統計学輪講 第24回

Deep neural networks (DNNs) have recently achieved outstanding performance in many large–scale learning tasks. From a statistical viewpoint, an important question is whether such models can overcome the classical curse of dimensionality in nonparametric regression. Schmidt–Hieber (2020) [1] shows that, when the target regression function admits a hierarchical compositional structure, deep ReLU networks can achieve near–optimal convergence rates that depend only on low effective dimensions rather than the full input dimension. Oga–Koike (2024) [2] extend this idea to nonparametric drift estimation for multi–dimensional diffusion processes observed at high frequency, establishing an oracle inequality and (almost) minimax optimal rates. In this presentation, we overview the main ideas of these two papers and briefly discuss possible extensions to L´evy–driven SDEs.

[1] J. Schmidt–Hieber. Nonparametric regression using deep neural networks with ReLU activation function. Annals of Statistics, 48(4):1875–1897, 2020.
[2] A. Oga and Y. Koike. Drift estimation for a multi-dimensional diffusion process using deep neural networks. Stochastic Processes and their Applications, 170:104240, 2024.