Markov chain analysis of evolutionary algorithms on OneMax function – From coupon collector's problem to (1 + 1) EA

Abstract

The theoretical investigation of Evolutionary Algorithms (EAs) has increased our understanding of the computational mechanism of algorithms. OneMax is a test function most frequently and deeply studied in the field of EAs. In this work, a method is presented for describing the runtime properties of (1+1) EA on OneMax. This method is motivated by the work of Erdös and Rényi treating the coupon collector's problem. They showed that the success probability of the coupon collector's problem is given by a function of double exponential form, and that the number of uncollected coupons follows the Poisson distribution. Today, the double exponential function is called Gumbel function, which is one of three fundamental functions in extreme value statistics. We introduce an algorithm that is a variant of the (1+1) EA, First Order Evolutionary Algorithm (FO-EA). FO-EA takes into account only the effect of single-bit mutations in the (1+1) EA, which in general includes multiple-bit mutations. We modified the method of Erdös and Rényi to apply FO-EA. We apply the Gumbel distribution for calculating the success probability of the (1+1) EA on OneMax. This method turns out to give a sufficiently reliable estimation for success probabilities, even in the tail region.