1. Introduction

In this tutorial, we’ll explain the Candidate Elimination Algorithm (CEA), which is a supervised technique for learning concepts from data. We’ll work out a complete example of CEA step by step and discuss the algorithm from various aspects.

2. Concept Learning

A concept is a well-defined collection of objects. For example, the concept “a bird” encompasses all the animals that are birds and includes no animal that isn’t a bird.

Each concept has a definition that fully describes all the concept’s members and applies to no objects that belong to other concepts. Therefore, we can say that a concept is a boolean function defined over a set of all possible objects and that it returns \boldsymbol{true} only if a given object is a member of the concept. Otherwise, it returns \boldsymbol{false}.

In concept learning, we have a dataset of objects labeled as either positive or negative. The positive ones are members of the target concept, and the negative ones aren’t. Our goal is to formulate a proper concept function using the data, and the Candidate Elimination Algorithm (CEA) is a technique for doing precisely so.

3. Example

Let’s say that we have the weather data on the days our friend Aldo enjoys (or doesn’t enjoy) his favorite water sport:

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We want a boolean function that would be \boldsymbol{true} for all the examples for which \boldsymbol{EnjoySport = Yes}. Each boolean function defined over the space of the tuples of <Sky, AirTemp, Humidity, Wind, Water, Forecast> is a candidate hypothesis for our target concept “days when Aldo enjoys his favorite water sport”. Since that’s an infinite space to search, we restrict it by choosing an appropriate hypothesis representation.

For example, we may consider only the hypotheses that are conjunctions of the terms A = v, where A \in \{Sky, AirTemp, Humidity, Wind, Water, Forecast\} and v is:

  • a single value A can take
  • or one of two special symbols:
    • ?, which means that any admissible value of A is acceptable
    • \emptyset, which means that A should have no value

For instance, the hypothesis (Sky = Rainy) \land (AirTemp = Normal) can be represented as follows:

    \[<Rainy, Normal, ?, ? ,? ?>\]

and would correspond to the boolean function:

    \[f(x) = \begin{cases} true,& (x.Sky = Rainy) \land (x.AirTemp = Normal) \\ false,& \text{otherwise} \end{cases}\]

We call the hypothesis space \boldymbol{H} the set of all candidate hypotheses that the chosen representation can express.

4. The Candidate Elimination Algorithm

There may be multiple hypotheses that fully capture the positive objects in our data. But, we’re interested in those also consistent with the negative ones. All the functions consistent with positive and negative objects (which means they classify them correctly) constitute the version space of the target concept. The Candidate Elimination Algorithm (CEA) relies on a partial ordering of hypotheses to find the version space.

4.1. Partial Ordering

The ordering CEA uses is induced by relation “more general than or equally general as”, which we’ll denote as \geq. We say that hypothesis h_1 is more general than or equally general as hypothesis h_2 if all the examples h_2 labels as positive are also classified as such by h_1:

(1)   \begin{equation*} h_1 \geq h2 \equiv (\forall x) (h_2(x) = true \implies h_1(x) = true) \end{equation*}

This relation has a nice visual interpretation:

The most general hypothesis is the one that labels all examples as positive. Similarly, the most specific is the one that always returns false. The accompanying relations: > (strictly more general), < (strictly less general, strictly more specific), and \leq are analogously defined. For brevity, we’ll refer to \geq as “more general than” from now on.

4.2. Finding the Version Space

CEA finds the version space \boldsymbol{V} by identifying its general and specific boundaries, \boldsymbol{G} and \boldsymbol{S}. G contains the most general hypotheses in V, whereas the most specific ones are in S. We can obtain all the hypotheses in V by specializing the ones from G until we reach S or by generalizing those from S until we get G.

Let’s denote as V(G, S) the version space whose general and specific boundaries are G and S. At the start of CEA, G contains only the most general hypothesis in H. Similarly, S contains only the most specific hypothesis in H. CEA processes the labeled objects one by one. If necessary, it specializes \boldsymbol{G} and generalizes \boldsymbol{S} so that the \boldsymbol{V(G, S)} correctly classifies all the objects processed so far:

The Flowchart of Candidate Elimination Algorithm

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5. An Example of CEA in Practice

Let’s show how CEA would handle the example with our friend Aldo.

5.1. Initialization

In the beginning, CEA initializes G to G_0=\{<?,?,?,?,?,?>\} and S to S_0=\{<\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset>\}. The space bounded by S_0 and G_0, V(G_0, S_0), is the whole hypothesis space H:

5.2. Processing the First Positive Object

Since the only hypothesis in G_0, <?,?,?,?,?,?>, is consistent with the object <Sunny, Warm, Normal, Strong, Warm, Same>, CEA doesn’t change the general boundary in the first iteration, so it stays the same: G_1 = G_0.

However, <\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset> as the only hypothesis in S is not consistent with the object, so we remove it. The minimal generalization of <\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset>  that covers the object <Sunny, Warm, Normal, Strong, Warm, Same> is precisely the hypothesis <Sunny, Warm, Normal, Strong, Warm, Same>. It’s less general than the hypothesis in G_1. So, the updated specific boundary is S_1 = \{<Sunny, Warm, Normal, Strong, Warm, Same>\}:

5.3. Processing the Second Positive Object

As the hypothesis <?,?,?,?,?,?> is consistent with the positive example <Sunny, Warm, High, Strong, Warm, Same>, no changes to the general boundary are necessary. So, it will hold that G_2=G_1.

However, hypothesis <Sunny, Warm, Normal, Strong, Warm, Same> from S_1 is overly specific, so we remove it from the boundary. Its minimal generalization that covers the second positive object is <Sunny, Warm, ?, Strong, Warm, Same>, which is less general than <?,?,?,?,?,?> we have in G_2, so S_2=\{<Sunny, Warm, ?, Strong, Warm, Same>\}:

5.4. Processing the First Negative Object

The third example in our data (<Rainy, Cold, High, Strong, Warm, Change>) is negative, so we execute the else-branch in the main loop of CEA. Since S_2 is not inconsistent with the object, we don’t change it, so we’ll have S_3=S_2. Inspecting G_2, we see that <?,?,?,?,?,?> is too general and remove it from the boundary. Next, we see that there are six minimal specializations of the removed hypothesis that are consistent with the example:

 

To specialize a hypothesis so that it correctly classifies a negative object, we replace each \boldsymbol{?} in the hypothesis with values other than the one in the object. So, we have Sunny instead of Rainy, Warm instead of Cold, and so on.

However, only <Sunny, ?, ?, ?, ?, ?>, <?, Warm, ?, ?, ?, ?>, and <?, ?, ?, ?, ?, Same> are more general than the hypothesis <Sunny, Warm, ?, Strong, Warm, Same> we have in S_3. A hypothesis that isn’t more general than at least one hypothesis in the specific boundary doesn’t classify all the positive examples as positive. That’s why we keep only the minimal specializations that are \geq than the hypothesis in S_3:

5.5. Processing the Third Positive Object

Since the hypothesis <?,?,?,?,?,Same> is inconsistent with the object <Sunny, Warm, High, Strong, Cool, Change>, we remove it. We can’t specialize it because all its specializations wouldn’t classify the object as positive. Similarly, we can’t generalize it as that wouldn’t be consistent with the negative object we previously processed. Therefore, the only option is to remove the hypothesis.

The object <Sunny, Warm, High, Strong, Cool, Change> shows that S_3 is overly specific, so CEA generalizes the hypothesis in S_3 to <Sunny, Warm, ?, Strong, ?, ?>, and we get a new boundary S_4.

Finally, CEA returns the version space V(G_4, S_4):

6. Convergence of CEA

If H contains a hypothesis that correctly classifies all the training objects and the data have no errors, then the version space CEA finds will include that hypothesis. If the output version space contains only a single hypothesis h, then we say that CEA converges to h. However, if the data contain errors or H doesn’t contain any hypothesis consistent with the whole dataset, then CEA will output an empty version space.

7. Speeding up CEA

The order in which CEA processes the training objects doesn’t affect the resulting version space, but it affects the convergence speed. CEA attains the maximal speed when it halves the size of the current version space in each iteration, as that reduces the search space the most.

A good approach is to allow CEA to query the user for a labeled object to process next. Since CEA can inspect its version space, it can ask for an object whose processing would halve the space’s size.

8. Partially Learned Concepts

If the version space V contains multiple hypotheses, then we say that CEA has partially learned the target concept. In that case, we must decide how to classify new objects.

One strategy could be to classify a new object x only if all the hypotheses in V agree on its label. If there’s no consensus, we don’t classify x. An efficient way of doing so is to consider the space’s boundaries \boldsymbol{G} and \boldsymbol{S}. If all the hypotheses in S classify x as positive, then we know that all the other hypotheses in V will too. That’s because they’re all at least as general as those in S. Conversely, we may test if the hypotheses in G agree that x is negative. Hence, there’s no need to look beyond the boundaries if we seek unanimity.

On the other hand, if we can tolerate disagreement, then we can output the majority vote. If we take all the hypotheses in V to be equally likely, then the majority vote represents the most probable classification of x. Additionally, the percentage of hypotheses classifying x as positive can be considered the probability that x is positive. The same goes for negative classification.

9. Unrestricted Hypothesis Spaces

If H contains no hypothesis consistent with the dataset, then CEA will return an empty version space. To prevent this from happening, we can choose a representation that makes any concept a member of H. If X is the space the objects come from, then H should represent every possible subset of X. In other words, H should correspond to the power set of X.

One way to do that is to allow complex hypotheses that consist of disjunctions, conjunctions, or negations of simpler hypotheses. To do so, we’d first design atomic hypotheses, which are the simplest of all. For instance, we’d say that the definition of hypotheses we gave in Section 3 describes the atomic hypotheses in our working example. Then, we’d add two more rules to allow combinations:

  • The atomic hypotheses are hypotheses.
  • If h_1 and h_2 are hypotheses, then \neg h_1, h_1 \land h_2, and h_1 \lor h_2 are also hypotheses.

9.1. Overfitting the Data

But, the problem is that a so expressive space H will cause CEA to overfit the data! The returned version spaces specific boundary will memorize all the positive training objects (D^+), whereas its general boundary will memorize all the observed negative instances (D^-):

    \[\begin{aligned} S &=&& \left\{\left(\bigvee_{x \in D^+}x\right)\right\}  \\ G &=&& \left\{\neg \left(\bigvee_{x\in D^-}x\right)\right\} \end{aligned}\]

9.2. Classification in Unrestricted Spaces

As a result, such a version space provides the unanimous vote only when asked about the training objects. For any other object x, half the hypotheses in V(G,S) classify it as positive, while the other half classifies it as negative. Since H corresponds to the power set of X, we know that for any h\in V(G,S) that classifies x as positive, there must be an h' \in H that agrees with h on all the objects except for x, which it labels as negative. But since h and h' disagree only on a non-training object, h' is also in V(G, S). So, unrestricted hypothesis spaces are useless, and some sort of restriction is necessary.

10. From Inductive Bias to Deductive Reasoning

That restriction biases the search towards a specific type of hypothesis and is therefore called the inductive bias. We can describe bias as a set of assumptions that allow deductive reasoning. That means that we can prove that the version spaces output correct labels. Essentially, we regard the labels as theorems and the training data, the object to classify, and the bias as the input to a theorem prover.

If we require the unanimous vote of hypotheses in a version space, then the inductive bias of CEA is simply:

The hypothesis space contains the target concept.

That’s the weakest assumption we’d have to adopt in order to have formal proofs for labels. However, we should note that the proofs will be semantically correct only if the bias truly holds.

11. Conclusion

In this article, we explained the Candidate Elimination Algorithm (CEA). We showed how to use it for concept learning and also discussed its convergence, speed, and inductive bias.

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