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Title: LEARNED STEP SIZE QUANTIZATION

Paper content:
Deep networks run with low precision operations at inference time offer power and space advantages over high precision alternatives, but need to overcome the challenge of maintaining high accuracy as precision decreases.
Here, we present a method for training such networks, Learned Step Size Quantization, that achieves the highest accuracy to date on the ImageNet dataset when using models, from a variety of architectures, with weights and activations quantized to 2-, 3- or 4-bits of precision, and that can train 3-bit models that reach full precision baseline accuracy.
Our approach builds upon existing methods for learning weights in quantized networks by improving how the quantizer itself is configured.
Specifically, we introduce a novel means to estimate and scale the task loss gradient at each weight and activation layer's quantizer step size, such that it can be learned in conjunction with other network parameters.
This approach works using different levels of precision as needed for a given system and requires only a simple modification of existing training code.
Deep networks are emerging as components of a number of revolutionary technologies, including image recognition (Krizhevsky et al., 2012) , speech recognition , and driving assistance (Xu et al., 2017) .
Unlocking the full promise of such applications requires a system perspective where task performance, throughput, energy-efficiency, and compactness are all critical considerations to be optimized through co-design of algorithms and deployment hardware.
Current research seeks to develop methods for creating deep networks that maintain high accuracy while reducing the precision needed to represent their activations and weights, thereby reducing the computation and memory required for their implementation.
The advantages of using such algorithms to create networks for low precision hardware has been demonstrated in several deployed systems (Esser et al., 2016; Jouppi et al., 2017; Qiu et al., 2016) .
It has been shown that low precision networks can be trained with stochastic gradient descent by updating high precision weights that are quantized, along with activations, for the forward and backward pass (Courbariaux et al., 2015; Esser et al., 2016) .
This quantization is defined by a mapping of real numbers to the set of discrete values supported by a given low precision representation (often integers with 8-bits or less).
We would like a mapping for each quantized layer that maximizes task performance, but it remains an open question how to optimally achieve this.
To date, most approaches for training low precision networks have employed uniform quantizers, which can be configured by a single step size parameter (the width of a quantization bin), though more complex nonuniform mappings have been considered (Polino et al., 2018) .
Early work with low precision deep networks used a simple fixed configuration for the quantizer (Hubara et al., 2016; Esser et al., 2016) , while starting with Rastegari et al. (2016) , later work focused on fitting the quantizer to the data, either based on statistics of the data distribution (Li & Liu, 2016; Cai et al., 2017; McKinstry et al., 2018) or seeking to minimize quantization error during training (Choi et al., 2018c; Zhang et al., 2018) .
Most recently, work has focused on using backpropagation with (Jung et al., 2018) , FAQ (McKinstry et al., 2018) , LQ-Nets (Zhang et al., 2018) , PACT (Choi et al., 2018b) , Regularization (Choi et al., 2018c) , and NICE (Baskin et al., 2018 stochastic gradient descent to learn a quantizer that minimizes task loss (Zhu et al., 2016; Mishra & Marr, 2017; Choi et al., 2018b; a; Jung et al., 2018; Baskin et al., 2018; Polino et al., 2018) .
While attractive for their simplicity, fixed mapping schemes based on user settings place no guarantees on optimizing network performance, and quantization error minimization schemes might perfectly minimize quantization error and yet still be non optimal if a different quantization mapping actually minimizes task error.
Learning the quantization mapping by seeking to minimize task loss is appealing to us as it directly seeks to improve on the metric of interest.
However, as the quantizer itself is discontinuous, such an approach requires approximating its gradient, which existing methods have done in a relatively coarse manner that ignore the impact of transitions between quantized states (Choi et al., 2018b; a; Jung et al., 2018) .
Here, we introduce a new way to learn the quantization mapping for each layer in a deep network, Learned
Step Size Quantization (LSQ), that improves on prior efforts with two key contributions.
First, we provide a simple way to approximate the gradient to the quantizer step size that is sensitive to quantized state transitions, arguably providing for finer grained optimization when learning the step size as a model parameter.
Second, we propose a simple heuristic to bring the magnitude of step size updates into better balance with weight updates, which we show improves convergence.
The overall approach is usable for quantizing both activations and weights, and works with existing methods for backpropagation and stochastic gradient descent.
Using LSQ to train several network architectures on the ImageNet dataset, we demonstrate significantly better accuracy than prior quantization approaches (Table 1 ) and, for the first time that we are aware of, demonstrate the milestone of 3-bit quantized networks reaching full precision network accuracy (Table 4) .
The results presented here demonstrate that on the ImageNet dataset across several network architectures, LSQ exceeds the performance of all prior approaches for creating quantized networks.
We found best performance when rescaling the quantizer step size loss gradient based on layer size and precision.
Interestingly, LSQ does not appear to minimize quantization error, whether measured using mean square error, mean absolute error, or Kullback-Leibler divergence.
The approach itself is simple, requiring only a single additional parameter per weight or activation layer.
Although our goal is to train low precision networks to achieve accuracy equal to their full precision counterparts, it is not yet clear whether this goal is achievable for 2-bit networks, which here reached accuracy several percent below their full precision counterparts.
However, we found that such 2-bit solutions for state-of-the-art networks are useful in that they can give the best accuracy for the given model size, for example, with an 8MB model size limit, a 2-bit ResNet-50 was better than a 4-bit ResNet-34 (Figure 3 ).
This work is a continuation of a trend towards steadily reducing the number of bits of precision necessary to achieve good performance across a range of network architectures on ImageNet.
While it is unclear how far it can be taken, it is noteworthy that the trend towards higher performance at lower precision strengthens the analogy between artificial neural networks and biological neural networks, which themselves employ synapses represented by perhaps a few bits of information (Bartol Jr et al., 2015) and single bit spikes that may be employed in small spatial and/or temporal ensembles to provide low bit width data representation.
Analogies aside, reducing network precision while maintaining high accuracy is a promising means of reducing model size and increasing throughput to provide performance advantages in real world deployed deep networks.

assistant: text: A method for learning quantization configuration for low precision networks that achieves state of the art performance for quantized networks.