A Note on Crossover with Interval Representations Christopher Stone and Larry Bull UWE Learning Classifier System Technical Report UWELCSG03-00 Faculty of Computing, Engineering and Mathematical Sciences University of the West of England Bristol, BS16 1QY, United Kingdom christopher.stone@uwe.ac.uk larry.bull@uwe.ac.uk Abstract. We analyse crossover in Learning Classifier Systems with three different interval representations, using both niche and panmictic Genetic Algorithms. We find that the choice of interval representation has a significant bearing on the characteristics of offspring interval produced and the means by which evolutionary search is carried out. 1 Introduction Learning Classifier Systems [] typically use a ternary representation to encode the environmental condition that a classifier matches. Bits in the condition string of a classifier are allocated to represent the state of a single environmental variable, x i. Exact matching in this way is generally not suitable for a continuousvalued environment X R n, where real-valued data over a range must be represented. The representations considered here replace the {0,1,#} classifier predicate with one representing a half-open interval [p i,q i ) [4, 6, 7]. This interval matches the environment if p i x i < q i. The classifier condition is a vector of length n, each element of which encodes such an interval. A classifier with such a representation describes a hyper-rectangle in solution space. This paper studies the effects of crossover in Learning Classifier Systems that use such an interval representation. The interval representations analysed are Centre-Spread Representation (CSR) [6], Ordered Bound Representation (OBR) [3, 7] and Unordered Bound Representation (UBR) [4]. We do this by analysing the effects of crossover on a single interval predicate. In this situation, crossover occurs within an interval predicate, between alleles representing the interval. We refer to this as crossover within interval predicates. Crossover between interval predicates is also possible, where crossover points can occur only between interval predicates. As this type of crossover constructs offspring by exchanging entire intervals and cannot affect the composition of individual intervals, we do not consider it further in the present work. To avoid confusion and aid precision, we adopt the following notation:
Christopher Stone and Larry Bull Table 1. Phenotype regions and their structural forms p i Region [p min, q i) p Region 4 [p min, q max) p Region 1 [p i, q i) p Region 3 [p i, q max) p q i Table. Phenotype proportions for crossover within interval predicates with k = 8 Region 1 3 4 Form [p i, q i) p [p min, q i) p [p i, q max) p [p min, q max) p Parent proportion 0.5 0.5 0.5 0.5 CSR niche 0.5 0.5 0.5 0.75 CSR panmictic 0.48 0.5 0.5 0.5 OBR niche 0.984 0.008 0.008 0 OBR panmictic 0.984 0.008 0.008 0 UBR niche 0.984 0.008 0.008 0 UBR panmictic 0.984 0.008 0.008 0 1. Intervals in phenotype space are tagged with the subscript p, e.g., [0,1) p. Intervals in genotype space are tagged with the subscript g, e.g., [0,n] g 3. Tuples are distinguished from intervals by the absence of a subscript. Analysis Previous work [4] studied the effects of crossover with respect to the type of interval (classified by region) generated by the crossover operator. Regions are partitions of the phenotype space corresponding to the four different structural forms of interval predicate shown in Table 1 where p min < p i q max p min q i < q max The effects of crossover were studied for XCS [5] with a niche Genetic Algorithm (GA) [1] only. Other researchers use an interval representation with a panmictic GA [3]. Table extends the results presented in [4] to show the effects of crossover with Centre-Spread Representation, Ordered Bound Representation and Unordered Bound Representation using both niche and panmictic GAs. For consistency with earlier work, we use a binary encoding of length k = 8 bits for real numbers. From the table, it is clear that the distribution of intervals amongst regions depends on the representation in use, but is not materially affected by whether a niche GA or a panmictic GA is used. Analysing the intervals resulting from crossover by region is useful, but it does not tell the whole story of the effects of crossover. It is possible for region 1, and 3 intervals to be generated by crossover with varying widths and to understand the effects of crossover further, we must study the widths of offspring intervals.
A Note on Crossover with Interval Representations 3 Using CSR, an interval is represented as a centre and spread tuple (c i,s i ) where c i encodes the centre (or position) of the interval and s i encodes its spread (or width). During crossover within predicates, centre and spread alleles are exchanged: [c 1 i,s 1 i] g [c i,s i] g [c 1 i,s i] g,[c i,s 1 i ] g This process can be viewed one of two ways either that the interval positions are invariant from parent to offspring with the interval widths changing, or that the interval widths are invariant from parent to offspring with the position of the intervals changing. In practice, both of these transformations occur simultaneously. From the perspective of interval width, crossover has the effect of specializing or generalizing intervals by altering their widths (but not positions), whereas from the perspective of interval position, crossover alters the position, but maintains the widths of intervals. The epistasis between centre and spread alleles means that the degree of difference in position and width between parents and offspring depends on the amount of variance between the centres and spreads of the respective parents. Where there is a large difference between parental centres and/or spreads, offspring will have little in common with their parents, so large jumps in interval position or width are possible under crossover. A metric that can be readily observed experimentally is mean interval width. This may be measured for the entire population, for the match set or, as we use it here, for the two parental intervals undergoing crossover and for the resulting two offspring. For CSR, all pairs of offspring produced with CSR preserve the mean interval width of the parents. With OBR, an interval is represented by the tuple (l i,u i ) where l i is the lower bound and u i is the upper bound of the interval. Crossover within predicates swaps the two alleles representing an interval: (l 1 i,u 1 i) (l i,u i) (l 1 i,u i) (l i,u 1 i ) The mean width of the parental intervals is (u 1 i l1 i ) + (u i l i ) The mean width of the offspring intervals for OBR is (u i l1 i ) + (u1 i l i ) = (u1 i l1 i ) + (u i l i ) So, the mean width of the two offspring generated by OBR crossover within predicates is the same as that of the two parental intervals. For UBR, the situation is more complex. Because, in general, there are two possible genotypes, [l i,u i ] g and [u i,l i ] g for a particular phenotype, intervals produced by crossover with UBR depend on the ordering of the parental genotypes.
4 Christopher Stone and Larry Bull Table 3. Parental and offspring genotypes for UBR. All four parental genotypes express to the same pair of phenotypes. Offspring genotypes express to two distinct phenotype pairs, A and B Parent 1 Parent Offspring 1 Offspring 1 [li 1, u 1 i] g [li, u i] g [li 1, u i] g [li, u 1 i] g A [li 1, u 1 i] g [u i,li ] g [li 1, li ] g [u i,u 1 i] g B 3 [u 1 i,li 1 ] g [li, u i] g [u 1 i,u i] g [li, li 1 ] g B 4 [u 1 i,li 1 ] g [u i,li ] g [u 1 i,li ] g [u i,li 1 ] g A Table 3 shows the possible parental genotype orderings for a crossing of parental phenotypes [l 1 i,u1 i ) p [l i,u i ) p and the resulting offspring genotypes. The lack of an ordering restriction on genotypes with UBR means that offspring genotypes 1 and 4 in Table 3 are equivalent, as are offspring genotypes and 3, so there are exactly two forms of offspring phenotype, one where parental lower and upper bounds are paired and the other where a parental lower (upper) bound is paired with another parental lower (upper) bound. Note that crossover with OBR can only produce genotypes of type A. The only ordering restriction imposed at the phenotype level is l 1 i u 1 i l i u i We cannot assume anything about the relative values of the bounds of one interval compared to the interval with which it is paired. To proceed, we must consider the possible basic configurations of pairs of intervals. Fig. 1 shows the three possible configurations of parental interval and the offspring configurations that result after crossover if the relative orderings of parental interval bounds are taken into account. Other symmetries of the basic configurations are possible. Parental configurations 1 and occur with both niche and panmictic GAs, as the parental intervals share at least one point in common, namely the environmental variable x i. Configuration 3 can only occur with a panmictic GA as the parental intervals do not overlap. Offspring configurations in column A represent offspring intervals with phenotypes of the form [l 1 i,u i ) p or [l i,u1 i ) p, as in Table 3. As already explained, these can occur with both OBR and UBR. The additional offspring configurations resulting from the use of unordered tuples with UBR appear in offspring column B. These configurations cannot occur with OBR as they have the form [l 1 i,l i ) p or [u 1 i,u i ) p. Fig. shows the possible transitions from the three parental configurations to offspring configurations, where offspring are also classified according to their configuration, 1-3. This shows that only OBR with a niche GA and UBR with a panmictic GA are symmetrical and allow all possible transitions between parental and offspring configurations. It is straightforward to compute the gain or loss of mean interval width from parents to offspring for each of the configurations that appear in Fig. 1. These are shown in Table 4, which, for completeness, also shows this information for
A Note on Crossover with Interval Representations 5 Parent Intervals A. OBR & UBR Offspring B. UBR Offspring 1. Niche & Panmictic. Niche & Panmictic 3. Panmictic Fig. 1. Crossover configurations. Parental configurations are rows in the diagram and offspring configurations are columns. Each entry shows two intervals (shaded) in a single dimension of the solution space. The dotted line represents the environmental variable x i 1 1 1 1 3 3 3 3 OBR Niche UBR Niche OBR Panmictic UBR Panmictic Fig.. Transitions possible from parental to offspring configurations. Numbered states correspond to the configurations shown in Fig. 1 CSR. This makes it apparent that certain configurations of parental and offspring intervals produce a gain or loss of mean interval width during crossover. We can see that crossover with OBR and a niche GA is neutral with respect to mean interval width, but with a panmictic GA it possible to produce offspring intervals that are on average, wider than the parents. Crossover with UBR and a niche GA can produce intervals that are narrower, on average, than the parents, whilst with a panmictic GA it is also possible to produce intervals that are wider, on average, than the parents. These results can also be seen intuitively in Fig. 1 and Fig.. Table 4. Gain or loss of mean interval width from parent to offspring. A gain in mean interval width is shown as positive and a loss as negative Parent CSR A. OBR & UBR B. UBR 1. Niche & panmictic 0 0 (u 1 i li ). Niche & panmictic 0 0 (u i li ) 3. Panmictic 0 +(li u 1 i) +(li u 1 i)
6 Christopher Stone and Larry Bull Table 5. Mean interval width for parent and offspring over all possible crossings Representation and GA type Mean parental width Mean offspring width CSR niche 0.699 0.699 CSR panmictic 0.665 0.665 OBR niche 0.400 0.400 OBR panmictic 0.333 0.401 UBR niche 0.401 0.301 UBR panmictic 0.335 0.335 To investigate these effects further, we enumerated all possible combinations of interval for two parents. For niche GA results, we enumerated only those parental intervals that have at least one point in common with each other. For consistency with the results presented in Table and [4], we used an encoding of length k = 8. For each combination, we noted the mean interval width of the parents and offspring and averaged these over all combinations of parental interval to provide an indication of the overall effects of crossover with different representations and types of GA. Results are shown in Table 5. Table 5 shows that over all possible intervals crossover with CSR is neutral with respect to mean interval width for both niche and panmictic GAs. We can also see from these results that the mean interval width for CSR is quite wide (around 0.7), in contrast to OBR and UBR which have a mean interval width of around 0.33 for a niche GA and 0.4 for a panmictic GA. As expected from Fig. and Table 4, crossover with OBR and a niche GA causes no change to the mean interval width of parents and offspring, while crossover with OBR and a panmictic GA tends to generate wider intervals. Conversely, crossover with UBR and a niche GA tends to generate narrower intervals, whereas with UBR and a panmictic GA, no change occurs to mean interval width. Because UBR with a panmictic GA can generate offspring that are both wider and narrower, on average, than those of the parents, it is more difficult to predict the overall direction of the effect. Although this is neutral over all possible crossings, this may not be true for intervals drawn from arbitrary populations. 3 Conclusions There are differences in the characteristics of intervals produced by crossover with the three representations considered for real numbers in Learning Classifier Systems. CSR preserves interval position or width from parents to offspring, which, in general, causes the endpoints of parental intervals to be lost under crossover. OBR and UBR preserve the endpoints of parental intervals, but shuffle them into new combinations to produce offspring intervals. UBR offers additional possibilities for endpoint shuffling over OBR, due to the lack of ordering of bounds in this representation.
A Note on Crossover with Interval Representations 7 The GA in a Learning Classifier System with an interval representation is searching for useful hyper-rectangular decision surfaces. Crossover with OBR and UBR appear to facilitate this search by preserving elements of the boundaries of hyper-rectangles represented by high fitness classifiers. In contrast, crossover with CSR constructs offspring by generalizing, specializing or shifting the position of high fitness hyper-rectangles. However, there would seem to be a degree of randomness involved in this process, due to epistatic interactions between centre and spread alleles that do not exist between alleles in OBR and UBR representations. Thus, the search mechanisms of CSR and OBR/UBR work in different ways, both of which are able to produce solutions to problems in continuousvalued environments [4, 6, 7]. References 1. Holland, J. H. (1975). Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor, MI. Republished by MIT Press, 199.. Holland, J. H. (1986). Escaping brittleness: the possibilities of general-purpose learning algorithms applied to parallel rule-based systems. In R. S. Michalski, J. G. Carbonell & T. M. Mitchell (eds.), Machine Learning, an Artificial Intelligence Approach. Volume II. Los Altos, California: Morgan Kauffmann, pages 593 63. 3. Hurst, J., Bull, L. & Melhuish, C. (00). TCS Learning Classifier System controller on a real robot. In J. Merelo, P. Adamidis, H-G. Beyer, J-L. Fernandez-Villacanas & H-P. Schwefel (eds.), Parallel Problem Solving from Nature - PPSN VII, Berlin: Springer, pages 588 600. 4. Stone, C. & Bull, L. (003) For real! XCS with continuous-valued inputs. To appear in Evolutionary Computation. 5. Wilson, S. W. (1995). Classifier fitness based on accuracy. Evolutionary Computation, 3():149 175. 6. Wilson, S. W. (000). Get real! XCS with continuous-valued inputs. In P. L. Lanzi, W. Stolzmann and S. W. Wilson (eds.), Learning Classifier Systems. From Foundations to Applications, Lecture Notes in Artificial Intelligence (LNAI-1813), Berlin: Springer, pages 09 19. 7. Wilson, S. W. (001). Mining oblique data with XCS. In P. L. Lanzi, W. Stolzmann and S. W. Wilson (eds.), Advances in Learning Classifier Systems. Proceedings of the Third International Workshop (IWLCS-000), Lecture Notes in Artificial Intelligence (LNAI-1996), Berlin: Springer, pages 158 174.