T RAINING AND I NFERENCE WITH I NTEGERS IN D EEP N EURAL N ETWORKS
Anonymous authors
Paper under double-blind review
A BSTRACT
Researches on deep neural networks with discrete parameters and their deployment in embedded systems have been active and promising topics.
Although previous works have successfully reduced precision in inference, transferring both training and inference processes to low-bitwidth integers has not been demonstrated simultaneously.
In this work, we develop a new method termed as “WAGE” to discretize both training and inference, where weights (W), activations (A), gradients (G) and errors (E) among layers are shifted and linearly constrained to low-bitwidth integers.
To perform pure discrete dataﬂow for ﬁxed-point devices, we further replace batch normalization by a constant scaling layer and simplify other components that are arduous for integer implementation.
Improved accuracies can be obtained on multiple datasets, which indicates that WAGE somehow acts as a type of regularization.
Empirically, We demonstrates the potential to deploy training on hardware systems such as integer-based deep learning accelerators and neuromorphic chips with comparable accuracy and higher energy efﬁciency, which is crucial for future AI applications in variable scenarios with transfer and continual learning demands.
I NTRODUCTION
Recently deep neural networks (DNNs) are being widely used for numerous AI applications (Krizhevsky et al., 2012; Hinton et al., 2012; Silver et al., 2016).
Depending on the massive tunable parameters, DNNs are considered to have powerful multi-level feature extraction and representation abilities.
However, training DNNs needs energy-intensive devices such as GPU and CPU, with high precision (ﬂoat32) processing units and abundant memory capacity, which has greatly challenged their extensive applications in portable devices.
In addition, a state-of-art network often has far more weights and effective capacity to shatter all training samples (Zhang et al., 2016), which leads to overﬁtting easily.
As a result, there is much interest in reducing the size of network during inference(Hubara et al., 2016; Rastegari et al., 2016; Li et al., 2016), as well as dedicated hardware for commercial solutions (Jouppi et al., 2017; Chen et al., 2017; Shi et al., 2015).
Due to the accumulation in stochastic gradient descent (SGD) optimization, the precision demand for training is usually higher than inference (Hubara et al., 2016).
Therefore, most of the existing techniques only focus on the deployment of a well-trained compressed network, while still keeping high precision and computational complexity during training.
In this work, we address the problem as how to process both training and inference with low-bitwidth integers, which is essential for implementing DNNs in dedicated hardware.
To this end, two fundamental issues are addressed for discretely training DNNs: i) how to quantize all the operands and operations, and ii) how many bits or states are needed for SGD computation and accumulation.
With respect to the issues, we propose a framework termed as “WAGE” that constrains weights (W), activations (A), gradients (G) and errors (E) among all layers to low-bitwidth integers in both training and inference.
Firstly, for operands, linear mapping and orientation-preserved shifting operations are applied to achieve ternary weights, 8 or less bits integers for activations and gradients accumulation.
Secondly, for operations, batch normalization (Ioffe & Szegedy, 2015) is replaced by a constant scaling factor.
Other techniques for ﬁne-tuning such as SGD optimizer with momentum and L2 regularization are simpliﬁed or abandoned with little performance degradation.
Considering the full bidirectional propagation, we completely streamline inference into accumulate-compare cycles and training into low-bitwidth multiply-accumulate (MAC) cycles with alignment operations, respectively.
Since low-bitwidth arithmetic operations will not only speed up computation but also reduce the energy consumption and the communication (memory accesses) cost that are required during training (Horowitz, 2014).
We heuristically explore the bitwidth requirements of integers for error computation and gradient accumulation, which have rarely been discussed in previous works.
Experiments indicate that it is the relative values (orientations) rather than absolute values (orders of magnitude) in error that leads previous layers to converge.
Moreover, small values in errors have negligible effects on previous orientations though propagated layer by layer, which can be partially discarded in quantization.
We leverage these phenomena and use an orientation-preserved shifting operator to constrain errors.
As for the gradient accumulation, though weights are quantized to ternary values in inference, a relatively higher bitwidth is indispensable to store and accumulate gradient updates.
The proposed framework is evaluated on MNIST, CIFAR10, SVHN, ImageNet datasets.
Comparing to those who only discretize weights and activations at inference time, it has comparable convergence accuracy and can further alleviate overﬁtting, indicating some type of regularization.
To the best of our knowledge, WAGE is the ﬁrst work to produce pure bidirectional low-precision integer dataﬂow for DNNs, which can be applied for training and inference in dedicated hardware neatly.
We publish the code on GitHub 1 .
R ELATED W ORK
We mainly focus on reducing precision of operations and operands in both training and inference.
Orthogonal and complementary techniques for reducing complexity like network compression, pruning (Han et al., 2015; Zhou et al., 2017) and compact architectures (Howard et al., 2017) are impressively efﬁcient but outside the scope this paper.
Weight and activation Courbariaux et al.
; Hubara et al.
(2016) propose methods to train
DNNs with binary weights (BC) and activations (BNN) successively.
They add noises to weights and activations as a form of regularization but real-valued gradients are accumulated in real-valued variables, suggesting that high precision accumulation is likely required for SGD optimization.
XNORNet (Rastegari et al., 2016) has a ﬁlter-wise scaling factor for weights to improve the performance.
Convolutions in XNOR-Net can be implemented efﬁciently using XNOR logical units and bit-count operations.
However, these factors are calculated simultaneously during training with ﬂoat32, which generally aggravates the training effort.
In TWN (Li et al., 2016) and TTQ (Zhu et al., 2016) two symmetric thresholds are introduced to constrain the weights to be ternary-valued: { +1 , 0 , − 1 } .
They claimed a tradeoff between model complexity and expressive ability.
Gradient computation and accumulation DoReFa-Net (Zhou et al., 2016) quantizes gradients to low-bitwidth ﬂoating-point numbers with discrete states in the backward pass.
TernGrad (Wen et al., 2017) quantizes gradients to ternary values to reduce the overhead of gradient synchronization in distributed training.
Nevertheless, weights in DoReFa-Net and TernGrad are stored and updated with ﬂoat32 like previous works.
Besides, the quantization of batch normalization and its derivative are ignored.
Thus, the overall computation graph for the training process is still presented with ﬂoat32 and more complex with external quantization.
Generally, it is difﬁcult to apply DoReFa-Net training in an integer-based hardware directly, but it shows potential of exploring high-dimensional discrete spaces with discrete gradient descent directions.
WAGE Q UANTIZATION
The main idea of WAGE quantization is to constrain four operands to low-bitwidth integers: weight
W and activation a in inference, error e and gradient g in backpropagation training, see Figure 1.
We extend the original deﬁnition of errors to multi-layer: error e is the gradient of activation a for the perspective of each convolution or fully-connected layer, while gradient g particularly refers to the gradient accumulation of weight W .
Considering the i -th layer of a feed-forward network, we have:
[Equation]
where L is the loss function.
We separate these two terms that are mixed up in most existing schemes.
The gradient of weight g and the gradient of activation e ﬂows to different paths in each layer, which is a fork both in inference and in backward training and generally acts as node of MAC operations.
For the forward propagation in the i -th layer, assuming that weights are stored and accumulated with k G -bit integers, then numerous works strive for a better quantization function Q W ( · ) that maps higher precision weights to their k W -bit reﬂections, for example, [ − 0 .
7] to [ − 1 , 0 , 1] .
Although weights are accumulated in high precision like ﬂoat32, deployment of the reﬂections in dedicated hardware are much more memory efﬁcient after training.
Activations are quantized with function Q A ( · ) to k A bits to align the increased bitwidth caused by MACs.
Weights and activations are discretized to even binary values in previous works, then MACs degrade into logical and bit-count operations that are extremely efﬁcient (Rastegari et al., 2016).
For the backward propagation in the i -th layer, the gradients of activations and weights are calculated by the derivatives of MACs that are generally considered to be in 16-bit ﬂoating-point precision at least.
As illustrated in Figure 1, after the MACs between k A -bit inputs and k W -bit weights are performed, the bitwidth of outputs increases to [ k A + k W − 1] in signed integer representation, and the similar broadening happens in errors e as well.
In consideration of training with only low-bitwidth integers, we propose additional functions Q E ( · ) and Q G ( · ) to constrain bitwidth of e and g to k E bits and k G bits, respectively.
In general, where there is a MAC operation, there are quantization operators named Q W ( · ) , Q A ( · ) , Q G ( · ) and Q E ( · ) in inference and backpropagation.
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Forward the - th layer ! "
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Backward the - th layer [ ! + " #1] [ ! + $ # 1] [ ! + $ #1] : operand : operation : dataflow [ ! + " # 1] [ ]: bit width
3. S HIFT - BASED L INEAR M APPING AND S TOCHASTIC R OUNDING
In WAGE quantization, we adopt a linear mapping with k -bit integers for simplicity, where continuous and unbounded values are discretized with uniform distance σ : [Equation]
[Equation]
Then the basic quantization function that converts a ﬂoating-point number x to its k -bitwidth signed integer representation can be formulated as:
[Equation]
where round approximates continuous values to their nearest discrete states.
Clip is the saturation function that clips unbounded values to [ − 1 + σ, 1 − σ ] , where the negative maximum value − 1 is removed to maintain symmetry.
For example, Q ( x, 2) quantizes {− 1 , 0 . 2 , 0 . 6 } to {− 0 .
Equation 3 is only for simulation in ﬂoating-point hardware like GPU, whereas in a fully ﬁxed-point device, quantization and saturation is satisﬁed automatically.
Before applying linear mapping in some operands (e.g., errors), we introduce an additional monolithic scaling factor for shifting values distribution to an appropriate order of magnitude, otherwise values will be all saturated or cleared by Equation 3.
The scaling factor is calculated by Shift function and then divided in later steps:
[Equation]
[Equation]
Finally, we propose stochastic rounding to substitute small and real-valued updates for gradient accumulation in training.
Section 3.3.4 will detail the implementation of operator Q G ( · ) , where high bitwidth gradients are constrained to k G -bit integers stochastically by a 16-bit random number generator.
Figure 2 summarizes quantization methods used in WAGE.
- -    1e1e-  -
[Figure]
[Equation] & ’ |&| ! "
[Equation] ’ $|&|% [Figure]
[Figure]
3. W EIGHT I NITIALIZATION
In previous works, weights are binarized directly by sgn function or ternarized by threshold parameters calculated during training.
However, BNN fails to converge without batch normalization because weight values ± 1 are rather big for a typical DNN.
Batch normalization not only efﬁciently avoids the problem of exploding and vanishing gradients, but also alleviates the demand for proper initialization.
However, normalizing outputs for each layer and computing their gradients are quite complex without ﬂoating point unit (FPU).
Besides, the moving averages of batch outputs occupy external memory.
BNN shows a shift-based variation of batch normalization but it is hard to transform all of the elements to the ﬁxed-point representations.
As a result, weights should be cautiously initialized in this work where batch normalization is simpliﬁed to a constant scaling layer.
A modiﬁed initialization method based on MSRA (He et al., 2015) can be formulated as:
[Equation]
where n in is the layer fan-in number, and the original limit (cid:112) 6 /n in in MSRA is calculated to keep same variance between inputs and outputs of the same layer theoretically.
The additional limit L min is a minimum value that the uniform distribution U should reach, and β is a constant greater than 1 to create overlaps between minimum step size σ and maximum value L .
In case of k W -bit linear mapping, if weights W are quantized directly with original limits, we will get all-zero tensors when bitwidth k W is small enough, e.g., 4 , or fan-in n in is wide enough, where initialized weights may never reach the minimum step σ presented by ﬁxed-point integers.
So L min ensures that weights can go beyond σ and quantized to non-zero values after Q W ( · ) when initialized randomly.
3. Q UANTIZATION D ETAILS 3.3. W EIGHT Q W ( · )
The modiﬁed initialization in Equation 5 will amplify weights holistically and guarantee their proper distribution, then W is quantized directly with Equation 3:
[Equation]
It should be noted that the variance of weights now are scaled compared to the original limit, which will cause exploding of network’s outputs.
To alleviate the ampliﬁcation effect, XNOR-Net proposed a ﬁlter-wise scaling factor calculated continuously with full precision.
In consideration of integer implementation, we introduce a layer-wise shift-based scaling factor α to attenuate the ampliﬁcation effect:
[Equation]
where α is a pre-deﬁned constant for each layer determined by the network structure.
The modiﬁed initialization and attenuation factor α together approximate ﬂoating-point weights to their integer representations, except that α takes effect after activations to maintain precision of weights presented by k W -bit integers.
3.3. A CTIVATION Q A ( · )
As stated above, the bitwidth of operands increases after MACs.
Then a typical CNN is usually followed with pooling, normalization and activation.
Average pooling is avoided because mean operations will increase precision demand.
Besides, we hypothesize that batch outputs of each hidden layer approximately have zero-mean, then batch normalization degenerates into to a scaling layer where trainable and batch-calculated scaling parameters are replaced by α mentioned in Equation 7.
If activations are presented in k A bits, the overall quantization of activations can be formulated as: [Equation] [Equation] [Equation]
E RROR Q
[Equation]
Errors e are calculated layer by layer using the chain rule during training.
Although the computation graph of backpropagation is similar to the inference, the inputs are the gradients of L , which are relatively small compared to actual inputs for networks.
More importantly, the errors are unbounded and might have signiﬁcantly larger ranges than that of activations, e.g., [10 − 9 , 10 − 4 ] .
DoReFa-Net ﬁrst applies an afﬁne transform on e to map them into [ − 1 , 1] , and then inverts the transform after quantization.
Thus, the quantized e are still presented as ﬂoat32 numbers with discrete states and mostly small values.
However, experiment uncovers that it is the orientation rather than order of magnitude in errors that leads previous layers to converge, then the inverse transformation after quantization in DoReFa-Net is no longer needed.
The orientation-only preservation prompts us to propagate errors with integer thoroughly, where error distribution is ﬁrstly scaled into [ −√ 2 , + √ 2] by dividing a shift factor as shown in Figure 2 and then quantized by Q ( e , k E ) :
[Equation]
where max | e | extracts the layer-wise maximum absolute value among all elements in error e , multichannel for convolution and multi-sample for batch training.
The quantization of error discards large proportion of values smaller than σ , we will discuss the inﬂuence on accuracy later.
3.3. G RADIENT Q G ( · )
Since we only preserve relative values of error after shifting, the gradient updates g derived from
MACs between backward errors e and forward activations a are shifted consequently.
We ﬁrst rescale gradients g with another scaling factor and then bring in shift-based learning rate η :
[Equation]
[Equation]
where η is an integer power of 2.
The shifted gradients g s represent for minimum step numbers and directions for updating weights.
If weights are stored with k G -bit numbers, the minimum step of modiﬁcation will be ± 1 for integers and ± σ ( k G ) for ﬂoating-point values, respectively.
The implement of learning rate η here is quite different from that in training a vanilla DNN based on ﬂoat32.
In WAGE, there only remains directions for weights to change and the step sizes are integer multiples of minimum step σ .
Shifted gradients g s may get greater than 1 if η is 2 or bigger to accelerate training at the beginning, or smaller than 0 .
5 during latter half of training when learning rate decay is usually applied.
As illustrated in Figure 2, to substitute accumulation of small gradients in latter case, we separate g s into integer parts and decimal parts, then use a 16-bit random number generator to constrain high bitwidth g s to k G -bit integers stochastically:
[Equation]
where Bernoulli (Zhou et al., 2016) stochastically samples decimal part to either 0 or 1 .
With proper setting of k G , quantization of gradients will restrict the minimum step size, which may avoid local minimum and overﬁtting.
Furthermore, the gradients will be ternary values when η is not greater than 1 , which reduces communication cost for distributed training (Wen et al., 2017).
At last, weights W might exceed the range [ − 1 + σ, 1 − σ ] presented by k G -bit integers after updating with discrete increments ∆ W .
So Clip function is indispensable to saturate and make sure there are only 2 k G − 1 − 1 states for weights accumulation.
In case of t -th iteration, we have:
[Equation]
3. M ISCELLANEOUS
From the above, we have illustrated our quantization methods for weights, activations, gradients and errors.
See Algorithm 1 for the detailed computation graph.
There remains some issues to specify in an overall training with integers.
Gradient descent optimizer like Momentum, RMSProp and Adam contains at least one copy of gradient updates ∆ W or their moving average, doubling memory consumption for weights during training, which is partially equivalent to use bigger k G .
Since the weight updates ∆ W are quantized to integer multiple of σ and scaled by η , we adopt pure mini-batch SGD without any form of momentum or adaptive learning rate to show the potential of reducing storage demands.
Although L2 regularization works quite well for many large-scale DNNs where overﬁtting occurs commonly, WAGE removes small values in Equation 3 and introduces randomness in Equation 11, acting as certain types of regularization and can get comparable accuracy in later experiments.
Thus, we leave out L2, dropout or other regularization methods.
The Softmax layer and cross-entropy criterion are widely adopted in classiﬁcation tasks but the calculation of e x can hardly be applied in low-bitwidth linear mapping occasions.
For tasks with small number of categories, we avoid Softmax layer and apply mean-square-error criterion but omit mean operation to form a sum-square-error (SSE) criterion since shifted errors will get the same values in Equation 9.
E XPERIMENTS
In this section, we set W-A-G-E bits to 2-8-8-8 as default for all layers in a CNN or MLP.
The bitwidth k W is 2 for ternary weights, which implies that there are no multiplication during inference.
Constant parameter β is 1 .
5 to make equal probabilities for ternary weights when initialized randomly.
Activations and errors should be of the same bitwidth since computation graph of backpropagation is similar to inference and might be applied in the same partition of hardware or memristor array (Sheridan et al., 2017).
Although XNOR-Net achieves 1-bit activations, reducing errors to 4 or less bits dramatically degenerates accuracies in our tests, so bitwidth k A and k E are increased to 8 simultaneously.
Weights are stored with 8-bit integers during training and ternarized by two constant symmetrical thresholds during inference.
Last but not the least, we make no exception for the ﬁrst or the last layer, whereas previous works mostly compromise to the last layer for fear of severe accuracy drop caused by large variation range of outputs fed into the Softmax function (Tang et al., 2017).
We ﬁrst build the computation graph for a vanilla network, then insert quantization nodes in forward propagation and override gradients in backward propagation for each layer on Tensorﬂow (Abadi et al., 2016).
Our method is evaluated on MNIST, SVHN, CIFAR10 and
ILSVRC12 (Russakovsky et al., 2015) and Table 1 shows the comparison results.
4. I MPLEMENT D ETAILS
MNIST A variation of LeNet-5 (LeCun et al., 1998) with 32C5-MP2-64C5-MP2-512FC-10SSE is adopted.
The input grayscale images are regarded as activations and quantized by Equation 8 where α equals to
WAGE
We report average accuracy of 10 runs on
SVHN & CIFAR
We use a VGG-like network (Simonyan & Zisserman, 2014) with 2 × (128C3)-
MP2-2 × (256C3)-MP2-2 × (512C3)-MP2-1024FC-10SSE.
For CIFAR10 dataset, we follow the data augmentation in Lee et al.
(2015) for training: 4 pixels are padded on each side, and a 32 × 32 patch is randomly cropped from the padded image or its horizontal ﬂip.
For testing, only single view of the original 32 × 32 image is evaluated.
The model are trained with mini-batch size of 128 and totally 300 epochs.
Learning rate η is set to 8 and divided by 8 at epoch 200 and epoch 250.
The original images are scaled and biased to the range of [ − 1 , +1] for 8-bit integer activation representation.
As for SVHN dataset, we leave out randomly ﬂip augmentation and reduce training epochs to 40 since it is a rather big dataset.
The error rate is evaluated in the same way as MNIST.
ImageNet WAGE framework is evaluated on ILSVRC12 dataset with AlexNet (Krizhevsky et al., 2012) model but removes dropout and local response normalization layers.
Images are ﬁrstly resized to 256 × 256 then randomly cropped to 224 × 224 and horizontally ﬂipped, followed by bias subtraction as CIFAR10.
For testing, the single center crop in validation set is evaluated.
Since ImageNet task is much difﬁcult than CIFAR10 and has 1000 categories, it is hard to converge when applying
SSE or hinge loss criterion in WAGE, so we compromise to Softmax and remove Q A ( · ) in the last layer.
Besides, k G and k E are increased to 12 to form 28CC pattern for stable convergence.
The model are trained with mini-batch size of 256 and totally 80 epochs.
Learning rate η is set to 32 and divided by 8 at epoch 40 and epoch 60. [Table] [Table]
T RAINING C URVES AND R EGULARIZATION
We further compare WAGE with two variations and a vanilla CNN on CIFAR10.
The vanilla CNN has the same VGG-like architecture described above except that none quantization of any operand or operation is applied.
We add batch normalization in each layer and Softmax for the last layer, replace SSE with cross-entropy criterion [Table] and then use a L 2 weight decay of 1e-4 and momentum of 0 . 9 for training.
The learning rate is set to 0.1 and divided by 10 at epoch 200 and epoch 250.
For variations of WAGE, pattern 28ff has no quantization nodes in backpropagation and pattern 28ff-BN has batch normalization.
Figure 3 shows the training curves of four counterparts.
It can be seen that the 2888 pattern has comparable convergence rate to the vanilla CNN, better accuracy than those who only discretize weights and activations in inference time, though slightly more volatile.
The discretization of backpropagation somehow acts as another type of regularization and have signiﬁcant error rate drop when decreasing learning rate η .
Moreover, comparison of 28ff and 28ffBN curves reveals that the scaling factor α introduced in Equation 7 reduces the demand for batch normalization.
[Figure]
Vanilla 28FF-BN 28FF
4. B ITWIDTH OF E RRORS
The bitwidth k E is set to 8 as default in previous experiments.
To further explore a proper bitwidth and its truncated boundary, we ﬁrstly export errors from vanilla CNN for CIFAR10 after 100 training epochs.
The histogram of errors in the last convolution layer among 128 mini-batch data is shown in Figure 4.
It is obvious that errors approximately obey logarithmic normal distribution where values are relatively small and have signiﬁcantly large range.
When quantized with k E -bit integers, a proper window function should be chosen to truncate the distribution while retaining the approximate orientations for backpropagation.
Firstly, the upper (right) boundary is immobilized to the maximum absolute value among all elements in errors as described in Equation 9.
Then the left boundary will be based on the bitwidth k E .
We conduct a series of experiments for k E ranging from 4 to 15.
The boxplot in Figure 4 indicates that 4-8 bits of errors represented by integers are enough for CIFAR10 classiﬁcation task.
Bitwidth 8 is chosen as default to match the 8-bit image color levels and most operands in the micro control unit (MCU).
The histogram of errors in the same layer of WAGE-2888 shows that after being shifted and quantized layer by layer, the distribution of errors reshapes and mostly aggregates into truncated window.
Thus, most information of orientations are retained.
Besides, the smaller values in errors have negligible effects on previous orientations though accumulated layer by layer, which are partially discarded in quantization.
Since the width of the window has been optimized, we left-shift the window with factor γ to explore its horizontal position.
The right boundary can be formulated as max {| e |} /γ .
Table 2 shows the effect of shifting errors: although large values are in the minority, they play critical roles for backpropagation training while the majority with small values actually act as noises.
[Table]
4. B ITWIDTH OF G RADIENTS
The bitwidth k G is set to 8 as default in previous experiments.
Although weights are propagated with ternary values in inference and achieve 16 × compression rate than ﬂoat32 weights, they are saved and accumulated in a relatively higher bitwidth (8 bits) for backpropagation training.
Therefore, the overall compression rate is only 4 × .
The inconsistent bitwidth between weight updates k G and their effects in inference k W provides indispensable buffer space.
Otherwise, there might be too many weights changing their ternary values in each iteration, making training very slow and unstable.
To further explore a proper bitwidth for gradients, we use WAGE 2-8-8-8 in CIFAR10 as
[Figure]
baseline and range k G from 2 to 12, the learning rate η is divided by 2 every time the k G decreases 1 bit to keep approximately equal weights accumulation in large number of iterations.
Results in
Table 3 show the effect of k G and indicate the similar bitwidth requirement as previous experiments for k E .
For ImageNet implementation, we conduct six patterns to show bitwidth requirements:
[Table]
28CC from Table 1, 288C for smaller buffer space, 28Cf for no quantization of errors and 28fC for no quantization of gradients, 28ff for errors and gradients in ﬂoat32 as unlimited case and its BN counterpart.
The accuracy of original Alexnet reproduction is reported as baseline.
Learning rate η is set to 4 and divided by 8 in 288C pattern, 0.1 and divided by 10 in 28ff counterparts and vanilla Alexnet.
Table 4 indicates higher bits errors and gradients are preferred for ImageNet classiﬁcation task.
Besides, the comparison between pattern 28Cf and 28fC reveal that it might be more important to keep high-resolution orientation k E than to make more buffer space k G for gradient accumulation.
To avoid external memory consumption for full-precision hidden weights in the training, Deng et al.
(2017) achieved overall 1-bit weights representation in both training and inference.
They use a much larger mini-batch size of 1000 and ﬂoat32 backpropagation dataﬂow to accumulate more precise weight updates, equally compensating the buffer space in WAGE provided by external bits of k G .
However, large batch size will dramatically increase total training time, counteracting the speed beneﬁts brought by integer arithmetic units.
Besides, intermediate variables like feature maps in hidden layer often consume much more memory than weights and linearly correlated with mini-batch size.
Therefore, we compromise to bigger inconsistent k G for better convergence rate, accuracy and smaller mini-batch size.
[Table]
D ISCUSSION AND F UTURE W ORK
The goal of this work is to demonstrate potentials of applying training and inference with ﬁxed-point integers in DNNs.
We mainly reduce bitwidths of operations and operands for an overall integer dataﬂow and introduce many techniques to simplify the training process.
However, there are some points not involved in this work but yet to be improved or solved in future algorithm developments and hardware deployment.
MAC Operation : WAGE framework is mainly tested with 2-8-8-8 bitwidth conﬁguration, which means that though there are no multiplications during inference with ternary weights, MACs are still needed to calculate g in training.
Possible solution is 2-2-8-8 pattern if we do not consider the matching of bitwidths between a and e .
However, ternary a will dramatically slow down convergence and hurt accuracy since Q ( x, 2) has two relatively high thresholds and clear most outputs of each layer at the beginning of training, this phenomenon is also observed in our BNN replication.
Non-linear quantization : The linear mapping with uniform distance is adopted in WAGE for its simplicity.
However, non-linear quantization method like logarithmic representation (Miyashita et al., 2016) might be more efﬁcient because the weights and activations in a trained network naturally have logarithmic normal distributions as shown in Figure 4.
Besides, values in logarithmic representation have much larger range in fewer bits than ﬁxed-point representation and are naturally encoded in digital hardware.
It is promising to training DNNs with integers encoded with logarithmic representation.
Normalization : Normalization layers like Softmax and batch normalization are avoided or removed in some WAGE demonstrations.
We think normalizations are essential for end-to-end multichannel perception where sensors with different modalities have different input distributions, as well as cross-model features encoding and cognition where information from different branches gather to form higher-level representations.
Therefore, a better way to quantize normalization is of great interest in further studies.
C ONCLUSION
WAGE empowers pure low-bitwidth integer dataﬂow in DNNs for both training and inference.
We introduce a new initialization method and a layer-wise constant scaling factor to replace batch normalization, which is a pain spot for network quantization.
Many other components for training are also considered or simpliﬁed by alternative solutions.
In addition, the bitwidth requirements for error computation and gradient accumulation are explored.
Experiments reveal that we can quantized relative values of gradients, as well as discard the majority of small values and their orders of magnitude in backpropagation.
Although the accumulation for weights updates are indispensable for stable convergence and ﬁnal accuracy, there are still some space for compression and memory consumption can be further reduced in training.
WAGE achieves state-of-art accuracies on multiple datasets with 2-8-8-8 bitwidth conﬁguration.
It is promising for incremental works via ﬁne-tuning, more efﬁcient mapping methods, quantization of batch normalization, etc.
Overall, we introduce a framework without ﬂoating-point representation and demonstrate the potential to implement both discrete training and inference on integer-based lightweight ASIC or FPGA with on-site learning capability.
A A LGORITHM
We assume that network structures are deﬁned and initialized with Equation 5.
The annotations after pseudo codes are potential corresponding operations for implementation in a ﬁxed-point dataﬂow.
Require: a mini-batch of inputs and targets ( a 0 q , a ∗ ) which are quantized to k A -bit integers, shiftbased α for each layer, learning rate scheduler η , previous weight W saved in k G bits.
Ensure: updated weights W t + 1.
Forward propagation: 1: for i = 1 to I do 2:
W iq ← Q W ( W i )
#Clip 3: a i ← ReLU ( a i − 1 q W iq )
#MAC, Clip 4: a iq ← Q A ( a i ) #Shift, Clip 5: end for 2.
Back propagation:
Compute e I ← ∂ L ∂ a
I knowing a I and a ∗ #Substrate 6: for i
= I to 1 do 7: e iq ← Q E ( e i )
#Max, Shift, Clip 8: e i − 1 ← e iq W iq #MAC, Clip 9: g i ← e iq T a
i − 1 q #MAC, Clip 10: ∆ W i ← Q G ( g i )
#Max, Shift, Random, Clip 11:
Update and Clip W i according to Equation#Update, Clip 12: end for