## Effective Distributed Learning with Random Features: Improved Bounds and Algorithms

### Yong Liu · Jiankun Liu · Shuqiang Wang

Keywords: [ kernel methods ] [ statistical learning theory ] [ Risk bound ]

Abstract: In this paper, we study the statistical properties of distributed kernel ridge regression together with random features (DKRR-RF), and obtain optimal generalization bounds under the basic setting, which can substantially relax the restriction on the number of local machines in the existing state-of-art bounds. Specifically, we first show that the simple combination of divide-and-conquer technique and random features can achieve the same statistical accuracy as the exact KRR in expectation requiring only $\mathcal{O}(|\mathcal{D}|)$ memory and $\mathcal{O}(|\mathcal{D}|^{1.5})$ time. Then, beyond the generalization bounds in expectation that demonstrate the average information for multiple trails, we derive generalization bounds in probability to capture the learning performance for a single trail. Finally, we propose an effective communication strategy to further improve the performance of DKRR-RF, and validate the theoretical bounds via numerical experiments.