Though distribution shifts have caused growing concern for machine learning scalability, solutions tend to specialize towards a specific type of distribution shift. We learn that constructing a Compressed Parameter Subspaces (CPS), a geometric structure representing distance-regularized parameters mapped to a set of train-time distributions, can maximize average accuracy over a broad range of distribution shifts concurrently. We show sampling parameters within a CPS can mitigate backdoor, adversarial, permutation, stylization and rotation perturbations. Regularizing a hypernetwork with CPS can also reduce task forgetting.