5. RESULTADOS Y DISCUSIÓN
5.2. Análisis comparativo
5.2.2. Comparación entre grupos taxonómicos
Abstract
A class of genetic algorithms (GAs) for learning bounded Boolean conjuncts and disjuncts is presented and analyzed in the context of computational learning theory. Given any reasonable recombination operator, and any confidence and accuracy level, the results in this article provide the number of generations and the size of the population sufficient for the genetic algorithm to become a polynomial-time probably approximately correct (PAC) learner for the target classesk-CNF (conjunctive normal form) andk-DNF (disjunctive normal form) Boolean formulas ofν variables.
B2.8.1 Introduction
In this article, a class of genetic algorithms (GAs) for learning bounded Boolean conjuncts and disjuncts is presented and analyzed in the context of computational learning theory. Given any reasonable recombination operator, and any confidence and accuracy level, the results in this article provide the C3.3 number of generations and the size of the population sufficient for the GA to become a polynomial-time probably approximately correct (PAC) learner for the target classesk-CNF (conjunctive normal form) and k-DNF (disjunctive normal form) Boolean formulas ofν variables. The set ofk-CNF formulas comprises all Boolean functions that can be expressed as the conjunction of clauses, where each clause is a disjunct of at most kliterals. Similarly, the set ofk-DNF formulas is all Boolean functions that can be expressed as the disjunction of terms, where each term is a conjunct of at mostk literals.
The results in this article are based on the work on PAC learning analysis by Ros (1992), where further details can be found. To enhance the presentation of these results, we have ignored the constants (which can be obtained from lemmas A.16 and A.17 of Ros (1992)), and have used the O-notation to denote asymptotic upper bounds, the-notation to denote asymptotic lower bounds, and the 2-notation to denote asymptotic tight bounds (Cormen et al 1991).
B2.8.2 Computational learning theory
Computational learning theory deals with theoretical issues in machine learning from a quantitative point of view. The goal is to produce practical learning algorithms in non-trivial situations based on assumptions that capture the essential aspects of the learning process. In this context, the learning model consists of a world W of concepts, a learning machine M, and an oracle that presents M with a sequence of labeled examples (e.g. strings of bits) from world W. For any concept c from W, it is M’s task, after having received a sufficiently large sample from the oracle, to produce a hypothesis that adequately describesc, assuming such a hypothesis exists.
The PAC model of computational learning theory was founded by Valiant in 1984 when he introduced a new formal definition of concept learning, the distribution-free learnability model (Valiant 1984). In this model, the learning algorithm produces with high probability an accurate description of the target concept within polynomial time in the number of training examples and the size of the smallest possible representation for the target concept. The training examples are independently selected at random from
Probably approximately correct (PAC) learning analysis
unknown (but fixed) probability distributions. The lack of prior knowledge about these distributions justifies the term distribution free. The hypotheses produced under such a model are PAC, since with high probability (1−δ) they representε-close approximations to the target concepts (whereε, δ∈ <(0,1)).
There are a number of aspects of the PAC model that are notably different from other formal learning paradigms. First, the learner is allowed to approximate the target concept rather than to identify it exactly. For example, the inductive inference of classes of recursive functions typically requires exact identification. Second, the model insists on polynomial-time learning algorithms, which is necessary for practical learning systems. Third, the probability distribution over the training examples is not a parameter of the model: there is no prior knowledge about this distribution available to the learner. This contrasts with certain statistical pattern recognition techniques where the input distributions are sometimes restricted to certain classes. Finally, the PAC model is not restricted to certain knowledge representations: it treats the representation of hypothesis spaces as a parameter.
The performance of a PAC learner is measured by its sample complexity (i.e. number of training examples) and its computational complexity (i.e. running time). Clearly, the computational complexity is bounded from below by the sample complexity. While for some classes the computational complexity is close to the sample complexity, the complexity of computing a hypothesis from a sample often dominates the total running time. Indeed, for certain hypothesis spaces the computational complexity is believed to be exponential in l, the maximum size of an example.
For finite hypothesis spaces, a simple counting argument provides an upper bound for the sample complexity: d(1/ε)ln(|Hl|/δ)e, whereεis the desired accuracy level,δis the desired confidence level and |Hl|denotes the number of possible hypotheses. For all those concept classes where the counting method does not yield optimal results or does not apply (e.g. infinite classes), the sample complexity may be obtained via the Vapnik–Chervonenkis (VC) dimension of the hypothesis space (Vapnik and Chervonenkis 1971, Blumer et al 1989).
B2.8.3 The genetic probably approximately correct learner
This section describes the population, the fitness function, and the genetic plan of the genetic PAC learner for the classes ofk-CNF and k-DNF Boolean formulas. These formulas are the largest Boolean classes known to be PAC learnable in polynomial time in ν (number of Boolean variables), 1/ε, and 1/δ. In particular, Valiant (1984) showed that k-CNF is PAC learnable in polynomial time using only positive examples, and k-DNF using only negative examples. Hence, our GA will use positive examples for learningk-CNF, and negative examples fork-DNF.
B2.8.3.1 The population
The GA maintains a population of bit strings of length l that represent the sets of k-DNF and k-CNF C1.2 Boolean formulas overv Boolean variables. More specifically, the set of k-DNF formulas is represented by Dν,k = {0,1}l, where l = O(νk) is the number of terms, and every bit represents the presence or absence of a term. For example,D3,2 needs 18 bits to represent all its possible terms:
x1, x2, x3,x¯1,x¯2,x¯3, x1x2, x1x3, x2x3,x¯1x2,x¯1x3,x¯2x3, x1x¯2, x1x¯3, x2x¯3,x¯1x¯2,x¯1x¯3,x¯2x¯3.
Similarly, the set of k-CNF formulas is represented byCν,k= {0,1}l, where l=O(νk)is the number of clauses. Since the main result in the next section applies to bothCν,kandDν,k(by the duality principle), we define0≡ {0,1}, and use0l to denote either class, withl being the number of attributes (clauses or terms).
The size of the population is denoted byµ, and is fixed during the entire learning process.
B2.8.3.2 The fitness function
The fitness function F takes three arguments: representation r ∈ 0l, training example e∈ {0,1}ν, and the classification of this example: c ∈ {+,−}. The total credit is the sum of the credit values that are generated by each individual attribute in the formula. These credit values are positive integers and take the following values:
• G01, if attribute is absent fromr, buteshows that it may have to be present
• G10, if attribute is present in r, buteshows that it must be absent
• G11, if attribute is present in r, andeshows that it may have to be present.
For example, suppose that out of five attributes (i.e.l =5), a1 anda3 are present in representation
r, and all others are absent (i.e.r =10100). Suppose further thate and c show that attributesa3 and
a5 must be absent and that a1, a2, and a4 may have to be present. Then, r receives a total credit of
F (r, e,c)=G11+G01+G10+G01+G00, for a1, . . . , a5, respectively.
We say thatF (r, e,c)is a productive fitness function if the following credit relations are satisfied: 1≤G10 < G01< G11=G00.
B2.8.3.3 The crossover operator
For any p≥2, the crossover operatorχ is a (possibly randomized) function from(0l)p to0l that maps C3.3.1 p parental structuresw1, . . . , wp into one offspring structurew(i.e.w←χ (w1, . . . , wp)), by having the parents donate their symbols to the offspring structure. The shuffle factor of χ, SF(χ ), is defined as the smallest probability of the event that any two positions in the offspring structure receive their symbols from two different parental structures. The shuffle factor of a crossover operator characterizes its disruptiveness: a larger shuffle factor corresponds to a more disruptive operator. For two parents (i.e.p=2) the uniform crossover operator has a shuffle factor of 0.5, since for any two positions in the offspring structure the probability that they receive their symbol from two different parents is 0.5. Similarly, the one-point crossover operator has a shuffle factor of only2(1/ l)(wherelis the length of the structure), since for any two positions in the structure the probability that the cut will be made between these two positions can be as small as 1/ l (or of that magnitude, depending on the implementation). A shuffle factor with the value of unity can be obtained with a deterministic crossover operator that, on input of l parental structures, produces an offspring structure of lengthl in which each bit is copied from a different parent. Therefore, this operator is the most disruptive crossover within our framework, and produces the most efficient GA in terms of the population size and the number of generations, as is shown by the corollary in the next section.
B2.8.3.4 The genetic plan
The main result is based on the genetic plan at the top of the next page. For all ν, k∈ ℵ+, let0l denote the hypothesis class k-CNF (k-DNF) such thatl≥2. Then, for all target functions f ∈k-CNF (k-DNF), all accuracy parameters ε∈ <(0,1), and all confidence parametersδ∈ <(0,1), the genetic plan at the top of the next page computes the function GF0,χ ,m,F,µ(ν, l, ε, δ), wheremis the number of generations.
The population is initialized by selecting uniformly at random elements from the set0l(statement 1). Then, formgenerations, a positive or negative training example is obtained from the oracle (statement 2), and a new population is formed by assigning credit and applying proportional selection and the crossover C2.2 operator to the current population (statement 3). After m generations, the final hypothesis is selected at random from the last population (statements 4, 5).
The probability distribution function (pdf) DBt ∈ D0l provides the probability that formular ∈0
l
occurs in populationBt. In other words,DBt(r)≡ Pr
RAND(DBt)=r
. For each offspring structure, the genetic plan randomly draws p≥2 parents under the proportional selection scheme before applying the crossover operator. Proportional selection implies that the probability of each individualrin the population is weighted based on the amount of credit it receives from fitness function F (r, e,c)relative to the total amount of credit as collected by the entire population, which is characterized by pdfDBt,F,e,c(r). On top
of that, the probability of producing offspring structure r is also determined by the crossover operator, which is characterized by pdfDBt,F,e,c,χ(r):
DBt,F,e,c,χ(r)≡Pr
χ (RAND1(DBt−1,F,e,c), . . . ,RANDp(DBt−1,F,e,c))=r
. (B2.8.1)
B2.8.4 Main result
This section contains the main result, which provides an upper bound on the size of the population and the number of generations for the GA to be a polynomial-time PAC learner for the target classesk-CNF andk-DNF.
Probably approximately correct (PAC) learning analysis
function GF0,χ ,m,F,µ Input: ν, l, ε, δ
Output: r: an individual (representation) selected at random from final population {Bt denotes the t-th population}
{Bt(j )denotes the j-th member of the t-th population} {Duniform∈D0l is the uniform distribution over elements in0l}
{DBt ∈D0l is a pdf over elements in 0
l based on population B t} begin
{initialize the first population} forj ←1 toµdo
1 B0(j )←RAND(Duniform);
od
{apply genetic operators to mpopulations} fort ←1 tomdo
{obtain a v-bit positive example ( c= ‘+’) if0l represents k-CNF formulas} {obtain a v-bit negative example ( c= ‘−’) if0l represents k-DNF formulas} 2 e←ORACLEν;
forj ←1 toµdo{obtainBt by applying credit, selection, and crossover†}
3 Bt(j )←RAND(DBt−1,F,e,c,χ);{see equation (B2.8.1)}
od od
{select final hypothesis at random from final population} 4 r←RAND(DBm)
5 output(r); end.
Theorem. For all ν, k∈ ℵ+, let 0l represent the hypothesis space fork-CNF (k-DNF) formulas where l, the number of attributes, is polynomial in the number of Boolean variables ν. Then, for all target functions f ∈ k-CNF (k-DNF), all accuracy parameters ε ∈ <(0,1) (such that ε/ l < 0.25), and all confidence parameters δ∈ <(0,1), letGbe a GA that computes the function GF0,χ ,m,F,µ(ν, l, ε, δ), where χ is the crossover operator such that 0<SF(χ )≤1, mis the number of generations,F is a productive fitness function with fitness ratiosa≡G11/ (G11−G01)andb≡G00/ (G00−G10), andµis the size of
the population. Define ω≡1−SF(χ ).
(i) Ifω≥ε/[l2ln(4l/δ)] (such that 1/(1−ω)is polynomial inν, 1/ε, and 1/δ), and if
m= l 4ωln(l/δ) ε2(1−ω) lnl 5ωln(l/δ) ε2δ(1−ω) 2! a=2 l3ω(ln(lm/δ))2 ε2(1−ω) (B2.8.2) b=2 l2ωln(lm/δ) ε(1−ω) µ= l11ω2(ln(lm/δ))5 ε4δ3(1−ω)2 (B2.8.3) thenGis a (polynomial-time) PAC learner for the classesk-CNF andk-DNF.
(ii) If 0< ω < ε/[l2ln(4l/δ)] (such that 1/ωis polynomial inν, 1/ε, and 1/δ), and if
m= 1 ωln(l/δ) ln l δωln(l/δ) 2! a=2 (ln(lm/δ))2 lω(ln(l/δ))2 (B2.8.4) b=2 ln(lm/δ) ln(l/δ) µ= l3(ln(lm/δ))5 δ3ω2(ln(l/δ))4 (B2.8.5) thenGis a (polynomial-time) PAC learner for the classesk-CNF andk-DNF.
(iii) For all rational numbersb >1, if m= l2b2ln(l/δ) ε(b−1+1/ l)ln l3b2ln(l/δ) εδ(b−1+1/ l) a=2 lb2ln(lm/δ) ε(b−1+1/ l) (B2.8.6) † Mutation is not considered here, since a fixed mutation rate is not expected to significantly affect the main result (Ros 1992, Chapter 10).
0≤ω≤ b(b−1) b(b−1)+4la µ= l7b4(ln(lm/δ))3(b/(b−1))3(l+1)/ l ε2δ3(b−1+1/ l)2 (B2.8.7) thenGis a (polynomial-time) PAC learner for the classesk-CNF andk-DNF.
Proof. The proof of this theorem can be found in the dissertation of Ros (1992). Corollary. Ifω=0 and m= l2ln(l/δ) ε ln l3ln(l/δ) εδ a=2 lln(lm/δ) ε (B2.8.8) b= 1+2(1) µ= l7(ln(lm/δ))3 ε2δ3 (B2.8.9)
thenGis a (polynomial-time) PAC learner for the classesk-CNF andk-DNF. B2.8.5 Discussion
The main problem for any GA is to prevent its population from converging prematurely to a small set of genotypically identical individuals. Training examples that will point out the fallacy of these dominating individuals may arrive too late due to their low but yet significant probability (i.e. greater thanε).
One way to limit premature convergence is to reduce the amount of credit given to well performing individuals relative to the amount given to weaker ones (which is obtained by increasing the fitness ratios a andb). However, this slows down the overall rate of growth, which means that more generations are necessary to accomplish the same amount of total growth. In addition, by leveling the relative credit values, selection noise may become more significant unless the population size increases.
It turns out that a better way to avoid premature convergence is to improve the effectiveness of the crossover operator, which can be accomplished by increasing the operator’s shuffle factor. By definition, a larger shuffle factor increases the chances for every bit of being exchanged. This makes it more difficult for any structure to dominate the population prematurely, because it will be more likely that the crossover operator will quickly break the dominating structures up and redistribute their (well performing) parts among other members of the population.
Parts (i) and (iii) of the theorem show the effect of the various crossover operators. According to (i), the fitness ratios, the number of generations, and the population size decrease if the crossover operator is made more disruptive, all other things being equal. We see a similar behavior in (iii), except for the fact thatωdoes not appear explicitly in the expressions. Instead, one is able to manipulate the upper bound of ωwith fitness ratiob: ifb→ ∞,ω’s upper bound increases (weakening the crossover operator), as does the fitness ratioa, the number of generations, and the population size. On the other hand, ifb →1, ω’s upper bound decreases (strengthening the crossover operator), and so do the other variables. Part (ii) of the theorem covers the case where (i) and (iii) do not apply.
As mentioned in section B2.8.3.3, the corollary shows that the l-parental deterministic crossover operator obtains the best results within our framework in terms of the number of generations and the size of the population.
Finally, the analytical tools that were developed to obtain the above results should also carry over to other applications of GAs (e.g. function optimization).
References
Blumer A, Ehrenfeucht A, Haussler D and Warmuth M K 1989 Learnability and the Vapnik–Chervonenkis dimension
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Cormen T H, Leiserson C E and Rivest R L 1991 Introduction to Algorithms (New York: McGraw-Hill)
Ros J P 1992 Learning Boolean Functions with Genetic Algorithms: a PAC Analysis Doctoral Dissertation, University of Pittsburgh
Valiant L G 1984 A theory of the learnable Commun. ACM 27 1134–42
Vapnik V N and Chervonenkis A Y 1971 On the uniform convergence of relative frequencies of events to their probabilities Theor. Prob. Appl. 16 264–80
Theoretical Foundations and Properties of Evolutionary Computations