On the application of McDiarmid's inequality to complex systems

TL;DR

This paper compares McDiarmid's inequality bounds with absolute bounds for complex systems, showing that the inequality is more useful when the effective number of inputs is large and independent.

Contribution

It provides a detailed analysis of when McDiarmid's inequality offers advantages over absolute bounds in complex system margin setting.

Findings

McDiarmid's bound is less effective with small effective input numbers.

Absolute bounds are preferable when few inputs dominate uncertainty.

McDiarmid's inequality is most useful with many weakly dependent inputs.

Abstract

McDiarmid's inequality has recently been proposed as a tool for setting margin requirements for complex systems. If $F$ is the bounded output of a complex system, depending on a vector of $n$ bounded inputs, this inequality provides a bound $B_F(\epsilon)$, such that the probability of a deviation exceeding $B_F(\epsilon)$ is less than $\epsilon$. I compare this bound with the absolute bound, based on the range of $F$. I show that when $n_{eff}$, the effective number of independent variates, is small, and when $\epsilon$ is small, the absolute bound is smaller than $B_F(\epsilon)$, while also providing a smaller probability of exceeding the bound, i.e., zero instead of $\epsilon$. Thus, for $B_F(\epsilon)$ to be useful, the number of inputs must be large, with a small dependence on any single input, which is consistent with the usual guidance for application of concentration-of-measure results. When the number of inputs is small, or when a small number of inputs account for much of the uncertainty, the absolute bounds will provide better results. The use of absolute bounds is equivalent to the original formulation of the method of Quantification of Margins and Uncertainties (QMU).