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We demonstrate the ability of hybrid regularization methods to automatically avoid the double descent phenomenon arising in the training of random feature models (RFM). The hallmark feature of the double descent phenomenon is a spike in the regularization gap at the interpolation threshold, i.e. when the number of features in the RFM equals the number of training samples. To close this gap, the hybrid method considered in our paper combines the respective strengths of the two most common forms of regularization: early stopping and weight decay. The scheme does not require hyperparameter tuning as it automatically selects the stopping iteration and weight decay hyperparameter by using generalized cross-validation (GCV). This also avoids the necessity of a dedicated validation set. While the benefits of hybrid methods have been well-documented for ill-posed inverse problems, our work presents the first use case in machine learning. To expose the need for regularization and motivate hybrid methods, we perform detailed numerical experiments inspired by image classification. In those examples, the hybrid scheme successfully avoids the double descent phenomenon and yields RFMs whose generalization is comparable with classical regularization approaches whose hyperparameters are tuned optimally using the test data. We provide our MATLAB codes for implementing the numerical experiments in this paper at https://github.com/EmoryMLIP/HybridRFM.