Adaptive Hopfield Network

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Adaptive Hopfield Network

• Dr. Gürsel Serpen
• Associate Professor
• Electrical Engineering and Computer Science
  Department
• University of Toledo
• Toledo, 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

FOR MORE INFO...

• Serpen et al., Upcoming Journal Article
  (Insallah!)
• http//www.eecs.utoledo.edu/serpen

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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 pseudo
  values for desired neuron outputs (much like the
  way desired 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
Number of Neurons
Neuron Dynamics
Sigmoid function
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Weights (interconnection) - Redefined
Liapunov Function
Generic
Decomposed
Weights Defined
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Adaptive Hopfield NetBlock Diagram
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Adaptive Hopfield NetPseudoCode

• Initialization
• Initialize network constraint weighting
  coefficients.
• Initialize weights.
• Initialize Hopfield net neuron outputs
  (randomly).
• Adaptive Search
• Relaxation
• 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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Adaptation of WeightsAdjoint Hopfield Network
Adjoint Network
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Adaptation of WeightsRecurrent BackProp
Weight Update Recurrent BackProp
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AdaptationConstraint Weighting Coefficients
Gradient Descent Adaptation Rule
Error Function Problem Specific and Redefined
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AdaptationConstraint Weighting Coefficients
Partial Derivative Readily Computable
Final Form of Coefficient Update Rule
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Mapping A Static Optimization Problem
Generic Partial
Problem-Specific Partial
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Simulation Study

• Traveling Salesman Problem
• A preliminary work at this time
• Up to 100 cities performed
• Computing Resources Ohio Supercomputing Center
• Preliminary findings suggest that the theoretical
  framework is sound and projections are valid
• Computational cost (weight matrix size) poses
  significant challenge for simulation purposes
  on going research effort
• Currently 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.