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

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

3 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

4 Maximize Minimize Subject to and with m i  {0, 1}, i = 1,.., n where  ij denotes a covariance between stocks i and j,  i is an expected return rate of stock i, x i is investing rate to stock i, n denotes the total number of stocks and S denotes the number of stocks selected, and finally, m i denotes a selection variable of investing stocks. The portfolio selection problem

5 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 K u, K l are weights of the expected return rate for each layer and s i is the output value of the ith unit of the Meta-Controlling layer.

6 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.

7 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.

8 Chart of behaviors of Meta-controlled Boltzmann Machine Disturb back value = 80%

9 Chart of behaviors of Meta-controlled Boltzmann Machine Disturb back value = 1 %

10 Comparison of computing time between a Conventional Boltzmann machine and a Meta-controlled Boltzmann Machine (1286 units)

11 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. 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. 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.

12 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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15 Comparing performance

16 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.

17 THANK YOU!