I’ve always been curious about the statistical physics approach to problems from computer science. The physics-inspired algorithm survey propagation is the current champion for random 3SAT instances, statistical-physics phase transitions have been suggested as explaining computational difficulty, and statistical physics has even been invoked to explain why deep learning algorithms seem to often converge to useful local minima.
Unfortunately, I have always found the terminology of statistical physics, “spin glasses”, “quenched averages”, “annealing”, “replica symmetry breaking”, “metastable states” etc… to be rather daunting
Jaan Altosaar’s guided translation is great.
Phase transitions in statistical inference
There is a deep analogy between statistical inference and statistical physics; I will give a friendly introduction to both of these fields. I will then discuss phase transitions in two problems of interest to a broad range of data sciences: community detection in social and biological networks, and clustering of sparse high-dimensional data. In both cases, if our data becomes too sparse or too noisy, it suddenly becomes impossible to find the underlying pattern, or even tell if there is one. Physics both helps us locate these phase transitions, and design optimal algorithms that succeed all the way up to this point. Along the way, I will visit ideas from computational complexity, random graphs, random matrices, and spin glass theory.
There is an overview lecture by Thomas Orton, which cites lots of the good stuff
Last week, we saw how certain computational problems like 3SAT exhibit a thresholding behavior, similar to a phase transition in a physical system. In this post, we’ll continue to look at this phenomenon by exploring a heuristic method, belief propagation (and the cavity method), which has been used to make hardness conjectures, and also has thresholding properties. In particular, we’ll start by looking at belief propagation for approximate inference on sparse graphs as a purely computational problem. After doing this, we’ll switch perspectives and see belief propagation motivated in terms of Gibbs free energy minimization for physical systems. With these two perspectives in mind, we’ll then try to use belief propagation to do inference on the the stochastic block model. We’ll see some heuristic techniques for determining when BP succeeds and fails in inference, as well as some numerical simulation results of belief propagation for this problem. Lastly, we’ll talk about where this all fits into what is currently known about efficient algorithms and information theoretic barriers for the stochastic block model.
See Igor Carron’s “phase diagram” list, and stuff like (Oymak and Tropp 2015). Likely there are connections to Erdős-Renyi giant components and other complex network things in probabilisitic graph learning. Read (Barbier 2015; Poole et al. 2016).
Replicator equations and evolutionary processes
Gentle intro lecture by John Baez, Biology as Information Dynamics.
Achlioptas, Dimitris, and Amin Coja-Oghlan. 2008. “Algorithmic Barriers from Phase Transitions.” October 2008. https://doi.org/10.1109/FOCS.2008.11.
Baez, John C. 2011. “Renyi Entropy and Free Energy,” February. https://arxiv.org/abs/1102.2098.
Baldassi, Carlo, Christian Borgs, Jennifer T. Chayes, Alessandro Ingrosso, Carlo Lucibello, Luca Saglietti, and Riccardo Zecchina. 2016. “Unreasonable Effectiveness of Learning Neural Networks: From Accessible States and Robust Ensembles to Basic Algorithmic Schemes.” Proceedings of the National Academy of Sciences 113 (48): E7655–E7662. https://doi.org/10.1073/pnas.1608103113.
Barbier, Jean. 2015. “Statistical Physics and Approximate Message-Passing Algorithms for Sparse Linear Estimation Problems in Signal Processing and Coding Theory.” November 5, 2015. http://arxiv.org/abs/1511.01650.
Barbier, Jean, Florent Krzakala, Nicolas Macris, Léo Miolane, and Lenka Zdeborová. 2017. “Phase Transitions, Optimal Errors and Optimality of Message-Passing in Generalized Linear Models.” August 10, 2017. http://arxiv.org/abs/1708.03395.
Braunstein, A., M. Mezard, and R. Zecchina. 2002. “Survey Propagation: An Algorithm for Satisfiability.” December 4, 2002. http://arxiv.org/abs/cs/0212002.
Castellani, Tommaso, and Andrea Cavagna. 2005. “Spin-Glass Theory for Pedestrians.” Journal of Statistical Mechanics: Theory and Experiment 2005 (05): P05012. https://doi.org/10.1088/1742-5468/2005/05/P05012.
Chang, Bo, Lili Meng, Eldad Haber, Lars Ruthotto, David Begert, and Elliot Holtham. 2018. “Reversible Architectures for Arbitrarily Deep Residual Neural Networks.” In. http://arxiv.org/abs/1709.03698.
Choromanska, Anna, MIkael Henaff, Michael Mathieu, Gerard Ben Arous, and Yann LeCun. 2015. “The Loss Surfaces of Multilayer Networks.” In Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, 192–204. http://proceedings.mlr.press/v38/choromanska15.html.
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———. 2016b. “Why Does Deep and Cheap Learning Work so Well?” August 29, 2016. http://arxiv.org/abs/1608.08225.
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Moore, Cristopher. 2017. “The Computer Science and Physics of Community Detection: Landscapes, Phase Transitions, and Hardness.” Bulletin of the EATCS, February. http://arxiv.org/abs/1702.00467.
Oymak, Samet, and Joel A. Tropp. 2015. “Universality Laws for Randomized Dimension Reduction, with Applications.” November 30, 2015. http://arxiv.org/abs/1511.09433.
Poole, Ben, Subhaneil Lahiri, Maithreyi Raghu, Jascha Sohl-Dickstein, and Surya Ganguli. 2016. “Exponential Expressivity in Deep Neural Networks Through Transient Chaos.” In Advances in Neural Information Processing Systems 29, edited by D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, 3360–8. Curran Associates, Inc. http://papers.nips.cc/paper/6322-exponential-expressivity-in-deep-neural-networks-through-transient-chaos.pdf.
Ruthotto, Lars, and Eldad Haber. 2018. “Deep Neural Networks Motivated by Partial Differential Equations.” April 11, 2018. http://arxiv.org/abs/1804.04272.
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Shwartz-Ziv, Ravid, and Naftali Tishby. 2017. “Opening the Black Box of Deep Neural Networks via Information.” March 2, 2017. http://arxiv.org/abs/1703.00810.
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