We show a connection between sampling and optimization on discrete domains. For a family of distributions $\mu$ defined on size $k$ subsets of a ground set of elements that is closed under external fields, we show that rapid mixing of natural local random walks implies the existence of simple approximation algorithms to find $\max \mu(\cdot)$. More precisely we show that if (multi-step) down-up random walks have spectral gap at least inverse polynomially large in $k$, then (multi-step) local search can find $\max \mu(\cdot)$ within a factor of $k^{O(k)}$. As the main application of our result, we show a simple nearly-optimal $k^{O(k)}$-factor approximation algorithm for MAP inference on nonsymmetric DPPs. This is the first nontrivial multiplicative approximation for finding the largest size $k$ principal minor of a square (not-necessarily-symmetric) matrix $L$ with $L+L^\intercal\succeq 0$. We establish the connection between sampling and optimization by showing that an exchange inequality, a concept rooted in discrete convex analysis, can be derived from fast mixing of local random walks. We further connect exchange inequalities with composable core-sets for optimization, generalizing recent results on composable core-sets for DPP maximization to arbitrary distributions that satisfy either the strongly Rayleigh property or that have a log-concave generating polynomial.
翻译:在离散域中, 我们显示取样和优化之间的关联。 对于在外部字段中封闭的一组元素的大小为美元=mu$的分布式组合, 外部字段中根据一组基块的大小定义的美元=mu$( cdot), 我们显示, 快速混合自然本地随机行走意味着存在简单的近似算法, 以寻找 $max\ mu( mu( cdot) $) 。 更准确地说, 我们显示, 如果( 多步) 下调随机行道的光谱差异至少以美元为单位, 那么( 多步) 本地的直径搜索可以在一个系数的大小为美元=mu( mus) 范围内找到 $\ mus( mus) 的基数子子组 。 我们展示了一个简单的近距离近似最优化算算法算法, 将本地的极值的极值的极值排序比值比值( ) 。