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Endogeneity, i.e. the dependence between noise and covariates, is a commonphenomenon in real data due to omitted variables, strategic behaviours,measurement errors etc. In contrast, the existing analyses of stochastic onlinelinear regression with unbounded noise and linear bandits depend heavily onexogeneity, i.e. the independence between noise and covariates. Motivated bythis gap, we study the over-and just-identified Instrumental Variable (IV)regression for stochastic online learning. IV regression and the Two-StageLeast Squares approach to it are widely deployed in economics and causalinference to identify the underlying model from an endogenous dataset. Thus, wepropose to use an online variant of Two-Stage Least Squares approach, namelyO2SLS, to tackle endogeneity in stochastic online learning. Our analysis showsthat O2SLS achieves $\mathcal{O}\left(d_x d_z \log ^2 T\right)$ identificationand $\tilde{\mathcal{O}}\left(\gamma \sqrt{d_x T}\right)$ oracle regret after$T$ interactions, where $d_x$ and $d_z$ are the dimensions of covariates andIVs, and $\gamma$ is the bias due to endogeneity. For $\gamma=0$, i.e. underexogeneity, O2SLS achieves $\mathcal{O}\left(d_x^2 \log ^2 T\right)$ oracleregret, which is of the same order as that of the stochastic online ridge.Then, we leverage O2SLS as an oracle to design OFUL-IV, a stochastic linearbandit algorithm that can tackle endogeneity and achieves$\widetilde{\mathcal{O}}\left(\sqrt{d_x d_z T}\right)$ regret. For differentdatasets with endogeneity, we experimentally show efficiencies of O2SLS andOFUL-IV in terms of regrets.

Online Instrumental Variable Regression: Regret Analysis and Bandit Feedback