A Modified A Modified Meta-controlled Boltzmann MachineMeta-controlled Boltzmann Machine

Tran Duc Minh, Le Hai Khoi (*), Junzo Watada (**), Teruyuki Watanabe (**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

03/200403/2004

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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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Maximize

Minimize

Subject to and

with mi {0, 1}, i = 1, .., nwhere ij denotes a covariance between stocks i and j, i is an expected return rate of sto

ck 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.

The portfolio selection problem

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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 layer’s energy function are very small, while the lower layer’s energy function’s 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)

Comparison of the computing time between a

Conventional Boltzmann Machine and a Meta-

controlled Boltzmann Machine

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100150200250300350

Tem

pera

ture

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Com

puti

ng t

ime (

sec)

Conventional BM

Meta-controlled BM

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Some hints on accelerating the Meta-controlled Boltzmann machine

• Trying to use only a layer of Boltzmann MachiTrying to use only a layer of Boltzmann Machine, modify the algorithm of original Boltzmann ne, modify the algorithm of original Boltzmann Machine by removing the discouraged units befMachine by removing the discouraged units before goes into final computation.ore goes into final computation.

• Modify the original Boltzmann Machine by repModify the original Boltzmann Machine by replacing deterministic neurons by stochastic neurlacing deterministic neurons by stochastic neurons since the disturb from the lower layer to thons since the disturb from the lower layer to the upper layer may not be worth.e 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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Actual Hopfield layer's energy function

- 3

- 2.5

- 2

- 1.5

- 1

- 0.5

010 70 130

190

250

310

370

430

490

550

610

670

730

790

Timing

Val

ue

Actual Boltzmann layer's energy function

0.001365

0.00137

0.001375

0.00138

0.001385

1 10 19 28 37 46 55 64 73 82 91 100

Timing

Val

ue

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Comparing performance

Compare time to converge

7.67.8

88.28.48.68.8

99.2

Se

q 3 6 9 12

15

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Sequence number

Se

con

ds

TimeHPBM

TimeModifiedHPBM

Compare Energy values

- 1.0009

- 1.00085

- 1.0008

- 1.00075

- 1.0007

- 1.00065

1 5 9 13

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Sequence number

En

erg

y v

alu

es Energy

HPBM

EnergyModifiedHPBM

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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!THANK YOU!