  We show that many definitions of stability found in the learning theory
literature are equivalent to one another. We distinguish between two families
of definitions of stability: distribution-dependent and
distribution-independent Bayesian stability. Within each family, we establish
equivalences between various definitions, encompassing approximate differential
privacy, pure differential privacy, replicability, global stability, perfect
generalization, TV stability, mutual information stability, KL-divergence
stability, and R\'enyi-divergence stability. Along the way, we prove boosting
results that enable the amplification of the stability of a learning rule. This
work is a step towards a more systematic taxonomy of stability notions in
learning theory, which can promote clarity and an improved understanding of an
array of stability concepts that have emerged in recent years.
