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Sequential complexities and uniform martingale LLN

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Uniform laws of large numbers play a key role in statistics and learning theory. In this talk, we describe martingale analogues of the uniform laws and introduce new notions of ``sequential complexities’’. Extending the symmetrization technique to sequences of dependent random variables leads us to a notion of a tree. We then introduce a definition of a tree covering number, extend the chaining analysis, introduce an analogue of the VC dimension, and prove a counterpart to the classical combinatorial result of Vapnik-Chervonenkis-Sauer-Shelah. Our definitions and results can be seen as non-i.i.d. extensions of some of the key notions in empirical process theory.

This talk is part of the Probability Theory and Statistics in High and Infinite Dimensions series.

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