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Title: Per-Tensor Fixed-Point Quantization of the Back-Propagation Algorithm

Paper content:
The high computational and parameter complexity of neural networks makes their training very slow and difficult to deploy on energy and storage-constrained comput- ing systems.
Many network complexity reduction techniques have been proposed including fixed-point implementation.
However, a systematic approach for design- ing full fixed-point training and inference of deep neural networks remains elusive.
We describe a precision assignment methodology for neural network training in which all network parameters, i.e., activations and weights in the feedforward path, gradients and weight accumulators in the feedback path, are assigned close to minimal precision.
The precision assignment is derived analytically and enables tracking the convergence behavior of the full precision training, known to converge a priori.
Thus, our work leads to a systematic methodology of determining suit- able precision for fixed-point training.
The near optimality (minimality) of the resulting precision assignment is validated empirically for four networks on the CIFAR-10, CIFAR-100, and SVHN datasets.
The complexity reduction arising from our approach is compared with other fixed-point neural network designs.
Though deep neural networks (DNNs) have established themselves as powerful predictive models achieving human-level accuracy on many machine learning tasks BID12 , their excellent performance has been achieved at the expense of a very high computational and parameter complexity.
For instance, AlexNet BID17 requires over 800 × 10 6 multiply-accumulates (MACs) per image and has 60 million parameters, while Deepface (Taigman et al., 2014) requires over 500 × 10 6 MACs/image and involves more than 120 million parameters.
DNNs' enormous computational and parameter complexity leads to high energy consumption BID4 , makes their training via the stochastic gradient descent (SGD) algorithm very slow often requiring hours and days BID9 , and inhibits their deployment on energy and resource-constrained platforms such as mobile devices and autonomous agents.A fundamental problem contributing to the high computational and parameter complexity of DNNs is their realization using 32-b floating-point (FL) arithmetic in GPUs and CPUs.
Reduced-precision representations such as quantized FL (QFL) and fixed-point (FX) have been employed in various combinations to both training and inference.
Many employ FX during inference but train in FL, e.g., fully binarized neural networks BID13 use 1-b FX in the forward inference path but the network is trained in 32-b FL.
Similarly, BID10 employs 16-b FX for all tensors except for the internal accumulators which use 32-b FL, and 3-level QFL gradients were employed (Wen et al., 2017; BID0 to accelerate training in a distributed setting. Note that while QFL reduces storage and communication costs, it does not reduce the computational complexity as the arithmetic remains in 32-b FL.Thus, none of the previous works address the fundamental problem of realizing true fixed-point DNN training, i.e., an SGD algorithm in which all parameters/variables and all computations are implemented in FX with minimum precision required to guarantee the network's inference/prediction accuracy and training convergence. The reasons for this gap are numerous including: 1) quantization Step 1: Forward PropagationStep 2: Back PropagationStep 3: Update errors propagate to the network output thereby directly affecting its accuracy (Lin et al., 2016) ; 2) precision requirements of different variables in a network are interdependent and involve hard-toquantify trade-offs (Sakr et al., 2017) ; 3) proper quantization requires the knowledge of the dynamic range which may not be available (Pascanu et al., 2013) ; and 4) quantization errors may accumulate during training and can lead to stability issues BID10 .Our
work makes a major advance in closing this gap by proposing a systematic methodology to obtain close-to-minimum per-layer precision requirements of an FX network that guarantees statistical similarity with full precision training. In
particular, we jointly address the challenges of quantization noise, inter-layer and intra-layer precision trade-offs, dynamic range, and stability. As
in (Sakr et al., 2017) , we do assume that a fully-trained baseline FL network exists and one can observe its learning behavior. While
, in principle, such assumption requires extra FL computation prior to FX training, it is to be noted that much of training is done in FL anyway. For
instance, FL training is used in order to establish benchmarking baselines such as AlexNet BID17 , VGG-Net (Simonyan and Zisserman, 2014) , and ResNet BID12 , to name a few. Even
if that is not the case, in practice, this assumption can be accounted for via a warm-up FL training on a small held-out portion of the dataset BID6 .Applying
our methodology to three benchmarks reveals several lessons. First and
foremost, our work shows that it is possible to FX quantize all variables including back-propagated gradients even though their dynamic range is unknown BID15 . Second, we
find that the per-layer weight precision requirements decrease from the input to the output while those of the activation gradients and weight accumulators increase. Furthermore
, the precision requirements for residual networks are found to be uniform across layers. Finally, hyper-precision
reduction techniques such as weight and activation binarization BID13 or gradient ternarization (Wen et al., 2017) are not as efficient as our methodology since these do not address the fundamental problem of realizing true fixed-point DNN training.We demonstrate FX training on three deep learning benchmarks (CIFAR-10, CIFAR-100, SVHN) achieving high fidelity to our FL baseline in that we observe no loss of accuracy higher then 0.56% in all of our experiments. Our precision assignment
is further shown to be within 1-b per-tensor of the minimum. We show that our precision
assignment methodology reduces representational, computational, and communication costs of training by up to 6×, 8×, and 4×, respectively, compared to the FL baseline and related works.
In this paper, we have presented a study of precision requirements in a typical back-propagation based training procedure of neural networks.
Using a set of quantization criteria, we have presented a precision assignment methodology for which FX training is made statistically similar to the FL baseline, known to converge a priori.
We realized FX training of four networks on the CIFAR-10, CIFAR-100, and SVHN datasets and quantified the associated complexity reduction gains in terms costs of training.
We also showed that our precision assignment is nearly minimal.The presented work relies on the statistics of all tensors being quantized during training.
This necessitates an initial baseline run in floating-point which can be costly.
An open problem is to predict a suitable precision configuration by only observing the data statistics and the network architecture.
Future work can leverage the analysis presented in this paper to enhance the effectiveness of other network complexity reduction approaches.
For instance, weight pruning can be viewed as a coarse quantization process (quantize to zero) and thus can potentially be done in a targeted manner by leveraging the information provided by noise gains.
Furthermore, parameter sharing and clustering can be viewed as a form of vector quantization which presents yet another opportunity to leverage our method for complexity reduction.

assistant: text: We analyze and determine the precision requirements for training neural networks when all tensors, including back-propagated signals and weight accumulators, are quantized to fixed-point format.