An Extended Hybrid Genetic Algorithm for Exploring a Large Search Space

Transcription

1 2nd Internatonal Conference on Autonomous Robots and Agents Abstract An Extended Hybrd Genetc Algorthm for Explorng a Large Search Space Hong Zhang and Masum Ishkawa Graduate School of L.S.S.E., Kyushu Insttute of Technology, 2-4 Hbkno, Wakamatsu, Ktakyushu , Japan {zhang, shkawa}@bran.kyutech.ac.jp Recently, a hybrd methodology for combnng genetc algorthms and local search algorthms has receved consderable attenton. Ths paper proposes an extended hybrd genetc algorthm to further mprove the performance of fndng the optmal soluton n a large search space. Three key deas,.e. the eltsm, nonredundant search, and steepest-ascent hll clmbng, are ntroduced nto a standard genetc algorthm. The frst one s to copy superor ndvduals to the next generaton for mprovng convergence. The second one s to ncrease the effcency of fndng the best ndvdual, and the thrd one s to ncrease the effcacy of fndng the best ndvdual. Through the combnaton of these deas, the proposed method s well suted to fnd the optmal soluton n a large search space correspondng to a gven problem. To evaluate the effectveness of the proposed method, computer experments that estmate the weghts of connectons n neural networks for solvng XOR problem are carred out. The results well demonstrate ts effectveness. Keywords: hybrd genetc algorthm, non-redundant search, steepest-ascent hll clmbng, learnng, neural network. 1 Introducton Genetc algorthms (GAs) proposed by J.H. Holland [3], whch use evolutonary computaton, have been appled to real world problems n many applcaton dscplnes. Recent studes have shown ts effectveness over conventonal search and optmzaton methods for solvng complex and llposed problems. In spte of the smplcty of operators used, the standard genetc algorthm (SGA) [1] s good at solvng complex optmzaton problems [4]. Although the SGA has advantages of adaptablty and robustness, t stll has dsadvantages such as no guarantee for convergence and nstablty n computaton. These greatly affect the ablty of fndng the optmal soluton n a large search space. To mprove the performance of the SGA, heurstc algorthms have been ntroduced nto the SGA [5, 6, 7]. Although these approaches are effectve, they are not suted to combnatoral optmzaton problems wth tme-varyng parameters due to heavy computaton. In prevous work, we proposed a hybrd genetc algorthm (HGA) as a knd of global search [8]. The pont ncludes adopton of the eltsm for mprovng convergence and a non-redundant search for quckly fndng the optmal soluton n a large space. Due to the smplcty of operators used such as selecton, comparson, and conservaton, the HGA s well suted to quck acquston of a sequence of optmal solutons correspondng to the changng envronment. Many computatonal and cogntve tasks nvolve a search n a large space. It s well known that GAs are good at global search, but weak at local search. For mprovng the performance of fndng the optmal soluton, we propose to extend the HGA by ncorporatng local search. Hll-clmbng restrcts the search around the best soluton found so far by explorng ts drect neghbourhood [10, 11]. Although hll-clmbng has a drawback whch has nablty to detect the unsolvablty of a problem nstance, t provdes a powerful strategy for explorng the optmal soluton. In addton to ths, local search requres less workng memory, and s easy to mplement. As an example, hll-clmbng technque has been appled to SGA for solvng a toy optmzaton problem [12]. Based on these characterstcs of hll-clmbng, we expect the extended hybrd genetc algorthm to renforce the effcacy of fndng the optmal soluton n a large search space by performng global search and local search smultaneously. The rest of the paper s organzed n the followng way: Secton 2 descrbes steepest-ascent hll clmbng, rankng, and non-redundant search algorthms. Secton 3 gves a flow chart of the proposed algorthm and ts features. Secton 4 provdes results of computer experments, whch demonstrates the effectveness of the proposed method. Secton 5 concludes the paper. 244

2 2nd Internatonal Conference on Autonomous Robots and Agents 2 Algorthms for mprovement 2.1 SAHC algorthm Exhaustve local search algorthm provdes a local optmal soluton n the neghbourhood of an ndvdual. We, here, refer to the algorthm as steepest-ascent hll clmbng (SAHC). The procedure of the SAHC s as follows. Step 1: Select an ntal ndvdual, x, randomly. We refer to the ndvdual as a core one. Step 2: Generate the neghbourhood ndvduals of x, y, wth Hammng dstance, d H (x, y), less than a threshold. For clarty, Fgure 1 shows the set of neghbourhood ndvduals of x={0000} wth d H =1 and 2. Consequently, non-redundant search mproves the effcency of search for fndng the optmal soluton by the expanson of search range. 3 Proposed method By carryng out global search (NRS) and local search (SAHC) smultaneously, we can avod the problem that falls nto local mnmum, and keep the dversty of ndvduals n search process to realze an effcent search. Fgure 2 shows the flow chart of the extended hybrd genetc algorthm (EHGA) whch mproves convergence and ncreases the capablty of fndng the optmal soluton. Fgure 1: The core ndvdual, x={0000}, and ts neghbourhood ndvduals. Step 3: Calculate the ftness of each neghbourhood ndvdual. Let the ndvdual wth maxmum ftness value be the core ndvdual, x. Step 4: Repeat Step 2 through Step 3 untl the core ndvdual s not changed. The SAHC algorthm can fnd out the best ndvdual n the neghbourhood by a seres of comparson wthn the set of neghbourhood ndvduals. 2.2 Rankng algorthm Instead of the roulette wheel selecton, we use the eltsm for mprovng convergence. Accordngly, the ndvduals n the current populaton are sorted accordng to ther ftness. The sorted populaton s referred to as a new one. Then the ndex of ndvdual n the populaton wll be used to determne whch ndvduals are allowed to reproduce drectly to the next generaton. 2.3 NRS algorthm Non-redundant search (NRS) s a knd of global search. The basc constructon of ts algorthm s as follows. Frst acton s deleton of ndvduals wth the same chromosome n the current populaton. Second acton s addton of new ndvduals selected randomly nstead of these redundant ones. Fgure 2: Flow chart of the extended hybrd genetc algorthm. The EHGA uses the followng operators n addton to those n the SGA. Operator 1: Select some ndvduals n populaton P t, and determne the best ndvdual n ther neghborhood set correspondng to core ndvdual through the SAHC algorthm. Operator 2: Adjust order of the ndvduals n the current P t based on ther ftness. Let the ordered populaton be P t. Operator 3: Delete redundant ndvduals from the populaton P t, whch are generated by the SGA and add new ndvduals selected randomly. Let the resultng populaton be P t. Operator 4: Merge the number of the best ndvduals from the populaton P t and the number of obtaned ndvduals from the populaton P t n a certan proporton for generatng the next generaton P t+1. 4 Computer experments To demonstrate the effectveness of the proposed method, we dscuss how to estmate the weghts of connectons n neural networks for solvng the exclusve-or (XOR) problem, well-known problem [9] by the EHGA. 245

3 2nd Internatonal Conference on Autonomous Robots and Agents 4.1 Expresson of neural network Fgure 3 ndcates a structure of neural network wth two nputs, two hdden-layer unts, and a sngle output unt for solvng XOR problem n whch four nputs of {0, 0}, {0, 1}, {1, 0}, and {1, 1} must be mapped to outputs of 0, 1, 1, and 0, respectvely. Each of the three unts (2-hdden, 1-output) has three weghts (one of them represents the bas term of the unt). And each unt s nonlnear one whch has the z standard logstc output functon, f ( z) = 1 /(1 + E ), where z s net nput of the unt, that s the sum of an adjustable bas term, w 0, and dot product of the ncomng actvatons and ther assocated weghts,.e. 2 z = = w x x = 1. 0, 0 Fgure 3: A structure of neural network for solvng XOR problem. We, here, represent each weght of connectons n the neural network by a m-bt bnary strng, then scalng the weght to a number between -5 and 5. Fnally, concatenate all of the weghts together nto one long strng whch length s 9x m-bt, w={w 1, w 2, w 3 }={w 11, w 12, w 10, w 21, w 22, w 20, w 31, w 32, w 30 } as one ndvdual for evolutonary computaton. Notce that w 1 and w 2 denote the weghts of hdden unts, and w 3 denotes the weghts of output unt. We use drectly the value that s the nverse of the sum of mean squared error (MSE) of the neural network outputs and a constant as the objectve functon. Hence, the followng ftness functon, g k (w), s consdered to evaluate the ndvdual of the weght of connectons n kth neural network, = 1 g k ( w ) = MSE k µ ( t 4 = 1 1 y ) 2 + µ (1) where t and y s the target output and the actual output accordng to the weghts of connectons n kth neural network, respectvely. µ s a mnute constant for preventng the denomnator of Equaton (1) becomes zero. 4.2 Expermental results In our computer experments, the major parameters of the genetc algorthms appled to solve XOR problem are as follows. Populaton sze: 100; the number of generatons: 100; selecton: roulette wheel selecton; crossover: 6-pont crossover (p c =0.8); bt-wse mutaton (p m =0.1). Fgure 4: The MSE of the neural network outputs correspondng to each generaton for solvng XOR problem. For computatonal convenence, Make m-bt=6 bnary strng to represent each weght of the neural network n Fgure 3. Nevertheless, we need to fnd the optmal soluton (best weghts) n the large search space, the order of 2 54 ponts, for solvng XOR problem. Computer experments are mplemented. As a result, Fgure 4 shows a convergence process of the neural network correspondng to each generaton for solvng XOR problem by the EHGA. From Fgure 4, t s clear that the approxmate perfect mappng functon s evolved wthn 100 generatons allowed. The estmated weghts of each unt of the neural network are: w 1 ={5., -5., }; w 2 ={-5, 5, }; w 3 ={5., 5., }. Fgure 5 shows the curved plane determned by these weghts for dstngushng 0/1 classes of XOR problem. Fgure 5: The curved plane for solvng XOR problem. To compare the performance of the proposed method and other algorthms and confrm a role of the parameters such as Nc, Ns ( Nc s the number of coped superor ndvduals to the next generaton, and Ns s the number of selected ndvduals appled to the SAHC algorthm.) and so on, Fgure 6 ndcates the hstograms of the results whch reflect probabltes related to ftness dstrbuton obtaned. 246

4 2nd Internatonal Conference on Autonomous Robots and Agents 5 Dscusson In fact, as we know that solvng XOR problem tself s not a bg search problem. However, due to satsfyng hgher estmate accuracy to solve XOR problem, a bnary-coded GAs has to explore a large search space whch represents the weghts of connectons n the neural network for searchng the optmal soluton. For reducng the search space n scale, a smple soluton to the problem s the use of floatng-pont representaton of the parameters as an ndvdual. In real-coded GAs [2, 13], an ndvdual s coded as a fnte-length strng of the real number correspondng to each weght of connectons n the neural network. We, here, make a concesson about the expanson to real-coded GAs of the proposed method. Fgure 6: Probabltes related to ftness dstrbuton. (a) The EHGA wth Nc=1, Ns=3, and d H =2. (b) The EHGA wth Nc=1, Ns=1, and d H =2. (c) The EHGA wth Nc=10, Ns=10, and d H =1. (d) The EHGA wth Nc=1, Ns=10, and d H =1. (e) The SGA wth Nc=10. (f) The SGA. Obvously, n comparson wth the result of the SGA n Fgure 6(f), the result of the SGA wth the eltsm n Fgure 6(e) s better for solvng XOR problem. However, t cannot fnd much better weghts as stated above than the results of the EHGA n Fgure 6(a) and Fgure 6(b). Furthermore, through comparng the results between Fgure 6(a) and Fgure 6(b), we can confrm the fact,.e. the greater the number of selected ndvduals s, the bgger the probablty of the obtaned best ftness s. For examnng the role of parameters n the EHGA, Table 1 ndcates the statstc results correspondng to ftness n the dfferent cases. Table 1: The maxmum, mean, mnmum values of ftness n the dfferent cases. From the results of Table 1, t s clear that the larger the neghbourhood range taken s, the bgger the probablty of the obtaned best ftness s. These results ndcate that the SAHC algorthm s very powerful for explorng the large search space, and provde evdence of the effcacy for fndng the optmal soluton by the proposed method. 6 Conclusons We have proposed an extended hybrd genetc algorthm wth three strateges,.e. the eltsm, nonredundant search and steepest-ascent hll clmbng. Based on the combnaton of advantages of three key ponts appled to t, the proposed method further mproves the effcency and effcacy for fndng the optmal soluton n a large search space. Applcatons of the proposed method to estmate the weghts of the neural network for solvng XOR problem well demonstrate ts effectveness. Obtaned results ndcate that the EHGA performs better than the other approaches such as the SGA and the SGA wth the eltsm. Because of the gven problem, XOR problem, n our computer experments s qute smple, and due to enforcement of the SAHC algorthm, shortenng ts computatonal tme to be necessary, t s left for further study to apply the proposed method to real world problem. 7 References [1] Goldberg, D.E., Genetc Algorthms n Search, Optmzaton, and Machne Learnng, Addson- Wesley, Boston, USA (1989). [2] Goldberg, D.E., Real-coded genetc algorthms, vrtual alphabets, and blockng, Complex Systems, No.5, pp (1991). [3] Holland, J.H., Adaptaton n natural and artfcal systems, Reprnted, MIT press, Cambrdge, USA (1992). [4] Man, K.F., Tang, K.S. and Kwong S., Genetc Algorthm, Sprnger-Verlage, London, UK (1999). [5] Therens, D., Scalablty Problems of Smple Genetc Algorthms, Evolutonary Computaton 7(4): pp (1999). 247

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