论文标题
迈向非concave采样理论：langevin monte carlo的一阶平稳性保证
Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo
论文作者
论文摘要
对于从$ \ Mathbb {r}^d $上的密度$π\ propto \ exp（-v）$进行取样的任务，其中$ v $可能是非convex，但$ l $ -lob-gradient Lipschitz，我们证明，我们证明，Langevin Monte carlo a Samplos a Sampam a Sampam a Sampam a Sampam a Sampam a Sampam a Sampam a pamplos a Sampam a pampame a pampame a pamplops $ \ varepsil $ -oy（ d^2/\ varepsilon^2）$迭代。这是在非凸优化中找到$ \ varepsilon $ approximate的一阶固定点的复杂性界限的采样类似物，因此构成了迈向非concave采样的一般理论的第一步。我们讨论结果的许多扩展和应用；特别是，它为满足庞加莱不平等的分布提供了新的最新保证。
For the task of sampling from a density $π\propto \exp(-V)$ on $\mathbb{R}^d$, where $V$ is possibly non-convex but $L$-gradient Lipschitz, we prove that averaged Langevin Monte Carlo outputs a sample with $\varepsilon$-relative Fisher information after $O( L^2 d^2/\varepsilon^2)$ iterations. This is the sampling analogue of complexity bounds for finding an $\varepsilon$-approximate first-order stationary points in non-convex optimization and therefore constitutes a first step towards the general theory of non-log-concave sampling. We discuss numerous extensions and applications of our result; in particular, it yields a new state-of-the-art guarantee for sampling from distributions which satisfy a Poincaré inequality.
