Power-of-Two Quantization for Low Bitwidth and Hardware Compliant Neural Networks

Abstract

Deploying Deep Neural Networks in low-power embedded devices for real time-constrained applications requires optimization of memory and computational complexity of the networks, usually by quantizing the weights. Most of the existing works employ linear quantization which causes considerable degradation in accuracy for weight bit widths lower than 8. Since the distribution of weights is usually non-uniform (with most weights concentrated around zero), other methods, such as logarithmic quantization, are more suitable as they are able to preserve the shape of the weight distribution more precise. Moreover, using base-2 logarithmic representation allows optimizing the multiplication by replacing it with bit shifting. In this paper, we explore non-linear quantization techniques for exploiting lower bit precision and identify favorable hardware implementation options. We developed the Quantization Aware Training (QAT) algorithm that allowed training of low bit width Power-of-Two (PoT) networks and achieved accuracies on par with state-of-the-art floating point models for different tasks. We explored PoT weight encoding techniques and investigated hardware designs of MAC units for three different quantization schemesuniform, PoT and Additive-PoT (APoT) -to show the increased efficiency when using the proposed approach. Eventually, the experiments showed that for low bit width precision, non-uniform quantization performs better than uniform, and at the same time, PoT quantization vastly reduces the computational complexity of the neural network.

Reducing Computational Complexity of Neural Networks in Optical Channel Equalization: From Concepts to Implementation

“…The PoT quantization is a logarithmic quantizer [87] designed to approximate the weights to the closest power of two in the range defined by the considered number of bits. Mathematically, we can represent the PoT quantization considering 2 BW elements as [84], [87], [88]:…”

“…The latter is done by assuming that ∂W qc ∂W = 1. As stated in [88], this process is known as a Straight-Through Estimator, and it results in a smoother transition between consecutive quantization levels in the learning process.…”

“…The forward propagation uses the quantized alphabet c q to generate the quantized weights W qc and, in the backpropagation, such a weight is "skipped" by imposing its gradient to be one (straight-through estimator, Ref. [88]).…”

Computational Complexity Evaluation of Neural Network Applications in Signal Processing

“…5 . In the case of Power-of two (PoT) quantization, we have: X = 0, because each multiplication costs just a shift [42], [53]. Lastly, for the Additive Powers-of-Two (APoT) quantization, we have: X = n, where n denotes the number of additive terms.…”

“…Since APoT is a quantization scheme represented by a sum of PoT terms, APoT provides a smooth transition between PoT and uniform quantization. In various works, PoT was claimed to have very low complexity because the multiplications are replaced by just shifts [53], [69], [70]. However, when we consider that the multiplication in the uniform quantization can be represented by shifts and adders, and we have a fair metric like NABS to compare between different quantization techniques, the NABS when applying PoT is only around an order of magnitude lower than the NABS when using the uniform quantization.…”