A Universal Hierarchy of Shift-Stable Distributions and the Tradeoff Between Stability and Performance. (arXiv:1905.11374v3 [stat.ML] UPDATED)

Many methods which find invariant predictive distributions have been
developed to learn models that can generalize to new environments without using
samples from the target distribution. However, these methods consider differing
types of shifts in environment and have been developed under disparate
frameworks, making their comparison difficult. In this paper, we provide a
unifying graphical representation of the data generating process that can
represent all such shifts. We show there is a universal hierarchy of
shift-stable distributions which correspond to operators on a graph that
disable edges. This provides the ability to compare current methods and derive
new algorithms that find optimal invariant distributions, all of which can be
mapped to the hierarchy. We theoretically and empirically show that the degree
to which stability is desirable depends on how concerned we are about large
shifts: there is a tradeoff between stability and average performance.

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