At the PAC-Bayes workshop earlier this week, Olivier Catoni described a result that I hadn’t believed was possible: a deviation bound depending only on the variance of a random variable.
For people not familiar with deviation bounds, this may be hard to appreciate. Deviation bounds, are one of the core components for the foundations of machine learning theory, so developments here have a potential to alter our understanding of how to learn and what is learnable. My understanding is that the basic proof techniques started with Bernstein and have evolved into several variants specialized for various applications. All of the variants I knew had a dependence on the range, with some also having a dependence on the variance of an IID or martingale random variable. This one is the first I know of with a dependence on only the variance.
The basic idea is to use a biased estimator of the mean which is not influenced much by outliers. Then, a deviation bound can be proved by using the exponential moment method, with the sum of the bias and the deviation bounded. The use of a biased estimator is clearly necessary, because an unbiased empirical average is inherently unstable—which was precisely the reason I didn’t think this was possible.
Precisely how this is useful for machine learning isn’t clear yet, but it opens up possibilities. For example, it’s common to suffer from large ranges in exploration settings, such as contextual bandits or active learning.