We introduce a framework for obtaining tight mixing times for Markov chains based on what we call restricted modified log-Sobolev inequalities. Modified log-Sobolev inequalities (MLSI) quantify the rate of relative entropy contraction for the Markov operator, and are notoriously difficult to establish. However, infinitesimally close to stationarity, entropy contraction becomes equivalent to variance contraction, a.k.a. a Poincare inequality, which is significantly easier to establish through, e.g., spectral analysis. Motivated by this observation, we study restricted modified log-Sobolev inequalities that guarantee entropy contraction not for all starting distributions, but for those in a large neighborhood of the stationary distribution. We show how to sample from the hardcore and Ising models on $n$-node graphs that have a constant $\delta$ relative gap to the tree-uniqueness threshold, in nearly-linear time $\widetilde O_{\delta}(n)$. Notably, our bound does not depend on the maximum degree $\Delta$, and is therefore optimal even for high-degree graphs. This improves on prior mixing time bounds of $\widetilde O_{\delta, \Delta}(n)$ and $\widetilde O_{\delta}(n^2)$, established via (non-restricted) modified log-Sobolev and Poincare inequalities respectively. We further show that optimal concentration inequalities can still be achieved from the restricted form of modified log-Sobolev inequalities. To establish restricted entropy contraction, we extend the entropic independence framework of Anari, Jain, Koehler, Pham, and Vuong to the setting of distributions that are spectrally independent under a restricted set of external fields. We also develop an orthogonal trick that might be of independent interest: utilizing Bernoulli factories we show how to implement Glauber dynamics updates on high-degree graphs in $O(1)$ time, assuming standard adjacency array representation of the graph.
翻译:我们引入了一个基于我们称之为限制修改的对数-对数不平等的马可夫链获取紧密混合时间的框架。 修改的对数- Sobolev 不平等( MLSI) 将Markov 操作员相对的增缩速度量化, 并且非常难以建立。 然而, 极接近于固定状态, 增缩相当于差异收缩, a.k.a. 。 诗意不平等, 通过( 例如) 光谱分析来建立。 受此观察的驱动, 我们研究限制的对数- Sobolev 不平等, 保证不是所有启动的分布, 而是在固定分布的较大区域 。 然而, 我们展示如何从硬核和Ising模型样本 $- node 图形中保持恒定的 $dela 相对差距, 在几乎线性时间 $loopilde Ocrealtile 时间( n) 。 我们的约束并不取决于最大值 $( Delta) 的递增缩缩缩缩缩, 因此, Ordeal- dealal limaildalalalalalalal drealmograde, rodude rodude.