Study of DSMGA2

許泓崴 莊佾霖

2016.01.20

Outline

  • Pasting Problem
  • Trials of improving DSMGA2

Pasting Problem

Properties of Good Topologies

  • Every bit belongs to no more than six subfunctions

  • Non-hierachical

We only discuss on problems with following properties

Proof of O(1) Solution

Counter Example in 2D Spin Glass

Counter Example in 2D Spin Glass

Trial of Probability Model

  • Show that difficult cases are rare enough

  • Prove that in average GA performs well

  • Be able to solve any good topologies

We hope this model can do

But what is

probability model

based on?

Trial of Different Metrics

  • Simple metric

  • Chi-square metric

  • Estimated group mutual information

Simple Metric 

2\left(P_{00}P_{11} + P_{01}P_{10}\right)
2(P00P11+P01P10)2\left(P_{00}P_{11} + P_{01}P_{10}\right)

Chi-square Metric 

P_{ij}^{ex} = P_{i}^{1} \cdot P_{j}^{2}
Pijex=Pi1⋅Pj2P_{ij}^{ex} = P_{i}^{1} \cdot P_{j}^{2}
P_{i}^{1} = P_{i0} + P_{i1}
Pi1=Pi0+Pi1P_{i}^{1} = P_{i0} + P_{i1}
\chi ^{2} = \sum_{i,j=0}^{1} \frac{\left(P_{ij}^{ex}-P_{ij}\right)^{2}}{P_{ij}^{ex}}
χ2=∑i,j=01(Pijex−Pij)2Pijex\chi ^{2} = \sum_{i,j=0}^{1} \frac{\left(P_{ij}^{ex}-P_{ij}\right)^{2}}{P_{ij}^{ex}}

Estimated Group Mutual Information

P_{000}\sim P_{111}
P000∼P111P_{000}\sim P_{111}

  • Predict probability

  • Evaluate with multivariate mutual information

Predict Probability

P_{000} \leq \min\left(P_{00-}, P_{0-0}, P_{-00}\right)
P000≤min(P00−,P0−0,P−00)P_{000} \leq \min\left(P_{00-}, P_{0-0}, P_{-00}\right)

Upper Bound

P_{000} = 1-P_{1--} - P_{-1-} - P_{--1}
P000=1−P1−−−P−1−−P−−1P_{000} = 1-P_{1--} - P_{-1-} - P_{--1}
\leq 1 - P_{1--} - P_{-1-} - P_{--1}
≤1−P1−−−P−1−−P−−1\leq 1 - P_{1--} - P_{-1-} - P_{--1}
+ P_{11-} + P_{1-1} + P_{-11}
+P11−+P1−1+P−11+ P_{11-} + P_{1-1} + P_{-11}
+ P_{11-} + P_{1-1} + P_{-11}
+P11−+P1−1+P−11+ P_{11-} + P_{1-1} + P_{-11}
P_{-11} - P_{111}
P−11−P111P_{-11} - P_{111}

Predict Probability

P_{000} \geq max\left({P_{00-}-P_{001},P_{0-0}-P_{010},P_{-00}-P_{100}}\right)
P000≥max(P00−−P001,P0−0−P010,P−00−P100)P_{000} \geq max\left({P_{00-}-P_{001},P_{0-0}-P_{010},P_{-00}-P_{100}}\right)

Lower Bound

Ability to Predict Group Relation

  1. Compute group relation

  2. Compute the consistency of these two orders

The order of prediction should agree to the order of group relation

Result

metric value
simple 0.143
MI 0.406
Estimated MI 0.276

Reweight the order in RM

A wiser way to pick the order may reduce NFE

When building masks, preserve second possible candidate

f([1,2,3]) = 7 ; f([1,2,4]) = 6; \delta = \frac{1}{3}
f([1,2,3])=7;f([1,2,4])=6;δ=13f([1,2,3]) = 7 ; f([1,2,4]) = 6; \delta = \frac{1}{3}

Use the order of differences as the order we build choose in RM

Result

  • Can only recognize BB but not order

  • Useless when RM-NFE is not dominant

Summary

metric mean std mean std
original 51k 25k 545k 251k
simple 56k 23k 708k 316k
kai 48k 19k 556k 241k
mi3 85k 46k - -
mi3ex 70k 32k - -
dRank 51k 23k 567k 219k

Future Prospectives 

  • Proper assumptions to derive Pasting Problem

  • New methods to reweight

  • Balance the effect of RM and BM

End

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