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A Restart CMA Evolution Strategy with Increasing Population Size

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A Restart CMA Evolution Strategy with Increasing Population Size 1769 A Restart CMA Evolution Strategy With Increasing Population Size Anne Auger Nikolaus Hansen CoLab Computational Laboratory, CSE Lab, ETH Zurich, Switzerland ETH Zurich, Switzerland [email protected] [email protected] Abstract- In this paper we introduce a restart-CMA- 2 The restart CMA-ES evolution strategy, where the population size is increased for each restart (IPOP). By increasing the population The (,uw, )-CMA-ES In this paper we use the (,uw, )- size the search characteristic becomes more global af- CMA-ES thoroughly described in [3]. We sum up the gen- ter each restart. The IPOP-CMA-ES is evaluated on the eral principle of the algorithm in the following and refer to test suit of 25 functions designed for the special session [3] for the details. on real-parameter optimization of CEC 2005. Its perfor- For generation g + 1, offspring are sampled indepen- mance is compared to a local restart strategy with con- dently according to a multi-variate normal distribution stant small population size. On unimodal functions the ) fork=1,... performance is similar. On multi-modal functions the -(9+1) A-()(K ( (9))2C(g)) local restart strategy significantly outperforms IPOP in where denotes a normally distributed random 4 test cases whereas IPOP performs significantly better N(mr, C) in 29 out of 60 tested cases. vector with mean m' and covariance matrix C. The , best offspring are recombined into the new mean value (Y)(0+1) = Z wix(-+w), where the positive weights 1 Introduction w E JR sum to one. The equations for updating the re- The Covariance Matrix Adaptation Evolution Strategy maining parameters of the normal distribution are given in (CMA-ES) [5, 7, 4] is an evolution strategy that adapts [3]: Eqs. 2 and 3 for the covariance matrix C, Eqs. 4 and the full covariance matrix of a normal search (mutation) 5 for the step-size (cumulative step-size adaptation / path distribution. Compared to many other evolutionary algo- length control). On convex-quadratic functions, the adap- rithms, an important property of the CMA-ES is its invari- tation mechanisms for and C allow to achieve log-linear ance against linear transformations of the search space: the convergence after an adaptation time which can scale be- CMA-ES exhibits the same performances on a given objec- tween 0 and n2. The default strategy parameters are given tive function f : x E RI f(x) E R, where n E N, in [3, Eqs. 6-8]. Only (x) (°) and (0) have to be set depend- and on the same function where a linear transformation is ing on the problem.1 applied, i.e. fR : x E Rn H-4 f(Rx) E R where R de- The default population size prescribed for the (,w, notes a full rank linear transformation. This is true only CMA-ES grows with log n and equals to = 10, 14,15 for if a corresponding transformation of the strategy (distribu- n = 10,30,50. On multi-modal functions the optimal pop- tion) parameters is made. In practice this transformation is ulation size can be considerably greater than the default learned by the CMA algorithm. population size [3]. The CMA-ES efficiently minimizes unimodal objective functions and is in particular superior on ill-conditioned and The restart ([Lw, )-CMA-ES with increasing pop- non-separable problems [7, 6]. In [3], Hansen and Kern ulation (IPOP-CMA-ES) For the restart strategy the show that increasing the population size improves the per- (pw, )-CMA-ES is stopped whenever one stopping crite- formance of the CMA-ES on multi-modal functions. Con- rion as described below is met, and an independent restart sequently, they suggest a CMA-ES restart strategy with suc- is launched with the population size increased by a factor of cessively increasing population size. Such an algorithm, re- 2. For all parameters of the ([uw, )-CMA-ES the default ferred to as IPOP-CMA-ES in the following, is introduced values are used (see [3]) except for the population size, start- and investigated here for optimizing the test suit of the CEC ing from the default value but then repeatedly increased. To special session on real-parameter optimization [8]. our intuition, for the increasing factor, values between 1.5 The remainder of this paper is organized as follows: Sec- and 5 could be reasonable. Preliminary empirical investiga- tion 2 presents the main lines ofthe algorithm. Section 3 ex- tions on the Rastrigin function reveal similar performance plains the experimental procedure. Section 4 presents and for factors between 2 and 3. We chose 2, to conduct a larger comments the experimental results. The IPOP-CMA-ES is number of restarts per run. compared to a restart strategy with constant population size. To decide when to restart the following (default stop- ping) criteria do apply.2 1 A more elaborated algorithm description can be accessed via http:l/www.bionik.tu-berlin.de/user/niko/cmatutorial.pdf. 2These stopping criteria were developed before the benchmark function 0-7803-9363-5/05/$20.00 ©2005 IEEE. Authorized licensed use limited to: UNIVERSITY OF NOTTINGHAM. Downloaded on December 11, 2009 at 07:14 from IEEE Xplore. Restrictions apply. 1770 * Stop if the range of the best objective function val- functions 6-12 are multi-modal. Functions 13-25 result ues of the last 10 + [30n/ 1 generations is zero from the composition of several functions. To prevent ex- (equalfunvalhist), or the range of these func- ploitation of symmetry of the search space and of the typi- tion values and all function values of the recent gen- cal zero value associated with the global optimum, the local eration is below To1 fun= 10-12. optimum is shifted to a value different from zero and the function values of the global optima are non-zero. * Stop if the standard deviation of the normal distribu- tion is smaller than TolX in all coordinates and Pc (the evolution path from Eq. 2 in [3]) is smaller than Success performances Evaluating and comparing the TolX in all components. We set TolX= 10-12 (O). performances of different algorithms on multi-modal prob- lems implies to take into account that some algorithms * Stop if adding a 0.1-standard deviation vector in may have a small probability of success but converge fast a principal axis direction of C(M) does not change whereas others may have a larger probability of success but (x) () (noef fectaxi S)3 be slower. To evaluate the performances of an algorithm A the fol- * Stop if adding 0.2-standard deviation in each coordi- lowing success performance criterion has been defined in nate does change (x)(g) (noef fectcoord). [8] E * Stop if the condition number of the covariance matrix = (TI) SPI .11 (1) exceeds 1014 (conditioncov). Ps The distributions ofthe starting points (x) (°) and the ini- where E(TI) is an estimator of the expected number of ob- tial step-sizes (0) are derived from the problem dependent jective function evaluations during a successful run of A, initial search region and their setting is described in the next Nbr. evaluations in all successful runs E section, as well as the overall stopping criteria for the IPOP- (AI) Nbr. successful runs CMA-ES. and Pi an estimator of the probability of success5 of A 3 Experimental procedure Nbr. successful runs Ps Nbr. runs The restart-(pw, )-CMA-ES with increasing population size, IPOP-CMA-ES, has been investigated on the 25 test In [1] we have shown that this performance criterion is a functions described in [8] for dimension 10, 30 and 50. For particular case of a more general criterion. More precisely, each function a bounded subset [A, B]n ofRI is prescribed. consider T the random variable measuring the overall run- For each restart the initial point (x)(°) is sampled uniformly ning time (or number of function evaluations) before a suc- within this subset and the initial step-size (0) is equal to cess criterion is met by independent restarts of A, then the (B - A)/2. The stopping criteria before to restart are de- expectation of T is equal to scribed in the last section. The overall stopping criteria for Ps the algorithm prescribed in [8] are: stop after n x 104 func- E(T) = (1 )E(T(s) ±E(TA) (2) tion evaluations or stop if the objective function error value is below 10-8. where p, is the probability of success of A, and E(TAj) The boundary handling is done according to the standard and E(TI) are the expected number of function evalua- implementation of CMA-ES and consists in penalizing the tions for unsuccessful and successful runs, respectively (see individuals in the infeasible region.4 For each test function, [1, Eq. 2]). For E(TA8) = E(TI), Eq. 2 simplifies to 25 runs are performed. All performance criteria are evalu- E(T) = E(Tl)/p5. Therefore SP1 estimates the expected ated based on the same runs. In particular, the times when to running time T under this particular assumption. measure the objective function error value (namely at 103, Taking now into account that the algorithm investigated 104, 105 function evaluations) were not used as input pa- in this paper is a restart strategy and that the maximum num- rameter to the algorithm (e.g., to set the maximum number ber of function evaluations allowed is FEmax = n x 104, of function evaluations to adjust an annealing rate). we can derive another success performance criterion (see [1] for the details). Indeed for a restart strategy it is reasonable Test functions The complete definition of the test suit is to assume that a run is unsuccessful only because it reaches available in [8].
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