Towards a theory of out-of-distribution learning
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What is learning? 20^st century formalizations of learning theory – which precipitated revolutions in artificial intelligence – focus primarily on 𝑖𝑛-𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 learning, that is, learning under the assumption that the training data are sampled from the same distribution as the evaluation distribution. This assumption renders these theories inadequate for characterizing 21^st century real world data problems, which are typically characterized by evaluation distributions that differ from the training data distributions (referred to as out-of-distribution learning). We therefore make a small change to existing formal definitions of learnability by relaxing that assumption. We then introduce 𝐥𝐞𝐚𝐫𝐧𝐢𝐧𝐠 𝐞𝐟𝐟𝐢𝐜𝐢𝐞𝐧𝐜𝐲 (LE) to quantify the amount a learner is able to leverage data for a given problem, regardless of whether it is an in- or out-of-distribution problem. We then define and prove the relationship between generalized notions of learnability, and show how this framework is sufficiently general to characterize transfer, multitask, meta, continual, and lifelong learning. We hope this unification helps bridge the gap between empirical practice and theoretical guidance in real world problems. Finally, because biological learning continues to outperform machine learning algorithms on certain OOD challenges, we discuss the limitations of this framework vis-á-vis its ability to formalize biological learning, suggesting multiple avenues for future research.
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