Learning Geometric Concepts with an Evolutionary Algorithm. Andreas Birk. Universitat des Saarlandes, c/o Lehrstuhl Prof. W.J.

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1 Learnin Geometric Concepts with an Evolutionary Alorithm Andreas Birk Universitat des Saarlandes, c/o Lehrstuhl Prof. W.J. Paul Postfach , Saarbrucken, Germany Abstract We present a system that is able to learn descriptions of pictures with an evolutionary alorithm approach. The descriptions are prorams in a turtle-raphics lanuae and the described pictures are scenes from an environment with a robot-arm actin in a blocks-world. A measure of similarity of pictures is presented which can be computed fast and supplies radient information with reard to translation, rotation and expansion of objects. The alorithm is very qualied for the classication of lare sets of pictures as objects which, once reconized, are re-found quickly. Even if they are in dierent shapes and orientations or in lare composed scenes. The approach diers from enetic prorammin as three simple problem specic operators are used instead of crossover. 1 Introduction An important property of intellience is the ability to learn. Therefore, it is interestin to build and investiate articial learnin systems. At the Universitat des Saarlandes systems are developed, that are able to build autonomously theories as short descriptions of lare amounts of perceived data [1, 11, 2, 3]. One result is a system, that learns from scratch to control a real world robot-arm in a blocks-world environment. The controlmechanism learned is a kind of classier-system i.e., in response to a classication of the camera-input, an appropriate action of the robot-arm is chosen and executed [5, 4]. The environment of the system consists of a black robot-arm with a red trianular ripper, colored buildin blocks on a black backround and a camera which looks perpendicularly at the workin-area (ure 1). Pictures from the camera are therefore composed of simple eometric objects. Resolutions of the pictures rane from to pixels with 8 colors. The task of the evolutionary alorithm presented in this paper is to nd prorams in a turtle-raphics lanuae that describe the pictures from the camera. Because a descriptive proram is in eneral immensely shorter as the described picture, this is a data compression. The remainder of the paper is oranized as follows. In section two evolutionary alorithms are dened in a eneral way. In section three a measure of the similarity of pictures is presented and discussed. The particular alorithm is introduced in section four. In section ve results are iven. Section six concludes the paper and presents further work. 2 Evolutionary Alorithms Evolutionary alorithms can be viewed as the most eneral term for all forms of heuristic search techniques, that imitate the principle of evolution in nature i.e., they work with a set of competin potential solutions of the problem which evolve accordin to rules of selection and transformation. More precisely, an evolutionary alorithm operatin on a problem space P proceeds in iterations called enerations. In each eneration i 2 IN a set S i P of xed cardinality s, called population, is enerated. The members of a population are called individuals. The tness-function f fit : P! IR assins a value called tness to each individual. The rst population S 1 is enerated in a random fashion. Population S i is enerated from population S i?1 usin selection and transformation operators. Given an individual and its tness, a selection operator Sel : P R I! f0; 1 decides if an individual from population S i?1 is used as input of a transformation operator. A transformation operator T r : 2 P! P creates a new element of S i from an arbitrary number of elements of S i?1. As the transformation operators are focusin attention on hih tness, it is expected that durin the search process increasinly better solutions are found. Very common transformation operators are reproduction (passin an individual from S i?1 to S i unaltered), mutation (an individual is chaned in a random fashion) and crossover (subparts of two individuals are swapped). We prefer above described view of an evolutionary alorithm as process of selection and transformation with arbitrary operators to the more common view

2 Camerapicture Robotripper (red) Robotarm (black) Buildinblock (yellow) Backround (black) Fiure 1: Block-world with robot-arm based on the usae of special operators for followin reason. The intension of pushin some operators as e.. crossover to a kind of must is to et one alorithm that is universal i.e., independent of the tasks to be solved. Thouh this intension is nice, it seems not to be practical. This is indicated for example by the many dierent versions of crossover used in dierent applications. There are four commonly used classes of evolutionary alorithms: enetic alorithms [7, 8], evolutionary prorammin [6], evolutionary strateies [13, 12] and enetic prorammin [10, 9]. As the domain of the problem described in this paper consists of prorams, enetic prorammin would be the standard way to solve the task with an evolutionary alorithm. But as some experiments with enetic prorammin had very poor results, we decided to develop a new alorithm. So, we invented operators that are more problem specic than crossover, the main operator in enetic prorammin. As mentioned above, we believe that this viewpoint to use problem-specic operators rather then universal ones can be fruitful for most applications. The main operator used in our alorithm is hillclimbin on constants of the prorams. It is based on our measure of similarity of pictures, which supplies radient information with reard to translation, rotation and expansion of objects. In addition to hill-climbin an operator for concatenatin two prorams and an operator for splittin one proram into two is used. 3 Similarity of Pictures The concrete task of learnin picture descriptions can be stated as follows. Given a pixel array a, the current camera output, nd a (short) proram p that describes a i.e., the output a p of p is equal to a. To solve this problem with an evolutionary alorithm, a measure for the similarity of pictures a p and a has to be provided. The similarity is the main factor in determinin the tness of proram p. The task of measurin the similarity of pictures is nontrivial. For instance the naive trial of countin the number of pixels with the same color leads to a function which is in most cases { when objects are not overlappin { meaninless. A useful measure is described in [2], which is based on the idea that a pixel in picture a is attracted from a pixel of same color c in picture a 0 by a \force" accordin to a potential-eld (as known from physics). Unfortunately, the function depends on some parameters, which must be carefully set, and cannot be computed fast. Therefore, our picture-distance-function D was developed as a measure of the dierence between two pixel arrays a and a 0 : 2

3 d(a; a 0 ; c) = where D(a; a 0 ) = X c2c d(a; a 0 ; c) + d(a 0 ; a; c) Pa[p 1]=c minfmd(p 1 ; p 2 )ja 0 [p 2 ] = c # c (a) C denotes the set of colors of constant size, a[p] denotes the color c of pixel array a at position p = (x; y), md(p 1 ; p 2 ) = jx 1? x 2 j + jy 1? y 2 j is the Manhattandistance between p 1 and p 2, # c (a) = #fp 1 ja[p 1 ] = c is the number of pixels in a with color c. So, the picture-distance-function is a sum over all colors of the averae Manhattan-distance of the pixels with color c in picture a to the nearest pixel with color c in picture a 0 ( d(a; a 0 ; c) ) and the averae distance vice versa ( d(a; a 0 ; c) ). Note, that the picture-distance-function is symmetrical, but d(a; a 0 ; c) miht be not equal to d(a; a 0 ; c). Consider picture a bein a part of picture a 0 in reard to color c i.e., 8i : a[i] = c =) a 0 [i] = c and 9j : a 0 [j] = c ^ a[i] 6= c. Then d(a; a 0 ; c) = 0 and d(a; a 0 ; c) > 0. The picture-distance-function has two important properties: It reects the \natural feelin" of the similarity of pictures as it has the expected radients accordin to translation, rotation and expansion of objects. More precisely, assume a picture a showin an object and a picture a 0 showin the same object translated, rotated or expanded. A decrease in the euclidian distance, the rotation anle or expansion factor of the object in a 0 with reard to its representation in a leads frequently to a decrease of D(a; a 0 ). It can be computed in a time that is linear in the number of pixels. The less observable part of the linear time implementation of the picture-distance-function is the computation of the numerator in the d(a; a 0 ; c)-equation. It is based on a so-called distance-map d-map c for a color c. The distance-map is an array of the Manhattan-distances to the nearest point with color c in picture a 0 for all positions p 1 = (x 1 ; y 1 ): d-map c [x 1 ][y 1 ] = minfmd(p 1 ; p 2 )ja 0 [p 2 ] = c The distance-map d-map c for a color c is used as lookup-table for the computation of the sum over all pixels in a with color c. Fiure 2 shows an example of a distance-map. The alorithm shown in ure 3 computes a distancemap for a picture of size nn in time linear in the number of pixel. Fiure 4 shows the workin principle of the alorithm, which is based on relaxation. The alorithm consists of three parts with two basic loops in each case. The rst part is an initialization of d-map c : the values of positions with color c are set to Zero, the values of others are set to Innity. The next two parts of the alorithm are relaxation steps: every position is visited and updated; one time oin from \upper left" corner to \lower riht" and next time vice versa. The updatin is done as follow: the value of a visited position is set to the Minimum of its current value and the value of one of its neihbors plus One. 4 The particular Alorithm 4.1 The Fitness-Function A crucial point in desinin an evolutionary alorithm is the choice of the tness function. As an evolutionary alorithm may iterate many enerations on a possibly lare population, the tness function has to be calculated very often. Therefore, it should be quickly computable. The picture-distance-function, which is the main element of the soon dened tness-function, has this property as explained in the previous section. However, it is even possible to speed it up. Let a denote the taret picture which is to be described the current camera output and a 0 the output picture of a potential descriptive proram. The idea of the speed-up is based on the fact, that a remains xed durin the run of the evolutionary alorithm. The modied picture-distance-function D 0 is dened as D 0 (a; a 0 ) = X c d(a; a 0 ; c) + maxf0; # c(a 0 )? # c (a) # c (a 0 ) So the computation of d-map c is only necessary for the taret picture a. This must only be done once (for every color c) before the start of the evolutionary alorithm. The run-time of the computation of D 0 (a; a 0 ) is now linear in the number of pixels of a 0 with color c, which is in eneral much smaller than the number of all pixels. The speedup is achieved by a decrease in the power of exploitin useful radient-information. Various experiments revealed that for this particular task a positive trade-o is ained. Finally, the tness-function f fit : L T P ic! IR is dened as f fit (pro; a) =? (D 0 (a; a pro ; c) + len(pro)) 3

4 = pixel with color c d-map c Fiure 2: A distance-map d-map c for y = 0 to n-1 f for x = 0 to n-1 f if a 0 ((x,y)) = c d-map c [x][y] = 0 else d-map c [x][y] = 1 for y = 0 to n-1 f for x = 0 to n-1 f h = Minfd-map c [x-1][y] + 1, d-map c [x][y-1] + 1 d-map c [x][y] = Minfd-map c [x][y], h for y = n-1 to 0 step -1 f for x = n-1 to 0 step -1 f h = Minfd-map c [x+1][y] + 1, d-map c [x][y+1]+1 d-map c [x][y] = Minfd-map c [x][y], h Fiure 3: The alorithm for computin d-map c 4

5 Init Relaxation Step 1 = visited position Relaxation Step 2 = neihbor Fiure 4: The workin principle for computin d-map c where L T denotes the space of prorams, P ic the space of pixel-arrays, pro a proram with output a pro, a the taret picture and len(pro) the number of commands in pro. Note that for keepin the convention, that hiher tness means better solution, f fit is dened to be neative. 4.2 The Turtle-Graphics Lanuae The turtle-raphics lanuae L T which is used to construct the prorams consists of the followin simple commands: Move x y c : moves the turtle to point (x; y) and assins color c to it. Draw l c : draws a line in color c from the current position of the turtle with anle and lenth l and moves the turtle to the end of the line. Line l c : Similar to Draw, but the position of the turtle remains unaltered. For i = k 1 to k 2 step s f command-block : is used as for-loop command. The loop-variable can only be used in the command-block of the loop. End : terminates a proram. Fiure 5 shows an example L T -proram which produces a red trianle as output. 4.3 The concrete Operators As selection operator we use tournament selection i.e., two individuals are chosen randomly from population S i?1 and the one with hiher tness is selected for transformation. This is repeated as lon as individuals are needed by transformation operators to enerate population S i. As transformation operators we use reproduction and the new developed operators hill-climbin on constants, conc and split. Hill-climbin on constants : L T! L T Randomly chooses a constant in a proram and performs a hill-climbin-step on it, accordin to minimization of the picture-distance-function. In doin so, the probability of steppin width W is inversely proportional to W. Conc : L T L T! L T Takes two prorams and concatenates them to one. Split : L T! L T L T Splits a proram into two new ones at a randomly chosen place. Command-blocks of a for-loop cannot be split. 5 Results The system is set up as follows. The population-size s is set to 50. The rst population is enerated by randomly eneratin prorams with lenths between 1 and 10 commands. The followin populations are enerated usin the selection-operator and the transformation-operators. In the most successful experiments we randomly chose the transformation-operators with followin probabilities: hill-climbin 0:6, conc 0:2, split 0:1 and reproduction 0:1. Due to the more frequent use of conc than split the number of individuals enerated by transformation is less then #S i?1, hence S i is lled up with randomly enerated prorams. In experiments on a 40MHz SPARCstation 10/51 usin pictures from simulations and the real world as input, various eometric ures were found. Usin e.. 5

6 move 0 1 red for i = 11 to 1 step - 2 { O line 30 i red } end O draw 75 1 red : move : line : draw Fiure 5: A proram producin a red trianle Fiure 6: A partially hidden block 6

7 pixel-pictures, a quadranle was found on the averae in 7 hours and a trianle in 23 hours, both in arbitrary form and orientation. A very interestin feature of the alorithm is, that as soon as one particular eometric ure is found, any eometric ure of this kind is reconized very fast. If e.. the input picture contains a quadranle and an individual in the population is already a quadranle-proram (with wron constants), then adaption to the input picture is done in 3 minutes on averae. The same holds true for trianles which are adapted in 5 minutes on averae 1 and any other ure. This property can be credited to the expressive radients of the picture-distance-function and the hill-climbin-operator. This view is supported by the fact that turnin down the probability of the hillclimbin-operator leads to a decrease in this ability. Another very interestin feature of the alorithm is the capability of fast reconition of lare scenes composed of \known" objects i.e., objects for which the population already contains prorams. A scene consistin of a trianle and two quadranles is reconized on averae in 14 minutes (if trianle and quadranle are known), which is only a few minutes loner than it takes to adapt the constants for the three objects. This property is the eect of the introduction of the conc-operator. Both features are very useful for the task of learnin classications of scenes from a blocks-world. As described in section 1 the alorithm is used in a system that controls a robot-arm. In this task many often only slihtly dierent snapshots are taken, to which the picture-description-alorithm is repeatedly applied. By transferrin ood prorams i.e. prorams for known eometric concepts from the previous run to the rst population of the next run, an immense speedup is ained. 6 Conclusion and further Work We presented a measure of similarity between two pictures which miht be useful for a broad class of problems, namely the learnin of classication of vision data. The measure is computable fast and supplies radient information with reard to translation, rotation and expansion of objects. The transformation operators hill-climbin, split and conc used in the presented evolutionary alorithm are non-standard. It is not clear whether they are useful for other problem classes. This can only be tested by appropriate experiments. However, even if these concepts fail on other problem classes, their introduction is still advantaeous as classication of vision data is a frequent (real-world-)problem. Therefore, a fast problemdependent-desin for this task seems to be justied. 1 These times are stronly dependent on the kind of dierence between the related objects i.e., how many and which type of constants have to be xed. An interestin subject for future research is the development of hierarchical operators, in addition to the analysis of further applications of the present operators to other problem classes. Operators supportin (parameterized) subroutine calls would enable the system to dynamically adjust inductive bias in an ecient way. Some recent experiments indicate, that the described approach is very qualied for reconition of partially hidden objects. Fiure 6 shows an example: a buildinblock (rectanle) is partially covered by a robot-ripper (trianle). The evolutionary alorithm describes this scene by a proram which rst draws a rectanle and then draws a trianle paintin over a part of the rectanle. This information can be used to identify the buildin-block for automatic imae understandin. References [1] Wolfan J. Paul Andreas Birk. Schemas and enetic prorammin. In Conference on Interation of Elementary Functions into Complex Behavior, [2] P. Bermann, J. Keller, and W.J. Paul. A selforanizin system for imae reconition. In Proc. of the IASTED Intl. Symp. Machine Learnin and Neural Networks, paes 33{36. ACTA Press, New York, [3] P. Bermann, W.J. Paul, and L. Thiele. An information theoretic approach to computer vision. In Proc. of Dynamical Networks Workshop. Akademie- Verla, Berlin, [4] Andreas Birk. Stimulus Response Lernen, ein neues Machine Learnin Paradima. PhD thesis, Universitat des Saarlandes, Saarbrucken, [5] Andreas Birk. Robotic control via stimulus response learnin. In Proc. of The Sixth International Symposium on Robotics and Manufacturin. The MIT Press, Cambride, [6] L.J. Foel, A.J. Owens, and M.J. Walsh. Articial Intellience throuh Simulated Evolution. Wiley, New York, [7] David Goldber. Genetic Alorithms in Search Optimization and Machine Learnin. Addison-Wesley, Readin, [8] John H. Holland. Adaption in Natural and Articial Systems. The University of Michian Press, Ann Arbor, [9] John R. Koza. Genetic prorammin. The MIT Press, Cambride, [10] John R. Koza. Genetic prorammin II. The MIT Press, Cambride,

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