Optimizing Stochastic and Multiple Fitness Functions

How does one optimize a tness function when the values it generates have a stochastic component How does one simultaneously optimize mul tiple tness criteria These questions are important for many applications of evolutionary computation in an experimental environment Solutions to these problems are presented along with discussion of situations where they arise such as modeling and genetic programming A detailed numer ical example from control theory is also provided In the process we nd that population based search algorithms are well suited to such problems Stochastic Fitness Functions In the theoretical development of evolutionary computation the tness func tions considered are almost invariably deterministic functions of a prede ned parameter set In the real world application of optimization techniques we must often account for stochastic tness functions A simple example of a stochas tic tness function is one measurement drawn from a distribution of readings obtained from an experimental apparatus The tness may be viewed as on realization of a random variable with a mean corresponding to the presumed true tness In some problems the uncertainties can be signi cant Several approaches to optimizing stochastic tness functions are possible but the least successful method is to ignore the uncertainty If we follow that path we could easily be left at the end of the search with a parameter set which To appear inProceedingsof the Fourth Annual Conferenceon EvolutionaryProgramming This paper assumes that the reader has a basic understandingof population based stochas tic search techniques such as genetic algorithms GA evolutionary programming EP or evolution strategies ES is suboptimal but lucky enough in the single tness realization to score higher than all other sets To go beyond the simplistic approach it helps to present some possible situations Alternative Criteria In many applications supplementary tness criteria are available For example we can de ne a generic alternative criterion for noisy tness functions similar to a signal to noise ratio by using the height of a peak relative to the surrounding region of the tness landscape as a tness measure This is supplementary to the raw tness of the peak We will examine in the next section how to utilize multiple criteria Uncertainty Measures Another common alternative is the standard deviation of the tness over some a small neighborhood We can generate a mean tness and the uncertainty in that mean by computing the tness over a small neighborhood in parameter space This assumes real valued parameters or a tness function which varies relatively smoothly over a range of integers We can then incorporate the uncertainty estimate into our tness evaluation Segmented Fitness Functions In some situations we may want to segment the data and compute the tness on each segment This is an option in situations such as time series analysis Rather than minimize the overall error in a parameterized model over a data set we can segment the data so that we produce one tness estimate for each data segment Segmentation increases the uncertainty in the tness estimate of a single segment but provides multiple measures instead of a single combined measure This is most bene cial when we are concerned about drift in the tness estimates as we scan along the data set The classic example is non stationarity in time series As a more concrete example imagine that we are conducting a scattering experiment and recording the exit angle for each particle scattered from the target While collecting the data the target is brie y perturbed causing a strong peak at one angle If we average all the data together we discover a dominant From the author s point of view non stationarity is a property of a model describing a time series not the data itself Say we had a model for the water usage in San Diego on January based upon the usage patterns earlier in the month From our errors we might conclude that the data is non stationary whereas a more insightful model one which included the NFL Superbowl halftime would have no such problem though erroneous peak By segmenting the data the spurious nature of this peak becomes more apparent Segmentation is a natural way to produce models which are stable with re spect to perceived non stationarity It is conceptually similar to cross validation except that in multi stage cross validation the search considers each data seg ment sequentially They are not optimized in parallel as we intend to do Multiple Fitness Functions All of the approaches described above involve the use of vector valued tness functions Thus a secondary problem has been created The original scalar tness function has been replaced with a vector valued tness function which is no longer trivial to optimize By addressing this problem we expand the range of problems being addressed to any that require the simultaneous optimization of multiple criteria In the following sections we will discuss the pros and cons of various opti mization methods At some level all these approaches are intended to produce a single scalar value such that logical comparison of di erent parameter sets is possible Signi cance Measures When the vector tness is composed of the raw tness f and the uncertainty in that value they are commonly combined as s f to produce a signi cance estimate in units of standard deviation Such a quantity is useful but provides di erent information from f Optimizing s returns the most clearly de ned peak not necessarily the highest peak In most situations the two quantities f and s must still be simultaneously optimized