Table 1.

Estimated mean outcome under linear generative models for functional Q-learning, Q-learning with ad hoc features, and random guessing. Results are computed from 1000 datasets of size n = 250. Monte Carlo standard errors are in the second decimal place.

Example	Sparsity	$\mathbb{E}{Y}^{\ast }\left({\pi }^{\text{opt}}\right)$	$\mathbb{E}{Y}^{\ast }\left({\widehat{\pi }}^{FQ}\right)$	$\mathbb{E}{Y}^{\ast }\left({\widehat{\pi }}^{Q_{\text{last}}}\right)$	$\mathbb{E}{Y}^{\ast }\left({\widehat{\pi }}^{Q_{\text{mean}}}\right)$	$\mathbb{E}{Y}^{\ast }\left({\pi }^{\text{Rand}}\right)$
(S1)	Sparse	3.0	2.81	1.83	2.74	0.90
(S1)	Moderate	3.0	2.86	1.79	2.86	0.90
(S2)	Sparse	3.0	2.61	1.88	1.75	0.92
(S2)	Moderate	3.0	2.81	1.88	1.80	0.92
(S3)	Sparse	3.0	2.43	2.29	1.86	0.94
(S3)	Moderate	3.0	2.58	2.40	1.87	0.94