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arXiv:1410.3831

An exact mapping between the Variational Renormalization Group and Deep Learning

Pankaj Mehta, David J. Schwab

Published Oct 14, 2014• 195 citations
arXivHTMLPDF

Abstract

Deep learning is a broad set of techniques that uses multiple layers of representation to automatically learn relevant features directly from structured data. Recently, such techniques have yielded record-breaking results on a diverse set of difficult machine learning tasks in computer vision, speech recognition, and natural language processing. Despite the enormous success of deep learning, relatively little is understood theoretically about why these techniques are so successful at feature learning and compression. Here, we show that deep learning is intimately related to one of the most important and successful techniques in theoretical physics, the renormalization group (RG). RG is an iterative coarse-graining scheme that allows for the extraction of relevant features (i.e. operators) as a physical system is examined at different length scales. We construct an exact mapping from the variational renormalization group, first introduced by Kadanoff, and deep learning architectures based on Restricted Boltzmann Machines (RBMs). We illustrate these ideas using the nearest-neighbor Ising Model in one and two-dimensions. Our results suggests that deep learning algorithms may be employing a generalized RG-like scheme to learn relevant features from data.

DOI
10.48550/arxiv.1410.3831

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Deep learning is a broad set of techniques that uses multiple layers of representation to automatically learn relevant features directly from structured data. Recently, such techniques have yielded record-breaking results on a diverse set of difficult machine learning tasks in computer vision, speech recognition, and natural language processing. Despite the enormous success of deep learning, relatively little is understood theoretically about why these techniques are so successful at feature learning and compression. Here, we show that deep learning is intimately related to one of the most important and successful techniques in theoretical physics, the renormalization group (RG). RG is an iterative coarse-graining scheme that allows for the extraction of relevant features (i.e. operators) as a physical system is examined at different length scales. We construct an exact mapping from the variational renormalization group, first introduced by Kadanoff, and deep learning architectures based on Restricted Boltzmann Machines (RBMs). We illustrate these ideas using the nearest-neighbor Ising Model in one and two-dimensions. Our results suggests that deep learning algorithms may be employing a generalized RG-like scheme to learn relevant features from data.
The renormalization group (RG) is an essential technique in statistical physics and quantum field theory, which considers scale-invariant properties of physical theories and how these theories' parameters change with scaling. Deep learning is a powerful computational technique that uses multi-layered neural networks to solve a myriad of complicated problems. Previous research suggests the possibility that unsupervised deep learning may be a form of RG flow, by being a layer-by-layer coarse graining of the original data. We examined this connection on a more rigorous basis for the simple example of Kadanoff block renormalization of the 2D nearest-neighbor Ising model, with our deep learning accomplished via Restricted Boltzmann Machines (RBMs). We developed extensive renormalization techniques for the 1D and 2D Ising model to provide a baseline for comparison. For the 1D Ising model, we successfully used Adam optimization on a correlation length loss function to learn the group flow, yielding results consistent with the analytical model for infinite N. For the 2D Ising model, we successfully generated Ising model samples using the Wolff algorithm, and performed the group flow using a quasi-deterministic method, validating these results by calculating the critical exponent ν. We then examined RBM learning of the Ising model layer by layer, finding a blocking structure in the learning that is qualitatively similar to RG. Lastly, we directly compared the weights of each layer from the learning to Ising spin renormalization, but found quantitative inconsistencies for the simple case of nearest-neighbor Ising models.
Although there has been a rapid development of practical applications, theoretical explanations of deep learning are in their infancy. Deep learning performs a sophisticated coarse graining. Since coarse graining is a key ingredient of the renormalization group (RG), RG may provide a useful theoretical framework directly relevant to deep learning. In this study we pursue this possibility. A statistical mechanics model for a magnet, the Ising model, is used to train an unsupervised restricted Boltzmann machine (RBM). The patterns generated by the trained RBM are compared to the configurations generated through an RG treatment of the Ising model. Although we are motivated by the connection between deep learning and RG flow, in this study we focus mainly on comparing a single layer of a deep network to a single step in the RG flow. We argue that correlation functions between hidden and visible neurons are capable of diagnosing RG-like coarse graining. Numerical experiments show the presence of RG-like patterns in correlators computed using the trained RBMs. The observables we consider are also able to exhibit important differences between RG and deep learning.
A theory explaining how deep learning works is yet to be developed. Previous work suggests that deep learning performs a coarse graining, similar in spirit to the renormalization group (RG). This idea has been explored in the setting of a local (nearest-neighbor interactions) Ising spin lattice. We extend the discussion to the setting of a long-range spin lattice. Markov-chain Monte Carlo (MCMC) simulations determine both the critical temperature and scaling dimensions of the system. The model is used to train both a single restricted Boltzmann machine (RBM) network, as well as a stacked RBM network. Following earlier Ising model studies, the trained weights of a single-layer RBM network define a flow of lattice models. In contrast to results for nearest-neighbor Ising, the RBM flow for the long-ranged model does not converge to the correct values for the spin and energy scaling dimension. Further, correlation functions between visible and hidden nodes exhibit key differences between the stacked RBM and RG flows. The stacked RBM flow appears to move toward low temperatures, whereas the RG flow moves toward high temperature. This again differs from results obtained for nearest-neighbor Ising.
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At low energies, the microscopic characteristics and changes of physical systems as viewed at different distance scales are described by universal scale invariant properties investigated by the Renormalization Group (RG) apparatus, an efficient tool used to deal with scaling problems in effective field theories. We employ an information-theoretic approach in a deep learning setup by introducing an artificial neural network algorithm to map and identify new physical degrees of freedom. Using deep learning methods mapped to a genuine field theory, we develop a mechanism capable to identify relevant degrees of freedom and induce scale invariance without prior knowledge about a system. We show that deep learning algorithms that use an RG-like scheme to learn relevant features from data could help to understand the nature of the holographic entanglement entropy and the holographic principle in context of the AdS/CFT correspondence.
The key idea behind the renormalization group (RG) transformation is that properties of physical systems with very different microscopic makeups can be characterized by a few universal parameters.However, finding a systematic way to construct RG transformation for particular systems remains difficult due to the many possible choices of the weight factors in the RG procedure.Here we show, by identifying the conditional distribution in the restricted Boltzmann machine and the weight factor distribution in the RG procedure, that a valid real-space RG transformation can be learned without prior knowledge of the physical system.This neural Monte Carlo RG algorithm allows for direct computation of the RG flow and critical exponents.Our results establish a solid connection between the RG transformation in physics and the deep architecture in machine learning, paving the way for further interdisciplinary research.