# Near-optimal inference in adaptive linear regression

arXiv: https://arxiv.org/abs/2107.02266 · HTML: https://mlsanity.com/paper/2107.02266/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 39 references that are in the map.
- First posted on arXiv 5 Jul 2021.

## Builds on (most cited first, 12 of 12)

- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [lil' UCB : An Optimal Exploration Algorithm for Multi-Armed Bandits](https://mlsanity.com/paper/1312.7308/) (2013)
- [Multi-armed Bandit Models for the Optimal Design of Clinical Trials: Benefits and Challenges](https://mlsanity.com/paper/1507.08025/) (2015)
- [Confidence Intervals for Policy Evaluation in Adaptive Experiments](https://mlsanity.com/paper/1911.02768/) (2019)
- [Mixture Martingales Revisited with Applications to Sequential Tests and Confidence Intervals](https://mlsanity.com/paper/1811.11419/) (2018)
- [Inference for Batched Bandits](https://mlsanity.com/paper/2002.03217/) (2020)
- [Statistical Inference with M-Estimators on Adaptively Collected Data](https://mlsanity.com/paper/2104.14074/) (2021)
- [Accurate Inference for Adaptive Linear Models](https://mlsanity.com/paper/1712.06695/) (2017)
- [Why Adaptively Collected Data Have Negative Bias and How to Correct for It](https://mlsanity.com/paper/1708.01977/) (2017)
- [On the bias, risk and consistency of sample means in multi-armed bandits](https://mlsanity.com/paper/1902.00746/) (2019)
- [Are sample means in multi-armed bandits positively or negatively biased?](https://mlsanity.com/paper/1905.11397/) (2019)
- [Online Debiasing for Adaptively Collected High-dimensional Data with Applications to Time Series Analysis](https://mlsanity.com/paper/1911.01040/) (2019)

## Built on by (most cited first, 21 of 21)

- [Anytime-valid off-policy inference for contextual bandits](https://mlsanity.com/paper/2210.10768/) (2022)
- [UCB algorithms for multi-armed bandits: Precise regret and adaptive inference](https://mlsanity.com/paper/2412.06126/) (2024)
- [Inference with the Upper Confidence Bound Algorithm](https://mlsanity.com/paper/2408.04595/) (2024)
- [Off-policy estimation of linear functionals: Non-asymptotic theory for semi-parametric efficiency](https://mlsanity.com/paper/2209.13075/) (2022)
- [Online Statistical Inference for Contextual Bandits via Stochastic Gradient Descent](https://mlsanity.com/paper/2212.14883/) (2022)
- [Stable Thompson Sampling: Valid Inference via Variance Inflation](https://mlsanity.com/paper/2505.23260/) (2025)
- [Semi-parametric inference based on adaptively collected data](https://mlsanity.com/paper/2303.02534/) (2023)
- [Online covariance estimation for stochastic gradient descent under Markovian sampling](https://mlsanity.com/paper/2308.01481/) (2023)
- [Contextual Online Uncertainty-Aware Preference Learning for Human Feedback](https://mlsanity.com/paper/2504.19342/) (2025)
- [Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency](https://mlsanity.com/paper/2301.06240/) (2023)
- [High-dimensional scaling limits and fluctuations of online least-squares SGD with smooth covariance](https://mlsanity.com/paper/2304.00707/) (2023)
- [Adaptive Linear Estimating Equations](https://mlsanity.com/paper/2307.07320/) (2023)
- [Optimal Adaptive Experimental Design for Estimating Treatment Effect](https://mlsanity.com/paper/2410.05552/) (2024)
- [Doubly Robust Interval Estimation for Optimal Policy Evaluation in Online Learning](https://mlsanity.com/paper/2110.15501/) (2021)
- [Adaptive Off-Policy Inference for M-Estimators Under Model Misspecification](https://mlsanity.com/paper/2509.14218/) (2025)
- [Statistical Limits of Adaptive Linear Models: Low-Dimensional Estimation and Inference](https://mlsanity.com/paper/2310.00532/) (2023)
- [Avoiding the Price of Adaptivity: Inference in Linear Contextual Bandits via Stability](https://mlsanity.com/paper/2512.20368/) (2025)
- [SMI: Statistical Membership Inference for Reliable Unlearned Model Auditing](https://mlsanity.com/paper/2602.01150/) (2026)
- [High-dimensional online learning via asynchronous decomposition: Non-divergent results, dynamic regularization, and beyond](https://mlsanity.com/paper/2603.20696/) (2026)
- [Bandit Simulation for Average Reward Inference](https://mlsanity.com/paper/2606.00913/) (2026)
- [Statistical Inference for Misspecified Contextual Bandits](https://mlsanity.com/paper/2606.22639/) (2026)

## Abstract (arXiv)

When data is collected in an adaptive manner, even simple methods like ordinary least squares can exhibit non-normal asymptotic behavior. As an undesirable consequence, hypothesis tests and confidence intervals based on asymptotic normality can lead to erroneous results. We propose a family of online debiasing estimators to correct these distributional anomalies in least squares estimation. Our proposed methods take advantage of the covariance structure present in the dataset and provide sharper estimates in directions for which more information has accrued. We establish an asymptotic normality property for our proposed online debiasing estimators under mild conditions on the data collection process and provide asymptotically exact confidence intervals. We additionally prove a minimax lower bound for the adaptive linear regression problem, thereby providing a baseline by which to compare estimators. There are various conditions under which our proposed estimators achieve the minimax lower bound. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.