Abstract: In clinical trials and other applications, we often see regions of the feature space that appear to exhibit interesting behaviour, but these observed phenomena may not reflected at the population level. Focusing on a regression setting, we consider the subgroup selection challenge of identifying a region of the feature space on which the regression function exceeds a pre-determined threshold. We formulate the problem as one of constrained optimisation, where we seek a low-complexity, data-dependent selection set on which, with a guaranteed probability, the regression function is uniformly at least as large as the threshold; subject to this constraint, we would like the region to contain as much mass under the marginal feature distribution as possible. This leads to a natural notion of regret, and we determine the minimax optimal rate for this regret in both the sample size and the Type I error probability. The rate involves a delicate interplay between the different problem parameters. Finally, we expand the scope of our previous results by illustrating how they may be generalised to a treatment and control setting, where interest lies in the heterogeneous treatment effect.

https://www.statslab.cam.ac.uk/~rjs57/