  Enhancing the stability of machine learning algorithms under distributional
shifts is at the heart of the Out-of-Distribution (OOD) Generalization problem.
Derived from causal learning, recent works of invariant learning pursue strict
invariance with multiple training environments. Although intuitively
reasonable, strong assumptions on the availability and quality of environments
are made to learn the strict invariance property. In this work, we come up with
the ``distributional stability" notion to mitigate such limitations. It
quantifies the stability of prediction mechanisms among sub-populations down to
a prescribed scale. Based on this, we propose the learnability assumption and
derive the generalization error bound under distribution shifts. Inspired by
theoretical analyses, we propose our novel stable risk minimization (SRM)
algorithm to enhance the model's stability w.r.t. shifts in prediction
mechanisms ($Y|X$-shifts). Experimental results are consistent with our
intuition and validate the effectiveness of our algorithm. The code can be
found at https://github.com/LJSthu/SRM.
