Adaptive Confidence Sets for Binary Regression without Design Smoothness

We study honest adaptive confidence sets for the regression function in random-design binary regression under L^2(dx) loss. Assuming only known bounds 0 < c <= g <= C < infinity on the unknown design density, we construct asymptotically honest, rate-adaptive confidence sets without requiring g to be smooth. Full adaptation is possible when the range of regression-function smoothness spans at most a factor of two. Over wider smoothness ranges, adaptation is achieved on the usual separated classes at the corresponding testing rates n^{-2s/(4s+d)}. A lower bound under the uniform design shows that these separation rates are rate-optimal. This answers a question raised by Mukherjee and Sen (2018).