We study the problem of sampling from a distribution $p^*(x) \propto
\exp\left(-U(x)\right)$, where the function $U$ is $L$-smooth everywhere and
$m$-strongly convex outside a ball of radius $R$, but potentially nonconvex
inside this ball. We study both overdamped and underdamped Langevin MCMC and
establish upper bounds on the number of steps required to obtain a sample from
a distribution that is within $\epsilon$ of $p^*$ in $1$-Wasserstein distance.
For the first-order method (overdamped Langevin MCMC), the iteration complexity
is $\tilde{\mathcal{O}}\left(e^{cLR^2}d/\epsilon^2\right)$, where $d$ is the
dimension of the underlying space. For the second-order method (underdamped
Langevin MCMC), the iteration complexity is
$\tilde{\mathcal{O}}\left(e^{cLR^2}\sqrt{d}/\epsilon\right)$ for an explicit
positive constant $c$. Surprisingly, the iteration complexity for both these
algorithms is only polynomial in the dimension $d$ and the target accuracy
$\epsilon$. It is exponential, however, in the problem parameter $LR^2$, which
is a measure of non-log-concavity of the target distribution.

本研究研究了在采样中采用了过阻尼和欠阻尼 Langevin MCMC，证明了算法的迭代复杂度在维度和目标准确度方面均是多项式级别的，但在问题参数 LR ^ 2 中是指数级别的，从而可以更好地进行非凸优化。

非凸背景下 Langevin 动力学算法的尖锐收敛速率

Sharp convergence rates for Langevin dynamics in the nonconvex setting

Given samples from an unknown distribution $p$, is it possible to distinguish
whether $p$ belongs to some class of distributions $\mathcal{C}$ versus $p$
being far from every distribution in $\mathcal{C}$? This fundamental question
has received tremendous attention in statistics, focusing primarily on
asymptotic analysis, and more recently in information theory and theoretical
computer science, where the emphasis has been on small sample size and
computational complexity. Nevertheless, even for basic properties of
distributions such as monotonicity, log-concavity, unimodality, independence,
and monotone-hazard rate, the optimal sample complexity is unknown.
We provide a general approach via which we obtain sample-optimal and
computationally efficient testers for all these distribution families. At the
core of our approach is an algorithm which solves the following problem: Given
samples from an unknown distribution $p$, and a known distribution $q$, are $p$
and $q$ close in $\chi^2$-distance, or far in total variation distance?
The optimality of our testers is established by providing matching lower
bounds with respect to both $n$ and $\varepsilon$. Finally, a necessary
building block for our testers and an important byproduct of our work are the
first known computationally efficient proper learners for discrete log-concave
and monotone hazard rate distributions.

研究如何通过样本来鉴别未知分布是否属于某个分布类，提出了一种可获得最优样本和计算效率的通用方法，并为分布的基本属性提供了测试方法。

分布属性的最佳测试

Optimal Testing for Properties of Distributions

We review and formulate results concerning log-concavity and
strong-log-concavity in both discrete and continuous settings. We show how
preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under
convolution follows from a fundamental monotonicity result of Efron (1969). We
provide a new proof of Efron's theorem using the recent asymmetric
Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review
connections between log-concavity and other areas of mathematics and
statistics, including concentration of measure, log-Sobolev inequalities,
convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

该论文回顾和阐述了离散和连续设置中的对数凹性和强对数凹性问题，探讨了 Efron 定理和引理，证明了 Otto 和 Menz 提出的不对称 Brascamp-Lieb 不等式的基本单调性结论，同时回顾了对数凹性与其他数学和统计学领域的关系，包括测度集中，对数 Sobolev 不等式，凸几何，MCMC 算法，Laplace 近似和机器学习。

对数凹性和强对数凹性综述

Log-concavity and strong log-concavity: a review

In this paper, we present a comprehensive study of the monotonicity and
log-concavity of the generalized Marcum and Nuttall Q-functions. More
precisely, a simple probabilistic method is firstly given to prove the
monotonicity of these two functions. Then, the log-concavity of the generalized
Marcum Q-function and its deformations is established with respect to each of
the three parameters. Since the Nuttall Q-function has similar probabilistic
interpretations as the generalized Marcum Q-function, we deduce the
log-concavity of the Nuttall Q-function. By exploiting the log-concavity of
these two functions, we propose new tight lower and upper bounds for the
generalized Marcum and Nuttall Q-functions. Our proposed bounds are much
tighter than the existing bounds in the literature in most of the cases. The
relative errors of our proposed bounds converge to 0 as b tends to infinity.
The numerical results show that the absolute relative errors of the proposed
bounds are less than 5% in most of the cases. The proposed bounds can be
effectively applied to the outage probability analysis of interference-limited
systems such as cognitive radio and wireless sensor network, in the study of
error performance of various wireless communication systems operating over
fading channels and extracting the log-likelihood ratio for differential
phase-shift keying (DPSK) signals.

本文研究了 Marcum Q 函数和 Nuttall Q 函数的单调性和对数凸性，通过概率方法证明了这两个函数的单调性并建立了其对数凸性，利用这些函数的对数凸性，提出了新的上下界，这些界可用于干扰限制系统的服务质量分析和无线通信系统的误码性能研究以及差分相移键控信号的对数似然比提取。