty$ for some $p>1$, while remaining effective in heavy-tailed regimes where no finite moment of order $p>1$ exists.

翻译：我们建立了一个基于收缩的框架，用于证明马尔可夫链蒙特卡洛算法的混合时间界。该框架围绕马尔可夫核在$\mathsf E_γ$-散度（其中$γ\ge1$）下的全局与局部收缩系数构建。对于紧凸域上的投影朗之万蒙特卡洛，我们证明高斯平滑化可为$\mathsf E_γ$-散度提供一个显式全局收缩系数。这直接证明了对于一般光滑（可能非凸）势能，离散化平稳分布的指数收敛性。该收敛速率是显式的，适用于任意随机批采样方案，并为KL散度、$χ^2$散度及Rényi散度等多种散度提供了收敛保证。对于具有目标分布$π$、提议分布$q$及无界重要性权重$w=dπ/dq$的独立Metropolis–Hastings算法，全局收缩系数通常平凡。因此，我们在核心集$C_R=\{w\le R\}$上引入局部收缩系数，并证明该系数控制核心集上的拒绝分布。由此得到由局部收缩系数与尾部剖面$H_R=π(w>R)$共同支配的暖启动收敛界；当存在$p>1$使得$\mathbb E_q[w^p]<\infty$时，该方法可恢复已有的尖锐矩收敛率，并在不存在任何阶数$p>1$有限矩的重尾情形下仍保持有效。