SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We conclude this section by comparing our results with those of the (non-adaptive) synthetic interventions framework of Agarwal et al. [8]. Since we are regressing over time, our method for estimating  $\widehat{\mathbb{E}}\bar{Y}_{n,\text{post}}^{(a)}$  is known as a horizontal regression method in the panel data literature. This is in contrast to vertical regression methods, which regress over units. See Shen et al. [75] for more details on the similarities and differences between horizontal and vertical regression methods in panel data settings. While we do not exactly match the bound of Agarwal et al. [8] since their synthetic interventions framework of uses a vertical regression method, the two bounds are similar, with the notable differences being as follows: (1) The bound of [8] contains a "slow"  $\widetilde{O}(r^{1/2}T_0^{-1/4})$  term which does not appear in our analysis. (2) The non-adaptive SI bound also contains a term which scales as  $\widetilde{O}\left(\frac{\sqrt{n}}{\sqrt{T-T_0}}\right)$  in the worst case, while our bound has no such dependence. However, this comes at the price of slow rate whenever  $L$  is large compared to  $\sqrt{T-T_0}$ .
We obtain the first adaptive bounds for principal component regression and apply them to the problem of online experiment design in the context of panel data, where we allow for interventions to be assigned according to an adaptive policy. Exciting directions for future work include applications of our results to domains such as differential privacy, and using our bounds to obtain contextual bandit algorithms (e.g. based on LinUCB [3]) capable of regret minimization when given access to noisy contexts.
KH is supported in part by an NDSEG Fellowship. KH, JW, and ZSW are supported in part by NSF FAI Award #1939606. The authors would like to thank the anonymous NeurIPS reviewers for valuable feedback.
[82] Roman Vershynin. High-dimensional probability: An introduction with applications in data science, volume 47. Cambridge university press, 2018.
[83] Martin J Wainwright. High-dimensional statistics: A non-asymptotic viewpoint, volume 48. Cambridge university press, 2019.
[84] Tom Wansbeek and Erik Meijer. Measurement error and latent variables. A companion to theoretical econometrics, pages 162-179, 2001.
[85] Justin Whitehouse, Aaditya Ramdas, Steven Z Wu, and Ryan M Rogers. Brownian noise reduction: Maximizing privacy subject to accuracy constraints. Advances in Neural Information Processing Systems, 35:11217-11228, 2022.
[86] Justin Whitehouse, Aaditya Ramdas, Ryan Rogers, and Steven Wu. Fully-adaptive composition in differential privacy. In International Conference on Machine Learning, pages 36990-37007. PMLR, 2023.
[87] Justin Whitehouse, Zhiwei Steven Wu, and Aaditya Ramdas. On the sublinear regret of GP-UCB. Advances in Neural Information Processing Systems, 2023.
[88] Justin Whitehouse, Zhiwei Steven Wu, and Aaditya Ramdas. Time-uniform self-normalized concentration for vector-valued processes. arXiv preprint arXiv:2310.09100, 2023.
[89] Yiqing Xu. Generalized synthetic control method: Causal inference with interactive fixed effects models. Political Analysis, 25(1):57-76, 2017. doi: 10.1017/pan.2016.2.
[90] Jingwen Zhang, Yifang Chen, and Amandeep Singh. Causal bandits: Online decision-making in endogenous settings. arXiv preprint arXiv:2211.08649, 2022.
[91] Kelly Zhang, Lucas Janson, and Susan Murphy. Statistical inference with m-estimators on adaptively collected data. Advances in neural information processing systems, 34:7460-7471, 2021.
[92] Kelly W Zhang, Lucas Janson, and Susan A Murphy. Statistical inference after adaptive sampling in non-markovian environments. arXiv preprint arXiv:2202.07098, 2022.