As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.

A Survey of Quantization Methods for Efficient Neural Network Inference

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

This chapter provides approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. Over the past decade, people have observed significant improvements in the accuracy of Neural Networks (NNs) for a wide range of problems, often achieved by highly over-parameterized models. Achieving efficient, real-time NNs with optimal accuracy requires rethinking the design, training, and deployment of NN models. Model distillation involves training a large model and then using it as a teacher to train a more compact model. Loosely related to NN quantization is work in neuroscience that suggests that the human brain stores information in a discrete/quantized form, rather than in a continuous form. Gray and Neuhoff have written a very nice survey of the history of quantization up to 1998. 

We consider the problem of model compression for deep neural networks (DNNs) in the challenging one-shot/post-training setting, in which we are given an accurate trained model, and must compress it without any retraining, based only on a small amount of calibration input data. This problem has become popular in view of the emerging software and hardware support for executing models compressed via pruning and/or quantization with speedup, and well-performing solutions have been proposed independently for both compression approaches. In this paper, we introduce a new compression framework which covers both weight pruning and quantization in a unified setting, is time- and space-efficient, and considerably improves upon the practical performance of existing post-training methods. At the technical level, our approach is based on an exact and efficient realization of the classical Optimal Brain Surgeon (OBS) framework of [LeCun, Denker, and Solla, 1990] extended to also cover weight quantization at the scale of modern DNNs. From the practical perspective, our experimental results show that it can improve significantly upon the compression-accuracy trade-offs of existing post-training methods, and that it can enable the accurate compound application of both pruning and quantization in a post-training setting.

Optimal Brain Compression: A Framework for Accurate Post-Training Quantization and Pruning

Deep Neural Networks (DNNs) are very popular because of their high performance in various cognitive tasks in Machine Learning (ML). Recent advancements in DNNs have brought levels beyond human accuracy in many tasks, but at the cost of high computational complexity. To enable efficient execution of DNN inference, more and more research works, therefore, are exploiting the inherent error resilience of DNNs and employing Approximate Computing (AC) principles to address the elevated energy demands of DNN accelerators. This article provides a comprehensive survey and analysis of hardware approximation techniques for DNN accelerators. First, we analyze the state of the art, and by identifying approximation families, we cluster the respective works with respect to the approximation type. Next, we analyze the complexity of the performed evaluations (with respect to the dataset and DNN size) to assess the efficiency, potential, and limitations of approximate DNN accelerators. Moreover, a broad discussion is provided regarding error metrics that are more suitable for designing approximate units for DNN accelerators as well as accuracy recovery approaches that are tailored to DNN inference. Finally, we present how Approximate Computing for DNN accelerators can go beyond energy efficiency and address reliability and security issues as well.

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Hardware Approximate Techniques for Deep Neural Network Accelerators: A Survey

Pruning and quantization for deep neural network acceleration: A survey

Deep neural networks have been applied in many applications exhibiting extraordinary abilities in the field of computer vision. However, complex network architectures challenge efficient real-time deployment and require significant computation resources and energy costs. These challenges can be overcome through optimizations such as network compression. Network compression can often be realized with little loss of accuracy. In some cases accuracy may even improve. This paper provides a survey on two types of network compression: pruning and quantization. Pruning can be categorized as static if it is performed offline or dynamic if it is performed at run-time. We compare pruning techniques and describe criteria used to remove redundant computations. We discuss trade-offs in element-wise, channel-wise, shape-wise, filter-wise, layer-wise and even network-wise pruning. Quantization reduces computations by reducing the precision of the datatype. Weights, biases, and activations may be quantized typically to 8-bit integers although lower bit width implementations are also discussed including binary neural networks. Both pruning and quantization can be used independently or combined. We compare current techniques, analyze their strengths and weaknesses, present compressed network accuracy results on a number of frameworks, and provide practical guidance for compressing networks.

Pruning and Quantization for Deep Neural Network Acceleration: A Survey

High bandwidth requirements in edge devices can be a bottleneck for many systems - especially for accelerating both training and inference of deep neural networks. In this paper, we analyze feature map sparsity for several popular convolutional neural networks. When considering run-time behavior, we find a good probability of dynamically disabling many feature maps. By evaluating the number of 0-valued activations within feature maps, we find many feature maps that can be dynamically pruned. This is particularly effective when a ReLU activation function is used. To take advantage of inactive feature maps, we present a novel method to dynamically prune feature maps at runtime reducing bandwidth by up to 11.5% without loss of accuracy for image classification. We further apply this method on Non-ReLU activation functions by allowing the output of the activation function to be within an epsilon of 0. Additionally, we also studied how video streaming applications could benefit from bandwidth reduction.

Xiaotong Zhang

Papers

Pruning and quantization for deep neural network acceleration: A survey

Pruning and Quantization for Deep Neural Network Acceleration: A Survey

Dynamic Runtime Feature Map Pruning