Single-molecule localization microscopy is a widely used technique in biological research for measuring the nanostructures of samples smaller than the diffraction limit. This study uses multifocal plane microscopy and addresses the three-dimensional (3D) single-molecule localization problem, where lateral and axial locations of molecules are estimated. However, when multifocal plane microscopy is used, the estimation accuracy of 3D localization is easily deteriorated by the small lateral drifts of camera positions. A 3D molecule localization problem was presented along with the lateral drift estimation as a compressed sensing problem. A deep neural network (DNN) was applied to solve this problem accurately and efficiently. The results show that the proposed method is robust to lateral drift and achieves an accuracy of 20 nm laterally and 50 nm axially without an explicit drift correction.