A strong converse bound for multiple hypothesis testing, with applications to high-dimensional estimation
This provides improved theoretical guarantees for high-dimensional estimation problems, though it is incremental as it refines an existing technique.
The paper tackles the problem of obtaining sharp lower bounds on minimax risk in statistical inference by adapting a binary hypothesis testing technique from information theory, showing it yields tighter and asymptotically sharp bounds compared to Fano's inequality, with applications in density estimation, active learning, and compressed sensing.
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 obtain risk lower bounds involves the use of Fano's inequality. In an information-theoretic setting, it is known that Fano's inequality typically does not give a sharp converse result (error lower bound) for channel coding problems. Moreover, recent work has shown that an argument based on binary hypothesis testing gives tighter results. 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.