Sequential data collection has emerged as a widely adopted technique for
enhancing the efficiency of data gathering processes. Despite its advantages,
such data collection mechanism often introduces complexities to the statistical
inference procedure. For instance, the ordinary least squares (OLS) estimator
in an adaptive linear regression model can exhibit non-normal asymptotic
behavior, posing challenges for accurate inference and interpretation. In this
paper, we propose a general method for constructing debiased estimator which
remedies this issue. It makes use of the idea of adaptive linear estimating
equations, and we establish theoretical guarantees of asymptotic normality,
supplemented by discussions on achieving near-optimal asymptotic variance. A
salient feature of our estimator is that in the context of multi-armed bandits,
our estimator retains the non-asymptotic performance of the least square
estimator while obtaining asymptotic normality property. Consequently, this
work helps connect two fruitful paradigms of adaptive inference: a)
non-asymptotic inference using concentration inequalities and b) asymptotic
inference via asymptotic normality.

本文提出了一种构建无偏估计器的普遍方法，使用自适应线性估计方程的思想，并建立渐近正常性的理论保证，以及实现接近最优渐近方差的讨论。

自适应线性估计方程

Adaptive Linear Estimating Equations

This paper is devoted to studying the semi-supervised sparse statistical
inference in a distributed setup. An efficient multi-round distributed debiased
estimator, which integrates both labeled and unlabelled data, is developed. We
will show that the additional unlabeled data helps to improve the statistical
rate of each round of iteration. Our approach offers tailored debiasing methods
for $M$-estimation and generalized linear model according to the specific form
of the loss function. Our method also applies to a non-smooth loss like
absolute deviation loss. Furthermore, our algorithm is computationally
efficient since it requires only one estimation of a high-dimensional inverse
covariance matrix. We demonstrate the effectiveness of our method by presenting
simulation studies and real data applications that highlight the benefits of
incorporating unlabeled data.

本研究旨在研究半监督稀疏统计推断在分布式环境中的应用，提出了一种高效的多轮分布式去偏估计方法，有效地整合了有标记和无标记的数据，并应用于 M-estimation 和广义线性模型等不同损失函数形式。通过模拟研究和真实数据应用，证明了这种方法整合了未标记数据的好处。

分布式半监督稀疏统计推断

Distributed Semi-Supervised Sparse Statistical Inference

The stochastic gradient descent (SGD) algorithm has been widely used in
statistical estimation for large-scale data due to its computational and memory
efficiency. While most existing works focus on the convergence of the objective
function or the error of the obtained solution, we investigate the problem of
statistical inference of true model parameters based on SGD when the population
loss function is strongly convex and satisfies certain smoothness conditions.
Our main contributions are two-fold. First, in the fixed dimension setup, we
propose two consistent estimators of the asymptotic covariance of the average
iterate from SGD: (1) a plug-in estimator, and (2) a batch-means estimator,
which is computationally more efficient and only uses the iterates from SGD.
Both proposed estimators allow us to construct asymptotically exact confidence
intervals and hypothesis tests. Second, for high-dimensional linear regression,
using a variant of the SGD algorithm, we construct a debiased estimator of each
regression coefficient that is asymptotically normal. This gives a one-pass
algorithm for computing both the sparse regression coefficients and confidence
intervals, which is computationally attractive and applicable to online data.

研究了在 SGD 下如何进行统计推断以及使用其构建渐近无偏估计和置信区间，最终提出了一种高维线性回归算法，可以计算稀疏回归系数和置信区间。