Semi-continuous challenge

Transcription

1 Maurice Clerc Last upate: Why? Semi-continuous challenge The aim is to improve my parameter free (aaptive) particle swarm optimizer Tribes [CLE 03]. In orer to o that, I have also written a parametric one, calle OEP (means simply PSO in French), with a lot of options. By analyzing what are the best options/parameters for several problems, my hope is I coul fin better aaptations rules for Tribes. However, there are now so many PSO variants that I simply can't incorporate all of them in OEP. I have to choose the most promising ones. So the aim of this "challenge" is to ask people if they are aware of a PSO version that gives better results, an, if so, if they can eplain how they choose the parameters, if any. Tribes (an OEP) can easily cope with hybri problems (with some iscrete imensions), but it is not the case for all PSO versions. So the problems efine below are all continuous (or semi-continuous for the first one). In such a case, almost any kin of search space can be transforme into a -cubical one, or, simpler, into a set of such - cubes (you may see the "For amatheurs only" section). I on't say it is always easy, I just say it is possible. That is why, in orer to compare algorithms, you just nee to use interval constraints... Si continuous or semi-continuous problems I have carefully chosen these si problems so that it shoul be quite ifficult to solve them all efficiently with a given parametric optimisation metho, that is to say by keeping the same parameter set. They are place in ascening orer of theoretical ifficulty. For each case, the minimal value is zero, an the objective is to fin a function value smaller than 0-5. I also give the C source coe in the appeni.. Name Formula Search Objective ifficulty space Tripo p( )( + p( ) [-00,00] 0± p( )( p( ) Alpine p with ( p( ) ( u) = si u 5 sin( ) 0 = 0 si u < 0 + 0, [-0,0] 0 0± 0-5 Parabola 30 [-0,0] 30 0± Griewank 30 [-300,300] 335 ( 00) 30 0± cos 4000 Rosenbrock 30 f = 370 ( ) ( ) ( [-0,0] 30 0± 0-5 Ackley , 0e cos( π ) e + 0 [-30,30] 30 0± 0-5

2 .. Representations an comments on t forget that these figures give just a faint iea of the problem when the imension is in fact 0 or 30. z y Figure. Tripo. The minimum 0 is on (0, -50). Theoretically easy, this problem is in fact ifficult for a lot of algorithms that are trappe in the two local minima. Note the function is not a continuous one. It has been publishe first in [GAC 0] Figure. Alpine. A lot of local an global minima. It is not really symmetrical. This problem is nevertheless quite easy, an can be seen as a kin of pons asinorum for optimisation algorithms. The first version (this one is a bit ifferent, so it shoul be name Alpine ), has been publishe in [CLE 99]

3 Figure 3. Parabola. Well known. Just one minimum 0 in (0, 0). Sometimes calle Sphere, noboy knows why, maybe because of its equation, but, fortunately, cricket balls on't really have this shape. It is very easy to ajust its ifficulty just by moifying the imension. With such a function, algorithms like graient methos usually work very well, so it is by itself a challenge for stochastic methos like PSO z y Figure 4. Griewank. Well known an more ifficult. The minimum 0 is on (00, 00), an almost not ifferent from the numerous local minima aroun. On the one han, of course, it increases the ifficulty, but, on the other han, as there are so many small local minima, it is still quite easy to escape from them. So, increasing the imension not necessarily increases the ifficulty

4 Figure 5. Rosenbrock. Well known. eceptively flat, this function is here shown on [-0 0]. The global minimum in on (, ), an quite ifficult to fin as soon as the imension is high. z y Figure 6. Rosenbrock again, but on [0 ][0 ], so that you can see the minimum.

5 Figure 7. Ackley. Well known. Looks a bit like Alpine, but in fact more ifficult. The "attraction basin" of the global minimum is quite narrow, so ranom moves can't easily fin it. The challenge itself To initiate this challenge, I give in the table some results obtaine by OEP 5. The parameters are the followings: N, number of eplorer particles M, number of memory particles (for some ifficult problems, it is better to have M>N), or memory swarm size K, the mean number of information links from a memory to eplorers (note that in all cases each eplorer has just one link towars the memory swarm) info_option, which efines the topology of the information graph (fie or not, ranom or not, etc.) mouv_option, which efines the kin of proimity probabilistic istribution (-parallelepi as in classical PSO, -sphere, -gauss, istorte variants, etc.) For some mouv_option values (typically for the classical PSO), there is an aitional parameter ϕ. Note that for the moment the program oes not incorporate any kin of eplicit clustering/niching. It is probably a promising approach, but I am still not sure. For each function, I ran 00 times the program with a given "search effort" T, that is to say a given maimum number of objective function evaluations, an the failure rate is then compute. Of course, it is just an estimation. However, it is easy to prove there is 95 % of chance that it is correct by less than 5 %. It is usually enough to ecie whether a result is really better than another one or not. If the intrinsic ifficulty is δ you can then erive that the ifficulty for a given T is about δ-ln(t), as soon as T is big enough. So, T has been chosen to keep the same ifficulty orer for the si problems, but also big enough so that it is inee possible to solve most of them with OEP. First, I trie to carefully tune the parameters ifferently for each function, in orer to know what is possible to o with the program. It appears that just one problem (Rosenbrock) can absolutely not be solve (in less than evaluations). After that, I choose a given parameter set, an use it for all functions. For the time being, one of the best one I have foun is the following: N=M=45, K=3 info_option: ranom choice of K eplorers for each memory, if there has been no global improvement after the iteration mouv_option: if no global improvement after the previous iteration, use istorte positive spherical -sectors, else use the pivot metho (aapte from [SER 97]). ϕ=.7

6 As epecte, I on't have been able to fin a parameter set that give goo result for all the si functions, so I choose to accept worse results for easy problems, in orer to still have quite goo ones for ifficult problems. The opposite is possible, but, interestingly, the mean failure rate seems to be then a bit higher. Problem Search effort Failure rate when Failure rate with the (ma. number of parameters are tune for same parameter set for evaluations) an each problem all problems ifficulty Tripo () 0 % 9 % Alpine () 0 % 5 % Parabola (64) 0 % 0 % Griewank (35) 0 % 5 % Rosenbrock (360) 00 % (8,) 00 % (9,0) Ackley (460) 0 % 0 % Mean failure rate 5 % Table. Failure rates in ifferent cases, for 00 runs, when using OEP5. For Rosenbrock, as the failure rate is 00%, I have ae in parenthesis the mean of the 00 best values foun. When using just one parameter set, in orer to obtain the best mean failure rate, I have to accept a quite big one for the easiest problems 3. For amatheurs only 3.3. Anything is a cube, or Hanling constraints by homeomorphism An optimisation problem is usually given as follows, when the search space is a continuous one: minimise ( ) f, with [ ], { }, min,, ma,..., After that, it is often sai something like "Now, let's a some constraints g i ( ) 0 an ( ) 0 be sai about such a representation. The most important ones are: - the initial problem is alreay a constraine one, for we have [, ],min,ma h j,min =,ma ". A lot of remarks can - all inequality constraints can be transforme into equalities, for we have ( ) ( ) ( ) 0 g i g + g = i i. It is particularly interesting when they are just inicative constraints, to satisfy "as well as possible", an not imperative ones. The problem can then be seen as a multiobjective one. - the initial problem can be rewritten: minimise φ ( y), with y [ 0, ] just by the bijective continuous transformation (homeomorphism) y,min =, an by efining φ by,ma,min φ ( y ) = f ( ). So the initial search space, which was a -parallelepi, is now the unit -cube ( ) C. The last point can be generalise. First, note that the constraints efine a subspace of the initial search space H. Let us call it "restricte search space" 0, H. Secon, there is a strange theorem saying that there are the same "number" of points in [ ] r than in any other real interval (of course, it is not true for iscrete finite search spaces). More generally, there is a theorem saying that it is possible to map H r to C ( ) by using a homeomorphism, if the topological genus of H r is zero. In practice, this last conition is not really important, for, as long as H r is not too mathematical monstrous, you can always cut it into a C. finite number of parts whose genus is inee 0, an then map each of them to ( ) I give here just two eamples, so that you get the iea. Eample. (isc/4)=square minimise f (, ), search space = [ 0,] H, constraint +.

7 So H is the positive quarter of the unit isc centre in (,0) r 0. mapping y y µ (, ) H ( y, y ) ( ) = r + = atan π π π φ = The function φ is then efine by ( y, y ) C( ), ( y, y ) f y cos y, y sin y an the equivalent problem is: minimise φ ( y, y ), search space ( ) Eample. triangle = square C. minimise f (, ), restricte search space efine by H is then the triangle {(,0), (,0),( 0,) } r mapping y y µ (, ) H ( y, y ) ( ) r = + = + The function φ is then efine by ( y, y ) C( ), ( y, y ) = f ( y( y ), y y ) minimise φ ( y, y ), search space C ( ). φ. Again, the equivalent problem is: Remarks As you have certainly note, Eample an Eample are in fact the same (use the intermeiate variables z = an z = to transform the first one into the secon one). For most real problems, constraints are (or can be transforme into) linear ones. The restricte search space is then a polyheron. Any polyheron can be cut into -triangles (real triangles if =, tetraherons if =3), an any -triangle can be mappe to the unit -cube. Theoretically, you coul then unify these unit cubes by mapping them to a single one, but in practice, for optimisation, it is not necessary for you can look for the optimum successively insie each -cube. 3.. Some ifficulty level estimations The theoretical ifficulty is given by ln( σ ), where σ is the probability to fin a solution by choosing a point at ranom (uniform istribution) in the search space Tripo Let ε be the require accuracy. It is suppose smaller than, so that there is no nee to take local minima into account. The solution space it then a square whose surface is by ln ε ifficulty = = ln ε 00 ( 00) ln( ) ln( ) ε. As the whole search space is [ ] 00 00, the ifficulty level is given

8 for ε = 0 5, the value is then about Rosenbrock Here, it has just been statistically estimate. Note that the ifficulty is almost an increasing linear function of the imension, as shown in the table below. on't forget, though, it is a logarithm, so the real ifficulty is eponentially increasing. imension ifficulty Just for fun, it is also possible to perform an analytical estimation, by using the Taylor's formula. On the position ρ = (,...,), where the minimum is, first orer an secon orer partial erivatives are equal to 0. So, an estimation of the f ρ + h = h + 0. As we want that to fin a value smaller than ε, it give us function aroun this point is given by ( ) ( ( ) ) the ege of the -cube where are all the solution points, = ( + ( 0 ) ) So, in our eample, with ε = 0 5, = 30 is given by h ε., an the search space [ ] whose volume is 30 0, the theoretical ifficulty ifficulty = ln Using this metho, the value is necessarily smaller than the true one. So, we see that the statistical estimation 370 is quite acceptable. 3.. ifficulty epening on the search effort The theoretical ifficulty is of course ecreasing if you accept several ranom choices for the position. Let T be the number of these choices. As the success probability for T= is σ, the failure probability is ( σ ), an the probability to on't having foun a solution after T raws is ( σ ) T. So, the probability to have foun a solution is its complement to. Finally, by taking the logarithm, we obtain the theoretical ifficulty epening on the search effort ifficulty T ( ) ln( Tσ ) = ln( ) ln( T ) ( T ) ln ( σ ) = σ 4. Appeni 4.. Some C source coe // Tripo =.[0]; =.[]; if(<0) {f=fabs()+fabs(+50);} else { if(<0) f=+fabs(+50)+fabs(-50); else f=+fabs(-50)+fabs(-50);} // Parabola f=0; for(=0;<;++) f=f+.[]*.[]; // Alpine f=0; for( =0;<;++) { zz=.[]; f=f+fabs(zz*sin(zz)+0.*zz);}

9 // Griewank f=0; p=; for (=0;<;++) {=.[]-00; f=f+*; p=p*cos(/sqrt(+));} f=f/4000 -p +; // Rosenbrock f=0; for (=0;<-;++) {=-.[]; f=f+*; =.[]*.[]-.[+]; f=f+00**;} // Ackley E=ep(); two_pi=*acos(-); sum=0;sum=0; for (=0;<;++) {=.[]; sum=sum+*; sum=sum+cos(two_pi*);} f=(-0*ep(-0.*sqrt(sum/(ouble)))-ep(sum/(ouble))+0+e); 4.. Goo parameter sets for each function The following parameter sets are reffering to OEP v. 5. Ecept for Rosenbrock, they give a failure rate equal to zero. Tripo N=95, M=30, K=3, ϕ=.3 info_option=ranom, moifie if no improvement mouv_option=classical constricte PSO or N=40, M=40, K=3, ϕ=.08 info_option= circular, fie mouv_option=classical constricte PSO Alpine N=8, M=8, K=3, ϕ=. info_option=ranom, moifie if no improvement mouv_option=classical constricte PSO or N=8, M=8, K=3, ϕ=.09 info_option=circular, fie reorg=. Each memory checks its two neighbours. If they are both better, it choose one at ranom an take its position. mouv_option=classical constricte PSO Parabola N=8, M=8, K=3, ϕ=. info_option=ranom, moifie if no improvement mouv_option=classical constricte PSO or N=8, M=8, K=3, ϕ=.09 info_option=circular, fie reorg=. Each memory checks its two neighbours. If they are both better, it choose one at ranom an take its position. mouv_option=classical constricte PSO or N=45, M=45, K=3, ϕ=.7 info_option=ranom, moifie if no improvement mouv_option= if no improvement, use istorte positive spherical -sectors, else use the pivot metho Griewank N=95, M=95, K=3, ϕ=.7 info_option=ranom, moifie if no improvement mouv_option= if no improvement, use istorte positive spherical -sectors, else use the pivot metho

10 Rosenbrock N=45, M=45, K=5, ϕ=.07 info_option=ranom mouv_option=spherical istributions (equivalent volume metho). Instea of -parallelepis as in classical constricte PSO, it uses spheres that has the same volume. Ackley N=45, M=45, K=3, ϕ=.7 info_option=ranom, moifie if no improvement mouv_option= if no improvement, use istorte positive spherical -sectors, else use the pivot metho or N=45, M=45, K=3, ϕ=.08 info_option=ranom, moifie if no improvement mouv_option=classical constricte PSO 5. References [CLE 99] Clerc M., "The Swarm an the Queen: Towars a eterministic an Aaptive Particle Swarm Optimization", Congress on Evolutionary Computation, Washington C, 999. [CLE 03] Clerc M., "TRIBES - Un eemple 'optimisation par essaim particulaire sans paramètres e contrôle", OEP'03 (Optimisation par Essaim Particulaire), Paris, 003. [GAC 0] Gacôgne L., "Steay state evolutionary algorithm with an operator family", EISCI, Kosice, Slovaquie, 00. [SER 97] Serra P., Stanton A. F., Kais S., "Pivot metho for global optimization", Physical Review, vol. 55, 997, p