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This work initiates a general study of learning and generalization withoutthe i.i.d. assumption, starting from first principles. While the traditionalapproach to statistical learning theory typically relies on standardassumptions from probability theory (e.g., i.i.d. or stationary ergodic), inthis work we are interested in developing a theory of learning based only onthe most fundamental and necessary assumptions implicit in the requirements ofthe learning problem itself. We specifically study universally consistentfunction learning, where the objective is to obtain low long-run average lossfor any target function, when the data follow a given stochastic process. Weare then interested in the question of whether there exist learning rulesguaranteed to be universally consistent given only the assumption thatuniversally consistent learning is possible for the given data process. Thereasoning that motivates this criterion emanates from a kind of optimist'sdecision theory, and so we refer to such learning rules as being optimisticallyuniversal. We study this question in three natural learning settings:inductive, self-adaptive, and online. Remarkably, as our strongest positiveresult, we find that optimistically universal learning rules do indeed exist inthe self-adaptive learning setting. Establishing this fact requires us todevelop new approaches to the design of learning algorithms. Along the way, wealso identify concise characterizations of the family of processes under whichuniversally consistent learning is possible in the inductive and self-adaptivesettings. We additionally pose a number of enticing open problems, particularlyfor the online learning setting.

Learning Whenever Learning is Possible: Universal Learning under General Stochastic Processes