# Statistical Inference for Online Decision-Making: In a Contextual Bandit Setting

arXiv: https://arxiv.org/abs/2010.07283 · HTML: https://mlsanity.com/paper/2010.07283/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2020.

## Built on by (most cited first, 17 of 17)

- [Statistical Inference with M-Estimators on Adaptively Collected Data](https://mlsanity.com/paper/2104.14074/) (2021)
- [Anytime-valid off-policy inference for contextual bandits](https://mlsanity.com/paper/2210.10768/) (2022)
- [Online Statistical Inference for Contextual Bandits via Stochastic Gradient Descent](https://mlsanity.com/paper/2212.14883/) (2022)
- [Linear Contextual Bandits with Interference](https://mlsanity.com/paper/2409.15682/) (2024)
- [Optimal Multitask Linear Regression and Contextual Bandits under Sparse Heterogeneity](https://mlsanity.com/paper/2306.06291/) (2023)
- [Contextual Online Uncertainty-Aware Preference Learning for Human Feedback](https://mlsanity.com/paper/2504.19342/) (2025)
- [Doubly Robust Interval Estimation for Optimal Policy Evaluation in Online Learning](https://mlsanity.com/paper/2110.15501/) (2021)
- [Online Estimation and Inference for Robust Policy Evaluation in Reinforcement Learning](https://mlsanity.com/paper/2310.02581/) (2023)
- [Allocation Stability and Wald Inference under Variance-Aware UCB](https://mlsanity.com/paper/2412.08843/) (2024)
- [Online Policy Learning and Inference by Matrix Completion](https://mlsanity.com/paper/2404.17398/) (2024)
- [Asymptotic Time-Uniform Inference for Parameters in Averaged Stochastic Approximation](https://mlsanity.com/paper/2410.15057/) (2024)
- [Sequential Estimation under Multiple Resources: a Bandit Point of View](https://mlsanity.com/paper/2109.14703/) (2021)
- [Incentivized Exploration with Stochastic Covariates: A Two-Stage Mechanism Design for Recommender System](https://mlsanity.com/paper/2406.04374/) (2024)
- [Robust Tests in Online Decision-Making](https://mlsanity.com/paper/2208.09819/) (2022)
- [Kernel ε-Greedy for Multi-Armed Bandits with Covariates](https://mlsanity.com/paper/2306.17329/) (2023)
- [When Mining Electric Locomotives Meet Reinforcement Learning](https://mlsanity.com/paper/2311.08153/) (2023)
- [BOCO: Bayesian Online Contextual Optimization for Decision-Focused Online Learning](https://mlsanity.com/paper/2511.20413/) (2025)

## Abstract (arXiv)

Online decision-making problem requires us to make a sequence of decisions based on incremental information. Common solutions often need to learn a reward model of different actions given the contextual information and then maximize the long-term reward. It is meaningful to know if the posited model is reasonable and how the model performs in the asymptotic sense. We study this problem under the setup of the contextual bandit framework with a linear reward model. The varepsilon-greedy policy is adopted to address the classic exploration-and-exploitation dilemma. Using the martingale central limit theorem, we show that the online ordinary least squares estimator of model parameters is asymptotically normal. When the linear model is misspecified, we propose the online weighted least squares estimator using the inverse propensity score weighting and also establish its asymptotic normality. Based on the properties of the parameter estimators, we further show that the in-sample inverse propensity weighted value estimator is asymptotically normal. We illustrate our results using simulations and an application to a news article recommendation dataset from Yahoo!.