Transfer learning for piecewise-constant mean estimation: Optimality, $\ell_1$- and $\ell_0$-penalisation

We study transfer learning in the context of estimating piecewise-constant signals when source data, which may be relevant but disparate, are available in addition to the target data. We initially investigate transfer learning estimators that respectively employ $\ell_1$- and $\ell_0$-penalties for unisource data scenarios and then generalise these estimators to accommodate multisource data. To further reduce estimation errors, especially in scenarios where some sources significantly differ from the target, we introduce an informative source selection algorithm. We then examine these estimators with multisource selection and establish their minimax optimality under specific regularity conditions. It is worth emphasising that, unlike the prevalent narrative in the transfer learning literature that the performance is enhanced through large source sample sizes, our approaches leverage higher observation frequencies and accommodate diverse frequencies across multiple sources. Our theoretical findings are empirically validated through extensive numerical experiments, with the code available online, see https://github.com/chrisfanwang/transferlearning