Generalization error bounds for Neural Networks with ReLU activation

Harsh Pandey, Amitabha Bagchi, Srikanta J. Bedathur, Arindam Bhattacharya

Keywords: relu stability, sgd stability, non smooth neural network stability

TL;DR: We show that generalization error of Neural Netowrks with ReLU activations approaches zero with proabbility 1 as we increase the training points

Abstract: We show rigorous bounds on the generalization error for Neural Networks with ReLU activation under the condition that the network size doesn't grow with the training set size. In order to prove these bounds we weaken the notion of uniform stability of a learning algorithm in a probabilistic way by positing the notion of almost sure (a.s.) support stability and proving that if an algorithm has low enough a.s. support stability its generalization error tends to 0 as the training set size increases. Further we show that for Stochastic Gradient Descent to be almost surely support stable we only need the loss function to be locally Lipschitz and locally smooth with probability 1, thereby showing low generalization error with weaker conditions than have been used in the literature. We then show that Neural Networks with ReLU activation and a doubly differentiable loss function possess these properties, thereby proving low generalization error. The caveat is that the size of NN must not grow with the size of the training set. Finally we present experimental evidence to validate our theoretical results.

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