2020. In figure3(a), we generalize this process for a convolutional layer, but we also include an activation function to make it more realistic. Quantization is the process to convert a floating point model to a quantized model. Use, Smithsonian The code is working perfectly for float32 to int8 but not for what I want. The visualization step can reveal the source of the tensors sensitivity to quantization. Together, CLE and bias absorption followed by per-tensor quantization yield better results than per-channel quantization. However, we could use finer granularity to further improve performance. The second term, however, depends on the input data x. But the lack of offset restricts the mapping between integer and floating-point domain. For both solutions, we provide tested pipelines based on existing literature and extensive experimentation that lead to state-of-the-art performance for common deep learning models and tasks. Applying analytical bias correction improves quantized model performance from random to over 50%, indicating that the biased error introduced by quantization significantly harms model performance. This frees the neural network designer from having to be an expert in quantization and thus allows for a much wider application of neural network quantization. In this section, we start by discussing various common methods used in practice to find good quantization parameters. In this white paper, we. In this section, we describe AdaRound (Nagel et al., 2020), a systematic approach to finding good weight rounding choices for PTQ. To make PTQ still work, we identified these layers using our debugging procedure outlined in section 3.7 and kept them in 16 bit. Notice, Smithsonian Terms of In both papers, the authors observe that for many common activation functions (e.g., ReLU, PreLU), a positive scaling equivariance holds: for any non-negative real number s. This equivariance holds for any homogeneous function of degree one and can be extended to also hold for any piece-wise linear function by scaling its parameterization (e.g. This allows the model to find more optimal solutions than post-training quantization. The modified quantization function is defined as: The gradient w.r.t. The calculation starts by loading the accumulators with the bias value bn. In figure 11 we show the full training curve of this experiment. We further assume that the model has converged, implying that the contribution of the gradient term in the approximation can be ignored, and that the Hessian is block-diagonal, which ignores cross-layer correlations. quantization. (2018) both updates the running statistics during QAT and applies BN-folding using a correction. Quantizing deep convolutional networks for-1806.08342.pdf 858KB. On one hand, asymmetric quantization is more expressive because there is an extra offset parameter, but on the other hand there is a possible computational overhead. approximately 4% drop for EfficientNet lite with per-tensor quantization. As discussed in previous sections, we always start from a pre-trained model and follow some PTQ steps in order to have faster convergence and higher accuracy. We use the MSE based criteria for most of the layers, which requires a small calibration set to find the minimum MSE loss. Computer Science - Artificial Intelligence; Computer Science - Computer Vision and Pattern Recognition. The activations stored in the 32-bit accumulators need to be written to memory before they can be used by the next layer. A fundamental step in the PTQ process is finding good quantization ranges for each quantizer. For both solutions, we provide tested pipelines based on existing literature and extensive experimentation that lead to state-of-the-art performance for common deep learning models and tasks. We use the Adam optimizer for all models. This scheme is called uniform quantization and it is the most commonly used quantization scheme because it permits efficient implementation of fixed-point arithmetic. We find that, perhaps surprisingly, this is not the best we can do. (2019) introduces a similar scaling factor that also takes the intermediate activation tensor into account. W4A8 stays within 1% of the original GLUE score, indicating that low bit weight quantization is not a problem for transformer models. 2013), which approximates the gradient of the rounding operator as 1: Using this approximation we can now calculate the gradient of the quantization operation from equation (7). We then load the weight values Wn,m and the input values xm into the array and compute their product in the respective processing elements Cn,m=Wn,mxm in a single cycle. To absorb c from layer one (followed by a ReLU activation function f) into layer two, we can do the following reparameterization: where b(2)=W(2)c+b(2), h=hc, and b(1)=b(1)c. Neural network quantization is one of the most effective ways of achieving these savings but the additional noise it induces can lead to accuracy degradation. I'm trying to use the same procedure to quantize float32 to int16. To this end, we consider two main classes of quantization algorithms: Post-Training Quantization (PTQ) and Quantization-Aware Training (QAT). Quantization-aware training models the quantization noise during training through simulated quantization operations. ), A. Wang, A. Singh, J. Michael, F. Hill, O. Nagel et al. We demonstrate that with our QAT pipeline we can achieve 4-bit quantization of weights, and for some models even 4-bit activations, with only a small drop of accuracy compared to floating-point. One of the most impactful ways to decrease the computational time and energy consumption of neural networks is quantization. Per-channel quantization of activations is much harder to implement because we cannot factor the scale factor out of the summation and would, therefore, require rescaling the accumulator for each input channel. arXiv Vanity renders academic papers from They are very effective and fast to implement because they do not require retraining of the network with labeled data. To remove dependence on data, the authors propose to estimate the right hand side of (24) by the shift and scale parameters of the batch normalization layer which results111Assuming x is normally distributed, the equality will hold for approximately 99.865% of the inputs. In later work (Esser et al., 2020; Jain et al., 2019; Bhalgat et al., 2020), the STE is used to calculate the gradient w.r.t. However, determining parameters for activation quantization often requires a few batches of calibration data. 2018; A white-paper: . If we were to perform inference in FP32, the processing elements and the accumulator would have to support floating-point logic, and we would need to transfer the 32-bit data from memory to the processing units. In section 3.1, we explore in more detail how to choose the quantization parameters to achieve the right trade-off between clipping and rounding errors. Assuming a weight matrix WRnm we apply batch normalization to each output yk for k={1,,n}: In our naive quantized accelerator introduced in section 2.1, we saw that the requantization of activations happens after the matrix multiplication or convolutional output values are calculated. Levy, and S. Bowman (2018), GLUE: a multi-task benchmark and analysis platform for natural language understanding. So far, we have defined a single set of quantization parameters (quantizer) per tensor, one for the weights and one for activations, as seen in equation (3). DeeplabV3 (MobileNetV2 backbone) is evaluated on Pascal VOC (mean intersection over union), EfficientDet-D1 on COCO 2017 (mean average precision), BERT-base on the GLUE benchmark and other models on ImageNet (accuracy). In most cases, PTQ is sufficient for achieving 8-bit quantization However, the higher accuracy comes with the usual costs of neural network training, i.e., longer training times, need for labeled data and hyper-parameter search. While neural networks have advanced the frontiers in many applications, they often come at a high computational cost. These methods can be data-free or may require a small calibration set, which is often readily available. If we want to reduce the clipping error we can expand the quantization range by increasing the scale factor. In this section, we present a best-practice pipeline for QAT based on relevant literature and extensive experimentation. This is called quantization simulation. Batch normalization normalizes the output of a linear layer before scaling and adding an offset (see equation9). We then explore common issues observed during PTQ and introduce the most successful techniques to overcome them. We start with a hardware motivated introduction to quantization and then consider two main classes of algorithms: Post-Training Quantization (PTQ) and Quantization-Aware-Training (QAT). A floating-point vector x can be expressed approximately as a scalar multiplied by a vector of integer values: where sx is a floating-point scale factor and xint is an integer vector, e.g., INT8. To avoid error accumulation across layers of the neural network and to account for the non-linearity, the authors propose the following final optimization problem. Similar to PTQ, we introduce a standard training pipeline utilizing the latest algorithms in the field. While the MSE initialized model has a significantly higher starting accuracy, the gap closes after training for 20 epochs. Symmetric quantization is a simplified version of the general asymmetric case. Impact of cross-layer equalization (CLE) for MobileNetV2. Reducing the power and latency of neural network inference is key if we want to integrate modern networks into edge devices with strict power and compute requirements. N/A implies that the corresponding experiment was computationally . Our toy example in figure 1 has 16 processing elements arranged in a square grid and 4 accumulators. It is common practice in literature to start from a pre-trained FP32 model (Esser et al., 2020; Krishnamoorthi, 2018; Jacob et al., 2018; Jain et al., 2019). AdaRound provides a theoretically sound, computationally fast weight rounding method. The two branches that are being concatenated generally do not share the same quantization parameters. 3. If the problem is fixed and the accuracy recovers, we continue to the next quantizer. While this is not strictly an algorithm, these debugging steps can provide insights on why a quantized model underperforms and help to tackle the underlying issues. This requires a requantization step which is shown in figure 2. As shown in Figure 1, a standard neural network consists of layers of interconnected neurons, each with its own weight, bias, and activation function associated with it. The learning rate adjustment can be avoided if we use optimizers with adaptive learning rates such as Adam or RMSProp. For everything else, email us at [emailprotected]. However, Nagel et al. Both approaches are an essential part of any model efficiency toolkit and we hope that our proposed pipelines will help engineers deploy high-performing quantized models with less time and effort. Average ImageNet validation accuracy (%) over 5 runs. . (quantization-aware training QAT) A White Paper on Neural Network Quantization12 A White Paper on Neural Network Quantization3PTQ This process is. Note that in this white paper we only consider homogeneous bit-width. quantization noise on the network's performance while maintaining low-bit In most cases, PTQ is sufficient for achieving 8-bit quantization with close to floating-point accuracy. To illustrate this the authors quantized the weights of the first layer ofResNet18 to 4 bits using 100 different stochastic rounding samples (Gupta et al., 2015) and evaluated the performance of the network for each rounding choice. ai study (2020) is, where is annealed during the course of optimization to initially allow free movement of \mathnormalh(Vi,j) and later to force them to converge to 0 or 1. Thank you for the reply. While neural networks have advanced the frontiers in many applications, they often come at a high computational cost. Average ImageNet validation accuracy (%) over 3 runs. Training competitive binary neural networks from scratch. If this support is not available, we need to add a quantization step before and after the non-linearity in the graph. This poses an issue because the gradient of the round-to-nearest operation in equation (4) is either zero or undefined everywhere, which makes gradient-based training impossible. QAT requires fine-tuning and access to labeled training data but enables lower bit quantization with competitive results. weights and activations. The regularizer used in Nagel et al. Most existing fixed-point accelerators do not currently support such logic and for this reason, we will not consider them in this work. For on-device inference, these operations are folded into the previous or next linear layers in a step called batch normalization folding (Krishnamoorthi, 2018; Jacob et al., 2018). A simple but effective approach to modeling BN-folding in QAT is to statically fold the BN scale and offset into the linear layers weights and bias, as we saw in equations(11) and(12). In table 9 we compare the effect of other PTQ improvements such as CLE and bias correction. Therefore, it is important to check if it is possible in your intended target device. Neural network When quantizing neural networks, assigning each floating-point weight to Generative adversarial networks (GANs) have an enormous potential impact Quantization is wildly taken as a model compression technique, which obt Neural networks are essential components of learning-based software syst Data clipping is crucial in reducing noise in quantization operations an Neural Network Quantization with AI Model Efficiency Toolkit (AIMET), Up or Down? DeeplabV3 (MobileNetV2 backbone) is evaluated on Pascal VOC (mean intersection over union), EfficientDet-D1 on COCO 2017 (mean average precision), BERT-base on the GLUE benchmark and all other models on ImageNet (accuracy). (2019) showed that this is especially prevalent in depth-wise separable layers since only a few weights are responsible for each output feature and this might result in higher variability of the weights. More formally, during inference, batch normalization is defined as an affine map of the output x: where and are the mean and variance computed during training as exponential moving average over batch-statistics, and and are learned affine hyper-parameters per-channel. Set the quantized model bit-width to 32 bits for both weights and activation, or by-pass the quantization operation, if possible, and check that the accuracy matches that of the FP32 model. Neural networks are commonly trained using FP32 weights and activations. QAT models the quantization noise source (see section 2.3) during training. A schematic of matrix-multiply logic in an neural network accelerator for quantized inference. However, if after following the steps of our pipeline, the models performance is still not satisfactory, we recommend a set of diagnostics steps to identify the bottlenecks and improve the performance. Post-training quantization (PTQ) algorithms take a pre-trained FP32 network and convert it directly into a fixed-point network without the need for the original training pipeline. Post-training techniques may not be enough to mitigate the large quantization error incurred by low-bit quantization. This is likely due to the catastrophic performance drop caused by per-tensor quantization, which we discussed in section 3.2. [width=0.75]figures/On_Device_Fixed_Point_Inference. Let us see how this is possible by revisiting the BN folding equation from section2.3.1, but this time introduce per-channel quantization of the weights, such that Wk,:=q(Wk,:;sw,k)=sw,kWintk,:. To counteract this shift we can substract it from the output: Note, this correction term is a vector with the same shape as the bias and can thus be absorbed into the bias without any additional overhead at inference time. For each weight and activation quantization, we have to choose a quantization scheme. While some networks are robust to this noise, other networks require extra work to exploit the benefits of quantization. This kind of error is more pronounced in depth-wise separable layers with only a few elements per output channel (usually 9 for a 33 kernel). Another approach is to optimize the network by tying the quantization grids of the inputs. CF18ACM; UNIQ: Uniform noise injection for non-uniform quantization of neural networks. This means that their quantization grids may not overlap making a requantization step necessary. Deep learning has become an integral part of many machine learning applications and can now be found in countless electronic devices and services, from smartphones and home appliances to drones, robots and self-driving cars. Markus Nagel, Marios Fournarakis, Rana Ali Amjad, Yelysei Bondarenko, Mart van Baalen, Tijmen Blankevoort. This can have a big impact on the accuracy of quantized model. to z is calculated by applying the STE once again to the rounding operator: In section 2.3.1, we introduced batch normalization folding that absorbs the scaling and addition into a linear layer to allow for more efficient inference. In this white paper, we introduce state-of-the-art algorithms for mitigating the impact of quantization noise on the network's performance while maintaining low-bit weights and activations. Some common solutions involve custom range setting for this quantizer or allowing a higher bit-width for problematic quantizer, e.g., BERT-base from table6. The algorithm ofMeller et al. In the specific case of per-channel quantization, using the min-max method can be favorable in some cases. Quantization range setting refers to the method of determining clipping thresholds of the quantization grid, qmin and qmax (see equation7).
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