迈向朗之万蒙特卡洛的完整分析：超越庞加莱不等式
Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincaré Inequality
AI总结：
本文在弱庞加莱不等式下推导朗之万扩散与LMC的收敛上下界，明确初始误差对算法性能的三阶段相变依赖，拓展了LMC的收敛性分析边界。
摘要：
朗之万扩散在适当的泛函不等式假设下可快速收敛。因此，人们自然会预期，若引入额外的光滑性条件以处理离散化误差，朗之万蒙特卡洛（Langevin Monte Carlo, LMC）等离散化版本也会以类似方式收敛。这一研究方向由Vempala和Wibisono（2019）开创，他们在对数索伯列夫（log-Sobolev）不等式下建立了相关结果；Chewi等人（2022）将结果拓展至庞加莱（Poincaré）不等式的情形。本文超越庞加莱不等式，将该研究方向推进至极限：我们在弱庞加莱不等式下建立了朗之万扩散与LMC的上下界，该不等式被包括多项式衰减的重尾密度（即柯西型）在内的一大类密度满足。我们的结果明确量化了初始化值对LMC算法性能的影响：具体而言，我们证明当尾部从次高斯、次指数最终变为类柯西时，算法对初始误差的依赖关系从对数级、多项式级最终变为指数级。我们的下界结果表明，这种三阶段相变是不可避免的，清晰界定了LMC的能力边界。
英文摘要：
Langevin diffusions are rapidly convergent under appropriate functional inequality assumptions. Hence, it is natural to expect that with additional smoothness conditions to handle the discretization errors, their discretizations like the Langevin Monte Carlo (LMC) converge in a similar fashion. This research program was initiated by Vempala and Wibisono (2019), who established results under log-Sobolev inequalities. Chewi et al. (2022) extended the results to handle the case of Poincaré inequalities. In this paper, we go beyond Poincaré inequalities, and push this research program to its limit. We do so by establishing upper and lower bounds for Langevin diffusions and LMC under weak Poincaré inequalities that are satisfied by a large class of densities including polynomially-decaying heavy-tailed densities (i.e., Cauchy-type). Our results explicitly quantify the effect of the initializer on the performance of the LMC algorithm. In particular, we show that as the tail goes from sub-Gaussian, to sub-exponential, and finally to Cauchy-like, the dependency on the initial error goes from being logarithmic, to polynomial, and then finally to being exponential. This three-step phase transition is in particular unavoidable as demonstrated by our lower bounds, clearly defining the boundaries of LMC.
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