Optimization with the Hopfield network based on correlated noises: Experimental approach☆
Introduction
In this paper two simple optimization techniques based on combining the Langevin equation-based optimization with the Hopfield model are introduced. Proposed models – referred to as stochastic model (SM) and pulsed noise model (PNM) – can, in short, be regarded as straightforward stochastic extensions of the Hopfield optimization circuit. Instead of using ordinary differential equations like in the classical Hopfield's approach, which for a given starting point deterministically describe a trajectory in the search space, new models are defined by stochastic differential equations, obtained by adding a noise term to the Hopfield model. Similarly to the simulated annealing method [13], a noise term is multiplied by the coefficient (temperature), which decreases in time.
Both models follow the idea of stochastic neural network [14] and diffusion machine [24].
Optimization with stochastic model, unlike in referred works, in which δ-correlated Gaussian noises were considered, is based on Gaussian noises with positive autocorrelation times. This is a reasonable assumption from the hardware implementation point of view. Unfortunately, theoretical investigations of SM performance are limited by the lack of appropriate mathematical methodology, since in this case transitions of the stochastic process describing the model are not Markovian.
Hence, the paper is focused on comparison between experimental results obtained for three types of Gaussian noises that were tested in computer simulations. Distinctions between noises are based on the relation between the length of the noise autocorrelation time τ, and the RC time constant, which governs the relaxation time of the Hopfield electrical circuit.
In the other model – pulsed noise model, Gaussian noises are injected to the system only at certain time instances, as opposite to continuously maintained δ-correlated noises used in the previous related approaches.
In both models (SM and PNM), intensities of noises added to the system are independent of neurons’ potentials. Finally, instead of impractically long inverse logarithmic cooling schedules, the linear cooling is tested.
With the above strong simplifications neither SM nor PNM is expected to rigorously maintain thermal equilibrium (TE). However, numerical tests based on the canonical Gibbs–Boltzmann distribution show that differences between the rigorous and estimated values of TE parameters are relatively low (within a few percent). In this sense both models are said to perform quasithermal equilibrium.
Efficacy of SM and PNM is presented for two instances of the 10-city travelling salesman problem (TSP). Numerical results show that both models solve the problem efficiently. Moreover, it should be noted that no effort has been devoted to selection of suitable energy function or finding the optimal or sub-optimal set of energy coefficients. Finally, in both models improvement is expected with longer cooling schedules.
Results were presented in part at ICNN [16] – PNM and at ICCIN [17] – SM. Preliminary version of this paper appeared as ICSI Technical Report [18].
The paper is organized as follows: the next two sections briefly introduce the background of this work: the Hopfield model and the Langevin (diffusion) equation – both with respect to solving NP-Hard optimization problems. In Section 4 previous related papers are presented and their main conclusions discussed. The next Section describes stochastic model, and presents numerical results of solving TSP and of TE tests. Section 6 covers description and simulation results of pulsed noise model. Final remarks and conclusions are placed in the last section.
Notation remark: usually, the term stochastic model (SM in short) will address the idea of the optimization method proposed, whereas the plural term stochastic models (or SMs) will refer to various realizations based on white, moderate or quasistatic noise. The distinction will also be clearly indicated by the context.
Section snippets
The Hopfield model
In 1982 Hopfield [9] introduced a neural network model of content addressable memory (CAM) composed of many, highly interconnected two-state, McCulloch–Pitts neurons [20]. Subsequent papers [10], [11] described continuous version of the model, which was composed of a collection of continuous (graded) response neurons. The application domain was either construction of CAM [10] or solving combinatorial optimization problems [11].
After the seminal paper of Hopfield and Tank [11] lots of
The Langevin equation
The main drawback of HM, as stated in the previous section, is the fact that HM essentially performs gradient-descent local optimization. In order to force the model to escape from local minima and perform global optimization in the entire search space several modifications based on stochastic behaviour have been proposed. For example, stochastic optimization over binary variables can be implemented by Boltzmann machine [1] in combination with the simulated annealing technique [13]. The
Previous related work on stochastic optimization with the Hopfield model
Mathematical proofs of the ability of minimization algorithms [7], [8] to eventually converge to the global minimum provide a strong basis for research on implementation of the Langevin equation-based minimization methods. One of the enticing possibilities is a combination of hardware implementation simplicity of HM with minimization power of the Langevin-equation-based method. However, the obstacle in simple combination of the two methods is the fact that dynamics of the ith neuron in HM is
Stochastic model
Trying to avoid some of implementation limitations of SNN mentioned in the previous section and to simplify computational model, we propose the stochastic model (SM), which is defined based on four postulates:
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the noise injected to each stochastic neuron has positive autocorrelation time τ,
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amplifier's gain α is kept constant (in a high limit),
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the annealing schedule is reasonably fast,
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the noise intensity is independent of a potential of a neuron, hence the noise is injected in a very simple way.
Pulsed noise model
The other stochastic extension of the Hopfield model considered in this paper is pulsed noise model (PNM). PNM is defined by the same set of differential equations as SM, namely (17), where as previously T(t) is decreasing linearly according to (18). The difference between PNM and SM is in the characteristics of noises added to the system. In PNM, at each time for each neuron i, the noise value γi(t) is generated independently from other neurons, according to Gaussian
Final remarks
Stochastic model and pulsed noise model presented in this paper are simple stochastic modifications of the deterministic Hopfield model. The main advantages of these approaches are simplicity and implementation feasibility. Unlike in the previous related works regarding stochastic neural network [14] and diffusion machine [24], in SM and PNM, intensities of Gaussian noises injected to the system are independent of neurons’ potentials.
Moreover, instead of impractically long inverse logarithmic
Acknowledgements
Special thanks are due to Prof. Eugene Wong from UC Berkeley for invaluable discussions and comments on this work and on the nature of stochastic optimization processes, and to Dr. Philippe Lalanne from IOTA for inspiration for this research. The author would also like to thank anonymous reviewers for their constructive comments, and Prof. Bernard C. Levy from UC Davis, and James Beck and Brian Kingsbury from ICSI for discussion on this subject.
Jacek Mańdziuk received his M.Sc. (Honors) and Ph.D. in Applied Mathematics from Department of Applied Physics and Mathematics, Warsaw University of Technology, Poland in 1989 and 1993, respectively. He is currently Assistant Professor at the Warsaw University of Technology and computer consultant at Polish Oil and Gas Company. His research interest include optimization problems, time-series analysis, pattern recognition, game playing, constructive learning/training methods for neural networks.
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Jacek Mańdziuk received his M.Sc. (Honors) and Ph.D. in Applied Mathematics from Department of Applied Physics and Mathematics, Warsaw University of Technology, Poland in 1989 and 1993, respectively. He is currently Assistant Professor at the Warsaw University of Technology and computer consultant at Polish Oil and Gas Company. His research interest include optimization problems, time-series analysis, pattern recognition, game playing, constructive learning/training methods for neural networks. He serves as the referee for several international journals. Dr Mańdziuk is a member of INNS and NYAS. In 1996/97 he has been awarded the Senior Fulbright Research Grant and visited the International Computer Science Institute in Berkeley and EECS Department Univerity of California at Berkeley.
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This work was supported by the Tempus Phare Mobility Grant no. IMG-95-PL-1014 and the Senior Fulbright Research Grant no. 20895. It was performed while the author was on leave at the International Computer Science Institute, Berkeley, USA and EECS Department University of California at Berkeley.