1 Introduction
The main objective of this article is to derive convergence properties of two elitist Evolutionary Algorithms (EAs) on OneMax and Royal Roads test functions. One of the analyzed algorithms uses the kBitSwap (kBS) operator introduced in [TerSarkisov et al., 2010]. We compare our results to computational findings and other research.
A population consists of a set of solution strings. We split them into two groups: there are elite strings, which have the same highest fitness value and the remaining nonelite strings. We use the standard notation for the population , recombination pool , the elite species , the nonelite .
1.1 Past work
Recently EA with
flip probability (
being the length of a chromosome) became a matter of extensive investigation. Sharp lower and upper bounds for OneMax and general linear functions were found in [Doerr et al., 2010c, Doerr et al., 2010a, Doerr et al., 2011, Droste et al., 2002] applying drift analysis and potential functions. Specifically, in [Doerr et al., 2011] the upper bound for EA solving OneMax was derived to be and in [Doerr et al., 2010a] the lower bound for the same setting was found to be . Drift (a form of super martingale) was introduced in [Hajek, 1982, He and Yao, 2003, He and Yao, 2004].1.2 Definitions and Assumptions
We analyze two fitness functions here, OneMax (simple counting 1’s test function) and Royal Roads (see Section 4 for additional definitions for it). The fitness of a population is defined as the fitness of an elite string. Since both functions have global solution at , we are interested in the following time parameter:
(1) 
that is, the minimum time when (for algorithm A) the best species in the population reaches the highest fitness value. Since the analysis is probabilistic, we need the expectation of this parameter: .
We assume that we do not need a large number of species for evolution. Though this sounds a bit vague, this justifies the choice of distributions with respective parameters. The expectation of Poisson random variable used here is 1, for Uniform it is
. We use the latter due to its simplicity.We restrict our attention only to elite pairs (kBS) or parents (RLS), to simplify the analysis, since otherwise we would have to make more assumptions about the fitness of nonelite parents .
1.3 kBitSwap Operator
This genetic recombination operator (see Figure 1) was introduced in [TerSarkisov et al., 2010] and proved to work efficiently both alone and together with mainstream operators (crossovers and mutation). Its efficiency was mostly visible on functions like Rosenbrock, Ackley, Rastrigin and Royal Roads. We also tested its performance on OneMax specifically for this article.
1.4 Our findings
The models we derive are complete, i.e. they are functions of just population size , recombination pool and length of the chromosome , i.e. the actual parameters of EAs, though we make some weak assumptions about the pairing of parents.
We derive the expectation of convergence time for the populationbased elitist EA with a recombination operator (1BitSwap Operator) and mutationbased RLS. Our theoretical and computational findings confirm that for OneMax the benefit of population is unclear, i.e. its effect is not always positive. For Royal Roads it is always positive. This problemspecific issue was noticed before (see e.g. Figures 4 and 5 in [He and Yao, 2002]).
We use two distributions of elite species in the population: Uniform( and Poisson(1). Since the expressions for the expected first hitting time of algorithms
we have found are quite complicated, we do the computational estimation and find some asymptotic results as well.
Algorithm 1:()EA  

1  Initialize population size 
repeat until condition fulfilled:  
2a  select species from the population using 
Tournament selection  
2b  apply 1BS operator to each pair in the 
recombination pool  
2c  keep best species in the population, 
replace the rest with the best species  
from the pool 
Algorithm 2:RLS  

1  Initialize population size 
repeat until condition fulfilled:  
2a  select species from the population using 
Tournament selection  
2b  flip exactly one bit per chromosome 
2c  keep best species in the population, 
replace the rest with the best species  
from the pool 
Tournament Selection Procedure
We use this selection because it is fairly straightforward in implementation and analysis.

Select two species uniformly at random

if , either or enters the pool at random

else the species with better fitness enters the pool
2 Analysis of Algorithm 1 on Onemax Problem
We start with Uniform distribution of elite species with parameter
, which gives the lower bound on convergence time, which is due to the assumption on the number of elite species needed for the evolution.The probability of selecting an elite pair is
(2) 
Since we restrict the analysis only to elite pairs, the probability of evolution (generation of a better offspring as a result of 1BitSwap) is
(3) 
where , which is due to the assumption that at the start of the algorithm .
2.1 Uniform distribution of elite species
We are deriving an upper bound on the probability (and, therefore, lower bound on the expectation of the first hitting time). We are interested in the probability of evolving at least 1 new elite species next generation, i.e. of at least 1 successful swap.
We define to be the event that no new species evolves over 1 particular generation. The number of elite pairs in the population varies from 0 to , and elite species in the population from 1 to . In this regard, probability to select a number of pairs given elite species in the population is
. By the Law of total probability,
(4) 
We assume Uniform probability of each number of elite species in the population: .
(5) 
to get 1 elite pair:
and
therefore, the probability of failure given 1 elite pair given improvements so far is
(6) 
For cases the logic is similar, so the full expression for the probability of failure is
Interchanging the sums and using the standard binomial expansion
The probability of evolution (obtaining a better species) is therefore
(7) 
for each we have
and therefore the expected first hitting time for the algorithm is
(8) 
Despite having two sums without closed forms, the convergence rate of this algorithms depends only on the size of the population, recombination pool and the length of the string, that is, the reallife parameters of EA. Therefore, the model is complete.
3 Analysis of Algorithm 2 Solving Onemax
For comparison, we derive for EA using similar approach (Law of total probability + sum of Geometric RVs). Changes apply mostly to the selection probability, as we have no pairs to form:
(9) 
3.1 Uniform distribution of elite species
We use the same assumptions of uniform distribution of elite species in the population as with the EA.
Failure event is defined in the same way: no successful flips in the recombination pool, so the probability thereof is defined in a similar way to the one in Equation 4.
(10) 
Only in this case is the number of elite parents in the pool and goes from 0 to .
and therefore (using the same idea with the binomial expansion)
(11) 
Unfortunately, the closed expression for this sum exists only for specific values of , so we have to keep it this way and later obtain the results computationally.
So the expected optimization time of the algorithm is
(12) 
As the case is with EA, this is a somewhat optimistic estimate, since it assigns fairly high probabilities to high proportions of elite species in the population. This is confirmed by numerical estimates.
4 Analysis of Algorithm 1 on Royal Roads Function
We use the setup for RR problem along the lines of [Mitchell, 1996] (referred to as in the book).The chromosome of length is split into blocks, each of length . The fitness of each block is 0 if there are any 0s in the block, and M if all of the bits in it have value 1. The fitness of the chromosome is the sum of the value of the blocks, so it can take values . We index the blocks using index
. Originally this problem was designed to test EA’s capacity for recombining building blocks compared to other heuristics (for details see
[Mitchell, 1996]).Additionally, we introduce an auxiliary function used to measure progress between improvements in the fitness (the idea is similar to that in, e.g. [Doerr et al., 2010b, He and Yao, 2004]), which in our case is since both functions achieve the global optimum at and .
There is an important difference from the standard OneMax problem: unlike it, when parents exchange genetic information, it doesn’t matter where the information comes from (which segment of the parent), but it matters where it is inserted, because it may mean that the fitness of the recipient segment has reached , and therefore the fitness of the whole parent increased by the same value.
The second important observation is that of all segments in the chromosome there is one, which evolves first, denote it (this is only possible due to the parameters of the EA in discussion). This means that segments evolve in a sequence: .
We pessimistically assume that the best auxiliary function value in the first generation is and fitness function is 0. We also assume that the starting value in each bin is . In the same way as with OneMax, we make assumptions about the distribution of elite species in the population, rather than their exact or approximate number.
We start with introducing the probability of failure:
where all variables are the same as in EA solving OneMax: is j’th elite pair in the recombination pool , is the number of elite species in the population with both highest fitness and auxiliary function values. The selection function is the same as Equation 2:
The successful event is defined as evolution of at least one more elite species in the population. The number of bits equal to 0 left to swap/flip in a segment we use . So now the probability of successful swap in Equation 2 becomes
(13) 
The auxiliary function for each bin lies between and and k between and , where K is the total number of bins to fill. The probability of failure is
The probability to fail to improve a bit in a bin given improvements so far is
(14) 
Therefore,
Expected time until improving the auxiliary function of a bin is
(15) 
and, finally,summing over all from 0 to we obtain (since G depends on both and )
(16) 
5 Analysis of Algorithm 2 on Royal Roads Function
Just as is the case with OneMax, we present the results for populationbased RLS on Royal Roads. This model is a bit simpler since we do not have to pair the parents, and the selection is just
Instead of Uniform distribution of elite species in the population, we try Poisson distribution with parameter 1, and normalizing constant
(17) 
since where is incomplete Gamma function. The sizes of populations used in the computational experiments, the values of the normalizing constant are set in Table 2.
4  

10  
20  
30 
The flip probability is just
(18) 
Probability of failure given successful flips so far is
Therefore, the probability of success is
and the expected time to fill the first bin is therefore
Since we have such bins and the probability of successful sampling does not depend on the number of 1’s in the parent (unlike EA), we obtain the expected first hitting time for the algorithm on RR:
(19) 
50  1  2  112.9801  113.14  190.7979  192.12 
2  2  144.6145  218.12  116.2812  184.6  
4  4  94.7621  145.4  73.124  193.54  
8  8  62.5691  121.86  48.596  184.52  
10  10  55.4784  116.98  43.488  197.92  
100  1  2  259.8688  265.84  449.9205  418.8 
2  2  332.6321  455.9048  271.56  393.39  
4  4  215.2445  314.1  168.03  410.39  
8  8  139.2885  267.72  108.826  420.416  
10  10  122.4884  266.9  96.3978  405.22  
1000  1  2  3743.2354  3682.6  6792.8  6715 
2  2  4791.3413  7072.7  4025.9876  6630  
4  4  3021.5468  4574.3  2413.0033  6990  
8  8  1872.3595  3866.5  1481.761  7016  
10  10  1616.4433  3807  1283.5502  6834 
6 Computational Results
Since the expressions derived in this article do not have a closed form, we find them computationally. To test our results, we run each algorithm with parameter set () with almost always for 50 independent runs, each run was 2000 generations long. The average of optimization time is denoted
. Probability distribution used for each model follow standard notation in Probability theory:
for Uniform and for Poisson.In general, results for Algorithm 1 tend to be better than for Algorithm 2 and for OneMax sharper than for RR. As we mentioned already, this is due to different patterns of dynamics of elite species and has to be investigated further. Apparently for both algorithms solving RR both Uniform and Poisson(1) distributions give a fairly rough approximation that we can improve both statically (using other parameters) and dynamically (modeling change in the number of elite species).
The other important result is that the increase of population size for both algorithms solving OneMax problem does not necessarily result in the improvement in performance, which we showed both theoretically and numerically. For RR the situation is much more clear: increase in the population always brings about the improvement in performance. Both models confirm this quite consistently.
K  M  

32  4  8  4  4  145  315.3077  64.8084  672.25 
10  10  72.4  268.2195  58.124  504.625  
20  20  44.2  192.2917  56.175  334.125  
30  30  34.5  173.5625  55.1  221  
64  8  8  4  4  570.625  612.46  249.959   
10  10  279.88  497.93  222.565  820.6667  
20  20  153.46  454.4681  212.452  715.92  
30  30  112.297  372.04  209.373  663.6744  
128  16  8  4  4  2264.36  1365  1021   
10  10  1048  1239  940.999    
20  20  570.44  1091.5  887.396  1612  
30  30  401.99  949.4  871.1131  1505 
7 Conclusions and Future Work
We have derived expected running time for EAs based on two different genetic operators on two relatively simple fitness functions. In the future there are two extensions that we particularly plan to focus on:
Approximate results for Equations 8, 12, 16, 19
This is the most obvious of developments. Although these equations give good estimates for optimization time, and we have found some asymptotic lower bounds, it is desirable to find sharper bounds in the form . A big problem here are the complicated expressions involving sums.
Evolution of elite species.
This area has seen little focus in EA community, and we are keen to develop a dynamic model for evolution of species. In this article the model is static, i.e. distribution of elite species in the population is fixed (Uniform or Poisson). As a result, some bounds, especially for RR, seem to be quite loose. If instead of assuming a probability distribution of elite species with fixed parameters we study the convergence of the distribution, we can derive sharper bounds on optimization time.
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