Deep Semi-Random Features for Nonlinear Function Approximation

TL;DR

This paper introduces semi-random features for nonlinear function approximation, bridging the gap between deep learning and kernel methods, with theoretical guarantees and practical advantages in performance and efficiency.

Contribution

It provides a new semi-random feature approach with proven universality, optimization, and generalization properties, outperforming traditional random features and matching neural networks.

Findings

Semi-random features achieve universality and good approximation with increasing width.

Deep semi-random models have strong theoretical guarantees including generalization bounds.

Experiments show semi-random features match neural networks and outperform random features.

Abstract

We propose semi-random features for nonlinear function approximation. The flexibility of semi-random feature lies between the fully adjustable units in deep learning and the random features used in kernel methods. For one hidden layer models with semi-random features, we prove with no unrealistic assumptions that the model classes contain an arbitrarily good function as the width increases (universality), and despite non-convexity, we can find such a good function (optimization theory) that generalizes to unseen new data (generalization bound). For deep models, with no unrealistic assumptions, we prove universal approximation ability, a lower bound on approximation error, a partial optimization guarantee, and a generalization bound. Depending on the problems, the generalization bound of deep semi-random features can be exponentially better than the known bounds of deep ReLU nets; our generalization error bound can be independent of the depth, the number of trainable weights as well as the input dimensionality. In experiments, we show that semi-random features can match the performance of neural networks by using slightly more units, and it outperforms random features by using significantly fewer units. Moreover, we introduce a new implicit ensemble method by using semi-random features.