Machine Learning Beyond Physics Jonathan Walsh from a Physicist’S Perspective How Do We Do Physics? Model Building: Parameters Model Data
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
A Relativistic Extension of Hopfield Neural Networks Via the Mechanical Analogy
A relativistic extension of Hopfield neural networks via the mechanical analogy Adriano Barra,a;b;c Matteo Beccariaa;b and Alberto Fachechia;b aDipartimento di Matematica e Fisica Ennio De Giorgi, Universit`adel Salento, Lecce, Italy bINFN, Istituto Nazionale di Fisica Nucleare, Sezione di Lecce, Italy cGNFM-INdAM, Gruppo Nazionale per la Fisica Matematica, Sezione di Lecce, Italy E-mail: [email protected], [email protected], [email protected] Abstract: We propose a modification of the cost function of the Hopfield model whose salient features shine in its Taylor expansion and result in more than pairwise interactions with alternate signs, suggesting a unified framework for handling both with deep learning and network pruning. In our analysis, we heavily rely on the Hamilton-Jacobi correspon- dence relating the statistical model with a mechanical system. In this picture, our model is nothing but the relativistic extension of the original Hopfield model (whose cost function is a quadratic form in the Mattis magnetization which mimics the non-relativistic Hamiltonian for a free particle). We focus on the low-storage regime and solve the model analytically by taking advantage of the mechanical analogy, thus obtaining a complete characterization of the free energy and the associated self-consistency equations in the thermodynamic limit. On the numerical side, we test the performances of our proposal with MC simulations, showing that the stability of spurious states (limiting the capabilities of the standard Heb- bian -
Neural Networks for Machine Learning Lecture 12A The
Neural Networks for Machine Learning Lecture 12a The Boltzmann Machine learning algorithm Geoffrey Hinton Nitish Srivastava, Kevin Swersky Tijmen Tieleman Abdel-rahman Mohamed The goal of learning • We want to maximize the • It is also equivalent to product of the probabilities that maximizing the probability that the Boltzmann machine we would obtain exactly the N assigns to the binary vectors in training cases if we did the the training set. following – This is equivalent to – Let the network settle to its maximizing the sum of the stationary distribution N log probabilities that the different times with no Boltzmann machine external input. assigns to the training – Sample the visible vector vectors. once each time. Why the learning could be difficult Consider a chain of units with visible units at the ends w2 w3 w4 hidden w1 w5 visible If the training set consists of (1,0) and (0,1) we want the product of all the weights to be negative. So to know how to change w1 or w5 we must know w3. A very surprising fact • Everything that one weight needs to know about the other weights and the data is contained in the difference of two correlations. ∂log p(v) = s s − s s i j v i j model ∂wij Derivative of log Expected value of Expected value of probability of one product of states at product of states at training vector, v thermal equilibrium thermal equilibrium under the model. when v is clamped with no clamping on the visible units Δw ∝ s s − s s ij i j data i j model Why is the derivative so simple? • The energy is a linear function • The probability of a global of the weights and states, so: configuration at thermal equilibrium is an exponential ∂E function of its energy. -
A Boltzmann Machine Implementation for the D-Wave
A Boltzmann Machine Implementation for the D-Wave John E. Dorband Ph.D. Dept. of Computer Science and Electrical Engineering University of Maryland, Baltimore County Baltimore, Maryland 21250, USA [email protected] Abstract—The D-Wave is an adiabatic quantum computer. It is an understatement to say that it is not a traditional computer. It can be viewed as a computational accelerator or more precisely a computational oracle, where one asks it a relevant question and it returns a useful answer. The question is how do you ask a relevant question and how do you use the answer it returns. This paper addresses these issues in a way that is pertinent to machine learning. A Boltzmann machine is implemented with the D-Wave since the D-Wave is merely a hardware instantiation of a partially connected Boltzmann machine. This paper presents a prototype implementation of a 3-layered neural network where the D-Wave is used as the middle (hidden) layer of the neural network. This paper also explains how the D-Wave can be utilized in a multi-layer neural network (more than 3 layers) and one in which each layer may be multiple times the size of the D- Wave being used. Keywords-component; quantum computing; D-Wave; neural network; Boltzmann machine; chimera graph; MNIST; I. INTRODUCTION The D-Wave is an adiabatic quantum computer [1][2]. Its Figure 1. Nine cells of a chimera graph. Each cell contains 8 qubits. function is to determine a set of ones and zeros (qubits) that minimize the objective function, equation (1). -
Boltzmann Machine Learning with the Latent Maximum Entropy Principle
UAI2003 WANG ET AL. 567 Boltzmann Machine Learning with the Latent Maximum Entropy Principle Shaojun Wang Dale Schuurmans Fuchun Peng Yunxin Zhao University of Toronto University of Waterloo University of Waterloo University of Missouri Toronto, Canada Waterloo, Canada Waterloo, Canada Columbia, USA Abstract configurations of the hidden variables (Ackley et al. 1985, Welling and Teh 2003). We present a new statistical learning In this paper we will focus on the key problem of esti paradigm for Boltzmann machines based mating the parameters of a Boltzmann machine from on a new inference principle we have pro data. There is a surprisingly simple algorithm (Ack posed: the latent maximum entropy principle ley et al. 1985) for performing maximum likelihood (LME). LME is different both from Jaynes' estimation of the weights of a Boltzmann machine-or maximum entropy principle and from stan equivalently, to minimize the KL divergence between dard maximum likelihood estimation. We the empirical data distribution and the implied Boltz demonstrate the LME principle by deriving mann distribution. This classical algorithm is based on new algorithms for Boltzmann machine pa a direct gradient ascent approach where one calculates rameter estimation, and show how a robust (or estimates) the derivative of the likelihood function and rapidly convergent new variant of the with respect to the model parameters. The simplic EM algorithm can be developed. Our exper ity and locality of this gradient ascent algorithm has iments show that estimation based on LME attracted a great deal of interest. generally yields better results than maximum However, there are two significant problems with the likelihood estimation when inferring models standard Boltzmann machine training algorithm that from small amounts of data. -
Stochastic Classical Molecular Dynamics Coupled To
Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 199 Sto chastic Classical Molecular Dynamics Coupled to Functional Density Theory: Applications to Large Molecular Systems K. C. Mundim Institute of Physics, Federal University of Bahia, Salvador, Ba, Brazil and D. E. Ellis Dept. of Chemistry and Materials Research Center Northwestern University, Evanston IL 60208, USA Received 07 Decemb er, 1998 Ahybrid approach is describ ed, which combines sto chastic classical molecular dynamics and rst principles DensityFunctional theory to mo del the atomic and electronic structure of large molecular and solid-state systems. The sto chastic molecular dynamics using Gener- alized Simulated Annealing GSA is based on the nonextensive statistical mechanics and thermo dynamics. Examples are given of applications in linear-chain p olymers, structural ceramics, impurities in metals, and pharmacological molecule-protein interactions. scrib e each of the comp onent pro cedures. I Intro duction In complex materials and biophysics problems the num- b er of degrees of freedom in nuclear and electronic co or- dinates is currently to o large for e ective treatmentby purely rst principles computation. Alternative tech- niques whichinvolvea hybrid mix of classical and quan- tum metho dologies can provide a p owerful to ol for anal- ysis of structure, b onding, mechanical, electrical, and sp ectroscopic prop erties. In this rep ort we describ e an implementation which has b een evolved to deal sp ecif- ically with protein folding, pharmacological molecule do cking, impurities, defects, interfaces, grain b ound- aries in crystals and related problems of 'real' solids. As in anyevolving scheme, there is still muchroom for improvement; however the guiding principles are simple: to obtain a lo cal, chemically intuitive descrip- tion of complex systems, which can b e extended in a systematic way to the nanometer size scale. -
Learning and Evaluating Boltzmann Machines
DepartmentofComputerScience 6King’sCollegeRd,Toronto University of Toronto M5S 3G4, Canada http://learning.cs.toronto.edu fax: +1 416 978 1455 Copyright c Ruslan Salakhutdinov 2008. June 26, 2008 UTML TR 2008–002 Learning and Evaluating Boltzmann Machines Ruslan Salakhutdinov Department of Computer Science, University of Toronto Abstract We provide a brief overview of the variational framework for obtaining determinis- tic approximations or upper bounds for the log-partition function. We also review some of the Monte Carlo based methods for estimating partition functions of arbi- trary Markov Random Fields. We then develop an annealed importance sampling (AIS) procedure for estimating partition functions of restricted Boltzmann machines (RBM’s), semi-restricted Boltzmann machines (SRBM’s), and Boltzmann machines (BM’s). Our empirical results indicate that the AIS procedure provides much better estimates of the partition function than some of the popular variational-based meth- ods. Finally, we develop a new learning algorithm for training general Boltzmann machines and show that it can be successfully applied to learning good generative models. Learning and Evaluating Boltzmann Machines Ruslan Salakhutdinov Department of Computer Science, University of Toronto 1 Introduction Undirected graphical models, also known as Markov random fields (MRF’s), or general Boltzmann ma- chines, provide a powerful tool for representing dependency structure between random variables. They have successfully been used in various application domains, including machine learning, computer vi- sion, and statistical physics. The major limitation of undirected graphical models is the need to compute the partition function, whose role is to normalize the joint probability distribution over the set of random variables. -
Optimization by Mean Field Annealing
91 OPTIMIZATION BY MEAN FIELD ANNEALING Griff Bilbro Reinhold Mann Thomas K. Miller ECE Dept. Eng. Physics and Math. Div. ECE Dept. NCSU Oak Ridge N atl. Lab. NCSU Raleigh, NC 27695 Oak Ridge, TN 37831 Raleigh, N C 27695 Wesley. E. Snyder David E. Van den Bout Mark White ECE Dept. ECE Dept. ECE Dept. NCSU NCSU NCSU Raleigh, NC 27695 Raleigh, NC 27695 Raleigh, NC 27695 ABSTRACT Nearly optimal solutions to many combinatorial problems can be found using stochastic simulated annealing. This paper extends the concept of simulated annealing from its original formulation as a Markov process to a new formulation based on mean field theory. Mean field annealing essentially replaces the discrete de grees of freedom in simulated annealing with their average values as computed by the mean field approximation. The net result is that equilibrium at a given temperature is achieved 1-2 orders of magnitude faster than with simulated annealing. A general frame work for the mean field annealing algorithm is derived, and its re lationship to Hopfield networks is shown. The behavior of MFA is examined both analytically and experimentally for a generic combi natorial optimization problem: graph bipartitioning. This analysis indicates the presence of critical temperatures which could be im portant in improving the performance of neural networks. STOCHASTIC VERSUS MEAN FIELD In combinatorial optimization problems, an objective function or Hamiltonian, H(s), is presented which depends on a vector of interacting 3pim, S = {81," .,8N}, in some complex nonlinear way. Stochastic simulated annealing (SSA) (S. Kirk patrick, C. Gelatt, and M. Vecchi (1983)) finds a global minimum of H by com bining gradient descent with a random process. -
7Network Models
7 Network Models 7.1 Introduction Extensive synaptic connectivity is a hallmark of neural circuitry. For ex- ample, a typical neuron in the mammalian neocortex receives thousands of synaptic inputs. Network models allow us to explore the computational potential of such connectivity, using both analysis and simulations. As illustrations, we study in this chapter how networks can perform the fol- lowing tasks: coordinate transformations needed in visually guided reach- ing, selective amplification leading to models of simple and complex cells in primary visual cortex, integration as a model of short-term memory, noise reduction, input selection, gain modulation, and associative mem- ory. Networks that undergo oscillations are also analyzed, with applica- tion to the olfactory bulb. Finally, we discuss network models based on stochastic rather than deterministic dynamics, using the Boltzmann ma- chine as an example. Neocortical circuits are a major focus of our discussion. In the neocor- tex, which forms the convoluted outer surface of the (for example) human brain, neurons lie in six vertical layers highly coupled within cylindrical columns. Such columns have been suggested as basic functional units, cortical columns and stereotypical patterns of connections both within a column and be- tween columns are repeated across cortex. There are three main classes of interconnections within cortex, and in other areas of the brain as well. Feedforward connections bring input to a given region from another re- feedforward, gion located at an earlier stage along a particular processing pathway. Re- recurrent, current synapses interconnect neurons within a particular region that are and top-down considered to be at the same stage along the processing pathway. -
Chapter 14 Stochastic Networks
14 Stochastic Networks 14.1 Variations of the Hopfield model In the previous chapter we showed that Hopfield networks can be used to provide solutions to combinatorial problems that can be expressed as the minimization of an energy function, although without guaranteeing global optimality. Once the weights of the edges have been defined, the network shows spontaneous computational properties. Harnessing this spontaneous dynamics for useful computations requires some way of avoiding falling into local minima of the energy function in such a way that the global minimum is reached. In the case of the eight queens problem, the number of local minima is much higher than the number of global minima and very often the Hopfield network does not stabilize at a correct solution. The issue to be investigated is therefore whether a certain variation of the Hopfield model could achieve better results in combinatorial optimization problems. One possible strategy to improve the global convergence properties of a network consists in increasing the number of paths in search space, in such a way that not only binary but also real-valued states become possible. Contin- uous updates of the network state become possible and the search for minima of the energy function is no longer limited to the corners of a hypercube in state space. All interior points are now legal network states. A continuous acti- vation function also makes it possible to describe more precisely the electrical circuits used to implement Hopfield networks. A second strategy to avoid local minima of the energy function consists in introducing noise into the network dynamics. -
Learning Algorithms for the Classification Restricted Boltzmann
JournalofMachineLearningResearch13(2012)643-669 Submitted 6/11; Revised 11/11; Published 3/12 Learning Algorithms for the Classification Restricted Boltzmann Machine Hugo Larochelle [email protected] Universite´ de Sherbrooke 2500, boul. de l’Universite´ Sherbrooke, Quebec,´ Canada, J1K 2R1 Michael Mandel [email protected] Razvan Pascanu [email protected] Yoshua Bengio [email protected] Departement´ d’informatique et de recherche operationnelle´ Universite´ de Montreal´ 2920, chemin de la Tour Montreal,´ Quebec,´ Canada, H3T 1J8 Editor: Daniel Lee Abstract Recent developments have demonstrated the capacity of restricted Boltzmann machines (RBM) to be powerful generative models, able to extract useful features from input data or construct deep arti- ficial neural networks. In such settings, the RBM only yields a preprocessing or an initialization for some other model, instead of acting as a complete supervised model in its own right. In this paper, we argue that RBMs can provide a self-contained framework for developing competitive classifiers. We study the Classification RBM (ClassRBM), a variant on the RBM adapted to the classification setting. We study different strategies for training the ClassRBM and show that competitive classi- fication performances can be reached when appropriately combining discriminative and generative training objectives. Since training according to the generative objective requires the computation of a generally intractable gradient, we also compare different approaches to estimating this gradient and address the issue of obtaining such a gradient for problems with very high dimensional inputs. Finally, we describe how to adapt the ClassRBM to two special cases of classification problems, namely semi-supervised and multitask learning. -
An Efficient Learning Procedure for Deep Boltzmann Machines
ARTICLE Communicated by Yoshua Bengio An Efficient Learning Procedure for Deep Boltzmann Machines Ruslan Salakhutdinov [email protected] Department of Statistics, University of Toronto, Toronto, Ontario M5S 3G3, Canada Geoffrey Hinton [email protected] Department of Computer Science, University of Toronto, Toronto, Ontario M5S 3G3, Canada We present a new learning algorithm for Boltzmann machines that contain many layers of hidden variables. Data-dependent statistics are estimated using a variational approximation that tends to focus on a sin- gle mode, and data-independent statistics are estimated using persistent Markov chains. The use of two quite different techniques for estimating the two types of statistic that enter into the gradient of the log likelihood makes it practical to learn Boltzmann machines with multiple hidden lay- ers and millions of parameters. The learning can be made more efficient by using a layer-by-layer pretraining phase that initializes the weights sensibly. The pretraining also allows the variational inference to be ini- tialized sensibly with a single bottom-up pass. We present results on the MNIST and NORB data sets showing that deep Boltzmann machines learn very good generative models of handwritten digits and 3D objects. We also show that the features discovered by deep Boltzmann machines are a very effective way to initialize the hidden layers of feedforward neural nets, which are then discriminatively fine-tuned. 1 A Brief History of Boltzmann Machine Learning The original learning procedure for Boltzmann machines (see section 2) makes use of the fact that the gradient of the log likelihood with respect to a connection weight has a very simple form: it is the difference of two pair-wise statistics (Hinton & Sejnowski, 1983). -
Machine Learning Algorithms Based on Generalized Gibbs Ensembles
Machine learning algorithms based on generalized Gibbs ensembles Tatjana Puˇskarov∗and Axel Cort´esCuberoy Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht, the Netherlands Abstract Machine learning algorithms often take inspiration from the established results and knowledge from statistical physics. A prototypical example is the Boltzmann machine algorithm for supervised learn- ing, which utilizes knowledge of classical thermal partition functions and the Boltzmann distribution. Recently, a quantum version of the Boltzmann machine was introduced by Amin, et. al., however, non- commutativity of quantum operators renders the training process by minimizing a cost function inefficient. Recent advances in the study of non-equilibrium quantum integrable systems, which never thermalize, have lead to the exploration of a wider class of statistical ensembles. These systems may be described by the so-called generalized Gibbs ensemble (GGE), which incorporates a number of “effective temperatures". We propose that these GGEs can be successfully applied as the basis of a Boltzmann-machine{like learn- ing algorithm, which operates by learning the optimal values of effective temperatures. We show that the GGE algorithm is an optimal quantum Boltzmann machine: it is the only quantum machine that circumvents the quantum training-process problem. We apply a simplified version of the GGE algorithm, where quantum effects are suppressed, to the classification of handwritten digits in the MNIST database. While lower error rates can be found with other state-of-the-art algorithms, we find that our algorithm reaches relatively low error rates while learning a much smaller number of parameters than would be needed in a traditional Boltzmann machine, thereby reducing computational cost.