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This chapter presents some foundational methods for bounding random processes. We begin with the chaining technique and prove the Dudley inequality, which bounds a random process using covering numbers. Applications include Monte Carlo integration and uniform bounds for empirical processes. We then develop VC (Vapnik– Chervonenkis) theory, offering combinatorial insights into random processes and applying it to statistical learning. Building on chaining, we introduce generic chaining to obtain optimal two-sided bounds using Talagrand’s g2 functional. A key consequence is the Talagrand comparison inequality, a generalization of the Sudakov–Fernique inequality for sub-Gaussian processes. This is used to derive the Chevet inequality, a powerful tool for analyzing random bilinear forms over general sets. Exercises explore the Lipschitz law of large numbers in higher dimensions, one-bit quantization, and the small ball method for heavy-tailed random matrices.

This article examines the results of [6] in a purely tree-theoretic context. We show how this rephrased Chase–Freitag Lemma is sufficient to prove the Sauer–Shelah and Bhaskar Lemmas. We also use this approach to expand these two lemmas to the context of arbitrary finite label sets.

The last few years have witnessed important new developments inthe theory and practice of pattern classification. We intend tosurvey some of the main new ideas that have led to theserecent results.

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8 - Chaining

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ON THE CHASE–FREITAG LEMMA

Theory of Classification: a Survey of Some Recent Advances

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3 results

8 - Chaining

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ON THE CHASE–FREITAG LEMMA

Theory of Classification: a Survey of Some Recent Advances