Boris Mitavskiy

A schema-based version of geiringer's theorem for nonlinear genetic programming with homologous crossover

Geiringers theorem is a statement which tells us something about the limiting frequency of occurrence of a certain individual when a classical genetic algorithm is executed in the absence of selection and mutation. Recently Poli, Stephens, Wright and Rowe extended the original theorem of Geiringer to include the case of variable length genetic algorithms and linear genetic programming. In the current paper a rather powerful version of Geiringers theorem which has been established recently by Mitavskiy is used to derive a schema-based version of the theorem for nonlinear genetic programming with homologous crossover.

An Extension of Geiringer's Theorem for a Wide Class of Evolutionary Search Algorithms.

The frequency with which various elements of the search space of a given evolutionary algorithm are sampled is affected by the family of recombination (reproduction) operators. The original Geiringer theorem tells us the limiting frequency of occurrence of a given individual under repeated application of crossover alone for the classical genetic algorithm. Recently, Geiringers theorem has been generalized to include the case of linear GP with homologous crossover (which can also be thought of as a variable length GA). In the current paper we prove a general theorem which tells us that under rather mild conditions on a given evolutionary algorithm, call it A, the stationary distribution of a certain Markov chain of populations in the absence of selection is unique and uniform. This theorem not only implies the already existing versions of Geiringers theorem, but also provides a recipe of how to obtain similar facts for a rather wide class of evolutionary algorithms. The techniques which are used to prove this theorem involve a classical fact about random walks on a group and may allow us to compute and/or estimate the eigenvalues of the corresponding Markov transition matrix which is directly related to the rate of convergence towards the unique limiting distribution.

Exploiting quotients of markov chains to derive properties of the stationary distribution of the markov chain associated to an evolutionary algorithm

In this work, a method is presented for analysis of Markov chains modeling evolutionary algorithms through use of a suitable quotient construction. Such a notion of quotient of a Markov chain is frequently referred to as “coarse graining” in the evolutionary computation literature. We shall discuss the construction of a quotient of an irreducible Markov chain with respect to an arbitrary equivalence relation on the state space. The stationary distribution of the quotient chain is “coherent” with the stationary distribution of the original chain. Although the transition probabilities of the quotient chain depend on the stationary distribution of the original chain, we can still exploit the quotient construction to deduce some relevant properties of the stationary distribution of the original chain. As one application, we shall establish inequalities that describe how fast the stationary distribution of Markov chains modelling evolutionary algorithms concentrates on the uniform populations as the mutation rate converges to 0. Further applications are discussed.

Preliminary theoretical analysis of a local search algorithm to optimize network communication subject to preserving the total number of links

A variety of phenomena such as world wide web, social or business interactions are modeled by various kinds of networks (such as the scale free or preferential attachment networks). However, due to the model-specific requirements one may want to rewire the network to optimize the communication among the various nodes while not overloading the number of channels (i.e. preserving the number edges). In the current paper we present a formal framework for this problem and a simple heuristic local search algorithm to cope with it. We estimate the expected single-step improvement of our algorithm, establish the ergodicity of the algorithm (i.e. that the algorithm never gets stuck at a local optima) with probability 1) and we also present a few initial empirical results for the scale free networks.

Estimating the ratios of the stationary distribution values for markov chains modeling evolutionary algorithms

The evolutionary algorithm stochastic process is well-known to be Markovian. These have been under investigation in much of the theoretical evolutionary computing research. When the mutation rate is positive, the Markov chain modeling of an evolutionary algorithm is irreducible and, therefore, has a unique stationary distribution. Rather little is known about the stationary distribution. In fact, the only quantitative facts established so far tell us that the stationary distributions of Markov chains modeling evolutionary algorithms concentrate on uniform populations (i.e., those populations consisting of a repeated copy of the same individual). At the same time, knowing the stationary distribution may provide some information about the expected time it takes for the algorithm to reach a certain solution, assessment of the biases due to recombination and selection, and is of importance in population genetics to assess what is called a genetic load (see the introduction for more details). In the recent joint works of the first author, some bounds have been established on the rates at which the stationary distribution concentrates on the uniform populations. The primary tool used in these papers is the quotient construction method. It turns out that the quotient construction method can be exploited to derive much more informative bounds on ratios of the stationary distribution values of various subsets of the state space. In fact, some of the bounds obtained in the current work are expressed in terms of the parameters involved in all the three main stages of an evolutionary algorithm: namely, selection, recombination, and mutation.

Propagation time in stochastic communication networks

Dynamical processes taking place on networks have received much attention in recent years, especially on various models of random graphs (including ldquosmall worldrdquo and rdquoscale freerdquo networks). They model a variety of phenomena, including the spread of information on the Internet; the outbreak of epidemics in a spatially structured population; passage of services or queries in a digital ecosystem network; and communication between randomly dispersed processors in an ad hoc wireless network. Typically, research has concentrated on the existence and size of a large connected component (representing, say, the size of the epidemic) in a percolation model, or uses differential equations to study the dynamics using a mean-field approximation in an infinite graph. Here we investigate the time taken for information to propagate from a single source through a finite network, as a function of the number of nodes and the network topology. We assume that time is discrete, and that nodes attempt to transmit to their neighbors in parallel, with a given probability of success. We solve this problem exactly for several specific topologies, and use a large-deviation theorem to derive general asymptotic bounds, which apply to any family of networks where the diameter grows at least logarithmically in the number of nodes. We use these bounds, for example, to show that a scale-free network has propagation time logarithmic in the number of nodes, and inversely proportional to the transmission probability.

Invariant Subsets of the Search Space, and the Universality of a Generalized Genetic Algorithm

In this paper we shall give a mathematical description of a general evolutionary heuristic search algorithm which allows to see a very special property which slightly generalized binary genetic algorithms have comparing to other evolutionary computation techniques. It turns out that such a generalized genetic algorithm, which we call a binary semi-genetic algorithm, is capable of encoding virtually any other reasonable evolutionary heuristic search technique.

NP-Completeness of deciding binary genetic encodability

In previous work of the second author a rigorous mathematical foundation for re-encoding one evolutionary search algorithm by another has been developed. A natural issue to consider then is the complexity of deciding whether or not a given evolutionary algorithm can be re-encoded by one of the standard classical evolutionary algorithms such as a binary genetic algorithm. In the current paper we prove that, in general, this decision problem is NP-complete.