{"type":"ScholarlyArticle","title":"Stochastic Fractional Hamiltonian Monte Carlo","authors":[{"name":"Nanyang Ye","slug":"nanyang-ye","given":"Nanyang","family":"Ye","affiliation":"University of Cambridge , Cambridge , United Kingdom"},{"name":"Zhanxing Zhu","slug":"zhanxing-zhu","given":"Zhanxing","family":"Zhu","affiliation":"Center for Data Science , Peking University , Beijing , China; Beijing Institute of Big Data Research (BIBDR)"}],"abstract":"In this paper, we propose a novel stochastic fractional Hamiltonian Monte Carlo approach which generalizes the Hamiltonian Monte Carlo method within the framework of fractional calculus and L\\'evy diffusion. Due to the large ``jumps'' introduced by L\\'evy noise and momentum term, the proposed dynamics is capable of exploring the parameter space more efficiently and effectively.  We have shown that the fractional Hamiltonian Monte Carlo could sample the multi-modal and high-dimensional target distribution more efficiently than the existing methods driven by Brownian diffusion. We further extend our method for optimizing deep neural networks. The experimental results show that the proposed stochastic fractional Hamiltonian Monte Carlo  for training deep neural networks could converge faster than other popular optimization schemes and generalize better.","doi":"10.24963/ijcai.2018/419","url":"https://ijcai.org/proceedings/2018/419","pdf":"https://ijcai.org/proceedings/2018/0419.pdf","markdown":"https://ijcai.org/proceedings/2018/419.md","bibtex":"https://ijcai.org/proceedings/2018/bibtex/419","volume":{"slug":"2018","title":"Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2018","url":"https://ijcai.org/proceedings/2018/","year":2018,"location":"Stockholm","dates":"13-19 July 2018","editors":"Jérôme Lang","isbn":"978-0-9992411-2-7","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2018"},"section":"Machine Learning","track":"Main track","number":419,"pages":{"first":3019,"last":3025},"keywords":["Machine Learning: Deep Learning","Machine Learning: Machine Learning"],"videos":[],"references":[{"position":1,"raw":"[Ahn et al., 2012] S. Ahn, A. Korattikara, and M. Welling. Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring. arXiv preprint arXiv:1206.6380, 2012.","title":"Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring","authors":"S Ahn, A Korattikara, M Welling","year":"2012","venue":null,"doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Barndorff-Nielsen and Shephard, 2003] Ole Barndorff- Nielsen and Neil Shephard. Impact of jumps on returns and realised variances: econometric analysis of time-deformed levy processes. Economics Papers 2003-W12, Economics Group, Nuffield College, University of Oxford, 2003.","title":"Ole Barndorff-Nielsen and Neil Shephard. Impact of jumps on returns and realised variances: econometric analysis of time-deformed levy processes","authors":"Barndorff-Nielsen, Shephard","year":"2003","venue":null,"doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[C ¸elik and Duman, 2012] Cem C ¸elik and Melda Duman. Crank-nicolson method for the fractional diffusion equa- tion with the riesz fractional derivative. Journal of Compu- tational Physics, 231(4):1743-1750, 2012.","title":"Crank-nicolson method for the fractional diffusion equation with the riesz fractional derivative","authors":"; Duman, C Cem, Melda ¸elik, Duman","year":"2012","venue":"Journal of Computational Physics","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Chaudhari et al., 2016] Pratik Chaudhari, Anna Choromans- ka, Stefano Soatto, Yann LeCun, Carlo Baldassi, Christian Borgs, Jennifer T. Chayes, Levent Sagun, and Riccardo Zecchina. Entropy-sgd: Biasing gradient descent into wide valleys. CoRR, abs/1611.01838, 2016.","title":"Entropy-sgd: Biasing gradient descent into wide valleys","authors":"Pratik Chaudhari, Anna Choromanska, Stefano Soatto, Yann Lecun, Carlo Baldassi, Christian Borgs, Jennifer T Chayes, Levent Sagun, Riccardo Zecchina","year":"2016","venue":null,"doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Chen et al., 2014] T. Chen, E. B. Fox, and C. Guestrin. S- tochastic gradient Hamiltonian Monte Carlo. In Proceed- ings of the 31st International Conference on Machine Learning, pages 1683-1691, 2014.","title":"Stochastic gradient Hamiltonian Monte Carlo","authors":"T Chen, E B Fox, C Guestrin","year":"2014","venue":"Proceedings of the 31st International Conference on Machine Learning","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Chen et al., 2016] C. Chen, D. Carlson, Z. Gan, C. Li, and L. Carin. Bridging the gap between stochastic gradient MCMC and stochastic optimization. In AISTATS, 2016.","title":"Bridging the gap between stochastic gradient MCMC and stochastic optimization","authors":"C Chen, D Carlson, Z Gan, C Li, L Carin","year":"2016","venue":"AISTATS","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Dauphin et al., 2014] Y. N. Dauphin, R. Pascanu, C. Gul- cehre, K. Cho, S. Ganguli, and Y. Bengio. Identifying and attacking the saddle point problem in high-dimensional non-convex optimization. In NIPS, 2014.","title":"Identifying and attacking the saddle point problem in high-dimensional non-convex optimization","authors":"Y N Dauphin, R Pascanu, C Gulcehre, K Cho, S Ganguli, Y Bengio","year":"2014","venue":"NIPS","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Goodfellow et al., 2016] Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning. MIT Press, 2016. http://www.deeplearningbook.org.","title":"Deep Learning","authors":"Ian Goodfellow, Yoshua Bengio, Aaron Courville","year":"2016","venue":null,"doi":null,"url":"http://www.deeplearningbook.org","internal_url":null},{"position":9,"raw":"[Keskar et al., 2016] N. S. Keskar, D. Mudigere, J. Nocedal, M. Smelyanskiy, and P. T. Tang. On large-batch training for deep learning: Generalization gap and sharp minima. arXiv preprint arXiv:1609.04836, 2016.","title":"On large-batch training for deep learning: Generalization gap and sharp minima","authors":"N S Keskar, D Mudigere, J Nocedal, M Smelyanskiy, P T Tang","year":"2016","venue":null,"doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Kingma and Welling, 2013] Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. CoRR, ab- s/1312.6114, 2013.","title":"Auto-encoding variational bayes","authors":"P Diederik, Max Kingma, Welling","year":"2013","venue":"CoRR","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Max and Whye, 2011] Welling Max and Teh Yee Whye. Bayesian learning via stochastic gradient langevin dynam- ics. In ICML, 2011.","title":"Bayesian learning via stochastic gradient langevin dynamics","authors":"Max, Whye ;, Welling Max, Teh Yee, Whye","year":"2011","venue":"ICML","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Rapaport, 2004] D. C. Rapaport. The art of molecular dy- namics simulation. Cambridge university press, 2004.","title":"The art of molecular dynamics simulation","authors":"D C Rapaport","year":"2004","venue":null,"doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Simsekli, 2017] Umut Simsekli. Fractional Langevin Monte carlo: Exploring Levy driven stochastic differential equa- tions for Markov chain Monte Carlo. In Proceedings of the 34th International Conference on Machine Learning, volume 70, pages 3200-3209, 2017.","title":"Fractional Langevin Monte carlo: Exploring Levy driven stochastic differential equations for Markov chain Monte Carlo","authors":"Umut Simsekli","year":"2017","venue":"Proceedings of the 34th International Conference on Machine Learning","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Tripuraneni et al., 2017] Nilesh Tripuraneni, Mark Rowland, Zoubin Ghahramani, and Richard Turner. Magnetic Hamil- tonian Monte Carlo. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Ma- chine Learning Research, pages 3453-3461, International Convention Centre, Sydney, Australia, 06-11 Aug 2017. PMLR.","title":"Magnetic Hamiltonian Monte Carlo","authors":"Nilesh Tripuraneni, Mark Rowland, Zoubin Ghahramani, Richard Turner","year":"2017","venue":"Proceedings of the 34th International Conference on Machine Learning","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Wolpert and Taqqu, 2005] Robert L. Wolpert and Murad S. Taqqu. Fractional ornstein-uhlenbeck lévy processes and the telecom process: Upstairs and downstairs. Signal Pro- cess., 85(8):1523-1545, August 2005.","title":"Fractional ornstein-uhlenbeck lévy processes and the telecom process: Upstairs and downstairs","authors":"Robert L Wolpert, Murad S Taqqu","year":"2005","venue":"Signal Process","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}