Diversity Loss in General Estimation of Distribution Algorithms J. L. Shapiro PPSN (Parallel Problem Solving From Nature) ’06 BISCuit 2 nd EDA Seminar

Abstract A class of EDAs on which universal results on the rate of diversity loss can be derived.  The class (SML-EDA) requires two restrictions.  In each generation, the new probability model is built using only sampled from the current probability model.  Maximum likelihood is used to set model parameters. The algorithm will never find the optimum unless the population size grows exponentially.

EDA Initialize the EDA probability model (Start with a random population of M vectors.) Repeat Select N vectors using a selection method Learn a new probability model from the selected population Sample M vectors from the probability model Until stopping criteria are satisfied

SML-EDA A restricted class of EDAs needs 3 assumptions.  1. Data in generation t can only affect data in generation t+1 through the probability model.  2. The parameters of the estimated model are chosen using maximum likelihood.  3. The sample size M and the size of the population N are of a constant ratio independent of the number of variables L. SML-EDA (Simple, Maximum-Likelihood EDA)  The class of EDAs for which assumptions 1 and 2 hold are called SML-EDAs.  This class of EDAs includes BOA, MIMIC, FDA, UMDA, etc.

The Empirical frequency (the component i has the value A through the population) One diversity measure The trace of the empirical co-variance matrix Some Definitions for diversity loss Population member Population size Expectation that 2 components both take the value A Expectation that they take value A independently. the value is fixates, 0 random population, maximum value

Diversity Loss on a Flat Fitness Landscape Theorem 1  For EDAs in class SML-EDA on a flat landscape, the expected value of v is reduced in each generation by the following.  Because the parameters are set by ML, empirical variance is reduced by a factor (1-1/N) from the parent population. Universal “expected diversity loss” for all SML-EDA  decays with characteristic time approximately equal to the population size for large N.

A universal bound for the minimum population size in the needle problem Needle in a haystack problem  There is one special state (the needle), which has a high fitness value, and all others have the same low fitness value.

Probability that the needle is never sampled. Theorem 2  In the limit that such that for any EDA in SML-EDA searching on the Needle problem.  The population size must grow at least as fast as for the optimum to be found. The time that the needle is first sampled.

Proof of theorem 2  Lemma 1  Let t* be defined as If the needle has not been found after a time t > t*, the probability that the needle will never be found is greater that 1 – ε, where.

 Proof of Lemma 1  We can write the probability about the diversity loss with the formula that the variance is small.  Choosing t* to be large enough so that if v t is so small, there must be fixation at every component except possibly one component. If L-1 components are fixed, the needle will only be sampled if they are fixed at values found in the needle. We are not certain that v t appropriately small, it is just probable so. Take  t* is calculated by solving simultaneous equations.

 Lemma 2  The probability that the needle is not found after t* steps obeys  Proof –The probability that SML-EDA does not find the needle in time t* the result is found by putting into this equation the value for t* The Prob. of not finding the needle

 So, the probability of never finding the needle can be decomposed into  Combining Lemma 1, 2 gives the following  If in the limit, N grows sufficiently slowly that the third term vanishes, then the probability of never finding the needle will go to 1.  Thus, if, as, the needle will never be found.

Expected Runtime for the Needle Problem and the Limits of Universality Corollary 1.  If the needle is found, the time to find it is bounded above by t* with probability Convergence conditions will not be universal, but will be particular to the EDA.