The phase diagram of kernel interpolation in large dimensions
This work addresses the problem of understanding benign overfitting in neural networks for researchers in machine learning theory, though it is incremental as it builds on existing kernel regression frameworks.
The authors tackled the generalization ability of kernel interpolation in high dimensions, fully characterizing the variance and bias to obtain a phase diagram that determines regions of minimax optimality, sub-optimality, and inconsistency.
The generalization ability of kernel interpolation in large dimensions (i.e., $n \asymp d^γ$ for some $γ>0$) might be one of the most interesting problems in the recent renaissance of kernel regression, since it may help us understand the 'benign overfitting phenomenon' reported in the neural networks literature. Focusing on the inner product kernel on the sphere, we fully characterized the exact order of both the variance and bias of large-dimensional kernel interpolation under various source conditions $s\geq 0$. Consequently, we obtained the $(s,γ)$-phase diagram of large-dimensional kernel interpolation, i.e., we determined the regions in $(s,γ)$-plane where the kernel interpolation is minimax optimal, sub-optimal and inconsistent.