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1 Tailoring Mutation to Landscape Properties William G. Macready? Bios Group L.P. 317 Paseo de Peralta Santa Fe, NM Abstract. We present numerical results on Kauman's NK landscape family indicating that the optimal distance at which to search for tter variants depends on both the current tness and the sampling that can be aorded at each distance. The optimal search distance from average tness congurations is large to escape local correlation limits and decreases as tness increases. An analytic derivation of the optimal search distance as a function the landscape correlation, the current tness f, and the number of samples n is determined by introducing a new landscape family { -landscapes. The utility of -landscapes is demonstrated by determining a few of their simple properties. 1 Introduction Evolutionary metaphors have re-invigorated research into the design of optimization algorithms. An important next step in this research is to gain a deeper understanding of the subtle connection between mutation operators (or equivalently landscape properties) and eective search algorithms. Until we systematically understand this connection progress in evolutionary-based search will be haphazard at best [WM97,MW96]. This paper moves towards a deeper understanding of this issue by investigating the optimal distance at which to search for tter variants as a function of landscape properties. Landscapes consist of both the tness function to be optimized and a neighborhood relationship amongst congurations in the input space. This neighborhood relation is dened by the mutation operator. The most familiar continuous landscapes are dened over IR n with the natural topology induced by IR n.combinatorial optimization is concerned with discrete input spaces and in this case, the neighborhood relation is expressed by a graph whose nodes consist of all congurations in the input space and whose edges connect congurations which are mutants of one another. The prototypical example of a discrete landscape might be spin glass models [FH91,MPV87] (including the NK model we study here) dened over bit strings with Boolean hypercube topology dened by single bit-ip Hamming neighbors. Recent work has elucidated many properties of landscapes in terms of their underlying neighborhood graph (see [Sta95] for a detailed view of the state of the? I thank both Bios Group L.P. and the Santa Fe Institute for support.

2 art in landscape theory). However, our understanding of the relationship between properties of landscapes and the design of eective optimization algorithms over those landscapes goes little beyond the obvious observation that the number of local peaks (under whatever mutation operator is being used) serves as a good indicator of the diculty of the problem. To aid in understanding the connection between landscape properties and effective algorithms, a family of tunably dicult landscape models proves useful. Presently, the most popular landscape family is Kauman's NK model [Kau93]. This family of landscapes, inspired by spin glass models, is parameterized by N the number of bits, and K the epistasis between bits. By tuning K, landscapes can generated with varying degrees of ruggedness. The decision problem associated with optimization on the NK family of landscapes is NP-complete for K>[Wei96] sothenk model generates dicult optimization tasks. The NK family has generated some useful results connecting landscape properties to optimization strategies. Particularly interesting is the result that recombination is most eective on landscapes of intermediate ruggedness, i.e K (seethe early work of Kauman [Kau93] onnk landscapes and later work by Bergman et al. [BOF95a,BOF95b] using population genetic approaches). The NK model has contributed to our empirical understanding of algorithm design but it is too complex (for intermediate K) to address deeper theoretical insights. What is needed is a landscape family which is complex enough to encompass dicult optimization problems and yet simple enough to permit analytic insight. To generally understand optimal search distances on landscapes we present a family of landscapes parameterized by thenumber of bits N and the nearest neighbor-correlation coecient coecient between tnesses. We demonstrate the utility of this landscape model by determining the optimal mutation rate as a function of N and. The paper is organized as follows. In Section we demonstrate using the NK model that optimal search distances (or equivalently, mutation rates) should be adjusted according to current tness. Mutation rates should be higher when the current conguration is of average tness, and decrease as the conguration increases in tness. In Section 3weintroduce -landscapes and determine the -landscape best matching a particular NK landscape. Section 4 calculates the mutation rate as a function of current tness f, landscape correlation, and the problem size N. Section 5 ties together the results of previous section and suggests directions for further research. Search distances on NK landscapes: numerical results Mutation rate is an important parameter in evolutionary optimization algorithms. Set too high, the search wanders over congurations never settling into a region of high tness. Set too low, little exploration is done and the algorithm settles onto the nearest local maximum. In this section we present a new method for optimizing mutation rates by determining the optimal search distance as a function of landscape correlation.

3 We imagine an algorithm that at each stage samples congurations at distance d from the current conguration. If congurations at this distance are sampled n times, what is the optimal distance at which to search for tter variants? Simulation with the NK model provides an interesting answer d Fig. 1. The mean one standard deviation of tnesses at distance d from starting congurations having average, below average, and above average tness for an N = 100 and K = 0 landscape Figures 1 and present results for N = 100 landscapes with K = 0 and K = 4. The simulation begins by picking a conguration of average, below average, or better than average tness, and then random congurations at specied distances from the initial conguration are generated and their tnesses recorded. Figures 1 and plot the average tness plus or minus its standard deviation at each distance. In all cases, the variance increases as we move away from the initial conguration and the mean moves toward the mean of the landscape. Though the mean decreases with distance, it may well be better to sample further since the variance is higher and there is a good chance one of the samples may have quite high tness. If the landscape is correlated, nearby congurations are constrained to have similar tness, while points further away have no such constraint and there is the possibility of nding a much tter variant. However, if the current conguration is already of higher than average tness, the higher variability at further distances may not overcome the decay of the mean towards the landscape average. If prior knowledge exists that the landscape is isotropic (meaning that its statisical properties are constant across the input space) then these results nat-

4 d Fig.. The mean one standard deviation of tnesses at distance d from starting congurations having slightly above average, much above average, and much below average tnesses for an N = 100 and K = 4 landscape urally suggest an optimization strategy. If current tness is about average, then it is best to search futher than a correlation length away on the landscape where much tter variants could conceivably be found. As current tness increases, conne search closer to the current conguration to exploit the gains made thus far. At high tnesses it is best to search in the immediate neighborhood, since increased variance will not compensate for the rapid drop in the mean with increasing distance. These general results depend on the amount of sampling that can be done at each distance. If more sampling is allowed then the order statistics improve and the optimal search distance increases. To understand this result so that it can be applied in other situations we need a deeper quantitative understanding. The next section denes a new family of landscapes in which optimal search distances can be determined. 3 A new family of landscapes A fundamental characteristic of a landscape is the correlation between tness values at varying distances. We start from this observation to parameterize a family of landscapes based on the correlation coecient between tnesses of nearest-neighbor congurations. In keeping with most landscape theory, which is statistical in nature, we approximate a landscape by a joint probability distribution. Under this annealed approximation, a landscape is characterized entirely by the distribution P (f ;f )

5 where f and f are the tnesses of neighboring congurations x and x. A remarkable number of landscape predictions can be made from this simple starting point. 3.1 Preliminaries A tness landscape consists of a tness function f : X 7! R and a metric structure over a search space X. In this paper we focus on discrete search spaces in which case the metric structure over X can be represented with a directed graph G. Thevertices in G consist of all points in the search space with directed edges (; ) indicating that conguration x is a neighbor of x. If the neighbor relation is symmetric, then the edges of G are undirected. The distance of two congurations in X is given by their distance in G. Nearest neighbors in G are at distance 1, while the maximal distance is given by the diameter of G. As a familiar example the neighborhood graph G for bit-strings of length N with single bit ip mutation is the hypercube graph. As discussed above landscapes are approximated by the joint probability P (f ;f )wherex and x are connected by anedgeing. Formally this probability is dened from P (f ;f )= P hx i;x ji (f, f(x i ))(f, f(x j )) P hx i;x ji 1 where the notation hx i ;x j i requires that congurations x i and x j are neighbors in G and () is the Dirac delta function 1.From P (f ;f )wemay calculate both P (f ), the probability that a randomly chosen conguration x has tness f, and P (f jf ) the probability ofa conguration having tness f given that a neighboring conguration x has tness f.formally these quantities are dened by P (f )= Z df P (f ;f ) and P (f jf )= P (f ;f ) P (f ) The joint distribution P (f jf ) is particularly useful as it gives the distribution of the tnesses of the neighbors of a conguration having tness f. 3. An application Having dened the probabilistic approximation to landscapes, we give one simple example of an application. More interesting (and complex) applications can be found in [Mac98a]. Under a rather crude simplication it is a simple matter to estimate the number of local maxima in a tness landscape. The probability 0 (f ) that a 1 The delta function (x) is dened to be zero away forx6= 0 and satises R dx (x) = I 1 if the region of integration I includes x =0.

6 conguration x with tness f is of higher tness than all of its n neighbors is 0 (f )= Z f df n,1 Z f df n,1,1 Z f,1 df 1 P (f 1 ; ;fn jf ) where P (f 1 ; ;fn jf ) is the joint probability that the n neighbors of a site with tness f have tnesses f 1 ; ;fn.ifwe assume that given f the tnesses f 1 ; ;fn are otherwise independentofeach other then P (f 1 ; ;fn jf ) factors as P (f 1 ; ;fn jf )= Q n i=1 P (f i jf )andwe can write 0 (f )= "Z f df P (f jf ),1 # n (f) n (1) (f ) is the probability that a randomly chosen neighbor has a tness lower than f. Given this result, the expected number, N p, of local maxima is then N p = jx j Z df 0 (f )P (f ) where jx j is the size of the conguration space. 3.3 NK landscapes under the annealed approximation In this section we use the annealed approximation to approximate Kauman's NK landscapes. The NK model [Kau93] generates a family of tunably rugged energy landscapes over bit-strings b = fb 1 b N g of length N. Each bitb i makes a contribution to the total tness dependent onitsown state and the state of K other bits. These K other bits may be selected at random or according to some specied connectivity. The total tness of a bit-string, b, is dened by: ffbg = 1 N NX i=1 f i (b i ; b i1 ; ;b ik ) By analogy with spin glasses [FH91] the local tness contributions, f i, for each of the K+1 local bit congurations, fb i ;b i1 ; ;b ik g, are chosen at random. Normally this underlying distribution is uniform over [0,1) but for calculational simplicity we will assume it is Gaussian with mean 0 and variance : P (f i )= exp [,f i = ] p : By specializing the f i and the selection of the K neighbors, the NK model encompasses many optimization problems, including spin-glasses, graph-coloring, number partitioning etc. Much is known about the NK model in the limit K = N, 1 [Der80b,Der80a,MP89,KW89] andk = 0 [Kau93].

7 In [Mac98a] the relevant probabilities for the NK model are calculated as: p N P (f )= p exp, N f () N P (f ;f )= p exp N, 1, s N P (f jf )= (1, exp, ) The nearest-neighbor correlation coecient is given by, f (1, ) + f, f f (3) N, f, (1, ) f : (4) =1, K +1 N and ranges from very correlated when =(N, 1)=N at K = 0 to completely uncorrelated when =0atK = N, 1. Within the NK family the correlation is constrained to be positive. In Section 4, we shall use Eq. (3) as dening a more general family of landscapes with correlation ranging from,1 1. A constructive procedure to generate landscapes satisfying Eq. (3) is described in [Mac98b]. Applying Equation (1) we see the probability that a bit-string with tness f is a local maximum is where the error function is dened as 0 (f )= 1 N erfn N(1, ) (1 + ) f Z erf[x] = p x dt exp[,t ]: 0 With this brief introduction to the landscape family we move on to determine optimal search distances within this family. 4 Search distances on -landscapes: analytical results Determination of the optimal mutation distance is straightforward using landscapes. In [Mac98a] it is shown that tnesses at distance d away from a conguration of tness f are distributed Gaussianly with mean and variance given by: (f ;d)=f d (5) (d) =1, d : (6) The error function is an odd function and has the limiting value lim x!1 erf[x] =1.

8 If the algorithm samples n times at each distance and goes with the highest tness obtained, f, the distribution in maximal tnesses P 1 (f jf ;n;d) is given by the highest order statistic: P 1 (f jf ;n;d)=np (f jf ;d) "Z f n,1 df P(fjf ;d)# f Cn (f jf ;d),1 where P (f jf ;d) is the conditional density of tness f found at distance d from a bit-string having tness f and C(f jf ;d) is its cumulative distribution. As noted above this distribution is Gaussian, i.e. P (f jf ;d)=n, (f ;d); (d). The expected maximal tness f max (f ;n;d) is just the mean of P 1 (f jf ;n;d). The mean is dicult to calculate exactly but we can approximate it by f max (f ;n;d)= (f ;d)+(n)(d). The best choice for (n) is determined in the appendix and is found to be (n) = C,1 ( np 1=). In the above equation C(f ) is the cumulative distribution function for N (0; 1) and C,1 (x) is its inverse. Explicitly C(x) = (1 + erfc[x= p ])=. The optimal search distance is that distance at which f max (f ;n;d) is greatest. The function f max (f ;n;d) is well behaved as a function of d having only a single maximum and consequently the optimal choice of d opt is the integer nearest to the d which d=d d=d (n) =, (f ;d) (f Using the results in Eq. (5) and (6) we see that d satises f d ln = d=d (n) p 1, d d ln : (7) Solving this quadratic in d yields an explicit solution for d as d (f ;n)=, 1 ln [1 + ((n)=f ) ] : ln z In the case when f 0 the only solution to Eq. (7) is found by driving both sides to zero as d!1. This corresponds to a situation in which tnessisworse than average so it is best to search maximally far away where the variance is greatest. The expected tness at the optimal distance is easily calculated as f max (f ;n;d= d )=f q 1+, (n)=f A plot of the optimal search distance as a function of f for various n on landscapes with dierent correlations is shown in Figure 3. A slight variant of this algorithm might also be applied in practice. Rather than have the algorithm continue to sample n times at each distance, it may be more ecient to terminate the sampling if a very high tness is encountered. The analysis of this situation is more complicated and requires appeal to dynamic programming. A complete solution to this problem (including the possibility of a cost of search) is given in [KLM98] in an economic context.

9 5 0 d f n 30 Fig. 3. Optimal search distance for a = 0:9 landscape as a function of the initial tness f and the number of samples n. 5 Conclusions We have outlined the framework for a new family of landscapes that allows analytic insight into dicult optimization problems. The aim in designing a new landscape family is to attempt an understanding of the connection between eective optimization algorithms and prior properties of the problem. In this simple case we have accomplished that goal by deriving an optimal parameter setting within a narrow class of algorithms based on mutation alone. We described the new family of -landscapes motivated by an annealed approximation and results for the NK model. A constructive technique for generating such landscapes based on Gaussian processes in described in [Mac98b]. We then described a family of algorithms which search at tness dependent distances and determined the optimal search distance for the -landscape family. These explorations are preliminary but encourage further research. For more complicated algorithms like evolutionary based algorithms, and simulated annealing we may be able to determine parameter settings exactly as a function of. Moreover, a greater understanding of -landscapes might suggest new - dependent optimization techniques. Acknowledgements I would like to thank Stuart Kauman and Jose Lobo for ideas and encouragement. References [BOF95a] A. Bergman, S. P. Otto, and M. W. Feldman. On the evolution of recombination in haploids and diploids i. deterministic models. Complexity, 1:57{67, [BOF95b] A. Bergman, S. P. Otto, and M. W. Feldman. On the evolution of recombination in haploids and diploids ii. stochastic models. Complexity, :49{57, 1995.

10 [Der80a] B. Derrida. The random energy model. Phys. Rep., 67:9{35, [Der80b] B. Derrida. Random energy model: Limit of a family of disordered models. Phys. Rev. Lett., 45:79{8, [FH91] K. H. Fischer and J. A. Hertz. Spin Glasses. Cambridge University Press, [Kau93] S. A. Kauman. The Origin of Order. Oxford University Press, New York, Oxford, [KLM98] S. A. Kauman, J. Lobo, and W. G. Macready. Optimal search onatech- nology landscape. In review at Econometrica, [KW89] S. A. Kauman and E. D. Weinberger. The nk model of rugged tness landscapes and its application to maturation of the immune response. J. Theor. Biol., 141:11, [Mac98a] W. G. Macready. An annealed theory of landscapes: part 1. In preparation, [Mac98b] W. G. Macready. An annealed theory of landscapes: part. In preparation, [MP89] C. A. Macken and A. S. Perelson. Protein evolution on rugged landscapes. Proc. Natl. Acad. Sci. USA, 86:6191{6195, [MPV87] M. Mezard, G. Parisi, and M.A. Virasoro. Spin Glass Theory and Beyond. [MW96] World Scientic, Singapore, W. G. Macready and D. H. Wolpert. What makes an optimization problem? Complexity, 5:40{46, [Sta95] P. F. Stadler. Towards a theory of landscapes. In R. Lopez-Pe~na, R. Capovilla, R. Garcia-Pelayo, H Waelbroeck, and F. Zertuche, editors, Complex systems and binary networks. Springer Verlag, Berlin, [Wei96] E. D. Weinberger. Np completeness of kauman's nk model, a tuneable rugged tness landscape. Santa Fe Institute technical report SFI , [WM97] D. H. Wolpert and W. G. Macready. No free lunch theorems for optimization. IEEE Trans. Evol. Comp., 1:67{83, A Determination of (n) Here we present a simple motivation for our approximation to the largest order statistic. Let P (x) be a probability distribution and C(x) be its cumulative distribution. The probability distribution for the largest of n samples from P (x) is given by np (x)c n,1 (x) =@ x C n (x). Integrating by parts we nd the expected value for the largest of the n samples, n max is then n max = xc n (x) 1,1 + Z dx C n (x) For large n, C(x) n approaches a step function at some value x c (n) that increases with n. If we make this assumption then C n (x) =(x, x c (n)) for x>0and the above integral gives n max = x c (n). So how do we determine x c (n). We simply assume that the cuto is at the value where C n (x) =1= givingx c (n) = C,1 ( np 1=). For Gaussian P (x) this estimation to n max gives an approximation which isoby3%atn = and becomes more accurate with larger n.

Optimal Search on a Technology Landscape

Optimal Search on a Technology Landscape Stuart A. Kauffman José Lobo William G. Macready SFI WORKING PAPER: 1998-10-091 SFI Working Papers contain accounts of scientific work of the author(s) and do not