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arXiv · 2501.03488

A Simple and Combinatorial Approach to Proving Chernoff Bounds and Their Generalizations

William Kuszmaul
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Abstract

The Chernoff bound is one of the most widely used tools in theoretical computer science. It's rare to find a randomized algorithm that doesn't employ a Chernoff bound in its analysis. The standard proofs of Chernoff bounds are beautiful but in some ways not very intuitive. In this paper, I'll show you a different proof that has four features: (1) the proof offers a strong intuition for why Chernoff bounds look the way that they do; (2) the proof is user-friendly and (almost) algebra-free; (3) the proof comes with matching lower bounds, up to constant factors in the exponent; and (4) the proof extends to establish generalizations of Chernoff bounds in other settings. The ultimate goal is that, once you know this proof (and with a bit of practice), you should be able to confidently reason about Chernoff-style bounds in your head, extending them to other settings, and convincing yourself that the bounds you're obtaining are tight (up to constant factors in the exponent).

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William Kuszmaul. 2025-01-07. A Simple and Combinatorial Approach to Proving Chernoff Bounds and Their Generalizations. https://arxiv.org/abs/2501.03488

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