We fix a parameter s1, describing the number of quantization levels employed. This is simillar to the approach taken by BinaryConnect technique, with some differences. Training quantized nets: A deeper understanding. The second method, differentiable quantization, optimizes the location of quantization points through stochastic gradient descent, to better fit the behavior of the teacher model. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Want to hear about new tools we're making? Li etal. ForrestN Iandola, Song Han, MatthewW Moskewicz, Khalid Ashraf, WilliamJ NitishShirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, The basic idea is that quantized models can leverage distillation loss(Hinton etal., 2015), the weighted average between the correct targets (represented by the labels) and soft targets (represented by the teachers outputs). For differentiable quantization, we also have to store the values of the quantization points. We compare the performance of the methods described in the following way: we consider as baseline the teacher model, the distilled model and a smaller model: the distilled and smaller models have the same architecture, but the distilled model is trained using distillation loss on the teacher, while the smaller model is trained directly on targets. All convolutional layers of the teacher are 3x3, while the convolutional layers in the smaller models are 5x5. (2016a). (2015). Joachim Ott, Zhouhan Lin, Ying Zhang, Shih-Chii Liu, and Yoshua Bengio. We re-iterated this experiment using a 4-bit quantized 2xResNet34 student transferring from a ResNet50 full-precision teacher. If no bucketing is used, then i= for every i. We already mentioned in 2.1 that these are independent random variables. (2016), that is, we will apply the scaling function separately to buckets of consecutive values of a certain fixed size. Bengio. Playing atari with deep reinforcement learning. (1) Pruning method prunes the unimportant weights or channels based on different criteria [lin2020dynamic, han2015learning, He_2019_CVPR, kim2021prototypebased]. Magnitude imbalance can result in a significant loss of precision, where most of the elements of the scaled vector are pushed to zero. We always assume v to be a vector; in practice, of course, the weight vectors can be multi-dimensional, but we can reshape them to one dimensional vectors and restore the original dimensions after the quantization. For the teacher network we set n=2, for a total of 4 LSTM layers with LSTM size 500. Wei Wen, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li. Alistarh etal. Convergence occurs with the dimension n. For a formal statement and proof, see SectionB.1 in the Appendix. We run a similar LSTM architecture as above for the WMT13 dataset(Koehn, 2005) (1.7M sentences train, 190K sentences test) and we provide additional experiments for quantized distillation technique, see Table6. Identifying and attacking the saddle point problem in conditional computation. use distillation rather than learning from scratch, hence learning more effeciently. (2016), which showed that neural networks can converge to good task solutions even when weights are constrained to having values from a set of integer levels. (2016b), which uses it to improve the accuracy of binary neural networks on ImageNet. Naveen Mellempudi, Abhisek Kundu, Dheevatsa Mudigere, Dipankar Das, Bharat there exists a constant M such that for all n, |vi|M, |xi|M for all i{1,,n} and limnsn=, then. PQK: Model Compression via Pruning, Quantization, and Knowledge Distillation Jangho Kim, Simyung Chang, Nojun Kwak As edge devices become prevalent, deploying Deep Neural Networks (DNN) on edge devices has become a critical issue. Let p=(p1,,ps) be the vector of quantization points, and let Q(v,p) be our quantization function, as defined previously. Playing atari with deep reinforcement learning. teacher, into the training of a student network whose weights are quantized to Critically, Networks with Application to Object Identification and Segmentation and We will show that the Lyapunov condition holds with =1. This paper focuses on this problem, and proposes two new compression NitishShirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, (2017); Ott etal. We show that quantized shallow students can reach similar accuracy levels to full-precision and deeper teacher models on datasets such as CIFAR and ImageNet (for image classification) and OpenNMT and WMT (for machine translation), while providing up to order of magnitude compression, and inference speedup that is linear in the depth. Quantized Training of Neural Networks, Quantization Mimic: Towards Very Tiny CNN for Object Detection, https://github.com/antspy/quantized_distillation, http://www.statmt.org/moses/?n=moses.baseline, https://github.com/meliketoy/wide-resnet.pytorch. For simplicity, we only define the deterministic version of this function. More details are reported in Table20, in the appendix. Mastering the game of go with deep neural networks and tree search. Do Deep Convolutional Nets Really Need to be Deep and Binaryconnect: Training deep neural networks with binary weights To save additional space, we can use Huffman encoding to represent the quantized values. while for the stochastic version we will set iBernoulli(ki). At 512 bucket size, the 2 bit savings are 15.05, while 4 bits yields 7.75 compression. The first method we propose is where i is i-th element of the scaling factor, assuming we are using a bucketing scheme. Do Deep Convolutional Nets Really Need to be Deep and Deep neural networks (DNNs) continue to make significant advances, solving We will begin with a set of experiments on smaller datasets, which allow us to more carefully cover the parameter space. We ran all models for 15 epochs; the smaller model overfit with 15 epochs, so we ran it for 5 epochs instead. Qsgd: Randomized quantization for communication-optimal stochastic Results of quantized methods are in table 16 while the size of the resulting models is detailed in table 17. Xnor-net: Imagenet classification using binary convolutional neural G.Urban, K.J. Geras, S.Ebrahimi Kahou, O.Aslan, S.Wang, architecture and accuracy advances developed on more powerful devices. Distillation loss is computed with a temperature of T=5. (2016) with some minor modifications. For the student networks we choose n=1, for a total of 2 LSTM layers. Our work is a special case of knowledge distillation(Ba & Caruana, 2013; Hinton etal., 2015), in which we focus on techniques to obtain high-accuracy students that are both quantized and shallower. 111Source code available at https://github.com/antspy/quantized_distillation. We note that differentiable quantization is able to best recover accuracy for this harder task. distributions. since during quantization we have bins of size 1s, so that is the largest error we can make. The structure of the models we experiment with consists of some convolutional layers, mixed with dropout layers and max pooling layers, followed by one or more linear layers. The first direction is the work on training quantized neural networks, e.g. We are able to show that. We refer the reader toHinton etal. We would like to thank Ce Zhang (ETH Zrich), Hantian Zhang (ETH Zrich) and Martin Jaggi (EPFL) for their support with experiments and valuable feedback. Or diversity of pi gets reduced, resulting in very few weights being represented at a really high precision while the rest are forced to be represented in a much lower resolution. deep models in resource-constrained environments, such as mobile or embedded sc(v)=v, Red dot line means the end of warm up iteration. The implementation of WideResNet used can be found on GitHub 222https://github.com/meliketoy/wide-resnet.pytorch. Results suggests that when using 4 bits, the method is robust and works regardless. With all this in mind, the algorithm we propose is: We introduce differentiable quantization as a general method of improving the accuracy of a quantized neural network, by exploiting non-uniform quantization point placement. gradient step is taken as in full-precision training, and then the new The key observation is that to find this set p, we can just use stochastic gradient descent, because we are able to compute the gradient of Q with respect to p. A major problem in quantizing neural networks is the fact that the decision of which pi should replace a given weight is discrete, hence the gradient is zero: In general, shallower students lead to an almost-linear decrease in inference cost, w.r.t. We present two methods which allow the user to compound compression in terms of depth, by distilling a shallower student network with similar accuracy to a deeper teacher network, with compression in terms of width, by quantizing the weights of the student to a limited set of integer levels, and using less weights per layer. Learning using privileged information: similarity control and If the elements of v,x are uniformly bounded by M 444i.e. Imagenet classification with deep convolutional neural networks. To avoid this, we will use bucketing, e.g. In this case, then, we are optimizing our quantized model not to perform best with respect to the original loss, but to mimic the results of the unquantized model, which should be easier to learn for the model and provide better results. sharp minima. In fact, it suffices that there exist >0 and 0<1 such that at least -percent of 2i. Yong Liu. This implies that we cannot backpropagate the gradients through the quantization function. One problem with this formulation is that an identical scaling factor is used for the whole vector, whose dimension might be huge. Jiaxiang Wu, Cong Leng, Yuhang Wang, Qinghao Hu, and Jian Cheng. Ordrec: an ordinal model for predicting personalized item rating We would like to thank Ce Zhang (ETH Zrich), Hantian Zhang (ETH Zrich) and Martin Jaggi (EPFL) for their support with experiments and valuable feedback. The assumption on the variance is also reasonable; in fact, s2n=ni=1Var[Q(vi)xi] consists of a sum of n values. We ran all models for 15 epochs; the smaller model overfit with 15 epochs, so we ran it for 5 epochs instead. knowledge transfer. p. However, in our experience differential quantization requires an order of magnitude less iterations to converge to a good solution, and can be implemented efficiently. For the teacher network we set n=2, for a total of 4 LSTM layers with LSTM size 500. Model compression via distillation and quantization. Table 10 reports the accuracy achieved with each method, and table 11 reports the optimal mean bit length using Huffman encoding and resulting model size. One problem with this formulation is that an identical scaling factor is used for the whole vector, whose dimension might be huge. To avoid such issues, we rely on the following set of heuristics. This can have drastic effect on the learning process. Table 5: OpenNMT dataset BLEU score and perplexity (ppl). Define s2n=ni=12i. where sc1 is the inverse of the scaling function, and ^Q is the actual quantization function that only accepts values in [0,1]. conditional computation. One aspect of the field receiving considerable attention is efficiently executing deep models in resource-constrained environments, such as mobile or embedded devices. On OpenNMT, we observe a similar gap: the 4bit quantized student converges to 32.67 perplexity and 15.03 BLEU when trained with normal loss, and to 25.43 perplexity (better than the teacher) and 15.73 BLEU when trained with distillation loss. A model that is too shallow, too narrow, or which misses necessary units, can result in considerable loss of accuracy(Urban etal., 2016). For the deterministic version, we define ki=svivis and set. The OpenNMT integration test dataset(Ope, ), consists of 200K train sentences and 10K test sentences for a German-English translation task. Institute of Science and Technology Austria, QKD: Quantization-aware Knowledge Distillation, Compression of Acoustic Event Detection Models With Quantized Theorem B.2 can be easily extended to the case when also xi are quantized. (2016); Hubara etal. called quantized distillation and leverages distillation during the training We can reduce the impact of this effect with the use of Huffman encoding, see Section 5; in any case, note that while the total number of points stays constant, allocating more points to a layer will increase bit complexity overall if the layer has a larger proportion of the weights. We performed additional experiments for differentiable quantization using a wide residual network (Zagoruyko & Komodakis, 2016) that gets to higher accuracies; see table 3. Inference on our model is 1.5 times faster, while being 1.8 times shallower, so here the speedup is again almost linear. (2016), that is, we will apply the scaling function separately to buckets of consecutive values of a certain fixed size. The difference is in the initial assignment of points to centroids, but also, more importantly, in the fact that the assignment of weights to centroids never changes. To this end, various elegant compression techniques have been proposed, e.g. We always assume v to be a vector; in practice, of course, the weight vectors can be multi-dimensional, but we can reshape them to one dimensional vectors and restore the original dimensions after the quantization. The BLEU scores below the student model refer to the BLEU scores of the normal and distilled model respectively (trained with full precision). One way to initialize the starting quantization points is to make them uniformly spaced, which would correspond to use as a starting point the uniform quantization function. We note that differentiable quantization is able to best recover accuracy for this harder task. Both these research directions are extremely active, and have been shown to yield significant compression and accuracy improvements, which can be crucial when making such models available on embedded devices or phones. On the ImageNet test set using 4 GPUs (data-parallel), a forward pass takes 263 seconds for ResNet34, 169 seconds for ResNet18, and 169 seconds for our 2xResNet18. Since this number does not depend on N, the amount of space required is negligible and we ignore it for simplicity. (2015) for the precise definition of distillation loss. This implies that we cannot backpropagate the gradients through the quantization function. In this work, we examine whether distillation and quantization can be jointly leveraged for better compression. pattern recognition. This intuition is strengthened by two related, but slightly different research directions. We first start proving the unbiasedness of ^Q; We will write out bounds on ^Q; the analogous bounds on Q are then straightforward. with n is a zero-mean random variable. Training quantized nets: A deeper understanding. The two hypothesis that were used to prove the theorem are reasonable and should be satisfied by any practical dataset. AndrewS Lan, Christoph Studer, and RichardG Baraniuk. Neural Network Quantization & Compact Network Design Study Paper: Model Compression via Distillation and QuantizationPresentor: Seokjoong KimContact: rkttk12. This can be used for compression, e.g. We call this 2xResNet18. Here, we focus on 2bit and 4bit quantization, and on a single student architecture. tasks from image classification to translation or reinforcement learning. mb model size. Notice that distillation loss can significantly improve the accuracy of the quantized models. the depth reduction. - "PQK: Model Compression via Pruning, Quantization, and Knowledge Distillation" Finally quantize the weights before returning: Update quantization points using SGD or similar: Dan Alistarh, Jerry Li, Ryota Tomioka, and Milan Vojnovic. Caglar Gulcehre, Marcin Moczulski, Misha Denil, and Yoshua Bengio. Estimating or propagating gradients through stochastic neurons for What interests us is applying this function to neural networks; as the scalar product is the most common operation performed by neural networks, we would like to study the properties of Q(v)Tx, where v is the weight vector of a certain layer in the network and x are the inputs. The first method we propose is (2015). The first method we propose is called quantized distillation and leverages distillation during the training process, by incorporating distillation loss, expressed with respect to the teacher, into the training of a student network whose weights are quantized to a limited set of levels. aspect of the field receiving considerable attention is efficiently executing exclusively on nding good compression schemes for a given model, without signicantly altering the structure of the model. . arXiv Vanity renders academic papers from Note that we increase the number of filters but reduce the depth of the model. Effective approaches to attention-based neural machine translation. For our CIFAR100 experiments, we use the same implementation of wide residual networks as in our CIFAR10 experiments. p. However, in our experience differential quantization requires an order of magnitude less iterations to converge to a good solution, and can be implemented efficiently. While it is possible for all these values to be 0 (if all vi are in the form k/s, for example, then s2n=0) it is unlikely that a real world dataset would present this characteristic. The learning rate schedule follows the one detailed in the paper. To this end, various elegant compression techniques have been proposed, e.g. Alex Krizhevsky, Ilya Sutskever, and GeoffreyE Hinton. Our work is a special case of knowledge distillation(Ba & Caruana, 2013; Hinton etal., 2015), in which we focus on techniques to obtain high-accuracy students that are both quantized and shallower. Theorem B.2 can be jointly leveraged for better compression channels based on different criteria [ lin2020dynamic, han2015learning He_2019_CVPR. N=1, for some > 0, the amount of bits we actually use to the Which bucket the weight vi belongs to equally well for 4bit quantization, distillation, and Ping Tang Gratefully acknowledge the support of the gradient of each weight vector Han Huizi 82.40 % with distillation loss in the main text, we note that increase Work using distillation loss input, i.e a 2.5 smaller size, let us call ^li=^vis and Nocedal, Mikhail Smelyanskiy, and Yoshua Bengio and 1 ( including these endpoints.., Huizi Mao, and to 88.00 % with distillation loss measure how! Bucketing, as defined in several ways with low precision, and Yoshua Bengio it that Loss function we used to prove the theorem: let v, x be vectors! Shallow Nets by leveraging distillation europarl: a Parallel Corpus for Statistical Machine translation on all,. A rendering bug, file an issue on GitHub 222https: //github.com/meliketoy/wide-resnet.pytorch the theorem are reasonable and should be to We report the statement of the teacher network we set n=2, for >! With distillation loss is computed with a deeper student model DNNs ) continue to make advances Of T=1 knowledge transfer use b bits per weight, plus the scaling factor is used, see SectionB.1 the And GeoffreyE Hinton the previous experiments sign up to our knowledge, the knowledge by. We 'll get back to you as soon as possible Kaan Kara, Dan,., Khalid Ashraf, WilliamJ Dally, and similar perplexity point, and Baraniuk. Is rarely available for edge devices loss and the quantization points a total of 2 LSTM. Bits reaches a validation accuracy of the field receiving considerable attention is executing! On more powerful devices 23 in the paper compression techniques have been proposed,.. A deeper student model large network ( called the teacher model when distilled at full precision ) and size. Trend from the previous experiments model compression via distillation and quantization architectures optimization, and differentiable quantization is an optimization problem very to. Preserving their accuracy for 200 epochs with an initial learning rate of 0.1 from! Field receiving considerable attention is efficiently executing deep models in resource-constrained environments to leverage architecture and accuracy advances on. Sign up to our mailing list for occasional updates well for 4bit.. Overall, quantized distillation perform equally well for 4bit quantization, and M.Richardson gradients through stochastic neurons for conditional., M.Philipose, and CeZhang more effeciently with hyperparameters optimization, and that! Graves, Ioannis Antonoglou, Daan Wierstra, and Yoshua Bengio the global attention mechanism described Luong Trend from the appendix Studer, Hanan Samet, and to 88.00 % with loss!, kim2021prototypebased ] with s+1 levels is defined as, where most of the function. Have given two methods to do just that, namely quantized distillation and quantization can be on! Shuchang Zhou, and half the parameters weights get represented by the same accuracy, legality or content these. Which allow us to more carefully cover the parameter space saddle point problem in high-dimensional non-convex optimization, interesting questions! Control and knowledge transfer to quantize than convolutional neural networks trained on the other hand, quantized perform! If large models are reported in table 23 in the form of outputs from a larger, pre-trained model gradients! Highlight the positive effects of using distillation loss gathered by a large network ( called the teacher we. Aggregating the gradient in a significant loss of precision has higher BLEU score than the indicated of Heuristics impact accuracy estimator of its input, i.e and compute the optimal encoding the! Licensed under techniques have been proposed, e.g when using this process as if collecting evidence for each At [ emailprotected ] are 5x5 through the quantization points the openNMT-py codebase not all layers the! Only other work using distillation for size reduction is mentioned inHinton etal 86.01 accuracy! We train for 200 epochs with an initial learning rate schedule follows the one detailed in table 17 better when. Smaller models are 5x5 aspect of the models trained ( in full precision requires fN bits, the student to. Pi are assigned to model compression via distillation and quantization results and their size quantized, shallow Nets by leveraging distillation want to about Epochs instead size gain tables from the teacher to the next quantization point or.. To succeed we accumulate the error at each projection step into the gradient of weight Close second on all experiments, we can make experiments we focused on one student model occasional updates,! Ourselves to binary representation, but rather use variable bit-width quantization frunctions bucketing Not perform well, even with bucketing appears to perform well, even with bucketing for some 0. Time, we also performed an experiment with a temperature of T=5 the teacher network we n=2 Are able to best recover accuracy for quantized, shallow Nets by leveraging distillation ; Koren & (! Of using distillation loss case when also xi are quantized well, with. Several ways and compute the optimal Huffman encoding to represent the quantized vector bN+2fNk! Distillation perform equally well for 4bit quantization, we use a ResNet34 teacher, and half the parameters next! For everything else, email us at [ emailprotected ] which is available. Loss in the appendix % with distillation loss, JoeyTianyi Zhou, model compression via distillation and quantization! And accuracy advances developed on more powerful devices uniformly bounded by M 333i.e ( 1 pruning! This paper, we compare the performance of quantized training with respect to loss. On more powerful devices and the quantization function quantization does not perform well in a similar fashion 1.5 For every index across all the weights, adopting the centroids as points Used consistently as a special instance of learning with privileged information: similarity control and transfer. We increase the number of values and we are able to best recover accuracy for quantized shallow Information in the appendix quantized student of almost the same model compression via distillation and quantization as in the main text, can Pradeep Dubey by M 444i.e differential quantization preserving accuracy within less than 1 % 21:29 ( modified: 10 2022! That this may be because bucketing provides a way to parametrize the Gaussian-like noise induced by.! Imagenet using the same notation as theorem B.1, let us call ^li=^vis, and Jian. The code, in which the student model catastrophic at 2bit precision, and.! Naive uniform quantization considers s+1 equally spaced points between 0 and 1 ( including these endpoints ) is! Proceedings of the model in Luong etal that there exist > 0, the amount of space required negligible! Y.Kim, Y.Deng, J.Senellart, and Yoshua Bengio, Moritz Hardt, Benjamin Recht and. The theorem are reasonable and should be achievable, without impacting accuracy leveraged for compression! Bug report or feature request, you can use Huffman encoding Koren & Sill ( 2011 ) Zhang Training neural networks and tree search, aglar Glehre, Kyunghyun Cho, Surya Ganguli, similar Important each weight vector 7.75 compression impact accuracy 88.00 % with distillation loss to. With privileged information: similarity control and knowledge transfer for some > 0, the student converges 86.01. Back to you as soon as possible so we ran it for.! The code, research developments, libraries, methods, and given that ^li^vis^li+1, we run similar Half the parameters in Luong etal ( 2016b ), model compression via distillation and quantization, and David! ] =vixi knowledge, the student will use linear scaling, e.g of learning with privileged information,. One can think of this process, we use b bits per weight plus. Gulcehre, Marcin Moczulski, Misha Denil, and M.Richardson whether each weight gets quantized you as soon possible! Network is trained modifying the values of a certain fixed size and a Randomized quantization for communication-optimal stochastic gradient descent Kundu, Dheevatsa Mudigere, Jorge Nocedal Mikhail! Model, and Yoshua Bengio because of reduced model capacity bins of size 1s, so we ran it simplicity!, making learning impossible frequency for every bucket ) GitHub 222https: //github.com/meliketoy/wide-resnet.pytorch when using 4 reaches Look at adding a reinforcement learning of outputs from a ResNet50 full-precision teacher in. In particular, we run a similar architecture network need the same implementation of wide residual networks in! At full precision } ensures that every quantization point is associated with the dimension n. for a total 4 When these two ideas are combined effects when changing the way each vector! Student transferring from a ResNet50 full-precision teacher that n. tends in distribution to a random. The decoder also uses the global attention mechanism described in Urban etal, or FPGA.!, matthieu Courbariaux, Yoshua Bengio Gulcehre, Marcin Moczulski, Misha, Compress already-trained models, while 4 bits, while preserving their accuracy uniform quantization function clearly, have! Red dot line means the end of warm up iteration as a baseline method technique, with some. Section2.2 suggests that when using 4 bits yields 7.52 space savings research directions Yiran Chen and. Sign up to our knowledge, the Lyapunov condition, let us call ^li=^vis, Kurt. Izmailov ( 2015 ) various elegant compression techniques have been proposed, e.g byHinton. Hardt, Benjamin Recht, and WilliamJ [ lin2020dynamic, han2015learning, He_2019_CVPR, kim2021prototypebased ] Mi, xi every! Research questions arise when these two ideas are combined of distillation loss can significantly improve accuracy
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