4.4 Exploring Network Learning
4.4.1 Network Structure learning on a single Trap5 Problem
From this initial test case, the BOA case produced a vast amount of data, which can be overwhelming. There is a need to understand how the BOA responds to the decomposable portions of the objective function. To explore the networks created by the optimzer, it may be best to step back and examine a series of increasingly more complex and difficult problems. To do this, a methodical examination beginning with Pelikan’s Decomposable work and transitioning to a series of stiffened T-Panels is presented. The work seeks to highlight how the optimizer creates networks first at the truly decomposable level and then at the global level as decomposable elements are combined to form a complete design.
The most eye-catching portion of Pelikan’s work begins with the trap 5 problem. As previously discussed, this is a binary logic trap, with a size of 5 bits. This problem is interesting in that it has a strongly preferred local optima away from the global
optima which will lead a gradient optimizer away from the global solution. The problem may be mathematically summarized as :
f (u) = 5 if u = 5
When used the the BOA, this problem is represented by 5 variables or nodes each with discrete probability bins of 0 or 1 to be assessed by the aforementioned equations 4.9 and 4.10.
As you can see in figure 4.2 the problem is not smooth, causing both ends of the
0 1 2 3 4 5
Figure 4.2: Graphical depiction of the Trap 5 problem. u is the sum of the binary bit values and f (U ) is the corresponding fitness value. Notice the global maximum value, f (5) = 5 is located away from the gradient dominated solution f (0) = 4 making the problem more difficult to solve.
problem to be preferred by a gradient optimizer and locating the global solution as far as possible from the local solution. Sampling of this problem is also an interesting challenge. The problem has 25 potential outputs, with f (u) value of 4 and 5
occur-ring for only 2 of the 32 results. Therefore a high quality sample to capture this is needed. Both 4 and 5 point solutions must be in the training data, else the network is mapping a relatively random solution space (as location in the design variable vector is unimportant in this case).
To examine the network formation given a set of data, a series of 10 stoichastic simulations were conducted. Training data size continues to be an area of exploration.
The designer seeks to examine a training set that is not random, else the network will lack edges due to the nature of the network scoring K2 metric. For a single 5 bit trap problem 100 random designs were created, of which a portion most elite designs were selected as training data. A sample of 4 heuristic parameters will be shown. These samples seek to recreate stages of the optimization process. Ideally the optimizer goes through 3 major stages: feasibility, refinement, and optimality convergence. The network edges at each stage will be examined.
The first simulation takes the training set and selects 50% of the population to train the network on. The results are reported in Table4.7. Due to the random nature of the simulation it is difficult to visualize the edge data in compact way. Typically matrices are used to show edge connection. With 10 simulations to represent, this becomes difficult. Instead a tabular output is presented. When an edge exists in the network, the parent is recorded with a + symbol followed by its child. Likewise, the Child records a - symbol followed by its parent. In this way even number of sets must be reported in each line of the table. What we see from the initial sample is that the network edges are essentially random with no preferred form. This is to be expected as the sample is still relatively random and unrefined. There are 3 distinct structures to pay attention to in these tables. Chains within the network will be represented in elements that have both + and - symbols recorded. Downward facing Vs significant for Bayesian inference are represented by two or more + symbols for the same node. Likewise upward facing Vs, important to denote independence in
Sim No. X1 X2 X3 X4 X5
Table 4.7: Network edge representations of a single trap 5 problem at a population size of 100, with a selection pressure of 50% without addition of any additional highly fit solutions. Note the lack of defined structure. For notation, the + symbol indicates the node is a parent of the designated neighbor, the - symbol indicates it is a child.
networks are identified by two - symbols on the same node. In Table 4.7, no defined structure exists, as edges are haphazardly scattered.
To improve the quality of the sample and remove some of the randomness in the simulation the selection pressure is double. Instead only 25 designs of the 100 member population pool are selected for training. Table 4.8 shows the results of this change. What begins to emerge are a series of downward and upward facing Vs.
This is an improvement as the algorithm begins to identify the bifurcation in results.
Again though, this sample space is 32 members large and the 25 member training population is likely insufficient to capture enough globally fit solutions to fully resolve the network edges.
To combat this, the known globally and locally fit solutions, uniform 0 bits and uniform 1 bits are added to the training pool prior to selection. Now 25 % of the 110 members are selected, ensuring roughly 40% of the sample is high quality solutions.
the results, shown in Table4.9highlight the single dominant structure, the downward facing V appearing in every network. This is ideal, as the bifurcation is controlled by the root node in the network. If the root node is chosen to be a value of 0, the entire design should uniformly be 0. Likewise if the root node is chosen to be 1, the entire
design should be chosen uniformly as 1.
The final progression is to increase the selectivity of the training pool such that a
Sim No. X1 X2 X3 X4 X5
Table 4.8: Network edge representations of a single trap 5 problem at a population size of 100, with a selection pressure of 25% without addition of any additional highly fit solutions. Note the number of edges in the network has increased considerably, as the sample is less random/more heavily pressured when selected. Additionally, downward facing V structures, denoted by 2 or more + signs for the same node, begin to appear.
These network structures fit the trap problem as they allow influence from one node to be passed to another.
majority of the solutions are highly elite. To do this, the selection pressure is again doubled such that only 12.5% of the solutions are selected for the training data. The results are very encouraging, as the simulations converge to a single ideal structure.
To graphically visualize this transition, figure 4.3 shows representations of an average network at each of the 4 groups of parameters used. The transition from random to clearly defined and usable network form is stark. This trend likely exists in the refinement of a structural design network, where the quality of the data supplied to the optimizer controls the resulting quality of the network. As previously postulated, the network becomes more valuable for analysis when high quality training sets are used for network learning.
Sim No. X1 X2 X3 X4 X5
Table 4.9: Network edge representations of a single trap 5 problem at a population size of 100, with a selection pressure of 25%. Now five solutions of uniform bits = 1 and five solutions of uniform bits = 0 are added. This begins to crystallize a distinct single network structure, consisting of multiple downward facing Vs originating from a single node.
Table 4.10: Network edge representations of a single trap 5 problem at a population size of 100, with a selection pressure of 12.5%. Again five solutions of uniform bits = 1 and five solutions of uniform bits = 0 are added. This simulation seeks highly fit training data. The sample size is the same as that of tables 4.8 and 4.9. The highly fit data produces a uniform structure of a single root node with 4 children. This is incredibly useful to designers.