2.3.1 Model
We have seen in the previous section, that the general model is not capable to reflect one -the biggest and most serious- problem of local stochastic searches: the local extrema.
We have to add this crucial property into our model, without specializing it into practical problems. We have to stay as general as possible. To do this we have to consider the position of the elements in the state space in general, without creating a special state space related to some specific model. To avoid the implementation of a specific state space we can assume that every element in the state space will have a propertyp1, to stuck in a local extremum25and withp0it will not find local bounds around itself in the state-space.
This is of course again a generalization, because in case of a real problem, the state space
23and on every commonly used cellular architecture, this can be done easily
24considering that in case of the global method we would need a large global memory.
25which can be also global extremum
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is not homogeneous, and elements in different points will have different probabilities to stuck in local extrema. But in this dissertation my scope is not to compare or classify different state spaces, but to compare selection mechanisms, and this model is general but useful enough to reveal one main aspects of local stochastic searches: the diversity of the population. I also have to note that the other operations (recombination, mutation) are the same for both the global and local version of the algorithm, so eventual better results can only be explained by the selection mechanism.
We can also assume that elements close to each other will most likely be altered in the same way: If an element will stuck in a local extremum, we can presume, that all the elements around it will also stuck at the same place in the state space. Also if an element can move/mutate toward a better solution (without getting stuck) all the surrounding elements have the potential to move as well.
To achieve our goal we have to consider somehow the position of the elements in the state space, without specifying the state space itself. If we can label the positions in the state space we can calculate which elements will get stuck together. We also know that when an element with a relatively high weight is selected and copied multiple times there will be more identical elements (at the same position in the stat space) in the next generation before the mutation operator is applied. Based on this I have divided the elements into groups and each group is determined by the identity in the previous generation it was copied from. Thep0 andp1probabilities are applied for every group and they can determine whether a group can mutate (increase the weights of the elements) or not (they stuck in a local extrema)
During the resampling step I divide the elements into groupsαj, j = 1, . . . M according to the previously described rules.Gj-s are binary random (Bernoulli) variables with values zero or one. Withp0 probability αj = 0 and withp1= 1−p0 probability αj = 1.
Then for everyiifwt,i ∈Gj and αj = 0
wt+1,i=wt,i+ηt,i (2.4)
ifwt,i∈Gj andαj = 1
wt+1,i=wt,i (2.5)
Where ηt,i ∼ N(0, σ2) are independent and Gj represents a group of the elements in proximity.
Of course this model is a simplification of the position of the elements in the state space, and I have to note that elements can be in proximity caused by the mutation step as well, not only by the selection mechanism. The elements migrate in the state space and perform stochastic walks and even without a selection mechanism every element should visit all the points of the state space once in a while if the mutation factor is relatively large and we have to note that this model is not considering this temporal ’moving’ of the elements. However in case of a relatively large state space and if the distribution of the
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elements is uniform in this state space the effect of resampling step driving the elements to the same places is much stronger.
2.3.2 Results
The parameters of the model were:σ1= 10, σ2 = 3,p0= 0.70, p1 = 1−p0 = 0.30 To compare the global and local selection mechanism I have generated a sample with K = 400 elements, and these elements were placed on a 20×20 two-dimensional grid in case of the local sampling. In every experiments I examined the 10th population26 and I have repeated every experiments 1000 time, to reduce the noise on the simulations caused by the stochastic operators.
After the 10th iteration I have selected the best element 27. The results of both the global and local selection as the average of 1000 experiments can be seen in Table 2.3 The graphical interpretation of these results can be seen on figure 2.2.
These result are very important regarding the local selection. This model is the most general, that is capable to represent the most challenging problem of local stochastic optimization. Namely, when we would like to maximize (or minimize) a function we have to have the most diverse set, and cover the whole state space to avoid all the local extrema except the only global extremum.
We can derive two conclusions from the results: It can be seen that the local algorithm outperforms its global counterpart. It can also be seen that the values of the fitness of the local selection for different parameters have a concave shape. It is true, that the extreme cases28 are usually not efficient. This implies, that with the proper setting of the neighborhood radius we can optimize the performance of the localized algorithm. We can alter how the information spreads out through the processing elements and this way we can set the exploitation/exploration ratio of the algorithm. This way we can set either our aim is to maximize the fitness function and move towards the best value (case of global interactions) or to maintain the most diverse population possible (no interaction at all).
In case of a specific problem we can set the best parameter considering the probability of finding a local extrema. Setting this parameter to an optimal value we can solve practical problems more easily and efficiently. Because this dissertation investigates primarily the implementation and design of many-core algorithms, I stress the importance of the fact, that the local selection mechanism can be easily parallelized and its execution time can be decreased to a fraction of the execution time of the global method.
26the population after 10 selection and mutation steps
27the element with the highest fitness weight
28small local interactions or large ,global interactions affecting the whole population
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Figure 2.2: On this figure we can see the results of the local (red, continuous) and global selection (blue, dashed) mechanism. The numbers on the X-axis are the parameters, in every bracket the first is the neighborhood radius for the local method and the other is the number of the selected elements for the global method. The experiments were done withK = 400 elements. On the Y-axis we can see the fitness values, higher values mean better solutions. It can also be seen that the values of the fitness of the local selection for different parameters have a concave shape. This implies, that with the proper setting of the neighborhood radius we can optimize the performance of our algorithm by picking out the parameter value corresponding to the tip of this curve.
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Table 2.3: Results of the specific model for local and global selection
The first column of the table shows the neighborhood radius for local selection r. The second column is the result of the local method , the maximal value of the fitness function w (as an average of 1000 independent simulations). The third column is the result of the global method The last column represents the parameter of the global selection, the number of selected elements. As it can be seen the best result can be obtained with the local method, however the comparison of the two methods for one parameter-setup is not straightforward, because the parameters in the algorithms (the neighborhood radius and the number of preserved elements) are different.
neighborhood radius fitness(L) fitness(G) selected elements
1 45.00 49.50 20
2 49.14 44.61 40
3 51.04 42.28 60
4 52.82 40.85 80
5 53.59 37.95 100
6 54.19 36.24 120
7 55.75 34.88 140
8 55.27 33.85 160
9 55.36 32.72 180
10 54.20 31.82 200
11 54.34 30.86 220
12 51.63 29.95 240
13 51.40 29.08 260
14 49.19 28.21 280
15 48.29 31.15 300
16 48.88 33.68 320
17 45.56 35.51 340
18 38.29 37.89 360
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