Covariant Tarpeian Method for Bloat Control in Genetic Programming

Abstract

In this paper a simple modification of the Tarpeian bloat-control method is presented which allows one dynamically set the parameters of the method in such a way to guarantee that the mean program size will either keep a particular value (e.g., its initial value) or will follow a schedule chosen by the user. The mathematical derivation of the technique as well as its numerical and empirical corroboration are presented. BackgroundMany techniques to control bloat have been proposed in the last two decades (for recent reviews see Luke and Panait, 2006;Alfaro-Cid et al., 2010;Silva, 2008)). One of the very few with a theoretically-sound basis is the Tarpeian method introduced in . This is the focus of this paper.The Tarpeian method is extremely simple in its implementation. All that is needed is a wrapper for the fitness function like the following algorithm:Tarpeian Wrapper: if size(program) > average program size and random() < p t then return( f low ); else return( fitness(program) );were p t is a real number between 0 and 1, random() is a function which returns uniformly distributed random numbers in the range [0, 1) and f low is a constant which represents an extremely low (or high, if minimising) fitness value such that individuals with such fitness are almost guaranteed not to be selected. A feature of this algorithm is that it does not require a priori knowledge of the size of the potential solutions to a problem. If programs need to grow in order to improve fitness, the Tarpeian method will not prevent this. It will occasionally hit some individuals that, if evaluated, would result in being fitter than average and this may slow down a little the progress of a run. However, because the wrapper does not evaluate the individuals being given a low fitness, very little computation is wasted. Even at a high anti-bloat intensity, p t , a better-than-average longerthan-average individual has still a chance of making it into the population. If enough individuals of this kind are produced (w.r.t. the individuals which are better-than-average but also shorter-thanaverage), eventually the average size of the programs in the population may grow. However, when this happens the Tarpeian method will immediately adjust so as to discourage further growth.

Theoretical results in genetic programming: the next ten years?

“…The anti-bloat intensity of Tarpeian control can be modulated by changing how frequently large individuals are penalised. As recently shown in [65] this can be done dynamically to gain complete control over the dynamics of mean program sizes.…”

Evolution of an Effective Brain-Computer Interface Mouse via Genetic Programming with Adaptive Tarpeian Bloat Control

“…An earlier method -the covariant Tarpeian method introduced last year (Poli, 2010) -can solve this problem analytically. However, the method is only applicable to generational systems using fitness proportionate selection and relying only on crossover as their genetic operator.…”

“…To model the effects on program size of the Tarpeian method in a GP system (Poli, 2010) specialised this equation and derived the following approximation:…”

“…Recent research (Poli, 2010) has developed a technique, called covariant Tarpeian method, that allows one to dynamically and optimally set the rate of application of the Tarpeian method, p t , in such a way as to completely control the evolution of the mean program size. The method has excellent properties but also some important drawbacks which we seek to overcome in this paper.…”

A Hyper-Heuristic Approach to Evolving Algorithms for Bandwidth Reduction Based on Genetic Programming

“…We also used elitism to ensure that the best individual in one generation was transferred unaltered to the next. In addition, in order to control excessive code growth or bloat, the Tarpeian method (which artificially weeds out a proportion of above-average-size trees) [37,38] was utilised in the GP system. The termination criterion used was based on the predetermined maximum number of generations to be run.…”