Our framework may be thought of as a generalization of the synthetic control and synthetic interventions frameworks, where data is collected via an adaptive intervention assignment policy.

Our framework may be thought of as a generalization of the synthetic control and synthetic interventions frameworks, where data is collected via an adaptive intervention assignment policy.

Learning the Koopman operator and its spectral decomposition from data is enabled by a number of algorithms.

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However,thisoperation can greatly impair the performance of stochastic methods, e.g.

Principal component regression (PCR) is a popular technique for fixed-design error-in-variables regression, a generalization of the linear regression setting in which the observed covariates are corrupted with random noise. We provide the first time-uniform finite sample guarantees for online (regularized) PCR whenever data is collected adaptively. Since the proof techniques for PCR in the fixed design setting do not readily extend to the online setting, our results rely on adapting tools from modern martingale concentration to the error-in-variables setting. As an application of our bounds, we provide a framework for counterfactual estimation of unit-specific treatment effects in panel data settings when interventions are assigned adaptively. Our framework may be thought of as a generalization of the synthetic interventions framework where data is collected via an adaptive intervention assignment policy.

Consider the setting of Linear Regression where the observed response variables, in expectation, are linear functions of the p-dimensional covariates. Then to achieve vanishing prediction error, the number of required samples scales faster than pσ2, where σ2 is a bound on the noise variance. In a high-dimensional setting where p is large but the covariates admit a low-dimensional representation (say r p), then Principal Component Regression (PCR), cf.