1. Introduction to SaDE adaptive differential evolution algorithm
(self-adaptive Differential Evolution)
1. Pseudo code:
2. Specific process:
(1) Mutation (mutation)
Each mutation uses roulette to select a strategy from the candidate strategy pool, which contains four mutation strategies
(2) Crossover (cross)
After the mutation stage is over, a crossover operation is performed on each pair of target vectors Xi, G and their corresponding mutation vectors Vi, G to generate a test vector Ui, G. In the basic version, the binomial crossing method is used as follows :
(3) Selection
Before choosing, you need to evaluate the experimental vectors Ui, G and calculate the Fes value
The choice here is definitely the better one. Whether it is greater than or less than is based on whether the evaluation function is to be maximized or minimized.
1. There is a problem here. The SaDE algorithm is parameter-adaptive, that is, the parameters are automatically adjusted. It may take a long time for the parameters to appear once, and it will change again next time, which may not be so good. It is a waste of time and evaluation times. . In other words, using such F and Cr to generate a promising test vector may cost a lot of Fes, which results in a low efficiency of the evolution process. Many other adaptive DE variants have the same disadvantages. With this in mind, we will introduce a new ARM-based parameter adaptation strategy to accelerate the adjustment process of F and Cr.
2. Most adaptive technologies use random number generators or automatically generate new F and Cr based on previously successfully archived calculated values. This means that the potentially effective correlation between F and Cr hidden in previous successful combinations is rarely noticed. If we can mine and use F and Cr couples with better fitness values in the iterative process, we can avoid many tedious experiments on F and Cr couples and make the search process more effective. Next, we will introduce the Apriori algorithm widely used in ARM.
2. Adjust parameters F and Cr based on association rule mining in DE algorithm
The most classic Apriori algorithm in association rule mining algorithms is quoted here
1. Association rule mining
Item set : In association analysis, a set containing zero or more items is called an item set. If an itemset contains k items, it is called a k-item set. For example: {beer, diapers, milk, peanuts} is a 4-item set. An empty set is an item set that does not contain any items.
Association rule (association rule): It is an implicit expression of the form X → Y, where X and Y are disjoint itemsets, namely: X∩Y=∅. The strength of an association rule can be measured by its support and confidence.
Support : How often an item set or rule appears in all things, and determines how often the rule can be used in a given data set. σ(X): indicates the support count of item set X
Support for item set X : s(X)=σ(X)/N; Support for rule X → Y: s(X → Y) = σ(X∪Y) / N
Popular explanation : Simply put, the support of X==>Y refers to the probability that item set X and item set Y appear at the same time.
Probability description : the support degree of item set X to item set Y support (X==>Y)=P(X n Y)
Example description : A total of 1,000 customers went to the mall to purchase items one day, and 150 of them bought ballpoint pens and notebooks at the same time, so the support for the above association rules is 15%.
Confidence : Determine how often Y appears in transactions that include X. c(X → Y) = σ(X∪Y)/σ(X)
Popular explanation : Simply put, credibility refers to the probability that the item set Y will also appear in the transaction T where the item set X appears at the same time.
Probability description : the confidence of item set X to item set Y (X==>Y)=P(X|Y)
Example explanation : The credibility of the above association rules answers the question: if a customer buys a ballpoint pen, how likely is it that he will also buy a notebook? In the above example, 65% of the customers who bought the ballpoint pen bought the notebook, so the credibility is 65%.
Desired confidence level (Expected confidence)
Definition: Let W e% have sets of transaction support articles B, e% confidence referred to the desired degree of association rule A → B.
Popular explanation : Expected credibility describes the probability of item set B appearing in all transactions when there is no conditional influence.
Example description : If a total of 1000 customers go to the mall to purchase items on a certain day, and 250 of them have purchased ballpoint pens, the expected credibility of the above association rules is 25%.
Probability description : The expected confidence of item set A to item set B is support(B)=P(B)
Lift (Lift)
Definition: lift reliability is desired ratio credibility
Popular explanation : The degree of improvement reflects how much the "appearance of item set A" has changed the probability of occurrence of item set B.
Example description : the promotion of the above association rules=65%/25%=2.6
Probability description : The expected confidence of item set A to item set B is lift(A==>B)=confidence(A==>B)/support(B)=p(B|A)/p(B)
Support is an important measure, because rules with very low support may only appear by accident, and rules with low support are mostly meaningless. Therefore, support is usually used to delete those meaningless rules;
confidence measure is the reliability of reasoning through rules. For a given rule X → Y, the higher the confidence level, the greater the probability that Y will appear in things that contain X. That is, the conditional probability P(Y|X) of Y under a given X is larger.
2. Use association rule mining to adjust parameters
Here, the Apriori algorithm in ARM can be used to find the interval between F and Cr (the most frequent item set). We use this method to adapt F and Cr to the DE algorithm.
It should be noted that the Apriori algorithm has few limitations. This algorithm requires multiple scans of the database. When the database is large, it takes a long time to obtain frequent itemsets. In this article, we will use the Apriori algorithm to find association rules based on successful records, which are small data sets in the iterative process. In addition, we only need the F and Cr association rules, that is, frequent 2-itemset. Therefore, the extra CPU time generated by the Apriori algorithm is acceptable. Another disadvantage of the Apriori algorithm is that it derives a lot of trivial rules, and it is difficult to extract the most interesting rules. In this article, we will only select the generated association rules with the most support.
(1) Prerequisite :
In order to obtain satisfactory performance for adaptive DEs, trial and error methods are usually used to adjust control parameter values. This mechanism will consume a lot of function evaluation (Fes). At the same time, in most of the most advanced adaptive DE variables,Previously successful parameter values, which may contain potentially useful information and knowledge, are often underutilized. Based on these observations, we incorporated ARM into adaptive DEs to generate explicit and promising parameter pairs to accelerate evolution.
(2) Steps :
Step 1 :Record each successful F and Cr into the original data set Do.Here, if the experimental vectors Ui and G generated by the pair of F and Cr are successfully selected, it means that it is a successful parameter pair.
Step 2 : == Discretization. ==Before ARM, Do should be preprocessed into a binary matrix, named Db. Different combinations of F and Cr may cause different search behaviors. For example, F = 1 and Cr = 0.9 are mainly to maintain population diversity, while F = 0.8 and Cr = 0.2 are to encourage exploration.
This paper divides these two parameters into 5 different intervals ((0,0.1), [0.1,0.4), [0.4,0.6), [0.6,0.9), [0.9,1]) to represent different behaviors. Therefore, according to the value of F and Cr, it is discretized into 5 different intervals ((0,0.1), [0.1,0.4), [0.4,0.6), [0.6,0.9), [0.9,1]). Therefore, Db is a ten-column binary matrix, the first 5 columns represent F, and the last 5 columns represent Cr.
Step 3 :Use Apriori algorithm to find the most frequent 2 item sets.Based on the original frequent k-itemset Apriori algorithm, an improved Apriori-2 algorithm is developed, which can obtain the most successful combination of F and Cr so far. Here, because we only need the association rules of F and Cr, so k is set to 2. In this article, the most successful combination of F and Cr is represented by the Apriori-2 algorithm to represent the 2 most frequent itemsets of Db. In other words, in Db, the row with the most repetitions represents the most frequent F and Cr interval. That is, the most successful combination of F and Cr. Figure 5 shows an example. One Db has 5 rows. Using the Apriori-2 algorithm, output the most successful F and Cr pairs (F [0.6, 0.9), Cr [0.6, 0.9)).
Step 4 :Generate new values according to the above association rules.
Through the Apriori-2 algorithm, we can get the number of two columns representing different F and Cr distribution intervals. For example, if L2=[3,9] is returned, it means that the most common association rules between F and Cr distribution domains are F∈[0.4,0.6) and Cr∈[0.6,0.9). The recommended upper and lower limits for simultaneous output are Fu, Fl, Cru and Crl respectively. According to these explicit association rules between F and Cr, new F and Cr can be generated, as follows:
Each particle in each generation uses the following formula to produce F and Cr
Step 5 :Use the greedy operator to select better candidate parameters.In order to maintain the good search performance of the existing algorithms, a greedy operator is proposed by combining the arm-based parameter adaptation strategy with the original algorithm . In each iteration, two test vectors Ui and Ui_ARM are evaluated respectively, representing each test vector generated by the original parameter adaptive strategy and the parameter adaptive strategy based on arm. Then, select a test vector with a better fitness value, and try to replace Xi in the selection operation. This greedy selection operator can be described as follows:
(3) Supplement:
There are two algorithms for finding K frequent itemsets. The second algorithm can directly generate 2 frequent itemsets that are not repeated. It is an improved algorithm and is suitable for self-adjustment of F and Cr parameters.
三. SaDE-ARM algorithm
Overall knowledge points:
1. In fact, here is to change the update method of parameters F and Cr on the basis of the traditional SaDE parameter self-maintenance differential evolution algorithm. The self-maintenance parameter self-adjustment does not consider the influence between F and Cr. Introducing association rule mining here, the purpose is to learn the information of historical success parameters, through the classic 2-frequent item set search, find the range of F and Cr with the highest support, and search for the value of F and Cr in this range.
2. However, this method is not directly used to replace the original parameter self-maintenance method, but the two are combined and carried out at the same time. In the selection stage, the optimal particle is selected and the successful parameter information is saved.
3. Not every generation performs the ARM algorithm to find the optimal range of F and Cr. Here, the optimal range is calculated only once every NP generation.
4. At the end of this experiment, experimental tests were carried out on the division of population size, dimension size, F and Cr range to verify the correctness or suitability of the parameters used in this experiment.