Online Instrumental Variable Regression: Regret Analysis and Bandit Feedback

The independence of noise and covariates is a standard assumption in online linear regression and linear bandit literature. This assumption and the following analysis are invalid in the case of endogeneity, i.e., when the noise and covariates are correlated. In this paper, we study the online setting of instrumental variable (IV) regression, which is widely used in economics to tackle endogeneity. Specifically, we analyse and upper bound regret of Two-Stage Least Squares (2SLS) approach to IV regression in the online setting. Our analysis shows that Online 2SLS (O2SLS) achieves $O(d^2 \log^2 T)$ regret after $T$ interactions, where d is the dimension of covariates. Following that, we leverage the O2SLS as an oracle to design OFUL-IV, a linear bandit algorithm. OFUL-IV can tackle endogeneity and achieves $O(d \sqrt{T} \log T)$ regret. For datasets with endogeneity, we experimentally demonstrate that O2SLS and OFUL-IV incur lower regrets than the state-of-the-art algorithms for both the online linear regression and linear bandit settings.