Correction

This article refers to:

Statistical Inference for Online Decision Making via Stochastic Gradient Descent

Article title: Statistical Inference for Online Decision Making via Stochastic Gradient Descent

Authors: Haoyu Chen,Wenbin Lu, and Rui Song

Journal: Journal of the American Statistical Association

DOI: 10.1080/01621459.2020.1826325

In Section 4, Assumption A5 should be

A5. The features vector X satisfies $\left|\right|X\left|\right|<\infty$. The second moment of reward exists for any given covariates and action, that is,

${\theta }^2(X,A)=\mathbb{E}(Y^2|X,A)<\infty .$

The only change is the removal of the expectation notation $\mathbb{E}$ before $\left|\right|X\left|\right|$ in the first sentence. That is, we would require a bound on all realizations of $\left|\right|X\left|\right|$ instead of the expectation of $\left|\right|X\left|\right|$. The main theorems built upon this assumption and conclusions of the article will remain unchanged.

In the supplement, Section A.3 “Proof of Theorems 4.1 and 4.2”, are modified as follows

Note that ${\Delta }_{s-1}(X)=X^T\{(-1,1)\otimes I_p\}({\overline{\beta }}_{s-1}-{\beta }_0)$ under our model setting. By Assumption A5, there exists a constant $L_X>0$ such that $\left|\right|X\left|\right|<L_X$. By Cauchy-Schwarz inequality

$\int |{\Delta }_{s-1}(X)|d{\mathcal{P}}_X\le \sqrt{2}{L}_X||{\overline{\beta }}_{s-1}-{\beta }_0||.$

Then using Assumption A6 we have,

$J_1\le \frac{1}{\sqrt{t}}{\sum }_{s=1}^t\left[\int I\{0<|\delta (X;{\beta }_0)|\le t^{-\frac{1}{4}}\}|{\Delta }_{s-1}(X)|d{\mathcal{P}}_X\right]\le \sqrt{2}C{L}_X{t}^{-\frac{1}{2}-\frac{\tau }{4}}{\sum }_{s=1}^t\left|\right|{\overline{\beta }}_{s-1}-{\beta }_0\left|\right|.$