The Multiplicative Version of Azuma's Inequality, with an Application to Contention Analysis
William Kuszmaul, Qi Qi
TL;DR
The paper introduces a multiplicative analogue of Azuma's inequality for supermartingales with bounded increments, yielding tail bounds that scale multiplicatively with the per-step means. It extends the result to adaptive adversaries and to lower-tail bounds, broadening applicability to online and adversarial settings. The authors demonstrate the utility of these bounds by giving a simple, correct analysis of the $(P, M)$-recycling game in work-stealing, achieving near-optimal high-probability delay bounds and showing the superiority of the multiplicative approach over standard Azuma. This tool simplifies and strengthens concentration analyses in randomized algorithms with dependencies, enabling more robust guarantees for concurrent and online systems.
Abstract
Azuma's inequality is a tool for proving concentration bounds on random variables. The inequality can be thought of as a natural generalization of additive Chernoff bounds. On the other hand, the analogous generalization of multiplicative Chernoff bounds does not appear to be widely known. We formulate a multiplicative-error version of Azuma's inequality. We then show how to apply this new inequality in order to greatly simplify (and correct) the analysis of contention delays in multithreaded systems managed by randomized work stealing.