The key insights and highlights of the content are:
The author generalizes McDiarmid's inequality to handle functions that have bounded differences only on a high probability subset of the input space, rather than the entire input space.
The generalized inequality shows that such functions concentrate around their conditional expectations given the high probability subset, rather than their unconditional expectations. This is important when the function's behavior outside the high probability subset is arbitrary.
The author provides several illustrative examples, including the concentration of the number of triangles in random graphs, the error of maximum likelihood estimation, and the empirical risk minimization. These examples demonstrate how the generalized inequality can provide tighter bounds compared to the basic McDiarmid's inequality.
The results are further extended to the general setting of Lipschitz continuous functions on metric spaces, which includes Gaussian concentration as a special case.
The proofs rely on an extension argument to construct a Lipschitz extension of the original function, which allows applying McDiarmid's inequality on the extended function.
Overall, the content presents a useful generalization of McDiarmid's inequality that can be applied to a broader class of problems where the function of interest has bounded differences only on a high probability subset of the input space.
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by Richard Comb... at arxiv.org 05-01-2024
https://arxiv.org/pdf/1511.05240.pdfDeeper Inquiries