Multiple Descent in the Multiple Random Feature Model

Recent works have demonstrated a double descent phenomenon in over-parameterized learning: as the number of model parameters increases, the excess risk has a $\mathsf{U}$-shape at beginning, then decreases again when the model is highly over-parameterized. Although this phenomenon has been investigated by recent works under different settings such as linear models, random feature models and kernel methods, it has not been fully understood in theory. In this paper, we consider a double random feature model (DRFM) consisting of two types of random features, and study the excess risk achieved by the DRFM in ridge regression. We calculate the precise limit of the excess risk under the high dimensional framework where the training sample size, the dimension of data, and the dimension of random features tend to infinity proportionally. Based on the calculation, we demonstrate that the risk curves of DRFMs can exhibit triple descent. We then provide an explanation of the triple descent phenomenon, and discuss how the ratio between random feature dimensions, the regularization parameter and the signal-to-noise ratio control the shape of the risk curves of DRFMs. At last, we extend our study to the multiple random feature model (MRFM), and show that MRFMs with $K$ types of random features may exhibit $(K+1)$-fold descent. Our analysis points out that risk curves with a specific number of descent generally exist in random feature based regression. Another interesting finding is that our result can recover the risk peak locations reported in the literature when learning neural networks are in the "neural tangent kernel" regime.