When working with statistical data, it is often desirable to be able to quantify the probability of certain undesirable events taking place. In this post we discuss an interesting connection between convex optimization and extreme event analysis. We start with the classical Chernoff bound for the empirical average.
Let be data samples in drawn independently and identically from the distribution with mean . An important class of an undesirable events can be expressed as the empirical average of data realizing in a convex set . When for instance have an interpretation as losses, then knowing the probability of the average loss exceeding a critical value is paramount. In that case, knowing the probability that the empirical average realizes in the half space would be of great interest. Chernoff’s classical inequality quantifies the probability of such events quite nicely.
with the convex dual of the log moment generating function and the distribution of .
Proof: Let and , and consider the positive function . If the function satisfies for all in , then we may conclude that
Using the independence of distinct samples and taking the logarithm on both sides of the previous inequality establishes
It is clear that for all in if and only if for all in . From this we obtain the general form of Chernoff’s bound
The last inequality follows from the minimax theorem for convex optimization.
The Chernoff bound (1) expresses the probability of the extreme event in terms of the convex conjugate of the log moment generating function. This makes that computing Chernoff’s bound can be done by solving a convex optimization problem.
The function is the log moment generating function of the recentered distribution generating the data and is always convex. We give the cumulant generating function for some common standardized (zero mean, unit variance) distributions in the table below.
Furthermore, it comes with a nice geometric interpretation as well. The function is positive and convex and thus defines a distance between the mean of the distribution generating the data and its empirical mean.
The minimum distance as measured by the convex dual of the log moment generating function between the mean of the distribution generating the data and the set bounds the probability of the event taking place.
Chernoff’s bound is furthermore exponentially tight as it correctly identifies the exact exponential rate with which the probability of the event diminishes to zero. The last surprising fact is codified in Cramer’s theorem.
Cramer’s Theorem : Assume that the distribution satisfies . Then the probability that the empirical average of realizes in a open set diminishes with exponential rate
The lower bound in Cramer’s thorem makes that the Chernoff inequality accurately quantifies the probability of extreme events taking place as the number of samples tends toward infinity. Notice that for the Cramer’s theorem convexity of the set is not a requirement. Note that as by definition of being a probability distribution it follows that . The condition requires the distribution to be light-tailed. The tails of the distribution must diminish at an exponential rate.
- A. Dembo, and O. Zeitouni. “Large Deviations Techniques and Applications”, Springer (2010).
- S. Boyd, and L. Vandenberghe. “Convex Optimization”, Cambridge University Press (2004).