Implicit Regularization of Random Feature Models
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Random Feature (RF) models are used as efficient parametric approximations of kernel methods. We investigate, by means of random matrix theory, the connection between Gaussian RF models and Kernel Ridge Regression (KRR). For a Gaussian RF model with P features, N data points, and a ridge λ, we show that the average (i.e. expected) RF predictor is close to a KRR predictor with an effective ridge λ̃. We show that λ̃ > λ and λ̃λ monotonically as P grows, thus revealing the implicit regularization effect of finite RF sampling. We then compare the risk (i.e. test error) of the λ̃-KRR predictor with the average risk of the λ-RF predictor and obtain a precise and explicit bound on their difference. Finally, we empirically find an extremely good agreement between the test errors of the average λ-RF predictor and λ̃-KRR predictor.
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