A strong converse bound for multiple hypothesis testing, with applications to high-dimensional estimation

In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A standard technique to do this involves the use of Fano’s inequality. However, recent work in an information-theoretic setting has shown that an argument based on binary hypothesis testing gives tighter converse results (error lower bounds) than Fano for channel coding problems. We adapt this technique to the statistical setting, and argue that Fano’s inequality can always be replaced by this approach to obtain tighter lower bounds that can be easily computed and are asymptotically sharp. We illustrate our technique in three applications: density estimation, active learning of a binary classifier, and compressed sensing, obtaining tighter risk lower bounds in each case.
MSC 2010 subject classifications: 62G05, 62B10, 62G07.
Keywords and phrases: minimax lower bounds, Fano’s inequality, compressed
sensing, density estimation, active learning.