Abstract
Training Binarized Neural Networks (BNNs) is challenging due to the discreteness. In order to efficiently optimize BNNs through backward propagations, real-valued auxiliary variables are commonly used to accumulate gradient updates. Those auxiliary variables are then directly quantized to binary weights in the forward pass, which brings about large quantization errors. In this paper, by introducing an appropriate proxy matrix, we reduce the weights quantization error while circumventing explicit binary regularizations on the full-precision auxiliary variables. Specifically, we regard pre-binarization weights as a linear combination of the basis vectors. The matrix composed of basis vectors is referred to as the proxy matrix, and auxiliary variables serve as the coefficients of this linear combination. We are the first to empirically identify and study the effectiveness of learning both basis and coefficients to construct the pre-binarization weights. This new proxy learning contributes to new leading performances on benchmark datasets.
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Notes
- 1.
h, w, n and c are kernel height, width, kernel number and input channel number, respectively. For \(1\times 1\) convolutions and FC layers, \([h\times w\times n]\times c\) degrades into \(n\times c\).
- 2.
Further details in appendix 1.
References
Alizadeh, M., Fernández-Marqués, J., Lane, N.D., Gal, Y.: A systematic study of binary neural networks’ optimisation. In: International Conference on Learning Representations (2019). https://openreview.net/forum?id=rJfUCoR5KX
Anderson, A.G., Berg, C.P.: The high-dimensional geometry of binary neural networks. In: International Conference on Learning Representations (2018). https://openreview.net/forum?id=B1IDRdeCW
Andoni, A., Indyk, P.: Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. Commun. ACM 51(1), 117–122 (2008). http://doi.acm.org/10.1145/1327452.1327494
Bahou, A.A., Karunaratne, G., Andri, R., Cavigelli, L., Benini, L.: XNORBIN: A 95 top/s/w hardware accelerator for binary convolutional neural networks. In: 2018 IEEE Symposium in Low-Power and High-Speed Chips, COOL CHIPS 2018, Yokohama, Japan, 18–20 Apr 2018, pp. 1–3. IEEE Computer Society (2018). https://doi.org/10.1109/CoolChips.2018.8373076
Bethge, J., Yang, H., Bornstein, M., Meinel, C.: Back to simplicity: How to train accurate BNNs from scratch? CoRR abs/1906.08637 (2019). http://arxiv.org/abs/1906.08637
Bulat, A., Tzimiropoulos, G.: XNOR-Net++: Improved binary neural networks. In: British Machine Vision Conference, BMVC 2019 (2019)
Bulat, A., Tzimiropoulos, G., Kossaifi, J., Pantic, M.: Improved training of binary networks for human pose estimation and image recognition. CoRR abs/1904.05868 (2019). http://arxiv.org/abs/1904.05868
Cheng, J., Leng, C., Wu, J., Cui, H., Lu, H.: Fast and accurate image matching with cascade hashing for 3d reconstruction. In: 2014 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2014, Columbus, OH, USA, 23–28 Jun 2014, pp. 1–8 (2014). https://doi.org/10.1109/CVPR.2014.8
Courbariaux, M., Bengio, Y., David, J.: Binaryconnect: Training deep neural networks with binary weights during propagations. In: Advances in Neural Information Processing Systems 28: Annual Conference on Neural Information Processing Systems 2015, 7–12 Dec 2015, Montreal, Quebec, Canada, pp. 3123–3131 (2015). http://papers.nips.cc/paper/5647-binaryconnect-training-deep-neural-networks-with-binary-weights-during-propagations
Darabi, S., Belbahri, M., Courbariaux, M., Nia, V.P.: BNN+: improved binary network training. CoRR abs/1812.11800 (2018). http://arxiv.org/abs/1812.11800
Ding, R., Chin, T., Liu, Z., Marculescu, D.: Regularizing activation distribution for training binarized deep networks. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2019, Long Beach, CA, USA, 16–20 June 2019, pp. 11408–11417 (2019)
Gong, Y., Lazebnik, S., Gordo, A., Perronnin, F.: Iterative quantization: A procrustean approach to learning binary codes for large-scale image retrieval. IEEE Trans. Pattern Anal. Mach. Intell. 35(12), 2916–2929 (2013). https://doi.org/10.1109/TPAMI.2012.193
Gu, J., et al.: Projection convolutional neural networks for 1-bit CNNs via discrete back propagation. In: The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, 27 Jan–1 Feb 2019, pp. 8344–8351 (2019). https://doi.org/10.1609/aaai.v33i01.33018344
Gu, J., et al.: Bayesian optimized 1-bit CNNs. In: IEEE Proceedings of the IEEE International Conference on Computer Vision ICCV 2019, Seoul, South Korea (2019)
He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: 2016 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2016, Las Vegas, NV, USA, 27–30 Jun 2016, pp. 770–778 (2016). https://doi.org/10.1109/CVPR.2016.90
He, X., Wang, P., Cheng, J.: K-nearest neighbors hashing. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2019, Long Beach, CA, USA, 16–20 Jun 2019, pp. 2839–2848 (2019). http://openaccess.thecvf.com/content_CVPR_2019/html/He_K-Nearest_Neighbors_Hashing_CVPR_2019_paper.html
Helwegen, K., Widdicombe, J., Geiger, L., Liu, Z., Cheng, K., Nusselder, R.: Latent weights do not exist: Rethinking binarized neural network optimization. In: Wallach, H.M., Larochelle, H., Beygelzimer, A., d’Alché-Buc, F., Fox, E.B., Garnett, R. (eds.) Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, 8–14 Dec 2019, Vancouver, BC, Canada, pp. 7531–7542 (2019). http://papers.nips.cc/paper/8971-latent-weights-do-not-exist-rethinking-binarized-neural-network-optimization
Hu, Q., Wang, P., Cheng, J.: From hashing to CNNs: Training binary weight networks via hashing. In: Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), the 30th innovative Applications of Artificial Intelligence (IAAI-18), and the 8th AAAI Symposium on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA, 2–7 Feb 2018, pp. 3247–3254 (2018). https://www.aaai.org/ocs/index.php/AAAI/AAAI18/paper/view/16466
Hu, Q., Wu, J., Bai, L., Zhang, Y., Cheng, J.: Fast k-means for large scale clustering. In: Proceedings of the 2017 ACM on Conference on Information and Knowledge Management, CIKM 2017, Singapore, 06–10 Nov 2017, pp. 2099–2102 (2017). https://doi.org/10.1145/3132847.3133091
Hubara, I., Courbariaux, M., Soudry, D., El-Yaniv, R., Bengio, Y.: Binarized neural networks. In: Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, 5–10 Dec 2016, Barcelona, Spain, pp. 4107–4115 (2016). http://papers.nips.cc/paper/6573-binarized-neural-networks
Ji, C., Psaltis, D.: Capacity of two-layer feedforward neural networks with binary weights. IEEE Trans. Inf. Theory 44(1), 256–268 (1998). https://doi.org/10.1109/18.651033
Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. In: Bengio, Y., LeCun, Y. (eds.) 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, 7–9 May 2015, Conference Track Proceedings (2015). http://arxiv.org/abs/1412.6980
Krizhevsky, A., Sutskever, I., Hinton, G.E.: ImageNet classification with deep convolutional neural networks. Commun. ACM 60(6), 84–90 (2017). http://doi.acm.org/10.1145/3065386
Leng, C., Dou, Z., Li, H., Zhu, S., Jin, R.: Extremely low bit neural network: Squeeze the last bit out with ADMM. In: Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), the 30th innovative Applications of Artificial Intelligence (IAAI-18), and the 8th AAAI Symposium on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA, 2–7 Feb 2018, pp. 3466–3473 (2018). https://www.aaai.org/ocs/index.php/AAAI/AAAI18/paper/view/16767
Lin, X., Zhao, C., Pan, W.: Towards accurate binary convolutional neural network. In: Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4–9 Dec 2017, Long Beach, CA, USA, pp. 345–353 (2017). http://papers.nips.cc/paper/6638-towards-accurate-binary-convolutional-neural-network
Lin, Z., Courbariaux, M., Memisevic, R., Bengio, Y.: Neural networks with few multiplications. In: 4th International Conference on Learning Representations, ICLR 2016, San Juan, Puerto Rico, 2–4 May 2016, Conference Track Proceedings (2016). http://arxiv.org/abs/1510.03009
Liu, C., Ding, W., Hu, Y., Zhang, B., Liu, J., Guo, G.: GBCNs: Genetic binary convolutional networks for enhancing the performance of 1-bit DCNNs. In: The Thirty-Fourth AAAI Conference on Artificial Intelligence (AAAI-20) (February 2020)
Liu, Z., Wu, B., Luo, W., Yang, X., Liu, W., Cheng, K.: Bi-real net: Enhancing the performance of 1-bit CNNs with improved representational capability and advanced training algorithm. In: Computer Vision - ECCV 2018–15th European Conference, Munich, Germany, 8–14 Sept 2018, Proceedings, Part XV, pp. 747–763 (2018). https://doi.org/10.1007/978-3-030-01267-0_44
Martinez, B., Yang, J., Bulat, A., Tzimiropoulos, G.: Training binary neural networks with real-to-binary convolutions. In: International Conference on Learning Representations (2020). https://openreview.net/forum?id=BJg4NgBKvH
Mayoraz, E., Aviolat, F.: Constructive training methods for feedforward neural networks with binary weights. Int. J. Neural Syst. 7(2), 149–66 (1995)
Mishra, A.K., Nurvitadhi, E., Cook, J.J., Marr, D.: WRPN: wide reduced-precision networks. In: 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, 30 Apr–3 May 2018, Conference Track Proceedings (2018). https://openreview.net/forum?id=B1ZvaaeAZ
Oliveira, A.L., Sangiovanni-Vincentelli, A.L.: Learning complex Boolean functions: Algorithms and applications. In: Advances in Neural Information Processing Systems 6, [7th NIPS Conference, Denver, Colorado, USA, 1993], pp. 911–918 (1993). http://papers.nips.cc/paper/857-learning-complex-boolean-functions-algorithms-and-applications
Pagallo, G., Haussler, D.: Boolean feature discovery in empirical learning. Mach. Learn. 5, 71–99 (1990). https://doi.org/10.1007/BF00115895
Paszke, A., et al.: Pytorch: An imperative style, high-performance deep learning library. In: Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, 8–14 Dec 2019, Vancouver, BC, Canada, pp. 8024–8035 (2019). http://papers.nips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library
Peters, J.W., Genewein, T., Welling, M.: Probabilistic binary neural networks (2019). https://openreview.net/forum?id=B1fysiAqK7
Qin, H., Gong, R., Liu, X., Wei, Z., Yu, F., Song, J.: IR-Net: Forward and backward information retention for highly accurate binary neural networks. In: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Seattle Wastington, USA (June 2020)
Rastegari, M., Ordonez, V., Redmon, J., Farhadi, A.: XNOR-Net: Imagenet classification using binary convolutional neural networks. In: Computer Vision - ECCV 2016–14th European Conference, Amsterdam, The Netherlands, 11–14 Oct 2016, Proceedings, Part IV, pp. 525–542 (2016). https://doi.org/10.1007/978-3-319-46493-0_32
Rublee, E., Rabaud, V., Konolige, K., Bradski, G.R.: ORB: An efficient alternative to SIFT or SURF. In: IEEE International Conference on Computer Vision, ICCV 2011, Barcelona, Spain, 6–13 Nov 2011, pp. 2564–2571 (2011). https://doi.org/10.1109/ICCV.2011.6126544
Santurkar, S., Tsipras, D., Ilyas, A., Madry, A.: How does batch normalization help optimization? In: Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, 3–8 Dec 2018, Montréal, Canada, pp. 2488–2498 (2018). http://papers.nips.cc/paper/7515-how-does-batch-normalization-help-optimization
Schonemann, P.H.: A generalized solution of the orthogonal procrustes problem. Psychometrika 31(1), 1–10 (1966)
Shayer, O., Levi, D., Fetaya, E.: Learning discrete weights using the local reparameterization trick. In: 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, 30 Apr–3 May 2018, Conference Track Proceedings. OpenReview.net (2018). https://openreview.net/forum?id=BySRH6CpW
Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recognition. In: 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, 7–9 May 2015, Conference Track Proceedings (2015). http://arxiv.org/abs/1409.1556
Soudry, D., Hubara, I., Meir, R.: Expectation backpropagation: Parameter-free training of multilayer neural networks with continuous or discrete weights. In: Advances in Neural Information Processing Systems 27: Annual Conference on Neural Information Processing Systems 2014, 8–13 Dec 2014, Montreal, Quebec, Canada, pp. 963–971 (2014). http://papers.nips.cc/paper/5269-expectation-backpropagation-parameter-free-training-of-multilayer-neural-networks-with-continuous-or-discrete-weights
Soudry, D., Meir, R.: Mean field Bayes backpropagation: scalable training of multilayer neural networks with binary weights (2013)
Tang, W., Hua, G., Wang, L.: How to train a compact binary neural network with high accuracy? In: Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, 4–9 Feb 2017, San Francisco, California, USA, pp. 2625–2631 (2017). http://aaai.org/ocs/index.php/AAAI/AAAI17/paper/view/14619
Wang, P., He, X., Li, G., Zhao, T., Cheng, J.: Sparsity-inducing binarized neural networks. In: The Thirty-Fourth AAAI Conference on Artificial Intelligence (AAAI-20) (February 2020)
Wen, Y., Zhang, K., Li, Z., Qiao, Y.: A discriminative feature learning approach for deep face recognition. In: Computer Vision-ECCV 2016–14th European Conference, Amsterdam, The Netherlands, 11–14 Oct 2016, Proceedings, Part VII, pp. 499–515 (2016). https://doi.org/10.1007/978-3-319-46478-7_31
Yang, J., et al.: Quantization networks. In: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Long Beach CA, USA (June 2019)
Yazdani, M.: Linear backprop in non-linear networks. In: Compact Deep Neural Network Representation with Industrial Applications Workshop, Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, 3–8 Dec 2018, Montréal, Canada (2018)
Zagoruyko, S., Komodakis, N.: Wide residual networks. In: Proceedings of the British Machine Vision Conference 2016, BMVC 2016, York, UK, 19–22 Sept 2016 (2016). http://www.bmva.org/bmvc/2016/papers/paper087/index.html
Zhao, T., He, X., Cheng, J., Hu, J.: Bitstream: Efficient computing architecture for real-time low-power inference of binary neural networks on CPUs. In: Boll, S., Lee, K.M., Luo, J., Zhu, W., Byun, H., Chen, C.W., Lienhart, R., Mei, T. (eds.) 2018 ACM Multimedia Conference on Multimedia Conference, MM 2018, Seoul, Republic of Korea, 22–26 Oct 2018, pp. 1545–1552. ACM (2018). https://doi.org/10.1145/3240508.3240673
Zhou, S., Ni, Z., Zhou, X., Wen, H., Wu, Y., Zou, Y.: DoReFa-Net: Training low bitwidth convolutional neural networks with low bitwidth gradients. CoRR abs/1606.06160 (2016). http://arxiv.org/abs/1606.06160
Acknowledgement
This work was supported in part by National Natural Science Foundation of China (No.61972396, 61876182, 61906193), National Key Research and Development Program of China (No. 2019AAA0103402), the Strategic Priority Research Program of Chinese Academy of Science(No.XDB32050200), the Advance Research Program (No. 31511130301), and Jiangsu Frontier Technology Basic Research Project (No. BK20192004).
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He, X. et al. (2020). ProxyBNN: Learning Binarized Neural Networks via Proxy Matrices. In: Vedaldi, A., Bischof, H., Brox, T., Frahm, JM. (eds) Computer Vision – ECCV 2020. ECCV 2020. Lecture Notes in Computer Science(), vol 12348. Springer, Cham. https://doi.org/10.1007/978-3-030-58580-8_14
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