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1603.06147#43 | A Character-Level Decoder without Explicit Segmentation for Neural Machine Translation | David Vilar, Jan-T Peter, and Hermann Ney. 2007. In Proceedings of the Can we translate letters? Second Workshop on Statistical Machine Transla- tion, pages 33â 39. Association for Computational Linguistics. Philip Williams, Rico Sennrich, Maria Nadejde, Matthias Huck, and Philipp Koehn. 2015. Edin- burghâ s syntax-based systems at wmt 2015. In Pro- ceedings of the Tenth Workshop on Statistical Ma- chine Translation, pages 199â 209. 2016. Efï¬ cient character-level document classiï¬ | 1603.06147#42 | 1603.06147#44 | 1603.06147 | [
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1603.06147#44 | A Character-Level Decoder without Explicit Segmentation for Neural Machine Translation | cation by combin- ing convolution and recurrent layers. arXiv preprint arXiv:1602.00367. Xiang Zhang, Junbo Zhao, and Yann LeCun. 2015. Character-level convolutional networks for text clas- siï¬ cation. In Advances in Neural Information Pro- cessing Systems, pages 649â 657. | 1603.06147#43 | 1603.06147 | [
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1603.05279#0 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | 6 1 0 2 g u A 2 ] V C . s c [ 4 v 9 7 2 5 0 . 3 0 6 1 : v i X r a # XNOR-Net: ImageNet Classiï¬ cation Using Binary Convolutional Neural Networks Mohammad Rastegariâ , Vicente Ordonezâ , Joseph Redmonâ , Ali Farhadiâ â Allen Institute for AIâ , University of Washingtonâ {mohammadr,vicenteor}@allenai.org {pjreddie,ali}@cs.washington.edu Abstract. We propose two efï¬ | 1603.05279#1 | 1603.05279 | [
"1602.07360"
] |
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1603.05279#1 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | cient approximations to standard convolutional neural networks: Binary-Weight-Networks and XNOR-Networks. In Binary-Weight- Networks, the ï¬ lters are approximated with binary values resulting in 32à mem- ory saving. In XNOR-Networks, both the ï¬ lters and the input to convolutional layers are binary. XNOR-Networks approximate convolutions using primarily bi- nary operations. This results in 58à faster convolutional operations (in terms of number of the high precision operations) and 32à memory savings. XNOR-Nets offer the possibility of running state-of-the-art networks on CPUs (rather than GPUs) in real-time. Our binary networks are simple, accurate, efï¬ cient, and work on challenging visual tasks. We evaluate our approach on the ImageNet classiï¬ - cation task. The classiï¬ | 1603.05279#0 | 1603.05279#2 | 1603.05279 | [
"1602.07360"
] |
1603.05279#2 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | cation accuracy with a Binary-Weight-Network version of AlexNet is the same as the full-precision AlexNet. We compare our method with recent network binarization methods, BinaryConnect and BinaryNets, and out- perform these methods by large margins on ImageNet, more than 16% in top-1 accuracy. Our code is available at: http://allenai.org/plato/xnornet. # 1 Introduction Deep neural networks (DNN) have shown signiï¬ cant improvements in several applica- tion domains including computer vision and speech recognition. In computer vision, a particular type of DNN, known as Convolutional Neural Networks (CNN), have demon- strated state-of-the-art results in object recognition [1,2,3,4] and detection [5,6,7]. Convolutional neural networks show reliable results on object recognition and de- tection that are useful in real world applications. Concurrent to the recent progress in recognition, interesting advancements have been happening in virtual reality (VR by Oculus) [8], augmented reality (AR by HoloLens) [9], and smart wearable devices. Putting these two pieces together, we argue that it is the right time to equip smart portable devices with the power of state-of-the-art recognition systems. However, CNN- based recognition systems need large amounts of memory and computational power. While they perform well on expensive, GPU-based machines, they are often unsuitable for smaller devices like cell phones and embedded electronics. For example, AlexNet[1] has 61M parameters (249MB of memory) and performs 1.5B high precision operations to classify one image. These numbers are even higher for deeper CNNs e.g.,VGG [2] (see section 4.1). These models quickly overtax the limited storage, battery power, and compute capabilities of smaller devices like cell phones. | 1603.05279#1 | 1603.05279#3 | 1603.05279 | [
"1602.07360"
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1603.05279#3 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | 2 Rastegari et al. Network Variations Operations | Memory | Computation | accuracy on used in Saving Saving ImageNet Convolution | (Inference) | (Inference) | (AAlexNet) Real-Value inputs Standard Convolution [o.1-021 -034 +,7-,% 1x 1x %56.7 } Input wo2s061 082 pal f 7 Real-Value Inputs ho A) Binary Weights p.|. «Binary Weight |o.11.021 2038 irr +,7- ~32x ~2x %56.8 Neight 0.25 0.61... 052°) Ad Win - 2 Binary inputs ye BinaryWeight nay wees | NOR Binary input || 4-14 TE "| 32x ~58x 44.2 (KNORNet) | ttt Ee bitcount Fig. 1: We propose two efï¬ | 1603.05279#2 | 1603.05279#4 | 1603.05279 | [
"1602.07360"
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1603.05279#4 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | cient variations of convolutional neural networks. Binary- Weight-Networks, when the weight ï¬ lters contains binary values. XNOR-Networks, when both weigh and input have binary values. These networks are very efï¬ cient in terms of memory and computation, while being very accurate in natural image classiï¬ - cation. This offers the possibility of using accurate vision techniques in portable devices with limited resources. In this paper, we introduce simple, efï¬ cient, and accurate approximations to CNNs by binarizing the weights and even the intermediate representations in convolutional neural networks. Our binarization method aims at ï¬ nding the best approximations of the convolutions using binary operations. We demonstrate that our way of binarizing neural networks results in ImageNet classiï¬ cation accuracy numbers that are comparable to standard full precision networks while requiring a signiï¬ cantly less memory and fewer ï¬ oating point operations. We study two approximations: Neural networks with binary weights and XNOR- Networks. In Binary-Weight-Networks all the weight values are approximated with bi- nary values. A convolutional neural network with binary weights is signiï¬ cantly smaller (â ¼ 32à ) than an equivalent network with single-precision weight values. In addition, when weight values are binary, convolutions can be estimated by only addition and subtraction (without multiplication), resulting in â ¼ 2à | 1603.05279#3 | 1603.05279#5 | 1603.05279 | [
"1602.07360"
] |
1603.05279#5 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | speed up. Binary-weight ap- proximations of large CNNs can ï¬ t into the memory of even small, portable devices while maintaining the same level of accuracy (See Section 4.1 and 4.2). To take this idea further, we introduce XNOR-Networks where both the weights and the inputs to the convolutional and fully connected layers are approximated with binary values1. Binary weights and binary inputs allow an efï¬ cient way of implement- ing convolutional operations. If all of the operands of the convolutions are binary, then the convolutions can be estimated by XNOR and bitcounting operations [11]. XNOR- Nets result in accurate approximation of CNNs while offering â | 1603.05279#4 | 1603.05279#6 | 1603.05279 | [
"1602.07360"
] |
1603.05279#6 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | ¼ 58à speed up in CPUs (in terms of number of the high precision operations). This means that XNOR-Nets can enable real-time inference in devices with small memory and no GPUs (Inference in XNOR-Nets can be done very efï¬ ciently on CPUs). To the best of our knowledge this paper is the ï¬ rst attempt to present an evalua- tion of binary neural networks on large-scale datasets like ImageNet. Our experimental 1 fully connected layers can be implemented by convolution, therefore, in the rest of the paper, we refer to them also as convolutional layers [10]. XNOR-Net: ImageNet Classiï¬ cation Using Binary Convolutional Neural Networks results show that our proposed method for binarizing convolutional neural networks outperforms the state-of-the-art network binarization method of [11] by a large margin (16.3%) on top-1 image classiï¬ cation in the ImageNet challenge ILSVRC2012. | 1603.05279#5 | 1603.05279#7 | 1603.05279 | [
"1602.07360"
] |
1603.05279#7 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | Our contribution is two-fold: First, we introduce a new way of binarizing the weight val- ues in convolutional neural networks and show the advantage of our solution compared to state-of-the-art solutions. Second, we introduce XNOR-Nets, a deep neural network model with binary weights and binary inputs and show that XNOR-Nets can obtain sim- ilar classiï¬ cation accuracies compared to standard networks while being signiï¬ cantly more efï¬ cient. | 1603.05279#6 | 1603.05279#8 | 1603.05279 | [
"1602.07360"
] |
1603.05279#8 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | Our code is available at: http://allenai.org/plato/xnornet # 2 Related Work Deep neural networks often suffer from over-parametrization and large amounts of re- dundancy in their models. This typically results in inefï¬ cient computation and memory usage[12]. Several methods have been proposed to address efï¬ cient training and infer- ence in deep neural networks. Shallow networks: Estimating a deep neural network with a shallower model re- duces the size of a network. Early theoretical work by Cybenko shows that a network with a large enough single hidden layer of sigmoid units can approximate any decision boundary [13]. In several areas (e.g.,vision and speech), however, shallow networks cannot compete with deep models [14]. [15] trains a shallow network on SIFT features to classify the ImageNet dataset. | 1603.05279#7 | 1603.05279#9 | 1603.05279 | [
"1602.07360"
] |
1603.05279#9 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | They show it is difï¬ cult to train shallow networks with large number of parameters. [16] provides empirical evidence on small datasets (e.g.,CIFAR-10) that shallow nets are capable of learning the same functions as deep nets. In order to get the similar accuracy, the number of parameters in the shallow net- work must be close to the number of parameters in the deep network. They do this by ï¬ rst training a state-of-the-art deep model, and then training a shallow model to mimic the deep model. These methods are different from our approach because we use the standard deep architectures not the shallow estimations. Compressing pre-trained deep networks: Pruning redundant, non-informative weights in a previously trained network reduces the size of the network at inference time. Weight decay [17] was an early method for pruning a network. Optimal Brain Damage [18] and Optimal Brain Surgeon [19] use the Hessian of the loss function to prune a network by reducing the number of connections. Recently [20] reduced the number of parameters by an order of magnitude in several state-of-the-art neural net- works by pruning. [21] proposed to reduce the number of activations for compression and acceleration. Deep compression [22] reduces the storage and energy required to run inference on large networks so they can be deployed on mobile devices. They remove the redundant connections and quantize weights so that multiple connections share the same weight, and then they use Huffman coding to compress the weights. HashedNets [23] uses a hash function to reduce model size by randomly grouping the weights, such that connections in a hash bucket use a single parameter value. Matrix factorization has been used by [24,25]. We are different from these approaches because we do not use a pretrained network. We train binary networks from scratch. | 1603.05279#8 | 1603.05279#10 | 1603.05279 | [
"1602.07360"
] |
1603.05279#10 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | 4 Rastegari et al. Designing compact layers: Designing compact blocks at each layer of a deep net- work can help to save memory and computational costs. Replacing the fully connected layer with global average pooling was examined in the Network in Network architec- ture [26], GoogLenet[3] and Residual-Net[4], which achieved state-of-the-art results on several benchmarks. The bottleneck structure in Residual-Net [4] has been proposed to reduce the number of parameters and improve speed. | 1603.05279#9 | 1603.05279#11 | 1603.05279 | [
"1602.07360"
] |
1603.05279#11 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | Decomposing 3 Ã 3 convo- lutions with two 1 Ã 1 is used in [27] and resulted in state-of-the-art performance on object recognition. Replacing 3 Ã 3 convolutions with 1 Ã 1 convolutions is used in [28] to create a very compact neural network that can achieve â ¼ 50Ã reduction in the number of parameters while obtaining high accuracy. Our method is different from this line of work because we use the full network (not the compact version) but with binary parameters. ing high performance in deep networks. [29] proposed to quantize the weights of fully connected layers in a deep network by vector quantization techniques. They showed just thresholding the weight values at zero only decreases the top-1 accuracy on ILSVRC2012 by less than %10. [30] proposed a provably polynomial time algorithm for training a sparse networks with +1/0/-1 weights. | 1603.05279#10 | 1603.05279#12 | 1603.05279 | [
"1602.07360"
] |
1603.05279#12 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | A ï¬ xed-point implementation of 8-bit integer was compared with 32-bit ï¬ oating point activations in [31]. Another ï¬ xed-point net- work with ternary weights and 3-bits activations was presented by [32]. Quantizing a network with L2 error minimization achieved better accuracy on MNIST and CIFAR-10 datasets in [33]. [34] proposed a back-propagation process by quantizing the represen- tations at each layer of the network. To convert some of the remaining multiplications into binary shifts the neurons get restricted values of power-of-two integers. In [34] they carry the full precision weights during the test phase, and only quantize the neu- rons during the back-propagation process, and not during the forward-propagation. Our work is similar to these methods since we are quantizing the parameters in the network. But our quantization is the extreme scenario +1,-1. Network binarization: These works are the most related to our approach. Several methods attempt to binarize the weights and the activations in neural networks.The per- formance of highly quantized networks (e.g.,binarized) were believed to be very poor due to the destructive property of binary quantization [35]. Expectation BackPropaga- tion (EBP) in [36] showed high performance can be achieved by a network with binary weights and binary activations. This is done by a variational Bayesian approach, that infers networks with binary weights and neurons. A fully binary network at run time presented in [37] using a similar approach to EBP, showing signiï¬ | 1603.05279#11 | 1603.05279#13 | 1603.05279 | [
"1602.07360"
] |
1603.05279#13 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | cant improvement in energy efï¬ ciency. In EBP the binarized parameters were only used during inference. Bi- naryConnect [38] extended the probablistic idea behind EBP. Similar to our approach, BinaryConnect uses the real-valued version of the weights as a key reference for the binarization process. The real-valued weight updated using the back propagated error by simply ignoring the binarization in the update. BinaryConnect achieved state-of-the- art results on small datasets (e.g.,CIFAR-10, SVHN). Our experiments shows that this method is not very successful on large-scale datsets (e.g.,ImageNet). BinaryNet[11] propose an extention of BinaryConnect, where both weights and activations are bi- narized. Our method is different from them in the binarization method and the net- XNOR-Net: ImageNet Classiï¬ cation Using Binary Convolutional Neural Networks | 1603.05279#12 | 1603.05279#14 | 1603.05279 | [
"1602.07360"
] |
1603.05279#14 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | work structure. We also compare our method with BinaryNet on ImageNet, and our method outperforms BinaryNet by a large margin.[39] argued that the noise introduced by weight binarization provides a form of regularization, which could help to improve test accuracy. This method binarizes weights while maintaining full precision activa- tion. [40] proposed fully binary training and testing in an array of committee machines with randomized input. [41] retraine a previously trained neural network with binary weights and binary inputs. # 3 Binary Convolutional Neural Network We represent an L-layer CNN architecture with a triplet (Z, W, x). Z is a set of ten- sors, where each element I = Z)(j~1,...,:) is the input tensor for the 1 layer of CNN (Green cubes in figure 1). W is a set of tensors, where each element in this set W = cos x?) is the kâ ¢ weight filter in the 7" layer of the CNN. K' is the number of weight filters in the /'" layer of the CNN. * represents a convolutional operation with I and W as its operandsâ | 1603.05279#13 | 1603.05279#15 | 1603.05279 | [
"1602.07360"
] |
1603.05279#15 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | . I â ¬ R°*â â ¢*"m, where (c, win, hin) represents channels, width and height respectively.W â ¬ R°*â *", where w < win, h < hin. We propose two variations of binary CNN: Binary-weights, where the elements of W are binary tensors and XNOR-Networks, where elements of both Z and W are binary tensors. Wik(k=1 # 3.1 Binary-Weight-Networks In order to constrain a convolutional neural network (Z, W, *) to have binary weights, we estimate the real-value weight filter W â ¬ W using a binary filter B â ¬ {+1, â 1}**â *" and a scaling factor a â ¬ R* such that W ~ aB. A convolutional operation can be ap- priximated by: I â W â (I â B) α (1) where, © indicates a convolution without any multiplication. Since the weight values are binary, we can implement the convolution with additions and subtractions. The bi- nary weight filters reduce memory usage by a factor of ~ 32x compared to single- precision filters. We represent a CNN with binary weights by (Z, B, A, ©), where B is a set of binary tensors and A is a set of positive real scalars, such that B = Bix is a binary filter and a = Aj, is an scaling factor and Wiz © AinBir Estimating binary weights: Without loss of generality we assume W, B are vectors in Rn, where n = c à w à h. To ï¬ nd an optimal estimation for W â αB, we solve the following optimization: a*, B* = argminJ(B, a) 2) aB 2 In this paper we assume convolutional ï¬ lters do not have bias terms 6 Rastegari et al. by expanding equation 2, we have J(B, α) = α2BTB â 2αWTB + WTW (3) since B â {+1, â 1}n, BTB = n is a constant . WTW is also a constant because W is a known variable. | 1603.05279#14 | 1603.05279#16 | 1603.05279 | [
"1602.07360"
] |
1603.05279#16 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | Lets deï¬ ne c = WTW. Now, we can rewrite the equation 3 as follow: J(B, α) = α2n â 2αWTB + c. The optimal solution for B can be achieved by maximizing the following constrained optimization: (note that α is a positive value in equation 2, therefore it can be ignored in the maximization) Bâ = argmax {WTB} s.t. B â {+1, â 1}n B (4) This optimization can be solved by assigning Bi = +1 if Wi â ¥ 0 and Bi = â 1 if Wi < 0, therefore the optimal solution is Bâ = sign(W). In order to ï¬ nd the optimal value for the scaling factor αâ , we take the derivative of J with respect to α and set it to zero: αâ = WTBâ n (5) By replacing Bâ with sign(W) , W' sign(W W; 1 a gn(W) _ de |Wil Whe 6) n n n therefore, the optimal estimation of a binary weight ï¬ lter can be simply achieved by taking the sign of weight values. The optimal scaling factor is the average of absolute weight values. Training Binary-Weights-Networks: Each iteration of training a CNN involves three steps; forward pass, backward pass and parameters update. To train a CNN with binary weights (in convolutional layers), we only binarize the weights during the forward pass and backward propagation. | 1603.05279#15 | 1603.05279#17 | 1603.05279 | [
"1602.07360"
] |
1603.05279#17 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | To compute the gradient for sign function sign(r), we fol- low the same approach as [11], where â sign â r = r1|r|â ¤1. The gradient in backward after n + â sign the scaled sign function is â C ( 1 α). For updating the parameters, we â Wi â Wi use the high precision (real-value) weights. Because, in gradient descend the parameter changes are tiny, binarization after updating the parameters ignores these changes and the training objective can not be improved. [11,38] also employed this strategy to train a binary network. Algorithm | demonstrates our procedure for training a CNN with binary weights. First, we binarize the weight filters at each layer by computing B and A. Then we call forward propagation using binary weights and its corresponding scaling factors, where all the convolutional operations are carried out by equation 1. Then, we call backward propagation, where the gradients are computed with respect to the estimated weight filters W. Lastly, the parameters and the learning rate gets updated by an update rule e.g.,SGD update with momentum or ADAM [42]. Once the training ï¬ nished, there is no need to keep the real-value weights. Because, at inference we only perform forward propagation with the binarized weights. | 1603.05279#16 | 1603.05279#18 | 1603.05279 | [
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1603.05279#18 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | XNOR-Net: ImageNet Classiï¬ cation Using Binary Convolutional Neural Networks Algorithm 1 Training an L-layers CNN with binary weights: Input: A minibatch of inputs and targets (I, Y), cost function C(Y, Ë Y), current weight W t and current learning rate ηt. Output: updated weight W t+1 and updated learning rate ηt+1. 1: Binarizing weight ï¬ lters: 2: for l = 1 to L do 3: 4: 5: 6: 7: Ë Y = BinaryForward(I, B, A) // standard forward propagation except that convolutions are computed for kâ filter in I" layer do 1 Atk = £\|Wieller # 1 Atk = £\|Wieller Bu = sign(Wix) Win = An Bir using equation 1 or 11 8: 2& â BinaryBackward( aw _ W instead of Wt ac ay? W) // standard backward propagation except that gradients are computed | 1603.05279#17 | 1603.05279#19 | 1603.05279 | [
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1603.05279#19 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | 8: 2& â BinaryBackward( aw _ using W instead of Wt 9: W'** = UpdateParameters(Wâ , xs, me) / Any update rules (e.g.,SGD or ADAM) 10: 7'*! = UpdateLearningrate(7â , t) // Any learning rate scheduling function # 3.2 XNOR-Networks So far, we managed to find binary weights and a scaling factor to estimate the real- value weights. The inputs to the convolutional layers are still real-value tensors. Now, we explain how to binarize both weigths and inputs, so convolutions can be imple- mented efficiently using XNOR and bitcounting operations. This is the key element of our XNOR-Networks. In order to constrain a convolutional neural network (Z, W, *) to have binary weights and binary inputs, we need to enforce binary operands at each step of the convolutional operation. A convolution consist of repeating a shift operation and a dot product. Shift operation moves the weight filter over the input and the dot product performs element-wise multiplications between the values of the weight filter and the corresponding part of the input. If we express dot product in terms of binary operations, convolution can be approximated using binary operations. Dot product be- tween two binary vectors can be implemented by XNOR-Bitcounting operations [11]. In this section, we explain how to approximate the dot product between two vectors in R" by a dot product between two vectors in {+1, â | 1603.05279#18 | 1603.05279#20 | 1603.05279 | [
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1603.05279#20 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | 1}". Next, we demonstrate how to use this approximation for estimating a convolutional operation between two tensors. Binary Dot Product: To approximate the dot product between X, W â Rn such that XTW â βHTαB, where H, B â {+1, â 1}n and β, α â R+, we solve the following optimization: aâ , B*, 8°, H« = argmin||X © W â baH © B]| (7) a,B,6,H where © indicates element-wise product. We define Y â ¬ Râ such that Y; = X;,W,, Ce {+1, -1}â such that C; = H;B; and y â ¬ Rt such that y = Ba. The equation 7 can be written as: y*,C* = argmin|/Y â yC|| (8) an 7 8 Rastegari et al. | 1603.05279#19 | 1603.05279#21 | 1603.05279 | [
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1603.05279#21 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | (1) Binarizing Weight = - he f . foao-w 5 |Wla-a ine pB sign(W) (2) Binarizing Input saver nlX Inefficient aman MXalla= Bo" Redundant computations in overlapping areas . fn Efficient B ~p2 (4) Convolution with XNOR-Bitcount 0.2 0.1 3 01" meee , ~ dtn dln nines â 1405. 0.2 2", * Ree = 1.40, « 1alay y fea -0.5 3... -1.2 0.2" aA1â war Tf Ww . sign(W) K iif sign(I) Fig. 2: This ï¬ gure illustrates the procedure explained in section 3.2 for approximating a convo- lution using binary operations. the optimal solutions can be achieved from equation 2 as follow C* = sign(Y) = sign(X) © sign(W) = H* © B* (9) Since |Xi|, |Wi| are independent, knowing that Yi = XiWi then, E [|Yi|] = E [|Xi||Wi|] = E [|Xi|] E [|Wi|] therefore, Y; X,||W; 1 1 _ y= ERLE INS (2x) (Zila) =r" 0 n n Binary Convolution: Convolving weight filter W â | 1603.05279#20 | 1603.05279#22 | 1603.05279 | [
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1603.05279#22 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | ¬ RXâ ! (where win > w, hin > h) with the input tensor I â ¬ R°*â i» requires computing the scaling factor ( for all possible sub-tensors in I with same size as W. Two of these sub-tensors are illustrated in figure 2 (second row) by X; and X»g. Due to overlaps between subtensors, comput- ing £ for all possible sub-tensors leads to a large number of redundant computations. To overcome this redundancy, first, we compute a matrix A = Dlbsl which is the average over absolute values of the elements in the input I across the channel. Then we convolve A with a 2D filterk â | 1603.05279#21 | 1603.05279#23 | 1603.05279 | [
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] |
1603.05279#23 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | ¬ Râ *", K = A «xk, where Vij kij = oe K contains scaling factors 6 for all sub-tensors in the input I. K;; corresponds to 6 for a sub-tensor centered at the location ij (across width and height). This procedure is shown in the third row of figure 2. Once we obtained the scaling factor a for the weight and for all sub-tensors in I (denoted by K), we can approximate the convolution between input I and weight filter W mainly using binary operations: I* W & (sign(I) @ sign(W)) © Ka dd) where ® indicates a convolutional operation using XNOR and bitcount operations. This is illustrated in the last row in figure 2. Note that the number of non-binary operations is very small compared to binary operations. XNOR-Net: ImageNet Classiï¬ cation Using Binary Convolutional Neural Networks | 1603.05279#22 | 1603.05279#24 | 1603.05279 | [
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1603.05279#24 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | >| | 2s > f= >| la â ¬)|/6 mile ||) Be} SHEHE angip a oy} fal) ty} y* 3||s| |e Atypical block in CNN A block in XNOR-Net Fig. 3: This ï¬ gure contrasts the block structure in our XNOR-Network (right) with a typical CNN (left). Training XNOR-Networks: A typical block in CNN contains several different layers. Figure 3 (left) illustrates a typical block in a CNN. This block has four layers in the following order: 1-Convolutional, 2-Batch Normalization, 3-Activation and 4-Pooling. Batch Normalization layer[43] normalizes the input batch by its mean and variance. The activation is an element-wise non-linear function (e.g.,Sigmoid, ReLU). The pool- ing layer applies any type of pooling (e.g.,max,min or average) on the input batch. Applying pooling on binary input results in signiï¬ | 1603.05279#23 | 1603.05279#25 | 1603.05279 | [
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1603.05279#25 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | cant loss of information. For exam- ple, max-pooling on binary input returns a tensor that most of its elements are equal to +1. Therefore, we put the pooling layer after the convolution. To further decrease the information loss due to binarization, we normalize the input before binarization. This ensures the data to hold zero mean, therefore, thresholding at zero leads to less quanti- zation error. The order of layers in a block of binary CNN is shown in Figure 3(right). The binary activation layer(BinActiv) computes K and sign(I) as explained in sec- tion 3.2. In the next layer (BinConv), given K and sign(I), we compute binary convo- lution by equation 11. Then at the last layer (Pool), we apply the pooling operations. We can insert a non-binary activation(e.g.,ReLU) after binary convolution. This helps when we use state-of-the-art networks (e.g.,AlexNet or VGG). Once we have the binary CNN structure, the training algorithm would be the same as algorithm 1. Binary Gradient: The computational bottleneck in the backward pass at each layer is computing a convolution between weight ï¬ lters(w) and the gradients with respect of the inputs (gin). Similar to binarization in the forward pass, we can binarize gin in the backward pass. | 1603.05279#24 | 1603.05279#26 | 1603.05279 | [
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1603.05279#26 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | This leads to a very efï¬ cient training procedure using binary operations. Note that if we use equation 6 to compute the scaling factor for gin, the direction of maximum change for SGD would be diminished. To preserve the maximum change in all dimensions, we use maxi(|gin i |) as the scaling factor. k-bit Quantization: So far, we showed 1-bit quantization of weights and inputs using sign(x) function. One can easily extend the quantization level to k-bits by using qk(x) = 2( [(2kâ 1)( x+1 2 )] 2 ) instead of the sign function. Where [.] indicates rounding operation and x â [â 1, 1]. # 4 Experiments We evaluate our method by analyzing its efï¬ ciency and accuracy. We measure the ef- ï¬ ciency by computing the computational speedup (in terms of number of high preci- sion operation) achieved by our binary convolution vs. standard convolution. To mea- 9 10 Rastegari et al. 4cB = Double Precision "Binary Precision v 4008 HeMe_ews1.sqe | W7.4Me. ° VoG-419 ResNet18 AlexNet (a) (b) (c) Fig. 4: This ï¬ gure shows the efï¬ | 1603.05279#25 | 1603.05279#27 | 1603.05279 | [
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] |
1603.05279#27 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | ciency of binary convolutions in terms of memory(a) and computation(b-c). (a) is contrasting the required memory for binary and double precision weights in three different architectures(AlexNet, ResNet-18 and VGG-19). (b,c) Show speedup gained by binary convolution under (b)-different number of channels and (c)-different ï¬ lter size sure accuracy, we perform image classiï¬ cation on the large-scale ImageNet dataset. This paper is the ï¬ rst work that evaluates binary neural networks on the ImageNet dataset. Our binarization technique is general, we can use any CNN architecture. We evaluate AlexNet [1] and two deeper architectures in our experiments. We compare our method with two recent works on binarizing neural networks; BinaryConnect [38] and BinaryNet [11]. | 1603.05279#26 | 1603.05279#28 | 1603.05279 | [
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1603.05279#28 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | The classiï¬ cation accuracy of our binary-weight-network version of AlexNet is as accurate as the full precision version of AlexNet. This classiï¬ cation ac- curacy outperforms competitors on binary neural networks by a large margin. We also present an ablation study, where we evaluate the key elements of our proposed method; computing scaling factors and our block structure for binary CNN. We shows that our method of computing the scaling factors is important to reach high accuracy. # 4.1 Efï¬ ciency Analysis | 1603.05279#27 | 1603.05279#29 | 1603.05279 | [
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1603.05279#29 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | In an standard convolution, the total number of operations is cNWNI, where c is the number of channels, NW = wh and NI = winhin. Note that some modern CPUs can fuse the multiplication and addition as a single cycle operation. On those CPUs, Binary- Weight-Networks does not deliver speed up. Our binary approximation of convolution (equation 11) has cNWNI binary operations and NI non-binary operations. With the current generation of CPUs, we can perform 64 binary operations in one clock of CPU, therefore the speedup can be computed by S = # cNWNI 1 64 cNWNI+NI The speedup depends on the channel size and ï¬ lter size but not the input size. In ï¬ g- ure 4-(b-c) we illustrate the speedup achieved by changing the number of channels and ï¬ lter size. While changing one parameter, we ï¬ x other parameters as follows: c = 256, nI = 142 and nW = 32 (majority of convolutions in ResNet[4] architecture have this structure). Using our approximation of convolution we gain 62.27à theoretical speed up, but in our CPU implementation with all of the overheads, we achieve 58x speed up in one convolution (Excluding the process for memory allocation and memory ac- cess). With the small channel size (c = 3) and ï¬ lter size (NW = 1 à 1) the speedup is not considerably high. This motivates us to avoid binarization at the ï¬ | 1603.05279#28 | 1603.05279#30 | 1603.05279 | [
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1603.05279#30 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | rst and last XNOR-Net: ImageNet Classiï¬ cation Using Binary Convolutional Neural Networks layer of a CNN. In the ï¬ rst layer the chanel size is 3 and in the last layer the ï¬ lter size is 1 à 1. A similar strategy was used in [11]. Figure 4-a shows the required memory for three different CNN architectures(AlexNet, VGG-19, ResNet-18) with binary and double precision weights. Binary-weight-networks are so small that can be easily ï¬ | 1603.05279#29 | 1603.05279#31 | 1603.05279 | [
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1603.05279#31 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | tted into portable devices. BinaryNet [11] is in the same order of memory and computation efï¬ ciency as our method. In Figure 4, we show an analysis of computation and memory cost for a binary convolution. The same analysis is valid for BinaryNet and BinaryCon- nect. The key difference of our method is using a scaling-factor, which does not change the order of efï¬ ciency while providing a signiï¬ cant improvement in accuracy. # Image Classiï¬ cation We evaluate the performance of our proposed approach on the task of natural im- age classiï¬ cation. So far, in the literature, binary neural network methods have pre- sented their evaluations on either limited domain or simpliï¬ ed datasets e.g.,CIFAR-10, MNIST, SVHN. To compare with state-of-the-art vision, we evaluate our method on ImageNet (ILSVRC2012). ImageNet has â ¼1.2M train images from 1K categories and 50K validation images. The images in this dataset are natural images with reasonably high resolution compared to the CIFAR and MNIST dataset, which have relatively small images. | 1603.05279#30 | 1603.05279#32 | 1603.05279 | [
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1603.05279#32 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | We report our classiï¬ cation performance using Top-1 and Top-5 accuracies. We adopt three different CNN architectures as our base architectures for binarization: AlexNet [1], Residual Networks (known as ResNet) [4], and a variant of GoogLenet [3].We compare our Binary-weight-network (BWN) with BinaryConnect(BC) [38] and our XNOR-Networks(XNOR-Net) with BinaryNeuralNet(BNN) [11]. BinaryConnect(BC) is a method for training a deep neural network with binary weights during forward and backward propagations. Similar to our approach, they keep the real-value weights during the updating parameters step. Our binarization is different from BC. The bina- rization in BC can be either deterministic or stochastic. We use the deterministic bina- rization for BC in our comparisons because the stochastic binarization is not efï¬ cient. The same evaluation settings have been used and discussed in [11]. BinaryNeural- Net(BNN) [11] is a neural network with binary weights and activations during infer- ence and gradient computation in training. In concept, this is a similar approach to our XNOR-Network but the binarization method and the network structure in BNN is dif- ferent from ours. Their training algorithm is similar to BC and they used deterministic binarization in their evaluations. CIFAR-10 : BC and BNN showed near state-of-the-art performance on CIFAR- 10, MNIST, and SVHN dataset. BWN and XNOR-Net on CIFAR-10 using the same network architecture as BC and BNN achieve the error rate of 9.88% and 10.17% re- spectively. In this paper we explore the possibility of obtaining near state-of-the-art results on a much larger and more challenging dataset (ImageNet). [1] is a CNN architecture with 5 convolutional layers and two fully- connected layers. This architecture was the ï¬ rst CNN architecture that showed to be successful on ImageNet classiï¬ | 1603.05279#31 | 1603.05279#33 | 1603.05279 | [
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1603.05279#33 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | cation task. This network has 61M parameters. We use AlexNet coupled with batch normalization layers [43]. Train: In each iteration of training, images are resized to have 256 pixel at their smaller dimension and then a random crop of 224 à 224 is selected for training. We run 12 Rastegari et al. Top-1, Binary-Weight â Top-5, Binary-Weight-Input 0 10 20 Number of epochs Number of epochs Number of epochs Number of epochs 20 Fig. 5: This ï¬ gure compares the imagenet classiï¬ cation accuracy on Top-1 and Top-5 across training epochs. Our approaches BWN and XNOR-Net outperform BinaryConnect(BC) and Bi- naryNet(BNN) in all the epochs by large margin(â | 1603.05279#32 | 1603.05279#34 | 1603.05279 | [
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] |
1603.05279#34 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | ¼17%). Classiï¬ cation Accuracy(%) Binary-Weight Binary-Input-Binary-Weight Full-Precision BNN[11] XNOR-Net AlexNet[1] Top-1 Top-5 Top-1 Top-5 Top-1 Top-5 Top-1 Top-5 Top-1 Top-5 56.8 79.4 35.4 61.0 80.2 BWN BC[11] 50.42 56.6 44.2 69.2 27.9 Table 1: This table compares the ï¬ nal accuracies (Top1 - Top5) of the full precision network with our binary precision networks; Binary-Weight-Networks(BWN) and XNOR-Networks(XNOR- Net) and the competitor methods; BinaryConnect(BC) and BinaryNet(BNN). the training algorithm for 16 epochs with batche size equal to 512. We use negative-log- likelihood over the soft-max of the outputs as our classiï¬ cation loss function. In our implementation of AlexNet we do not use the Local-Response-Normalization(LRN) layer3. We use SGD with momentum=0.9 for updating parameters in BWN and BC. For XNOR-Net and BNN we used ADAM [42]. ADAM converges faster and usually achieves better accuracy for binary inputs [11]. The learning rate starts at 0.1 and we apply a learning-rate-decay=0.01 every 4 epochs. | 1603.05279#33 | 1603.05279#35 | 1603.05279 | [
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] |
1603.05279#35 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | Test: At inference time, we use the 224 à 224 center crop for forward propagation. Figure 5 demonstrates the classiï¬ cation accuracy for training and inference along the training epochs for top-1 and top-5 scores. The dashed lines represent training ac- curacy and solid lines shows the validation accuracy. In all of the epochs our method outperforms BC and BNN by large margin (â ¼17%). Table 1 compares our ï¬ nal accu- racy with BC and BNN. We found that the scaling factors for the weights (α) is much more effective than the scaling factors for the inputs (β). Removing β reduces the ac- curacy by a small margin (less than 1% top-1 alexnet). Binary Gradient: Using XNOR-Net with binary gradient the accuracy of top-1 will drop only by 1.4%. Residual Net : We use the ResNet-18 proposed in [4] with short-cut type B.4 Train: In each training iteration, images are resized randomly between 256 and 480 pixel on the smaller dimension and then a random crop of 224 à 224 is selected for training. We run the training algorithm for 58 epochs with batch size equal to 256 3 Our implementation is followed by https://gist.github.com/szagoruyko/dd032c529048492630fc 4 We used the Torch implementation in https://github.com/facebook/fb.resnet.torch | 1603.05279#34 | 1603.05279#36 | 1603.05279 | [
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1603.05279#36 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | # XNOR-Net: ImageNet Classiï¬ cation Using Binary Convolutional Neural Networks (a) (b) Fig. 6: This ï¬ gure shows the classiï¬ cation accuracy; (a)Top-1 and (b)Top-5 measures across the training epochs on ImageNet dataset by Binary-Weight-Network and XNOR-Network using ResNet-18. Network Variations Binary-Weight-Network XNOR-Network Full-Precision-Network ResNet-18 top-5 83.0 73.2 89.2 top-1 60.8 51.2 69.3 GoogLenet top-5 86.1 N/A 90.0 top-1 65.5 N/A 71.3 | 1603.05279#35 | 1603.05279#37 | 1603.05279 | [
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] |
1603.05279#37 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | Table 2: This table compares the ï¬ nal classiï¬ cation accuracy achieved by our binary precision networks with the full precision network in ResNet-18 and GoogLenet architectures. images. The learning rate starts at 0.1 and we use the learning-rate-decay equal to 0.01 at epochs number 30 and 40. Test: At inference time, we use the 224 à 224 center crop for forward propagation. Figure 6 demonstrates the classiï¬ cation accuracy (Top-1 and Top-5) along the epochs for training and inference. The dashed lines represent training and the solid lines repre- sent inference. Table 2 shows our ï¬ nal accuracy by BWN and XNOR-Net. GoogLenet Variant : We experiment with a variant of GoogLenet [3] that uses a similar number of parameters and connections but only straightforward convolutions, no branching5. It has 21 convolutional layers with ï¬ lter sizes alternating between 1 à 1 and 3 à 3. Train: | 1603.05279#36 | 1603.05279#38 | 1603.05279 | [
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] |
1603.05279#38 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | Images are resized randomly between 256 and 320 pixel on the smaller di- mension and then a random crop of 224 à 224 is selected for training. We run the training algorithm for 80 epochs with batch size of 128. The learning rate starts at 0.1 and we use polynomial rate decay, β = 4. Test: At inference time, we use a center crop of 224 à 224. # 4.3 Ablation Studies There are two key differences between our method and the previous network binariaza- tion methods; the binararization technique and the block structure in our binary CNN. 5 We used the Darknet [44] implementation: http://pjreddie.com/darknet/imagenet/#extraction | 1603.05279#37 | 1603.05279#39 | 1603.05279 | [
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] |
1603.05279#39 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | 13 14 14 Rastegari et al. Binary-Weight-Network top-1 56.8 46.2 Strategy for computing α Using equation 6 Using a separate layer top-5 79.4 69.5 XNOR-Network top-1 30.3 44.2 Block Structure C-B-A-P B-A-C-P top-5 57.5 69.2 (a) (b) Table 3: In this table, we evaluate two key elements of our approach; computing the optimal scaling factors and specifying the right order for layers in a block of CNN with binary input. (a) demonstrates the importance of the scaling factor in training binary-weight-networks and (b) shows that our way of ordering the layers in a block of CNN is crucial for training XNOR- Networks. C,B,A,P stands for Convolutional, BatchNormalization, Acive function (here binary activation), and Pooling respectively. For binarization, we ï¬ nd the optimal scaling factors at each iteration of training. For the block structure, we order the layers in a block in a way that decreases the quantiza- tion loss for training XNOR-Net. Here, we evaluate the effect of each of these elements in the performance of the binary networks. Instead of computing the scaling factor α using equation 6, one can consider α as a network parameter. In other words, a layer after binary convolution multiplies the output of convolution by an scalar parameter for each ï¬ | 1603.05279#38 | 1603.05279#40 | 1603.05279 | [
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] |
1603.05279#40 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | lter. This is similar to computing the afï¬ ne parameters in batch normalization. Table 3-a compares the performance of a binary network with two ways of computing the scaling factors. As we mentioned in section 3.2 the typical block structure in CNN is not suitable for binarization. Table 3-b compares the standard block structure C-B-A-P (Convolution, Batch Normalization, Activation, Pooling) with our structure B-A-C-P. (A, is binary activation). | 1603.05279#39 | 1603.05279#41 | 1603.05279 | [
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] |
1603.05279#41 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | # 5 Conclusion We introduce simple, efï¬ cient, and accurate binary approximations for neural networks. We train a neural network that learns to ï¬ nd binary values for weights, which reduces the size of network by â ¼ 32à and provide the possibility of loading very deep neural networks into portable devices with limited memory. We also propose an architecture, XNOR-Net, that uses mostly bitwise operations to approximate convolutions. This pro- vides â ¼ 58à | 1603.05279#40 | 1603.05279#42 | 1603.05279 | [
"1602.07360"
] |
1603.05279#42 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | speed up and enables the possibility of running the inference of state of the art deep neural network on CPU (rather than GPU) in real-time. # Acknowledgements This work is in part supported by ONR N00014-13-1-0720, NSF IIS- 1338054, Allen Distinguished Investigator Award, and the Allen Institute for Artiï¬ cial Intelligence. XNOR-Net: ImageNet Classiï¬ cation Using Binary Convolutional Neural Networks | 1603.05279#41 | 1603.05279#43 | 1603.05279 | [
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] |
1603.05279#43 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | # References 1. Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classiï¬ cation with deep convolutional neural networks. In: Advances in neural information processing systems. (2012) 1097â 1105 1, 10, 11, 12 2. Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recog- nition. arXiv preprint arXiv:1409.1556 (2014) 1 | 1603.05279#42 | 1603.05279#44 | 1603.05279 | [
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] |
1603.05279#44 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | 3. Szegedy, C., Liu, W., Jia, Y., Sermanet, P., Reed, S., Anguelov, D., Erhan, D., Vanhoucke, V., Rabinovich, A.: Going deeper with convolutions. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. (2015) 1â 9 1, 4, 11, 13 4. He, K., Zhang, X., Ren, S., Sun, J.: | 1603.05279#43 | 1603.05279#45 | 1603.05279 | [
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] |
1603.05279#45 | XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks | Deep residual learning for image recognition. CoRR (2015) 1, 4, 10, 11, 12 5. Girshick, R., Donahue, J., Darrell, T., Malik, J.: Rich feature hierarchies for accurate object detection and semantic segmentation. In: Proceedings of the IEEE conference on computer vision and pattern recognition. (2014) 580â 587 1 6. Girshick, R.: Fast r-cnn. In: Proceedings of the IEEE International Conference on Computer Vision. (2015) 1440â 1448 1 7. Ren, S., He, K., Girshick, R., Sun, J.: Faster r-cnn: Towards real-time object detection with region proposal networks. | 1603.05279#44 | 1603.05279#46 | 1603.05279 | [
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1603.05027#0 | Identity Mappings in Deep Residual Networks | # Identity Mappings in Deep Residual Networks 6 1 0 2 l u J 5 2 ] V C . s c [ Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun Microsoft Research Abstract Deep residual networks [1] have emerged as a family of ex- tremely deep architectures showing compelling accuracy and nice con- vergence behaviors. In this paper, we analyze the propagation formu- lations behind the residual building blocks, which suggest that the for- ward and backward signals can be directly propagated from one block to any other block, when using identity mappings as the skip connec- tions and after-addition activation. A series of ablation experiments sup- port the importance of these identity mappings. This motivates us to propose a new residual unit, which makes training easier and improves generalization. We report improved results using a 1001-layer ResNet on CIFAR-10 (4.62% error) and CIFAR-100, and a 200-layer ResNet on ImageNet. Code is available at: https://github.com/KaimingHe/ resnet-1k-layers. | 1603.05027#1 | 1603.05027 | [
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1603.05027#1 | Identity Mappings in Deep Residual Networks | 3 v 7 2 0 5 0 . 3 0 6 1 : v i X r a # 1 Introduction Deep residual networks (ResNets) [1] consist of many stacked â Residual Unitsâ . Each unit (Fig. 1 (a)) can be expressed in a general form: yl = h(xl) + F(xl, Wl), xl+1 = f (yl), where xl and xl+1 are input and output of the l-th unit, and F is a residual function. In [1], h(xl) = xl is an identity mapping and f is a ReLU [2] function. | 1603.05027#0 | 1603.05027#2 | 1603.05027 | [
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1603.05027#2 | Identity Mappings in Deep Residual Networks | ResNets that are over 100-layer deep have shown state-of-the-art accuracy for several challenging recognition tasks on ImageNet [3] and MS COCO [4] compe- titions. The central idea of ResNets is to learn the additive residual function F with respect to h(xl), with a key choice of using an identity mapping h(xl) = xl. This is realized by attaching an identity skip connection (â shortcutâ ). In this paper, we analyze deep residual networks by focusing on creating a â directâ path for propagating information â not only within a residual unit, but through the entire network. Our derivations reveal that if both h(xl) and f (yl) are identity mappings, the signal could be directly propagated from one unit to any other units, in both forward and backward passes. Our experiments empirically show that training in general becomes easier when the architecture is closer to the above two conditions. To understand the role of skip connections, we analyze and compare various types of h(xl). | 1603.05027#1 | 1603.05027#3 | 1603.05027 | [
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1603.05027#3 | Identity Mappings in Deep Residual Networks | We ï¬ nd that the identity mapping h(xl) = xl chosen in [1] 2 2-4 20 x) x) H â , ResNetâ 1001, original (error: 7.61%) sen ResNetâ 1001, proposed (error: 4.92%) ~~ ~~ \ weight BN 15 + + BN RelU 02 M 4 Fa RelU weight 8 4 + 2 weight BN = e £ BN RelU a 1 0.02 ac addition weight 5 a "y â RelU addition My ths : Â¥ v th VW at x xi Teg PNA Why tt 1 . 0.002 te 0 (a) original (b) proposed o 1 2 3 4 5 6 Iterations x10" Figure 1. Left: (a) original Residual Unit in [1]; (b) proposed Residual Unit. The grey arrows indicate the easiest paths for the information to propagate, corresponding to the additive term â | 1603.05027#2 | 1603.05027#4 | 1603.05027 | [
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1603.05027#4 | Identity Mappings in Deep Residual Networks | xlâ in Eqn.(4) (forward propagation) and the additive term â 1â in Eqn.(5) (backward propagation). Right: training curves on CIFAR-10 of 1001-layer ResNets. Solid lines denote test error (y-axis on the right), and dashed lines denote training loss (y-axis on the left). The proposed unit makes ResNet-1001 easier to train. achieves the fastest error reduction and lowest training loss among all variants we investigated, whereas skip connections of scaling, gating [5,6,7], and 1Ã 1 convolutions all lead to higher training loss and error. These experiments suggest that keeping a â | 1603.05027#3 | 1603.05027#5 | 1603.05027 | [
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1603.05027#5 | Identity Mappings in Deep Residual Networks | cleanâ information path (indicated by the grey arrows in Fig. 1, 2, and 4) is helpful for easing optimization. To construct an identity mapping f (yl) = yl, we view the activation func- tions (ReLU and BN [8]) as â pre-activationâ of the weight layers, in contrast to conventional wisdom of â post-activationâ . This point of view leads to a new residual unit design, shown in (Fig. 1(b)). Based on this unit, we present com- petitive results on CIFAR-10/100 with a 1001-layer ResNet, which is much easier to train and generalizes better than the original ResNet in [1]. We further report improved results on ImageNet using a 200-layer ResNet, for which the counter- part of [1] starts to overï¬ | 1603.05027#4 | 1603.05027#6 | 1603.05027 | [
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1603.05027#6 | Identity Mappings in Deep Residual Networks | t. These results suggest that there is much room to exploit the dimension of network depth, a key to the success of modern deep learning. # 2 Analysis of Deep Residual Networks The ResNets developed in [1] are modularized architectures that stack building blocks of the same connecting shape. In this paper we call these blocks â Residual # z a 10e g & Unitsâ . The original Residual Unit in [1] performs the following computation: (1) yl = h(xl) + F(xl, Wl), xl+1 = f (yl). (2) Here xl is the input feature to the l-th Residual Unit. Wl = {Wl,k|1â | 1603.05027#5 | 1603.05027#7 | 1603.05027 | [
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1603.05027#7 | Identity Mappings in Deep Residual Networks | ¤kâ ¤K} is a set of weights (and biases) associated with the l-th Residual Unit, and K is the number of layers in a Residual Unit (K is 2 or 3 in [1]). F denotes the residual function, e.g., a stack of two 3Ã 3 convolutional layers in [1]. The function f is the operation after element-wise addition, and in [1] f is ReLU. The function h is set as an identity mapping: h(xl) = xl.1 If f is also an identity mapping: xl+1 â ¡ yl, we can put Eqn.(2) into Eqn.(1) and obtain: xl+1 = xl + F(xl, Wl). (3) Recursively (xl+2 = xl+1 + F (xl+1, Wl+1) = xl + F (xl, Wl) + F (xl+1, Wl+1), etc.) we will have: L-1 Xr =x + > F(xi,M), (4) i=l for any deeper unit L and any shallower unit 1. Eqn.(4) exhibits some nice properties. (i) The feature xy, of any deeper unit L can be represented as the feature x; of any shallower unit | plus a residual function in a form of ae F, indicating that the model is in a residual fashion between any units L and 1. (ii) The feature x, = xo + are F (xi, Wi), of any deep unit L, is the summation of the outputs of all preceding residual functions (plus xo). This is in contrast to a â plain networkâ where a feature xz is a series of matrix-vector products, say, en W;Xo (ignoring BN and ReLU). i=0 Wix0 (ignoring BN and ReLU). Eqn.(4) also leads to nice backward propagation properties. Denoting the loss function as E, from the chain rule of backpropagation [9] we have: Ox; Oxy Ox, Oxy, â g L-1 ae gee = HE 1a ) (5) Dag De F(x Mi) i=l | 1603.05027#6 | 1603.05027#8 | 1603.05027 | [
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1603.05027#8 | Identity Mappings in Deep Residual Networks | Eqn.(5) indicates that the gradient ge can be decomposed into two additive terms: a term of ee that propagates information directly without concern- ing any weight layers and another term of 2& (or wig! F) that propagates Oxp, through the weight layers. The additive term of 2 oa ensures that information is directly propagated back to any shallower unit I. Eqn. (5) also suggests that it 1 It is noteworthy that there are Residual Units for increasing dimensions and reducing feature map sizes [1] in which h is not identity. In this case the following derivations do not hold strictly. But as there are only a very few such units (two on CIFAR and three on ImageNet, depending on image sizes [1]), we expect that they do not have the exponential impact as we present in Sec. 3. One may also think of our derivations as applied to all Residual Units within the same feature map size. | 1603.05027#7 | 1603.05027#9 | 1603.05027 | [
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1603.05027#9 | Identity Mappings in Deep Residual Networks | 3 4 is unlikely for the gradient â E to be canceled out for a mini-batch, because in â xl general the term â i=l F cannot be always -1 for all samples in a mini-batch. â xl This implies that the gradient of a layer does not vanish even when the weights are arbitrarily small. # Discussions Eqn.(4) and Eqn.(5) suggest that the signal can be directly propagated from any unit to another, both forward and backward. The foundation of Eqn.(4) is two identity mappings: (i) the identity skip connection h(xl) = xl, and (ii) the condition that f is an identity mapping. These directly propagated information ï¬ ows are represented by the grey ar- rows in Fig. 1, 2, and 4. And the above two conditions are true when these grey arrows cover no operations (expect addition) and thus are â | 1603.05027#8 | 1603.05027#10 | 1603.05027 | [
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1603.05027#10 | Identity Mappings in Deep Residual Networks | cleanâ . In the fol- lowing two sections we separately investigate the impacts of the two conditions. # 3 On the Importance of Identity Skip Connections Letâ s consider a simple modiï¬ cation, h(xl) = λlxl, to break the identity shortcut: xl+1 = λlxl + F(xl, Wl), (6) where 4; is a modulating scalar (for simplicity we still assume f is identity). Recursively applying this formulation we obtain an equation similar to Eqn. (4): xp = a 1 "Nix + ear 7 (jn ini As) F (xi, Wi), or simply: - L-1 xp = di d)xi + SO F(«:,Wi), (7) i=l i=l where the notation Ë F absorbs the scalars into the residual functions. Similar to Eqn.(5), we have backpropagation of the following form: | 1603.05027#9 | 1603.05027#11 | 1603.05027 | [
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1603.05027#11 | Identity Mappings in Deep Residual Networks | ae ae ( a eo ae, (UID â mF â mm. ®) Unlike Eqn.(5), in Eqn.(8) the first additive term is modulated by a factor We 7) di. For an extremely deep network (L is large), if 4; > 1 for all i, this factor can be exponentially large; if A; < 1 for all i, this factor can be expo- nentially small and vanish, which blocks the backpropagated signal from the shortcut and forces it to flow through the weight layers. This results in opti- mization difficulties as we show by experiments. In the above analysis, the original identity skip connection in Eqn.(3) is re- placed with a simple scaling h(x;) = A;x:. If the skip connection h(x;) represents more complicated transforms (such as gating and 1x1 convolutions), in Eqn.(8) the first term becomes Wa hi, where hâ is the derivative of h. This product may also impede information propagation and hamper the training procedure as witnessed in the following experiments. â fBxsconv â fBxsconv yeu Trew 3x3 conv 3x3 conv i - addition addition y= (a) original (b) constant scaling (sa conv Rel ~ss conv Trew ixicony| [Bx conv ixtcony] [SxS conv emod = â addition | . . addition . ye (c) exclusive gating ney (d) shortcut-only gating 3x conv ~{BxS conv Raw TRaw ixi conv 3x3 conv dropout 3x3 conv addition addition Rely (e) conv shortcut Ret (f) dropout shortcut Figure 2. Various types of shortcut connections used in Table 1. The grey arrows indicate the easiest paths for the information to propagate. The shortcut connections in (b-f) are impeded by diï¬ erent components. For simplifying illustrations we do not display the BN layers, which are adopted right after the weight layers for all units here. | 1603.05027#10 | 1603.05027#12 | 1603.05027 | [
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1603.05027#12 | Identity Mappings in Deep Residual Networks | # 3.1 Experiments on Skip Connections We experiment with the 110-layer ResNet as presented in [1] on CIFAR-10 [10]. This extremely deep ResNet-110 has 54 two-layer Residual Units (consisting of 3à 3 convolutional layers) and is challenging for optimization. Our implementa- tion details (see appendix) are the same as [1]. Throughout this paper we report the median accuracy of 5 runs for each architecture on CIFAR, reducing the impacts of random variations. Though our above analysis is driven by identity f , the experiments in this section are all based on f = ReLU as in [1]; we address identity f in the next sec- tion. Our baseline ResNet-110 has 6.61% error on the test set. The comparisons of other variants (Fig. 2 and Table 1) are summarized as follows: Constant scaling. We set λ = 0.5 for all shortcuts (Fig. 2(b)). We further study two cases of scaling F: (i) F is not scaled; or (ii) F is scaled by a constant scalar of 1 â λ = 0.5, which is similar to the highway gating [6,7] but with frozen gates. The former case does not converge well; the latter is able to converge, but the test error (Table 1, 12.35%) is substantially higher than the original ResNet-110. Fig 3(a) shows that the training error is higher than that of the original ResNet-110, suggesting that the optimization has diï¬ culties when the shortcut signal is scaled down. | 1603.05027#11 | 1603.05027#13 | 1603.05027 | [
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1603.05027#13 | Identity Mappings in Deep Residual Networks | 5 6 Table 1. Classiï¬ cation error on the CIFAR-10 test set using ResNet-110 [1], with diï¬ erent types of shortcut connections applied to all Residual Units. We report â failâ when the test error is higher than 20%. case Fig. on shortcut on F error (%) remark original [1] Fig. 2(a) 1 1 6.61 constant scaling Fig. 2(b) 0 0.5 1 1 fail fail This is a plain net 0.5 0.5 12.35 frozen gating exclusive gating Fig. 2(c) 1 â g(x) 1 â g(x) g(x) g(x) fail 8.70 init bg =0 to â 5 init bg =-6 1 â g(x) g(x) 9.81 init bg =-7 shortcut-only gating Fig. 2(d) 1 â g(x) 1 â g(x) 1 1 12.86 6.91 init bg =0 init bg =-6 1à 1 conv shortcut Fig. 2(e) 1à 1 conv 1 12.22 dropout shortcut Fig. 2(f) dropout 0.5 1 fail Exclusive gating. Following the Highway Networks [6,7] that adopt a gating mechanism [5], we consider a gating function g(x) = Ï (Wgx + bg) where a transform is represented by weights Wg and biases bg followed by the sigmoid function Ï (x) = 1 1+eâ x . In a convolutional network g(x) is realized by a 1à 1 convolutional layer. The gating function modulates the signal by element-wise multiplication. | 1603.05027#12 | 1603.05027#14 | 1603.05027 | [
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1603.05027#14 | Identity Mappings in Deep Residual Networks | We investigate the â exclusiveâ gates as used in [6,7] â the F path is scaled by g(x) and the shortcut path is scaled by 1â g(x). See Fig 2(c). We ï¬ nd that the initialization of the biases bg is critical for training gated models, and following the guidelines2 in [6,7], we conduct hyper-parameter search on the initial value of bg in the range of 0 to -10 with a decrement step of -1 on the training set by cross- validation. The best value (â 6 here) is then used for training on the training set, leading to a test result of 8.70% (Table 1), which still lags far behind the ResNet-110 baseline. Fig 3(b) shows the training curves. Table 1 also reports the results of using other initialized values, noting that the exclusive gating network does not converge to a good solution when bg is not appropriately initialized. The impact of the exclusive gating mechanism is two-fold. When 1 â g(x) approaches 1, the gated shortcut connections are closer to identity which helps information propagation; but in this case g(x) approaches 0 and suppresses the function F. To isolate the eï¬ ects of the gating functions on the shortcut path alone, we investigate a non-exclusive gating mechanism in the next. Shortcut-only gating. In this case the function F is not scaled; only the shortcut path is gated by 1 â g(x). See Fig 2(d). | 1603.05027#13 | 1603.05027#15 | 1603.05027 | [
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1603.05027#15 | Identity Mappings in Deep Residual Networks | The initialized value of bg is still essential in this case. When the initialized bg is 0 (so initially the expectation of 1 â g(x) is 0.5), the network converges to a poor result of 12.86% (Table 1). This is also caused by higher training error (Fig 3(c)). # 2 See also: people.idsia.ch/~rupesh/very_deep_learning/ by [6,7]. 02 (09) 2011350 Training Loss (99) 101131891 San as i â 5 a Wailea laa 110 original iy â â 110, const scaling (0.5, 0.5) 0.002 ° ° T 2 3 4 5 Iterations (a) (6) 0013 3891 Training Loss (6) 0013891 a 5 AW Ay Mi My 110, original Hethoe 110, original â ia ; â 110, shortcut-only gating (nit b=0) [Whit â 110, 1x1 conv shortcut HA pat 0.002 Jo 0.002 Jo 0 1 2 3 4 5 6 0 7 2 3 4 5 6 erations 10! erations ero! (c) (d) # Training Loss # Training Loss | 1603.05027#14 | 1603.05027#16 | 1603.05027 | [
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1603.05027#16 | Identity Mappings in Deep Residual Networks | Figure 3. Training curves on CIFAR-10 of various shortcuts. Solid lines denote test error (y-axis on the right), and dashed lines denote training loss (y-axis on the left). When the initialized bg is very negatively biased (e.g., â 6), the value of 1 â g(x) is closer to 1 and the shortcut connection is nearly an identity mapping. Therefore, the result (6.91%, Table 1) is much closer to the ResNet-110 baseline. 1Ã 1 convolutional shortcut. Next we experiment with 1Ã 1 convolutional shortcut connections that replace the identity. This option has been investigated in [1] (known as option C) on a 34-layer ResNet (16 Residual Units) and shows good results, suggesting that 1Ã 1 shortcut connections could be useful. | 1603.05027#15 | 1603.05027#17 | 1603.05027 | [
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1603.05027#17 | Identity Mappings in Deep Residual Networks | But we ï¬ nd that this is not the case when there are many Residual Units. The 110-layer ResNet has a poorer result (12.22%, Table 1) when using 1à 1 convolutional shortcuts. Again, the training error becomes higher (Fig 3(d)). When stacking so many Residual Units (54 for ResNet-110), even the shortest path may still impede signal propagation. We witnessed similar phenomena on ImageNet with ResNet-101 when using 1à 1 convolutional shortcuts. Dropout shortcut. Last we experiment with dropout [11] (at a ratio of 0.5) which we adopt on the output of the identity shortcut (Fig. 2(f)). The network fails to converge to a good solution. Dropout statistically imposes a scale of λ with an expectation of 0.5 on the shortcut, and similar to constant scaling by 0.5, it impedes signal propagation. | 1603.05027#16 | 1603.05027#18 | 1603.05027 | [
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1603.05027#18 | Identity Mappings in Deep Residual Networks | 7 8 Table 2. Classiï¬ cation error (%) on the CIFAR-10 test set using diï¬ erent activation functions. case Fig. ResNet-110 ResNet-164 original Residual Unit [1] Fig. 4(a) 6.61 5.93 BN after addition Fig. 4(b) 8.17 6.50 ReLU before addition Fig. 4(c) 7.84 6.14 ReLU-only pre-activation Fig. 4(d) 6.71 5.91 full pre-activation Fig. 4(e) 6.37 5.46 x x xX xX) xX ~~ a â . se weight weight weight ReLU BN 1 1 t t BN BN BN weight ReLU t 4 1 ReLU ReLU ReLU BN weight t 1 1 weight weight weight ReLU BN _BN addition BN weight ReLU a 1 addition BN ReLU BN weight a me a ReLU ReLU addition addition addition v Â¥ Xt Xt Xt Xr Xie oe (b) BN after (c) ReLU before (d) ReLU-only age (a) original addition addition pre-activation (©) full pre-activation Figure 4. Various usages of activation in Table 2. All these units consist of the same components â only the orders are diï¬ erent. | 1603.05027#17 | 1603.05027#19 | 1603.05027 | [
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1603.05027#19 | Identity Mappings in Deep Residual Networks | # 3.2 Discussions As indicated by the grey arrows in Fig. 2, the shortcut connections are the most direct paths for the information to propagate. Multiplicative manipulations (scaling, gating, 1Ã 1 convolutions, and dropout) on the shortcuts can hamper information propagation and lead to optimization problems. It is noteworthy that the gating and 1Ã 1 convolutional shortcuts introduce more parameters, and should have stronger representational abilities than iden- tity shortcuts. In fact, the shortcut-only gating and 1Ã 1 convolution cover the solution space of identity shortcuts (i.e., they could be optimized as identity shortcuts). However, their training error is higher than that of identity short- cuts, indicating that the degradation of these models is caused by optimization issues, instead of representational abilities. # 4 On the Usage of Activation Functions Experiments in the above section support the analysis in Eqn.(5) and Eqn.(8), both being derived under the assumption that the after-addition activation f | 1603.05027#18 | 1603.05027#20 | 1603.05027 | [
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1603.05027#20 | Identity Mappings in Deep Residual Networks | is the identity mapping. But in the above experiments f is ReLU as designed in [1], so Eqn.(5) and (8) are approximate in the above experiments. Next we investigate the impact of f . We want to make f an identity mapping, which is done by re-arranging the activation functions (ReLU and/or BN). The original Residual Unit in [1] has a shape in Fig. 4(a) â BN is used after each weight layer, and ReLU is adopted after BN except that the last ReLU in a Residual Unit is after element- wise addition (f = ReLU). Fig. 4(b-e) show the alternatives we investigated, explained as following. # 4.1 Experiments on Activation In this section we experiment with ResNet-110 and a 164-layer Bottleneck [1] architecture (denoted as ResNet-164). A bottleneck Residual Unit consist of a 1Ã 1 layer for reducing dimension, a 3Ã 3 layer, and a 1Ã 1 layer for restoring dimension. As designed in [1], its computational complexity is similar to the two-3Ã 3 Residual Unit. More details are in the appendix. | 1603.05027#19 | 1603.05027#21 | 1603.05027 | [
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1603.05027#21 | Identity Mappings in Deep Residual Networks | The baseline ResNet- 164 has a competitive result of 5.93% on CIFAR-10 (Table 2). BN after addition. Before turning f into an identity mapping, we go the opposite way by adopting BN after addition (Fig. 4(b)). In this case f involves BN and ReLU. The results become considerably worse than the baseline (Ta- ble 2). Unlike the original design, now the BN layer alters the signal that passes through the shortcut and impedes information propagation, as reï¬ ected by the diï¬ culties on reducing training loss at the beginning of training (Fib. 6 left). ReLU before addition. A na¨ıve choice of making f into an identity map- ping is to move the ReLU before addition (Fig. 4(c)). However, this leads to a non-negative output from the transform F, while intuitively a â residualâ func- tion should take values in (â â , +â ). | 1603.05027#20 | 1603.05027#22 | 1603.05027 | [
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1603.05027#22 | Identity Mappings in Deep Residual Networks | As a result, the forward propagated sig- nal is monotonically increasing. This may impact the representational ability, and the result is worse (7.84%, Table 2) than the baseline. We expect to have a residual function taking values in (â â , +â ). This condition is satisï¬ ed by other Residual Units including the following ones. Post-activation or pre-activation? In the original design (Eqn.(1) and Eqn.(2)), the activation xl+1 = f (yl) aï¬ ects both paths in the next Residual Unit: yl+1 = f (yl) + F(f (yl), Wl+1). Next we develop an asymmetric form where an activation Ë f only aï¬ ects the F path: yl+1 = yl + F( Ë f (yl), Wl+1), for any l (Fig. 5 (a) to (b)). By renaming the notations, we have the following form: xl+1 = xl + F( Ë f (xl), Wl), . (9) It is easy to see that Eqn.(9) is similar to Eqn.(4), and can enable a backward formulation similar to Eqn.(5). For this new Residual Unit as in Eqn.(9), the new after-addition activation becomes an identity mapping. This design means that if a new after-addition activation Ë f is asymmetrically adopted, it is equivalent to recasting Ë f as the pre-activation of the next Residual Unit. This is illustrated in Fig. 5. 9 10 | 1603.05027#21 | 1603.05027#23 | 1603.05027 | [
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1603.05027#23 | Identity Mappings in Deep Residual Networks | - act Pact original Fo + T Residual walght asymmet weight weight Unit output q act. activation act â weight weight weight = â pre-activation | ~~. Ge I Residual Unit a weight weight 4 act. act. act. t weight weight weight el ee 7 addition addition i addi act. M + adopt output activation . â PE output equivalent to â only to weight path (a) (b) () Figure 5. Using asymmetric after-addition activation is equivalent to constructing a pre-activation Residual Unit. Table 3. Classiï¬ | 1603.05027#22 | 1603.05027#24 | 1603.05027 | [
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1603.05027#24 | Identity Mappings in Deep Residual Networks | cation error (%) on the CIFAR-10/100 test set using the original Residual Units and our pre-activation Residual Units. dataset network ResNet-110 (1layer skip) 9.90 8.91 CIFAR-10 ResNet-110 ResNet-164 6.61 5.93 6.37 5.46 ResNet-1001 7.61 4.92 ResNet-164 ResNet-1001 25.16 27.82 24.33 22.71 The distinction between post-activation/pre-activation is caused by the pres- ence of the element-wise addition. For a plain network that has N layers, there are N â 1 activations (BN/ReLU), and it does not matter whether we think of them as post- or pre-activations. But for branched layers merged by addition, the position of activation matters. and (ii) full pre-activation (Fig. 4(e)) where BN and ReLU are both adopted be- fore weight layers. Table 2 shows that the ReLU-only pre-activation performs very similar to the baseline on ResNet-110/164. This ReLU layer is not used in conjunction with a BN layer, and may not enjoy the beneï¬ ts of BN [8]. Somehow surprisingly, when BN and ReLU are both used as pre-activation, the results are improved by healthy margins (Table 2 and Table 3). In Table 3 we report results using various architectures: (i) ResNet-110, (ii) ResNet-164, (iii) a 110-layer ResNet architecture in which each shortcut skips only 1 layer (i.e., | 1603.05027#23 | 1603.05027#25 | 1603.05027 | [
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1603.05027#25 | Identity Mappings in Deep Residual Networks | â â 164, original â 164, proposed (pre-activation) wei . si SiN apa * - aL err â waa, 5 110 original â thy © Se om â â 110, BNafter add eH it oth ae i eid - 0 5 6 10° 0.002 0 0.002 ° T 2 3 4 5 6 ° 1 2 3 4 x10" Iterations x Iterations Figure 6. Training curves on CIFAR-10. Left: BN after addition (Fig. 4(b)) using ResNet-110. Right: pre-activation unit (Fig. 4(e)) on ResNet-164. Solid lines denote test error, and dashed lines denote training loss. a Residual Unit has only 1 layer), denoted as â ResNet-110(1layer)â , and (iv) a 1001-layer bottleneck architecture that has 333 Residual Units (111 on each feature map size), denoted as â ResNet-1001â | 1603.05027#24 | 1603.05027#26 | 1603.05027 | [
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1603.05027#26 | Identity Mappings in Deep Residual Networks | . We also experiment on CIFAR- 100. Table 3 shows that our â pre-activationâ models are consistently better than the baseline counterparts. We analyze these results in the following. # 4.2 Analysis We ï¬ nd the impact of pre-activation is twofold. First, the optimization is further eased (comparing with the baseline ResNet) because f is an identity mapping. Second, using BN as pre-activation improves regularization of the models. Ease of optimization. | 1603.05027#25 | 1603.05027#27 | 1603.05027 | [
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