Optimal subgroup selection

TL;DR

This paper addresses the challenge of identifying regions in feature space where a regression function exceeds a threshold, providing minimax optimal rates for the regret in subgroup selection with guarantees.

Contribution

It formulates subgroup selection as a constrained optimisation problem and derives the minimax optimal rate for regret, extending previous results to treatment effect settings.

Findings

Derived minimax optimal rate for regret in subgroup selection.

Established the interplay between smoothness and approximation parameters.

Extended results to treatment and control scenarios.

Abstract

In clinical trials and other applications, we often see regions of the feature space that appear to exhibit interesting behaviour, but it is unclear whether these observed phenomena are 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 our main contribution is to 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 parameters that control the smoothness of the regression function, as well as exponents that quantify the extent to which the optimal selection set at the population level can be approximated by families of well-behaved subsets. 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.