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An empirical study of the efficiency of learning ... - ResearchGate

I (M, N, z) 3.0E+06 2.0E+06 1.0E+06 0.0E+00 geometry 2 3 4 6 8 10 20 30 40 50 pop. size 4 x 4 7 x 7 10 x 10 Figure 8: Variation **of** I(M,N,z) with population size, and different geometries (breeding rate=0%) Figs. 2, 3, 4, 6 demonstrate very clearly that when large numbers **of** gates are free to be used, **the** computational effort for correctly evolving **the** parity functions largely increases with increasing population size. It doesn’t seem to depend on **the** arity. For **the** 3-bit parity problem **the** optimum population size seems to be about 6 (Fig. 2). However Fig. 5 shows that when **the** maximum number **of** allowed nodes is much smaller, **the** dependence **of** computational effort with population size is reversed. Figs. 7 and 8 show **the** variation **of** I(M,N, z) with population size for **the** 2-bit multiplier. Three geometries were examined for 100% breeding and 0%. Again **the** growth **of** effort with increasing population size is observed. The smaller geometry 4 x 4 doesn’t show **the** inverse dependency with population, which was seen in Fig 5. It may be that 4 x 4 is still large enough for **the**re to be a reasonable density **of** solutions (**the** minimum number **of** gates required to build **the** multiplier is 7). Comparing Figs. 7 and 8 with each o**the**r reveals that **the** use **of** recombination reduces computational effort, but only marginally. PROBABILISTIC HILLCLIMBER RESULTS In all **the** experiments with **the** probabilistic hillclimber algorithm, **the** mutation rate per chromosome was set at 1%, thus for **the** parity functions and an array **of** 16 x 16 gates **the** number **of** genes mutated per chromosome is 10. In **the** case **of** **the** 2-bit multiplier with 10 x 10 geometry, this figure becomes 4. The levels-back parameter was set to 2 for all experiments. For all parity experiments **the** gate set is {and, or, nand, nor} as in **the** GA experiments. For **the** parity functions **the** geometry was fixed at 16 x 16. EVEN PARITY FUNCTIONS I (M, N, z) Table 9: 3-bit even parity N Pop. size, M R(z) I(M, N, z) 6,000 2 1 (100) 3,122 5,000 3 1 (100) 1,803 4,000 4 1 (100) 2,564 4,000 6 1 (100) 1,926 6,000 8 1 (100 ) 2,888 6,000 10 1 (100 ) 2,410 6,000 20 1 (100 ) 3,220 6,000 30 2 (100 ) 4,860 1,000 40 1 (100 ) 4,040 1,000 50 4 (100 ) 4,200 6000 5000 4000 3000 2000 1000 0 2 3 4 6 8 10 20 30 40 50 pop. size Figure 9: Variation **of** I(M,N,z) with population size for 3- bit even parity function (16 x 16) Table 10: Pop. size, M 4-bit even parity (4,000 generations) R(z) I(M, N, z) 2 3 (100) 20,346 3 1 (100) 11,013 4 1 (100) 9,884 5 1 (100) 7,005 6 1 (100 ) 9,726 8 1 (100 ) 8,328 10 1 (100 ) 9,010 20 1 (100 ) 9,220 30 1 (100 ) 19,830 40 1 (100 ) 17,640 50 1 (100 ) 21,550

I (M, N, z) 25000 20000 15000 10000 5000 0 2 3 4 5 6 8 10 20 30 40 50 pop. size Figure 10: Variation **of** I(M,N,z) with population size for 4-bit even parity function (16 x 16) Table 11: 5-bit even parity (10,000 generations) I (M, N, z) Pop. size, M R(z) I(M, N, z) 1.4E+05 1.2E+05 1.0E+05 8.0E+04 6.0E+04 4.0E+04 2.0E+04 0.0E+00 2 5 (65) 93,010 3 2 (96) 49,206 4 2 (98) 44,008 5 1 (99) 33,005 6 1 (99 ) 46,812 8 2 (99 ) 46,416 10 1 (99 ) 38,010 20 1 (100 ) 66,020 30 1 (100 ) 66,030 40 2 (100 ) 120,080 50 1 (100 ) 85,050 2 3 4 5 6 8 10 20 30 40 50 pop. size Figure 11: Variation **of** I(M,N,z) with population size for 5-bit even parity function (16 x 16) 2-BIT MULTIPLIER For all experiments **the** geometry = 10 x 10, gate set = {all}, **the** maximum number **of** generations was 80,000. R(z) = 1 in all **the**se cases. All 100 runs were successful in all cases. I (M,N,z) 2.0E+05 1.5E+05 1.0E+05 5.0E+04 0.0E+00 Table 12: 2-bit multiplier Pop. size, M I(M,N,z) 2 55,042 3 39,363 4 37,124 6 42,246 10 60,810 20 83,220 30 134,430 40 128,040 50 160,050 2 3 4 6 10 20 30 40 50 pop. size Figure 12: Variation **of** I(M,N,z) with population size for 2-bit multiplier (10 x 10) Problem Table 13: Previous published results GP Koza 94 EP Chellapilla 98 3-bit even parity 64,000 * 63,000 4-bit even parity 176,000 * 118,500 * 5-bit even parity 464,000 * 126,000 2-bit multiplier - - Table 14: Results with z = 1.0 Problem GA PH 3-bit even parity 7,810 (10) 1,926 (6) 4-bit even parity 21,604 (4) 8,205 (5) 5-bit even parity 60,004 (4) 34,010 (20) 2-bit multiplier 132,002 (2) 49,924 (4) Table 15: Results with z = 0.99 Problem GA PH 3-bit even parity 4,326 (6) 1,803 (3) 4-bit even parity 18,404 (4) 7,005 (5) 5-bit even parity 49,204 (4) 33,005 (5) 2-bit multiplier 124,002 (2) 37,124 (4)

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