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Learning From Data Yaser S. Abu Mostafa Pdf. Free PDF ebooks user's guide, manuals, sheets about Abu mostafa learning from data ready for download. This is an introductory course in machine learning. Pick a tolerance level 8. We will also make the contrast between a training set and a test set more precise.

Generalization error. E direction of the bound assures us that we couldn't do much better because every hypothesis with a higher Ein than the g we have chosen will have a comparably higher Eout. Generalization is a key issue in learning.

This can be rephrased as follows. We may now 2Me 2NE2. Notice that the other side of IEout Ein l The error bound ln in 2. We have already discussed how the value of Ein does not always generalize to a similar value of Eout. A word of warning: this chapter is the heaviest in this book in terms of mathematical abstraction. Once we properly account for the overlaps of the different hypotheses.

In a typical learning model. To do this. The union bound says that the total area covered by If h1 is very similar to h2 for instance. We then over-estimated the probability using the union bound.

The mathematical theory of generalization hinges on this observation. If the events B1. If you take the perceptron model for instance. Such an N-tuple is called a dichotomy since it splits x1. The definition o f the growth function i s based on the number o f different hypotheses that 1-l can implement. The dichotomies generated by 1-l on these points are defined by 1-l x1.

If h E 1-l is applied to a finite sample x1. The growth function is defined for a hypothesis set 1-l by where I I denotes the cardinality number of elements of a set. For any 1-l. Definition 2. A larger 1-l x1. This signifies that 1-l is as diverse as can be on this particular sample.

Each h E 1-l generates a dichotomy on x1. We will focus on binary target functions for the purpose of this analysis. If 1-l is capable of generating all possible dichotomies on x1. Let x1. These three steps will yield the generalization bound that we need. To compute mH N. Let us find a formula for mH N in each of the following cases. In the case of 4 points. Figure 2. Example 2. D Let us now illustrate how to compute mH N for some simple hypothesis sets. These examples will confirm the intuition that m1-l N grows faster when the hypothesis set 1-l becomes more complex.

The dichotomy of red versus blue on the 3 colinear points in part a cannot be generated by a perceptron. At most 14 out of the possible 16 dichotomies on any 4 points can be generated.

The most a perceptron can do on any 4 points is 14 dichotomies out of the possible One can verify that there are no 4 points that the perceptron can shatter.

Since this is the most we can get for any points. N which is allowed. Per the next :figure. If both end values fall in the same region. Notice that m1-l N grows as the square of of the 'simpler' positive ray case.

As we vary a. To compute m1-l N in this case. If you connect the 1 points with a polygon. For the dichotomies that have less than three 1 points.

Adding up these possibilities. This does since it is defined based on the maximum 2. Each hypothesis is specified by the two end values of that interval. The dichotomy we get is decided Nil by which two regions contain the end values of the interval. To compute m1-l N. The dichotomy we get on the points is decided by which region contains the value a. Getting a good bound on mH N will prove much easier than computing m1l N itself.

If no data set of size k can be shattered by 1-l. D It is not practical to try to compute m11 N for every hypothesis set we use. Verify that m If k is a break point. In general. We now use the break point k to derive a bound on the growth function m11 N for all values of N. Exercise 2. We will exploit this idea to get a significant bound on m1-l N in general.

If m1-l N replaced M in Equa- tion 2. The fact that the bound is polynomial is crucial. A similar green box will tell you when rejoin. Since B N. To prove the polynomial bound. The notation B comes from ' Binomial' and the reason will become clear shortly. This bound will therefore apply to any 1-l. To evaluate B N. The definition of B N. This means that we will generalize well given a sufficient number of examples. Absent a break point as is the case in the convex hypothesis example.

We now assume N 2: 2 and k 2: 2 and try t o develop a recursion. A second different dichotomy must differ on at least one point and then that subset of size 1 would be shattered. We list these dichotomies in the following table. XN in the table are labels for the N points of the dichotomy. Consider the dichotomies on xi. We collect these dichotomies in the set S1. The remaining dichotomies on the first N 1 points appear twice. Since no subset of k of these first N 1 points can - Let S1 have a rows.

St and S-. We collect these dichotomies in the set S2 which can be divided into two equal parts. We have chosen a convenient order in which to list the dichotomies. Consider the B N. Since the total number of rows in the table is B N. N0 and all k. If there existed such a subset. Assume the statement is true for all N Lemma 2. We can also use the recursion to bound B N. The proof is by induction on N. II It turns out that B N.

We have thus proved the induction step. The implication of Theorem 2. Theorem 2. For a given break point k. The RHS is polynomial in N of degree k. The form of the polynomial bound can be further simplified to make the dependency on dvc more salient. It is easy to see that no smaller break point exists since ti can shatter dvc points.

It is also the best we can do using this line of reasoning. The smaller the break point. We state a useful form here. If dvc i s the VC dimension o f ti. The Vapnik. Chervonenkis dimension of a hypothesis set ti. Note: you can use the break points you found in Exercise 2.

Because of its significant role. For any finite value of dvc. One implication of this discussion is that there is a division of models into two classes. Ein will be close to Eout. This is done in two steps. There is a logical difference in arguing that dvc is at least a certain value. The smaller dvc is. One way to gain insight about dvc is to try to compute it for learning models that we are familiar with.

The 'bad models' have infinite dvc. The 'good models' have finite dvc. With a bad model. Perceptrons are one case where we can compute dvc exactly. If we manage to do that. This is because dvc 2. If we were to directly replace M by mH N in 2.

The perceptron case provides a nice intuition about the VC dimension. No set of N points can be shattered by 1-l. Diversity is not necessarily a good thing in the context of generalization. One can view the VC dimension as measuring the 'effective' number of parameters.

The VC dimension measures these effective parameters or 'degrees of freedom' that enable the model to express a diverse set of hypotheses. Wd In other models. This is consistent with Figure 2.

In the case of perceptrons. Based only on this information. Conclude that there is some dichotomy that cannot be implemented. Any set of N points can be shattered by 1-l. There is a set of N points that cannot be shattered by 1-l. This means that some vector is a linear combination of all the other vectors. The more parameters a model has. Show that the dimension of the perceptron with d 1 para m eters. Eout g :s. Each V is a point on that canvas.

Let 's think of probabilities of different events as areas on that canvas. Let us think of this space as a 'canvas' Figure 2. The probability of a point is determined by which Xn 's in X happen to be in that particular V. The quantities in red need to be technically modified to make 2. Consider the space of all possible data sets. It establishes the feasibility of learning with infinite hypothesis sets.

This means that. Sketch of the proof. The data set V is the source of randomization in the original Hoeffding Inequality. The correct bound. If you compare the blue items in 2.

The VC generalization bound is the most important mathematical result in the theory of learning. There are two parts to the proof. The key is that the effective number of hypotheses. Since the formal proof is somewhat lengthy and technical.

Even if each h contributed very little. The argument goes as follows. This is the essence of the VC bound as illustrated in Figure 2. Here is the idea. What the basic Hoeffding Inequality tells us is that the colored area on the canvas will be small Figure 2. If you were told that the hypotheses in 1-i are such that each point on the canvas that is colored will be colored times because of different h's.

This is the worst case that the union bound considers. For a given hypothesis h E 1-i. The bulk of the VC proof deals with how to account for the overlaps. Let us paint these points with a different color. If we keep throwing in a new colored area for each h E 1-i. This was the problem with using the union bound in the Hoeffding Inequality 1.

The area covered by all the points we colored will be at most the sum of the two individual areas. For a particular h.

When you put all this together. It can be extended to other types of target functions as well. It accounts for the total size of the two samples D and D'. This breaks the main premise of grouping h's based on their behavior on D.

Given the generality of the result. What the growth function enables us to do is to account for this kind of hypothesis redundancy in a precise way. This is where the 2N comes from. Any statement based on D alone will be simultaneously true or simultaneously false for all the hypotheses that look the same on that particular D. The reason m 1-l 2N appears in the VC bound instead of m 1-l N is that the proof uses a sample of 2N points instead of N points. If it happens that the number of dichotomies is only a polynomial.

When 1-l is infinite. This is the essence of the proof of Theorem 2. Why do we need 2N points? Using mH N to quantify the number of dichotomies on N points. The basic Hoeffding Inequality used in the proof already has a slack. Among them. The slack in the bound can be attributed to a number of technical factors. With this understanding. The reality is that the VC line of analysis leads to a very loose bound. Why did we bother to go through the analysis then?

Two reasons. XN and used I H x1. In real applications. Bounding mH N by a simple polynomial of order dvc. Some effort could be put into tightening the VC bound. The inequality gives the same bound whether Eout is close to 0.

This is an observation from practical experience. E and 8. We can obtain a numerical value for N using simple iterative methods.

The constant of proportionality it suggests is The performance is specified by two parameters. If dvc were 4. From Equation 2. We can use the VC bound to estimate the sample complexity for a given learning model. This gives an implicit bound for the sample complexity N.

How big a data set do we need? Using 2. The error tolerance E determines the allowed generalization error. D 4 The term 'complexity' comes from a similar metaphor in computational complexity.

How fast N grows as E and 8 become smaller4 indicates how much data is needed to get good generalization. If we replace m1-l 2N in 2. Eout may still be close to 1. In most practical situations. If we use the polynomial bound based on dvc instead of m1-l 2N. If someone manages to fit a simpler model with the same training The first part is Ein. The bound in 2. D Let us look more closely at the two parts that make up the bound on Eout in 2.

We could ask what error bar can we offer with this confidence. One way to think of rl N. A combination of the two.

If you are developing a system for a customer. The final hypothesis g is evaluated on the test set. When we report Etest as our estimate of Eout.

Etest is just a sample estimate like Ein. Let us call the error we get on the test set Etest. Although O N. The optimal model is a compromise that minimizes a combination of the two terms.

An alternative approach that we alluded to in the beginning of this chapter is to estimate Eout by using a test set. While the estimate can be useful as a guideline for the training process. How do we know Etest g. The bigger the test set you use. You use a learning model with 1. Had the choice of g been affected by the test set in any shape or form.

There is a price to be paid for having a test set. The VC generalization bound implicitly takes that bias into consideration. The test set just tells us how well we did. Another aspect that distinguishes the test set from the training set is that the test set is not biased. There is only one hypothesis as far as the test set is concerned. This hypothesis would not change if we used a different test set as it would if we used a different training set.

We wish to estimate Eout g. We have access to two estimates: Ein g. The test set just has straight finite-sample variance. This is a much tighter bound than the VC bound. We can answer this question with authority now that we have developed the theory of generalization in concrete mathematical terms. The training set has an optimistic bias. Both sets are finite samples that are bound to have some variance due to sample size. When you report the value of Etest to your customer and they try your system on new data.

The test set does not affect the outcome of our learning process. To properly test the performa nce of the fin a l hypothesis. We will address that tradeoff in more detail and learn some clever tricks to get around it in Chapter 4. If we take a big chunk of the data for testing and end up with too few examples for training.

We may end up reporting to the customer. The approach is based on bias-variance analysis. There is thus a tradeoff to setting aside test examples. If f and h are real-valued. We can define in-sample and out-of-sample versions of this error measure.

The proofs in those cases are quite technical. When we report experimental results in this book. An error measure that is commonly used in this case is the squared error e h x. We defined Ein and Eout in terms of binary error. Since the training set is used to select one of the hypotheses in 1-l.

Etest is used as synonymous with Eout. In some of the learning literature. In fact. In order to deal with real-valued functions. When you select your hypothesis set. The same issues of the data set size and the hypothesis set complexity come into play just as they did in the VC analysis.

The new way provides a different angle. If 1-l is too simple. If 1-l is too complex. The ideal 1-l is a singleton hypothesis set containing only the target function. Generalization Tradeoff The VC analysis showed us that the choice of 1-l needs to strike a balance between approximating f on the training data and generalizing on new data. Since we do not know the target function. There is another way to look at the approximation-generalization tradeoff which we will present in this section.

The VC generalization bound is one way to look at this tradeoff. It is particularly suited for squared error measures. The out-of-sample error is 2. We then get the expected out-of-sample error for our learning model. The term g x. We can rid Equation 2. The term lEv [g D x ] gives an 'average function'. We have made explicit the dependence of the final hypothesis g on the data V.

One can interpret g x in the following operational way. The function g is a little counterintuitive. Generate many data sets V1. V K and apply the learning algorithm to each data set to produce final hypotheses The var is large the target f. One can also view the variance as a measure of 'instability' in the learning model. The spread around f in the red region. The term 1Ev [ g V x g x 2] is the variance of the random variable g V x.

Our derivation assumed that the data was noiseless. The noise term is unavoidable no matter what we do. The target only one hypothesis. To illustrate. Since there is Very large model. The approximation-generalization tradeoff is captured in the bias-variance decomposition. We thus arrive at the bias-variance decomposition of out-of-sample error. A similar derivation with noise in the data would lead to an additional noise term in the out-of-sample error Problem 2. Very small model. Different data sets erage function g and the f nal hy will lead to different hypotheses that pothesis g D will be the same.

With the simpler Y1 and x 2. The figures which follow show the resulting fits on the same random data sets for both models. For Ho. The bias-var analysis is summarized in the next figures. We sample x uniformly in [. Repeating this process with many data sets. D The learning algorithm plays a role in the bias-variance analysis that it did not play in the VC analysis. The total out-of-sample error has a much smaller expected value of 0. These curves are only conceptual.

Although the bias-variance analysis is based on squared-error measure. It can use any criterion to produce g V based on V. The first is to try to lower the variance without significantly increasing the bias. By design. In the bias-variance analysis. Two points are worth noting. We have one data set. With the same 1-l. These goals are achieved by different techniques. Regularization is one of these techniques that we will discuss in Chapter 4.

The simpler model wins by significantly decreasing the var at the expense of a smaller increase in bias. Reducing the bias without increasing the variance requires some prior information regarding the target function to steer the selection of 1-l in the direction of f. There are two typical goals when we consider bias and variance.

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