A Modified Meta-controlled Boltzmann Machine

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A Modified Meta-controlled Boltzmann Machine

• Tran Duc Minh, Le Hai Khoi (), Junzo Watada
  (), Teruyuki Watanabe ()
• () Institute Of Information Technology-Viet Nam
  Academy of Science Technology
• () Graduate School of Information, Production
  and System, Waseda University, Japan
• () Osaka Institute of Technology

03/2004
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CONTENT

• Introduction
• The portfolio selection problem
• Inner behaviors of the Meta-controlled Boltzmann
  machine
• A Modified Meta-controlled Boltzmann machine
• Conclusion

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Introduction

• H. Markowitz proposed a method to allocate an
  amount of funds to plural stocks for investment
• Model of Meta-controlled Boltzmann Machine
• The ability of Meta-controlled Boltzmann Machine
  in solving the quadratic programming problem

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The portfolio selection problem

• Maximize
• Minimize
• Subject to and
• with mi ? 0, 1, i 1, .., n
• where ?ij denotes a covariance between stocks i
  and j, ?i is an expected return rate of stock i,
  xi is investing rate to stock i, n denotes the
  total number of stocks and S denotes the number
  of stocks selected, and finally, mi denotes a
  selection variable of investing stocks.

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The portfolio selection problem

• Convert the objective function into the energy
  functions of the two components that are
  Meta-controlling layer (Hopfield Network) and the
  Lower-layer (Boltzmann Machine) as described
  below
• Meta-Controlling layer
• Lower Layer
• where Ku, Kl are weights of the expected return
  rate for each layer and si is the output value of
  the ith unit of the Meta-Controlling layer.

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Algorithm of the Meta-controlled Boltzmann
machine

• Step 1. Set each parameter to its initial value.
• Step 2. Input the values of Ku and Kl.
• Step 3. Execute the Meta-controlling layer.
• Step 4. If the output value of a unit in the
  Meta-controlling layer is 1, add some
  amount of value to the corresponding unit in the
  lower layer. Execute the lower layer.
• Step 5. After executing the lower layer the
  constant number of times, decreases the
  temperature.
• Step 6. If the output value is sufficiently
  large, add a certain amount of value to the
  corresponding unit in the Meta-controlling layer.
• Step 7. Iterate from Step 3 to Step 6 until the
  temperature reaches the restructuring
  temperature.
• Step 8. Restructure the lower layer using the
  selected units of the Meta-controlling
  layer.
• Step 9. Execute the lower layer until reaching at
  the termination.

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Inner behaviors of the Meta-controlled Boltzmann
machine

• Some times, the Hopfield layer may converge to a
  local minimum but the disturb values make it to
  get over
• The changes of Meta layers energy function are
  very small, while the lower layers energy
  functions is quite large
• The number of cycles to execute the Meta layer is
  much smaller than the cycles for the lower layer
• Similar to the simulated annealing that we will
  try to go downhill most of the time instead of
  always going downhill
• The time to converge is much shorter than a
  conventional Boltzmann machine
• All the neurons that are encouraged will be
  selected before the system goes to the final
  computation.

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Chart of behaviors of Meta-controlled Boltzmann
Machine
Disturb back value 80
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Chart of behaviors of Meta-controlled Boltzmann
Machine
Disturb back value 1
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Comparison of computing time between a
Conventional Boltzmann machine and a
Meta-controlled Boltzmann Machine (1286 units)

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Some hints on accelerating the Meta-controlled
Boltzmann machine

• Trying to use only a layer of Boltzmann Machine,
  modify the algorithm of original Boltzmann
  Machine by removing the discouraged units before
  goes into final computation.
• Modify the original Boltzmann Machine by
  replacing deterministic neurons by stochastic
  neurons since the disturb from the lower layer to
  the upper layer may not be worth.

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A Modified Meta-controlled Boltzmann machine

• Step 1. Set each parameter to its initial value.
• Step 2. Input the values of Ku , Kl.
• Step 3. Execute the Meta-controlling layer.
• Step 4. If the output value of a unit in the
  Meta-controlling layer is 1, add some
  amount of value to the corresponding unit in the
  lower layer. Execute the lower layer.
• Step 5. After executing the lower layer the
  constant number of times, decreases the
  temperature.
• Step 6. Iterate Step 4, 5 until the temperature
  reaches the restructuring temperature.
• Step 7. Restructure the lower layer using the
  selected units of the Meta- controlling
  layer.
• Step 8. Execute the lower layer until reaching at
  the termination.
• Algorithm of the Modified Meta-controlled
  Boltzmann machine

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Comparing performance
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CONCLUSION

• The trend of accelerating algorithms is focused
  mainly on heuristic modification and numeric
  optimization technique, i.e. toward the faster
  convergence of algorithms whereas keeping the
  correctness for them.
• The Meta-controlled Boltzmann Machine can be used
  to solve quadratic programming problems.
• Future works
• Try the model with other quadratic programming
  problem.
• Evaluate the modified Meta-controlled Boltzmann
  Machine.

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THANK YOU!