Two-sample Mendelian randomization (MR) uses genetic variants as instrumental variables to estimate causal effects from observational data using summary association statistics. However, horizontal pleiotropy can invalidate standard MR estimators and lead to biased causal inference. Pleiotropy-robust methods have been proposed to address this issue, including regularization-based approaches such as MR-Lasso. However, MR-Lasso may fail to identify invalid instruments consistently, and its post-selection inference can be unreliable. In this paper, we develop two lasso-type procedures for two-sample MR with summary-level data. The first, MR-ALasso, extends MR-Lasso by introducing adaptive penalty weights for pleiotropic effects in order to improve the identification of valid and invalid instruments. The second, MR-ALasso-B, combines adaptive lasso selection with bootstrap smoothing to improve post-selection inference. We establish theoretical results for MR-ALasso under the two-sample summary data framework, including invalid instrument identification consistency and oracle-type post-selection behavior. Simulation studies show that MR-ALasso generally improves upon MR-Lasso in estimation accuracy and invalid-instrument identification, whereas MR-ALasso-B substantially improves coverage and type-I error control relative to naive post-selection inference. A real-data application based on bidirectional analyses of multiple complex traits further illustrates the practical usefulness of the proposed methods. We provide an R package, MRAlasso, to facilitate implementation.

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