Epochal dynamics, in which long periods of stasis in an evolving population
are punctuated by a sudden burst of change, is a common behavior in both n
atural and artificial evolutionary processes. We analyze the population dyn
amics for a class of fitness functions that exhibit epochal behavior using
a mathematical framework developed recently, which incorporates techniques
from the fields of mathematical population genetics, molecular evolution th
eory, and statistical mechanics. Our analysis predicts the total number of
fitness function evaluations to reach the global optimum as a function of m
utation rate, population size, and the parameters specifying the fitness fu
nction. This allows us to determine the optimal evolutionary parameter sett
ings for this class of fitness functions.
We identify a generalized error threshold that smoothly bounds the two-dime
nsional regime of mutation rates and population sizes for which epochal evo
lutionary search operates most efficiently. Specifically, we analyze the dy
namics of epoch destabilization under finite-population sampling fluctuatio
ns and show how the evolutionary parameters effectively introduce a coarse
graining of the fitness function. More generally, we find that the optimal
parameter settings for epochal evolutionary search correspond to behavioral
regimes in which the consecutive epochs are marginally stable against the
sampling fluctuations. Our results suggest that in order to achieve optimal
search, one should set evolutionary parameters such that the coarse graini
ng of the fitness function induced by the sampling fluctuations is just lar
ge enough to hide local optima.