adaptive hopfield network gürsel serpen dr. gürsel serpen associate professor electrical...
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Adaptive Hopfield Network
Dr. Gürsel SerpenGürsel SerpenAssociate Professor
Electrical Engineering and Computer Science DepartmentUniversity of ToledoToledo, Ohio, USA
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Presentation Topics
Motivation for research Classical Hopfield network (HN) Adaptation – Gradient Descent Adaptive Hopfield Network (AHN) Static Optimization with AHN Results and Conclusions
Serpen et al., Upcoming Journal Article (Insallah!)
http://www.eecs.utoledo.edu/~serpen
FOR MORE INFO...
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Motivation Classical Hopfield neural network (HN) has been shown to
have the potential to address a very large spectrum of static optimization problems.
Classical HN is NOT trainable: implies that it can NOT learn from prior search attempts.
A hardware realization of the Hopfield network is very attractive for real-time, embedded computing environments.
Is there a way (e.g., training or adaptation) to incorporate
the experience (gained as a result of prior search attempts) into the network dynamics (weights) to help the network focus on promising regions of the overall search space?
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Research Goals Propose gradient-descent based procedures to “adapt”
weights and constraint weighting coefficients of HN. Develop an indirect procedure to define Develop an indirect procedure to define “pseudo” values “pseudo” values
for desired neuron outputsfor desired neuron outputs (much like the way desired (much like the way desired output values for hidden layer neurons in an MLP).output values for hidden layer neurons in an MLP).
Develop space-efficient schemes to store the symmetric weight matrix (upper/lower triangular) for large-scale problem instances.
Apply (through simulation) the adaptive HN algorithm to (large-scale) static optimization problems.
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Classical Hopfield Net Dynamics
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Neuron Dynamics Sigmoid functionNumber of Neurons
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Weights (interconnection) - Redefined
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Weights Defined
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Adaptive Hopfield NetBlock Diagram
(Classical)
Hopfield Network
Adapt Constraint Weighting Coefficients
Adapt Weights
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Initial weight values
Initial weight coefficient values
Initial neuron outputs
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Adjoint Hopfield Network
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Adaptive Hopfield NetPseudoCode
Initialization• Initialize network constraint weighting coefficients.• Initialize weights.• Initialize Hopfield net neuron outputs (randomly).
Adaptive SearchRelaxation• Relax Hopfield dynamics until convergence to a fixed
point.Adaptation• Relax Adjoint network until convergence to a fixed point.• Update weights.• Update constraint weighting coefficients.
Termination Criteria • if not satisfied, continue with Adaptive Search.
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Hopfield Network Relaxation
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Hopfield Network
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Adaptation of WeightsAdjoint Hopfield Network
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Adjoint Network
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Adaptation of WeightsRecurrent BackProp
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Weight Update – Recurrent BackProp
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AdaptationConstraint Weighting Coefficients
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1zGradient Descent
Adaptation Rule
Error Function – Problem Specific and Redefined
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AdaptationConstraint Weighting Coefficients
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, ' zz
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Partial Derivative – Readily Computable
Final Form of Coefficient Update Rule
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Mapping A Static Optimization Problem
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Generic Partial Problem-Specific Partial
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Simulation Study
Traveling Salesman ProblemA preliminary work at this timeUp to 100 cities performedComputing Resources – Ohio Supercomputing CenterPreliminary findings suggest that the theoretical framework is sound and projections are validComputational cost (weight matrix size) poses significant challenge for simulation purposes – on going research effortCurrently in progress
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Conclusions
An adaptation mechanism, which modifies constraint weighting coefficient parameter values and weights of the classical Hopfield network, was proposed.
A mathematical characterization of the adaptive Hopfield network was presented.
Preliminary simulation results suggest the proposed adaptation mechanism to be effective in guiding the Hopfield network towards high-quality feasible solutions of large-scale static optimization problems.
We are also exploring incorporating a computationally viable stochastic search mechanism to further improve quality of solutions computed by the adaptive Hopfield network while preserving parallel computation capability.
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Thank You !
Questions ?We gratefully acknowledge the computing resources grant provided by the State of Ohio Supercomputing Center (in USA) in facilitating the simulation study.
We appreciate the support provided by the Kohler Internationalization Awards Program at the University of Toledo to facilitate this conference presentation.