The minimax risk is often considered as a gold standard against which we can
compare specific statistical procedures. Nevertheless, as has been observed
recently in robust and heavy-tailed estimation problems, the inherent reduction
of the (random) loss to its expectation may entail a significant loss of
information regarding its tail behaviour. In an attempt to avoid such a loss,
we introduce the notion of a minimax quantile, and seek to articulate its
dependence on the quantile level. To this end, we develop high-probability
variants of the classical Le Cam and Fano methods, as well as a technique to
convert local minimax risk lower bounds to lower bounds on minimax quantiles.
To illustrate the power of our framework, we deploy our techniques on several
examples, recovering recent results in robust mean estimation and stochastic
convex optimisation, as well as obtaining several new results in covariance
matrix estimation, sparse linear regression, nonparametric density estimation
and isotonic regression. Our overall goal is to argue that minimax quantiles
can provide a finer-grained understanding of the difficulty of statistical
problems, and that, in wide generality, lower bounds on these quantities can be
obtained via user-friendly tools.

最小最大分位数是一种对比特定统计程序的黄金标准，通过引入该概念及其依赖分位水平的表达方式，作者开发了新的方法来研究鲁棒性估计问题，得到了多个重要结果，包括协方差矩阵估计、稀疏线性回归、非参数密度估计、保序回归等。作者的目标是通过最小最大分位数提供对统计问题难度的更仔细理解，并通过用户友好的工具获得这些数量的下界。

高概率极小化下界

High-probability minimax lower bounds

This paper investigates asymptotic behaviors of gradient descent algorithms
(particularly accelerated gradient descent and stochastic gradient descent) in
the context of stochastic optimization arising in statistics and machine
learning where objective functions are estimated from available data. We show
that these algorithms can be computationally modeled by continuous-time
ordinary or stochastic differential equations. We establish gradient flow
central limit theorems to describe the limiting dynamic behaviors of these
computational algorithms and the large-sample performances of the related
statistical procedures, as the number of algorithm iterations and data size
both go to infinity, where the gradient flow central limit theorems are
governed by some linear ordinary or stochastic differential equations like
time-dependent Ornstein-Uhlenbeck processes. We illustrate that our study can
provide a novel unified framework for a joint computational and statistical
asymptotic analysis, where the computational asymptotic analysis studies
dynamic behaviors of these algorithms with the time (or the number of
iterations in the algorithms), the statistical asymptotic analysis investigates
large sample behaviors of the statistical procedures (like estimators and
classifiers) that the algorithms are applied to compute, and in fact the
statistical procedures are equal to the limits of the random sequences
generated from these iterative algorithms as the number of iterations goes to
infinity. The joint analysis results based on the obtained gradient flow
central limit theorems can identify four factors - learning rate, batch size,
gradient covariance, and Hessian - to derive new theory regarding the local
minima found by stochastic gradient descent for solving non-convex optimization
problems.

本研究探讨随机优化中梯度下降算法（尤其是加速梯度下降和随机梯度下降）的渐近行为，并建立了渐近分析的计算和统计统一框架。基于时间依赖奥恩斯坦 - 乌伦贝克过程等建立梯度流中心极限定理，最终识别学习率、批处理大小、梯度协方差和黑塞矩阵等四个因素，以解决非凸优化问题。

梯度下降算法在统计和计算范式中的随机微分方程渐近分析

Asymptotic Analysis via Stochastic Differential Equations of Gradient  Descent Algorithms in Statistical and Computational Paradigms

How should statistical procedures be designed so as to be scalable
computationally to the massive datasets that are increasingly the norm? When
coupled with the requirement that an answer to an inferential question be
delivered within a certain time budget, this question has significant
repercussions for the field of statistics. With the goal of identifying
"time-data tradeoffs," we investigate some of the statistical consequences of
computational perspectives on scability, in particular divide-and-conquer
methodology and hierarchies of convex relaxations.

该论文探讨了统计过程如何设计以在计算能力上可扩展以处理大规模的数据集，重点考虑了分而治之的方法和基于凸松弛的层次结构等算法以及它们对统计学的影响。