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

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

I (M, N, z) 3.0E+06

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 there 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 other 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 these 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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