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Adaptive collection of data is commonplace in applications throughout science and engineering. From the point of view of statistical inference however, adaptive data collection induces memory and correlation in the samples, and poses significant challenge. We consider the high-dimensional linear regression, where the samples are collected adaptively, and the sample size $n$ can be smaller than $p$, the number of covariates. In this setting, there are two distinct sources of bias: the first due to regularization imposed for consistent estimation, e.g. using the LASSO, and the second due to adaptivity in collecting the samples. We propose online debiasing, a general procedure for estimators such as the LASSO, which addresses both sources of bias. In two concrete contexts $(i)$ time series analysis and $(ii)$ batched data collection, we demonstrate that online debiasing optimally debiases the LASSO estimate when the underlying parameter $theta_0$ has sparsity of order $o(sqrt{n}/log p)$. In this regime, the debiased estimator can be used to compute $p$-values and confidence intervals of optimal size.
Learning optimal policies from historical data enables the gains from personalization to be realized in a wide variety of applications. The growing policy learning literature focuses on a setting where the treatment assignment policy does not adapt t
From scientific experiments to online A/B testing, the previously observed data often affects how future experiments are performed, which in turn affects which data will be collected. Such adaptivity introduces complex correlations between the data a
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