We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger class includes functions whose second derivatives may be singular or unbounded at their minima. Our methods are discretizations of conformal Hamiltonian dynamics, which generalize the classical momentum method to model the motion of a particle with non-standard kinetic energy exposed to a dissipative force and the gradient field of the function of interest. They are first-order in the sense that they require only gradient computation. Yet, crucially the kinetic gradient map can be designed to incorporate information about the convex conjugate in a fashion that allows for linear convergence on convex functions that may be non-smooth or non-strongly convex. We study in detail one implicit and two explicit methods. For one explicit method, we provide conditions under which it converges to stationary points of non-convex functions. For all, we provide conditions on the convex function and kinetic energy pair that guarantee linear convergence, and show that these conditions can be satisfied by functions with power growth. In sum, these methods expand the class of convex functions on which linear convergence is possible with first-order computation.

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Machine Learning in High Energy Physics Community White Paper
Jul 08, 2018
Kim Albertsson, Piero Altoe, Dustin Anderson, Michael Andrews, Juan Pedro Araque Espinosa, Adam Aurisano, Laurent Basara, Adrian Bevan, Wahid Bhimji, Daniele Bonacorsi, Paolo Calafiura, Mario Campanelli, Louis Capps, Federico Carminati, Stefano Carrazza, Taylor Childers, Elias Coniavitis, Kyle Cranmer, Claire David, Douglas Davis, Javier Duarte, Martin Erdmann, Jonas Eschle, Amir Farbin, Matthew Feickert, Nuno Filipe Castro, Conor Fitzpatrick, Michele Floris, Alessandra Forti, Jordi Garra-Tico, Jochen Gemmler, Maria Girone, Paul Glaysher, Sergei Gleyzer, Vladimir Gligorov, Tobias Golling, Jonas Graw, Lindsey Gray, Dick Greenwood, Thomas Hacker, John Harvey, Benedikt Hegner, Lukas Heinrich, Ben Hooberman, Johannes Junggeburth, Michael Kagan, Meghan Kane, Konstantin Kanishchev, Przemysław Karpiński, Zahari Kassabov, Gaurav Kaul, Dorian Kcira, Thomas Keck, Alexei Klimentov, Jim Kowalkowski, Luke Kreczko, Alexander Kurepin, Rob Kutschke, Valentin Kuznetsov, Nicolas Köhler, Igor Lakomov, Kevin Lannon, Mario Lassnig, Antonio Limosani, Gilles Louppe, Aashrita Mangu, Pere Mato, Narain Meenakshi, Helge Meinhard, Dario Menasce, Lorenzo Moneta, Seth Moortgat, Mark Neubauer, Harvey Newman, Hans Pabst, Michela Paganini, Manfred Paulini, Gabriel Perdue, Uzziel Perez, Attilio Picazio, Jim Pivarski, Harrison Prosper, Fernanda Psihas, Alexander Radovic, Ryan Reece, Aurelius Rinkevicius, Eduardo Rodrigues, Jamal Rorie, David Rousseau, Aaron Sauers, Steven Schramm, Ariel Schwartzman, Horst Severini, Paul Seyfert, Filip Siroky, Konstantin Skazytkin, Mike Sokoloff, Graeme Stewart, Bob Stienen, Ian Stockdale, Giles Strong, Savannah Thais, Karen Tomko, Eli Upfal, Emanuele Usai, Andrey Ustyuzhanin, Martin Vala, Sofia Vallecorsa, Mauro Verzetti, Xavier Vilasís-Cardona, Jean-Roch Vlimant, Ilija Vukotic, Sean-Jiun Wang, Gordon Watts, Michael Williams, Wenjing Wu, Stefan Wunsch, Omar Zapata

Machine learning is an important research area in particle physics, beginning with applications to high-level physics analysis in the 1990s and 2000s, followed by an explosion of applications in particle and event identification and reconstruction in the 2010s. In this document we discuss promising future research and development areas in machine learning in particle physics with a roadmap for their implementation, software and hardware resource requirements, collaborative initiatives with the data science community, academia and industry, and training the particle physics community in data science. The main objective of the document is to connect and motivate these areas of research and development with the physics drivers of the High-Luminosity Large Hadron Collider and future neutrino experiments and identify the resource needs for their implementation. Additionally we identify areas where collaboration with external communities will be of great benefit.

* Editors: Sergei Gleyzer, Paul Seyfert and Steven Schramm
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