Universal Approximation Theorems of Fully Connected Binarized Neural Networks
This work provides theoretical guarantees for the approximation power of BNNs, which are important for researchers and practitioners interested in resource-efficient neural networks.
This paper investigates the universal approximation capabilities of fully connected Binarized Neural Networks (BNNs). It demonstrates that BNNs can universally approximate functions with one hidden layer for binarized inputs, and with two hidden layers for real-valued inputs, specifically for Lipschitz-continuous functions.
Neural networks (NNs) are known for their high predictive accuracy in complex learning problems. Beside practical advantages, NNs also indicate favourable theoretical properties such as universal approximation (UA) theorems. Binarized Neural Networks (BNNs) significantly reduce time and memory demands by restricting the weight and activation domains to two values. Despite the practical advantages, theoretical guarantees based on UA theorems of BNNs are rather sparse in the literature. We close this gap by providing UA theorems for fully connected BNNs under the following scenarios: (1) for binarized inputs, UA can be constructively achieved under one hidden layer; (2) for inputs with real numbers, UA can not be achieved under one hidden layer but can be constructively achieved under two hidden layers for Lipschitz-continuous functions. Our results indicate that fully connected BNNs can approximate functions universally, under certain conditions.