DOCSLIB.ORG

NTT Technical Review, Vol. 17, No. 12, Dec. 2019

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
NTT Technical Review, Vol. 17, No. 12, Dec. 2019 Feature Articles: Establishment of NTT Research, Inc. toward the Strengthening and Globalization of Research and Development Mission of Physics & Informatics Laboratories Yoshihisa Yamamoto Abstract At the Physics & Informatics Laboratories (PHI Labs) of NTT Research, Inc., we explore a new prin- ciple that will bring about a revolution in information processing technology in the interdisciplinary area between quantum physics and brain science, and it is here where we have positioned our research field. We will focus on quantum-classical crossover physics and critical phenomena in neural networks. We will concentrate, in particular, on optical implementation as a means of achieving a simple, elegant, and practical implementation of a computer that is based on this new principle, specifically on the optical parametric oscillator that can achieve quantum neural networks at room temperature. This article intro- duces the concepts, technologies, and target applications making up this research field at PHI Labs. Keywords: combinatorial optimization problem, quantum neural network, optical parametric oscillator 1. Quantum neural networks using optical waves for practical use) in 1968 by Stephen Harris parametric oscillators and Robert Byer of Stanford University [2]. The foundation of optical communication technol- The development of new oscillators for generating ogy, which blossomed in the 20th century, was the coherent electromagnetic waves and the expansion of laser, but it is our future vision that the foundation of their application fields has a history marked by a optical information processing technology of the 21st rivalry between oscillators based on two different century will be the optical parametric oscillator. This principles. In electrical engineering terms, one is a type of parametric oscillator behaves like an analog negative-resistance oscillator requiring no pump- device with strong quantum properties in the pump source coherence, and the other is a nonlinear reac- region below an oscillation threshold and like a digi- tance oscillator requiring coherent pump waves. The tal device with strong classical properties in the pump development of oscillators for various frequency region above the threshold. As described below, bands began with the emergence of negative-resis- future information processing technology will require tance oscillators that are easy to achieve. This was both quantum computing resources and classical followed by the development of nonlinear reactance computing resources, and the optical parametric oscillators that generate a coherent wave with less oscillator is practically the only device that can noise. simultaneously achieve this dual quantum-classical The development of optical oscillators covering a nature at room temperature. broad wavelength band from ultraviolet to infrared is We have been researching the neural networks no exception to this historical rivalry. A laser, which using optical parametric oscillators as neurons and is an optical negative-resistance oscillator, was devel- using such a configuration in an attempt to solve oped in 1960 through the work of Theodore Maiman combinatorial optimization problems and quantum of Hughes Research Laboratories [1]. This was fol- many-body problems that have posed a challenge to lowed by the development of an optical nonlinear the modern computer [3] (Fig. 1). Regarding another reactance oscillator as an optical parametric oscillator element (synapse connections) making up a neural (in particular, an oscillator generating continuous network, there are two methods of implementing a NTT Technical Review Vol. 17 No. 12 Dec. 2019 9 Feature Articles Brain Quantum (Critical phenomena (Quantum-classical in neural networks) crossover) Optics (Optical parametric Focus on the interdisciplinary field oscillators) of quantum, brain, and optics Fig. 1. Research fields at Physics & Informatics Laboratories (PHI Labs). Pump optical Signal optical pulses Pump optical Signal optical pulses LO pulses pulses pulses SHG pulse #2 #2 #1 #1 SHG SHG Homodyne Optical parametric Optical parametric detection amplifier PSA amplifier Optic fiber Output coupler Optic fiber ring resonator ring resonator Targeted Targeted MOD1 MOD2 … MODN–1 optical pulse optical #pulsei #i ADC FPGA Feedback optical pulse DAC Feedback optical pulse IM/PM (a) Optical delay line coupling (b) Measurement feedback coupling ADC: analog-to-digital converter LO: local oscillator DAC: digital-to-analog converter MOD: optical modulator FPGA: field-programmable gate array PSA: phase sensitive amplifier IM/PM: intensity modulation/phase modulation SHG: second harmonic generation Fig. 2. Neural network configuration based on optical parametric oscillators. fully connected neural network in which synapse [Jij] through (N–1) optical delay lines and the same connections are formed between all neurons (Fig. 2). number of optical modulators [4]. With this method, These methods use N optical parametric oscillator (N–1) input pulses j simultaneously couple with a pulses circulating in an optical fiber ring cavity of 1 single target pulse i at each time point via coupling to 10 km for N neurons instead of using N optical coefficient Jij, so N (N–1) Jij coefficients are imple- parametric oscillators, a scheme that achieves N mented for each round trip to achieve the above for all defect-free uniform neurons simultaneously within a N target pulses. This method has the advantage of single cavity [4–6]. simplifying the implementation of synapse coupling The optical delay line coupling method (Fig. 2(a)) having directionality (Jij ≠ Jji). can achieve N (N–1) synapse coupling coefficients The measurement feedback coupling method NTT Technical Review Vol. 17 No. 12 Dec. 2019 10 Feature Articles (Fig. 2(b)), on the other hand, can achieve N (N–1) diagnosis (community detection) and portfolio opti- all-synapse coupling with just a single measurement mization in the Fintech field. feedback circuit [5, 6]. The former method is thought Still another type of problem representative of com- to be applicable to the implementation of large-scale, binatorial optimization problems is the satisfiability high-speed, and sparsely coupled neural networks (SAT) problem. A coherent SAT solver that can solve and to quantum many-body problems. The latter this problem with good efficiency can be achieved by method, however, is thought to be applicable to the configuring a recurrent neural network with degener- implementation of medium-scale, high-order nonlin- ate optical parametric oscillators [10]. One specific ear coupling and densely coupled neural networks type of problem that could be solved by a coherent and to combinatorial optimization problems. SAT solver is hardware/software verification. The concept of quantum-inspired optimization has 2. Application fields recently been attracting attention as a practical solver targeting combinatorial optimization problems. This section introduces the application fields envi- Instead of actually constructing an optical (or super- sioned for quantum neural networks using optical conducting) parametric oscillator network, we apply parametric oscillators. this concept, which is aimed at obtaining optimal solutions by programming the quantum mechanical 2.1 Combinatorial optimization problems equations of motion that describe the experimental The Ising model is highly representative of combi- quantum neural network as an algorithm in a standard natorial optimization problems. This model can be digital circuit such as a field-programmable gate implemented through the use of degenerate optical array and conducting a type of numerical simulation parametric oscillators in which the signal wave and [8, 11]. idler wave have identical frequencies [3]. As a result of recent improvements in hardware technology for 2.2 Quantum many-body problems achieving a degenerate optical parametric oscillator The development of new tools for achieving effi- network together with the advancement in associated cient numerical simulations with good accuracy of algorithms, the performance of the coherent Ising electron behavior within a solid, especially of corre- machine, a type of quantum neural network, is mak- lated electron systems having strong interactions ing significant gains. In fact, it is becoming superior among electrons, is essential to searching for new to the quantum annealing machine—a type of quan- materials (material informatics) and discovering new tum computer specialized for combinatorial optimi- phenomena (topological physics). In particular, two- zation problems—and von Neumann computers dimensional systems in which electrons are confined implementing advanced algorithms such as Breakout by two-dimensional potential forces exhibit novel Local Search [7, 8]. Expectations are high that the era quantum phenomena not found in ordinary three- of solving a variety of combinatorial optimization dimensional systems. However, unlike three-dimen- problems will eventually arrive by mapping them to sional systems, numerical simulations based on the Ising model to achieve a general-purpose optimi- mean-field approximation tend to break down in two- zation solver. Application algorithms are now being dimensional systems, and it is also difficult to obtain developed for specific problems such as lead optimi- exact solutions in two-dimensional systems as in one- zation in drug discovery and the development of dimensional systems. optimum biocatalysts, resource allocation in wireless This state of affairs
Load more

Recommended publications
  • Opportunities for US-Israeli Collaborations in Computational Neuroscience
    Opportunities for US-Israeli Collaborations in Computational Neuroscience Report of a Binational Workshop* Larry Abbott and Naftali Tishby Introduction Both Israel and the United States have played and continue to play leading roles in the rapidly developing field of computational neuroscience, and both countries have strong interests in fostering collaboration in emerging research areas. A workshop was convened by the US-Israel Binational Science Foundation and the US National Science Foundation to discuss opportunities to encourage and support interdisciplinary collaborations among scientists from the US and Israel, centered around computational neuroscience. Seven leading experts from Israel and six from the US (Appendix 2) met in Jerusalem on November 14, 2012, to evaluate and characterize such research opportunities, and to generate suggestions and ideas about how best to proceed. The participants were asked to characterize the potential scientific opportunities that could be expected from collaborations between the US and Israel in computational neuroscience, and to discuss associated opportunities for training, applications, and other broader impacts, as well as practical considerations for maximizing success. Computational Neuroscience in the United States and Israel The computational research communities in the United States and Israel have both contributed significantly to the foundations of and advances in applying mathematical analysis and computational approaches to the study of neural circuits and behavior. This shared intellectual commitment has led to productive collaborations between US and Israeli researchers, and strong ties between a number of institutions in Israel and the US. These scientific collaborations are built on over 30 years of visits and joint publications and the results have been extremely influential.
    [Show full text]
  • Surya Ganguli: Innovator in Artificial Intelligence
    Behind the Tech Kevin Scott Podcast EP-10: Surya Ganguli: Innovator in artificial intelligence SURYA GANGULI: (VOICEOVER) A lot of people, of course, listening to your podcast and out there in the world have become incredibly successful based on the rise of digital technology. But I think we need to think about digital technology now as a suboptimal legacy technology. To achieve energy efficient artificial intelligence, we've really got to take cues from biology. [MUSIC] KEVIN SCOTT: Hi, everyone. Welcome to Behind the Tech. I'm your host, Kevin Scott, Chief Technology Officer for Microsoft. In this podcast, we're going to get behind the tech. We'll talk with some of the people who have made our modern tech world possible and understand what motivated them to create what they did. So join me to maybe learn a little bit about the history of computing and get a few behind- the-scenes insights into what's happening today. Stick around. [MUSIC] CHRISTINA WARREN: Hello, and welcome to the show. I'm Christina Warren, Senior Cloud Advocate at Microsoft. KEVIN SCOTT: And I'm Kevin Scott. CHRISTINA WARREN: Today, we're excited to have with us Surya Ganguli. KEVIN SCOTT: Yeah, Surya is a professor at Stanford working at the intersection of a bunch of different, super interesting fields. So, he's a physicist. He's a neuroscientist. He is actually quite familiar with modern psychology, and he's a computer scientist. And so, he's applying all of this interest that he has across all of these different fields, trying to make sure that we're building better and more interesting biologically and psychologically inspired machine learning systems in AI.
    [Show full text]
  • Predictive Coding in Balanced Neural Networks with Noise, Chaos and Delays
    Predictive coding in balanced neural networks with noise, chaos and delays Jonathan Kadmon Jonathan Timcheck Surya Ganguli Department of Applied Physics Department of Physics Department of Applied Physics Stanford University,CA Stanford University,CA Stanford University, CA [email protected] Abstract Biological neural networks face a formidable task: performing reliable compu- tations in the face of intrinsic stochasticity in individual neurons, imprecisely specified synaptic connectivity, and nonnegligible delays in synaptic transmission. A common approach to combatting such biological heterogeneity involves aver- aging over large redundantp networks of N neurons resulting in coding errors that decrease classically as 1= N. Recent work demonstrated a novel mechanism whereby recurrent spiking networks could efficiently encode dynamic stimuli, achieving a superclassical scaling in which coding errors decrease as 1=N. This specific mechanism involved two key ideas: predictive coding, and a tight balance, or cancellation between strong feedforward inputs and strong recurrent feedback. However, the theoretical principles governing the efficacy of balanced predictive coding and its robustness to noise, synaptic weight heterogeneity and communica- tion delays remain poorly understood. To discover such principles, we introduce an analytically tractable model of balanced predictive coding, in which the de- gree of balance and the degree of weight disorder can be dissociated unlike in previous balanced network models, and we develop a mean field theory of coding accuracy. Overall, our work provides and solves a general theoretical framework for dissecting the differential contributions neural noise, synaptic disorder, chaos, synaptic delays, and balance to the fidelity of predictive neural codes, reveals the fundamental role that balance plays in achieving superclassical scaling, and unifies previously disparate models in theoretical neuroscience.
    [Show full text]
  • Surya Ganguli
    Surya Ganguli Address Department of Applied Physics, Stanford University 318 Campus Drive Stanford, CA 94305-5447 [email protected] http://ganguli-gang.stanford.edu/~surya.html Biographic Data Born in Kolkata, India. Currently US citizen. Education University of California Berkeley Berkeley, CA Ph.D. Theoretical Physics, October 2004. Thesis: \Geometry from Algebra: The Holographic Emergence of Spacetime in String Theory" M.A. Mathematics, June 2004. M.A. Physics, December 2000. Massachusetts Institute of Technology Cambridge, MA M.Eng. Electrical Engineering and Computer Science, May 1998. B.S. Physics, May 1998. B.S. Mathematics, May 1998. B.S. Electrical Engineering and Computer Science, May 1998. University High School Irvine, CA Graduated first in a class of 470 students at age 16, May 1993. Research Stanford University Stanford, CA Positions Department of Applied Physics, and by courtesy, Jan 12 to present Department of Neurobiology Department of Computer Science Department of Electrical Engineering Assistant Professor. Google Mountain View, CA Google Brain Deep Learning Team Feb 17 to present Visiting Research Professor, 1 day a week. University of California, San Francisco San Francisco, CA Sloan-Swartz Center for Theoretical Neurobiology Sept 04 to Jan 12 Postdoctoral Fellow. Conducted theoretical neuroscience research. Lawrence Berkeley National Lab Berkeley, CA Theory Group Sept 02 to Sept 04 Graduate student. Conducted string theory research; thesis advisor: Prof. Petr Horava. MIT Department of Physics Cambridge, MA Center for Theoretical Physics January 97 to June 98 Studied the Wigner-Weyl representation of quantum mechanics. Discovered new admissibility conditions satisfied by all Wigner distributions. Also formulated and implemented simulations of quantum dynamics directly in phase space.
    [Show full text]
  • Surya Ganguli Associate Professor of Applied Physics and , by Courtesy, of Neurobiology and of Electrical Engineering Curriculum Vitae Available Online
    Surya Ganguli Associate Professor of Applied Physics and , by courtesy, of Neurobiology and of Electrical Engineering Curriculum Vitae available Online Bio ACADEMIC APPOINTMENTS • Associate Professor, Applied Physics • Associate Professor (By courtesy), Neurobiology • Associate Professor (By courtesy), Electrical Engineering • Member, Bio-X • Associate Director, Institute for Human-Centered Artificial Intelligence (HAI) • Member, Wu Tsai Neurosciences Institute HONORS AND AWARDS • NSF Career Award, National Science Foundation (2019) • Investigator Award in Mathematical Modeling of Living Systems, Simons Foundation (2016) • McKnight Scholar Award, McKnight Endowment Fund for Neuroscience (2015) • Scholar Award in Human Cognition, James S. McDonnell Foundation (2014) 4 OF 9 PROFESSIONAL EDUCATION • Ph.D., UC Berkeley , Theoretical Physics (2004) • M.A., UC Berkeley , Mathematics (2004) • M.Eng., MIT , Electrical Engineering and Computer Science (1998) • B.S., MIT , Mathematics (1998) 4 OF 6 LINKS • Lab Website: http://ganguli-gang.stanford.edu/index.html • Personal Website: http://ganguli-gang.stanford.edu/surya.html • Applied Physics Website: https://web.stanford.edu/dept/app-physics/cgi-bin/person/surya-gangulijanuary-2012/ Research & Scholarship CURRENT RESEARCH AND SCHOLARLY INTERESTS Theoretical / computational neuroscience Page 1 of 2 Surya Ganguli http://cap.stanford.edu/profiles/Surya_Ganguli/ Teaching COURSES 2019-20 • Artificial Intelligence, Entrepreneurship and Society in the 21st Century and Beyond: CS 28 (Aut) • Introduction
    [Show full text]
  • A Mathematical Theory of Semantic Development in Deep Neural Networks
    A mathematical theory of semantic development in deep neural networks Andrew M. Saxe ∗, James L. McClelland †, and Surya Ganguli †‡ ∗University of Oxford, Oxford, UK,†Stanford University, Stanford, CA, and ‡Google Brain, Mountain View, CA An extensive body of empirical research has revealed remarkable ganization of semantic knowledge powerfully guides its deployment regularities in the acquisition, organization, deployment, and neu- in novel situations, where one must make inductive generalizations ral representation of human semantic knowledge, thereby raising about novel items and properties [2, 3]. Indeed, studies of children a fundamental conceptual question: what are the theoretical prin- reveal that their inductive generalizations systematically change over ciples governing the ability of neural networks to acquire, orga- development, often becoming more specific with age [2, 3, 17–19]. nize, and deploy abstract knowledge by integrating across many individual experiences? We address this question by mathemat- Finally, recent neuroscientific studies have begun to shed light on ically analyzing the nonlinear dynamics of learning in deep lin- the organization of semantic knowledge in the brain. The method of ear networks. We find exact solutions to this learning dynamics representational similarity analysis [20,21] has revealed that the sim- that yield a conceptual explanation for the prevalence of many ilarity structure of neural population activity patterns in high level disparate phenomena in semantic cognition, including the hier-
    [Show full text]
  • The Statistical Physics of Deep Learning on Infant Category Learning
    The statistical physics of deep learning On infant category learning, dynamic criticality, random landscapes, and the reversal of time. Surya Ganguli Dept. of Applied Physics, Neurobiology, and Electrical Engineering Stanford University Neural circuits and behavior: theory, computation and experiment with Baccus lab: inferring hidden circuits in the retina with Clandinin lab: unraveling the computations underlying fly motion vision from whole brain optical imaging with the Giocomo lab: understanding the internal representations of space in the mouse entorhinal cortex with the Shenoy lab: a theory of neural dimensionality, dynamics and measurement with the Raymond lab: theories of how enhanced plasticity can either enhance or impair learning depending on experience Statistical mechanics of high dimensional data analysis N = dimensionality of data! α = N / M! M = number of data points! Classical Statistics Modern Statistics N ~ O(1)! N -> " ! M -> "! M -> "! α -> 0 α ~ 0(1) Machine Learning and Data Analysis! Statistical Physics of Quenched ! Disorder! Learn statistical parameters by! ! maximizing log likelihood of data ! Energy = - log Prob ( data | parameters)! given parameters. ! ! Data = quenched disorder! Parameters = thermal degrees of freedom! Applications to: 1) compressed sensing! 2) optimal inference in high dimensions! 3) a theory of neural dimensionality and measurement ! The project that really keeps me up at night Motivations for an alliance between theoretical neuroscience and theoretical machine learning: opportunities for statistical physics •# What does it mean to understand the brain (or a neural circuit?) •# We understand how the connectivity and dynamics of a neural circuit gives rise to behavior. •# And also how neural activity and synaptic learning rules conspire to self- organize useful connectivity that subserves behavior.
    [Show full text]
  • Arxiv:1907.08549V2 [Q-Bio.NC] 4 Dec 2019 1 Introduction
    Universality and individuality in neural dynamics across large populations of recurrent networks Niru Maheswaranathan∗ Alex H. Williams∗ Matthew D. Golub Google Brain, Google Inc. Stanford University Stanford University Mountain View, CA Stanford, CA Stanford, CA [email protected] [email protected] [email protected] Surya Ganguli David Sussilloy Stanford University and Google Brain Google Brain, Google Inc. Stanford, CA and Mountain View, CA Mountain View, CA [email protected] [email protected] Abstract Task-based modeling with recurrent neural networks (RNNs) has emerged as a popular way to infer the computational function of different brain regions. These models are quantitatively assessed by comparing the low-dimensional neural rep- resentations of the model with the brain, for example using canonical correlation analysis (CCA). However, the nature of the detailed neurobiological inferences one can draw from such efforts remains elusive. For example, to what extent does training neural networks to solve common tasks uniquely determine the network dynamics, independent of modeling architectural choices? Or alternatively, are the learned dynamics highly sensitive to different model choices? Knowing the answer to these questions has strong implications for whether and how we should use task-based RNN modeling to understand brain dynamics. To address these foundational questions, we study populations of thousands of networks, with com- monly used RNN architectures, trained to solve neuroscientifically motivated tasks and characterize
    [Show full text]
  • Statistical Mechanics of Deep Learning
    CO11CH22_Ganguli ARjats.cls February 13, 2020 10:27 Annual Review of Condensed Matter Physics Statistical Mechanics of Deep Learning Yasaman Bahri,1 Jonathan Kadmon,2 Jeffrey Pennington,1 Sam S. Schoenholz,1 Jascha Sohl-Dickstein,1 and Surya Ganguli1,2 1Google Brain, Google Inc., Mountain View, California 94043, USA 2Department of Applied Physics, Stanford University, Stanford, California 94035, USA; email: [email protected] Annu. Rev. Condens. Matter Phys. 2020. 11:501–28 Keywords First published as a Review in Advance on neural networks, machine learning, dynamical phase transitions, chaos, spin December 9, 2019 glasses, jamming, random matrix theory, interacting particle systems, The Annual Review of Condensed Matter Physics is nonequilibrium statistical mechanics online at conmatphys.annualreviews.org https://doi.org/10.1146/annurev-conmatphys- Abstract 031119-050745 The recent striking success of deep neural networks in machine learning Copyright © 2020 by Annual Reviews. raises profound questions about the theoretical principles underlying their All rights reserved success. For example, what can such deep networks compute? How can we train them? How does information propagate through them? Why can they generalize? And how can we teach them to imagine? We review recent work Annu. Rev. Condens. Matter Phys. 2020.11:501-528. Downloaded from www.annualreviews.org in which methods of physical analysis rooted in statistical mechanics have begun to provide conceptual insights into these questions. These insights yield connections between deep learning and diverse physical and mathe- Access provided by Stanford University - Main Campus Robert Crown Law Library on 06/23/20. For personal use only. matical topics, including random landscapes, spin glasses, jamming, dynam- ical phase transitions, chaos, Riemannian geometry, random matrix theory, free probability, and nonequilibrium statistical mechanics.
    [Show full text]
  • We Live in a World
    We live in a world in which science, from mathematics through to biology and computer science, affects us all. By encouraging the full development of science across disciplines we help to shape the future. –Michael Atiyah TATA INSTITUTE OF FUNDAMENTAL RESEARCH CONTENTS FROM THE DIRECTOR’S DESK PAGE 2 THE ROAD TRAVELLED PAGE 4 ICTS AT TEN PAGE 5 RESEARCH PAGE 6 PROGRAMS PAGE 12 OUTREACH PAGE 18 ORGANISATION PAGE 22 CAMPUS PAGE 24 BrochurePAGE 21 Editorial Team–Ananya Dasgupta, Parul Sehgal, Renu Singh Design–Anisha Heble, Juny K Wilfred I would describe ICTS with three words: Bold, Deliberate, Effective. Bold in its vision for its role in the Indian and international scientific community. Deliberate in its assessments of opportunity and in charting its path. Effective in supporting its faculty to achieve the full potential. –Stan Whitcomb, Caltech FROM THE DIRECTOR'S DESK “In our acquisition of knowledge of the universe (whether As an institution in its childhood, at ICTS there is a mathematical or otherwise) that which renovates the quest notion of playfulness and, perhaps, naivety in our is nothing more or less than complete innocence ... approach to all matters. We have a brand new campus It alone can unite humility with boldness so as to allow us to like a new toy in the hands of our youthful faculty and penetrate to the heart of things ... ” students as well as enthusiastic admin staff - who are all Alexander Grothendieck (From “Reaping and Sowing”) eager to try new things, to do things differently. In short, to create a different ethos.
    [Show full text]
  • Arxiv:2012.04728V2 [Cs.LG] 29 Mar 2021
    Published as a conference paper at ICLR 2021 NEURAL MECHANICS: SYMMETRY AND BROKEN CON- SERVATION LAWS IN DEEP LEARNING DYNAMICS Daniel Kunin*, Javier Sagastuy-Brena, Surya Ganguli, Daniel L.K. Yamins, Hidenori Tanaka*y Stanford University Physics & Informatics Laboratories, NTT Research, Inc. y ABSTRACT Understanding the dynamics of neural network parameters during training is one of the key challenges in building a theoretical foundation for deep learning. A central obstacle is that the motion of a network in high-dimensional parameter space undergoes discrete finite steps along complex stochastic gradients derived from real-world datasets. We circumvent this obstacle through a unifying theoretical framework based on intrinsic symmetries embedded in a network’s architecture that are present for any dataset. We show that any such symmetry imposes strin- gent geometric constraints on gradients and Hessians, leading to an associated conservation law in the continuous-time limit of stochastic gradient descent (SGD), akin to Noether’s theorem in physics. We further show that finite learning rates used in practice can actually break these symmetry induced conservation laws. We apply tools from finite difference methods to derive modified gradient flow, a differential equation that better approximates the numerical trajectory taken by SGD at finite learning rates. We combine modified gradient flow with our frame- work of symmetries to derive exact integral expressions for the dynamics of certain parameter combinations. We empirically validate our analytic expressions for learning dynamics on VGG-16 trained on Tiny ImageNet. Overall, by exploiting symmetry, our work demonstrates that we can analytically describe the learning dynamics of various parameter combinations at finite learning rates and batch sizes for state of the art architectures trained on any dataset.
    [Show full text]
  • Arxiv:1910.05929V1 [Cs.LG] 14 Oct 2019 Multaneously Accounts for All 4 of These Sur- the Journey to These Minima
    Emergent properties of the local geometry of neural loss landscapes Stanislav Fort Surya Ganguli Stanford University Stanford University Stanford, CA, USA Stanford, CA, USA Abstract 1 Introduction The geometry of neural network loss landscapes and The local geometry of high dimensional neu- the implications of this geometry for both optimiza- ral network loss landscapes can both chal- tion and generalization have been subjects of intense lenge our cherished theoretical intuitions as interest in many works, ranging from studies on the well as dramatically impact the practical suc- lack of local minima at significantly higher loss than cess of neural network training. Indeed recent that of the global minimum [1, 2] to studies debating works have observed 4 striking local prop- relations between the curvature of local minima and erties of neural loss landscapes on classifi- their generalization properties [3, 4, 5, 6]. Fundamen- cation tasks: (1) the landscape exhibits ex- tally, the neural network loss landscape is a scalar loss actly C directions of high positive curvature, function over a very high D dimensional parameter where C is the number of classes; (2) gradi- space that could depend a priori in highly nontriv- ent directions are largely confined to this ex- ial ways on the very structure of real-world data itself tremely low dimensional subspace of positive as well as intricate properties of the neural network Hessian curvature, leaving the vast major- architecture. Moreover, the regions of this loss land- ity of directions in weight space unexplored; scape explored by gradient descent could themselves (3) gradient descent transiently explores in- have highly atypical geometric properties relative to termediate regions of higher positive curva- randomly chosen points in the landscape.
    [Show full text]