Deep quantization of neural networks below eight bits can lead to superlinear benefits in storage and compute efficiency. However, homogeneously quantizing all the layers to the same level does not account for the distinction of the layers and their individual properties. Heterogenous assignment of bitwidths to individual layers is attractive but opens an exponentially large non-contiguous hyperparameter space (${Available Bitwidths}^{\# Layers}$). As such finding the bitwidth while also quantizing the network to those levels becomes a major challenge. This paper addresses this challenge through a sinusoidal regularization mechanism, dubbed WaveQ. Adding our parametrized sinusoidal regularizer enables us to not only find the quantized weights but also learn the bitwidth of the layers by making the period of the sinusoidal regularizer a trainable parameter. In addition, the sinusoidal regularizer itself is designed to align its minima on the quantization levels. With these two innovations, during training, stochastic gradient descent uses the form of the sinusoidal regularizer and its minima to push the weights to the quantization levels while it is also learning the period which will determine the bitwidth of each layer separately. As such WaveQ is a gradient-based mechanism that jointly learns the quantized weights as well as the heterogeneous bitwidths. We show how WaveQ balance compute efficiency and accuracy, and provide a heterogeneous bitwidth assignment for quantization of a large variety of deep networks (AlexNet, CIFAR-10, MobileNet, ResNet-18, ResNet-20, SVHN, and VGG-11) that virtually preserves the accuracy. WaveQ is versatile and can also be used with predetermined bitwidths by fixing the period of the sinusoidal regularizer. In this case. WaveQ enhances quantized training algorithms (DoReFa and WRPN) with about 4.8% accuracy improvements on average, and outperforms multiple state-of-the-art techniques. Finally, WaveQ applied to quantizing transformers

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