Title: Which Invariance Should We Transfer? A Causal Minimax Learning Approach. Abstract: A major barrier to deploy current machine learning models lies in their sensitivity to dataset shifts. To resolve this problem, most existing studies attempted to transfer stable information to unseen environments. Among these, graph-based methods causally decomposed the data generating process into stable and mutable mechanisms. By removing the effect of mutable generation, they identified a set of stable predictors. However, a key question regarding robustness remains: which subset of the whole stable information should the model transfer, in order to achieve optimal generalization ability? To answer this question, we provide a comprehensive minimax analysis that fully characterizes conditions for a subset to be optimal. Particularly in general cases, we propose to maximize over mutable mechanisms (i.e., the source of dataset shifts), which is provable to identify the worst-case risk over all environments. This ensures us to select the optimal subset with the minimal worst-case risk. To reduce computational costs, we propose to search over only equivalent classes in terms of worst-case risk, instead of over all subsets. In cases when the searching space is still large, we turn this subset selection problem into a sparse min-max optimization scheme, which enjoys the simplicity and efficiency of implementation. The utility of our methods is demonstrated on the diagnosis of Alzheimer's Disease and gene function prediction. 